Commun. Math. Phys. 289, 1–44 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0805-5
Communications in
Mathematical Physics
On the Persistence of Invariant Curves for Fibered Holomorphic Transformations Mario Ponce Pontificia Universidad Católica de Chile, Santiago, Chile. E-mail:
[email protected] Received: 18 December 2007 / Accepted: 19 February 2009 Published online: 18 April 2009 – © Springer-Verlag 2009
Abstract: We consider the problem of the persistence of invariant curves for analytical fibered holomorphic transformations. We define a fibered rotation number associated to an invariant curve. We show that an invariant curve with a prescribed fibered rotation number persists under small perturbations on the dynamics provided that the pair of rotation numbers verifies a Brjuno type arithmetical condition. Nevertheless, an extra complex parameter is added to the problem and the persistence becomes a one-complex codimension property. 1. Introduction Skew-product transformations over irrational rotations have been widely studied as a source of examples of interesting dynamics and for modeling fundamental phenomena in mathematical physics (cf. quasi-periodic Schrödinger cocycle, see [1,16]). In the former direction, Furstenberg [7] constructed the first example of a minimal non-ergodic conservative diffeomorphism of the torus as a fibered circle diffeomorphism. Later on, the basis for the study of these maps was established by M. Herman [10], who defined in particular the notion of the fibered rotation number (see also [13]). In recent years, this theory was relaunched mainly by the works of G. Keller, J. Stark and T. Jäger [11,12], who established (among other things) a Poincaré-like classification relating the fibered rotation number to the existence of invariant graphs. In his doctoral thesis [27] (see also [28]), O. Sester studied hyperbolic fibered polynomials, successfully generalizing the classical notions of Julia set, Green’s function, and the principal cardioid of the Mandelbrot set in the parameter space. Closely related works due to M. Jonsson [14,15] are also important contributions to the subject. In this work we study fibered holomorphic dynamics. More precisely, given an irrational number α > 0, an open simply connected subset U of C, and a real number δ > 0, we consider transformations of the form
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M. Ponce
F : Bδ × U −→ Bδ × C, (θ, z) −→ (θ + α, f θ (z)) , where Bδ denotes the strip θ ∈ C/Z |I m(θ )| < δ . We also assume that this transformation is a holomorphic function as a two variables complex function. In particular, the maps f θ : U → C are holomorphic for every θ ∈ Bδ . We call F an analytic fibered holomorphic transformation, and we refer to it as a fht. Note that a map F as above has neither fixed nor periodic point. The natural object that plays this role for the local dynamics is an invariant curve, that is, a holomorphic curve u : Bδ → U such that, for every θ ∈ Bδ , F (θ, u(θ )) = (θ + α, u(θ + α)) ,
(1)
which is equivalent to f θ (u(θ )) = u(θ + α). Indeed, the dynamics of F is organized around this “fibered fixed point”, thus generalizing the role of a fixed point for the local dynamics of an holomorphic germ g : (C, 0) → (C, 0) (see [21,22]). We define an appropriate notion of fibered rotation number, which appears to be quite useful for studing this dynamics (see Definition 3.1). Fibered holomorphic dynamics are the skew-product version of holomorphic germs. We can imagine the skew product version of area preserving maps having an elliptic fixed point and the related KAM problems (see for instance [19,24]). This last situation appears in a natural way when we consider the so-called Melnikov’s problem on the persistence of elliptic lower dimensional invariant tori for integrable Hamiltonian systems. Let us recall that, concerning this problem, celebrated results due to H.Eliasson [6] and J.Bourgain [3] establish the persistence of lower-dimensional invariant tori provided that the tangential frequencies verify a diophantine condition. The Pöschel paper [23] treats the case of a Rüssmann type arithmetic condition. However, a diophantine condition is not optimal for this result. In fact, Gentile [9] shows that Melnikov’s persistence holds under a Brjuno condition on the tangential frequencies notwithstanding a degeneracy condition on the normal frequencies. For further references about the Brjuno condition on this problem we refer the reader to the excellent introduction by Gentile in [9] and references therein. Also, the Brjuno condition appears as a sufficient condition on the base dynamics for the problem of reducibility of skew-product cocycles (see Lopes Dias [18], Gentile [8]). In this work we investigate how the Brjuno condition appears in fht’s in the problem of the persistence of an invariant curve with prescribed fibered rotation number. Let us point out that the problem of the persistence of invariant curves for C ∞ fibered holomorphic maps was settled by the author in [20], where it is proven that the optimal arithmetic condition in that setting is a diophantine type condition on the small divisors nα − β. The real number β corresponds to the fibered rotation number. The optimality of this arithmetical condition tells us that the persistence of an invariant curve depends on the pair (α, β) and not only on the base frequency α. The main theorem of this work states that invariant curves persist under small perturbations under a Brjuno type arithmetical condition B1 (which is more general than any diophantine condition) between the rotation on the base α and the fibered rotation number (which in our setting corresponds to the tangential and normal frequencies, respectively). Nevertheless, an extra complex parameter is added to the problem and the persistence becomes a one-complex codimension property.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
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Although fht’s correspond to higher-dimensional systems, due to their skewproduct structure they have a one-dimensional flavor. In this direction, let us recall that the Brjuno condition is optimal for the linearization problem of both holomorphic germs and analytic circle diffeomorphisms close to rotations (see [30,31]). In view of this, we conjecture that condition B1 is optimal for our problem. However, as in the classical case, the use of KAM techniques seems to be inapropriate to prove optimality. 2. Arithmetical Conditions and Linearized Equation Throughout this work, α > 0 will be an irrational number (it will correspond to the rotation angle on the base for our fibered transformation). For a real number x denote by x the distance to the nearest integer, that is, x = min p∈Z |x − p|. 2.1. Arithmetical conditions for a real number. A complete treatment of what follows in this section may be found in [5,17,26]. For each N ∈ N we define the worst divisor of α up to the order N by α (N ) = max nα−1 . 0<|n|≤N
Worst divisors may be completely described in terms of the denominators of the reduced continuous fraction of α. Indeed, the sequence q0 = 1, q1 , q2 , . . . of these denominators has an important property, namely, for every n ≥ 1, qn+1 = min k > qn kα < qn α . Therefore, if k ∈ N is such that qk ≤ N < qk+1 , then α (N ) = qk α−1 . We define the set ⎫ ⎧ ⎬ ⎨ log α (N ) < ∞ . B = α ∈ T1 \ Q ⎭ ⎩ N2 N ≥1
If α belongs to B we say that α verifies the Brjuno (arithmetical) condition B (as defined by A. Brjuno, see [4]). According to H.Rüssmann (see [25, §8]), this condition is equivalent to the convergence of the series log qn+1 qn
n≥1
.
It is well known that B has full Lebesgue measure, but is a small set from the topological point of view (Baire’s category). The following lemma contains a useful equivalence of Brjuno condition, which we will use throughout this work. Lemma 2.1 (Rüssmann[25]). The number α belongs to B if and only if log α (2n ) n≥0
2n
< ∞.
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Proof. For every n ≥ 2 one has 2n+1 −1 i=2n
2 −1 2 −1 1 1 1 1 1 − < − , < 2 i i +1 i i − 1 i n n n+1
n+1
i=2
i=2
and therefore 2 −1 1 1 1 1 1 1 − n+1 < n. = n − n+1 < < n 2 2 2 i 2 −1 2 −1 2 n n+1
1 2n+1
i=2
Since α (·) is increasing, this implies that 2 −1 1 log α (2n ) log α (i) α (2n+1 ) ≤ ≤ 2 , 2 2n i2 2n+1 n n+1
i=2
which concludes the proof.
2.2. Arithmetical conditions for a pair of real numbers. Given a real number β, we say that the pair (α, β) is rational if there exists k ∈ Z such that kα ≡ β mod (1). We denote by T2I the set of non-rational pairs. If (α, β) belongs to T2I then, for a given N ∈ N, we define its worst divisor up to the order N by α,β (N ) = max nα − β−1 . 0≤|n|≤N
This represents the quality of the approximations of β by multiples of α. Let B1 be the set defined by ⎧ ⎨
⎫ ⎬ log α,β (n) B1 = (α, β) ∈ T2I α ∈ B and < ∞ , ⎩ ⎭ n2 n≥1
and let Bα1 be the set of β such that (α, β) lies in B1 . If α belongs to B then Bα1 is a set of full Lebesgue measure which is meager (that is, equal to a countable union of empty interior closed sets). Lemma 2.2. Assuming that α lies in B, the pair (α, β) belongs to B1 if and only if log α,β (2n ) n≥0
2n
Proof. Analogous to the proof of Lemma 2.1.
< ∞.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
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2.3. Cohomological equation. For more information on this classical subject see for instance [29]. Given an analytic function φ : Bδ → C, we look for an analytical solution ψ : Bδ → C to the cohomological equation ψ(θ + α) − ψ(θ ) = φ(θ ).
(2)
By integrating both sides of this equality, one readily checks that a necessary condition for the existence of a continuous solution to this equation is that the mean value of φ (with respect to the Lebesgue measure in T1 ⊂ Bδ ) is zero, that is, T1 φ(θ )dθ = 0. A continuous solution to the equation is not unique, but any two solutions differ by a constant. Proposition 2.3. If α belongs to B and the mean value of φ is zero, then there exists an analytic function ψ : Bδ → C which is a solution to (2). Moreover, such a solution may be taken so that T1 ψ(θ )dθ = 0 Although condition B is not optimal for this proposition, it is optimal for the problem of linearization of holomorphic germs in the neighborhood of indifferent irrational fixed points [30]. Remark 2.4. An easyhomotopy type argument shows that the mean value of φ on the circle T1c = θ ∈ Bδ I m(θ ) = c is constant as a function of c ∈ (−δ, δ). If the value of this constant is non-zero then we may envisage Eq. (2) replacing φ by φ − T1 φ(θ )dθ c for some (any) c. 3. Definitions and Normal Forms In order to state properly the main theorem of this work we need to introduce some definitions. Let F be a fht and u an invariant curve for F. We consider the circle T1 natu rally embedded in Bδ . In the remaining part of this work we will always suppose that T1 log |∂z f θ (u(θ ))| dθ = 0. (Notice that, since F is injective, the differential ∂z f θ is always non-zero.) This condition says that the curve is neither attracting nor repulsive at the infinitesimal level. We say that the curve is indifferent. We also suppose that the application θ → ∂z f θ (u(θ )) is homotopic in C \ {0} to a constant. Definition 3.1. We define the fibered rotation number by 1 tr (u) = log ∂z f θ (u(θ )) dθ. 2πi T1 This number represents the average rotation speed of the dynamics around the invariant curve. Notice that the log above is well defined mod 2πi, and the number tr (u) is well defined mod 1. Remark 3.2. Notice that, since all of the circles T1c are homotopic between them and F is holomorphic, one has 1 log ∂z f (θ, u(θ )) dθ tr (u) = 2πi T1c for every circle T1c . Thus, the indifferent nature of the curve (resp. the fibered rotation number) may be detected (resp. computed) on any circle T1c , for |c| < δ.
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Along this work we will often deal with holomorphic changes of coordinates H˜ defined from a tubular neighborhood of the zero section {z ≡ 0} Bδ to a tubular neighborhood of the invariant curve, say H˜
(θ, z) −→ (θ, h θ (z)) . The functions h θ will be biholomorphic transformations between two topological discs sending the origin to the curve, that is, h θ (0) = u(θ ). Moreover, the application θ → h θ (0) will be homotopic to a constant in C \ {0} (in other words, it will have zero topological degree). Therefore, when conjugating our transformation F by such a H we will get a new transformation F˜ = H −1 ◦ F ◦ H having the zero section u˜ = {z ≡ 0} Bδ as an indifferent invariant curve and having the same fibered rotation number, that is, tr (u) ˜ = tr (u). In particular, the indifferent nature of the curve and the fibered rotation number are invariant by this type of conjugacy. Assume F : Bδ × U → Bδ × C is a fht, α is a Brjuno number, and u is an indifferent invariant curve for F. For β = tr (u) we will solve the (cohomological) equation v(θ ) − v(θ + α) = 2πiβ − log ∂z f θ (u(θ )) on the strip Bδ (see Proposition 2.3). Then letting u 1 = exp(v) and H (θ, z) = (θ, u(θ )+ u 1 (θ )z), and performing the corresponding change of coordinates, one gets the following normal form for F:
N F (z, θ ) = H −1 ◦ F ◦ H (θ, z) = θ + α, e2πiβ z + ρ(θ, z) . Here ρ is an analytical function vanishing up to the second order at z = 0. This function ρ, and so the fht N F , is defined on the strip Bδ for the θ variable, and ρ(θ, ·) is defined on a non-constant open set Uθ = H −1 ({θ } × U ). Assume now that F is a fht that may be put in a normal form
H −1 ◦ F ◦ H −1 (θ, z) = θ + α, e2πiβ z + ρ(θ, z) by a change of coordinates of the form H (θ, z) = (θ, u 0 (θ ) + u 1 (θ )z) , where u 0 : Bδ → U , u 1 : Bδ → C are analytical functions for some δ ∈ (0, δ], and u 1 has zero topological degree. In this case one can readily check that u 0 is an indifferent invariant curve, and its fibered rotation number equals (u 0 ) = β. Therefore, the existence of an indifferent invariant curve with transversal rotation number β is equivalent, under the Brjuno condition over α, to the existence of an analytical fibered affine change of coordinates that puts F in an appropriate normal form. 4. Statement of the Problem and Result Let F be a fht with an indifferent invariant curve having a given fibered rotation number, say β ∈ R. As we claimed in the previous section, if α ∈ B then F can be written in a neighborhood of the curve in a normal form θ + α, e2πiβ z + ρ(θ, z) , where ρ is an analytic function. The function ρ(θ, ·) vanishes up to the order 2 at z = 0, and is
Persistence of Invariant Curves for Fibered Holomorphic Transformations
7
convergent for |z| < r for some positive radius r . A small perturbation of such a transformation is defined as being a fht of the form
(θ, z) −→ θ + α, ρ0 (θ ) + e2πiβ + ρ1 (θ ) z + ρ(θ, ˜ z) , where ρ0 , ρ1 are small analytic functions, where ρ˜ is an analytic function whose size is comparable to that of ρ, and where ρ(θ, ˜ ·) vanishes up to the order 2 at z = 0 and is convergent for |z| < r . Notation for sizes. Every function that will appear in this work is a bounded holomorphic function with respect to its variables (defined in some complex open set). Let f : U ⊂ Ck → C be a bounded holomorphic function from the open set U ⊂ Ck , k ∈ {1, 2, 3}, to C. We define the norm f U of f as f U = sup | f (z)|. z∈U
In general, U can be thought of as a product Bδ × D Z × D t , where D Z , D t are discs in the complex plane. We then may define the oscillation of f with respect to the z variable as osc ( f ) Bδ ,D Z ,D t =
sup (θ,t)∈Bδ ×D t z 1 ,z 2 ∈D Z
| f (θ, z 1 , t) − f (θ, z 2 , t)|.
Note that if z = 0 belongs to D Z and f vanishes at this point, then its oscillation is a bound for its norm, that is, f Bδ ,D Z ,Dt ≤ osc ( f ) Bδ ,D Z ,Dt ≤ 2 f Bδ ,D Z ,D t . We will currently omit the corresponding domains of definition when this does not lead to confusion. 4.1. 1−parameter families. In general KAM results, a perturbation over an elliptical dynamics (rotations of the circle, completely integrable Hamiltonians, etc.) leads to a perturbation of the frequencies which control the dynamics. Thus, in the perturbed situation, we can not expect to retrieve the same dynamical properties as in the unperturbed elliptic situation. In order to retrieve these original properties, one introduces a 1−parameter correction, where the parameter is a vector value of the same nature (and dimension) as the related frequencies. In this way, one usually shows that some dynamical properties (linearization, existence of invariant curves with a given rotation number, etc.) are persistent in codimension 1 (see for instance [2]). In our work, we will only perturb the holomorphic part of the transformations. Therefore, our frequency corresponds to the fibered rotation number, that is, a complex number. Let F be a fht and u an indifferent invariant curve, with tr (u) = β ∈ R. Let be an open set in C. A transversal small perturbation of F is a 1-complex parameter family {Ft }t∈ of fht’s (i.e., a complex curve in the space of fht’s) verifying the following properties: – every element Ft is a small perturbation of F, – the fibered rotation number (even if an invariant curve does not exist) changes along the complex curve {Ft }t∈ (see Theorem 4.1 for more explanations).
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We say that the curve u is persistent if for any transversal small perturbation {Ft }t∈ there exists a parameter t ∗ ∈ such that Ft ∗ has an invariant indifferent curve u ∗ satisfying tr (u ∗ ) = β. Roughly speaking, our result says that, except for a complex correction, any small perturbation of F has an indifferent invariant curve having β as its fibered rotation number. However, an arithmetical condition over the pair (α, β) is required. We say that a 1-parameter family Ft of fht’s is analytic if the application × Bδ × D(0, r ) −→ C (t, θ, z) −→ Ft (θ, z) is holomorphic. We consider the usual development for the elements of this family, namely,
Ft (θ, z) = θ + α, ρ0,t (θ ) + e2πiβ + ρ1,t (θ ) z + ρt (θ, z) .
(3)
To fix ideas, we will suppose that ρt (θ, ·) is convergent in the unit disc D. With this notation we can finally state our main result. Theorem 4.1. For every pair (α, β) verifying the Brjuno condition B1 , and for every positive constants L > 1, M, T and δ, there exists a real number ε∗ (L , M, T, δ, (α, β)) > 0 and a positive constant K R (L , M, T, δ, (α, β)) such that the following holds: if ε belongs to (0, ε∗ ] and for an analytic family {Ft }t∈ there exists a disc D(t0 , K R ε) ⊂ such that L > ∂t ρ1,t (θ )dθ > L −1 (4) T1 t=t0 and such that, for every t in D(t0 , K R ε), ⎧ ρ ≤ ε ⎪ ⎨ ρ0,t Bδ ≤ ε 1,t Bδ 2ρ ∂ ⎪ t Bδ ,D ≤ M z ⎩ ∂t ∂z ρt Bδ ,D + ∂t2 ρ1,t Bδ ≤ T, then there exists a parameter t¯ in D(t0 , K R ε) such that Ft has an indifferent invariant curve u which is analytic on the strip B δ . Moreover, the fibered rotation number of this 2 curve equals β. Furthermore, the size of u goes to 0 when ε goes to 0. Note that the transversality condition is given by (4). For example, take L = M = T = 2, some analytic function f : Bδ × D → C such that f (θ, z) − e2πiβ z Bδ ×D ≤ ε∗ (L , M, T, δ, (α, β)) and the family of fht {Ft (θ, z) = (θ + α, et f (θ, z))}t∈C . In [20] we show an analogous result in the case of C ∞ transformations. Indeed, we show that a diophantine condition over the pair (α, β) is sufficient in order to get persistence of the invariant curve. We also show that this arithmetical condition is optimal. For a nondiophantine pair (α, β) we present a smooth fht having a non-persistent indifferent invariant curve with β as fibered rotation number. We conjecture that in the analytic setting, the Brjuno B1 arithmetical condition is optimal in this sense.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
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Plan of the proof. The proof of Theorem 4.1 will be obtained as an application of the successive conjugacy method (Newton’s algorithm). We will find an analytic fibered affine coordinates change that will put one of the fht’s Ft in the family in the adequate normal form. As usual in this type of proof, we must keep control on several quantities at the same time and perform many different operations. Thus, the whole proof is divided in various parts that we describe now. In Sect. 5 we a review some simple but useful results in complex analysis that will be used in the demonstration of Theorem 4.1. Some conventions about the names of the constants arising in the work are done in that section. In Sect. 6 we show two lemmas which represent the core of the proof. These lemmas define the convergence speeds of the iterative process. Moreover, they give the size of the width strip losses that we must introduce at each stage. These lemmas establish the relations between the arithmetical properties of the rotations numbers and the characteristics of the iterative process to be applied, and therefore, between arithmetics and dynamics. In Sect. 7 we introduce the actual iterative process. In 7.1 we define a constant ℵ which determines the values given by the lemmas in Sect. 6. In 7.2 we make a detailed description of the algorithm. We introduce and describe the different operations used at each stage. We also list the order in which these operations will be applied. In 7.3 we make some preliminary estimates and we assume two additional hypothesis in order to reach the adequate situation which enables us to start with the iterative process. In 7.4, 7.5, 7.6 we perform in a detailed way all the operations and estimates of a single stage of the algorithm. We show indeed that these operations allow us to get the good estimates in order to iterate the process and thus, to get closer to the normal form. This is the more delicate part of the proof, since we must control many quantities that arise as the result of the different operations. In 7.7 we show that our algorithm converges, and thus, it gives the desired normal form for one of the elements Ft in the family. In 7.8 we exhibit a modification of the first stage of the algorithm, in order to obtain more precise results when ε is very small. In Sect. 8 we remove the additional hypothesis introduced in Sect. 7.3. We show that we can get the desired situation as a consequence of the original hypothesis of Theorem 4.1 by means of a previous preparative procedure. Finally, in Sect. 9 we prove a parametrized version of the main theorem. We expect that this version of the theorem should be useful in many other applications. 5. Some Results on Analysis 5.1. Constants. In this work, several positive constants with different origin will appear. Some of them have a fixed numerical value. These constants are called universal and are denoted by C j , where the index represents the order of apparition of the constant in the work. If we denote a constant by C j then this means that this constant is universal. There exist constants which make part of the statement of the theorem; these constants are L , M, T, δ, (α, β). In a third group, there are constants whose values depend on the values of the constants L , M, T, δ, (α, β), and these are denoted by K j . The index represents the order of apparition or the feature of the constant. The special notation ℵ is reserved for a particular constant belonging to this last group. To some extent, this constant determines completely the proof.
5.2. Estimations for truncated analytical functions. Let f be an analytical function defined on the strip Bδ , and let N be a natural number. We define respectively the truncation up to the order N and the remainder up to the order N of f by
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M. Ponce
f | N (θ ) =
fˆ(n)e2πinθ ,
|n|≤N
f | N (θ ) =
fˆ(n)e2πinθ ,
|n|>N
where fˆ(n) is the n th coefficient of the Fourier series of f . These two functions are analytical in the strip Bδ . If δ ∈ (0, δ) and δ − δ is small then there exists a constant C1 such that | fˆ(n)| ≤ f Bδ e−2π |n|δ , f | N Bδ ≤ C1 f Bδ
e−2π N (δ−δ )
δ − δ
(5) .
(6)
The last inequality says that a small loss in the width of the definition strip gives an estimate on the norm of the remainder of f (Cauchy estimate). We recall here the following simple fact. Affirmation 5.1. If f is an analytical function whose differential ∂θ f is bounded in Bδ then f Bδ ≤ f (θ )dθ + ∂θ f Bδ . (7) T1
5.3. Estimates for the solutions of some truncated cohomological equations. We consider the following cohomological equations on g1 and g2 : g1 (θ ) − g1 (θ + α) = p1 (θ ) , e2πiβ g2 (θ ) − g2 (θ + α) = p2 (θ ),
(8) (9)
where α, β are real numbers, p1 , p2 are trigonometric polynomials of orders smaller than N + 1, and T1 p1 (θ )dθ = 0. Using the Fourier coefficients method we obtain the expressions for the solutions (as explained in the Introduction, for the first equation we just retain the solution with zero mean): g1 (θ ) =
0<|n|≤N
g2 (θ ) =
|n|≤N
pˆ 1 (n)e2πinθ , 1 − e2πinα
pˆ 2 (n)e2πinθ . e2πiβ − e−2πinα
Let f 1 , f 2 be analytic functions defined on Bδ , with T1 f 1 = 0. We put p j = f j N , for j ∈ {1, 2}. Denoting ϕ1 (N ) = α (N ), ϕ2 (N ) = α,β (N ) (see Sect. 2), we get a constant C2 such that g j Bδ ≤ C2 N f j Bδ ϕ j (N ) , ∂θ g j Bδ ≤ C2 N 2 f j Bδ ϕ j (N ) for j ∈ {1, 2}. Note that the factor C2 N is far from being optimal, but it will be enough for our purposes.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
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6. Two Technical Lemmas In this section we show two technical lemmas which will provide the definition of two sequences of positive numbers {ln }, {wn } converging to 0 in a very particular way. These sequences will be used as a measure of the rate of convergence of our procedure. These two lemmas will also provide two summable sequences of positive numbers {dn0 }, {dn1 }. These sequences will measure losses in the width of the analyticity strip in order to get some Cauchy estimates. This section is the most important part from the point of view of the relations between dynamics and arithmetic. Let ℵ > 2 be a real parameter and {An }n≥0 , {Bn }n≥0 be the non-decreasing sequences defined by An = α,β (2n ) and Bn = α (2n ). Under the Brjuno B1 condition these sequences verify log An n
2n
log Bn
< ∞,
2n
n
< ∞.
Lemma 6.1. For e−n 16ℵAn Bn 2n
(10)
ℵln e−2π 2 dn1
(11)
2
ln = the following holds: 1. 2. 3. 4.
limn→∞ ln = 0. 1 n n ℵ2 Bn ln < 8 . 1 ℵ2n An ln < 16 . The relation ln+1 =
n d1 n
defines a sequence {dn1 }n≥0 of real positive numbers such that dn1 < K 1 (1 + log ℵ) n
for a constant K 1 depending only on {An }, {Bn }. Proof. The proof of Statements 1, 2, 3 are straightforward. For 4 we put σˆ n = − log(ℵ2n ln ) = log An + log Bn + log 16 + n 2 , and we define a sequence {d˜n1 }n≥0 by ln+1 = ℵln e−2π 2
n d˜ 1 n
.
We obtain that 2π 2n d˜n1 = σˆ n+1 − σˆ n + log 2ℵ.
(12)
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M. Ponce
σˆ n As σˆ n is increasing, the numbers d˜n1 are positive. Since the series 2n is convergent, 1 ˜ the series dn is also convergent, and its value is bounded by K 1 (1 + log ℵ), where K 1 only depends on {An }, {Bn }. We construct now a sequence {xn1 }n≥0 in such a way that the perturbed numbers dn1 = d˜n1 + xn1 verify point 4. For this, we define xn1 as being the solution of the equation e−2π 2 xn = 1. d˜n1 + xn1 n 1
(13)
It is easy to see that (13) has a unique solution xn1 . Moreover, we have xn < sn , where sn is the solution to e−2π 2
ns n
= sn . This equality easily implies that the series sn converges, thus concluding the proof. Lemma 6.2. For wn =
ln , ℵ4n An
(14)
where the sequence {ln }n≥0 is defined as in Lemma 6.1, the following holds: 1 1. ℵwn (2n An )2 < 32 . 1 n 2. n ℵwn 2 An < 16 . 3. The relation
wn+1 wn ℵe−2π 2 = 4 dn0
n d0 n
defines a sequence {dn0 }n≥0 of positive numbers such that dn0 ≤ K 2 (1 + log ℵ),
(15)
(16)
n
for a constant K 2 depending only on {An }, {Bn }. Proof. The proof of Statements 1, 2 are straightforward. Let us show 3. As in the proof of Lemma 6.1 we define a sequence {d˜n0 }n≥0 by the relation wn+1 n ˜0 = wn ℵe−2π 2 dn , 4 that is, d˜0n =
1 2n + 1 + log(A2n+1 Bn+1 ) − log(A2n Bn ) + log 32ℵ . n 2π 2
Therefore, we have a positive and summable sequence. As in Lemma 6.1, we can find a real sequence {xn0 }n≥0 such that the perturbed numbers dn0 = d˜n0 + xn0 verify 3.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
13
7. An Iterative Process 7.1. The choice of ℵ. The sequences defined in the previous section depend on the value of the constant ℵ. In order to use these sequences in our process we need to choose the value ℵ being larger than many quantities which depend only on the values of constants of type K (cf. Sect. 5.1). This choice can be made a priori since the type K constants depend only on the values of the constants L , M, T, δ, (α, β) that appear in the statement of Theorem 4.1. However, along the process we will write explicitly the bounds assumed for ℵ. As in the previous section, by fixing the value of ℵ, we obtain two positive sequences {dn0 }n≥0 , {dn1 }n≥0 . These sequences verify
max dn0 , dn1 < (K 1 + K 2 )(1 + log ℵ) < ∞. n≥0
We choose a natural number n ∗ such that δ
max dn0 , dn1 ≤ , 4 ∗ n≥n
where δ is the initial width of the strip for the θ variable. The proof of Theorem 4.1 will be made using an algorithm divided in stages. The number n ∗ represents the starting stage for our algorithm. At each stage the functions will be analytic on a strip (for the θ variable) whose width equals ∗
3δ , 4 = δ n − max(dn0 , dn1 ).
δn = δ n+1
(17) (18)
The choice of n ∗ implies that for every n ≥ n ∗ one has δ n > 2δ . 7.2. Description of the algorithm. This section is devoted to describe the algorithm in an informal way and the reader interested in the precise estimates of the scheme might want to skip to Sect. 7.3. The proof of Theorem 4.1 will be obtained by showing that there exists an analytic conjugacy of the form h(θ, z) = (θ, u 0 (θ ) + u 1 (θ )z) that puts one of the fht of the family {Ft } into the adequate normal form, with u 1 having zero degree. (As noted in Sect. 3, this implies the desired result.) Recall λ = e2πiβ . After conjugacy one obtains
1 h −1 ◦ Ft ◦ h = θ + α, u 1 (θ+α) ρ0,t (θ ) + λu 0 (θ ) − u 0 (θ + α) 1 + u 1 (θ+α) ρ1,t (θ )u 0 (θ ) + ρt (θ, u 0 (θ )) 1 (θ) λ + ρ1,t (θ ) + ∂z ρt (θ, u 0 (θ )) + z u 1u(θ+α) 1 {ρt (θ, u 0 (θ ) + u 1 (θ )z) − ρt (θ, u 0 (θ )) + u 1 (θ+α) − zu 1 (θ )∂z ρt (θ, u 0 (θ ))} . We will perform a process in order to eliminate the terms ρ0 , ρ1 (this implies the desired normal form). This process is divided in many stages. The goal of each stage is to
14
M. Ponce
decrease the sizes of ρ0 and ρ1 , in such a way that at the limit these sizes are 0. However, the rate of convergence is very slow and deeply depends on the arithmetical conditions for the pair (α, β). At each stage we will apply four types of operations. The first two ones are KAM type (or Newton algorithm type) operations. That is, we solve the related equations but in a simplified way (we only keep terms which turn these equations into cohomological equations). These operations provide the functions u 0 , u 1 as solutions to cohomological equations, which give the corresponding coordinate change related to the stage. Obviously, this coordinate change does not conjugate our transformation into the desired normal form, since we only solve simplified equations. We must deal with some remainders, whose sizes are controlled via the other two operations. The third operation consists on introducing a loss in the width of the analyticity strip for the θ variable. The fourth operation consists on restricting the parameter space. 7.2.1. The four operations. We describe now the first two KAM type operations. These operations consist of solving a truncated cohomological equation (cf. Sect. 5.3): if we are in the stage n we solve the equations with truncation up to the order 2n . Case u 1 = 1 (solving the u 0 equation). After conjugacy by a map h for which u 1 ≡ 1 one obtains h −1 ◦ F ◦ h = θ + α, ρ0,t (θ ) + λu 0 (θ ) − u 0 (θ + α) + ρ1,t (θ )u 0 (θ ) + ρt (θ, u 0 (θ )) + z λ + ρ1,t (θ ) + ∂z ρt (θ, u 0 (θ )) + {ρt (θ, u 0 (θ ) + z) − ρt (θ, u 0 (θ )) − z∂z ρt (θ, u 0 (θ ))} . Choosing u 0 so that ρ0,t (θ )2n + λu 0 (θ ) − u 0 (θ + α) = 0,
(19)
the new dynamics is given by new = ρ1,t u 0 + ρt (·, u 0 ) + ρ0,t 2n ρ0,t new ρ1,t = ρ1,t + ∂z ρt (·, u 0 )
ρtnew (·, z) = ρt (·, u 0 + z) − ρt (·, u 0 ) − z∂z ρt (·, u 0 ). A crucial estimate will be u 0 ≤ C2 2n α,β (2n )ρ0,t ,
(20)
where norms are taken in a domain adapted to the current stage. Having in mind the sizes of ρ0,t and ρ1,t , using this inequality we will see (cf. Sect. 7.4.1) that the size new of ρ0,t − ρ0,t n decreases (“quadratic” effect). The sizes of the remaining involved 2 functions remain almost unchanged.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
15
Case u 0 = 0 (solving the u 1 equation). After conjugacy by a map h for which u 0 ≡ 0 one obtains h
−1
u 1 (θ ) ρ0,t (θ ) +z λ + ρ1,t (θ ) + ◦ Ft ◦ h = θ + α, u 1 (θ + α) u 1 (θ + α)
ρt (θ, u 1 (θ )z) . u 1 (θ + α)
It would be natural to choose u 1 so that u 1 (θ ) λ + ρ1,t (θ ) = λ˜ u 1 (θ + α)
(21)
for an adequate constant λ˜ (given by the zero mean condition in the cohomological equation). However, due to technical reasons, we will only deal with the corresponding 2n truncated cohomological equation. The resulting dynamics is then given by ρ0,t (·) , u 1 (· + α) = λ˜ − λ + Rest (θ, t) ,
new ρ0,t = new ρ1,t
ρtnew =
ρt (·, u 1 (·)z) , u 1 (· + α)
where the expression of Rest (θ, t) will be made precise in the next sections. For u 1 we will get an estimate of the form u 1 − 1 ≤ ℵ2n α (2n )∂θ ρ1,t .
(22)
new and ρ new remains essentially unchanged. The effect of The bound for the sizes of ρ0,t t solving an equation as the truncated version of (21) may be interpreted as an attempt for turning ρ1 independent of θ . Indeed, this operation allows us to control the size of ∂θ ρ1,t . However, to accomplish this task we must control the size of Rest (θ, z).
Loss in the width of the strip. The third operation consists of applying Cauchy estimates on the sizes of the functions. For this, we take the sup on a smaller strip. We call this operation a loss on the analyticity strip. By this way, we control the resulting remainder from the two precedent operations. At the stage n we will lose max(dn0 , dn1 ). The new new width of the strip will be as in (18). This operation provides a control on the size of ρ0,t new and ∂θ ρ1,t . Reduction of the parameter space. Due to Claim 5.1, in order to control size of the new we need to control the size of ∂ ρ new and the complex number new (θ )dθ . ρ1,t ρ 1 θ 1,t T 1,t The latter complex number may be changed with the parameter t due to the transversality condition on the family {Ft }t∈ . Thus, this operation consists of localizing a disc in the parameter space where complex number is small enough. We pick a disc centered this new (θ )dθ , and with a very small radius. at a simple zero of T1 ρ0,t
16
M. Ponce
7.2.2. Description of one stage of the iterative scheme. This Section only presents an informal description, and the complete details will be postponed until Sect. 7.4. At the n , ρ n , ρ n , and the estimates beginning of the n th stage we have the functions ρ0,t t 1,t n ρ0,t ≤ wn ,
(23)
n ρ1,t
(24)
≤ K 3 ln
for a constant K 3 . The sequences ln , wn are those defined in Sect. 6, and the norms are taken on the strip Bδ n . We also have a parameter space disc D(tn , pn ). The goal at the end of the stage is to achieve the corresponding necessary estimates to begin with the n + 1th stage. Each stage splits into 4 parts: First part. In this part we apply many times the “solving the u 0 equation” operation. Hence, this part is composed by many steps. We introduce a supplementary index indin,i+1 n,i+1 , ρ1,t , ρtn,i+1 as the cating the corresponding step. We will obtain the functions ρ0,t result of the step i. At each step the sizes of the functions ρ1,t , ρt remain essentially n,i unchanged. Function ρ0,t is the sum of a quadratic term whose size decreases in half at each step, plus the series of remainders having orders greater than 2n (Fourier series orders). The first part ends when the size of the quadratic term is smaller than wn+1 4 . Second part. This part consists in applying one time the “solving the u 1 equation” operation. This enables to relate the control on the size of ∂θ ρ1,t to a control of a remainder having order greater than 2n . Third part. This part consists in introducing a loss in the width of the strip. The first two parts provide some remainders arising essentially from truncated functions whose sizes are smaller than wn , ln , respectively. Inequality (6), the relations (15), (11), and a n+1 ≤ loss on the strip width of an amount of max(dn0 , dn1 ), provide the estimates ρ0,t n+1 ≤ l wn+1 , ∂θ ρ1,t n+1 , where the norms are taken in a new strip Bδ n+1 . Fourth part. This part consists in applying the operation reduces the parameter which n+1 . Hence, we obtain all the space. We get the bound ln+1 for the complex number T1 ρ1,t necessary estimates to start with the n + 1th stage. Therefore, the proof of Theorem 4.1 depends on the possibility of showing that the hypotheses on the family {Ft }t∈ allow, on the one hand, to obtain good estimates to start the first stage (stage n ∗ ), and on the other hand, to complete our plan for any stage. We will see in Section 7.7 that the properties ensured by Lemmas 6.1 and 6.2 for the sequences ln , wn allow to show, for only one parameter t ∗ in , the convergence of our process. In other words, the successive composition of coordinates changes provided each stage converges to an analytical coordinates change which puts the fibered holomorphic transformation Ft ∗ in an adequate normal form. 7.3. Initializing the algorithm. At the beginning of our process we will deal with functions defined on the following domains: the unitary disc D = D(0, 1) ⊂ C for the z variable; the strip Bδ of width δ for the θ variable; and the disc D(0, K ε) ⊂ C for the parameter t, where the constant K will be explicit very soon (see Eq. (28)). Before starting the iterative process we need to estimate the derivatives with respect to θ of some
Persistence of Invariant Curves for Fibered Holomorphic Transformations
17
functions. Allowing a loss of δ/4 in the width of the strip, classical Cauchy estimates give 24ε , δ 24M =: N , ∂θ ∂z ρt ≤ δ where the last equality provides the definition of the constant N . We may already exhibit the minimal value for the starting sizes, namely, ∂θ ρ1,t ≤
δ ln ∗ . (25) 24 These sizes only depend on δ and n ∗ , which in their turn only depend on L , M, T, δ, (α, β). ε¯ =
Additional hypothesis. We need to assume the following additional hypothesis: 1. ρ0 ≤ wn ∗ . n ∗ ,0 2. t = 0 is a simple zero for T1 ρ1,t (θ )dθ . n ∗ ,0 dθ − 3. There exists a complex number 0 such that L > |0 | > L −1 and T1 ∂t ρ1,t −1
L 0 ≤ 1000 . Notice that these hypotheses are stronger than those required in the statement of Theorem 4.1. However, we will see that we can reduce the general case to this one by a preliminary process whose description we delay up to Sects. 8, 8.1, 8.2. We consider the functions ∗
∗
n ,0 n ,0 ρ0,t = ρ0,t , ρ1,t = ρ1,t , ρtn
∗ ,0
= ρt
(26)
∗
defined on the following domain: the strip of width δ n = 3δ/4 for θ , and the disc ∗ ∗ D(0, R n ,0 ) of radius R n ,0 = 1 for z. The parameter space is the disc D(tn ∗ , pn ∗ ), centered at tn ∗ = 0. The radius of the parameter space disc is pn ∗ = 100ln ∗ =
2400¯ε . δ
(27)
Hence we pick 2400 . (28) δ Since ℵ is a large number, we have that pn ∗ < 1. So far, we have the following estimates to start with the stage n ∗ : K =
∗
n ,0 ≤ ln ∗ , ∂θ ρ1,t n ∗ ,0 ρ0,t ∗ ∂z2 ρtn ,0
T1
≤ wn ∗ ,
≤
n ∗ ,0 osc ∂θ ∂z ρt ≤
∗ osc ∂t ∂z ρtn ,0 ≤ n ∗ ,0 ∂t ρ1,t dθ − 0 ≤
(29) (30)
M,
(31)
2N ,
(32)
2T ,
(33)
L −1 . 1000
(34)
18
M. Ponce
Remark 7.1 (On the starting stage and size of ε). We will see in Sect. 7.8 that a subtle modification in the first stage yields the last claim of Theorem 4.1, which concerns the sizes of u and ε.
7.4. Realization of the iterative scheme. To control the many functions arising during the realization of the iterative scheme, we introduce some real sequences. (The justification of some claims on these sequences made below will be deferred until they become transparent from the point of view of the iterative process.) At the beginning of the stage n we have the family {Ft } written in the form
n,0 n,0 (θ ) + zρ1,t (θ ) + λz + ρtn,0 (θ, z) , Ftn,0 (θ, z) = θ + α, ρ0,t where ρtn,0 vanishes up to order 2 at z = 0. The definition domain for Ftn,0 is the disc D(0, R n,0 ) for the z variable, where the sequence {R n,i }n≥n ∗ ,i∈N∪{∞} verifies 3 ∗ ∗ ∗ ∗ ∗ ∗ < · · · < R n +1,1 < R n +1,0 < R n ,∞ < · · · < R n ,2 < R n ,1 < R n ,0 = 1, (35) 8 The parameter space and the strip Bδ n of width δ n defined by (18) for the θ variable. n,0 is the disc D(tn , pn ) which is centered at a simple zero tn of T1 ρ1,t (θ )dθ , and whose radius equals pn = 100ln .
(36)
In this section we will deal only with the estimates which do not concern the parameter t, and we defer to Sect. 7.5 the estimates involving this parameter. At the beginning of the stage we have the estimates n,0 ρ0,t ≤ wn ,
∂z2 ρtn,0 ≤ Mn , osc ∂θ ∂z ρtn,0 ≤ Nn,0 ,
n,0 ∂θ ρ1,t
≤ ln ,
(37) (38) (39) (40)
where the real sequences {Mn }n≥n ∗ and {Nn,i }n≥n ∗ ,i∈N∪{∞} are bounded from above by some constants K M and K N respectively. Using Claim 5.1, Inequality (40), and an n,0 estimate on the norm T1 ∂t ρ1,t − 0 (see Inequality (94) in Sect. 7.5), we obtain n,0 ρ1,t ≤ K 3 ln
(41)
for some constant K 3 > 0. The goal of Sects. 7.4.1, 7.4.2, 7.4.3 is to show. Lemma 7.2. By applying a finite number of u0 equation operations followed by one application of the u1 equation operation and one application of the loss in the width of the strip operation, we can get the estimates (37), (38), (39) and (40) for the n + 1 index, where norms are taken in domains Bδ n+1 , D(0, R n+1,0 ) for the θ and z variables.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
19
7.4.1. First part: the u 0 equation. Suppose that at the first part of the stage n we have n,i n,i , ρ1,t , and ρtn,i , where the last one is defined on the disc D(0, R n,i ) the functions ρ0,t for the z variable. We will describe the i th step. In this step we search for a coordinate change of the form h tn,i (θ, z) = (θ, u n,i 0,t (θ ) + z) by solving the truncated equation n,i n,i n λu n,i 0,t (θ ) − u 0,t (θ + α) = −ρ0,t |2 .
(42)
By the results of Sect. 5.3, the solution u n,i 0,t satisfies n,i n n u n,i 0,t ≤ C 2 2 α,β (2 )ρ0,t ,
(43)
n,i n n ∂θ u n,i 0,t ≤ C 2 4 α,β (2 )ρ0,t .
(44)
Notice that the resulting new ρ0 is n,i+1 n,i n,i n,i = ρ0,t |2n + ρ1,t u 0,t + ρtn,i (·, u n,i ρ0,t 0,t ),
(45)
n,i |2n satisfies which due to the definition of ρ0,t
n,i+1 n,i n,i |2n = ρ1,t u 0,t + ρtn,i (·, u n,i ρ0,t 0,t ) n .
(46)
n,i+1 n,i = ρ1,t + ∂z ρtn,i (·, u n,i ρ1,t 0,t ),
(47)
n,i n,i n,i n,i ρtn,i+1 (·, z) = ρtn,i (·, u n,i 0,t + z) − ρt (·, u 0,t ) − z∂z ρt (·, u 0,t ).
(48)
2
We also get a new ρ1 ,
as well as a new ρ,
The function ρtn,i+1 (θ, ·) is defined on D(0, rtn,i+1 (θ )), with rtn,i+1 (θ ) = R n,i − |u n,i 0,t (θ )|.
(49)
The sequence {R n,i } verifies R n,i+1 ≤ rtn,i+1 (θ )
(50)
for every θ in Bδ n . From now on the domain for z is D(0, R n,i+1 ). We recall that Eq. (42) is solved without losing width on the analyticity strip. In fact, we will introduce a loss of width of size d0n only at the moment when we will need to estimate the remainders n,i ρ0,t |2n . This will be made only once per stage, and at the third part of each stage.
20
M. Ponce
Estimates at the first part of the stage. We define n,0 t = ρ0,t , ηn,0
(51)
n,i n,i t = ρ1,t u 0,t + ρtn,i (θ, u n,i ηn,i+1 0,t ).
(52)
Since (46) yields n,i+1 t ρ0,t |2n = ηn,i+1 |2 n ,
(53)
t each time that we solve Eq. (42) we actually obtain a better bound on ηn,i+1 . This allows to improve (43) and (44) to n n t u n,i 0,t ≤ C 2 2 α,β (2 )ηn,i ,
∂θ u n,i 0,t
(54)
t ≤ C2 4n α,β (2n )ηn,i .
(55)
Lemma 7.3. There exists a constant K 4 ≥ K 3 such that ∂z2 ρtn,i ≤ Mn ,
osc ∂θ ∂z ρtn,i ≤ Nn,i , wn t ≤ i , ηn,i 2 n,i ∂θ ρ1,t ≤ 2ln ,
(56)
n,i ρ1,t
(60)
(57) (58) (59)
≤ K 4 ln
hold for every i ≥ 0. The proof is by induction on i ≥ 0. During the proof we will also provide an inductive definition for the sequences {R n,i }i≥0 and {Nn,i }i≥0 . Notice that inequalities (54), (55) and (58) give C2 ln , ℵ2n 2i C2 ln . ∂θ u n,i 0,t ≤ ℵ2i u n,i 0,t ≤
(61) (62)
The case i = 0 for the induction is furnished by the hypothesis (37), . . . ,(41). We now assume that the desired relations hold for every j ∈ {0, . . . , i} and that we have at our disposal the constants R n, j and Nn, j for every j ∈ {0, . . . , i}. We let R
n,i+1
=R
n,i
− C2 2 α,β (2 )wn n
n
R n,∞ = R n,0 − 2C2 2n α,β (2n )wn .
1 2i
,
(63) (64)
Persistence of Invariant Curves for Fibered Holomorphic Transformations
21
If ℵ is larger than C2 then inequality (50) follows from (49) and (61). Inequality (60) follows from (47), (41), (56) and (61), since i n, j n, j n,i+1 n,0 ∂z ρt (θ, u 0,t ) ρ1,t ≤ ρ1,t + j=0 ≤ K 3 l n + Mn
i
n, j
u 0,t
j=0
≤ K 3 ln + K M C2 ≤ K 4 ln .
ln 2n
Notice that the constant K 4 only depends on C2 , K 3 and K M . From relation (48) we obtain ∂z2 ρtn,i+1 = ∂z2 ρtn,i (·, z + u n,i 0 ) ≤ Mn , which implies (56). Differentiating (48) we get n,i n,i 2 n,i ∂θ ∂z ρtn,i+1 = ∂θ ∂z ρtn,i (·, z + u n,i 0,t ) + ∂z ρt (·, z + u 0,t )∂θ u 0,t n,i n,i 2 n,i −∂θ ∂z ρtn,i (·, u n,i 0,t ) − ∂z ρt (·, u 0,t )∂θ u 0,t .
By estimating the corresponding terms we can bound the oscillation
wn osc ∂θ ∂z ρtn,i+1 ≤ Nn,i + 2K M K 2 4n α,β (2n ) i . 2 The above inequality allows us to give the definition of the sequence {Nn,i }i≥0 : letting Nn ∗ ,0 = 2N and assuming that Nn,i is already defined, we put wn (65) Nn,i+1 = Nn,i + 2K M C2 4n α,β (2n ) i . 2 We also define Nn,∞ as the supremum of the Nn,i ’s, that is, Nn,∞ = Nn,0 + 4K M C2 4n α,β (2n )wn . It is not hard to check that
osc ∂θ ∂z ρtn,i+1 ≤ Nn,i+1 .
Relation (47) allows us to (compute and) estimate ∂θ ρ1n,i+1 by i
n, j n, j n,i+1 n,0 = ∂θ ∂z ρt (θ, u 0,t ) ∂θ ρ1,t ∂θ ρ1,t + j=0 i n, j n, j n, j n, j 2 n, j ≤ ln + ∂θ ∂z ρt (θ, u 0,t ) + ∂z ρt (θ, u 0,t )∂θ u 0,t j=0 ≤ ln +
i j=0
≤ ln +
n, j
n, j
Nn, j u 0,t + Mn ∂θ u 0,t
K5 ln , ℵ
(66)
22
M. Ponce
where K 5 is a constant which only depends on C2 , K M and K N . If we pick ℵ larger than K 5 we get n,i+1 ∂θ ρ1,t ≤ 2ln . t Finally, we estimate ηn,i+1 using point 3 of Lemma 6.1 and point 1 of Lemma 6.2, t ηn,i+1 ≤ K 4 ln C2 2n α,β (2n )
1 wn 1 wn + i i 16 2 2 16 2i wn ≤ i+1 . 2
2 wn n n wn C + M 2 (2 ) n 2 α,β 2i 2i
≤
We used here that ℵ > K 4 C2 and ℵ > K M C22 .
End of the first part. We apply the operation “solving the u 0 equation” until the size of t becomes smaller than wn+1 , more precisely, when wn is smaller than wn+1 . At this ηn,i 4 4 2i time i n we stop and we pass to the second part of the stage. 7.4.2. Second part: the u 1 equation. This part consists in solving the equation for u 1 only once. More precisely, we search for a coordinates change of the form h tn (θ, z) = (θ, u˜ n1,t (θ )z) such that u˜ n1,t (θ ) u˜ n1,t (θ
+ α)
n,i n ρ1,t (θ ) + λ = λtn .
(67)
Notice that if we put ev˜t (θ) = u˜ n1,t (θ ), Eq. (67) reduces to solving n
n,i n (θ ) + λ). v˜tn (θ ) − v˜tn (θ + α) = log λtn − log(ρ1,t
(68)
However, due to technical reasons we will just deal with the following truncated equation: n,i n vtn (θ ) − vtn (θ + α) = log λtn − log ρ1,t (θ ) + λ n . (69) 2
The complex number λtn is the unique one for which a continuous solution to this equation exists (see Sect. 5.3). More precisely, it corresponds to the unique complex number for which the mean of the r.h.s. vanishes, that is, n,in ρ 1,t n,i log λtn = + 1 dθ. (70) log ρ1,t n (θ ) + λ dθ = log λ + log λ T1 T1 2n We have
n,i ρ1 n + 1 dθ ≤ C3 ρ1n,in log T1 λ
Persistence of Invariant Curves for Fibered Holomorphic Transformations
23
for a constant C3 . To estimate the solution of (69) we need an estimate on the size of the r.h.s. For this, we rewrite this expression as n,in n,in
ρ1,t ρ1,t n,i n t + 1 dθ − log +1 log λn − log ρ1,t (θ ) + λ = log λ λ T1 and we obtain
n,i n n,i n log λtn − log ρ1,t (θ ) + λ ≤ C4 ρ1,t for a constant C4 . We have n,i n vtn ≤ C2 C4 2n α (2n )ρ1,t ,
(71)
n,i n )ρ1,t ,
(72)
∂θ vtn
≤ C2 C4 4 α (2 n
n
n ) and conjugating by since we choose the zero mean solution. Letting u n1,t = exp(v1,t h tn = (θ, u n1,t (θ )z) we get the new functions
n+1,0 ρ0,t =
ρtn+1,0
n,i n (·) ρ0,t
, u n1,t (· + α) 1 ρ n,in (·, u n1,t z). = n u 1,t (· + α) t
(73) (74)
The last one is defined on a disc D(0, rtn+1,0 (θ )) for the z variable, with rtn+1,0 (θ ) =
R n,in . |u n1,t (θ )|
(75)
The sequence {R n,i } verifies R n+1,0 ≤ rtn+1,0 (θ )
(76)
for every θ in the strip Bδ n . From now on, the domain for the z variable is the disc D(0, R n+1,0 ). The untouched remainder in Eq. (69) gives rise to a new function ρ1 defined by n,i n (θ)+λ) n log(ρ1,t n+1,0 2 − λ. (77) ρ1,t (θ ) = λtn e Estimates at the second part of the stage. The goal of this paragraph is Lemma 7.4. The following estimates hold: ∂z2 ρtn+1,0 ≤ Mn+1 , osc ∂θ ∂z ρtn+1,0 ≤ Nn+1,0 .
24
M. Ponce
By (71) and (60) we have u n1,t − 1 ≤ C3 C2 C4 2n α (2n )K 4 ln .
(78)
Therefore, there exists a constant K 6 such that u n1,t ≤ e K 6 2 (u n1,t )−1
≤e
n (2n )l α n
,
(79)
.
(80)
K 6 4n α (2n )ln
(81)
K 6 2n α (2n )ln
Using the equality ∂θ u n1 = ∂θ v n u n1 and (72) we obtain ∂θ u n1,t ≤ e K 6 2
n (2n )l α n
≤ 3K 6 4 α (2 )ln , n
n
(82)
provided that ℵ is larger than K 6 . We remark that the estimate (78) says that the topological degree of u 1 is zero. We may now proceed to conclude the definition of the sequence {R n,i } and to prove the lower bound 3/8 for it. The sequence {R n,i }. Assume that R n,∞ is already defined. We put R n+1,0 =
R n,∞ eℵ2
n (2n )l α n
.
(83)
With this definition, inequality (76) follows from (75) and (80). The sequence R n,i also satisfies the monotonicity properties announced in (35). Finally, to check that R n,0 (and therefore each R n,i ) is bounded from below by 38 , we first note that R n+1,0 = 1 −
n
R j,0 − R j+1,0 .
j=0
Each of the differences above can be estimated by R j,0 − R j+1,0 = R j,0 (1 − e−ℵ2
j (2 j )l α j
)+
2C2 2 j α,β (2 j )w j
eℵ2 α (2 ≤ ℵ2 j α (2 j )l j + 2C2 2 j α,β (2 j )w j . j
j )l
j
Thus, from point 2 of Lemma 6.1 and point 2 of Lemma 6.2, n 0
5 R j,0 − R j+1,0 ≤ ℵ2n α (2n )ln + 2 C2 2n α,β (2n )wn < . 8 n n
Relation (74) gives ∂z2 ρtn+1,0 (θ, z) =
u n1,t (θ )2 u 1,t (θ + α)
∂z2 ρtn,in (θ, u n1,t (θ )z).
Therefore, according to the definition below of the sequence {Mn } we get the desired estimate for ∂z2 ρt , namely, ∂z2 ρtn+1,0 ≤ Mn e3K 6 2 = Mn+1 .
n (2n )l α n
(84)
Persistence of Invariant Curves for Fibered Holomorphic Transformations
25
The sequence {Mn }. We first let Mn ∗ = M, and assuming that Mn is defined, we let Mn+1 = Mn e3ℵ2
n (2n )l α n
.
(85)
According to this definition, one easily checks the existence of an upper bound K M for the sequence Mn . In order to compute the oscillation ∂θ ∂z ρtn+1,0 we first note that ∂θ ∂z ρtn+1,0 (θ, z) =
∂θ u n1,t (θ )u n1,t (θ + α) − ∂θ u n1,t (θ + α)u n1,t (θ ) u n1,t (θ + α)2
+
u n1,t (θ ) u n1,t (θ
+ α)
∂z ρtn,in (θ, u n1,t (θ )z)
∂θ ∂z ρtn,in (θ, u n1,t (θ )z)
+ ∂z2 ρtn,in (θ, u n1,t (θ )z)z∂θ u n1,t (θ ) , which implies
osc ∂θ ∂z ρtn+1,0 ≤ 4∂θ u n1,t u n1,t (u n1,t )−1 2 K M + Nn,in u n1,t (u n1,t )−1 + 2(u n1,t )−1 u n1,t K M ∂θ u n1,t .
(86)
At this point we can conclude the definition of the sequence {Nn,i } and show the existence of an upper bound K N for it. The sequence {Nn,i }. Assuming that Nn,∞ is already defined, we put Nn+1,0 = e2K 6 2
n (2n )l α n
Nn,∞ + 144K M K 6 4n α (2n )ln .
(87)
By replacing the value of Nn,∞ (see (66)) in the definition of Nn+1,0 we obtain Nn+1,0 ≤ e2K 6 2
n (2n )l α n
≤ e2K 6 2
n (2n )l α n
≤ e2K 6
Nn,0 + 8K M C2 4n α,β (2n )wn + 144K M K 6 4n α (2n )ln
Nn,0 + K 7 4n α (2n )ln
2n α (2n )ln 4n α (2n )ln . 2N + K 7
(88)
These last inequalities hold for a well defined constant K 7 . We see that (88) and the properties of the sequence {ln } imply an upper bound K N for the sub-sequence Nn,0 . Finally, due to the following monotonicity relations, 2N = Nn ∗ ,0 < Nn ∗ ,1 < · · · < Nn ∗ ,∞ < Nn ∗ +1,0 < . . . , this implies the existence of an upper bound for the whole sequence {Nn,i }. This definition together with inequality (86) finally give us
osc ∂θ ∂z ρtn+1,0 ≤ Nn+1,0 .
(89)
We conclude this second part of the stage by writing out the functions whose expressions involve some truncated functions and for which we need to allow a loss in the width of the strip in order to get good estimates. First of all, relation (77) implies
n,i n +λ) n log(ρ1,t n,i n n+1,0 t 2 ∂θ log(ρ1,t ∂θ ρ1,t = λn e + λ) n . 2
26
M. Ponce
Since the truncation operator commutes with the derivation with respect to θ , we have
n,i n 1 +λ) n log(ρ1,t n,i n n+1,0 t 2 ∂θ ρ1,t (90) ∂θ ρ1,t = λn e . n,i n ρ1,t + λ 2n Finally, the expression for ρ0 is n+1,0 ρ0,t
⎞ ⎛ i n −1 1 t t ⎠ ⎝ηn,i = n + ηn, = n j |2 n . n u 1,t (· + α) u 1,t (· + α) n,i n ρ0,t
(91)
j=0
7.4.3. Third part: loss on the width of the strip. In order to estimate the size of the truncated functions appearing in (90) and (91), we introduce a loss in the width of the strip of size max(dn0 , dn1 ). The new domain for the θ variable is δ n+1 . We can estimate n+1,0 by ρ0,t ⎛ ⎞ n d0 i n −1 −2π 2 n wn+1 wn e n+1,0 ⎠ + ≤ 2⎝ C1 ρ0,t 4 2 j dn0 j=0
wn+1 wn e−2π 2 ≤ + 4C1 2 dn0 ≤ wn+1 ,
n d0 n
provided that ℵ is larger than 2C1 . Note that here we used the estimate (u n1,t )−1 < 2. This estimate is not sharp but it is enough for our purposes. To estimate the expression n,i n in (90) we first notice that, since ρ1,t is uniformly small, there exists a constant C5 such that n+1,0 ∂θ ρ1,t
≤
n,i n −2π 2 e C5 ∂θ ρ1,t
dn1
2C5ln e−2π 2 dn1 ≤ ln+1 . ≤
n d1 n
n d1 n
(92)
This last inequality holds thanks to the definition of the sequence {ln }n≥0 (see Lemma 6.1) and the fact that ℵ is greater than 2C5 . In this way, we get the family of inequalities (37). . . (40) for the index n + 1. This finishes the third part and shows Lemma 7.2. 7.5. Estimates with respect to the parameter. Before dealing with the fourth and last part of the stage, we treat again the first three parts in order to get some estimates on the parameter t. These estimates will be essential in the fourth part, where we will reduce the parameter space in an important way. However, in this section we will also reduce the parameter space in order to get some useful Cauchy estimates. At the beginning of the stage n we have the following bounds:
osc ∂t ∂z ρtn,0 ≤ Tn,0 , (93) n,0 ∂t ρ1,t dθ − 0 (94) ≤ sn . T1
Persistence of Invariant Curves for Fibered Holomorphic Transformations
27
The sequence of real numbers {Tn,i }n≥n ∗ ,∈N∪{∞} verifies Tn+1,0 ≤ n K T 2n α (2n )
(95)
for a constant K T . The sequence of real numbers {sn }n≥n ∗ verifies L −1 L −1 = sn ∗ < sn ∗ +1 < sn ∗ +2 < · · · < . 1000 100
(96)
The goal of this section is to show. Lemma 7.5. By applying a finite number of u0 equation operations followed by one application of the u1 equation operation and one application of the loss in the width of the strip operation, we can get the estimates (93) and (94) for the n + 1 index, where norms are taken in domains Bδ n+1 , D(0, R n+1,0 ) for the θ and z variables. Estimates at the first part of the stage. We will show inductively on i ≥ 0 that
osc ∂t ∂z ρtn,i ≤ Tn,i .
(97)
Suppose that this estimate holds for 0 ≤ j < i + 1. We start by estimating ∂t u n,i 0,t . For this we use a Cauchy estimate, that is, we lose a small quantity in the radius of the disc D(tn , pn ) (the parameter space) in order to get an estimate for ∂t u n,i 0,t . More precisely, we construct a sequence of radius { pn − }i∈N∪∞ defined by i
pn − = pn , −1 i 2 pn − = pn − − l n , i i−1 3 pn −∞ = pn − 3ln . i This says that at each step we lose a very small quantity of radius of size ln 23 (we use this loss in the estimate (98) that follows). We note that at the limit, the total loss has size 3ln . From (61) the Cauchy estimate gives 6C2 3 i ∂t u n,i ≤ . (98) 0,t ℵ2n 4 We can now estimate the oscillation of ∂t ∂z ρ by n,i n,i 2 n,i ∂t ∂z ρtn,i+1 = ∂t ∂z ρtn,i (θ, u n,i 0,t + z) + ∂z ρt (θ, u 0,t + z)∂t u 0,t n,i n,i 2 n,i −∂t ∂z ρtn,i (θ, u n,i 0,t ) − ∂z ρt (θ, u 0,t )∂t u 0 ,
osc ∂t ∂z ρtn,i+1 ≤ Tn,i + 2Mn ∂t u n,i 0,t .
This enables us to define the sequence {Tn,i }i≥0 by letting Tn ∗ ,0 = 2T and defining recursively 12K M C2 3 i Tn,i+1 = Tn,i + . (99) ℵ2n 4
28
M. Ponce
We define Tn,∞ by Tn,∞ = Tn,0 +
48K M C2 . ℵ2n
(100)
Notice that Tn,0 is an upper bound for the Tn,i , and the following inequality holds:
osc ∂t ∂z ρtn,i+1 ≤ Tn,i+1 . Estimates at the second part of the stage. First, we estimate the size of ∂t u n1 = u n1 ∂t vn . By taking derivatives with respect to t in both sides of (69) we obtain ∂t vtn (θ ) − ∂t vtn (θ
+ α) = −
2n . +λ
n,i n ∂t ρ1,t n,i n ρ1,t
(101)
1
2n
Recall that the notation |1 means that we only keep the Fourier series terms with orders between 1 and 2n . The best estimate that we can get for the right-hand side function (in his original untruncated form) is of the order of a constant. More precisely, by introducing a loss in the radius of the disc D(tn , pn −∞ ) of size ln , we get a Cauchy estimate ∂t ρ n,in 1,t n,in ≤ 12K 4 . ρ + λ
(102)
1,t
The new parameter space disc is D(tn , pn − ), where pn − = pn −∞ − ln = pn − 4ln . Then we have ∂t u n1,t ≤ C2 24K 4 2n α (2n ).
(103)
We may also estimate the oscillation of ∂t ∂z ρtn+1,0 by ∂t ∂z ρtn+1,0 (θ, z) =
∂t u n1,t (θ )u n1,t (θ + α) − ∂t u n1,t (θ + α)u n1,t (θ ) u n1,t (θ + α)2
+
u n1,t (θ ) u n1,t (θ + α)
∂z ρtn,in (θ, u n1,t (θ )z)
∂t ∂z ρtn,in (θ, u n1,t (θ )z)
+ ∂z2 ρtn,in (θ, u n1,t (θ )z)z∂t u n1,t (θ ) ,
osc ∂t ∂z ρtn+1,0 ≤ 4∂t u n1,t u n1,t (u n1,t )−1 2 Mn + u n1,t (u n1,t )−1 Tn,in + 2u n1,t (u n1,t )−1 Mn ∂t u n1,t .
(104)
We introduce now the definition of the sequence {Tn,0 }n≥n ∗ and show the claimed upper bound (95).
Persistence of Invariant Curves for Fibered Holomorphic Transformations
29
The sequence {Tn,i }. Suppose Tn,∞ is defined. We put Tn+1,0 = K˜ T 2n α (2n ) + 2Tn,∞ ,
(105)
where the constant K˜ T is given by K˜ T = 32K M C2 24K 4 . If we replace the value of Tn,∞ (see (100)) in the definition of Tn+1,0 we get K˜ T Tn+1,0 ≤ K˜ T 2n α (2n ) + 2Tn,0 + n . 2 This gives the desired estimate (in fact, we get a sharper estimate). By looking at the inequality (104) and the definition above we obtain
osc ∂t ∂z ρtn+1,0 ≤ Tn+1,0 . Estimates at the third part of the stage. We can estimate the distance between T1 ∂t ρ1,t and 0 and show that this distance is small. In Sect. 7.6 this will allow us to show that T ∂t ρ1,t (θ )dθ grows almost like a linear map. In order to simplify the notation, we introduce the constants K 8 , K 9 in such a way that ∂t
i
n, j n, j ∂z ρt (θ, u 0,t (θ ))
=
0
i
n, j
n, j
n, j
n, j
n, j
∂t ∂z ρt (θ, u 0,t ) + ∂z2 ρt (θ, u 0,t )∂t u 0,t ,
0
i K8 n, j n, j ∂z ρt (θ, u 0,t (θ )) ≤ Tn,∞ 2C2 2n α,β (2n )wn + n ∂t ℵ2 0
≤
K9 . ℵ2n
(106)
We will also need some estimates on the size of ∂t (λtn ). For this, it will be very useful to have a more developed expression for it. From (70) we have ∂t (λtn ) = λtn ∂t (log λtn ) ⎞ ⎛ n,i n ∂t ρ1,t λtn ⎝ = dθ ⎠ n,i n ρ1,t λ T1 1+ λ
⎞ ⎛ ∂t ρ n,0 + ∂t in −1 ∂z ρ n, j (θ, u n, j ) t 0,t 0 1,t λtn ⎝ = dθ ⎠ . i,i n ρ1,t λ T1 1+ λ By differentiating (77) we get n+1,0 ∂t ρ1,t
=
n,i n
∂t (λtn )e{log(ρ1
+λ)}|2n
n,i
n + λtn e{log(ρ1,t +λ)}|2n
. +λ
n,i n ∂t ρ1,t n,i n ρ1,t
2n
30
M. Ponce
n+1,0 The difference between the integral of ∂t ρ1,t and 0 may be written as the sum I1 + I2 + I3 of three terms corresponding to the expressions in the three lines below: ⎞ ⎛ n,0 {log(ρ1n,i n +λ)}|2n t ∂ ρ λn e t 1,t n+1,0 ∂t ρ1,t (θ )dθ − 0 = dθ ⎠ − 0 dθ ⎝ n,i n 1 1 ρ1,t λ T T T 1+ λ
⎞ ⎛ n, j n, j i n −1 n,i n ∂t ∂z ρt (θ, u 0,t ) 0 λtn e{log(ρ1 +λ)}|2n dθ ⎝ + dθ ⎠ n,i n ρ1,t λ T1 T1 1+ λ n,i n ∂t ρ1,t n,i n + λtn e{log(ρ1,t +λ)}|2n dθ. T1 ρ n,in + λ 1,t
2n
Notice that the term I1 itself equals the sum of the following three terms: ⎞ ⎛ n,0 {log(ρ1n,i n +λ)}|2n t ∂ ρ − 1 λn e t 1,t dθ ⎠ dθ ⎝ n,i n 1 1 ρ1,t λ T T 1+ λ ⎛ ⎞⎛ ⎞ n,i n n,i n ρ ρ1,t 1 λtn e{log( λ +1)}|2n ⎟ ⎝ ⎜ λ +⎝ dθ ⎠ dθ ⎠ n,i n ρ1,t λ T1 T1 1+ λ n,i n,i ρ1 n ρ1 n + e log( λ +1)+{log( λ +1)}|2n − 1dθ 0 . T1
Hence, there exists a constant K s such that I1 ≤ e K s ln sn + K s ln .
(107)
n,i n λtn , ρ1,t
The terms are bounded from above by a universal constant. This together with (106) implies that there exists a constant C6 such that C6 K 9 . (108) ℵ2n The term I3 equals the integral of a universally bounded term times the truncated function whose size is also bounded by a type K constant. Notice that these bounds exist even before the introduction of any loss on the width of the strip, see (102)). When introducing the loss on the width, we obtain a constant K 10 for which the following bound holds: I2 ≤
e−2π 2 I3 ≤ K 10 dn1
n d1 n
=
K 10 ln+1 K 10 < 2n+1 . ℵln ℵe
(109)
By considering these three bounds for the integrals I1 , I2 and I3 , we get the following estimate at the third part of the stage: C6 K 9 K 10 n+1,0 K s ln sn + K s l n + + 2n+1 . (110) 1 ∂t ρ1,t dθ − 0 ≤ e n ℵ2 ℵe T In the next paragraph we will give an explicit definition of the sequence {sn } and also prove its claimed properties.
Persistence of Invariant Curves for Fibered Holomorphic Transformations
31
The sequence {sn }. We define this sequence by recurrence by letting L −1 , 1000 C6 K 9 K 10 + K s ln + + 2n+1 . ℵ2n ℵe
sn ∗ = sn+1 = sn e K s ln
(111) (112)
By iterating the definition we get the estimate ∞ ∞ ∞ " " K C K 6 9 10 Ks l j K l sn < e e sj Ks l j + + 2 j+1 sn ∗ + ℵ2n ℵe 0 0 0 ∞
2C6 K 9 K 10 Ks j l j Ks j l j + ≤ e sn ∗ + e lj + Ks . ℵ eℵ 0
If ℵ is large enough then this gives (96) as desired.
Relation (110) implies (tautologically) that n+1,0 ∂ ρ dθ − t 1,t 0 ≤ sn+1 , T1
and the proof of the Lemma 7.5.
(113)
7.6. Fourth part, reduction on the parameter space. Now we arrive to a different stage of our process. Indeed, the estimates here do not arise either as a consequence of the remainders of a linearized equation (quadratic residue) or as the result of some Cauchy estimates associated to some loss in the strip. In this section, we get our estimates as a consequence of the transversality properties of the family {Ft }. Let’s see the necessity of such a hypothesis. From relation (77), at this moment we have a constant C7 such that n,i n n+1,0 ρ1,t ≤ C7 ρ1,t ≤ C7 K 4 ln .
(114)
n+1,0 In other words, the size of ρ1,t is still of order ln . This fact is not a surprise, since we have made operations which deal only with the size of the derivative ∂θ ρ1 . We will use Affirmation 5.1 to control ρ1 . Thus, we need to control the size of the complex number T1 ρ1,t (θ )dθ . To do this we will strongly use the transversality property for the family n+1,0 {Ft }. We will find a parameter tn+1 such that T1 ρ1,t = 0. Then, we will exploit the n+1 n+1,0 fact that the size of T1 ρ1,t is very small if we are placed very near this special parameter, due to the continuity of the functions. More precisely, as in (36) we will reduce the parameter space D(tn , p n − ) by finding a new disc D(tn+1 , pn+1 ) ⊂ D(tn , pn − ) with n+1,0 tn+1 being a simple zero of T1 ρ1,t (θ )dθ and pn+1 small enough.
7.6.1. An application of Rouché Lemma. Lemma 7.6. There exists a complex parameter tn+1 which is a simple zero for n+1,0 − ρ 1 T 1,t (θ )dθ and such that D(tn+1 , pn+1 ) ⊂ D(tn , pn ).
32
M. Ponce
Proof. We will use the classical Rouché Lemma to find tn+1 . To do this, we need to show that the hypotheses of the Rouché Lemma are satisfied on the boundary of some disc around tn . We develop t n+1,0 ρ1,t (θ ) λn {log(ρ n,in (θ)+λ)}| n 1,t 2 − 1 dθ dθ = e λ λ T1 T1 n,i ⎛ ⎞ ρ1,t n (θ) n,i n log +1 dθ+{log(ρ (θ)+λ)}| 1 1,t λ 2n ⎜ T ⎟ = − 1⎠ dθ. ⎝e T1
We then define g+ (t) as being the value of this last expression. We compare this function ρ n,0 (θ) with the function g(t) = T1 1,tλ dθ on the boundary of some disc D(tn , r ): n,in ρ1,t (θ ) + 1 dθ − g(t) log |g+ (t) − g(t)| ≤ λ T1 T1 n,i n 2 Z + log(ρ1,t (θ ) + λ) |2n + Z e dθ, n,in ρ1,t (θ) n,i n + 1 dθ + log(ρ1,t (θ ) + λ) |2n . Then there exists a where Z = T1 log λ constant K 11 such that Z < K 11ln < 1, provided that ℵ is large enough. Thus, using the Taylor series expansion of log
n,i n ρ1,t (θ) λ
+ 1 , we have
⎛ ⎞ n,in 2 n,i n n,0 ρ1,t (θ ) ω(θ, t) ρ1,t (θ ) (θ ) ρ1,t ⎝ ⎠ dθ − − |g+ (t) − g(t)| ≤ λ 2 λ λ T1 n,i + log ρ1,t n (θ ) + λ n + Z 2 e Z dθ 2 T1 i n −1 ω(θ, t) ρ n,in (θ ) 2 n, j n, j 1,t dθ < ∂z ρt (θ, u 0,t (θ )) + 1 2 λ T 0 n,i + log ρ1,t n (θ ) + λ n + Z 2 e Z dθ, T1
2
where the absolute value of ω(θ, t) lies between 0 and 1. One can easily check the following estimates: i n −1 2C2 K M ln n, j n, j , (115) ∂z ρt (θ, u 0,t (θ )) dθ ≤ 1 ℵ T 0 n,in 2 2 2 ω(θ, t) ρ1,t (θ ) dθ ≤ K 4 ln , (116) 2 λ 2 T1 T1
n,i log ρ1,t n (θ ) + λ
−2π 2n dn1 C3 K 4 ln+1 dθ ≤ C3 K 4 ln e . ≤ 1 n 2 dn ℵ
(117)
Persistence of Invariant Curves for Fibered Holomorphic Transformations
33
Therefore, if ℵ is large enough, for every r ≤ pn − and every t ∈ D(tn , r ) we have |g+ (t) − g(t)| < 3ln L −1 . (118) The transversality condition on the derivative ∂t T ρ1,t (θ )dθ implies that g(t) grows almost as a linear application, namely, t 0 (t − tn ) 0 − ∂t g(t) = − g(t) + g(tn ) . sn |t − tn | ≥ λ λ tn This implies that the inequality
99 −1 99 −1 |g(t)| ≥ L −1 − sn |t − tn | ≥ L |t − tn | = L r 100 100 holds on the boundary of a disc D(tn , r ). Therefore, if the radius r is at least 100 99 3ln , then the hypothesis in the Rouché Lemma is satisfied. So, we get a simple zero, tn+1 , for g+ (t) inside D(tn , r ). Now we need to show that if we choose r as being this minimal value, then the new zero tn+1 lies effectively inside the parameter space D(tn , pn − ). To do this, it is enough to see that pn − − 3ln
100 > 92ln > 0. 99
(119)
Moreover, we need to show that we can start with the stage n + 1 having a parameter space disc of radius pn+1 . For this we notice that pn+1 = 100ln+1 100α,β (2n )α (2n ) = ln α,β (2n+1 )α (2n+1 )2e2n+1 < 25ln .
(120)
From (119) and (120) we conclude that the disc D(tn+1 , pn+1 ) is contained in D(tn , pn − ). Thus, the new parameter space is the disc D(tn+1 , pn+1 ). 7.7. Convergence of the method. As we have seen in the previous sections, we can iterate the algorithm infinitely many times for the fht corresponding to the parameter t¯ = ∩n≥n ∗ D(tn , pn ). (Of course, t¯ is the parameter claimed in Theorem 4.1.) In what follows, we will omit this parameter in all of the notation, and we will deal with F = Ft¯. We need to show that the successive compositions of the conjugacies constructed in the course of our process gives rise to an analytical change of coordinates. The decreasing rates for the functions and all of the estimates arising in our process are sufficient for this. We let δ
∞
∗
=δ −
∞ n∗
max(dn0 , dn1 ) >
δ , 2
and we denote Hn (θ, z) = h n,0 ◦ h n,1 ◦ · · · ◦ h n,in −1 ◦ h n (θ, z) = θ, u n0 (θ ) + u n1 (θ )z ,
(121)
34
M. Ponce
where we write u n0 (θ ) = up to the stage n is
in −1 i=0
u n,i 0 (θ ). The successive compositions of the conjugacies
∗
∗ ∗ ∗ Hn ∗ ◦ Hn ∗ +1 ◦ · · · ◦ Hn (θ, z) = θ, u n0 + u n1 u n0 +1 + u n1 +1 (. . . (u n0 + u n1 z) . . .) ⎛ ⎛ ⎞ ⎞ n−1 n " " ∗ ∗ ∗ j j u 1⎠ u n0 +⎝ u 1⎠z . = θ, u n0 +u n1 u n0 +1 + · · · + ⎝ j=n ∗
j=n ∗
We need to show that the constant term and the coefficient of z above, converge uni# j formly on θ when n goes to infinity. For this, it is enough to see that the product ∞ n ∗ u 1 (θ ) ∞ j converges uniformly and the series n ∗ u 0 (θ ) converges absolutely and uniformly. For the product we have ⎛ ⎞ i i " j log ⎝ u 1 (θ )⎠ = v j (θ ). j=n ∗
j=n ∗
The above series converges absolutely and uniformly by the estimate (71) and point 2 of Lemma 6.1. The series of u 0 is n j=n ∗
j
u 0 (θ ) =
j −1 n i
j=n ∗ i=0
j,i
u 0 (θ ),
which allows to obtain the estimate j −1 n i ∞ n 1 1 j j j wj u ≤ C 2 (2 ) < . 2 α,β 2 0 2i 8 4jej j=n ∗ j=n ∗ i=0 j=n ∗
(122)
Bδ ∞
Thus, the following limit exists: H (θ, z) = lim Hn ∗ ◦ Hn ∗ +1 ◦ · · · ◦ Hn (θ, z). n→∞
We write this limit in the form H (θ, z) = (θ, u 0 (θ ) + u 1 (θ )z), where the functions u 0 , u 1 are analytical on the strip Bδ ∞ . The function u 1 has zero topological degree since the topological degree is a homomorphism and each function j u 1 has zero degree. The conjugacy of F by H takes the form H −1 ◦ F ◦ H (θ, z) = (θ + α, λz + ρ ∞ (θ, z)). Indeed, the sequences ln , wn which control the size of ρ0n,0 , ρ1n,0 go to 0 when n goes to infinity. The function ρ ∞ vanishes up to order 2 at z = 0 and it is defined on the disc D(0, R ∞ ), with 38 ≤ R ∞ . The invariant curve is u 0 (θ ), and due to (122) it verifies u 0 Bδ∞ ≤
∞ 1 1 . j e j2 8 4 ∗ j=n
(123)
Persistence of Invariant Curves for Fibered Holomorphic Transformations
35
7.8. A modification at the first stage and the size of ε. Let n¯ ≥ n ∗ be a natural number and ε be a real positive number such that δ δ ln+1 < ε ≤ ln¯ . ¯ 24 24
(124)
Let {Ft }t∈ be an analytical family verifying the hypothesis of our Main Theorem and ρ1 ≤ ε , ρ0 ≤
24ε δℵ4n¯
α,β (2
n¯ )
.
(125)
We start the algorithm at the stage n. ¯ We notice that in order to pass to the stage n¯ + 1 we need to show the following facts at the end of the stage n: ¯ In the first part. The inequalities (58), (59) must be 24ε , δℵ4n¯ α,β (2n¯ )2i 24ε n,i ¯ . ∂θ ρ1,t ≤2 δ t ηn,i ¯ ≤
(126) (127)
In the second part. Inequality (78) must be ¯ u n1,t − 1 ≤ C3 C2 C4 2n¯ α (2n¯ )K 4
24ε . δ
(128)
Now the first two inequalities follow by repeating adequately the computations of the first part, while the third one follows immediately. In this way, we reach the third part and we obtain successfully the corresponding bounds for ln+1 and wn+1 when introducing the loss in the strip. ¯ ¯ In the estimates with respect to the parameter. In order to get (98) it is enough at each 2 i step to allow a loss of size 24ε δ 3 for the radius of the parameter space disc. Moreover, to obtain (102) it is enough to allow a loss of size 24ε δ . At the fourth part we can see that (115), (116), (117), are bounded from above by 2 −2π 2n¯ d 1 n¯ K 42 24ε C3 K 4 24ε 2C2 K M 24ε C3 K 4 ln+1 ¯ δ δ e , , (129) ≤ 1 ℵδ 2 ℵ dn¯ respectively. If ℵ is large enough then (118) is bounded from above by 3L −1 24ε δ . This allows us to finish the stage with an appropriate loss of width for the strip. The above considerations show that in order to start with the stage n¯ we just need a parameter space disc of radius pn¯ = 100
24ε . δ
(130)
For such a disc, we also verify that the parameter t¯ proving the existence of the invariant curve is located at a distance ε times a constant from the origin in the complex plane, as desired. If ε goes to 0, then the level n¯ of the first stage grows and (123) allows to conclude that the size of the invariant curve tends to 0.
36
M. Ponce
Actually, we have shown that Theorem 4.1 holds under the stronger hypothesis ρ0 ≤ wn ∗ (or more precisely under hypothesis (125)). In the next section we will see that we only need to impose the hypothesis ρ0 ≤ ε. We will see that even under this weaker hypothesis we can reduce the problem to the stronger hypothesis used so far. Indeed, we will describe a previous preparative process showing this reduction. 8. A Previous Preparative Process and the End of the Proof 8.1. An initial decrease in the size of ρ0 . We know that solving the u 0 equation gives rise to a diminution on the size of ρ0 . Thus, we will solve many times this equation to get the adequate size for ρ0 in order to start the algorithm. Although each time we solve the equation we slightly increase the size of ρ1 , the total amount of the increase is well behaved (and may be effectively controlled). We will omit many details when manipulating these estimates, the corresponding justifications being contained in the previous sections. We fix a constant ℵ by using the data L , M, 2T + (2000L)−1 , δ, (α, β) as described at the beginning of Sect. 7.1. This choice gives us our sequences {ln }, {wn }, {dn0 }, {dn1 } and a natural number n ∗ representing the lowest starting stage for the algorithm. Suppose that there exists n¯ ≥ n ∗ and ε > 0 such that δln+1 δln¯ ¯ < ε(wn ∗ )−1 ≤ . 24 24
(131)
Assume also that the hypotheses of Theorem 4.1 hold and that for every parameter t in the disc D(0, K R ε) we have ρ0,t ≤ ε , ρ1,t ≤ ε,
(132)
where K R is a constant which will be defined in a sequel. The value of the radius for the parameter space disc corresponds exactly to the sum of what we need to start this previous preparative process and the necessary radius to start with the algorithm at the stage n. ¯ This previous preparative process also has stages (indexed by the natural numbers between n ∗ and n¯ − 1). Each one of these stages contains two parts. The first part of the stage n is at some moment divided into steps. Each step consists in solving the equation of u 0 once, with truncation up to the order 2n . The second part consists in introducing a loss for the width of the strip of size dn0 . For notation reasons we will use the sequence of real numbers Wn = C2 M2n α,β (2n ).
(133)
We present the first stage n ∗ separately. The reason for doing this is twofold. On the one hand, at this stage the estimates are slightly different. On the other hand, this stage allows us to understand the manipulations and estimates that arise in the remaining stages. As n ∗ , ρ n ∗ , ρ n ∗ . We introduce an usual (in this work), we denote the starting functions as ρ0,t t 1,t index to indicate the steps. We define the sequence of functions ηnt ∗ ,i by ∗
n ,0 , ηnt ∗ ,0 = ρ0,t
ηnt ∗ ,i+1
=
n ∗ ,i n ∗ ,i ρ1,t u 0,t
∗ ∗ + ρtn ,i (·, u n0,t,i ).
(134) (135)
Persistence of Invariant Curves for Fibered Holomorphic Transformations
37
The hypotheses of Theorem 4.1 say that ηnt ∗ ,0 ≤ ε. We will show inductively that ηn ∗ ,0 ε ≤ i, 2i 2 n ∗ ,0 n ∗ ,0 ρ1,t ≤ ρ1,t + 2ηnt ∗ ,0 Wn ∗ ≤ ε(1 + 2Wn ∗ ) < ε(wn∗ )−1 . ηnt ∗ ,i ≤
(136) (137)
To do this, assume that these inequalities hold for 0 ≤ j < i + 1. Then we can estimate the new η by Wn ∗ Wn ∗ 2 t t t ηn ∗ ,i+1 ≤ ε(1 + 2Wn ∗ )ηn ∗ ,i + M ηn ∗ ,i . M M Since ε(1 + 2Wn ∗ ) < ε(wn ∗ )−1 , ηnt ∗ ,i ≤ ε and n¯ ≥ n ∗ we have ε(1 + 2Wn ∗ )
W 2∗ W 2∗ δWn ∗ Wn ∗ 1 1 < ln¯ < , ηnt ∗ ,i n ≤ ε n < . M 24M 4 M M 4 ∗
n ,i+1 is bounded from above This implies that ηnt ∗ ,i+1 ≤ ηn ∗ ,i 2−1 . The size of ρ1,t by ∗
∗
n ,i+1 n ,0 ρ1,t ≤ ρ1,t + Wn ∗
i
∗
n ,0 ηnt ∗ , j < ρ1,t + 2Wn ∗ ηnt ∗ ,0 ≤ ε(1 + 2Wn ∗ ).
0 εw
∗
n +1 We continue the steps until the moment when the size of η is smaller than 2ℵw , or more n∗ εwn ∗ +1 ε precisely when 2i ≤ 2ℵw ∗ (this subtle difference will be very important in Sect. 9). n Let i n ∗ be such a moment. In the second part of the stage we introduce a loss in the width of the strip of size dn0∗ . The final size of ρ0 is thus bounded by ∗
n ∗ +1 ρ0,t
≤
e ηnt ∗ ,in∗ + 2ηnt ∗ ,0
εwn ∗ +1 2εwn ∗ +1 + 2ℵwn ∗ 4ℵwn ∗ εwn ∗ +1 ≤ . ℵwn ∗
−2π 2n dn0∗
dn0∗
≤
8.1.1. An arbitrary stage n ∗ < n ≤ n¯ − 1. At the beginning of the stage we have εwn n,0 ≤ , (138) ρ0,t ℵwn ∗ 2wn ∗ +1 Wn ∗ +1 2wn−1 Wn−1 n,0 (139) ≤ ε 1 + 2Wn ∗ + + ··· + ρ1,t ℵwn ∗ ℵwn ∗ < ε(wn ∗ )−1 .
(140)
t } we will show inductively that For the usual sequence {ηn,i t ηn,i ≤
t ηn,0
, 2i 2wn Wn n,i < ε(wn ∗ )−1 . ≤ ε 1 + 2Wn ∗ + · · · + ρ1,t ℵwn ∗
(141) (142)
38
M. Ponce
So, suppose these inequalities hold for 0 ≤ j < i + 1. The size of η is bounded from above by Wn 2 −1 Wn t t ∗ ε(wn ) . η + M ηn,i M n,i M Moreover, we have Wn δWn 1 < ln < , M 24M 4 Mεwn Wn2 δM 1 t ln wn Wn2 < . Wn2 ≤ < Mηn,i ℵwn ∗ 24ℵ 4
ε(wn ∗ )−1
t t 2−1 . The size of ρ n,i+1 is bounded by That implies ηn,i+1 ≤ ηn,i 1,t n,i+1 n,0 ρ1,t ≤ ρ1,t + Wn
i
t ηn, j
0
<
n,0 ρ1,t +
2Wn εwn ℵwn ∗
2Wn wn < ε(wn ∗ )−1. ≤ ε 1 + 2Wn ∗ + · · · + ℵwn ∗ εwn+1 We continue the steps until the size of η is smaller than 2ℵw , or more precisely when n∗ εwn+1 εwn ≤ . Let i be such a moment. At the second part of the stage we introduce n 2ℵw ∗ ℵw ∗ 2i n
n
a loss in the width of the strip of size dn0 . The final size of ρ0 is bounded by e−2π 2 dn0 εwn εwn+1 +2 ℵwn ∗ 4ℵwn
n+1 t t ρ0,t ≤ ηn,i + 2ηn,0 n
εwn+1 2ℵwn ∗ εwn+1 < . ℵwn ∗ ≤
n d0 n
Hence, at the end of the (n¯ − 1)th stage we obtain n¯ ρ0,t ≤
εwn¯ δε(wn ∗ )−1 , < ℵwn ∗ ℵ4n¯ α,β (2n¯ )
(143)
n¯ ρ1,t ≤ ε(wn ∗ )−1 . (144) ¯ 0 n¯ These functions are defined on the strip δ − n−1 n ∗ dn . The function ρt (θ, ·) is defined, for the z variable, on a disc of radius
1−
in n−1 ¯ n=n ∗ i=0
u n,i 0,t .
(145)
If we start the algorithm at the stage n¯ with this parameter space disc, we will reach the end of the process with a radius even larger than 3/8. The size of ρt does not change with the operation of resolution of the equation of u 0 .
Persistence of Invariant Curves for Fibered Holomorphic Transformations
39
8.1.2. Estimates with respect to the parameter. We will need to estimate the values of the following two sums: in n−1 ¯ n=n ∗ i=0
in n−1 ¯
u n,i 0,t ,
n=n ∗ i=0
∂t u n,i 0,t .
The first one is bounded by in n−1 ¯ n=n ∗ i=0
Wn εwn Wn ∗ ε +2 M Mℵwn ∗ n>n ∗ 2Wn ∗ ε 4Wn ∗ 1 ln ε. ≤ < 1+ ∗ M 2ℵ l M ∗ n
u n,i 0,t ≤ 2
n>n
By introducing a loss of size 96000L Wn ∗ ε on the radius of the parameter space disc, we obtain a Cauchy estimate in n−1 ¯ n=n ∗ i=0
∂t u n,i 0,t ≤
1 . 4000L M
(146)
Now, we can estimate the final oscillation of ∂t ∂z ρt by in n−1 ¯
osc ∂t ∂z ρtn¯ ≤ 2T + 2M u n,i 0,t ≤ 2T + n=n ∗
i=0
1 . 2000L
(147)
We will need the following estimates: n−1 in ¯ 1 n,i n,i u 0 + M ∂z ρt (·, u 0,t ) ≤ 2T + ∂t u 0 ∂t 2L n=n ∗ i=0 1 1 4Wn ∗ ε+ ≤ 2T + 2L M 4000L 1 , ≤ 2000L where the last inequality follows since the size of ℵ is very large. We define the complex number 0 by 0 = ∂t
n∗ ρ1,t dθ 1
T
. t=t0
(148)
40
M. Ponce
8.2. A simple zero for T1 ρ1,t . There is only one remaining ingredient we miss in order n¯ to apply the algorithm to our family, namely, the existence of a simple zero for T1 ρ1,t at the center of the parameter space disc. We will find this zero by using the Rouché n¯ with the linear part of n∗ Lemma. Thus, we need to compare T1 ρ1,t T1 ρ1,t at t0 on the boundary of a disc of radius R and centered at t0 : ∗ ∗ n¯ n n ρ1,t − ∂t ρ1,t (t − t0 ) ≤ ρ1,t − ∂t ρ1,t (t − t0 ) T1 1 1 1 T T T t=t0 t=t0 n¯ n∗ + ρ1,t − ρ1,t 1 1 T
T
T ≤ ε + R 2 + 4Wn ∗ ε. 2 We have the linear estimate n∗ ρ1,t (t − t0 ) > L −1 R ∂t T1 t=t0 over this boundary. We choose the radius $ L −1 − L −2 − 2T ε(1 + 4Wn ∗ ) , (149) Rε = T which corresponds to the smallest root of the equation between two precedent bounds. the n¯ inside D(t , R ). We In this way we assure the existence of a simple zero t ∗ of T1 ρ1,t 0 ε easily verify that Rε is a positive real number. Furthermore, we also verify that L −1 2T ε(1 + 4Wn ∗ ) = 2Lε(1 + 4Wn ∗ ), (150) Rε < T L −2 √ because x > 1 − 1 − x for 0 < x < 1. In order to start the iterative part of our process (the algorithm described in Sects. 24ε(wn ∗ )−1 and centered at t ∗ . 7.4. . . 7.6) we need a parameter space disc of radius 100 δ All we have seen allows us to define the constant K R as being precisely what is needed. That is, to find t ∗ , and to lose a little bit of radius to obtain the estimates with respect to the parameter in the previous preparative process. Thus we have to choose 2400(wn ∗ )−1 . (151) δ To finish this previous preparative process and to start with the algorithm, it is enough to notice that the following estimate: n¯ n∗ 0 − ∂t ≤ 0 − ∂t ρ ρ 1,t 1,t T1 T1 n−1 in ¯ n,i n,i + ∂t ∂z ρt (·, u 0,t ) n=n ∗ i=0 L −1 2400(wn ∗ )−1 ≤ T ε 2L(1 + 4Wn ∗ ) + + δ 2000 K R = 2L(1 + 4Wn ∗ ) + 96000L Wn ∗ +
≤
L −1 1000
Persistence of Invariant Curves for Fibered Holomorphic Transformations
41
holds inside the parameter space disc centered at t0 with radious ε (2L(1 + 4Wn ∗ )+ 2400(wn ∗ )−1 . Notice that this disc contains the disc centered at t ∗ with radius δ ε
2400(wn ∗ )−1 . δ
The invariant curve is u=
in n−1 ¯ n=n ∗
i=0
u n,i 0,t¯
+ lim
n→∞ ˜
n˜ n=n¯
⎛ u n0,t¯ ⎝
n " j=n¯
⎞ j u 1,t¯⎠ .
(152)
It has zero degree because all the coordinate changes performed in this previous preparative process have zero degree. 9. A Parametrized Version of the Persistence of the Invariant Curve Theorem for the Fibered Holomorphic Dynamics Let ∈ C be an open set. We consider an application from to the set of analytical 1 complex parameter families of fibered holomorphic dynamics, s ∈ −→ {Fts }t∈ . We say that this application is analytic if (s, t, θ, z) −→ Fts (θ, z) is an analytical function. Let s¯ be in . Suppose that the family {Fts¯ }t∈ verify the hypothesis of Theorem 4.1. Hence, for every t in D(t0 , K R ε) ⊂ one has s¯ • L > ∂t T1 ρ1,t > L −1 t=t0
s¯ < ε, ρ s¯ < ε • ρ0,t 1,t s¯ < T • ∂z2 ρts¯ < M, ∂t ∂z ρts¯ + ∂t2 ρ1,t
for some ε ∈ (0, ε∗ ], where ε∗ is given by Theorem 4.1. Then, there exists a neighborhood V (¯s ) of s¯ in such that the following holds: every family {Fts }t∈ , with s in V (¯s ) verifies the above hypothesis. Theorem 4.1 gives us an application from V (¯s ) to D(t0 , K R ε) s → t¯s and an application from V (¯s ) to the set of analytical curves from the circle T1 to C, s −→ u s : Bδ → C. The curve u s is invariant by the fibered holomorphic dynamics Ft¯s , has zero degree and s its fibered rotation number equals β. The goal of this section is to show Theorem 9.1. The applications s → t¯s , s → u s defined above are holomorphic functions.
42
M. Ponce
The proof is obtained by making a more careful study of the method (previous process and iterative algorithm), and more precisely by the ability of performing this method in an uniform way for the whole set of families {Fts }t∈ , s ∈ V (¯s ). Let n¯ be the unique natural number verifying δln+1 δln¯ ¯ < ε(wn ∗ )−1 ≤ . 24 24 We can apply the previous process to every family {Fts }t∈ , s in V (¯s ). This gives us s ∗ the functions {(u n,i ¯ n ) defined for every t belonging to D(t0 , K R ε) (here 0,t ) }(n ≤n
n ,0 s ) is also holomorphic with respect to s and t. function (u 0,t We denote by < ˜ the (direct) lexicographical order on N × N. Notice that an index (m, j) appears in the algorithm before another index (n, i) if and only if (m, j)<(n, ˜ i). t )s is holoLet (n, i) be an index. If for every index (m, j)<(n, ˜ i) we have that (ηm, j
s morphic with respect to s and t, then (u n,i 0,t ) is holomorphic with respect to s and t. By
Persistence of Invariant Curves for Fibered Holomorphic Transformations
43
s ∗ an inductive reasoning we obtain that all of the functions {(u n,i ¯ n ) , as 0,t ) }(n ≤n
9.3. An arbitrary stage n ≥ n. ¯ Suppose that for every index (m, j)<(n, ˜ 0) all of n )s , (ρ n )s , (ρ n )s , {(u m, j )s } are holomorphic on s and t. Suppose the functions (ρ0,t t 0,t 1,t also that for every m ≤ n the centers of the parameter space discs tms are holomorphic functions on s. For every s in V (¯s ), when solving the first equation of u 0 s in the stage (the step i = 0), the trigonometrical polynomial solution (u n,0 0,t ) also depends in an holomorphic way on s and t. As a consequence we have that the funcn,1 s n,1 s tions (ρ0,t ) , (ρ1,t ) , (ρtn,1 )s are holomorphic. By repeating the argument we can get n, j
the same conclusion for every function (u 0,t )s0≤ j≤in , (u n1,t )s . This implies that the same n+1 )s , (ρ n+1 )s , (ρ n+1 )s . In particular, the functions holds for the functions (ρ0,t t 1,t T1
n+1 s (ρ1,t ) dθ,
(153)
which are holomorphic on t ∈ D(tnn¯ , pn ), are also holomorphic on s. Then there exists s such that a holomorphic application s → tn+1 T1
s s ) dθ = 0. (ρ1,tn+1
(154)
In this way, we show inductively on n that the center applications s → tns are holomorphic. Thus, the limit application s → t¯s also corresponds to a holomorphic function on s. The invariant curve u s is written as (see 152) ⎛ ⎞ in n−1 ¯ n n˜ " j us = (u n,i )s + lim (u n0,t¯s )s ⎝ (u 1,t¯ )s ⎠ . 0,t¯ n=n ∗ i=0
s
n→∞ ˜
n=n¯
j=n¯
s
(155)
The first term corresponds to the previous preparative process. This is a finite sum of holomorphic functions on s. The functions arising from the iterative algorithm are also holomorphic on s. We may then conclude that the application s → u s is holomorphic. This completes the proof of Theorem 9.1. Acknowledgements. This work corresponds to the central part of the author’s Thèse de Doctorat, prepared at the Université Paris-Sud XI. The author thanks Jean-Christophe Yoccoz for the suggestion of this subject, his advice and constant support to carry out this work. The author also thanks Jesús Zapata for a useful comment, as well as Andrés Navas and the anonymous referee for several pertinent remarks and corrections. This work was partially supported by CONICYT-Chile and the French government. The article was written during a postdoctoral position of the author at Pontificia Universidad Católica de Chile and supported by Proyecto Post-Doctorado FONDECYT 3080055.
44
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References 1. Avila, A., Krikorian, R.: Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2) 164(3), 911–940 (2006) 2. Bost, J.B.: Tores invariantes des systèmes dynamiques hamiltoniens. Asterisque, Séminaire Bourbaki 639(133–134), 113–157 (1986) 3. Bourgain, J.: On Melnikov’s persistency problem. Math. Res. Lett. 4(4), 445–458 (1997) 4. Bruno, A.D.: Analytic form of differential equations I. Trans. Moscow Math. Soc. 25, 131–288 (1971) 5. Cassels, J.W.: An Introduction to Diophantine Approximation. Volume 45 of Cambridge tracts. Cambridge: Cambridge University Press, 1957 6. Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15(1), 115–147 (1989) 7. Furstenberg, H.: Strict ergodicity and transformation of the torus. Amer. J. Math. 83, 573–601 (1961) 8. Gentile, G.: Resummation of perturbation series and reducibility for Bryuno skew-product flows. J. Stat. Phys. 125(2), 321–361 (2006) 9. Gentile, G.: Degenerate lower-dimensional tori under the Bryuno condition. Erg. Th. Dyn. Sys. 27(2), 427–457 (2007) 10. Herman, M.: Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58, 453–502 (1983) 11. Jäger, T., Keller, G.: The Denjoy type of arguments for quasiperiodically forced circle diffeomorphisms. Erg. Th. Dyn. Sys. 26, 447–465 (2006) 12. Jäger, T., Stark, J.: Towards a classification for quasiperiodically forced circle homeomorphisms. J. LMS 73(3), 727–744 (2006) 13. Johnson, R., Moser, J.: The rotation number for almost periodic potentials. Commun. Math. Phys. 84(3), 403–438 (1982) 14. Jonsson, M.: Dynamics of polynomial skew products on C 2. Math. Ann. 314, 403–447 (1999) 15. Jonsson, M.: Ergodic properties of fibered rational maps. Ark. Mat. 38, 281–317 (2000) 16. Krikorian, R.: Réductibilité des systèmes produits-croisés à valeurs dans des groupes compacts. Astérisque 259, vi+216 (1999) 17. Lang, S.: Introduction to Diophantine Approximation. Reading, MA: Addison-Wesley, 1966 18. Lopes Dias, J.: A normal form theorem for Brjuno skew systems through renormalization. J. Diff. Eqs. 230(1), 1–23 (2006) 19. Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962, 1–20 (1962) 20. Ponce, M.: Sur la persistance des courbes invariantes pour les dynamiques holomorphes fibrées lisses. To appear in Bull. Soc. Math. France 21. Ponce, M.: Courbes invariantes pour les dynamiques holomorphes fibrées. PhD thesis, Université ParisSud, 2007 22. Ponce, M.: Local dynamics for fibred holomorphic transformations. Nonlinearity 20(12), 2939–2955 (2007) 23. Pöschel, J.: On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202(4), 559–608 (1989) 24. Rüssmann, H.: Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1970, 67–105 (1970) 25. Rüssmann, H.: Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition. Erg. Th. Dyn. Sys. 22, 1551–1573 (2002) 26. Schmidt, W.M. Diophantine Approximations. Volume 785 of Lecture Notes in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 1980 27. Sester, O.: Étude dynamique des polynômes fibrés. PhD thesis, Université de Paris-Sud, 1997 28. Sester, O.: Hyperbolicité des polynômes fibrés. Bull. Soc. Math. France 127(3), 393–428 (1999) 29. Yoccoz, J.-C.: An introduction to Small Divisors Problems, in From Number Theory to Physics. Berlin-Heidelberg-New York: Springer-Verlag, 1992, pp. 659–679 30. Yoccoz, J.-C.: Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Asterisque 231, 3–88 (1995) 31. Yoccoz, J.-C.: Analytic linearization of circle diffeomorphisms. In: Dynamical systems and small divisors (Cetraro, 1998), Volume 1784 of Lecture Notes in Math., Berlin: Springer, 2002, pp. 125–173 Communicated by G. Gallavotti
Commun. Math. Phys. 289, 45–73 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0806-4
Communications in
Mathematical Physics
Existence of Weak Solutions for a Diffuse Interface Model for Viscous, Incompressible Fluids with General Densities Helmut Abels Max Planck Institute for Mathematics in Science, Inselstr. 22, 04103 Leipzig, Germany. E-mail:
[email protected] Received: 21 January 2008 / Accepted: 23 February 2009 Published online: 21 April 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: We study a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain. The fluids are assumed to be macroscopically immiscible, but a partial mixing in a small interfacial region is assumed in the model. Moreover, diffusion of both components is taken into account. In contrast to previous works, we study the general case that the fluids have different densities. This leads to an inhomogeneous Navier-Stokes system coupled to a Cahn-Hilliard system, where the density of the mixture depends on the concentration, the velocity field is no longer divergence free, and the pressure enters the equation for the chemical potential. We prove existence of weak solutions for the non-stationary system in two and three space dimensions. 1. Introduction and Main Result In this article we consider a so-called diffuse interface model for two viscous, incompressible Newtonian fluids of different density. In the model a partial mixing of the macroscopically immiscible fluids is considered and diffusion effects are taken into account. Such models have been successfully used to describe flows of two or more fluids macroscopically beyond the occurrence of topological singularities of the separating interface (e.g. coalescence or formation of drops). We refer to Anderson and McFadden [5] for a review on that topic. The model which we are considering leads (after a reduction) to the system ρ∂t v + ρv · ∇v − div S(c, Dv) + ρ∇g0 = ρµ0 ∇c in Q,
(1.1)
∂t ρ + div(ρv) = 0 in Q,
(1.2)
ρ∂t c + ρv · ∇c = div(m(c)∇µ0 ) in Q,
(1.3)
1 q
ρµ0 + ρ 2 p¯ = βρ 2 g0 − a (c)q A(c) + φ(c) in Q,
(1.4)
46
H. Abels
together with
µ0 (t) dx =
g0 (t) dx = 0,
t ∈ (0, ∞),
(1.5)
where Q = × (0, ∞) and ⊂ Rd , d = 2, 3, is a bounded domain with C 2 -boundary. Here v and ρ = ρ(c) ˆ are the (mean) velocity and the density of the mixture of the two fluids, g0 is a modified pressure, c is the concentration difference of the two fluids, µ is the chemical potential associated to c, µ0 is the mean-value free part of µ, and p¯ is a constant (depending on time), which is related to the mean values of the pressure and the chemical potential. Moreover, S(c, Dv) = 2ν(c)Dv + η(c) div v I,
1
ˆ Λˆ (s) = a q (s), Λ(0) = 0,
(1.6)
where S(c, Dv) is the stress tensor, Dv = 21 (∇v+∇v T ), ν(c), η(c) > 0 are two viscosity coefficients, q u = div(|∇u|q−2 ∇u) is the q-Laplacian, ∇ A ⊗ ∇ A = (∂ j A∂k A)dj,k=1 , ˆ A = A(c) is an auxiliary function related to a(c) > 0, which arises in the free energy density of the system, and m(c) > 0 is a mobility coefficient. Furthermore, (c) is the homogeneous free energy density for the mixture and φ(c) = (c). Finally, we assume that the fluids behave like a simple mixture, cf. Lowengrub and Truskinovski [13], which 1 means that the excess volume of the mixture is zero. This implies that ρ(c) ˆ = α+βc for some α > 0 and |β| < α. We close the system by adding the boundary and initial conditions n · v|∂ = n · S(c, Dv)τ + γ vτ |∂ = 0 on S,
(1.7)
∂n c|∂ = ∂n µ0 |∂ = 0 on S,
(1.8)
(v, c)|t=0 = (v0 , c0 ) in ,
(1.9)
where S = ∂ × (0, ∞), n is the exterior normal on ∂ and 0 < γ < ∞ is a friction coefficient. In the case a(c) = ρ(c), ˆ q = 2 the model was proposed by Lowengrub and Truskinovski [13] as a generalization of a well-known diffuse interface model in the case of matched densities which corresponds to the case β = 0, ρ(c) ˆ ≡ const., respectively, cf. e.g. Gurtin et al. [10]. The present modification can be derived in the same way, cf. e.g. [2, Chap. II]. We note that systems (1.1)–(1.5) are obtained from the original system by a reduction, which is explained in Sect. 3 below and which will be essential for the following analysis. We will only consider the case ρ(c) ≡ const., i.e., β = 0. Results for the case of matched densities (ρ(c) ≡ const.) were obtained by Starovoitov [17], Boyer [6], Liu and Shen [12], and the author [1]. Moreover, in [7] Boyer considered a different diffuse interface model for fluids with non-matched densities. He proved existence of strong solutions, locally in time, and existence of global weak solutions if the densities of the fluids are sufficiently close. Finally, A. and Feireisl [3] constructed weak solutions globally in time for a corresponding diffuse interface model for compressible fluids. We note that the total energy of the system is E(c, v) = E free (c) + E kin (c, v), where E free (c) =
(c) dx +
a(c)
|∇c|q dx, q
E kin (c, v) =
ρ(c)
|v|2 dx, 2
(1.10)
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
47
and that every sufficiently smooth solution of (1.1)–(1.9) satisfies d E(c(t), v(t)) dt =− 2ν(c)|Dv(t)|2 + η(c)| div v(t)|2 dx − m(c)|∇µ|2 dx − γ |vτ |2 dσ,
∂
cf. (3.12) below. Moreover, note that = |∇ A(c)|q . We will only consider the case β = 0 since ρ(c) ≡ const. if β = 0. Furthermore, we can assume w.l.o.g. that −α < β < 0. Otherwise we replace c with −c. Then ρ(s) ˆ is a strictly increasing function. Let us comment on the new difficulties that arise for the construction of weak solutions (in comparison with the case of matched densities): a(c)|∇c|q
1. Since ρ = ρ(c) ˆ ≡ const. in general, div v ≡ 0 too. Therefore it is not sufficient to work in function spaces of divergence free vector fields and we cannot simply apply the Helmholtz projection to (1.1) to get an evolution equation for v independent of g0 . 2. In the case of non-matched densities β = 0 the (modified) pressure g0 enters the equation for the chemical potential (1.4). But g0 has low regularity, in particular with respect to time as will be discussed below. Therefore we will not be able to obtain an a priori estimate of φ(c) − c in some L r (Q T ), r ≥ 2, (even not for r > 1). This would be essential to be able to choose φ = , where is a singular free energy density as e.g.
(c) =
θ θc ((1 + c) ln(1 + c) + (1 − c) ln(1 − c)) − c2 , 2 2
(1.11)
where θ, θc > 0, a = −1, b = 1. In the case of matched densities β = 0 and with q = 2, it is possible to show existence of weak solutions such that c(t, x) ∈ (−1, 1) a.e.[1], see also [4]. Since we cannot use these free energy densities, we need some other mechanism to keep the concentration c ∈ [−1, 1] or at least in a suitable neighborhood c ∈ [−1 − ε, 1 + ε], ε > 0. Otherwise, ρ = ρ(c) ˆ becomes singular or non-positive. In order to overcome the last difficulty, it will be essential to choose the exponent of the gradient term in the free energy as q > d and to choose the free energy density (c) “steep enough” outside the interval [−1, 1], cf. Lemma 2.3 below. The following proofs will not work for the standard case q = 2. For the following analysis it is essential to use a suitable decomposition of g0 , namely: g0 = g1 − ∂t G(v),
(1.12)
where G(v) = div v
and
in ,
∂n G(v) = 0 on ∂,
(1.13) (1.14)
G(v) dx = 0, which implies that ∇G(v) = (I − Pσ )v.
(1.15)
48
H. Abels
Hence (1.1) is equivalent to ρ∂t Pσ v + ρv · ∇v − div S(c, Dv) + ρ∇g1 = ρµ0 ∇c in Q.
(1.16)
Here the part g1 has relatively good regularity, e.g., g1 ∈ L 2 (0, ∞; L p ()) with d , cf. Lemma 6.1 below. It is the part ∂t G(v), which makes the analy1 < p < d−1 sis difficult and which does not allow to use a singular free energy as in (1.11). Finally, let us note that for our estimates of g1 it is important to consider Navier boundary conditions for v and not no-slip boundary conditions. The article is organized as follows: In Sect. 2 we define weak solutions of the system (1.1)–(1.9) and state the main result on existence of weak solutions. In Sect. 3 we state the original form of the system (1.1)–(1.9) as derived in [13], explain the reduction to (1.1)–(1.9), and discuss the conserved quantities. Moreover, in Sect. 4, we introduce the used function spaces and summarize some preliminary results. The existence is proved with the aid of a two-level approximation. Firstly, we add some extra terms depending on a parameter δ > 0, which yield an additional a priori bound for g0 ∈ L 2 (Q). The corresponding system is presented in Sect. 5 and existence of weak solutions to the latter system is proved with the aid of an implicit time discretization, which is the second level of approximation. Then, in Sect. 6, we consider the limit δ → 0 and prove our main result on existence of weak solutions to (1.1)–(1.9). Finally, we note the present results are part of the author’s Habilitation thesis [2]. 2. Weak Solutions and the Main Result First of all, let us make some basic assumptions: Assumption 2.1. We assume that a ∈ C 1 (R), ν, η, m ∈ C 0 (R) with a(s), ν(s), m(s) ≥ m 0 > 0, η(s) ≥ 0 for all s ∈ R, let ρ(s) ˆ = (α + βs)−1 for some 0 < −β < α, and let S(c, Dv), A(c) be defined as in (1.6). Moreover, let 0 < γ < ∞ and let ⊂ Rd , d = 2, 3, be a bounded domain with C 2 -boundary. In the following, we frequently use the spaces Hn1 () = { f ∈ H 1 ()d : n · f |∂ = 0} and Hn−1 () = (Hn1 ()) . For more definitions of the used function spaces we refer to Sect. 4. Moreover, we denote Q (s,t) = × (s, t), Q t = Q (0,t) , S(s,t) = ∂ × (s, t), St = S(0,t) and A : B = dj,k=1 a jk b jk for two matrices A = (a jk ), B = (b jk ) ∈ Rd×d . Having in mind the decomposition (1.12), we define weak solutions of (1.1)–(1.9) as follows: Definition 2.2. Let Assumption 2.1 hold, let v0 ∈ L 2 ()d , c0 ∈ Wq1 (), q > d, let : R → [0, ∞) be twice continuously differentiable and set φ = . Then (v, g1 , c, µ0 , p) ¯ with v ∈ BCw ([0, ∞); L 2 ()d ) ∩ L 2 (0, ∞; Hn1 ()), g1 ∈ L 2 (0, ∞; L 1(0) ()),
c ∈ BCw ([0, ∞); Wq1 ()),
µ0 ∈ L 2 (0, ∞; H 1 ()),
1 p¯ ∈ L loc (0, ∞),
and such that 0 < ρ = ρ(c) ˆ ∈ L ∞ (Q) is called a weak solution of (1.1)–(1.9) if the following conditions are satisfied:
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
49
1. For every ϕ ∈ C0∞ (0, ∞; Hn1 () ∩ L ∞ ()d ), −(Pσ v, ∂t ϕ) Q + (v · ∇v, ϕ) Q + (ρ −1 S(c, Dv), Dϕ) Q + γ (ρ −1 vτ , ϕτ ) S = (g1 , div ϕ) Q + (µ0 ∇c, ϕ) Q − (∇ρ −1 · S(c, Dv), ϕ) Q . (2.1) 2. For every ψ ∈ C0∞ (0, ∞; C 1 ()), (ρ, ∂t ψ) Q + (ρv, ∇ψ) Q = 0,
(2.2)
(ρc, ∂t ψ) Q + (ρcv, ∇ψ) Q = (m(c)∇µ, ∇ψ) Q ,
(2.3)
and −1 −1 a(c) q (ρµ0 + ρ 2 p¯ − φ(c)), ψ = β a(c) q ρ 2 g1 , ψ Q Q − q1 2 q−2 −β G(v), ∂t a(c) ρ ψ + |∇ A(c)| ∇ A(c), ∇ψ . (2.4) Q
Q
3. (v, c)|t=0 = (v0 , c0 ). 4. (v, c, µ) satisfy the energy inequality E(c(t), v(t))+ S(c, Dv) : Dv + m(c)|∇µ|2 d(x, τ ) + γ vτ 2L 2 (S
(s,t) )
Q (s,t)
(2.5) ≤ E(c(s), v(s))
for all t ∈ [s, ∞) and almost all 0 ≤ s < ∞ including s = 0. We note that the boundary conditions (1.8) and the second condition in (1.7) are part of the weak formulations (2.1)–(2.3) because of − (div S(c, Dv), ϕ) = (S(c, Dv), ∇ϕ) − (n · S(c, Dv), ϕ)∂ = (S(c, Dv), ∇ϕ) + γ (ν(c)v, ϕ)∂ ,
(2.6)
for all v ∈ H 2 () satisfying (1.7) and all ϕ ∈ Hn1 (). The following lemma shows that, given ∈ C 2 ([−1, 1]) and R, ε > 0, we can “trap” all c(x) with E free (c) ≤ R in an arbitrary small neighborhood (−1 − ε, 1 + ε) by extending “suitably steep” outside of [−1, 1]. This is essential for everything that follows. Lemma 2.3. Let R, ε > 0, q > d, and let ∈ C 2 ([−1, 1]) with (c) > 0, c ∈ [−1, 1], be given. Then there is an extension ∈ C 2 (R), (c) ≥ 0, (c) ≥ −M > −∞ such that for all c ∈ Wq1 (),
a(c)|∇c|q dx + q
(c) dx ≤ R
⇒
c(x) ∈ (−1 − ε, 1 + ε).
Proof. Since Wq1 () → C α (), α = 1 − qd , there is a constant C1 such that |c(x) − c(x0 )| ≤ C1 ∇c L q () |x − x0 |α
(2.7)
50
H. Abels
for all x, x0 ∈ and c ∈ Wq1 () with c dx = 0. Since we can change c(x) by a constant in the estimate above, it also holds for all c ∈ Wq1 (). Moreover, if E free (c) ≤ R, then ∇c L q ≤ R for some R > 0. Hence
ε |c(x) − c(x0 )| ≤ C1 R |x − x0 | ≤ 2
α
for all x ∈ Br (x0 ) ∩ , r :=
ε 2C1 R
1
α
.
Furthermore, let κ = inf x0 ∈ |Br (x0 ) ∩ | > 0. Hence for every x0 ∈ we have |c(x)| ≥ |c(x0 )| − 2ε on a set of at least measure κ. Now we choose the extension of ∈ C 2 ([−1, 1]), (s) > 0, to : R → [0, ∞) such that (s) ≥ M := κ −1 (R + 1) and (s) ≥ 0 for all |s| ≥ 1 + 2ε . In order to prove (2.7), we assume that for c ∈ Wq1 () there is some x0 ∈ such that |c(x0 )| ≥ 1 + ε. By the observations before, we conclude that |c(x)| ≥ |c(x0 )| − 2ε ≥ 1 + 2ε on a set S = Br (x0 ) ∩ of measure at least κ > 0. Hence E free (c) ≥ (c) dx ≥ κ M = R + 1, S
which proves the implication (2.7). Now we are able to state our main result on existence of weak solutions. Theorem 2.4. Let q > d, ε, R > 0. Moreover, let ∈ C 2 (R), (c) ≥ 0, (c) ≥ −M, be given such that (2.7) holds. Then for every v0 ∈ L 2 ()d , c0 ∈ Wq1 () with E(c0 , v0 ) ≤ R there exists a weak solution (v, g1 , c, µ0 , p) ¯ of (1.1)–(1.9) in the sense of Definition 2.2 and with the property that c(t, x) ∈ [−1 − ε, 1 + ε]
for all (x, t) ∈ Q,
g1 ∈ L (0, ∞; L ()),
p¯ ∈ L 2uloc ([0, ∞))
2
for every 1 < p <
p
d d−1 .
3. Reformulation of the System and Conserved Quantities Our starting point is the system ρ∂t v + ρv · ∇v − div S(c, Dv) + ∇ p = − div(|∇ A(c)|q−2 ∇ A(c) ⊗ ∇ A(c)),
ρµ = −ρ
−1 ∂ρ
∂c
(3.1)
∂t ρ + div(ρv) = 0,
(3.2)
ρ∂t c + ρv · ∇c = div(m(c)∇µ),
(3.3)
|∇ A(c)|q p + (c) + q
1
+ φ(c) − a(c) q q A(c),
(3.4)
which is a variant of the model proposed in [13] for an interfacial energy of the form |∇c|q E free (c) =
(c) dx + a(c) dx, q
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
51
where the choice q = 2, a(c) = ρ(c) was proposed in the latter article. A derivation of the latter system can also be found in [2, Chap. II]. We close the system by adding initial and boundary conditions (1.7)–(1.9). We note that the term |∇ A(c)|q−2 ∇ A(c) ⊗ ∇ A(c) comes from an extra contribution to the stress tensor, which models capillary forces in an interfacial region. In order to derive (1.1)–(1.5), we define
(c) |∇ A(c)|q p + + − µc, ¯ ρ ρq ρ 1 where µ = µ0 + µ¯ and µ¯ = || µ dx. Then |∇ A(c)|q |∇ A(c)|q ∂ρ ρ∇g = φ(c)∇c + ∇ + ∇ p − ρ −1 p + (c) + ∇c − ρ µ∇c, ¯ q ∂c q g=
and therefore 1
ρ∇g − ρµ0 ∇c = ∇ p + a(c) q q A(c)∇c + ∇
|∇ A(c)|q q
= ∇ p + div(|∇ A(c)|q−2 ∇ A(c) ⊗ ∇ A(c)). Hence we see that (3.1)–(3.4) is equivalent to the system ρ∂t v + ρv · ∇v − div S(c, Dv) + ρ∇g = ρµ0 ∇c in Q, ∂t ρ + div(ρv) = 0
in Q,
ρ∂t c + ρv · ∇c = div(m(c)∇µ) and
(3.5) (3.6)
in Q,
1 ∂ρ ∂ρ c µ¯ = − g − a q (c)q A(c) + φ(c). ρµ0 + ρ + ∂c ∂c
(3.7)
(3.8)
For the following mathematical analysis it will be essential to consider the case of a so-called simple mixture. This is expressed in the relation 1 c1 c2 = + , ρ ρ¯1 ρ¯2 which means that the volume of the mixed fluids is the same as the sum of the volume filled by the separated fluids. Here c j is the concentration and ρ¯ j is the mass density of the fluid j = 1, 2. Therefore ρˆ can be written in the form ρ(c) ˆ =
1 α + βc
where β =
1 1 1 1 − ,α = + 2ρ¯1 2ρ¯2 2ρ¯2 2ρ¯1
(3.9)
with α > 0 and |β| < α. One easily calculates that ∂ρ = −βρ 2 , ∂c
ρ+
∂ρ c = αρ 2 . ∂c
(3.10)
Therefore (3.8) reduces to 1
ρµ0 + ρ 2 (α µ¯ + β g) ¯ = βρ 2 g0 − a q (c)q A(c) + φ(c),
(3.11)
52
H. Abels
1 where g = g0 + g¯ and g¯ = || ¯ g¯ are not uniquely g(x) dx. Hence one sees that µ, determined by the system. But p¯ = α µ¯ + β g¯ is determined uniquely by (3.11) if µ0 , g0 and c are known. A closely related identity is ∂t ρ = −βρ 2 ∂t c = −
β α
∂ρ β ρ+ c ∂t c = − ∂t (ρc). ∂c α
In particular, this implies d dt
ρc dx = 0
⇒
d dt
ρ dx = 0.
Summing up, we have derived (1.1)–(1.5) from the original system (3.1)–(3.4) assuming the case of a simple mixture. Now we discuss the conserved quantities and the energy estimate for solutions of the system. Because of (3.2), (3.3) is equivalent to ∂t (ρc) + div(ρcv) = div(m(c)∇µ). Therefore the total mass difference ρc and the total mass are conserved: ρ(x, t)c(x, t) dx = ρ0 (x)c0 (x) dx for all t ∈ (0, ∞), ρ(x, t) dx = ρ0 (x) dx for all t ∈ (0, ∞).
Moreover, multiplying (3.5) with v, (3.7) with µ, and (3.8) with ∂t c, integrating with respect to , and using the boundary conditions and (3.6), one sees that every sufficiently smooth solution of (1.1)–(1.9) satisfies 2ν(c(t))|Dv(t)|2 + η(c(t))| div v(t)|2 dx − m(c)|∇µ|2 dx − γ |v|2 dσ,
d E(c(t), v(t)) = − dt
∂
(3.12)
where E(c, v) = E free (c) + E kin (c, v) and E free (c), E kin (c, v) are defined as in (1.10). Integrating (3.12) yields (2.5). Finally, we note that (3.6) and (3.10) imply − βρ 2 (∂t c + v · ∇c) = −ρ div v.
(3.13)
Combining this with (3.7), we obtain the simple identity div v = β div(m(c)∇µ0 ).
(3.14)
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
53
4. Preliminaries Notation. Let us fix some notation. If X is a Banach space and X is its dual, then f, g ≡ f, g X ,X = f (g),
f ∈ X , g ∈ X,
denotes the duality product. The inner product on a Hilbert space H is denoted by (., .) H . Moreover, we use the abbreviation (., .) M = (., .) L 2 (M) . Function spaces. If M ⊆ Rd is measurable, L q (M), 1 ≤ q ≤ ∞ denotes the usual Lebesgue-space and . q its norm. Moreover, L q (M; X ) denotes its vector-valued variant of strongly measurable q-integrable functions/essentially bounded functions, where X is a Banach space. If M = (a, b), we write for simplicity L q (a, b; X ) and L q (a, b). q Furthermore, f ∈ L loc ([0, ∞); X ) if and only if f ∈ L q (0, T ; X ) for every T > 0. q Moreover, L uloc ([0, ∞); X ) denotes the uniformly local variant of L q (0, ∞; X ) consisting of all measurable f : [0, ∞) → X such that f L q
uloc ([0,∞);X )
= sup f L q (t,t+1;X ) < ∞. t≥0
Recall that, if X is a Banach space with the Radon-Nikodym property, then
L q (M; X ) = L q (M; X ) for every 1 ≤ q < ∞ by means of the duality product f, g = M f (x), g(x) X ,X dx for f ∈ L q (M; X ), q g ∈ L (M; X ). If X is reflexive or X is separable, then X has the Radon-Nikodym property, cf. Diestel and Uhl [9]. Moreover, recall the lemma of Aubin-Lions: If X 0 →→ X 1 → X 2 are Banach spaces, 1 < p < ∞, 1 ≤ q < ∞, and I ⊂ R is a bounded interval, then dv ∈ L q (I ; X 2 ) →→ L p (I ; X 1 ). v ∈ L p (I ; X 0 ) : (4.1) dt See J.-L. Lions [11] for the case q > 1 and Simon [16] or Roubíˇcek [14] for q = 1. Let ⊂ Rd be a domain. Then Wqm (), m ∈ N0 , 1 ≤ q ≤ ∞, denotes the usual m () the closure of C ∞ () in W m (), W −m () = (W m ()) , L q -Sobolev space, Wq,0 q q 0 q ,0 −m and Wq,0 () = (Wqm ()) . The L 2 -Bessel potential spaces are denoted by H s (), s ∈ R, which are defined by restriction of distributions in H s (Rd ) to , cf. Triebel [19, Sect. 4.2.1]. We note that, if ⊂ Rd is a bounded domain with C 0,1 -boundary, then there is an extension operator E which is a bounded linear operator E : W pm () → W pm (Rd ), 1 ≤ p ≤ ∞ for all m ∈ N and E f | = f for all f ∈ W pm (), cf. Stein [18, Chap. VI, Sect. 3.2]. This extension operator extends to E : H s () → H s (Rn ), which shows that H s () is a retract of H s (). Therefore all results on interpolation spaces of H s (Rn ) carry over to H s (). Furthermore, we note that f g W p1 ≤ C p,q f Wq1 g W p1
for all 1 < p ≤ q ≤ ∞, q > d,
(4.2)
which is easily proved using well-known Sobolev embeddings. Let I = [0, T ] with 0 < T < ∞ or let I = [0, ∞) and let X be a Banach space. Then BC(I ; X ) is the Banach space of all bounded and continuous f : I → X equipped with the supremum norm and BU C(I ; X ) is the subspace of all bounded and uniformly
54
H. Abels
continuous functions. Moreover, we define BCw (I ; X ) as the topological vector space of all bounded and weakly continuous functions f : I → X . By C0∞ (0, T ; X ) we denote the vector space of all smooth functions f : (0, T ) → X with supp f ⊂⊂ (0, T ). Finally, f ∈ W p1 (0, T ; X ), 1 ≤ p < ∞ if and only if f, ddtf ∈ L p (0, T ; X ), where ddtf denotes the 1 vector-valued distributional derivative of f . Furthermore, W p,uloc ([0, ∞); X ) is defined p p 1 by replacing L (0, T ; X ) by L uloc ([0, ∞); X ) and we set H (0, T ; X ) = W21 (0, T ; X ). Finally, we note: Lemma 4.1. Let X, Y be two Banach spaces such that Y → X and X → Y densely and let 0 < T < ∞. Then L ∞ (0, T ; Y ) ∩ BU C([0, T ]; X ) → BCw ([0, T ]; Y ). Proof. If f ∈ L ∞ (0, T ; Y ) ∩ BU C([0, T ]; X ), then f (t), ϕ X,X ∈ BU C([0, T ]) for all ϕ ∈ X . Now let ϕ ∈ Y . Since X → Y densely, we can find a sequence ϕk ∈ X such that ϕk →k→∞ ϕ in Y . Because of f ∈ L ∞ (0, T ; Y ), this implies that f (t), ϕk Y,Y → f (t), ϕY,Y uniformly in [0, T ]. Hence f (t), ϕY,Y ∈ BU C([0, T ]). 1 Neumann-Laplace equation. Given f ∈ L 1 (), we denote by m( f ) = || f (x) dx its mean value. Moreover, for m ∈ R we set q
L (m) () := { f ∈ L q () : m( f ) = m}, Then 1 P0 f := f − m( f ) = f − ||
1 ≤ q ≤ ∞.
f (x) dx
is the orthogonal projection onto L 2(0) (). Furthermore, we define 1 1 ≡ H(0) () = H 1 () ∩ L 2(0) (), H(0)
(c, d) H 1
(0) ()
:= (∇c, ∇d) L 2 () .
1 () is a Hilbert space due to Poincaré’s inequality. Moreover, let Then H(0) −1 −1 1 () . Furthermore, we define div : L 2 ()d → H −1 () by H(0) ≡ H(0) () = H(0) n (0)
divn u, ϕ H −1 ,H 1 = −(u, ∇ϕ) L 2 () (0)
1 for all ϕ ∈ H(0) ().
(0)
(4.3)
Note that this operator should not be confused with the distributional divergence extended to an operator div : L 2 ()d → H −1 () = (H01 ()) , since the latter is defined only for C0∞ ()- resp. H01 ()-vector fields. The operator divn acts on vector fields, which do not vanish on the boundary necessarily, and involves boundary conditions in a weak sense. 1 () → Let m ∈ L ∞ () such that, m(x) ≥ m 0 > 0 a.e. Then divn (m(x)∇·) : H(0) −1 H(0) (), defined by
−divn (m(x)∇u), ϕ H 1
−1 (0) ,H(0)
= (m(x)∇u, ∇ϕ)
1 for all ϕ ∈ H(0) ()
is an isomorphism because of the lemma of Lax-Milgram. In particular, this is true for 1 (), the weak Neumann-Laplace operator N := divn ∇ and N u = f for u ∈ H(0) −1 f ∈ H(0) () implies
u H 1
(0) ()
≤ f H −1 () . (0)
(4.4)
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
55 q
1 () solves u = f for some f ∈ L (), 1 < q < Finally, we note that, if u ∈ H(0) N (0) 2 ∞, and is a bounded domain with C -boundary, then it follows from standard elliptic theory that u ∈ Wq2 () and u = f a.e. in and ∂n u|∂ = 0 in the sense of traces. Moreover,
u Wq2 () ≤ Cq f L q ()
q
for all f ∈ L (0) ()
(4.5)
with a constant Cq depending only on 1 < q < ∞, d, and . For the following we denote
2 W p,N () = u ∈ W p2 () : ∂n u|∂ = 0 , where 1 < p < ∞. By a simple duality argument one can define so-called very weak solutions of the Neumann-Laplace operator, which is the content of the next lemma. Lemma 4.2. Let 1 < p < ∞. Then for every f ∈ (W p2 ,N ()) there is a unique u ∈ L p () such that (u, ϕ) = f, ϕ(W 2
p ,N
(4.6)
) ,W p2 ,N
for all ϕ ∈ W p2 ,N (). Moreover, there is some C p such that u L p () ≤ C p f (W 2
p ,N
(4.7)
())
for all u ∈ L p () and f ∈ (W p2 ,N ()) , solving (4.6).
Proof. Since N : W p2 ,N () → L p () is a bijection, the adjoint N : L p () → (W p2 ,N ()) is a bijection too. Hence for every f ∈ (W p2 ,N ()) there is a unique u ∈ L p () such that N u = f . Moreover, (4.7) holds since (N )−1 is continuous. Then (u, ϕ) = (u, N ϕ) = N u, ϕ(W 2
p ,N
()) ,W p2 ,N ()
= f, ϕ(W 2
p ,N
()) ,W p2 ,N ()
for all ϕ ∈ W p2 ,N () by the definition of the adjoint. A result related to energy inequalities. The following lemma will be useful for passing to the limit in energy inequalities. Lemma 4.3. Let E : [0, T ) → [0, ∞), 0 < T ≤ ∞, be a lower semi-continuous function and let D : (0, T ) → [0, ∞) be an integrable function. Then T T E(t)ϕ (t) dt ≥ D(t)ϕ(t) dt (4.8) E(0)ϕ(0) + 0
holds for all ϕ ∈
W11 (0, T )
0
with ϕ(T ) = 0 if and only if t D(τ ) dτ ≤ E(s) E(t) + s
holds for all s ≤ t < T and almost all 0 ≤ s < T including s = 0.
(4.9)
56
H. Abels
Proof. First assume that (4.8) holds. If s = 0 and 0 ≤ t < T , then we choose ⎧ ⎪ if 0 ≤ τ < t ⎨1 1 ϕε,t (τ ) = 1 − ε (τ − t) if t ≤ τ ≤ t + ε ⎪ ⎩0 otherwise in (4.8). Then
T 0
E(τ )ϕε (τ ) dτ
1 = ε
t+ε
t
E(τ ) dτ →ε→0 E(t)
for every Lebesgue point t of E (hence almost everywhere). This implies that (4.9) holds for s = 0 and all Lebesgue points 0 ≤ t < ∞. Since E is lower semi-continuous, we have E(t) ≤ lim inf E(t ). t →t
Therefore, choosing a sequence of Lebesgue points t j → t, we conclude that (4.9) holds for all 0 ≤ t < ∞. In order to prove (4.9) one chooses ϕ = ϕε,t − ϕε,s in (4.8) and proceeds as before. Conversely, assume that (4.9) holds for all s ∈ M with |[0, T ] \ M| = 0. Then (4.8) holds for every ϕ ∈ W11 (0, T ) such that ϕ(T ) = 0 and ϕ (τ ) is piecewise constant on some intervals [tk , tk+1 ], k = 0, . . . , N , 0 = t0 < t1 < · · · < t N +1 = T , where tk ∈ M, k = 0, . . . , N +1. Since every ϕ ∈ L 1 (0, T ) can be approximated by piecewise constant functions of the latter form, we conclude that (4.8) holds. Monotone Operators. Let X be a real-valued Banach space. Recall that A : X → X is a monotone operator if and only if A(x) − A(y), x − y X ,X ≥ 0
for all x, y ∈ X.
Moreover, a monotone operator A is maximal monotone if for every w, x ∈ X such that w − A(y), x − y X ,X ≥ 0
for all y ∈ X
(4.10)
w = A(x) holds. An important consequence of (4.10) is the following lemma. Lemma 4.4. Let A : X → X be a maximal monotone operator. Assume that xn , x ∈ X , y ∈ X , n ∈ N, satisfy xn n→∞ x, A(xn ) n→∞ y, and lim supn→∞ A(xn ), xn ≤ y, x. Then y = A(x). The result follows from the fact that a maximal monotone operator is of “type M”, cf. [15, Sect. II.2]. −1 We define the q-Laplacian q : Wq1 () → Wq,0 () = (Wq1 ()) , 1 < q < ∞, by − q u, ϕW −1 ,W 1 = |∇u|q−2 ∇u · ∇ϕ dx for all ϕ ∈ Wq1 (). (4.11) q ,0
q
1 () → R is defined by It is easy to check that −q u ∈ ∂ϕ(u), where ϕ : Wq,0
ϕ(u) =
1 q ∇u L q () q
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
57
−1 and w ∈ ∂ϕ(u) ⊆ Wq,0 () if and only if
w, v − uW −1 ,W 1 ≤ ϕ(v) − ϕ(u) q,0
for all v ∈ Wq1 ().
q
(Consider e.g. the function f (t) = ϕ((1 − t)u + tv), t ∈ R, and use that f (t) is non-decreasing since f (t) is convex.) In particular, this implies that −q : Wq1 () → −1 Wq,0 () is a monotone operator. Moreover, −q is even strictly monotone, i.e., q u − q v, u − vW −1 ,W 1 = 0 if and only if u = v, q ,0
which follows from the fact that x → that
q
1 q q |x|
is strictly convex. Furthermore, we note q
λ(u, u) L 2 + −q u, uW −1 ,W 1 = |λ| u 2L 2 () + ∇u L q () . q ,0
q
Hence λ − q is coercive on Wq1 () for every λ > 0 if q ≥ 2, i.e., λ(u, u) L 2 + −q u, uW −1 ,W 1 q ,0
u Wq1 ()
q
→∞
as u Wq1 () → ∞.
(4.12)
Therefore we obtain from [15, Cor. 2.2, Chap. II] the following lemma. Lemma 4.5. Let 2 ≤ q < ∞. Then −q : Wq1 () → Wq−1 ,0 () is maximal monotone
and λ − q : Wq1 () → Wq−1 ,0 () is bijective with strongly continuous inverse for every λ > 0. Proof. The fact that −q is maximal monotone follows from [15, Prop. 2.2, Chap. II]. Moreover, λ − q is coercive as seen above. Hence [15, Cor. 2.2 and Lemma 2.1] yield that λ − q is onto. Since λ − q is strictly monotone, λ − q is a bijection. Finally, if f n →n→∞ f in Wq−1 ,0 () and (λ − q )u n = f n , (λ − q )u = f , then λ u n − u 2L 2 − q u n − q u, u n − u = f n − f, u n − u →n→∞ 0
(4.13)
since f n →n→∞ f strongly and u n L 2 + ∇u n L q ≤ C because of (4.12) and the boundedness of f n W −1 . Hence u n →n→∞ u in L 2 (). Since u n , n ∈ N, is bounded q ,0
in Wq1 (), u n n→∞ u in Wq1 (). Finally, since q
q
lim ∇u n L q = lim −q u n , u n = −q u, u = ∇u L q ,
n→∞
n→∞
u n →n→∞ u strongly in Wq1 ().
Remark 4.6. If we consider −q : L q (0, T ; Wq1 ()) → L q (0, T ; Wq1 ,0 ()), then −q is still maximal monotone since T −q u − f, u − vW −1 ,W 1 dt ≥ 0 0
q ,0
q
58
H. Abels
for all u ∈ L q (0, T ; Wq1 ()) and some f ∈ L q (0, T ; Wq1 ,0 ()), v ∈ L q (0, T ; Wq1 ()) implies −q v(t) = f (t) for almost all t ∈ (0, T ), which can be seen as follows: Choosing u = v + tw above with w ∈ L q (0, T ; Wq1 ()), t ∈ R, arbitrary, we obtain T −q (v + tw) − f, wW −1 ,W 1 dt ≥ 0. t q ,0
0
q
Dividing by t and passing t → 0, we obtain T −q v − f, wW −1 ,W 1 dt ≥ 0 q ,0
0
q
for all w ∈ L q (0, T ; Wq1 ()), which implies −q v = f almost everywhere. 5. Approximate System and Implicit Time Discretization In this section we construct weak solutions of the approximate system v ρ∂t v + ρv · ∇v − div S(c, Dv) + ρ∇g0 − δg0 = ρµ0 ∇c in Q, 2 ρt + div(ρv) + δg0 = 0 in Q,
(5.1) (5.2)
P0 (ρct + ρv · ∇c) = div(m(c)∇µ0 ) in Q,
(5.3)
∂A q A(c), ∂c
(5.4)
and ρµ0 + ρ 2 p(t) ¯ = βρ 2 g0 + φ(c) −
together with (1.5)–(1.9). Here as before P0 f = f − f , where f denotes the mean value of f on . Because of the damping term δg0 in (5.2), every sufficiently smooth solution of the latter system satisfies d E(c(t), v(t)) = − S(c(t), Dv(t)) : Dv(t) dx dt − m(c)|∇µ|2 dx − γ |v|2 dσ − δ |g0 |2 dx,
∂
L 2 (Q)-bound
which implies an additional of g0 . The latter energy identity can be obtained by multiplying (5.1) by v, (5.3) by µ, (5.4) by ∂t c and using (5.2) together with −βρ 2 ∂t c = ∂t ρ. Similar calculations are contained in the following proofs. The main result of this section is: Theorem 5.1. Let q > d, d = 2, 3, δ > 0, let Assumption 2.1 be satisfied and let R, ε > 0 and ∈ C 2 (R), (s) ≥ 0, (s) ≥ −M be such that (2.7) holds. Then for every v0 ∈ L 2 ()d , c0 ∈ Wq1 () with E(v0 , c0 ) < R, there are some v ∈ BCw ([0, ∞); L 2 ()d ) ∩ L 2 (0, ∞; Hn1 ()), 1 ()), c ∈ BCw ([0, ∞); Wq1 ()), µ0 ∈ L 2 (0, ∞; H(0)
g0 ∈ L 2 (0, ∞; L 2(0) ()),
p¯ ∈ L 2uloc ([0, ∞))
with (v, c)|t=0 = (v0 , c0 ) solving (5.1)–(5.4) together with (1.6)–(1.8) in the following weak sense:
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
59
1. For every ψ ∈ C0∞ (0, ∞; Hn1 () ∩ L ∞ ()d ), −(v, ∂t ψ) Q + (v · ∇v, ψ) Q + (ρ −1 S(c, Dv), ∇ψ) Q + (S(c, Dv), ∇ρ −1 ⊗ ψ) Q δ +γ (ρ −1 v, ψ) S − (g0 , div ψ) Q − (ρ −1 g0 v, ψ) Q = (µ0 ∇c, ψ) Q . (5.5) 2 2. For every ϕ ∈ C0∞ (0, ∞; C 1 ()), −(ρ, ∂t ϕ) Q − (ρv, ∇ϕ) Q + δ(g0 , ϕ) Q = 0,
(5.6)
(ρc, P0 ∂t ϕ) Q + (ρcv, ∇ϕ) Q − δ(g0 c, P0 ϕ) Q = (m(c)∇µ0 , ∇ϕ) Q , 1 − a q (c) ρµ0 + ρ 2 ( p¯ − βg0 ) − φ(c) , ϕ = |∇Λ|q−2 ∇Λ, ∇ϕ ,
(5.7)
Q
Q
(5.8)
where A = A(c). Moreover, E(c(t), v(t)) + γ v 2L 2 (S ) + δ g0 2L 2 (Q ) (s,t) (s,t) S(c(t), Dv(t)) : Dv(t) + m(c)|∇µ0 |2 d(x, τ ) ≤ E(c(s), v(s)) +
(5.9)
Q (s,t)
for all t ∈ [s, ∞) and almost all s ∈ [0, ∞) including s = 0. We note that for the weak formulation (5.7) one uses that (5.2) implies ρ∂t c + ρv · ∇c = ∂t (ρc) + div(ρcv) + δg0 c. In order to prove the latter theorem, we use an implicit time discretization. To this end, let h = N1 , N ∈ N. Then for given vk , ck and ρk = ρ(c ˆ k ), k ∈ N0 , we determine (vk+1 , g0,k+1 , ck+1 , µ0,k+1 , p¯ k+1 ) as a solution of the non-linear elliptic system v − vk ρk + ρk v · ∇v, ψ + (S(ck , Dv), ∇ψ) h δ +γ (v, ψ)∂ + (g0 , div(ρk ψ)) = (ρk µ0 ∇c, ψ) + (g0 v, ψ) (5.10) 2 for all ψ ∈ Hn1 (), ρ − ρk + div(ρk v) + δg0 = 0 in , h c − ck + ρk v · ∇c, ϕ ρk = − (m(ck )∇µ0 , ∇ϕ) h
(5.11) (5.12)
1 (), and for all ϕ ∈ H(0)
|∇ A(c)|q−2 ∇ A(c), ∇ϕ c − ck c + ck ρk µ0 + ρρk p¯ − βρρk g0 − φ0 (c) + κ ,ϕ = A(c) − A(ck ) 2
(5.13)
60
H. Abels
for all ϕ ∈ Wq1 (). Here ρ = ρ(c), ˆ ρk = ρ(c ˆ k ), and κ ∈ R is chosen such that κ 2
(c) = 0 (c) − 2 c with 0 (c) convex, which is possible since (c) ≥ −M > c−ck −∞. Moreover, we note that, if c = ck , then A(c)−A(c above has to be replaced by k) −1
c−ck A (c)−1 = a q (c). For simplicity, we will write A(c)−A(c even in the case c = ck , k) having the latter replacement in mind. Finally, we note that (5.11) implies ρ dx = ρk dx. (5.14)
|α| Lemma 5.2. Let q > d = 2, 3, let R, h, δ > 0, 0 < ε < |β| − 1, and let 2
∈ C (R), (c) ≥ 0, (c) ≥ −M be chosen such that (2.7) holds. Then for every (vk , ck ) ∈ L 2 ()d × Wq1 () with E(vk , ck ) < R there are some
(v, g0 , c, µ0 , p) ¯ ∈ Hn1 () × L 2(0) () × Wq1 () × H 2 () ∩ L 2(0) () × R solving (5.10)–(5.13) and which satisfy the discrete energy estimate |v − vk |2 E(c, v) + dx + δh g0 2L 2 () + γ h v 2L 2 (∂) ρk 2 +h S(ck , Dv) : Dv dx + h m(ck )|∇µ0 |2 dx ≤ E(ck , vk ).
(5.15)
Moreover, there is a constant C(R) independent of (vk , ck ) such that | p| ¯ ≤ c. Proof. We first show the a priori estimate (5.15) for any (v, g0 , c, µ, p) ¯ ∈ H 1 ()d × L 2 () × Wq1 () × H 2 () × R solving (5.10)–(5.13) and satisfying E free (c) ≤ R. First of all, because of (5.11) multiplied with 21 |v|2 we obtain |v|2 |v|2 dx = − div(ρk v) 2 2 2 2 ρ − ρk |v| |v| +δ . g0 = h 2 2
Thus
ρk (v · ∇v) · v dx =
ρk v · ∇
ρk |v|2 − ρk vk · v |v|2 dx + dx ρk (v · ∇v) · v dx − δ g0 h 2 |v|2 |vk |2 |v − vk |2 ρ − ρk |v|2 = dx − dx + dx + dx ρk ρk ρk 2h 2h 2h h 2 |vk |2 |v − vk |2 |v|2 = dx − dx + dx, ρ ρk ρk 2h 2h 2h
where we have used the simple algebraic relation a · (a − b) =
|a|2 |b|2 |a − b|2 − + , 2 2 2
a, b ∈ Rd .
(5.16)
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
61
Hence, choosing ψ = v ∈ Hn1 () in (5.10), we derive |v − vk |2 |v|2 ρ ρk S(ck , Dv) : Dv dx + γ |v|2 dσ dx + dx + 2h 2h ∂ |vk |2 = dx + ρk ρk µ0 ∇c · v dx + g0 div(ρk v) dx. (5.17) 2h Moreover, choosing ϕ = µ0 in (5.12), we conclude c − ck µ0 dx + ρk ρk µ0 ∇c · v dx = − m(ck )|∇µ0 |2 dx, h
(5.18)
where
2 c − ck2 c − ck c − ck c − ck µ0 dx + dx p¯ = dx − κ dx ρρk φ0 (c) h h h h 1 ρ − ρk + g0 dx |∇ A(c)|q−2 ∇ A(c) · ∇(A(c) − A(ck )) dx − h h 1 ≥ (E free (c) − E free (ck )) + g0 div(ρk v) dx (5.19) h ρk
because of (5.13) with ϕ = simple relation
1 h (A(c)
− A(ck )). Here we have used (5.11), (5.16), the
ρ − ρk = βρρk (ck − c) , and that
(5.20)
φ0 (c)(c − ck ) ≥ 0 (c) − 0 (ck ), |∇u|q−2 ∇u, ∇(u − u k )
1 q q ∇u L q − ∇u k L q , ≥ q q
since 0 is convex, −q u is the subgradient of u → q1 ∇u L q () , cf. Sect. 4. Moreover, because of (5.20) and (5.14), c − ck −β dx p¯ = (ρ − ρk ) dx p¯ = 0. ρρk h Combining (5.17)–(5.19), we conclude (5.15). Furthermore, we note that, since c(x) ∈ [−1 − ε, 1 + ε] due to E free (c) ≤ R by assumption and (2.7), we conclude c Wq1 () ≤ C(ε, R). k ) −1 −1 ρ ρk . Finally, the estimate | p| ¯ ≤ C follows from (5.13) with ϕ = A(c)−A(c c−ck In order to show existence of weak solutions, we use a homotopy argument based on the Leray-Schauder degree, cf. e.g. [8]. To this end we introduce the operators Lk , Fk : X → Y where
1 X = Hn1 () × L 2(0) () × Wq1 () × H(0) () × R, −1 () × Wq−1 Y = Hn−1 () × L 2(0) () × H(0) ,0 (),
62
H. Abels
and for w = (v, g0 , c, µ0 , p) ¯ ∈ X, ⎛
⎞ L k (v, g0 ) ⎜div(ρk v) + δg0 + p¯ ⎟ , Lk w = ⎝ divn (m(ck )∇µ0 ) ⎠ A(c) − q A(c) L k (v, g0 ), ϕ Hn−1 ,H 1 = (S(ck , Dv), ∇ϕ) + γ (v, ϕ)∂ − (g0 , div(ρk ϕ)) n
for all ϕ ∈ Hn1 () and ⎛
⎞ k ρk µ0 ∇c + 2δ g0 v − ρk v−v − ρk v · ∇v h ⎜ ⎟ ρ(c)−ρ ˆ c−ck ˆ ⎜−P ( h k ) − ρ(c)ρ k h dx + p¯ ⎟ Fk w = ⎜ 0 ⎟ , where k P0 ρk c−c ⎝ ⎠ h + ρk v · ∇c c−ck F (g , c, µ , p) ¯ + A(c) k 0 0 A(c)−A(ck ) c + ck Fk (c, g0 , µ) = ρk µ0 + ρk ρ p¯ − βρρk g0 − φ0 (c) + κ . 2 −1 () is defined as in (4.3). Here we have to modify Note that divn : L 2 ()d → H(0) 1 ρ(c) ˆ = α+βc outside of [−1 − ε, 1 + ε] to some ρˆ ∈ C 1 (R) in order to have Fk (w) well defined for all c ∈ Wq1 (). Using the lemma of Lax-Milgram,
L k (v, g0 ) div(ρk v) + δg0
Hn−1 () f × = 1 ∈ f2 L 2(0) ()
has a unique solution v ∈ Hn1 (), g0 ∈ L 2(0) () since L k (v, g0 ), v H 1 ,Hn−1 + (div(ρk v), g0 ) + δ(g0 , g0 ) ≥ c0 v 2H 1 () + δ g0 2L 2 () . n
2 Moreover, we note that div(ρ k v) + p¯ = f ∈ L () is equivalent to div(ρk v) = P0 f and p¯ = m( f ), since div(ρk v) dx = 0 due to n · v|∂ = 0. The invertibili1 () → H −1 () is a simple consequence of the lemma ty of divn (m(ck )∇·) : H(0) (0)
of Lax-Milgram too, the invertibility of I − q : Wq1 () → Wq−1 ,0 () follows from 1
Lemma 4.5, and c → A(c) is an invertible mapping on Wq1 (), since A (c) = a q (c) ≥ 1
m 0q > 0. Altogether it follows that Lk : X → Y is invertible and L−1 k : Y → X is continuous. Therefore w = (v, g0 , c, µ0 , p) ¯ is a solution of (5.10)–(5.13) if and only if Lk (w) − Fk (w) = 0, where we note that Lk (w) = Fk (w) implies −1 ρ(c)ρ ˆ (ρ(c) ˆ − ρk ) dx 0= k (c − ck ) dx = −β
due to (5.20). Moreover, Lk (w) − Fk (w) = 0 is equivalent to w − L−1 k (Fk (w)) = 0,
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
63
since L−1 k (0) = 0 due to A(0) = 0. Moreover, it is easy to observe that Fk : X → Y is a continuous and bounded mapping, where we note that Wq1 () c → is continuous since A(c) − A(ck ) = c − ck
1
c − ck ∈ Wq1 () A(c) − A(ck )
A (τ c + (1 − τ )ck ) dτ =
0
1
1
a q (τ c + (1 − τ )ck ) dτ
(5.21)
0
and a(s) ≥ m 0 > 0. Moreover, Kk (w) := L−1 k (Fk (w)) defines a compact operator on X since 3
Fk (w) ∈ L 2 ()d × Wq1 () × L 2(0) () × L 2 () →→ Y for all w ∈ X . Now we are able to apply a homotopy argument in order to show that the LeraySchauder degree of I − Kk at 0 is 1 in some suitable open set U . To this end, let ρˆτ (c) = (α + (1 − τ )βc)−1 , τ ∈ [0, 1]. Replacing vk , ck , ρ, ρk , β, κ, and φ0 in (5.10)– (5.13) by vkτ = (1 − τ )vk , ckτ = (1 − τ )ck , ρτ = ρˆτ (c), ρkτ = ρˆτ (ckτ ), βτ = (1 − τ )β, κτ = (1 − τ )κ, φ0τ (s) = (1 − τ )φ0 (s) for every τ ∈ [0, 1], and denoting by Lτk , Fkτ , Kkτ the corresponding operators, we get a family of compact operator Kkτ , depending continuously on τ ∈ [0, 1] such that Kk0 = Kk and deg(I + Kk0 , 0, U ) = deg(I + Kk1 , 0, U ), provided that (I + Kkτ )(w) = 0
for all w ∈ ∂U, τ ∈ [0, 1].
Because of the energy estimate (5.15) and the estimate of p, ¯ we have w X < C(R, δ) for any solution of (5.10)–(5.13) such that E free (c) ≤ R, which implies the latter condition for U = {w ∈ X : w X < C(R, δ), E free (c) < R}. In order to show that deg(I + Kk1 , 0, U ) = 1, we define a second homotopy by Kkτ (w) = (L1k )−1 (2 − τ )Fk1 (w),
τ ∈ [1, 2].
¯ is a solution of Then w − Kkτ (w) = 0 if and only if w = (v, g0 , c, µ0 , p) v δ λ ρ0 + ρ0 v · ∇v − g0 v, ϕ + (ν(0)Dv, Dϕ) + γ (v, ϕ)∂ , h 2 = (ρ0 g0 , div ϕ) + λ(ρ0 µ0 ∇c, ϕ) , ρ0 div v + δg0 = 0, c λρ0 + λρ0 v · ∇c = div(m(0)∇µ0 ), h c (ρ0 µ0 + ρ02 p), (1 − λ)A(c) − q A(c) = λ ¯ A(c) c dx (1 − λ) p¯ = −ρ02 h
(5.22) (5.23) (5.24) (5.25) (5.26)
64
H. Abels
for all ϕ ∈ Hn1 () together with n · v|∂ = ∂n c|∂ = ∂n µ0 |∂ = 0, where ρ0 = ρ(0) ˆ = α −1 and λ = (2 − τ ). Choosing ϕ = v in (5.22), multiplying (5.24) by µ0 and (5.25) by A(c) h , we obtain by similar calculations as before q
λρ0
A(c) 22 v 22 ∇ A(c) q + (1 − λ) +λ + (1 − λ)λ p¯ 2 2h 2h qh + ν(0) Dv 22 + γ v 2L 2 (∂) + δ g0 22 + m(0) ∇µ0 22 = 0,
(5.27)
where one uses that δ |v|2 2 dx = 0, (ρ0 v · ∇v · v − g0 |v| ) dx = − (ρ0 div v + δg0 ) 2 2 q q q ∇ A(c) q ∇ A(c) q ∇ A(0) q = − q q q ≤ −q A(c), A(c) − A(0) = −q A(c), A(c), c dx p¯ = −λ(1 − λ) p¯ 2 λρ02 h q
∇u
because of (5.23), and since −q is the subgradient of u → q q . Hence (5.27) implies that w = Kkτ (w), τ ∈ [1, 2], if and only if w = 0. Therefore any solution of w − Kkτ (w) remains in U and deg(I + Kk , 0, U ) = deg(I + Kk2 , 0, U ) = deg(I, 0, U ) = 1 since Kk2 = 0. Therefore (5.10)–(5.13) has a solution in U .
Now let N ∈ N be given and let (vk+1 , g0,k+1 , ck+1 , µ0,k+1 , p¯ k+1 ), k ∈ N0 := N∪{0}, be chosen successively as a solution of (5.10)–(5.13) with h = N1 and (vk , ck ) as initial value. Moreover, define f N (t) : [−h, ∞) by f N (t) = f k for t ∈ [(k − 1)h, kh) and f ∈ {v, g0 , c, µ0 } (setting g0,0 = µ0,0 = 0). For the following we denote + (5.28) h f (t) = f (t + h) − f (t), (− h f )(t) = f (t) − f (t − h), 1 ± gh ≡ (τh∗ g)(t) = g(t − h), ∂t,h f = ± f. (5.29) h h k(h+1) Then, choosing ψ = ρk−1 kh ϕ(x, t) dt in (5.10), where ϕ ∈ C0∞ ((0, ∞); Hn1 ()), and summing over all k ∈ N0 , gives − N v + v N · ∇v N , ϕ) Q + ( S(chN , Dv N ) − g0N I, Dϕ) Q + γ ((ρhN )−1 v N , ϕ) S (∂t,h δ − ((ρhN )−1 g0N v N , ϕ) Q = (µ0N ∇c N , ϕ) Q − (∇(ρhN )−1 · S(chN , Dv N ), ϕ) Q , 2 (5.30)
S(c, Dv) = ρˆ −1 (c)S(c, Dv). Morewhere ϕ ∈ C0∞ ((0, ∞); Hn1 ()) is arbitrary and over, since − + v N , ϕ) Q = −(v N , ∂t,h ϕ) Q + (v0 , ϕ|t=0 ) , (∂t,h
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
65
we conclude + −(v N , ∂t,h ϕ) Q + (v N · ∇v N , ϕ) Q + (( S(chN , Dv N ) − g N I, Dϕ) Q + γ ((ρhN )−1 v N , ϕ) S δ − ((ρhN )−1 g0N v N , ϕ) Q = (µ0N ∇c N , ϕ) Q − (∇(ρhN )−1 · S(chN , Dv N ), ϕ) Q , (5.31) 2
for all ϕ ∈ C0∞ (0, ∞; Hn1 ()). In the same way, one obtains that + − (ρ N , ∂t,h ψ) Q − (ρhN v N , ∇ψ) Q + δ(g0 , ψ) = 0,
(ρ c
N N
+ , ∂t,h ψ) Q
+ (ρhN c N v N
− m ∇µ , ∇ψ) Q = N
N
(5.32)
δ(g0N c N , ψ) Q
(5.33)
1 ()), where m N = m(τ ∗ c N ) and for all ψ ∈ C0∞ (0, ∞; H(0) h
|∇ A(c N )|q−2 ∇ A(c N ), ∇ψ Q − N h c κ N N N N N N N N N ρh µ0 + ρ ρh ( p¯ − βg0 ) − φ0 (c ) + (c + ch ) , ψ = N 2 − h A(c )
(5.34)
Q
for all ψ ∈ C0∞ (0, ∞; Wq1 ()). Here we have used that (5.11) implies ρk
c − ck ρc − ρk ck + ρk v · ∇c = + div(ρk vc) + δg0 c. h h
Finally, let E N (t) be the piecewise linear interpolation of E(ck , vk ) at tk = kh, k ∈ N0 , and let D N (t) = γ vk+1 2L 2 (∂) + δ g0,k+1 2L 2 () S(ck , Dvk+1 ) : Dvk+1 + m(ck )|∇µ0,k+1 |2 dx +
for all t ∈ (tk , tk+1 ), k ∈ N0 . Then (5.15) implies −
E(ck , vk ) − E(ck+1 , vk+1 ) d E N (t) = ≥ D N (t) for all t ∈ (tk , tk+1 ), k ∈ N0 , dt h
and therefore
E(c0 , v0 )ϕ(0) +
∞
∞
E N (t)ϕ (t) dt ≥
0
D N (t)ϕ(t) dt
(5.35)
0
for all ϕ ∈ W 1 (0, ∞) with ϕ ≥ 0. In particular, we have 2ν(chN )|Dv N |2 + η(chN )| div v N |2 d(x, τ ) + δ g0N 2L 2 (S E(c N (t), v N (t)) +
(s,t) )
Q (s,t)
+ γ v N 2L 2 (S
(s,t) )
+ Q (s,t)
m(chN )|∇µ0N |2 d(x, τ ) ≤ E(c N (s), v N (s))
for all 0 ≤ s ≤ t < ∞ with s, t ∈ hN0 .
(5.36)
66
H. Abels
Using the bounds given by the energy estimate above, we can pass to a subsequence, again denoted by (v N , g0N , c N , µ0N , p¯ N ), such that (v N , µ0N , g0N , p¯ N ) N →∞ (v, µ0 , g0 , p) ¯ in L 2 (0, ∞; H 1 ()d+1 × L 2 () × R), (v N , c N ) ∗N →∞ (v, c) in L ∞ (0, ∞; L 2 ()d × Wq1 ()). In order to pass to the limit in all non-linearities, we show strong convergence of v N , c N , ∇ A(c N ). To this end, we define ρ˜ N and v˜ N as piecewise linear interpolation of ρ N (tk ), v N (tk ), resp., where tk = kh, k ∈ N0 . More precisely, ρ˜ N = h1 χ[0,h] ∗t ρ N and v˜ N = h1 χ[0,h] ∗t v N , where the convolution is only taken with respect to the time variable t. − N Then ∂t v˜ N = ∂t,h v and v˜ N − v N H −s () ≤ h ∂t v˜ N H −s () , ρ˜ N − ρ N H −s () ≤ h ∂t ρ˜ N H −s () (5.37) for all s ≥ 0. Because of (5.30), ∂t v˜ N is bounded in L 2 (0, ∞; H −s ()) for any s > d2 , where one uses that v N · ∇v N ∈ L 2 (0, ∞; L 1 ()), µ0N ∇c N ∈ L 2 (Q) and ∇(ρhN )−1 · S(chN , Dv N ) ∈ L 2 (0, ∞; L 1 ()) are bounded due to (5.36). Hence, applying the lemma of Aubin-Lions, we conclude that v˜ N → N →∞ v
in L 2 (0, T ; H s ()),
for all 0 ≤ s < 1, 0 < T < ∞, and for a suitable subsequence. On the other hand, (5.37) implies v˜ N − v N → N →0 0
in L 2 (0, ∞; H −s ()) if s >
d . 2
Since v˜ N , v N are bounded in L ∞ (0, ∞; L 2 ()), we also obtain v˜ N − v N → N →0 0
in L p (0, T ; H −s ())
for any 1 < p < ∞ and s, T > 0 by interpolation. Moreover, since v˜ N , v N are bounded in L 2 (0, ∞; H 1 ()), we conclude v N → N →∞ v
in L 2 (0, T ; H s ()) for all T > 0, 0 ≤ s < 1.
Finally, since v˜ N ∈ L ∞ (0, ∞; L 2 ()) is bounded and v˜ N converges weakly in H 1 (0, ∞; H −s ()) → BU C([0, ∞); H −s ()) for s > d2 , and v˜ N |t=0 = v0 , we conclude v ∈ BCw ([0, ∞); L 2 ()) and v|t=0 = v0 , cf. Lemma 4.1. Similarly, because of (5.32), ∂t ρ˜ N is bounded in L 2uloc ([0, ∞); H −1 ()). On the other hand, ρ˜ N ∈ L ∞ (0, ∞; Wq1 ()) because of c ∈ L ∞ (0, ∞; Wq1 ()) and c(t, x) ∈ [−1 − ε, 1 + ε]. Therefore the lemma of Aubin-Lions yields that for a suitable subsequence, ρ˜ N → N →∞ ρ˜
in L 2 (Q T ) for all T > 0,
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
67
and almost everywhere for some ρ˜ ∈ L ∞ (Q). Furthermore, by (5.37) ρ˜ N − ρ N → N →∞ 0
in L 2uloc ([0, ∞); H −1 ()).
As before we conclude that also ρ N = ρ(c ˆ N ) converges strongly in L 2 (Q T ) for every T > 0 and almost everywhere. Furthermore, since α+βc N = (ρ N )−1 , also c N converges strongly in L 2 (Q T ), T > 0, and almost everywhere and we conclude ρ˜ = ρ(c). ˆ Since 1 ([0, ∞); H −1 ()) → BU C([0, ∞); H −1 ()) and ρ˜ N ∈ L ∞ (0, ∞; W 1 ()) ρ˜ N ∈ Huloc q are bounded, ρ ∈ BCw ([0, ∞); Wq1 ()) due to Lemma 4.1. Moreover, since ρ N ∈ H 1 (0, T ; H −1 ()), 0 < T < ∞, converges weakly, ρ|t=0 = ρ0 = lim N →∞ ρ N |t=0 weakly in H −1 (). Therefore also c ∈ BCw ([0, ∞); Wq1 ()) and c|t=0 = c0 . Using these convergence results, it is easy to pass to the limit in Eqs. (5.32), (5.33) to obtain (5.6), (5.7). Moreover, the right-hand side of (5.34) converges to −1 f = a(c) q ρµ0 + ρ 2 ( p(t) ¯ − βg0 ) − φ(c) weakly in L 2 (Q T ) for every T < ∞, where one can use (5.21) in order to pass to the c−ck limit in A(c)−A(c . k) N Hence q A(c ) converges weakly in L 2 (Q T ) to f for every T < ∞ and therefore −q A(c N ), A(c N ) X T ,X T = ( f N , A(c N )) Q T → N →∞ ( f, A(c)) Q T with X T = L q (0, T ; Wq1 ()), where we note that −q is a maximal monotone operator on X T due to Remark 4.6. Hence Lemma 4.4 implies f = −q A(c) and A(c N ) → A(c) strongly in L q (0, T ; Wq1 ()) for every 0 < T < ∞ because of T T q N q ∇ A(c ) L q () dt → N →∞ ∇ A(c) L q () dt. 0
Since and
A−1
∈
0
C 1 (R), c N
converges strongly in
L q (0, T ; Wq1 ()) too. Hence (5.8) holds
∇(τh∗ ρ N )−1 · S(c N , Dv N ) N →∞ ∇ρ −1 · S(c, Dv)
in L 1 (Q T ), T < ∞.
Using the convergence results above, it is easy to show that (5.31) converges to (5.5). Finally, since v N (t) → N →∞ v(t) in L 2 () and c N (t) → N →∞ c in Wq1 () for almost every t ∈ (0, ∞) (for a suitable subsequence), E N (t) → N →∞ E(c(t), v(t))
for almost all t ∈ (0, ∞).
Moreover, by lower semi-continuity of norms and almost everywhere convergence of c N to c, ∞ ∞ D N (t)ϕ(t) dt ≥ D(t)ϕ(t) dt lim inf N →∞
0
0
for all ϕ ∈ W11 (0, ∞) with ϕ ≥ 0, where D(t) = γ v(t) 2L 2 (∂) + δ g0 (t) 2L 2 () 2ν(c(t))|Dv(t))| + η(c(t))| div v(t)|2 + m(c(t))|∇µ0 (t)|2 dx. +
68
H. Abels
Hence, passing to the limit in (5.35), we obtain ∞ E(c0 , v0 )ϕ(0) + E(c(t), v(t))ϕ (t) dt ≥ 0
∞
D(t)ϕ(t) dt
(5.38)
0
for all ϕ ∈ W11 (0, ∞) with ϕ ≥ 0. Because of Lemma 4.3, this implies (5.9). 6. Existence of Weak Solutions for the General System Let R, ε, c0 , v0 , and be as in the assumptions of Theorem 2.4. Moreover, for 0 < δ ≤ 1 let (v, g0 , c, µ0 , p) ¯ ≡ (vδ , g0,δ , cδ , µ0,δ , p¯ δ ) be a solution of (5.1)–(5.4) together with (1.5)–(1.9) due to Theorem 5.1. In order to pass to the limit δ → 0, we need a good representation of g0,δ and a suitable estimate of p(t). ¯ As in (1.12), we use the decomposition g0,δ = g1,δ − ∂t G(vδ ), where G(v) = −1 N div v due to (1.13)–(1.14). Then (5.1) is equivalent to ∂t Pσ vδ + vδ · ∇vδ − ρδ−1 div S(cδ , Dvδ ) + ∇g1,δ − δρ −1 g0,δ
vδ = µ0,δ ∇cδ 2
(6.1)
and its weak formulation (5.5) is equivalent to −(Pσ vδ , ∂t ψ) Q +(vδ · ∇vδ , ψ) Q +(ρδ−1 S(cδ , Dvδ ), ∇ψ) Q +(S(cδ , Dvδ ), ∇ρδ−1 ⊗ψ) Q δ + γ (ρδ−1 vδ , ψ) S − (g1,δ , div ψ) Q − (ρδ−1 g0,δ vδ , ψ) Q = (µ0,δ ∇cδ , ψ) Q (6.2) 2 for all ψ ∈ C0∞ (0, ∞; Hn1 () ∩ L ∞ ()d ). Moreover, we have the following estimates uniformly in 0 < δ ≤ 1. d and let g1,δ , G(vδ ) be as above. Then there is a constant Lemma 6.1. Let 1 < p < d−1 C = C (R, ε, q) independent of 0 < δ ≤ 1 such that
G(vδ ) L ∞ (0,∞;H 1 ()) + G(vδ ) L 2 (0,∞;H 2 ()) + g1,δ L 2 (0,∞;L p ()) ≤ C. Proof. The estimate of the first two terms simply follows from G(vδ (t)) H 1 () ≤ C v(t) L 2 () , G(vδ (t)) H 2 () ≤ C v(t) H 1 () , cf. (4.4) and (4.5). In order to estimate g1,δ , we choose ϕ = η(t)∇ψ, ψ ∈ W p2 ,N () = {u ∈ W p2 () : ∂n u|∂ = 0}, η ∈ C0∞ (0, ∞) in (6.2) and obtain
∞ 0
=
g1,δ ψ ∞ ∞
0
+ 0
dx η(t) dt
ρδ−1 S(cδ , Dvδ ) − v ⊗ v +
|v|2 I 2
: ∇ 2 ψ dx η dt
S(cδ , Dvδ ) · ∇ρδ−1 − µ0,δ ∇cδ − δρ −1 g0,δ
vδ · ∇ψ dx η dt 2
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
69
for all ψ ∈ W p2 ,N (), η ∈ C0∞ (0, ∞). Here we have used the identity v ·
∇v = div(v ⊗ v) − ∇ |v|2 . Hence g1,δ (t) is a very weak solution of the NeumannLaplace equation, cf. Sect. 4, for almost every 0 < t < ∞ with right-hand side Fδ (t) satisfying 2
Fδ (t) (W 2 ()) p ,N ≤ C S(cδ (t), Dvδ (t)) L 2 + v(t) 2L s vδ + S(cδ , Dvδ ) · ∇ρδ−1 (t) − µ0,δ ∇cδ (t) − δρ −1 g0,δ 2 L1 2 ≤ C(R, q) vδ (t) H 1 () + ∇µ0,δ (t) L 2 () ∈ L (0, ∞), where we have used that to W p1 () → L ∞ () as well as v 2L s ≤ v 2 C v H 1 v L 2 with s = 3 if d = 3 and s = 4 if d = 2. Therefore g1,δ L 2 (0,∞;L p ()) ≤ C(R, ε, q) Fδ (t) L 2 (0,∞;(W 2
p ,N
()) )
1
H2
≤
≤ C (R, ε, q)
uniformly in 0 < δ ≤ 1. Next we estimate p¯ δ (t). Lemma 6.2. There is a constant C = C(R, ε, q) such that p¯ δ L 2
uloc ([0,∞))
≤C
uniformly in δ > 0.
1 Proof. Choosing ϕ = η(t)a q (cδ )ρδ−2 , η ∈ C0∞ (0, ∞) in (5.8) and using g0,δ dx = 0, we obtain 1 1 ρδ−2 φ(cδ ) − ρ −1 µ0,δ dx + p¯ δ (t) = f (cδ )|∇ A(cδ )|q dx || ||
for almost all t ∈ (0, ∞), where f (s) = a p¯ δ L 2
− q1
1 d a q (s)ρ −2 (s) . Hence (s) ds
uloc ([0,∞))
≤C
uniformly in 0 < δ ≤ 1 since cδ ∈ L ∞ (0, ∞; Wq1 ()) and µ0,δ ∈ L 2 (Q) are uniformly bounded and Wq1 () → L ∞ (). Using these bounds, we obtain the following essential compactness result: Lemma 6.3. There is a subsequence, again denoted by (vδ , g0,δ , cδ , µ0,δ , p¯ δ )δ>0 , such that (vδ , G(vδ )) →δ→0 (v, G(v)) for all 0 < T < ∞.
in L 2 (0, T ; L 2 () × H 1 ())
70
H. Abels 1
Proof. Choosing ϕ = a q (cδ )ρδ−2 ψ, ψ ∈ C0∞ (Q), in (5.8), we obtain 1
− β∂t G(vδ ) = ρδ−1 µ0,δ + p¯ δ (t) − φ(cδ ) − g1,δ + ρδ−2 a q (cδ )q A(cδ ) in D (Q), (6.3) where 1 ρδ−2 a q (cδ )q A(cδ ), ϕ D (Q),D(Q) ∂ A (cδ )|∇ A(cδ )|q−2 ∇ A(cδ ) · ∇ϕ dx + =− ρδ−2 f (cδ )|∇ A(cδ )|q ϕ dx ∂c 1 −1 d a q (s)ρ −2 (s) as before. Hence the last for all ϕ ∈ C0∞ (Q) and f (s) = a q (s) ds term in (6.3) is uniformly bounded in L ∞ (0, ∞; H −s ()) if s >
d 2
+ 1, and therefore
β∂t G(vδ ) ∈ L 2uloc ([0, ∞); H −s ()) is uniformly bounded due to Lemma 6.1, Lemma 6.2 and since µ0,δ ∈ L 2 (Q) and φ(cδ ) ∈ L ∞ (Q) are bounded. On the other hand by (6.2) and Lemma 6.1, ∂t Pσ vδ ∈ L 2uloc ([0, ∞); H −s ())
for s >
d , 2
see also proof of Lemma 6.1. Therefore ∂t vδ = ∂t Pσ vδ + ∂t ∇G(vδ ) ∈ L 2uloc ([0, ∞); H −s−1 ()) for s >
d +1 2
is uniformly bounded. Hence vδ →δ→0 v in L 2 (Q T ) for every T < ∞ by the AubinLions lemma. Since G(vδ (t)) ∈ H 1 () depends continuously on vδ (t) ∈ L 2 (), we obtain the second part. As an important convergence result we also need: Lemma 6.4. Let F ∈ C 1 (R). Then (G(vδ ), ∂t (F(cδ )ϕ)) Q →δ→0 (G(v), ∂t (F(c)ϕ)) Q for all ϕ ∈ C 1 (Q) and a suitable subsequence. Proof. First of all ∂t (F(cδ )ϕ) = F (cδ )∂t cδ ϕ + F(cδ )∂t ϕ. Since cδ →δ c almost everywhere, cδ ∈ L ∞ (Q), and ∂t cδ = −β −1 ρ −2 div(ρδ vδ ) ∈ L 2 (Q) is uniformly bounded, cf. (4.2), we obtain F (cδ )∂t cδ ϕ + F(cδ )∂t ϕ δ→0 F (c)∂t c ϕ + F(c)∂t ϕ = ∂t (F(c)ϕ) in L 2 (Q) for all C 1 (Q). Moreover, due to Lemma 6.3, G(vδ ) →δ→0 G(v) strongly in L 2 (Q T ) for every T < ∞. Thus the statement of the lemma follows. As a corollary we obtain strong convergence of ∇cδ in L q (Q T ):
Diffuse Interface Model for Viscous Incompressible Newtonian Fluids
71
Corollary 6.5. There is a subsequence, again denoted by (cδ )0<δ≤1 , such that ∇cδ →δ→0 ∇c in L q (Q T ) for every T < ∞. Proof. By (5.8) we have ∞ −q A(cδ (t)), ϕW −1 ,W 1 η(t) dt = −(G(vδ ), ∂t (F(cδ )ϕη)) Q + ( f δ , ϕη) Q q ,0
0
q
∞ (), η ∈ C ∞ (0, ∞), where F(c ) = β ρˆ 2 (c )a for all ϕ ∈ C(0) δ δ 0
fδ = a
− q1
− q1
(6.4)
(cδ ) and
(cδ ) ρδ µ0,δ + ρδ2 p¯ δ (t) − βρδ2 g1,δ − φ(cδ ) .
By the convergence of cδ almost everywhere and the weak convergence of µ0,δ , p¯ δ , and g1,δ , we obtain f δ δ→0 f
in L 2 (0, T ; L p ()) for all 1 < p <
for all T < ∞, where f =a
− q1
d , d −1
(c) ρµ0 + ρ 2 p(t) ¯ − βρ 2 g1 − φ(c) .
Hence lim ( f δ , A(cδ )η) Q = ( f, A(c)η) Q
δ→0
for all η ∈ C0∞ (0, ∞). On the other hand, by Lemma 6.4, lim (G(vδ ), ∂t (F(cδ )ϕη)) Q = (G(v), ∂t (F(c)ϕη)) Q ,
δ→0
lim (G(vδ ), ∂t (F(cδ )A(cδ )η)) Q = (G(v), ∂t (F(c)A(c)η)) Q
δ→0
(6.5)
∞ (). Moreover, because of (6.4), F(c )∂ G(c ) is for all η ∈ C0∞ (0, ∞) and ϕ ∈ C(0) δ t δ
uniformly bounded in L q (0, T ; Wq−1 ,0 ()) for every 0 < T < ∞, which implies F(cδ )∂t G(cδ ) δ→0 F(c)∂t G(c)
in L q (0, T ; Wq−1 ,0 ())
for all 0 < T < ∞. Hence −q A(cδ ) δ→0 F(c)∂t G(c) + f
in L q (0, T ; Wq−1 ,0 ()) =: X T .
Similarly, because of (6.4), (F(cδ )∂t G(cδ ), A(cδ )) is uniformly bounded in L 2 (0, T ) for every 0 < T < ∞. Therefore we obtain from (6.5) lim F(cδ )∂t G(cδ ), A(cδ ) X T ,X T = F(c)∂t G(c), A(c) X T ,X T
δ→0
for all 0 < T < ∞ by passing η → 1. Thus − lim q A(cδ ), A(cδ ) X T ,X T = F(c)∂t G(c) + f, A(c) X T ,X T δ→0
72
H. Abels
1 ()) → L q (0, T ; W −1 ()) is maxifor all 0 < T < ∞. Since −q : L q (0, T ; Wq,0 q ,0 mal monotone, cf. Remark 4.6, and because of Lemma 4.4, we conclude −q A(c) = F(c)∂t G(c) + f , and therefore q
lim ∇ A(cδ ) L q (Q T ) = − lim q A(cδ ), A(cδ ) X T ,X T
δ→0
δ→0
q
= −q A(c), A(c) X T ,X T = ∇ A(c) L q (Q T ) for all T < ∞, which proves strong convergence of ∇ A(cδ ) in L q (Q T ). Since −1
∇cδ = a q (cδ )∇ A(cδ ) and cδ ∈ L ∞ (Q) converges almost everywhere, this implies the strong convergence of ∇cδ in L q (Q T ). Proof of Theorem 2.4. Using the convergence results above, one can easily show that (6.2) converges to (2.1) and that (5.6)–(5.8) converge to (2.2)–(2.4) for a suitable subsequence δ j → j→∞ 0. Let us only mention the following points: Note that 1
1
δg0 = δ 2 δ 2 g0 →δ→0 0
in L 2 (Q)
1
since δ 2 g0 ∈ L 2 (Q) is uniformly bounded according to (5.9). Moreover, passing to the limit δ → 0 in (5.7), one first only obtains (P0 ρc, ∂t ψ) Q + (ρcv, ∇ψ) Q = (m(c)∇µ, ∇ψ) Q for all ψ ∈ C0∞ (0, ∞; C 1 ()). But, because of (2.2), we have β 0= ∂t ρ dx = − βρ 2 ∂t c dx = − ∂t (ρc) dx, α ∂(ρc) ∂ρ 2 2 since ∂ρ ∂c = −βρ and ∂c = ρ + ∂c c = αρ , cf. (3.10). Hence (2.3) follows. By the same arguments as in the proof of Theorem 5.1 one shows that v ∈ BCw ([0, ∞); L 2 ()), c ∈ BCw ([0, ∞); Wq1 ()) and (v, c)|t=0 = (v0 , c0 ). Finally, the energy inequality (2.5) follows from passing to the limit (5.38) using the strong convergence of vδ (t) ∈ L 2 (), cδ (t) ∈ Wq1 () for almost all t ∈ (0, ∞) and Lemma 4.3 as in the proof of Theorem 5.1.
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5. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. In: Annual review of fluid mechanics, Vol. 30, Palo Alto, CA: Annual Reviews, 1998, pp. 139–165 6. Boyer, F.: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999) 7. Boyer, F.: Nonhomogeneous Cahn-Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2), 225–259 (2001) 8. Deimling, K.: Nonlinear Functional Analysis. Berlin: Springer-Verlag, 1985 9. Diestel, J., Uhl, Jr., J.J.: Vector Measures. Providence, RI: Amer. Math. Soc., 1977 10. Gurtin, M.E., Polignone, D., Viñals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996) 11. Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod, 1969 12. Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D 179(3–4), 211–228 (2003) 13. Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454(1978), 2617–2654 (1998) ˇ 14. Roubíˇcek, T.: A generalization of the Lions-Temam compact imbedding theorem. Casopis Pˇest. Mat. 115(4), 338–342 (1990) 15. Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Volume 49 of Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc., 1997 16. Simon, J.: Compact sets in the space L p (0, T ; B). Ann. Mat. Pura Appl. (4), 146, 65–96, (1987) 17. Starovo˘ıtov, V.N.: On the motion of a two-component fluid in the presence of capillary forces. Mat. Zametki 62(2), 293–305 (1997) 18. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton, NJ: Princeton Univ. Press, 1970 19. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Amsterdam-New YorkOxford: North-Holland Publishing Company, 1978 Communicated by P. Constantin
Commun. Math. Phys. 289, 75–106 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0811-7
Communications in
Mathematical Physics
Compressible Flow and Transonic Shock in a Diverging Nozzle Shuxing Chen Institute of Mathematics, Fudan University, Shanghai 200433, P.R.China. E-mail:
[email protected] Received: 27 February 2008 / Accepted: 11 March 2009 Published online: 21 April 2009 – © Springer-Verlag 2009
Abstract: The paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible Euler system. Consider a nozzle having a shape as a diverging truncated sector with generic opening angle: if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow, and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned, then a transonic shock appears in the nozzle. To determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation. We prove that the receiver pressure can uniquely determine the location of the transonic shock, as well as the flow behind the shock. Such a conclusion was conjectured by Courant and Friedrichs, and is confirmed theoretically in this paper for the divergent nozzles. The main advantage of this paper compared with the previous studies on this subject is that the section of the nozzle is allowed to vary substantially, while the transonic shock is not assumed to pass a fixed point. The situation coincides with the requirement in Courant-Friedrichs’ conjecture. To describe the compressible flow we use the full Euler system, which is purely hyperbolic in the supersonic region and is elliptic-hyperbolic in the subsonic region. Solving the free boundary value problem of an elliptic-hyperbolic problem forms the main part of this paper. In our demonstration some new approaches, including the introduction of a pseudo-free boundary problem and the corresponding relaxation, design of a delicate double iteration scheme, are developed to overcome the difficulties caused by the divergence of the nozzle. 1. Introduction It is known that the study on nozzle gas flows plays a fundamental role in the operation of turbines, wind tunnels and rockets. When a nozzle is in its practical use, a compressible flow comes into the nozzle at the entrance, then changes its flow parameters in the nozzle and finally leaves it at the exit. There are various types of nozzle
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flows depending on the shape of the nozzle, the parameters of the upstream flow at the entrance and the condition given at the exit. As early in 1948 Courant and Friedrichs gave a first systematical analysis on this problem from the viewpoint of nonlinear partial differential equations in [14]. They analyzed a different wave pattern in the de Laval nozzle, which consists of a converging part near the entry and a diverging part near the exit. It is indicated that for a given subsonic flow at the entry of the nozzle, if the pressure of the flow is greater than a critical value, then the flow behind the throat of the nozzle may become supersonic. Moreover, if the pressure at the exit, called receiver pressure, is suitably assigned, then at a certain place in the diverging part of the nozzle a transonic shock front intervenes, and the position and the strength of the shock front are automatically adjusted so that the pressure at the exit coincides with the given receiver pressure. Such a conclusion is verified by physical experiments (see [14,27] and the references therein), while some corresponding approximate methods were also developed. It is desirable to establish an exact mathematical analysis based on the theory of partial differential equations. In the 70’s by using a quasi one-dimensional system as a model, T.P.Liu and his collaborators [17,21–23] studied the nozzle gas flow. For the de Laval nozzle they proved the stability of the steady flow in the diverging part of the nozzle, as well as the stability of the transonic shock in the corresponding case. In recent years the progress of studying the system of multi-dimensional conservation laws and shock waves (e.g. [7,10,25,26]) stimulated mathematician’s interest to the classical problem again. Recently, by taking the potential flow equation as a model G.Q.Chen and M.Feldman [3–6], Z.Xin and H.Yin [29,30] proved the existence and stability of the compressible flow with a transonic shock in the model of the potential equation. In [2,12,13,15,31] this problem is also studied in the model of the stationary Euler system. However, in these works the nozzle is assumed to be a duct with a constant section or slowly varying sections. A remarkable phenomenon is that for a nozzle with a constant section, if the upstream flow at the entrance is a uniform supersonic flow and the transonic shock is a plane shock perpendicular to the direction of the motion of the fluid, then the receiver pressure at the exit has been determined by RankineHugoniot conditions. Therefore, any other value of the receiver pressure may cause nonexistence. Meanwhile, the location of the transonic shock can have a shift in the nozzle, the uncertainty of the location of the shock front causes non-uniqueness of solutions to the problem. Obviously, such a non-existence and non-uniqueness in the simplest case also bring troubles to the perturbed case. Since the nozzle in practical use does not have constant section or slowly varying sections, people are more interested in studying the motion of compressible flow in general ducts. The de Laval nozzle usually has a converging part at the entrance and a diverging part at the exit. It has a throat with a minimum section in the middle. Physical experiments show that a possible flow pattern is as follows. A given upstream subsonic gas flow enters the converging part of the nozzle and speeds up due to the contraction of the area of sections of the nozzle, then becomes a supersonic flow after passing through the throat of the nozzle under suitable conditions. When the pressure at the exit is well controlled, there will appear a transonic shock in the divergent part of the nozzle, so that the supersonic flow will become subsonic again due to passing through a transonic shock. In this paper we will concentrate our attention to consider the flow in the diverging part of the nozzle. Therefore, we assume that the flow runs in a diverging truncated sector, while the flow at the entrance is supersonic and near an axial symmetric flow. Since in
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any supersonic flow its upstream part is not influenced by its downstream part, then the main problem is to find the shock surface and the solution in the subsonic region, which is surrounded by the lateral wall of the nozzle and located between the transonic shock front and the exit. Our study shows that for the flow in a diverging truncated sector if the parameters of the upstream flow at the entrance and the receiver pressure are suitably given, then any receiver pressure can uniquely determine the location of the transonic shock, as well as the flow behind the transonic shock. Particularly, the position and the strength of the shock front will be automatically adjusted so that the pressure at the exit takes the value of the given receiver pressure as Courant-Friedrichs predicted in [14]. Moreover, the transonic shock and the flow behind the shock are stable with respect to the perturbation of the parameters of the upstream flow and the receiver pressure. Therefore, the conclusion in our paper confirms Courant-Friedrichs conjecture on nozzle flow when the nozzle is a truncated sector. In this paper the flow is described by the two dimensional full Euler system. Since the full Euler system is hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region, then the related boundary value problem arising in our analysis is more complicated than the discussion for the problem for the potential flow equation in [3,4,29]. Moreover, we could not make an extension to eliminate the lateral boundary, as in [3,12], so that more sophisticated analysis on the existence and estimates of the solution to related boundary value problems is required. Another crucial point is that the background solution of our problem is radial, which does not have translational symmetry. Particularly, the derivatives of the parameters of the flow are not always small. Since the location and the profile of the shock front are unknown before we know the whole flow, the mathematical problem is a free boundary value problem for the Euler system called (FB) in this paper. To treat such free boundary value problems a usual method is to determine the location of the free boundary S and the solution in the domain with S as its part of the boundary alternatively, then to prove that the process is convergent to a right solution of (FB). Since in our problem both the profile and the shift of S is unknown, the above-mentioned method failed to construct a convergent process. To overcome this difficulty we first give a relaxation of the receiver pressure. That is, we temporarily allow the pressure at the exit to satisfy a condition up to a constant difference. Meanwhile, we ask the transonic shock to pass an assigned point P, while the shape of the shock is still to be determined. Such a problem is called pseudo-free boundary value problem (PsFB). Later, we will prove that the relaxation of the receiver pressure monotonically depends on the shift of S, so that the monotonicity can eliminate the relaxation and leads to the solvability of the problem (FB). Here we emphasize that it is crucial to introduce the problem (PsFB) in our analysis. To solve the problem with a free boundary passing a given point on the x-axis, we employ the Schauder fixed point theorem. Here the crucial point is to solve the boundary value problem of an elliptic-hyperbolic composite system in a fixed domain. However, because of the essential change of the area of the sections of the nozzle the coefficients of the system are not a perturbation of constants, so that the construction of the iterative scheme and the proof of its convergence in this paper are more delicate than the previous cases. Indeed, the iterative scheme includes double iterations, and the iterative process for nonlinear boundary value problems is a full Newton iteration. Then a careful analysis for all coefficients produced in the iteration help us to find the convergence mechanism (see Sect. 5).
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The remaining part of this paper is arranged as follows. In Sect. 2 we formulate the physical problem and reduce it to a free boundary problem (FB) in the subsonic region with a shock front as its free boundary. Due to the geometry of the nozzle it is convenient to describe the motion of the flow in the polar coordinate system. As a preparation the background solution and its some properties are described there. In Sect. 3 we introduce a problem (PsFB), in which a relaxation of the receiver pressure is added and the shock front is assumed to pass a given point on the symmetrical axis. As mentioned above, to solve the problem (PsFB) we temporarily fix the whole profile of the shock front further and then concentrate our attention on solving the fixed boundary value problem. In order to solve the latter we first separate the elliptic part and the hyperbolic part of the system. Then we introduce a Lagrange transformation to straighten all stream lines, so that two transport equations take the simplest form. In Sect. 4 we make linearization of the nonlinear problem including the linearization of both the system and the boundary conditions. In Sect. 5 we solve the linearized problem. Here a subordinate boundary value problem for a second order elliptic equation is introduced. The whole linearized problem is also solved via an iterative process in Sect. 5. In the process some delicate estimates are established. Based on these estimates we confirm in Sect. 6 that the iterative process is convergent, so that the nonlinear fixed boundary problem is solvable. Then in Sect. 7 we solve the problem (PsFB), so that the profile of the shock front can be determined together with the approximate solution. Finally, the real free boundary problem (FB) and the original physical problem are solved in Sect. 8 by adjusting the shift of the shock front and eliminating the relaxation of the receiver pressure.
2. Mathematical Formulation 2.1. System in polar coordinate system. The two dimensional stationary compressible flow can be described by the Euler system in Cartesian coordinates as 2
∂x j (ρu j ) = 0,
(2.1)
∂x j (ρu k u j + pδk j ) = 0, k = 1, 2,
(2.2)
j=1 2 j=1 2
∂x j (ρ Eu j ) = 0,
(2.3)
j=1
where δk j = 1 if k = j and δk j = 0 otherwise, u i stands for the velocity component of the flow along xi axis (we also write (u 1 , u 2 ) by (u, v) in some cases), p, ρ, E and S stand for the pressure, density, total energy and entropy respectively. For a polytropic gas, S − S0 γ )ρ , cν p 1 2 1 γp E = e + + | u| = + |u|2 , ρ 2 (γ − 1)ρ 2 p = A(s)ρ γ = (γ − 1) exp(
(2.4) (2.5)
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1 p is the internal energy, | u |2 = u 21 + u 22 , ρ can be written as a function γ −1ρ 1 ∂ p(ρ, S) 2 of p and S as ρ = ρ( p, S). Later on, we also use a = to denote the sound ∂ρ speed and use (u, v) to denote the components of the velocity instead of (u 1 , u 2 ). In this paper we are going to discuss the compressible flow in a nozzle, which is a two dimensional domain with the shape as a truncated sector. In this case it is convenient to work in the polar coordinate system (r, θ ). The nozzle is assumed to be located in r1 ≤ r ≤ r2 , θ1 ≤ θ ≤ θ2 . Therefore, in (r, θ ) coordinates, the nozzle can be described by Q = (r1 , r2 ) × (θ1 , θ2 ). The transformation from Cartesian coordinates to spherical coordinates is where e =
r = (x 2 + y 2 )1/2 0 ≤ r < ∞, y θ1 ≤ θ ≤ θ2 . θ = arctan x Its inverse transformation is x = r cos θ, y = r sin θ. The system of conservation laws (2.1)–(2.3) in the polar coordinate system takes the form ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ ρuv ( p + ρu 2 ) ρ(u 2 − v 2 ) 2 ∂ ⎜ ρuv ⎟ 1 ∂ ⎜ ( p + ρv ) ⎟ 1 ⎜ 2ρuv ⎟ (2.6) ⎠+ ⎠ = 0, ⎠+ ⎝ ⎝ ⎝ ρu ρu ρv ∂r r ∂θ r ρu E ρu E ρv E where (u, v) = (u r , u θ ) is the component of velocity in the coordinate directions. The system can also be written in a matrix form as Ar
∂U ∂U + Aθ + Hs = 0, ∂r ∂θ
(2.7)
where ⎛ ⎜ Ar = ⎝
ρu
⎛ Aθ =
ρu
1
1⎜ ⎝ r
ρv
1
⎞
⎟ , a −2 ρ −1 u ⎠ 2 −a u u ⎞
⎞ −ρv 2 1 ⎜ ρuv ⎟ ρv 1 ⎟ , , Hs = ⎝ 1 a −2 ρ −1 v ⎠ u ⎠ r 0 −a 2 v v ⎛
(2.8)
and U = (u, v, p, ρ)t is the unknown function. Here in the process of deriving (2.8) from (2.7), we have applied the equality 1 1 de = d S − pd (2.9) ρ ρ
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in thermodynamics. For any possible shock r = f (θ ), the Rankine-Hugoniot conditions on the shock are ⎤ ⎡ ⎡ ⎤ ρuv ( p + ρu 2 ) ⎢ ρuv ⎥ 1 ∂ f ⎢ ( p + ρv 2 ) ⎥ −⎣ (2.10) ⎦+ ⎣ ⎦ = 0. ρu ρv r ∂θ ρu E ρv E 2.2. Background solution. If the nozzle is a truncated sector, and if the flow in the nozzle is radial, then all the flow parameters only depend on r , so that v = 0, p = Aρ γ and u = u r satisfy ⎧ du dp ⎪ ⎨ρu + = 0, dr dr (2.11) d(ρu) ⎪ ⎩r + ρu = 0. dr Besides, Bernoulli’s relation 1 2 a 2 (r ) u (r ) + = C0 2 γ −1
(2.12)
is also satisfied, where C0 is a given constant determined by upstream flow. In a flow with parameters smoothly depending on the coordinates, the following proposition holds (see [14,32]). Lemma 2.1. Given an symmetrical flow in a nozzle as mentioned above, if the flow is subsonic in the diverging part of a nozzle, then the velocity is decreasing and the pressure is increasing, while if the flow is supersonic there, then the velocity is increasing and the pressure is decreasing. Conversely, in the converging part of the nozzle, if the flow is subsonic, then the velocity is increasing and the pressure is decreasing, while if the flow is supersonic, then the velocity is decreasing and the pressure is increasing. For radially symmetric flows in an diverging truncated sector θ1 ≤ θ ≤ θ2 , r1 < r < r2 , a direct proof is as follows. Indeed, from (2.12) we can obtain −ρa 2 u 2 dp = . dr r (a 2 − u 2 )
(2.13)
du dp dp > 0, < 0. On the other hand, if u > a, then < 0, Hence if u < a, then dr dr dr du > 0. That coincides with the conclusion of the first part of the proposition. dr Now if a transonic shock appears in the flow with its location r = rs , then the Rankine-Hugoniot condition holds there, i.e. ρ+ u + = ρ− u − = m, p+ + ρ+ u 2+ = p− + ρ− u 2− , a+2
(2.14)
2 a−
1 1 2 u + = u 2− + = C0 . 2 + γ −1 2 γ −1 For a radially symmetric flow with a transonic shock the following proposition holds.
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Lemma 2.2. For a radially symmetric flow U = U (r ) with one transonic shock r = rs (r1 < rs ), the receiver pressure p(r2 ) is a monotone decreasing function of rs . Therefore, the location rs is uniquely determined by p(r2 ). 21 a( p)2+ Proof. In fact, the Bernoulli relation implies u + = 2 C0 − , then denote γ −1 21 a( p)2 L( p) = 2 C0 − m + p, and we then have L( p+ ) = L( p− ) from (2.14). γ −1 Looking at p+ as a function of p− we have dp+ = L ( p− ), (2.15) L ( p+ ) dp− dp+ so that > 0. Denote by p (i) (r ) (i = 1, 2) the function p(r ) in the divergent dp− truncated sector with transonic shock at r = rsi , then (1)
(2)
p− (r ) = p− (r )
in r1 < r < min(rs1 , rs2 ). (2)
(2)
(1)
Now if rs2 > rs1 , then by using Lemma 2.1 we have p− (rs2 ) < p− (rs1 ) = p− (rs1 ), (2) (1) which implies p+ (rs2 ) < p+ (rs1 ), i.e. the pressure p+ (rs ) behind the shock front at rs is a decreasing function of rs . On the other hand, since the flow behind the shock dp (2) > 0 because of (2.13). Therefore, the curve p = p (1) (r ) front is subsonic, then dr (1) through the point (rs1 , p+ (rs1 ) is located above the curve p = p (2) (r ). This implies (1) (2) p (r2 ) > p (r2 ). Hence the receiver pressure p(r2 ) is a monotone decreasing function of the coordinate rs . This gives the conclusion of the lemma. Write p(r2 ) by pr , the above conclusion can also be obtained by writing dpr dp+ (r2 ) dp+ (rs ) dp− (rs ) · · = . (2.16) drs dp+ (rs ) dp− (rs ) drs The first factor is the derivative of the solution of p+ (r ) at r2 with respect to its initial value. Its value is positive due to Proposition 2.1. The negativity of the last factor is also shown by (2.13). Moreover, the second factor is positive because of (2.15). Therefore, we have dpr = κ < 0. (2.17) drs Remark. Lemma 2.2 indicates that for any receiver pressure pr in a given interval ( pr 1 , pr 2 ), one can find a unique location r = rs of transonic shock. Then the solution (u(r ), p(r )) of (2.11) in (r1 , rs ) and (rs , r2 ) gives a required flow in the truncated sector. Conversely, if we fix the location rs of transonic shock, then the system (2.11) generally does not have a solution satisfying the condition p(r2 ) = pr . Instead, we can have a solution satisfying the condition p(r2 ) = pr + const, where the const is a constant determined by rs . For given p(r2 ) = pr b and the corresponding transonic shock at r = rsb , we call the above solution (u(r ), p(r )) ( or U (r ) = (u(r ), 0, p(r ), ρ(r )) ) with the transonic shock r = rsb by background solution. The main purpose is to prove the solution is stable under perturbation of upstream flow and the receiver pressure. In the sequel we always assume rsb = 0 for notational simplicity.
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2.3. Perturbed problem. When the upstream flow and the receiver pressure are perturbed, then the whole flow in the nozzle will also be perturbed. Next we are going to prove that in this case the global flow pattern holds, and the perturbed flow as well as the new location of the transonic shock can be uniquely determined. Here the essential difference of the conclusion in this paper from previous results as [3,4,12,13] is that the location of the transonic shock is uniquely determined by the receiver pressure, and the shock front is not asked to pass a given point. Let us first give the mathematical formulation of the perturbed problem. The above background solution (u(r ), 0, p(r ), ρ(r )) with shock r = rsb given in the last subsection (in the sequel we simply assume rsb = 0) is denoted by Ub . Sometimes, we also write Ub as (Ub− , Ub+ ; b ), where b is the shock front r = 0, while Ub− (r ) and Ub+ (r ) are defined in r1 < r < 0 and 0 < r < r2 , respectively. Assume that the perturbed supersonic upstream flow at the entrance r = r1 and the receiver pressure at the exit r = r2
of the nozzle Q are given. On the lateral wall ∂1 Q = {θ1 } × (r1 , r2 ) ∪ {θ2 } × (r1 , r2 ) the flow satisfies the impermeability condition v = u · n = 0. Therefore, to determine the whole perturbed flow in the nozzle, we have to solve the following mathematical problem: ⎧ (2.7) in Q, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨R − H conditions (2.10) on , (P) : U = U− (r1 , θ ) (2.18) on r = r1 , ⎪ ⎪ ⎪ on r = r2 , p = pr (θ ) ⎪ ⎪ ⎩ u · n = 0 on ∂1 Q. Here U− (r1 , θ ) is the given data describing a supersonic flow near Ub− (r1 ), is the transonic shock front in Q, which is to be determined together with U , pr (θ ) is a C 2,α function near pb+ . The main result of this paper is the following theorem. Theorem 2.3. Assume that U− (r1 , θ ) ∈ H 4 (θ ) satisfies consistence conditions on {r = r1 , θ = θ1 or θ = θ2 }, U− (r1 , θ ) − Ub− (r1 ) H 4 < ; pr (θ ) ∈ C 2,α (θ1 , θ2 ), ∂ pr |θ=θ1 ,θ2 = 0, pr (θ ) − pb+ C 2,α < , then the problem (P) has a unique entropy ∂θ weak solution (U− , U+ ; ), where is a surface defined by the equation r = f (θ ), U− is defined in Q −f = {θ1 ≤ θ ≤ θ2 , r1 < r < f (θ )}, U+ is defined in Q f = {(θ1 ≤ θ ≤ θ2 , f (θ ) < r < r2 }. Furthermore, the following estimates hold: U− − Ub− C 2,α (Q − ) ≤ Cε,
(2.19)
U+ − Ub+ C 1,α (Q f ) ≤ Cε,
(2.20)
f C 2,α ( ) ≤ Cε,
(2.21)
f
where α is a constant determined later, the constant C depends only on the background solution Ub and α. ∂ pr |θ=θ1 ,θ2 = 0 is a ∂θ consistence condition. Indeed, the boundary condition u · n = 0 on the lateral boundary ∂p = 0 on ∂1 Q. Therefore, if ∂1 Q is v = 0, then the second equation of (2.6) implies ∂θ
Remark. In the assumptions of the above theorem the condition
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one expects to have a C 1,α solution of the problem (P), the prescribed pressure pr (θ ) ∂ pr = 0 at the corners {r = r2 , θ = θ1 or θ = θ2 }. should also satisfy ∂θ It is well known that for supersonic flow the upstream part is not influenced by its downstream part. Hence the supersonic flow ahead of the transonic shock can be completely determined by the data of the upstream flow at the entrance. Therefore, if U− (r1 ) is a small perturbation of U−b , then the solution U− of the system ahead of the shock front can be determined by using the theory of quasilinear hyperbolic systems (see e.g. [24]). It turns out that we only need to concentrate our attention on seeking the solution U+ in the subsonic region, as well as the location of the shock front r = f (θ ). Hence the problem (P) can be reduced to a free boundary value problem as follows: ⎧ ⎪ (2.7) in Q f : θ1 ≤ θ ≤ θ2 , f (θ ) < r < r2 , ⎪ ⎪ ⎨R − H conditions (2.10) on S : r = f (θ ), f (FB) : (2.22) ⎪ on r = r2 , p = pr (θ ) ⎪ ⎪ ⎩u · n = 0 on ∂1 Q f , where the state in the left on S f is known, while the surface S f is to be determined together with the solution U . Obviously, once the problem (FB) is solved, the problem (P) is also solved. Therefore, Theorem 2.3 can be deduced from the following theorem. Theorem 2.4. Under the assumption of Theorem 2.3 the problem (FB) has a unique solution U+ defined in Q f = {θ1 ≤ θ ≤ θ2 , f (θ ) < r < r2 }, where S f : r = f (θ ) is a part of the boundary of the domain for the unknown functions U+ . Furthermore, the following estimates of the solution hold: U+ − Ub+ C 1,α (Q f ) ≤ Cε,
(2.23)
f C 2,α ( ) ≤ Cε
(2.24)
with C depending only on the background solution Ub and α. 3. A Related Fixed Boundary Value Problem 3.1. Problem (PsFB) and Problem (NL). In the problem (FB) the shock front S f is to be determined as many problems in studying the compressible flow involving shock waves. However, a main difference from the previous study on nozzle flows is that the shock front is not assumed to pass a fixed point. This causes some difficulty even in looking for approximate location of the transonic shock. A new approach is that we give a relaxation of the receiver pressure, while temporarily requiring the shock front to pass an assigned point. That is, we introduce a so-called pseudo-free boundary value problem as follows: ⎧ (2.7) in Q f , ⎪ ⎪ ⎪ ⎪ ⎪ R − H conditions (2.10) on S f , ⎨ (PsFB) : p = pr (θ ) + const (3.1) on r = r2 , ⎪ ⎪ ⎪ Q , u · n = 0 on ∂ 1 f ⎪ ⎪ ⎩ f (0) = f 0 ,
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where f 0 is a given small number, and const is a constant to be determined. When the unique existence of the solution to the problem (PsFB) is established, we then try to eliminate the relaxation by adjusting the shift f 0 , so that we finally reach the solution of (FB). To solve the problem (PsFB) we consider a fixed boundary value problem determining the downstream flow field behind a fixed approximate shock front. Suppose that r = f (θ ) is given, then the fixed boundary value problem is defined in the domain Q f with the left boundary S f : r = f (θ ) satisfying f (0) = f 0 . The boundary condition on S f comes from Rankine-Hugoniot conditions (2.10), but we prefer to use a new form, in which the derivative f (θ ) does not appear. To this end we define G 1 (U+ , U− ) = [ρuv][ρv] − [ p + ρv 2 ][ρu],
(3.2)
G 2 (U+ , U− ) = [ p + ρu ][ p + ρv ] − [ρuv] ,
(3.3)
G 3 (U+ , U− ) = [ρuv][ρu E] − [ p + ρv 2 ][ρu E],
(3.4)
2
2
2
then the boundary condition on S f can be written as G i (U, U− ) = 0, i = 1, 2, 3.
(3.5)
The boundary condition on the lateral boundary ∂1 Q is still v = u · n = 0. Therefore, we introduced a fixed boundary value problem (NL) as ⎧ ⎪ System (2.7) in Q f , ⎪ ⎪ ⎨ G (U, U ) = 0, (i = 1, 2, 3) on S f , i − (NL) : ⎪ v=0 on ∂1 Q f , ⎪ ⎪ ⎩ p = p (θ ) + const on r = r2 , r
(3.6)
where const is a constant to be determined with the solution. For given small positive numbers , ζ, δ, we define O = {U ∈ C 2,α (Q) : U − Ub− C 2,α (Q) ≤ }, (3.7) ∂p O = { p(θ ) ∈ C 2,α (θ1 , θ2 ) : |θ=θ1 ,θ2 = 0, p(θ ) − pb+ C 2,α (θ1 ,θ2 ) ≤ }, (3.8) ∂θ K ζ = { f (θ ) ∈ C 2,α (θ1 , θ2 ) : f (θ1 ) = f (θ2 ) = 0, f C 2,α (θ1 ,θ2 ) ≤ ζ }, (3.9) δ = {U ∈ C 1,α (Q f ) : U − Ub+ C 1,α (Q f ) ≤ δ}.
(3.10)
Then we are going to prove Theorem 3.1. Assume that f (θ ) ∈ K ζ , U− (x, y) ∈ O , pr (θ ) ∈ O with ≤ 0 , ζ ≤ ζ0 and 0 , ζ0 being sufficiently small, then there is a unique solution U+ (x, y) of the problem (NL) in Q f , satisfying U+ (x, y) − Ub+ C 1,α (Q f ) < C , where C only depends on 0 , ζ0 .
(3.11)
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3.2. Decomposition. In the subsonic region the stationary Euler system in the problem (PsFB) is an elliptic-hyperbolic system. To look for the solution to such systems we are going to separate its elliptic part and hyperbolic part. From det (Aθ − λAr ) = 0 we can determine the eigenvalues λ, which is the roots of v v 1 det (Aθ − λAr ) = ρ( − λu)2 (a −2 ( − λu)2 − 2 − λ2 ) = 0. r r r
(3.12)
Its four eigenvalues are √ uv ± a u 2 + v 2 − a 2 λ± = , r (u 2 − a 2 ) v . λ3,4 = ru In the subsonic region u 2 + v 2 < a 2 the eigenvalues λ± are complex, and λ3,4 is real with multiplicity 2. Write λ± = λ R ± iλ I , the real part and imaginary part of λ± are √ uv a a2 − u 2 − v2 , λI = , (3.13) λR = r (u 2 − a 2 ) r (u 2 − a 2 ) respectively. The left eigenvectors corresponding to λ = λ± are 1 v ± = λ± , − , ρ − λ± u , 0 . r r Multiplying the system (2.7) by = ± we obtain ∂ ∂ +λ U + Hs = 0. Ar ∂r ∂θ
(3.14)
∂ ∂ ∂ + λR , DI = λI and decomposing the real part and imaginary Denoting D R = ∂r ∂θ ∂θ part of (3.14) we obtain 1 λ R ρ(u 2 + v 2 ), r 1 ρv D I u − ρu D I v − h D R p = λ I ρ(u 2 + v 2 ), r ρv D R u − ρu D R v + h D I p =
(3.15) (3.16)
1 2 a − u 2 − v 2 . On the other hand, the left eigenvector corresponding to where h = a λ3,4 is 3 = (u, v, 0, 0), 4 = (0, 0, 0, 1). Denoting D = u
1 ∂ ∂ +v , and multiplying (2.7) by 3 or 4 we obtain ∂r r ∂θ ρu Du + ρv Dv + Dp = 0. D( pρ −γ ) = 0.
(3.17) (3.18)
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We notice that (3.17) is nothing but the Bernoulli’s relation D
1 2 (u + v 2 ) + i 2
= 0,
(3.19)
γp for polytropic gas. Equations (γ − 1)ρ (3.15)–(3.18) are the canonical form for the system (2.7). In the new form of the system the hyperbolic part (3.17),(3.18) and the elliptic part (3.15),(3.16) have been decomposed in the level of the principal part, and these two parts are only weakly coupled in their coefficients. To simplify our computation in solving the system (3.15)–(3.18) we would like to use v unknown functions w = instead of v. Therefore, the all unknown functions become u (u, w, p, ρ), so that the system (3.15)–(3.18) are replaced by where i is the entropy, which satisfies i =
⎧ λR ⎪ ⎪ DR w − h1 DI p + (1 + w 2 ) = 0, ⎪ ⎪ r ⎪ ⎪ ⎪ λI ⎪ 2 ⎪ ⎪ ⎨ D I w + h 1 D R p + r (1 + w ) = 0, 1 2 2 ⎪ u D (1 + w ) + i = 0, ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ p ⎪ ⎪ = 0, ⎩D ργ where h 1 =
(3.20)
h . ρu 2
3.3. Lagrange transformation. Now we transform the system (3.20) to a more convenient form. For a fixed f (θ ) ∈ K ζ we introduce a transformation ⎧ ⎨r˜ = r − f (θ ) r , 2 T˜ : r2 − f (θ ) ⎩˜ θ = θ,
(3.21)
which transforms S f : r = f (θ ) to S0 : r˜ = 0, and transforms r = r2 to S1 : r˜ = r2 . Meanwhile, the domain Q f is also transformed to Q + : 0 ≤ r˜ ≤ r2 , θ1 ≤ θ˜ ≤ θ2 . Evidently, ∂ r2 ∂ = · , ∂r ∂ r˜ r2 − f ∂ ∂ r˜ − r2 ∂ + = · f · . ∂θ ∂ r˜ ∂ θ˜ r2 − f
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The differential operators D R , D I , D are transformed to ∂ r˜ − r2 ∂ r2 + f λR + λR , D˜ R = r2 − f r2 − f ∂ r˜ ∂ θ˜ ∂ ∂ r ˜ − r 2 f λI + λI , D˜ I = r2 − f ∂ r˜ ∂ θ˜ ∂ r ˜ − r r 1 v ∂ 2 2 ˜ D= u , +v f + r2 − f r r2 − f ∂ r˜ r ∂ θ˜ respectively. Hence (3.20) is reduced to ⎧ ˜ ˜ ⎪ ⎪ D R w − h 1 D I p + E 1 = 0, ⎪ ⎪ ⎪ ⎪ D˜ I w + h 1 D˜ R p + E 2 = 0, ⎪ ⎪ ⎨ ˜ 1 u 2 (1 + w 2 ) + i = 0, D ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ p ⎪ ⎩ D˜ = 0, ργ
(3.22)
where λ R (1 + w 2 )r2 , r˜ (r2 − f (θ )) + f (θ )r2 λ I (1 + w 2 )r2 . E2 = r˜ (r2 − f (θ )) + f (θ )r2 E1 =
Next we again denote (˜r , θ˜ ) by (r, θ ) to alleviate the notational burden, and use the Lagrange transformation to simplify the system (3.21). The transformation can straighten all stream lines, so that the differential operator D˜ will become a simplest differential ∂ . The application of Lagrange transformation to unsteady compressible flow operator ∂ξ can be found in [16,28]. Assume that U is known in a domain on the (r, θ ) plane. Let θˆ = θˆ (r, h) denote the stream line passing through (0, h), i.e. ⎧ ˆ ⎪ ⎨ d θ = 1 w(r, θˆ (r, h)), dr r (3.23) ⎪ ⎩ˆ θ(0, h) = h. Then ∂ ∂r
ˆ θ(r,h)
rρudθ = [rρu]θ=θ(r,h) ˆ
ˆ θ(r,θ 1) θˆ (r,h)
+
θˆ (r,θ1 )
d θˆ d θˆ (r, θ ) − [rρu]θ= θˆ (0,h) (r, θ1 ) dr dr
∂ (rρu)dθ ∂r
= ρv(r, θˆ (r, h) − ρv(r, θˆ (r, θ1 )) − = 0.
ˆ θ(r,h)
ˆ θ(r,θ 1)
∂ (ρv)dθ ∂θ
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Therefore,
ˆ θ(r,h) ˆ θ(r,θ 1)
rρudθ is independent of r . Denote
denoting its inverse by h(η) we obtain η=
ˆ θ(r,h)
ˆ θ(r,θ 1)
rρudθ by η(h). Then
θˆ (r,h(η))
θˆ (r,θ1 )
rρudθ.
(3.24)
Based on the above discussion, we can introduce the transformation T :
r = ξ, θ = θ¯ (ξ, η),
(3.25)
¯ η) = θ(ξ, ˆ H(η)). where θ(ξ, ∂r ∂r = 1, = 0, ∂ξ ∂η ∂θ w ∂θ 1 = , = , ∂ξ r ∂η rρu and ∂ ∂ w ∂ ∂ 1 ∂ = + , = , ∂ξ ∂r r ∂θ ∂η rρu ∂θ ∂ ∂ ∂ ∂ ∂ = − ρv , = rρu . ∂r ∂ξ ∂η ∂θ ∂η Under such a transformation the system (3.22) becomes ⎧ Dˆ R w − h 1 Dˆ I p + E 1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ Dˆ w + h 1 Dˆ R p + E 2 = 0, ⎪ ⎪ I ⎪ ⎪ ⎨ ∂ 1 u 2 (1 + w 2 ) + i = 0, ∂ξ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p ∂ ⎪ ⎪ ⎪ ⎩ ∂ξ ρ γ = 0,
(3.26)
where ∂ ∂ r2 r − r2 r2 r − r2 + f λR + −ρv + f λ R + rρuλ R r2 − f r2 − f ∂ξ r2 − f r2 − f ∂η ∂ ∂ = d R1 + d R2 , ∂ξ ∂η ∂ r − r2 ∂ r − r2 f + −ρv f + rρuλ I Dˆ I = λ I r2 − f ∂ξ r2 − f ∂η ∂ ∂ + dI 2 , = dI 1 ∂ξ ∂η Dˆ R =
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The boundary conditions for (3.26) keep the form as listed in (3.6). Then the problem (NL) is reduced to the following equivalent problem: ⎧ ⎪ in Qˆ + , ⎪ System (3.26) ⎪ ⎨ G i (U, U− ) = 0, (i = 1, 2, 3) on Sˆ0 , ˆ : (NL) (3.27) + ˆ ⎪ w=0 on ∂1 Q , ⎪ ⎪ ⎩ p = pr (θ ) + const on ξ = r2 , where (3.28) Qˆ + = {0 ≤ ξ ≤ r2 , 0 ≤ η ≤ η2 }, Sˆ0 = {ξ = 0, 0 ≤ η ≤ η2 } θ(r,θ ˆ 2) with η2 = rρudθ. To shorten the notations we will omit “hat” for Qˆ + and Sˆ 0 ˆ θ(r,θ 1)
in the following discussion. 4. Linearization Next we linearize the problem (3.27) at any point in a C 1,α neighborhood of Ub+ . First the linearization of the system (3.26) is a system for the perturbation δU as follows: ⎧ Dˆ R δw − h 1 Dˆ I δp + M1 δU = F1 , ⎪ ⎪ ⎪ ⎪ ⎪ Dˆ I δw + h 1 Dˆ R δp + M2 δU = F2 , ⎨ 1 1 (4.1) 2 2 2 2 u(1 + w )δu + u wδw + δp = u(1 + w )δu + u wδw + δp , ⎪ ⎪ ρ ρ ⎪ (0,η) ⎪ (ξ,η) ⎪ ⎩δp − a 2 δρ 2 δρ = δp − a , (ξ,η) (0,η) where F1 , F2 are terms independent of δU and will be given in the iterative process. M1 δU, M2 δU are linear forms of δU , which are the Frechet derivatives of the left-hand side of the first two equations of (3.26) with respect to U . More precisely, ∂w ∂w δd R1 + δd R2 − Dˆ I pδh 1 ∂ξ ∂η ∂ E1 ∂p ∂p δd I 1 + δd I 2 + δU, − h1 ∂ξ ∂η ∂U ∂w ∂w M2 δU = δd I 1 + δd I 2 + Dˆ R pδh 1 ∂ξ ∂η ∂p ∂ E2 ∂p δd R1 + δd R2 + δU. +h 1 ∂ξ ∂η ∂U M1 δU =
(4.2)
Furthermore, the linearization of the boundary conditions (3.5) on S0 are αi δu + βi δw + γi δp + χi δρ = g˜i , i = 1, 2, 3, where αi , βi , γi , χi are derivatives of G i with respect to u, w, p, ρ. Let us compute the matrix ⎛ ⎞ α1 β1 γ1 χ1 ⎝α2 β2 γ2 χ2 ⎠ α3 β3 γ3 χ3
(4.3)
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at U− = Ub− , U+ = Ub+ . Noticing v+ = v− = 0, we have u [ p], χ1 = −u[ p], a2 u2 α2 = 2[ p]ρu, β2 = 0, γ2 = [ p](1 + 2 ), χ2 = u 2 [ p], a uE ∂E ∂E ), χ3 = −[ p](ρ E + ρu ). α3 = −ρ E[ p], β3 = 0, γ3 = −[ p]( 2 + ρu a ∂p ∂ρ α1 = −ρ[ p], β1 = 0, γ1 = −
Direct computation shows ⎛
α1 det ⎝α2 α3
γ1 γ2 γ3
⎛
⎞
ρ
χ1 ⎜ ρu χ2 ⎠ = [ p]3 det ⎜ ⎝ χ3 0
u a2 1 ∂E ρu ∂p
u
⎞
⎟ 0 ⎟. ∂E ⎠ ρu ∂ρ
From the expression E=
γp 1 2 + | u| (γ − 1)ρ 2
we have ∂E γ = > 0, ∂p (γ − 1)ρ Then
⎛
α1 det ⎝α2 α3
γ1 γ2 γ3
∂E ∂E = a2 . ∂ρ ∂p
⎞ χ1 ∂E χ2 ⎠ = [ p]3 ρ 2 ua 2 >0 ∂p χ3
(4.4)
at U− = U−b , U+ = Ub+ . Therefore, for any U ∈ δ the boundary conditions (4.4) on S˜0 for the linearized problem can be written as δp + β˜1 δw = g1 , δu + β˜2 δw = g2 , δρ + β˜3 δw = g3 ,
(4.5) (4.6) (4.7)
where β˜i = 0 for i = 1, 2, 3, if U− = Ub− , U = Ub+ . Moreover, β˜i C 2,α (S0 ) < Cδ due to U ∈ δ . The boundary conditions on other boundaries are linear, then we are led to the following linear problem: ⎧ ⎪ System (4.1) in Q + , ⎪ ⎪ ⎨ (4.5)–(4.7) on S0 , (4.8) (L) : ⎪ δw = 0 on ∂1 Q + , ⎪ ⎪ ⎩ δp = const. on ξ = r2 . Evidently, solving the boundary value problem (L) for the linear elliptic-hyperbolic coupled system is a necessary step to solve the problem (NL). Later we establish existence of the linearized problem (L) and corresponding a priori estimates. That is:
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Theorem 4.1. Assume that U− ∈ O , pr ∈ O , f ∈ K ζ , U ∈ δ with ≤ 0 , ζ ≤ ζ0 , δ ≤ δ0 sufficiently small, then the problem (L) has a unique solution δU ∈ C 1,α (Q + ). Meanwhile, the constant δp on ξ = r2 is also uniquely determined. Moreover, the following estimate: δU C 1,α (Q + ) ≤ C( F C 0,α (Q + ) + g C 1,α (S0 ) ) holds, where C is a constant independent of 0 , ζ0 , δ0 , F and g are 3 i=1 gi , respectively.
(4.9) 2
i=1 Fi and
Remark. The value of δp on the boundary ξ = r2 is determined together with the solution δU inside the domain. It is also dominated by the right side of (4.9). 5. Existence of the Linearized problem In order to solve (L), we derive a second order elliptic equation for δp from the first two equations of the linearized system (4.1) and construct a boundary value problem of the elliptic equation. The problem will be applied to looking for δp under the assumption that δw, δu and δρ are known. Then a suitable iterative process can be established, which will lead us to the solution of the problem (L). For the commutator [ Dˆ R , Dˆ I ] we have ∂d I 1 ∂d I 1 ∂d R1 ∂d R1 ∂ + d R2 − dI 1 − dI 2 [ Dˆ R , Dˆ I ] = d R1 ∂ξ ∂η ∂ξ ∂η ∂ξ ∂d I 2 ∂d I 2 ∂d R2 ∂d R2 ∂ + d R2 − dI 1 − dI 2 + d R1 ∂ξ ∂η ∂ξ ∂η ∂η
= µ R Dˆ R + µ I Dˆ I .
(5.1)
Dˆ R acting on the second equation in (4.1) and Dˆ I acting on the first equation in (4.1), then subtracting one from another, we obtain Dˆ R (h 1 Dˆ R δp) + Dˆ I (h 1 Dˆ I δp) + Dˆ R (M2 δU ) − Dˆ I (M1 δU ) + µ R Dˆ R δw + µ I Dˆ I δw = D R F2 − D I F1 . Again applying (4.1) to replace Dˆ R δw and Dˆ I δw in the above equality we have Dˆ R (h 1 Dˆ R δp) + Dˆ I (h 1 Dˆ I δp) + Dˆ R (M2 δU ) − Dˆ I (M1 δU ) +µ R h 1 Dˆ I δp − µ R M1 δU − µ I h 1 Dˆ R δp − µ I M2 δU = F∗ ,
(5.2)
where F∗ = Dˆ R F2 − Dˆ I F1 − µ R F1 − µ I F2 . Lemma 5.1. Equation (5.2) can be written as Lδp + w δw + u δu + ρ δρ = F∗ ,
(5.3)
where L is a second order elliptic operator, w , u , ρ are first order operators, and the coefficient of w is a small quantity of type O( + ζ + δ).
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Proof. The ellipticity of the operator L can be confirmed by directly checking the positivity of the symbol of the operator Dˆ 2R + Dˆ 2I . Hence we only need to estimate the coefficient of w . In fact, if we can verify w = 0 for the background solution, then w = O( + ζ + δ) is obtained. For the notational simplicity we will denote all small quantities O( + ζ + δ) by o(1). ∂U = 0. Hence λ R = 0. For the background solution we have f = 0, w = 0 and ∂η ∂h 1 Meanwhile, since λ I and h 1 depend on w with the form of functions of w 2 , then = ∂w ∂λ I = 0 for the background solution. Therefore, ∂w ∂ E2 = 0, ∂w d R1 = 1, d R2 = 0, d I 1 = 0, d I 2 = rρuλ I , ∂ ∂ , Dˆ I = rρuλ I , Dˆ R = ∂ξ ∂η ∂ ) ∂(rρuλ I , [ Dˆ R , Dˆ I ] = ∂ξ ∂η 1 ∂(rρuλ I ) . µ R = 0, µ I = rρuλ I ∂ξ E 1 = 0,
Hence the coefficients of δw in M1 δU and M2 δU vanish for the background solution. Then we obtain the conclusion of the lemma. To estimate the solution δp of (5.3) we use C −1,α norm. If a function b(r, θ ) can ∂b1 ∂b2 + with b1 , b2 ∈ C 0,α with 0 < α < 1, then we say be written as b(r, θ ) = ∂r ∂θ b ∈ C −1,α . Moreover, then the C −1,α norm is defined by b C −1,α =
min
b=(b1 )r +(b2 )θ
( b1 C 0,α + b2 C 0,α ).
By using the notation C −1,α norm and the conclusion of the above lemma, we have F∗ − w δw − u δu − ρ δρ C −1,α ≤ C( F∗ C −1,α + o(1) δw C 0,α + δu C 0,α + δρ C 0,α ).
(5.4)
Moreover, if we define δφ = u(1 + w 2 )δu + u 2 wδw +
1 δp, δψ = δp − a 2 δρ, ρ
then δu =
uw 1 1 δφ − δp, δw − u(1 + w 2 ) 1 + w2 ρu(1 + w 2 ) 1 1 δρ = 2 δp − 2 δψ. a a
(5.5)
By using this transformation of δU Eq. (5.3) can be written as L δp + w δw + φ δφ + ψ δψ = F∗ ,
(5.6)
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where L is a second order elliptic operator, w , φ , ψ are first order operators, and the coefficient of w is o(1). Moreover, F∗ − w δw − φ δφ − ψ δψ C −1,α ≤ C( F∗ C −1,α + o(1) δw C 0,α + δφ C 0,α + δψ C 0,α ).
(5.7)
Now let us derive the boundary conditions satisfied by δp. On the shock front and the exit of the nozzle the boundary conditions are δp = g1 − β˜1 δw on ξ = 0, δp = const on ξ = r2 ,
(5.8) (5.9)
where const is to be determined. On the lateral boundary we have w = 0, then the first equation implies that p should ∂p ∂p = 0, which leads to = 0 there. Hence the boundary condition for δp on satisfy ∂θ ∂η the lateral boundary is ∂ δp = 0 on ∂ Q + . ∂η
(5.10)
We emphasize here that one additional condition should be added to let δp be well determined. In fact, the second equation of the system (4.1) on ξ = r2 is rρuλ I
∂ δw = F2 − h 1 Dˆ R δp − M2 δU. ∂η
Dividing the equality by rρuλ I and then integrating along ξ = r2 , we have 1 (F2 − h 1 Dˆ R δp − M2 δU )dη = 0. S1 λ I
(5.11)
This is the condition we have to add. Therefore, we set a boundary value problem for δp as follows: ⎧ + ⎪ ⎪ L δp = F∗ − w δw − φ δφ − ψ δψ in Q , ⎪ ⎪ ⎪ ⎪δp = g1 − β˜1 δw on ξ = 0, ⎪ ⎪ ⎨ ∂δp = 0 on η = 0, η2 , (L1 ) : ∂η (5.12) ⎪ ⎪ ⎪ δp = const on ξ = r , 2 ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ (F2 − h 1 Dˆ R δp − M2 δU )dη = 0. λ S1 I Lemma 5.2. Assume that U− ∈ O , f ∈ K ζ , U ∈ δ with ≤ 0 , ζ ≤ ζ0 , δ ≤ δ0 sufficiently small. Then for given F∗ ∈ C 0,α (Q + ) , δw, δφ, δψ ∈ C 1,α (Q + ), the problem (5.12) admits a unique solution δp ∈ C 1,α (Q + ). Moreover, the following estimate: δp C 1,α (Q + ) ≤ C( g1 C 1,α (S0 ) + o(1) δw C 0,α (Q + ) + δφ, δψ C 0,α (Q + ) + F∗ C −1,α (Q + ) ) holds, where C is independent of η, δ, .
(5.13)
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Proof. In the problem (5.12) the condition for δp on ξ = r2 is called an equi-valued boundary condition. The variable δρ takes an constant to be determined on ξ = r2 means that the variable δp is given with one dimensional freedom. Then the integral condition just gives a supplement to fix the freedom. In order to solve this problem, we introduce two related problems: ⎧ L δpa = F∗ − w δw − φ δφ − ψ δψ in Q + , ⎪ ⎪ ⎪ ⎪ ⎨δpa = g1 − β˜1 δw on ξ = 0, (La ) : ∂δpa (5.14) ⎪ = 0 on η = 0, η2 , ⎪ ⎪ ∂η ⎪ ⎩ δpa = 0 on ξ = r2 , ⎧ L δpb = 0 in Q + , ⎪ ⎪ ⎪ ⎪ ⎨δpb = 0 on ξ = 0, (Lb ) : ∂δpb (5.15) ⎪ = 0 on η = 0, η2 , ⎪ ⎪ ∂η ⎪ ⎩ δpb = 1 on ξ = r2 , where δw, δφ, δψ are given as stated in the assumptions of this lemma. Next we first derive the existence of the C 1,α solution to the problems (La ), (Lb ) and the corresponding estimates. Indeed, these two boundary value problems for the second order elliptic equation are given in a domain with corners. Then in order to use the localization method to derive a priori estimate as was done in [18], the neighborhood of corners has to be treated carefully. Assume that the domain Q + is covered by a set of subdomains derived from localization method. Consider the neighborhood ω of the corner O1 with coordinate (ξ, η) = (0, 0). In view of v = 0 and f = 0 on η = 0 we have Dˆ R =
∂ r2 , r2 − f ∂ξ
ρau ∂ ∂ Dˆ I = rρuλ I = −√ , 2 2 ∂η ∂η a −u
so that the second order term of L is r22 ∂2 (ρau)2 ∂ 2 h1 + . (r2 − f )2 ∂ξ 2 a 2 − u 2 ∂η2 Therefore, we can reflect the neighborhood ω of O1 on η > 0 with respect to η = 0 to obtain a domain ω, ˜ and make an even extension of the function δp, the right hand side ∂ . Meanwhile, we make an odd and the all coefficients of L except the coefficient of ∂η ∂ ∂ , so that the product of this coefficient with δp is extension of the coefficient of ∂η ∂η still an even function. Hence the equation and the boundary conditions still hold after extension. It turns out that for the extended operator L˜ defined in ω, ˜ the coefficients of second order derivatives are in C α , and its coefficients of lower order terms are in L ∞ at least. Therefore, we can use the theorem on the estimates for the weak solution of the second order elliptic equation ([18], Sect. 8.11) to obtain a C 1,α estimate of δp in ω˜ by the right-hand side of (5.13). We remark here that the result in Sect. 8.11 of [18] is for the equation in the divergence form, while the operator L is in the “divergence form” with respect to Dˆ R and Dˆ I . Fortunately, the coefficients of the second derivatives in L
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are regular enough, so that L can be rewritten in a divergence form, for which the result in [18] is available. The neighborhood of (0, η2 ), (r2 , 0), (r2 , η2 ) can be treated similarly. Since the estimates in all other subdomains derived from the localization have been established in [18], then the standard summation procedure gives us the existence of C 1,α solutions δpa , δpb for (La ), (Lb ) and the corresponding estimates as shown in (5.13). Now consider the condition (5.11). Write M2 δU as follows: M2 δU = M2 p δp + M2w δw + M2u δu + M2ρ δρ = M2 p δp + M2w δw + M2φ δφ + M2ψ δψ. Denote
Ca =
S1
Cb = S1
(5.16)
1 (F2 − h 1 Dˆ R (δp) − M2 δU )δp=δpa dη, λI 1 (−h 1 Dˆ R (δp) − M2 p δp)δp=δpb dη, λI
Ca δpb is the solution of (L1 ). Cb Now we verify Cb = 0. Indeed, we only need to verify the fact for the background solution, because all solutions considered in this paper are small perturbation of the background solution. By using the expression of M2 in (4.2) and the computations in Lemma 5.1 we have we see that if Cb = 0, then δpa −
∂ h 1 Dˆ R δp = h 1 δp, ∂ξ ∂ p ∂h 1 1 ∂λ I ∂ p ∂h 1 ∂ E 2 + = + , M2 p = ∂ξ ∂ p ∂p ∂ξ ∂ p r ∂ p ∂ p ∂h 1 1 ∂λ I M2u = + , ∂ξ ∂u r ∂u ∂ p ∂h 1 1 ∂λ I + M2ρ = ∂ξ ∂ρ r ∂ρ for the background solution. Applying the transformation (5.5) we have M2 p = M2 p −
1 1 M2u − 2 M2ρ ρu a ∂δp > 0 on the boundary ∂ξ > 0 on ξ = r2 is obtained.
as w = 0. According to the Hopf maximum principle we know ξ = r2 . Therefore, we are led to Cb = 0, once M2 p When w = 0,
√ − 21 1 ρu 2 a a2 − u2 =− 1− λI = , r (u 2 − a 2 ) r γp 1 u2 2 h 1 1− 2 h1 = = . ρu 2 ρu 2 a
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Then − 21 − 23 ∂λ I 1 ∂ ρu 2 ρu 2 ρu 2 1 =− 1− 1− = > 0, ∂p r ∂p γp 2r γp γ p2 − 21 h ∂h 1 ρu 2 1 ρu 2 ∂ > 0, = 1− = 2 2 ∂p ∂ p ρu 2ρu γp γ p2 1 ρu 2 2ρu ∂λ I = 1− − < 0, ∂u 2r γp γp 1 − 21 ∂h 1 u2 2 u2 2u 2 1 1− 2 − 2 < 0, =− 3 1− 2 + ∂u ρu a 2ρu 2 a a ∂λ I p ∂λ I =− , ∂ρ ρ ∂p 1 u2 2 ∂h 1 1 p ∂h 1 =− 2 2 1− 2 . − ∂ρ ρ u a ρ ∂p ∂p > 0 in the subsonic region. Hence ∂ξ 1 ∂h 1 1 ∂h 1 1 ∂λ I 1 ∂λ I 1 ∂λ I ∂ p ∂h 1 − − 2 + − − 2 = ∂ξ ∂ p ρu ∂u a ∂ρ r ∂p ρu ∂u a ∂ρ ⎞ ⎛ 1 23 2 2 2 2 ∂ p ⎝ γ − 1 ∂h 1 u u u 1 1 = 2− 2 1− 2 + + 2 2 2 1− 2 ⎠ ∂ξ γ ∂ p ρ2u4 a a ρ u a a 1 ∂λ 1 γ − 1 ∂λ − + r γ ∂ p ρu ∂u > 0. (5.17)
Moreover, Lemma 2.1 indicates M2 p
The fact implies Cb > 0, so that the solvability of the problem (L1 ) is obtained. According to Lemma 5.1 the coefficient of w is o(1) small quantity. Then we have (5.13). Remark 1. In the above lemma we could not expect to have the C 2,α solution to the problem (L1 ), because Q + is a domain with corners (see [19]). In (5.13) the number α is determined by U− , but is independent of , ζ, δ. Remark 2. According to the theory of elliptic partial differential equations of second order [18] to establish the C 1,α estimates only the boundedness of C 0,α norm of the coefficients of the second order terms and the boundedness of L ∞ norm of the coefficients of other terms are required. Therefore, we can choose α , which depends on θ0 , U− and is independent of , ζ, δ, so that the estimate (5.13) also holds with α replaced by α ∈ (0, α). Lemma 5.3. Under the assumptions of Lemma 5.2, if F1 vanishes at η = 0 and η2 , δp is a solution to the problem (L1 ), then the system Dˆ R δw − h 1 Dˆ I (δp) + M1 δU = F1 , (5.18) Dˆ I δw + h 1 Dˆ R (δp) + M2 δU = F2
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has a unique solution δw satisfying δw = 0 at η = 0, η2 . Moreover, δw C 1,α (Q + ) ≤ C( g C 1,α (S0 ) + δp, δφ, δψ C 1,α (Q + ) + F C 0,α (Q + ) ).
(5.19)
Proof. Equation (5.18) is a system of first order differential equations. The consistency condition for its solvability is nothing but Eq. (5.2). By assigning the initial value δw = 0 at (ξ, η) = (r2 , 0) one can integrate (5.17) to obtain its solution. ∂ ∂ δp = 0, Dˆ R = and F1 = 0 Look the first equation of (5.18) on η = 0. Since ∂η ∂ξ on η = 0, we then have r2 ∂ δw + M1 δU = 0. r2 − f ∂ξ ∂ E1 δU . Moreover, Using (4.2) to M1 δU on η = 0 we find all terms vanish except ∂U ∂ E1 ∂ E1 ∂ E1 ∂ E1 due to w = 0 on η = 0, among the derivatives , , , we only have ∂ p ∂w ∂u ∂ρ ∂ E1 = 0. Hence the first equation (5.18) on η = 0 becomes ∂w ∂ δw + mδw = 0 ∂ξ on η = 0. Then δw = 0 at (ξ, η) = (r2 , 0) implies δw = 0 on the whole line η = 0. Furthermore, in view of the last condition in (5.12) and δw = 0 at (r2 , 0) we can obtain δw = 0 at (r2 , η2 ) by integration. Then we also have δw = 0 on the whole wall η = η2 as explained above. The estimate (5.19) can be directly derived from (5.14) and (5.18). Lemma 5.4. Under the assumptions of Lemma 5.1, and assuming that F1 , F2 ∈ C 1,α (Q + ), g1 , g2 , g3 ∈ C 1,α (S0 ), then the problem (L) admits a unique solution δU ∈ C 1,α (Q + ). Proof. We are going to construct an iterative process to seek the solution to (L). Take δU (0) = 0, construct a sequence {δU (n) } by induction, and then prove the convergence of the sequence {δU (n) }. Suppose δU (k) has been obtained; we look for δU (k+1) by the following steps. First, we solve δp (k+1) by using (5.12), where δw, δφ, δψ are taken as the k-times approximation. Then substituting δp (k+1) into (5.18) we solve δw (k+1) . The value of δφ (k+1) and δψ (k+1) on S0 can be determined by using (4.5)–(4.7) with δw = δw (k+1) on S0 . The last two equations in (4.1) give us
δφ (k+1) (ξ, η) = δφ (k+1) (0, η), δψ (k+1) (ξ, η) = δψ (k+1) (0, η).
(5.20)
Then δu (k+1) and δρ (k+1) can be solved from (5.20). Hence the sequence {δU (k )} is established.
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Now we prove the convergence of the iterative process. Denote δU = δU (k+1) − (k+1) (k) satisfies δU , then δp ⎧ (k+1) (k) (k) (k) ⎪ L δp + w δw + φ δφ + ψ δψ = 0 in Q + , ⎪ ⎪ ⎪ (k+1) (k) ⎪ ⎪ δp = −β˜1 δw on ξ = 0, ⎪ ⎪ ⎪ ⎨ ∂ (k+1) δp = 0 on η = 0, η2 , (5.21) ∂η ⎪ ⎪ (k+1) ⎪ ⎪ = const on ξ = r2 , ⎪δp ⎪ ⎪ 1 ⎪ (k+1) (k+1) ⎪ (k) )dη = 0, (−h 1 Dˆ R δp − M2 p δp − M2U ⎩ δU λ I S1 where δU means (δw, δφ, δψ). (k+1) δw satisfies (k) (k+1) (k+1) (k+1) − h 1 Dˆ I (δp ) + M1 p δp + M1U δU = 0, Dˆ R δw (k) (k+1) (k+1) (k+1) Dˆ I δw + h 1 Dˆ R (δp ) + M2 p δp + M2U δU = 0,
(5.22)
(k+1)
with δw = 0 at (r2 , 0). (k+1) (k+1) δu and δρ on S0 satisfy
due to (4.6),(4.7). Then δφ δφ
(k+1)
(k+1)
δu
(k+1)
δρ
(k+1)
, δψ
C 1,α (S0 ) , δψ
+ β˜3 δw
(k+1)
(k+1)
(k+1)
= 0,
(5.23)
(k+1)
= 0,
(5.24)
+ β˜2 δw
on S0 are also determined; they satisfy
C 1,α (S0 ) ≤ o(1) δw
(k+1)
C 1,α (S0 ) .
(5.25)
Moreover, we have δφ
(k+1)
δψ
(ξ, η) = δφ
(k+1)
(k+1)
(ξ, η) = δψ
(0, η),
(k+1)
(5.26)
(0, η).
Therefore, Lemma 5.2 indicates δp
(k+1)
C 1,α (Q + ) ≤ C(o(1) δw
(k)
C 1,α (Q + ) + δφ
(k)
, δψ
(k)
C 1,α (Q + ) ).
(5.27)
Combining with (5.19) we have δw
(k+1)
C 1,α (Q + ) ≤ C(o(1) δw
(k)
C 1,α (Q + ) + δφ
(k)
, δψ
(k)
C 1,α (Q + ) ).
(5.28)
By substituting (5.25) into (5.26) we obtain δw
(k+1)
C 1,α ( + ) ≤ C · o(1) δw
(k)
C 1,α ( + ) .
(5.29)
(k) δw is a convergent series. This implies the Therefore, when 0 , ζ0 , δ0 are small, convergence of {δw(k) }. Correspondingly, other components of {δU (k) } are also convergent. Obviously, the limit of {δU (k) } solves problem (L).
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The proof of Theorem 4.1. The existence of the solution to problem (L) has been obtained in Lemma 5.4. Next we prove estimate (4.9). Substituting estimate (5.13) into (5.19) we obtain δw C 1,α (Q + ) ≤ C( g C 1,α (S0 ) +o(1) δw C 1,α (Q + ) + δφ, δψ C 1,α (Q + ) + F C 0,α (Q + ) ). (5.30) The conditions (4.5)–(4.7) indicate δp, δu, δρ C 1,α (S0 ) ≤ C( g C 1,α (S0 ) + o(1) δw C 1,α (S0 ) ).
(5.31)
δφ, δψ C 1,α (S0 ) ≤ C( g C 1,α (S0 ) + o(1) δw C 1,α (S0 ) ).
(5.32)
Then
From the last two equations in (4.1) we have δφ, δψ C 1,α (Q) ≤ C( g C 1,α (S0 ) + o(1) δw C 1,α (S0 ) ).
(5.33)
Substituting it into (5.30) we obtain δw C 1,α (Q + ) ≤ C( g C 1,α (S0 ) + o(1) δw C 1,α (Q + ) + F C 0,α (Q + ) ).
(5.34)
1 Therefore, for sufficiently small 0 , ζ0 , δ0 we have C · o(1) < . Then (4.9) is easily 2 obtained from (5.34). 6. Existence of the Nonlinear Problem (NL) In this section we use Newton’s iterative scheme to solve the nonlinear problem (NL). For the reader’s convenience we present the details here. We still assume the shock front r = f (θ ) is temporarily fixed. The sequence of the approximate solution δU
is defined by induction. Let U <0> = Ub+ , δU <0> = 0. Assume that U = Ub+ + δU has been given, then δU is defined as follows. In the domain Q + , δU satisfies ⎧ Dˆ δw − h δp + M1 δU Dˆ ⎪ 1 I ⎪ ⎪ R ⎪ ⎪ = − Dˆ w + h p − E 1 + Dˆ δw Dˆ ⎪ 1 R I R ⎪ ⎪ ⎪ ˆ ⎪ −h 1 D I δp + M1 δU , ⎪ ⎪ ⎪ ⎪ ˆ δw + h Dˆ δp + M δU ⎪ D ⎪ 1 2 I R ⎪ ⎪ ⎪ w − h D ˆ p − E + Dˆ δw ⎪ ⎪ = − Dˆ I 1 2 R I ⎪ ⎨ D ˆ δp + M δU , + h 1 2 R (6.1) ⎪ ⎪ ⎪ u (1 + (w )2 )δu + (u )2 w δw + δp ⎪ ⎪ ⎪ ρ ⎪ ⎪ (ξ,η) ⎪ ⎪ ⎪ ⎪ δp ⎪ 2 2 ⎪ ⎪ − u (1 + (w ) )δu + (u ) w δw + = 0, ⎪ ⎪ ρ ⎪ ⎪ (0,η) ⎪ ⎪ ⎪ ⎩ δp − (a )2 δρ − δp − (a )2 δρ = 0. (ξ,η)
(0,η)
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Denote A = (A , A ) 1 2 = (− Dˆ w + h Dˆ p − E 1 , 1 R
I
w + h p − E 2 ), Dˆ − Dˆ 1 I R then (6.1) can be simply written as F δU = F ,
(6.2)
where F stands for the operator in the left-hand side of the system (6.1), F is a (n) column vector with components Fi (1 ≤ i ≤ 4), Fi = −Ai + Fi δU (i = 1, 2) and F3 = F4 = 0. The boundary condition on S0 is αi δu + βi δw + γi δu + δi δρ = −G i (U , U− ) + αi δu + βi δw + γi δu + δi δρ (6.3) for i = 1, 2, 3. Denoting by Gi the linear operator in the left-hand side, the condition (6.3) can be written as Gi δU = gi , i = 1, 2, 3,
(6.4)
where gi = −G i + Gi δU . Besides, on the lateral boundary the boundary condition is δw = 0, and on the boundary ξ = r2 the boundary condition is δp = const, which is to be determined. Therefore, when we know U , the variation δU can be determined by the following boundary value problem: ⎧ ⎪ F δU = F in Q + , ⎪ ⎪ ⎨G δU (n+1) = g on S , i 0 i (L) : (6.5) (n+1) = 0 on η = 0, η , ⎪ δw 2 ⎪ ⎪ ⎩δp = const to be determined on ξ = r . 2 Once δU is solved, then U is defined by U = U + δU . Lemma 6.1. Assume that U− ∈ O , pr (θ ) ∈ O , f (θ ) ∈ K ζ , ≤ 0 , ζ ≤ ζ0 , 0 , ζ0 are sufficiently small. Then there is a sequence {δU } of solutions to problem (L) , which is convergent in the space C 1,α (Q + ). Proof. Since the coefficients of (6.2) satisfy the condition of Theorem 4.1, then the solvability of (L) is given by this theorem. Meanwhile, (4.9) indicates δU C 1,α (Q + ) ≤ C( F C 0,α (Q + ) + g C 1,α (S0 ) ).
(6.6)
According to the construction of the Newton iteration scheme 2 <0> Fi C 0,α (Q + ) ≤ C( δU C C 1,α (Q + ) ), 1,α (Q + ) + Ai (0)
2 gi C 1,α (S0 ) ≤ C( δU C 1,α (S ) + G i C 1,α (S0 ) ), 0
(6.7) (6.8)
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where Ai<0> is the value obtained by substituting U+b into Ai and vanishes in fact. Therefore, we have 2 <0> δU C 1,α (Q + ) ≤ C( δU C C 1,α (S0 ) ). 1,α (Q + ) + G
(6.9)
Since G <0> C 1,α (S0 ) is a small quantity o(1), then δU C 1,α (Q + ) = o(1) for all n uniformly, provided 0 , η0 , δ0 are sufficiently small. Correspondingly, we have the boundedness of the sequence {U } in C 1,α ( ). In order to prove the convergence of the sequence {U }, we derive the problem for U¯ = U − U . That is ⎧ F = −A + A + F (U − U ) in Q + , U¯ ⎪ ⎪ ⎪ ⎨G U¯ (n+1) = −g + g + G (U −U ) (i = 1, 2, 3) on S , 0 i i i i (6.10) (n+1) = 0 on η = 0, η , ⎪ w ¯ ⎪ 2 ⎪ ⎩ = const on ξ = r2 . p¯
Applying the estimates (4.9) we have U¯ C 1,α (Q + ) ≤ C( − A + A + F (U − U ) C 1,α (Q + ) +
3
− gi + gi + Gi (U − U ) C 1,α (S0 ) )
i=1 2 ≤ C U¯ C 1,α (Q + ) .
(6.11)
Then for sufficiently small 0 , η0 , δ0 we can establish U¯ C 1,α (Q + ) ≤
1 ¯ U C 1,α (Q + ) . 2
Hence the sequences {δU } and {U } are convergent.
(6.12)
Theorem 6.2. Under the assumptions of Lemma 6.1 the problem (NL) admits a unique solution U+ ∈ Q + , satisfying U+ − Ub+ C 1,α (Q + ) ≤ C .
(6.13)
Proof. By using Lemma 6.1 we find a convergent sequence of solutions δU to ˆ Then by the inverse transform (L) . The limit is the solution to the problem (NL). −1 T of (3.25) we obtain the solution U+ to (NL). The estimates (6.12) and (6.9) imply U+ − Ub+ C 1,α (Q + ) ≤ C G <0> C 1,α (S0 ) .
(6.14)
The right-hand side is dominated by C U− − Ub− C 2,α (Q) ≤ C . Then we obtain (6.13).
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7. Solution to the Pseudo-Free Boundary Problem PsFB Now let us turn to solve the pseudo-free boundary problem (3.1) for sufficiently small 0 . To this end we decompose (PsFB) into two problems. One is a fixed boundary value problem to determine U in the domain with a temporarily fixed boundary r = f (θ ), and the other one is an initial value problem for an ordinary differential equation, which is applied to update the location of the approximate shock front. These two steps form a map in a suitable Banach space, and the existence of the solution to (PsFB) amounts to prove the existence of the fixed point of this map. ζ0 Let f 0 be a number satisfying | f 0 | < . Let B be a bounded set in C 2,α (θ1 , θ2 ): 2 B = { f ; f − f 0 C 2,α (θ1 ,θ2 ) <
ζ0 }. 2
(7.1)
For any f (θ ) ∈ B, the existence of the solution U to the problem (NL) is given by Theorem 3.1. Then the new location of the shock front is determined by ⎧ ⎪ ρuv − ρ− u − v− ⎨ d f˜ = r · , 2 (Sh) : dθ (7.2) p + ρv 2 − p− − ρ− v− ⎪ ⎩ f˜(0) = f . 0 Lemma 7.1. For any given f (θ ) ∈ B, U ∈ C 1,α (Q + ), then there is a unique f˜, which satisfies (7.2) and f˜ − f 0 C 2,α (θ1 ,θ2 ) ≤ C ,
(7.3)
where C is independent of and ζ . Proof. Since the solution U of the problem (NL) is only defined in Q + , we first extend it to the whole domain Q, so that U is well defined in the right-hand side of (7.2) and satisfies (e.g. see [1]) U (x, y) C 1,α (Q) ≤ 2 U (x, y) C 1,α (Q + ) .
(7.4)
By the standard theory on ordinary differential equations we know that problem (Sh) admits a unique solution f˜(θ ) ∈ C 2,α (θ1 , θ2 ), which satisfies f (θ1 ) = f (θ2 ) = 0 in view of v = v− = 0 on θ = θ1 , θ2 . Denote the right-hand side of (7.2) by H (U, U− ), it can be written as H (U, U− ) = H (U, U− ) − H (U, Ub− ) + H (U, Ub− ) − H (Ub+ , Ub− ),
(7.5)
due to H (Ub+ , Ub− ) = 0. Therefore, d f˜ 1,α f˜(θ ) − f 0 C 2,α (θ1 ,θ2 ) ≤ C dθ C (θ1 ,θ2 ) ≤ C( U − Ub+ C 1,α (Q) + U− − Ub− C 1,α (Q) ). Then Theorem 6.2 implies the validity of (7.3).
(7.6)
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Theorem 7.2. Assume that ≤ 0 , ζ ≤ ζ0 and 0 , ζ0 are sufficiently small, ζ0 U ∈ O , pr (θ ) ∈ O ( ), | f 0 | ≤ , then the problem (PsFB) admits a unique solution 2 (U (r, θ ), f (θ )), satisfying U − Ub+ C 1,α (Q + ) ≤ C1 ,
f (θ ) − f 0 C 2,α (θ1 ,θ2 ) < C1 .
(7.7)
Proof. As we mentioned in the beginning of this section, the process of solving the problems (NL) and (Sh) forms a map T f from f (θ ) to f˜(θ ). Lemma 7.1 implies that for sufficiently small , T f is an inner map on the convex set B of the Banach space C 2,α (θ1 , θ2 ). Notice that the index α is determined by the domain and the upstream U− . In fact, we can determine an index α0 , so that for any α ≤ α0 all the above propositions involving α0 the index α hold. On the other hand, take f ∈ C 2, 2 (θ1 , θ2 ), problem (L1 ) still admits a solution δp ∈ C 1,α0 (see Remark 2 in Sect. 5). Then Theorem 4.1 and Theorem 3.1 are still valid when α is replaced by α0 . Therefore, the solution f˜(θ ) of (7.2) belongs α0 to C 2,α0 . In view of the embedding map C 2,α0 (θ1 , θ2 ) to C 2, 2 (θ1 , θ2 ) is compact, then α0 T is a compact map on C 2, 2 (θ1 , θ2 ). Thus by the Schauder fixed point theorem the α0 map T has a fixed point f (θ ) in C 2, 2 (θ1 , θ2 ). The function f (θ ) and the corresponding U (r, θ ) gives the solution to problem (PsFB). Next we prove uniqueness. Suppose that the pseudo-free boundary value problem (PsFB) has two solutions. The corresponding free boundaries are 1A : r = f A (θ ) and 1B : r = f B (θ ) respectively, while the solution U A (r, θ ) is defined in Q f A = { f A (θ ) < r < r2 , θ1 < θ < θ2 }, and the solution U B (r, θ ) is defined in Q f B = { f B (θ ) < r < r2 , θ1 < θ < θ2 }. U A (r, θ ) and U B (r, θ ) satisfy Eq. (3.1) and the estimate in Theorem 7.2, then we have to prove f A (θ ) = f B (θ ) and U A (r, θ ) = U B (r, θ ). By using the dilation transformation (3.21) and the Lagrange transformation (3.26) we obtain U A (ξ, η) and U B (ξ, η) defined in Q + . (Here with some abuse of notations we still use U A (ξ, η), U B (ξ, η) to stand for U A , U B as the functions of ξ and η.) U A (ξ, η) satisfies ⎧ ⎪ A[U A (ξ, η)] = 0 in Q + , ⎪ ⎪ ⎨G (U (ξ, η), U ) = 0, i = 1, 2, 3 on S , i A − 0 (7.8) (NL)A : +, ⎪ w = 0 on ∂ Q A 1 ⎪ ⎪ ⎩ p = p (θ ) + const on r = r2 , A r where A[U A (ξ, η)] is the abbreviation for the operators in (3.26). Similarly, U B (ξ, η) solves a corresponding problem (NL)B with all index A in (7.8) replaced by B. Denoting U B (ξ, η)−U A (ξ, η) by U B A (ξ, η), i.e. u B A = u B −u A , w B A = w B −w A , p B A = p B − p A , ρ B A = ρ B − ρ A , we have FA UBA = F # , (GA )i UBA = gi# , where F # = A[U B ] − A[U A ] − F A U B A , gi# = G i (U B , U− ) − G i (U A , U− ) − (GA )i U B A .
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Therefore, U B A is the solution of the problem ⎧ ⎪ FA U B A = F # in Q + , ⎪ ⎪ ⎨(G ) U = g # , i = 1, 2, 3 on S , A i BA 0 i (NL)AB : +, ⎪ = 0 on ∂ Q w BA 1 ⎪ ⎪ ⎩ p = const on r = r2 . BA
(7.9)
Due to the expression of F # and G i# , we have 2 F # C 0,α (Q + ) ≤ C U B − U A C 1,α (Q + ) ,
G i# C 1,α (S0 )
≤
2 C U B − U A C 1,α (S ) . 0
(7.10) (7.11)
Theorem 4.1 indicates that the solution U B A of (7.8) satisfies 2 U B A C 1,α (Q + ) ≤ C( F # C 0,α (Q + ) + G # C 1,α (S0 ) ) ≤ C U B A C (7.12) 1,α (Q + ) .
Since U B , U A and then U B A are small quantities as O( ), then U B A C 1,α (Q + ) ≤ C U B A C 1,α (Q + ) . Therefore, for sufficiently small we must have U B A = 0 or U B = U A . Moreover, U A (r, θ ) and U B (r, θ ) can be obtained from U A (ξ, η) and U B (ξ, η) via T −1 , the inverse transformation of T in (3.25), we then have f B (θ ) − f A (θ ) C 1,α (θ1 ,θ2 ) ≤ C U B A C 1,α (Q + ) .
Hence f A (θ ) = f B (θ ) = f (θ ), and U A (r, θ ),U B (r, θ ) are defined in the same domain f (θ ) ≤ r ≤ r2 , θ1 ≤ θ ≤ θ2 . Moreover, U B (r, θ ) − U A (r, θ ) C 1,α (Q f ) ≤ C U B A C 1,α (Q + ) = 0. Thus the uniqueness is proved.
8. Solution to the Free Boundary Problem (FB) In the last section we will finally prove that by suitably choosing the value f 0 in (PsFB), we can let the solution U (r, θ ) satisfy the condition p(r2 ) = pr (θ ) precisely. Correspondingly, the free boundary value problem (FB) is also solved. Lemma 8.1. If U (r, θ ) is the solution of the problem (PsFB), then the pressure p(r2 , θ ) is a uniformly continuous functional of (U− , pr ) in C 2,α × C 2,α (θ1 , θ2 ). Proof. Assume that U−A , U−B ∈ O , prA (θ ), prB (θ ) ∈ O satisfy U−A − U−B C 2,α (Q − ) ≤ , prA (θ ) − prB (θ ) C 2,α (θ1 ,θ2 ) ≤ ,
(8.1)
then according to Theorem 7.2 there are solutions (U+A , f A (θ )) and (U−B , f B (θ )) to problem (PsFB). Like the proof of uniqueness in Theorem 7.2 we can derive U+A − U+B C 1,α ( + ) ≤ C
(8.2)
p A (r2 , θ ) − p B (r2 , θ ) C 1,α (θ1 ,θ2 ) ≤ C .
(8.3)
and
This is the required continuity.
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The proof of Theorem 2.4. Now we can finally prove the main conclusion in this paper. Denote k = p(r2 , θ ) − pr (θ ), which is a constant and is determined by U− (r, θ ), pr (θ ) and f 0 . Such a dependence is written by k = k(U− , pr , f 0 ) in the sequel. When U− and pr are taken as functions solely depending on r , then the problem is reduced to a one dimensional problem, which has been discussed in Sect. 2. From Lemma 2.2 we dk know < 0, provided U− = Ub− , p2 (θ ) = pb+ and f 0 = 0. Then there is a constant d f0 c0 , such that (8.4) k Ub− , pb+ , f 0(1) − k Ub− , pb+ , f 0(2) > c0 f 0(2) − f 0(1) , provided f 0(1) − f 0(2) < 0 and | f 0(1) |, | f 0(2) | < 0 are sufficiently small. In view of the uniform continuity we have k U− , pr , f 0(1) − k U− , pr , f 0(2) > c0 f 0(2) − f 0(1) , (8.5) provided U− ∈ O 0 , pr (θ ) ∈ O 0 , f 0(1) − f 0(2) < 0, | f 01 |, | f 02 | < 0 and 0 is sufficiently small. c0 Let C1 and c0 be the constants defined in (7.7) and (8.4) respectively. Take ≤ 0 , C1 U− ∈ O , pr (θ ) ∈ O and f 0 = 0; we obtain a solution U (r, θ ) of (PsFB) by using Theorem 7.2. Then (7.7) indicates |k(U− , pr , 0)| ≤ C1 .
(8.6)
According to the monotonicity of k(U− , pr , f 0 ) with respect to f 0 as shown in (8.4), C1 we can find f 0∗ satisfying | f 0∗ | ≤ , such that k(U− , pr , f 0∗ ) = 0. c0 Therefore, we find > 0, so that the corresponding neighborhoods O and O have the following property: for any U− ∈ O and pr (θ ) ∈ O , there exists f 0∗ so that the solution (U (r, θ ), f (θ )) to problem (PsFB) with f (0) = f 0∗ also satisfies the condition k(U− , pr (θ ), f 0∗ ) = 0. Hence (U (r, θ ), f (θ )) also solves problem (FB), and the proof of Theorem 2.4 is thus complete. As mentioned in Sect. 2, the Theorem 2.4 implies the stability of problem (P) and the validity of Theorem 2.3. Acknowledgement. The paper is partially supported by the National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902, the Doctorial Foundation of National Educational Ministry 20050246001 and STCSM 06JC14005.
References 1. Adams, R.A.: Sobolev Spaces. New York: Academic Press, 1975 2. Chen, G.Q., Chen, J., Song, K.: Transonic nozzle flows and free boundary problems for the full Euler equation. J. Diff. Eqs. 229, 93–120 (2006) 3. Chen, G.Q., Feldman, M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Amer. Math. Soc. 16, 461–494 (2003) 4. Chen, G.Q., Feldman, M.: Steady transonic shocks and free boundary problems in infinite cylinders for the euler equations. Comm. Pure. Appl. Math. 57, 310–356 (2004) 5. Chen, G.Q., Feldman, M.: Free boundary problems and transonic shocks for the Euler equations in unbounded domains. Ann. Sc. Norm. Sup. Pisa CI. Sci. 3, 827–869 (2004)
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6. Chen, G.Q., Feldman, M.: Existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross-sections. Arch. Rat. Mech. Anal. 184, 185–242 (2007) ˘ c, 7. Cani ´ S., Keyfitz, B.L., Lieberman, G.M.: A Proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm. Pure. Appl. Math. 53, 484–511 (2000) ˘ c, 8. Cani ´ S., Keyfitz, B.L., Kim, E.H.: A free boundary problems for unsteady transonic small disturbance equation: transonic regular reflection. Meth. Appl. Anal. 7, 313–336 (2000) ˘ c, 9. Cani ´ S., Keyfitz, B.L., Kim, E.H.: A free boundary problems for a quasilinear degenerate elliptic equation: transonic regular reflection. Comm. Pure Appl. Math. 55, 71–92 (2002) 10. Chen, S.X.: Existence of stationary supersonic flow past a pointed body. Arch. Rat. Mech. Anal. 156, 141–181 (2001) 11. Chen, S.X.: Stability of oblique shock fronts. Science in China 45, 1012–1019 (2002) 12. Chen, S.X.: Stability of transonic shock fronts in two-dimensional Euler systems. Trans. Amer. Math. Soc. 357, 287–308 (2005) 13. Chen, S.X.: Transonic shock in 3-D compressible flow passing a duct with a general section for Euler systems. Trans. Amer. Math. Soc. 360, 5265–5289 (2008) 14. Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves. Interscience Publishers Inc., 1948 15. Chen, S.X., Yuan, H.R.: Transonic shocks in compressible flow passing a duct for three-dimensional Euler system, submitted 16. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuums Physics. Fundamental Principles of Mathematical Sciences, 325. Berlin: Springer, 2000 17. Glaz, H.M., Liu, T.P.: The asymptotic analysis of wave interactions and numerical calculations of transonic flow. Advances in Appl. Math. 5, 111–146 (1984) 18. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equation of Second Order. Second edition, Grundlehren der Mathematischen Wissenschaften, 224. Berlin-New York: Springer, 1983 19. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics 24, London: Pitman, 1985 20. Kuz’min, A.G.: Boundary Value Problems for Transonic Flow. New York: John Wiley & Sons LTD., 2002 21. Liu, T.P.: Nonlinear stability and instability of transonic flows through a nozzle. Commun. Math. Phys. 83, 243–260 (1982) 22. Liu, T.P.: Transonic gas flows in a variable area duct. Arch. Rat. Mech. Anal. 80, 1–18 (1982) 23. Li, C.Z., Liu, T.P.: Asymptotic states for hyperbolic conservation laws with a moving source. Adv. Appl. Math. 4, 353–379 (1983) 24. Li, T.T., Yu, W.C.: Boundary Value Problem for Quasilinear Hyperbolic System. Duke University Mathematical Series 5, Durham, NC: Duke Univ. Press, 1985 25. Majda, A.: The stability of multi-dimensional shock fronts. Memoir Amer. Math. Soc. 273 (1983) 26. Majda, A.: One perspective on open problems in multi-dimensional conservation laws. IMA Vol. Math. Appl. 29, 217–237 (1991) 27. Pirumov, U.G., Roslyakov, G.S.: Gas Flow in Nozzles. Berlin Heidelberg New York Tokyo: SpringerVerlag, 1978 28. Smoller, J.: Shock Waves and Reaction-diffusion Equations. Second edition, New York: Springer-Verlag, 1994 29. Xin, Z.P., Yin, H.C.: Transonic shock in a nozzle I: two-dimensional case. Commun. Pure Appl. Math. 58, 999–1050 (2005) 30. Xin, Z.P., Yin, H.C.: 3-Dimensional transonic shocks in a nozzle. Pacific J. Math. 236, 139–193 (2008) 31. Xin, Z.P., Yin, H.C.: The transonic shock in a nozzle, 2-D and 3-D complete Euler systems. J. Diff. Eqs. 245, 1014–1085 (2008) 32. Yuan, H.R.: On transonic shocks in two dimensional variable-area ducts for steady Euler system. SIAM J. Math. Anal. 38, 1343–1370 (2006) 33. Yuan, H.R.: Transonic shocks for steady Euler flows with cylindrical symmetry. Nonlinear Anal. 66, 1853–1878 (2007) 34. Yuan, H.R: Examples of steady subsonic flows in a convergent-divergent approximate nozzle. J. Diff. Eqs. 244, 1675–1691 (2008) Communicated by P. Constantin
Commun. Math. Phys. 289, 107–155 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0810-8
Communications in
Mathematical Physics
Spectral Action on SUq (2) B. Iochum1,2 , C. Levy1,2 , A. Sitarz3, 1 Centre de Physique Théorique , CNRS–Luminy, Case 907, 13288 Marseille Cedex 9,
France. E-mail: [email protected]; [email protected]
2 Also at Université de Provence, Aix-Marseille, France 3 Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland.
E-mail: [email protected] Received: 11 March 2008 / Accepted: 30 January 2009 Published online: 26 April 2009 – © Springer-Verlag 2009
Abstract: The spectral action on the equivariant real spectral triple over A SUq (2) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S3. Contents 1. 2. 3.
4.
5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Action in 3-Dimension . . . . . . . . . . . . . . . . . . . . . . . 2.1 Tadpole and cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Scale-invariant term of the spectral action . . . . . . . . . . . . . . . The SUq (2) Triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The spectral triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The noncommutative integrals . . . . . . . . . . . . . . . . . . . . . 3.3 The tadpole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reality Operator and Spectral Action on SUq (2) . . . . . . . . . . . . . . 4.1 Spectral action in dimension 3 with [F, A] ∈ O P −∞ . . . . . . . . . 4.2 Spectral action on SUq (2): main result . . . . . . . . . . . . . . . . . 4.3 Balanced components and Poincaré–Birkhoff–Witt basis of A . . . . 4.4 The reality operator J on SUq (2) . . . . . . . . . . . . . . . . . . . . 4.5 The smooth algebra C ∞ (SUq (2)) . . . . . . . . . . . . . . . . . . . 4.6 Noncommutative integrals with reality operator and one-forms on SUq (2) 4.7 Proof of Theorem 4.3 and corollaries . . . . . . . . . . . . . . . . . . Differential Calculus on SUq (2) and Applications . . . . . . . . . . . . .
108 109 109 112 113 113 118 119 120 120 121 122 123 128 129 133 134
UMR 6207, – Unité Mixte de Recherche du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l’Université du Sud Toulon-Var, Laboratoire affilié à la FRUMAM – FR 2291. Partially supported by Polish Government grants 189/6.PRUE/2007/7; 115/E-343/SPB/6.PR UE/DIE and N 201 1770 33.
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5.1 The sign of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The ideal R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Operators L q and Mq . . . . . . . . . . . . . . . . . . . . . 5.2.2 Automorphisms of the algebra and symmetries of integrals. 5.2.3 The noncommutative integrals at |D|−2 . . . . . . . . . . . . Examples of Spectral Action . . . . . . . . . . . . . . . . . . . . . . . The Commutative Sphere S3 . . . . . . . . . . . . . . . . . . . . . . . 7.1 Translation of Dirac operator . . . . . . . . . . . . . . . . . . . . 7.2 Tadpole and spectral action on S3 . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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134 136 139 140 143 144 145 146 147 148 149
1. Introduction The quantum group SUq (2) has already a rather long history of studies [32] being one of the finest examples of quantum deformation. This includes an approach via the noncommutative notion of spectral triple introduced by Connes [10,15] and various notions of Dirac operators were introduced in [2,4,6,13,28]. Finally, a real spectral triple, which was exhibited in [21], is invariant by left and right action of Uq (su(2)) and satisfies almost all postulated axioms of triples except the commutant and first-order properties. These, however, remain valid only up to infinitesimal of arbitrary high order. The last presentation generalizes in a straightforward way all geometric construction details of the spinorial spectral triple for the classical three-sphere. In particular, both the equivariant representation and the symmetries have a q → 1 proper classical limit. The goal of this article is to obtain the spectral action defined in [7] by S(DA , , ) := Tr ((DA /)),
(1)
where D is the Dirac operator, A is a selfadjoint one-form, DA = D + A + J AJ −1 and J is the reality operator. Here, is any even positive cut-off function, which could be replaced by a step function up to some mathematical difficulties investigated in [23]. This means that S counts the spectral values of |DA | less than the mass scale . Actually, as shown in [8], S(DA , , ) = k k − |DA |−k + (0) ζ DA (0) + O(−1 ), (2) 0
ζ DA (0) = ζ D (0) +
d (−1)k −1 k k −(AD ) ,
(3)
k=1
where DA := DA + PA , PA the projection on Ker DA , D := D + P0 , P0 the projection ∞ on Ker D, k = 21 0 (t) t k/2−1 dt, d is the spectral dimension of the triple and Sd + is the strictly positive part of the dimension spectrum Sd of (A, H, D). This spectral action has been computed on few examples: [3,8,9,15,22,24–26,30, 33]. Here, we compute (4) on SUq (2), which is a spectral triple with an invertible Dirac operator. The main difficulty here lies in the control of the differential calculus generated by the Dirac operator. This question of computation of spectral action was addressed in the epilogue of [36].
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In the case of SUq (2), we have Sd + = Sd = { 1, 2, 3 }, so S(DA , , ) = k k − |DA |−k + (0) ζ DA (0).
(4)
1≤k≤3
Recall that the tadpole of order k is the linear term in A ∈ 1D (A) in the k part of (4). Note that in the case of SUq (2) there are no terms in −k , k > 0 because the dimension spectrum is bounded from below by 1. To proceed with the computation of (4), we introduce two presentations of one-forms. The main ingredient is F = sign (D), which appears to be a one-form up to O P −∞ . In Sect. 2, we discuss the spectral action of an arbitrary 3-dimensional spectral triple using cocycles. In Sects. 3 and 4 we recall the main results on SUq (2) of [21] and show that the full spectral action with reality operator given by (4) is completely determined by the terms (here D is invertible) − Aq |D|− p , 1 ≤ q ≤ p ≤ 3, where A is a linear combination of terms of the form a[|D|, b] with a, b ∈ A. In Sect. 5, we establish a differential calculus up to some ideal in pseudodifferential operators and apply these results to the precise computation of previous noncommutative integrals. Section 6 is devoted to explicit examples, while in the next section are given different comparisons with the commutative case of the 3-sphere corresponding to SU (2). 2. Spectral Action in 3-Dimension 2.1. Tadpole and cocycles. Let (A, H, D) be a spectral triple of dimension 3. We refer to [8,15,16] for the definition of the O P α spaces and the algebra of pseudodifferential operators (A) on a spectral triple (see also [22] when reality operator J is present.) For n ∈ N∗ and ai ∈ A, define φn (a0 , . . . , an ) := − a0 [D, a1 ]D −1 · · · [D, an ]D −1 . We also use notational integrals on the universal n-forms nu (A) defined by a0 da1 . . . dan := φn (a0 , a1 , . . . , an ). φn
and the reordering fact that (da0 )a1 = d(a0 a1 ) − a0 da1 . We use the b − B bicomplex defined in [10]: b is the Hochschild coboundary map (and b is the truncated one) defined on n-cochains φ by bφ(a0 , . . . , an+1 ) := b φ(a0 , . . . , an+1 ) + (−1)n+1 φ(an+1 a0 , a1 , . . . , an ), n b φ(a0 , . . . , an+1 ) := (−1) j φ(a0 , . . . , a j a j+1 , . . . , an+1 ). j=0
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Recall that B0 is defined on the normalized cochains φn by B0 φn (a0 , a1 , . . . , an−1 ) := φn (1, a0 , . . . , an−1 ), thus ω for ω ∈ n−1 = u (A).
φn
dω
B0 φn
Then B := N B0 , where N := 1 + λ + . . . λn is the cyclic skewsymmetrizer on the n-cochains and λ is the cyclic permutation λφ(a0 , . . . , an ) := (−1)n φ(an , a0 , . . . , an−1 ). We will also encounter the cyclic 1-cochain N φ1 : a0 da1 := N φ1 (a0 , a1 ). N φ1 (a0 , a1 ) := φ1 (a0 , a1 ) − φ1 (a1 , a0 ) and N φ1
Remark 2.1. Assume the integrand of is in O P −3 . Since [D −1 , a] = −D −1 [D, a] D −1 ∈ O P −2 , this commutator introduces an integrand in O P −4 so has a vanishing integral: under the integral, we can commute D −1 with all a ∈ A, but not with oneforms. Note also that since P0 ∈ O P −∞ , any integrand containing P0 has a vanishing integral. Lemma 2.2. We have (i) bφ1 = −φ2 , (ii) bφ2 = 0, (iii) bφ3 = 0, (iv) Bφ1 = 0, (v) B0 φ2 = −(1 − λ)φ1 , (vi) bB0 φ2 = 2φ2 + B0 φ3 , (vii) Bφ2 = 0, (viii) B0 φ3 = N b φ1 , (ix) Bφ3 = 3B0 φ3 . Proof. (i) bφ1 (a0 , a1 , a2 ) = − a0 a1 [D, a2 ]D −1 − − a0 (a1 [D, a2 ] + [D, a1 ]a2 ) D −1 + − a2 a0 [D, a1 ]D −1 −1 −1 = − − a0 [D, a1 ]D −1 [D, a2 ]D −1 = − a0 [D, a1 ] D a2 − a2 D = −φ2 (a0 , a1 , a2 ), where we have used the trace property of the noncommutative integral. (ii) bφ2 (a0 , a1 , a2 , a3 ) −1 −1 = − a0 a1 [D, a2 ]D [D, a3 ]D − − a0 (a1 [D, a2 ] + [D, a1 ]a2 ) D −1 [D, a3 ]D −1 + − a0 [D, a1 ]D −1 (a2 [D, a3 ] + [D, a2 ]a3 )D −1 − − a3 a0 [D, a1 ]D −1 [D, a2 ]D −1
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= − a0 [D, a1 ] D −1 a2 − a2 D −1 [D, a3 ]D −1 + − a0 [D, a1 ]D −1 [D, a2 ] a3 D −1 − D −1 a3 = − − a0 [D, a1 ]D −1 [D, a2 ]D −1 [D, a3 ]D −1 + − a0 [D, a1 ]D −1 [D, a2 ]D −1 [D, a3 ]D −1 = 0. (iii) Using Remark2.1, similar computations as for bφ 2 gives bφ3 = 0. (iv) B0 φ1 (a0 ) = [D, a0 ]D −1 = Da0 D −1 − a0 = 0. −1 −1 −1 (v) B0 φ2 (a0 , a1 ) = −[D, a0 ]D [D, a1 ]D = − a0 D [D, a1 ] − − a0 [D, a1 ]D −1 = − a0 a1 − − a0 D −1 a1 D − − a0 [D, a1 ]D −1 −1 = − − a1 [D, a0 ]D − − a0 [D, a1 ]D −1 = −φ1 (a1 , a0 ) − φ1 (a0 , a1 ). (vi) Since −bλφ1 (a0 , a1 , a2 ) = φ1 (a2 , a0 a1 ) − φ1 (a1 a2 , a0 ) + φ1 (a1 , a2 a0 ), one obtains that −bλφ1 (a0 , a1 , a2 ) = − a0 a1 D −1 a2 D + a0 D −1 a1 Da2 − a0 D −1 a1 a2 D − a0 a1 a2 . So by direct expansion, this is equal to − a0 D −1 [D, a1 ]D −1 [D, a2 ], which means that −bλφ1 (a0 , a1 , a2 ) = −[D −1 , a0 ][D, a1 ]D −1 [D, a2 ] − a0 [D, a1 ]D −1 [D, a2 ]D −1 = − B0 φ3 (a0 , a1 , a2 ) − φ2 (a0 , a1 , a2 ). Now the result follows from (i), (v). (vii) Bφ2 = N B0 φ2 = −N (1 − λ)φ1 = 0 since N (1 − λ) = 0. (viii) B0 φ3 (a0 , a1 , a2 ) = −[D, a0 ]D −1 [D, a1 ]D −1 [D, a2 ]D −1 −1 −1 = − a0 D [D, a1 ]D [D, a2 ] − − a0 [D, a1 ]D −1 [D, a2 ]D −1 = − a0 a1 D −1 [D, a2 ] − − a0 D −1 a1 [D, a2 ] − − a0 [D, a1 ]D −1 [D, a2 ]D −1 −1 −1 = − a0 a1 a2 − − a0 a1 D a2 D − − a0 D a1 Da2 + − a0 D −1 a1 a2 D − − a0 [D, a1 ]D −1 [D, a2 ]D −1 = − a0 a1 a2 − a2 Da1 a0 D −1 + a1 a2 Da0 D −1 + a2 Da0 a1 D −1 − a0 Da1 a2 D −1 − a0 Da1 D −1 − a0 a1 Da2 D −1 + a0 a1 a2 .
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Expanding (id + λ + λ2 )b φ1 (a0 , a1 , a2 ), we recover the previous expression. (ix) Consequence of (viii). 2.2. Scale-invariant term of the spectral action. We know from [8] that the scale-invariant term of the action can be written as ζ DA (0) − ζ D (0) = − − AD −1 + 21 − AD −1 AD −1 − 13 − AD −1 AD −1 AD −1 . (5) In fact, this action can be expressed in dimension 3 as contributions corresponding to tadpole and the Yang–Mills and Chern–Simons actions in dimension 4: Proposition 2.3. For any one-form A, ζ DA (0) − ζ D (0) = − 21 A+ N φ1
1 2
φ2
(dA + A2 ) −
1 2
φ3
(AdA + 23 A3 ).
(6)
To prove this, we calculate now each term of the action. Lemma 2.4. For any one-form A, we have (i) φ2 dA = B0 φ2 A = − φ1 A + λφ1 A, (ii) AD −1 = φ1 A = 21 N φ1 A − 21 φ2 dA, (iii) AD −1 AD −1 = − φ3 AdA + φ2 A2 , (iv) AD −1 AD −1 AD −1 = φ3 A3 . Proof. (i) and (ii) follow directly from Lemma 2.2 (v). (iii) With the shorthand A = ai dbi (summation on i), − AD −1 AD −1 = − a0 [D, b0 ]D −1 a1 [D, b1 ]D −1 −1 =− AdA + − a0 [D, b0 ]a1 b1 D − − a0 [D, b0 ]a1 D −1 b1 . φ3
We calculate further the remaining terms − a0 [D, b0 ]a1 b1 D −1 − − a0 [D, b0 ]a1 D −1 b1 = − a0 Db0 a1 b1 D −1 − − a0 b0 Da1 b1 D −1 −1 −− a0 Db0 a1 D b1+−a0 b0 Da1 D −1 b1, which are compared with φ2 A2 = φ2 a0 (db0 )a1 db1 = φ2 a0 d(b0 a1 )db1 −a0 b0 da1 db1 : 2 −1 −1 A = − a0 [D, b0 a1 ]D [D, b1 ]D − − a0 b0 [D, a1 ]D −1 [D, b1 ]D −1 φ2 −1 −1 −1 = − a0 Db0 a1 b1 D − − a0 Db0 a1 D b1 − − a0 b0 a1 Db1 D + − a0 b0 a1 b1 −− a0 b0 Da1 b1 D −1 + − a0 b0 Da1 D −1 b1 + − a0 b0 a1 Db1 D −1 − − a0 b0 a1 b1 −1 −1 = − a0 Db0 a1 b1 D − − b1 a0 Db0 a1 D − − a0 b0 Da1 b1 D −1 + − b1 a0 b0 Da1 D −1 .
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(iv) Note that A3 = a0 (db0 )a1 (db1 )a2 db2 = φ3
φ3
φ3
a0 d(b0 a1 )d(b1 a2 )db2 − a0 b0 da1 d(b1 a2 )db2
−a d(b0 a1 b1 )d(a2 db2 + a0 b0 d(a1 b1 )da2 db2 0 = − a0 [D, b0 a1 ]D −1 [D, b1 a2 ]D −1 [D, b2 ]D −1 − a0 b0 [D, a1 ]D −1 [D, b1 a2 ]D −1 [D, b2 ]D −1 − a0 [D, b0 a1 b1 ]D −1 [D, a2 ]D −1 [D, b2 ]D −1 + a0 b0 [D, a1 b1 ]D −1 [D, a2 ]D −1 [D, b2 ]D −1 . Summing up the first two terms and the last two ones gives A3 = − a0 [D, b0 ]a1 D −1 [D, b1 a2 ]D −1 [D, b2 ]D −1 φ3
− a0 [D, b0 ]a1 b1 D −1 [D, a2 ]D −1 [D, b2 ]D −1 . Using Remark 2.1, we can commute under the integral D −1 with all a ∈ Abb and similarly −1 −1 −1 − AD AD AD = − a0 [D, b0 ]a1 D −1 [D, b1 ]a2 D −1 [D, b2 ]D −1 , which proves (iv).
We deduce Proposition 2.3 from (5) using the previous lemma. 3. The SUq (2) Triple We briefly recall the main facts of the real spectral triple 3.1. The spectral triple. A(SUq (2)), H, D introduced in [21], see also [4,5,13]. The algebra. Let A := A(SUq (2)) be the ∗ -algebra generated polynomially by a and b, subject to the following commutation rules with 0 < q < 1: ba = q ab, b∗ a = q ab∗ , bb∗ = b∗ b, a ∗ a + q 2 b∗ b = 1, aa ∗ + bb∗ = 1.
(7)
We recall the following lemma from [38, Lemma A2.1]: Lemma 3.1. For any representation π of A, Spect π(bb∗ ) = { 0, q 2k | |k ∈ N } or π(b) = 0, Spect π(aa ∗ ) = { 1, 1 − q 2k | k ∈ N } or π(b) = 0 and π(a) is a unitary. This result is interesting since it shows the appearance of discreteness for 0 ≤ q < 1 while for q = 1, SUq (2) = SU (2) S3 and the spectrum of the commuting operator π(aa ∗ ) and π(bb∗ ) are equal to [0, 1]. Moreover, all foregoing results on noncommutative integrals will involve q 2 and not q. Any element of A can be uniquely decomposed as a linear combination of terms of the form a α bβ b∗ γ , where α ∈ Z, β, γ ∈ N, with the convention a −|α| := a ∗ |α| .
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The spinorial Hilbert space. H = H↑ ⊕ H↓ has an orthonormal basis consisting of vectors | jµn↑ with j = 0, 21 , 1, . . . , µ = − j, . . . , j and n = − j + , . . . , j + , together with | jµn↓ for j = 21 , 1, . . . , µ = − j, . . . , j and n = − j − , . . . , j − (here x± := x ± 21 ). It is convenient to use a vector notation, setting: (8) | jµn
:= || jµn↑ jµn↓ and with the convention that the lower component is zero when n = ±( j + 21 ) or j = 0. The representation π and its approximate π . It is known that representation theory of SUq (2) is similar to that of SU (2) [38]. The representation π given in [21] is: − + + π(a) | jµn
:= α +jµn | j + µ+ n +
+ α − jµn | j µ n
, − + − π(b) | jµn
:= β +jµn | j + µ+ n −
+ β − jµn | j µ n
, − − − π(a ∗ ) | jµn
:= α˜ +jµn | j + µ− n −
+ α˜ − jµn | j µ n
, − − + π(b∗ ) | jµn
:= β˜ +jµn | j + µ− n +
+ β˜ − jµn | j µ n
,
where α +jµn α− jµn β +jµn β− jµn
(9)
j+n+3/2] q − j−1/2 [[2 0 j+2] µ+n−1/2 √ √ := q [ j + µ + 1] , j−n+1/2] j+n+1/2] q − j [[2 q 1/2 [2 [j+1][2 j+2] j+1] √ √
j−n+1/2] q j+1 [ [2 −q 1/2 [2[ j+n+1/2] j+1] j][2 j+1] µ+n+1/2 √ := q [ j − µ] , 0 q j+1/2 [ j−n−1/2] [2 j] √
[ j−n+3/2] 0 [2 j+2] µ+n−1/2 √ √ := q [ j + µ + 1] , [ j+n+1/2] −1/2 [ j−n+1/2] −q − j−1 [2 j+1][2 j+2] q [2 j+1] √ √
−1/2 [ j+n+1/2] −q j [ j−n+1/2] −q [2 j+1] [2 j][2 j+1] √ := q µ+n−1/2 [ j − µ] 0 − [ j+n−1/2] [2 j] √
∓ ∓ ∗ ˜± ∗ with α˜ ± jµn := (α j ± µ− n − ) , β jµn := (β j ± µ− n + ) and with the q-number of α ∈ R defined as
[α] :=
q α −q −α . q−q −1
For the purpose of this paper it is sufficient to use the approximate spinorial ∗ -representation π of SUq (2) presented in [21,37] instead of the full spinorial one π. This approximate representation is π (a) := a+ + a− , π (b) := b+ + b− with the following definitions, where qn := 1 − q 2n , q+ + 0 a+ | jµn
:= q j + +µ+ j0 +n +1 q + | j + µ+ n +
, j +n 1 a− | jµn
:= q 2 j+µ+n+ 2 q0 10 | j − µ+ n +
, 1 b+ | jµn
:= q j+n− 2 q j + +µ+ q0 10 | j + µ+ n −
, q + 0 − + − | j µ n
. b− | jµn
:= −q j+µ j 0+n q j − +n
(10)
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115
All disregarded terms are trace-class and do not influence residue calculations. More precisely, π(x) − π (x) ∈ Kq , where Kq is the principal ideal generated by the operator Jq | jµn
:= q j | jµn
.
(11)
Actually, Kq is independent of q and is contained in all ideals of operators such that µn = o(n −α ) (infinitesimal of order α) for any α > 0, and Kq ⊂ O P −∞ . j↑ j↓ We define the alternative orthonormal basis vm,l and vm,l and the vector notation j
vm,l :=
j↑ vm,l j↓
vm,l
where vm,l := | j, m − j, l − j + , ↑ , vm,l := | j, m − j, l − j − , ↓ . j↑
j↓
↓, j
Here j ∈ 21 N, 0 ≤ m ≤ 2 j, 0 ≤ l ≤ 2 j + 1 and vm,l is zero whenever j = 0 or l = 2 j or 2 j + 1. The interesting effect is that now, the operators a± and b± assume simpler form: j+
j
j−
j
a+ vm,l = qm+1 ql+1 vm+1,l+1 , a− vm,l = q m+l+1 vm,l , j+
j
b+ vm,l = q l qm+1 vm+1,l ,
j−
j
b− vm,l = −q m ql vm,l−1 .
(12)
Thus j−
∗ a− vm,l = q m+l+1 vm,l ,
j−
∗ b− vm,l = −q m ql+1 vm,l+1 .
a+∗ vm,l = qm ql vm−1,l−1 , j
b+∗ vm,l = q l qm vm−1,l , j
j+
j
j+
j
(13)
Moreover, we have a− a+ = q 2 a+ a− , b− b+ = q 2 b+ b− , b+ a+ = q a+ b+ , b− a− = q a− b− ,
∗ ∗ ∗ ∗ a− a+ = q 2 a+ a− , a− a− = a− a− , ∗ ∗ ∗ ∗ a− b+ = q b+ a− , a− b− = q b− a− , ∗ ∗ b+ = b+ b− , a+∗ a− = q 2 a− a+∗ , b− ∗ ∗ b− a+ = q a+ b− , a− b+ = q b+ a− .
(14)
Note for instance that 2 2 a+ a+∗ vm,l = qm2 ql2 vm,l , a+∗ a+ vm,l = qm+1 ql+1 vm,l , j
j
j
j
2 vm,l , b+ b+∗ vm,l = q 2l qm2 vm,l , b+∗ b+ vm,l = q 2l qm+1 j
j
j
j
j
so applied to vm,l , we get the first relation (and similarly for the others) a+∗ a+ − q 2 a+ a+∗ + q 2 (b+∗ b+ − b+ b+∗ ) = 1 − q 2 , ∗ ∗ a+ a+∗ + a− a− + b+ b+∗ + b− b− = 1, ∗ ∗ a− + q 2 (b+∗ b+ + b− b− ) = 1, a+∗ a+ + a−
∗ ∗ ∗ ∗ a− − q 2 a− a− + q 2 b− b− − q 2 b− b− = a− ∗ ∗ ∗ 2 ∗ a+ a− + b− b+ = 0, a− a+ + q b− b+ = 0, a− a+∗ + b+∗ b− = 0, a+∗ a− + q 2 b+∗ b− = 0, ∗ ∗ b+ b+∗ − b+∗ b+ + b− b− − b− b− = 0,
q a+ b− − b− a+ + q a− b+ − b+ a− = 0.
(15) (16) (17)
0,
(18) (19) (20) (21) (22)
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The irreducible representations. Note that we also use two other infinite dimensional ∗ -representations π of A on 2 (N) defined on the orthonormal basis { ε : n ∈ N } of ± n 2 (N) by π± (a) εn := qn+1 εn+1 ,
π± (b) εn := ±q n εn .
(23)
These representations are irreducible but not faithful as, for instance, π± (b − b∗ ) = 0. The Dirac operator: It is chosen to be the same as in the classical case of a 3-sphere with the round metric: D | jµn
:=
0 | jµn
, −2 j− 21
2 j+ 23 0
j
which means, with our convention, that D vml =
0 j vml . −2 j− 21
2 j+ 23 0
(24)
Note that this oper-
ator is invertible (D = D, P0 = 0). Moreover, it is asymptotically diagonal with linear spectrum and, the eigenvalues 2 j + 21 for j ∈ 21 N, have multiplicities (2 j + 1)(2 j + 2), whereas the eigenvalues −(2 j + 21 ) for j ∈ 21 N∗ , have multiplicities 2 j (2 j + 1). Note that this Dirac operator coincides exactly with the classical one for the 3-sphere (see [1,31]) with a gap around 0. Let D = F|D| be the polar decomposition of D, thus |D| | jµn
= F | jµn
=
d j+ 0 0
1 0
dj
| jµn
, d j := 2 j + 21 ,
(25)
0 −1 | jµn
,
(26)
and it follows from (10) and (26) that F commutes with a± , b± .
(27)
The reality operator. This antilinear operator J is defined on the basis of H by J | j, µ, n, ↑ := i 2(2 j+µ+n) | j, −µ, −n, ↑ , J | j, µ, n, ↓ := i 2(2 j−µ−n) | j, −µ, −n, ↓ ,
(28)
thus it satisfies J −1 = −J = J ∗ and D J = J D, j↑
j↑
J vm,l = i 2(m+l)−1 v2 j−m,2 j+1−l ,
J vm,l = i −2(m+l)+1 v2 j−m,2 j−1−l . j↓
j↓
In the paper we use the following notation: – B the ∗ -subalgebra of B(H) generated by the operators in δ k (π(A)) for all k ∈ N, – 00 (A) the algebra generated by δ k (π(A)) and δ k ([D, π(A)]) for all k ∈ N, – X the ∗ -subalgebra of B(H) algebraically generated by the set { a± , b± }.
Spectral Action on SUq (2)
117
Note that 00 (A) is a subalgebra of 0 (A) (the space of pseudodifferential operators of order less or equal to zero). The Hopf map r . For the explicit calculations of residues, we need a ∗ -homomorphism r : X → π+ (A) ⊗ π− (A) defined by the tensor product in the sense of Hopf algebras of representations π+ and π− : r (a+ ) := π+ (a) ⊗ π− (a), r (a− ) := −q π+ (b) ⊗ π− (b∗ ), r (b+ ) := −π+ (a) ⊗ π− (b), r (b− ) := −π+ (b) ⊗ π− (a ∗ ).
(29)
In fact, A is a Hopf ∗ -algebra under the coproduct (a) := (b) := a ⊗ b + b ⊗ a ∗ . These homomorphisms appeared in
a ⊗ a − q b ⊗ b∗ , [38] with the trans a b is the canonical generator of lation α ↔ a ∗ , γ ↔ −b. In particular, if U := −qb ∗ a∗ ˙ U, where ⊗ ˙ means the matrix product of tensors the K 1 (A)-group (a, b) = (a, b)⊗ of components. The grading. A Z-grading on the algebra X is naturally determined by the shift in j, j → j ± in formulae (12), (13), thus a+ , b+ , a− ∗ , b− ∗ are of degree 1 and a− , b− , a+ ∗ , b+ ∗ of degree −1. Any operator T ∈ X can be (uniquely) decomposed as T = j∈J ⊂Z T j , where T j is homogeneous of degree j. For T ∈ X , T ◦ will denote the 0-degree part of T for this grading and by a slight abuse of notations, we write r (T )◦ instead of r (T ◦ ). The symbol map. We also use the ∗ -homomorphism σ : π± (A) → C ∞ (S 1 ) defined for z ∈ S 1 on the generators by σ (π± (a)) (z) := z, σ π± (a ∗ ) (z) := z¯ , σ (π± (b)) (z) = σ π± (b∗ ) (z) := 0. The application (σ ⊗ σ ) ◦ r is defined on X (and so on B) with values in C ∞ (S 1 ) ⊗ C ∞ (S 1 ). We define dT := [D, T ] and δ(T ) := [|D|, T ]. Lemma 3.2. a± , b± are bounded operators on H such that for all p ∈ N, (i) δ(a± ) = ±a± , δ(b± ) = ±b± , (ii) δ p (π (a)) = a+ + (−1) p a− , δ p (π(b)) = b+ + (−1) p b− , p p p p (iii) δ(a± ) = ± p a± , δ(b± ) = ± p b± . Proof. (i) By definition, a± | jµn
= α0± β±0 | j ± µ+ n +
, where the numbers α± and β± depend on j, µ, n and q, so we get by (25), (d )α 0 (d + )α 0 δ(a± )| jµn
= 0j +± d± β | j ± µ+ n +
− 0j ±d β | j ± µ+ n +
± j ± ± j ±α± 0 ± + + = 0 ±β± | j µ n
= ±a± | jµn
, and similar proofs for b± . (ii) and (iii) are straightforward consequences of (i) and definition of π.
Remark 3.3. By Lemma 3.2, we see that, modulo O P −∞ , X is equal to B and in particular contains π(A). Using (27), we get that B ⊂ 00 (A) ⊂ algebra generated by B and BF. Note that, despite the last inclusion, F is not a priori in 00 (A).
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integrals. Recall that for any pseudodifferential operator T , 3.2. The noncommutative T (s), where ζ T (s) := Tr(T |D|−s ). T := Res ζD D s=0
Theorem 3.4. The dimension spectrum (without reality structure given by J ) of the spectral triple A(SUq (2)), H, D is simple and equal to {1, 2, 3}. Moreover, the corresponding residues for T ∈ B are − T |D|−3 = 2(τ1 ⊗ τ1 ) r (T )◦ , − T |D|−2 = 2 (τ1 ⊗ τ0 + τ0 ⊗ τ1 ) r (T )◦ , − T |D|−1 = 2 τ0 ⊗ τ0 − 21 τ1 ⊗ τ1 r (T )◦ , − F T |D|−3 = 0, − F T |D|−2 = 0, − F T |D|−1 = (τ0 ⊗ τ1 − τ1 ⊗ τ0 ) r (T )◦ , where the functionals τ0 , τ1 are defined for x ∈ π± (A) by τ0 (x) := lim (Tr N x − (N + 1) τ1 (x)),
τ1 (x) :=
N →∞
with Tr N x =
1 2π
N
2π
σ (x)(eiθ ) dθ,
0
n=0 εn , x εn .
Proof. Consequence of [37, Theorem 4.1 and (4.3)].
Remark 3.5. Since F is not in B, the equations of Theorem 3.4 are not valid for all T ∈ 00 (A). Nevertheless when T ∈ 00 (A), T |D|−k = 0 for k ∈ / { 1, 2, 3 } since the dimension spectrum is { 1, 2, 3 } [37]. Compared to [37] where we had ↑
τ0 (x) := lim Tr N x − (N + 23 ) τ1 (x), N →∞
↓
τ0 (x) := lim Tr N x − (N + 21 ) τ1 (x), N →∞
we replaced them with τ0 : ↑
τ0 = τ0 − 21 τ1 ,
↓
τ0 = τ0 + 21 τ1 .
1 Note that τ1 is a trace on π± (A) such that τ1 (1) = 1 and τ1 (π+ (aa ∗ )) = 2π while τ0 is not since τ0 (1) = 0 and ∞ 1 τ0 π± (aa ∗ ) = lim (1 − q 2n ) − (N + 1) = − 1−q 2, N →∞
2π 0
1 dθ = 1,
(30)
n=0
so, because of the shift, the replacement a ↔ a ∗ gives τ0 π± (a ∗ a) = q 2 τ0 π± (aa ∗ ) .
(31)
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119
3.3. The tadpole. Lemma 3.6. For SUq (2), the condition of the vanishing tadpole (see [15]) is not satisfied. 2 Proof. For example, an explicit calculation gives π(b)[D, π(b∗ )]D−1 = 1−q 2 : Let x, y ∈ π(A). Since [F, x] = 0, we have − x[D, y]D−1 = − xδ(y)|D|−1 = τ (r (xδ(y))0 ), where τ := 2 τ0 ⊗ τ0 − 21 τ1 ⊗ τ1 . By Lemma 3.2, π (b)δ π (b∗ ) =(b+ +b− ) ((b− )∗ −(b+ )∗ ) = − b+ b+ ∗ +b− b− ∗ + b+ b− ∗ − b− b+ ∗ . Since only the first two terms have degree 0, we get, using the formulae from Theorem 3.4, τ r (−b+ b+ ∗ ) = −τ π+ (aa ∗ ) ⊗ π− (bb∗ ) = −2τ0 π+ (aa ∗ ) τ0 π− (bb∗ ) + 21 τ1 π+ (aa ∗ ) τ1 π− (bb∗ ) and τ1 (π− (bb∗ )) = 0. Similarly, using (31), τ r (b− b− ∗ ) = 2τ0 π+ (bb∗ ) τ0 π− (a ∗ a) = 2q 2 τ0 π− (aa ∗ ) τ0 π+ (bb∗ . 2n = 1 and (30), Since τ0 (π± (bb∗ )) = Tr (π± (bb∗ )) = ∞ n=0 q 1−q 2 2 −1 1 1 1 − π(b)[D, π(b∗ )]D−1 = 2 1−q 2 1−q 2 + 2q 1−q 2 1−q 2 =
2 . 1−q 2
Remark 3.7. Other examples, with the shortcut notation x instead of π (x): (τ1 ⊗ τ1 ) r (aδ(a ∗ )◦ ) = −1, (τ1 ⊗ τ1 )r (a ∗ δ(a)◦ ) = 1, 2 (τ0 ⊗ τ0 ) r (aδ(a ∗ )◦ ) = q 21−1 , (τ0 ⊗ τ0 ) r (a ∗ δ(a)◦ ) = q 2q−1 , ∗ 2 3q 2 +1 aδ(a ∗ )|D|−1 = 2(qq 2+3 , a δ(a)|D|−1 = 2(q 2 −1) , −1) ∗ ∗ −1 −1 = 0, bδ(b)|D| b∗ δ(b )|D|−1 = 0, bδ(b∗ )|D|−1 = q−2 b δ(b)|D| = q−2 2 −1 , 2 −1 .
In particular, N φ1 does not vanish on 1-forms since N φ1 ada ∗ = N φ1 (a, a ∗ ) = −1. The pairing of the tadpole cyclic cocycle φ1 with the generator of K 1-group is a b nontrivial: Let U be the canonical generator of the K 1 (A)-group, U = −qb ∗ a∗ acting on H ⊗ C2 . Then for AU := 2k,l=1 π(Ukl ) dπ(U ∗ kl ), using the above remark, 1 φ1 AU = −2 as obtained in [37, p. 391]: in fact, with P := 2 (1 + F), ∗ ∗ −1 ψ1 (U, U ) := 2 − Ukl δ(Ukl )P|D| − − Ukl δ 2 (Ukl∗ )P|D|−2 k,l
+ − Ukl δ 3 (Ukl∗ )P|D|−3 2 3
satisfies ψ1 (U, U ∗ ) = 2
k,l
Ukl δ(Ukl∗ )P|D|−1 =
φ1
AU .
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4. Reality Operator and Spectral Action on SUq (2) 4.1. Spectral action in dimension 3 with [F, A] ∈ O P −∞ . Let (A, H, D) be a real −1 spectral triple of dimension 3. Assume that [F, A] ∈ O P −∞ , where F := D|D| (we suppose D invertible). Let A be a selfadjoint one form, so A is of the form A = i ai dbi , where ai , bi ∈ A. Thus, A AF mod O P −∞ , where A := i ai δ(bi ) is the δ-one-form associated to A. Note that A and F commute modulo O P −∞ . We define DA := DA + PA , PA the projection on Ker DA , DA := D + A, A := A + J AJ −1 . Theorem 4.1. The coefficients of the full spectral action (with reality operator) on any real spectral triple (A, H, D) of dimension 3 such that [F, A] ∈ O P −∞ are (i) |DA |−3 = |D|−3 , (ii) |DA |−2 = |D|−2 − 4 A|D|−3 , (iii) |DA |−1 = |D|−1− 2 A|D|−2 + 2 A2 |D|−3 + 2 AJ A J −1 |D|−3 , (iv) ζ DA (0) = ζ D (0)−2 A|D|−1 + A(A+ J A J −1 )|D|−2 + δ(A)(A+ J A J −1 )|D|−3 − 23 A3 |D|−3 − 2 A2 J A J −1 |D|−3 . Proof. (i) We apply [22, Prop. 4.9]. (ii) By [22, Lemma 4.10 (i)], we have |DA |−2 = |D|−2 − ( AD + D A+ A2 )|D|−4 . 2 −4 By the trace property of thenoncommutative integral and the fact that A |D| is traceAD|D|−4 = |D|−2 − 4 AD|D|−4 . Since class, we get |DA |−2 = |D|−2 − 2 AD ∼ A|D| mod O P −∞ , we get the result. (iii) By [22, Lemma 4.10 (ii)], we have − |DA |−1 = − |D|−1 − 21 −( AD + D A+ A2 )|D|−3 + 38 −( AD + D A+ A2 )2 |D|−5 . Following arguments of (ii), we get −( AD + D A+ A2 )|D|−3 = 4 − A|D|−2 + 2 − A2 |D|−3 + 2 − A J A J −1 |D|−3 , 2 2 −5 2 −3 −(AD + DA + A ) |D| = 8 − A |D| + 8 − A J A J −1 |D|−3 , and the result follows. j (iv) By [22, Lemma 4.5] gives ζ DA (0) = 3j=1 (−1) ( AD−1 ) j . j AD−1 = 2 A|D|−1 and we have Moreover, −1 2 −1 −1 −1 . Since δ(A) ∈ O P 0 , we can (AD ) = 2 (A|D|−1)2 + 2 A|D| JAJ |D| −1 2 2 −2 −3 (A|D| ) = A |D| + δ(A)A|D| and, with the same argument, that check that A|D|−1 J A J −1 |D|−1 = A J A J −1 |D|−2 + δ(A)J A J −1 |D|−3 . Thus, we get −( AD−1 )2 = 2 − A(A + J A J −1 )|D|−2 + 2 − δ(A)(A + J A J −1 )|D|−3 .
(32)
Spectral Action on SUq (2)
121
The third term to be computed is −( AD−1 )3 = 2 −(A|D|−1 )3 + 4 −(A|D|−1 )2 J A J −1 |D|−1 + 2 − A|D|−1 J A J −1 |D|−1 A|D|−1 . Any operator in O P −4 being trace-class here, we get −1 3 3 −3 2 −1 −3 −(AD ) = 2 − A |D| + 4 − A J A J |D| + 2 − A J A J −1 A|D|−3 . (33) Since A J A J −1 A|D|−3 = A2 J A J −1 |D|−3 by trace property and the fact that δ(A) ∈ O P 0 , the result follows then from (32) and (33). Corollary 4.2. For the spectral action of A without the reality operator (i.e. DA = D + A), we get − |DA |−2 = − |D|−2 − 2 − A|D|−3 , −1 −1 −2 − |DA | = − |D| − − A|D| + − A2 |D|−3 , ζ DA (0) = ζD (0) − − A|D|−1 + 21 − A2 |D|−2 + 21 − δ(A)A|D|−3 − 13 − A3 |D|−3 .
4.2. Spectral action on SUq (2): main result. On SUq (2), since F commutes with a± and b± , Theorem 4.1 can be used for the spectral action computation. Here is the main result of this section: Theorem 4.3. In the full spectral action (4) (with the reality operator) of SUq (2) for a one-form A and A its associated δ-one-form, the coefficients are: − |DA |−3 = 2, − |DA |−2 = −4 − A|D|−3 ,
2 −1 2 −3 −2 1 + − A|D|−3 , − |DA | = − 2 + 2 − A |D| − − A|D| −1 2 −2 2 ζ DA (0) = −2 − A|D| + − A |D| − 3 − A3 |D|−3
−2 2 −3 −3 1 1 −3 + 2 − A|D| − A|D|−2 . + − A|D| − − A |D| 2 − A|D| In order to prove this theorem (in Sect. 4.7), we will use a decomposition of one-forms in the Poincaré-Birkhoff–Witt basis of A with an extension of previous representations to operators like T J T J −1 , where T and T are in X .
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4.3. Balanced components and Poincaré–Birkhoff–Witt basis of A. Our objective is to compute all integrals containing A and express the results using functions of A, which capture certain symmetries of this object. For convenience, let us first introduce these functions. Let A = i π(xi )dπ(y i ) on SUq (2) be one-form and A the associated δ-one-form. The xi and y i are in A and as such they can be uniquely written as finite sums xi = α xiα m α and y i = β yβi m β , where m α := a α1 bα2 b∗ α3 is the canonical monomial of A with α, β ∈ Z × N × N based on a fixed Poincaré–Birkhoff–Witt type basis of A. Remark 4.4. Any one-form A = i π(xi )dπ(y i ) on SUq (2) is characterized by a com β plex valued matrix Aα = i xiα yβi , where α, β ∈ Z × N × N. This matrix is such that A = Aβα Mβα ,
where Mβα := π(m α )δ π(m β ) . In the following, we denote A¯ := A¯ βα Mβα ¯ − p = A|D|− p . so for any p ∈ N, A|D| This presentation of one-forms is not unique modulo O P −∞ since, as we will see in Sect. 5 that, F = i xi dyi , where xi , yi ∈ A, thus for any generator z we get, [F, z] = i xi d(yi z) − xi yi dz − zxi dyi = 0 mod O P −∞ . We do not know, however, if this presentation is unique when the O P −∞ part is taken into account. The δ-one-forms Mβα are said to be canonical. Any product of n canonical δ-one forms, where n ∈ N∗ , is called a canonical δ n -one-form. Thus, if A is a δ-one-form, β¯ An = (An )α¯ Mβα¯¯ , where α¯ = (α, α , . . . , α (n−1) ), β¯ = (β, β , . . . , β (n−1) ) are in β¯
β
β (n−1)
Zn × Nn × Nn , (An )α¯ := Aα · · · Aα (n−1) by Mβα¯¯ is the canonical δ n -one form equal (n−1)
to Mβα · · · Mβα(n−1) .
Definition 4.5. A canonical δ n -one-form is a-balanced if it is of the form (n−1) (n−1) a α1 δ(a β1 ) · · · a α1 δ a β1 , where
n−1 i=0
(i)
(i)
α1 + β1 = 0. For any δ-one-form A, the a-balanced components of An β¯
are denoted Ba (An )α¯ . Note that β¯
β 00
1 Ba (A)α¯ = A−β δ δ . 1 00 α1 +β1 ,0 α2 +α3 +β2 +β3 ,0
Definition 4.6. A canonical δ n -one-form is balanced if it is of the form (n−1)
(n−1)
m α δ(m β ) · · · m α δ(m β ), n−1 (i) (i) n−1 (i) (i) n−1 (i) (i) where i=0 α1 +β1 = 0 and i=0 α2 +β2 = i=0 α3 +β3 . For any δ-one-form A, β¯
the balanced components of An are denoted B(An )α¯ .
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Note that β¯
β β β
1 2 3 B(A)α¯ = A−β δ δ . 1 α2 α3 α1 +β1 ,0 α2 +β2 ,α3 +β3
As we will show, a contribution to the k th -coefficient in the spectral action, is only brought by one-forms A such that Ak is balanced (and even a-balanced in the case k = 1). Note also that if A is balanced, then Ak for k ≥ 1 is also balanced, whereas the converse is false. 4.4. The reality operator J on SUq (2). Let for any n, p ∈ N, q−n := 0, qn := 1 − q 2n , ↑ ↓ qn, p := qn+1 · · · qn+ p , qn, p := qn · · · qn−( p−1) , ↑
↓
with the convention qn,0 = qn,0 := 1. Thus, we have the relations ↑ π± (a p ) εn = qn, p εn+ p ,
↓ π± (a ∗ p ) εn = qn, p εn− p ,
π± (b p ) εn = (±q n ) p εn , π± (b∗ p )εn = (±q n ) p εn , where εk := 0 if k < 0. The sign of x ∈ R is denoted ηx . By convention, a j := a, a±, j := a± if j ≥ 0 and ∗ if j < 0. Note that, with convention a j := a ∗ , a±, j := a± ↑α
↑α
↑
↑ ↓ 1 0 qn, p1 := qn, p if α1 > 0, qn, p := qn, p if α1 < 0, and qn, p := 1, ↑α
p
we have for any α1 ∈ Z and p ≤ α1 , π± (aα1 ) εn = qn, p1 εn+ηα1 p . Recall that the reality operator J is defined by j↑
j↑
J vm,l = i 2(m+l)−1 v2 j−m,2 j+1−l ,
J vm,l = i −2(m+l)+1 v2 j−m,2 j−1−l , j↓
j↓
thus the real conjugate operators a± := J a± J −1 , b± := J b± J −1 satisfy j+ q 2 j+2−l 0 j − j j vm−1,l−1 , a− vm,l := −q 2 j−m a+ vm,l := −q2 j+1−m q20j+2−l q 0 vm,l , 0 q 2 j−l 2 j−l q 2 j+1−l 0 j + j− j j b+ vm,l := q2 j+1−m 0 q 2 j−1−l vm,l+1 , b− vm,l := −q 2 j−m q2 0j+1−lq 0 vm−1,l . 2 j−1−l
Therefore, the real conjugate operator behaves differently on the up and down part of the Hilbert space. The difference comes from the fact that the index l is not treated uniformly by J on up and down parts. We denote by X the algebra generated by { a± , b± }, X the algebra generated by 2 2 { a± , b± , a± , b± } and H := (N) ⊗ (Z) and we construct two ∗-representations π± of A: The representation π+ gives bounded operators on H while π− represents A into B(H ⊗ C2 ).
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The representation π+ is defined on the generators by: π+ (a) εm ⊗ ε2 j := q2 j+1−m εm ⊗ ε2 j+1 ,
π+ (b) εm ⊗ ε2 j := −q 2 j−m εm+1 ⊗ ε2 j+1
while π− is defined by: π− (a) εl ⊗ ε2 j ⊗ ε↑↓ := −q2 j+1±1−l εl ⊗ ε2 j+1 ⊗ ε↑↓ , π− (b) εl ⊗ ε2 j ⊗ ε↑↓ := −q 2 j±1−l εl+1 ⊗ ε2 j+1 ⊗ ε↑↓ , where ε↑↓ is the canonical basis of C2 and the + in ± corresponds to ↑ in ↑↓. The link between π± and π± , which explains the notations about these intermediate objects and the fact that π± are representations on different Hilbert spaces, is in the parallel between Eqs. (29), (34) and (35). Let us give immediately a few properties (xβ equals x if the sign β is positive and equals x∗ otherwise) ↑
β π+ (aβ ) p εm ⊗ ε2 j = q2 j−m, p εm ⊗ ε2 j+ηβ p ,
↑
β π− (aβ ) p εl ⊗ ε2 j ⊗ ε↑↓ = (−1) p q2 j±1−l, p εl ⊗ ε2 j+ηβ p ⊗ ε↑↓ ,
π+ (bβ ) p εm ⊗ ε2 j = (−1) p q (2 j−m) p εm+ηβ p ⊗ ε2 j+ηβ p , π− (bβ ) p εl ⊗ ε2 j ⊗ ε↑↓ = (−1) p q (2 j±1−l) p εl+ηβ p ⊗ ε2 j+ηβ p ⊗ ε↑↓ . Note that the π± representations still contain the shift information, contrary to representations π± . Moreover, π± (b) = π± (b∗ ) while π± (b) = π± (b∗ ). The operators a± , b± are coded on H ⊗ H ⊗ C2 as the correspondence a+ ←→ π+ (a) ⊗ π− (a), a− ←→ −q π+ (b∗ ) ⊗ π− (b∗ ), π+ (a) ⊗ π− (b), b− ←→ − π+ (b∗ ) ⊗ π− (a ∗ ). b+ ←→ −
(34)
We now set the following extension to B(H ) of π+ and to B(H ⊗ C2 ) of π− by π+ (a) := π+ (a) ⊗ V, π+ (b) := π+ (b) ⊗ V (V is the shift of 2 (Z)), π− (a) := π− (a) ⊗ V ⊗ 12 , π− (b) := π− (b) ⊗ V ⊗ 12 . So, we can define a canonical algebra morphism ρ from X into the bounded operators 2 on H ⊗ H ⊗ C . This morphism is defined on the generators part { a± , b± } of X by the preceding correspondence and on the generators part { a± , b± } by (compare (29)): a+ ←→ π+ (a) ⊗ π− (a),
a− ←→ −q π+ (b∗ ) ⊗ π− (b∗ ),
b+ ←→ −π+ (a) ⊗ π− (b), b− ←→ −π+ (b∗ ) ⊗ π− (a ∗ ).
(35)
We denote by S the canonical surjection from H ⊗ H ⊗ C2 onto H. This surjection is associated to the parameters restrictions on m, j, l, j . In particular, the index j associated to the second 2 (N) in H ⊗ H ⊗ C2 is set to be equal to j. Any vector in H ⊗ H ⊗ C2 not satisfying these restrictions is sent to 0 in H. Denote by I the canonical injection of H into H ⊗ H ⊗ C2 (the index j is doubled). T functions, we Thus, S ρ (·)I is the identity on X . In the computation of residues of ζD can therefore replace the operator T by S ρ (T )I .
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We now extend τ0 on π± (A) π± (A): For x, y ∈ A, we set N Tr N π+ (x) π+ (y) := εm ⊗ ε N , π+ (x) π+ (y) εm ⊗ ε N , m=0 N ↑ Tr N π− (x) π− (y) := εl ⊗ ε N −1 ⊗ ε↑ , π− (x) π− (y) εl ⊗ ε N −1 ⊗ ε↑ , l=0 N ↓ Tr N π− (x) π− (y) := εl ⊗ ε N +1 ⊗ ε↓ , π− (x) π− (y) εl ⊗ ε N +1 ⊗ ε↓ . l=0
Actually, a computation on monomials of A shows that ↓ ↑ Tr N π− (x) π− (y) = Tr N π− (x) π− (y) . π− (y) . For convenience, we shall denote this functional by Tr N π− (x) Lemma 4.7. Let x, y ∈ A. Then, π± (y) := lim N →∞ U N exists where (i) τ0 π± (x) π± (y) − (N + 1)τ1 (π± (x)) τ1 (π± (y)) . U N := Tr N π± (x) (ii) U N = τ0 π± (x) π± (y) + O(N −k ) for all k > 0. Proof. (i) We can suppose that x and y are monomials, since the result will follow by linearity. We will give a proof for the case of the π+ representations, the case π− being similar, with minor changes. We have π+ (y) = ( π+ aβ1 )|β1 | ( π+ b)β2 ( π+ b∗ )β3 . A computation gives ↑β
1 ε ⊗ ε2 j−β3 +β2 +β1 π+ (y) εm ⊗ ε2 j = (−1)β2 +β3 q (2 j−m)(β2 +β3 ) q2 j−m,|β 1 | m−β3 +β2
and with the notation t2 j,m := εm ⊗ε2 j , π± (x) π± (y) εm ⊗ε2 j and T2 j := we get ↑β
2 j
m=0 t2 j,m ,
↑α
1 q 1 q (m+β2 −β3 )(α2 +α3 ) δα1 +β2 −β3 ,0 t2 j,m = (−1)β2 +β3 q (2 j−m)(β2 +β3 ) q2 j−m,|β 1 | m−β3 +β2 ,|α1 |
×δ−α3 +α2 +β1 ,0
↑β
↑α
1 = (−1)α1 q (2 j−m)(β2+β3 )+(m−α1 )(α2+α3 ) q2 j−m,|β q 1 δ δ 1 | m−α1 ,|α1 | α1 +β2 −β3 ,0 α2 −α3 +β1 ,0
=: f α,β q 2 jλ t2 j,m =: f α,β q 2 jκ t2j,2 j−m , where ↑β
↑α
1 q 1 , t2 j,m := q m(κ−λ) q2 j−m,|β 1 | m−α1 ,|α1 |
↑β
↑α
1 q 1 , t2j,m := q m(λ−κ) qm,|β 1 | 2 j−m−α1 ,|α1 |
(36) (37)
with λ := β2 + β3 ≥ 0 and κ := α2 + α3 ≥ 0. We will now prove that if λ = κ, then 2 j (T2 j ) is a convergent sequence. Suppose κ > λ. Let us note U2 j := m=0 t2 j,m . Since
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the t2 j,m are positive and t2 j+1,m ≥ t2 j,m for all j, m, U2 j is an increasing real sequence. The estimate U2 j
≤
2j
q m(κ−λ) ≤
1 1−q κ−λ
<∞
m=0
proves then that U2 j is a convergent sequence. With T2 j = f α,β q 2 jλ U2 j , we obtain 2 j our result. Suppose now that λ > κ. Let us note U2j := m=0 t2j,m . Since the t2j,m are positive and t2j+1,m ≥ t2j,m for all j, m, U2j is an increasing real sequence. The estimate U2j ≤
2j
q m(λ−κ) ≤
1 1−q λ−κ
<∞
m=0
proves then that U2j is a convergent sequence. With T2 j = f α,β q 2 jκ U2j , we have again our result. Moreover, note that if λ and κ are both different from zero, the limit of (T2 j ) is zero and more precisely, T2 j = O(q 2 jλ ) if κ > λ > 0,
(38)
T2 j = O(q
(39)
2 jκ
) if λ > κ > 0.
Suppose now that λ = κ = 0. In that case, (T2 j ) also converges rapidly to zero. 2 j Indeed, let us fix q < ε < 1. We have ε−2 jλ T2 j = m=0 cm d2 j−m = c ∗ d(2 j), ↑ ↑ β1 α 1 . Since both m cm where cm := f α,β (q/ε)λm qm−α1 ,|α1 | and dm := (q/ε)λm qm,|β 1| −2 jλ and m dm are absolutely convergent series, their Cauchy product 2 j ε T2 j is −2 jλ T2 j = 0, and convergent. In particular, lim j→∞ ε T2 j = O(ε2 jλ ).
(40)
Finally, T2 j has a finite limit in all cases except possibly when λ = κ = 0, which is the case when α1 = α2 = α3 = β1 = β2 = β3 = 0. In that case, t2 j,m = 1. A straightforward computation gives τ1 (π± (x)) τ1 (π± (y)) =δα1 ,0 δβ1 ,0 δα2 ,0 δβ2 ,0 δα3 ,0 δβ3 ,0 . Thus, U2 j = T2 j − (2 j + 1)δα1 ,0 δβ1 ,0 δα2 ,0 δβ2 ,0 δα3 ,0 δβ3 ,0 has always a finite limit when j → ∞. (ii) The result is clear if λ = κ = 0 (in that case U N = τ0 = 0). Finally, suppose that neither λ nor κ are zero. In that case U2 j = T2 j . By (39), (38) and (40), we see that if λ > κ > 0 or κ > λ > 0 or κ = λ, (T2 j ) converges to 0 with a rate in O(ε2 jα ) where α > 0 and q ≤ ε < 1. Thus, it only remains to check the cases (κ > 0, λ = 0) and (κ = 0, λ > 0). In the first one, we get from (36), 2 j ↑ β1 . If β1 = 0, we are done. U2 j = f α,β m=0 q mκ q2 j−m,|β 1|
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∞ ↑ β1 r p 2| p|1 (2 j−m) , where Suppose β1 > 0. We have q2 j−m,|β = p=0 l p q q 1| 1 p = ( p1 , . . . , pβ1 ) and l p = (−1)| p|1 p2 , r p := 2 p1 + · · · + 2β1 pβ1 . Thus, cutting 2 j the sum in two, we get, noting L 2 j := f α,β m=0 q mκ , 4| p|1 j −q (2 j+1)κ−2| p|1 l p q r p q 1−q U2 j − L 2 j = f α,β κ−2| p|1 | p|1 >κ/2
+ f α,β
lp q
rp
q
4| p|1 j
0=| p|1 ≤κ/2
2j
q m(κ−2| p|1 ) .
m=0
2 j j
r p 4| p|1 m(κ−2| p|1 ) is in O 4j Since j→∞ ( jq ), we have, 0=| p|1 ≤κ/2 l p q q m=0 q modulo a rapidly decreasing sequence, 4| p|1 j −q (2 j+1)κ−2| p|1 l p q r p q 1−q =: f α,β q 2κ j V2 j U2 j − L 2 j ∼ f α,β κ−2| p|1 | p|1 >κ/2
with V2 j =
| p|1 >κ/2
l p qr p
1−q (2| p|1 −κ)(2 j+1) 1−q 2| p|1 −κ
=
2j
l p q r p q (2| p|1 −κ)m .
| p|1 >κ/2 m=0
The family vm, p := (l p q r p q (2| p|1 −κ)m )( p,m)∈I , where I = { ( p, m) ∈ Nβ1 × N : | p|1 > κ/2 } is (absolutely) summable. Indeed |vm, p | ≤ |l p |q r p q m , so |vm, p | is summable as the product of two summable families. As a consequence, lim j→∞ V2 j exists and is finite, which proves that (q 2κ j V2 j ), and thus (U2 j − L 2 j ) converge rapidly to 0. ↑β
↓
↑
1 =q2 j−m,|β1 | =q2 j−(m+|β1 |),|β1 | and by Suppose now that β1 < 0. In that case, q2 j−m,|β 1| 2 j 2 j+|β | ↑ ↑ mκ (36), we get U2 j = f α,β m=0 q q2 j−(m+|β1 |),|β1 | = f α,β q −|β1 |κ m=|β11| q mκ q2 j−m,|β1 | , so the same arguments as in case β1 > 0 apply here, the summation on m simply shifted of |β1 |. The same proof can be applied for the other case (κ = 0, λ > 0). This time, we only need to use (37) instead of (36) and the preceding arguments follow by replacing κ by λ and β1 by α1 .
Remark 4.8. Contrary to the preceding τ0 , the new functional contains the shift information. In particular, it filters the parts of nonzero degree. π+ (A) ⊗ π− (A) π− (A). If T ∈ X X, ρ (T ) ∈ π+ (A) For notational convenience, we define τ1 on π± (A) π± (A) as τ1 π± (x) π± (y) := τ1 (π± (x)) τ1 (π± (y)) . In the following, the symbol ∼e means equals modulo an entire function. Theorem 4.9. Let T ∈ X X . Then T (s) ∼ 2(τ ⊗ τ ) ( ρ (T )) ζ (s − 1) (i) ζD e 1 1 ρ (T )) ζ (s − 2) + 2(τ0 ⊗ τ1 + τ1 ⊗ τ0 ) ( + 2(τ0 ⊗ τ0 − 21 τ1 ⊗ τ1 ) ( ρ (T )) ζ (s), (ii) T |D|−3 = 2(τ1 ⊗ τ1 ) ( ρ (T )),
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(iii) T |D|−2 = 2(τ0 ⊗ τ1 + τ1 ⊗ τ0 ) ( ρ (T )), (iv) T |D|−1 = 2(τ0 ⊗ τ0 − 21 τ1 ⊗ τ1 ) ( ρ (T )). Proof. (i) Since T ∈ X X, ρ (T ) is a linear combination of terms like π+ (x) π+ (y) ⊗ π− (z) π− (t), where x, y, z, t ∈ A. Such a term is noted in the following T+ ⊗ T− . Linear combination of these terms is implicit. j,↓ With the shortcut Tc1 ,...,c p := εc1 ⊗ · · · ⊗ εc p , T εc1 ⊗ · · · ⊗ εc p , recalling that vm,l is 0 when j = 0, or l ≥ 2 j, we get j+1 2 j 2 ∞ v j,↑ −s 0 0 v j,↑ m,l d ζD (s) = , S ρ (T )I ρ (T )I v j,↓ d −s m,l j j + + v j,↓ , S 0 0 T
m,l
2 j=0 m=0 l=0
= =
j+1 2 j 2 ∞
ρ (T )m,2 j,l,2 j,↑ d −s j+ +
2 j=0 m=0 l=0 ∞
j−1 2 j 2 ∞
m,l
ρ (T )m,2 j,l,2 j,↓ d −s j
2 j=1 m=0 l=0
↑ ↓ Tr 2 j (T+ ) Tr 2 j+1 (T− ) + Tr 2 j+1 (T+ ) Tr 2 j (T− ) d −s j+ .
2 j=0
By Lemma 4.7 (ii), for all k > 0, Tr 2 j (T± ) = (2 j + 23 )τ1 (T± ) + τ0 (T± ) − 21 τ1 (T± ) + O (2 j)−k , Tr 2 j+1 (T± ) = (2 j + 23 )τ1 (T± ) + τ0 (T± ) + 21 τ1 (T± ) + O (2 j)−k . The result follows by noting that the difference of the Hurwitz zeta function ζ (s, 23 ) and Riemann zeta function ζ (s) is an entire function. (ii,iii,iv) are direct consequences of (i). 4.5. The smooth algebra C ∞ (SUq (2)). In [13,37], the smooth algebra C ∞ (SUq (2) is defined by pulling back the smooth structure C ∞ (Dq2± ) into the C ∗ -algebra generated by A, through the morphism ρ and the application λ (the compression, which gives an operator on H from an operator on l 2 (N) ⊗ l 2 (N) ⊗ l 2 (Z) ⊗ C2 ). The important point is that with [13, Lemma 2, p. 69], this algebra is stable by holomorphic calculus. By defining ρ := ρ ◦ c and λ(·) := S(·)I , the same lemma (with the same notation) can be applied to our setting, with c := π(x) → π (x) and C := C ∞ (Dq2+ ) ⊗ C ∞ (S 1 ) ⊗ C ∞ (Dq2+ ) ⊗ C ∞ (S 1 ) ⊗ M2 (C) as algebra stable by holomorphic calculus containing the image of ρ . Here, we use Schwartz sequences to define the smooth structures. We finally obtain C ∞ (SUq (2)) with real structure as a subalgebra stable by holomorphic calculus of the C ∗ -algebra generated by π(A) ∪ J π(A)J −1 and containing π(A) ∪ J π(A)J −1 . Corollary 4.10. The dimension spectrum of the real spectral triple C ∞ (SUq (2)), H, D is simple and given by {1, 2, 3}. Its KO-dimension is 3.
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Proof. Since F commutes with π (A), the pseudodifferential operators of order 0 (without the real structure and in the sense of [22]) are exactly (modulo O P −∞ ) the operators in B + B F. From Theorem 3.4 we see that the dimension spectrum of SUq (2) without taking into account the reality operator J is { 1, 2, 3 }, in other words, the possible poles b : s → Tr(bF ε |D|−s ) (with ε ∈ { 0, 1 }, b ∈ B) are in { 1, 2, 3 }. Theorem 4.9 (i) of ζD shows that the possible poles are still { 1, 2, 3 } when we take into account the real structure of SUq (2), that is to say, when B is enlarged to B J B J −1 . Indeed, any element of B J B J −1 is in X X and it is clear from the preceding proof that adding F in the previous zeta function does not add any pole to { 1, 2, 3 }. All arguments go true from the polynomial algebra A(SUq (2)) to the smooth pre-C∗ algebra C ∞ (SUq (2)). KO-dimension refers just to J 2 = −1 and D J = J D since there is no chirality because spectral dimension is 3. 4.6. Noncommutative integrals with reality operator and one-forms on SUq (2). The goal of this section is to obtain the following suppression of J : Theorem 4.11. Let A and B be δ-one-forms. Then (i) A J B J −1 |D|−3 = 21 A|D|−3 B|D|−3 , (ii) A J B J −1 |D|−2 = 21 A|D|−2 B|D|−3 + 21 A|D|−3 B|D|−2 , 2 (iii) A J B J −1 |D|−3 = 21 A2 |D|−3 B|D|−3 , (iv) δ(A)A|D|−3 = δ(A)J A J −1 |D|−3 = 0. We gather at the beginning of this section the main notations for the technical lemmas that will follow. For any pair (k, p) ∈ N3 ×N3 such that ki ≤ |αi |, pi ≤ |βi |, where α, β ∈ Z×N×N, we define vk, p := g( p) |αk11 | 2ηα1 αk22 αk33 |βp11 | 2ηβ1 βp22 βp33 (−1)k1 + p1 +α2 +α3 +β2 +β3 q σk, p , q
q
h k, p := α1 + α2 − α3 − 2(ηα1 k1 + k2 − k3 ) + g( p), g( p) := β1 + β2 − β3 − 2(ηβ1 p1 + p2 − p3 ), t u σk, p := k1 + p1 + σk, p + σk, p , t σk, p1 |k|1 + p2 (|k|1 + p1 ) − p3 (|k|1 + p1 + p2 ), p := k1 k2 − k3 (k1 + k2 ) + ηβ1 u σk, p1 − p2 + p3 )(k1 + k2 + k3 ) − k2 (k1 + k 2 ) + ( p1 + p2 )(− p2 + p3 ) p := (k3 + ηβ1 + p3 p3 ,
p
p2 ∗ a p3 b|k|1 +| p|1 , tk, p = aαk11 a k2 a ∗ k3 aβ11 a
p
p |1 , u k, p = aαk11 a ∗ k2 a k3 aβ11 a ∗ p2 a p3 b|k|1 +|
where we used the notation ki := |αi | − ki , pi := |βi | − pi , so 0 ≤ ki ≤ |αi |, 0 ≤ pi ≤ |βi |. We will also use the shortcut k := (k1 , k2 , k3 ).
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For β1 ∈ Z and j ∈ N, we define ∞ ↑β w1 (β1 , j) := q 2 jn (qn,|β1 1 | )2 − δ j,0 , n=0
wβα := 2β1 q β1 (2α3 +β3 −β2 ) w1 (β1 , α3 + β3 ). We introduce the following notations: ↑α
↑ ↓ qn−k +η qn+η k −η , k p + p − p , k p2 − p3 , k3 α β β1 p1 + 1 1 3 1 2 3 2 k, p 1 1
+ n(|k|1 +| p|1 ) qk, qn+r1+ p,n := q ↑β
↑
↓
1 qn+ p2 − p3 , p1 qn− p2 qn, p3 , p3 ,
↑α
− n(|k|1 +| p |1 ) qk, qn+r1− p,n := q
↓ ↑ p1 − p2 + p3 ,k2 qn+ηβ1 p1 − p2 + p3 ,k3 qn+k3 +ηβ1
k, p −ηα1 k1 ,k1
↑β
↓
↑ |k|1 +| p |1 ×qn−1p2 + p3 , , p1 qn+ p3 , p2 qn, p3 (−1)
+ p1 + p2 − p3 , rk, p := ηα1 k1 + k2 − k3 + ηβ1 − p1 − p2 + p3 . r := ηα1 k1 − k2 + k3 + ηβ1 k, p
− + + Thus, π+ (tk, p )εn = qk, p,n εn+rk, p and π− (u k, p )εn = qk, p,n εn+r − . k, p
Lemma 4.12. We have r (Mβα )◦ = δh k, p ,0 vk, p π+ (tk, p ) ⊗ π− (u k, p ), k, p
where the summation is done on ki , pi in N such that ki ≤ |αi |, pi ≤ |βi | for i ∈ { 1, 2, 3 }. α2 α3 ∗ )α3, with v := |α1 | Proof. Since π (m α ) = (a+ + a− )α1 (b+ + b− )α2 (b+∗ + b− k 2ηα1 k k1 k3 , 2 q
π (m α ) =
|α |−k1
k
vk ck where ck := a+,α1 1
k1 k2 ∗ α3 −k3 ∗ k3 a−,α bα2 −k2 b− b+ b− . 1 +
By Lemma 3.2 (iii) we see that δ(π (m β )) = p w p d p where we introduce |β |− p p1 β −p p ∗ p3 . b 2 2 b−2 b+∗ β3 − p3 b− w p := |βp11| 2ηβ1 βp22 βp33 and d p := g( p) a+,β1 1 1 a−,β 1 + q As a consequence, (Mβα )◦ = k, p δh(k, p),0 g( p) vk w p ck, p , where
p
p
k1 k2 ∗ k3 ∗ k3 p3 ∗ p 3 1 a k1 bk2 b− b+ b− a+,β a 1 b 2 b−2 b+∗ b− . ck, p = a+,α 1 −,α1 + 1 −,β1 + p
p
(41)
) ⊗ π (u ), where With (41), we get r (ck, p ) = (−1)k1 + p1 +α2 +α3 +β2 +β3 q k1 + p1 π+ (tk, − k, p p
p
p
p3 p3 k 1 k 1 k 2 k 2 ∗ k 3 k 3 1 p1 b aβ1 b a p2 b p2 a ∗ b , tk, p = aα1 b a b a p2 ∗ p2 a b p3 a p3 . u k, p = aαk11 bk1 bk2 a ∗ k2 bk3 a k3 aβ11 b p1 b
A recursive use of relation ba j = q η j a j b yields the result.
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Lemma 4.13. We have (i) (τ1 ⊗ τ1 ) r (Mβα )◦ = β1 δα1 ,−β1 δα2 ,0 δα3 ,0 δβ2 ,0 δβ3 ,0 . (ii) (τ1 ⊗ τ0 + τ0 ⊗ τ1 ) r (Mβα )◦ = δα1 ,−β1 δα2 +β2 ,α3 +β3 wβα . In particular, if A is a δ-one-form, we have β1 00 − A|D|−3 = 2β1 A−β , 1 00 − A|D|−2 = 2wβα B(A)βα , where we implicitly summed on all α, β indices. Proof. See Appendix A.
With notations of Lemma 4.12, it is direct to check that for given α¯ = (α, α , . . . , α (n−1) ) and β¯ = (β, β , . . . , β (n−1) ), δh K ,P ,0 v K ,P π+ (t K ,P ) ⊗ π− (u K ,P ), (42) r (Mβα¯¯ )◦ = K ,P
( j)
where K = (k, k , . . . k (n−1) ), P = ( p, p , . . . , p (n−1) ) with 0 ≤ ki ( j) ( j) 0 ≤ pi ≤ |βi |, t K ,P := tk, p tk , p · · · tk (n−1) , p(n−1) , v K ,P := vk, p vk , p · · · vk (n−1) , p(n−1) ,
( j)
≤ |αi |,
u K ,P := u k, p u k , p · · · u k (n−1) , p(n−1) , h K ,P := h k, p + h k , p + · · · h k (n−1) , p(n−1) . (n−1)
In the following, we will use the shortcuts Ai := αi + αi + · · · + αi , (n−1) ± ± ± . In the case n = 2, we also note r K ,P := rk, p + rk , p Bi := βi + βi + · · · + βi ± ± ± and q K ,P,n := qk , p ,n qk, p,n+r ± . Thus, we have π+ (t K ,P ) εm = q K+ ,P,m εm+r K+ ,P and π− (u K ,P )εm =
k , p
q K−,P,n εm+r − . K ,P
vβ1 ,α1 ,β1 (l, j) :=
We also introduce, still for n = 2,
∞ ↑−β −α −β ↑α ↑β ↑β q l+2n j qn+β 1+α1 +β1 ,|β +α +β | qn+β1 +α ,|β | qn+β1 ,|α | qn,|β1 | −δ j,0 , n=0
1
1
1
1
1
1
Vβα¯¯ := 2[β1 β1 + (β2 − β3 )(β2 − β3 )] q
1
1
1
1
1
1
2β1 (α2 +α3 )+2β1 (α2 +α3 )
×vβ1 ,α1 ,β1 ((α2 + β2 + α3 + β3 )(α1 + β1 ), A3 + B3 ).
Lemma 4.14. We have (i) (τ1 ⊗ τ1 ) r (Mβα Mβα )◦ = β1 β1 δ A1 ,−B1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0. (ii) (τ1 ⊗ τ0 + τ0 ⊗ τ1 ) r (Mβα Mβα )◦ = δ A2 +B2 ,A3 +B3 δ A1 ,−B1 Vβα¯¯ . (iii) (τ1 ⊗ τ1 ) r (Mβα Mβα Mβα )0 = β1 β1 β1 δ A1 ,−B1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 . (iv) (τ1 ⊗ τ1 ) r (δ(Mβα )Mβα )0 = −(α1 + β1 )β1 β1 δ A1 ,−B1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 .
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(v) In particular, if A is a δ-one-form, β¯ − A2 |D|−3 = 2β1 β1 Ba (A2 )α¯ , β¯ − A2 |D|−2 = 2Vβα¯¯ B(A2 )α¯ , β¯ − A3 |D|−3 = 2β1 β1 β1 Ba (A3 )α¯ , −3 − δ(A)A|D| = − Aδ(A)|D|−3 = 0. Proof. See Appendix B.
For a given δ-1-form A, we say that A is homogeneous of degree in a equal to n ∈ Z if it is a linear combination of Mβα such that α1 + β1 = n. From Lemma 4.14 (iv) we get, Corollary 4.15. Let A, A be two δ-1-forms, then −(A|D|−1 )2 = − A2 |D|−2 , −1 −1 −2 − A|D| A |D| = − A A |D| −n − A A |D|−3 , when A homogenous of degree n. Lemma 4.16. We have (i) (τ1 ⊗ τ1 ) ρ Mβα J Mβα J −1 = β1 β1 δα1 ,−β1 δα1 ,−β1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 . (ii) (τ0 ⊗ τ1 + τ1 ⊗ τ0 ) ρ Mβα J Mβα J −1 = δα1 ,−β1 δα1 ,−β1 β1 wβα δα2 +β2 +α3 +β3 ,0 δα2 +β2 ,α3 +β3 + β1 wβα δα2 +β2 +α3 +β3 ,0 δα2 +β2 ,α3 +β3 . (iii) (τ1 ⊗ τ1 ) ρ Mβα Mβα J Mβα J −1 = β1 β1 β1 δα1 +α1 ,−β1 −β1 δα1 ,−β1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 .
δ(Mβα )J Mβα J −1 = −(α1 + β1 )β1 β1 δα1 ,−β1 δα1 ,−β1 δ A2 ,0 (iv) (τ1 ⊗ τ1 ) ρ δ A3 ,0 δ B2 ,0 δ B3 ,0 . (v) In particular, if A and A are δ-one forms, β1 00 β1 00 )(β1 A¯ −β ), − A J A J −1 |D|−3 = 2(β1 A−β 1 00 1 00 β1 00 β1 00 − A J A J −1 |D|−2 = 2(β1 A¯ −β )(wβα B(A)βα ) + 2(β1 A−β )(wβα B( A¯ )βα ), 1 00 1 00 β1 00 β¯ )(β1 β1 Ba (A2 )α¯ ), − A2 J A J −1 |D|−3 = 2(β1 A¯ −β 1 00 − δ(A)J A J −1 = 0.
Proof. See Appendix C.
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Lemma 4.17. Let β, β ∈ Z. Then, lim
2 j→∞
2j ∞ ∞ ↑β ↑β ↑β ↑β 2 2 2 (qm,|β| (q (qm,|β q2 j−m,|β ) − 1 = ) − 1 + | |) − 1 . m,|β| m=0
m=0
m=0
Proof. See Appendix D.
Proof of Theorem 4.11. The result follows from Lemmas 4.13, 4.14 (v) and 4.16 (v). 4.7. Proof of Theorem 4.3 and corollaries. Lemma 4.18. We have on SUq (2), (i) |D|−3 = 2, (ii) |D|−2 = 0, (iii) |D|−1 = − 21 , (iv) ζD (0) = 0. Proof. (iv) We have by definition ζD (s) := Tr(|D|−s ) =
j+1 2 j 2 ∞
vm,l , |D|−s vm,l . j
j
2 j=0 m=0 l=0
Since |D|−s vm,l = j
ζD (s) =
∞ 2 j=0
d −s j+ 0
0 d −s j
j
vm,l , where d j := 2 j + 21 , we get
(2 j + 1)(2 j + 2) d −s j+ +
∞
(2 j + 1)(2 j) d −s j =2
2 j=1
∞
(2 j + 1)(2 j) d −s j .
2 j=0
With the equalities (2 j + 1)(2 j) = d 2j − 41 and ζ (s, 21 ) = (2s − 1)ζ (s) 1 (here ζ (s, x) := n∈N (n+x) s is the Hurwitz zeta function and ζ (s) := ζ (s, 1) is the Riemann zeta function) we get ζD (s) = 2(2s−2 − 1)ζ (s − 2) − 21 (2s − 1)ζ (s), which entails that ζD (0) = 0. (i,ii,iii) are direct consequences of Eq. (43).
(43)
Proof of Theorem 4.3. It is a consequence of Lemma 4.18 and Theorems 4.1, 4.11.
As we have seen, the computation of the noncommutative integral on SUq (2) leads to certain function of A, which filter some symmetry on the degree in a, a ∗ , b, b∗ of the canonical decomposition. Precisely, it is this “balance” feature that appears in the following functionals of An , n ∈ { 1, 2, 3 }: − An |D|− p , (44) where 1 ≤ n ≤ p ≤ 3. In the next section we will demonstrate a direct method for the computation of these integrals.
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Corollary 4.19. Let u be a unitary in C ∞ SUq (2) and γu (A):=π(u)Aπ(u ∗ )+π(u)dπ(u ∗ ) be a gauge-variant of a one-form A. Then the following terms of Theorem 4.3 are gauge invariant − A|D|−3 , − A2 |D|−3 − − A|D|−2 , −2 − A|D|−1 + − A2 |D|−2 − 23 − A3 |D|−3 . Proof. It is sufficient to remark that all terms |DA |−k and ζDA (0) in spectral action (4) are gauge invariant. This can also be seen via the computation Dγu (A) =Vu DVu∗ +Vu P0 Vu∗ , where P0 is the projection on Ker D and Vu =π(u)J π(u)J −1 and |DA |n−k = Ress=n−k Tr |DA |n−k (see [22, Prop. 5.1 (iii) and Prop. 4.8].) Corollary 4.20. In the case of the spectral action without the reality operator (i.e. DA = D + A), we get − |DA |−2 = −2 − A|D|−3 , − |DA |−3 = 2, −1 −2 1 − |DA | = − 2 − − A|D| + − A2 |D|−3 , ζ DA (0) = − − A|D|−1 + 21 − A2 |D|−2 − 13 − A3 |D|−3 . As a consequence, if A is a one-form such that A|D|−3 = 0, then the scale-invariant term of the spectral action with or without J is exactly the same modulo a global factor of 2. 5. Differential Calculus on SUq (2) and Applications 5.1. The sign of D . There are multiple differential calculi on SUq (2), see [32,38]. Thanks to [35, Theorem 3], the 3D and 4D± differential calculi do not coincide with the one considered here: the right multiplication of one-forms by an element in the algebra A is a consequence of the chosen Dirac operator that was introduced according to some equivariance properties with respect to the duality between the two Hopf algebras SUq (2) and Uq (su(2)). It is known that the Fredholm module associated to (A, H, D) is one-summable since [F, π(x)] is trace-class for all x ∈ A. In fact, more can be said about F1 : Proposition 5.1. Since ∗ 2 ∗ 2 ∗ 2 ∗ 1 F = 1−q π (a ) dπ (a) + q π (b) dπ (b ) + q π (a) dπ (a ) + q π(b ) dπ (b) 2 mod O P −∞ ,
(45)
F is a central one-form modulo O P −∞ . 1 Note that a similar result for a different spectral triple over SU (2) when q = 0 was obtained in [13, Eq. q (48)]
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135
Proof. Forgetting π, this follows from a ∗ δa + q 2 b δb∗ + q 2 a δa ∗ + q 2 b∗ δb ∗ ∗ ∗ = (a+∗ + a− )(a+ − a− ) + q 2 (b+ + b− )(b− − b+∗ ) + q 2 (a+ + a− )(a− − a+∗ ) ∗ )(b+ − b− ) + q 2 (b+∗ + b−
= [a+∗ a+ − q 2 a+ a+∗ + q 2 b+∗ b+ − q 2 b+ b+∗ ] + R = (1 − q 2 ) + R
(46)
by (15) where we check that the remainder R is zero: ∗ ∗ a+ + q 2 b− b+ ] R = −[a+∗ a− + q 2 b+∗ b− ] + [a−
∗ ∗ ∗ ∗ a− − q 2 a− a− + q 2 b− b− − q 2 b− b− ] −[a−
∗ ∗ + q 2 q− b+ ) − (a+∗ a− + q 2 b+∗ b− ), +(q 2 a+ a− ∗ + q 2 q ∗ b ) − (a ∗ a + q 2 b∗ b ) = 0 thus, applying (18), (19), (20), R = + (q 2 a+ a− − + + − + − using commutation relations (14). Now, multiplying by F in (46) (that means that we replace δ by d) gives (45) since F commute with a± , b± and F is central by (27).
Proposition 5.2. The one-form in (45) is in fact exactly a function of the Dirac operator D: π(a ∗ ) dπ(a) + q 2 π(b) dπ(b∗ ) + q 2 π(a) dπ(a ∗ ) + q 2 π(b∗ ) dπ(b) = ξq (D) = F ξq (|D|),
(47)
[2s]−2s where ξq (s) := q [s+1/2][s−1/2] . Moreover, F = limq→0 ξq (D).
Proof. First, let us observe that the one-form ω in (47) is invariant under the action of the Uq (su(2)) × Uq (su(2)): h ω = (h) ω for any h ∈ Uq (su(2)) × Uq (su(2)). For instance, using notations of [21],
1 1 1 1 −2 −1− 2 ∗ −1 ∗ 2 ∗ ∗ 2 2 bda + q bda − q a db = 0 = (e) ω. e ω = q a db + q −q Therefore, since both the representation π as well as the operator D are equivariant, the image of ω must be diagonal in the spinorial base. A tedious computation with the full spinorial representation π given in (9) yields j↑
j↑
q 8 j+8 −q 8 j+6 −(4 j+3) q 4 j+6 +(8 j+6) q 4 j+4 −(4 j+3) q 4 j+2 −q 2 +1 (q 4 j+4 −1)(q 4 j+2 −1)
= ξq (2 j + 23 ),
j↓
j↓
−q 8 j+4 +q 8 j+2 +(4 j+1) q 4 j+4 −(8 j+2) q 4 j+2 +(4 j+1) q 4 j +q 2 −1 (q 4 j+2 −1)(q 4 j −1)
= −ξq (2 j + 21 ).
vml , ω vml = vml , ω vml =
These expressions have a clear q = 0 limit equal respectively to 1 and -1, so ω → F as q → 0. In the q = 1 limit, these expressions yield identically 0, which is confirmed by the fact that all one-forms are central, then ω could be expressed as d(aa ∗ + bb∗ ) = d 1 = 0. Note that since the invariant one-form we constructed differs by O P −∞ from F, hence any commutator with it will be itself in O P −∞ . We do not know if a central form ω is automatically invariant by the action of both Uq (su(2)), that is: h ω = (h)ω.
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Proposition 5.3. The order one calculus up to O P −∞ is not universal. Proof. Let us take the one-form ω F from (45), which gives F. Then, for any x ∈ A(SUq (2)) we have π (xω F − ω F x) = 0. Note that since ω F ∼ (1 − q 2 )−1 ω mod O P −∞ , we get 1 ∼ (1 − q 2 )−1 ξq (|D|) mod O P −∞ . Corollary 5.4. Still modulo O P −∞ , 1 ∈ π 2u (A) . Proof. 1 = F 2 is by definition in π 2u (A) . In fact, one checks, using (15), (18), (21) that q 2 da da ∗ − da ∗ da = 1 − q 2 2 showing again that 1 ∈ π u (A) . Similarly, using (14) and (16), (21), (22), we get still up to O P −∞ , q da db = db da, q da db∗ = db∗ da, da ∗ db = q db da ∗ , da ∗ db∗ = q db∗ da ∗ , db db∗ = db∗ db, da da ∗ + db db∗ = −1.
(48)
(49)
The use of the last equality of (49) and (48) gives Proposition 5.5. Up to O P −∞ , F is not a (universal) closed one-form, as da ∗ da + q 2 da da ∗ + q 2 db∗ db + q 2 db db∗ = −1 − q 2 .
(50)
5.2. The ideal R. In order to perform explicit calculations of all terms of the spectral action, we observe that each δ-one-form could be expressed in terms of xδ(z)y, where z is one of the generators a, a ∗ , b, b∗and x, y are some elements of the algebra A(SUq (2)). Then, for the computation of xdzy|D|−1 we can use the trace property of the noncommutative integral to get: −1 −1 − xδ(z)y |D| = − yxδ(z) |D| + − xδ(z) |D|−1 δ(y) |D|−1 . Therefore, the problem of calculating the tadpole-like integral could be in effect reduced to the calculation of much simpler integrals: xδ(z) for all generators z and the integrals of higher order in |D|−1 . However, it appears that the calculations of higher-order terms simplify a lot, when we further restrict the algebra by introducing an ideal, which is invisible to the parts of the integral at dimension 2 and 3. For instance, consider the space of pseudodifferential operators T ∈ 0 (A) of order less or equal to zero (see [16]), that satisfy −2 −2 −3 − T t |D| = − t T |D| = − T t |D| = − t T |D|−3 = 0, ∀t ∈ 00 (A). (51) The elements a− , b− b+ , b− b+∗ and their adjoints are in this space up to O P −∞ : this is due to the fact that in Theorem 3.4, τ1 ⊗ τ1 (r (x)) = 0 when r (x) ∈ π± (A) ⊗ π± (A) mod O P −∞ contains tensor products of π± (b) or π± (b∗ ) since these elements are in the kernel of the grading σ .
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Definition 5.6. Let R be the kernel in X of (σ ⊗ σ ) ◦ r , where r is the Hopf-map defined in (29) and σ is the symbol map and let R be the vector space generated by R and R F. Note that R is a ∗ -ideal in X and a− , b− b+ (= q 2 b+ b− ), b− b+∗ are in R. By construction and Theorem 3.4, any T ∈ R satisfies (51) and R is invariant by F. ∗ ] ∈ R, so by (15) and (21), a ∗ a −q 2 a a ∗ −(1−q 2 ) ∈ R Moreover, by (18), [b− , b− + + + + and by (22), q a+ b− − b− a+ ∈ R. It is interesting to quote, thanks to Theorem 3.4 that if x ∈ R, then F x |D|−1 = 0 while a priori, x |D|−1 = 0. Note that F ∈ 0 (A) also satisfies (51) by Theorem 3.4 while F ∈ / R since F 2 = 1. Moreover other elements are ∗ ∗ in R like for instance d(b b) = d(bb ): ∗ ∗ δ(bb∗ ) = −δ(aa ∗ ) = −δa a ∗ − a δa ∗ = −(a+ − a− )(a+∗ + a− ) − (a+ + a− )(a− − a+∗ ) ∗ = 2(a+ a− − a− a+∗ )
is in R since a− ∈ R yielding d(bb∗ ) ∈ R F. We do not know if R is equal to the subset of the algebra generated by B and B F satisfying (51). Lemma 5.7. R is a ∗-ideal in 0 (A), which is invariant by F, d, δ. Proof. Since R is an ideal in X = B mod O P −∞ (see Remark 3.3), R appears to be an ideal in the 0 (A) ⊂ algebra generated by B and B F. Since R is invariant by F, its invariance by d follows from its invariance by δ that is true on the generators of R. that, according to Theorem 4.13, da |D|−2 = da |D|−3 = 0 whereas Note a ∗ da |D|−3 = 2, which emphasizes the role of “for all t” in (51). Lemma 5.8. For any t ∈ 00 (A) and T ∈ R, we have t T |D|−1 = T t |D|−1 . Proof. For any t ∈ B, we have T t |D|−1 = t T|D|−1 + T |D|−1 δ(t) |D|−1 and, moreover, T |D|−1 δ(t) |D|−1 = T δ(t) |D|−2 − T δ 2 (t) |D|−3 , which comes from |D|−1 δ(t)|D|−1 = δ(t)|D|−2 + [|D|−1 , δ(t)]|D|−1 = δ(t)|D|−2 − |D|−1 δ 2 (t)|D|−2 = δ(t)|D|−2 − δ 2 (t)|D|−3 + |D|−1 δ 3 (t)|D|−3 . So we get the result because T satisfies (51).
Lemma 5.9. If means equality up to the ideal R, the following rules with d(.) = [D, .] of the first-order differential calculus hold (forgetting π ) a da da a, a da ∗ −da a ∗ , a db q −1 db a, a db∗ q −1 db∗ a,
a ∗ da −da ∗ a, a ∗ da ∗ da ∗ a ∗ , a ∗ db q db a ∗ , a ∗ db∗ q db∗ a ∗ ,
b da q da b, b da ∗ q −1 da ∗ b, b db db b, b db∗ db∗ b −b∗ db,
b∗ da q da b∗ , b∗ da ∗ q −1 da ∗ b∗ , b∗ db db b∗ −b db∗ , b∗ db∗ db∗ b∗ .
Moreover a ∗ da − q 2 da a ∗ (1 − q 2 ) F,
q 2 a da ∗ − da ∗ a (1 − q 2 ) F.
(52)
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Proof. The table follows from relations (7) and Lemma 3.2 with (27) (one can also use (14).) For instance, since a− ∈ R, using the fact that R is invariant by F, b da = (b+ + b− )(a+ − a− ) F (b+ + b− )(a+ + a− ) F = ba F = q ab F q (a+ − a− ) F b = q da b ∗ )(a − a )F (a ∗ − a ∗ )(a + a )F = −da ∗ a. or similarly, a ∗ da = (a+∗ + a− + − + − + − The second equivalence of (52) is just the adjoint of the first one that we prove now: ∗ ∗ a ∗ da − q 2 da a ∗ = (a+∗ + a− )(a+ − a− )F − q 2 (a+ − a− )F(a+∗ + a− ) ∗ ∗ )(a+ + a− )F − q 2 (a+ + a− )(a+∗ + a− )F (a+∗ + a−
= (a ∗ a − q 2 aa ∗ ) F = (1 − q 2 ) F.
Remark 5.10. The above written rules remain valid if dx is replaced by δ(x) and F by 1. Working modulo R simplifies the writing of a one-form: Lemma 5.11.
(i) Every one-form A can be, up to elements from R, presented as A xa da + da ∗ xa ∗ + xb db + db∗ xb∗ ,
where all x∗ are the elements of A. (ii) When A is selfadjoint, A can be written up to R (not in a unique way, though) as A xa da − da ∗ (xa )∗ + xb db − db∗ (xb )∗ , where xa , xb are arbitrary elements of A.
Proof. (i) A basis for one-forms consists of the following forms: a α bβ (b∗ )γ d(a α bβ (b∗ )γ ), where α, α ∈ Z and β, γ , β , γ ∈ N. Using the Leibniz rule and the commutation rules within the algebra (up to the R according 5.9), we reduce to Lemma the problem to the case of the forms: a α bβ (b∗ )γ dx a α bβ (b∗ )γ , where x can be either of the generators a, a ∗ , b, b∗ . If x = b or x = b∗ , the straightforward application of the rules of the differential calculus leads to the answer that the one-form could be expressed as: a α bβ (b∗ )γ db and db∗ a α bβ (b∗ )γ . Similar considerations for the case x = a, a ∗ lead to the remaining terms. Note that the presentation is not unique, since there still might remain terms, which are in R, for example: b∗ db + db∗ b = d(bb∗ ) ∈ R. (ii) is direct. Next we can start explicit calculation of the integrals, beginning with the tadpole terms. Application of the Leibniz rule yields a presentation of one-forms that is different from the one of the previous lemma. Each δ-one-form could be expressed as a finite sum of the terms xδ(z)y, where z isone of the generators a, a ∗ , b, b∗ and x, y are some elements of the algebra A SUq (2) .
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Proposition 5.12. For all x, y ∈ A SUq (2) and z ∈ {a, a ∗ , b, b∗ } we have − xδ(z)y |D|−1 = − yxδ(z) |D|−1 + − xδ(z)δ(y) |D|−2 − − xδ(z)δ 2 (y) |D|−3 . Proof. This is just the application of the trace property of the noncommutative integral, together with the identity: |D|−1 δ(z)|D|−1 = − |D|−1 , z . The computation of tadpole-like integrals is reduced to the integrals Remark 5.13. xδ(z)|D|−1 for all generators z and the integrals of higher order in |D|−2 . However, the calculations of higher-order terms simplify a lot after we use the relations that hold up to this erases parts of the integral depending on |D|−2 and |D|−3 . Thus, beside the ideal R:−1 xδ(z) |D| , we only need to compute xδ(z)δ(z ) |D|−2 , where z and z are generators, since all the |D|−3 integrals have already been explicitly computed in Sect. 4.6 (these integrals do not depend on q.) Besides the tadpole, the only integrals that need to be computed are A |D|−2 and A2 |D|−2 , where A is a δ-1-form. Working modulo R and using again Leibniz rule, we only need to compute xδ(z) |D|−2 and the previous integrals xδ(z)δ(z ) |D|−2 . j
5.2.1. Operators L q and Mq . In the notation vl,m of H, we have already used the j j
j
dependence in (11) with Jq vm,l := q j vm,l . Let L q and Mq be the similar diagonal operators j
j
L q vm,l := q 2l vm,l , j
j
Mq vm,l := q 2m vm,l . We immediately get Lemma 5.14. For n ∈ N∗ , (L q )n |D−2 | = (Mq )n |D−2 | =
2 . 1−q 2n
Proof. We have 2 j 2 j+1 ∞ j j vm,l , L qn |D|−2−s vm,l Tr L qn |D|−2−s =
=
2 j=0 m=0 l=0 ∞
2n(2 j+2)
(2 j + 1) 1−q1−q 2n
2 j=0
∼0 1−q1 2n
d −2−s + j+
∞
2n(2 j+2)
(2 j + 1) 1−q1−q 2n
d −2−s j
2 j=0
ζ (s + 1, 23 ) + ζ (s + 1, 21 ∼e
2 ζ (s 1−q 2n
+ 1),
where ∼0 means modulo a function holomorphic at 0. This gives the result for L qn and a similar computation can be done for Mqn . These operators become useful due to the following lemma:
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Lemma 5.15. We have L q Mq ∈ R. Moreover, b δb∗ Mq − L q , b∗ δb L q − Mq , bb∗ L q + Mq , a δ(a ∗ ) − aa ∗ L q + Mq − 1, a ∗ δa a ∗ a 1 − q 2 (L q + Mq ), da da ∗ L q + Mq − 1, da ∗ da q 2 (L q + Mq ) − 1, bn−2 (b∗ )n db db (L q )n + (Mq )n , bn−1 (b∗ )n−1 db db∗ −(L q )n − (Mq )n , bn (b∗ )n−2 db∗ db∗ (L q )n + (Mq )n . ∗ ∈ R, we compute up to the ideal R, Proof. Since L q Mq = q 2 a− a− ∗ ∗ b δb∗ = (b+ + b− )(b− − b+∗ ) −b+ b+∗ + b− b− = Mq − L q + L q Mq (1 − q 2 ) Mq − L q ,
and similarly for the other relations.
5.2.2. Automorphisms of the algebra and symmetries of integrals. Proposition 5.16. For any n ∈ N∗ , 2n ) −(bb∗ )n |D|−1 = −2(1+q , (1−q 2n )2 −(bb∗ )n b∗ δb |D|−1 = −(bb∗ )n b δb∗ |D|−1 = 1−q22n+2 , 4n+2 4n −2q 2n+2 +6q 2n −(bb∗ )n a da ∗ D−1 = −2q (1−q−2q , 2n )2 (1−q 2n+2 ) 2n+2 2n −2q 2 −2 . −(bb∗ )n a ∗ da D−1 = 6q(1−q 2n−2q )2 (1−q 2n+2 ) Note thatthe knowledge of these integrals is enough for the computation of any term of the form xδ(z)|D|−1 , where z is a generator, since any other δ-one-form will be unbalanced. To show this proposition, we will use few symmetries, properties of the ideal R and replacement of δ-one-forms in terms of L q , Mq as above. Let U be the following unitary operator on the Hilbert space: j↑
j +↓
j↓
j −↑
U vm,l = (−1)m+l vl,m , U vm,l = (−1)m+l vl,m . Then, by explicit computations we have U ∗ aU = a, U ∗ b∗ U = b,
U ∗ a ∗U = a ∗ , U ∗ bU = b∗ , ∗ and U DU = −D.
Lemma 5.17. Each noncommutative integral (44) of an element of the algebra or differential forms is (up to sign) invariant under the algebra automorphism ρ defined by ρ(a) := a, ρ(a ∗ ) := a ∗ , ρ(b) := b∗ , ρ(b∗ ) := b.
(53)
Spectral Action on SUq (2)
141
Proof. For any homogeneous polynomial p and any k ∈ N, − p(a, a ∗ , b, b∗ , D) D−k = − U ∗ p(a, a ∗ , b, b∗ , D) D−k U k = (−1) − p(U ∗ aU, U ∗ a ∗ U, U ∗ bU, U ∗ b∗ U, U ∗ DU ) D−k = (−1)k+d − p(ρ(a), ρ(a ∗ ), ρ(b), ρ(b∗ ), D) D−k , where d is the degree of p with respect to D. Corollary 5.18. For any n ∈ N, (bb∗ )n b∗ db D−1 = (bb∗ )n b db∗ D−1 . Lemma 5.19. For any x, y ∈ 0 (A), −1 + xδ(y)|D|−2 − xδ 2 (y) |D|−2 . (i) xy|D|−1 = yx|D| (ii) z x D−1 y D−1 = z xy D−2 , where z ∈ A contains b or b∗ . Proof. (i) is direct consequence of the trace property of and the fact that O P −4 operators are trace-class. ii) We calculate: − z x D−1 y D−1 = − z x y D−1 − D−1 [D, y] D−1 D−1 −2 −1 −2 = − z xy D − − z xD [D, y] D = − z xy D−2 . The last step is based on the observation that any integral with D−3 vanishes if the expression integrated contains b or b∗ . Lemma 5.20. For any n ∈ N, (i) (bb∗ )n b∗ db D−1 = 1−q22n+2 , (ii) (bb∗ )n d(bb∗ ) D−1 = 0, 2n ) . (iii) (bb∗ )n |D|−1 = −2(1+q (1−q 2n )2
Proof. (i) With n > 1, we begin with d ((b b∗ )n ) D−1 = 0, which follows directly from the trace property of the noncommutative integral. Expanding the expression using Leibniz rule and the commutation xD−1 = D−1 x + D−1 [D, x]D−1 ,
(54)
we obtain n−1 n−1 0= − bk db bn−k−1 (b∗ )n D−1 + − bn (b∗ )k db∗ (b∗ )n−k−1 D−1 k=0
k=0
= n − bn−1 (b∗ )n db D−1 + − bn (b∗ )n−1 db∗ D−1 +
n−1 − bk db D−1 d(bn−k−1 (b∗ )n )D−1 + bn (b∗ )k db∗ D−1 d((b∗ )n−k−1 )D−1 . k=0
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Using Lemma 5.19, 0 = n −(bb∗ )n−1 (b∗ db + b db∗ ) D−1 + − 21 n(n − 1)bn−2 (b∗ )n db db + n 2 bn−1 (b∗ )n−1 db db∗ + 21 n(n − 1)bn (b∗ )n−2 db∗ db∗ D−2 . The integrals with D−2 could be easily calculated when we restrict ourselves to calculations modulo ideal R: n −(bb∗ )n−1 (b∗ db + b db∗ ) D−1 = −2 n(n − 1) − 2n 2 + n(n − 1) 1−q1 2n = 4n 1−q1 2n . Hence (bb∗ )n−1 (b∗ db + b db∗ ) D−1 = 1−q4 2n , that together with Corollary 5.18 proves i). (ii) In a similar way, (bb∗ )n−1 d(bb∗ ) D−1 = 0 = (bb∗ )n−1 d(aa ∗ ) D−1 implies: 0=
n−1
(bb∗ )n−k−1 d(bb∗ )(bb∗ )k D−1
k=0
= n −(bb∗ )n−1 d(bb∗ ) D−1 + 21 n(n − 1) −(bb∗ )n−2 d(bb∗ ) d(bb∗ ) D−2 = n −(bb∗ )n−1 d(bb∗ ) D−1 , where in the last step we used that d(bb∗ ) ∈ R. The identity (ii) now follows from the equality aa ∗ = 1 − bb∗ . (iii) Using Lemma 5.19, we get ∗ n −1 An := −(bb ) |D| = −(bb∗ )n (aa ∗ + bb∗ )|D|−1 , and we push now a ∗ through |D|−1 and from cyclicity of the trace through (bb∗)n , = An+1 + −(bb∗ )n q 2n a ∗ a |D|−1 + −(bb∗ )n q 2n aδ(a ∗ ) |D|−2 , the last term being calculated explicitly, since up to ideal R, aδ(a ∗ ) L q + Mq − 1, = An+1 (1 − q 2n+2 ) + q 2n An + 4 1−q12n+2 − 1−q1 2n , which leads to An (1 − q 2n ) + Assuming An = 1+q 2
fn (1−q 2n )2
4 1−q 2n
= An+1 (1 − q 2n+2 ) +
f n +4 f n+1 +4 = 1−q 2n+2 , 1−q 2n 2n 1+q −2 (1−q 2n )2 .
we have
A0 = −2 (1−q 2 )2 , we obtain An =
Finally, to get Proposition 5.16, it remains to prove
4 . 1−q 2n+2
and taking into account that
Spectral Action on SUq (2)
Lemma 5.21. For n ≥ 1, −(bb∗ )n a da ∗ D−1 = −(bb∗ )n a ∗ da D−1 =
143
−2q 4n+2 −2q 4n −2q 2n+2 +6q 2n , (1−q 2n )2 (1−q 2n+2 ) 6q 2n+2 −2q 2n −2q 2 −2 . (1−q 2n )2 (1−q 2n+2 )
Proof. First, using Leibniz rule, (54) and Lemma 5.19 we have (for n ≥ 1) −(bb∗ )n a da ∗ D−1 = −q 2n −(bb∗ )n a ∗ da − −(bb∗ )n da da ∗ D−2 . Further, we use the identity (45): ∗ n ∗ 2 ∗ 2 ∗ 2 ∗ −1 2 −(bb ) a da + q a da + q b db + q b db D = (1 − q ) −(bb∗ )n |D|−1 , taking into account that F D = |D|. These equations give together a system of linear equations −(bb∗ )n a da ∗ D−1 + q 2n −(bb∗ )n a ∗ da D−1 = −4 1−q12n+2 − 1−q1 2n , 1+q 2n 4q 2 q 2 −(bb∗ )n a da ∗ D−1 + −(bb∗ )n a ∗ da D−1 = −2(1 − q 2 ) (1−q 2n )2 − 1−q 2n+2 , which is solved by the expressions stated in the lemma.
5.2.3. The noncommutative integrals at |D|−2 . We need to separate this task into two problems. First, we shall calculate all integrals of the type x δ(z)|D|−2 , with x ∈A(SUq (2)) and z being one of the generators. The second problem is to calculate x δ(y) δ(z) |D|−2 , with both y and z being the generators {a, a ∗ , b, b∗ }. Lemma 5.22. The only a priori non-vanishing integrals of the type x δ(z) |D|−2 are for n ∈ N: −(bb∗ )n b∗ δ(b) |D|−2 = −(bb∗ )n bδ(b∗ ) |D|−2 = 0, 4q 2n (1−q 2 ) −(bb∗ )n aδ(a ∗ ) |D|−2 = (q 2n+2 , n > 0, −1)(1−q 2n ) 2) −(bb∗ )n a ∗ δ(a) |D|−2 = (1−q4(1−q 2n+2 )(1−q 2n ) . Proof. Since aδ(a ∗ ) L q + Mq − 1 and (bb∗ )n L qn + Mqn , we get (bb∗ )n aδ(a ∗ ) L qn+1 + Mqn+1 − L qn − Mqn , and the second result is obtained from Lemma 5.14. The other integrals are computed in a similar way.
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Lemma 5.23. The only a priori non-vanishing integrals of the type x dy dz |D|−2 are for n ∈ N: −(bb∗ )n (b∗ )2 db db |D|−2 = 1−q42n+4 , −(bb∗ )n db db∗ |D|−2 = 1−q42n+2 , −(bb∗ )n (a ∗ b∗ )(da db) |D|−2 = 0, −(bb∗ )n (ab∗ )(da ∗ db) |D|−2 = 0, −(bb∗ )n (a ∗ b)(da db∗ ) |D|−2 = 0, −(bb∗ )n (ab)(da ∗ db∗ ) |D|−2 = 0, 4(q 2n+2 −q 2n ) −(bb∗ )n (da da ∗ ) |D|−2 = (1−q 2n+2 )(1−q 2n ) , n > 0, 2 −1) −(bb∗ )n (da ∗ da) |D|−2 = (1−q4(q 2n+2 )(1−q 2n ) . Proof. This follows from Lemma 5.14 with the equivalences up to R gathered in Lemma 5.15. 6. Examples of Spectral Action It is clear from Theorem 4.3 that no one-form of the type ada, bdb, adb, a ∗ db, etc., contributes to the spectral action. Indeed, only the balanced parts of one-forms give nonzero term in the coefficients. Let us now give the values of the terms a possibly An |D|− p and the full ζDA (0) for a few examples: 1) Clearly the spectral action depends on q: for instance, S(Da ∗ da , , ) = 2 3 3 − 8 2 2 +
q 2 +15 2(1−q 2 )
1 1 +
11q 4 +36q 2 +13 3(q 4 −1)
(0).
Table 1. Values of noncommutative integrals A
A |D|−3
A2 |D|−3
A3 |D|−3
A |D|−2
a ∗ da
2
2
2
4q 2 q 2 −1
b∗ db
0
0
0
0
ada ∗
−2
2
−2
bdb∗
−4 q 2 −1
0
0
0
0
A2 |D|−2
4q 2 (q 2 +2) q 4 −1 −4 q 4 −1 4(2q 2 +1) q 4 −1 −4 q 4 −1
A |D|−1
3q 2 +1 2(q 2 −1) −2 q 2 −1 q 2 +3 2(q 2 −1) −2 q 2 −1
ζDA (0) 11q 4 +36q 2 +13 3(q 4 −1) 4q 2 q 4 −1 13q 4 +36q 2 +11 3(q 4 −1) 4q 2 q 4 −1
Spectral Action on SUq (2)
145
2) Moreover, for B := a δa ∗ and A := B + B ∗ , we get since B B ∗ mod R, p −k p − A |D| = 2 − B p |D|−k , 1 ≤ p ≤ k ≤ 3. (55) Thus the spectral action of the selfadjoint one-form A := ada ∗ + (ada ∗ )∗ is S(DA , , ) = 2 3 3 + 16 2 2 +
q 2 −33 2(1−q 2 )
122q 4 +168q 2 −2 3(q 4 −1)
1 1 +
(0).
3) When Bn := (bb∗ )n b δb∗ , then by Lemma (5.15), Bn Bn∗ , so for An := Bn + Bn∗ , p the equation (55) is still valid and Bn |D|−k are all zero but Bn |D|−1 = 1−q22n+2 and Bn2 |D|−2 = 1−q44n+4 , so S(DAn , , ) = 2 3 3 −
1 2
1 1 +
8 1+q 2n+2
(0).
(56)
Remark that this spectral action still exists as q → 1! Note however that the symmetrization process (55) is not true in general, for instance 2 −1 4 −q 2 −1) B |D| = 0 if B := a δb and A := B + B ∗ , then A2 |D|−1 = 8(q(1−q 4 )2 , while 4 ∗ −1 or [B, B ]|D| = 1−q 4 . 4) The spectral action can be also independent of q: for instance, if A = the q-dependent selfadjoint one-form given in (47), then, S(DA , , ) = 2 3 3 − 8 2 2 +
15 2
1 1 −
1 1−q 2
ξ(D) is
13 3 .
7. The Commutative Sphere S3 Since SU (2) S3 , we get a concrete spinorial representation of the algebra A := C ∞ (S3 ) on the same Hilbert space H and same Dirac operator D with (9), where q = 1, that means that q-numbers are trivially the “usual” numbers: [α] = α. So, − + + π(a) | jµn
:= α +jµn | j + µ+ n +
+ α − jµn | j µ n
, − + − π(b) | jµn
:= β +jµn | j + µ+ n −
+ β − jµn | j µ n
, − − − π(a ∗ ) | jµn
:= α˜ +jµn | j + µ− n −
+ α˜ − jµn | j µ n
, − − + π(b∗ ) | jµn
:= β˜ +jµn | j + µ− n +
+ β˜ − jµn | j µ n
,
where α +jµn :=
α− jµn
:=
⎛
√
j + µ + 1⎝
√
⎛√ j − µ⎝
j+n+3/2 2 j+2
j−n+1/2 (2 j+1)(2 j+2
j−n+1/2 2 j+1
0
−
√
j+n+1/2] 2 j+1
√
√
⎞
0
j+n+1/2 2 j (2 j+1)
j−n−1/2 2j
⎠,
⎞ ⎠,
(57)
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β +jµn :=
β− jµn :=
⎛
j
√
j−n+3/2 0 2 j+2 ⎝ +µ+1 √ √ j+n+1/2 j−n+1/2 − (2 j+1)(2 j+2) 2 j+1
⎛
j − µ⎝
−
√
j+n+1/2 2 j+1
0
√
j−n+1/2 2 j (2 j+1 √ − j+n−1/2 2j
−
⎞ ⎠,
⎞ ⎠
∓ ∓ ∗ ˜± ∗ with α˜ ± jµn := (α j ± µ− n − ) , β jµn := (β j ± µ− n + ) . j
Note that the representation on the vectors vm,l is not as convenient as in (10). One can check that the generators π(a), π(b) and their adjoint commute and that [x, [D, y]] = 0 for any x, y ∈ A. 7.1. Translation of Dirac operator. In general the Dirac operator is defined in a more symmetric way than we did. So, although not absolutely necessary here, we define for the interested reader the unbounded self-adjoint Dirac operator translated by the constant λ as: D := D + λ. For instance, this gives for λ = − 21 in the case of S3 , see [31], D vm,l = (2 j + 1) 1 0 j j 0 −1 vm,l so vm,l is an eigenvector of |D |. We define D := D + P0 and D := D + P0 , where P0 is the projection on Ker D and P0 is the projection on Ker D . As the following lemma shows, the computation of noncommutative integrals involving D can be reduced to the computation of certain noncommutative integrals involving D : Lemma 7.1. If T := Ress=0 Tr T |D |−s , then for any 1-form A on a spectral triple of dimension n, −A |D|−(n−2) = − A |D |−(n−2) +λ (n − 2) − AD |D |−n +λ2 (n−1)(n−2) − A |D |−n , 2 −(n−2) −(n−2) −(n−1) 2 (n−1)(n−2) −AD − A D −n . =− AD + λ (n − 2) − A D +λ 2 j
Proof. Recall from [22, Prop. 4.8] that for any pseudodifferential operator P, − P|D|−r = Ress=0 Tr P|D|−r |D |−s . Moreover, by [22, Lemma 4.3], for any s ∈ C and N ∈ N∗ , |D|−s = |D |−s +
N
K p,s Y p |D |−s
mod O P −N −1− (s) ,
(58)
p=1
N (−1)k+1 where Y = k=1 (−2λD + λ2 )k D −2k mod O P −N −1 and K p,s are complex k numbers that can be explicitly computed. Precisely, we find K p,s = (− 2s ) p V ( p) where
Spectral Action on SUq (2)
147
V ( p) is the volume of the p-simplex. Since the spectral dimension is n, we work modulo O P −(n+1) , and for s = n − 2, we get from (58): |D|−(n−2) = |D |−(n−2) + λ(n − 2)D |D |−n + λ2 (n−1)(n−2) |D |−n mod O P −(n+1) . 2 As a consequence, we have for P ∈ O P 0 (the O P 0 spaces are the same for D or D ), −(n−2) −(n−2) −n 2 (n−1)(n−2) − P|D| − P|D |−n . = − P|D | +λ(n − 2) − PD |D | + λ 2 Since A and AF are in O P 0 , we get both formulae.
7.2. Tadpole and spectral action on S3 . We consider now the commutative spectral triple (C ∞ (S3 ), H, D). It is 1-summable since jµn s | [F, π(x)] | jµn s = 0 when x = a, a ∗ , b, b∗ for any j, µ, n, s = ↑, ↓. All integrals of the above lemma are zero for S3 : Proposition 7.2. There is no tadpole on the commutative real spectraltriple (C ∞ (S3 ), H, D). More generally, for any one-form A and any p ∈ N, A|D|− p = AF|D|− p = 0. Proof. Since the representation is real, that is any matrix elements of the generators are − p when A is assumed to be a real combination of real, so must be the trace of AF|D| the generating one forms. Hence AF|D|− p = A∗ F|D|− p . The reality operator J introduced in (28) satisfies, when q = 1, the commutative −1 = x∗ for x ∈ A. So J AJ −1 = −A∗ and relation J−xJ AF|D| p = J A∗ F|D|− p J −1 = − AF|D|− p and AF|D|− p = 0. This extends to arbitrary one-form A. The equality A|D|− p = 0 is obtained by the substitution AF → A in the above proof. For any selfadjoint one-form A, D A := D + A = D. Thus, the spectral action for the real spectral triple C ∞ (S3 ), H, D for DA is trivialized by S(DA , , ) = 2 3 3 −
1 2
1 1 + O(−1 ).
(59)
But it is more natural to compare with the spectral action of D + A. This is obtained respectively from Lemma 4.18 and the general heat kernel approach [27]: S(D + A, , ) = 2 3 3 + − |D + A|−1 1 1 + O(−1 ), since all termsof (2) in n−k are zero for k odd and ζD+A (0) = 0 when n is odd: as a verification, |D + A|−2 is zero according to [22, Lemma 4.10], Lemmas 4.18 and Proposition 7.2. Similarly, ζD+A (0) = 0 because in (3), all terms with k odd are zero proof as in Proposition 7.2) but for k even, it is not that easy to show that (the same AD−1 AD−1 = 0. Moreover, the curvature term does not depend on A: Lemma 7.3. For any one-form A on a commutative spectral triple of dimension n based on a compact Riemannian spinc manifold without boundary, we have (60) − |D + A|−(n−2) = − |D|−(n−2) .
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B. Iochum, C. Levy, A. Sitarz
Proof. Follows from [29, first formula p. 511] with ρ := A = A∗ , N (ρ) = ρ (the constraint Jρ J −1 = ±ρ is not used.) One can also use [15, Prop. 1.149]. Few comparisons with the case q = 1: – Equations (45) and (47) are trivialized for q = 1 since ξ1 (D) = 0. 2 – For A = a ∗ da, Table 1 gives AFD−2 = A|D|−2 = q4q 2 −1 , while the same expression is zero for q = 1 according to Proposition 7.2: no continuity! – From [22, Lemma 4.10] 2 |D + A|−(n−2) = |D|−(n−2) + n(n−2) (AF)2 |D|−3 + (n−2) A2 |D|−3 using 4 4 2 0 X := AD + DA + A and [|D|, A] ∈ O P , but again, it is not that easy to show that the last two terms cancel: for instance here, for B = b[D, b∗ ], we obtain by direct computation (using the easiest translated Dirac operator D ) Tr B2 |D |−3−s = Tr (B∗ )2 |D |−3−s = 21 Tr BB∗ |D |−3−s = 21 Tr B∗ B|D |−3−s j+1 , = 43 (2 j+1)2+s 2 j∈N
so B2 |D |−3 = 23 . Similarly, one checks that (BF)2 |D|−3 = 21 BFB∗ F|D|−3 = −29 . Thus if A := B + B∗ , A2 |D|−3 = A2 |D |−3 = 4 and (AF)2 |D|−3 = − 43 , which yields (60). So we have shown
Proposition 7.4. For any one-form A on the commutative real spectral triple (C ∞ (S3 ), H, D), S(D + A, , ) = 2 3 3 −
1 2
1 1 + O(−1 , A).
As for (59), this is not identical to (56), which contains a nonzero constant term 0 for q = 1. 8. Conclusion We computed in this paper the full spectral action on the SUq (2) spectral triple of [21] with the reality operator J (notice the change of definition for pseudodifferential operators). The main results are: – The dimension spectrum being a finite set, there is only a finite number of terms in the spectral action expansion. – The tadpole hypothesis is not satisfied. – The action depends on q and the limit q → 1 does not exist automatically. When it exists, such limit does not lead to the associated action on the commutative sphere S3 . This means that the family of spectral actions associated to a continuous field of smooth pre-C∗ -algebras parameterized by q is not continuous. – The sign F of the Dirac operator has special properties: first, it commutes modulo O P −∞ with elements of the algebra, and second, it can be seen as a one-form (still modulo O P −∞ ), giving terms independent of q in the spectral action. Here, we were interested in the computation of the spectral action of a quantum group. Naturally, it would be interesting to investigate other related cases like the Podle´s spheres [17,19] or the Euclidean quantum spheres [20,34], especially the 4-sphere [18].
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149
9. Appendix A. Proof of Lemma 4.13. (i) Using the same notations of Lemma 4.12, we obtain by definition of τ1 , τ1 π+ (tk, p ) = δk,0 δ p,0 δα1 +α2 −α3 +β1 +β2 −β3 ,0 , (61) (62) τ1 π− (u k, p ) = δ k,0 δ p ,0 δα1 −α2 +α3 +β1 −β2 +β3 ,0 . We get τ1 π+ (tk, p ) τ1 π− (u k, p ) = δk,0 δ p,0 δα2 ,0 δα3 ,0 δβ2 ,0 δβ3 ,0 δα1 ,−β1 , so Lemma 4.12 gives the result. − + + (ii) Since π+ (tk, p )εn = qk, p,n εn+rk, p and π− (u k, p )εn = qk, p,n εn+r − , we get, k, p
τ0 π+ (tk, p ) = δrk,+ p ,0 τ0 π− (u k, p ) = δr −
∞
+ qk, p,n − δk,0 δ p,0 δα1 +α2 −α3 +β1 +β2 −β3 ,0 ,
n=0 ∞
− qk, p ,0 δα1 −α2 +α3 +β1 −β2 +β3 ,0 . k,0 δ p,n − δ
k, p ,0
(63)
(64)
n=0
With (61) and (64) we get τ1 π+ (tk, p ) τ0 π− (u k, p ) = δk,0 δ p,0 δα2 +β2 ,α3 +β3 δα1 ,−β1 ∞ − δk,0 δ p,0 qk, × − δ α +β ,0 3 3 p,n n=0
= δk,0 δ p,0 δα2 +β2 ,α3 +β3 δα1 ,−β1 w1 (β1 , α3 + β3 ). Using (62) and (63), τ0 π+ (tk, p ) τ1 π− (u k, p ) = δ k,0 δ p ,0 δα2 +β2 ,α3 +β3 δα1 ,−β1 ∞ + δ k,0 δ × p ,0 qk, p,n − δα3 +β3 ,0 n=0
= δ k,0 δ p ,0 δα2 +β2 ,α3 +β3 δα1 ,−β1 w1 (β1 , α3 + β3 ). Lemma 4.12 yields the result. B. Proof of Lemma 4.14. We have τ1 (π+ (t K ,P )) = δ K ,0 δ P,0 δ A1 +A2 −A3 +B1 +B2 −B3 ,0 , τ1 (π− (u K ,P )) = δ K ,0 δ P,0 δ A1 −A2 +A3 +B1 −B2 +B3 ,0 ,
(65) (66)
and τ0 (π+ (t K ,P )) = δr K+ ,P ,0 τ0 (π− (u K ,P )) = δr −
K ,P ,0
∞ + q K ,P,n − δ K ,0 δ P,0 δ A1 +A2 −A3 +B1 +B2 −B3 ,0 , n=0 ∞
(67)
q K−,P,n − δ K ,0 δ P,0 δ A1 −A2 +A3 +B1 −B2 +B3 ,0 . (68)
n=0
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(i) Equations (65) and (66) give (τ1 ⊗ τ1 ) r (A A )0 = δ A1 ,−B1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 λ0,0 . A computation of v0,0 with δ A1 ,−B1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 = 1 gives the result. (ii) Equations (65) and (68) yield τ1 (π+ (t K ,P )) τ0 (π− (u K ,P )) = δ K ,0 δ P,0 δ A2 +B2 ,A3 +B3 δ A1 ,−B1 ×vβ1 ,α1 ,β1 ((α2 + β2 + α3 + β3 )(α1 + β1 ), A3 + B3 ). Equations (67) and (66) yield τ0 (π+ (t K ,P )) τ1 (π− (u K ,P )) = δ K ,0 δ P,0 δ A2 +B2 ,A3 +B3 δ A1 ,−B1
×vβ1 ,α1 ,β1 ((α2 + β2 + α3 + β3 )(α1 + β1 ), A3 + B3 )
and the result follows. (iii) With (42) a direct computation gives τ1 (π+ (t K ,P )) = δ K ,0 δ P,0 δ A1 +A2 −A3 +B1 +B2 −B3 ,0 , (69) τ1 (π− (u K ,P )) = δ K ,0 δ P,0 δ . (70) A1 −A2 +A3 +B1 −B2 +B3 ,0 Using (69) and (70), (τ1 ⊗ τ1 ) r (A A A )◦ = δ A1 ,−B1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 v0,0 . A computation of v0,0 with δ A1 ,−B1 δ A2 ,0 δ A3 ,0 δ B2 ,0 δ B3 ,0 = 1 gives the result. (iv) We have δ(Mβα )Mβα = δ(x)δ(y)x δ(y ) + xδ 2 (y)x δ(y ), where x, x , y, y are monomials (π omitted). Since α k α k2 k2 ∗ k1 k3 ∗ k3 1 π (x) = k a+,α1 a−,α1 b+ b− b+ b− =: k ck , k
k
we get δ(π(x)) = k g(k) αk ck . Similarly, δ(π (y)) = p g( p) βp c p and δ 2 (π (y)) = p g( p)2 βp c p . Thus, with c K ,P := ck c p ck c p , δ(x)δ(y)x δ(y ) = g(k) g( p) g( p ) Kα βP c K ,P , K ,P
xδ (y)x δ(y ) = 2
g( p)2 g( p )
α β K
P
c K ,P ,
K ,P
r (δ(Mβα )Mβα )0 = =:
δh K ,P ,0 (g(k) + g( p)) g( p) g( p )
α β K
P
r (c K ,P )
K ,P
λ K ,P r (c K ,P ).
K ,P
Since r (ck ) = (−q)k1 (−1)α2 +α3 π+ (tk ) ⊗ π− (u k ) with tk , u k defined by
tk := aαk11 bk1 a k2 bk2 a ∗ k3 bk3 and u k := aαk11 bk1 bk2 a ∗ k2 bk3 a k3 , we get
r (δ(Mβα )Mβα )0 =
K ,P
λ K ,P (−q)k1 +k1 + p1 + p1 (−1) A2 +A3 +B2 +B3 π+ (t K ,P ) ⊗ π− (u K ,P ),
Spectral Action on SUq (2)
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where t K ,P = tk t p tk t p and u K ,P = u k u p u k u p . Direct computations yield τ1 π+ (t K ,P ) = δ K ,0 δ P,0 δ A1 +A2 −A3 +B1 +B2 −B3 ,0 , τ1 π− (u K ,P ) = δ K ,0 δ P,0 δ A1 −A2 +A3 +B1 −B2 +B3 ,0 . The result follows. (v) For the last equality, note that by (iv) β 00 β 00 (α1 + β1 )β1 β1 Aα11 000 Aα1 00 δα1 +α1 +β1 +β1 ,0 . − δ(A)A|D|−3 = −2 1
α1 ,α1 ,β1 ,β1
The following change of variables α1 ↔ α1 , β1 ↔ β1 , implies by symmetry that this is equal to zero.
C. Proof of Lemma 4.16. (i) Following notations of Lemma 4.12, we have
Mβα J Mβα J −1 =
v K ,P ck, p J ck , p J −1 ,
K ,P
where K = (k, k ), P = ( p, p ), λ K ,P = g( p)g( p )vk vk w p w p . Thus,
ρ (Mβα J Mβα J −1 ) = (−1) A2 +A3 +B2 +B3
K ,P
(−q)k1 +k1 + p1 + p1 λ K ,P TK+,P ⊗ TK−,P ,
where TK+,P := π+ (tk t p ) π+ (tk t p ) and TK−,P := π− (u k u p ) π− (u k u p ) with
tk := aαk11 b∗ αk11 a k2 b∗ k2 a ∗ k3 bk3 , u k := aαk11 b∗ αk11 bk2 a ∗ k2 b∗ k3 a k3 . A direct computation leads to τ1 (TK+,P ) = δ K ,0 δ P,0 δα1 +α2 −α3 +β1 +β2 −β3 ,0 δα1 +α2 −α3 +β1 +β2 −β3 ,0 ,
τ1 (TK−,P ) = δ K ,0 δ P,0 δα1 −α2 +α3 +β1 −β2 +β3 ,0 δα1 −α2 +α3 +β1 −β2 +β3 ,0 ,
which gives the result. (ii) Using the commutation relations on A, we see that there are real functions of (K , P), denoted σ Kt ,P and σ Ku ,P such that π+ (tk , p ), TK+,P = q σ K ,P π+ (tk, p ) t
π− (u k , p ), TK−,P = q σ K ,P π− (u k, p ) u
p
p
p
p
p2 ∗ a p3 b∗ αk11 b∗ β11 b∗ k2 + p2 bk3 + p3 , tk, p := aαk11 a k2 a ∗ k3 aβ11 a
p3 p2 ∗ k3 + b . u k, p := aαk11 a ∗ k2 a k3 aβ11 a ∗ p2 a p3 b∗ αk11 b∗ β11 bk2 +
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We have, under the hypothesis τ1 (TK−,P ) = 1,
↑α
↑β
1 1 π+ (tk , p )εm,2 j = (−1)λ q (2 j−m)λ q2 j−m−s+β ,|α | q2 j−m−s,|β | εm+s,2 j , 1
s := −α2 + α3 − β2 + β3 = α1 + β1 , λ := α2 + α3 + β2 + β3 , λ := α2 + α3 + β2 + β3 , τ1 (TK+,P ) = δλ,0 δλ ,0 ,
1
1
and then,
(TK+,P )m,2 j = q σ K ,P +sλ (−1)λ q (2 j−m)λ +mλ Fm F2 j−m δ A1 +B1 ,0 , t
↑α
↑β
1 1 F2 j−m := q2 j−m−α ,|α | q2 j−m−α −β ,|β | , 1
Fm :=
1
1
1
1
↑α1 ↑β qm−α q 1 . 1 ,|α1 | m−β1 −α1 ,|β1 |
Following the proof of Lemma 4.7, we see that τ0 (TK+,P ) is possibly nonzero only in the two cases λ = 0 or λ = 0. Suppose first λ = λ = 0. In that case, we have τ0 (TK+,P ) = lim
2 j→∞
2j ↑β ↑ β1 2 1 (qm,|β q ) − 1 1 | 2 j−m,|β | 1
m=0
∞ ∞ ↑ ↑ β1 β1 2 2 (qm,|β | ) − 1 , (qm,|β1 | ) − 1 + = m=0
1
m=0
where the second equality comes from Lemma 4.17. In the case (λ = 0, λ > 0), we get α1 = −β1 and thus, ↑β
↑β
2 1 (TK+,P )m,2 j = q σ K ,P q mλ (qm,|β q 1 ) δα1 +β1 ,0 . 1 | 2 j−m,|β | t
1
↑ 2 j mλ (q ↑β1 q β1 2 mλ (q ↑β1 )2 . m=0 q m,|β1 | 2 j−m,|β1 | ) and L 2 j = m,|β1 | ↑β 1 2 | p|1 q r p q 2(2 j−m)| p|1 , Suppose β1 > 0. Since (q2 j−m,|β ) −1 = | p|1 =0, pi ∈{ 0,1 } (−1) 1| where we have r p = 2 + · · · + 2β1 . As in the proof of Lemma 4.7 (ii), we can conclude that U2 j − L 2 j converges to 0. The case β1 ≤ 0 is similar. In the other case (λ > 0, λ = 0), the arguments are the same, replacing λ by λ and α1 , β1 by α1 , β1 . Finally,
Let us note U2 j =
2 j
m=0 q
τ0 (TK+,P )τ1 (TK−,P ) = δ K ,0 δ P,0 δα1 ,−β1 δα1 ,−β1 ×(δλ ,0 δα2 +β2 ,α3 +β3 sα,β + δλ,0 δα2 +β2 ,α3 +β3 sα ,β ), sαβ : = q β1 (α3 −α2 )
∞ ↑ β1 2 q mλ (qm,|β . ) − δ λ,0 1|
m=0
Spectral Action on SUq (2)
153
A similar computation of τ0 (TK−,P ) can be done following the same arguments. We find eventually τ1 (TK+,P )τ0 (TK−,P ) = δ K ,0 δ P,0 δα1 ,−β1 δα1 ,−β1 ×(δλ ,0 δα2 +β2 ,α3 +β3 sα,β + δλ,0 δα2 +β2 ,α3 +β3 sα ,β ) and the result follows. (iii) The same arguments of (i) apply here with minor changes. (iv) follows from a slight modification of the proof of Lemma 4.14 (iv). (v) is a straightforward consequence of (i, ii, iii, iv). D. Proof of Lemma 4.17. We give a proof for β and β > 0, the other cases being similar. ↑β | p|1 q r p q 2| p|1 m , where p = ( p , · · · , p ) and Since (qm,|β| )2 = 1 β pi ∈{ 0,1 } (−1)
r p := 2( p1 + · · · + βpβ ), we get, with the notations λ p, p := (−1)| p+ p |1 q r p +r p and 2 j ↑β ↑β 2 U2 j := m=0 (qm,|β| q2 j−m,|β | ) − 1, U2 j =
2j
λ p, p q 2| p|1 m+2| p |1 (2 j−m)
m=0 | p+ p |1 >0
= | p|1
≥| p |
λ p, p V2 j, p, p +
1 ,| p|1 >0
| p|1
<| p |
1
,| p |
λ p, p V2 j, p, p , 1 >0
where
V2 j, p, p = q 4 j| p |1
2j
q 2(| p|1 −| p |1 )m ,
V2 j, p, p = q 4 j| p|1
m=0
2j
q 2(| p |1 −| p|1 )m .
m=0
It is clear that V2 j, p, p has 0 for limit when j → ∞ when | p |1 > 0, and V2 j, p, p has 0 for limit when j → ∞ when | p|1 > 0. As a consequence, U2 j =
λ p,0 V2 j, p,0 +
| p |
| p|1 >0
λ0, p V2 j,0, p + o(1).
1 >0
The result follows as 2j
↑β (qm,|β| )2
2j ↑β 2 (qm,|β −1 = λ p,0 V2 j, p,0 and |) − 1 | p|1 >0
m=0
=
| p |1 >0
m=0
λ0, p V2 j,0, p .
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References 1. Bär, C.: The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces. Arch. Math. 59, 65–79 (1992) 2. Bibikov, P.N., Kulish, P.P.: Dirac operators on the quantum group SUq (2) and the quantum sphere. Zap. Nauchn. Sem. St. Petersburg. Otdel. Mat. Inst. Steklov. 245, 49 (1997); Vopr. Kvant. Teor. Polya i Stat. fiz. 14, 49–65 (1997); translated in J. Math. Sci. 100, 2039–2050 (2000) 3. Carminati, L., Iochum, B., Schücker, T.: Noncommutative Yang-Mills and noncommutative relativity: a bridge over troubled water. Eur. Phys. J. C 8, 697–709 (1999) 4. Chakraborty, P.S., Pal, A.: Equivariant spectral triples on the quantum SU (2) group. K-Theory 28, 107–126 (2003) 5. Chakraborty, P.S., Pal, A.: On equivariant Dirac operator for SUq (2). Proc. Indian Acad. Sci. 116, 531–541 (2003) 6. Chakraborty, P.S., Pal, A.: Spectral triples and associated Connes-de Rham complex for the quantum SU (2) and the quantum sphere. Commun. Math. Phys. 240, 447–456 (2003) 7. Chamseddine, A., Connes, A.: The spectral action principle. Commun. Math. Phys. 186, 731–750 (1997) 8. Chamseddine, A., Connes, A.: Inner fluctuations of the spectral action. J. Geom. Phys. 57, 1–21 (2006) 9. Chamseddine, A., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991–1089 (2007) 10. Connes, A.: Noncommutative Geometry. London, San Diego: Academic Press, 1994 11. Connes, A.: Geometry from the spectral point of view. Lett. Math. Phys. 34, 203–238 (1995) 12. Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194–6231 (1995) 13. Connes, A.: Cyclic cohomology, quantum group symmetries and the local index formula for SUq (2). J. Inst. Math. Jussieu 3, 17–68 (2004) 14. Connes, A., Landi, G.: Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221, 141–159 (2001) 15. Connes, A., Marcolli, M.: Noncommutative geometry, quantum fields and motives. To appear, available at http://www.alainconnes.org/docs/bookwebfinal.pdf 16. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 5, 174–243 (1995) 17. D’Andrea, F., D¸abrowski, L.: Local index formula on the equatorial Podle´s sphere. Lett. Math. Phys. 75, 235–254 (2006) 18. D’Andrea, F., D¸abrowski, L., Landi, G.: The isospectral Dirac operator on the 4-dimensional orthogonal quantum sphere. Commun. Math. Phys. 279, 77–116 (2008) 19. D’andrea, F., D¸abrowki, L., Landi, G., Wagner, E.: Dirac operators on all Podle´s spheres. J. Noncommut. Geom. 1, 213–239 (2007) 20. D¸abrowski, L.: Geometry of quantum spheres. J. Geom. Phys. 56, 86–107 (2005) 21. D¸abrowski, L., Landi, G., Sitarz, A., van Suijlekom, W., Várilly, J.: The Dirac operator on SUq (2). Commun. Math. Phys. 259, 729–759 (2005) 22. Essouabri, D., Iochum, B., Levy, C., Sitarz, A.: Spectral action on noncommutative torus. J. Noncommut. Geom. 2, 53–123 (2008) 23. Estrada, R., Gracia-Bondía, J.M., Várilly, J.C.: On summability of distributions and spectral geometry. Commun. Math. Phys. 191, 219–248 (1998) 24. Gayral, V., Iochum, B.: The spectral action for Moyal plane. J. Math. Phys. 46, no. 4, 043503, 17 pp. (2005) 25. Gayral, V., Iochum, B., Várilly, J.C.: Dixmier traces on noncompact isospectral deformations. J. Funct. Anal. 237, 507–539 (2006) 26. Gayral, V., Iochum, B., Vassilevich, D.V.: Heat kernel and number theory on NC-torus. Commun. Math. Phys. 273, 415–443 (2007) 27. Gilkey, P.B.: Asymptotic Formulae in Spectral Geometry. Boca Raton, FL: Chapman & Hall/CRC, 2004 28. Goswami, D.: Some noncommutative geometric aspects of SUq (2). http://arxiv.org/abs/math-ph/ 018003v4, 2002 29. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser Advanced Texts, Boston: Birkhäuser, 2001 30. Grosse, H., Wulkenhaar, R.: 8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory. http://arxiv.org/abs/0709.0095v1[hep-th], 2007 31. Homma, Y.: A representation of Spin(4) on the eigenspinors of the Dirac operator on S 3. Tokyo J. Math. 23, 453–472 (2000) 32. Klimyk, A., Schmüdgen, K.: Quantum Groups and their Representations. Text and Monographs in Physics, Berlin: Springer-Verlag, 1997 33. Knecht, M., Schücker, T.: Spectral action and big desert. Phys. Lett. B 640, 272–277 (2006)
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Commun. Math. Phys. 289, 157–204 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0782-8
Communications in
Mathematical Physics
Surface Tension in the Dilute Ising Model. The Wulff Construction Marc Wouts Université Paris 13, CNRS, UMR 7539 LAGA, 99, Avenue Jean-Baptiste Clément, F-93 430 Villetaneuse, France. E-mail: [email protected] Received: 22 April 2008 / Accepted: 19 January 2009 Published online: 7 April 2009 – © Springer-Verlag 2009
Abstract: We study the surface tension and the phenomenon of phase coexistence for the Ising model on Zd (d 2) with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations: upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value τ q of surface tension to maximal flows (first passage times if d = 2). For a broad class of distributions of the couplings we show that the inequality τ a τ q – where τ a is the surface tension under the averaged Gibbs measure – is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models. Contents 1.
2.
The Model and Our Main Results . . . . . . . . . . . 1.1 The dilute Ising model . . . . . . . . . . . . . . 1.2 The Fortuin-Kasteleyn representation . . . . . . . 1.3 Surface tension . . . . . . . . . . . . . . . . . . 1.4 Low temperature asymptotics . . . . . . . . . . . 1.5 Phase coexistence . . . . . . . . . . . . . . . . . 1.6 Phase coexistence under averaged Gibbs measures 1.7 Organization of the paper . . . . . . . . . . . . . Surface Tension . . . . . . . . . . . . . . . . . . . . 2.1 Sub-additivity and convergence . . . . . . . . . . 2.2 Upper large deviations . . . . . . . . . . . . . . 2.3 Lower large deviations . . . . . . . . . . . . . . 2.4 Surface tension under averaged Gibbs measures .
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158 158 160 162 166 167 169 171 171 171 176 178 180
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2.5 Concentration at low temperatures . . . . . . . . . . . . . 2.6 Concentration in a general setting . . . . . . . . . . . . . . 2.7 Low temperatures asymptotics . . . . . . . . . . . . . . . 3. Phase Coexistence . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Profiles of bounded variation and surface energy . . . . . . 3.2 Covering theorems for BV profiles . . . . . . . . . . . . . 3.3 Lower bound for phase coexistence . . . . . . . . . . . . . 3.4 Upper bound for phase coexistence . . . . . . . . . . . . . 3.5 Localization of the Wulff crystal under averaged measures Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A considerable amount of work permitted to understand on a rigorous basis the phenomenon of phase coexistence in models of statistical mechanics like the Ising model. Phase coexistence in the Ising model was first described in the pioneer work [20], in the two dimensional case and at low temperatures. The construction was then simplified [37] and extended up to the critical temperature [28–30], still in the two dimensional case. The generalization to higher dimensions was achieved later thanks to the L 1 -approach [5,9,11]. The interested reader will find pedagogical presentations of the problem and the methods in the course [10] and the review [6]. The present work is concerned with the phenomenon of phase coexistence for the dilute model with random (ferromagnetic) couplings. The random couplings model either rare defects in the media, or intrinsic randomness. As an example, quenched alloys made of magnetic materials have intrinsic randomness since the strength of the interaction between two spins depends on the nature of the two corresponding atoms. In order to describe rigorously the phenomenon of phase coexistence in presence of random media, we followed the same plan as in the above mentioned works. In a first step we established a coarse graining for the model [44]. In a second step – the present one – we study surface tension. The combination of these tools allow us to describe the phenomenon of phase coexistence in the presence of random media. Before we turn to the presentation of the model and of our results, we would like to stress two consequences of the media randomness on the phenomenon of phase coexistence: first, it is the case that the shape of crystals are smoother than in the presence of uniform couplings. Second, we give an insight to the expected localization phenomenon of the crystal which is determined by the realization of the media under averaged Gibbs measure. The organization of the paper is as follows. In Sect. 1 below we introduce the model and give a complete summary of our results on surface tension, its low temperature asymptotics (maximal flows) and phase coexistence. Proofs and intermediate results are given in Sects. 2 and 3.
1. The Model and Our Main Results 1.1. The dilute vectors of Rd are denoted (ei )i=1...d and for n Ising model. The canonical d any x = i=1 xi ei = (x1 , . . . , xd ) ∈ R we consider the following norms on Rd : x1 =
d i=1
|xi |, x2 =
d i=1
1/2 xi2
d
and x∞ = max |xi |. i=1
(1.1)
Surface Tension in the Dilute Ising Model. The Wulff Construction
159
Given x, y ∈ Zd we say that x, y are nearest neighbors (which we denote x ∼ y) if they are at Euclidean distance 1, i.e. if x − y2 = 1. To any domain ⊂ Zd we associate the edge sets E() = {{x, y} : x, y ∈ and x ∼ y}, and E w () = {x, y} : x ∈ , y ∈ Zd and x ∼ y .
(1.2) (1.3)
We consider in this paper the dilute Ising model on Zd for d 2. It is defined in two steps : first, the couplings between adjacent spins are represented by a random sequence J = (Je )e∈E(Zd ) of law P, such that the (Je )e∈E(Zd ) are independent, identically distributed in [0, 1] under P. Then, given ⊂ Zd a finite domain and a spin configuration + , where σ ∈ + = σ : Zd → {±1} : σz = 1, ∀z ∈ / , we let HJ,+ (σ ) = −
Je σx σ y ,
(1.4)
e={x,y}∈E w ()
the Hamiltonian with plus boundary condition on . The dilute Ising model on with plus boundary condition, given a realization J of the couplings, is the probability meaJ,+ + that satisfies on sure µ 1 β J,+ + µ ({σ }) = J,+ exp − HJ,+ (σ ) , ∀σ ∈ , (1.5) 2 Z ,β J,+ where β 0 is the inverse temperature and Z ,β is the partition function J,+ Z ,β =
+ σ ∈
β exp − HJ,+ (σ ) . 2
(1.6)
Consider m β = lim Eµ J,+ (σ0 ) ˆ N →∞
N ,β
(1.7)
ˆN = ˆ N is the symmetric box the magnetization in the thermodynamic limit, where {−N , . . . , N }d and E the expectation associated with P. When m β > 0 the boundary condition has an influence on the spins at an arbitrary distance. In the region m β > 0 we J,+ say that the Ising model has two phases because the structure of the spins under Eµ J,− (the plus phase) is not the same as the structure of the spins under Eµ (the minus J,− phase), where µ corresponds to the minus boundary condition. It is shown in [1] that the dilute Ising model undergoes a phase transition at low temperature when the random interactions percolate. In our settings, this means that the critical inverse temperature
βc = inf β 0 : m β > 0 , (1.8)
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M. Wouts pure
which is never smaller than βc – the critical inverse temperature for the pure Ising model (J ≡ 1) – is finite if and only if P(Je > 0) > pc (d), where pc (d) is the threshold for bond percolation on Zd . The aim of the paper is to understand the mechanism of phase coexistence in the dilute Ising model, hence we will consider in the following a distribution P of the couplings such that P(Je > 0) > pc (d) and an inverse temperature β > βc . However, some of our results hold on a possibly stronger assumption β > βˆc βc , where βˆc is the critical inverse temperature for slab percolation – see (1.14) below – as this assumption allows us to use the renormalization framework of [44]. 1.2. The Fortuin-Kasteleyn representation. The study of surface tension for the dilute Ising model will be led under the random-cluster model that corresponds to the measure J,+ . We call µ,β = ω : E(Zd ) → {0, 1} the set of cluster configurations on E(Zd ), and for any ω ∈ and E ⊂ E(Zd ) we call ω|E the restriction of ω to E, defined by ωe if e ∈ E (ω|E )e = 0 else. The set of cluster configurations on E is E = {ω|E , ω ∈ }. Given a parameter q 1 and an inverse temperature β 0, a realization of the random couplings J : E(Zd ) → [0, 1], a finite edge set E ⊂ E(Zd ) and a boundary condition π ∈ E c we consider the J,π,q random cluster model E,β on E defined by J,π,q
E,β ({ω}) =
1
J,π,q Z E,β e∈E
π
peωe (1 − pe )1−ωe × q C E (ω) ,
∀ω ∈ E ,
(1.9)
where pe = 1 − exp(−β Je ), C Eπ (ω) is the number of clusters of the set of vertices in Zd J,π,q attained by E under the wiring ω ∨ π such that (ω ∨ π )e = max(ωe , πe ), and Z E,β is J,π,q
the renormalization constant making E,β a probability measure. J,π,q
For convenience we use the same notation for the probability measure E,β and for its expectation. Most often we will take either π = f , where f is the free boundary condition : f e = 0, ∀e ∈ E c , or π = w where w is the wired boundary condition : we = 1, ∀e ∈ E c . When the parameters q and β are clear from the context we omit them. Given R a compact subset of Rd (usually a rectangular parallelepiped) we denote J,π ˙ ∩ Zd ), where R ˙ the measure J,π˙ d on the cluster configurations on E(R by R E(R∩Z )
J,π stands for the interior of R. In particular, for any g, h : → R the quantities R (g) 1
J,π and R (h) are independent under P. 2
J,+ The connection between the dilute Ising model µ,β and the random-cluster model was made explicit in [22]. Consider the joint probability measure 1{σ ≺ω} J,+ +
,β × E() , ({(σ, ω)}) = ( pe )ωe (1 − pe )1−ωe , ∀(σ, ω) ∈ J,+ Z˜ ,β w e∈E ()
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where pe = 1−exp(−β Je ), σ ≺ ω is the event that σ and ω are compatible, namely that J,+ is the corresponding normalizing ωe = 1 ⇒ σx = σ y , ∀e = {x, y} ∈ E w (), and Z˜ ,β factor. Then, J,+ J,+ i. The marginal of ,β on the variable σ is the Ising model µ .
J,w,2 ii. Its marginal on the variable ω is the random-cluster model E(),β with wired boundary condition w and parameter q = 2. iii. Conditionally on ω, the spin σ of each connected component of for ω (now cluster) is constant, and equal to +1 if the cluster is connected to c . The spin of all clusters not touching c are independent and equal to +1 with a probability 1/2. iv. Conditionally on σ , the edges are open (i.e. ωe = 1 for e = {x, y}) independently, with respective probabilities pe δσx ,σ y .
According to point ii and iii we can study surface tension for the Ising model under the Fortuin-Kasteleyn representation. This representation allows to study at the same time the surface tension and the phenomenon of phase coexistence for the dilute Ising model (q = 2), but also for dilute percolation (q = 1) and for the dilute Potts model (q ∈ {3, 4, . . .}). An important benefit of the representation is that it makes possible the use of the comparison inequalities for the random cluster model. We say that a function f : E → R+ is increasing if, for all ω, ω ∈ E one has ω ω ⇒ f (ω) f (ω ), where stands for the product order on E . It was shown in [2] that: J,π,q
i. For any h : E → R+ increasing, E,β (h) is a non-decreasing function of J , β and π . ii. (FKG inequality) For any g, h : E → R+ increasing, J,π,q
J,π,q
J,π,q
E,β (gh) E,β (g) E,β (h).
(1.10)
iii. (DLR Equation) For any E E and ω ∈ E , J,π,q
E,β
J,π ∨ω ,q .|ω|E = ω = E\E ,β .
(1.11)
Finally, let us recall the assumption of slab percolation, that is the basis for a renormalization framework in the dilute Ising model [44]. When d 3, we say that slab J, f,q percolation occurs under E β if, for large enough H , inf
inf
L∈N x,y∈SL ,H
ω J, f E SL ,H x ↔ y > 0,
(1.12)
where SL ,H is the slab SL ,H = {1, . . . , L}d−1 × {1, . . . H }. When d = 2, we say that slab percolation occurs when there exists κ : N → N with lim N →∞ κ(N )/N = 0 such that J, f
lim E SN ,κ(N ) (there is an horizontal crossing for ω) > 0.
(1.13)
The critical inverse temperature for slab percolation is J, f,q , βˆc = inf β 0 : slab percolation occurs under E β
(1.14)
N →∞
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Fig. 1. The rectangular parallelepiped Rx,L ,H (S, n)
it satisfies βˆc βc , where βc is the critical inverse temperature for phase transition in the dilute Ising (resp. Potts) model. We believe that βˆc and βc do coincide. Upper bounds on βˆc are derived in [44] from the argument of [1]. The technical assumption β > βˆc allows us to use a coarse graining, which is a fundamental tool at the moment of defining the local phase of the dilute Ising model (see Theorem 5.7 in [44], or Sect. 3 below). 1.3. Surface tension. One of the main issues we address in this paper is the behavior of surface tension and the influence of the random couplings. We consider the surface tension in a large rectangular parallelepiped oriented along some direction n ∈ S d−1 , where S d−1 is the set of unit vectors of Rd . The other axes of the parallelepiped are represented by S ∈ Sn , where d−1 d Sn = [±1/2]uk ; (u1 , . . . , uk−1 , n) is an orthonormal basis of R k=1
is the set of d − 1 dimensional hypercubes of side-length 1, centered at 0, orthogonal to n ∈ S d−1 . Finally, we call x ∈ Rd the center of the rectangular parallelepiped and L , H its side-lengths, and denote finally by Rx,L ,H (S, n) = x + LS + [−H, H ]n
(1.15)
the rectangular parallelepiped centered at x, with basis x + LS and extension 2H in the ˆ =R ˙ ∩ Zd and the inner discrete direction n (see Fig. 1). The discrete version of R is R boundary of R is ˆ = y∈R ˆ : ∃z ∈ Zd \R, ˆ z∼y . ∂R ˆ = {y ∈ ˆ into its upper and lower parts ∂ + R For any R as in (1.15) we decompose ∂ R − ˆ : (y − x) · n 0} and ∂ R ˆ = {y ∈ ∂ R ˆ : (y − x) · n < 0}. ∂R In the context of statistical physics, the surface tension is the excess free energy per surface unit due to the presence of an interface. The surface tension in R thus quantifies the probability of observing the plus phase in the upper part of R and the minus phase J with free boundary condition. It is more conin the opposite part under the measure µR venient to formulate the definition under the random cluster model, where we translate the event of phase coexistence into an event of disconnection.
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Definition 1.1. Let R be a rectangular parallelepiped as in (1.15). The event of disconˆ is nection between the upper and lower parts of ∂ R ω − ˆ ˆ (1.16) ∂ R DR = ω ∈ : ∂ + R and the surface tension in R is J =− τR
1 J,w log R (DR ) . L d−1
(1.17)
Although the surface tension depends on the inverse temperature β, we have chosen to keep this dependence implicit in the notation. We denote by J min and J max the lowest and largest values of the couplings according to the support of P, that is to say: J min = inf{λ 0 : P(Je < λ) > 0}, and J max = sup{λ 0 : P(Je > λ) > 0}. min (resp. τ max ) – which again depend on β – the value of the surface We also denote by τR R tension in R corresponding to the constant couplings J ≡ J min (resp. J ≡ J max ). We have: √ Proposition 1.2. Let R be a rectangular parallelepiped as in (1.15), with L , H 2 d. J is a non-decreasing function of J and β. It is a non-increasing The surface tension τR function of H . With probability one under P, min J max 0 τR (n) τR (n) τR (n) cd β J max ,
where cd < ∞ depends on d only. As in the uniform case [36], surface tension is sub-additive (see Theorem 2.1), and J (n). this implies convergence in probability of τR Theorem 1.3. There exists τ q (n) 0 (a function of β), the quenched surface tension, such that, for all β 0 and n ∈ S d−1 , lim τ J N →∞ R0,N ,δ N (S ,n)
= τ q (n) in P-probability
whatever is S ∈ Sn and δ > 0. Similarly, the surface tension for the constant couplings J min and J max also converge and we denote by τ min (n) and τ max (n) their respective limits. The sub-additivity is of much help for controlling the order of deviations from the quenched value of the surface tension τ q (n). The large deviation results below imply in particular that surface tension does in fact converge almost-surely for all but at most countably many β. Upper large deviations happen at a volume order, hence they have no influence on the phenomena we study here: Theorem 1.4. For any β 0, any ε > 0 and δ > 0, 1 J q lim sup d log P τR τ (n) + ε < 0. ( S , n ) 0,N ,δ N N N
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Fig. 2. Large deviations of surface tension
We will be more concerned with lower large deviations. These are possible when τ min (n) < τ q (n). The inequality is known to be strict only in two specific cases: when J min = 0 and β > βˆc , it is the case that τ min (n) < τ q (n) because τ min (n) = 0, while the coarse graining [44] implies: Proposition 1.5. Assume β > βˆc . For any n ∈ S d−1 , τ q (n) > 0. When P(Je > J min ) > pc (d) and β is large enough, the strict inequality τ min (n) < also holds, cf. Corollary 2.16 below. When lower large deviations occur, they have at most surface order. It is another consequence of sub-additivity that: τ q (n)
Theorem 1.6. For every β 0 and τ > τ min (n), the limit 1 J In (τ ) = lim − d−1 log P τR τ 0,N ,δ N (S ,n) N N
(1.18)
exists in [0, +∞), depends on β but not on δ > 0, nor on S ∈ Sn . In is continuous, convex non-increasing, and In (τ ) = 0 for τ τ q (n). For convenience, we extend the definition of In letting +∞ for τ < τ min (n) In (τ ) = limε→0+ In (τ + ε) at τ = τ min (n).
(1.19)
In order to show that lower deviations are exactly of surface order, we need to prove that In is positive on the left of τ q . We developed an argument based on measure concentration coupled with a control of the length of the interface. In the case of the Ising model at low temperatures we could establish a first control: Theorem 1.7. Assume q = 2 and J min > 0. Then, for β large enough there exists c > 0 such that, for all r > 0, In (τ q (n) − r ) cr 2 .
(1.20)
In the general case a careful adaptation of the method yields: Theorem 1.8. For every n ∈ S d−1 , for Lebesgue-almost all β 0, lim sup r →0+
In (τ q (n) − r ) > 0. r2
(1.21)
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These quadratic lower bounds on the rate function generalize common controls for directed polymers models [8,16], which were introduced in order to represent interfaces in the two-dimensional Ising model with random couplings at low temperatures [16,27]. Its is probable however that the quadratic order in the former theorems is not optimal in two dimensions, as the comparison with directed polymers suggests that In (τ q (n) − r ) ∼ + cr 3/2 . r →0
(1.22)
This scaling has been established rigorously for the zero-temperature limit of directed polymers: last passage percolation, for a geometric distribution of the passage times, see Theorem 1.1 and (2.23) in [31]. Some of our results on phase coexistence and on the dynamics of the dilute Ising model [45] require that lower large deviations are actually of surface order. An easy but important consequence of Theorem 1.8 is: Corollary 1.9. The lower large deviations are of surface order when β → τ q (n) is left continuous. Hence the set (1.23) N I = β 0 : ∃n ∈ S d−1 and r > 0 such that In (τ q (n) − r ) = 0 is at most countable. We end the presentation on surface tension with the definition of the surface tension under the averaged Gibbs measure. It is the Fenchel-Legendre transform of In , τ λ (n) = inf {λτ + In (τ )} , τ ∈R
(1.24)
which coincides with the surface tension under an average of Gibbs measures: λ 1 J,w λ , λ > 0, n ∈ S d−1 , (1.25) τ (n) = lim − d−1 log E R N DR N N →∞ N where R N = R0,N ,δ N (S, n), as shown in Proposition 2.3. The particular case λ = 1 corresponds to the usual notion of surface tension under the averaged measure (or annealed surface tension) and we denote τ a (n) = τ λ=1 (n). The asymptotics of τ λ /λ as λ → 0 or +∞ are given in Proposition 2.4. An important question about the surface tension under the averaged Gibbs measure is whether the random media is able to turn Jensen’s inequality τ λ (n) λτ q (n)
(1.26)
into a strict inequality. A partial answer to this question is given in the next section. Let us make explicit the connection between the strict inequality τ λ (n) λτ q (n) and the asymptotics of In on the left of τ q (n): the opposite of the slope of In on the left of τ q (n) is exactly
αn = sup λ > 0 : τ λ (n) = λτ q (n) , (1.27) with the convention that sup ∅ = 0. Finally, as in the non-random case, the homogeneous extension of each of these notions of surface tension τ q , τ λ , τ min and τ max are convex and continuous (Proposition 2.5).
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1.4. Low temperature asymptotics. The low temperature asymptotics of surface tension permit to give a more precise insight into the properties of surface tension in random media. First we need to introduce the concept of maximal flow through the capacities J , where J = (Je )e∈E(Zd ) is the family of random couplings introduced in the former section. Here we give only a brief overview of maximal flows. The reader is invited to consult [32,33] for a pedagogical introduction. Recent results on maximal flows, including large deviations, can be found in [39,42,47]. We will use an analogy for describing maximal flows. Imagine a liquid which has to cross a lattice made of tubes with limited capacity. Then, the maximal flow, in a given direction, is the quantity of liquid that can flow through the lattice, per unit of surface. ˆ we consider the Given a rectangular parallelepiped R as in (1.15) and I ⊂ E(R), event that ω is closed on I : Z I = {ωe = 0, ∀e ∈ I }. We say that I is an interface for R if Z I ⊂ DR and ∀e ∈ I, Z I \{e} ⊂ DR . In other ˆ words, I is an interface for R if the disconnection on I is enough for disconnecting ∂ + R − ˆ from ∂ R and if there is no superfluous edge in I . This notion of interface corresponds to the geometrical notion of interface if, to the edges of I we associate their dual d − 1 dimensional facets. ˆ to According to the max-flow min-cut Theorem [7], the maximum flow from ∂ − R + ˆ by the edges of capacities Je is also the flow through the interface of minimal capac∂ R ity. We use this characterization for our definition. Given a rectangular parallelepiped R = Rx,L ,H (S, n) as in (1.15) we call I(R) the set of interfaces for R and define the maximal flow in R, for a realization J of the media, as 1 J µR = d−1 inf Je . (1.28) L I ∈I (R) e∈I
This quantity has the same properties as surface tension since it is as well sub-additive. In particular, the maximal flow in R0,N ,δ N (S, n) converges in P-probability, upper deviations occur at volume order and lower deviations occur at surface order [15,41]. We will make use of the following results: Theorem 1.10. There exists µ(n) ∈ [0, +∞), the maximal flow for the distribution P in the direction n ∈ S d−1 , such that, for any δ > 0 and S ∈ Sn , J −→ µ(n) in P-probability. µR 0,N ,δ N (n,S ) N →∞
It is positive if and only if P(Je > 0) > pc (d). Furthermore, J min n1 µ(n)
(1.29)
and the inequality is strict when P(Je > J min ) > pc (d). The convergence of the maximal flow is a consequence of the sub-additivity. It is shown in [46] that the maximal flow is 0 when P(Je > 0) pc (d), while its positivity was established in [13], under the conjecture that the critical threshold for percolation and slab percolation do coincide, proved later on in [26]. The inequality (1.29) is easily obtained from the remark that minimal interfaces have cardinal of order N d−1 n1 . When P(Je > J min ) > pc (d), for small ε > 0 there is a percolating net of edges with
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values Je J min + ε [26,38], which is responsible for the strict inequality. See also Proposition 4.1 in [39]. It turns out that the maximal flow determines the asymptotics of the quenched value of surface tension at low temperatures, as τ q (n) = µ(n) β→+∞ β lim
(1.30)
when P(Je > 0) = 1 (see Sect. 2.7 and references therein). On the other hand, the asymptotics of the surface tension under averaged Gibbs measure are determined by J min : for any P and any λ > 0, τ λ (n) = λJ min n1 . β→+∞ β lim
(1.31)
These asymptotics imply that the inequality τ λ (n) λτ q (n) is strict at low temperatures in a number of cases (see Corollary 2.16). They also have consequences on the shape of crystals under both the quenched and the averaged Gibbs measure (see below). 1.5. Phase coexistence. We describe finally the phenomenon of phase coexistence in the dilute Ising model. Phase coexistence occurs when both the plus and the minus phase are present at the same time and occupy (distinct) regions of the domain. This phenomenon does not occur naturally in the Ising model. One way of obtaining phase coexistence is J on the event that the overall magnetization by conditioning the measure µ m =
1 σx ||
(1.32)
x∈
is smaller than m < m β . Under this conditional measure, we will show that the two phases do coexist and that the minus phase occupies a fraction of the volume v such that (1 − 2v)m β = m. Furthermore, the shape U of the region containing the minus phase is deterministic : if τ is the surface tension of the model, the observed shape minimizes the surface energy F(U ) = τ (n)ds, ∂U
under the volume constraint Vol(U ) v, and this implies that U is a translate of v 1/d W, where W is the renormalized Wulff crystal associated to τ : (1.33) W = λ x ∈ Rd : x · n τ (n), ∀n ∈ S d−1 where λ > 0 is chosen such that Vol(W) = 1. Before we state our results, let us recall that N I stands for the at-most-countable set of β at which lower large deviations for surface tension are possible at less than surface order (Corollary 1.9) and βˆc is the slab percolation threshold (1.14). Another important notation is J, f , (1.34) N = β 0 : lim E ˆ = lim E J,w ˆ N →∞
N
N →∞
N
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the set of β such that infinite volume averaged FK measures are not unique. N is at most countable (Theorem 2.3 in [44]). We denote by W q (resp. W λ ) the Wulff crystal associated with the surface tension q τ (resp. τ λ ) as in (1.33), and v = α d the fraction of the volume occupied by the minus phase. The Wulff crystal αW of volume v fits into the unit box [0, 1]d only if α diam∞ (W) 1, where diam∞ (A) = sup x − y∞ ,
A ⊂ Rd .
x,y∈A
Our first theorem concerns the cost of the lower large deviations for the magnetization. In the sequel, N = {1, . . . , N }d . / N . Then, for all 0 α < 1/ diam∞ (W q ), Theorem 1.11. Assume β > βˆc with β ∈
1 N d−1
J,+ log µ N
m N 1 − 2α d mβ
−→ −F q (αW q ) in P-probability. (1.35)
N →∞
Then we describe the geometry of the two phases. We consider a mesoscopic scale K ∈ N and define the magnetization profile M K as M K : [0, 1]d −→ x −→
1 Kd
[−1, 1]
z∈ N ∩i(x)
σz ,
(1.36)
where i(x) =
N xd N x1 ,..., and i = K i + {1, . . . , K }d . K K
(1.37)
Hence, unless x is too close to the border of [0, 1]d , M K (x) is the magnetization in a block of side-length K that contains N x. Theorem 5.7 in [44] provides a strong stochastic control on M K when β > βˆc . In particular, when K is large enough, at every x the probability that M K (x) is close to either m β or −m β is close to one under the J,+ . Hence M K /m β describes the geometry of the phases in the averaged measure Eµ N Ising model: it is close to one on the plus phase region, close to minus one on the minus phase region. We need a few more notations. To U ⊂ Rd Borel measurable, we associate the profile χU : x ∈ R → d
1 −1
if x ∈ /U else
(1.38)
and denote by . L 1 the norm of the L 1 -space L 1 [0, 1]d ; R . We also consider the set of vectors z such that the translate z + U fits into [0, 1]d : (1.39) T (U ) = z ∈ Rd : z + U ⊂ [0, 1]d . Our second theorem describes the geometrical structure of the two phases when they coexist:
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Theorem 1.12. Assume that β > βˆc and β ∈ / N . For all 0 α < 1/ diam∞ (W q ) and ε > 0, for any K large enough one has MK m N J,+ d q inf = 1 (1.40) lim µ − χ ε 1 − 2α z+α W m N z∈T (α W q ) m N →∞ β β L1 in P-probability (P-a.s. when β ∈ / N I ). Note that, although we state our theorems for the Ising model, they could easily be adapted to the Potts model with random interactions, or to random-cluster models, as the two fundamental tools for the study of phase coexistence, the coarse graining [44] and the study of surface tension, were developed in the more general setting of the random-cluster model (q 1) with random couplings. In the former theorem we have taken K finite. In general, one can take any K = K N such that 1 K N N because on the one hand, M K N is close to the local mean of M K as K N 1, and this local mean is close to χz+α W because the K N -blocks intersecting N ∂(z + αW) contribute to a negligible volume as K N N . Let us conclude this paragraph on a first consequence of the presence of random couplings : the limit shape of the droplet at low temperatures is modified. In the case of the pure Ising model, the Wulff crystal converges to the unit hypercube [±1/2]d as the temperature goes to zero. Here, W q converges to the Wulff crystal W µ associated with the maximal flow µ when P(Je > 0) = 1 (Proposition 2.17). Little is known on that crystal, yet, Durrett and Liggett [21] have shown that W µ is not a square when d = 2 and, for instance, P (Je = 1/2) = p and P (Je = 1) = 1 − p → → pc is the critical threshold for oriented bond percolation. with − pc < p < 1, where − 1.6. Phase coexistence under averaged Gibbs measures. Now we consider the issue of phase coexistence under averaged Gibbs measures, that is, when phase coexistence is imposed on both the spin configuration and the random couplings. Before we go further, we would like to remark that averaged Gibbs measures do not have the physical meaning of the quenched measure: in quenched ferromagnets, the disorder is frozen and thus cannot be influenced by the spin configuration itself. However, the analysis presented here gives an insight on the phenomenon of localization which can occur in models with media randomness. First we remark that the cost for phase coexistence is here determined by the surface tension τ λ (n): Theorem 1.13. For all λ > 0 and 0 α < 1/ diam∞ (W λ ), λ 1 J,+ m N d log E µ N 1 − 2α −→ −F λ (αW λ ). N →∞ N d−1 mβ
(1.41)
The inequality τ λ < λτ q at low temperatures (Corollary 2.16) implies that F λ (W λ ) F λ (W q ) < λF q (W q ), in other words the cost for phase coexistence is strictly smaller under the averaged Gibbs measure than under the quenched Gibbs measure. One can go further and analyze the
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cost for reducing the cost for phase coexistence under averaged Gibbs measures: the functional J ( f ) = sup{F λ (W λ ) − λ f } ∈ [0, ∞], λ>0
f ∈R
is the rate function for lower deviations of the cost for phase coexistence. If W min and F min stand respectively for the Wulff crystal and the surface energy associated to τ min , then J is infinite on the left of F min (W min ), finite on the right of F min (W min ) and zero on the right of F q (W q ), and: Corollary 1.14. For any f = F min (W min ) and α 0 small enough, 1 1 J,+ m N d d−1 lim d−1 log P log µ N 1 − 2α −α f = −α d−1 J ( f ). N N N d−1 mβ Upper deviations for the cost of phase coexistence, on the other hand, happen at volume order (cf. the proof of Proposition 3.14). The shape of crystals under averaged Gibbs measures is as well determined by the surface tension τ λ (n): Theorem 1.15. For any 0 α one has J,+ lim Eµ inf N N →∞
< 1/ diam∞ (W λ=1 ) and ε > 0, for any K large enough
MK m N d = 1. − χ ε 1 − 2α λ=1 z+α W m 1 m z∈T (α W λ=1 ) β β L (1.42)
This result extends in fact to all λ > 0, at the price however of heavier notations, J,+ because E[(µ (.))λ ] is not a measure when λ = 1: N Theorem 1.16. For any λ > 0, any 0 α < 1/ diam∞ (W λ ) and ε > 0, for any K large enough one has λ mN MK J,+ d E µ N inf z∈T (α W λ ) m β − χz+α W λ 1 ε and m β 1−2α L lim = 1. m λ N →∞ J,+ N d E µ N m β 1−2α We conclude the summary of our results with a description of a phenomenon of J under localization. First, let us characterize the typical value of the surface tension τR the averaged measure, conditioned to phase coexistence. For any β 0, λ > 0 and n ∈ S d−1 , this value stands between
τˆ λ,− (n) = inf τ 0 : In (τ ) + λτ = τ λ (n) , (1.43)
λ,+ λ and τˆ (n) = sup τ 0 : In (τ ) + λτ = τ (n) . (1.44) The equality τˆ λ,− (n) = τˆ λ,+ (n) holds whenever there is at most one τ at which the slope of In equals λ, that is, for all but at most countably many values of λ > 0. Note also that the strict inequality τ λ (n) < λτ q (n) implies τˆ λ,+ (n) < τ q (n). We also consider a similar quantity for the quenched value of surface tension: τ˜ q (n) = inf {τ : In (τ ) = 0} ,
(1.45)
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which coincides with τ q (n) for all n ∈ S d−1 , for all but at most countably many β 0, see Corollary 1.9. Our last theorem describes the value of the surface tension τ J conditionally on the position of the crystal: we prove that the typical value of surface tension is τ q outside the boundary of the crystal, while on the boundary it is reduced to τˆ λ . When τ λ (n) < λτ q (n) for some n ∈ S d−1 , the location of the Wulff crystal under averaged Gibbs measures is thus determined by the realization of the media: the boundary of the crystal coincides with the place where surface tension is reduced. Theorem 1.17. Let λ > 0, 0 α < 1/ diam∞ (W λ ) and z ∈ T (αW λ ). Consider h, δ, γ > 0 and a parallelepiped rectangle R = Rx,h,δh (n, S) ⊂ (0, 1)d as in (1.15). Call R N = N R + z N (R), where z N (R) ∈ (−1/2, 1/2]d is chosen such that the center of R N belongs to Zd . For ε > 0 small enough and K large enough, λ J,+ M K E 1 J × µ N m β − χz+α W λ 1 ε τ N ∈A L R lim =1 λ N →∞ J,+ M K E µ N m β − χz+α W λ 1 ε L
when
i. R ∩ z + α∂W λ = ∅ and A = τ˜ q (n) − γ , τ q (n) + γ , λ ii. or x ∈ z + α∂W λ , n is the outer local normal to z + αW at x, h is small enough λ,− λ,+ and A = τˆ (n) − γ , τˆ (n) + γ .
1.7. Organization of the paper. The remainder of the paper is organized as follows. In Sect. 2 we present the proofs of our results on surface tension. The sub-additivity and convergence are detailed in Sect. 2.1, upper and lower large deviations in Sects. 2.2 and 2.3. The surface tension under averaged Gibbs measures is discussed in Sect. 2.4, while the concentration properties appear in Sects. 2.5 (for low temperatures) and 2.6 (in a more general settings). The low temperature asymptotics of surface tension are given in Sect. 2.7. In Sect. 3 the phase coexistence theorems are established. After some preliminaries we establish detailed lower and upper bounds for phase coexistence in Sects. 3.3 and 3.4, and conclude with the proof of the localization properties in Sect. 3.5. 2. Surface Tension As announced in the former section, surface tension is a fundamental tool for understanding the mechanism of phase coexistence. It quantifies the free energy per surface unit of an interface separating the plus and minus phases in the dilute Ising model. In this section, we prove the convergence of surface tension in dilute models and study its large deviations. 2.1. Sub-additivity and convergence. In many aspects the surface tension for the dilute Ising model is similar to the one of the Ising model with deterministic couplings. It has the crucial property of being sub-additive, as in the uniform case [36]: this is shown in Theorem 2.1 below. We present here the proof of Proposition 1.2, Theorem 2.1 and finally
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Theorem 1.3. We also explain why surface tension is positive under the assumption that β > βˆc (Proposition 1.5). J is a non-decreasing function of J and Proof (Proposition 1.2). The surface tension τR J stochastically increases with β because DR is a decreasing event while the measure R p = 1 − exp(−β Je ). Now we consider H H and call R = Rx,L ,H (S, n) and R = Rx,L ,H (S, n). In J,π J,w view of the DLR equation and of the monotonicity of R along π , the measure R
ˆ is stochastically smaller than J,w . On the other hand, it is clear that restricted to E(R) R DR ⊂ DR , and because DR is a decreasing event we conclude that J,w J,w J,w R (DR ) R
(DR ) R (DR ), J is a non-increasing function of H . which shows that τR min 0. The inequality τ min τ J τ max is It is clear from the definition that τR R R R max a consequence of the monotonicity in J . We conclude with the upper bound on τR . √ Because of the monotonicity in H we can take H = 2 d (which ensures that dismax τ max , where R = R √ connection is still possible). We have: τR x,L ,2 d (S, n). It is R . The DLR equation, to realize the disconnection in R enough to close all the edges of R J max ,π along the boundary condition π yields: combined with the monotonicity of {e} max τR
−
1 L d−1
log
max ,w
) e∈E(R
J {e}
({ωe = 0}) = β J max
)| |E(R . d−1 L
, which is itself not larger )| is not larger than 2d times the cardinal of R Finally, |E(R
d ⊂R √ √ (S, n). Consequently, than the volume of V = x∈R x + [0, 1] 0,L+2 d,3 d
max
J τR
β J max × 2d ×
√ d−1 √ L +2 d ×6 d L d−1
β J max × 6 × 2d d 3/2 .
Now we address the issue of sub-additivity. It is a fundamental tool not only for proving the convergence of surface tension, but also for establishing the large deviations principles in the next section. √ √ Theorem 2.1. Consider n ∈ S d−1 , S, S ⊂ Sn and H, l 2 d, L 4 dl. Let R = R0,L ,H +√d/2 (S, n). There is a collection (Ri )i∈C of rectangular parallelepipeds Ri = Rzi ,l,H (S , n) that are disjoint subsets of R, centered at z i ∈ Zd , with d−1 l 1 l + 1 − cd |C| 1 (2.1) L l L ˆ → [0, 1]: such that, for any J : E(R) J τR
1 J τRi + βcd |C| i∈C
l 1 , + L l
where cd < ∞ is a constant that depends on d only.
(2.2)
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Fig. 3. The rectangular parallelepiped R and the collection (Ri )i∈C in Theorem 2.1
Let us make a few comments on this theorem. First, a key feature of the sub-additivity J under P since the R are as formulated in Theorem 2.1 is the independence of the τR i i J disjoint. Note that as well, the τRi have the same law as the Ri are all centered at lattice points. Three error terms appear in Theorem 2.1. Their origins are as follows (see also Fig. 3): i. the term βcd /l stands for the cost of disconnection in the middle section of R between adjacent Ri , ii. the term βcd l/L represents the cost of disconnection in the area not covered by the Ri , √ iii. and the increase of H by d/2 for R with respect to the Ri is a consequence of the requirement that the Ri be all centered at lattice points. The last error term could be avoided for rational directions n ∈ S d−1 , yet (as the two others) it will soon disappear when we take the limit H → ∞. The reader will notice that the use of the FK representation permits to give a relatively short proof of Theorem 2.1. Proof (Theorem 2.1). We begin with the definition of z i and C. We call (e k )k=1...d−1 the edges of S and e d = n, so that (e k )k=1...d is an orthonormal basis of Rd . For all i = (i k )k=1...d−1 ∈ Zd−1 we define z i as the unique point of Zd such that d−1 √ 1 1 d l+ d i k e k ∈ z i + − , 2 2 k=1
and call
C = i ∈ Zd−1 : Ri ⊂ R ,
letting R = R0,L ,H +√d/2 (S, n) and Ri = Rzi ,l,H (S , n).
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We proceed with the proof of (2.1) first. We call Hn the hyperplane of Rd orthogonal to n that contains 0 and remark that the orthogonal projections of z i + lS (for all i ∈ C) on Hn are disjoint and all included in LS. Hence their total surface |C|l d−1 does not exceed the surface of LS, namely L d−1 , and the upper bound in (2.1) follows. Reusing the previous notations we call d−1 √ z i = l + d i k e k ,
∀i ∈ Zd−1
k=1
so that z i ∈ Hn . We consider then √ C = i ∈ Zd−1 : z i + l + d S ⊂ LS . √ In view of the inequality d(z i , z i ) d/2 it follows that z i + lS ⊂ R, for all i ∈ C , √ hence C ⊂ C. On the other hand, for any i ∈ Zd−1 such that z i + (l + d)S ∩ (L − √ √ 2 d(l + d))S = ∅ we have i ∈ C , hence
√ d−1 √ L − 2 d(l + d) |C | |C| √ l+ d
and d−1 √ l d−1 l l |C| √ −2 d L L l+ d d−1 √ √ l d −2 d 1− l L √ √ l d +2 d 1 − (d − 1) , l L which yields the lower bound for (2.1). We pass now to the proof of (2.2) and call √ ! ˆ " i ) : d(e, Hn ) d , E(R E = e ∈ E(R)\ 2 i∈C
where d(e, Hn ) stands for the shortest distance between one extremity of e and Hn . The inclusion # # {ωe = 0, ∀e ∈ E} ⊂ DR DRi i∈C
holds: consider ω that belongs to the left-hand side and let c be an ω-open path issued ˆ Every time c enters some R ˆ i by the upper boundary ∂ + R, ˆ it also exits by the from ∂ + R. same upper boundary since ω ∈ D ˆ . As c cannot use the edges of E it is not able to cross Ri
ˆ i , and in particular it cannot reach the middle hyperplane Hn elsewhere than in the R
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ˆ Since the D ˆ as well as the {ωe = 0} are decreasing events, the DLR equations ∂ − R. R i
and the monotonicity along the boundary condition for J imply that J,w J,w J,w (DR ) Ri (DRi ) × {e} ({ωe = 0}) ˆ R
i∈C
i∈C
e∈E
J,w
Ri (DRi ) × exp (−β|E|)
(2.3)
J,w as {e} ({ωe = 0} = 1 − pe = exp (−β Je ) exp(−β). We proceed then with an esti
mate over the cardinality of E: we call F = x ∈ Zd : ∃y, {x, y} ∈ E the set of extremities of some e ∈ E and remark that |E| d Vol (V ), where V = x∈F x + [0, 1]d . We have
V ⊂ R0,L+2√d,3√d/2 (S, n) while V ∩ Rzi ,l−2√d,∞ (S , n) = ∅, ∀i ∈ C, hence
√ √ d−1 √ d−1 l 1 3 d |E| d × cd L d−1 × L +2 d + − |C| l − 2 d 2 L l
in view of the lower bound in (2.1). Taking logarithms in (2.3) and dividing by −L d−1 we obtain the inequality d−1 1 l l J J + τR + c β τR d i L L l i∈C
and (2.2) follows from the upper bound in (2.1). J ) We conclude with a word on the structure of the sequence (τR i∈C . The Ri are i ˆ i ), hence the τ J are independisjoint by construction, hence so are the edge sets E(R Ri
dent. They are identically distributed as the Ri are all centered at lattice points, P being translation invariant as a product measure.
Now we establish the convergence for surface tension and prove theorem 1.3. The proof of this theorem is based on the sub-additivity of surface tension. We do not apply directly Kingman’s Sub-additive Theorem [34] as we want to show that τ q does not depend on S, nor on δ. Proof. Taking the expectation E in the sub-additivity inequality (2.2) we get l 1 J J EτR + . EτR0,l,H (S ,n) + βcd √ 0,L ,H + d/2 (S ,n) L l Applying lim sup L→∞ , then lim inf l→∞ and taking the decreasing limit in H we obtain J J lim lim sup EτR lim lim inf EτR ,
0,L ,H (S ,n) 0,L ,H (S ,n)
H →∞ L→∞
H →∞ L→∞
which proves that J J = lim lim sup EτR τ q (n) = lim lim inf EτR 0,L ,H (S ,n) 0,L ,H (S ,n) H →∞ L→∞
exists and does not depend on S ∈ Sn .
H →∞ L→∞
(2.4)
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J q N = We prove now the convergence τR N → τ (n) in P-probability, where R R0,N ,δ N (S, n). The sub-additivity (2.2) yields: for any δ > 0 and N large enough,
1 J J J τRz ,L ,H + βcd τR √ N τR ( S , n ) 0,N ,H + d/2 i |C| i∈C
L 1 . + N L
Taking lim sup N →∞ and applying the strong law of large numbers give: J J lim sup τR N EτR0,L ,H (S ,n) + N →∞
βcd L
P-a.s.,
and after lim inf L→∞ and lim H →∞ we see that, for all S ∈ Sn and δ > 0, J q lim sup τR N τ (n) N →∞
P-a.s.
(2.5)
On the other hand, the sub-additivity (2.2) is also responsible for the convergence of J : remark that EτR N EτR J
√ 0,L ,δ N + d/2 (S ,n)
EτR N + βcd J
N 1 + , L N
hence lim sup L→∞ followed by lim inf N →∞ give: J τ q (n) lim inf EτR N. N →∞
(2.6)
J ensures the convergence in probTogether with (2.5) and (2.6), the boundedness of τR N ability.
Remark 2.2. The forthcoming large deviation analysis shows that surface tension converges P-a.s. at any β such that β → τ q (n) is left continuous (cf. Theorems 1.4 to 1.8 and Corollary 1.9). Let us sketch now a proof of Proposition 1.5, namely that the quenched surface tension τ q (n) is positive for any β > βˆc : thanks to the renormalization argument of [44], one can compare the surface tension τ a = τ λ=1 under the averaged Gibbs measure to the surface tension of high density site percolation, which is positive. The claim follows as τ q τ a by Jensen’s inequality.
2.2. Upper large deviations. Due to the presence of the random couplings, surface tension can fluctuate around its typical value. The sub-additivity permits to study the order of the cost of large deviations. First, we examine upper deviations and prove Theorem 1.4. The proof is based on the following argument: we split R0,N ,δ N (S, n) into cN J rectangular parallelepipeds Ri with finite height H . In order to increase τR 0,N ,δ N (S ,n)
J is one has to increase surface tension in each Ri , but the cost of increasing one τR i already of surface order by sub-additivity.
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Proof (Theorem 1.4). As a first step towards the proof we estimate the cost for upper deviations of surface tension in a rectangular parallelepiped of fixed height, using the sub-additivity of τ J . From the definition of τ q (n) at (2.4) it follows that for any H large enough, ε J lim sup EτR τ q (n) + . 0,L ,H (S ,n) 6 L J Given such an H we fix l large enough such that EτR τ q (n) + ε/3 and 0,l,H (S ,n) cd β/l ε/4, where cd refers to the constant in the sub-additivity equation. With the notations of Theorem 2.1 we have: l 1 J ε J τR (2.7) τRz ,l,H (S ,n) + + βcd , √ ( S , n ) 0,L ,H + d/2 i |C| 4 L i∈C
J and the τR are i.i.d. variables of mean not larger than τ q (n) + ε/3. Hence, z i ,l,H (S ,n) Cramér’s Theorem tells that 1 J ε q P τRz ,l,H (S ,n) τ (n) + exp(−c|C|) i |C| 2 i∈C
for some c > 0. Reporting in (2.7) proves that for any ε > 0, for any H large enough: 1 J q τ (n) + ε <0 (2.8) lim sup d−1 log P τR 0,L ,H (S ,n) L→∞ L J – that is, the cost for increasing τR is of surface order. We fix such an H and 0,L ,H (S ,n) decompose now the rectangular parallelepiped R = R0,N ,δ N (S, n) in the direction n. Precisely, we let √ d √ ˜i = R x˜i = 2 H + i n, ∀i ∈ Z and R x˜i ,N ,H + d/2 (S, n). 2
˜ i ⊂ R and consider, for all i ∈ G, xi the point of We call G the set of i ∈ Z such that R d d Z such that x˜i ∈ xi + [−1/2, 1/2) and let Ri = Rxi ,N −√d,H (S, n). The rectangular parallelepipeds Ri are disjoint subsets of R = R0,N ,δ N (S, n), all cenˆ with one tered at lattice points. Furthermore, if we call Elat the set of edges in E(R) √ extremity at distance at most d from the lateral boundary of R, we have: ! ω∈ DRi and ωe = 0, ∀e ∈ Elat ⇒ ω ∈ DR . i∈G
Hence the DLR equation yields: J,w J,w ωe = 0, ∀e ∈ Elat and ω ∈ DRi R (DR ) max R i∈G J,w ω ∈ DRi . e−β|Elat | × max R i i∈G
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As |Elat | cd δ N d−1 we conclude finally to the inequality J J cd δβ + min τR . τR i i∈G
(2.9)
J , one must increase each Inequality (2.9) states that in order to increase significantly τR J J J are τRi . Yet, the cost for increasing one of the τRi is of surface order (2.8), and the τR i independent variables. Hence for any δ > 0 such that cd δβ < ε,
lim sup N →∞
1 J q log P τ τ (n) + 2ε < 0. R ( S , n ) 0,N ,δ N Nd
J decreases with δ, the claim follows for arbitrary δ > 0. As τR 0,N ,δ N (S ,n)
2.3. Lower large deviations. Contrary to upper deviations, lower large deviations occur at surface order. Here we consider the rate function In for lower large deviations. The fact that deviations occur at the same order as the disconnecting event defining surface tension is responsible for the distinct behavior of surface tension under quenched and averaged measures. Explicit bounds on the rate function In will be derived in Sects. 2.5 and 2.6. Proof (Theorem 1.6). We begin with the definition of the rate function IR in a rectanJ to τ : gular parallelepiped R = R0,L ,H (S, n) as the surface cost for reducing τR IR (τ ) = −
1 L d−1
J log P τR τ .
(2.10)
According to Proposition 1.2, IR0,L ,H (S ,n) (τ ) is a non-increasing function of τ and H . Hence the limit I(S ,n) (τ ) = lim+ inf lim sup IR0,L ,H (S ,n) (τ + ε) ∈ [0, ∞] ε→0
H
(2.11)
L
exists – we introduce the parameter ε > 0 in order to compensate for the error terms in (2.2). It is clearly a non-increasing function of τ . We prove now that it is also convex in τ and that it does not depend on S ∈ Sn : let S ∈ Sn , ε > 0 and α ∈ [0, 1]. Using the notations R = R0,L ,H +√d/2 (S, n), Ri = Rzi ,l,H (S , n) and C of the Sub-additivity Theorem (Theorem 2.1), we have l 1 |C 1 | 1 |C 2 | 2 J τ + τ + ε + βcd + τR |C| |C| L l if C 1 C 2 is a partition of C such that 1 τ +ε J τR i τ2 + ε
if i ∈ C 1 if i ∈ C 2 .
The probability for realizing the independent conditions in (2.12) is exp −|C 1 |l d−1 IR0,l,H (S ,n) τ 1 + ε − |C 2 |l d−1 IR0,l,H (S ,n) τ 2 + ε
(2.12)
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as the rectangles Ri are identical to R0,l,H (S , n) up to a lattice translation. Letting |C 1 |/|C| → α and L → ∞ we see that
lim sup L IR0,L ,H +√d/2 (S ,n) ατ 1 + (1 − α)τ 2 + 2ε + βcd /l
(2.13) α IR0,l,H (S ,n) τ 1 + ε + (1 − α)IR0,l,H (S ,n) τ 2 + ε . Taking the superior limit in l, then the limit in H , then ε → 0+ , we obtain I(S ,n) ατ 1 + (1 − α)τ 2 α I(S ,n) τ 1 + (1 − α)I(S ,n) τ 2 , which proves both the independence of I(S ,n) with respect to S (take α = 1) and the convexity along τ . We let now In = I(S ,n) and postpone the proof of (1.18) for a while. The continuity of In on the interior of its effective domain is a consequence of convexity. Hence we examine the effective domain of In . Let first τ < τ min (n). If ε > 0 is small J enough, the event τR τ + ε < τ min (n) has a probability zero and conse0,L ,H (S ,n) quently, In (τ ) = +∞. The second easy regime is τ τ q (n): from Proposition 1.4 we J infer that lim L→∞ P(τR τ + ε) = 1 provided that H is large enough and 0,L ,H (S ,n) this implies In (τ ) = 0. At last we assume that τ > τ min (n) in order to prove that In is finite. We show first that, for some finite H and δ > 0, min
J +δ lim sup τR < τ. 0,L ,H (S ,n)
(2.14)
L
The definition of the surface tension τ min (n) yields H such that min
J lim sup τR
√ 0,l,H − d/2 (S ,n)
l
< τ.
On the other hand, the sub-additivity of surface tension (Theorem 2.1) gives min
min +δ
J +δ J lim sup τR τR 0,L ,H (S ,n)
√ 0,l,H − d/2 (S ,n)
L
+ βcd /l. min +δ
J Thus (2.14) is a consequence of the continuity of δ → τR
large enough. Now we write, for any L large enough: IR0,L ,H (S ,n) (τ ) = −
1
√ 0,l,H − d/2 (S ,n)
once we fix l
J log P τR τ 0,L ,H (S ,n) ˆ 0,L ,H (S, n) log P Je J min + δ, ∀e ∈ E R
L d−1 1 − d−1 L cd H × (− log P(Je ∈ [J min , J min + δ])),
which is finite thanks to the definition of J min . This ends the proof that In (τ ) < ∞, for any τ > τ min (n). We address at last the convergence (1.18). The inequality IR0,N ,δ N (S ,n) (τ ) IR0,N ,H (S ,n) (τ ) when N δ H yields an upper bound on the superior limit: lim sup IR0,N ,δ N (S ,n) (τ ) inf lim sup IR0,L ,H (S ,n) (τ ) In (τ − ) = In (τ ) N
H
L
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for all τ > τ min (n), thanks to the continuity of In . For the lower bound we use the sub-additivity of surface tension. Applying (2.13) with α = 1, l = N , H = δ N yields: for any ε > 0 and N large enough, lim sup IR0,L ,δ N +√d/2 (S ,n) (τ + 3ε) IR0,N ,δ N (S ,n) (τ + ε), L
and replacing τ + ε with τ , we obtain after the limits N → ∞ and ε → 0+ the lower bound In (τ ) lim inf IR0,N ,δ N (S ,n) (τ ) , ∀τ ∈ R. N
2.4. Surface tension under averaged Gibbs measures. The rate function In can be analyzed through a dual quantity: the surface tension under the averaged Gibbs measure defined at (1.24). The duality of Fenchel-Legendre transforms for convex functions (Lemma 4.5.8 in [18]) implies that λ → τ λ (n) is concave and that In (τ ) = sup{τ λ (n) − λτ }. λ>0
(2.15)
As we said at (1.25), τ λ (n) can be interpreted as the surface tension under an average J,w . Indeed, if we let of R λ 1 1 J,w λ J = − d−1 log E exp −λL d−1 τR , τR = − d−1 log E R (DR ) L L (2.16) for any rectangular parallelepiped R of side-length L as in (1.15), then Varadhan’s Lemma yields: Proposition 2.3. For any λ > 0 and n ∈ S d−1 , for any sequence of rectangular paralλ converges lelepipeds R N = R0,N ,δ N (S, n) with δ > 0 and S ∈ Sn , the quantity τR N λ to τ (n): λ λ lim τR N = τ (n). N
(2.17)
Thus, the limit does not depend on δ > 0 nor on S ∈ Sn . We defined at (1.45) the value τ˜ q (n) of the surface tension at which In (τ ) becomes zero. Below are some immediate consequences of the definition of τ λ (n) at (1.24) together with (2.17), which allow to sketch the graph of λ → τ λ (n) on Fig. 4: Proposition 2.4. The following inequalities hold: λτ min (n) τ λ (n) λτ˜ q (n), ∀n ∈ S d−1 , λ > 0,
(2.18)
τ λ (n) τ λ (n) −→+ τ˜ q (n) and −→ τ min (n), ∀n ∈ S d−1 . λ λ→0 λ λ→+∞
(2.19)
while
Hence, τ λ (n) is positive if and only if τ˜ q (n) > 0. Furthermore: τ λ (n) −→ lim+ In (τ ) ∈ [0, ∞]. λ→+∞ τ →0
(2.20)
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181
min
Fig. 4. The graph of λ → τ λ (n) in the case of dilution (τ min = 0 and In (0) < ∞, left) and distributions with τ min > 0 (right)
Another important yet classical fact is the convexity of surface tension [36]. The J (see [36] or proposition below is a consequence of the weak triangle inequality for τR [10] for the uniform case, or Appendix 2.5.2 in [45]). Proposition 2.5. Let f q be the homogeneous extension of τ q to Rd , namely: xτ q (x/x) if x ∈ Rd \{0} f q (x) = 0 if x = 0, and let f λ (resp. f˜q ) be the homogeneous extension of τ λ (resp. τ˜ q ) to Rd . Then, f q , f λ and f˜q are convex and τ q , τ λ and τ˜ q are continuous on S d−1 . 2.5. Concentration at low temperatures. In this section and the next one we establish respectively Theorems 1.7 and 1.8. In both cases we use concentration of measure theory, which is a very efficient tool for analyzing the fluctuations of product measures. In the case of polymers or even spin glasses it yields relevant bounds on the probabilities of deviations, see [35] for a review. Concerning the Ising (or random-cluster) model with random couplings, its application to the deviations of surface tension requires a control over the surface of the interface, and this is the point where the proofs of Theorems 1.7 and 1.8 differ: at low temperatures one can control rather easily the length of the interface, while under the only assumptions of Theorem 1.8 the same control is not immediate. The surface tension τ λ (n) under averaged Gibbs measure plays an important role here, as well as the modified measure Eλ defined at (2.26) below. We will obtain lower bounds on τ λ (n), which correspond to lower bounds on In (τ ) by (2.15). Rather than making the assumption that the product measure P satisfies a logarithmic Sobolev inequality as in [45]1 , we use general bounds on product measure (Corollary 5.8 in [35]). The author thanks Raphaël Rossignol for pointing out this improvement. The proof of Theorem 1.7 is made of four steps, the first three being common with the proof of Theorem 1.8. λ (n) in a The first step consists in relating the derivative of the surface tension τR rectangular parallelepiped R as in (1.15), with a basis of side-length L, to the entropy of the positive function exp( f λ ), where J . f λ = −λL d−1 τR 1 Usual measures such as dilution P(J ∈ {0, 1}) = 1, or J with positive density on [0, 1] do satisfy a e e logarithmic Sobolev inequality, cf. [35] or Theorems 4.2, 6.6 and Sect. 6.3 in [12].
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We recall that the entropy of a positive measurable function f with E( f log(1+ f )) < ∞ is Ent P ( f ) = E( f log f ) − E( f ) log E( f ).
(2.21)
With these notations, it is immediate that: Lemma 2.6. For any λ > 0, ∂ − ∂λ
λ τR λ
=
Ent P (exp( f λ )) 1 . λ2 L d−1 E (exp ( f λ ))
(2.22)
As a second step we study the quantity aeJ =
J L d−1 ∂τR . β ∂ Je
(2.23)
The proposition below provides an interpretation of aeJ as the probability that the disconnecting interface due to the event DR passes through the edge e. We prove also, and this is crucial for our construction, that the actual value of Je does not influence too much that of aeJ : Proposition 2.7. For any e, aeJ is a C ∞ function of the Je . For any J ∈ [0, 1] E(Z ) , one has 1 J,w J,w aeJ = R (ωe ) − R (ωe |DR ) if Je > 0 (2.24) pe d
together with the following inequalities: 0 aeJ 1 and
sup aeJ eβ inf aeJ . Je
Je
(2.25)
The controls (2.25), together with Corollary 5.8 in [35], permit to establish the third step. Given a rectangular parallelepiped R as in (1.15) and λ 0, we introduce the probability measure Pλ that to any bounded measurable h : J → h(J ) ∈ R gives expectation
J exp −λL d−1 τR .
(2.26) Eλ (h(J )) = E h(J ) J E exp −λL d−1 τR Proposition 2.8. Denote m P = E(Je ). For any λ > 0, we have both ⎧ J ⎨ 2 β(1+λ) a E λ ˆ) e EntP (exp( f λ )) β e e∈E(R λ2 . λ ⎩ 1 L d−1 1 ∂τR . E (exp ( f λ )) 4 mβ
(2.27)
λ ∂β
The second majoration leads to Theorem 1.8, while the first one yields Theorem 1.7 as, thanks to the usual contour argument in the Ising model (q = 2) with couplings Je ∈ [ε, 1], the length of the interface is known to be of order N d−1 at low temperatures.
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Proposition 2.9. Let q = 2 and ε > 0. There exists cd < ∞ such that, for β large enough, for R = R N = R0,N ,δ N (S, n) with δ ∈ (0, 1), for any realization J of the random couplings such that Je ε and N large enough, cd d−1 N aeJ . (2.28) ε RN ) e∈E(
We give now the proofs of all the propositions, followed by that of Theorem 1.7. Proof (Proposition 2.7). The fact that aeJ is a C ∞ function of Je is a consequence of J , the quantity J,w (D ) being always positive. We introduce the same property for τR R R next a few notations: we let pe ωe C w (ω) J J J ˆ (ω) = q E(R) and Z R (A) = wR (ω) (2.29) wR 1 − pe ω∈A
ˆ) e∈E(R
for any ω ∈ E(Rˆ ) and A ⊂ E(Rˆ ) , see (1.9) for the definition of C w ˆ (ω). For all J E(R) with Je > 0, we have J (ω) ∂ log wR ωe =β , ∂ Je pe
and as a consequence, for all J with Je > 0, Z J (DR ) 1 ∂ log J R β ∂ Je Z R ( E(Rˆ ) ) 1 J,w J,w R (ωe ) − R (ωe |DR ) . = pe
aeJ = −
J,w Under this formulation, the FKG inequality and the bound R (ωe ) pe imply that 0 aeJ 1 for any J ∈ J with Je > 0, and the inequality extends by continuity to the whole of J . We now calculate the derivative of aeJ along Je for Je > 0 and obtain, as J,w J,w J,w R (ωe |A) R (ωe |A) R (ωe |A)2 ∂ − =β , ∂ Je pe pe pe2
that, for any J ∈ J with Je > 0, J,w J,w R (ωe ) R (ωe |DR ) ∂aeJ J = βae 1 − − . ∂ Je pe pe This implies in particular that J ∂ae J ∂ J βae , e and the comparison sup Je ∈[0,1] aeJ eβ inf Je ∈[0,1] aeJ follows.
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Proof (Proposition 2.8). According to Corollary 5.8 in [35] and to the Mean Value Theorem, we have 2 ∂ fλ E sup Je ∈[0,1] ∂ Je exp(sup Je ∈[0,1] f λ ) 1 Ent P (exp( f λ )) . E (exp ( f λ )) 4 E (exp ( f λ )) ˆ) e∈E(R
It is clear that ∂ fλ = −λβaeJ . ∂ Je On the other hand, Proposition 2.7 yields sup (aeJ )2 sup aeJ eβ
Je ∈[0,1]
inf
Je ∈[0,1]
Je ∈[0,1]
aeJ
and sup exp( f λ ) eβλ inf
Je ∈[0,1]
Je ∈[0,1]
exp( f λ ),
hence
λ2 β 2 eβ(1+λ) inf Je ∈[0,1] exp( f λ ) Ent P (exp( f λ )) , E inf aeJ × Je ∈[0,1] E (exp ( f λ )) 4 E (exp ( f λ )) ˆ) e∈E(R
and the first bound follows. For the second one, remark that as we take infimums over Je we in fact obtain a quantity that is independent of Je . Thus λ2 β 2 eβ(1+λ) Ent P (exp( f λ )) Je inf Je ∈[0,1] exp( f λ ) E inf aeJ × E (exp ( f λ )) 4 m P Je ∈[0,1] E (exp ( f λ )) ˆ) e∈E(R ⎛ ⎞ λ2 β 2 eβ(1+λ) ⎝ Eλ Je aeJ ⎠ 4m P ˆ) e∈E(R
which ends the proof as
⎛ ⎞ λ 1 ∂τR 1 = d−1 Eλ ⎝ Je aeJ ⎠. λ ∂β L ˆ) e∈E(R
Proof (Proposition 2.9). This bound is based on the usual contour representation of spin configurations in Ising models. We describe here how we relate the left-hand side of (2.28) to the length of the spin interface. As R = R N = R0,N ,δ N (S, n) is centered at the origin, we consider + ˆ R ˆ , R = σ : Zd → {±1} : σx = 1, ∀x ∈ / R\∂ 1 if x · n 0 ± ˆ R ˆ , R = σ : Zd → {±1} : σx = , ∀x ∈ / R\∂ −1 else
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ˆ with plus or mixed boundary conditions. The corthe set of spin configurations on R respondence between the random-cluster representation (with q = 2) and Ising model gives J τR =
1 N d−1
log
J,+ J,± and Z R are the partition functions where Z R ⎛ ⎜β J,+ ZR = exp ⎝ 2 + σ ∈R
J,± and Z R =
± σ ∈R
⎛ ⎜β exp ⎝ 2
J,+ ZR J,± ZR
,
⎞
⎟ Je σx σ y ⎠
ˆ) e={x,y}∈E(R
⎞ ⎟ Je σx σ y ⎠,
ˆ) e={x,y}∈E(R
leading thus to J,+ J,± ˆ (σx σ y ) − µR (σx σ y ), ∀e = {x, y} ∈ E(R), (2.30) aeJ = µR J,± ˆ minus ˆ with mixed boundary condition (plus on ∂ + R, where µR is the Ising model on R − ˆ on ∂ R). The remaining part of the proof is standard [20,37]. Conditionally on a realization J,± of the spin interface under the measure µR , the correlation between any two spins that J,+ do not cross that interface is larger than under µR . Hence the sum J,+ J,± aeJ = µR (σx σ y ) − µR (σx σ y ) N) e∈E(R
N) e∈E(R
is not larger than twice the average length of the interface, which does not exceed cd N d−1 / for large β. Proof (Theorem 1.7). The combination of Lemma 2.6, Propositions 2.8 and 2.9 implies that in the setting of Theorem 1.7, λ ∂ τR cd β 2 eβ(1+λ) − ∂λ λ 4J min for β large enough, R = R N = R0,N ,δ N (S, n), δ ∈ (0, 1) and N large enough. λ /λ = Eτ J , the inequality Integrating over λ we obtain, as limλ→0+ τR R λ J λEτR − λ2 τR
cd β 2 eβ(1+λ) . 4J min
Letting N → ∞ gives τ λ (n) λτ q (n) − λ2
cd β 2 eβ(1+λ) 4J min
and the duality formula (2.15) yields the claim with c = cd J min /(β 2 exp(2β)), for large enough β.
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2.6. Concentration in a general setting. We give now the proof of Theorem 1.8, which is based on Herbst’s argument, together with the controls of Lemma 2.6 and Proposition 2.8. We will then give the proof of Corollary 1.9. First we give an immediate consequence of the duality formula (2.15): Lemma 2.10. Assume that lim sup λ→0+
τ λ (n) − λτ q (n) −c for some c ∈ [0, ∞]. λ2
(2.31)
In (τ q (n) − r ) 1 ∈ [0, ∞]. r2 4c
(2.32)
Then, lim sup r →0+
Proof (Theorem 1.8). Given δ > 0 and S ∈ Sn , we denote R N the rectangular parallelepiped R N = R0,N ,δ N (n, S) and introduce P,β Kn
1 = lim inf lim inf + N →∞ λ λ→0
λ
∂τR N dλ ∈ [0, ∞]. ∂β λ
λ
0
In view of Theorem 1.3 and Proposition 2.3 we have λ J τ λ (n) − λτ q (n) = lim τR N − λEτR N N →∞ λ λ τR N ∂ = lim λ dλ
N →∞ ∂λ λ 0 J = lim λ as EτR λ→0+ τR N /λ for any N finite. Lemma 2.6 and Proposition 2.8 yield, for N any ε > 0:
τ λ (n) − λτ q (n) 1 β 2 eβ(1+ε) lim sup − lim inf lim inf 2 + N →∞ λ λ→0 λ 4m P λ→0+ =−
β 2 eβ(1+ε)
λ
0
λ ∂τR N dλ ∂β λ
P,β
Kn ,
4m P
and an immediate application of Lemma 2.10 gives, after the limit ε → 0, the lower bound: q
lim sup r →0+
Iβ,n (τβ (n) − r ) r2
mP
P,β
β 2 eβ K n
.
(2.33)
P,β
The lower bound is positive when K n < ∞. In order to show that this is the case for P,β Lebesgue almost all β, we evaluate the integral of K n on some interval [β1 , β2 ]. For any δ > 0 and S ∈ Sn , Fatou’s Lemma and Fubini’s Theorem imply that
β2 β1
P,β K n dβ
1 lim inf lim inf + N →∞ λ λ→0 1 = lim inf lim inf + N →∞ λ λ→0
λ β2
β1
0 λ 0
λ
∂τR N dλ ∂β λ
τβλ2 ,R N − τβλ1 ,R N λ
dλ .
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The convergence as N → ∞ is uniformly dominated (by Jensen’s inequality and Propλ λc β) hence we finally obtain osition 1.2, 0 τR d N
β2 β1
1 lim inf λ→0+ λ
P,β K n dβ
q
λ
τβλ2 (n) − τβλ1 (n) λ
0 q
dλ
= τ˜β2 (n) − τ˜β1 (n), P,β
(2.34)
is finite for Lebesgue almost all β 0.
in view of (2.19). In particular, K n
P,β
We would like to make a remark on K n . In view of Corollary 1.9, for Lebesgue q q almost every β1 , β2 with β1 β2 , one can replace τ˜β2 (n) − τ˜β1 (n) in (2.34) with q q q τβ2 (n) − τβ1 (n). As a consequence, whenever β → τβ (n) is differentiable on some P,β
interval, K n
q
∂τβ (n)/∂β for Lebesgue almost every β in that interval.
Proof (Corollary 1.9). We denote by q
q
τβ − (n) = lim+ τβ−ε (n) ε→0
q
q
the left limit of β → τβ (n). For any τ ∈ R, β → In (τ ) is non-decreasing, hence τ˜β (n) q (defined at (1.45)) does not decrease with β. According to Theorem 1.8, τ˜β (n) coincides q with τβ (n) for almost all β, hence q
q
q
τβ − (n) τ˜β (n) τβ (n), ∀β 0, q
q
q
hence the left continuity of τβ (n) at a particular β implies that τ˜β (n) = τβ (n), in other words that lower deviations are (at least) of surface order. This is the first part of the claim. Now we consider q q D = β ∈ R+ , ∃n ∈ S d−1 : τβ − (n) = τβ (n) , q
and prove that D is at most countable. The homogeneous extension of τβ − (n) to Rd q
q
is convex as the pointwise limit of the f β−ε , hence τβ − (n) is a continuous function of n ∈ S d−1 . Consequently, for any dense sequence (nn )n∈N in S d−1 , we have ! q q β ∈ R+ : τβ (nn ) = τβ − (nn ) , D⊂ n∈N
which is at most countable. 2.7. Low temperatures asymptotics. Here we describe the low temperatures asymptotics of surface tension. Only a sketch of the corresponding (easy) proofs will be given, and the interested reader is invited to consult the PhD Thesis [45] for further detail. The low temperatures asymptotics of quenched surface tension is determined by the maximal flow µ(n) defined at Theorem 1.10:
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Proposition 2.11. Assume that P(Je > 0) = 1. Then, uniformly over n ∈ S d−1 , τ q (n) = µ(n). β→+∞ β
(2.35)
lim
Remark 2.12. The upper bound τ q (n) βµ(n) is immediate and necessitates no assumption on P. The lower bound uses a Peierls control on the length of the interface which is possible as P(Je > 0) = 1 and β → ∞. If J min > 0, one even has τ q (n) βµ(n) − C for some C < ∞, uniformly in β > 0 and n ∈ S d−1 . The convergence (2.35) also holds in the trivial case P(Je > 0) pc (d) since τ q (n) βµ(n) = 0. In the remaining case P(Je > 0) > pc (d) a renormalization argument allows to prove: Proposition 2.13. Assume that P(Je > 0) > pc (d). Then, lim inf β→+∞
τ q (n) > 0, β
(2.36)
uniformly over n ∈ S d−1 . On the other hand, the surface tension under the averaged Gibbs measure is asymptotically determined by J min : Proposition 2.14. For all λ > 0, uniformly over n ∈ S d−1 , τ λ (n) = λJ min n1 . β→+∞ β lim
(2.37)
Remark 2.15. The upper bound in (2.37) is a consequence of the easy inequality τ λ (n) n1 × log
1 . E exp (−λβ Je )
For the lower bound one may assume that P(Je > 0) = 1 and show that, uniformly over n ∈ S d−1 , τ λ (n) (1 − oβ→∞ (1)) × n1 × log
1 . E exp (−λβ Je )
Note that this provides an equivalent to τ λ (n) when J min = 0 and P(Je > 0) = 1. An important consequence of the former propositions together with Theorem 1.10 is: Corollary 2.16. Assume that P(Je > J min ) > pc (d). Then, for any λ > 0 there is βcλ < ∞ such that τ λ (n) < λτ q (n), ∀n ∈ S d−1 , ∀β > βcλ .
(2.38)
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This corollary has several implications. First, when J min = 0 and P(Je > 0) > pc (d) (there is still a phase transition), the inequality between surface tension under averaged Gibbs measures and its quenched value is always strict at low temperatures. Second, under the same assumptions as in the corollary, one has τ min (n) < τ q (n) at low temperatures (see Proposition 2.4), which means that lower deviations are possible. About this point let us mention the case of the directed polymer model in 1 + 1 dimensions: for that model, it was proved recently [17] that the Lyapunov exponent is positive at all β 0, which corresponds in our settings to the strict inequality τ a (n) = τ λ=1 < τ q (n). Another consequence of the above uniform limits (or equivalents) is the convergence, at low temperature, of Wulff crystals (1.33): Proposition 2.17. Assume that P(Je > 0) = 1. i. The Wulff crystal W q converges to the Wulff crystal W µ associated with the maximal flow µ. ii. For any λ > 0, the Wulff crystal W λ converges to the hypercube W .1 = [±1/2]d . Here we see another consequence of the media randomness on the phenomenon of phase coexistence : instead of being an hypercube as in the uniform Ising model, the low temperature limit of the quenched Wulff crystal might be influenced by the media, cf. [21] and our discussion in Sect. 1.5. 3. Phase Coexistence 3.1. Profiles of bounded variation and surface energy. The coarse graining for the dilute Ising model (Theorem 5.10 in [44]) implies that at every position, the local magnetization M K is close to ±m β with large probability. In order to describe the geometrical structure of the phases, we estimate the probability that M K /m β be close, in L 1 -distance, to a given Borel measurable function u : [0, 1]d → {±1}. As a first step towards the description of phase coexistence, we define here the set of profiles we consider, define surface energy and the associated isoperimetric problem. In the following, Ld stands for the Lebesgue measure on Rd and Hd−1 for the d − 1 dimensional Hausdorff measure, which gives to any Borel set X ⊂ Rd the weight ! αd−1 d−1 d−1 [diam(E i )] : sup diam(E i ) < δ, X ⊂ Ei , H (X ) = lim+ d−1 inf δ→0 2 i∈I i∈I
i∈I
where the infimum takes into account finite or countable coverings (E i )i∈I , and αd−1 is the volume of the unit ball of Rd−1 . The L 1 -distance between two Borel measurable functions u, v : [0, 1]d → R is |u − v|dLd , u − v L 1 = [0,1]d
and the set L 1 is u : [0, 1]d → R Borel measurable, u L 1 < ∞ .
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In order that L 1 be a Banach space for the L 1 -norm, we identify u : [0, 1]d → R with the class of functions v : u − v L 1 = 0 that coincide with u on a set of full measure. We also denote by V(u, δ) the neighborhood of radius δ > 0 in L 1 around u ∈ L 1 . For the study of phase coexistence, we have to consider virtually any u ∈ L 1 taking values in {±1}. Before we can define the surface energy for such profiles, a description of the boundary of these profiles is necessary. It is done conveniently in the framework of bounded variation profiles (Chap. 3 in [4]). Given a Borel subset U ⊂ Rd , the variation (or perimeter) of U is d ∞ d P(U ) = sup div f dL , f ∈ Cc (R , B(0, 1)) ∈ [0, ∞], U
where Cc∞ (Rd , B(0, 1)) is the set of C ∞ vectors fields with compact support in Rd , taking values in the Euclidean unit ball B(0, 1), and div is the divergence operator: div f =
∂ f1 ∂ fd + ··· + . ∂ x1 ∂ xd
To U ⊂ Rd Borel measurable, we associate u = χU as in (1.38) and define the set of bounded variation profiles BV as follows: BV = u = χU : U ⊂ (0, 1)d is a Borel set and P(U ) < ∞ . Bounded variations profiles u = χU ∈ BV have a reduced boundary ∂ u and an outer normal nu. : ∂ u → S d−1 with, in particular, Hd−1 (∂ u) = P(U ). This allows us to define the surface energy of bounded variation profiles. As the outer normal nu. defined on ∂ u is Borel measurable, we can consider q F (u) = τ q (nux )dHd−1 (x), ∀u ∈ BV (3.1) ∂ u
and F λ (u) =
∂ u
τ λ (nux )dHd−1 (x),
∀u ∈ BV, ∀λ > 0,
(3.2)
where τ q (resp. τ λ ) stands for the quenched surface tension of the dilute Ising model (resp. surface tension under the averaged Gibbs measure), see Theorem 1.3 and (1.24). Because the homogeneous extension of the surface tensions τ q and τ λ are convex (Proposition 2.5), F q and F λ are lower semi-continuous with respect to the L 1 -norm. See Chapter 14 in [10] or Theorem 2.1 in [3]. For commodity, when u = χU ∈ BV we also denote by F q (U ) (resp. F λ (U )) the surface energy of u. When surface tension is positive, the level sets of F q and F λ are compact since, for all a 0, the set BVa = {u = χU ∈ BV : P(U ) a}
(3.3)
is itself compact for the cf. Theorem 3.23 in [4]. Consequently, and F λ are good rate functions. Let us conclude with a word on the solutions to the isoperimetric problem of finding the u ∈ BV such that m u dLd and F q (u) is minimal ? (3.4) mβ [0,1]d L 1 -norm,
Fq
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The renormalized Wulff crystal W q (1.33) is known to be the solution to the same problem without the constraint that U ⊂ (0, 1)d . Precisely, the solutions to U ⊂ Rd Borel set with Ld (U ) = 1 and F q (U ) minimal are the translates of W q , as the homogeneous extension of τ q is convex (Proposition 2.5) – see [24,40 and 25]. For m < m β not too small, W q determines as well the optimal profiles in the cube (3.4). Consider α > 0 with m 1 d . (3.5) 1− α = 2 mβ The .quantity α d is precisely the least volume of U corresponding to u = χU ∈ BV with [0,1]d udLd m/m β . If some translate of αW q fits into the unit cube [0, 1]d , that is if α diam∞ (W q ) 1, then T (αW q ) .defined at (1.39) is not empty and therefore the infimum of F q (u) for u ∈ BV with [0,1]d udLd m/m β is exactly F q (αW q ). As a consequence, for all α satisfying α diam∞ (W q ) 1 the optimal phase profiles correspond to the translates of αW q that belong to [0, 1]d , which are the z + αW q , for z ∈ T (αW q ). The same remains true if we replace F q and W q with F λ and W λ , for any λ > 0. 3.2. Covering theorems for BV profiles. Covering theorems play an essential role in the study of phase coexistence, as they allow to pass from the macroscopic scale (the phase profile u) to the microscopic scale (the dilute Ising model). The reader familiar with former works on phase coexistence will note that we have replaced the usual polyhedral covering theorem for Wulff crystals with a generalization of the BV covering Theorem (Theorem 3.4 below). This permits to establish the lower bound for phase coexistence for a wide class of profiles, which is useful for controlling the dynamics of dilute Ising systems (Chap. 4 in [45]). We begin with two definitions: Definition 3.1. Let u ∈ BV, τ : S d−1 → [0, ∞] continuous, δ > 0 and R a rectangular parallelepiped as in (1.15), included in [0, 1]d . We say that R is δ-adapted to u and τ at x ∈ ∂ u if the following holds: i. If n = nux is the outer normal to u at x, there are S ∈ Sn and h ∈ (0, δ] such that, if R ⊂ (0, 1)d (we say that R is interior), then R = x + hS + [±δh]n, and if R ∩ ∂[0, 1]d = ∅ (we say that R is on the border), then x ∈ ∂[0, 1]d , n is also the outer normal to [0, 1]d at x and R = x + hS + [−δh, 0]n. ii. We have
Hd−1 ∂ u ∩ ∂R = 0,
1 − 1 Hd−1 ∂ u ∩ R δ, h d−1
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and u d−1 τ (n) − 1 τ (n. )dH δ. h d−1 ∂ u∩R iii. If χ : Rd → {±1} is the characteristic function of R defined by +1 if (z − x) · n 0 χ (z) = , ∀z ∈ Rd , −1 else then 1 |χ − u|dHd δ. 2δh d R Definition 3.2. Let u ∈ BV, τ : S d−1 → [0, ∞] continuous and δ > 0. A finite sequence (Ri )i=1...n of disjoint rectangular parallelepipeds included in [0, 1]d is said to be a δ-covering for ∂ u and τ if each Ri is δ-adapted to u and τ and if n ! d−1 H Ri δ. ∂ u\ (3.6) i=1
The Vitali covering theorem (Theorem 13.3 in [10]) is especially well adapted to our purpose. Given a Borel set E ⊂ Rd , we say that a collection of sets U is a Vitali class for E if, for each x ∈ E and δ > 0, there is U ∈ U with 0 < diam U < δ containing x. Theorem 3.3. [Vitali]. Let E ⊂ Rd be Hd−1 -measurable and consider U a Vitali class of closed sets for E. Then, there is a countable disjoint sequence (Ui )i∈I in U such that ! d−1 d−1 either (diam Ui ) = ∞ or H Ui = 0. E\ i∈I
i∈I
The Vitali Theorem allows us to state a short proof of the following: Theorem 3.4. For any u ∈ BV, τ : S d−1 → [0, ∞] continuous and δ, h > 0, there is a δ-covering (Ri )i=1...n for ∂ u and τ . Before we give the proof of Theorem 3.4 we recall a property of the reduced boundary (see Theorem 3.59 in [4]): Lemma 3.5. Let u ∈ BV. For all x ∈ ∂ u, for all δ ∈ (0, 1), all S ∈ Snux one has lim+
1
h→0 h d−1
˙ x,h,δh (S, nux ) = 1. Hd−1 ∂ u ∩ R
Proof (Theorem 3.4). We design a set E that has zero Hd−1 -measure and such that the collection of closed rectangular parallelepipeds
Uδ = R δ-adapted to u and τ at x ∈ ∂ u\E
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193
is a Vitali class for ∂ u\(E). This is enough to prove the claim: thanks to the Vitali Covering Theorem, this implies the existence of a countable disjoint sequence (Ri )i∈I of δ-adapted rectangular parallelepipeds with either ! (diam Ri )d−1 = ∞ or Hd−1 ∂ u\ Ri = 0. i∈I
i∈I
The first case is in contradiction with the inequalities 1/ h id−1 Hd−1 (∂ u ∩ Ri ) 1 − δ and Hd−1 (∂ u) < ∞, hence the second is realized and the theorem is proved. We define the set E by its complement in ∂ u: ∂ u\E is the set of all x ∈ ∂ u such that, for all S ∈ Snux , the following holds: d d nux is the outer normal to [0, If x ∈ ∂[0, 1] at x. 1] , thend−1 u ∂ u ∩ ∂Rx,h,δh (S, n x ) > 0 has zero Lebesgue measure. The set h > 0 : H
1 ˙ x,h,δh (S, nux ) = 1. Hd−1 ∂ u ∩ R lim h→0+ h d−1 . 1 u d−1 = τ (nu ). lim h→0+ h d−1 ˙ x,h,δh (S ,nu ) τ (n. )dH x ∂ u∩R x . v. lim h→0+ h1d Rx,h,δh (S ,nu ) χx,nux − u dLd = 0.
i. ii. iii. iv.
x
This definition for E implies that Uδ is a Vitali class of closed sets for ∂ u\E. We conclude the proof of Theorem 3.4 showing that E has zero Hd−1 -measure, and more precisely that each of conditions (i)-(v) is true for (at least) Hd−1 -almost all x ∈ ∂ u: i. This condition holds for all x ∈ ∂ u because of the inclusion U ⊂ (0, 1)d if u = χU , cf. Theorem 3.59 in [4]. ii. Since the volume of ∂ u is zero, (ii) holds for all x. iii. Condition (iii) holds for all x ∈ ∂ u in view of Lemma 3.5. iv. It is a consequence of the strong form of the Besicovitch derivation theorem (Theorem 5.52 in [4]) together with Lemma 3.5, that condition (iv) holds for Hd−1 -almost all x ∈ ∂ u. v. Condition (v) holds for all x ∈ ∂ u, cf. Theorem 3.59 in [4]. 3.3. Lower bound for phase coexistence. Here we establish lower bounds for the probability of phase coexistence. In view of the applications, in particular to the control of the dynamics [43] or Chapter 4 in [45], we establish it for a large class of profiles, that include Wulff crystals and shapes with C 1 boundary. Proposition 3.8 below relates the probability of an event of disconnection along the boundary of a given profile to the surface tension τ J , for a given realization of the media. In Proposition 3.9 we show that conditionally on this event of disconnection, phase coexistence has large probability. Then we state in Proposition 3.10 a lower bound on the probability of phase coexistence for both quenched and averaged measures. Given some region U ⊂ (0, 1)d , N ∈ N and δ > 0, we consider EUN ,δ the set of √ edges at distance at most N dδ from N ∂U : √ EUN ,δ = e ∈ E w ( N ), d(e, N ∂U ) N dδ (see Fig. 5) and call ∀x ∈ N \N U, √ y ∈ N ∩ N U with ω √ DUN ,δ = ω ∈ E w ( N ) : x y, d(x/N , ∂U ) > dδ and d(y/N , ∂U ) > dδ
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Fig. 5. The scales K and L N
the event that disconnection occurs around ∂U . In order to be able to control the probability of DUN ,δ , we introduce the following definition: Definition 3.6. We say that a profile u = χU is regular if i. U is open and at positive distance from the boundary ∂[0, 1]d of the unit cube, ii. ∂U is d − 1 rectifiable, and iii. for small enough r > 0, [0, 1]d \ (∂U + B(0, r )) has exactly two connected components. We recall that E ⊂ Rd is a d − 1 rectifiable set if there exists a Lipschitzian function mapping some bounded subset of Rd−1 onto E (Definition 3.2.14 in [23]). It is the case in particular of the boundary of non-empty Wulff crystals (Theorem 3.2.35 in [23]) and of bounded polyhedral sets. It follows from Proposition 3.62 in [4] that any u = χU regular belongs to BV and that ∂U = ∂ u up to a Hd−1 -negligible set, so that the covering theorem applies as well to ∂U . Assumption (ii) in Definition 3.6 has the following consequence: Lemma 3.7. Let u = χU ∈ BV be a regular profile. Then, for any δ > 0, for any δ-covering (Ri )i=1...n of u, one has
n Ld ∂U \ i=1 Ri + B (0, r ) lim sup 2δ. r r →0 Proof. Clearly, the set E = ∂U \
n !
˙i R
i=1
is a closed, d − 1 rectifiable set. Thus, the d − 1 Minkowski content of E equals the d − 1 dimensional Hausdorff measure of E (Theorem 3.2.39 in [23]). In other words: Ld (E + B (0, r )) = Hd−1 (E) δ, r →0 2r lim
and the claim follows.
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195
Before we state Propositions 3.8 and 3.9, we give one more notation. The analysis of surface tension has been done for a rectangular parallelepiped centered at lattice points. Changing the center of the parallelepipeds does not modify the behavior of surface tension, but this would have led to heavier notations. We prefer to proceed to a small adjustment here: given a macroscopic rectangular parallelepiped R ⊂ (0, 1)d and N ∈ N , we let R N = N R + z N (R),
(3.7)
where z N (R) ∈ (−1/2, 1/2]d is chosen such that the center of R N belongs to Zd . Still, for any finite collection (Ri )i=1...n of disjoint rectangular parallelepipeds in (0, 1)d and large enough N , the collection (RiN )i=1...n is disjoint and included in (0, N )d . Proposition 3.8. Consider a regular u = χU . For any δ > 0 and any δ-covering (Ri )i=1...n for u, we have 1 N d−1
n J,w N ,δ J D − log h id−1 τR N − cβδ U N
(3.8)
i
i=1
for any N large enough, where c < ∞ depends on d and u. / N , and let u = χU regular. For any ε > 0, Proposition 3.9. Assume β > βˆc and β ∈ for small enough δ > 0, there are K ∈ N and c > 0 such that, for large enough N : √ √ M 1 K ω = π on E N ,δ − e−c N e−c N . P inf J,w,+ ∈ V(u, ε) U N mβ 2 π ∈DUN ,δ (3.9) Proof (Proposition 3.8). To realize the event of disconnection DUN ,δ , it is enough to √ realize all the DR N and to close all the edges that are at distance less than 1 + d from i
N
∂U \
n ! i=1
˙i R
∪
n !
∂lat Ri ,
i=1
where ∂lat R stands for the lateral boundary of R, that is the faces of ∂R that are parallel to the orientation n of R. Thanks to Lemma 3.7 and Definition 3.1, there are at most
δcd N d−1 1 + Hd−1 (∂U ) such edges for large enough N . An immediate application of the DLR equation yields (3.8). Proof (Proposition 3.9). In order to obtain the claim for a mesoscopic scale K that does not depend on √ N , we proceed to a coarse grained analysis at two characteristic scales K and L N = [ N ]. Given K ∈ N , we consider (i , i )i∈I N ,K the (K , K )-covering of N as in Definition 5.1 in [44] as well as the phase indicator (φi )i∈I N ,K given by Theorem 5.10 in [44], for the tolerance δ. We call F = {0, 1} I N ,K the set of site configurations on the index of blocks I N ,K . In order to apply the stochastic domination theorem 5.10 (iv) in [44], we will define an increasing function f : F → {0, 1} with the
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˜ j, ˜ ) j∈JN ,K appropriate properties. First, we need to describe the L N -blocks: we call ( j the (L N , L N )-covering for I N ,K as in Definition 5.1 in [44]. Then we let
˜ j , E w i ∩ E N ,δ = ∅ J = j ∈ J N ,K : ∀i ∈ U and I =
!
˜ j .
j∈J
˜ Given ρ ∈ F a site configuration on I N ,K and j ∈ J , we say that the L N -block j ˜ , of density at least 1 − δ. is good if there is a crossing cluster of open sites for ρ in j Then we define f : F → {0, 1} letting ˜ is good . For all j ∈ J , j
f (ρ) = 1
Clearly, f is an increasing function. We prove now that its expectation is close to 1 under high-parameter site percolation. Consider B Ip the site percolation process on I of density p ∈ (0, 1). According to Theorem 1.1 in [19], for large enough p < 1 there is c > 0 such that, for large enough N , for all j ∈ J : ˜ j is good 1 − exp −2cL d−1 , B Ip N and consequently (the cardinal of J is bounded by N d ), for p < 1 close enough to 1, for large enough N , √ B Ip ( f ) 1 − exp −c N . Consequently, the stochastic domination for (|φi |)i∈I N ,K (see Theorem 5.10 (iv) in [44]) yields the same lower bound on the expectation of f ((|φi |)i∈I ): for large enough K (depending on δ), there is c > 0 such that, for any N large enough: √ !
J,+ w w
E (i ) 1 − e−c N . E inf N ,β f (|φi |)i∈I ω = π on E ( N )\ π i∈I
(3.10)
The event that f (|φi |)i∈I = 1 gives a control on the magnetization. For large enough N , the blocks (i )i∈I cover a fraction of N that is close to 1−Ld (∂U + B(0, cd δ)) −→+ δ→0
1. This and the properties of (φi )i∈I N ,K (Theorem 5.10 (i) and (ii) in [44]) imply that, for small enough δ > 0, for large enough N :
MK MK f (|φi |)i∈I = 1 ⇒ ∈ V(u, ε) or ∈ V(1, ε). mβ mβ We now consider a boundary condition π ∈ DUN ,δ . Because of the ω-disconnection, the spin of the clusters touching some i ⊂ N U with i ∈ I has a symmetric distribution under the conditional measure N ,δ f
J,+ . . = 1 and ω = π on E |) (|φ i i∈I U N ,β
Surface Tension in the Dilute Ising Model. The Wulff Construction
Hence, one has inf π ∈D N ,δ J,+ N ,β U
1 2
inf π ∈D N ,δ U
MK mβ J,+
N ,β
197
∈ V(u, ε) ω = π on EUN ,δ f (|φi |)i∈I ω = π on EUN ,δ .
The claim follows as (3.10) implies, as EUN ,δ ⊂ E w ( N )\ i∈I E w (i ), that √ √ J,+ N ,δ −c/2 N 1−e P inf N ,β f (|φi |)i∈I ω = π on EU e−c/2 N . π ∈DUN ,δ
The final formulation of the lower bound for phase coexistence is the following: Proposition 3.10. Assume β > βˆc and β ∈ / N . For any 0 α < 1/ diam∞ (W q ) and ε > 0 there exists K ∈ N such that, 1 J,+ M K q ∈ V(χ , ε) −F q (χα W q ) P-a.s., lim inf d−1 log µ z 0 +α W N N →∞ N mβ (3.11) where z 0 = (1/2, . . . 1/2). Similarly, for any λ > 0 and 0 α < 1/ diam∞ (W λ ), λ 1 J,+ M K lim inf d−1 log E µ N ∈ V(χz 0 +α W λ , ε) −F λ (χα W λ ). N →∞ N mβ (3.12) Proof. Let U = z 0 + According to Theorem 3.2.35 in [23], ∂U is rectifiable, hence the profile u = χU is regular. Let ε, δ > 0. Thanks to Theorem 3.4 there exists a n δ-covering (Ri )i=1 adapted to the profile χU and τ q . Proposition 3.8 applies and gives, for δ > 0 small enough: J,+ M K J,w,+ M K N ,δ µ ∈ V(χ , ε) inf
∈ V(χ , ε)|ω = π on E U U N U N mβ mβ π ∈DUN ,δ n d−1 J d−1 × exp −N h i τR N + cβδ , (3.13) αW q .
i=1
i
where c < ∞ depends on d and u. An important remark is that the two factors are independent under the product measure P. Proposition 3.9 yields: √ 1 J,w,+ M K N ,δ P inf N ∈ V(χU , ε)|ω = π on EU e−c N . (3.14) mβ 3 π ∈DUN ,δ We prove first (3.11) and consider γ , ξ > 0. If δ > 0 is small enough, Theorem 1.4 tells that the P-probability that τ J N > τ q (ni ) + γ for some i ∈ {1, . . . , n} decays like Ri
√
exp(−cN d ) where c > 0. Hence, with P-probability at least 1 − e−c N we have n 1 1 J,+ M K τ , ε) − log µ ∈V(χ h id−1 (τ q (ni ) + γ )−cβδ− d−1 log 3 z +α W 0 N d−1 N mβ N i=1
−F q (χα W q ) − ξ
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for small enough δ > 0 and γ > 0. Borel-Cantelli Lemma ensures that P-almost surely, 1 J,+ M K lim inf d−1 log µ N ∈ V(χz 0 +α W τ , ε) −F q (χα W q ) − ξ N →∞ N mβ and (3.11) follows letting ξ → 0+ . We conclude with the proof of (3.12), take λ > 0 n and denote here U = z 0 + αW λ . Again, there exists a δ-covering (Ri )i=1 adapted to N λ the profile χU and τ . For N large enough, the Ri are disjoint and hence the τ J N are Ri
independent under P. Consequently, for N large enough and λ > 0, (3.13) and (3.14) give λ l 1 J,+ M K d−1 d−1 J ∈ V(χU , ε) × E exp −λN h τ E µ N N i Ri mβ 2 × 3λ i=1 × exp −λN d−1 cβδ . In view of Proposition 2.3, this means λ n 1 J,+ M K lim inf d−1 log E µ N ∈ V(χU , ε) h id−1 τ λ (ni ) − λcβδ, − N →∞ N mβ i=1
and the claim follows as δ → 0. 3.4. Upper bound for phase coexistence. Here we address the opposite problem of providing an upper bound on the probability of phase coexistence along a given phase profile. Our analysis follows the same line as [5,6,11]. The cost of phase coexistence is easily related (Proposition 3.11) to another notion of surface tension (3.15), that uses a L 1 -characterization of phase coexistence. Then the L 1 -notion of surface tension is related to a percolative definition of surface tension with free boundary conditions, with the help of the minimal section argument (Proposition 3.12). As in the uniform setting [11], the surface tension with free boundary condition differs very slightly from the usual notion of surface tension (Proposition 3.13). The L 1 -definition of surface tension is as follows. Given δ > 0, a rectangular parallelepiped R ⊂ [0, 1]d as in Definition 3.1 (i) and K , N ∈ N we define MK 1 J,δ,K J,σ¯ d log sup µ − χ 2δL (R) , (3.15) τ˜N R = − 1 N R mβ (h N )d−1 σ¯ ∈ + L (R) NR
J,σ¯ where χ is the characteristic function of R as in Definition 3.1 (iii), and µ the Gibbs NR
measure on N R with boundary condition σ¯ . We have:
Proposition 3.11. Let u ∈ BV, δ > 0 and assume that (Ri )i=1...n is a δ-covering for u. Then, for any ε > 0 small enough, any K , N ∈ N one has: n 1 J,+ M K log µ ∈ V(u, ε) − h id−1 τ˜NJ,δ,K (3.16) N Ri . N d−1 mβ i=1
Surface Tension in the Dilute Ising Model. The Wulff Construction
199
Proof. For ε > 0 small enough, the implication MK MK ∈ V(u, ε) ⇒ − u δLd (Ri ), ∀i ∈ {1, . . . , n} 1 mβ mβ L (Ri ) holds. Thanks to (iii) in Definition 3.1, for such ε we have MK MK ∈ V(u, ε) ⇒ − χi 2δLd (Ri ), 1 mβ mβ L (Ri )
∀i ∈ {1, . . . , n}.
J,+ implies that Now, the Gibbs property for µ N
J,+ µ N
MK MK J,+ d ∈ V(u, ε) µ N − χi 2δL (Ri ), ∀i ∈ {1, . . . , n} mβ mβ L 1 (Ri ) n J,σ MK J,+ d = µ N µ − χi 2δL (Ri ) 1 N Ri m β L ( R ) i i=1 exp −h id−1 N d−1 τ˜NJ,δ,K Ri ,
thanks to (3.15), and the claim is proved. Using the minimal section argument as in [5] one can compare the L 1 -surface tension to the surface tension under free boundary condition in R = Rx,L ,H (S, n), defined as J τ˜R =−
1 L d−1
J, f
log R
DR˜ ,
(3.17)
˜ = Rx,L ,H/2 (S, n) is a rectangular parallelepiped twice finer than R. where R Proposition 3.12. Assume β > βˆc with β ∈ / N . Then, there exists cd,δ ∈ (0, ∞) with limδ→0 cd,δ = 0 such that, for any R as in Definition 3.1 (i), for any δ > 0, if K is large enough then: lim sup N
1 J,δ,K J < 0. log P τ ˜ τ ˜ − c d,δ NR RN Nd
(3.18)
We do not detail here the proof of Proposition 3.12 as it is easily adapted from [5]. Then, the argument of [11] let us quantify the influence of the boundary condition on the value of surface tension: Proposition 3.13. Assume β > βˆc and β ∈ / N . Let R be a rectangular parallelepiped R as in Definition 3.1 (i), with δ ∈ (0, 1). Then, lim sup N
1 J J log P τ ˜ τ − c δ < 0, d N N R R Nd
where cd < ∞ depends on d only.
(3.19)
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We cannot afford to give here the proof of Proposition 3.13 as the generalization to the random case of the argument of [11] makes it far too long. However, no new ingredient needs to be introduced with respect to the original construction [11], and the interested reader can consult the PhD thesis [45] for a complete development of the proofs of both Propositions 3.12 and 3.13. The consequence of the three last propositions, together with Varadhan’s Lemma, is a lower bound on the probability of phase coexistence along a given profile under quenched and averaged measures: Proposition 3.14. For all β > βˆc with β ∈ / N , for every u ∈ BV and ξ, λ > 0, there exists ε > 0 such that, for K ∈ N large enough,
lim sup N
1 N d−1
J,+ log µ N
MK ∈ V(u, ε) −F q (u) + ξ mβ
(3.20)
in P-probability (and P-almost surely if β ∈ / N I ) and
lim sup N
1 N d−1
λ J,+ M K log E µ ∈ V(u, ε) −F λ (u) + ξ. N mβ
(3.21)
Proof. We fix δ ∈ (0, 1) and a δ-covering (Ri )i=1...n for u as in Definition 3.2. We examine first the quenched convergence: according to Propositions 3.12 and 3.13 there is c > 0 such that J P τ˜NJ,δ,K τ − c − c δ 1 − exp(−cN d ), ∀i = 1 . . . n d,δ d N Ri R
(3.22)
for K and N large enough. On the other hand, for any ε > 0 small enough Propositions 3.11 yields 1 N d−1
J,+ log µ N
n MK ∈ V(u, ε) − h id−1 τ˜NJ,δ,K Ri , mβ i=1
and hence, for K and N large enough, 1 N d−1
J,+ log µ N
n MK J ∈ V(u, ε) − h id−1 τR N − cd,δ − cd δ mβ i=1
with P-probability greater than 1 − n exp(−cN d ). This implies (3.20) for δ > 0 small enough in view of the convergence τ J N → τ q (ni ) in P-probability (Theorem 1.3) or of Ri
the almost-sure convergence if β ∈ / N I (Corollary 1.9). We examine now the averaged convergence: consider λ > 0 and again, a δ-covering (Ri )i=1...n for u. For K , N large
Surface Tension in the Dilute Ising Model. The Wulff Construction
201
enough and ε > 0 small enough we have
λ MK E ∈ V(u, ε) mβ n d−1 J,δ,K λ(h i N ) τ˜N Ri E exp −
J,+ µ N
i=1
n exp(−cN d ) + E exp − × exp λ
n
J λ(h i N )d−1 τR N
i=1
n
d−1 d−1 cd,δ + cd δ hi N
i=1
in view of (3.22). Varadhan’s Lemma (Proposition 2.3) yields: for any ε > 0 small enough, any K large enough, lim sup N
−
1 N d−1
l
J,+ µ N
log E
λ MK ∈ V(u, ε) mβ
h id−1 τ λ (ni ) − cd,δ − cd δ ,
i=1
and the conclusion follows for δ > 0 small enough. The way to the proof of 1.11 and 1.12 is now standard : The proof of Bodineau, Ioffe and Velenik given in [6] for the exponential tightness property applies in the present case: (the compact set BVa was defined at (3.3)) / N , there exists C > 0 and for every δ > 0, Proposition 3.15. For any β > βˆc with β ∈ for any K ∈ N large enough one has lim sup N
1 N d−1
J,+ log Eµ N
MK ∈ / V(BVa , δ)c mβ
−Ca.
(3.23)
Then, Theorems 1.11 and 1.12 are consequences of the large deviations estimates (Propositions 3.10 and 3.14) together with the exponential tightness (Proposition 3.15) in view of the compactness of BVa . The case of averaged Gibbs measures (Theorems 1.13, 1.15 and 1.16) presents complete similarity with the non-random case.
3.5. Localization of the Wulff crystal under averaged measures. One consequence of the introduction of the random media is the localization of the Wulff crystal if the volume constraint acts on the media as well: the surface tension appears to be reduced on the contour of the crystal. Here we give the proof of Theorem 1.17 after we state the following immediate consequence of the lower large deviations described in Theorem 1.6:
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Lemma 3.16. Let R N = R0,N ,δ N (S, n) and γ > 0, A = [τˆ λ,− (n) − γ , τˆ λ,+ (n) + γ ]. Then, 1 × exp −λN d−1 τ J < τ λ (n). lim sup d−1 log E 1 J RN τ N ∈Ac N N R Proof (Theorem 1.17). According to Theorems 1.13 and 1.16, it is enough to prove that λ M 1 K J,+ × µ N lim sup d−1 log E 1 J m − χz+α W λ 1 ε τ N ∈Ac β N →∞ N R L < −F λ (αW λ ).
(3.24) + α∂W λ ,
for δ > 0 In the case that the parallelepiped R does not intersect the crystal z small enough any δ-covering (Ri )i=1...n for z + αW λ and τ q does not intersect R. For ε > 0 small enough and K large enough, Propositions 3.11, 3.12 and 3.13, the definition J from the τ˜ J,δ,K under the product measure of the δ-covering and the independence of τR N Ri P imply that the right-hand side of (3.24) is bounded from above by 1 J c , −F λ (αW λ ) + o (1) + lim sup d−1 log P τR N ∈ A δ→0 N N which is strictly smaller than −F λ (αW λ ) for small enough δ, as the last term is strictly negative. Now we consider the case when the parallelepiped R is tangent to the crystal. For h > 0 small enough, for ε > 0 small enough, the strict inequality ⎤ ⎡ ⎛ ⎞λ 1 ⎥ ⎢ ⎝ sup µ J,σ¯ M K − χz+α W λ lim sup d−1 log E ⎣1 J ε ⎠⎦ 1 m τ N ∈Ac N R + N β N R σ¯ ∈ L (R) NR
<−
∂(z+α W λ )∩R
τ λ (n. )dH
(3.25)
holds according to Propositions 3.12 and 3.13, and Lemma 3.16. Let (Ri )i=1...n be a η-covering for z + αW λ and τ λ . Propositions 3.11, 3.12 and 3.13 and the properties of the η-covering imply that for ε > 0 small enough (depending on η) and large enough K , the cost of phase coexistence outside of R is bounded above by ⎡⎛ ⎞ ⎤ λ MK 1 ⎢⎜ ⎟ ⎥ J,σ¯ lim sup d−1 log E ⎣⎝ sup µ − χz+α W λ ε ⎠ ⎦ 1 N Ri + N m i:Ri ∩R=∅ σ¯ ∈ N Ri
N
−
∂(z+α W λ )\R
β
L (R)
τ λ (n. )dH + o (1). η→0
Thus, choosing h > 0 small enough then ε > 0 small enough and K large enough, the strict inequality holds in (3.24) and the claim follows. Acknowledgements. During the elaboration of this work the author enjoyed numerous stimulating discussions with Thierry Bodineau. Most of the results presented here were obtained during a PhD Thesis at Université Paris Diderot [45]. The author is grateful to Marie Théret and Raphaël Rossignol for useful and pleasant discussion about maximal flows and concentration, and acknowledges the very helpful work done by the referees on the paper.
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32. Kesten, H.: Aspects of first passage percolation. In: École d’été de probabilités de Saint-Flour, XIV—1984, Volume 1180 of Lecture Notes in Math., Berlin: Springer, 1986, pp. 125–264 33. Kesten, H.: Surfaces with minimal random weights and maximal flows: a higher-dimensional version of first-passage percolation. Illinois J. Math. 31(1), 99–166 (1987) 34. Kingman, J.F.C.: The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30, 499–510 (1968) 35. Ledoux, M.: The concentration of measure phenomenon. Volume 89 of Mathematical Surveys and Monographs. Providence, RI: Amer. Math. Soc., 2001 36. Messager, A., Miracle-Solé, S., Ruiz, J.: Convexity properties of the surface tension and equilibrium crystals. J. Stat. Phys. 67(3-4), 449–470 (1992) 37. Pfister, C.-E.: Large deviations and phase separation in the two-dimensional Ising model. Helv. Phys. Acta 64(7), 953–1054 (1991) 38. Pisztora, A.: Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Related Fields 104(4), 427–466 (1996) 39. Rossignol, R., Théret, M.: Lower large deviations for maximal flows through a box in first passage percolation. http://arXiv.org/abs/0801.0967v1[math.PR], 2008 40. Taylor, J.E.: Crystalline variational problems. Bull. Amer. Math. Soc. 84(4), 568–588 (1978) 41. Théret, M.: On the small maximal flows in first passage percolation. Annales de la faculté Des Sciences de Toulouse 17(1), 207–219 (2008) 42. Théret, M.: Upper large deviations for the maximal flow in first passage percolation. Stoch. Proc. Appl. 117(9), 1208–1233 (2007) 43. Wouts, M.: Glauber dynamics in the dilute Ising model below Tc . In preparation 44. Wouts, M.: A coarse graining for the Fortuin-Kasteleyn measure in random media. Stoch. Procs. Appl. 118(11), 1929–1972 (2008) 45. Wouts, M.: The dilute Ising model : phase coexistence at equilibrium & dynamics in the region of phase transition. Ph.D. thesis, Université Paris 7 - Paris Diderot, available at http://tel.archives-ouvertes.fr/tel00272899, 2007 46. Zhang, Y.: Critical behavior for maximal flows on the cubic lattice. J. Stat. Phys. 98(3-4), 799–811 (2000) 47. Zhang, Y.: Limit theorems for maximum flows on a lattice. http://arXiv.org/abs/0710.4589[math.PR], 2007 Communicated by F. Toninelli
Commun. Math. Phys. 289, 205–252 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0814-4
Communications in
Mathematical Physics
Exact Results for Topological Strings on Resolved Y p,q Singularities Andrea Brini1,2 , Alessandro Tanzini1,2 1 Mathematical Physics sector, International School for Advanced Studies (SISSA/ISAS),
Via Beirut 2, I-34014, Trieste, Italy. E-mail: [email protected]; [email protected] 2 Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste, Trieste, Italy Received: 1 June 2008 / Accepted: 17 February 2009 Published online: 24 April 2009 – © Springer-Verlag 2009
Abstract: We obtain exact results in α for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y p,q manifolds and give rise via M-theory compactification to SU ( p) gauge theories on R4 × S 1 . As an application we present a detailed study of the local F2 case and compute open and closed genus zero GromovWitten invariants of the C3 /Z4 orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A1 Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y p,q geometries. Contents 1. 2. 3.
4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cones over Y p,q . . . . . . . . . . . . . . . . . . . . . . . . . . Mirror Symmetry for Local CY and Integrable Systems . . . . . 3.1 Period integrals . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Mirror symmetry for open strings. . . . . . . . . . . . . 3.2 The B-model moduli space . . . . . . . . . . . . . . . . . . 3.3 Relation with integrable systems and five-dimensional gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving the G K Z System in the Full Moduli Space . . . . . . . 4.1 Period integrals and Lauricella functions . . . . . . . . . . . Warm-up Tests of the Formalism . . . . . . . . . . . . . . . . . 5.1 Local F0 : mirror map at large radius . . . . . . . . . . . . . 5.2 Local F0 : orbifold point . . . . . . . . . . . . . . . . . . . 5.3 Local F2 at large radius . . . . . . . . . . . . . . . . . . . .
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Local F2 and [C3 /Z4 ] Orbifold Gromov-Witten Invariants . . . . . . . 6.1 Orbifold mirror map and genus zero invariants . . . . . . . . . . . 6.2 Adding D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Modular structure of topological amplitudes . . . . . . . . . . . . 6.3.1 The change of basis from large radius. . . . . . . . . . . . . . 6.3.2 Local F2 and (2) modular forms . . . . . . . . . . . . . . . 6.4 Higher genus amplitudes . . . . . . . . . . . . . . . . . . . . . . 6.4.1 One loop partition function for Y p,q and genus 1 orbifold GW. 6.4.2 Generalities on g > 1 free energies. . . . . . . . . . . . . . . 6.4.3 g > 1 and modular forms. . . . . . . . . . . . . . . . . . . . 6.4.4 The g = 2 case in detail. . . . . . . . . . . . . . . . . . . . . 7. Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . A. Euler Integral Representations, Analytic Continuation and Generalized Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . B. Lauricella Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Analytic continuation formulae for Lauricella FD . . . . . . . . .
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1. Introduction Since their formulation, topological theories have been a most fruitful source of results and ideas both in physics and mathematics. Topological amplitudes naturally arise in the BPS sector of superstrings [1,2] and supersymmetric gauge theories [3] and as such have a wide range of applications, from the evaluation of BPS protected terms in lowenergy effective actions to black hole microstates counting [4]. Moreover, topological theories have provided new and powerful tools for the computation of global properties of manifolds, e.g. Donaldson polynomials, Gromov-Witten invariants, revealing at the same time surprising relationships between seemingly very different areas of mathematics. One of the most appealing features of topological strings is that the calculation of its amplitudes can be pushed to high orders, sometimes to all orders, in perturbation theory. To this end, one exploits symmetries and recursion relations coming either from the underlying N = (2, 2) supersymmetric sigma-models - as mirror symmetry and holomorphic anomaly equations [2] - or from the properties of some specific class of target manifolds - as localization and geometric transitions for the A-model on toric Calabi-Yaus [5], or W-algebras and integrable hierarchies on the corresponding B-model side [6]. These methods have been mostly applied in the large volume region of the Calabi-Yau, where the perturbative expansion in α is well-behaved and the topological string partition function has a clear geometric interpretation as a generating functional of Gromov-Witten invariants. However, away from the large volume region, the perturbative series diverges and the corresponding geometrical interpretation breaks down. Very few exact results are known outside this perturbative regime, although significant progress has been recently obtained by using modular invariance [7] and new matrix-model inspired techniques [8]. In this paper, we obtain exact results in α for a large class of toric Calabi-Yau threefolds, and calculate the corresponding topological string amplitudes in the full moduli space of closed and open strings. The basic idea is to resort to the correspondence with five-dimensional gauge theories via M-theory compactification on the Calabi-Yau times a circle [9]. More precisely, the geometries that we consider are obtained from minimal resolution of Y p,q singularities, and M-theory compactification over them give rise
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to SU ( p) gauge theories on R4 × S 1 with a q-dependent five dimensional Chern-Simons term [10,11]. The mirror geometry can be written as a fibration over an hyperelliptic curve, whose periods provide a basis for the solutions of the B-model Picard-Fuchs equations. Our main result is that we get a closed form for the (derivatives of the) periods on the whole B-model moduli space. We then expand them in different patches and calculate in this way topological amplitudes not only in the large volume region, but in all phases, including orbifold and conifold points. The analytic continuation properties and modular structure underlying higher genus amplitudes can be easily worked out in our approach. As an application, we give predictions for Gromov-Witten invariants for the orbifold C3 /Z4 , which corresponds to the blow-down of local F2 geometry ( p = q = 2). We observe that, for p = q, the hyperelliptic curve appearing in the mirror geometry can be identified with the spectral curve of an integrable system, given by the relativistic generalization of the A p−1 Toda chain [12]. The fact that with our method we can find closed formulae for any value of the parameter q suggests the existence of a wider class of relativistic integrable systems. The structure of the paper is the following: in Sect. 2 we review the toric geometry of Y p,q singularities and their minimal resolutions, in Sect. 3 we discuss mirror symmetry and the relation with integrable systems, in Sect. 4 we outline our procedure to find topological amplitudes in the whole B-model moduli space, in Sect. 5 we provide some preliminary checks of our formalism. In Sect. 6 we present a detailed study of the local F2 case: we first compute open and closed genus zero Gromov-Witten invariants of the C3 /Z4 orbifold, then analyze the modular properties of the topological wave function and use them to predict higher genus invariants. We conclude in Sect. 7 with some comments and future perspectives. Some technical details on the analytic continuation of topological amplitudes are collected in the Appendix. 2. Cones over Y p,q The toric geometry of Y p,q singularities [13] has been extensively studied in the context of AdS/CFT correspondence [14], with the aim to provide non-trivial checks1 for superconformal theories with reduced amount of supersymmetry. We observe here that minimal resolution of such singularities gives rise precisely to the local Calabi-Yau geometries that one usually considers to “geometrically engineer” gauge theories via M-theory compactifications [11]. The manifolds Y p,q , with p and q integers such that 1 < q < p, are an infinite class of five-dimensional manifolds on which explicit Sasaki-Einstein metrics can be constructed [13]; the two extremal cases q = 0 and q = p may be formally added to the family, corresponding to Z p quotients respectively of T 1,1 (the base of the conifold) and of S 5 /Z2 . Since Y p,q are Sasaki-Einstein, the metric cone C(Y p,q ) constructed over them is Kähler Ricci-flat; moreover, given that the base has a T3 of isometries effectively acting by (Hamiltonian) symplectomorphisms, the cone is a toric threefold [14], that is, it contains an algebraic three-torus (C∗ )3 as a dense open subset acting on the full variety through an extension of the natural action on itself (for an introduction to toric geometry see for example [16,17]). As any toric CY threefold, its geometry is fully codified by a three dimensional fan whose rays end on an affine hyperplane, say r3 = 1, in the three dimensional space R3 with coordinates (r1 , r2 , r3 ). For C(Y p,q ), this is given by 1 See also [15] for related work.
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the following four lattice vectors in Z3 : ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 −1 v1 = ⎝ 0 ⎠ , v2 = ⎝ 0 ⎠ , v3 = ⎝ p ⎠ , v4 = ⎝ p − q ⎠ . 1 1 1 1
(2.1)
In the following we will be interested in investigating the (G K Z extended) Kähler moduli space of toric and canonical class preserving complete resolutions of C(Y p,q ). At the level of the toric diagram, this amounts [17] to add the p − 1 internal points v4+ j = (0, j, 1) for j = 1, . . . , p − 1 and declare that the set of three dimensional cones in the fan is given by the simplicial cones whose projection on the r3 = 1 hyperplane yields a triangulation of the polyhedron {v1 , v2 , v3 , v4 }. p,q ) and the cor The resolved geometry will be henceforth denoted as X p,q ≡ C(Y responding fan as p,q . It might also be described as a holomorphic quotient (C p+3 \ Z )/(C∗ ) p with Z a co-dimension > 0 locus determined by the toric data [16] and the k th C∗ factor (k) (k) acting on the coordinates of C p+3 as z i → λQi z i , where Qi ∈ Z is a set of integers such that p+3
(k)
Qi vi = 0
k = 1, . . . , p.
(2.2)
i=1
The set of charges for X p,q is given by (see Fig. 2) Q1 Q2 Q3 Q4 .. .
Qp
= = = =
(A, −2 A − B, (0, 1, (0, 0, (0, 0, .. .. . . = (0, 0,
B, 0, 0, 0, .. .
A, 0, 0, 0, 0, −2, 1, 0, 0, 1, −2, 1, 0, 0, 1, −2, .. .. .. .. . . . . 1, 0, 0, 0, 0,
0, 0, 0, . . . , 0, . . . , 1, . . . , .. .. . . 0, . . . ,
0, 0, 0, 0, .. .
0) 0) 0) 0) .. .
(2.3)
1, −2)
with A and B coprime numbers solving the Diophantine equation ( p − q)A + p B = 0 for q < p, while A = 1 and B = 0 for q = p. In real polar coordinates (|z i |, θi ), this corresponds to the Higgs branch of a N = (2, 2)d = 2 gauged linear σ -model (GLSM) [18] with p + 3 chiral fields z i . The D−term equation of motion is p+3
Qkj |z i |2 = tk
k = 1, . . . , p
(2.4)
k = 1, . . . , p,
(2.5)
j=1
and
U (1) p
acts as (k)
z j → e2πiQ j
θj
zj
where θ j = arg(z j ). The Fayet-Iliopoulos parameters tk are complexified Kähler parameters of X p,q . Indeed, the full cohomology ring of the smooth CY manifold thus obtained can be easily read off from the fan [16,19]. For example, Betti numbers are b0 = 1,
b2 = p,
b4 = p − 1,
b6 = 0.
(2.6)
Exact Results for Topological Strings on Resolved Y p,q Singularities
209
(0, p)
(−1, p−q)
(0, 0)
(1, 0)
Fig. 1. The fan of C(Y p,q ) for p = 5, q = 2
(0, p)
(−1, p−q)
(0, 0)
(1, 0)
p,q ) for p = 5, q = 2 Fig. 2. The fan of C(Y
Various aspects of these geometries have been considered in the context of topological strings. First of all, notice from Figs. 1–3 and formula (2.6) that for p = 1 we encounter the two most studied local curves: the conifold (q = 0) and C × K P1 (q = 1), whose enumerative geometry [20], phase structure [21] and local mirror symmetry properties [22] have been extensively studied. For p = 2, the local CY in question is the total space of the canonical line bundle K Fq over the q th Hirzebruch surface, q = 0, 1, 2. For higher p we have the ladder geometries considered in [11,23–25] in the context
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(p, 1)
(q−p, −1) (0, −1)
p,q ) for p = 5, q = 2 Fig. 3. The pq-web diagram of C(Y
Table 1. Orbifold degenerations of X p,q into C3 /Z p+q for the first few values of p and q. The fourth column lists the weights of the Z p+q action on the coordinates (z 1 , z 2 , z 3 ) of C3 p 1 1 2 2 2
q 0 1 0 1 2
p+q 1 2 2 3 4
Weights (0, 0, 0) (0, 1, 1) (0, 1, 1) (1, 1, 1) (1, 1, 2)
of geometric engineering of pure SY M theories with eight supercharges. In a suitable field theory limit described in [23], the Gromov-Witten large radius expansion for these geometries was shown to reproduce for all q the weak coupling instanton expansion of the prepotential for N = 2 SU ( p) pure Yang-Mills in d = 4. Subsequently, they were shown [9,10,26], to geometrically engineer N = 1 SU ( p) SY M on R4 × S 1 with Chern-Simons coupling k = p − q, with the field theory limit above interpreted now as the limit in which the fifth-dimensional circle shrinks to zero size. The Kähler moduli space of these geometries presents a manifold richness of phenomena which provide a natural testing ground for A-twisted topological string theory away from the large radius phase, as well as for the search of direct evidence for open/closed dualities in the strongly coupled α regime. The pivotal example of the latter is given by the case q = 0, which is the large N dual background of the open A-model on T ∗ L( p, 1) obtained via geometric transition [27]. Moreover, it was noticed in [14,19] that X p,q can be “blown-down” to orbifolds of flat space of the form C3 /Z p+q (see Table 1); here the geometric picture becomes singular, though leaving still open the possibility for extracting enumerative results in terms of orbifold Gromov-Witten invariants. These are precisely the cases we will turn to study in Sect. 4. 3. Mirror Symmetry for Local CY and Integrable Systems 3.1. Period integrals. A procedure for constructing mirror duals of (among others) toric CY threefolds has been provided in [28], elaborating on previous results of [29,30]. The 2 ∗ 2 mirror geometry X p,q of X p,q is an affine hypersurface in C × (C ) , x1 x2 = H p,q (u, v)
(3.1)
with (x1 , x2 ) ∈ C2 and (u, v) ∈ (C∗ )2 . In (3.1) H p,q (u, v) is the Newton polynomial [31] of the polytope p,q ∩ {r3 = 1} in Z3 given by the intersection of the fan with the affine hyperplane r3 = 1, a2 u p−q − ai+3 u i . v p
H p,q (u, v) = a1 v +
i=0
(3.2)
Exact Results for Topological Strings on Resolved Y p,q Singularities
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The geometry is therefore that of a quadric fibration over the H p,q (u, v) = λ ∈ C plane, which degenerates to a node above the punctured Riemann surface H p,q (u, v) = 0. We will call the latter the mirror curve p,q . Now, mirror symmetry in the compact case prescribes to reconstruct the A-model prepotential from the computation of the (properly normalized2 ) periods of the holomorphic (3, 0) form [] ∈ H 3,0 (X p,q ) on a symplectic basis of homology three-cycles = , X p,q ,Z) ∈H 3 (
where in this case would be the residue form on H p,q = 0 of the holomorphic 4-form in H 4,0 (C2 × (C∗ )2 \ X p,q ), dx1 dx2 du/udv/v = Res H p,q (u,v)=x1 x2 . (3.3) x1 x2 − H p,q (u, v) Special geometry then ensures [32] that the periods are related as = (t0 (a), ti (a), ∂ti F(a), 2F − ti ∂ti F),
(3.4)
i
where ti (a) defines a local isomorphism between the A model moduli space of X p,q and the B model moduli space of X p,q , while F(t) is the prepotential, i.e. the sphere amplitude. They have respectively single and double logarithmic singularities at the large complex structure point [29]. In the local case under scrutiny we must actually cope with the absence of a symplectic basis for H3 (X p,q , Z); according to [22,33], the formalism carries through to the local setting by considering non-compact cycles as well and defining the periods along them via equivariant localization.3 We will denote the corresponding extended homology as H3(ext) (X p,q , Z). The usual procedure that (ti , ∂i F) are found is via integration of the associated G K Z hypergeometric system [29,32] (k) (k) ∂ Qi ∂ Qi = ∂ai ∂ai (k) (k)
Qi >0
k = 1, . . . , p
(3.5)
Qi <0
with Qi(k) as in (2.3). In a patch of the B-model moduli space, these are the Picard-Fuchs (P F) equations for X p,q . Typically, solutions of (3.5) are obtained via series integration by Frobenius method, i.e. solving recursion relations for the coefficients of a series expansion for (ti , ∂i F). This has to be done patch by patch in the moduli space though, and it turns out to be hardly practicable as soon as the number of Kähler classes increases. To our knowledge, there is no explicit solution in the literature when h 11 (X p,q ) > 2. is via direct integration. This not viable in the A possible alternative way to find general case, but notice that due to the particular form (3.1) of X p,q , the integration of 2 In this section, we will not be careful about normalization factors. This will be of course our concern in the calculations of Sect. 4. 3 We bother with this solely for the case of local curves. For p ≥ 2, as we will see, the integration over compact cycles is sufficient to extract enumerative information.
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(p, 1)
L (q−p, −1) (0, −1)
Fig. 4. The pq-web for p = 5, q = 2 with a lagrangian brane on an inner leg
over three-cycles boils down to that of a 1-differential dλ over a basis of cycles in (ext) H1 ( p,q , Z). As shown in [22], the periods of solve the P F system (3.5) if and only if those of dudv log H p,q (u, v) (3.6) dλ p,q ≡ Res H p,q (u,v)=0 uv do on a basis of H1(ext) ( p,q , Z). Picking up the residue gives dλ = log v
du , u
and the periods are thus computed as γ p,q =
(ext) γ ∈H1 ( p,q ,Z)
(3.7)
log v
du . u
(3.8)
Unfortunately, the integrals are typically too awkward to carry out and no expression is known except for the simplest case of local curves; a perturbative evaluation of them, though clearly possible, has no real advantages compared to tackling the P F system upfront. However, we will see in Sect. 4 how to handle them in a direct way. 3.1.1. Mirror symmetry for open strings. Recently, the open string sector of the A-model on toric CY has been subject to deeper investigation, following the insight of [34,35], where a class of special Lagrangian submanifolds was constructed generalizing [36]. The prescription of [34] relies on the realization of a toric CY as a (degenerate) T3 fibration, parameterized by θi , over the |z i | base, see (2.4, 2.5). The authors of [34] consider a 3 − k real dimensional linear subspace W of the base qiα |z i |2 = cα α = 1, . . . , k; qiα ∈ Q, (3.9) i
Tk
fibration L over this subspace in such a way that the Kähler form and then specify a ω = i d|z i |2 ∧ dθi vanishes on it, ω| L = 0.
(3.10)
The total space of this fiber bundle L is then Lagrangian by construction; moreover, it turns out that it is volume minimizing in its homology class (special Lagrangian) if and only if i qiα = 0. In this case (3.10) implies θi = 0. (3.11) i
Exact Results for Topological Strings on Resolved Y p,q Singularities
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In the case in which cα in (3.9) are such that W intersects the edges of the toric web, i.e. the loci where one S 1 of the toric fibration shrinks, L splits into two Lagrangians L ± with topology R2 × S 1 : the open modulus z is then given by the size of the circle, complexified with the holonomy of a U (1) connection along it. The mirror symmetry construction of [28] has been extended to these brane configurations in [34]. When k = 2, L + (resp. L − ) gets mirror mapped to a curve parameterized by x2 (resp. x1 ) H p,q (u, v) = 0 = x1 (resp. = x2 ).
(3.12)
The moduli space of the mirror brane is then simply the mirror curve p,q . Picking a parametrization thereof (for instance the projection on the u or v-lines) by a complex variable z leads us to write the open topological partition function Fopen ({ti }, z) = Fg,h ({ti })z h ; (3.13) g,h
the sum is both over the genus of the source curve and the number of connected components of its boundary. The choice of a “good” parametrization is dictated by mirror symmetry and is related to phase transitions in the open string moduli space for branes ending on toric curves meeting at a vertex in the web (see [8] for details). A very important fact is that the meromorphic differential dλ turns out to have a significant role for open string amplitudes as well. The dimensional reduction of the holomorphic Chern-Simons action on the mirror brane indeed yields a particularly simple expression for the disc amplitude g = 0, h = 1. It is simply given by the “Abel-Jacobi” map z du F0,1 (t, z) = (3.14) log v(u) , u where the integral on the r.h.s. is a chain integral [z ∗ , z], with z ∗ a fixed point on the mirror curve. In [37] it was noticed that the disc amplitude (3.14) in a suitable parameterization gets the form n F0,1 (t, z open ) = Nm,n Li 2 (e−t·m z open ). (3.15) m,n
In (3.15) Nm,n are integer numbers counting open string BPS states and z open is the dressed open coordinate [35] wi − ti z open = z + , (3.16) ri i
where wi are combinations of gauge-invariant sigma model variables vanishing at the point of maximally unipotent monodromy wi =
(i)
Qj
aj
= ti + O(e−ti )
(3.17)
j
and ri are rational numbers. Notice that the open flat coordinates get corrected by closed worldsheet instantons only. As discovered in [38],(see also [39]) an extended PicardFuchs system may be constructed such that (3.14), (3.16) are in its kernel and this can be used for determining the ri in (3.16).
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3.2. The B-model moduli space. A few remarks are in order at this point. The Riemann surfaces p,q come in a family parameterized by {ai } in (3.1), which are the complex moduli of the mirror geometry. The curve p,q will be generically smooth in the B-model moduli space: we will denote the open set where this happens as M Bp,q . However, a compactification of M Bp,q will lead to loci where this is no longer true. Indeed, p,q degenerates to a singular curve on the so-called principal discriminant locus of the P F system (3.5). In correspondence with this, one of the homology cycles of p,q shrinks to zero size; G K Z solutions have then singularities and are subject to logarithmic monodromy transformations around these loci, to which we will refer as conifold loci. Moreover, there are also regions in the B-model moduli space where the curve p,q stays smooth, but the periods have finite monodromy because the moduli space itself is singular, locally looking like C p /Zn ; at the conformal field theory level, this would be reflected by the appearance of a discrete quantum symmetry. We will refer to the latter as an orbifold phase, in which we still retain a geometric picture, though involving a singular (orbifold) target space in the A-model side. To see why this happens from the mirror perspective one might argue as follows. The ai ’s are sort of homogeneous coordinates for the complex structure moduli space p,q . Indeed, only p out of p + 3 are really independent, as an overall rescaling of of X them and scalings of u and v in (3.1) leave invariant the symplectic form du dv ∧ u v
(3.18)
in (C∗ )2 . That is, the moduli space of the mirror theory might be seen as arising from a holomorphic quotient of C p+3 by a (C∗ )3 action with charges (see (3.2)) a1 a3 a p+3 a2 Q1 1, 0, 0, −1, Q2 0, 0, p, p − q, Q3 1, 1, 1, 1,
a4 0, 1, 1,
. . . a p+2 ..., 0 ..., p − 1 ..., 1
(3.19)
By subtracting a suitable codimension >0 locus to C p+3 , we thus end up with a toric compactification of the family M Bp,q , which we call M B,tor p,q . Remarkably enough, inspection shows [32] that the skeleton of a fan for the above system of charges is simply given by the columns of the G L S M (2.3), and the toric variety associated with it is complete. This fan is called the secondary fan of X p,q . In fact, strictly speaking we are not dealing with a toric variety, as typically the secondary fan will contain non-smooth simplicial cones, perhaps with marked points along their facets. In the latter case, this would mean that the patch parameterized by the corresponding ai ’s looks like C p /Zn rather than C p ; as such, the periods of the holomorphic three-form will inherit the finite monodromy from the monodromy of the ai themselves. This will be of fundamental importance in our study of the [C3 /Z4 ] orbifold in sect. 4. Additional, but somewhat milder phase transitions involve the purely open string sector as well and are related, as already anticipated, to a choice of a parametrization of p,q . See [8], to which we refer for a complete discussion of this subject. 3.3. Relation with integrable systems and five-dimensional gauge theories. Interestingly, the mirror curves of X p,q geometries are related to the Seiberg-Witten curve of five dimensional gauge theories and the related integrable systems. Since, as we will
Exact Results for Topological Strings on Resolved Y p,q Singularities
215
show in the next section, this observation will prove to be very fruitful in the study of the topological string moduli space of X p,q , we describe it here in some detail. First of all, let us rewrite (3.2) as Y 2 = Pp (X )2 − 4a1 a2 X p−q
(3.20)
Y = a1 v − a2 u p−q /v, X = u, p Pp (X ) = ai+3 X i .
(3.21)
upon setting
i=0
In the a1 = a2 = ( R) p , a3 = a p = 1 patch the curve (3.20) and the differential (3.7) are precisely the Seiberg-Witten curve and differential of SU ( p)N = 1 SYM theory on R4 × S 1 with a q-dependent Chern-Simons term [11,26]. Moreover, the SW curve and differential in the case p = q were shown in [40] to coincide with the spectral curve and action differential of the A p−1 Ruijsenaars model [12], i.e. the A p−1 periodic relativistic Toda chain. More precisely, setting ζ = R, (3.2) reads for p = q, 1 du ζ p (v + ) = 1 + u l Sl + u p , dλ p, p = log v v u p−1
p, p :
l=1
which can be rewritten as det(L(z) − w) =
p (−w) p− j σ j (z) = 0
(3.22)
j=0
with the Lax matrix defined as L i j = e Rpi f i (li j + bi j ), li j = δi, j+1 (1 + ζ p z)ξi − δi,1 δ j, p (1 + ζ − p z −1 )ξ1 , −(iζ ) p i ≤ j − 1 , bi j = 1 i > j −1
(3.23)
f i2 = (1 − ζ 2 eqi+1 −qi )(1 − ζ 2 eqi −qi−1 ), ξi−1 = 1 − ζ −2 eqi−1 −qi , where q p+1 = q1 , q0 = q p , σ j are the elementary symmetric functions of L(z), S j their z-independent factor, and we have made the change of variables [40] −wu = 1 + ζ p z, z = v. An identification of the curves for q < p as the spectral curves of some finite dimensional integrable mechanical system seems to be presently not known, and it would be interesting to understand the role of the q parameter in this context. A second important remark is about the “field theory limit” discussed in [23]. From the mechanical system point of view, the parameter ζ in (3.23) is essentially the inverse
216
A. Brini, A. Tanzini
of the speed of light, while in the field theory perspective ζ = R, where is the strong coupling scale and R is the radius of the fifth-dimensional circle. This means that the four-dimensional limit might be achieved as the non-relativistic limit of the Toda chain. Denoting with eφi , i = 1 . . . , p the roots of the polynomial Pp (x) (3.22) and introducing the new set of variables [11], y = Y, X = e2Rx , eφi = e R(ai −ai+1 ) ,
(3.24)
we have that in the R → 0 limit (3.20) reduces to y 2 = P˜ p (x)2 − 4 2 p ,
(3.25)
which is the Seiberg-Witten curve of N = 2 SU ( p) Super Yang-Mills in d = 4. Notice that in the R → 0 limit we completely lose track of the Chern-Simons parameter q, which has disappeared in formula (3.25). More importantly, the variable x in (3.25) takes now values in C in contrast with the C∗ −variable X . Because of this, the peculiarly fivedimensional form of the differential (3.7), i.e. dλ = log vd log u, is replaced in the R → 0 limit by that of the usual (non-relativistic) Toda differential dλ = xd ln( P˜p + y). In fact, as we will see in the following, the relativistic system and its non-relativistic counterpart - which by the discussion above coincide respectively with the A−model on X p,q and with its 4d Seiberg-Witten limit - bear still deep structural resemblances and our aim will be to try to exploit this to our advantage. 4. Solving the G K Z System in the Full Moduli Space In this section we provide a method for finding the mirror map, as well as the sphere and disc amplitude, for the A-model on X p,q to all orders in α without resorting to solving the G K Z system directly. This will be accomplished by finding closed forms for deriv atives of the period integrals γ p,q w.r.t. the bare moduli as generalized hypergeometric functions. First of all, let us resume what the ingredients at our disposal are. According to (3.20) the mirror curve p,q is a two-fold covering of the X plane branched at Y (X ) = 0, that is the locus Pp (X )2 = 4a1 a2 X p−q .
(4.1)
The resulting curve has genus p − 1 and four punctures corresponding to the two inverse 2p images of X = 0, X = ∞. Let us denote the solutions to (4.1) as {bi }i=1 . A basis for (ext) H1 ( p,q , Z) might be taken as the circles Ai , Bi encircling the intervals I Ai = [b2i−1 , b2i ]
I Bi = [b2i , b2i+1 ]
(4.2)
for i = 1, . . . , p − 1, plus a circle A0 around one of the punctures at X = 0 and a contour B0 connecting the two punctures at X = 0 and X = ∞. The 1-differential dλ p,q is given, in an affine patch parameterized by X , as
⎛ ⎞ P (X ) ± Pp (X )2 − 4a1 a2 X p−q d X p du ⎠ = log ⎝ , (4.3) dλ p,q (X ) = log v(u) u 2a1 X
Exact Results for Topological Strings on Resolved Y p,q Singularities
217 X=
A0
B
A
X=0
X=b1
X=b2
X=b3
X=b4
Fig. 5. Cuts and punctures of the X plane in the genus 1 case
and a complete set of periods can be obtained by integrating it over the A/B-cycles dλ p,q . (4.4) A/B = A/B
More explicitly, Ai
Bi
A0
B0
⎛
⎞ Pp (X )2 − 4a1 a2 X p−q d X ⎠
, = log ⎝ X b2i−1 Pp (X ) − Pp (X )2 − 4a1 a2 X p−q
⎛ ⎞ b2i+1 Pp (X ) + Pp (X )2 − 4a1 a2 X p−q d X ⎠
, = log ⎝ X b2i Pp (X ) − Pp (X )2 − 4a1 a2 X p−q
⎛ ⎞ Pp (X ) ± Pp (X )2 − 4a1 a2 X p−q d X ⎠ , = log ⎝ 2a1 X X =0
⎛ ⎞ ∞ Pp (X ) ± Pp (X )2 − 4a1 a2 X p−q d X ⎠ . = log ⎝ 2a1 X 0
b2i
Pp (X ) +
(4.5)
(4.6)
(4.7)
(4.8)
We now make the following observation. As we have already noticed, the curve (3.20) and the differential (4.3) are the Seiberg-Witten (SW) curve and differential of a five dimensional theory compactified on a circle. In Seiberg-Witten theory, the gauge coupling matrix τi j =
∂ Bi ∂u k
∂ Ak ∂u j
−1
,
(4.9)
where u i are Weyl-invariant functions of the scalar fields, is known to be the period matrix of the compactified SW curve, that is a ratio of periods of holomorphic differentials. We then expect that derivatives of dλ p,q with respect to suitable functions of the bare moduli are holomorphic differentials on the compactified p,q , [∂ f (ai ) dλ] ∈ H 1,0 ( p,q ).
(4.10)
This is substantiated by the fact that, for p = q = 2, the relativistic Toda system and the non-relativistic one share the same oscillation periods [41]; more precisely, the derivatives of the action with respect to the energy are the same (elliptic) functions of the bare
218
A. Brini, A. Tanzini
parameters. This was also noticed in [40] in the study of the singularities of the moduli space of N = 1 SU (2) SY M in d = 5. Explicitly, we indeed have ∂dλ p,q Xj = d X, ∂a j+4 Pp2 (X ) − 4a1 a2 X p−q
(4.11)
i.e., for j = 0, . . . , p−2, a basis of holomorphic 1-forms on the 4-point compactification p,q = p,q ∪ {0+ , 0− , ∞+ , ∞− } of the spectral curve p,q . 4.1. Period integrals and Lauricella functions. This last observation allows us to give a straightforward recipe for computing series expansions of solutions of the G K Z system (3.5) in the full B-model moduli space. The procedure is the following: 1. Start with Ai /Bi and consider its a j+4 derivative for 0 ≤ j ≤ p − 2, ei+1 ∂ Ai /Bi Xj
= dX ∂a j+4 2p ei (X − b ) i i=1
(4.12)
with ei = b2i−1 , ei = b2i for the A and B cycles respectively. The hyperelliptic integral (4.12) has a closed expression given in terms of multivariate generalized hypergeometric functions of Lauricella type [42], ∂ Ai /Bi = eiϕ π ∂a j+4
(ei ) j k =i,i+1 (ek
(2 p−1)
× FD
− ei )
1 1 1 ei+1 − ei , ; , . . . , , j; 1; x1 , . . . , xi , xi+1 , . . . , x2 p , 2 2 2 ei (4.13)
where x j = (ei+1 − ei )/(e j − ei ), 2ϕ = lπ , l ∈ Z is a phase depending on xi and FD(n) is the hypergeometric series (n)
FD (α; {βi }; γ ; {δi }) =
∞ m 1 ...m n =0
(α)m + ···+m n (β1 )m 1 . . . (βn )m n δ1m 1 . . . δnm n (γ )m + ···+m n m 1 ! . . . m n ! (4.14)
which converges when |δi | < 1 for every i. In the above formula we used the standard Pochhammer symbol (α)m = (α + m)/ (α). There are many alternative ways to express (4.12), for instance in terms of hyperelliptic θ functions; however, the above expression proves to be useful due to the fact that Lauricella FD(n) has good analytic continuation properties outside the unit polydisc |δi | < 1; some formulae, as well as asymptotic expansions around singular submanifolds, are collected in the Appendix, while others can be found in [42,43]. Notice that, as opposed to the usual situation in solving P F equations by the Frobenius method, we are not dealing here with hypergeometric functions of the bare moduli, but rather of the relative distance xi between ramification points; they have singular values precisely when the latter becomes 0,
Exact Results for Topological Strings on Resolved Y p,q Singularities
219
1 or infinity, that is when we encounter a pinching point of p,q . This shift in perspective is definitely an advantage compared to other expressions for hyperelliptic integrals, involving for instance the F4 Appell function for genus 2 [44,45]. These are simpler functions of the bare moduli, but have worse analytic continuation properties and are less suited for a more complete study of the moduli space, regarding for instance intersecting submanifolds of the principal discriminant locus. The above fact was already pointed out in [46], where the properties of FDn were exploited to study the Z3 point of N = 2 SU (3) SY M. (n) In many cases, FD can be reduced to a more familiar form. For instance, for p = 2 we have the expected complete elliptic integrals of the first kind (b1 − b2 )(b3 − b4 ) ∂ A 2 , (4.15) K = √ ∂a4 (b1 − b3 )(b2 − b4 ) (b1 − b3 )(b2 − b4 ) ∂ B (b1 − b3 )(b2 − b4 ) 2 . (4.16) K = √ ∂a4 (b1 − b2 )(b3 − b4 ) (b1 − b2 )(b3 − b4 ) 2. Once we have a representation for the derivatives of the periods in the form (4.13), (4.15)-(4.16) we can use the formulae in Appendix B to analytically continue them in any given patch of the B-model moduli space and find a corresponding power series expansion in the bare moduli ai . Integrating back with respect to a j yields Ai and Bi up to a constant of integration, independent of a j for 0 ≤ j ≤ p − 2. This has to be fixed either by some indirect consideration (for instance, by imposing a prescribed asymptotic behavior around a singular point) or by plugging it inside the P F system and imposing that the period be in the kernel of the G K Z operators. This operation leads to a closed O D E integrable by quadratures, which completes the solution of the problem of finding expansions for Ai /Bi everywhere in the B-model moduli space. 3. The procedure provides us with p − 1 flat coordinates as well as p − 1 conjugate periods out of which to extract the prepotential. In order to find the p th modulus, we pick up the residue (4.7), ⎧ a3 ⎪ log ± for q < p ⎪ ⎨ a1 2 dλ = , (4.17) a3 ± a3 −4a1 a2 ⎪ X =0± for q = p ⎪ ⎩ log 2a1 which are manifestly solutions of (3.5). In the following, we will choose an appropriate combination of them in order to have a prescribed behavior around the expansion point under scrutiny. 4. Closed form computation of derivatives with respect to a j can be done for open string amplitudes as well, which might be used to trade an expansion in terms of the z parameter in (3.16) with one in a j , completely resummed w.r.t. z. In this case, dealing with chain integrals instead of period integrals leads one to consider indefinite integrals and thus incomplete hyperelliptic integrals. The latter can still be given the form of a multivariate Lauricella function, but with order increased by one [42], √ ∂F0,1 ({ak }, z) (ei ) j = eiϕ π (4.18) 2 z ∂a j+4 k =i,i+1 (ek − ei )
1 1 3 e2 − e1 (2 p) 1 1 × FD ; , . . . , , j, ; ; x3 , . . . , x2 p , ,z . 2 2 2 2 2 e1
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A. Brini, A. Tanzini
(−1,2)
(0,2)
(0,1)
(0,0)
(1,0)
Fig. 6. The fan of local F0
As before, for p = 2 (4.18) boils down to an incomplete elliptic integral of the first kind in the form ∂F0,1 ({ak }, z) 1 (b1 − b4 )(b2 − z) a˜ −1 F sin , (4.19) =2 ∂a4 (b2 − b4 )(b1 − z) b˜ b˜ where a˜ = (b2 − b4 )(b1 − b3 )
b˜ = (b2 − b3 )(b1 − b4 ).
Another important advantage of this method is that, instead of integrating back patchwise with respect to a j , we can get our hands dirty and work directly with an Euler-type integral representation of the periods. The fact that FD(n) has a single integral representation saves us most of the pain in the problem of finding the explicit analytic continuation of Ai /Bi , which in the multi-parameter case involves the use of multi-loop MellinBarnes integrals. The details for the case p = q = 2 which will be of interest later on for the computation of orbifold Gromov-Witten invariants are reported in Appendix A, where also a closed expression for the A-period can be found in terms of a generalized Kampé de Fériet hypergeometric function. 5. Warm-up Tests of the Formalism Let us show how the steps described in Sect. 4.1 allow to quickly recover some known results about mirror symmetry for local surfaces. 5.1. Local F0 : mirror map at large radius. Local mirror symmetry for K F0 has been studied in [47] in the check of the large N duality with Chern-Simons theory on S 3 /Z2 . The mirror curve in this case can be written as a1 v + a2 /v = a3 /u + a4 + a5 u.
(5.1)
Exact Results for Topological Strings on Resolved Y p,q Singularities
221
Good variables around the large complex structure point [29] are given by a1 a2 a3 a5 zB = 2 , zF = 2 . a4 a4
(5.2)
Let us use the scaling freedom (3.19) to set a3 = a5 = 1,
a1 = a2 .
(5.3)
By using the change of variables (3.20) the curve (5.1) is then given by
2 X 4z B 2 2 2 X , Y = X + √ +1 − zF zF which is a double covering of the X −plane branched at √ √ −1 + 2 z B − 1 − 4 z B + 4z B − 4z F b1 = , √ 2 zF √ √ −1 − 2 z B − 1 − 4 z B + 4z B − 4z F b2 = , √ 2 zF √ √ −1 + 2 z B + 1 − 4 z B + 4z B − 4z F b3 = , √ 2 zF √ √ −1 − 2 z B + 1 − 4 z B + 4z B − 4z F b4 = . √ 2 zF
(5.4)
(5.5) (5.6) (5.7) (5.8)
We choose the A-cycle as the loop encircling [b1 , b2 ]. The asymptotics of the corresponding period will indeed identify it as the flat coordinate around z B = z F = 0. By expanding (4.15) in (z B , z F ) we have √ ∂ A = z F (20z 3B + 6(30z F + 1)z 2B ∂a4 + 2(90z 2F + 12z F + 1)z B + 20z 3F + 6z 2F + 2z F + 1) + · · ·
(5.9)
which integrates to 20z 3B + 60z F z 2B + 3z 2B + 60z 2F z B . 3 From (4.17) and (5.2) we can compute the remaining flat coordinate as A = log(z F ) +
zB 1 0 = − log . 2 zF
(5.10)
(5.11)
It is then easy to see that the combinations of periods that have the right asymptotics at large radius are given by − t B ≡ −2 0 (z B , z F ) + A (z B , z F ), Inversion of (5.10) and (5.11) reads, setting Q B =
−t F ≡ A (z B , z F ). e−t B ,
QF =
(5.12)
e−t F ,
z B = 6Q3B − 2Q2B + 6Q2F Q B − 2Q F Q B + Q B + · · · , z F = 6Q3F − 2Q2F + 6Q2B Q F − 2Q B Q F + Q F + · · · , which is the mirror map as written in [48].
(5.13)
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A. Brini, A. Tanzini
5.2. Local F0 : orbifold point. Analogously, we can write down the expansion for the orbifold point √ [47], which corresponds to a1 = a2 = a3 = a5 = 1, a4 = 0. Setting a1 = a2 = 1 − x1, a4 = x1 x2 , a3 = a5 = 1 as in [47], we have s1 ≡ 0 = − log(1 − x1 ), 1 s2 ≡ A + B /2 = [x2 (35(32(x1 − 2)x1 (x1 (11x1 − 96) + 96)E(x1 ) 61931520π + x1 (x1 (x1 (x1 (105x1 − 1856) + 8000) − 12288) + 6144)K (x1 ))x82 + · · · .
Upon introducing s˜1 = s1 and s˜2 = s1 /s2 we have x1 (˜s1 ) = 1 − e−˜s1 ,
s˜23 s˜23 s˜2 s˜2 49˜s2 2 3 − − s˜1 + − s˜1 + − 192 192 256 768 737280 5˜s23 7˜s25 7˜s25 7˜s23 − − (5.14) + s˜14 + s˜ 5 + · · · 73728 245760 98304 983040 1
s˜2 x2 (˜s1 , s˜2 ) = s˜2 + s˜1 + 4
in perfect agreement with [47]. Needless to say, the prepotential computation can be checked exactly the same way. We have Fs2 ≡ A =
1 x1 x2 (75x31 x62 − 56x21 (10x22 + 9)x42 + 64x1 (10x42 + 21x22 53760
+70)x22 − 107520 K (1 − x1 ) + · · ·
x x32 x2 x32 2 21 5 3 3 1 s2 − x1 + ( + )x1 + − x2 + x x = log 16 12 4 48 128 768 2 1 5x32 185x2 + + (5.15) x41 + · · · 1536 1024 which reproduces the analogous formula in [47], modulo the ambiguity in the degreezero contribution. 5.3. Local F2 at large radius. We might proceed along the same lines for the case of local F2 . The curve is given by a1 v +
a2 = a3 + a4 u + a5 u 2 . v
(5.16)
Branch points are located at ⎧ √ √ 2 ⎪ ⎨ ± a4 −4a3 a5 −8 a1 a2 a5 ≡ ±c a4 1 2a5
u+ = , 2 −4a a +8√a √a a ⎪ a 2a5 3 5 1 2 5 ⎩ ± 4 ≡ ±c 2 2a5
(5.17)
Exact Results for Topological Strings on Resolved Y p,q Singularities
223
and we have accordingly ∂ A = ∂a4
c2
dX (X 2
c1
K 1−
− c12 )(X 2
− c22 )
=
c22 c12
,
c1
(5.18)
∂ B = ∂a4
c1
−c1
dX = (X 2 − c12 )(X 2 − c22 )
c12 c22
2K c2
.
(5.19)
In this case good coordinates associated to the base P1 and the P1 fiber are zB =
a1 a2 , a32
zF =
a3 a5 . a42
Upon setting a1 = a2 , a3 = a5 = 1, periods take the form √ 16 z B z F √ 2K − ∂t F ∂ A −8 z B z F −4z F +1
= =− √ , ∂z F ∂z F πzF 1 − 4 2 zB + 1 zF √ −8 z z −4z +1 4K 8√z Bz F−4z F+1 ∂ B ∂ 2F B F F = =− √ , ∂z F ∂t F ∂z F zF 1 − 4 1 − 2 zB zF t B = 0+ − 0− = 2i tan−1 4z B − 1 ,
(5.20)
(5.21)
where the normalization has been chosen in order to get the right asymptotics. Integration and inversion yields the mirror map at the large radius point z B (Q B ) =
QB , (Q B + 1)2
z F (Q B , Q F ) = (1 + Q B ) Q F + −2 − 4Q B − 2Q2B Q2F + 3 + 3Q B + 3Q2B + 3Q3B Q3F + · · · ,
(5.22)
with Q B = e−t B , Q F = e−t F and therefore ∂t F F(Q B , Q F ) = (log(Q F ) log(Q B Q F )) + (4 + 4Q B ) Q F + (1 + 16Q B 4Q3B 4 2 2 2 + 36Q B + 36Q B + + Q B )Q F + Q3F 9 9
1 + 260Q2B + 64(Q B + Q3B ) Q4F + · · · + (5.23) 4 as in [29].
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A. Brini, A. Tanzini
(0, 2)
(0, 1)
(−1, 0)
(0, 0)
(1, 0)
Fig. 7. The fan of local F2
6. Local F2 and T C3 /Z4 U Orbifold Gromov-Witten Invariants 6.1. Orbifold mirror map and genus zero invariants. We will now apply the considerations above to the study of the tip of the classical Kähler moduli space for local F2 , where the compact divisor collapses to zero size. The resulting geometry [19] is a Z4 orbifold of C3 by the action (ω; z 1 , z 2 , z 3 ) → (ωz 1 , ωz 2 , ω−2 z 3 ), with ω ∈ Z4 . In the orbifold phase, the genus zero closed amplitude computes [49] the generating function of genus-zero correlators of twist fields F or b (s1/4 , s1/2 ) =
1 m n m n O1/4 O1/2 s1/4 s1/2 . n!m! n,m
(6.1)
In (6.1) the sum is over the generators s1/4 , s1/2 of the orbifold cohomology ring and they are associated respectively with the twisted sectors 1/4 and 1/2 under the Z4 action. The corresponding topological observables are denoted respectively as O1/4 and O1/2 . The m O n compute genus-zero orbifold Gromov-Witten invariants N or b correlators O1/4 1/2 0,(m,n) with m insertions of weight 1/4 and n of weight 1/2. From Fig. 7 we see that Mori vectors for local F2 are Q1 = (0, 1, 1, 0, −2), Q2 = (1, −2, 0, 1, 0),
(6.2)
and the mirror curve 2,2 has the form (5.16) a1 v +
a2 = a3 + a4 u + a5 u 2 . v
Following [50] we argue that the point we are looking for in the B-model moduli space is given by a3 = a4 = 0. This would amount to shrinking to zero size the compact divisor given, in the homogeneous coordinates z i introduced in Sect. 2, by z 5 = 0. When resolving C3 /Z4 , the latter corresponds to the extra divisor in the blow-up procedure: indeed, dropping z 5 from the G L S M (6.2) leads one to the system of charges of the base F2 inside local F2 .
Exact Results for Topological Strings on Resolved Y p,q Singularities
225
(0, 2)
(0, 1)
(−1, 0)
(0, 0)
(1, 0)
Fig. 8. The fan of [C3 /Z4 ]
a1
Z 2 POINT
a2 2
(0, 1) LARGE RADIUS
(C x C /Z 2 ) a4 (−2, 0)
(1, 0) (−1, 0)
a5
(0, 0)
Z 2 POINT Z 4 POINT 3
(C /Z4 ) a3 (1, −2) Fig. 9. The secondary fan of local F2
This argument is strengthened by the following remark. The secondary fan of (6.2) is shown in Fig. 9 and has the set of charges (see (3.19)) a1 a3 a5 a2 a4 Q1 1, 0, 0, −1, 0 Q2 0, 0, 2, 0, 1 Q3 1, 1, 1, 1, 1
(6.3)
B,tor B is simplicial but with marked The fan of the toric compactification M2,2 of M2,2 B,tor points: M2,2 is thus a toric orbifold. Its orbifold patches are, as shown in Fig. 9, a 2 smooth C patch containing the large complex structure point, two non-smooth [C2 /Z2 ]
226
A. Brini, A. Tanzini
cones, and finally a [C2 /Z4 ] patch parameterized by (a3 , a4 ). Inspection shows that the latter is a toric orbifold of C2 by the action Z4 × C2 → C2 λ(x, y) → (λx, λ2 y).
(6.4)
(a3 , a4 ) = (0, 0) is therefore the only Z4 point in the compactified moduli space as expected. From (6.4) we see that good coordinates around (a3 , a4 ) = (0, 0) are given by √ a3 = de, (6.5) a4 = d 1/4 . Let us then find a complete basis of solutions for the G K Z system around this point. Picard-Fuchs operators are written in this patch as 1 L1 = a3 ∂a24 + θa4 θa3 , 2 1 2 1 2 L2 = ∂a3 − (θa4 − 4θa23 ) − θa3 θa4 , 16 4 and the branch points (5.17) here read
1 2 ± c1 = ± a − 4a3 − 8, 2 4
1 2 ± c2 = ± a − 4a3 + 8, 2 4 while the period integrals (5.18),(5.19) and (4.17) become
a42 −4a3 −8 a42 −4a3 +8 K a 2 −4a −8 K a 2 −4a +8 3 3 4 4 ∂a4 A = − , a42 − 4a3 + 8 a42 − 4a3 − 8
a42 −4a3 −8 K a 2 −4a +8 3 4 ∂a4 B = 2 , 2 a4 − 4a3 + 8
⎛ ⎞ 2−4 a a3 3 ⎠. 0± = log ⎝ ± 2 2
(6.6)
(6.7)
(6.8)
We want to find solutions of the P F system (6.6) with prescribed monodromy around (d, e) = (0, 0), in order to match them with the conjugacy classes of Z4 . Defining !
(8 − 8i)π 1/2 B A − , (6.9) s1/4 = 2 2 1−i 41 2 !
(4 + 4i) 41 B A s3/4 = + , (6.10) π 3/2 2 1+i s1/2 = −2i 0− + π,
(6.11)
Exact Results for Topological Strings on Resolved Y p,q Singularities
227
we then have
2 e e 1 25e3 2 − + d− d + ··· , s1/4 (d, e) = d 1/4 1 + 32 192 2560 18432 ! 3 5 2 7 3 3e 5e d d d e + + + ··· , s1/2 (d, e) = d 1/2 e + 24 640 7168
3 1 3e 3e2 9e 3 3/4 s3/4 (d, e) = d e− + − + − d + ··· . 12 32 128 2560 14336
(6.12) (6.13) (6.14)
The normalization of the mirror map has been fixed by imposing the correct asymptotics s1/4 ∼ a4 , s1/2 ∼ a3 as to reproduce the generators of the classical orbifold cohomology. These are given by a4 and a3 respectively for the weight 1/4 and 1/2 twisted sectors. Concerning the solution s3/4 , this is identified with the derivative of the generating function For b in (6.1) with respect to s1/4 ; in fact, the orbifold cohomology pairing modifies this relation by a factor of 4, i.e. s3/4 = 4∂s1/4 For b . Taking all this into account, inversion of (6.12) and (6.13) gives the following expression for the prepotential 4
∂For b (s1/2 , s1/4 ) ∂s1/4 3 5 7 9 s1/2 29s1/2 457s1/2 s1/2 11 + + + + O s1/2 = s1/2 + s1/4 48 960 430080 92897280 2 4 6 8 s1/2 11s1/2 49s1/2 601s1/2 1 10 3 + − − − − − + O s1/2 s1/4 12 96 9216 368640 41287680 3 5 7 s1/2 47s1/2 6971s1/2 7s1/2 9 5 + + + + O s1/2 + s1/4 3840 1920 460800 412876800 +....
(6.15)
As a check, the prepotential thus obtained is invariant under monodromy. The first few orbifold GW invariants are listed in Table 2. Our predictions exactly match the results4 obtained in [51] after the methods of [52].
6.2. Adding D-branes. Following the discussion of Sect. 3.1.1 we might want to turn on an open sector and add Lagrangian branes to the orbifold. The procedure of [34,35] is in principle valid away from the region of semi-classical geometry and has had a highly non-trivial check for the local F0 case in [8], where open amplitudes have been matched against Wilson lines in the large N dual Chern-Simons theory. First of all, we will consider the setups I and II of Fig. 10, with a D-brane ending respectively on the outer leg |z 1 | = |z 3 | and |z 1 | = |z 2 |. The choice of variables (3.2) we have made for the mirror curve 2,2 , in which the B-model coordinate mirror to |z 2 | was gauge-fixed to one, corresponds to phase I I . This means that v is the variable that goes to one on the brane and X (I I ) ≡ u is the good open string parameter to be taken as the independent 4 We are grateful to Tom Coates for sharing with us his computations and for enlightening discussions on this point.
228
A. Brini, A. Tanzini or b Table 2. Genus zero orbifold Gromov-Witten invariants N0,(m,n) of [C3 /Z4 ]
m n 0 1 2 3 4 5 6 7 8 9 10 11 12 13
2
4
6
8
0
− 18 0 1 − 32 0 11 − 256 0 147 − 1024 0 1803 − 2048 0 − 70271 8192 0 − 15933327 131072 0
0
9 − 64 0 143 − 512 0 159 − 128 0 − 157221 16384 0 − 3719949 32768 0 − 498785781 262144 0 − 11229229227 262144 0
1 4
0 1 32
0 1 32
0 87 1024
0 457 1024
0 7859 2048
0 801987 16384
7 128
0 3 32
0 47 128
0 20913 8192
0 1809189 65536
0 56072653 131072
0 2354902131 262144
10 0 1083 1024
0 85383 16384
0 360819 8192
0 73893099 131072
0 5312434641 524288
0 254697581847 1048576
0 31371782305803 4194304
I |z3| = 0 |z1| = 0
|z5| = 0
|z4| = 0
II |z2| = 0 Fig. 10. The pq-web of local F2 with lagrangian branes on an upper (I) and lower (II) outer leg
variable in (3.16) [8]. The transition from phase II to phase I is accomplished by the (exponentiated) S L(2, Z) transformation X (I I ) ≡ u →
1 ≡ X (I ) , u
(6.16)
v → vu 2 . Accordingly, the differential (3.7) has the form ⎧
2 +a X +1+ (a X 2 +a X +1)2 −4X 4 a3 X (I ⎪ 3 (I ) 4 (I ) ) 4 (I ) (I ) d X (I ) ⎪ ⎨ log 4 X (I ) phase I 2X (I )
dλ = . 2 2 2 ⎪ ⎪ ⎩ log a3 +a4 X (I I ) +X (I I ) + (a3 +a4 X (I I ) +X (I I ) ) −4 d X (I I ) phase II 2 X (I I )
(6.17)
We now turn to analyze the unframed A-model disc amplitude for a brane in phase I. In order to do that we have to compute the instanton corrected open modulus (3.16) and the
Exact Results for Topological Strings on Resolved Y p,q Singularities
229
Abel-Jacobi map (3.14). To determine the former, and more precisely the ri coefficients in (3.16), we use the result of [38], where the authors show that for this outer-leg configuration the large radius open flat variable solving the extended Picard-Fuchs system is given by LR z open =z+
tB tF + + πi, 4 2
(6.18)
where z = log X (I ) . In the (a3 , a4 ) patch containing the orbifold point this becomes LR z open = z + πi + O(a4 ) + O(a3 ).
(6.19)
L R and z + πi solve the extended P F system and Notice that in (6.18), (6.19), both z open L R does the job by construction, and the same is can then serve as a flat coordinate: z open true for z because it is a difference of solutions of the Picard-Fuchs system by (6.18). L R − t B − t F = z + πi is a global open Following [8], we have that the difference z open 4 2 flat variable and serves as the expansion parameter at the orbifold point. In terms of exponentiated variables, we then have: or b Z open = −X (I ) .
(6.20)
Having the mirror map and using (3.14) or (4.18) one can then mimic [8] and expand the chain integral, thus obtaining the disc amplitude F0,1 (a3 , a4 , z) as a function of the bare variables, or, using (6.12)–(6.13), of the flat variables. Notice that, since (a3 , a4 ) have non-trivial Z4 transformations, in order to preserve the fact that the curve (5.16) stays invariant we are forced to assign weights (1/4, 1/2) to (u, v) respectively, and so or b has weight −1/4. Eventually we get according to (6.20) Z open or b F0,1 (s1/4 , s1/2 , Z open )
= −
3 s1/2 s1/4
192
+
2 s s1/2 1/4
32
or b − s1/4 Z open
4 1 s1/4 or b 2 − + s1/2 − + ) (Z open 64 4 2 384 2 3 3 s1/4 7s1/2 s1/4 s1/2 s1/4 or b 3 − + + ) (Z open 576 9 3 2 s2 s1/2 1/4
+···
2 s1/4
(6.21)
which is monodromy invariant. The amplitude (6.21) should correspond to a generating function of open Gromov-Witten invariants of the C3 /Z4 orbifold. The situation for phase II appears to be more subtle. The resulting topological amplitude computed from the chain integral (3.14) picks up a sign flip under Z4 . This is not completely surprising, since it is known that disc amplitudes may have non-trivial monodromy [37], and it might also be seen to be related to the more complicated geometrical structure of the Z4 orbifold with respect to the Z3 case, due to the presence of non-trivial stabilizers for the cyclic group action.
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A. Brini, A. Tanzini
6.3. Modular structure of topological amplitudes. Higher genus amplitudes are associated to the quantization of the symplectic space spanned by the periods of the mirror curve [54]. The corresponding topological wave functional obeys recursion relations (BCOV equations [2]) that allow to compute higher genus amplitudes building on genus zero and one results, up to holomorphic ambiguities. It has been shown in [7] that this algorithm is made simpler and more efficient by exploiting modular properties of the topological amplitudes. Let us summarize very briefly the results of [7] relevant for our discussion. As recalled in Sect. 3, the choice of B-model complex structures can be parametrized in terms of the periods of the three-form in a chosen symplectic basis Ai ∩ B j = δ ij in H3 ( Xˆ , Z), which define "a so-called “real"polarization”. Special geometry relations between the periods xi = Ai and p j = B j are summarized in terms of a prepotential F0 (xi ) which turns out to be the genus zero free-energy of the topological string. The “phase space” (xi , p j ) can be endowed with a natural symplectic structure with symplectic form dxi ∧ dpi . The higher genus amplitudes Fg are associated to the quantization of this space, with the string coupling gs2 playing the rôle of . More precisely, the full topo 2g−2 logical string partition function Z (xi ) ∼ exp g gs Fg (xi ) is interpreted as a wave i function [54]. The periods (x , p j ) generically undergo an Sp(2 p−2, Z) transformation under a change of symplectic basis of the mirror curve. Correspondingly, the B-model topological amplitudes Fg have definite transformation properties that can be derived by implementing the canonical transformation at the quantum level on the topological wave function. The crucial observation of [7] is that there is a finite index subgroup ⊂ Sp(2 p − 2, Z) which is a symmetry of the theory. is precisely the group generated by the monodromies of the periods, which must leave invariant the topological wave-function. This symmetry constrains the topological amplitudes; in particular in the real polarization the Fg can be shown to be quasi-modular forms of [7], namely they transform with a shift. For example for the case of elliptic mirror curves, i.e. local surfaces, this amounts to say that the wave-function is a finite power series in the second Eisenstein series. We recall that one could also have chosen to parameterize the B-model moduli space with the Hodge decomposition of H 3 ( Xˆ , Z) in terms of a fixed background complex structure. The topological wave function in this holomorphic polarization can be shown [54] to obey the BCOV holomorphic anomaly equations. The topological amplitudes Fˆ g in this case turn out to be proper modular forms of weight zero under , namely invariant under , but they are non-holomorphic. For elliptic mirror curves, they can be written in terms of a polynomial in a canonical, non-holomorphic extension of the second Eisenstein series 3 1 E 2 (τ ) → Eˆ2 (τ, τ¯ ) := E 2 (τ ) − π mτ
(6.22)
with coefficients in the ring of holomorphic modular forms of . Thus one can pass from the real to the holomorphic polarization just by the above shift of variables. The advantage of the approach proposed in [7] is twofold. On one side it simplifies the solution of BCOV equations by restricting the functional dependence of the Fˆ g to the ring of modular forms. On the other it allows to relate the topological amplitudes in different patches of the B-model moduli space allowing in this way to extract enumerative invariants, e.g. at the orbifold point.
Exact Results for Topological Strings on Resolved Y p,q Singularities
231
As we will show in the following, our method is perfectly tailored to display the modular symmetry of the topological wave-function. In fact, the relation with the Seiberg-Witten curves greatly simplifies the analysis of the modular properties of higher genus amplitudes. Moreover, since we obtain explicit expressions for the periods of the mirror curve in terms of the branch points, it is enough to write the latter in terms of modular forms to make manifest the modular properties of genus zero and one topological amplitudes, thus providing the building blocks for the solution of BCOV equations. About the latter we point out however that there is a caveat: for the geometries under our study in addition to the modular dependence there is also a dependence on an extra parameter (independent of τ ), as in the discussion of [7] about the similar case of local F0 . This makes the solution of the BCOV equations at higher genus more involved computationally, since one has to fix a functional dependence on an extra datum. We choose to handle this problem with the approach developed in [8,58] in which the holomorphic Fg are defined via recurrence relations inspired by matrix-model techniques. This will allow us to display the general modular structure of the free energies in the local F2 case, in a way in which both the dependence on the modular variable and that on the extra parameter are completely fixed. In this section we first find the relevant change of basis from large radius to the orbifold point and then identify the ring of modular functions relevant for the local F2 case. These results provide the necessary tools to discuss higher genus invariants, which will be the subject of the next section. 6.3.1. The change of basis from large radius. We already saw in the last section that the mirror map at the orbifold point is obtained by choosing solutions of the G K Z system which diagonalize the monodromy of the periods. This implies that the solutions at large radius (1, t B , t F , ∂ F F) are related to those at the orbifold point (1, s1/4 , s1/2 , s3/4 ) by a linear transformation, which, for the subsector relating (t F , ∂ F F) and (s1/2 , s3/4 ) might be regarded as an (unnormalized)5 automorphism in H1 (2,2 , Z). In [7] it was shown that under a symplectic change of basis
A B → S = , S ∈ Sp(2 p − 2, Z) (6.23) C D the genus-g amplitudes Fg are subject to a transformation which can be derived by implementing the canonical transformation associated to (6.23) in the path integral defining the topological wave function. From saddle point expansion one then gets F˜ g = Fg + g (, Fr
(6.24)
where g is determined by the Feynman rules in terms of lower genus vertices ∂ n Fr
1 τ + C −1 D
(6.25)
up to normalization factors. In (6.25) τ is the period matrix of 2,2 . This means, for instance, that knowledge of Fg in the large radius region allows one to compute genus-g free energies in the full moduli space, provided that we know how the periods are transformed when going from one region to another. This was successfully exploited in [7] to 5 The determinant of the change of basis is equal to 2, see (6.28), due to the fact that we are dealing with a local threefold.
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A. Brini, A. Tanzini
predict higher genus orbifold Gromov-Witten invariants for the [C3 /Z3 ] orbifold from the ones of the large radius K P2 geometry. We underline that our method of solving the extended GKZ system in the full moduli space has the advantage to make much easier the study of the analytic continuation properties and the consequent computation of the linear change of basis (6.23). Indeed, instead of performing standard (but cumbersome) multiple Mellin-Barnes transforms, we can easily read off what S and are in our case from formulae (A.2-A.3) and (6.8). Let us define ⎞ ⎛ ⎛ ⎞ 1 1 tB ⎟ s1/2 ⎟ LR = ⎜ or b = ⎜ (6.26) ⎝ t ⎠, ⎝s ⎠. F 1/4 ∂F F s3/4 We can now simply compute the change of basis between the large radius and the orbifold point by using the Euler integral representation6 (A.2) (see also (6.9–6.11)). ˜ L R through We can relate or b = S ⎛ ⎞ 1 0 0 0 ⎜ π −i 0 0 ⎟ S˜ = ⎝ (6.27) ⎠, α β S γ δ where ⎛ ⎜ ⎜ S=⎜ ⎝
√ (1−i) π 2 41
2π 3/2 2 41
−
2 1 4
√
π
2 1 i 1 2+2 4
⎞ ⎟ ⎟ ⎟. ⎠
(6.28)
π 3/2
6.3.2. Local F2 and (2) modular forms The last formula in the previous section relates the physical periods of the large radius patch to those of the orbifold patch and represents one of the main ingredients to make predictions about orbifold Gromov-Witten invariants at higher genus. This should be already clear at this stage from (6.24), but it can in fact be brought to full power due to the beautiful results of [7]. Let us see how this works in detail for the p = 2, q = 2 case we have been considering, and in particular let us figure out what the relevant modular group is in this case. In fact, we have already answered this question: as we stressed in Sect. 3.3, the family of elliptic curves in this case is the same as its field theory limit, the only thing that changes being the symplectic structure defined on the elliptic fibration, i.e., the SW differential. This was already noticed in the strictly related case of local F0 in [7]. In particular we might argue as follows for the present case. By formulae (3.20), (5.17) the 2,2 family can be written as Y 2 = ( Xˆ 2 − c12 )( Xˆ 2 − c22 ),
(6.29)
6 Actually, the computation is simple only when one works with the a derivatives of the periods and then 4 integrates back. This gives rise to the unknown coefficients (α, β, γ , δ). A more careful inspection of the direct asymptotic expansion of the integrals (A.2), (A.3) would allow one to compute explicitly α, β, γ and δ in (6.27); anyway, all this will not overly bother us, as only the S subsector in (6.27) will be relevant for actual computations.
Exact Results for Topological Strings on Resolved Y p,q Singularities
233
where we have shifted the X variable in (3.20) by Xˆ = X + a4 /2. Through the following S L(2, C) automorphism of the Xˆ -plane: a X˜ + b , Y˜ = (c X˜ + d)2 Y, Xˆ = ˜ cX + d
%
a= √
c12 c2
− c2 c1 + 3c2 c2 (3c1 + c2 ) , d= , b= , c=− , 2(c + c ) 4c1 + 4c2 2 2 1 2 2 c2 c1 − c2 2 c2 c12 − c22
c1 c2 − c22
(6.30) we bring (6.29) to the celebrated Seiberg-Witten (2)-symmetric form Y˜ 2 = ( X˜ 2 − 1)( X˜ − u),
(6.31)
where u=
c12 + 6c2 c1 + c22 . (c1 − c2 )2
(6.32)
With (6.32) at hand we can re-express the quantities computed in the previous section as (2) modular forms, whose ring is generated by the Jacobi theta functions θ2 (τ ), θ3 (τ ), θ4 (τ ), all having modular weight 1/2. This goes as follows: the Klein invariant j (τ ) of the curve (6.31) is rationally related to u as j (u) = 64
(3 + u 2 )3 (u 2 − 1)2
(6.33)
while inversion of (6.32) gives, writing everything for definiteness in the (a3 , a4 ) patch, u+3 a42 − a3 = √ √ . 2 u+1
(6.34)
Combining the two formulae above and using the definition (5.17) of c1 , c2 we can write the latter as (2) modular forms as c1 (τ ) = 2
θ42 (τ ) θ22 (τ )
,
c2 (τ ) = 2
θ32 (τ ) θ22 (τ )
,
(6.35)
which, being coordinates on the moduli space, are correctly modular invariant. Given (6.35) it is then straightforward to write the building blocks of the BCOV recursion in terms of modular forms. According to [2], the recursion relies on knowledge of the Yukawa coupling C(τ ) and the genus one closed amplitude Fˆ 1 (τ, τ¯ ); having exact expressions for the genus zero data as functions of the branch points, one can use (6.35) to write down explicitly all the relevant quantities as modular functions. Let us analyze the large radius Yukawa coupling first. We have
∂ 3F 4 ∂a4 ∂t F −1 C≡ 3 = . π ∂τ ∂a4 ∂t F
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A. Brini, A. Tanzini
Using (5.18) we have ∂t F = ∂a4
K 1−
c22 c12
c1
=
π 2 θ (τ ), 4 2
(6.36)
while combining (6.35) and (5.17) yields 26 η12 (τ ) ∂a4 =− , ∂τ a4 (τ ) θ2 (τ )8
(6.37)
where η(τ ) is Dedekind’s function and we have used 2η3 (τ ) = θ2 (τ )θ3 (τ )θ4 (τ ) besides the modular expression of a4 from (5.17) θ 4 (τ ) a4 (τ ) = 2 4 44 + a3 + 2. θ2 (τ )
(6.38)
(6.39)
Putting it all together we arrive at C(τ ) = −
a4 (τ ) θ26 (τ ) . 64 η12 (τ )
(6.40)
Let us now address the issue of the genus 1 free energy Fˆ 1 (τ, τ¯ ). For g = 1 the holomorphic anomaly equation of [2] reads in the local case at hand7 ∂t¯2t Fˆ 1 (t, t¯) =
1 tt C Cttt , 2 t¯
(6.41)
where indices in (6.41) are raised with the Weyl-Petersson metric G t¯t = 2mτ . Rewriting everything as a function of τ , τ¯ (6.41) integrates immediately to 1 Fˆ 1 (τ, τ¯ ) = − log mτ − log |ψ(τ )|, 2
(6.42)
where we have denoted by ψ(τ ) the holomorphic ambiguity at genus one. The pole structure of the amplitude fixes it uniquely; we will do it in the next section by computing explicitly its holomorphic limit.
6.4. Higher genus amplitudes. In this section we will examine the higher genus amplitudes for K F2 . After discussing the genus one free energy, we will turn to the analysis of the g > 1 closed amplitudes, treating in detail the case g = 2, and we will give predictions for orbifold Gromov-Witten invariants of C3 /Z4 for g = 1, 2. 7 In the following we will suppress for notational simplicity the dependence on the t parameter, which is B an auxiliary parameter entering in the definition of the differential, and write t ≡ t F for the flat coordinate coming from an actual A-period integration.
Exact Results for Topological Strings on Resolved Y p,q Singularities
235
6.4.1. One loop partition function for Y p,q and genus 1 orbifold GW. Let us address the issue of the genus 1 free energy in slightly larger generality for the full Y p,q class. In this latter case it would be natural to guess that, again, the same structure as in SU ( p) Seiberg-Witten theory holds: monodromy invariance requires it to be written in terms of ⊂ Sp(2 p − 2, Z) modular forms, and this would lead to the appearance of Siegel modular forms with characteristic. However we happen to have already largely answered this question in the language of hypergeometric functions of the branch points bi . Indeed, ˆ t, t¯) denoting collectively with t the set of flat coordinates, the holomorphic t¯ limit of F( is given on general grounds [2] as 1 1 log , F1 (t) = − log det J − 2 12
(6.43)
where J is the Jacobian matrix of the A-periods (in the appropriate polarization) with respect to the bare variables and is a rational function of the branch points, with zeroes at the discriminant locus of the curve. But it turns out that the awkward Jacobian J in (6.43) is precisely the main object for which we have found a closed form expression in (4.13)! This is definitely an advantage with respect to finding a hyperelliptic generalization of the modular symmetry of local surfaces, and in the elliptic case it compendiates nicely the modular expression obtained in the previous section. In the following we will therefore use (6.43) in the local F2 case to obtain a closed expression for F1 in homogeneous (bare) coordinates, and then exploit (6.24) to compute genus 1 orbifold GW invariants of C3 /Z4 . From the considerations above we have at large radius
∂t F 1 LR + log c1a c2b (c1 − c2 )c (c1 + c2 )d , F1 (t F , t B ) = − log (6.44) 2 ∂a4 where the exponents of the second term are fixed by the topological vertex computation as a = −1/6, b = −1/6, c = −1/12, d = −1/12. Then, from (5.17) and (5.18),
c2 K 1 − c22 1 1 1 1 log c12 − c22 , (6.45) F1L R (Q F , Q B ) = − − log c1 c2 − 2 c1 6 12 and plugging in the mirror map (5.22) we can straightforwardly compute F1L R (Q F , Q B )
Q2 Q3 Q4 QF log(Q B ) log(Q F ) − − − F − F − F (6.46) = − 24 12 6 12 18 24 Q2 Q3 Q3 37Q4F Q2 QF − F − F QB + − F − F + + − Q2B + · · · , 6 3 2 12 2 6
which is the correct form predicted by the topological vertex computation [5]. In order to verify explicitly the assertions of the previous section, we can also use the modular expression of c1 , c2 and ∂a4 t F to obtain the holomorphic limit of (6.42) in quasi-modular form. We indeed get 1 F1L R (τ ) = − log η(τ ). 2
(6.47)
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A. Brini, A. Tanzini
Plugging in the expression for the modular parameter q = e2πiτ in exponentiated flat coordinates which can be computed from (5.22), (6.33) and (6.34), q(Q B , Q F ) = Q B Q2F + 4Q2B + 4Q B Q3F (6.48) + 10Q3B + 48Q2B + 10Q B Q4F + O Q5F , we recover precisely (6.46). Knowing F1 for local F2 we can straightforwardly obtain a prediction for genus one orbifold Gromov-Witten invariants of C3 /Z4 ; to relate (6.46) to the expansion of the g = 1 topological partition function at the orbifold point we just need to specialize the Feynman expansion (6.24) to the case at hand. The one loop term is given as 1 =
4 1 log , 2 τ + C −1 D
(6.49)
where the factor of 4 comes again from the orbifold cohomology pairing. The genus one orbifold free energy will be then given as F1or b = F1L R + 1 ,
(6.50)
where τ can be written as a function of the ai variables using (5.18), (5.19) or directly as an expansion in orbifold flat coordinates using (6.28). We can now rewrite everything in terms of s1/2 , s1/4 by plugging the orbifold mirror map (6.13) into the expression (6.44) of F1L R , yielding F1or b (s1/4 , s1/2 )
=
or b N1,(m,n) n,m
=−
n!m!
m n s1/4 s1/2
(6.51)
2 s 4 2 s5 6 s2 5s 2 s 3 7s12 13s14 31s12 s14 12 12 + 12 − 14 12 + − + + ··· . 384 192 9216 18432 163840 1105920
The same result would be obtained by using a modified expression for the Jacobian in (6.43) adapted to the orbifold patch, i.e. by replacing ∂a4 t F in (6.44) with ∂a4 s1/4 , without then considering the extra piece (6.49) from the Feynman expansion. The first few GW invariants are reported in Table 3. 6.4.2. Generalities on g > 1 free energies. Turning now to the case of g > 1 closed amplitudes, let us first of all recall the main statements put forward in [7,56] for the computation of higher genus free energies. As mentioned in Sect. 6.3.2, modular symmetry is an amazingly stringent constraint. For 1-parameter models with elliptic mirror curve (like SU (2) Seiberg-Witten theory, or local P2 ) the authors of [7,56] claim that solutions of the genus g holomorphic anomaly equations can be written for g ≥ 2 as Fˆ g (τ, τ¯ ) = C 2g−2 (τ )
n
(g) (g) Eˆ 2k (τ, τ¯ )ck (τ ) + C 2g−2 (τ )c0 (τ ),
(6.52)
k=1
where τ is the complex modulus of the mirror torus, C is the Yukawa coupling ∂t3 F0 , (g) ck (τ ) are -modular forms of weight 6(g − 1) − 2k and the full non-holomorphic
Exact Results for Topological Strings on Resolved Y p,q Singularities
237
or b Table 3. Genus one orbifold Gromov-Witten invariants N1,(m,n) of [C3 /Z4 ]
m n 0 1 2 3 4 5 6 7 8 9 10 11 12
0
0 1 96
0 7 768
0 31 1536
0 2219 24576
0 16741 24576
0 1530037 196608
2
4
6
8
10
0 1 - 192 0 5 - 768 0 39 - 2048 0 2555 - 24576 0 - 22523 24576 0 - 389975 32768 0
1 128
0 31 - 1024 0 485 - 4096 0 40603 - 49152 0 - 293685 32768 0
441 4096
0 71291 - 32768 0
0 35 3072
0 485 12288
0 2025 8192
0 240085 98304
0 54986255 1572864
0 1434341595 2097152
- 73017327 524288 0 - 18440181205 6291456 0
0 235 512
0 458295 131072
0 10768885 262144
0 1437926315 2097152
0 32280203275 2097152
0 7495469356455 16777216
- 2335165 131072 0 - 58775443 262144 0 - 522517275 131072 0 - 397762755193 4194304 0 - 12177409993695 4194304 0
dependence of Fˆ g is captured by the modular, non-holomorphic extension of the second Eisenstein series (6.22). Now, there are two ways to compute the expressions in (6.52). The first one consists in a direct study of the BCOV equations: in this context the holomorphic modular coef(g) ficients ck for k > 0 can either be fixed by the Feynman expansion (6.24) in terms of derivatives of lower genus Fg , or much more efficiently by exploiting the modular symmetry to perform a direct integration of the holomorphic anomaly equations as in [56]. Within this method, the only real issue is to fix the so-called “holomorphic ambiguity” (g) at k = 0, i.e. c0 (τ ). In the 1-parameter cases analyzed in [7,56], this is systematically (g) done by plugging into (6.52) an ansatz for c0 (τ ) which is then determined from extra (g) boundary data. In more detail, this works as follows: at fixed genus g, c0 (τ ) is a 8 weight w = 6g − 3 modular form. Now, the ring of weight w holomorphic modular forms Mw () is finitely generated, and the analytic behavior of Fg (τ, τ¯ ) at large radius (g) allows to write an ansatz for c0 (τ ) with only a finite number of unknown coefficients. (g) At the same time, c0 (τ ) is constrained to satisfy the so-called “gap condition” [57]: this imposes a sufficient number of constraints to completely determine (indeed, overdetermine) the conjectured form of the ambiguity as a function of the generators of Mw (). The discussion of Sect. 6.3.2 has shown that the case of local F2 is in many ways similar to the simpler examples of SU (2) Seiberg-Witten theory and local P2 . However there is an extra complication, given by the fact that the elliptic modulus τ is not the only variable in the game: here we actually have an extra bare parameter a3 , or z B , which is independent on τ and is related to the Kähler volume of the base P1 (see (4.17), (6.8)). That is, we deal here with a two-parameter model, even though with an elliptic mirror curve, and we have to properly take this into account. A first consequence of this fact is that the idea of using the gap condition to fix the holomorphic ambiguity 8 Recall that F (τ, τ¯ ) is modular invariant and that C(τ ) has weight −3 - see for example (6.40) for the g local F2 case.
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A. Brini, A. Tanzini
becomes computationally more complicated, since our task is no longer reduced to fix (g) simply a finite set of unknown numerical coefficients of c0 (τ ) as generated by a basis of Mw (): rather we should fix a finite set of unknown functions of a3 . A second possibility is to avail ourselves of the framework proposed in [8] for the computation of topological string amplitudes based on the Eynard-Orantin recursion for (g) matrix models. This is based on a sequence of polydifferentials Wh on the mirror curve , which are recursively computed in terms of residue calculus on and out of which it is possible to extract the free energies Fg at any given genus. Let us briefly review here this formalism in order to describe the general structure of higher amplitudes, referring the reader to [8,58] for further details. The ingredients needed are the same as for genus zero amplitudes, namely the family of Hori-Vafa mirror curves H (u, v) = 0 (3.2) with differential dλ (3.7). The genus g free energies are then recursively given as Fg =
1 (g) Res φ(u)W1 (u), u=bi 2 − 2g
(6.53)
bi
where φ(u) is any antiderivative of the Hori-Vafa differential dφ(u) = dλ(u) = log v(u)
du u
(6.54)
(g)
and Wh ( p1 , . . . , ph ) with g, h ∈ Z+ , h ≥ 1 is an infinite sequence of meromorphic differentials on the curve defined by the Eynard-Orantin recursion (0)
(0)
W1 ( p1 ) = 0, W2 ( p1 , p2 ) = B( p1 , p2 ), (6.55) d E q ( p) (g) (g−1) Wh+2 (q, q, Wh+1 ( p, p1 . . . , ph ) = Res ¯ p1 , . . . , p h ) q=bi dλ(q) − dλ(q) ¯ bi g (g−l) (l) W|J |+1 (q, p J )W|H |−|J |+1 (q, ¯ p H \J ) . (6.56) + l=0 J ⊂H
In the formulae above, q¯ denotes the conjugate point to q, B( p, q) is the Bergmann kernel, the one form d E q ( p) is given as 1 q¯ d E q ( p) = B( p, ξ )dξ, (6.57) 2 q and finally, given any subset J = {i 1 , . . . , i j } of H := {1, . . . , h}, we defined p J = { pi1 , . . . , pi j }. We refer the reader to [8,58] for an exhaustive description of the objects introduced above. At a computational level, the formalism of [8] is somewhat lengthier than the one of [56] for computing higher genus Fg . On the other hand, the recursion of [8] has the great (g) advantage of providing unambiguous results, with the holomorphic ambiguity c0 (τ ) automatically fixed. This precisely overcomes the problem raised above. In the next section, we will therefore follow this second path to complete the discussion of Sect. 6.3.2 by displaying explicitly the modular structure of the Fg obtained through (6.53). An explicit computation of the g = 2 case, as well as predictions at the orbifold point, will be left to Sect. 6.4.4.
Exact Results for Topological Strings on Resolved Y p,q Singularities
239
6.4.3. g > 1 and modular forms. Let us specialize the recursion to the case of local F2 . The Hori-Vafa differential (4.3) reads, in the (a3 , a4 ) patch,
P2 (u) ± Y (u) du dλ2,2 (u) = log , (6.58) 2 u where P2 (u) = a3 + a4 u + u 2 ,
Y (u) =
P22 (u) − 4
and the 2,2 family can be written in the Z2 symmetric form (6.29) as a two-fold branched covering of the compactified u-plane Y 2 = (u − b1 )(u − b2 )(u − b3 )(u − b4 ) = (uˆ 2 − c12 )(uˆ 2 − c22 ),
(6.59)
thanks to (5.17) and (6.29) and having defined uˆ = u + a4 /2. We have first of all that dλ(u) − dλ(u) ¯ = 2M(u)Y (u)du,
(6.60)
where the so-called “moment function” M(u) is given, after using the fact that log(P + Y ) − log(P − Y ) = 2 tanh−1 (Y/P), as Y (u) 1 M(u) = tanh−1 . (6.61) uY (u) P2 (u) Moreover, the one form d E( p, q) can be written as [58]
1 1 Y (w) d E w (u) = − LC(w) du, 2 Y (u) u − w where C(w) :=
1 2πi
A
1 du , Y (u) u − w
L −1 :=
1 2πi
A
(6.62)
du . Y (u)
(6.63)
We have assumed here that w stays outside the contour A; when w lies inside the contour A, C(w) in (6.62) should be replaced by its regularized version C reg (w) = C(w) −
1 . Y (w)
(6.64)
Since 2,2. is elliptic, it is possible to find closed form expressions for C(u), Cr eg (u), B(u, w) and L. We have 2(b2 − b3 ) u − b2 (n 4 , k) + C(u) = K (k) , √ b2 − b3 π(u − b3 )(u − b2 ) (b1 − b3 )(b2 − b4 ) (6.65) − b ) 2(b u − b 3 2 3 (n 1 , k) + C reg (u) = K (k) , √ b3 − b2 π(u − b3 )(u − b2 ) (b1 − b3 )(b2 − b4 ) (6.66) (b − b )(b − b ) 2 1 2 3 4 L −1 = √ , (6.67) K (b1 − b3 )(b2 − b4 ) (b1 − b3 )(b2 − b4 ) 1 Y 2 (u) A(u) (Y 2 ) (u) B(u, w) = + + Y (u) 2Y (w)(u − w)2 4Y (w)(w − u) 4Y (w) 1 (6.68) + 2(u − w)2
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A. Brini, A. Tanzini
where k=
(b1 − b2 )(b3 − b4 ) , (b1 − b3 )(b2 − b4 )
(b4 − b3 )(u − b2 ) , (b4 − b2 )(u − b3 ) (6.69) E(k) A(u) = (u − b1 )(u − b2 ) + (u − b3 )(u − b4 ) + (b1 − b3 )(b2 − b4 ) , (6.70) K (k) n4 =
(b2 − b1 )(u − b3 ) , (b3 − b1 )(u − b2 )
n1 =
and K (k), E(k) and (n, k) are the complete elliptic integrals of the first, second and third kind respectively. With these ingredients one can compute the residues as required in (6.56). Given that d E q ( p)/(dλ(q) − dλ(q)), ¯ as a function of q, is regular at the branch-points, all residues appearing in (6.56) will be linear combinations of the following kernel differentials:
d E q ( p) 1 (n) χi ( p) = Resq=xi dλ(q) − dλ(q) ¯ (q − xi )n
n−1 1 1 d 1 1 = − LC(q) . (6.71) (n − 1)! Y ( p) dq n−1 2M(q) p − q q=xi In (6.71), C( p) should be replaced by Creg ( p) when i = 1, 2. Let us then explicitly display the quasi-modular structure of the free energies Fg (a3 , τ ). We claim that the holomorphic limit of the 1-parameter examples (6.52) Fg (τ ) = C 2g−2 (τ )
n
(g)
(g)
E 2k (τ )ck (τ ) + C 2g−2 (τ )c0 (τ ),
k=1
gets replaced here by Fg (a3 , τ ) = C 2g−2 (a3 , τ )
n
(g)
(g)
E 2k (τ )ck (a3 , τ ) + C 2g−2 (a3 , τ )c0 (a3 , τ ),
(6.72)
k=1
i.e., as a polynomial in the second Eisenstein series having (algebraic) functions of a3 and θi (τ ), i = 2, 3, 4 as coefficients; moreover, these coefficients are completely determined in closed form from (6.56). Let us show in detail how this happens in general, leaving the concrete example of the g = 2 case to the next section. Formulae (6.53), (6.56), (6.68) and (6.71) imply that the final answer will be a polynomial in the following five objects: (n) Mi ,
(n) φi ,
(n) Ai ,
(n) 1 , Y i
(n)
Ci ,
(6.73)
where, for a function f (x) with meromorphic square f 2 (x), we denote with f i(n) the (n + 1) f th coefficient in a Laurent expansion of f (x) around bi , f (x) =
∞ n=−Ni
f i(n+Ni ) , ( p − bi )n/2
(6.74)
Exact Results for Topological Strings on Resolved Y p,q Singularities
and have defined (n)
Ci
=
⎧ (n) ⎨ Creg,i for i = 1, 2 ⎩
(n)
Ci
for i = 3, 4
(n)
241
.
(6.75)
(n)
Of the five building blocks in (6.73), Mi and φi are the ones which are computed most elementarily from (6.58), (6.54) and (6.61), the result being in any case an algebraic function of (a3 , a4 ). When re-expressed in modular form, we can actually say more about (n) (n) them: we have that the a3 -dependence in Mi (a3 , τ ) and φi (a3 , τ ) is constrained to come only through a4 (a3 , τ ) as written in formula (6.39). Indeed, from (5.17), (6.35) we have that the branch points bi have the form −
a4 ± c1 (τ ), 2
−
a4 ± c2 (τ ), 2
(6.76)
and therefore depend on a3 only through a4 (a3 , τ ). Moreover, since P2 (bi ) = 2 and the derivatives of P2 (u) do not depend explicitly on a3 , we have that the a3 dependence in Fg as obtained from the recursion may only come through a4 (a3 , τ ). Notice moreover that these are the only pieces bringing a dependence on the additional a3 variable: all the others do not depend on the form of the differential (6.58), and are functions only of differences of branch points bi . This means in particular that they only depend on the variables c1 and c2 introduced in (5.17) and whose modular expressions we already found in (6.35)! This is immediate to see for Ai(n) and (1/Y )i(n) from formulae (6.59) and (6.70). The case of Ci(n) is just slightly more complicated, but it is worth describing in detail for the discussion to come. For n = 1, we need the first derivative of (x, y) with respect to x: xE(y) + (y − x)K (y) + x2 − y (x, y) . ∂x (x, y) = 2(x − 1)x(y − x) The above formula implies that ∂x(n) (x, y) = An (x, y)K (y) + Bn (x, y)E(y) + Cn (x, y) (x, y),
(6.77)
where An , Bn and Cn are rational functions of x and y. From (6.69), to compute Ci(n) , we need to evaluate these expressions when n 1 (resp. n 4 ) equals either 0 or k. But using (0, y) = K (y),
(y, y) =
E(y) 1−y
(6.78)
we conclude that Ci(n) = R1(n) (c1 , c2 )K (k) + R2(n) (c1 , c2 )E(k)
(6.79)
for two sequences of rational functions Ri(n) . We now make the following basic observation: by (6.71), Ci(n) always appears multiplied by L in the recursion. By (6.67) (n)
LCi
E(k) (n) (n) . = R˜ 1 (c1 , c2 ) + R˜ 2 (c1 , c2 ) K (k)
(6.80)
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A. Brini, A. Tanzini
This last observation allows us to collect all the pieces together and state the following. By (6.56) and (6.53) we have that Fg (a3 , τ ) is a polynomial in Mi(n) , φi(n) , Ai(n) , (1/Y )i(n) , Ci(n) , and moreover the whole discussion above as well as formulae (6.70) and (6.80) imply that this takes the form of a polynomial in W (τ ) := E(k)/K (k), Fg (a3 , τ ) =
n
(g)
W k (τ )h k (a3 , τ )
(6.81)
k=0 (g)
with coefficients h k (a3 , τ ) in the ring of weight zero modular forms of (2), parametrically depending on a3 . To conclude, we can exploit the fact that [59] π 2 4E (2τ ) − E (τ ) 2 2 E(k)K (k) = (6.82) 2 3 and that from (6.35) and (6.36), K (k) =
π θ3 (τ )θ4 (τ ), 2
(6.83)
where we have used the fact that in our case % c22 c2 K 1− 2 , K (k) = c1 c1 as the reader can easily check. Moreover, the second Eisenstein series satisfies the duplication formula E 2 (2τ ) =
E 2 (τ ) θ44 (τ ) + θ34 (τ ) + . 2 4
(6.84)
Therefore, 1 4 4 (τ ) + θ (τ ) + θ (τ ) . E 2 3 4 3θ42 (τ )θ32 (τ )
W (τ ) =
(6.85)
This proves (6.72). 6.4.4. The g = 2 case in detail. Let us complete the discussion of this section by presenting the explicit formulae for the genus 2 case. By (6.53) and (6.56), we need the (0) (0) (1) (1) (2) complete expression of W2 , W3 , W1 , W2 and W1 . The first three were computed in [8] and are given by (0)
W2 ( p1 , p2 ) = B( p1 , p2 ), (0)
1 2 (1) (1) (1) M (bi )(Y 2 ) (bi )χi ( p1 )χi ( p2 )χi ( p3 ), (6.86) 2 i=1 ⎛ ⎞ 4 4 1 ⎠ (1) 1 (2) 1 ⎝ 2 A(bi ) (1) W1 ( p) = − χi ( p) + χi ( p). 16 8 (Y 2 ) (bi ) bi − b j 4
W 3 ( p1 , p2 , p3 ) =
i=1
i=1
j =i
Exact Results for Topological Strings on Resolved Y p,q Singularities
243
(1)
W2 is then given from (6.56) as W2(1) ( p, p1 ) =
bi
d E q ( p) W3(0) (q, q, ¯ p1 ) + 2W1(1) (q)W2(0) (q, ¯ p1 ) . q=bi dλ(q) − dλ(q) ¯
Res
(6.87) A very lengthy, but straightforward computation leads us to 1 Ai (q)χi(3) ( p) + Bi (q)χi(2) ( p) 8 i=1 ⎤ + Ci (q)χ (1) ( p) + Di j (q)χ (1) ( p)⎦ . 4
W2(1) ( p, q) = −
i
i
(6.88)
j =i
For the sake of notational brevity, we spare to the reader the very long expressions of the (n) (n) (n) rational functions Ai (q), Bi (q), Ci (q) and Di j (q). They involve Mi , Ai , (1/Y )i (n) and Ci up to the third order in a Taylor-Laurent expansion around the branch points. The next step is given by (2)
W1 ( p) =
bi
d E q ( p) (1) (1) (1) W2 (q, q) ¯ + W1 (q)W1 (q) ¯ . q=bi dλ(q) − dλ(q) ¯
Res
(6.89)
(2)
The pole structure of Ai (q), Bi (q), Ci (q) and Di j (q) dictates for W1 ( p) the following linear expression in terms of kernel differentials (2)
W1 ( p) =
4 5
(n) (n)
E i χi ( p)
(6.90)
n=1 i=1 (n)
for some (very complicated) coefficients E i . The recursion is finalized for g = 2 by (6.53) F2 = −
1 (2) Res φ( p)W1 ( p). p=bi 2
(6.91)
bi
It is useful to collect together terms involving the same powers of W (τ ). Taking the residues in (6.91) yields9 3
n h (2) n (a3 , τ )W (τ ),
(6.92)
n=0 9 It must be noticed that, in order to match exactly the asymptotics of the Gromov-Witten expansion at large radius, we have to subtract from (6.91) a constant term in τ , namely, a rational function of a3 of the form a32 −10 . It would be interesting to investigate the origin of this discrepancy further. 1440 a32 −4
244
A. Brini, A. Tanzini
where (2)
h 3 (a3 , τ ) =
5a42 (a3 , τ )θ24 (τ ) 24576θ32 (τ )θ42 (τ )
,
a4 (a3 , τ )2 1 − θ2 (τ )4 (15θ4 (τ )6 + 16θ3 (τ )2 θ4 (τ )4 1024 49152θ3 (τ )6 θ4 (τ )4 + θ2 (τ )4 (8θ3 (τ )2 + 15θ4 (τ )2 )) , θ2 (τ )4 + 2θ4 (τ )4 + 3θ3 (τ )2 θ4 (τ )2 a42 (a3 , τ ) 13θ2 (τ )12 (2) h 1 (a3 , τ ) = − + 3072θ4 (τ )2 θ3 (τ )2 294192 θ3 (τ )6 θ4 (τ )6 8 8 91θ2 (τ ) 48θ2 (τ ) 91θ4 (τ )2 θ2 (τ )4 96θ2 (τ )4 + + + + , θ3 (τ )6 θ4 (τ )2 θ3 (τ )4 θ4 (τ )4 θ3 (τ )6 θ3 (τ )4
1 1 θ2 (τ )8 − 5θ3 (τ )2 θ2 (τ )4 + 10θ3 (τ )6 1 − + (a , τ ) = h (2) 3 0 61440 a3 + 2 a3 − 2 30720θ3 (τ )4 θ4 (τ )4
a42 (a3 , τ )θ2 (τ )4 θ2 (τ )4 θ2 (τ )4 65θ4 (τ )2 175 12 − +4 − − 2949120 θ3 (τ )8 θ4 (τ )8 θ3 (τ )6 θ3 (τ )2 θ4 (τ )2 17 311 311 65θ3 (τ )2 + . (6.93) − − − 4 4 6 2θ3 (τ ) 2θ4 (τ ) θ4 (τ ) 46080
h (2) 2 (a3 , τ ) =
Plugging in the expression (6.48) of the modular parameter q in exponentiated flat coordinates reproduces as expected the topological vertex expansion at large radius F2L R (Q B , Q F )
1 − = − 120 1 + − − 40 1 + − − 24
Q2B QB QB 1 − QF + − − Q2F 120 60 60 60 2 3 Q2B Q3B Q Q QB Q 1 B − − − B − B Q4F Q3F + − − 40 40 40 30 30 6 30 299Q2B 299Q3B QB − − (6.94) Q5F + +O Q6F . 24 24 24
Finally, we can use (6.92) to make predictions for genus 2 orbifold Gromov-Witten invariants of C3 /Z4 by using the Feynman expansion method of [2,7] as we did for the genus 1 free energy; the same result would be obtained by analytically continuing (2) the holomorphic ambiguity h 0 (a3 , τ ) and taking the holomorphic limit of the physical amplitude directly at the orbifold point (see [60] for a detailed description of this method). The results are shown in Table 4.10 7. Conclusions and Outlook In this paper we have proposed an approach to the study of A-model topological amplitudes which yields exact results in α and as such applies to the full moduli space, 10 While the final version of this paper was under completion, a preprint appeared [65] where the same results have been obtained following a different method.
Exact Results for Topological Strings on Resolved Y p,q Singularities
245
or b Table 4. Genus two orbifold Gromov-Witten invariants N2,(m,n) of [C3 /Z4 ]
m n 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 − 960 0 7 − 7680 0 11 − 5120 0 2479 − 245760 0 19343 − 245760 0 604199 − 655360 0 − 59566853 3932160
2
4
61 − 30720 0 647 0 − 92160 257 0 92160 65819 0 − 1474560 23227 0 1474560 437953 0 − 983040 418609 0 2949120 0 − 303139073 47185920 1380551 0 737280 0 − 2982122587 23592960 200852963 0 5898240 0 − 818897894611 251658240
0
41 46080
6
8
9023 − 81920 0 1066027 0 − 1310720 168049 0 983040 0 − 18530321 1966080 43685551 0 23592960 0 − 9817250341 62914560 452348269 0 15728640 0 − 438364727389 125829120 25384681949 0 41943040 0 − 16896151842371 167772160 25012290702059 0 1509949440 0 − 1840152188554961 503316480
10
0
0
6061 245760
36213661 7864320
0 887800477 15728640
0 62155559923 62914560
0 5851085490887 251658240
0 355405937648809 503316480
0 54049855936801961 2013265920
0
including orbifold and conifold divisors, of closed and open strings on a large class of toric Calabi-Yau threefolds. One of the main virtues of this approach is that it provides us with a closed expression for the (derivatives of the) periods of the mirror curve, considerably simplifying the study of their analytic continuation in the various patches and of the modular properties of the Gromov-Witten generating functional. The local geometries that we have analyzed arise from the minimal resolution of Y p,q singularities. The general procedure to compute topological string amplitudes, outlined in Sect. 4, is based on the correspondence with five-dimensional gauge theories and the associated Seiberg-Witten curves; it has been fully exploited in Sect. 6 for the case p = 2, and used in particular to predict open and closed orbifold Gromov-Witten invariants of C3 /Z4 also at higher genus. Of course our strategy is completely general and could be adopted with no changes, though becoming technically more involved, to compute amplitudes for p > 2; moreover, it can be used to get some qualitative information about the behavior of the B-model moduli space, which for these cases displays a richer set of phenomena. Indeed, the mirror curves have higher genera and can be subject to more general degeneration limits, for example when the neck connecting two handles becomes infinitely long. In the underlying four-dimensional gauge theory this limit has been recognized as a new superconformal phase [61]; it would be interesting to explore its interpretation in the topological string moduli space. The computations of Sect. 4 have been based on extensively exploiting the holomorphic properties (4.11) of the B-model 1−differential, which came out by appealing to the relation with gauge theories and integrable systems. On the gauge theory side, one is able to obtain the Seiberg-Witten curve and the related differential in a suitable semiclassical limit involving a large number of instantons [62]. The considerations above suggest to reinterpret the transition to the mirror and (4.11) at the string theoretical level in terms of a semiclassical geometry in gs → 0 which resums a large number of world-sheet instantons. Some remarks are in order concerning the relation with integrable systems. First of all, as we have discussed in Sect. 3.3, the mirror geometry for resolved Y p, p singularities can be realized as a fibration over the spectral curve of the relativistic A p−1 Toda chain.
246
A. Brini, A. Tanzini
Actually our results for generic Y p,q singularities seems to indicate the existence of a larger class of integrable systems: it would be interesting to understand this better and see what kind of deformations of the Toda chain are associated to the q parameter. Moreover, the existence of a set of holomorphic differentials like (4.11) could be recognized as a signal of a relation with integrable hierarchies. More precisely, one could expect that a suitable generalization of the topological string prepotential - possibly including gravitational descendants - could be interpreted in terms of a Whitham deformation of the integrable system. This would correspond to an “uplift” to topological strings of similar notions developed in [63] for four-dimensional Seiberg-Witten theory. As a final comment, we might wonder how much of what we have learned might be extended to other cases. Moving beyond Y p,q , it is in fact straightforward to show that holomorphicity of (derivatives of) the differential can be shown exactly as for the Y p,q family, at least in the case in which the mirror curve is hyperelliptic;11 at a pictorial level, this class coincides with those toric CY whose toric diagram is contained into a vertical strip of width 2, modulo S L(2, Z) transformations. Our methods thus continue to hold and apply with no modification for this more general family as well; it would be very interesting to investigate the possibility to generalize our approach to all toric Calabi-Yau three-folds. Acknowledgements. We would like to thank G. Bonelli, H. L. Chang, B. Dubrovin, B. Fantechi, C. Manolache, F. Nironi, S. Pasquetti, E. Scheidegger for useful conversations, and we are particularly grateful to Tom Coates for enlightening discussions of Sect. 5.2. A.B. acknowledges I. Krichever, O. Ragnisco, S. Ruijsenaars, N. Temme and especially Ernst D. Krupnikov for kind email correspondence, and V. Bouchard, M. Mariño and S. Pasquetti for fruitful and stimulating discussions during the final phase of this project. The present work is partially supported by the European Science Foundation Programme “Methods of Integrable Systems, Geometry, Applied Mathematics” (MISGAM) and Marie Curie RTN “European Network in Geometry, Mathematical Physics and Applications” (ENIGMA).
A. Euler Integral Representations, Analytic Continuation and Generalized Hypergeometric Functions As we pointed out in Sect. 4.1, another important feature of our formalism is the fact that we can work directly with an Euler-type integral representation for the periods. We will focus here in the case p = q = 2, but the strategy is completely general and computationally feasible as long as xi is algebraically related to ai . For p = q = 2, the derivatives of the periods have the simple form (5.18), (5.19). Using the standard Euler integral representation for the complete elliptic integral K (x), 1 dθ 1 , (A.1) 2K (x) = √ √ √ θ 1 − θ 1 − xθ 0 we can integrate back a4 and get 1
2a5 dθ A (ai ) = (A.2) log a4 + c12 + (c22 − c12 )(1 − θ ) , √ √ θ 1−θ 0 ⎡ ⎤
1
1 2a5 dθ a4 θ + (c12 − c22 ) + c2 θ 2 ⎦ , B (ai ) = 4 log ⎣ √ 2 2 1 − θ 2θ 0 c1 − c2 (A.3) 11 For example, the canonical bundle over the second Del Pezzo d P falls into this category, though not 2 being part of the Y p,q class.
Exact Results for Topological Strings on Resolved Y p,q Singularities
247
where the constant factors in a4 are introduced as a constant of integration in order to satisfy (3.5). Formulae (A.2), (A.3) then yield simple and globally valid expressions for the periods and significantly ease the task of finding their analytic continuation from patch to patch. For small a4 , we can simply expand the integrand and integrate term by term. For large a4 A has the following asymptotic behavior:
5 1 2 1 4 1 + −3a32 a53 − 6a1 a2 a53 +O , A = 2a5 log (2a4 ) − 2 a3 a52 a4 a4 a4 (A.4) but an expansion for B is much harder to find. The leading order term can still be extracted, for example in the a2 = a3 = a5 = 1 patch using ⎡
1
log ⎣θa +
0
⎤
dθ a2 1+ b+ = 2Li 2 (−1 − a) + O(log a) θ 2⎦ √ 4 1 − θ 2θ (A.5)
which gives
2
1 1 B = 4 log √ − log + O (log a4 ) . 4 2 a1 a4
(A.6)
Single and double logarithmic behaviors as in (A.4, A.6) are characteristic of the large radius patch in the moduli space, which as we will see will be given precisely by a4 → ∞ (and a1 → 0). Lastly, a nice fact to notice is that the periods for this particular case take the form of known generalized hypergeometric functions of two variables. For example we have that, modulo a4 independent terms, the A period can be written as A =
π π log c1 + 4 4
a42 1,2,2 − c1 F1,1,1 c1
1 23 , 1 21 , 21 c 1 − 2 2 1 1
a42 c12
! c2 , c1 1 − 22 c1
(A.7) in terms of the Kampé de Fériet12 hypergeometric function of two variables.
B. Lauricella Functions We collect here a number of properties and useful formulae for Lauricella’s FD(n) functions. The interested reader might want to look at [42] for a detailed discussion of this topic. 12 See Eric Weinstein, “Kampé de Fériet Function”, http://mathworld.wolfram.com/ KampedeFerietFunction.html, or [42,43] for a more detailed account on such functions.
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A. Brini, A. Tanzini (n)
B.1. Definition. The usual power series definition of Lauricella FD of n complex variables is (n)
FD (a, b1 , . . . , bn ; c; x1 , . . . , xn ) ∞
=
···
m 1 =0
∞ (a)m 1 +···+m n (b1 )m 1 · · · (bn )m n m 1 n x1 · · · xm n , (c)m1+···+m n m 1 ! · · · m n !
(B.1)
m n =0
whenever |x1 |, . . . , |xn | < 1. For n = 1 this is nothing but Gauss’ hypergeometric function 2 F1 (a, b; c; x); for n = 2 it boils down to Appell’s F1 (a, b, c; d; x, y). It also satisfies the following system of P D E’s, which generalizes the n = 1 hypergeometric equation ∂ 2 FD ∂ 2 FD ∂ FD ab j FD = x j (1 − x j ) + (1 − x ) x + [c − (a + b j + a)x j ] j k ∂xk ∂x j ∂x j ∂x2j −b j
= j
k = j
∂ FD xk , ∂xk
j = 1, . . . , n.
(B.2)
The system (B.2) has regular singular points when xi = 0, 1, ∞ and xi = x j
i = 1, . . . , n, j = i.
(B.3)
The number of intersecting singular submanifolds in correspondence of the generic singular point (x1 , . . . , xn ) = (0, . . . , 0, 1 . . . , 1, ∞, . . . , ∞) * +, - * +, - * +, p
is
p+1 2
q +1 2
q
(B.4)
n− p−q
n− p−q +1 . 2
In contrast with the well-known n = 1 case, typically the Lauricella system does not close under analytic continuation around a singular point. As explained in [42], a complete set of solutions of the FDn system (B.2) away from the region of convergence p,q |xi | < 1 involves a larger set of functions, namely Exton’s Cnk and D(n) . We will report here a number of analytic continuation formulae valid for generic n, and refer to [42] for further results in this direction. See also [64] for further developments in finding asymptotic expressions for large values of the parameters. B.2. Analytic continuation formulae for Lauricella FD . In the following, results on analytic continuation for FD will be expressed in terms of Exton’s C and D functions (k)
Cn ({bi }, a, a ; {xi }) = m 1 ,...m n i (bi )m i (a) n
k k (a )− n i=k+1 m i − i=1 m i i=k+1 m i + i=1 m i
m
xi i i mi !
, (B.5)
p,q D(n) (a, b1 , . . . , bn ; c, c ; x1 , . . . , xn ) ∞ (a)m p+1 +···+m n −m1−···−m p (b1 )m 1 ···(bn )m n = ∞ m 1 =0 · · · m n =0 (c)m +···+m n −m −···−m p cm m !···m n ! q+1 1 p+1 +···+m q 1
mn 1 xm 1 · · · xn ,
(B.6)
Exact Results for Topological Strings on Resolved Y p,q Singularities
249
• Continuation around (0, 0, . . . , 0, ∞) (n)
FD (a, b1 , . . . , bn ; c; x1 , . . . , xn ) = c, bn − a (n) x (−xn )−a FD (a, b1 , . . . , bn−1 , 1 − c + a; 1 − bn + a; xxn1 , . . . , xn−1 , x1n ) n bn , c − a c, −bn + a (n−1) (−xn )−b Cn + (b1 , . . . , bn , 1 − c + bn ; a − bn ; −x1 , . . . , −xn−1 , x1n ). a, c − bn
(B.7) • Continuation around (0, 0, . . . , 0, 1) (n) FD (a, b1 , . . . , bn ; c; x1 , . . . , xn ) = ×(1 − x1 )−b1 . . . (1 − xn−1 )−bn−1
c, c − bn − a c − a, c − bn
(n−1)
xn−1 x1 n n ×x−b (b1 , . . . , bn , 1 + bn − c; c − a − bn ; 1−x , . . . , 1−x , 1−x n Cn xn ) 1 n−1 c, bn + a − c (1 − x1 )−b1 . . . (1 − xn−1 )−bn−1 (1 − xn )c−a−bn + a, bn 1−xn n ×FD(n) (c − a, b1 , . . . , bn−1 ; c − a − bn + 1; 1−x 1−x1 , . . . , 1−xn−1 , 1 − xn ).
(B.8) • Continuation around (0, 0, . . . , ∞, 1) FD(n) (a, b1 , . . . , bn ; c; x1 , . . . , xn ) = n−1 (1 − xi )−bi ×(1 − xn )c−a−bn i=1
c, bn + a − c a, bn
(n)
×FD (c − a, b1 , . . . , bn−1 ; c − b1 − · · · − bn ; c − a − bn + 1; 1−xn 1−xn 1−x1 , . . . , 1−xn−1 , 1 − xn )
c, c − a − bn , a − bn−1 n (1 − x1 )−b1 . . . (1 − xn−1 )−bn−1 x−b + n c − a, c − bn−1 − bn , a (B.9) . 1,2 ×D(n) (c − a − bn , bn , . . . , b1 ; c − bn−1 − bn ; bn−1 − a + 1;
xn−2 xn −1 1 xn , 1−xn−1 , 1−xn−2
+
c,
bn−1 − a
c − a,
bn−1
x1 . . . , 1−x ) 1 !
(1 − xn−1 )−a
×FD(n) (a, b1 , . . . , bn−2 ; c −
n
i=1 bi ; bn , a
− bn−1 + 1;
1−xn−2 1−xn 1−x1 1 1−xn−1 , . . . , 1−xn−1 , 1−xn−1 , 1−xn−1 ).
Notice that the formulae above are valid only for generic values of the parameters bi , a and c. Should one be confronted with singular cases, it would be necessary to take a suitable regularization (such as bi → bi + ) and after analytic continuation take the
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→ 0 limit. See Appendix B in [46] for more details; suffice it here to report as an example the case bn = a: FD(n) (a,b1 , . . . , bn−1 , a; c; x1 , . . . , xn ) ∞ c − a − |M|| c −a = (−xn ) M m n =0 c − a + |M| a, c − a (a)|M|+m n (1−c+a)2|M|+m n n−1 (bi )m i × i=1 m i ! (|M|+m n )!m n ! m n −1 m n x1 m 1 1 × log(−xn ) + h m n xn · · · xxn−1 xn n |M|−1 (a) n (|M|−m n ) c, c − a + (−xn )−a M m n =0 m nm!(c−a) |M|−m n a m n n−1 (bi )m i m 1 m n−1 1 × i=1 m i ! x1 · · · xn−1 xn ,
(B.10)
with h m n = ψ(1 + |M| + m n ) + ψ(1 + m n ) − ψ(a + |M| + m n ) − ψ(c − a − m n ), (B.11) m n and M = (m 1 , . . . , m n ) is a multindex (so that |M| ≡ i=1 m i ). References 1. Antoniadis, I., Gava, E., Narain, K.S., Taylor, T.R.: Topological amplitudes in string theory. Nucl. Phys. B 413, 162 (1994) 2. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B 405, 279 (1993); Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) 3. Witten, E.: Topological Quantum Field Theory. Commun. Math. Phys. 117, 353 (1988) 4. Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004) 5. Aganagic, M., Klemm, A., Marino, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005) 6. Aganagic, M., Dijkgraaf, R., Klemm, A., Marino, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451 (2006) 7. Aganagic, M., Bouchard, V., Klemm, A.: Topological Strings and (Almost) Modular Forms. Commun. Math. Phys. 277, 771 (2008) 8. Bouchard, V., Klemm, A., Marino, M., Pasquetti, S.: Remodeling the B-model. Commun. Math. Phys. 287, 117–178 (2009) 9. Lawrence, A.E., Nekrasov, N.: Instanton sums and five-dimensional gauge theories. Nucl. Phys. B 513, 239 (1998) 10. Intriligator, K.A., Morrison, D.R., Seiberg, N.: Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces. Nucl. Phys. B 497, 56 (1997) 11. Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix Models, Geometric Engineering and Elliptic Genera. JHEP 0803, 069 (2008) 12. Ruijsenaars, S.M.N.: Relativistic Toda systems. Commun. Math. Phys. 133(2), 217–247 (1990) 13. Gauntlett, J.P., Martelli, D., Sparks, J., Waldram, D.: Sasaki-Einstein metrics on S(2) x S(3). Adv. Theor. Math. Phys. 8, 711 (2004) 14. Martelli, D., Sparks, J.: Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals. Commun. Math. Phys. 262, 51 (2006) 15. Bertolini, M., Bigazzi, F., Cotrone, A.L.: New checks and subtleties for AdS/CFT and a-maximization. JHEP 0412, 024 (2004) 16. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, 131. The William H. Roever Lectures in Geometry. Princeton, NJ: Princeton University Press, 1993 17. Bouchard, V.: Lectures on complex geometry, Calabi-Yau manifolds and toric geometry. http://arxiv.org/ abshep-th/0702063v1, 2007
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18. Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159 (1993) 19. Benvenuti, S., Franco, S., Hanany, A., Martelli, D., Sparks, J.: An infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals. JHEP 0506, 064 (2005); Hanany, A., Kazakopoulos, P., Wecht, B.: JHEP 0508, 054 (2005) 20. Faber, C., Pandharipande, R.: Hodge integrals and Gromov-Witten theory. Invent. Math. 139(1), 173–199 (2000) 21. Caporaso, N., Griguolo, L., Marino, M., Pasquetti, S., Seminara, D.: Phase transitions, double-scaling limit, and topological strings. Phys. Rev. D 75, 046004 (2007) 22. Forbes, B., Jinzenji, M.: Extending the Picard-Fuchs system of local mirror symmetry. J. Math. Phys. 46, 082302 (2005) 23. Katz, S.H., Klemm, A., Vafa, C.: Geometric engineering of quantum field theories. Nucl. Phys. B 497, 173 (1997) 24. Iqbal, A., Kashani-Poor, A.K.: SU(N) geometries and topological string amplitudes. Adv. Theor. Math. Phys. 10, 1 (2006) 25. Eguchi, T., Kanno, H.: Five-dimensional gauge theories and local mirror symmetry. Nucl. Phys. B 586, 331 (2000) 26. Tachikawa, Y.: Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting. JHEP 0402, 050 (2004) 27. Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415 (1999); Halmagyi, N., Okuda, T., Yasnov, V.: Large N duality, lens spaces and the Chern-Simons matrix model. JHEP 0404, 014 (2004) 28. Hori, K., Vafa, C.: Mirror symmetry. http://arxiv.org/abs/hep-th/0002222v3, 2000 29. Chiang, T.M., Klemm, A., Yau, S.T., Zaslow, E.: Local mirror symmetry: Calculations and interpretations. Adv. Theor. Math. Phys. 3, 495 (1999) 30. Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Alg. Geom. 3, 493 (1994) 31. Feng, B., He, Y.H., Kennaway, K.D., Vafa, C.: Dimer models from mirror symmetry and quivering amoebae. Adv. Theor. Math. Phys. 12, 3 (2008) 32. Cox, D.A., Katz, S.: Mirror symmetry and algebraic geometry. Providence, RT: Amer. Math. Soc., 2000 33. Hosono, S.: Central charges, symplectic forms, and hypergeometric series in local mirror symmetry. In: Mirror Symmetry V, Yui, N., Yau, S.-T., Lewis, J.D. eds, Providence, RI: Amer. Math. Soc./Intl Press, 2006, pp. 405–439 34. Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. http://arxiv.org/ abs/hep-th/0012041v1, 2000 35. Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1 (2002) 36. Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47 (1982) 37. Ooguri, H., Vafa, C.: Knot invariants and topological strings. Nucl. Phys. B 577, 419 (2000) 38. Lerche,W., Mayr, P.: On N = 1 mirror symmetry for open type II strings. http://arxiv.org/abs/hep-th/ 0111113v2, 2002 39. Forbes, B.: Open string mirror maps from Picard-Fuchs equations on relative cohomology. http://arxiv. org/abs/hep-th/0307167v4, 2004 40. Nekrasov, N.: Five dimensional gauge theories and relativistic integrable systems. Nucl. Phys. B 531, 323 (1998) 41. Degasperis, A., Ruijsenaars, S.M.N.: Newton-equivalent Hamiltonians for the harmonic oscillator. Ann. Physics 293(1), 92–109 (2001) 42. Exton, H.: Multiple hypergeometric functions and applications. Mathematics and its Applications. New York-London-Sydney: Chichester: Ellis Horwood Ltd., Halsted Press [John Wiley and Sons, Inc.], 1976 43. Exton, H.: Handbook of Hypergeometric Integrals. Theory, Applications, Tables, Computer Programs. Mathematics and its Applications. Chichester: Ellis Horwood Ltd., Chichester-New York-Brisbane: Halsted Press [John Wiley and Sons], 1978 44. Klemm, A., Lerche, W., Theisen, S.: Nonperturbative effective actions of N = 2 supersymmetric gauge theories. Int. J. Mod. Phys. A 11, 1929 (1996) 45. Ohta, Y.: Picard-Fuchs ordinary differential systems in N = 2 supersymmetric Yang-Mills theories. J. Math. Phys. 40, 3211 (1999) 46. Akerblom, N., Flohr, M.: Explicit formulas for the scalar modes in Seiberg-Witten theory with an application to the Argyres-Douglas point. JHEP 0502, 057 (2005) 47. Aganagic, M., Klemm, A., Marino, M., Vafa, C.: Matrix model as a mirror of Chern-Simons theory. JHEP 0402, 010 (2004) 48. Marino, M.: Open string amplitudes and large order behavior in topological string theory. JHEP 0803, 060 (2008)
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49. Zaslow, E.: Topological Orbifold Models And Quantum Cohomology Rings. Commun. Math. Phys. 156, 301 (1993); Aspinwall, P.S.: Resolution of orbifold singularities in string theory. http://arxiv.org/abs/hepth/9403123v2, 1994; Aspinwall, P.S., Greene, B.R., Morrison, D.R.: Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory. Nucl. Phys. B 416, 414 (1994) 50. Dela Ossa, X., Florea, B., Skarke, H.: D-branes on noncompact Calabi-Yau manifolds: K-theory and monodromy. Nucl. Phys. B 644, 170 (2002) 51. Coates, T.: Wall-crossings in toric Gromov-Witten theory II: local examples. http://arxiv.org/abs/0804. 2592v1[math.AG], 2008 52. Coates, T., Corti, A., Iritani, H., Hsian-Hua Tseng: Computing Genus-Zero Twisted Gromov-Witten Invariants. http://arxiv.org/abs/math/0702234v3[math.AG], 2008 53. Bouchard, V., Cavalieri, R.: On the mathematics and physics of high genus invariants of [C 3 /Z 3 ]. http:// arxiv.org/abs/0709.3805v1[math.AG], 2007 54. Witten, E.: Quantum background independence in string theory. http://arxiv.org/abs/hep-th/9306122v1, 1993 55. Yamaguchi, S., Yau, S.T.: Topological string partition functions as polynomials. JHEP 0407, 047 (2004); Alim, M., Lange, J.D.: Polynomial Structure of the (Open) Topological String Partition Function. JHEP 0710, 045 (2007) 56. Grimm, T.W., Klemm, A., Marino, M., Weiss, M.: Direct integration of the topological string. JHEP 0708, 058 (2007) 57. Huang, M.X., Klemm, A.: Holomorphic anomaly in gauge theories and matrix models. JHEP 0709, 054 (2007) 58. Eynard, B.: Topological expansion for the 1-hermitian matrix model correlation functions. JHEP 0411, 031 (2004); Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. http:// arxiv.org/abs/math-ph/0702045v4, 2007 59. Somos, M.: Sequence A122858. In: N. J. A. Sloane (ed.), The On-Line Encyclopedia of Integer Sequences (2008), published electronically at http://www.research.att.com/~njas/sequences/A122858, 2006 60. Bouchard, V.: Orbifold Gromov-Witten invariants and topological strings. In: Modular Forms and String Duality, Fields Institute Communications, Vol. 54, Providence, RI: Amer. Math. Soc., 2008 61. Argyres, P.C., Douglas, M.R.: New phenomena in SU(3) supersymmetric gauge theory. Nucl. Phys. B 448, 93 (1995) 62. Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. http://arxiv.org/abs/hep-th/ 0306238v2, 2003 63. Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten exact solution. Phys. Lett. B 355, 466 (1995); Nakatsu, T., Takasaki, K.: Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory. Mod. Phys. Lett. A 11, 157 (1996); Marino, M.: The uses of Whitham hierarchies. Prog. Theor. Phys. Suppl. 135, 29 (1999) 64. Ferreira, C., López, J.L.: Asymptotic expansions of the Lauricella hypergeometric function FD . J. Comput. Appl. Math. 151(2), 235–256 (2003) 65. Alim, M., Lange, J.D., Mayr, P.: Global Properties of Topological String Amplitudes and Orbifold Invariants. http://arxiv.org/abs/0809.4253v1[hep-th], 2008 Communicated by N.A. Nekrasov
Commun. Math. Phys. 289, 253–289 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0753-0
Communications in
Mathematical Physics
Strange Non-Chaotic Attractors in Quasiperiodically Forced Circle Maps Tobias Jäger Collège de France, 3 rue d’Ulm, 75005 Paris, France. E-mail: [email protected] Received: 2 June 2008 / Accepted: 18 November 2008 Published online: 19 February 2009 – © Springer-Verlag 2009
Abstract: The occurrence of strange non-chaotic attractors (SNA) in quasiperiodically forced systems has attracted considerable interest over the last two decades, in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos. Their existence was first described by Million˘sc˘ ikov (and later by Vinograd and also Herman) for quasiperiodic SL(2, R)-cocycles and by Grebogi et al (and later Keller) for so-called pinched skew products. However, except for these two particular classes there are still hardly any rigorous results on the topic, despite a large number of numerical studies confirming the widespread existence of SNA in quasiperiodically forced systems. Here, we prove the existence of SNA in quasiperiodically forced circle maps under rather general conditions, which can be stated in terms of C 1 -estimates. As a consequence, we obtain the existence of SNA for parameter sets of positive measure in suitable parameter families. These SNA carry the unique physical measure of the system, which determines the behaviour of Lebesgue-almost all initial conditions. Finally, we show that the dynamics are minimal in the considered situations. The results apply in particular to a forced version of the Arnold circle map. For this example, we also describe how the first Arnold tongue collapses and looses its regularity due to the presence of strange non-chaotic attractors and a related unbounded mean motion property. 1. Introduction In 1984, Grebogi et al [1] introduced a class of quasiperiodically forced (qpf) interval maps which exhibit non-continuous invariant graphs with negative vertical Lyapunov exponents (see also [2]). As these objects attract a set of initial conditions of positive measure and combine a complicated structure with non-chaotic dynamics (in particular zero topological entropy), they are commonly referred to as strange non-chaotic attractors (SNA). This observation was spectacular for several reasons. First, the existence of a strange attractor is remarkable in itself – one just has to think of the seminal work of
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Bennedicks and Carleson on the Hénon map [3]. Secondly, as this example indicates, strange attractors usually come along with chaotic dynamics and strange non-chaotic attractors are very rare in other types of systems [4]. Finally, the phenomenon can be viewed as a particular kind of non-uniform hyperbolicity, which is again a very classical and important topic in dynamical systems. This latter point of view is also demonstated by quasiperiodic SL(2, R)-cocycles, which induce qpf circle diffeomorphisms by their projective action. For these, the existence of SNA had already been proved earlier (without using the name) by Million˘sc˘ ikov [5], Vinograd [6] and, in a more general way, Herman [7]. (In [8] these results are discussed in modern terminology.) In this context, the phenomenon is known as the non-uniform hyperbolicity of the cocycle (see [9]). In the following years, the topic attracted a considerable amount of interest, and a large number of numerical studies indicated that the existence of SNA is quite common in qpf systems ([10] gives a good overview). At the same time, the theory of quasiperiodic SL(2, R)-cocycles was developed further, and impressive progress was made in particular in recent years (see, for example, [11–13]). However, rigorous results on other classes of qpf systems remained rare, and in particular no further results were obtained that would confirm the existence of SNA on parameter sets of positive measure in the numerically studied parameter families. ([14] contains some results on non-smooth bifurcations, where SNA occur at isolated parameter values.) The aim of this article is two-fold. First, we show that once the skew-product structure is given, which is usually motivated by the physical context of the model, the existence of SNA in qpf circle maps is a phenomenon which is both ‘robust’ and ‘non-degenerate’. To make this more precise, we denote by Diff0 (T2 ) the set of all diffeomorphisms of the two-torus which are homotopic to the identity and by πi the projection to the respective coordinate. Further, for any ω ∈ T1 we let Rω (θ, x) = (θ +ω, x). Then, as a consequence of our results, we obtain the following: Let F := {F ∈ Diff 0 (T2 ) | π1 ◦ F = π1 }. Then there exists a non-empty set U ⊆ F, which is C 1 -open in F and has the following property: For any F ∈ U there exists a set F ⊆ T1 of positive Lebesgue measure, such that for any ω ∈ F the map f = Rω ◦ F has a SNA. This SNA carries the unique physical measure of the system and attracts Lebesgue-almost all initial conditions. Furthermore, the dynamics of f are minimal. A more precise characterisation of the set U in the above statement, in terms of explicit C 1 -estimates, is provided by Theorem 2.1 and/or Theorem 2.5 below. Our second objective is to apply our methods to a particular model that is well-known from the literature, namely the qpf Arnold circle map (θ, x) →
θ + ω, x + τ + a sin(2π x) + b cos(2π θ )d .
(1.1)
Here τ ∈ T1 , a ∈ [0, 1/2π ), b ∈ R and d is an odd positive integer. This example was proposed by Ding et al [15] as a simple model of an oscillator forced at two incommensurate frequencies and has been intensively studied numerically [16,17].1 Provided d is chosen sufficiently large, we show that there exist rotation numbers ω for which (1.1) exhibits SNA on a set of positive measure in the (τ, a, b)-parameter space (Corollary 2.8). 1 In the numerical studies usually d = 1. However, as mentioned in [15], any real-analytic forcing function is of more or less equal interest.
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Particular attention in the study of (1.1) has been given to the structure of the Arnold tongues, which are subsets of the parameter space on which the fibred rotation number stays constant. The authors of [18] observe that the Arnold tongue corresponding to zero seems to collapse in some regions of the parameter space. In Sect. 2.3, we prove that for sufficiently large d this does indeed happen. In addition, we show that the boundaries of the zero tongue do not depend analytically on b in this case. We want to mention that the approach employed here is inspired by the one of Bjerklöv in [19]. The latter was developed in the setting of quasiperiodic Schrödinger cocycles, but its techniques are basically non-linear, which allows us to adapt and to apply them to the non-linear setting. Similar ideas have also been used earlier by Young [20] to prove positive Lyapunov exponents for certain quasiperiodic SL(2, R)-cocycles. From a more distant point of view, all these works are inspired by the seminal work of Benedicks and Carleson on the Hénon map [3] and rely on the powerful tool of parameter exclusion. 1.1. Notation. Let T1 := R/Z and denote by πi : T2 → T1 the projection to the respective coordinate. A quasiperiodically forced (qpf) circle homeomorphism/diffeomorphism is a homeomorphism/diffeomorphism f : T2 → T2 of the form f : (θ, x) → (θ + ω, f θ (x)),
(1.2)
where ω ∈ T1 \ Q and the fibre maps f θ are defined by f θ (x) = π2 ◦ f (θ, x). Note that we do not assume that f is homotopic to the identity. (Sometimes this is included in the definition of a qpf cirlce homeomorphism, but we will not need this assumption here.) Derivatives with respect to θ or x will be denoted by ∂θ and ∂x , respectively. Further, we use the notation f θn (x) = π2 ◦ f n (θ, x)
∀n ∈ Z.
Note that this implies f θ−1 = ( f θ−ω )−1 . For a, b ∈ T1 we denote by [a, b] the interval of all points x ∈ T1 that lie between a and b in the counterclockwise direction, similarly for open intervals. Note that thus [b, a] = T1 \ (a, b). For two points x, y ∈ T1 , we denote the usual Euclidean distance on the circle by d(x, y). We will also use the notation y − x in order to denote the distance between x and y in the counterclockwise direction, i.e. the length of the interval [x, y]. If ϕ, ψ : T1 → T1 are two measurable functions, we let [ϕ, ψ] := {(θ, x) ∈ T2 | x ∈ [ϕ(θ ), ψ(θ )]}. For any initial point (θ0 , x0 ) we denote its iterates under f by (θk , xk ) := f k (θ0 , x0 ). 1.2. Some preliminaries. To any measurable function ϕ : T1 → T1 , we associate a Borel measure µϕ on T2 by µϕ (A) = Leb {θ ∈ T1 | (θ, ϕ(θ )) ∈ A} . (1.3) Let Tn∗ = {(x1 , . . . , xn ) ∈ Tn | xi = x j ∀i = j}. Then a measurable function ϕ = (ϕ1 , . . . , ϕn ) : T1 → Tn∗ is called a (multi-valued) invariant graph for the qpf circle homeomorphism f , if it satisfies f θ ({ϕ1 (θ ), . . . , ϕn (θ )}) = {ϕ1 (θ + ω), . . . , ϕn (θ + ω)}
∀θ ∈ T1 ,
(1.4)
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n and in addition the f -invariant measure µϕ = n1 i=1 µϕi is ergodic. (The ergodicity assumption ensures that ϕ cannot be further decomposed into invariant subgraphs.) Note that (1.4) implies that the corresponding point set := (θ, ϕi (θ )) | θ ∈ T1 , i = 1, . . . , n is f -invariant. The Lyapunov exponent of an invariant graph ϕ is defined as n 1 λ(ϕ) := log |∂x f θ (x)| dµϕ (θ, x) = log |∂x f θ (ϕi (θ ))| dθ. n T1 T1 i=1
We call a non-continuous invariant graph a strange non-chaotic attractor (SNA) if its Lyapunov exponent is negative and a strange non-chaotic repeller (SNR) if it is positive. A convenient criterion for the existence of SNA involves pointwise Lyapunov exponents, forwards and backwards in time. These are given by 1 λ+ (θ, x) = lim sup | log ∂x f θn (x)| n→∞ n and 1 λ− (θ, x) = lim sup | log ∂x f θ−n (x)|. n→∞ n A point (θ, x) ∈ T2 (or more precisely its orbit) which has a positive Lyapunov exponent both forwards and backwards in time is called a sink-source-orbit. The existence of such orbits implies the existence of SNAs. Proposition 1.1 ([14]). Suppose f is a quasiperiodically forced circle diffeomorphism which has a sink-source-orbit. Then f has both a SNA and a SNR. Proof. For qpf monotone interval maps the proof is given in [14], and it only remains to reduce the situation to that case. First, assume that f has no invariant graphs. Then by [21, Theorem 4.1] there exists a unique f -invariant measure µ. Due to the uniform ergodic theorem all pointwise forwards Lyapunov exponents are equal to λ(µ) = D f θ (x) µ(θ, x), whereas all backwards Lyapunov exponents are equal to −λ(µ). 2 T This contradicts the existence of a sink-source orbit. (In fact, λ(µ) = 0 in this situation.) Hence, we may assume that f has an invariant graph. In this case, it follows again from [21, Theorem 4.1] that all invariant ergodic measures are supported on invariant graphs (in the sense of (1.3)). If all these had non-negative Lyapunov exponents, then they would have non-positive Lyapunov exponents for the inverse f −1 . By the semiuniform ergodic theorem (see [22, Theorem 1.12]), this would imply that all pointwise backwards Lyapunov exponents were non-positive, again contradicting the assumptions. Consequently, there exists at least one invariant graph ϕ with negative Lyapunov exponent. If ϕ is non-continuous, then it is an SNA. Otherwise, we can ‘cut the torus open’ along the continuous curve ϕ and obtain a qpf monotone interval map f˜. The existence of a sink-source-orbit now follows from [14, Theorem 2.4].
The fibred rotation number of a qpf circle homeomorphism is defined as ρ( f ) = ρ(F) mod 1, where F : T1 × R ← − is a lift of f and ρ(F) := lim (Fθn (x) − x)/n. n→∞
(1.5)
This limit always exists and is independent of (θ, x) [7]. Concerning the behaviour of the fibred rotation number w.r.t. monotone perturbations, we will use the following:
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Proposition 1.2 ([23]). Suppose a qpf circle homeomorphism f is minimal. Let F be a lift of f and Fε (θ, x) := (θ + ω, Fθ (x) + ε). Then the mapping ε → ρ(Fε ) is strictly monotone in ε = 0. In fact, the statement given in [23] is more general: The assertion of the proposition is true whenever f has no invariant strip, which is the appropriate analogue of a periodic orbit in this context (see [24 or 25] for the precise definition). Since invariant strips are always compact invariant strict subsets of T2 , the above version follows immediately. Finally, we will need a result concerning the uniqueness of the minimal set. Proposition 1.3 ([26]). Suppose a qpf circle homeomorphism f is transitive. Then it has a unique minimal set. 2. Main Results 2.1. The existence of SNA and a first application. In the following, we formulate a number of assumptions that are used in the statements of our main results. It is important to note that none of them involves the rotation number ω on the base, since this will later be seen as a free parameter. Thus, all conditions below should be understood as assumptions on a collection of fibre maps ( f θ )θ∈T1 . Equivalently, the latter might be considered as a map F which satisfies π1 ◦ F = Id, as in the highlighted statement in the introduction, such that F(θ, x) = (θ, f θ (x)). I. Regions in the phase space. Suppose I0 ⊆ T1 is a finite union of N disjoint open intervals I01 , . . . , I0N . We will refer to I0 as the first critical region. Further, suppose that E = [e− , e+ ] and C = [c− , c+ ] are two non-empty, compact and disjoint intervals of positive length in T1 . We will call E the expanding and C the contracting interval, motivated by the bounds on the derivatives given below. The first condition we require is a strong forward invariance of the contracting interval outside of the critical region: f θ (cl(T1 \ E)) ⊆ int(C)
∀θ ∈ / I0 .
(A1)
Note that this implies f θ−1 (cl(T1 \ C)) ⊆ int(E)
∀θ ∈ / I0 + ω.
(A1 )
II. Bounds on the derivatives. Let α = (αl , αc , αe , αu ) ∈ R4 satisfy 0 < αl < αc < 1 < αe < αu and suppose the following estimates hold: αl < ∂x f θ (x) < αu ∀(θ, x) ∈ T2 ;
(A2)
∂x f θ (x) > αe ∀(θ, x) ∈ T1 × E;
(A3)
∂x f θ (x) < αc ∀(θ, x) ∈ T1 × C.
(A4)
Further, let S > 0 and suppose that |∂θ f θ (x)| < S
∀(θ, x) ∈ T2 .
(For example S = sup(θ,x)∈T2 |∂θ f θ (x)| + 1.)
(A5)
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III. Transversal Intersections. The last property we will need is the fact that for each connected component I0ι of the first critical region I0 , the set f (I0ι × C) crosses the expanding strip T1 × E in a ‘nice’ transversal intersection, either upwards or downwards. This is ensured by the following: First, we suppose that |∂θ f θ (x)| > s
∀(θ, x) ∈ I0 × T1
(A6)
for some constant s with 0 < s < S. This implies that the sign of ∂θ f θ (x) is constant on every connected component I0ι × T1 of I0 × T1 and we speak of an upwards crossing if it is positive and of a downwards crossing if it is negative. Secondly, we assume that ∃!θι1 ∈ I0ι ∃!θι2 ∈ I0ι
with f θι1 (c+ ) = e− and with f θι2 (c− ) = e+ .
(A7)
This ensures that the image of I0ι ×C crosses the strip (I0ι + ω)× E exactly once and does not ‘wind around the torus’ more than one time. Note that with respect to the canonical ordering inside the interval I0ι , the point θι1 lies on the left of θι2 if the crossing is upwards and on the right of θι2 if it is downwards. Now we can state the first main result. The proof is given in Sect. 3. Theorem 2.1. Suppose ( f θ )θ∈T1 satisfies (A1)–(A7). Further assume that 2
αc−1 = αe = α p
and αl−1 = αu = α p
ι for some p ∈ N. Let ε0 := maxN ι=1 |I0 | and fix δ > 0. Then there exist strictly positive constants c0 = c0 (δ, p, s, S, N ) and α0 = α0 (δ, p, s, S, N ) with the following property. If ε0 < c0 and α > α0 , then there exists a set ⊆ T1 of measure Leb() ≥ 1 − δ, such that for all ω ∈ the qpf circle diffeomorphism (θ, x) → (θ + ω, f θ (x)) satisfies ⎧ ⎨ (∗1) f has a sink-source orbit and thus a SNA ϕ − and a SNR ϕ + ; (∗2) ϕ − and ϕ + are one-valued and the only invariant graphs of f ; (∗) ⎩ (∗3) f is minimal.
Remark 2.2. (a) Property (∗2), together with the non-zero Lyapunov exponents, determines the dynamics of f quite effectively. In fact, for almost every θ ∈ T1 ,
Wθ = x ∈ T1 lim f θn (x) − ϕ − (θ + nω) = 0 n→∞
is a non-empty open interval. (When f is C 1+α this follows from Pesin Theory. For a short direct proof without the additional regularity assumption, see [27].) The endpoints of these intervals constitute invariant graphs and must therefore both coincide with ϕ + (θ ) (since there is no other invariant graph except ϕ + and ϕ − ). Hence, almost all orbits converge to ϕ − , which implies quite easily that the measure µϕ − is the unique physical measure of the system and determines the behaviour of Lebesgue-almost all initial onditions. In particular, almost every point has a negative Lyapunov exponent. Similarly, µϕ + is the unique physical measure for the inverse f −1 .
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(b) When f is given by the projective action of a quasiperiodic SL(2, R)-cocycle, then property (∗2) comes for free once the non-uniform hyperbolicity of the cocylce is established. (The latter is equivalent to the existence of SNA [9].) However, without the linear structure there is no obvious reason why this should be so, and in general it must clearly be expected that multi-valued SNA exist as well. (c) The highlighted statement in the introduction is an immediate consequence, since all the conditions of the theorem are C 1 -open in F and the subset of F which satisfies these hypotheses is obviously non-empty. The only possible obstruction for constructing examples is the relation ε0 ≥ (1 − |C|)/S, which follows from the fact that the strip f (I0ι × C) has to cross I0ι × (c+ , c− ) by (A7) and (A1), but cannot have slope greater than S. However, this does not lead to any conflict with the dependence of c0 on S, since no assumption is made on the size of the interval C. In any case, some explicit examples to which the theorem applies are given in Corollary 2.3 below. (d) There is an interesting relation to the general classification of qpf circle homeomorphisms, which we briefly want to mention. Suppose that a qpf circle diffeomorphism f is minimal and has a SNA, as in the assertion of the theorem. Then it also has the property that its ‘deviations from the average rotation’ Fθn (x) − x − nρ(F)
(2.1)
are unbounded in n. This follows quite easily from results in [24, Theorem 3.1], which states that if the quantities (2.1) are bounded then either f has an invariant strip and is therefore not minimal, or is semi-conjugate to an irrational rotation and therefore has zero Lyapunov exponents. There is a mechanism for the creation of SNA that is very similar to the one studied here, but leads to SNA which are the semi-continuous boundary graphs of invariant strips. In particular, the dynamics are not minimal and the quantities (2.1) remain bounded. This mechanism is described in [28 and 14]. In order to give some explicit examples to which the above theorem applies, denote by γ : T1 → (−1/2, 1/2] the lift of the identity map on T1 . Then π ◦ γ = IdT1 , where π : R → T1 is the canonical projection. Further, given any p ≥ 2, define a p : R → R by x 1 dξ. (2.2) a p (x) := 1 + |ξ | p 0 Of course, for p = 2 this just yields the arcus tangent. For α ∈ R+ and x ∈ T1 , let a p (αγ (x)) . (2.3) h α (x) := π 2a p (α/2) It is easy to check that for all α the map h α is a diffeomorphism of the circle. Finally, let g : T1 → T1 be a differentiable map, such that g −1 ({1/2}) is a finite and non-empty set;
(2.4)
g (θ ) = 0 ∀x ∈ g −1 ({1/2}).
(2.5)
For example, one could choose g(θ ) = β cos(2π θ ) for any β > 21 . Then Theorem 2.1 implies the following
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Corollary 2.3. Suppose h α and g are chosen as above and δ > 0 is fixed. Then there exists a constant α0 = α0 (δ, p, g) with the following property: If α ≥ α0 , then there exists a set ⊆ T1 of measure Leb() ≥ 1 − δ, such that for any ω ∈ the system (θ, x) → (θ + ω, h α (x) + g(θ ))
(2.6)
satisfies (∗). The proof is given in Sect. 3.8. Remark 2.4. Let c p := lim x→∞ a p (x) and suppose h˜ α is the map which is obtained ¯ ← by projecting the mapping R − , x → α 2 x to the circle via the change of variables x → a p (x)/2c p mod 1. Then the preceding corollary remains true if h α is replaced by h˜ α . The proof in Sect. 3.8 can be adapted easily. If p = 2, such that a p (x) = arctan(x), then the map (θ, x) → (θ + ω, h˜ α (x) + g(θ )) is the projective action of the SL(2, R)-cocycle − , (θ, v) → (θ + ω, A(θ )v) T1 × R2 ← with
A(θ ) = Rg(θ) ◦
α 0 0 α −1
,
where Rφ denotes the rotation matrix with angle φ. This means that, at least in the case of an analytic forcing function g and except for the minimality, similar statements can be derived from classical results on SL(2, R)-cocycles, for example in [7]. This is not true for the parameter family (2.6), and the fact that the results presented here also apply to the non-linear case should be considered as the the main point of this article. 2.2. A refined result for the quasiperiodically forced Arnold circle map. The statement of Theorem 2.1 can be circumscribed by saying that SNA occur for a large set of frequencies if the fibre maps are ‘sufficiently hyperbolic’, meaning that the expansion and contraction constants provided by (A3) and (A4) are large enough. However, for the forced Arnold circle map (1.1) this constitutes a problem. In the realm of invertibility (a ≤ 1/2π ), the derivative of the fibre maps is bounded by 2. For the contraction, the situation is similar. While the derivative at x = 21 goes to zero when a is close to 1/2π , a strong contraction only occurs on a very small neighbourhood of 21 . For any interval of fixed length, the uniform contraction rate always remain bounded. In order to overcome this obstruction and to obtain a result which applies to the qpf Arnold circle map, we have to make use of additional information on the forcing function θ → cos(2π θ )d , namely of the fact that for large d its derivative almost vanishes on a large part of the phase space. This is done via the following assumption. ˜ Suppose I0 ⊆ T1 is the disjoint union of at most N open intervals J 1 , . . . , J N ,
N˜ ≤ N , and let s ∈ (0, S). Then assume that I0 ⊆ I0 and |∂θ f θ (x)| < s ∀(θ, x) ∈ (T1 \ I0 ) × C. The refined version of Theorem 2.1 now reads as follows:
(A8)
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Theorem 2.5. Suppose ( f θ )θ∈T1 satisfies (A1)–(A8 ) and 2
αc−1 = αe = α p
and αl−1 = αu = α p
ι for some p ∈ N. Let ε0 := maxN ι=1 |J | and fix δ > 0. Further, assume there exist constants A, d > 0 such that
S < A · d, √ s > d/A, √ 3 ε0 < A/ d.
(2.7) (2.8) (2.9)
Then there exist constants c0 = c0 (δ, α, p, N ) > 0 and d0 = d0 (δ, α, p, N , A) > 0 with the following property.
If ss < c0 and d > d0 , then there exists a set ⊆ T1 of measure Leb() ≥ 1 − δ, such that for all ω ∈ the system (θ, x) → (θ + ω, f θ (x)) satisfies (∗). Now suppose h is an orientation-preserving diffeomorphism of the circle, such that there exist disjoint closed intervals C, E ⊆ T1 which satisfy sup h (x) < 1, x∈C
inf h (x) > 1,
x∈E
h(cl(E c )) ⊆ int(C).
(2.10) (2.11)
For example, this holds when h has exactly two fixed points and two points of inflexion. Corollary 2.6. Suppose h satisfies (2.10) and (2.11) and δ > 0 is fixed. Then there exist constants d0 = d0 (δ, h) > 0 and ε = ε(δ, h) > 0 with the following property. If d ≥ d0 and b ∈ (1 − ε, 1 + ε), then there exists a set ⊆ T1 of measure Leb() ≥ 1 − δ, such that for any ω ∈ the system (2.12) (θ, x) → θ + ω, h(x) + b cos(2π θ )d satisfies (∗). The proof is given in Sect. 4.2. Remark 2.7. (a) Corollary 2.6 applies in particular to h(x) = x + τ + a sin(2π x) whenever 0 ≤ τ < a < 1/2π . Thus, we obtain the existence of SNA for the qpf Arnold circle map (1.1). We reformulate the result in Corollary 2.8 below. (b) We remark that the above statement remains true if cos(2π θ )d is replaced by other forcing functions depending on a parameter d, as long as these show a similar scald with d ∈ R+ ing behaviour. For example, one could take gd (θ ) = 1+sin(2πθ) 2 very large in order to ensure the existence of SNA. The proof of the corollary in Sect. 4.2 can be adapted accordingly. However, the symmetry cos(2π(θ + 21 ))d = − cos(2π θ )d will play an important role in Sect. 2.3, such that we concentrate on this forcing function. In the literature, a typical point of view is to fix ω and d and to view (1.1) as a three-parameter family depending on τ, a and b. Using Fubini’s Theorem, we obtain
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Corollary 2.8. There is a constant d0 > 0, such that for any d ≥ d0 there exists a set of positive measure ⊆ T1 with the following property. For each ω ∈ , there exists a set of positive measure Bω ⊆ T1 × (0, 1/2π ) × R, such that for all (τ, a, b) ∈ Bω the qpf Arnold circle map (1.1) satisfies (∗). Of course, similar statements hold if one likes to consider (1.1) as a parameter family only depending on one or two parameters, while the other(s) are fixed. 2.3. Collapsing of the first Arnold tongue. In this section, we explain the consequences of our results for the structure of the first Arnold tongue. We denote the qpf Arnold circle map (1.1) with parameters τ, a and b by f τ,a,b . First of all, the following statement is an immediate consequence of Corollary 2.6, applied to h(x) = x + a sin(2π x), and Fubini’s Theorem. Corollary 2.9. Given any a ∈ (0, 1/2π ), there exists a constant d0 = d0 (a), such that for any d ≥ d0 there exists a set ⊆ T1 of positive measure with the following property. For any ω ∈ , there exists a set of positive measure Bω ⊆ R, such that for any b ∈ Bω the qpf Arnold circle map f 0,a,b satisfies (∗). Since we want to study the dependence of the first Arnold tongue on the parameter b, the following notation will be convenient: Aaρ := {(τ, b) ∈ T1 × R | ρ( f τ,a,b ) = ρ}.
(2.13)
As the rotation number depends monotonically on the parameter τ , there exist functions − , τ + : R → T1 , such that τa,ρ a,ρ − + Aaρ = {(τ, b) ∈ T1 × R | τ ∈ [τa,ρ (b), τa,ρ (b)]}.
(2.14)
± are continuous for all a, ρ and coincide (meaning τ − = τ + ) These functions τa,ρ a,ρ a,ρ whenever ρ does not depend rationally on ω, i.e. ρ ∈ / Q + Qω mod 1 [23]. The canonical lift of the qpf Arnold circle map is given by − , (θ, x) → θ + ω, x + τ + a sin(2π x) + b cos(2π θ )d . Fτ,a,b : T1 × R ←
Obviously there holds F0,a,b,θ (−x) = −F0,a,b,θ+1/2 (x). This symmetry immediately − + (b)] ∀b ∈ R. On the other hand, (b), τa,0 implies ρ(F0,a,b ) = 0, and therefore 0 ∈ [τa,0 if d ≥ d0 (a) and b ∈ Bω , where d0 (a) and Bω are chosen as in the above corollary, then f 0,a,b is minimal. Then by Proposition 1.2 the mapping τ → ρ( f τ,a,b ) is strictly monotone at τ = 0. Consequently, the first Arnold tongue is collapsed to a single point − + (b) = 0. As this happens on a set of b of positive at this b-value, meaning τa,0 (b) = τa,0 measure, and since the first Arnold tongue for a > 0 is clearly not collapsed at b = 0, ± the dependence of τa,0 on b cannot be real-analytic. We summarise our observations. Proposition 2.10. Suppose a ∈ (0, 1/2π ) is fixed, d ≥ d0 (a) and b ∈ Bω , where d0 (a) − + (b) = 0. and Bω are as in Corollary 2.9. Then for any b ∈ Bω , there holds τa,0 (b) = τa,0 ± Furthermore, the mappings b → τa,0 (b) are not real-analytic. Of course, this raises the question whether the dependence of the boundaries of the Arnold tongues is analytic in a. We have to leave this open here. However, by the same argument with the roles of a and b interchanged, one obtains the existence of parameters b, such that for a set of a’s of positive measure the first Arnold tongue is collapsed. Hence, if such a parameter b is fixed and the dependence on a was real-analytic, then the first tongue would have to be reduced to a single point for all a ∈ [0, 1/2π ].
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3. Creation of SNA: The Basic Mechanism The aim of this section is to prove Theorem 2.1. Thereby, we proceed in three steps. First, we place certain ‘imaginary’ conditions on the rotation number ω, and show that these imply the existence of a sink-source-orbit (Sects. 3.1 and 3.2). After this, it remains to show that there exist rotation numbers which satisfy these conditions. In order to do so, we first describe the geometry of certain critical sets, which were used before in the formulation of the conditions on ω (Sect. 3.3). Using the obtained information, we then perform a parameter exclusion, which still leaves a set of positive measure of ‘good’ ω’s, which have all the required properties. The technical statements for the parameter exclusion are contained in Sect. 3.4, the final step in the proof is then given in Sect. 3.5. The proof of the uniqueness of the invariant graphs is contained in 3.6, the one of the minimality statement in Sect. 3.7. 3.1. Critical sets and good frequencies. Critical sets. First we have to define a sequence of critical sets, which project down to critical regions and play a major part in all that follows. Definition 3.1. For ω ∈ T1 , I0 as in Sect. 2.1 and any monotonically increasing sequence (Mn )n∈N0 of integers with M0 ≥ 2 we inductively define nested sequences C0 , C1 , . . . of critical sets and I0 ⊇ I1 ⊇ I2 . . . of critical regions in the following way: If I0 , . . . , In have been defined, let An := {(θ, x) | θ ∈ In − (Mn − 1)ω, x ∈ C}, Bn := {(θ, x) | θ ∈ In + (Mn + 1)ω, x ∈ E}, Cn := f Mn −1 (An ) ∩ f −Mn −1 (Bn ) and In+1 := int(π1 (Cn )). We note that a priori it is certainly possible that In is empty for some n ∈ N and hence for all m ≥ n. To show that this does not happen for the selected frequencies will be a crucial point in the proof. Good frequencies. Further, we impose certain ‘Diophantine’ conditions on the frequency ω, which mainly state that the critical sets do not return too fast. Definition 3.2. Suppose (Mn )n∈N0 and (In )n∈N0 are chosen as above and let (K n)n∈N0 be a monotonically increasing sequence of positive integers. Further, assume that (εn )n∈N0 is a non-increasing sequence of positive real numbers which satisfy ε0 ≤ 1 and εn ≥ 3εn+1 ∀n ∈ N0 . Finally, let Xn :=
2K n Mn
(In + kω) and Yn :=
k=1
n
M j +1
(I j + kω).
j=0 k=−M j +1
Then we define Fn = Fn (M0 , . . . , Mn ) as the set of frequencies ω ∈ T1 which satisfy d(I j , X j ) > 3ε j
∀ j = 0, . . . , n
(F1)n
d((I j −(M j − 1)ω) ∪ (I j + (M j + 1)ω), Y j−1 ) > 0
∀ j = 1, . . . , n. (F2)n
and
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Further, let Zn :=
n
Mj
(I j + kω),
j=0 k=−M j +2
Z−1 := ∅ and F−1 := T1 . Finally let Vn :=
j +1 n M
(I j + kω) and Wn :=
j=0 k=1
n
0
(I j + kω)
j=0 k=−M j +1
and V−1 = W−1 = ∅. Remark 3.3. For an easier reading of the next sections, the reader should keep in mind the following ‘intuitive’ description of the relations between the sets Yn , Zn , Vn and Wn : Vn and Wn are just the ‘right’ and ‘left’ part of Yn , whereas Zn is just reduced by one iterate on either side in comparison with Yn , such that Zn ± ω is still contained in Yn . 3.2. Construction of the sink-source-orbits. Recall that for any given point (θ0 , x0 ), we denote its orbit by (θk , xk ) = f k (θ0 , x0 ). Lemma 3.4. Suppose (A1) holds. Then for all n ≥ 0, the following are true: Forwards iteration: If ⎧ ⎨ ω ∈ Fn−1 θ0 ∈ / Zn−1 ⎩x ∈ C 0
(B1)n
and L ≥ 0 is the first integer, such that θL ∈ In , then xm ∈ / C ⇒ θm ∈ Vn−1
∀m = 1, . . . , L.
(C1)n
Backwards iteration: If ⎧ ⎨ ω ∈ Fn−1 θ0 ∈ / Zn−1 ⎩x ∈ E 0
(B2)n
and R ≥ 0 is the first integer, such that θ−R ∈ In + ω, then x−m ∈ / E ⇒ θ−m ∈ Wn−1
∀m = 1, . . . , R.
(C2)n
Remark 3.5. Although there is a slight asymmetry between the forwards and the backwards iteration, the proofs of the respective statements are more or less the same. We will therefore omit the details for the backwards case. (However, full details are given in the preprint version of this article [29].) This applies equally to the proof of Lemma 3.8.
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Proof. First of all, note that (C1)0 follows directly from (A1). Now suppose that (B1)n implies (C1)n and fix ω ∈ Fn , θ0 ∈ / Zn and x0 ∈ C. Using (F1)n and (F2)n , it is easy to see that (In − (Mn − 1)ω) ∩ Vn = ∅, (In + (Mn + 1)ω) ∩ I0 = ∅, (In + (Mn + 2)ω) ∩ Zn = ∅ .
(3.1) (3.2) (3.3)
Let L be the first integer such that θL ∈ In+1 and let 0 < L 1 < L 2 < · · · < L J = L be those times 0 ≤ i ≤ L with θi ∈ In . If we denote condition (C1)n+1 with L replaced by L j by (C1)n+1 [L j ], then (C1)n+1 [L 1 ] follows from (C1)n (note that Zn−1 ⊆ Zn , Fn ⊆ Fn−1 and Vn−1 ⊆ Vn ). Assume now that (C1)n+1 [L j ] holds for some 1 ≤ j < J . As θ0 ∈ / Zn we have L j − Mn + 1 ≥ 0, and as θ L j −Mn +1 ∈ / Vn due to (3.1) it follows that x L j −Mn +1 ∈ C. Consequently (θ L j −Mn +1 , x L j −Mn +1 ) ∈ An , and as θ L j ∈ / In+1 we must have (θ L j +Mn +1 , x L j +Mn +1 ) ∈ / Bn , which means / E. x L j +Mn +1 ∈ / I0 by (3.2) we can apply (A1) and obtain x L j +Mn +2 ∈ C. Before, we As θ L j +Mn +1 ∈ could have had xk ∈ / C for some k ∈ {L j + 1, . . . , L j + Mn + 1}, but for such k θk ∈ In + ω ∪ . . . ∪ In + (Mn + 1)ω ⊆ Vn obviously holds. / Zn ⊇ Zn−1 by (3.3) and Fn ⊆ Fn−1 , we can now apply (C1)n Further, as θ L j +Mn +2 ∈ and obtain (C1)n+1 [L j+1 ]. As L J = L, this completes the proof of (C1)n+1 . As mentioned, the statement for the backwards iteration can be proved more or less in the same way (see Remark 3.5).
Remark 3.6. (a) Suppose (A1) holds, ω ∈ Fn and (θ0 , x0 ) ∈ An . Then (B1)n holds and L = Mn − 1. In order to see this, note that x0 ∈ C holds by definition of An , and θ0 ∈ / Zn−1 follows from (In − (Mn − 1)ω) ∩ Zn−1 = ∅,
(3.4)
which is a consequence of (F2)n . (b) Similarly, suppose ω ∈ Fn−1 and (θ0 , x0 ) ∈ Bn . Then (B2)n holds and R = Mn . This follows by the same argument as (a): x0 ∈ E holds by definition of Bn and θ0 ∈ / Zn−1 follows from (In + (Mn + 1)ω) ∩ Zn−1 = ∅,
(3.5)
which is again a consequence of (F2)n . Corollary 3.7. Suppose (A1) holds and ω ∈ Fn . Then f Mn −Mn−1 (An ) ⊆ An−1 and f −Mn +Mn−1 (Bn ) ⊆ Bn−1 .
(3.6)
Consequently C0 ⊇ C1 ⊇ C2 ⊇ . . . . Further f Mn −1 (An ) ⊆ In × C and f −Mn (Bn ) ⊆ (In + ω) × E.
(3.7)
266
T. Jäger
Proof. Let (θ0 , x0 ) ∈ An , such that, by the preceding remark, (B1)n holds. Also (In − (Mn−1 − 1)ω) ∩ Vn−1 = ∅ holds. This follows from (3.1), applied to n − 1 and using that In ⊆ In−1 . Therefore we have θ Mn −Mn−1 ∈ / Vn−1 , such that we can apply Lemma 3.4 and obtain that x Mn −Mn−1 ∈ C, which means that f Mn −Mn−1 (θ0 , x0 ) ∈ An−1 . This proves the first inclusion in (3.6), and the argument for the second one is similar. Finally, as In ∩ Vn = ∅, In + ω ∩ Wn = ∅,
(3.8) (3.9)
due to (F1)n , the inclusions in (3.7) follow in the same way.
Lemma 3.4 gives some first control about the time an orbit spends in the expanding and contracting region. In order to make use of this information, we need to quantify it. For given ω, θ0 , x0 and 0 ≤ m ≤ N , let PmN := #{k ∈ [m, N − 1] | xk ∈ C}, QmN
:= #{k ∈ [m, N − 1] | x−k ∈ E}.
(3.10) (3.11)
Further, let β0 = 1 and βn :=
n−1
1−
j=0
1 Kj
.
(3.12)
Lemma 3.8. Suppose (A1) holds. Then for all n ≥ 0 the following hold: Forwards iteration: Suppose (B1)n holds and let L be chosen as in Lemma 3.4. Then PmL ≥ βn · (L − m)
∀m = 0, . . . , L − 1.
(C3)n
Further xL ∈ C. Backwards iteration: Suppose (B2)n holds and let R be chosen as in Lemma 3.4. Then QR m ≥ βn · (R − m)
∀m = 0, . . . , R − 1.
(C4)n
Further x−R ∈ E. Proof. As V−1 is void, (C3)0 follows directly from (C1)0 . Suppose that (B1)n implies (C3)n and fix ω ∈ Fn , θ0 ∈ / Zn and x0 ∈ C. As in the proof of Lemma 3.4, let 0 < L 1 < L 2 < · · · < L J = L be those times 0 ≤ i ≤ L with θi ∈ In and denote condition (C3)n with L replaced by L j by (C3)n [L j ]. As βn+1 ≤ βn , condition (C3)n+1 [L 1 ] follows from (C3)n . Suppose (C3)n+1 [L j ] holds for some 1 ≤ j < J . Using (F1)n and (F2)n we see that (In + (Mn + 2)ω) ∩ Vn = ∅,
(3.13)
such that in particular θ L j +Mn +2 ∈ / Vn and consequently x L j +Mn +2 ∈ C by (C1)n+1 . As further θ L j +Mn +2 ∈ / Zn by (3.3), (C3)n implies that for any m ∈ [L j + Mn + 2, L j+1 ] there holds L
Pm j+1 ≥ βn · (L j+1 − m).
(3.14)
SNA in Quasiperiodically Forced Circle Maps
267
This proves (C3)n+1 [L j+1 ] for such m. Further, by (F1)n we have L j+1 − L j > 2K n Mn . Hence, for any m ∈ [L j , L j + Mn + 1] we obtain the estimate L
L
Pm j+1 ≥ P L jj+1 +Mn +2 ≥ βn · (L j+1 − L j − Mn − 2) L j+1 − L j − Mn − 2 · (L j+1 − m) L j+1 − L j M0 ≥2 Mn + 2 (L j+1 − m) ≥ βn+1 · (L j+1 − m). ≥ βn · 1 − 2K n Mn
≥ βn ·
L
Finally, if m ∈ [0, L j − 1] the statement follows by combining the estimate for P L jj+1 L
with the one for Pm j obtained from (C3)n+1 [L j ]. As before, we omit the details for the backwards iteration (see 3.5).
Let β := lim βn = inf βn n→∞
(3.15)
n
and α− := αcβ αu1−β , α+ := αeβ αl
1−β
.
(3.16)
Corollary 3.9. Suppose (A1)–(A4) hold and ω ∈ Fn . If (θ, x) ∈ cl( f Mn (An )), then for all k ∈ [0, Mn ], −k holds. ∂x f θ−k (x) ≥ α−
(3.17)
If (θ, x) ∈ cl( f −Mn (Bn )), then for all k ∈ [0, Mn ], ∂x f θk (x) ≥ α+k holds.
(3.18)
Proof. By continuity, it suffices to prove the above estimates on f Mn (An ) and f −Mn (Bn ), respectively. We start by proving (3.18). Suppose (θ, x) ∈ f −Mn (Bn ) and let (θ0 , x0 ) = f Mn (θ, x) ∈ Bn . Then due to Remark 3.6 we have that R = Mn and (B2)n holds. Using (A2), (A3) and the fact that x = x−R ∈ E (see Lemma 3.8) we obtain ∂x f θk (x)
=
R
QR R−k+1
∂x f θ− j (x− j ) ≥ αe · αe
k−1−QR R−k+1
· αl
.
(3.19)
j=R−k+1
Applying (C4)n and using that αe ≥ α+ yields the statement. The estimate in (3.17) can k be obtained in the same way by using that ∂x f θ−k (x) = (∂x f θ−kω ( f θ−k (x)))−1 .
−1 , α+ } > 1, ω ∈ n∈N Fn and all Proposition 3.10. Suppose (A1)–(A4) hold, min{α− critical sets In are non-void. Then f has a sink-source-orbit and consequently a SNA and a SNR.
268
T. Jäger
Proof. As all critical sets In are non-void, the same is obviously true for the sets cl(Cn ) and their images cl( f (Cn )) = cl( f Mn (An )) ∩ cl( f −Mn (Bn )). Due to Corollary 3.7, the latter form a nestedsequence of compact sets, such that their intersection is non-void as well. Let (θ, x) ∈ n∈N cl( f (Cn )). Then due to (3.17) and as Mn ∞, we obtain λ− (θ, x) = lim sup k→∞
1 log |∂x f θ−k (x)| ≥ − log α− > 0, k
and similarly (3.18) yields λ+ (θ, x) ≥ log α+ > 0. Consequently f has a sink-sourceorbit, and the existence of a SNA and a SNR follows from Proposition 1.1.
3.3. Geometry of the critical sets. In this section we turn to the description of the critical sets Cn and the corresponding critical regions In+1 . In particular, we want to obtain information about their size and their dependence on ω (which we have kept implicit so far). Suppose I = I (ω) = (a(ω), b(ω)) is a connected component of In . Then we use the notation |∂ω I | = max{|∂ω a(ω)|, |∂ω b(ω)|}, provided both derivatives on the right side exist. In this case we call I differentiable with respect to ω. We will use the following inductive assumption: ⎧ (i) For each j ∈ [0, n], I j consists of N disjoint open intervals ⎪ ⎪ ⎪ ⎪ I 1j , . . . , I N ⎪ j . ⎪ ⎪ ⎪ ⎨ (ii) For j ∈ [1, n], each connected component of I j−1 contains exactly one connected component of I j . Thus, by suitable (I)n ⎪ ι ⊆ Iι ⎪ labelling, I ∀ι = 1, . . . , N . ⎪ j j−1 ⎪ ⎪ ⎪ (iii) For all j ∈ [0, n] the set F j is open and all I ιj are differentiable ⎪ ⎪ ⎩ with respect to ω on F j . Note that (I)0 follows directly from the choice of I0 in Sect. 2.1 and the definition of F0 . (ii) is void for n = 0. The statement we aim for is the following. Proposition 3.11. Suppose (A1)–(A7) hold and let ω ∈ Fn = Fn (M0 , . . . , Mn ) for some n ≥ 0. Further, assume that α− < 1 < α+ and 1 1 s (3.20) + S := s − S · ≥ , −1 2 α− − 1 α+ − 1 γ := S ·
∞ S k kα− + (k + 1)α+−k ≤ . 4
(3.21)
k=1
Then (I)n+1 holds and 2 · max{α− , α+−1 } Mn , s 1 ι |∂ω In+1 |≤ . 4 ι |≤ |In+1
(3.22) (3.23)
We give the proof right here, but since it mainly depends on notions and statements introduced below, the reader might prefer to postpone it until the end of this section.
SNA in Quasiperiodically Forced Circle Maps
269
Proof. Suppose that (A1)–(A7) hold. We proceed by induction, using the additional inductive assumption (( /)n ) introduced below. The conditions (I)0 and ( /)0 follow directly from (A1) and the definition of F0 . Now, assume that (I)n and ( /)n hold for some n ≥ 0 and ω ∈ Fn . Due to Lemma 3.14 and (3.20) we have lnϕ − u ψ n ≥ S ≥ s/2 > 0.
(3.24)
Therefore we can apply Lemma 3.12, which implies that (I)n+1 and ( /)n+1 hold. The required estimates on |I ιj | and |∂ω I ιj | follow from Lemma 3.12, in combination with (3.24), Lemma 3.13, Lemma 3.15 and the estimates provided by (3.20) and (3.21).
Let us briefly motivate this. Under the assumptions of the proposition, the fact that ω ∈ Fn (M1 , . . . , Mn ) implies that the N connected components of In+1 are small and do not depend too much on ω. Roughly spoken, we will use this information during the parameter exclusion in the next section to deduce that ‘most’ ω ∈ Fn are also contained in Fn+1 , or more precisely that for ‘most’ ω ∈ Fn (M0 , . . . , Mn ) there exists a suitable integer Mn+1 (ω), such that ω ∈ Fn+1 (M0 , . . . , Mn , Mn+1 (ω)). (Recall here that the conditions (F1)n and (F2)n defining Fn are assumptions on the distances between the sets I j and their iterates and therefore more likely to be fulfilled when the connected components of these sets are small.) In the end, there remains aset of ω’s of positive measure, which admit a sequence (Mn (ω))n∈N , such that ω ∈ n∈N Fn (M0 (ω), . . . , Mn (ω)). For such ω, Proposition 3.10 then yields the existence of SNA. For any ι ∈ [1, N ] we let Aιn := {(θ, x) | θ ∈ Inι − (Mn − 1)ω, x ∈ C}, Bnι := {(θ, x) | θ ∈ Inι + (Mn + 1)ω, x ∈ E}.
(3.25) (3.26)
For θ ∈ In + ω, let −Mn ± ψι,n (θ, ω) := f θ+M (e± ), nω
(3.27)
− + f Mn (Aιn ) = {(θ, x) | θ ∈ Inι + ω, x ∈ [ϕι,n (θ, ω), ϕι,n (θ, ω)]},
(3.28)
−Mn
(3.29)
Mn ± (θ, ω) := f θ−M (c± ) ϕι,n nω
and
such that
f
(Bnι )
= {(θ, x) | θ ∈
Inι
+ ω, x ∈
− + [ψι,n (θ, ω), ψι,n (θ, ω)]}.
In order to start the induction, it is also convenient to define ± ϕ−1 (θ, ω) := f θ−ω (c± )
and
± ψ−1 (θ, ω) := e± .
(3.30)
In all proofs of this section, we will fix the index ι in order to concentrate on one component of In . In principle we would have to distinguish two cases, namely that of an upwards and that of a downwards crossing (see (A7)). However, as the two cases are completely symmetric we can always assume, without loss of generality, that the crossing between f Mn (An ) and f −Mn (Bn ) is ‘upwards’, that is ∂θ f θ (x) > s on Inι ⊆ I0ι . With the above notions, we can now state the second inductive assumption that will ϕ be used in this section. Suppose that Inι (ω) = (aι,n (ω), bι,n (ω)) and let Jn (θ, ω) := ψ (ϕn− (θ, ω), ϕn+ (θ, ω)) and Jn (θ, ω) := (ψn− (θ, ω), ψn+ (θ, ω)). Then we assume that ϕ
ψ
Jn−1 (aι,n (ω) + ω, ω) ∩ Jn−1 (aι,n (ω) + ω, ω) =
∅,
ψ ϕ Jn−1 (bι,n (ω) + ω, ω) ∩ Jn−1 (bι,n (ω) + ω, ω)
∅.
=
( /)n
270
T. Jäger
± ± Note that due to the definition of ϕ−1 and ψ−1 in (3.30), the statement ( /)0 is a consequence of (A1). The proof of Proposition 3.11 is not too difficult on a conceptual level, but requires a number of technical estimates. In order to separate the geometrical and more intuitive part from the analytical one, we introduce the following notions.
h ϕn := hψ n :=
inf |ϕn+ (θ, ω) − ϕn− (θ, ω)|,
(3.31)
inf |ψn+ (θ, ω) − ψn− (θ, ω)|,
(3.32)
θ∈In +ω θ∈In +ω
Hnϕ := sup |ϕn+ (θ, ω) − ϕn− (θ, ω)|,
(3.33)
Hnψ := sup |ψn+ (θ, ω) − ψn− (θ, ω)|,
(3.34)
θ∈In +ω θ∈In +ω
inf ∂θ ϕn± (θ, ω) , θ∈In +ω
ϕ u n := sup ∂θ ϕn± (θ, ω) , lnϕ :=
(3.35) (3.36)
θ∈In +ω
±
uψ n := sup ∂θ ψn (θ, ω) ,
(3.37)
γnϕ := sup ∂θ ϕn± (θ, ω) + ∂ω ϕn± (θ, ω) ,
(3.38)
γnψ := sup ∂θ ψn± (θ, ω) + ∂ω ψn± (θ, ω) .
(3.39)
θ∈In +ω θ∈In +ω θ∈In +ω
ι | and |∂ I ι | in terms of these quantities (Lemma 3.12) Now we can first control |In+1 ω n+1 and then derive the required analytic estimates later (Lemmas 3.13–3.15).
Lemma 3.12. Suppose that (A1) holds and ω ∈ Fn . Further assume that (I)n and ψ ϕ ( /)n hold and ln > u n . Then (I)n+1 and ( /)n+1 hold, and for all ι = 1, . . . , N , ϕ
ψ
ϕ
ψ
hn + hn un + un
ι ≤ |In+1 | ≤
ψ
ϕ
Hn + Hn ψ
ϕ
ln − u n
(3.40)
and ι |∂ω In+1 | ≤
ϕ
ψ
ϕ
ψ
γn + γn
ln − u n
.
Proof. As f Mn (Aιn ) ⊆ f Mn−1 (Aιn−1 ) and f −Mn (Bnι ) ⊆ Corollary 3.7), ( /)n implies
(3.41) ι f −Mn−1 (Bn−1 ) (see
Jnϕ (an (ω) + ω, ω) ∩ Jnψ (an (ω) + ω, ω) = ∅, Jnϕ (bn (ω) + ω, ω) ∩ Jnψ (bn (ω) + ω, ω) = ∅. j
ϕ
ψ
ϕ
ψ
As |∂θ (ϕni − ψn )| ≥ ln − u n for i, j = ±, and ln − u n > 0 by assumption, this ensures that the intersection has the geometry depicted in Fig. 3.1. Therefore it is obvious that ι Inι contains exactly one connected component In+1 of In+1 , which is not reduced to a
SNA in Quasiperiodically Forced Circle Maps
271
Fig. 3.1. The intersection of f Mn (Aιn ) and f −Mn (Bnι )
single point. Since In+1 is open by definition, this implies the first two statements of ι (ω) = (a (I)n+1 . In addition In+1 n+1 (ω), bn+1 (ω)) is characterised by the equations ϕn+ (an+1 (ω) + ω, ω) = ψn− (an+1 (ω) + ω, ω),
ϕn− (bn+1 (ω) + ω, ω) = ψn+ (bn+1 (ω) + ω, ω), which yields ( /)n+1 . Further, we have
+ − ϕ ψ h ϕn + h ψ n ≤ ψn (an+1 (ω) + ω, ω) − ϕn (an+1 (ω) + ω, ω) ≤ Hn + Hn ,
and − + ϕ ψ lnϕ − u ψ n ≤ ∂θ (ϕn − ψn ) ≤ u n + u n ,
which together implies (3.40). (See Fig. 3.1 for the geometric picture.) In order to prove (3.41), we apply the implicit function theorem to the identity ϕn+ (an+1 (ω) + ω, ω) − ψn− (an+1 (ω) + ω, ω) = 0, and obtain that an+1 is a differentiable function that satisfies ∂ω an+1 (ω) = −
(∂θ + ∂ω )ϕn+ (an+1 (ω) + ω, ω) − (∂θ + ∂ω )ψn− (an+1 (ω) + ω, ω) . ∂θ ϕn+ (an+1 (ω) + ω, ω) − ∂θ ψn− (an+1 (ω) + ω, ω) ϕ
ψ
ϕ
ψ
Therefore (3.41) follows from the definitions of γn , γn , ln and u n , with the same argument applied to bn+1 . Consequently In+1 depends differentially on ω ∈ Fn , and the fact that the set Fn+1 is open follows quite easily from its definition. Thus (I)n+1 (iii) holds as well, and this completes the proof.
It remains to obtain the required estimates on the quantities in (3.31)–(3.39).
272
T. Jäger
Lemma 3.13. Suppose (A1)–(A4) hold and ω ∈ Fn . Then ϕ
ϕ
ψ
ψ
Mn , |C| · αlMn ≤ h n ≤ Hn ≤ |C| · α−
|E| · αu−Mn ≤ h n ≤ Hn ≤ |E| · α+−Mn .
(3.42) (3.43)
Proof. As the vertical size of the sets An and Bn is |C| and |E|, respectively, the lower bounds are a direct consequence of (A2) and the upper bounds follow from Corollary 3.9.
Lemma 3.14. Suppose (A1)–(A7) hold α− < 1 < α+ and ω ∈ Fn . Then −1 −1 − 1) ≤ lnϕ ≤ u ϕn ≤ S + S/(α− − 1), s − S/(α−
uψ n ≤ S/(α+ − 1).
(3.44) (3.45)
Proof. In order to prove (3.44), note that for any L ∈ N and (θ0 , x0 ) ∈ T2 , ∂θ f θL0 +1 (x0 ) = ∂θ f θL1 (x1 ) + ∂x f θL1 (x1 ) · ∂θ f θ0 (x0 ) holds.
(3.46)
By induction, we thus obtain ∂θ f θL0 +1 (x0 ) = ∂θ f θL (xL ) +
L −1
−k ∂x f θLk+1 (xk+1 ) · ∂θ f θk (xk ).
(3.47)
k=0
Now suppose θ ∈ Inι + ω and let (θ0 , x0 ) = (θ − Mn ω, c± ) and L = Mn − 1, such that f θL0 +1 (x0 ) = ϕn± (θ ). Note that thus L coincides with the choice in Lemma 3.4 (see Remark 3.6). By (A5) and (A6) we have
s < ∂θ f θL (xL ) < S. Further, using (3.17) from Corollary 3.9 we obtain that
−1
−(L−k) L−k
≤ α L−k .
∂x f θk+1 (xk+1 ) = ∂x f θL+1 (xL+1 ) −
(3.48)
As |∂θ f θk | ≤ S ∀k by (A5), this yields the required estimates. The proof of (3.45) is slightly more intricate. First of all, similar to (3.47) we obtain that for any R ∈ N and (θ0 , x0 ) ∈ T2 , ∂θ f θ−0 R (x0 ) =
R
∂x f θ−−kR+k (x−k ) · ∂θ f θ−1 (x−k+1 ). −k+1
(3.49)
k=1
Let (θ0 , x0 ) = (θ + Mn ω, e± ) and R = Mn , such that f θ−0 R (x0 ) = ψn± (θ ). Again, this coincides with the choice of R in Lemma 3.4. In order to obtain an estimate on the second factor in the sum in (3.49), we note that 0 = ∂θ f θ−ω ◦ f θ−1 (x) = ∂θ f θ−ω ( f θ−1 (x)) + ∂x f θ−ω ( f θ−1 (x)) · ∂θ f θ−1 (x), such that
−1
∂ f
θ θ−ω ( f θ (x))
−1
.
∂θ f θ (x) ≤
∂x f θ−ω ( f −1 (x)) θ
(3.50)
SNA in Quasiperiodically Forced Circle Maps
273
Therefore ∂θ f θ−1 (x )
will be smaller than −k+1 −k+1 |∂θ f θ−k (x−k )| . αl
|∂θ f θ−k (x−k )| αe
whenever x−k ∈ E and
always smaller than Combining this with (3.19) yields
(x )
∂x f θ−−kR+k (x−k ) · ∂θ f θ−1 −k+1 −k+1
−1
R−k −1
= ∂x f θ−R (x−R ) · ∂θ f θ−k+1 (x−k+1 )
−QR k
≤ αe−1 · αe
−(R−k−QR k )
· αl
(3.51)
· |∂θ f θ−k (x−k )|
≤ α+−(R+1−k) · |∂θ f θ−k (x−k )| ≤ α+−(R+1−k) · S, and summing up over k proves (3.45).
Lemma 3.15. Suppose (A1)–(A7) hold and ω ∈ Fn . Then γnϕ ≤ S ·
∞
k kα−
(3.52)
(k + 1)α+−k .
(3.53)
k=1
and γnψ ≤ S ·
∞ k=1
Proof. For any k, L ∈ N (θ, x) ∈
T2
there holds
k+1 k ∂ω f θ−( L+1)ω (x) = −(L + 1 − k) · ∂θ f θ−(L+1−k)ω ( f θ−(L+1)ω (x)) k k +∂x f θ−(L+1−k)ω ( f θ−( L+1)ω (x)) · ∂ω f θ−(L+1)ω (x).
(3.54)
As in the preceding proof, let (θ0 , x0 ) = (θ − Mn ω, c± ) and L = Mn − 1. Then (3.54) simplifies to ∂ω f θk+1 (x0 ) = −(L + 1 − k) · ∂θ f θk (xk ) + ∂x f θk (xk ) · ∂ω f θk0 (x0 ), 0
(3.55)
and inductive application gives ∂ω f θL0 +1 (x0 ) = −∂θ f θL (xL )−
L −1
−k (L + 1 − k) · ∂x f θLk+1 (xk+1 ) · ∂θ f θk (xk ).
(3.56)
k=0
Combining this with (3.47) and using (3.48) yields
∂θ ϕ ± (θ, ω) + ∂ω ϕ ± (θ, ω) =
∂θ f L+1 (x0 ) + ∂ω f L+1 (x0 )
n n θ0 θ0
L−1
−k =
(L − k) · ∂x f θLk+1 (xk+1 ) · ∂θ f θk (xk )
k=0
≤
L −1
∞
k=0
k=1
L−k (L − k) · α− ·S ≤ S·
This proves (3.52).
k kα− .
(3.57) (3.58)
(3.59)
274
T. Jäger
Now let (θ0 , x0 ) = (θ + Mn ω, e± ) and R = Mn . Similar to (3.56), ∂ω f θ−0 R (x0 )
=
R −1
R+k+1 (R − k) · ∂x f θ−−k−1 (x−k−1 ) · ∂θ f θ−1 (x−k ) holds. (3.60) −k
k=0
Using (3.51) as in the proof of Lemma 3.14 we obtain
∂ω ψ ± (θ, ω) = |∂ω f −R (x0 )| n θ0 ≤
R −1
−(R−k)
(R − k) · α+
k=0
·S ≤ S·
∞
kα+−k .
k=1
Combined with (3.45), this yields (3.53).
3.4. Good frequencies. In order to prove Theorem 2.1, we will have to show that under the hypothesis of the theorem there exists a set ⊆ T1 of positive measure with the property that for any ω ∈ one can find a monotonically increasing sequence (Mn (ω))n∈N0 of positive integers, such that ω ∈ Fn (M0 (ω), . . . , Mn (ω)). n∈N
The problem is that in order to choose the sequences Mn (ω) inductively for a sufficiently large set of ω, we will have to make use of the estimates on the length of the connected components of In in Proposition 3.11. However, these estimates depend in turn on the choice of the sequence (Mn (ω))n∈N0 . In order to overcome this obstacle, we restrict ourselves to choosing the sequences (Mn (ω))n∈N0 from the set M := (Mn )n∈N0 | Mn ∈ [Nn , 2Nn ) ∀n ∈ N0 , where (Nn )n∈N0 is a sequence of positive numbers which is fixed a priori (for simplicity, we do not assume that the Nn are integers). In this way we can verify that all required estimates hold, independent of the particular choice of (Mn (ω))n∈N0 in M. We remark that the statements of this section are completely independent of those in the preceding ones. In fact, they do not even involve the dynamics of the system. We only assume that (In )n∈N0 is a family of subsets of T1 , such that In depends on the integers M0 , . . . , Mn−1 and on ω (as before, we keep this dependence implicit). While we will make use of the notation introduced in Definition 3.2, we do not use the fact that the sets In are dynamically defined as in Definition 3.1. Suppose that the sequences (K n )n∈N0 and (εn )n∈N0 in Definition 3.2 are given and (Nn )n∈N0 is a monotonically increasing sequence of integers which satisfies N0 ≥ 3
and
Nn+1 > 2K n Nn ∀n ∈ N0 .
(N 1)
Then we will use the following inductive assumption. Recall that F−1 = T1 . If n ≥ −1, M j ∈ [N j , 2N j ) ∀ j ∈ [1, n] and ω ∈ Fn (M0 , . . . , Mn ), then (i) (I)n+1 holds ; ι |≤ε (ii) |In+1 n+1 ∀ι ∈ [1, N ]; ι | ≤ 1 ∀ι ∈ [1, N ]. (iii) |∂ω In+1 4
(N 2)
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275
Lemma 3.16. Suppose (N 1) and (N 2) hold and let M j ∈ [N j , 2N j ) be fixed for j ∈ [0, n]. Further assume that ∞ 1 1 < . Kj 6N 2
(K)
j=0
Then for every ω ∈ Fn (M0 , . . . , Mn ) there exists an integer M ∈ [Nn+1 , 2Nn+1 ) such that d((In+1 − (M − 1)ω) ∪ (In+1 + (M + 1)ω), Yn ) > εn . Proof. If j ∈ [0, n] then In+1 ⊆ I j and εn ≤ ε j . Therefore ⎛ ⎞ M j +1 d ⎝In+1 − ( p − 1)ω, I j + kω⎠ ≤ εn
(3.61)
k=−M j +1
implies
⎛
⎞
M j +1
d ⎝I j − ( p − 1)ω,
I j + kω⎠ ≤ ε j .
(3.62)
k=−M j +1
We are going to estimate the number of integers in (Nn+1 , Nn+1 + 2K n Mn ] ⊆ [Nn+1 , 2Nn+1 ) for which (3.62) can happen. Due to (F1)n and (N 2)(ii), for any j ∈ [0, n], ι, κ ∈ [1, N ] and any interval J ⊆ Z of length |J | ≤ 2K j M j , there is at most one p ∈ J such that d(I ιj − ( p − 1)ω, I κj ) ≤ ε j . Hence, there are at most 2M j + 1 integers p in J , such that ⎞ ⎛ M j +1 I κj + kω⎠ ≤ ε j , (3.63) d ⎝ I ιj − ( p − 1)ω, k=−M j +1
and consequently, due to (I)n (i), at most N 2 (2M j + 1) integers p in J , such that ⎛ ⎞ M j +1 d ⎝I j − ( p − 1)ω, I j + kω⎠ ≤ ε j . (3.64) k=−M j +1
Dividing the interval (Nn+1 , Nn+1 + 2K n Mn ] into subintervals of length 2K j M j , plus maybe one shorter, we obtain that the number of p in (Nn+1 , Nn+1 + 2K n Mn ] for which (3.62) holds is bounded by K n Mn 6K n Mn N 2 + 1 N 2 (2M j + 1) ≤ . K j Mj Kj Summing up over all j, this yields that there are at most 2K n Mn · 3N 2 ·
n 1 Kj j=0
p in (Nn+1 , Nn+1 + 2K n Mn ] with d(In+1 − ( p − 1)ω, Yn ) ≤ εn . Repeating this argument yields the same bound for the number of p in (Nn+1 , Nn+1 + 2K n Mn ] with d(In+1 + ( p + 1)ω, Yn ) ≤ εn . Hence, due to (K) there must be at least one integer M ∈ (Nn+1 , Nn+1 + 2K n Mn ] ⊆ (Nn+1 , 2Nn+1 ] with the required property.
The following lemma is taken from [19].
276
T. Jäger
Lemma 3.17. Suppose I = I(ω) consists of exactly N connected components I 1, . . . ,I N of length |I ι | ≤ δ and satisfying |∂ω I ι | ≤ γ < 21 . Then given M ≥ 2 and ε > 0, the set ⎫ ⎧
⎛ ⎞
M ⎬ ⎨
I + jω⎠ ≤ ε ω ∈ T1
d ⎝I, ⎭ ⎩
j=1 δ+ε has measure ≤ 2N 2 M 1−2γ and consists of at most N 2 M 2 connected components.
For any n ∈ N0 let 2 · u n+1 := 64 · N 2 · K n+1 · Nn+1
vn+1 :=
εn+1 , εn
8 2 3 · N 2 · K n+1 · Nn+1 . εn
(3.65) (3.66)
Further, let u 0 := 32N 2 K 0 N0 ε0 and v0 =: 4N 2 K 02 N02 . Lemma 3.18. Suppose (N 1), (N 2) and (K) hold and n ≥ 0. Let M j ∈ [N j , 2N j ) be fixed for j ∈ [0, n] and assume ⊆ Fn (M0 , . . . , Mn ) is an interval. Then for some r ≤ vn+1 there exist disjoint intervals ν ⊆ and numbers M ν ∈ [Nn+1 , 2Nn+1 ), ν = 1, . . . , r , such that ν ⊆ Fn+1 (M0 , . . . , Mn , M ν ) and r Leb(ν ) ≥ Leb() − u n+1 .
(3.67) (3.68)
ν=1 n intervals κ of length ≤ Nεn+1 . Proof. Obviously can be divided into at most 2Nεn+1 n κ κ For each κ, let ω be the midpoint of . According to Lemma 3.16, there exist integers M κ ∈ [Nn+1 , 2Nn+1 ), such that
d((In+1 − (M κ − 1)ωκ ) ∪ (In+1 + (M κ + 1)ωκ ), Yn ) > εn . As M j < 2N j < Nn+1 ∀ j ∈ [0, n] and |∂ω I kj | ≤ γ ≤
1 4
∀k, j we obtain
d((In+1 − (M κ − 1)ω) ∪ (In+1 + (M κ + 1)ω), Yn ) > 0 ∀ω ∈ κ . Thus (F2)n+1 holds for all ω ∈ κ . Let ˜ κ be the set of those ω’s in κ that satisfy (F1)n+1 . We have to estimate the size and the number of connected components of ˜ κ . However, since it follows from (N 2)(i) and (ii) that In+1 consists of N connected components of length ≤ εn+1 and 1 ι | ≤ 1 ∀ι ∈ [1, N ] by (N 2)(iii), Lemma 3.17 with δ = ε |∂ω In+1 n+1 , ε = 3εn+1 , γ = 4 4 and M = 2K n+1 Nn+1 yields Leb( k \ ˜ k ) ≤ 32N 2 K n+1 Nn+1 εn+1 , 2 N 2 . Summing up and the number of connected components of ˜ κ is at most 4N 2 K n+1 n+1 over κ yields the statement.
n Let V−1 := 1 and Vn := i=0 vi ∀n ≥ 0.
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277
Proposition 3.19. Suppose (N 1), (N 2) and (K) hold and σ := 1 −
∞
Vn−1 u n .
(3.69)
n=0
Then there exists a set ⊆ T1 of measure Leb() ≥ σ , such that for each ω ∈ there exists a sequence (Mn (ω))n∈N0 with the property that Fn (M0 (ω), . . . , Mn (ω)). (3.70) ω ∈ n∈N0
Proof. We are going to construct a nested sequence of sets T1 ⊇ 0 ⊇ 1 ⊇ . . . with the following properties: ρ
(i) n consists of ρn ≤ Vn disjoint open intervals 1n , . . . , n n . n (ii) Leb(n ) ≥ 1 − i=0 Vi−1 u i . (iii) For each i = 1, . . . , ρn there exist numbers M0n,i , . . . , Mnn,i such that in ⊆ Fn (M0n,i , . . . , Mnn,i ). (iv) For each k ≤ n and each i ∈ [1, ρn ] there exists a unique κ ∈ [1, ρk ] such that k,κ in ⊆ κk and M n,i j = M j ∀ j = 0, . . . , k. For n = 0 we choose M0 ∈ [N0 , 2N0 ) arbitrary and let 0 = F0 . Recall that this is the set of all ω which satisfy condition (F1)0 , and the fact that this set has all required properties can be deduced from Lemma 3.17. Now suppose 0 , . . . , n with the above properties exists. Then for each i ∈ [1, ρn ] we can apply Lemma 3.18 to the component in and obtain a union of at most vn+1 intervals with overall measure ≥ m(in ) − u n+1 . Doing this for all the at most Vn components of n yields the required set n+1 , with at most Vn+1 = vn+1 · Vn connected n+1 Vi−1 u i . Thus (i) and (ii) hold for n + 1, (iv) is components and measure ≥ 1 − i=0 obvious from the construction and (iii) follows from (3.67). As the sets n form a nested sequence, their intersection has measure ≥ σ . Further, for any ω ∈ and n ∈ N there exists a unique i n ∈ [1, ρn ] with ω ∈ inn . If we let Mn (ω) = Mnn,in , then due to property (iv) we obtain (3.70).
3.5. Proof of Theorem 2.1, Part A: Existence of SNA. Suppose that the assumptions of Theorem 2.1 hold. First of all, we choose the sequence K n in a way that allows to obtain a lower bound on the asymptotic expansion and contraction rate, namely 1
−1 min{α− , α+ } ≥ α p .
In order to do so, we fix t ∈ N sufficiently large, such that t ≥ 4 and 2 2−t+2 p +2 . ≤ log N2 p2 + 1 Then we let K n := 2n+t N 2 . Note that this choice satisfies (K). We obtain ∞ ∞ 1 1 p2 + 1 1− ≥ exp −2 , β = ≥ 2 Kn Kn p +2 n=0
n=0
(3.71)
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T. Jäger
and this implies 2
−1 α− = αp
β− p(1−β)
1
≥ αp.
1
Similarly we obtain α+ ≥ α p , such that (3.71) holds. Now let N0 := 3 and Nn+1 := α Nn /16 p . As the sequence Nn grows super-exponentially, (N 1) holds whenever α is sufficiently large. Further, let N
ε0 := min |I0ι | ι=1
and
εn :=
2 −Nn−1 / p ·α . s
Again, if α is sufficiently large, then on the one hand εn ≥ 3εn+1 ∀n ∈ N0 (which is the only requirement on the sequence (εn )n∈N in Definition 3.2), and on the other hand (3.20) and (3.21) hold. Therefore we can apply Proposition 3.11 to see that (N 2) holds for the sets In given by Definition 3.1. This means that all assumptions of Proposition 3.19 are met, and we obtain a set ⊆ T1 of measure Leb() ≥ 1 −
∞
Vn−1 u n ,
(3.72)
n=0
with the property that for all ω ∈ there exists a sequence (Mn (ω))n∈N , such that ω ∈ n∈N0 Fn (M0 (ω) . . . Mn (ω)). Proposition 3.10 then implies that for all ω ∈ the system f (θ, x) = (θ + ω, f θ (x)) has a sink-source-orbit and hence a SNA and SNR. It remains to estimate the size of , i.e. to obtain a lower bound on the right side of (3.72). In all of the following estimates we assume that α is chosen sufficiently large, such that the sequence Nn grows sufficiently fast, and indicate the steps in which this fact is used by placing (α) over the respective inequality signs. For any n ∈ N0 we have 2 · u n+1 = 64N 2 · K n+1 · Nn+1
εn+1 εn (α)
= 64N 2 · K n+1 · α Nn /8 p−Nn / p+Nn−1 / p ≤ α −3Nn /4 p and vn+1 =
8 2 3 · N 2 · K n+1 · Nn+1 εn (α)
2 ≤ 4s · N 2 · K n+1 · α Nn−1 / p+3Nn /16 p ≤ α Nn /4 p .
Now note that (α)
V0 = v0 = N 2 · K 02 · N02 ≤ α N0 /4 p . Further, if we suppose that Vn ≤ α Nn /4 p ,
(3.73)
then (α)
Vn+1 = Vn · vn+1 ≤ α Nn /4 p+Nn /4 p ≤ α Nn+1 /4 p .
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279
Consequently, by induction, (3.73) holds for all n ≥ 1. We conclude that Vn u n+1 ≤ α −Nn /2 p and 1−
∞
Vn−1 u n ≥ 1 − u 0 −
n=0
∞
α −Nn /2 p .
(3.74)
n=0
As u 0 = 32N 2 K 0 N0 ε0 → 0 if ε0 → 0, the right side goes to 1 as α → ∞ and ε0 → 0. To summarise, this means that we can choose constants α0 and c0 in such a way that all the assumptions on α used above hold and (3.74) is larger than 1−δ whenever α > α0 and ε0 < c0 . Then Leb() ≥ 1 − δ, and for all ω ∈ Property (∗1) in Theorem 2.1 will be satisfied. 3.6. Proof of Theorem 2.1 B: Uniqueness of the invariant graphs. We choose α0 and c0 as at the end of the preceding section, and assume that α ≥ α0 and ε0 ≤ c0 . Further, we fix ω ∈ and the sequence (Mn )n∈N = (Mn (ω))n∈N and let f (θ, x) = (θ + ω, f θ (x)) as before. Recall that Mn ∈ [Nn , 2Nn ) and Nn+1 = α Nn /16 p . We start with some preliminary remarks and estimates. Let εn and Nn be chosen as in the last section. Since α ≥ α0 and ω ∈ Fn ∀n ∈ N0 , the assumptions of Proposition 3.11 are satisfied for all n ∈ N0 . Consequently, for all n ≥ 0 the statements (I)n and ( /)n hold and |Inι | ≤ εn = Let := T1 \
n∈N0
2 −Nn−1 / p ·α ∀ι ∈ [1, N ]. s
Zn . Then
Leb() = 1 −
∞
m(Zn \ Zn−1 ) ≥ 1 −
n=0
= 1 − 4N0 ε0 −
∞
4Nn εn
n=0 ∞ 8 −15Nn−1 /16 p · α . s
(3.75)
n=1
Further, let V :=
∞
Vn =
n=0
j +1 ∞ M
(I j + kω).
j=0 k=1
We obtain Leb(V) ≤
∞
2N j ε j ≤
j=0
∞ 4 −15Nn−1 /16 p · α . s
(3.76)
j=0
By increasing α0 and reducing c0 further if necessary, we may assume that for all α > α0 and ε0 < c0 , Leb() > 1 −
1 holds. 4(1 + p 2 )
(3.77)
280
T. Jäger
and L := Leb(V) ≤
1 hold. p2 + 2
(3.78)
In this case, the set ∩ ( − ω) ∩ ( + ω) has measure ≥ 1 − 1/(1 + p 2 ). Consequently, ˆ ⊆ ∩ ( − ω) ∩ ( + ω) of the same measure with the additional there exists a set ˆ property that for all θ ∈ , n−1 1 1V (θ + iω) = Leb(V) holds. n→∞ n
lim
(3.79)
i=0
ˆ and x ∈ ˆ ⊆ ⊆ Z c ⊆ I c , (A1) implies that Now assume θ ∈ / E. Since 0 0 x0 := f θ (x) ∈ C. Furthermore θ0 := θ + ω ∈ . This means that condition (B1)n in Lemma 3.4 holds for all n ∈ N, and consequently θm ∈ / V implies xm ∈ C for all m ∈ N. Using (3.79) together with (A2) and (A4), this yields that λ+ (θ, x) ≤ (L p − 2(1 − L)/ p) log α
(3.78)
≤
− log α/ p.
(3.80)
ˆ E c have strictly negative forward Lyapunov exponent. In In other words, all points in × ˆ × C c have strictly negative backward the same way, we may assume that all points in Lyapunov exponent (using an additional largeness assumption on α, which ensures that 2 the set W := ∞ n=0 Wn has measure ≤ 1/( p + 2)). − Now, suppose for a contradiction that ϕ is multi-valued, say ϕ − = (ϕ1− , . . . , ϕn− ). Note that µϕ − -almost all points have negative forward and positive backward Lyapunov ˆ and all exponents, and the converse is true for µϕ + . Consequently, for almost all θ ∈ i = 1, . . . , n, we must have ϕi− (θ ) ∈ E c . However, since points of ϕ − and ϕ + must ˆ × E c ) > 0, contradicting the alternate on the fibres [21], this would imply that µϕ + ( c fact that all points in × E have negative forward Lyapunov exponent. Similarly, assume that there exists a third invariant graph ψ. Without loss of generality, we may assume λ(ψ) < 0. (Note that λ(ψ) = 0 is not possible, since either ˆ × E c ) > 0, which implies λ(ψ) < 0, or µψ ( ˆ × C c ) > 0, which implies µψ ( − − λ(ψ) > 0.) In this case, both sets (ψ, ϕ ) and (ϕ , ψ) must contain an invariant graph with non-negative Lyapunov exponent [27, Cor. 3.4]. (In fact, such invariant graphs are given by the endpoints of the intervals Wθ defined in Remark 2.2.) Now, since ˆ × E) and µψ ( ˆ × E) must both be equal to zero, at least one of these invariµϕ − ( ant graphs intersects × E c on a set of positive measure, which again contradicts the negative Lyapunov exponents in this set. This proves property (∗2). Remark 3.20. (a) The above proof also yields an upper bound for the Lyapunov exponent of ϕ − (and an upper one for that of ϕ + ), provided by (3.80). In fact, for sufficiently large α this bound could be chosen arbitrarily close to log αc = −2 log α/ p. ˆ × (T1 \ (E ∪ C)) (b) Note that none of the invariant graphs can intersect the set on a set of positive measure, since all points in this set have negative Lyapunov exponents both forwards and backwards in time.
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281
3.7. Proof of Theorem 2.1, Part C: Minimality. We use the same notions and choose α0 and c0 as in the preceding sections, and suppose α ≥ α0 and ε0 ≤ c0 . Further, we let S ∗ := S +
S −1 α− −1
and choose a constant > 1 with the following property: If ⊆ T2 is the graph of a differentiable curve γ : I → T1 , defined on an interval I ⊆ T1 , and has slope at most S ∗ , then f n () has slope at most S ∗ · n . Due to the lower bound in (3.40) and the estimates provided by Lemmas 3.13 and 3.14, there exist constants B > 0 and λ > 0, such that for any n ≥ 0 and any connected component Inι of In , |Inι | ≥ B · λ−Nn−1 =: δn holds. Since
Leb
∞
Vk \ Vn
≤
k=n+1
∞ k=n+1
∞ 4 −15Nk−1 /16 p ·α (Mk + 1) · εk ≤ , s k=n+1
and due to the super-exponential growth of the sequence Nn , there exists n 0 ≥ 0, such that ∞ Leb Vk \ Vn < δn /2 Mn−1 ∀n ≥ n 0 . (3.81) k=n+1
By slightly reducing the set if necessary, it is therefore possible to find a set ∗ ⊆ with the following properties: (∗ 1) Leb(∗ ) > 1 − 2(1+1 p2 ) ; (∗ 2) For any θ ∈ ∗ , any n ≥ n 0 and any ι ∈ [1, N ], !the forward orbit" {θ + nω | ∞ n ≥ 0} is δn / Mn−1 -dense in (Inι − (Mn − 1)ω) \ k=n+1 Vk \ Vn . Now we come to the key point of the proof. The crucial observation is the fact that there is a large set of points with dense orbit - minimality will then follow by rather general arguments. More precisely, we prove the following: Claim 3.21. Suppose θ0 ∈ ∗ ∩ (∗ − ω) and x0 ∈ E c . Then the forward orbit of (θ0 , x0 ) is dense in T2 . Proof. For any point (θ0 , x0 ) ∈ T2 , denote its forward orbit by O+ (θ0 , x0 ) := {(θk , xk ) | k ≥ 0}. Suppose θ ∈ ∗ ∩ (∗ − ω) and x ∈ E c . Since ∗ ⊆ ⊆ Z0c ⊆ I0c , we can use (A1) to see that f θ (x) ∈ C. Therefore, it suffices to show that the forward orbit of any point (θ0 , x0 ) with θ0 ∈ ∗ and x0 ∈ C is dense. Fix such θ0 and x0 and any ι ∈ [1, N ]. Further, choose n 0 as in (3.81). We proceed in four steps: Step 1: If n ≥ n 0 , then π1 (O+ (θ0 , x0 ) ∩ Aιn ) is δn / Mn−1 -dense in Inι − (Mn − 1)ω. Since θ0 ∈ ∗ , it is not contained in Zk for any k ∈ N0 . Hence, it follows from Lemma 3.4 that xm ∈ / C implies θm ∈ Vk for some k ∈ N0 . Now Inι − (Mn − 1)ω is disjoint from Vn by (F1)n and (F2)n . Therefore θm ∈ Inι − (Mn − 1)ω and xm ∈ / C ∞ ι − (M − 1)ω) \ imply θ ∈ V \ V . In other words, x ∈ C whenever θ ∈ (I k k n m m n n k=n+1 " ! ∞ ∗ ∗ k=n+1 Vk \ Vn . The statement follows from property ( 2) of the set .
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T. Jäger
Step 2: There exists an integer n 1 ≥ n 0 , such that for all n ≥ n 1 the set π2 (O+ (θ0 , x0 )∩ ι + (M + 1)ω) × T1 ) is 2−n -dense in E. Let n ≥ n . With the notation of Sect. 3.3, (In+1 n 0 we have − ι + f Mn+1 (Aιn+1 ) = {(θ, x) | θ ∈ In+1 + ω, x ∈ [ϕn+1 (θ ), ϕn+1 (θ )]}.
Due to the estimates (3.42) and (3.44), this set is a small strip2 of vertical size at most α −Mn+1 / p and slope at most S ∗ . As described in the proof of Lemma 3.12, this strip crosses the strip f −Mn (Bnι ) from below to above (assuming again that the crossing is upwards), see Fig. 3.1. This implies that A := f Mn+1 +Mn (Aιn+1 ) crosses the horizontal strip Bnι = (Inι + (Mn + 1)ω) × E in the same way. From (A2) and αu = α p , it follows that A has vertical size at most α −Mn+1 / p+Mn p . Further, it has slope at most S ∗ · Mn . Since π1 (O+ (θ0 , x0 ) ∩ Aιn+1 ) is δn+1 / Mn -dense ι in In+1 − (Mn+1 − 1)ω by Step 1, it follows that π2 (A) is dn -dense in E, where dn = S ∗ · δn+1 + α −Mn+1 / p+Mn p . Given the super-exponential growth of the sequence Nn and Mn , there exists n 1 ≥ n 0 , such that dn ≤ 2−n ∀n ≥ n 1 . This completes Step 2. Step 3: cl(O+ (θ0 , x0 )) contains a vertical segment {ζ } × E for some ζ ∈ − ω. Due to compactness and since the size of the intervals Inι goes to zero as n goes to infinity, there exists a strictly increasing sequence (n i )i∈N of integers and a point ζ ∈ T1 , such that the intervals Inι i +1 + (Mn i + 1)ω converge to {ζ } in Hausdorff distance. It follows from Step 2 that {ζ } × E ⊆ cl(O+ (θ0 , x0 )). (F1)n and (F2)n imply that Inι + (Mn + 1)ω is contained in Znc − ω for all n ∈ N0 . c Since the sets n − ω form a nested sequence of compact sets, it follows that ζ is Z ∞ contained in i=0 Znci − ω = − ω. Step 4: O+ (θ0 , x0 ) is dense in T2 . Let x ± := f ζ (e± ). Since − ω is disjoint from I0 ⊆ Z0 − ω, (A1) implies x ± ∈ C. As ζ + ω ∈ , this means that (B1)n holds for all (ζ + ω, x) with x ∈ [x + , x − ] and all n ∈ N0 . Let L be the smallest positive integer such that ζ + (L + 1)ω ∈ In . Then we can use (C3)n together with (A2) and (A4) to conclude that L ∂x f ζL+ω (x) ≤ α− ∀x ∈ [x + , x − ]. L. It follows that f L ({ζ + ω} × [x + , x − ]) is a vertical segment of size smaller than α− Since L ≥ Mn − 1 (due to ζ + ω ∈ / Zn ) and n was arbitrary, this means that the length of the corresponding iterates of {ζ + ω} × [x + , x − ] goes to zero as n goes to infinity. Therefore, the orbit of the segment {ζ + ω} × [x − , x + ] = f ({ζ } × E) is dense in T2 . Since {ζ } × E ⊆ cl(O+ (θ0 , x0 )) by Step 3, this completes the proof of Step 4 and the claim.
The preceding claim implies in particular that f is topologically transitive. It follows from Proposition 1.3 that there is a unique minimal set M. Obviously, M cannot be a continuous invariant curve with positive Lyapunov exponent. Therefore, it follows from 2 By ‘strip’, we just mean a set which is the region between two continuous curves, defined on a subinterval of T1 . By the slope of a strip we mean the slope (or derivative) of its boundary curves.
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283
[22] that M must support at least one f -invariant measure µ with non-positive vertical Lyapunov exponent, that is λ(µ) := ∂x log f θ (x) dµθ (x) dθ ≤ 0. (3.82) T1 T1
(Here µθ are the conditional measures on the fibres, defined by the relation T2 f dµ = 0 2 T1 T1 f (θ, x) dµθ (x) dθ ∀ f ∈ C (T , R).) We claim that this is only possible if M intersects (∗ ∩ (∗ − ω)) × E c . In order to see this, note that due to (∗ 1) the set ∗ ∩ (∗ − ω) has measure > 1 − 1/(1 + p 2 ). If M is disjoint from ( ∩ ( − ω)) × E c , then supp(µ) ⊆ M together with (A2) and (A3) imply that λ(µ) >
1−
1 1 + p2
· log(α 1/ p ) −
1 · log(α p ) = 0, 1 + p2
contradicting (3.82). It follows that M intersects (∗ ∩ (∗ − ω)) × E c , and since all points from the latter set have dense orbits by Claim 3.21 we obtain M = T2 .
3.8. Proof of Corollary 2.3. Obviously, we just have to check that assumptions (A1)– (A7) of Theorem 2.1 with αl−1 = αu = α p are satisfied for large α. Here p is meant to be the same as in (2.2). In all of the following, we assume that α is chosen sufficiently large and just indicate by (α) whenever this fact is used. Now, due to (2.5), there exist ε > 0 and s > 0, such that |g (θ )| > s ∀θ ∈ g −1 (Bε (1/2)). We let I0 := g −1 (Bε (1/2)), such that (A6) holds by definition. Note that due to (2.4), I0 is the disjoint union of a finite number of open intervals. In addition, by reducing ε further if necessary, we can assume that all connected components have length smaller than ε0 , where ε0 = ε0 (δ, p, s, S, N ) from Theorem 2.1 with S := maxθ∈T1 |g (θ )|. Note that this choice of S automatically implies (A5). − 2 p−1
Further, we define e± := ±α 2 p and c± := ∓ε/2, and let E = [e− , e+ ] and C = [c− , c+ ] as before. Then for large α we have h α (T1 \ E) ⊆ Bε/2 (1/2), since h α (e± ) = ±π
a p (α 1/2 p ) 2a p (α/2)
α→∞
−→
1 . 2
/ I0 , such that (A1) holds. Similarly, the above Consequently f θ (T1 \ E) ⊆ C ∀θ ∈ choices imply that (A7) holds (provided we take ε < 21 ). Further, for any (θ, x) ∈ T2 we have ∂x f θ (x) = h α (x) ≥ h α (1/2) α · a p (α/2) α · (1 + (α/2) p )−1 (α) − p = = ≥ α , 2a p (α/2) 2a p (α/2) (α) α ∂x f θ (x) = h α (x) ≤ h α (0) = ≤ α p. a p (α/2)
284
T. Jäger
Thus (A2) holds. Finally, we check (A3) and (A4). Suppose x ∈ E. Then ∂x f θ (x) ≥ h α (x) =
αa p (αe) 2a p (α/2)
=
α · (1 + α 1/2 )−1 (α) 1/ p ≥ α . 2a p (α/2)
Similarly, if x ∈ C there holds α · (1 + (αε) p )−1 (α) −1/ p ≤ α . 2a p (α/2)
∂x f θ (x) ≤ h α (ε) =
Hence, all the assumptions of Theorem 2.1 are satisfied for sufficiently large α.
4. Proof of the Refined Statement for the qpf Arnold Circle Map 4.1. Proof of Theorem 2.5. In this section, we describe how the basic construction has to be modified in order to prove Theorem 2.5. In fact, only minor changes are needed. The only thing which has to be done is to improve some of the estimates in Sect. 3.3, taking advantage of the additional assumption (A8), and then adapt the proof from Sect. 3.5 accordingly. We remark that all results of Sects. 3.2 and 3.3 only depend on the assumptions (A1)–(A7) and not on the fact that the parameter α is chosen very large. Therefore, they all apply in the situation of Theorem 2.5. Similarly, we can still use all results of Sect. 3.4, since these were completely independent of the dynamics. First of all, we slightly modify the definition of the sets Fn : We replace condition (F1)0 by d
I0 ,
2K 0 M0
(I0
+ kω)
(F1 )0
> 3ε0
k=1
and define Fn as the set of all frequencies ω ∈ T1 which satisfy (F1 )0 , (F2)0 and (F1-2) j ∀ j = 1, . . . , n. Since I0 ⊆ I0 , condition (F1 )0 is stronger than (F1)0 , which means that Fn ⊆ Fn . Consequently, all the results from Sects. 3.2–3.4 remain true if Fn is replaced by Fn in the respective statements. Since the expansion and contraction rates in Theorem 2.5 are fixed, we have to improve the estimates from Sect. 3.3, by using the strengthened condition (F1 )0 together with the additional assumption (A8). As the proofs are just slight variations of those in Sect. 3.3, we keep the exposition rather brief and only describe the necessary modifications. First of all, Lemma 3.14 will be replaced by the following. Lemma 4.1. Suppose (A1)–(A8 ) hold, α− < 1 < α+ and ω ∈ Fn . Then s−
M0 S s + α− −1 α− −1
≤ lnϕ ≤ u ϕn ≤ S +
uψ n ≤
s + α+M0 S . α+ − 1
M0 S s + α− −1 α− −1
,
(4.1) (4.2)
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285
Proof. As in the proof of Lemma 3.14, we fix θ ∈ Inι + ω and first let (θ0 , x0 ) = (θ − Mn ω, c± ) and L = Mn − 1, such that f θL0 +1 (x0 ) = ϕn± (θ ). We obtain ∂θ ϕn± (θ )
∂θ f θL0 +1 (x0 )
= (3.47)
=
L −1
∂θ f θL (xL ) +
−k ∂x f θLk+1 (xk+1 ) · ∂θ f θk (xk )
k=0 (F 1 )0
≥
≥
s−
s
L −1
L−k α− s −
L−M 0 −1
k=L−M0 s + α M0 S − −1 − . α− − 1
L−k α− S
k=0
The second estimate in (4.1) follows in the same way. In order to prove (4.2), we can proceed similarly: We let (θ0 , x0 ) = (θ + (Mn + 1)ω, e± ) and R = Mn , such that f θ−0 R (x0 ) = ψn± (θ ), and obtain the required estimate from (3.49) and (3.51) by using (F1 )0 once more.
Next, we derive an improved version of Lemma 3.15. Lemma 4.2. Suppose (A1)–(A8 ) hold and ω ∈ Fn . Then γnϕ
≤s ·
γnψ ≤ s ·
∞ k=1 ∞ k=1
k kα−
+S·
∞
k kα− ,
(4.3)
k=M0 +1
(k + 1)α+−k + S ·
∞
(k + 1)α+−k .
(4.4)
k=M0 +1
Proof. The proof is almost identical to that of Lemma 3.15. For proving the upper ϕ bound on γn , the only difference is that (A8) is used instead of (A5) in order to estimate |∂θ f θk (xk )| in the last M0 terms of the sum in (3.58). ψ Similarly, the improved bound on γn is obtained by using (A8) instead of (A5) when the last M0 terms of the sum on the right side of (3.60) are estimated via (3.51).
Lemma 3.12 can be used without any modifications. Consequently, we arrive at the following conclusion, whose proof is identical to that of Proposition 3.11. Proposition 4.3. Suppose (A1)–(A8 ) hold, α− < 1 < α+ and ω ∈ Fn (M0 , . . . , Mn ) for some n ≥ 0. Further, assume that
+ α M0 S
+ α −M0 S s s s + − S := s − + (4.5) ≥ −1 α − 1 2 α− − 1 + and γ := s ·
∞ ∞ S k k . kα− kα− + (k + 1)α+−k + S · + (k + 1)α+−k ≤ 4 k=1
k=M0 +1
(4.6)
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Then (I)n+1 holds and for all ι = 1, . . . , N we have ι |In+1 |≤
2 · max{α− , α+−1 } Mn , s 1 ι |∂ω In+1 |≤ . 4
(4.7) (4.8)
In order to complete the proof of Theorem 2.5, we now choose the sequence (K n )n∈N0 −1 as in the proof of Theorem 2.1, such that α− , α+ ≥ α 1/ p . Further, we let N0 be the smallest integer larger than d 1/4 . In all of the following, we assume that d is chosen sufficiently large to ensure all the required estimates. As before, we define the sequence (Nn )n∈N recursively by Nn+1 = α Nn /16 p and let N
ε0 := min |J ι | ι=1
and
εn :=
2 −Nn−1 / p ·α , s
˜
where J 1 , . . . , J N are the connected components of I0 as in Sect. 2.2. If d0 (and consequently N0 ) is chosen large enough, then (N 1) holds and εn ≥ 3εn+1 ∀n ∈ N. Further, (4.5) and (4.6) hold if d0 is large and s /s is small (note that the product α −M0 S ≤ α −N0 S decays super-exponentially as d is increased). Thus (N 2) holds by Proposition 4.3. Therefore, we can apply Proposition 3.18 and obtain Leb() ≥ 1 −
∞
Vn−1 u n .
n=0
From now on the proof is identical to the one of Theorem 2.1, with the only difference that the largeness condition on α is replaced by a largeness condition on d (and thus N0 ) in all the respective estimates. In this way, we obtain Leb() ≥ 1 − u 0 −
∞
α −Nn /4 p .
n=0
If d goes to infinity, then due to (2.9) and the choice of N0 the right side tends to 1 (recall that u 0 = 32N 2 K 0 N0 ε0 ). The uniqueness of the invariant graphs and the minimality can now be proved exactly in the same way as in Sects. 3.6 and 3.7. The only thing which has to be noted is that the estimate in (3.77) also holds for fixed α, provided N0 ≈ d 1/4 is chosen sufficiently large. Hence, we can find constants c0 and d0 with the required property, which completes the proof.
4.2. Proof of Corollary 2.6. We place ourselves under the hypothesis of the corollary and let f θ (x) := h(x) + βgd (θ ), where gd (θ ) = cos(2π θ )d . Let C and E be chosen as in (2.10) and (2.11). First of all, we fix some α > 1 and choose p ∈ N such that supx∈C h (x) ≤ α −2/ p , inf x∈E h (x) > α 2/ p , and in addition h (x) ∈ (α − p , α p ) ∀x ∈ T1 . Then f satisfies (A2)–(A4).
SNA in Quasiperiodically Forced Circle Maps
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Let ε := 21 d(h(T1 \ E), T1 \ C) and suppose β ∈ [1 − ε, 1 + ε]. Define I0 := gd−1 ([−1 + ε, −ε] ∪ [ε, 1 − ε]). Then it is easy to see that ( f θ )θ∈T1 satisfies (A1) and (A7). Further, since |∂θ f θ (x)| = |βgd (θ )| = |2πβd · cos(2π θ )d−1 · sin(2π θ )| < 4π d, we can choose S in (A5) smaller than 4π d. Next, we check that s in (A6) can be chosen in accordance with (2.8). In order to obtain an estimate for gd on I0 , we check the endpoints of the connected components and the points where gd
(θ ) = 0. Due to the symmetry of gd , we can restrict to [0, 1/4]. √ First, assume gd (θ ) = ε. Then cos(2π θ ) = ε1/d and sin(2π θ ) = 1 − ε2/d . Hence # gd (θ ) = −2πβd · ε(d−1)/d 1 − ε2/d . Since a y = 1 + ln(a)y + O(y 2 ) we have # √ # d 1 − ε2/d = 2 ln ε + O(1/d), such that for sufficiently large d, |gd (θ )| > ε ·
# √ ln(ε) · d holds.
1/d # Secondly, assume that gd (θ ) = 1 − ε. Then cos(2π θ ) = (1 − ε) and sin(2π θ ) = 1 − (1 − ε)2/d . Thus # gd (θ ) = −2πβd · (1 − ε)(d−1)/d 1 − (1 − ε)2/d .
Similarly as above we conclude that for sufficiently large d, # √ |gd (θ )| > (1 − ε) · ln(1 − ε) · d holds. Thirdly, assume that gd
(θ ) = 0. In this case sin(2π θ )2 = 1/d and cos(2π θ )2 = (d − 1)/d. Therefore √ d − 1 (d−1)/2 1 1 (d−1)/2 gd (θ ) = −2πβd , √ = −2πβ d 1 − d d d and the last factor is bounded for all d. From the above analysis we conclude that √ there is
a constant A, depending only on ε, such that for all sufficiently large d, gd (θ ) > d/A, holds for all θ ∈ I0 . Finally, we let I0 := B √1 (0) ∪ B √1 ( 21 ). Since cos(2π θ ) ≤ 1 − |θ |2 in a neighbour3d
3d
hood of 0, we obtain that for any θ ∈ [0, 41 ] \ I0 , d d→∞ |gd (θ )| ≤ 1 − d −2/3 −→ 0 holds. By symmetry, the same estimate holds on all of T1 \ I0 . Therefore I0 ⊆ I0 for large d. Similarly, we obtain that for any θ ∈ T1 \ I0 , d−1 d→∞ −→ 0 holds. |gd (θ )| ≤ 2πβd 1 − d −2/3
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Consequently, we can choose s in (A8) as a fixed constant, independent of d, which implies that s /s converges to 0 as d is increased. This shows that for sufficiently large d all assumptions of Theorem 2.5 are satisfied, which completes the proof of the corollary. Acknowledgements. I would like to thank Kristian Bjerklöv for inspiration and stimulating discussions, Bassam Fayad for asking the question which led me to including property (∗2) in Theorem 2.1 and J.-C. Yoccoz and the Collège de France for their hospitality during a two-year visit. Finally, I would like to thank the three referees whose thoughtful remarks helped to improve the paper. This work was supported by a research fellowship Ja1721/1-1 of the German Research Council (DFG).
References 1. Grebogi, C., Ott, E., Pelikan, S., Yorke, J.A.: Strange attractors that are not chaotic. Physica D 13, 261–268 (1984) 2. Keller, G.: A note on strange nonchaotic attractors. Fundamenta Mathematicae 151(2), 139–148 (1996) 3. Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. (2) 133(1), 73–169 (1991) 4. Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, 177–195 (1985) 5. Million˘sc˘ ikov, V.M.: Proof of the existence of irregular systems of linear differential equations with quasi periodic coefficients. Differ. Uravn. 5(11), 1979–1983 (1969) 6. Vinograd, R.E.: A problem suggested by N.R. Erugin. Differ. Uravn. 11(4), 632–638 (1975) 7. Herman, M.: Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d’un théorème d’Arnold et de Moser sur le tore de dimension 2. Comment. Math. Helv. 58, 453–502 (1983) 8. Jorba, A., Núñez, C., Obaya, R., Tatjer, J.C.: Old and new results on SNAs on the real line. Int. J. Bifur. Chaos 17(11), 3895–3928 (2007) 9. Haro, A., Puig, J.: Strange non-chaotic attractors in Harper maps. Chaos 16, 033/27 (2006) 10. Prasad, A., Negi, S.S., Ramaswamy, R.: Strange nonchaotic attractors. Int. J. Bif. Chaos 11(2), 291–309 (2001) 11. Puig, J.: A nonperturbative Eliasson’s reducibility theorem. Nonlinearity 19, 355–376 (2006) 12. Avila, A., Krikorian, R.: Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. Math. 164(2), 911–940 (2006) 13. Avila, A., Jitomirskaya, S.: The Ten Martini Problem. To appear in Ann. Math. (2)., available at http:// annais.math.princeton.edu/issues/2006/FinalFiles/AvilaJitomirskayaFinal.pdf 14. Jäger, T.: The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations. To appear in Mem. Am. Math. Soc., 2008 15. Ding, M., Grebogi, C., Ott, E.: Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5), 2593–2598 (1989) 16. Feudel, U., Kurths, J., Pikovsky, A.: Strange nonchaotic attractor in a quasiperiodically forced circle map. Physica D 88, 176–186 (1995) 17. Stark, J., Feudel, U., Glendinning, P., Pikovsky, A.: Rotation numbers for quasi-periodically forced monotone circle maps. Dyn. Syst. 17(1), 1–28 (2002) 18. Glendinning, P., Feudel, U., Pikovsky, A., Stark, J.: The structure of mode-locked regions in quasi-periodically forced circle maps. Physica D 140, 227–243 (2000) 19. Bjerklöv, K.: Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations. Erg. Theory Dyn. Syst. 25, 1015–1045 (2005) 20. Young, L.-S.: Lyapunov exponents for some quasi-periodic cocycles. Erg. Theory Dyn. Syst. 17, 483–504 (1997) 21. Furstenberg, H.: Strict ergodicity and transformation of the torus. Am. J. Math. 83, 573–601 (1961) 22. Stark, J., Sturman, R.: Semi-uniform ergodic theorems and applications to forced systems. Nonlinearity 13(1), 113–143 (2000) 23. Bjerklöv, K., Jäger, T.: Rotation numbers for quasiperiodically forced circle maps – Mode-locking vs strict monotonicity. To appear in J. Am. Math. Soc., doi:10.1090/s0894-0347-08-00627-9,od. 2008 24. Jäger, T., Stark, J.: Towards a classification for quasiperiodically forced circle homeomorphisms. J. Lond. Math. Soc. 73(3), 727–744 (2006) 25. Jäger, T., Keller, G.: The Denjoy type-of argument for quasiperiodically forced circle diffeomorphisms. Erg. Theory Dyn. Syst. 26(2), 447–465 (2006) 26. Béguin, F., Crovisier, S., Jäger, T., LeRoux, F.: Denjoy constructions for fibered homeomorphisms of the two-torus. To appear in Trans. Am. Math. Soc., 2008
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27. Jäger, T.: Quasiperiodically forced interval maps with negative Schwarzian derivative. Nonlinearity 16(4), 1239–1255 (2003) 28. Bjerklöv, K.: Dynamics of the quasiperiodic Schrödinger cocycle at the lowest energy in the spectrum. Commun. Math. Phys. 272, 397–442 (2005) 29. Jäger, T.: Strange non-chaotic attractors in quasiperiodically forced circle maps: a slightly extended preprint version of this article is available at http://arxiv.org/abs/0709.0269v1[math.DS], 2007 Communicated by B. Simon
Commun. Math. Phys. 289, 291–310 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0664-5
Communications in
Mathematical Physics
Analytic Structure of Many-Body Coulombic Wave Functions Søren Fournais1, , Maria Hoffmann-Ostenhof2 , Thomas Hoffmann-Ostenhof3,4 , Thomas Østergaard Sørensen5 1 Department of Mathematical Sciences, University of Aarhus, Ny Munkegade,
Building 1530, DK-8000 Århus C, Denmark. E-mail: [email protected]
2 Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, A-1090 Vienna,
Austria. E-mail: [email protected]
3 Institut für Theoretische Chemie, Universität Wien, Währingerstrasse 17,
A-1090 Vienna, Austria
4 The Erwin Schrödinger International Institute for Mathematical Physics,
Boltzmanngasse 9, A-1090 Vienna, Austria. E-mail: [email protected]
5 Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G,
DK-9220 Aalborg East, Denmark. E-mail: [email protected] Received: 6 June 2008 / Accepted: 28 July 2008 Published online: 10 December 2008 – © The Author(s) 2008
Abstract: We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic many-particle systems. We prove the following: Let ψ(x) with x = (x1 , . . . , x N ) ∈ R3N denote an N -electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which x1 = 0 and the other electron coordinates do not coincide, and differ from 0, ψ can be represented locally as ψ(x) = ψ (1) (x) + |x1 |ψ (2) (x) with ψ (1) , ψ (2) real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.
1. Introduction and Results 1.1. Introduction. In quantum chemistry and atomic and molecular physics, the regularity properties of the Coulombic wavefunctions ψ, and of their corresponding oneelectron densities ρ, are of great importance. These regularity properties determine the convergence properties of various (numerical) approximation schemes (see [2,3,26,31–33] for some recent works). They are also of intrinsic mathematical interest. The pioneering work is due to Kato [20], who proved that ψ is Lipschitz continuous, i.e., ψ ∈ C 0,1 , near two-particle coalescence points. In a series of recent papers the present authors have studied these properties in detail. In [7] we deduced an optimal representation of ψ of the form ψ = F with an explicit F ∈ C 0,1 , such that ∈ C 1,1 , characterizing the singularities of ψ up to second derivatives; see [7, Theorem 1.1] for a precise statement. In particular, F contains logarithmic terms which stem from the singularities of the potential at three-particle coalescence © 2008 by the authors. This article may be reproduced in its entirety for noncommercial purposes. On leave from: CNRS and Laboratoire de Mathématiques d’Orsay, Univ Paris-Sud, F-91405 Orsay
CEDEX, France
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S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Østergaard Sørensen
points. This characterization has been applied in [8] and [9] in the study of the electron density ρ and (in the atomic case) its spherical average ρ close to the nuclei. Real analyticity of ρ away from the nuclei was proved in [6]; see also [4,5]. In this paper we derive a different representation of ψ which completely settles its analytic structure in the neighbourhood of two-particle coalescence points. The Kustaanheimo-Stiefel transform (KS-transform for short) and analytic hypoanalyticity of a certain degenerate elliptic operator are crucial for the proof. We start with the one-particle case. Theorem 1.1. Let ⊂ R3 be a neighbourhood of the origin and assume that W (1) , W (2) , F (1) , and F (2) are real analytic functions in . Let H = − +
W (1) + W (2) , |x|
(1.1)
and assume that ϕ ∈ W 1,2 () satisfies Hϕ =
F (1) + F (2) |x|
(1.2)
in in the distributional sense. ⊂ of the origin, and real analytic functions Then there exist a neighbourhood → C such that ϕ (1) , ϕ (2) : . ϕ(x) = ϕ (1) (x) + |x|ϕ (2) (x), x ∈
(1.3)
Remark 1.2. Theorem 1.1 is a generalization of an almost 25 years old result by Hill [16]. The present investigations were partly motivated by this work. Hill considered solutions to
−−
Z + V (1) (x) + |x|V (2) (x) ϕ = 0, |x|
(1.4)
with V (1) and V (2) real analytic near the origin, and proved that ϕ satisfies (1.3). The statement (1.3) can easily be seen to hold for Hydrogenic eigenfunctions. These have the form e−β|x| P(x) for some β > 0, where P(x) can be written as linear combinations of polynomials in |x| times homogeneous harmonic polynomials. In particular, Hill’s result implies that ϕ satisfies (1.3) near the origin for a one-electron molecule with fixed nuclei, one of them at the origin. Remark 1.3. Hill’s proof is rather involved. Our proof is quite different, also not easy, but has the advantage that it can be generalized to treat the Coulombic many-particle case; see Theorem 1.4 and its proof below, and also Remark 1.6. The proof of Theorem 1.1 uses the KS-transform (see Sect. 2 for the definition). This transform was introduced in the 1960’s [25] to regularize the Kepler problem in classical mechanics (see also [22,24,30]) and has found applications in problems related to the Coulomb potential in classical mechanics and quantum mechanics, see [1,10,13– 15,19]. The KS-transform is a homogeneous extension of the Hopf map (also called the Hopf fibration), the first example of a map from S3 to S2 which is not null-homotopic, discovered in the 1930’s [17]. For more on the literature on the KS-transform, see [14,22].
Analytic Structure of Coulombic Wave Functions
293
We move to the N -particle problem. For the sake of simplicity we consider the atomic case and mention extensions in the remarks. Let H be the non-relativistic Schrödinger operator of an N -electron atom with nuclear charge Z > 0 in the fixed nucleus approximation, H=
N j=1
− j −
Z + |x j |
1≤i< j≤N
1 =: − + V. |xi − x j |
(1.5)
Here the x j = (x j,1 , x j,2 , x j,3 ) ∈ R3 , j = 1, . . . , N , denote the positions of the N electrons, and the j are the associated Laplacians so that = j=1 j is the 3N 3N -dimensional Laplacian. Let x = (x1 , x2 , . . . , x N ) ∈ R and let ∇ = (∇1 , . . . , ∇ N ) denote the 3N -dimensional gradient operator. The operator H is bounded from W 2,2 (R3N ) to L 2 (R3N ), and defines a bounded quadratic form on W 1,2 (R3N ) [21]. We investigate local solutions ψ of H ψ = Eψ,
E ∈ R,
(1.6)
in a neighbourhood of two-particle coalescence points. More precisely, let denote the set of coalescence points, N
|x j | := x = (x1 , . . . , x N ) ∈ R3N j=1
|xi − x j | = 0 .
(1.7)
1≤i< j≤N
If, for some ⊂ R3N , ψ is a distributional solution to (1.6) in , then [18, Sect. 7.5, pp. 177–180] ψ is real analytic away from , that is, ψ ∈ C ω (\ ). Let, for k, ∈ {1, . . . , N }, k = , k := x ∈ R3N
N
|x j |
j=1, j=k
(1.8)
1≤i< j≤N
N
k, := x ∈ R3N |x j | j=1
|xi − x j | = 0 ,
|xi − x j | = 0 .
(1.9)
1≤i< j≤N ,{i, j}={k, }
Then we denote k := \ k , k, := \ k,
(1.10)
the two kinds of ‘two-particle coalescence points’. The main result of this paper is the following. Theorem 1.4. Let H be the non-relativistic Hamiltonian of an atom, given by (1.5), let ⊂ R3N be an open set, and assume that ψ ∈ W 1,2 () satisfies, for some E ∈ R, H ψ = Eψ in
(1.11)
in the distributional sense. Let the sets k and k, be given by (1.10). Then, for all k ∈ {1, . . . , N }, there exist a neighbourhood k ⊂ of ∩ k , and real analytic functions ψk(1) , ψk(2) : k → C such that (1)
(2)
ψ(x) = ψk (x) + |xk |ψk (x), x ∈ k ,
(1.12)
294
S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Østergaard Sørensen
and for all k, ∈ {1, . . . , N }, k = , there exist a neighbourhood k, ⊂ of ∩ k, , (1) (2) and real analytic functions ψk, , ψk, : k, → C such that (1)
(2)
ψ(x) = ψk, (x) + |xk − x |ψk, (x), x ∈ k, .
(1.13)
Remark 1.5. The proof of Theorem 1.4 again uses the KS-transform. Due to the presence of the other electron coordinates we are confronted with additional problems. We have to deal with degenerate elliptic PDE’s where the corresponding operators (of Grušin-type) are analytic hypoelliptic, see [12]. Remark 1.6. Theorem 1.4 extends in the obvious way to electronic eigenfunctions of Hamiltonians of N -electron molecules with K nuclei fixed at positions (R1 , . . . , R K ) ∈ R3 , given by H=
N
− j −
K =1
j=1
Z + |x j − R |
1≤i< j≤N
1 . |xi − x j |
(1.14)
Furthermore we can replace in (1.14), as in Theorem 1.1, the potential terms by more general terms, and allow for inhomogeneities. For instance, the result holds for general Coulombic many-particle systems described by H=
n
−
j=1
j + 2m j
vi j (xi − x j ),
(1.15)
1≤i< j≤n
where the m j > 0 denote the masses of the particles, and vi j = vi,(1)j |xi − x j |−1 + vi,(2)j (k)
with vi j , k = 1, 2, real analytic. Remark 1.7. In separate work we will present additional regularity results (not primarily for Coulomb problems) obtained partly using the techniques developed in the present paper. 2. Proofs of the Main Theorems As mentioned in the Introduction our proofs are based on the Kustaanheimo-Stiefel (KS) transform. We will ‘lift’ the differential equations to new coordinates using that transform. The solutions to the new equations will be real analytic functions. By projecting to the original coordinates we get the structure results Theorem 1.1 and Theorem 1.4. In the present section we will introduce the KS-transform and show how it allows to obtain Theorems 1.1 and 1.4. The more technical verifications of the properties of the KS-transform and its composition with real analytic functions needed for these proofs are left to Sects. 3 and 4. Define the KS-transform K : R4 → R3 by ⎛ ⎞ y12 − y22 − y32 + y42 ⎜ ⎟ K (y) = ⎝ 2(y1 y2 − y3 y4 ) ⎠. (2.1) 2(y1 y3 + y2 y4 )
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It is a simple computation to verify that |K (y)| := K (y) R3 = y 2R4 =: |y|2 for all y ∈ R4 .
(2.2)
Let f : R3 → C be any C 2 -function, and define, with K as above, f K : R4 → C ,
f K (y) := f (K (y)).
(2.3)
Then for all y ∈ R4 \{0}, (see Lemma 3.1), ( f )(K (y)) =
1 4|y|2
f K (y).
(2.4)
2.1. Proof of Theorem 1.1. Assume ϕ ∈ W 1,2 () satisfies (see (1.1)–(1.2))
−+
F (1) W (1) + W (2) ϕ = + F (2) , |x| |x|
(2.5)
with W (1) , W (2) , F (1) , F (2) real analytic in ⊂ R3 . Assume without loss that = B3 (0, r ) for some r > 0. (Here, and in the sequel, Bn (x0 , r ) = {x ∈ Rn | |x − x0 | < r }.) Since ϕ ∈ L 2 (), Remark 3.2 in Sect. 3 below implies that ϕ K is well-defined, as an element of L 2 (K −1 (), π4 |y|2 dy). We will show that ϕ K satisfies (in the distributional sense) (1) (2) (1) (2) (2.6) − y + 4(W K + |y|2 W K ) ϕ K = 4(FK + |y|2 FK ) √ in K −1 () = B4 (0, r ), with W K(i) , FK(i) , i = 1, 2, defined as in (2.3). Since W (i) , F (i) , i = 1, 2, are real analytic in B3 (0, r ) by assumption, and K : R4 → R3 (see (2.1)) and y → |y|2 are√real analytic, the coefficients in the elliptic equation in (2.6) are real analytic in B4 (0, r ). It follows from elliptic regularity for √ equations with real analytic coefficients [18, Sect. 7.5, pp. 177–180] that ϕ K : B4 (0, r ) → C is real analytic. The statement of Theorem 1.1 then follows from Proposition 4.1 in Sect. 4 below. It therefore remains to prove that ϕ K satisfies (2.6). By elliptic regularity, ϕ ∈ W 2,2 ( ) for all = B3 (0, r ), r < r . (To see this, use Hardy’s inequality [28, Lemma p. 169] and that ϕ ∈ W 1,2 () to conclude that ϕ = G with G ∈ L 2 ( ). Then use [11, Theorem 8.8]). It follows that both (ϕ) K and (| · |−1 ϕ) K are well-defined, as elements of 2 L (K −1 ( ), π4 |y|2 dy) (see Remark 3.2 in Sect. 3 below; see also (3.5)). This and √ (2.5) imply that, as functions in L 2 (K −1 ( )) = L 2 (B4 (0, r )), |y|(ϕ) K = |y| (W ϕ) K − FK , (2.7) with W (x) =
W (1) (x) + W (2) (x), |x|
F(x) =
F (1) (x) + F (2) (x). |x|
(2.8)
Let f ∈ C0∞ (K −1 ()); then there exists r < r such that supp( f ) ⊂ K −1 ( ), := B3 (0, r ); choose {ϕn }n∈N ⊂ C ∞ ( ) such that ϕn → ϕ and ϕn → ϕ in L 2 ( )-norm. This is possible since ϕ ∈ W 2,2 ( ). Note that both f and 4|y|2 f
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belong to C0∞ (K −1 ()) when f does. Using (2.4) for ϕn ∈ C ∞ ( ), Remark 3.2 in Sect. 3 below therefore implies that ( f )(y) ϕ K (y) dy = lim ( f )(y) (ϕn ) K (y) dy K −1 ()
n→∞ K −1 ()
= lim
n→∞ K −1 ()
f (y) [ y (ϕn ) K ](y) dy = lim
n→∞ K −1 ()
4|y|2 f (y) (x ϕn ) K (y) dy
4|y|2 f (y) (x ϕ) K (y) dy.
= K −1 ()
It follows from this and (2.7) that ( f )(y) ϕ K (y) dy = K −1 ()
4|y|2 f (y) (W ϕ) K − FK (y) dy.
K −1 ()
Since (W ϕ) K = W K ϕ K , and, by (2.8) and (2.2), W K (y) = |y|−2 W K(1) (y) + |y|2 W K(2) (y) , FK (y) = |y|−2 FK(1) (y) + |y|2 FK(2) (y) , this implies that, for all f ∈ C0∞ (K −1 ()), (1) (2) ϕ K (y) − y f (y) + 4(W K (y) + |y|2 W K (y)) f (y) dy K −1 ()
(1) (2) 4 FK (y) + |y|2 FK (y) f (y) dy,
= K −1 ()
√ which means that ϕ K satisfies (2.6) in the distributional sense, in K −1 () = B4 (0, r ).
2.2. The N -particle problem. In this section we prove Theorem 1.4. We only prove the statement (1.12), the proof of (1.13) is completely analogous, after an orthogonal transformation of coordinates. We assume k = 1, the proof for other k’s is the same. Let H be given by (1.5). Then, with (x, x ) ≡ (x1 , x ) ∈ R3 × R3N −3 , x = (x2 , . . . , x N ), H − E = −x − x −
Z + VE (x, x ), |x|
(2.9)
where
VE (x1 , x ) =
N j=2
−
Z + |x j |
is real analytic in \ 1 (see (1.8) for 1 ).
1≤i< j≤N
1 −E |xi − x j |
(2.10)
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Assume ψ ∈ W 1,2 () satisfies (H − E)ψ = 0 in ,
(2.11)
and let (x0 , x0 )
∈ ∩ 1 ; then (see (1.10)) x0 = 0. We will first prove that there exist a (1) (2) neighbourhood 1 (P) of P = (0, x0 ) and real analytic functions ψ P , ψ P : 1 (P) → C such that (1)
(2)
ψ(x) = ψ P (x) + |x|ψ P (x), x ∈ 1 (P).
(2.12)
By the above, VE is real analytic in a neighbourhood of (0, x0 ), say, in U (R) = (x, x ) ∈ R3 × R3N −3 |x| < R, |x − x0 | < R ⊂ for some R > 0, R small. Let √ U K (R) := (y, x ) ∈ R4 × R3N −3 |y| < R, |x − x0 | < R .
(2.13)
Define now, with K : R4 → R3 as in (2.1), u : U K (R) → C , u(y, x ) := ψ(K (y), x ),
(2.14)
W : U K (R) → R , W (y, x ) := VE (K (y), x ).
(2.15)
Since (by (2.2)) (K (y), x ) ∈ U (R) for (y, x ) ∈ U K (R), it follows that u and W are well-defined, and W is real analytic in U K (R) since K is real analytic and VE is real analytic in U (R). As in the proof of Theorem 1.1, we get that (2.11) implies that u satisfies Q(y, x , D y , Dx )u = 0 in U K (R),
(2.16)
Q(y, x , D y , Dx ) := − y − 4|y|2 x + 4|y|2 W (y, x ) − 4Z
(2.17)
where
is a degenerate elliptic operator, a so-called ‘Grušin-type operator’. Since |y|2 W (y, x ) is real analytic in U K (R), the operator Q is (real) analytic hypoelliptic due to [12, Theorem 5.1]. Therefore (2.16) implies that u is real analytic in some neighbourhood of (0, x0 ) ∈ R4 × R3N −3 . It follows from Proposition 4.4 in Sect. 4 below that there exist a neighbourhood (1) (2) 1 (P) ⊂ R3N of P = (0, x0 ) ∈ R3 × R3N −3 and real analytic functions ψ P , ψ P : 1 (P) → C such that (2.12) holds. Let now 1 := 1 (P) ⊂ ⊂ R3N , P∈∩ 1 (1)
(2)
and define ψ1 , ψ1 : 1 → C by (i)
(i)
ψ1 (x) = ψ P (x) when x ∈ 1 (P) (i = 1, 2).
(2.18)
To see that this is well-defined, we need to verify that if x ∈ 1 (P) ∩ 1 (Q), then (i) (i) (i) = ψ (i) − ψ (i) , i = 1, 2, then ψ P (x) = ψ Q (x), i = 1, 2. Let therefore ψ P Q (2) (x) = 0, x ∈ 1 (P) ∩ 1 (Q), (1) (x) + |x|ψ ψ
(2.19)
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(1) , ψ (2) real analytic in 1 (P) ∩ 1 (Q). Let x˜ 0 = (0, x˜ ) ∈ 1 (P) ∩ 1 (Q). with ψ 0 (i) , i = 1, 2, are real analytic, there exist δ > 0 and Pn(i) , i = 1, 2, Then, since ψ homogeneous polynomials of degree n such that (i) (x) = ψ
∞
Pn(i) (x, x − x˜0 ), i = 1, 2,
(2.20)
n=0
for x ∈ B3N (˜x0 , δ). It follows from (2.19) (by homogeneity) that, for all n ∈ N and x = (x, x ) ∈ B3N (˜x0 , δ), (2)
Pn(1) (x, x − x˜0 ) + |x|Pn−1 (x, x − x˜0 ) = 0. (2) (2) , and therefore |x|Pn−1 , is odd. But for n even, Pn(1) is an even function, while Pn−1 (1) (2) (1) (2) Therefore, Pn = Pn−1 = 0. Similarly for n odd. It follows that ψ = ψ = 0 in (1) , ψ (2) are real analytic. B3N (˜x0 , δ), and therefore also in 1 (P) ∩ 1 (Q), since ψ (1)
(2)
This proves that ψ1 and ψ1 in (2.18) are well-defined. Since they are obviously real analytic, this finishes the proof of Thereom 1.4. 3. The Kustaanheimo-Stiefel Transform The KS-transform turns out to be a very useful and natural tool for the investigation of Schrödinger equations with Coulombic interactions. In particular (2.2) and the following lemma are important for our proofs. Most of the facts stated here are well-known (see e. g. [13, Appendix A]). Lemma 3.1. Let K : R4 → R3 be defined as in (2.1), let f : R3 → C be any C 2 -function, and define f K : R4 → C by (2.3). Finally, let L(y, D y ) := y1
∂ ∂ ∂ ∂ − y4 + y3 − y2 . ∂ y4 ∂ y1 ∂ y2 ∂ y3
(a) Then, with [A; B] = AB − B A the commutator of A and B, L(y, D y ) f K = 0, ; L(y, D y ) = 0,
(3.1)
(3.2)
and (2.4) holds. (b) Furthermore, for a function g ∈ C 1 (R4 ), the following two statements are equivalent: (i) There exists a function f : R3 → C such that g = f K . (ii) The function g satisfies Lg = 0.
(3.3)
(c) Finally, let U = B3 (0, r ) ⊂ R3 for r ∈ (0, ∞]. Then, for φ ∈ C0 (R3 ) (continuous with compact support), |φ(K (y))|2 dy = K −1 (U )
π 4
U
|φ(x)|2 d x. |x|
(3.4)
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299
In particular, |y|φ K 2 2
L (K −1 (U ))
=
π φ 2L 2 (U ) . 4
(3.5)
Remark 3.2. By a density argument, the isometry (3.5) allows to extend the composition by K given by (2.3) (the pull-back K ∗ by K ) to a map K ∗ : L 2 (U, d x) → L 2 (K −1 (U ), π4 |y|2 dy) φ → φ K in the case when U = B3 (0, r ), r ∈ (0, ∞]. This makes φ K well-defined for any φ ∈ L 2 (U ). Furthermore, if φn → φ in L 2 (U ), then, for all g ∈ C ∞ (K −1 (U )) (g ∈ C0∞ (K −1 (U )), if r = ∞) lim g(y)(φn ) K (y) dy = g(y)φ K (y) dy. (3.6) n→∞ K −1 (U )
K −1 (U )
This follows from Schwarz’ inequality and (3.5), g(y) ((φn ) K (y) − φ K (y)) dy K −1 (U )
≤
K −1 (U )
√ π = 2
|g(y)|2 1/2 |y| (φn ) K − φ K 2 −1 dy L (K (U )) 2 |y|
K −1 (U )
|g(y)|2 1/2 dy φn − φ L 2 (U ) → 0, n → ∞. |y|2
Here the y-integral clearly converges since g ∈ C ∞ (R4 ) (g ∈ C0∞ (R4 ), if r = ∞). Remark 3.3. As a consequence of (2.2) and (3.2) (choose f (x) = |x| j ), we have that L(y, D y )|y|2 j = 0,
j ∈ N.
(3.7)
Proof of Lemma 3.1. The lemma is easier to prove in ‘double polar coordinates’ in R4 . Let (R, ) := (r1 , r2 , θ1 , θ2 ) ∈ (0, ∞)2 × [0, 2π )2
(3.8)
be defined by the relation y ≡ y(R, ) = (y1 (R, ), y2 (R, ), y3 (R, ), y4 (R, )) ,
(3.9)
(y1 , y4 ) = r1 (cos θ1 , sin θ1 ) , (y3 , y2 ) = r2 (cos θ2 , sin θ2 ).
(3.10)
Then it follows directly from (2.1) that ⎛
⎞ r12 − r22 ⎟ ⎜ K (y(R, )) = ⎝−2r1r2 sin(θ1 − θ2 )⎠ . 2r1r2 cos(θ1 − θ2 )
(3.11)
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We note in passing that the relation (2.2) is immediate from (3.11). In the double polar coordinates, L=
∂ ∂ + , ∂θ1 ∂θ2
(3.12)
and =
∂2 1 ∂ 1 ∂2 ∂2 1 ∂ 1 ∂2 + . + + + + ∂r12 r1 ∂r1 r12 ∂θ12 ∂r22 r2 ∂r2 r22 ∂θ22
Therefore, it is obvious that L and commute. Furthermore, from (3.11) we see that f K only depends on the angles through the expression θ1 − θ2 and therefore, L fK =
∂ ∂ f K = 0. + ∂θ1 ∂θ2
The proof of (2.4) is merely an elementary computation, which we leave to the reader. This finishes the proof of point (a) of the lemma. From (3.2) we infer that in order to prove point (b) we have to show that (ii) implies (i). To do so, we need to define a function f : R3 → C such that g = f K . If x = 0, let f (x) := g(0), then g(0) = f (0) = f (K (0)) = f K (0) by (2.2). Assume now that x ∈ R3 \{0}. We claim that the pre-image of x under K , K −1 ({x}), is a circle in R4 (in the literature called the ‘Hopf circle’) and that g is constant on this circle. Then, taking any y ∈ K −1 ({x}) and letting f (x) := g(y), we have that f is well-defined, and satisfies f K (y) = f (K (y)) = f (x) = g(y). This will finish the proof of point (b) of the lemma. To prove the claim, assume first that x ∈ R3 \{0}, x = (x1 , x2 , x3 ) with (x2 , x3 ) = (0, 0). Then the equations (see (3.11) and (2.2)) r12 − r22 = x1 , −2r1r2 sin ϑ = x2 , 2r1r2 cos ϑ = x3 , (r12 + r22 )2 = x12 + x22 + x32
(3.13)
uniquely determine r1 , r2 ∈ (0, ∞), and determine ϑ modulo 2π ; choose the solution ϑ ∈ [0, 2π ). That is, the pre-image of x under K is the set of points in R4 with double polar coordinates (r1 , r2 , θ1 , θ2 ), where (r1 , r2 ) is the unique solution to (3.13), and θ1 −θ2 = ϑ modulo 2π , with ϑ ∈ [0, 2π ). Defining new angles θ = θ1 +θ2 , ϑ = θ1 −θ2 ,
this set is the circle in R4 with centre at the origin and radius (x12 +x22 +x32 )1/4 = r12 + r22 , parametrized by θ ∈ [0, 2π ). Since, by (3.12), the function g (strictly speaking, g composed with the map in (3.9)) is independent of θ = θ1 + θ2 , g is, as claimed, constant on this circle. On the other hand, assume x = (t, 0, 0), t ∈ R\{0}. Then the equations r12 − r22 = t, (r12 + r22 )2 = t 2
(3.14)
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301
√ have√a unique solution (r1 , r2 ); in fact, (r1 , r2 ) = ( t, 0) if t > 0 and (r1 , r2 ) = (0, −t) if t < 0. In both cases, the pre-image of x under K is a circle, namely (see also (3.8)) √ √ C+ = {( t cos θ1 , 0, 0, t sin θ1 )} ∈ R4 | θ1 ∈ [0, 2π )} (t > 0), √ √ 4 C− = {(0, −t sin θ2 , −t cos θ2 , 0)} ∈ R | θ2 ∈ [0, 2π )} (t > 0). Since y2 = y3 = 0 for any y = (y1 , y2 , y3 , y4 ) ∈ C+ , (3.1) and (3.3) imply that ∂g/∂θ1 = 0, with θ1 the angle parametrizing C+ , and so g is, as claimed, constant on C+ ; similarly for C− . This finishes the proof of point (b) of the lemma. We finish by proving point (c); this is merely a calculation which we for simplicity also do in ‘double polar coordinates’: Recall that |y|2 = r12 + r22 = |x| (see (2.2)). By √ (3.10) and (3.11), and since U = B3 (0, r ) and K −1 (U ) = B4 (0, r ), r −r12
√ r
|φ(K (y))| dy = 2
K −1 (U )
2π
r1 dr1 0
0
2π
r2 dr2 0
dθ1 0
2 2 φ r −r , −2r1 r2 sin(θ1 −θ2 ), 2r1r2 cos(θ1 −θ2 ) 2 dθ2 . 1
2
In the triple integral inside {·} we make (for fixed θ2 ) the change of variables x = K θ2 (r1 , r2 , θ1 ) = r12 − r22 , −2r1 r2 sin(θ1 − θ2 ), 2r1r2 cos(θ1 − θ2 ) . From the foregoing (see after (3.14)) it follows that the image of K θ2 is U . The determinant of the Jacobian is 2r1 −2r2 0 det(D K θ2 ) = −2r2 sin(θ1 − θ2 ) −2r1 sin(θ1 − θ2 ) −2r1r2 cos(θ1 − θ2 ) 2r2 cos(θ1 − θ2 ) 2r1 cos(θ1 − θ2 ) −2r1r2 sin(θ1 − θ2 ) = 8r1r2 (r12 + r22 ). Recall that |y|2 = r12 + r22 = |x|. Therefore the integral is |φ(K (y)| dy = 2
K −1 (U )
2π 0
|φ(x)|2
dx π dθ2 = 8|x| 4
U
U
|φ(x)|2 d x. |x|
√ This proves (3.4); applying it to |x|φ gives (3.5). This finishes the proof of point (c), and therefore, of Lemma 3.1. Lemma 3.4. Let the differential operator L = L(y, D) be given by (3.1), and let P2k be a harmonic, homogeneous polynomial of degree 2k in R4 such that L P2k = 0. Then there exists a harmonic polynomial in R3 , Yk , homogeneous of degree k, such that P2k (y) = Yk (K (y)) for all y ∈ R4 , with K : R4 → R3 from (2.1).
(3.15)
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Proof. Using that L P2k = 0 we get from Lemma 3.1 the existence of a function Yk such that P2k (y) = Yk (K (y)). Since the KS-transform is homogeneous of degree 2, Yk is necessarily homogeneous of degree k. Furthermore, by (2.4), Yk is harmonic. So we only have left to prove that Yk is a polynomial. Let Ln be the (positive) Laplace-Beltrami operator on the sphere Sn−1 . Then one can express the Laplace operator in Rn as =
Ln n−1 ∂ ∂2 − 2. + 2 ∂r r ∂r r
(3.16)
Furthermore, σ (Ln ) = { ( + n − 2)}∞ =0 and the eigenspace corresponding to the eigenvalue ( + n − 2) is exactly spanned by the restrictions to Sn−1 of the harmonic, homogeneous polynomials in Rn of degree . Using the fact that Yk = 0 and that Yk is homogeneous of degree k in R3 we find that Yk S2 is an eigenfunction of L3 with eigenvalue k(k + 1). Thus there exists a k of degree k such that homogeneous, harmonic polynomial Y k S2 = Yk S2 . Y Since the functions have the same homogeneity, they are identical everywhere. This finishes the proof of the lemma. 4. Analyticity and the KS-Transform In this section we study the regularity of functions given as a composition with the Kustaanheimo-Stiefel transform. We start with the one-particle case. Proposition 4.1. Let U ⊂ R3 be open with 0 ∈ U , and let ϕ : U → C be a function. Let U = K −1 (U ) ⊂ R4 , with K : R4 → R3 from (2.1), and suppose that ϕK = ϕ ◦ K : U → C
(4.1)
is real analytic. Then there exist functions ϕ (1) , ϕ (2) , real analytic in a neighbourhood of 0 ∈ R3 , such that ϕ(x) = ϕ (1) (x) + |x|ϕ (2) (x).
(4.2)
Proof. Note that K (−y) = K (y) for all y ∈ R4 , so that ϕ K (−y) = ϕ K (y) for all y ∈ R4 . It follows that ϕ K can be written as an absolutely convergent power series containing only terms of even order. Furthermore, since the sum is absolutely convergent, the order of summation is unimportant, and so, for some R > 0, cβ ∈ C, ϕ K (y) =
cβ y β =
∞
cβ y β for |y| < R.
(4.3)
n=0 |β|=2n
β∈N4 ,|β|/2∈N
This implies (see e. g. [23, Sects. 2.1–2.2]) that there exist constants C1 , M1 > 0 such that |β|
|cβ | ≤ C1 M1
for all β ∈ N4 .
(4.4)
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303
Note that for fixed n ∈ N,
Q (2n) (y) :=
cβ y β
(4.5)
β∈N4 ,|β|=2n
is a homogeneous polynomial of degree 2n. By [29, Theorem 2.1], Q (2n) (y) =
n
(2n)
|y|2 j H2n−2 j (y),
(4.6)
j=0 (2n)
where H2n−2 j is a homogeneous harmonic polynomial of degree 2n−2 j, j = 0, 1, . . . , n. It follows that ϕ K (y) =
∞
Q (2n) (y) =
n=0
n ∞
(2n) |y|2 j H2n−2 j (y).
(4.7)
n=0 j=0
We need the following lemma. (2n)
Lemma 4.2. There exist harmonic polynomials Yn− j : R3 → C, homogeneous of degree n − j, such that (2n)
(2n)
H2n−2 j (y) = Yn− j (K (y)) for all y ∈ R4 ,
(4.8)
with K : R4 → R3 from (2.1). In particular, the function q (2n) (x) :=
n
(2n)
|x| j Yn− j (x)
(4.9)
j=0
satisfies q (2n) (K (y)) = Q (2n) (y) for all y ∈ R4 .
(4.10)
Proof of Lemma 4.2. Recall (see (3.2)) that, with L ≡ L(y, D y ) as in (3.1), Lϕ K = 0, and therefore, since power series can be differentiated termwise (see (4.7)), 0 = Lϕ K =
∞
L Q (2n) .
(4.11)
n=0
Since L Q (2n) is again a homogeneous polynomial of degree 2n, it follows that L Q (2n) = 0, n = 0, 1, . . . .
(4.12)
Since L is a first order differential operator, (3.7) implies that (2n)
(2n)
L[|y|2 j H2n−2 j ] = |y|2 j [L H2n−2 j ],
(4.13)
(2n)
where L H2n−2 j is again a homogeneous polynomial of order 2(n − j). Then (4.6), (4.12), and (4.13) imply that n j=0
(2n) |y|2 j [L H2n−2 j ](y) = 0
for all y ∈ R4 .
(4.14)
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(2n)
Since H2n−2 j is harmonic, and (see (3.2)) [; L] = 0, we get that L H2n−2 j is a homogeneous harmonic polynomial of degree 2(n − j). Note that for |y| = 1, the left side of (4.14) is a linear combination of spherical harmonics of different degrees. From the linear independence of such spherical harmonics it follows that (2n)
L H2n−2 j = 0
for all j = 0, . . . , n, and n ∈ N.
(4.15)
(2n) From Lemma 3.4 it follows that there exist harmonic polynomials in R3 , Yn− j , homogeneous of degree n − j, such that (4.8) holds. Now (4.10) follows from this and (2.2). This finishes the proof of the lemma.
Lemma 4.2, (4.7), and |K (y)| = |y|2 , imply that ϕ K (y) = ϕ(K (y)) =
n ∞
(2n) |K (y)| j Yn− j (K (y)).
(4.16)
n=0 j=0
Formally, we can now finish the proof of Proposition 4.1 by defining ϕ (1) (x) : = ϕ (2) (x) : =
∞
n
(2n)
|x| j Yn− j (x),
n=0 j=0, j even ∞ n
(2n)
|x| j−1 Yn− j (x).
(4.17)
(4.18)
n=0 j=1, j odd
However, it is not a priori clear that these sums converge and thus define real analytic functions. The remainder of the proof will establish the necessary convergence. Lemma 4.3. There exists r > 0 such that the two series in (4.17) and (4.18) converge for |x| < r . 1 := RC1 , More precisely, there exists a universal constant R > 0 such that with C 1 = 2M 2 (with C1 , M1 from (4.4)), M 1 (2n)
1 M 1n |x|n− j for |x| < r. |Yn− j (x)| ≤ C
(4.19)
Proof. Clearly, the convergence of the series in (4.17) and (4.18) is a consequence of 1 ). Thus we only have to prove the estimate (4.19). (4.19): take r < 1/(2 M We return to (4.3). For fixed β, with |β| = 2n > 0 we have (again using [29, Theorem (β) 2.1]) that, for some d j ∈ C, yβ =
n
(β)
(β)
|y|2 j d j P2n−2 j (y),
(4.20)
j=0 (β)
where P2n−2 j is a harmonic homogeneous polynomial of degree 2n −2 j, which depends (β)
on β, and satisfies P2n−2 j L 2 (S3 ) = 1. It follows from (4.5) and (4.20) that Q (2n) (y) =
n j=0
|y|2 j
|β|=2n
(β)
(β)
cβ d j P2n−2 j (y).
(4.21)
Analytic Structure of Coulombic Wave Functions
305
Comparing (4.6) with (4.21) we see that n
(2n) (β) (β) |y|2 j H2n−2 j (y) − cβ d j P2n−2 j (y) = 0.
(4.22)
|β|=2n
j=0
Restricting to |y| = 1, (4.22) becomes a sum of spherical harmonics with different degrees, which are linearly independent, implying that (see (4.8)) (β) (β) (2n) (2n) cβ d j P2n−2 j (y). (4.23) Yn− j (K (y)) = H2n−2 j (y) = |β|=2n (2n) ∞ 3 We are now going to bound the Yn− j ’s in L . Since the (restriction to S of the)
(β)
P2n−2 j ’s in (4.20) are orthogonal in L 2 (S3 ) (they are homogeneous of different degrees), we get (by Parseval’s identity), from setting |y| = 1 in (4.20), that n
(β)
|d j |2 =
j=0
S3
|y β |2 dω ≤
S3
|y|2|β| dω =
S3
1 dω = Vol(S3 ),
(4.24)
(β)
and so the d j ’s are bounded, uniformly in j and β, by Vol(S3 )1/2 . Due to homogeneity, and using [27, Lemma 8], we get, for any y ∈ R3 \{0} and j ≤ n, that (β) (β) 2n−2 j (β) P P2n−2 j (y/|y|) ≤ |y|2n−2 j P2n−2 j L ∞ (S3 ) 2n−2 j (y) = |y| ≤ |y|2n−2 j
2n − 2 j + 1 3n ≤ |y|2n−2 j . Vol(S3 )1/2 Vol(S3 )1/2
(4.25)
Note that (see [29, pp. 138–139]) k+ −1 , # σ ∈ Nk |σ | = = k−1
(4.26)
and so (4 + 2n − 1)! # β ∈ N4 |β| = 2n = (4 − 1)!(2n)! =
(4.27)
1 (2n + 3)(2n + 2)(2n + 1) ≤ 10n 3 . 6
It follows from (4.23), (4.4), (4.24), (4.25), and (4.27) that (with C1 and M1 the constants in (4.4)) (2n) (β) (β) Y |cβ | |d j | P2n−2 j (y) n− j (K (y)) ≤ |β|=2n
≤ 10C1 n 4 |y|2n−2 j M12n = 10C1 n 4 |K (y)|n− j M12n .
(4.28)
The desired estimate (4.19) clearly follows, using the surjectivity of K , with R := 10 maxn n 4 2−n .
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Recall that each term |x| j Yn− j (x) in the definition (4.17) of ϕ (1) is a homogeneous polynomial (of degree n) in x, and similarly for ϕ (2) . Therefore, the series (4.17) and (4.18) are convergent power series. This implies that ϕ (1) , ϕ (2) define real analytic functions on {|x| < r } (see [23, Sects. 2.1–2.2]). Finally, using (4.16), (4.17), and (4.18), ϕ (1) (K (y)) + |K (y)|ϕ (2) (K (y)) =
n ∞
(2n) |K (y)| j Yn− j (K (y)) = ϕ(K (y)).
(4.29)
n=0 j=0
This, and the surjectivity of K , imply (4.2) and therefore finishes the proof of Proposition 4.1. For the N -particle case, we have the following analogous result. Proposition 4.4. Let U ⊂ R3 , U ⊂ R3N −3 be open, with 0 ∈ U , x0 ∈ U and let ψ : U × U → C be a function. Let U = K −1 (U ) ⊂ R4 , with K : R4 → R3 from (2.1), and suppose that u : U × U → C (y, x ) → ψ(K (y), x )
(4.30)
is real analytic. Then there exist functions ψ (1) , ψ (2) , real analytic in a neighbourhood W of (0, x0 ) ∈ 3N R , such that ψ(x, x ) = ψ (1) (x, x ) + |x|ψ (2) (x, x ), (x, x ) ∈ W.
(4.31)
1 γ ∂x ψ(x, x ) x =x , ϕγ ,K (y) := ϕγ (K (y)). 0 γ!
(4.32)
Proof. Define ϕγ (x) :=
This is well defined by the assumption on u. Since, as in the proof of Proposition 4.1, u is√ even with respect to y ∈ R4 , and the series converges absolutely, we have, for |y| < R, |x − x0 | < R for some R > 0, cβγ ∈ C, u(y, x ) = cβγ y β (x − x0 )γ , β∈N4 ,|β|/2∈N,γ ∈N3N −3
with |(β,γ )|
|cβγ | ≤ C2 M2
|β|
|γ |
= C 2 M2 M2
for all β ∈ N4 , γ ∈ N3N −3 ,
for some constants C2 , M2 > 0. Clearly it follows that cβγ y β , ϕγ ,K (y) = β∈N4 ,|β|/2∈N
(4.33)
(4.34)
Analytic Structure of Coulombic Wave Functions
307
so that
u(y, x ) =
γ ∈N3N −3
=
cβγ y β (x − x0 )γ
β∈N4 ,|β|/2∈N
ϕγ ,K (y) (x − x0 )γ .
(4.35)
γ ∈N3N −3
Moreover, from (4.33) we have that, for all γ ∈ N3N −3 , |β|
|cβγ | ≤ C1 (γ )M2
where
|γ |
C1 (γ ) := C2 M2 .
(4.36)
In particular, (4.34) and (4.36) show that ϕγ ,K is real analytic near y = 0. Repeating the arguments in the proof of Proposition 4.1 for ϕγ ,K for fixed γ ∈ N3N −3 , we get that ϕγ (x) =
∞ [n/2]
(2n),γ
|x|2 Yn−2 (x)
n=0 =0
+|x|
∞ [(n−1)/2] n=0
(2n),γ
|x|2 Yn−(2 +1) (x),
(4.37)
=0
(2n),γ
where Yn−k : R3 → C are harmonic polynomials, homogeneous of degree n − k, depending on γ ∈ N3N −3 . Therefore, for some aα (γ ), bα (γ ) ∈ C, α ∈ N3 , [n/2]
(2n),γ
|x|2 Yn−2 (x) = (2n),γ
|x|2 Yn−(2 +1) (x) =
=0
aα (γ )x α ,
(4.38)
|α|=n
=0 [(n−1)/2]
bα (γ )x α ,
(4.39)
|α|=n−1
with (see (4.19)), aα (γ )x α ≤ RC1 (γ )n(2M22 )n |x|n ,
(4.40)
|α|=n
bα (γ )x α ≤ RC1 (γ )n(2M22 )n |x|n .
(4.41)
|α|=n−1
Recall that (see (4.26)) 3N + k − 4 . # γ ∈ N3N −3 |γ | = k = 3N − 4 By definition, discarding part of the denominator, 3N + k − 4 (3N + k − 4)! ≤ = (3N + k − 4) · . . . · (k + 1). 3N − 4 k!
(4.42)
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This last product contains (3N − 4) factors each of which are smaller than (3N + k). Thus 3N + k − 4 ≤ (3N + k)3N −4 ≤ C3 k 3N , 3N − 4 for some C3 (depending on N ) and all k ≥ 1. It follows that, for |x| < 1/(4M22 ), |x − x0 | < 1/(2M2 ),
∞
aα (γ )x α (x − x0 )γ
γ ∈N3N −3 n=0 |α|=n ∞
∞
≤ RC2
|γ |
(2M22 )n |x|n M2 |x − x0 ||γ |
k=0 γ ∈N3N −3 ,|γ |=k n=0
≤ RC2 C3
∞ 3N ∞ k n < ∞, 2k 2n n=0
k=0
and so, with aαγ := aα (γ ), ψ (1) (x, x ) :=
γ ∈N3N −3
α∈N3
aαγ x α (x − x0 )γ
(4.43)
defines a real analytic function in a neighbourhood of (0, x0 ). Similarly, with bαγ := bα (γ ), bαγ x α (x − x0 )γ (4.44) ψ (2) (x, x ) := γ ∈N3N −3 α∈N3
defines a real analytic function in a neighbourhood of (0, x0 ). From the above observations and from (4.35), (4.32), (4.37), (4.38), and (4.39) it follows that ψ (1) (K (y), x ) + |K (y)| ψ (2) (K (y), x )
=
∞ [n/2]
(2n),γ
|K (y)|2 Yn−2 (K (y))(x − x0 )γ
γ ∈N3N −3 n=0 =0
+
|K (y)|
γ ∈N3N −3
=
∞ [(n−1)/2] n=0
(2n),γ
|K (y)|2 Yn−(2 +1) (K (y))(x − x0 )γ
=0
ϕγ ,K (y)(x − x0 )γ = u(y, x ) = ψ(K (y), x ),
(4.45)
γ ∈N3N −3
and so, by the surjectivity of K , ψ(x, x ) = ψ (1) (x, x ) + |x| ψ (2) (x, x ), with ψ (i) , i = 1, 2, real analytic in (x, x ) ∈ R3N |x| < 1/(4M22 ), |x − x0 | < 1/(2M2 ) . This finishes the proof of Proposition 4.4.
(4.46)
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Acknowledgement. Financial support from the Danish Natural Science Research Council, under the grant Mathematical Physics and Partial Differential Equations (TØS), and from the European Science Foundation Programme Spectral Theory and Partial Differential Equations (SPECT), is gratefully acknowledged. SF is supported by a Skou Grant and a Young Elite Researcher Award from the Danish Research Council. The authors wish to thank Bernard Helffer (SF, TØS), Günther Hörmann (MHO), Gerd Grubb (TØS), Andreas Knauf (TØS), and Heinz Siedentop (TØS) for helpful discussions.
References 1. Castella, F., Jecko, T., Knauf, A.: Semiclassical Resolvent Estimates for Schrödinger Operators with Coulomb Singularities. Ann. Henri Poincaré 9(4), 775–815 (2008) 2. Flad, H.-J., Hackbusch, W., Schneider, R.: Best N -term approximation in electronic structure calculations. I. One-electron reduced density matrix. Math. Model. Numer. Anal. 40(1), 49–61 (2006) 3. Flad, H.-J., Hackbusch, W., Schneider, R.: Best N -term approximation in electronic structure calculations. II. Jastrow factors. Math. Model. Numer. Anal. 41(2), 261–279 (2007) 4. Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: The Electron Density is Smooth Away from the Nuclei. Comm. Math. Phys. 228(3), 401–415 (2002) 5. Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: On the regularity of the density of electronic wavefunctions. In: Mathematical results in quantum mechanics (Taxco, 2001), Contemp. Math., vol. 307, Providence, RI: Amer. Math. Soc. (2002) pp. 143–148 6. Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: Analyticity of the density of electronic wavefunctions. Ark. Mat. 42(1), 87–106 (2004) 7. Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: Sharp Regularity Results for Coulombic Many-Electron Wave Functions. Commun. Math. Phys. 255(1), 183–227 (2005) 8. Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Østergaard Sørensen, T.: Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei. Ann. Henri Poincaré 8(4), 731–748 (2007) 9. Fournais, S., Hoffmann-Ostenhof, M., Østergaard Sørensen, T.: Third Derivative of the One-Electron Density at the Nucleus. Ann. Henri Poincaré 9(7), 1387–1412 (2008) 10. Gérard, C., Knauf, A.: Collisions for the Quantum Coulomb Hamiltonian. Comm. Math. Phys. 143(1), 17–26 (1991) 11. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Berlin: Springer-Verlag, 2001 (reprint of the 1998 edition) 12. Grušin, V.V.: A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold. Mat. Sb. (N.S.) 84 (126), 163–195 (1971) English translation: Math. USSR-Sb. 13, 155–185 (1971) 13. Helffer, B., Knauf, A., Siedentop, H., Weikard, R.: On the absence of a first order correction for the number of bound states of a Schrödinger operator with Coulomb singularity. Comm. Part. Differ. Eq. 17(3–4), 615– 639 (1992) 14. Helffer, B., Siedentop, H.: Regularization of atomic Schrödinger operators with magnetic field. Math. Z. 218(3), 427–437 (1995) 15. Helffer, B., Siedentop, H.: A generalization of the Kustaanheimo-Stiefel transform for two-centre systems. Bull. London Math. Soc. 28(1), 33–42 (1996) 16. Hill, R.N.: On the analytic structure of the wave function for a hydrogen atom in an analytic potential. J. Math. Phys. 25(5), 1577–1583 (1984) 17. Hopf, H.: Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche. Math. Ann. 104(1), 637–665 (1931) 18. Hörmander, L.: Linear partial differential operators, Third revised printing. Die Grundlehren der mathematischen Wissenschaften, Band 116. Berlin: Springer-Verlag, 1976 19. Jost, R.: Das H-Atom nach Kustaanheimo-Stiefel-Scheifele. Lecture notes by H. Tschudi of a course by R. Jost on theoretical physics for mathematicians, Winter Semester 1974/75, ETH-Zürich, 1975 20. Kato, T.: On the eigenfunctions of many-particle systems in quantum mechanics. Comm. Pure Appl. Math. 10, 151–177 (1957) 21. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995 (reprint of the 1980 edition) 22. Knauf, A.: The n-centre problem of celestial mechanics for large energies. J. Eur. Math. Soc. 4(1), 1–114 (2008) 23. Krantz, S.G., Parks, H.R.: A primer of real analytic functions, second ed., Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Boston, MA: Birkhäuser Boston Inc., 2002
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24. Kustaanheimo, P.E.: Spinor regularization of the Kepler motion. Ann. Univ. Turku. Ser. A I No. 73, 7 (1964) 25. Kustaanheimo, P.E., Stiefel, E.L.: Perturbation theory of Kepler motion based on spinor regularization. J. Reine Angew. Math. 218, 204–219 (1965) 26. Le Bris, C., Lions, P.-L.: From atoms to crystals: a mathematical journey. Bull. Amer. Math. Soc. (N.S.) 42(3), 291–363 (2005) (electronic) 27. Müller, C.: Spherical harmonics. Lecture Notes in Mathematics, vol. 17. Berlin: Springer-Verlag, 1966 28. Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier Analysis, Self-Adjointness. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1975 29. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Math. Series, No. 32. Princeton, N.J.: Princeton Univ. Press, 1971 30. Stiefel, E.L., Scheifele, G.: Linear and regular celestial mechanics. Perturbed two-body motion, numerical methods, canonical theory. Die Grundlehren der mathematischen Wissenschaften, Band 174. New York: Springer-Verlag, 1971 31. Yserentant, H.: On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98(4), 731–759 (2004) 32. Yserentant, H.: Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math. 101(2), 381–389 (2005) 33. Yserentant, H.: The hyperbolic cross space approximation of electronic wavefunctions. Numer. Math. 105(4), 659–690 (2007) Communicated by B. Simon
Commun. Math. Phys. 289, 311–334 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0751-2
Communications in
Mathematical Physics
Strong Asymmetric Limit of the Quasi-Potential of the Boundary Driven Weakly Asymmetric Exclusion Process Lorenzo Bertini1 , Davide Gabrielli2 , Claudio Landim3,4 1 Dipartimento di Matematica, Università di Roma ‘La Sapienza’, P.le Aldo Moro 2,
00185 Roma, Italy. E-mail: [email protected]
2 Dipartimento di Matematica, Università dell’Aquila, 67100 Coppito, L’Aquila, Italy.
E-mail: [email protected]
3 IMPA, Estrada Dona Castorina 110, J. Botanico, 22460 Rio de Janeiro, Brazil.
E-mail: [email protected]
4 CNRS UMR 6085, Université de Rouen, Avenue de l’Université, BP.12,
Technopôle du Madrillet, F76801 Saint-Etienne-du-Rouvray, France Received: 13 June 2008 / Accepted: 25 November 2008 Published online: 18 February 2009 – © Springer-Verlag 2009
Abstract: We consider the weakly asymmetric exclusion process on a bounded interval with particles reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. In the case in which the bulk asymmetry is in the same direction as the drift due to the boundary reservoirs, we prove that the quasipotential can be expressed in terms of the solution to a one-dimensional boundary value problem which has been introduced by Enaud and Derrida [16]. We consider the strong asymmetric limit of the quasi-potential and recover the functional derived by Derrida, Lebowitz, and Speer [15] for the asymmetric exclusion process.
1. Introduction The study of steady states of non-equilibrium systems has motivated a lot of work over the last decades. It is now well established that the steady states of non-equilibrium systems exhibit in general long-range correlations and that the thermodynamic functionals, such as the free energy, are neither local nor additive. The analysis of the large deviations asymptotics of stochastic lattice gases with particle reservoirs at the boundary has proven itself to be an important step in the physical description of nonequilibrium stationary states and a rich source of mathematical problems. We refer to [6,14] for two recent reviews on this topic. We consider a boundary driven one-dimensional lattice gas whose dynamics can be informally described as follows. Fix an integer N ≥ 1, an external force E in R and boundary densities 0 < ρ− < ρ+ < 1. At any given time each site of the interval {−N + 1, . . . , N − 1} is either empty or occupied by one particle. In the bulk, each particle attempts to jump to the right at rate 1 + E/2N and to the left at rate 1 − E/2N . To respect the exclusion rule, the particle jumps only if the target site is empty, otherwise nothing happens. At the boundary sites ±(N − 1) particles are created and removed for
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the local density to be ρ± : at rate ρ± a particle is created at ±(N − 1) if the site is empty and at rate 1 − ρ± the particle at ±(N − 1) is removed if the site is occupied. The dynamics just described defines an irreducible Markov process on a finite state space which has a unique stationary state denoted by µ N E . Let ϕ± := log[ρ± /(1 − ρ± )] be the chemical potential of the boundary reservoirs and set E 0 := (ϕ+ − ϕ− )/2. When E = E 0 , the drift caused by the external field E matches the drift due to the boundary reservoirs, and the process becomes reversible. In the limit N ↑ ∞, the typical density profile ρ E under the stationary state µ N E can be described as follows. For each E ≤ E 0 there exists a unique J E ≤ 0 such that 1 ρ+ 1 dr = 1, 2 ρ− Eχ (r ) − J E where χ is the mobility of the system: χ (a) = a(1 − a). The profile ρ E is then obtained by solving ρ E − E χ (ρ E ) = −J E with the boundary condition ρ E (−1) = ρ− . In the same limit N ↑ ∞, the probability of observing a density profile γ different from ρ E can be expressed as µN E {γ } ∼ exp{−N V E (γ )}.
(1.1)
The large deviations functional VE , which also depends on ρ− , ρ+ , is an extension of the notion of free energy in the context of non-equilibrium systems. The free energy of a boundary driven lattice gas has first been derived for the symmetric simple exclusion process by Derrida, Lebowitz and Speer [15] based on the so-called matrix method, introduced by Derrida, which permits to express the stationary state µ N E as a product of matrices. Bertini et al. [4] derived the same result through a dynamical approach which we extend here to the weakly asymmetric case. We consider only the situation E < E 0 for the bulk asymmetry to be in the same direction as the drift due to the boundary. The reversible case E = E 0 lacks interest because the stationary state is product and does not exhibit long range correlations. In contrast, the analysis of the quasi-potential VE for E > E 0 , not treated here, appears a most interesting problem. For instance, a representation of VE as a supremum of trial functionals analogous to (2.14) below seems to be ruled out. In the boundary driven weakly asymmetric exclusion process, for E < E 0 , the quasipotential takes the following form: du γ log γ + (1 − γ ) log(1 − γ ) + (1 − γ )ϕ − log 1 + eϕ VE (γ ) := −1 1 (1.2) ϕ log ϕ − (ϕ − E) log(ϕ − E) − A E , + E
1
where A E is the constant given by AE
1 := log(−J E ) + 2
γ+
γ−
Eχ (r ) 1 ; log 1 − dr Eχ (r ) JE
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313
and where ϕ is the unique solution of the Euler-Lagrange equation ϕ 1 + = γ ϕ (ϕ − E) 1 + eϕ satisfying ϕ(±1) = ϕ± , ϕ > max{0, E}. This result, stated in a different form, has been proved by Enaud and Derrida [16] based on the matrix method. We prove this result in Sect. 4 below by the dynamical approach introduced in [4,5]. We also show that the quasi-potential is convex and lower semi-continuous. In Sect. 5, we show that VE -converges, as E ↓ −∞, to the free energy of the boundary driven asymmetric exclusion process, first derived by Derrida, Lebowitz and Speer [15]. This asymptotic behavior is somewhat surprising since the hydrodynamic time scales at which the weakly asymmetric exclusion process and the asymmetric exclusion process evolve are different. We also prove convergence of the solutions of the Euler-Lagrange equations as the external force E diverges. The dynamical approach followed here permits to compute the fluctuation probabilities (1.1) in great generality, in any dimension and for a large class of processes. However, it is only in dimension one and for very few interacting particle systems that an explicit expression of type (1.2) is available for the non-equilibrium free energy VE . 2. Notation and Results The boundary driven weakly asymmetric exclusion process. Fix an integer N ≥ 1, E ∈ R, 0 < ρ− ≤ ρ+ < 1 and let N := {−N + 1, . . . , N − 1}. The configuration space is N := {0, 1} N ; elements of N are denoted by η so that η(x) = 1, resp. 0, if site x is occupied, resp. empty, for the configuration η. We denote by σ x,y η the configuration obtained from η by exchanging the occupation variables η(x) and η(y), i.e. ⎧ ⎪ ⎨η(y) if z = x (σ x,y η)(z) := η(x) if z = y ⎪ ⎩η(z) if z = x, y, and by σ x η the configuration obtained from η by flipping the configuration at x, i.e. 1 − η(x) if z = x (σ x η)(z) := η(z) if z = x. The one-dimensional boundary driven weakly asymmetric exclusion process is the Markov process on N whose generator L N can be decomposed as L N = L 0,N + L −,N + L +,N , where the generators L 0,N , L −,N , L +,N act on functions f : N → R as (L 0,N (L −,N (L +,N
N2 f )(η) = 2
N −2 x=−N +1
e−E/(2N ) [η(x+1)−η(x)] f (σ x,x+1 η) − f (η) ,
N2 c− (η(−N + 1)) f (σ −N +1 η) − f (η) , f )(η) = 2 N2 c+ (η(N − 1)) f (σ N −1 η) − f (η) , f )(η) = 2
(2.1)
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where c± : {0, 1} → R are given by c± (ζ ) := ρ± e∓E/(2N ) (1 − ζ ) + (1 − ρ± )e±E/(2N ) ζ. Notice that the (weak) external field is E/(2N ) and, in view of the diffusive scaling limit, the generator has been speeded up by N 2 . We denote by ηt the Markov process on N with generator L N and by PηN its distribution if the initial configuration is η. Note that PηN is a probability measure on the path space D(R+ , N ), which we consider endowed with the Skorohod topology and the corresponding Borel σ -algebra. Expectation with respect to PηN is denoted by EηN . Since the Markov process ηt is irreducible, for each N ≥ 1, E ∈ R, and 0 < ρ− ≤ ρ+ < 1 there exists a unique invariant measure µ N E in which we drop the dependence on ρ± from the notation. Let ϕ± := log[ρ± /(1 − ρ± )] be the chemical potential of the boundary reservoirs and set E 0 := (ϕ+ − ϕ− )/2. A simple computation shows that if E = E 0 then the process ηt is reversible with respect to the product measure µN E 0 (η)
=
N −1 x=−N +1
e
ϕN E (x) η(x) 0
1+e
ϕN E (x)
,
(2.2)
0
where ϕN E 0 (x) := ϕ−
N −x N +x + ϕ+ . 2N 2N
On the other hand, for E = E 0 the invariant measure µ N E cannot be written in a simple form. The dynamical large deviation principle. We denote by u ∈ [−1, 1] the macroscopic space coordinate and by ·, · the inner product in L 2 ([−1, 1], du). We set M := {ρ ∈ L ∞ ([−1, 1], du) : 0 ≤ ρ ≤ 1} ,
(2.3)
which we equip with the topology induced by the weak convergence of measures, namely a sequence {ρ n } ⊂ M converges to ρ in M if and only if ρ n , G → ρ, G for any continuous function G : [−1, 1] → R. Note that M is a compact Polish space that we consider endowed with the corresponding Borel σ -algebra. The empirical density of the configuration η ∈ N is defined as π N (η), where the map π N : N → M is given by π (η) (u) := N
N −1 x=−N +1
η(x) 1
1 x 1 x − , + N 2N N 2N
(u),
(2.4)
in which 1{A} stands for the indicator function of the set A. Let {η N } be a sequence of configurations with η N ∈ N . If the sequence {π N (η N )} ⊂ M converges to ρ in M as N → ∞, we say that {η N } is associated to the macroscopic density profile ρ ∈ M. Given T > 0, we denote by D ([0, T ]; M) the Skorohod space of paths from [0, T ] to M equipped with its Borel σ -algebra. Elements of D ([0, T ], M) will be denoted by π ≡ πt (u) and sometimes by π(t, u). Note that the evaluation map D ([0, T ]; M) π → πt ∈ M is not continuous for t ∈ (0, T ) but is continuous for t = 0, T . We denote by π N also the map from D ([0, T ]; N ) to D ([0, T ]; M) defined by π N (η· )t := π N (ηt ). The notation π N (t, u) is also used.
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Fix a profile γ ∈ M and consider a sequence {η N : N ≥ 1} associated to γ . Let be the boundary driven weakly asymmetric exclusion process starting from η N . In [13,21,23] it is proven that as N → ∞ the sequence of random variables {π N (η·N )}, which take values in D ([0, T ]; M), converges in probability to the path ρ ≡ ρt (u), (t, u) ∈ [0, T ] × [−1, 1] which solves the viscous Burgers equation with Dirichlet boundary conditions at ±1, i.e. ⎧ 1 E ⎪ ⎪ ⎨∂t ρ + ∇χ (ρ) = ρ 2 2 , (2.5) ρt (±1) = ρ± ⎪ ⎪ ⎩ρ (u) = γ (u)
ηtN
0
where χ : [0, 1] → R+ is the mobility of the system, χ (a) = a(1 − a), and ∇, resp.
, denotes the derivative, resp. the second derivative, with respect to u. In fact the proof presented in [13,21] is in real line, while the one in [23] is on the torus. The arguments however can be adapted to the boundary driven case, see [18,19,22] for the hydrodynamic limit of different boundary driven models. A large deviation principle for the empirical density can also be proven following [23,25,26], adapted to the open boundary context in [5]. In order to state this result some more notation is required. Fix T > 0 and let T = (0, T ) × (−1, 1), T = [0, T ] × [−1, 1]. For positive integers m, n, we denote by C m,n (T ) the space of functions G ≡ G t (u) : T → R with m derivatives in time, n derivatives in space which are continuous up the the boundary. We improperly denote by C0m,n (T ) the subset of C m,n (T ) of the functions which vanish at the endpoints of [−1, 1], i.e. G ∈ C m,n (T ) belongs to C0m,n (T ) if and only if G t (±1) = 0, t ∈ [0, T ]. Let the energy Q : D([0, T ], M) → [0, ∞] be given by Q(π )
T
= sup H
0
1
1 dt du π(t, u) (∇ H )(t, u)− 2 −1
T
dt 0
1
−1
du H (t, u) χ (π(t, u)) , 2
where the supremum is carried over all smooth functions H : T → R with compact support. If Q(π ) is finite, π has a generalized space derivative, ∇π , and 1 1 T (∇πt )2 Q(π ) = · dt du 2 0 χ (πt ) −1 Fix a function γ ∈ M which corresponds to the initial profile. For each H in C01,2 (T ), let JˆH = JˆT,H,γ : D([0, T ], M) −→ R be the functional given by T JˆH (π ) := πT , HT − γ , H0 − dt πt , ∂t Ht 0
ρ+ T ρ− T 1 T dt πt , Ht + dt ∇ Ht (1) − dt ∇ Ht (−1) − 2 0 2 0 2 0 E T 1 T − dt χ (πt ), ∇ Ht − dt χ (πt ), (∇ Ht )2 . 2 0 2 0 Let IˆT ( · |γ ) : D([0, T ], M) −→ [0, +∞] be the functional defined by IˆT (π |γ ) :=
sup H ∈C01,2 (T )
JˆH (π ).
(2.6)
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The rate functional IT (·|γ ) : D([0, T ], M) → [0, ∞] is given by IˆT (π |γ ) if Q(π ) < ∞ , IT (π |γ ) = ∞ otherwise.
(2.7)
It is proven in [8], for any E in R, that the functional IT (·|γ ) is lower semicontinuous, has compact level sets and that a dynamical large deviations principle for the empirical measure holds. Theorem 2.1. Fix T > 0 and an initial profile γ in M. Consider a sequence {η N : N ≥ 1} of configurations associated to γ . Then, the sequence of probability measures {PηNN ◦ (π N )−1 : N ≥ 1} on D([0, T ]; M) satisfies a large deviation principle with speed N and good rate function IT (·|γ ). Namely, IT (·|γ ) : D ([0, T ]; M) → [0, ∞] has compact level sets and for each closed set C ⊂ D([0, T ]; M) and each open set O ⊂ D([0, T ]; M), 1 log PηNN N →∞ N 1 log PηNN lim N →∞ N lim
π N ∈ C ≤ − inf IT (π |γ ), π ∈C
π N ∈ O ≥ − inf IT (π |γ ). π ∈O
The quasi-potential. From now on we consider only the case E ≤ E 0 = (ϕ+ − ϕ− )/2, where ϕ± = log[ρ± /(1 − ρ± )]. Simple computations, which are omitted, show that the unique stationary solution ρ E ∈ M of the hydrodynamic equation (2.5) can be described as follows. For each E ≤ E 0 there exists a unique J E ≤ 0 such that 1 ρ+ 1 dr = 1. (2.8) 2 ρ− Eχ (r ) − J E The profile ρ E is then obtained by solving ρ E − E χ (ρ E ) = −J E
(2.9)
with the boundary condition ρ E (−1) = ρ− . Note that J E /2 is the current maintained by the stationary profile ρ E . The solution to (2.9) can easily be written in an explicit form, see [16]. We shall however only use, as can be easily checked, that ρ E is strictly increasing and that the inequality J E /E > maxr ∈[ρ− ,ρ+ ] χ (r ) holds for E < 0. Given E ≤ E 0 , the quasi-potential for the rate function IT is the functional VE : M → [0, +∞] defined by VE (ρ) := inf inf IT (π |ρ E ) , π ∈ D ([0, T ]; M) : πT = ρ (2.10) T >0
so that VE (ρ) measures the minimal cost to produce the profile ρ starting from ρ E . Recall that µ N E is the unique invariant measure of the boundary driven weakly asymmetric exclusion process. The following result, which states that the quasi-potential gives the rate function of the empirical density when particles are distributed according to µ N E is proven in [10,20] in the case E = 0. Thanks to Theorem 2.1, the proof applies also to the weakly asymmetric case, see [20] for more details on this topic.
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Theorem 2.2. For each E in R, the sequence of probability measures on M given by N −1 {µ N E ◦ (π ) } satisfies a large deviation principle with speed N and rate function V E . Namely, for each closed set C ⊂ M and each open set O ⊂ M, 1 N log µ N lim ≤ − inf VE (ρ), E π ∈C N →∞ N ρ∈C 1 N lim log µ N E π ∈ O ≥ − inf V E (ρ). N ρ∈O N →∞ In this paper we prove that the quasi-potential VE can be expressed in terms of the solution to a one-dimensional boundary value problem. This result has been obtained in [16] by analyzing directly the invariant measure µ N E through combinatorial techniques; while we here follow instead the dynamic approach [4,5] by characterizing the optimal path, as also described in [17], for the variational problem (2.10). For E < E 0 , let C 1+1 ([−1, 1]) be the set of continuously differentiable functions on [−1, 1] with Lipshitz derivative and set (2.11) F E := ϕ ∈ C 1+1 ([−1, 1]) : ϕ(±1) = ϕ± , ϕ > 0 ∨ E , where, given a, b ∈ R, the notation a ∨ b, resp. a ∧ b, stands for max{a, b}, resp. min{a, b}. Note that F E = F E for E, E < 0. For E < E 0 , E = 0, let G E : M × F E → R be given by 1 G E (ρ, ϕ) := du ρ log ρ + (1 − ρ) log(1 − ρ) + (1 − ρ)ϕ − log 1 + eϕ −1 1 ϕ log ϕ − (ϕ − E) log(ϕ − E) − A E , (2.12) + E where, by convention, 0 log 0 = 0 and A E is the constant given by Eχ (r ) 1 ρ+ 1 . log 1 − dr A E := log(−J E ) + 2 ρ− Eχ (r ) JE
(2.13)
The right-hand side is well defined as J E < 0 and J E /E > maxr ∈[ρ− ,ρ+ ] χ (r ) for E < 0. For E = 0, G0 : M × F E → R is defined by continuity as G0 (ρ, ϕ) 1 du ρ log ρ+(1 − ρ) log(1−ρ)+(1−ρ)ϕ− log 1+eϕ + log ϕ +1 − A0 , = −1
where A0 = log[(ρ+ − ρ− )/2] + 1. For E < E 0 , define the functional S E : M → R by S E (ρ) := sup G E (ρ, ϕ). ϕ∈F E
(2.14)
Note that S E is a positive functional because a simple computation relying on (2.9) shows that 1 ρ 1−ρ (2.15) S E (ρ) ≥ G E (ρ, ϕ E ) = du ρ log + (1 − ρ) log ρE 1 − ρE −1 if ϕ E := log[ρ E /(1 − ρ E )].
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In the special case E = E 0 , as already observed, the weakly asymmetric exclusion process is reversible and the stationary state µ N E 0 is a product measure. In particular, the rate functional S E 0 of the static large deviations principle for the empirical density can be explicitly computed. It is given by 1 ρ 1−ρ . (2.16) S E 0 (ρ) = du ρ log + (1 − ρ) log ρ E0 1 − ρ E0 −1 The Euler-Lagrange equation associated to the variational problem (2.14) is ϕ 1 + = ρ. ϕ (ϕ − E) 1 + eϕ
(2.17)
A function ϕ ∈ F E solves the above equation when it is satisfied Lebesgue a.e. Recalling that the stationary profile ρ E satisfies (2.9) and (2.8), it is easy to check that if ρ = ρ E then ϕ E solves (2.17) and G E (ρ E , ϕ E ) = 0. The analysis of the quasi-potential for the boundary driven symmetric exclusion process, i.e. the case E = 0 of the current setting, has been considered in [5]. In particular it is there shown that V0 coincides with S0 . We prove in this article an analogous statement for any E ≤ E 0 . Theorem 2.3. Let E ≤ E 0 and VE , S E : M → [0, +∞] be the functionals defined in (2.10), (2.14) and (2.16). (i) The functional S E is bounded, convex, and lower semicontinuous on M. (ii) Fix E < E 0 . For each ρ ∈ M there exists in F E a unique solution to (2.17) denoted by (ρ). Moreover S E (ρ) = max G E (ρ, ϕ) = G E (ρ, (ρ)). ϕ∈F E
(2.18)
(iii) The equality VE = S E holds on M. The proof of the last item of the previous theorem is achieved by characterizing the optimal path, as also described in [17], for the variational problem (2.10) defining the quasi-potential. For E < E 0 it is obtained by the following algorithm. Given ρ ∈ M let (ρ) ∈ F E be the solution to (2.17) and define G = e(ρ) /[1 + e(ρ) ]. Let F ≡ Ft (u) be the solution to the viscous Burgers equation (2.5) with initial condition G and set ψ = log[F/(1−F)], note that ψ0 = (ρ) and ψt → ϕ E as t → ∞. Let ρt∗ = −1 (ψt ), i.e. ρt∗ is given by the l.h.s. of (2.17) with ϕ replaced by ψt . Observe that ρ0∗ = ρ and ∗ ; the fact that it ρt∗ → ρ E as t → ∞. The optimal path for (2.10) is then πt∗ = ρ−t is defined on the time interval (−∞, 0] instead of [0, ∞) makes no real difference. As discussed in [4,6], this description of the optimal path π ∗ is related to the possibility of expressing the hydrodynamic limit for the process on N whose generator is the adjoint of L N in L 2 (dµ N E ) in terms of (2.5) via the nonlocal map . The asymmetric limit. Consider the boundary driven asymmetric exclusion process, that is the process on N with generator given by (2.1), where the external field E is replaced by N α and the generator is speeded up by N instead of N 2 . We consider only the case α < 0. According to the previous notation, denote by µ N N α the unique invariant measure of the boundary driven asymmetric exclusion process with external field α N . In the hydrodynamic scaling limit, it is proved in [1] that the empirical density converges to the
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unique entropy solution to the inviscid Burgers equation with Bardos-le Roux-Nédélec boundary conditions [2], namely (2.5) with E/2 replaced by sinh(α/2) and no viscosity. Let ρ a ∈ {ρ− , ρ+ , 1/2} be such that maxr ∈[ρ− ,ρ+ ] χ (r ) = χ (ρ a ). It is not difficult to check that the stationary profile ρ E converges, as E → −∞, to the constant density profile equal to ρ a , which is the unique stationary solution to the inviscid Burgers equation with the prescribed boundary conditions. By using combinatorial techniques, it is shown in [15] that the sequence of probabilN −1 ity measures {µ N N α ◦ (π ) } on M satisfies a large deviation principle with speed N and rate function Sa defined as follows. Let Fa := ϕ ∈ C 1 ([−1, 1]) : ϕ(±1) = ϕ± , ϕ > 0 . (2.19) Note that F E ⊂ Fa . Given ρ ∈ M and ϕ ∈ Fa set Ga (ρ, ϕ) :=
1
−1
du ρ log ρ + (1 − ρ) log(1 − ρ) + (1 − ρ)ϕ − log 1 + eϕ − Aa , (2.20)
in which the constant Aa is Aa :=
max
r ∈[ρ− ,ρ+ ]
log χ (r ) = log χ (ρ a ).
(2.21)
Let Sa (ρ) := sup Ga (ρ, ϕ). ϕ∈Fa
(2.22)
The functional Sa is written in a somewhat different form in [15]. The above expression is however simply obtained by replacing the trial function F in [15] by eϕ /(1 + eϕ ). The advantage of the above formulation is that for each ρ ∈ M the functional Ga (ρ, ·) is concave on Fa . By choosing ϕ = log[ρ a /(1 − ρ a )] as trial function in (2.22) we get a lower bound analogous to (2.15): ρ 1−ρ . du ρ log + (1 − ρ) log Sa (ρ) ≥ ρa 1 − ρa −1
1
Note finally that Sa does not depend on α < 0. We prove in Sect. 5 that the functional S E converges, as E ↓ −∞, to Sa . As discussed in [7, Lemma 4.3], the appropriate notion of variational convergence for rate functionals is the so-called -convergence. Referring e.g. to [11] for more details, we just recall its definition. Let X be a metric space. A sequence of functionals Fn : X → [0, +∞] is said to -converge to a functional F : X → [0, +∞] if the following two conditions hold for each x ∈ X . There exists a sequence xn → x such that limn Fn (xn ) ≤ F(x) (-limsup inequality) and for any sequence xn → x we have limn Fn (xn ) ≥ F(x) (-liminf inequality). Theorem 2.4. Let S E : M → [0, +∞] be as defined in (2.14). As E ↓ −∞, the sequence of functionals {S E } -converges in M to Sa defined in (2.22).
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While the above result deals only with the variational convergence of the quasipotential, it is reasonable to expect also the convergence of the dynamical rate functional. More precisely, the dynamic rate functional (2.7) of the weakly asymmetric exclusion process should converge, in the appropriate scaling, to the one for the asymmetric exclusion process. We refer to [9] for a discussion of this topic and we mention that the above result has been proven in [3] for general scalar conservation laws on the real line. -convergence implies an upper bound for the infimum over open sets and a lower bound for the infimum over compact sets: For each compact set K ⊂ M and each open set O ⊂ M, lim
inf S E (ρ) ≥ inf Sa (ρ),
lim
inf S E (ρ) ≤ inf Sa (ρ).
E→−∞ ρ∈K
E→−∞ ρ∈O
ρ∈K
ρ∈O
The proof of this statement is straightforward and can be found in [11]. Since M is compact, the previous fact and Theorems 2.2, 2.3 (iii), 2.4 provide the following asymptotics for the invariant measure µ N E. Corollary 2.5. For each closed set C ⊂ M and each open set O ⊂ M, 1 N log µ N π lim lim ∈ C ≤ − inf Sa (ρ), E E→−∞ N →∞ N ρ∈C 1 N lim log µ N lim E π ∈ O ≥ − inf Sa (ρ). ρ∈O E→−∞ N →∞ N The last topic we discuss is the asymptotic behavior as, E → −∞, of the solution to the Euler-Lagrange equation (2.17). More precisely, we show that it converges to the unique maximizer for (2.22). Consider the set Fa equipped with the topology inherited from the weak convergence 1 1 of measures on [−1, 1): ϕ n → ϕ in Fa if and only if −1 dϕ n G → −1 dϕ G for any function G in C0 ([−1, 1)), the set of continuous functions G : [−1, 1) → R such that limu↑1 G(u) = 0. The closure of Fa , denoted by F a , consists of all nondecreasing, càdlàg functions ϕ : [−1, 1) → [ϕ− , ϕ+ ] such that ϕ(−1) = ϕ− , limu↑1 ϕ(u) ≤ ϕ+ . By the Helly theorem F a is a compact Polish space. Moreover, if ϕ n → ϕ in Fa then ϕ n (u) → ϕ(u) Lebesgue a.e. Theorem 2.6. Fix ρ ∈ M. There exists a unique φ ∈ F a such that Sa (ρ) = maxϕ∈Fa Ga (ρ, ϕ) = Ga (ρ, φ). Let φ E := (ρ) ∈ F E be the optimal profile for (2.14). As E → −∞ the sequence {φ E } converges to φ in Fa . 3. The Nonequilibrium Free Energy In this section we analyze the variational problem (2.14) and prove items (i) and (ii) in Theorem 2.3. We start by proving an existence and uniqueness result for the EulerLagrange equation (2.17) together with a C 1 dependence of the solution with respect to ρ. We consider the space C 1 ([−1, 1]) endowed with the norm f C 1 := f ∞ + f ∞ , where g∞ := supu∈[−1,1] |g(u)|. For each E < E 0 the set F E defined in (2.11) is a convex subset of C 1 ([−1, 1]); we denote by F E = ϕ ∈ C 1 ([−1, 1]) : ϕ(±1) = ϕ± , ϕ ≥ 0 ∨ E its closure in C 1 ([−1, 1]).
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Theorem 3.1. Let E < E 0 . For each ρ ∈ M there exists in F E a unique solution to (2.17), denoted by (ρ). Furthermore, (i) If ρ ∈ C([−1, 1]; [0, 1]) then (ρ) ∈ C 2 ([−1, 1]). (ii) Let {ρ n } ⊂ M be a sequence converging to ρ in M. Then {(ρ n )} ⊂ F E converges to (ρ) in C 1 ([−1, 1]). Proof. The proof is divided into several steps. Existence of solutions. For E ≤ 0, resp. E ∈ (0, E 0 ), we formulate (2.17) as an integral-differential equation informally obtained multiplying (2.17) by ϕ − E, resp. by ϕ , and integrating the resulting equation. Existence of solutions will be deduced from the Schauder fixed point theorem. Given E < E 0 , ρ ∈ M, and ϕ ∈ F E , let 1 (1) ϕ (u) − E , R (ρ, ϕ; u) := ρ − ϕ(u) 1+e 1 (2) ϕ (u). R (ρ, ϕ; u) := ρ − 1 + eϕ(u) (i)
For a fixed ρ ∈ M and i = 1, 2 we define the integral-differential operators Kρ : F E → C 1 ([−1, 1]) by v u dv exp dw R(1) (ρ, ϕ; w) −1 Kρ(1) (ϕ) (u) := ϕ− + (ϕ+ − ϕ− ) −1 v , 1 (1) dv exp dw R (ρ, ϕ; w) −1
Kρ(2) (ϕ) (u) := ϕ− + E(u + 1)
−1
u
+ (ϕ+ − ϕ− − 2E) −1 1 −1
dv exp
v
−1 v
dv exp
−1
dw R(2) (ρ, ϕ; w) (2)
dw R
· (ρ, ϕ; w)
For E ≤ 0, resp. E ∈ (0, E 0 ), we formulate the boundary problem (2.17) as a fixed (1) (2) point on F E for the operator Kρ , resp. Kρ . Consider first the case E ≤ 0 corresponding to i = 1. Simple computations show (1) (1) that for each ρ ∈ M the map Kρ is a continuous on F E and Kρ F E ⊂ F E . It is also straightforward to check that there exists a constant C1 = C1 (ϕ− , ϕ+ , E) ∈ (0, ∞) such that for any ρ ∈ M, ϕ ∈ F E , and u, v ∈ [−1, 1], d (1) 1 d (1) d (1) Kρ (ϕ) (u) ≤ C1 , Kρ (ϕ) (v) − Kρ (ϕ) (u) ≤ C1 |u − v|. ≤ C1 du dv du (3.1) In particular Kρ(1) F E ⊂ F E . Notice that F E is a closed convex subset of C 1 ([−1, 1]) (1) and, by the previous bounds and the Ascoli-Arzelà theorem, Kρ F E has compact closure in C 1 ([−1, 1]). By the Schauder fixed point theorem we get that for each ρ ∈ M
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there exists ϕ ∗ ∈ F E such that Kρ (ϕ ∗ ) = ϕ ∗ . From (3.1) it follows that ϕ ∗ ∈ F E and standard manipulations show that ϕ ∗ satisfies (2.17) Lebesgue a.e. The case E ∈ (0, E 0 ), corresponding to a fixed point for Kρ(2) , is analyzed in the same way. In this case, it is indeed straightforward to check that there exists a constant C2 = C2 (ϕ− , ϕ+ , E) ∈ (0, ∞) such that for any ρ ∈ M, ϕ ∈ F E , and u ∈ [−1, 1], 1 d (2) K (ϕ) (u) − E ≤ C2 , ≤ du ρ C 2 d (2) K (ϕ) (v) − d K(2) (ϕ) (u) ≤ C2 |u − v|. ρ dv ρ du
(3.2)
Uniqueness of solutions. Let φ ∈ F E , E = 0, be a solution to (2.17); by chain rule the equation 1 φ − E 1 φ log = ρ− ≡ E φ φ (φ − E) 1 + eφ holds Lebesgue a.e. Hence, for each u ∈ [−1, 1], φ (u) − E 1 φ (−1) − E 1 log = log + E φ (u) E φ (−1)
1 . dv ρ(v) − 1 + eφ(v) −1 u
(3.3)
Let φ1 , φ2 ∈ F E be two solutions to (2.17). If φ1 (−1) = φ2 (−1) an application of the Gronwall inequality in (3.3) yields φ1 = φ2 . We next assume φ1 (−1) < φ2 (−1) and deduce a contradiction. Recall that φi > 0 ∨ E and let u := inf{v ∈ (−1, 1] : φ1 (v) = φ2 (v)}, which belongs to (−1, 1] because φ1 (±1) = φ2 (±1) and φ1 (−1) < φ2 (−1). By definition of u, φ1 (u) < φ2 (u) for any u ∈ (−1, u), φ1 (u) = φ2 (u) and φ1 (u) ≥ φ2 (u). Note that the real function (0 ∨ E, ∞) z → E −1 log[(z − E)/z] is strictly increasing. Therefore from (3.3) we obtain φ1 (u) < φ2 (u), which is a contradiction and concludes the proof of the uniqueness. The case E = 0, that was examined in [5], can be treated similarly with −(1/φ ) in place of {(1/E) log[(φ − E)/φ ]} . Claims (i) and (ii). Claim (i) follows straightforwardly from the previous analysis. To prove (ii), let φ n := (ρ n ) ∈ F E . By (3.1), (3.2) and the Ascoli-Arzelà theorem, the sequence {φ n } ⊂ F E is precompact in C 1 ([−1, 1]). It remains to show uniqueness of its limit points. Consider a subsequence n j and assume that {φ n j } converges to ψ in C 1 ([−1, 1]). Since {ρ n j } converges to ρ in M and {φ n j } converges to ψ in C 1 ([−1, 1]), (1) (1) for E ≤ 0, resp. for E ∈ (0, E 0 ), we have that Kρ n j (φ n j ) converges to Kρ (ψ), resp. (2)
(2)
(i)
Kρ n j (φ n j ) converges to Kρ (ψ). In particular, ψ = lim j φ n j = lim j Kρ n j (φ n j ) = (i)
Kρ (ψ) for i = 1, 2. By the uniqueness result, ψ = (ρ). This shows that (ρ) is the unique possible limit point of the sequence {φ n }, and concludes the proof of Claim (ii). Fix a path ρ ≡ ρt (u) ∈ C 1,0 ([0, T ] × [−1, 1]; [0, 1]) and let φ ≡ (ρt )(u) be the solution to (2.17). We prove below that φ belongs to C 1,2 ([0, T ] × [−1, 1]). Note that, by (3.1) and (3.2), for each E < E 0 there exists a constant C ∈ (0, ∞) such that for any (t, u) ∈ [0, T ] × [−1, 1], −1 C ≤ ∇φt (u) ≤ C if E ≤ 0, (3.4) C −1 ≤ ∇φt (u) − E ≤ C if 0 < E < E 0 .
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Lemma 3.2. Let E < E 0 , T > 0, ρ ∈ C 1,0 ([0, T ] × [−1, 1]; [0, 1]), and φ := (ρt ) be the solution to (2.17). Then φ ∈ C 1,2 ([0, T ] × [−1, 1]) and ψ := ∂t φ is the unique classical solution to the linear boundary value problem ⎧ eφt ∇ψt ⎪ ⎨∇ − (t, u) ∈ [0, T ] × (−1, 1) ψt = ∂t ρt φt 2 ∇φt (∇φt − E) (3.5) 1 + e ⎪ ⎩ t ∈ [0, T ]. ψt (±1) = 0 Proof. Fix t ∈ [0, T ]. For h = 0 such that t + h ∈ [0, T ] define ψth (·) by ψth (u) := [φt+h (u) − φt (u)] / h. By Theorem 3.1 (i), ψth (·) belongs to C 2 ([−1, 1]). Set Rth := [ρt+h − ρt ]/ h; from (2.17) it follows that ψ h solves
ψth
φt+h (∇φt + ∇φt+h − E) − ∇ψth ∇φt (∇φt − E) ∇φt (∇φt − E)∇φt+h (∇φt+h − E) eh ψt − 1 eφt − = Rth h 1 + eφt 1 + eφt+h h
(3.6)
for (t, u) ∈ [0, T ] × (−1, 1) with the boundary conditions ψth (±1) = 0, t ∈ [0, T ]. Multiplying the above equation by ψth and integrating in du, using the inequality x(e x − 1) ≥ 0 and an integration by parts we get that ∇ψth h ∇ψt , ≤ − ψth , Rth + ψth , F(φt , φt+h )∇ψth , (3.7) ∇φt (∇φt − E) where F(φt , φt+h ) :=
1 )2 (∇φ
E)2 ∇φ
(∇φt t − t+h (∇φt+h − E) × { φt ∇φt+h (∇φt+h − E)(2∇φt − E) − φt+h ∇φt (∇φt − E)(∇φt+h + ∇φt − E)} .
For each t ∈ [0, T ], lim F(φt , φt+h )∞ = 0.
h→0
(3.8)
Indeed, since ρ ∈ C 1,0 ([0, T ] × [−1, 1]), as h → 0, ρt+h (·) → ρt (·) in C([−1, 1]). By Theorem 3.1 (ii), φt+h (·) → φt (·) in C 1 ([−1, 1]). By the differential equation (2.17), φt+h (·) → φt (·) in C 2 ([−1, 1]). Together with (3.4) this concludes the proof of (3.8). By (3.4), Cauchy-Schwarz, and the Poincaré inequality for the Dirichlet Laplacian in [−1, 1], we obtain from (3.7) that 1 ∇ψth h h h (3.9) ∇ψt , ∇ψt ≤ ∇ψt , C2 ∇φt (∇φt − E) ≤ ψth , ψth 1/2 Rth , Rth 1/2 + F(φt , φt+h )∞ ∇ψth , ∇ψth 1/2 ≤ C ∇ψth , ∇ψth 1/2 Rth , Rth 1/2 + F(φt , φt+h )∞ ∇ψth , ∇ψth 1/2 for some constant C > 0.
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From (3.9) and (3.8) it follows that there exists a constant C > 0 such that lim ∇ψth , ∇ψth ≤ C ∂t ρt , ∂t ρt , t ∈ [0, T ].
h→0
(3.10)
Therefore for each t ∈ [0, T ] the sequence {ψth (·)} is precompact in C([−1, 1]). By taking the limit h → 0 in (3.6) and using (3.8), it is now easy to show that any limit point of {ψth (·)} is a weak solution to (3.5). By the classical theory on one-dimensional elliptic problems, see e.g. [24, IV, §2.1], there exists a unique weak solution to (3.5) which is in fact the classical solution because ∂t ρt (·) belongs to C([−1, 1]). This implies that there exists a unique limit point ψt (·) ∈ C 2 ([−1, 1]). Finally ψ ∈ C 0,2 ([0, T ] × [−1, 1]) by the continuous dependence in the C 2 ([−1, 1]) topology of the solution to (3.5) w.r.t. ∂t ρt (·) in the C([−1, 1]) topology. We are now in a position to prove two statements of the first main result of this article. Proof of Theorem 2.3 (i) and (ii). We start with Claim (i). The case E = E 0 follows from the definition (2.16) of the functional S E 0 . Assume E < E 0 . By the convexity of the map (ρ) = ρ log ρ + (1 − ρ) log(1 − ρ), for each ϕ ∈ F E the functional G E (·, ϕ) is convex and lower semicontinuous on M. Hence, by (2.14), the functional S E , being the supremum of convex lower semicontinuous functionals, is a convex lower semicontinuous functional on M. On theother hand, since the real function (0 ∨ E, ∞) x → x log x − (x − E) log(x − E) /E is strictly concave, the Jensen inequality and ϕ(±1) = ϕ± imply that G E (ρ, ϕ) is bounded by some constant depending only on ϕ± and E. This proves (i). Fix ρ ∈ M. The strict concavity mentioned above and the strict concavity of the real function R x → − log (1 + e x ) yield that the functional G E (ρ, ·) is strictly concave on F E . Thanks to Theorem 3.1, it easily follows that the supremum on the r.h.s. of (2.14) is uniquely attained when ϕ = (ρ). In the proof of the equality between the quasi-potential VE and the functional S E , we shall need the following simple observation. Lemma 3.3. For each ρ ∈ M there exists a sequence {ρ n } ⊂ M converging to ρ in M and such that: ρ n ∈ C 2 ([−1, 1]), ρ n (±1) = ρ± , 0 < ρ n < 1, S E (ρ n ) → S E (ρ). Proof. For E = E 0 , this is obvious from the definition of the functional S E 0 . For E < E 0 , given ρ ∈ M, it is enough to consider a sequence {ρ n } ⊂ C 2 ([−1, 1]) with ρ n (±1) = ρ± and 0 < ρ n < 1, which converges to ρdu a.e. By Theorem 2.3 (ii), Theorem 3.1 (ii), and dominated convergence, S E (ρ n ) = G E (ρ n , (ρ n )) −→ G E (ρ, (ρ)) = S E (ρ). 4. The Quasi-Potential In this section we characterize the optimal path for the variational problem (2.10) defining the quasi-potential VE and conclude the proof of Theorem 2.3 by showing the equality VE = S E . The heuristic argument is quite simple. To the variational problem (2.10) is associated the following Hamilton-Jacobi equation [4,6]. The quasi-potential VE is the maximal solution to δVE δVE δVE 1 E 1 ∇ , χ (ρ) ∇ + , ρ − ∇χ (ρ) = 0 (4.1) 2 δρ δρ δρ 2 2
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with the boundary condition that δVE /δρ vanishes at the endpoints of [−1, 1]. Few formal computations show that S E solves (4.1). To check that S E is the maximal solution one constructs a suitable path for the variational problem (2.10), [4,6]. Since it is not clear how to analyze (4.1) directly, we first approximate, as in [5], paths π ∈ D([0, T ]; M) with IT (π |ρ E ) < ∞ by smooth paths bounded away from 0 and 1 which satisfy the boundary conditions ρ± at the endpoints of [−1, 1]. For such smooth paths we can make sense of (4.1) and complete the proof. In the case E = E 0 , the process is reversible and the picture is well known. The path which minimizes the variational formula defining the quasi-potential is the solution of the hydrodynamic equation reversed in time. The identity between S E 0 and VE 0 follows easily from this principle. The proof presented below for E < E 0 can be adapted with several simplifications. It is enough to set (ρ) = log{ρ E 0 /1 − ρ E 0 } everywhere. Assume from now on that E < E 0 . We first need to recall some notation introduced in [8]. Fix a density profile γ : [−1, 1] → [0, 1] and a time T > 0. Denote by F2 = F2 (T, γ , ρ± ) the set of trajectories π in C([0, T ], M) bounded away from 0 and 1 in the sense that for each t > 0, there exists ε > 0 such that ε ≤ π ≤ 1 − ε on [t, T ], which satisfy the boundary conditions, π0 = γ , πt (±1) = ρ± , 0 ≤ t ≤ T , and for which there exists δ1 , δ2 > 0 such that πt follows the hydrodynamic equation (2.5) in the time interval [0, δ1 ], πt is constant in the time interval [δ1 , δ1 + δ2 ] and πt is smooth in time in the time interval (δ1 , T ]. If the density profile γ is the stationary profile ρ E , the trajectories π in F2 are in fact constant in the time interval [0, δ1 + δ2 ]. Since they are also smooth in time in (δ1 , T ], we deduce that they are smooth in time in the interval [0, T ]. Moreover, since ρ E is bounded away from 0 and 1, there exists ε > 0 such that ε ≤ π ≤ 1 − ε on [0, T ]. Assume that γ = ρ E and recall from the proof of [8, Theorem 4.6] the definition of the sequence of trajectories {πε : ε > 0}. Since a path π in F2 is in fact constant in the time interval [0, b], each πε is smooth in space and time. In particular, let D0 := C ∞,∞ ([0, T ] × [−1, 1]) ∩ F2 .
(4.2)
Theorem 4.6 in [8] can be rephrased in the present context as Theorem 4.1. For each π in D([0, T ], M) such that IT (π |ρ E ) < ∞, there exists a sequence {π n } ⊂ D0 converging to π in D ([0, T ]; M) such that IT (π n |ρ E ) converges to IT (π |ρ E ). The first two lemmata of this section state that, for smooth paths, the functional S E satisfies (4.1). Recall that for ρ ∈ M we denote by (ρ) ∈ F E the unique solution to (2.17). Lemma 4.2. Let E < E 0 , T > 0, π ∈ D0 , and : [0, T ] × [−1, 1] → R be defined by t := log
πt − (πt ). 1 − πt
(4.3)
Then
T
S E (πT ) − S E (π0 ) = 0
dt t , ∂t πt .
(4.4)
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Proof. Let φ ≡ φt (u) := (πt ) (u), (t, u) ∈ [0, T ]×[−1, 1]. By Lemma 3.2, φ belongs to C 1,2 ([0, T ] × [−1, 1]). Since φt (±1) = ϕ± , then ∂t φt (±1) = 0, t ∈ [0, T ]. By Theorem 2.3 (ii), dominated convergence, an explicit computation, and an integration by parts, d d S E (πt ) = G E (πt , (πt )) dt dt = t , ∂t πt + ∂t φt ,
1
φt . + − π t ∇φt (∇φt − E) 1 + eφt
The lemma follows, noticing that the last term vanishes by (2.17). Let M0 :=
ρ ∈ C 2 ([−1, 1]) : ρ(±1) = ρ± , 0 < ρ < 1 .
(4.5)
Lemma 4.3. Let E < E 0 , ρ ∈ M0 , and : [−1, 1] → R be defined by := log
ρ − (ρ). 1−ρ
(4.6)
Then, ∇ , χ (ρ) ∇ − ∇ρ − Eχ (ρ) , ∇ = 0.
(4.7)
Proof. As before we let φ ≡ φ(u) := (ρ) (u), u ∈ [−1, 1]. By Theorem 3.1 (i), φ belongs to C 2 ([−1, 1]). By the definition of in (4.6), statement (4.7) is equivalent to ∇ρ , −∇φ + E + −∇φ , χ (ρ) (−∇φ + E) = 0. The above equation holds if and only if eφ ∇ ρ− , ∇φ − E + ∇ 1 + eφ − ∇φ , χ (ρ)(∇φ − E) = 0.
eφ 1 + eφ
, ∇φ − E
Since eφ(±1) /[1 + eφ(±1) ] = eϕ± /[1 + eϕ± ] = ρ± = ρ(±1), integrating by parts the previous equation, it becomes !" # $ eφ eφ ρ− , φ − 2 − χ (ρ) ∇φ , ∇φ − E = 0. 1 + eφ 1 + eφ At this point the explicit expression for χ given by χ (ρ) = ρ(1 − ρ) plays a crucial role. Indeed, for such χ , eφ eφ eφ − χ (ρ) = − − ρ − (1 − ρ) , (1 + eφ )2 1 + eφ 1 + eφ so that (4.7) is equivalent to eφ ρ− , φ + ∇φ (∇φ − E) 1 + eφ
1−ρ−
which holds true because φ = (ρ) solves (2.17).
eφ 1 + eφ
= 0,
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We next prove the first half of the equality VE = S E . In fact the argument basically shows that any solution to the Hamilton-Jacobi equation (4.1) gives a lower bound on the quasi-potential. Proof of Theorem 2.3: the inequality VE ≥ S E . In view of the variational definition of VE in (2.10), to prove the lemma we need to show that for each ρ ∈ M we have S E (ρ) ≤ IT (π |ρ E ) for any T > 0 and any path π ∈ D ([0, T ]; M) such that πT = ρ. Assume firstly that ρ ∈ M0 and consider only paths π ∈ D0 . Of course the energy Q(π ) of such a path π is finite. In view of the variational definition of IT (π |ρ E ) given in (2.6), (2.7), to prove that S E (ρ) ≤ IT (π |ρ E ), it is enough to exhibit some function H ∈ C01,2 ([0, T ] × [−1, 1]) for which S E (ρ) ≤ JˆT,H,ρ E (π ). We claim that given in (4.3) fulfills these conditions. Let φ ≡ φt (u) := (πt ) (u). Since π ∈ D0 , by Lemma 3.2 ∈ C 1,2 ([0, T ] × [−1, 1]). On the other hand, since πt (±1) = ρ± and φt (±1) = ϕ± , t (±1) = 0, t ∈ [0, T ]; whence ∈ C01,2 ([0, T ] × [−1, 1]). Recalling the definition of the functional JˆT,,ρ E , after an integration by parts, we obtain that T 1 1 dt t , ∂t πt + ∇t , ∇πt − Eχ (πt ) − χ (πt ), (∇t )2 . JˆT,,ρ E (π ) = 2 2 0 By using Lemmata 4.2 and 4.3, since π0 = ρ E , S E (ρ E ) = 0, it follows that JˆT,,ρ E (π ) = S E (ρ), which proves the statement for ρ ∈ M0 and paths π ∈ D0 . Let now ρ ∈ M and consider an arbitrary path π ∈ D ([0, T ]; M) such that πT = ρ. With no loss of generality we can assume IT (π |ρ E ) < ∞. Let {π n } ⊂ D0 be the sequence given by Theorem 4.1. The result for ρ ∈ M0 and paths in D0 , together with the lower semicontinuity of S E , yield IT (π |ρ E ) = lim IT (π n |ρ E ) ≥ lim S E (πTn ) ≥ S E (πT ) = S E (ρ), n→∞
n→∞
which concludes the proof. To prove the converse inequality VE ≤ S E on M, we need to characterize the optimal path for the variational problem (2.10). The following lemma explains which is the right candidate. Denote by C K∞ (T ) the smooth functions H : T → R with compact support. For a trajectory π in D([0, T ], M), let H01 (χ (π )) be the Hilbert space induced by C K∞ (T ) endowed with the scalar product defined by T 1 G, H
1,χ (π ) = dt du (∇G)(t, u) (∇ H )(t, u) χ (π(t, u)). 0
−1
Induced means that we first declare two functions F, G in C K∞ (T ) to be equivalent if F − G, F − G
1,χ (π ) = 0 and then we complete the quotient space with respect to the scalar product. Denote by · 1,χ (π ) the norm associated to the scalar product ·, ·
1,χ (π ) . Repeating the arguments of the proof of Lemma 4.7 in [8], we obtain an explicit expression of the rate function IT (π |γ ) in terms of a solution to an elliptic equation. Lemma 4.4. Fix a trajectory π in D0 . For each 0 ≤ t ≤ T , let Ht be the unique solution to the elliptic equation ∂t πt = (1/2) πt − ∇ {χ (πt ) [(E/2) + ∇ Ht ]} , (4.8) Ht (±1) = 0.
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Then, H is smooth on [0, T ] × [−1, 1] and 1 H 21,χ (π ) . (4.9) 2 We could have used the next lemma to prove the inequality VE ≥ S E ; we presented the separate argument before for its simplicity. On the other hand, (4.11) clearly suggests that the optimal path for the variational problem (2.10) is obtained by taking a path which satisfies (4.10) with K = 0. Recall that (ρ) denotes the solution to (2.17). IT (π |π0 ) =
Lemma 4.5. Let E < E 0 , T > 0, γ ∈ M0 , and π ∈ D0 be such that IT (π |γ ) < ∞. Then, there exists K in C01,2 ([0, T ] × [−1, 1]) such that π is a classical solution to ⎧ E 1 ⎪ ⎪ ⎨∂t πt + ∇χ (πt ) = − πt + ∇ [χ (πt )∇ ((πt ) + K t )] 2 2 (4.10) ⎪πt (±1) = ρ± ⎪ ⎩π = γ . 0 Furthermore, 1 K 21,χ (π ) · (4.11) 2 Proof. Note that γ = π0 because we assume the rate function to be finite. Denote by H the smooth function introduced in Lemma 4.4 and let be as defined in (4.3). We claim that K := − H meets the requirements in the lemma. As before we have that belongs to C01,2 ([0, T ] × [−1, 1]). Hence, K also belongs to this space because H is smooth and vanishes at the boundary of [−1, 1]. The equation (4.10) follows easily from (4.8) replacing H by − K . To prove identity (4.11), consider (4.4) and express ∂t πt in terms of the differential equation in (4.8). Since H = − K , after an integration by parts we get that S E (πT ) − S E (γ ) is equal to T 1 T dt ∇t , ∇πt − E χ (πt ) + dt ∇t , χ (πt )∇ (t − K t ) . − 2 0 0 IT (π |γ ) = S E (πT ) − S E (γ ) +
By Lemma 4.3, the previous expression is equal to T 1 T dt ∇t , χ (πt )∇t − dt ∇t , χ (πt )∇ K t . 2 0 0 Since K = − H , we finally get that 1 1 K 21,χ (π ) = H 21,χ (π ) , 2 2 which, in view of (4.9), concludes the proof. S E (πT ) − S E (γ ) +
We next show how a solution to the (nonlocal) Eq. (4.10) with K = 0 can be obtained by the algorithm presented below the statement of Theorem 2.3. Recall that such algorithm requires to solve (2.17) only for the initial datum and then to solve the ∗ , where ρ ∗ is (local) hydrodynamic equation (2.5). Note indeed that by setting πt∗ := ρ−t ∗ defined in the next lemma, then π solves the differential equation in (4.10) with K = 0. Fix E < E 0 , γ ∈ M0 and set G := e(γ ) /[1 + e(γ ) ]. By Theorem 3.1 the profile G belongs to C 4 ([−1, 1]), it is strictly increasing and satisfies G(±1) = ρ± . Denote by F ≡ Ft (u) ∈ C 1,4 ([0, ∞) × [−1, 1]) the solution to the hydrodynamic equation (2.5) with γ replaced by G. By the maximum principle, ρ− ≤ F ≤ ρ+ .
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Lemma 4.6. Let ψ := log[F/(1 − F)]. Then, ψ belongs to C 1,4 ([0, ∞) × [−1, 1]) and satisfies ∇ψ > 0 ∨ E. Let ρ ∗ ≡ ρt∗ (u) be defined by ρ ∗ :=
1
ψ · + 1 + eψ ∇ψ(∇ψ − E)
(4.12)
Then, ρ ∗ belongs to C 1,2 ([0, ∞) × [−1, 1]), satisfies ρt∗ (±1) = ρ± , 0 < ρ ∗ < 1, and solves ⎧ E 1 ⎪ ∗ ∗ ∗ ∗ ∗ ⎪ ⎨∂t ρt − ∇χ (ρt ) = ρt − ∇ χ (ρt )∇(ρt ) 2 2 ∗ (4.13) ⎪ρt (±1) = ρ± ⎪ ⎩ρ ∗ = γ . 0 Proof. Let ψ : [0, ∞) × [−1, 1] → R be given by ψ = log{F/1 − F} and set τ := sup {t ≥ 0 : ∇ψs (u) > 0 ∨ E for all (s, u) ∈ [0, t] × [−1, 1]} . Since ∇ψ0 = ∇(γ ) > 0 ∨ E, τ > 0 by continuity. We show at the end of the proof that τ = ∞. A straightforward computation shows that ψ solves ⎧ 1 1 − eψ 1 ⎪ ⎪ ∇ψ (∇ψ − E) ⎨∂t ψ = ψ + 2 2 1 + eψ (4.14) ψt (±1) = ϕ± ⎪ ⎪ ⎩ ψ0 = (γ ). Since ψ ∈ C 1,4 ([0, ∞) × [−1, 1]), definition (4.12) yields ρ ∗ ∈ C 1,2 ([0, τ )×[−1, 1]) and ρ0∗ = γ . On the other hand, from (4.14) we deduce that for any t ∈ [0, τ ),
ψt (±1) +
1 − e ϕ± ∇ψt (±1) [∇ψt (±1) − E] = 0. 1 + e ϕ±
Whence, again by (4.12), ρt∗ (±1) =
1 1 − e ϕ± − = ρ± . 1 + e ϕ± 1 + e ϕ±
By using (4.14), a long and tedious computation that we omit shows that ρt∗ , t ∈ [0, τ ), solves the differential equation in (4.13). We next show that 0 < ρ ∗ < 1. Since γ ∈ M0 , there exists δ ∈ (0, 1) such that δ ≤ γ ≤ 1 − δ. We claim that min{ρ− , 1 − ρ+ , δ} ≤ ρ ∗ ≤ max{ρ+ , 1 − ρ− , 1 − δ}. Fix t ∈ (0, τ ) and assume that ρt∗ (·) has a local maximum at u 0 ∈ (−1, 1). Since ρ ∗ solves (4.13), since (ρ ∗ ) solves (2.17) and since ∇ρt∗ (u 0 ) = 0, ρt∗ (u 0 ) ≤ 0, 1 ∗
ρ (u 0 ) − χ (ρt∗ (u 0 )) (ρt∗ )(u 0 ) 2 t ≤ −χ (ρt∗ (u 0 )) ∇(ρt∗ )(u 0 ) ∇(ρt∗ )(u 0 ) − E ρt∗ (u 0 ) −
∂t ρt∗ (u 0 ) =
1 ∗
1 + e(ρt )(u 0 )
.
Assume now that ρt∗ (u 0 ) > 1 − ρ− . Since (ρt∗ ) ≥ ϕ− we deduce ρt∗ (u 0 ) − [1 + ∗ e(ρt )(u 0 ) ]−1 > 1−ρ− −[1+eϕ− ]−1 = 0. As ∇(ρt∗ )(u 0 ) > 0∨E, we get ∂t ρt∗ (u 0 ) < 0.
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In particular, by a standard argument, ρ ∗ ≤ max{ρ+ , 1 − ρ− , 1 − δ}. The proof of the lower bound is analogous. We conclude the proof showing that τ = ∞. Assume that τ < ∞. Since for each t ∈ [0, τ ), ρt∗ belongs to M0 , it follows from (4.12) that (ρt∗ ) = ψt , t ∈ [0, τ ). By Theorem 3.1 (ii), (ρτ∗ ) = ψτ so that ∇ψτ = ∇(ρτ∗ ) > E ∨ 0 because (ρτ∗ ) belongs to F E . By continuity, there exists δ > 0 such that ∇ψt > E ∨ 0 for τ ≤ t < τ + δ. This contradicts the definition of τ . Fix a density profile γ : [−1, 1] → [0, 1], a time T > 0 and consider the solution ρ ∗ to (4.13). Let λt (·) = ρT∗ −t (·). Clearly, λ is the solution to (4.10) in the time interval [0, T ] with K = 0 and initial condition λ0 = ρT∗ . In particular, by (4.11), IT (λ|ρT∗ ) = S E (γ ) − S E (ρT∗ ). In the next lemma we prove that ρT∗ converges to ρ E as T → ∞. Letting T ↑ ∞ in the previous formula, we see that the time reversed trajectory of (4.13) is the natural candidate to solve the variational formula defining the quasi-potential. This argument is made rigorous in the next paragraphs. By standard properties of parabolic equations on a bounded interval, see e.g. [12], as t → ∞, the solution to (2.5) converges, in a strong topology, to the unique stationary solution ρ E . Such convergence implies that the path ρ ∗ , as defined in Lemma 4.6, also converges to ρ E as t → ∞. This is the content of the next lemma. This result will permit to use the time reversal of ρ ∗ as a trial path in the variational problem (2.10). Lemma 4.7. Let E < E 0 , γ ∈ M0 , and ρ ∗ be defined as in Lemma 4.6. As t → ∞, the profile ρt∗ ∈ M0 converges to ρ E in the C 1 ([−1, 1]) topology, uniformly for γ ∈ M0 . Proof. Recall the notation introduced just before Theorem 4.6. Let ρ be the solution to (2.5). In [12, Theorem 4.9] it is shown that, as t → ∞, the profile ρt converges to ρ E in the C 1 ([−1, 1]) topology, uniformly for γ ∈ M0 . By the methods there developed, it is however straightforward to prove this statement in the C 3 ([−1, 1]) topology. In particular, Ft converges to ρ E in the C 3 ([−1, 1]) topology to ρ E so that ψt converges to log[ρ E /(1 + ρ E )] = ϕ E in the C 3 ([−1, 1]) topology uniformly in γ ∈ M0 . Since (ρ E ) = ϕ E , the statement now follows from (4.12). We next show that profiles close to ρ E in a strong topology can be reached with a small cost. Lemma 4.8. Let E < E 0 and δ ∈ (0, 1). Then, there exist T > 0 and constant C = C(E, ρ± , δ) > 0 such that the following hold. For each ρ ∈ C 1 ([−1, 1]) satisfying ρ(±1) = ρ± and δ ≤ ρ ≤ 1 − δ, there exists a path πˆ ∈ D ([0, T ]; M) such that πˆ T = ρ and % %2 IT (πˆ |ρ E ) ≤ C %ρ − ρ E %C 1 . Proof. Simple computations show that T = 1 and the straight path πˆ t = ρ E + t (ρ − ρ E ) meet the requirements. For E = 0, in [5, Lemma 5.7] a more clever path is chosen which yields a bound in terms of the L 2 norm of ρ − ρ E . We can now conclude the proof of Theorem 2.3.
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Proof of Theorem 2.3: the inequality VE ≤ S E . Given ρ ∈ M and δ > 0 we need to find T > 0 and a path π ∗ ∈ D ([0, T ]; M) such that πT∗ = ρ and IT (π ∗ |ρ E ) ≤ S E (ρ) + δ. By Lemma 3.3, there exists a sequence {ρ n } ⊂ M0 converging to ρ in M and such that S E (ρ n ) → S E (ρ). Let ρ ∗,n be the path constructed in Lemma 4.6 with γ replaced by ρ n and pick ε > 0 to be chosen later. By Lemma 4.7, there exists a time T1 = T1 (ε) > 0 independent of n such that ρT∗,n − ρ E C 1 ≤ ε. Whence, by 1 Lemma 4.8, there exists a time T2 > 0, still independent of n, and a path πˆ tn , t ∈ [0, T2 ] such that πˆ 0n = ρ E , πˆ Tn2 = ρT∗,n and IT2 (πˆ n |ρ E ) ≤ βε , where βε vanishes as ε → 0 1 and is independent of n. We now set T := T1 + T2 and let πt∗,n , t ∈ [0, T ] be the path defined by πˆ tn t ∈ [0, T2 ] ∗,n , πt := ρT∗,n −t t ∈ (T2 , T ] which satisfies π0∗,n = ρ E and πT∗,n = ρ n . The covariance of I w.r.t. time shifts, Lemmata 4.5 and 4.6 yield |ρ ∗,n ) IT (π ∗,n |ρ E ) = IT2 (πˆ n |ρ E ) + IT1 (ρT∗,n 1 −· T1
) ≤ βε + S E (ρ n ). ≤ βε + S E (ρ n ) − S E (ρT∗,n 1
(4.15)
Since S E (ρ n ) → S E (ρ) < ∞ and IT (·|ρ E ) has compact level sets, see Theorem 2.1, the bound (4.15) implies precompactness of the sequence {π ∗,n } ⊂ D ([0, T ]; M). Therefore a path π ∗ and a subsequence n j exist such that π ∗,n j → π ∗ in D ([0, T ]; M). In ∗,n particular πT∗ = lim j πT j = lim j ρ n j = ρ. The lower semicontinuity of IT (·|ρ E ) and (4.15) now yield IT (π ∗ |ρ E ) ≤ lim IT (π ∗,n j |ρ E ) ≤ βε + lim S E (ρ n j ) = βε + S E (ρ), j→∞
j→∞
which, by choosing ε so that βε ≤ δ, concludes the proof. 5. The Asymmetric Limit In this section we discuss the asymmetric limit E → −∞ and prove Theorems 2.4 and 2.6. Proof of Theorem 2.4: -liminf inequality. Fix ρ ∈ M and a sequence {ρ E } ⊂ M converging to ρ in M as E → −∞. We show that lim E S E (ρ E ) ≥ Sa (ρ). Let J E be such that (2.8) holds; it is straightforward to check that lim
E→−∞
JE = max χ (r ), r ∈[ρ− ,ρ+ ] E
whence, recalling that A E has been defined in (2.13) and Aa in (2.21), lim A E − log(−E) = max log χ (r ) = Aa . E→−∞
r ∈[ρ− ,ρ+ ]
(5.1)
Fix ϕ ∈ C 1+1 ([−1, 1]) such that ϕ(±1) = ϕ± and ϕ > 0. From (5.1) it easily follows that 1 1 ϕ log ϕ − (ϕ − E) log(ϕ − E) − (A E − Aa ) = 0. (5.2) lim du E→−∞ −1 E
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Recalling (2.14), (2.12) and (2.20), from the convexity of the real function F : [0, 1] → R, F(ρ) = ρ log ρ + (1 − ρ) log(1 − ρ) and (5.2) we get lim S E (ρ E ) ≥ lim G E (ρ E , ϕ) ≥ Ga (ρ, ϕ). E→−∞
E→−∞
The proof of the -liminf inequality is now completed by optimizing on ϕ. Note indeed that the supremum in (2.22) can be restricted to strictly increasing ϕ ∈ C 1+1 ([−1, 1]) such that ϕ(±1) = ϕ± . Proof of Theorem 2.4: -limsup inequality. Fix ρ ∈ M; we need to exhibit a sequence {ρ E } ⊂ M converging to ρ in M as E → −∞ such that lim E S E (ρ E ) ≤ Sa (ρ). We claim that the constant sequence ρ E = ρ meets this condition. Recalling item (ii) in Theorem 2.3, let φ E := (ρ) ∈ F E be the solution to (2.17) in which we indicated explicitly its dependence on E. From the concavity of the real function F : [0, ∞) → R, F(x) = E −1 x log x − (x − E) log(x − E) , E < 0, the Jensen inequality, and (5.1) we deduce 1 1 φ log φ E − (φ E − E) log(φ E − E) − (A E − Aa ) ≤ 0. (5.3) lim du E→−∞ −1 E E Since F E ⊂ Fa and F a is compact, the sequence {φ E } is precompact in Fa . Let now φ ∗ ∈ F a be any limit point of {φ E } and pick a subsequence E → −∞ such that φ E → φ ∗ in Fa . In particular φ E (u) → φ ∗ (u) Lebesgue a.e. Recalling Theorem 2.3 (ii), (2.12), (2.20), and using (5.3) we get that lim
E →−∞
S E (ρ) =
lim
E →−∞
G E (ρ, φ E ) ≤ Ga (ρ, φ ∗ ) ≤ Sa (ρ),
which concludes the proof. Proof of Theorem 2.6. Existence of a maximizer for (2.22) follows from the compactness of F a and from the continuity of Ga (ρ, ·) for the topology of F a . On the other hand, the strict concavity of the function F : [ϕ− , ϕ+ ] → R+ , F(ϕ) = − log(1 + eϕ ), gives the uniqueness of the maximizer. The proof of the convergence of the maximizers follows a variational approach. Given ρ ∈ M and E < 0 we define G E (ρ, ·) : F a → [−∞, +∞) by G E (ρ, ϕ) if ϕ ∈ F E G E (ρ, ϕ) := −∞ otherwise. By [11, Theorem 1.21], with all inequalities reversed since we focus on maximizers instead of minimizers, the convergence of the sequence {φ E } to φ in F a follows from the next three conditions. Fix ρ ∈ M and ϕ ∈ F a then: (i) for any sequence ϕ E → ϕ in Fa , lim E G E (ρ, ϕ E ) ≤ Ga (ρ, ϕ); (ii) there exists a sequence ϕ E → ϕ in F a such that lim E G E (ρ, ϕ E ) ≥ Ga (ρ, ϕ); (iii) φ is the unique maximizer for the functional Ga (ρ, ·) on F a . Proof of (i). We may assume that ϕ E ∈ F E ; the proof of (i) is then achieved by noticing that (5.3) holds also if φ E is replaced by ϕ E .
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Proof of (ii). Assume firstly that ϕ belongs to C 1 ([−1, 1]) and satisfies ϕ(±1) = ϕ± , ϕ > 0. Since (5.2) holds for such ϕ, it is enough to take the constant sequence ϕ E = ϕ. The proof of (ii) is completed by a density argument, see e.g. [11, Rem. 1.29]. More precisely, it is enough to show that for each ϕ ∈ F a there exists a sequence ϕ n ∈ C 1 ([−1, 1]) satisfying ϕ n (±1) = ϕ± , (ϕ n ) > 0, and such that ϕ n → ϕ in F a , Ga (ρ, ϕ n ) → Ga (ρ, ϕ). This is implied by classical results on the approximation of BV functions by smooth ones. As we have already shown (iii), the proof is completed. Acknowledgement. The results within this paper are a natural development of our collaboration with A. De Sole and G. Jona-Lasinio to whom we are in a great debt. L.B. acknowledges the kind hospitality at IMPA and the support of PRIN MIUR.
References 1. Bahadoran, C.: Hydrodynamics and hydrostatics for a class of asymmetric particle systems with open boundaries. Preprint, http://arXiv.org/abs/math/0612094v1, 2006 2. Bardos, C., le Roux, A.Y., Nédélec, J.-C.: First order quasilinear equations with boundary conditions. Comm. Part. Diff. Eqs. 4, 1017–1034 (1979) 3. Bellettini, G., Bertini, L., Mariani, M., Novaga, M.: -entropy cost for scalar conservation laws. Preprint, http://arXiv.org/abs/0712.1198v2 [math. AP], 2007 4. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Macroscopic fluctuation theory for stationary non equilibrium state. J. Stat. Phys. 110, 635–675 (2002) 5. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Large deviations for the boundary driven simple exclusion process. Math. Phys. Anal. Geom. 6, 231–267 (2003) 6. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Large deviation approach to nonequilibrium processes in stochastic lattice gases. Bull. Braz. Math. Soc., New Series 37, 611–643 (2006) 7. Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G., Landim, C.: Large deviations of the empirical current in interacting particle systems. Theory Prob. and Appl. 51, 2–27 (2007) 8. Bertini, L., Landim, C., Mourragui, M.: Dynamical large deviations of boundary driven weakly asymmetric exclusion processes. Preprint, http://arXiv.org/abs/0804.2458v1 [math.PR], 2008 9. Bodineau, T., Derrida, B.: Current large deviations for asymmetric exclusion processes with open boundaries. J. Stat. Phys. 123, 277–300 (2006) 10. Bodineau, T., Giacomin, G.: From dynamic to static large deviations in boundary driven exclusion particles systems. Stoch. Proc. Appl. 110, 67–81 (2004) 11. Braides, A.: -convergence for beginners. Oxford: Oxford University Press, 2002 12. De Groen, P.P.H., Karadzhov, G.E.: Exponentially slow travelling waves on a finite interval for Burgers’ type equation. Electronic J. Diff. Eqs. 1998, 1–38 (1998) 13. De Masi, A., Presutti, E., Scacciatelli, E.: The weakly asymmetric simple exclusion process. Ann. Inst. H. Poincaré, Probabilités 25, 1–38 (1989) 14. Derrida, B.: Non-equilibrium steady states: fluctuations and large deviations of the density and of the current. J. Stat. Mech. Theory Exp. 7, P07023 (2007) 15. Derrida, B., Lebowitz, J.L., Speer, E.R.: Exact large deviation functional of a stationary open driven diffusive system: the asymmetric exclusion process. J. Stat. Phys. 110, 775–810 (2003) 16. Enaud, C., Derrida, B.: Large deviation functional of the weakly asymmetric exclusion process. J. Stat. Phys. 114, 537–562 (2004) 17. Enaud, C.: Processus d’exclusion asymétrique: Effet du désordre, Grandes déviations et fluctuations. Ph.D. Thesis, 2005, available at http://tel.archives-ouvertes.fr/tel-00010955/en/ 18. Eyink, G., Lebowitz, J.L., Spohn, H.: Hydrodynamics of stationary nonequilibrium states for some lattice gas models. Commun. Math. Phys. 132, 253–283 (1990) 19. Eyink, G., Lebowitz, J.L., Spohn, H.: Lattice gas models in contact with stochastic reservoirs: local equilibrium and relaxation to the steady state. Commun. Math. Phys. 140, 119–131 (1991) 20. Farfan, J.: Stationary large deviations of boundary driven exclusion processes. Preprint, 2008 21. Gärtner, J.: Convergence towards Burger’s equation and propagation of chaos for weakly asymmetric exclusion processes. Stoch. Proc. Appl. 27, 233–260 (1988) 22. Kipnis, C., Landim, C., Olla, S.: Macroscopic properties of a stationary non-equilibrium distribution for a non-gradient interacting particle system. Ann. Inst. H. Poincaré, Probabilités 31, 191–221 (1995)
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23. Kipnis, C., Olla, S., Varadhan, S.R.S.: Hydrodynamics and large deviations for simple exclusion processes. Commun. Pure Appl. Math. 42, 115–137 (1989) 24. Mikha˘ılov, V.P.: Partial differential equations. Second edition. Moscow: Nauka, 1983 25. Quastel, J.: Large deviations from a hydrodynamic scaling limit for a nongradient system. Ann. Probab. 23, 724–742 (1995) 26. Quastel, J., Rezakhanlou, F., Varadhan, S.R.S.: Large deviations for the symmetric simple exclusion process in dimensions d ≥ 3. Probab. Th. Rel. Fields 113, 1–84 (1999) Communicated by H. Spohn
Commun. Math. Phys. 289, 335–382 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0755-y
Communications in
Mathematical Physics
Quantization of Diffeomorphism Invariant Theories of Connections with a Non-Compact Structure Group—an Example Andrzej Okołów1,2 1 Institute of Theoretical Physics, Warsaw University, ul. Ho˙za 69,
00-681 Warsaw, Poland. E-mail: [email protected]
2 Department of Physics and Astronomy, Louisiana State University,
Baton Rouge, LA 70803, USA Received: 17 June 2008 / Accepted: 21 October 2008 Published online: 17 March 2009 – © Springer-Verlag 2009
Abstract: A simple diffeomorphism invariant theory of connections with the noncompact structure group R of real numbers is quantized. The theory is defined on a four-dimensional ‘space-time’ by an action resembling closely the self-dual Pleba´nski action for general relativity. The space of quantum states is constructed by means of projective techniques by Kijowski [1]. Except for this point the applied quantization procedure is based on Loop Quantum Gravity methods.
1. Introduction General relativity (GR) expressed in the complex Ashtekar variables [2,3] is a diffeomorphism invariant1 (background independent) theory of connections with S L(2, C) as the structure group. Canonical quantization of GR based on these variables faces two serious obstacles related to the fact that S L(2, C) is non-compact and complex: 1. the non-compactness of the group makes the task of constructing the space of quantum states for the theory very difficult—by now to the best of our knowledge there is no satisfactory solution to this problem; 2. the fact that S L(2, C) is complex implies the existence of some complicated constraints called reality conditions [5] which have to be included in the structure of the resulting quantum theory but, again, by now it is not clear how this should be done. 1 A diffeomorphism invariant theory of connections means a theory in a Hamiltonian form such that (i) its configuration space is a space of connections on a principal bundle P(, G), where is a base manifold, and G is a Lie group (i.e. the structure group of the bundle and the connections), (ii) there exist Gauss and vector constraints imposed on the phase space which ensures, respectively, gauge and diffeomorphism invariance of the theory [4].
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The only solution to these problems known nowadays consists in removing the problems by removing its source, that is, the group S L(2, C): one formulates GR in terms of the Ashtekar-Barbero connection [6] whose structure group SU (2) is compact, thereby the space of quantum states can be easily constructed. Moreover, since the Lie algebra of SU (2) is real there are no reality conditions in this formulation. The above solution, however, cannot be considered as completely blameless. There are at least three reasons for that (see also [7]): (i) the resulting quantum model, i.e. Loop Quantum Gravity (LQG), lacks the Lorentz symmetry which was broken in a nonnatural way while passing from the S L(2, C)-connection to the SU (2) one; (ii) the passage between the connections is not unique—it depends on the so-called Immirzi parameter [8], which at the classical level labels a family of canonical transformations and thereby is physically irrelevant. It turns out, however, that there is no unitary implementation of these transformations at the quantum level. Thus one obtains a family of inequivalent models of LQG depending on the parameter. In particular, black hole entropy derived in LQG framework is known modulo the value of the parameter appearing as a factor in the formula describing the entropy [9–11]; (iii) the scalar constraint of GR expressed in terms of the SU (2)-connection is much more complicated than the one written in terms of S L(2, C) (see e.g. [12,13]). Consequently, the resulting scalar constraint operator [14] is given by quite complicated and implicit formulae; there is a hope that applications of the S L(2, C)-connection can simplify the form of the operator. In our opinion, this is sufficient motivation to look for solutions of both non-compactness and reality conditions problems. In our previous paper [15] we discussed some attempts to construct a space of quantum states for a theory of connections with a non-compact structure group, i.e. attempts to solve the non-compactness problem. We concluded there that slight modifications of the construction known from theories of connections with a compact structure group may be insufficient to obtain a satisfactory result in the non-compact case. Thus to solve the problem one needs rather a radically different idea. Such an idea was presented by Jerzy Kijowski in his review [16] on this author’s Ph.D. thesis and was originally applied [1] more than 25 years ago as an element of a procedure of quantization of a field theory—a procedure which does not depend either on the curvature of space-time or its global structure (in [1] there was quantized a scalar field theory; for an application of the idea to electrodynamics see [17], for an application to quantum scalar field theory on a lattice and quantum many-body systems see [18]). Before we will describe the idea in detail let us say briefly that it consists in using projective techniques to build the space of quantum states instead of inductive ones which are used in the compact case. The goal of this paper is to apply Kijowski’s proposal to quantization of a diffeomorphism invariant theory of connections with a non-compact structure group. However, because of the lack of any experience with quantization of theories of this sort we do not dare to apply it to GR right now. Instead we will find a very simple ‘toy example’ of such a theory and will quantize it combining Kijowski’s idea with the standard methods of LQG. We emphasize that in this paper we do not present any ideas which could solve the problem of reality conditions. The paper is organized as follows: Section 2 contains the definition and brief description of the (classical) ‘toy theory’, Sect. 3 is devoted for quantization of the theory, while in Sect. 4 we present a discussion of the results obtained in Sect. 3. Before we turn to the definition of the ‘toy theory’, let us first describe reasons for which the present LQG methods fail in the non-compact case and present Kijowski’s proposal.
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1.1. Failure of inductive techniques in the non-compact case. The canonical quantization procedure providing the LQG model can be easily generalized (as it was done in [4]) to one applicable to any diffeomorphism invariant theory of connections with a compact structure group. However, this general procedure has not been extended yet to one applicable to theories with non-compact structure groups. The reason is that in order to build the space of quantum states the procedure employs some inductive techniques which do not work well when the structure group is non-compact. Now we are going to describe the problem in detail.2 Let us consider a theory of connections with a (compact or non-compact) structure group G and with local degrees of freedom. Clearly, such a theory possesses an infinite number of degrees of freedom—the configuration space is the infinite dimensional space A of the connections defined on a spatial slice of the spacetime underlying the theory. Our task now is to find the space of kinematic3 quantum states for the theory. Because of complexity of the configuration space we proceed in two steps: first (i) we reduce the number of degrees of freedom to a finite one obtaining a reduced configuration space and construct the space of quantum states for this case, then (ii) we combine the spaces obtained in the first step into the desired space of quantum states of the full theory. The reduction of the infinite number of the degrees of freedom proceeds as follows (for more details see e.g. [4,19]). One fixes a graph4 γ embedded in and defines the following equivalence relation on A: connections A1 , A2 ∈ A are said to be equivalent, A1 ∼γ A2 , if for every edge e of the graph γ , h e (A1 ) = h e (A2 ), where h e (A) is the holonomy of (i.e. parallel transport defined by) the connection A along e. Then Aγ := A/ ∼γ
(1.1)
is a finite dimensional space considered to be a reduced configuration space. An important fact is that Aγ is a manifold isomorphic to G N [20], where N is the number of the edges of γ , hence Aγ is (non-)compact if and only if G is (non-)compact. Now, the space Hγ of kinematic states for the reduced case is the Hilbert space L 2 (Aγ , dµγ ),
(1.2)
where dµγ is a measure on Aγ induced by the Haar measure on the corresponding G N . Thus the first step of the procedure is done. Let us now consider the second step. An important fact here is that (under some technical assumption) graphs embedded in form a directed set (Gra, ≥). Thus the family {Hγ } of the Hilbert spaces is labeled by the directed set and it is very tempting to endow the family with the structure of an inductive family for the inductive limit H := lim Hγ − →
(1.3)
2 For an equivalent description in terms of measures on the configuration space of (generalized) connections see [15]. 3 The term ‘kinematic’ refers here to the fact that in general the quantum states we are going to construct cannot be called physical since we expect the existence of some constraints—the physical quantum states are supposed to be extracted from kinematic ones by the Dirac procedure. 4 A graph is here a collection of a finite number of oriented edges. In particular one edge e also forms a graph {e}.
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of such a family would also be a Hilbert space which could serve as the space of kinematic quantum states we are looking for. In order to transform the family {Hγ } into an inductive one we have to define, for every pair γ ≥ γ , a linear map pγ γ : Hγ → Hγ preserving the scalar product. Moreover, for every triple of graphs such that γ ≥ γ ≥ γ the corresponding maps have to satisfy the following consistency condition: pγ γ = pγ γ ◦ pγ γ . So far we know how to construct the embeddings { pγ γ } only in the case when the structure group G is compact. In a particular case, when an edge e does not intersect the edges of a graph γ and γ := γ ∪ {e} we have γ ≥ γ and the embedding pγ γ is of the following form: Hγ → pγ γ := ⊗ I ∈ Hγ ,
(1.4)
where I is a constant function on Ae of value equal to 1. The definition of pγ γ is correct because (i)Aγ = Aγ × Ae and (ii) the function ⊗ I is square integrable on Aγ for the space is compact. It is clear now that in the case of a non-compact structure group this construction has to break down since the function I is not square integrable any longer. The question now is whether we can use any other function instead of I . Consider then a graph γ and edges {ei } (i = 1, 2, 3) such that e3 = e1 ◦ e2 (where ‘◦’ means the composition of the edges) and each ei does not intersect the edges of γ . Denote γi := γ ∪ {ei }, i = 1, 2, 3 and γ4 := γ ∪ {e1 } ∪ {e2 }. Note that the graph γ4 can be obtained from γ in two ways: (i) by adding to γ the edge e3 in the first step (resulting in the graph γ3 ) and by splitting the edge e3 into e1 and e2 in the second step or (ii) by adding the edge e1 to γ in the first step and by adding the edge e2 to the just obtained graph γ1 in the second one. Since the directing relation on the set Gra is defined in such a way (see e.g. [4]) that γ4 ≥ γ3 ≥ γi ≥ γ , the two ways of transforming γ to γ4 give rise to pγ4 γ3 ◦ pγ3 γ = pγ4 γ1 ◦ pγ1 γ .
(1.5)
Let the map p(∪{ei }) corresponding to adding the edge ei to a graph be of the following form: H → p(∪{ei }) := ⊗ i ∈ H∪{ei } , where i = 0 is a square integrable function on A{ei } ∼ = G which is supposed to replace the function I appearing in (1.4). Some methods commonly used in LQG (see again [4]) allow us to transform (1.5) to the following condition5 3 (g1 g2 ) = 1 (g1 )2 (g2 )
(1.6)
which has to be satisfied for every g1 , g2 ∈ G. Now we restrict our considerations to edges {ei } which are pairwise diffeomorphic. Since our goal is the space of quantum states equipped with an action of diffeomorphisms of on it, it is difficult to avoid an assumption that 1 = 2 = 3 . This conclusion 5 The equation makes sense if we treat the functions { } as ones on G. i
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and Eq. (1.6) mean that 3 defines a homomorphism from G into the commutative multiplicative group C \ {0} that is, a representation of G on C. Moreover, to define some operators (like e.g. area operator [21]) on the space of quantum states we usually use derivations. Because of that it is reasonable to assume that the function 3 , that is, the homomorphism is also differentiable. Now one can show that if G is semi-simple (like e.g. S L(2, C) or SU (1, 1) appearing in 2 + 1 gravity) then the homomorphism has to be trivial which means that 3 = I . It seems then that if we insist to construct the space of quantum states via the inductive techniques we have to use the function I , which means that the measure dµγ has to be (proportional to) a probability measure on Aγ . The problem is that by now nobody was able to show that in the non-compact case (i) there exist probability measures {dµγ } on {Aγ } such that they would provide us with a space of quantum states equipped with a non-trivial unitary action of the Yang-Mills gauge group and the diffeomorphism group, which are symmetries of GR and that (ii) there is a unique (or at least ‘natural’) choice of such a measure. Thus, in the (most interesting for us) case of semi-simple non-compact structure groups, we are not able to use the inductive limit to ‘glue’ the spaces {Hγ } into the desired Hilbert space.6 1.2. Projective techniques. The cornerstone of Kijowski’s proposal is the relation between graphs γ ≥ γ seen as the relation ‘system-subsystem’. Once γ is recognized as a subsystem of γ the validity of application of the inductive techniques to the construction of the space of quantum states becomes not so obvious as before. Moreover, this new point of view suggests strongly that projective techniques should be used instead of the inductive ones. Kijowski explains his point of view as follows [16]: “(...) to construct7 the Hilbert space by the inductive limit means to associate in a unique way every state of a subsystem (a small graph) with a state of a system (a large graph). This is not a natural procedure: in order to define such an embedding we have to choose arbitrarily a physical state corresponding to those degrees of freedom of the large system which are neglected while defining its subsystem. If the configuration space encompassing all degrees of freedom is compact then the topology suggests a unique choice: the physical state should be described by a constant wave function on the neglected degrees of freedom. However, in a general case of a non-compact configuration space there is no natural choice of such a state.8 Thus, in my opinion, one should try to base the construction on projective limits. From the physical point of view it means that we associate in a unique way every state of the system with a (mixed, in general!) state of its subsystem—the latter state is obtained by ‘forgetting’ about the neglected degrees of freedom. Such a ‘forgetting’ operator is well defined and it was introduced for the first time 6 It turns out that in the non-compact case it is possible to use an orthogonal sum to ‘glue’ the spaces {Hγ } [22] into a large Hilbert space, but then there exist obstacles [15] which do not allow us to define any acceptable representation of classical observables on the large Hilbert space. 7 The original text in Polish was translated into English by this author. 8 This statement is, in fact, very important since it gives the most general argument against applying inductive techniques in the non-compact case. Recall that in the previous subsection we showed that inductive techniques require the constant function on the structure group to be square integrable. This conclusion is not general, since it is based on the assumption that an edge can be expressed as a composition of two other edges. If one drops this assumption (e.g. by using so-called almost analytic loops introduced in Subsection 3.2.2) then there is no way to get Equation (1.6) which is indispensable for the conclusion.
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in the well known analysis of the Einstein-Rosen-Podolsky ‘experiment’: a mixed state of one particle is obtained here from a two-particle state by ‘forgetting’ about the other particle. This procedure matches our basic physical intuition concerning the relation ‘system-subsystem’.” This suggests the following strategy of constructing the space of (kinematic) quantum states for a theory of connection with any (compact or non-compact) structure group: (i) given graph γ , define the space Dγ of mixed states as the space of density matrices on the Hilbert space Hγ , next (ii) by defining appropriate projections πγ γ : Dγ → Dγ (i.e. ‘forgetting’ operators) endow the set {Dγ } with the structure of a projective family and, finally, (iii) define the desired space D of kinematic quantum states as the projective limit of the family. It is important to note that the resulting space D of quantum states is not a Hilbert space—it is a convex set and can be interpreted as a space of mixed states of the quantized theory. 2. Classical Theory The present section is devoted for the definition and brief description of a classical theory of connections with a non-compact structure group—this theory will be quantized in the sequel of the paper. Since we do not have any experience in quantizing theories of this sort we would like to find a very simple example of such a theory. On the other hand, we would like to deal with an example related to general relativity (GR) as closely as possible. To satisfy the former wish we will choose a theory whose structure group is the group R of real numbers which seems to be the simplest non-compact Lie group. To fulfill the latter one we will define the theory by means of an action resembling the Pleba´nski action for GR. We will begin this section by introducing the Lagrangian formulation of the theory. Then, since the quantization procedure we are going to apply is closely related to canonical quantization, we will describe the Hamiltonian framework of the theory including the algebra of constraints. Finally, we will briefly compare the theory to GR. 2.1. Lagrangian formulation. Let us consider a trivial principle bundle P := M × R, where the set R of real numbers equipped with the addition plays the role of the (noncompact, commutative) structure group of the bundle, and the base manifold M is 4-dimensional and real-analytic. We assume also that there exists a 3-dimensional real analytic manifold such that M = × R. Let A be a connection on the bundle P. Clearly, it can be represented by a real oneform on M, which will be denoted by A also. Consider now a theory, called in the sequel the ‘toy theory’, given by the following action: 1 S[A, σ, ] := (2.1) σ ∧ F − σ ∧ σ, 2 M where F = d A is the curvature of the connection A, σ is a two-form on M valued in the Lie algebra of R (which will be naturally identified9 with R) and is a real valued function on the manifold. 9 In this case the map exp from the Lie algebra into the group R is a bijection which provides us with the identification.
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2.2. Hamiltonian formulation. The action (2.1) can be expressed as 1 1 S= (σαβ Fµν − σαβ σµν ) d x α ∧ d x β ∧ d x µ ∧ d x ν 4 M 2 1 1 = (σαβ Fµν − σαβ σµν )˜ αβµν d x 4 , 4 M 2 where (x α ), (α = 0, 1, 2, 3) are (local) coordinates on M, and ˜ αβµν is the Levi-Civita density on M. Taking advantage of the assumption M = × R and treating as a ‘space-like’ slice of M and R as a ‘time’ we assume that the coordinates (x i ) (i = 1, 2, 3) are (local) coordinates on , while x 0 ≡ t is a coordinate on R. Then 1 dt d x 3 [ E˜ i A˙ i − (−A0 ∂i E˜ i − σ0i ˜ i jk F jk + 2σ0i E˜ i )], (2.2) S= 2 R where A˙ i := ∂t Ai , ˜ i jk := ˜ 0i jk is the Levi-Civita density on and 1 E˜ i (t, x l ) := ˜ i jk σ jk (t, x l ) 2 is, for every fixed t, a vector density on . Thus we obtain a Hamiltonian 1 i ˜ H [ E , Ai , A0 , σ0i , ] = − d x 3 (A0 ∂i E˜ i + σ0i ˜ i jk F jk − 2σ0i E˜ i ). 2
(2.3)
(2.4)
It follows from (2.2) that E˜ i (t, x j ) is the momentum canonically conjugated to the configuration variable Ai (t, x j ) as an R-connection on (i.e. a connection on the trivial principle bundle ×R), while the variables A0 , σ0i and are just Lagrange multipliers. Thus the phase space of the theory is P := E˜ × A, where E˜ is the space of all vector densities on and A is the space of all R-connections on . The Poisson bracket between a pair (ξ, ζ ) of functions on the phase space is of the standard form δξ δζ δξ δζ {ξ, ζ } = . (2.5) − dx3 δ Ai (x) δ E˜ i (x) δ E˜ i (x) δ Ai (x) Evidently, the Hamiltonian (2.4) is a sum of constraints. One of the constraints turns out to be easily solvable. Indeed, the variation of the Hamiltonian with respect to gives us σ0i E˜ i = 0. The general solution of the equation is σ0i = –1 ˜i jk N j E˜ k , where –1 ˜i jk is the Levi-Civita density on of weight −1, and N i (t, x j ) is, for every fixed t, a vector field on . Setting the solution to (2.4) we obtain i i ˜ H [ E , Ai , A 0 , N ] = − d x 3 (A0 ∂i E˜ i + N i E˜ j Fi j ), (2.6)
Ni
where is a Lagrange multiplier, Fi j is the curvature two-form of Ai , while the role of the other variables remains unchanged. The quantization of the ‘toy theory’ will be based on the Hamiltonian (2.6).
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2.3. Constraints and gauge transformations. The Hamiltonian (2.6) is a sum of the following constraints: CG ( E˜ i , A j ) := ∂i E˜ i , Ck ( E˜ i , A j ) := E˜ j Fk j .
(2.7)
In canonical theory constraints play a double role [23]: (i) they define a physical subspace of the phase space of the theory and (ii) generate gauge transformations on the phase space. Here the physical subspace is given by the equations CG ( E˜ i , A j ) = 0, and Ck ( E˜ i , A j ) = 0,
(2.8)
which possess the following large class of solutions: {( E˜ i (A j , g), A j )},
(2.9)
E˜ i (A j , g) = g ˜ i jk F jk ,
(2.10)
where
and g is a smooth function on such that
˜ i jk F jk ∂i g = 0.
(2.11)
It is easy to give a natural interpretation for the gauge transformations resulting from the constraints (2.7) if they are combined into functions CG () := d x 3 CG ( E˜ i , A j ), : → R, (2.12) CDiff ( N ) := d x 3 [N k Ck ( E˜ i , A j ) − (N i Ai ) CG ( E˜ i , A j )]. (2.13)
Then the gauge transformations are described by the following differential equations, respectively: d Ai = {Ai , CG ()} = −∂i , dτ d ˜i E = { E˜ i , CG ()} = 0, dτ and d Ai = {Ai , CDiff ( N )} = (L N A)i , dτ d ˜i ˜ i, E = { E˜ i , CDiff ( N )} = (L N E) dτ where L N denotes the Lie derivative along the vector field N . Integrating the above formulas we obtain, respectively, ˜ E˜ (τ ) = E,
(2.14)
˜ E˜ (τ ) = χτ ∗ E,
(2.15)
A(τ ) = A − τ d, and ∗ A(τ ) = χ−τ A,
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where χτ is an element of the one-parameter family of diffeomorphisms on generated by the vector field N . We easily recognize the former transformations as a Yang-Mills gauge transformation generated by the function , and the latter ones—as an action of the diffeomorphism. Thus it is justified to call CG () the Gauss constraint, and CDiff ( N )—the diffeomorphism one. The Poisson bracket (2.5) between functions (2.12) and (2.13) are of the following form: {CG (), CG ( )} = 0, {CG (), CDiff ( N )} = −CG (L N ), = CDiff ([ N , M]). {CDiff ( N ), CDiff ( M)}
(2.16)
This means that the constraints are of the first class and, for the Hamiltonian it is the sum of the constraints; each of them is preserved by the time evolution. Let us finally comment on the physical degrees of freedom of the ‘toy theory’. Before the constraints (2.8) are solved there are 6 degrees of freedom per point in . Since there are 4 constraints then taking into account the gauge transformations the number of true physical degrees of freedom seems to be reduced to 6 − 2 · 4 = −2 per point which sounds worrisome.10 A more detailed analysis (see Appendix A) shows however, that the ‘toy theory’ possesses an uncountable number of physical degrees of freedom (the conclusion is true at least in the case when = R3 ). 2.4. The ‘toy theory’ versus general relativity . The form of the action (2.1) is fully analogous to the form of the well known Pleba´nski self-dual action [24]: 1 S[A AB , AB , ABC D ] = AB ∧ FAB − ABC D AB ∧ C D , (2.17) 2 M where F AB (the index11 A = 0, 1) is the curvature form of an S L(2, C)-connection A AB on M, AB is a two-form valued in the Lie algebra of S L(2, C), and ABC D = (ABC D) is a symmetric spinor field on the manifold. Performing the Legendre transformation and solving the constraint (AB 0i E˜ iC D) = 0,
E˜ i AB := ˜ i jk jkAB ,
one obtains a Hamiltonian [5,25]: H [ E˜ i AB , Ai AB , A0AB , N i , –1N˜ ] d x 3 [ A0AB (Di E˜ i BA ) + N i E˜ j AB FiBj A + –1N˜ E˜ i AB E˜ j BC FiCj A ], =−
(2.18)
Di E˜ i AB := ∂i E˜ i AB + Ai AC E˜ iCB − AiCB E˜ i AC .
(2.19)
where
The canonical variables called complex Ashtekar variables are the S L(2, C)-connection Ai AB on as the configuration variable and the vector density E˜ i AB valued in the Lie 10 This issue, overlooked originally by this author, was pointed out to him by Prof. Abhay Ashtekar. 11 To raise and lower the indices one uses the antisymmetric bilinear form
AB .
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algebra of S L(2, C) as the momentum, while the time component A0AB of the connection form A AB , the vector field N i on and the density –1N˜ on of weight −1 are Lagrange multipliers. The differences between the forms of the Hamiltonians (2.6) and (2.18) originate merely in the differences between the structure groups R and S L(2, C). More precisely, ˜ be a subgroup of S L(2, C) isomorphic to R. We restrict let R ˜ • the phase space of GR to fields ( E˜ i AB , A jCD ) valued in the Lie algebra of R; ˜ • the gauge group S L(2, C) to R. In this way we obtain a theory describing the 1 + 1 degenerate sector of GR [26]. Note that after the restriction the Ashtekar variables can be expressed as E˜ i AB = E˜ i M AB and A jCD = A j M CD , ˜ It is clear that the two last terms where M AB is a non-zero vector in the Lie algebra of R. in (2.19) reduce now to zero and the covariant derivative becomes equal to the usual one. Moreover, the scalar constraint vanishes: E˜ i AB E˜ j BC FiCj A = E˜ i E˜ j Fi j M AB M BC M CA = 0. Assuming that M AB M BA = 1 we see that the Hamiltonian (2.18) reduces to (2.6). We conclude that the ‘toy theory’ describes the 1 + 1 degenerate sector of GR. 3. Quantization of the ‘Toy Theory’ Combining ideas of [1] with the methods of LQG we propose the following sequence of steps as a strategy of quantization of the ‘toy theory’: 1. First, we will introduce a method aimed at reducing the degrees of freedom of the theory. (a) We will begin by constructing a Lie algebra of elementary classical variables obtaining as a result the so-called Ashtekar-Corichi-Zapata (ACZ) algebra A [27] consisting of some (Yang-Mills gauge invariant) functions on A and ‘momentum’ (flux) operators. This algebra will correspond to Yang-Mills gauge invariant functions on the phase space P = E˜ × A of the theory and will encompass all relevant degrees of freedom. (b) The reduction will consist in distinguishing a subalgebra Aλ ⊂ A generated by some smooth functions on a reduced configuration space AL˜ and ‘momentum’ operators (AL˜ will be defined analogously to the reduced configuration space Aγ given by (1.1)). We will call the algebra Aλ a reduced classical system. The result of this step will be a family {Aλ }λ∈ of reduced classical systems labeled by a directed set (, ≥). 2. In this step we will define the relation ‘system-subsystem’ among members of the family {Aλ }λ∈ : we will observe that if λ ≥ λ then Aλ ⊂ Aλ and consequently we will call Aλ a subsystem of Aλ . This step will also contain a detailed analysis of the relation whose conclusions will be necessary for the further steps of the procedure. 3. Then we will quantize canonically every system Aλ obtaining a quantum system Sλ which should be considered as a ‘reduced’ quantum ‘toy theory’. This step will ˆ λ, be done as follows: we will extend the Lie algebra Aλ to an operator ∗-algebra A
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ˆ λ on a Hilbert space Hλ , whose vectors will then we will define a representation of A represent pure states of Sλ . This will provide us with a space Dλ of mixed states of the quantum system represented by density matrices on Hλ . By the very construction, the states in Dλ will be invariant under the Yang-Mills gauge transformations. 4. Next, we will organize the systems {Sλ } into a large quantum system S corresponding to the infinite dimensional phase space P quotiented by the Yang-Mills gauge transformations (2.14): (a) For every pair λ ≥ λ we will construct a projection πλλ : Dλ → Dλ (that is, the ‘forgetting’ operator) promoting the family {Dλ } into a projective one {Dλ , πλλ }. The projective limit D of the family will give us the desired space of (Yang-Mills gauge invariant) quantum states for the large system S; (b) Given λ, elements of Dλ define positive linear functionals on the C ∗ -algebra Bλ ∗ : of bounded operators on Hλ .12 This implies the existence of dual maps πλλ ∗ Bλ → Bλ such that {Bλ , πλλ } is an inductive family. Its inductive limit B will play the role of the algebra of quantum observables, which can be evaluated on the states in D. Thus the quantum system S will consist of the space D and the C ∗ -algebra B. 5. Finally, we will show that there exists an action of the diffeomorphisms of on the space D which naturally corresponds to the action (2.15) of the diffeomorphism constraint on the phase space. The space Dph of physical states of the ‘toy theory’ will be defined as the set of diffeomorphism invariant elements of D. The result of this step will be a quantum system Sph := (Dph , B) considered as the quantum ‘toy theory’. It turns out that in the particular case of the ‘toy theory’ the algebras {Aλ } i.e. the reduced classical systems can be defined in two different ways: they can be obtained by reduction of either (i) only configuration degrees of freedom or (ii) both momentum and configuration ones. In the first case the algebra Aλ will describe an infinite number of degrees of freedom, while in the second one— a finite one. Quantization based on the first method as a simpler one will be presented in this paper, while quantization based on the second one will be described in the companion paper [36]. Let us emphasize that the quantum ‘toy theory’ will consist merely of the C ∗ -algebra B of quantum observables and the space Dph of diffeomorphism invariant states on it without any Hilbert space. In principle, one can obtain a Hilbert space for this theory once a state in Dph is distinguished—it is enough then to use the GNS construction to get a representation of B on a Hilbert space as was done in [1]. However, in the case of the ‘toy theory’ there seems to be no natural way to single out any state13 in Dph . 3.1. Step 1: Reduction of degrees of freedom. To obtain the full information about a field configuration we need to measure an infinite number of quantities corresponding to the degrees of freedom of a theory, hence we need an infinite number of measuring instruments. Thus, in general, the reduction of degrees of freedom is performed by restricting oneself to some (usually finite) number of the instruments. Let us then begin Step 1 of the quantization procedure by defining the instruments. 12 D does not coincide with the space of all states on B (i.e. D form a set of normal states), however λ λ λ the states in Dλ separate points in Bλ . 13 The theory analyzed in [1] possesses a non-vanishing Hamiltonian, which can be used to single out a state of minimal energy; in the ‘toy theory’ Hamiltonian is a sum of constraints and every state in Dph is annihilated by it.
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3.1.1. Measuring instruments. Let ϕ and h be instruments which can be used to measure values of the fields, respectively, E˜ and A on . We assume also that the instruments are imperfect in the sense that their sensitivity varies from point to point—the numerical values of corresponding measurements are given by [1] ˜ = ϕ( E) Ai h˜ i , E˜ i ϕi and h(A) =
where ϕ = ϕi d x i is a differential one-form on , and h˜ i is a vector density (in general, both ϕ and h˜ can be of distributional character). The class of instruments, as defined above, is too broad for quantization of our theory. Following LQG methods [21,27,28] we restrict ourselves to instruments described by the following formulas: ˜ = ∗ E˜ f, ϕ S, f ( E) S (3.1) h l (A) = A, l
where S is a bounded oriented surface (two-dimensional submanifold) in , ∗ E˜ is a two-form on defined by the Levi-Civita form –1 ˜i jk of weight −1, ∗ E˜ := E˜ i –1 ˜i jk d x j ∧ d x k , f is a real (smooth) function on S, and l is an oriented loop embedded in . The quanti˜ and h l (A) are nothing else but, respectively, a flux of the field E˜ across the ties ϕ S, f ( E) surface S [21] and the holonomy of the connection A along the loop l [15]. The latter one obviously satisfies h l◦l = h l + h l ,
(3.2)
where l ◦ l is the composition of the loops l and l . It turns out that, in order to force the quantization machinery to work, we have to impose some rather technical assumptions on the manifold , the surface and the loop. We assume then that 1. is a real analytic manifold; 2. the surface S is an analytic submanifold of ; 3. the loop l is piecewise analytic and bases at an arbitrary but fixed point14 y ∈ . These assumptions are common in the LQG literature (see e.g. [20,21,27])—they will allow us to define the ACZ algebra A. However, the quantization method we are going to apply requires stronger assumptions15 which will be imposed on the measuring instruments later on. One can easily realize that two different loops can define the same instrument h. To obtain an unambiguous labeling of this sort of instruments one introduces the notion of a hoop [20]. Denote by L y the set of all piecewise analytic loops based at y and consider 14 As it will be shown later, no particular choice of the point y effects the quantization procedure (see the discussion at the beginning of Subsect. 3.5). 15 The assumptions just presented are sufficient for the other quantization method which will be described in [36].
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the set of the instruments {h l } given by all the elements of L y . Now, two loops l, l ∈ L y are said to be holonomically equivalent, l ∼ l , if and only if for every connection A ∈ A: h l (A) = h l (A). Denote by l˜ the equivalence class of loop l and call it a hoop. The set of all hoops HG := L y /∼ is an Abelian group called the hoop group with the group action given by ◦ l , l˜ ◦ l˜ = l where l ◦ l is a composition of the loops l and l . Clearly, there is one-to-one correspondence between elements of HG and the instruments {h l }. Now we will use the symbol h l˜ ≡ h l also. Note finally that all the measuring instruments under consideration produce YangMills gauge invariant outcomes which easily follow from (2.14)—this is a reason why we have used loops to define the instrument {h l } instead of edges (of a graph). 3.1.2. ACZ algebra. In the present subsection we are going to define the algebra of elementary variables encompassing all the (Yang-Mills gauge invariant) degrees of freedom. From the technical point of view our task consists merely in adapting the original construction of ACZ-algebra [27] to the ‘toy theory’. Obviously, what we call measuring instruments are nothing else but functions on the phase space of the ‘toy theory’. Therefore we can calculate a Poisson bracket between the instruments [21,27]: {h l I , h l J } = 0 = {ϕ S I , f I , ϕ S J , f J }, fˇ {ϕ S, f , h l } = − , 2
(3.3) (3.4)
where fˇ is a real number (a method of finding it will be presented below). Let us also emphasize that the vanishing of the Poisson bracket between ϕ S I , f I and ϕ S J , f J (Eq. (3.3)) is not a naïve conclusion drawn from the fact that both functions depend only on the ˜ momentum variable E—the vanishing of the bracket is implied in fact by the commutativity of the structure group R as it can be shown by means of a rigorous method developed in [27] (see also the remark below Definition 3.5). To obtain the number fˇ we subdivide the loop l on a finite number of connected and oriented segments (the orientation of the segment is inherited from the orientation of the loop) such that each segment is either (i) contained in S (modulo its endpoints) or (ii) the intersection of the segment with S coincides with precisely one endpoint of the segment or (iii) the segment does not intersect16 S. Let f + := f (yk ), k 16 The subdivision is possible thanks to analyticity of l and S—if the loops and the surface were only smooth they could have an infinite number of intersection points which would make the bracket (3.4) ill defined.
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where {yk } are all the intersection points between S and those segments of the kind (ii) which either are ‘outgoing’ from S and placed ‘above’17 the surface or are ‘incoming’ to S and are placed ‘below’ the surface. Similarly, f − :=
f (yn ),
n
where {yn } are all the intersection points between S and those segments of the kind (ii) which either are ‘outgoing’ from S and are placed ‘below’ the surface or are ‘incoming’ to S and are placed ‘above’ the surface. Then fˇ := f + − f − . Now we are going to define a kind of cylindrical functions, which will be used to define the ACZ algebra A. Let us begin by recalling the definition of a tame subgroup of HG [19,20]. Definition 3.1. A finite subset L = {l1 , . . . , l M } of L y is called a set of independent loops if and only if (i) each loop l I contains an open segment which is traversed only once and which is shared by any other loop at most at a finite number of points and (ii) it does not contain any path of the form e ◦ e−1 , where e is a piecewise analytic path in . Let L = {l1 , . . . , l M } be a set of independent loops. A subgroup L˜ of the hoop group HG generated by hoops {l˜1 , . . . , l˜M } is said to be a tame subgroup of HG. In the sequel we will also say that the group L˜ is generated by the loops L which is not too precise ˜ there exist in general many holonomically but convenient. Given the tame group L, ˜ inequivalent sets of independent loops such that each of them generates L. ˜ Given the tame subgroup L, we define the following equivalence relation on the space A of the R-connections on : we say that connections A1 , A2 ∈ A are equivalent, A1 ∼L˜ A2 , if and only if h l˜(A1 ) = h l˜(A2 ) for every hoop l˜ ∈ L˜ [20]. Then AL˜ := A/ ∼L˜ is a finite dimensional space. We will denote by pr L˜ the natural projection from A onto AL˜ , A → pr L˜ (A) := [A],
(3.5)
where [A] is the equivalence class of the connection A. Evidently, the space AL˜ is an analogue of the reduced configuration space Aγ (1.1) considered earlier. The following lemma describes properties of AL˜ . 17 The term ‘‘above’ (‘below’) the surface’ refers to the orientation of S.
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Lemma 3.2. Suppose that L˜ is generated by independent loops L = {l1 , . . . , l M }. Then the map AL˜ [A] → IL ([A]) := ( h l1 (A), . . . , h l M (A) ) ∈ R M
(3.6)
is a bijection [20] which equips AL˜ with a structure of a linear space. A linear structure on AL˜ given by a map IL , where L is another set of independent loops generating ˜ coincides with that defined by L, i.e. the map IL ◦ I −1 : R M → R M is linear and L, L invertible [15]. Now we are going to recall the definition of Schwarz functions which will be used to define a kind of cylindrical functions. Let α denote a multi-label (α1 , . . . , αn ) such that every αi belongs to N = {0, 1, 2, . . .}. Given a smooth function ψ : R N → C, we denote by D α a partial derivative α
D ψ :=
∂
i
αi
∂ x1α1 . . . ∂ xnαn
ψ,
where (xi ) are the Cartesian coordinates on R N . A smooth function ψ : R N → C is said to be a Schwarz function if and only if for every m ∈ N and for every derivative D α ψ, lim (D α ψ) r m = 0,
r →∞
2 N where r = i x i . It is clear that Schwarz functions on R form a ∗-algebra which will be denoted by S N . We will also need another kind of functions defined on R N : Definition 3.3. We say that a smooth function ψ : R N → C is a multiplier of the ∗-algebra S N if and only if for every derivative D α ψ there exists a finite set {P1,α , . . . , Pn α ,α } of polynomials on R N such that α
|D ψ| ≤
nα
|Pi,α |.
i=1
We will denote the set of all multipliers of S N by M N . It is a simple exercise to show that (i) for every Schwarz function ψ on R N and for every multiplier ψ of S N the product ψψ is again a Schwarz function and (ii) M N is a unital ∗-algebra. Definition 3.4. The set CylL˜ (Cyl S˜ ) of cylindrical functions (Schwarz cylindrical funcL tions) compatible with the tame group L˜ is the set of all complex functions on A of the form ∗ = pr ∗˜ IL ψ,
L
where ψ is any element of M N (S N ), pr L˜ is the projection (3.5) and IL is the map (3.6).
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By virtue of Lemma 3.2 the spaces CylL˜ and Cyl S˜ do not depend on the choice of the L ˜ Both spaces are ∗-algebras, set L of independent loops generating the tame group L. and CylL˜ possesses a unit given by the function on R N of constant value equal to 1. Consider now a (complex) vector space Cyl spanned by all the cylindrical functions, Cyl := span{ CylL˜ | L˜ ⊂ HG },
(3.7)
and a set of all operators on Cyl given as finite linear combinations of operators of the form φ S, f := {ϕ S, f , },
(3.8)
where S and f run through all admissible surfaces and functions. To convince ourselves that the definition of is correct we have to check whether every {ϕ S, f , } is an element of Cyl. In fact, every operator (3.8) preserves each space CylL˜ . Indeed, let L˜ be generated by the set L = {l1 , . . . , l N } of independent loops. Then by virtue of (3.4) we get ∂ψ 1 ∗ {ϕ S, f , } = − pr ∗˜ IL [ fˇJ J ], L 2 ∂x N
(3.9)
J =1
where fˇJ := −2{ϕ S, f , h l J }, and (x J ) are the canonical coordinates on R N (see (3.6)). In fact, the r.h.s. of the above equation belongs to CylL˜ for the algebra M N is preserved by any derivative. Definition 3.5. The Ashtekar-Corichi-Zapata algebra [27] is a complex vector space A := Cyl × equipped with the Lie bracket [(, φ), ( , φ )] := (φ − φ , [φ, φ ]).
The commutativity of R implies [φ, φ ] = 0 for every φ, φ ∈ . This is, in fact, why the second equation of (3.3) holds for {ϕ S I , f I , ϕ S J , f J } := [φ S I , f I , φ S J , f J ].
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3.1.3. Reduced classical systems. In the previous subsection we have defined the ACZ algebra A encompassing all Yang-Mills gauge invariant degrees of freedom of the classical theory. Now we are ready to perform the reduction of the degrees of freedom. As mentioned earlier we will distinguish a reduced classical system by limiting our attention to outcomes provided by a restricted set of measuring instruments. This can be done by distinguishing a subalgebra of A generated by 1. an algebra of cylindrical function CylL˜ , 2. a linear subspace F of . In this paper we choose trivially F = :18 Definition 3.6. A subalgebra AL˜ = × CylL˜ of A is called a reduced classical system. 3.1.4. Kijowski’s definition of reduced classical systems versus the present ones. While quantizing a scalar field theory Kijowski [1] reduces the infinite dimensional phase space of the theory to a finite dimensional one by restricting his attentions to outcomes provided by a finite number of measuring instruments which (from a mathematical point of view) are defined as some integrals over three-dimensional regions. Following the LQG methods we decided to use instruments which are distributional in the sense that they are defined as integrals along curves and surfaces. This choice (as it is shown in [27]) leads naturally to the ACZ algebra A which replaces the phase space P with its Poisson structure. This is the main reason why we defined reduced classical systems as subalgebras of the ACZ algebra A instead of reduced phase spaces, though each of the subalgebras can be associated with a reduced phase space. The subalgebra AL˜ = × CylL˜ introduced by Definition 3.6 corresponds to the reduced phase space E˜ × AL˜ which is infinite dimensional. The difference between this reduction and that applied by Kijowski is that here we do not reduce momentum degrees of freedom at all, while Kijowski reduces both momentum and configuration ones. 3.1.5. The directed set (, ≥). Now we are going to label the set of infinite classical systems by elements of directed sets. We define ˜ ≥), (, ≥) := ({L}, ˜ of all tame subgroups of the hoop group HG where the directing relation in the set {L} is defined as follows: L˜ ≥ L˜ if L˜ is a subgroup19 of L˜ [20]. Later on, as we will see in Subsect. 3.2.2, it will be necessary to restrict ourselves to a directed subset of this . Thus we have finished Step 1 of the quantization procedure—the result of the step is the family of reduced classical systems labeled by the directed set. 18 Reduced systems obtained by choosing F as a proper subspace of will be described in [36]. 19 The existence of the directing relation on {L} ˜ is guaranteed by the analyticity of the loops generating the
hoop group HG [20].
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3.2. Step 2: Systems and subsystems. The present section describes the crucial step of the quantization procedure, namely the definition and analysis of the relation ‘systemsubsystem’ among the classical systems {Aλ }λ∈ —once this step is done the remaining steps of the procedure will be (modulo some technical difficulties) rather straightforward. Since a step like this does not occur in the standard quantization procedure leading to the LQG model we advise the reader to study this section carefully. 3.2.1. Preliminaries Consider infinite classical systems Aλ and Aλ such that λ = L˜ ≥ ˜ The essential observation is that the space Cyl ˜ describes more configuration λ = L. L degrees of freedom than CylL˜ including those in CylL˜ . Indeed, the spaces are functions on, respectively, AL˜ and AL˜ and AL˜ = pr L˜ L˜ (AL˜ ), where pr L˜ L˜ is a linear20 projection from AL˜ onto AL˜ such that pr L˜ = pr L˜ L˜ ◦ pr L˜ .
(3.10)
By virtue of Definition 3.4 and (3.10) we can write ∗ ψ = pr ∗˜ pr ∗˜ CylL˜ = pr ∗˜ IL
L
L
LL˜
−1∗ ∗ ∗ ∗ ∗ IL ψ = pr ∗˜ IL [IL pr ˜ ˜ IL ] ψ.
L
LL
−1 N ∼ A Note now that IL ◦ pr L˜ L˜ ◦ IL = L˜ is nothing else but a linear projection from R onto R N ∼ = AL˜ . Since the pull-back defined by the projection maps polynomials on R N −1∗ ∗ ∗ to polynomials on R N , the function [IL pr ˜ ˜ IL ] ψ is bounded by polynomials in the LL sense of Definition 3.3. Thereby the function is a multiplier of S N , hence ∈ CylL˜ and
CylL˜ ⊂ CylL˜ .
(3.11)
Aλ ⊂ Aλ .
(3.12)
In this way we see that
Definition 3.7. We say that a classical system Aλ is a subsystem of a classical system Aλ if and only if λ ≥ λ. However, in order to proceed with the quantization procedure the relation (3.12) has to be analyzed more carefully—this analysis will turn out to be necessary to define the projection πλλ from Dλ onto Dλ required by the procedure. As it was already said the space Dλ (Dλ ) of mixed states will be represented by the set of density matrices on the Hilbert space Hλ (Hλ ). Therefore πλλ will be defined according to the well known formula describing a projection from a quantum system onto its subsystem: given λ ≥ λ, we will split Hλ into a tensor product H˜ λ λ ⊗ Hλ λ , where Hλ λ will correspond to Hλ via a natural unitary map. Next, given ρλ ∈ Dλ we will evaluate the partial trace with respect to H˜ λ λ , obtaining a density matrix on Hλ λ which will be identified by the unitary map with a density matrix on Hλ . Thus the question ‘how to construct the projection πλλ ?’ can be reduced to ‘how to decompose Hλ into H˜ λ λ ⊗ Hλ λ ?’. 20 The linearity of the projection follows from (3.2).
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Since we are going to define the Hilbert space Hλ in a natural way as L 2 (AL˜ , dµL˜ ), where dµL˜ is a Haar measure21 on AL˜ the decomposition H˜ λ λ ⊗ Hλ λ can be obtained from a decomposition of AL˜ . Taking into account the fact that πλλ is a quantum counterpart of the projection pr L˜ L˜ the decomposition should be of the form AL˜ = ker pr L˜ L˜ ⊕ Aλ λ .
(3.13)
Now we can set H˜ λ λ = L 2 (ker pr L˜ L˜ , d µ˜ λ λ ) and Hλ λ = L 2 (Aλ λ , dµλ λ ) ∼ = Hλ , where (i) dµλ λ is a Haar measure on the linear space Aλ λ corresponding via the projection pr L˜ L˜ to the Haar measure dµL˜ on AL˜ and (ii) the product d µ˜ λ λ ×dµλ λ coincides with dµL˜ . We conclude then that the definition πλλ should be based on the decomposition (3.13). However, there are a lot of such decompositions which differ from each other by the choice of Aλ λ and, as it is shown in Appendix B, the projection πλλ depends essentially on the choice of Aλ λ . Thus our task is to find a criterion which will allow us to distinguish Aλ λ in a natural way. Note that a choice of Aλ λ appearing in (3.13) is equivalent to a choice of a linear embedding θλ λ : AL˜ → AL˜ such that pr L˜ L˜ ◦ θλ λ = id and Aλ λ = im θλ λ .
(3.14)
Indeed, (3.13) implies pr L˜ L˜ restricted to Aλ λ is a linear isomorphism onto AL˜ . that−1 Then θλ λ := (pr L˜ L˜ A ) . On the other hand, if θλ λ satisfies the first condition of λλ (3.14) then Aλ λ given by the second one satisfies (3.13). Thus the equivalence follows. In fact, in the sequel we will use rather the embedding instead of Aλ λ since the former will turn out to be more convenient. To finish the preliminary considerations let us emphasize that the choice of the subspace Aλ λ and the embedding θλ λ can be done by referring merely to the structure of the classical systems Aλ and Aλ . This is why we decided to place this construction before the systems have been quantized. 3.2.2. Systems and subsystems—detailed consideration. Assume that λ = L˜ ≥ λ = L˜ and consider the corresponding classical systems Aλ ⊃ Aλ . The only difference between ˜ Therefore while the systems comes from the difference between the groups L˜ and L. defining Aλ λ we should refer to the groups. Suppose then that the sets {l1 , . . . , l N } and {l1 , . . . , l N } (N ≥ N ) of independent ˜ Assume moreover that for loops generate, respectively, the group L˜ and L. I = 1, . . . , N , l˜I = l˜I .
(3.15)
Then we could define Aλ λ as follows: 21 Note that since A is a linear space, and hence a Lie group, the notion of a Haar measure on AL˜ is L˜ well defined.
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Definition 3.8. Aλ λ := { [A] ∈ AL˜ | h l (A) = 0 for J = N + 1, . . . , N }, J
(3.16)
where [A] is the equivalence class of A with respect to the relation ∼L˜ defining AL˜ . However, the space Aλ λ given by the definition is not unique because even if the condition (3.15) is satisfied the choice of the remaining loops {l N +1 , . . . , l N } is still ambiguous—in general there are many choices of such loops and each of them may provide us with a different measuring instrument {h l N +1 , . . . , h l } (in other words, the N choices of the loops are not holonomically equivalent in general). Therefore to remove the ambiguity we will restrict ourselves to tame groups generated by a special kind of piecewise analytical loops which will be called almost analytic loops. Definition 3.9. We say that a loop l (based at y ∈ ) is almost analytic if and only if l = e−1 ◦ l¯ ◦ e,
(3.17)
where (i) l¯ is an analytic loop based at y ∈ such that almost all points of the path defined by l¯ are traced precisely once22 and (ii) e is a piecewise analytic oriented path of the source at y and the target at y (e−1 denotes here the path obtained from e by the change of the orientation). Note that the only role of the path e is to ensure that the base point of the loops l is y—the path does not effect the value of the holonomy along l, i.e. h l = h l¯. Therefore we can modify Definition 3.1 describing the notion of a set of independent loops in a way which will be suitable for further considerations. Using the notation introduced by Definition 3.9 we formulate Definition 3.10. Suppose that a finite subset L = {l1 , . . . , l M } of L y consists of almost analytic loops. L is called a set of independent loops if and only if for each I = 1, . . . , N the set l¯I ∩ (l¯1 ∪ · · · ∪ l¯I −1 ∪ l¯I +1 ∪ · · · ∪ l¯N )
(3.18)
consists of a finite number of points. It follows that for a set L = {l1 , . . . , l M } of independent almost analytic loops l˜I = l˜±1 J , unless I = J . We have the following lemma Lemma 3.11. 1. Suppose {l1 , . . . , l N } is a set of almost analytic loops such that for every I = J , l˜I = l˜±1 J . Then {l1 , . . . , l N } is a set of independent almost analytic loops. 22 This means that the loop can have self-intersections, but it cannot trace its own path more than once.
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2. Assume that the tame groups L˜ and L˜ are generated, respectively, by the sets L = {l1 , . . . , l N } and L = {l1 , . . . , l N } of almost analytic independent loops. If L˜ is a subgroup of L˜ then there exist a unique map σ : {1, . . . , N } → {1, . . . , N } such that l˜I = (l˜σ (I ) )±1 . The map σ is injective. Proof. Statement 1. Given I = J , consider analytic loops l¯I and l¯J defining the loops l I and l J respectively. Assume that the set of intersections between l¯I and l¯J is infinite. Then analyticity of the loops implies immediately that l¯I = l¯±1 J and consequently ˜l I = l˜±1 , which contradicts the assumptions of the lemma. Thus l¯I ∩ l¯J consists of at J most finite number of points and thereby the set (3.18) is finite. Statement 2. It is clear that the set L satisfies the assumptions of Statement 1 of the lemma. Suppose that, given I ∈ {1, . . . , N }, L ∪ l I also satisfies the assumptions. Then it is a set of independent loops and the contradiction l˜I ∈ L˜ follows. Thus there has to exist J ∈ {1, . . . , N } such that l˜I = (l˜J )±1 . Moreover, this J has to be unique, otherwise L would not be a set of independent loops. This proves the existence and uniqueness of σ . The map has to be injective also, otherwise the set L would not be a set of independent loops. Statement 1 of the above lemma means that within the domain of almost analytic loops the measuring instruments {h l } are independent modulo the trivial dependence h l = ±h l . More precisely, if {l1 , . . . , l N } are almost analytic loops satisfying the assumption of the statement then the only function H : R N → R which satisfies H (h l1 , . . . , h l N ) = 0 is H = 0. Statement 2 of Lemma (3.11) implies the following conclusions concerning the tame hoop groups L˜ , L˜ appearing in the statement: 1. L is the unique (modulo the change of orientation and holonomical equivalence of ˜ the loops) set of independent almost analytic loops generating L. 2. If L˜ is a subgroup of L˜ then there is a unique split of L into two disjoint subsets such that the loops in one of them coincide (modulo the change of orientation and holonomical equivalence) with those in L. The last conclusion guarantees that by restricting ourselves to almost analytic loops the ambiguity in Definition (3.8) of Aλ λ is removed. Therefore since now we will consider only those infinite classical systems which are defined by tame hoop groups generated by sets of almost analytic independent loops, it is straightforward to check that these tame hoop groups also form a directed set with the directing relation defined by inclusion as before. This set will be still denoted by (, ≥). Now let us turn back to the groups L˜ , L˜ considered above. Without loss of generality we can assume that l˜I = l˜I , I = 1, . . . , N .
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Let (x J ) and (x J ) be linear coordinates on, respectively, AL˜ and AL˜ given by the maps IL and IL (see Eq. (3.6)). Then the map θλ λ corresponding to Aλ λ can be written as follows: θλ λ (x1 , . . . , x N ) := (x1 , . . . , x N , 0, . . . , 0), where the r.h.s. of the equation is expressed by means of the coordinates (x J ). Now it is easy to show that there holds the following important: Theorem 3.12. Given λ ≥ λ ≥ λ, θλ λ = θλ λ ◦ θλ λ .
(3.19)
3.3. Step 3: Quantization of classical systems. In the previous step we gave a precise description of the relation ‘system-subsystem’ between the members of the family {Aλ }λ∈ . Now we are going to focus on a fixed classical system Aλ and quantize the system according to the standard rules of canonical quantization. Thus we will obtain the quantum system Sλ considered as an approximation of the quantum ‘toy theory’. Let us extend the Lie algebra Aλ which defines the classical system to an operator ˆ λ generated by the following linear operators acting on the algebra Cyl S of ∗-algebra A L˜ ˜ Schwarz cylindrical functions compatible with the tame group L: ˆ := , ∈ Cyl ˜ , Cyl S˜ → L L
Cyl S˜ → φˆ S, f,λ := iφ S, f , φ S, f ∈ F.
(3.20)
L
Clearly, is a function in Cyl S˜ , and φˆ S, f,λ also belong to the space, which is L guaranteed by Eq. (3.9) and the fact that every derivative maps the algebra of Schwarz ˆ λ is defined as follow: functions into itself. The ∗ involution on A
ˆ φˆ ∗ ˆ ∗ := , ˆ S, f,λ := φ S, f,λ (the correctness of the definition can be proved by a method described in [30]). The next step of the quantization procedure is constructing a ∗-representation of ˆ λ on a Hilbert space Hλ . Let us begin this construction by defining the space. Let A (x 1 , . . . , x N ) be the canonical coordinates on R N and let d x N be the Lebesgue measure on R N which, obviously, coincides with one of the Haar measures on R N thought of as a Lie group. Given a set L of independent loops generating the tame group L˜ we define a measure dµL˜ on AL˜ by the formula −1∗ ψ dµL˜ := (IL ψ) d x N , (3.21) AL˜
RN
where IL˜ is given by (3.6). As it was shown in [15] the measure dµL˜ does not depend on the choice of the set L. Hence Hλ := L 2 (AL˜ , dµL˜ ) is defined unambiguously.
(3.22)
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Let us consider the following pull-back: C ∞ (AL˜ ) → pr ∗˜ .
(3.23)
L
is a well The map pr L˜ is a surjection, hence the pull-back is an injective map and pr ∗−1 L˜ defined map from im pr ∗˜ onto the space C ∞ (AL˜ ) of smooth functions on AL˜ . Taking L
(Cyl S˜ ) is a dense subset of advantage of Definition 3.4 one can easily check that pr ∗−1 L˜ L ˆ λ on Cyl S which can Hλ . Equations (3.20) provide us with a natural representation of A L˜
ˆ λ on the Hilbert be mapped by means of the map pr ∗˜ onto a ∗-representation Tλ of A L space Hλ : ˆ λ ∈ aˆ → Tλ (a) ˆ := pr ∗−1 ˆ ◦ pr ∗˜ . A ˜ ◦a L
L
(3.24)
Thus we have obtained the desired quantum system Sλ consisting of: 1. pure states given by vectors in the Hilbert space Hλ , 2. mixed states represented by the set Dλ of all density matrices on Hλ , ˆ λ of observables on Hλ . 3. the representation Tλ of the ∗-algebra A ˆ λ (like e.g. the ‘flux-like’ operators {φˆ S, f,λ }) are mapped Note that some elements of A by the representation Tλ to unbounded operators on Hλ , which cannot be evaluated on some elements of Dλ . On the other hand every element of Dλ can be evaluated on the C ∗ -algebra Bλ of bounded operators on Hλ . Therefore in the sequel we will use the ˆ λ )} to build the algebra of quantum observables for algebras {Bλ } rather than {Tλ (A the ‘toy theory’.
3.4. Step 4: The space of states for the ‘toy theory’. In the previous subsection we have obtained the family {Sλ } of quantum systems labeled by the directed set (, ≥). Now we are going to build from these systems a large quantum system S which will consist of 1. the space D of quantum states, 2. an algebra B of observables which can be evaluated on the states in D. The system S can be considered as the quantum ‘toy theory’ with the Gauss constraint solved and the diffeomorphism one unsolved. 3.4.1. Construction of the space D of quantum states As mentioned earlier we are going to equip the set {Dλ } with a structure of a projective family—this means that we will have to define, for every pair λ ≥ λ, the projection πλλ : Dλ → Dλ (the ‘forgetting’ operator) in such a way that for every triple λ ≥ λ ≥ λ, πλλ = πλλ ◦ πλ λ .
(3.25)
The projections will be constructed by the method described in Subsect. 3.2.1. Assume then that λ ≥ λ. This means either λ = λ or λ > λ (i.e. λ ≥ λ and λ = λ). Let us begin the construction of the projections {πλλ } by considering the case λ > λ.
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Lemma 3.13. Suppose that λ > λ. Then dim AL˜ > dim AL˜ .
This lemma (proven in Appendix C) means that, given λ > λ, the passage from the classical system Aλ to Aλ involves a non-trivial reduction of the configuration degrees of freedom—the reduction is described by the projection pr L˜ L˜ . Note now that the decomposition (3.13) can be rewritten in the following form: AL˜ = ker pr L˜ L˜ ⊕ im θλ λ ⏐θ ⏐ λ λ,
(3.26)
AL˜ where im θλ λ = Aλ λ and, by virtue of Lemma 3.13, ker pr L˜ L˜ = 0.
(3.27)
Clearly, im θλ λ describes the degrees of freedom which do not undergo the reduction and are identified by means of θλ−1 λ with the degrees of freedom of AL ˜ . Now we are going to use (3.26) to decompose the Hilbert space Hλ = L 2 (AL˜ , dµL˜ ) into a tensor product of two Hilbert spaces. To achieve the goal let us equip every space occurring in (3.26) with an appropriate measure. Of course, the spaces AL˜ and AL˜ are already equipped with measures dµL˜ and dµL˜ respectively which are given by Eq. (3.21). Let us define a Haar measure dµλ λ on im θλ λ by the push-forward, dµλ λ := θλ λ∗ dµL˜ .
(3.28)
It is easy to check that there exists a unique Haar measure d µ˜ λ λ on ker pr L˜ L˜ such that dµL˜ = d µ˜ λ λ × dµλ λ . Thus we have obtained the following diagram: dµL˜ = d µ˜ λ λ × dµλ λ ⏐θ ⏐ λ λ∗ ,
(3.29)
dµL˜ which corresponds to (3.26). The measures just introduced allow us to define the following Hilbert spaces: H˜ λ λ := L 2 (ker pr L˜ L˜ , d µ˜ λ λ ) and Hλ λ := L 2 (im θλ λ , dµλ λ ), and a unitary map Hλ λ → Uλ λ := θλ∗ λ ∈ Hλ , which will be used to identify the Hilbert spaces Hλ λ and Hλ .
(3.30)
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Thus we obtained the desired decomposition: Hλ = H˜ λ λ ⊗ Hλ λ ⏐ ⏐U = θ ∗ .
λ λ λ λ
(3.31)
Hλ Comparing the above result with (3.26) we see that the Hilbert space H˜ λ λ corresponds to classical degrees of freedom undergoing the reduction, while Hλ λ corresponds to classical degrees which survive the reduction. To define the projection πλλ we will also need an isomorphism u λ λ from the algebra Bλ λ of bounded operators on Hλ λ onto the algebra Bλ of bounded operators on Hλ given by Bλ λ a → u λ λ a := Uλ λ ◦ a ◦ Uλ−1 λ ∈ Bλ .
(3.32)
Now, having the decomposition (3.31) and the above isomorphism we are ready to ˜ n } and {n } be orthonormal bases of, respectively, define the projection πλλ . Let { ˜ Hλ λ and Hλ λ . Then the partial trace ˜ k ⊗ n | ρ ( ˜ k ⊗ m ) ] |n m | [ Dλ ρ → tr˜ λ λ ρ := nm
k
produces a density matrix on Hλ λ . In other words tr˜ λ λ reduces those degrees of freedom of the quantum system Sλ which correspond to the classical degrees of freedom contained in ker pr L˜ L˜ . What remains to do is to map the resulting density matrix into one in Dλ . Thus we have arrived at Definition 3.14. Given λ ≥ λ, the projection πλλ : Dλ → Dλ is defined by the following formula: u λ λ ◦ tr˜ λ λ if λ > λ πλλ := . id if λ = λ Before we prove that the projections just defined satisfy the consistency condition (3.25) let us make some preparatory technical considerations concerning the case λ > λ > λ. Let us note first that the decomposition (3.31) applied to the Hilbert spaces Hλ , Hλ and Hλ gives us the following diagram: Hλ = H˜ λ λ ⊗Hλ λ ⏐ ⏐U
λλ Hλ
= H˜ λ λ ⊗Hλ λ U . = H˜ λ λ ⊗Hλ λ λλ ⏐ ⏐ ⏐U ⏐
λλ
Hλ
(3.33)
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Now we are going to reconcile the two decompositions of Hλ occurring in the top row of the diagram. Note first that by applying twice the decomposition (3.26) we get ⊕ im θλ λ im θ . AL˜ = ker pr L˜ L˜ ⊕ im θλ λ ker pr L˜ L˜
λλ
Theorem 3.12 allows us to simplify the last term of the above formula. Thus we obtain AL˜ = ker pr L˜ L˜ ⊕ im θλ λ ker pr ⊕ im θλ λ L˜ L˜
im θλ λ
= ker pr L˜ L˜ ⊕ im θλ λ ker pr ⊕ im θλ λ . L˜ L˜
(3.34)
ker pr L˜ L˜
Next, let us decompose the measure dµL˜ on AL˜ in an analogous manner. We have from (3.29), dµL˜ = d µ˜ λ λ × dµλ λ . By virtue of (3.29) and Theorem 3.12 we get dµλ λ = θλ λ ∗ dµL˜ = θλ λ ∗ (d µ˜ λ λ × dµλ λ ) = (θλ λ ker pr )∗ (d µ˜ λ λ ) × (θλ λ im θ )∗ (dµλ λ ), L˜ L˜
λλ
(3.35)
where (θλ λ ker pr )∗ (d µ˜ λ λ ) is a Haar measure on im (θλ λ ker pr ). It follows from ˜ ˜ LL L˜ L˜ (3.28) that (θλ λ im θ )∗ (dµλ λ ) = dµλ λ . λλ
In this way we obtain dµL˜ = d µ˜ λ λ × (θλ λ ker pr
L˜ L˜
)∗ (d µ˜ λ λ ) × dµλ λ .
It is clear that the product of the two first measures on the r.h.s. of the above equation is a measure on ker pr L˜ L˜ , and Eq. (3.29) allows us to conclude that the product coincides with d µ˜ λ λ . The desired decomposition of dµL˜ reads dµL˜ = d µ˜ λ λ × (θλ λ ker pr )∗ (d µ˜ λ λ ) × dµλ λ L˜ L˜
dµλ λ
= d µ˜ λ λ
× (θλ λ ker pr )∗ (d µ˜ λ λ ) × dµλ λ . ˜ ˜
LL d µ˜ λ λ
The above result and the decomposition (3.34) give us immediately Hλ = H˜ λ λ ⊗ H˜ λ λ λ ⊗ Hλ λ = H˜ λ λ ⊗ H˜ λ λ λ ⊗ Hλ λ ,
Hλ λ
H˜ λ λ
(3.36)
Quantization of Diffeomorphism Invariant Theories of Connections
where
H˜ λ λ λ = L 2 ( im (θλ λ ker pr
L˜ L˜
), (θλ λ ker pr
361
L˜ L˜
)∗ (d µ˜ λ λ ) ).
It is easy to see that the following maps:
H˜ λ λ λ → V˜λ λ = (θλc λ ker pr )∗ ∈ H˜ λ λ , L˜ L˜ c Hλ λ → Vλ λ = (θλ λ im θ )∗ ∈ Hλ λ , λλ
are unitary and satisfy Uλ λ = V˜λ λ ⊗ Vλ λ and Uλ λ ◦ Vλ λ = Uλ λ
(3.37)
(the first equation follows from Eq. (3.30), while the last one does from Theorem 3.12). Thus we have finished the preparatory considerations and are ready to state and prove Theorem 3.15. For every triple λ ≥ λ ≥ λ, πλλ = πλλ ◦ πλ λ .
(3.38)
> > λ otherwise the proof is trivial. Consider the map tr˜ λ λ Proof. Assume that occurring in the definition of πλλ . This map reduces the degrees of freedom described by the Hilbert space H˜ λ λ , hence by virtue of (3.36) it can be expressed as λ
λ
tr˜ λ λ = tr˜ 0 ◦ tr˜ λ λ , where tr˜ 0 reduces the degrees of freedom described by H˜ λ λ λ . The first equation of (3.37) is equivalent to the following diagram: Hλ λ = H˜ λ λ λ ⊗ Hλ λ ⏐ ⏐ ⏐ ⏐U ⏐˜ ⏐V
λλ
Vλ λ
λ λ. Hλ = H˜ λ λ ⊗ Hλ λ Denote by vλ λ an isomorphism from the algebra of bounded operators on Hλ λ onto one of bounded operators on Hλ λ defined by the unitary map Vλ λ . It is clear now that ˜ λ λ ◦ u λ λ ; tr˜ 0 = vλ−1 λ ◦ tr therefore the projection πλλ can be expressed as follows: ˜ λ λ ◦ (u λ λ ◦ tr λ λ ) πλλ = u λ λ ◦ tr˜ λ λ = u λ λ ◦ tr˜ 0 ◦ tr˜ λ λ = (u λ λ ◦ vλ−1 λ ) ◦ tr ˜ λ λ ◦ πλ λ . = (u λ λ ◦ vλ−1 λ ) ◦ tr
The second equation of (3.37) implies that u λ λ ◦ vλ−1 λ = u λ λ , hence πλλ = u λ λ ◦ tr˜ λ λ ◦ πλ λ = πλλ ◦ πλ λ . Thus we have constructed the desired projective family {Dλ , πλλ }. This gives us the space D of states of the ‘toy theory’ as the projective limit of the family D := lim Dλ . ← −
(3.39)
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3.4.2. Algebra of observables Following the construction in [1] we will define the algebra B of observables for the quantum ‘toy theory’. Let Bλ denote the C ∗ -algebra of bounded operators on the Hilbert space Hλ . Given aλ ∈ Bλ and ρλ ∈ Dλ , there is a complex number associated with them: aλ , ρλ := tr(aλ ρλ ) ∈ C. One can prove [1] the following Theorem 3.16. (i) Given any aλ ∈ Bλ , there exists a unique operator aλ ∈ Bλ such that for every ρλ ∈ Dλ , aλ , ρλ = aλ , πλλ ρλ . (ii) The map Bλ aλ → πλ∗ λ aλ := aλ ∈ Bλ is a C ∗ -isomorphism of Bλ onto its image in Bλ . (iii) For any triple λ ≥ λ ≥ λ, πλ∗ λ = πλ∗ λ ◦ πλ∗ λ . We conclude now that ∗ { Bλ , πλλ }λ∈
is an inductive family of (unital) C ∗ -algebras. Consequently, the inductive limit of the family, B := lim Bλ , − →
(3.40)
is a unital C ∗ -algebra also. It is easy to realize that every element of D defines a positive linear functional (a state) on B (note however that since Dλ does not coincide with the set of all states on Bλ , the set D does not coincide with the set of all states on B either). We will consider the algebra B as an algebra of (Yang-Mills gauge invariant) observables for the quantum ‘toy theory’. 3.5. Step 5: Solutions of the diffeomorphism constraint . The goal of this section is to define an action of the analytic diffeomorphisms of on the space D of quantum states of the ‘toy theory’ and find states preserved by the action—these states will be considered as quantum solutions of the diffeomorphism constraint (2.13). Let us begin by describing the action of diffeomorphisms on the hoop group HG. Given an analytic diffeomorphism χ and a loop l ∈ L y , the loop χ (l) is based at the point χ (y). If e is an (piecewise) analytic edge originating in y and ending in χ (y) then for all our purposes we can identify χ (l) with the loop e−1 ◦ χ (l) ◦ e based at y since h χ (l) = h e−1 ◦χ (l)◦e .
(3.41)
Moreover, the above equation holds independently of the choice of e. Thus the loop ˜ ∈ HG such that e−1 ◦ χ (l) ◦ e defines a unique hoop χ (l) h χ (l)˜ = h χ (l) .
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In fact, although diffeomorphisms do not preserve the base point y occurring in the definition of the hoop group HG, the action ˜ l˜ → χ (l) is an automorphism of the group [15]. This is why the quantization procedure does not depend of the choice of the point y as it was mentioned earlier. Now we turn to the action of diffeomorphisms on the measuring instruments ϕ S, f and h l˜. The action (2.15) of an analytic diffeomorphism χ on the canonical variables induces the action of the diffeomorphism on the instruments ˜ := ϕ S, f (χ∗−1 E) ˜ and (χ˜ h ˜)(A) := h ˜(χ ∗ A). (χ˜ ϕ S, f )( E) l l Then it is easy to show that χϕ ˜ S, f = ϕχ (S),χ −1∗ f and χ˜ h l˜ = h χ (l)˜ . Using the above formulas we can easily define a linear action of diffeomorphism on the ACZ algebra A: ∗ CylL˜ = pr ∗˜ IL ψ → χ˜ := pr ∗
I∗ ψ χ (L˜ ) χ (L)
L
∈ Cylχ (L˜ ) ,
˜ S, f := φχ (S),χ −1∗ f ∈ . φ S, f → χφ
(3.42)
Since above we have used the maps IL , Iχ (L) 23 it is necessary to show that the definition does not depend on the choice of the independent loops L = {l1 , . . . , ln }. Suppose then that L = {l1 , . . . , ln } is another set of independent loops generating the tame group ˜ Then, as stated in Lemma 3.2 the map M := IL ◦ I −1 is a linear automorphism L. L of Rn . As it is shown in [15] M is fully determined by the decomposition of hoops {l˜1 , . . . , l˜n } in terms of {l˜1 , . . . , l˜n } which is of the same form as the decomposition of {χ (l˜1 ), . . . , χ (l˜n )} in terms of {χ (l˜1 ), . . . , χ (l˜n )}. This means that M is diffeomorphism invariant, i.e. M = Iχ (L) ◦ Iχ−1 (L ) . Consequently, the definition under consideration is correct. It is obvious that χ(A ˜ λ ) = Aχ˜ (λ) , ˜ where χ˜ (λ) := χ (L). Let us now describe how the diffeomorphisms act on the space of states D. Consider first the set of pure states of a system Aλ which can be identified with the Hilbert space Hλ . The action of diffeomorphism reads (compare with (3.42)) ∗ ψ → χ ˜ := Iχ∗ (L) ψ ∈ Hχ˜ (λ) , Hλ = IL
(3.43)
where ψ ∈ L 2 (R N , d x) with N = dim AL˜ . Thus we obtained a unitary map from Hλ onto Hχ˜ (λ) which defines an isomorphism χˇ : Bλ → Bχ˜ (λ) .
(3.44)
Since Dλ can be naturally seen as a subset of Bλ the map χˇ defines an isomorphism from Dλ onto Dχ˜ (λ) which will be denoted by the same symbol. 23 The loops χ (L) = {χ (l ), . . . , χ (l )} are based at χ (y) but they still can be used by virtue of (3.41). n 1
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Lemma 3.17. For every analytic isomorphism χ , χˇ ◦ πλλ ◦ χˇ −1 = πχ˜ (λ)χ˜ (λ ) . ˜ Denote by L (L) a set of independent loops generating Proof. Let λ = L˜ ≥ λ = L. ˜ ˜ L (L). Then the projection and the set pr := Iχ˜ (L) ◦ pr χ˜ (L˜ )χ˜ (L˜ ) ◦ Iχ−1 ˜ (L ) , Rλλ := Iχ˜ (L ) (Aχ˜ (λ )χ(λ) ˜ ) do not depend on the choice of χ . It is clear that
R N = ker pr ⊕ Rλ λ , where N denotes the number of loops in L . The above decomposition defines a projec tion π from the set of density matrices on L 2 (R N , d x) onto the set of density matrices 2 N on L (R , d x), where N is the number of loops in L. Denote by ηL an isomorphism from the algebra of bounded operators on L 2 (R N , d x) ∗ : L 2 (R N , d x) → H . We can conclude now onto Bλ induced by the unitary map IL λ that −1 π = ηχ( ˜ ) ◦ ηχ( ˜ L ) ˜ L) ◦ πχ˜ (λ)χ(λ
independently on the choice of χ . It means in particular that −1 ηL ◦ πλλ ◦ ηL = ηχ−1 ˜ (L) ◦ πχ˜ (λ)χ˜ (λ ) ◦ ηχ˜ (L ) . −1 Now it is enough to note that χˇ = ηχ( ˜ L) ◦ ηL .
The lemma just proven means that the following map ˇ λ} χˇ ρ = χˇ {ρλ } := {χρ
(3.45)
maps D onto itself. We say that a state ρ ∈ D is a solution of the diffeomorphism constraint if and only if it is diffeomorphism invariant, i.e. for every (analytic) diffeomorphism χ , χˇ ρ = ρ.
(3.46)
˜ 3.5.1. Diffeomorphism invariant states in D. Consider a Hilbert space Hλ with λ = L. We know that modulo the change of orientation and the holonomical equivalence the set {l1 , . . . , l N } of almost analytic independent loops generating the tame group L˜ is unique. Thus there is a distinguished isomorphism Hλ ∼ = HL˜ (l˜1 ) ⊗ . . . ⊗ HL˜ (l˜N ) ,
(3.47)
˜ l˜I ) is a tame group of hoops generated by the hoop l˜I . Clearly, this isomorphism where L( ˜ l˜I ). is generated by the maps {θλλ I }, where λ I = L( As we know already, λ = L˜ ≥ λ = L˜ if and only if the set {l1 , . . . , l N } of independent loops generating L˜ is (again modulo the change of orientation and the holonomical
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equivalence) a subset of the set {l1 , . . . , l N } of independent loops generating L˜ . Consider now the projection πλλ which reduces some quantum degrees of freedom described by the Hilbert space Hλ ∼ = HL˜ (l˜ ) ⊗ . . . ⊗ HL˜ (l˜ ) . 1
N
It is easy to see that degrees of freedom undergoing the reduction are described by those Hilbert spaces {HL˜ (l˜ ) } for which the corresponding loops {l I } do not belong (modulo I the change of orientation and the holonomical equivalence) to {l1 , . . . , l N }. Thus every Hilbert space Hλ can be naturally and unambiguously decomposed into the tensor product (3.47). Moreover, the decomposition provides us with a simple description of the projections πλλ . Let us start the construction of diffeomorphism invariant states by defining an equivalence relation on the set AL y of all almost analytic loops based at the fixed point y. We say that the loops l1 and l2 are equivalent if and only if there exist an analytic diffeomorphism χ on such that ˜ l˜1 )) = (L( ˜ l˜2 )). χ (L(
(3.48)
Now, denoting by [l] the equivalence class of an almost analytic loop l, consider the following map: [l] → ρ([l]), where ρ([l]) is a density matrix on the Hilbert space L 2 (R, d x) such that it is preserved by the unitary map on the Hilbert space defined by R x → −x. This density matrix ˜ l)) ˜ on H ˜ ˜ . Indeed, the Hilbert can be mapped naturally into a density matrix ρ(L( L(l) 2 space HL˜ (l)˜ is obviously isomorphic to L (R, d x). In fact, there exist two distinguished isomorphisms between the Hilbert spaces: one given by I{l} and another one given by I{l −1 } . Then it is easy to see that thanks to the assumed special property of ρ([l]) the two distinguished isomorphisms between L 2 (R, d x) and HL˜ (l)˜ map ρ([l]) to the same density matrix. Thus we have constructed a map ˜ l) ˜ l)) ˜ → ρ(L( ˜ ∈D˜ ˜ L( L(l) which commutes with the action of every diffeomorphism χ : ˜ l)) ˜ l)) ˜ = ρ( χ (L( ˜ ). χˇ ρ(L(
(3.49)
To construct a diffeomorphism invariant state it is enough to decompose every Hilbert space Hλ as was done in Eq. (3.47) and define ˜ l˜1 )) ⊗ . . . ⊗ ρ(L( ˜ l˜N )) ∈ D ˜ ˜ ⊗ . . . ⊗ D ˜ ˜ ⊂ Dλ . ρλ := ρ(L( L(l1 ) L(l N ) It is straightforward to check that for every pair λ ≥ λ, πλλ ρλ = ρλ . Thus ρ = {ρλ } is an element of D. Consider now the action of an analytic diffeomorphism χ on the density matrix ρλ . Equation (3.49) gives us immediately χρ ˇ λ = ρχ˜ (λ) , which means that the state ρ is diffeomorphism invariant.
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Of course, every convex combination of (a finite number of) diffeomorphism invariant states is a diffeomorphism invariant state also. Thus we have obtained a plenteous set Dph of diffeomorphism invariant states which are, of course, also Yang-Mills gauge invariant. Therefore we can treat the set Dph as the set of physical states of the ‘toy theory’. Moreover, since Dph ⊂ D Yang-Mills gauge invariant quantum observables in B can be evaluated on the states in Dph . This completes the quantization of the ‘toy theory’: the resulting quantum model Sph is the C ∗ -algebra B constructed in Subsect. 3.4.2 equipped with the set Dph of physical states.
4. Discussion In this paper we have quantized a diffeomorphism invariant theory of connection with a non-compact structure group using a strategy based on the projective techniques by Kijowski combined with the LQG methods. It should be stated clearly that our goal was not the quantum model of the ‘toy theory’ by itself—we rather wanted to check whether the strategy can be applied to quantization of a theory of connections. Although we were interested in the application to the non-compact case it seems that the strategy does not distinguish between the compact and non-compact case since the ‘forgetting’ operator (3.14) can be defined once a split of a (separable) Hilbert space into the tensor product of two Hilbert spaces is given—it is absolutely irrelevant whether the Hilbert spaces are defined as L 2 spaces over compact or non-compact spaces. However, the fact that the strategy passed the test cannot be considered as a success because we do not know whether it was because of the power of the strategy or because of simplicity of the ‘toy theory’. In other words, we still do not know whether the strategy can be applied to any non-trivial theory, in particular, to GR in terms of the complex24 Ashtekar or the real Ashtekar-Barbero variables. Such an application does not seem to be easy: when one tries to apply the strategy to GR he immediately encounters problems caused by non-commutativity of the structure groups, respectively, S L(2, C) or SU (2), e.g. the Yang-Mills gauge transformations are much more complicated than in the commutative case, the scalar constraint does not vanish. Thus it is not obvious that the strategy can be successfully applied in the case of any non-commutative structure group. It is possible that the strategy just exchanges the non-compactness problem into the non-commutativity one, i.e. that the inductive techniques work well only in the compact case, while the projective ones do well only in the commutative one.
4.1. Almost analytic loops. The first question one can ask is whether the application of the almost analytic loops (Definition 3.9) can be justified, i.e. whether these loops provide a sufficiently large set of functions {h l }. To answer the question it is enough to check whether the functions separate points on A/G, where G is the group of Yang-Mills gauge transformations given by Eq. (2.14). In our case the group G is a space of real smooth functions on with the group action defined as pointwise addition. The quotient A/G can be easily described when the first de Rham cohomology of is trivial. Indeed, consider then two connections A1 , A2 ∈ A and their curvature two-forms F1 , F2 respectively. Then F1 = F2 implies that d(A1 − A2 ) = 0 and, by virtue of the triviality 24 Of course, in this case we would also have to solve the problem of the reality conditions. Since we do not have any idea how to do this we will neglect the problem in the present discussion.
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of the cohomology, we have A1 = A2 + d for ∈ G. Thus under the assumption about the first de Rham cohomology two connections define the same element of the quotient space A/G if and only if their curvature two-forms coincide. Assume then that A1 , A2 define different points in A/G. Then there exists a point y ∈ such that F1 (y) = F2 (y). Let (y k ) be a local coordinate frame with the origin at y such that the components F1,12 (y) and F2,12 (y) of the curvature two-forms are not equal. Since a holonomy h l (Ai ) (i = 1, 2) along a small (analytic) loop l lying in the coordinate ‘plane’ given by y 3 = 0 and surrounding y approximates the components, the function h l separates the two points in A/G. 4.2. ACZ algebra A and the algebra B of quantum observables . Let us recall that the first step of the quantization strategy applied here to the ‘toy theory’ involves the construction of the ACZ algebra A (Definition 3.5) as the algebra of classical observables. At the end of the procedure we obtained the algebra B defined by Eq. (3.40) interpreted as the algebra of quantum observables. Now we are going to check whether the relation between the two algebras coincides with the relation between algebras of classical and quantum observables given by the canonical quantization procedure. Recall that according to this procedure a given algebra A0 of classical observables is extended to a ˇ 0 of (possibly abstract) operators; next one finds a ∗-representation T0 of A ˇ0 ∗-algebra A on a Hilbert space. At this point one defines the algebra B0 of quantum observables as ˇ0 the algebra of bounded operators on the Hilbert space. Thus if a ∈ A0 defines aˇ ∈ A 25 ∗ such that aˇ = aˇ then T0 (a) ˇ is a self-adjoint operator and consequently exp(it T0 (a)) ˇ (t ∈ R) belongs to B0 . Thus the canonical quantization defines the following sequence of operators: A0 a → aˇ → T0 (a) ˇ → exp(it T0 (a)) ˇ ∈ B0 .
(4.1)
Now let us try to build an analogous sequence leading from the ACZ algebra A to the algebra B. ˇ be an algebra of linear operators on Cyl generated by elements {, ˇ φˇ S, f }, Let A ˇ := , ∈ Cyl, Cyl → Cyl → φˇ S, f := iφ S, f , φ S, f ∈ ,
(4.2)
where and φ S, f run through all elements of A of this sort. Assume now that the seemingly natural formulae ˇ φˇ ∗ = φˇ ˇ ∗ = , S, f S, f ˇ Denote by A ˇ λ a subalgebra of A ˇ generated by provide a well defined ∗ involution on A. S ˇ Aλ which means, in particular, that Aλ preserves the space Cyl ˜ . Let Jλ be a (right L=λ ˇ λ defined as and left) ideal in A ˇ λ | aˆ Cyl S Jλ := { aˆ ∈ A ˜
L=λ
= 0 }.
25 In fact, T (a) 0 ˆ is in general only symmetric on some dense domain but let us assume that it is also self-adjoint.
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Then it is easy to see that ˇ λ /Jλ , ˆλ = A A
(4.3)
ˆ λ is the ∗-algebra defined by (3.20). where A Now we can write down the following sequence of operators: A a → aˇ → aˆ λ → Tλ (aˆ λ ) → exp(it Tλ (aˆ λ )) → Atλ ∈ B,
(4.4)
ˇ Assuming which should be understood as follows: an element a of A generates aˇ ∈ A. ∗ ˇ ˆ that a ∈ Aλ we can project (see (4.3)) aˇ ∈ A onto aˆ λ ∈ Aλ . If aˆ λ = aˆ λ then Tλ (aˆ λ ) is a self-adjoint operator on Hλ which defines a unitary operator exp(it Tλ (aˆ λ )) ∈ Bλ . The latter defines via the inductive limit (3.40) the operator Atλ as an element of B. Comparing the sequences (4.1) and (4.4) we see that, in the case of canonical quantization the classical observable a is associated with exactly one quantum observable exp(it T0 (a)) ˇ (or, more precisely, with exactly one family of quantum observables). In the case of the ‘toy theory’ if a belongs to Aλ0 then it does to any Aλ such that λ ≥ λ0 . Consequently, the operator aˇ undergoes many distinct projections given by distinct ideals {Jλ } resulting in many {aˆ λ } which, in general, are also distinct as we will see soon. It seems then that in this case the classical observable a is associated with many distinct quantum observables Atλ . The question now is whether the operators {Atλ } (or {Tλ (aˆ λ )}) can be combined in a way into a single operator which could be evaluated on D (or on a subset of D) and be interpreted as a quantum counterpart of a. The answer to that question is negative as we will show by setting a = φ S, f . Let us start by rewriting the sequence (4.4): A φ S, f → φˇ S, f → φˆ S, f,λ → Tλ (φˆ S, f,λ ) → exp(it Tλ (φˆ S, f,λ )) → tS, f,λ ∈ B. (4.5) Note that any flux operator φ S, f belongs to all subalgebras {Aλ }, hence the r.h.s. of the above sequence is well defined for every λ. ˜ the ideal Jλ contains inter alia all linear combinations of those flux Given λ = L, operators which vanish on holonomies along hoops belonging to L˜ but whose action on holonomies along the other hoops are arbitrary. The consequence is that the operator φˆ S, f,λ can ‘see’ only hoops being elements of L˜ and is ‘blind’ with respect to the others. It is clear now that for λ = λ we have Jλ = Jλ and, in general, φˆ S, f,λ = φˆ S, f,λ . Now let us focus our attention on the operator Tλ (φˆ S, f,λ ). It is unbounded and (essenS ˜ tially) self-adjoint and its (dense) domain is pr ∗−1 ˜ Cyl ˜ ⊂ Hλ (λ = L). Consequently, L
L
the operator cannot be evaluated on the whole Dλ . We define then DλS ⊂ Dλ as a set of finite convex combinations of density matrices of the form S | |, ∈ pr ∗−1 ˜ Cyl ˜ .
L
L
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Evidently, evaluation of Tλ (φˆ S, f,λ ) on DλS provides a finite result26 . Define D S := { ρ = {ρλ } ∈ D | ρλ ∈ DλS for all λ ∈ }, and consider ρ ∈ D S . Then, in general, for λ = λ, Tλ (φˆ S, f,λ ), ρλ = Tλ (φˆ S, f,λ ), ρλ . This inequality holds even if λ ≥ λ—the reason for this is twofold: (i) the projection from ρλ to ρλ annihilates some non-trivial information contained in ρλ and (ii) the operator Tλ (φˆ S, f,λ ) can ‘see’ the hoops belonging to L˜ = λ , while Tλ (φˆ S, f,λ ) can ‘see’ those belonging to L˜ = λ only. Thus the operators {Tλ (φˆ S, f,λ )} cannot be combined into a single one which can be consistently evaluated on D S . As the last resource, we can try to combine {Tλ (φˆ S, f,λ )} into a single operator by means of the limit lim Tλ (φˆ S, f,λ ), ρλ . λ
(4.6)
However, this limit does not exist. To convince ourselves about that consider a Schwarz function ψ on R such that ¯ d x = 1 and ¯ x ψ d x = 0. (4.7) ψψ ψ∂ R
R
Given an almost analytic loop l, choose its orientation in such a way that −2φ S, f h l = fˇl ≥ 0. Next, by means of I{l} map the density matrix |ψ ψ| to one in DL˜ (l)˜ 27 which will be ˜ l)). ˜ l))} ˜ Combine all {ρ(L( ˜ into a state ρ ∈ D S . By direct calculation denoted by ρ(L( we get N 1 ˇ ¯ x ψ d x, Tλ (φˆ S, f,λ ), ρλ = − [ fl I ] ψ∂ 2 R I =1
(4.8)
where the loops {l1 , . . . , l N } generate the group L˜ = λ. It is clear now that in this case the limit (4.6) is divergent unless φ S, f = 0. The above discussion can be summarized as follows: Corollary 4.1. There is no natural way to evaluate any flux operator φ S, f on D S . Instead there is a net of operators {Tλ (φˆ S, f,λ )}, each of which can be evaluated on DλS . 26 For ρ ∈ D S we have λ λ
| Tλ (φˆ S, f,λ ) , ρλ | = |
n i |Tλ (φˆ S, f,λ ) i λ | < ∞, i=1
where ·|· λ is the scalar product on Hλ , and {i } are vectors in the Hilbert space. 27 The symbol D ˜ was introduced in Subsect. 3.5.1. L˜ (l)
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Similarly, the unitary operators { exp( it Tλ (φˆ S, f,λ ) ) }
(4.9)
belonging respectively to {Bλ } do not define any single operator in B. To see this consider ∗ introduced in Theorem 3.16 we get λ ≥ λ. Then using the embedding πλλ ∗ ˆ S, f,λ ) ) = id ⊗ u −1 ˆ πλλ exp( it Tλ (φ λ λ exp( it Tλ (φ S, f,λ ) ),
(4.10)
where id is the identity operator on the Hilbert space H˜ λ λ given by the decomposition (3.31), and u λ λ is defined by (3.32). In general, the r.h.s. of (4.10) does not coincide with exp( it Tλ (φˆ S, f,λ ) ). In other words, the net (4.9) of exponentiated operators does not define any single element of B but each of them defines via the inductive limit a distinct operator tS, f,λ = lim πλ∗ λ exp( it Tλ (φˆ S, f,λ ) ) ∈ B, − →
(4.11)
appearing at the end of the sequence (4.5). One can still ask whether the net {tS, f,λ } is convergent in a topology on B. Here it is natural to consider two topologies: the norm topology, i.e. one defined by the C ∗ -algebra norm on B) and the weak topology. The convergence of the net {tS, f,λ } in the weak topology would mean that the net tS, f,λ , ρ = exp( it Tλ (φˆ S, f,λ ) ), ρλ
(4.12)
is convergent for every ρ ∈ D. The answer to the question we are able to give now is not complete and can be formulated as the following lemma which is proven in Appendix D. Lemma 4.2. Suppose φ S, f = 0. Then: 1. The corresponding net {tS, f,λ } is divergent in the norm topology. 2. For every ρ ∈ D the net (4.12) is bounded |tS, f,λ , ρ | ≤ 1. Moreover, the equality holds if and only if tS, f,λ is the unit of B. 3. If ρ = {ρλ } is such that for every λ ∈ , ˜ 1 )) ⊗ . . . ⊗ ρ(L(l ˜ N )), ρλ = ρ(L(l ˜ I ) is a tame group where {l1 , . . . , l N } are independent loops generating L˜ = λ, L(l ˜ generated by the loop l I and ρ(L(L I )) ∈ DL˜ (l ) , then I
lim tS, f,λ , ρ = 0. λ
(4.13)
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Although we have to leave open the issue of the weak convergence of the net {tS, f,λ } the result (4.13) suggests that the net may be convergent to zero. The discussion presented above can be summarized by saying that the relation between the ACZ algebra and B seems to be essentially different from the relation between the algebras of classical and quantum observables obtained by applying canonical quantization: the sequences (4.1) and (4.4) appear to be not reconcilable. Let us finally note that above we were considering some operators evaluated on the spaces D and D S of states which are only Yang-Mills gauge invariant. Let us check what would happen if we restricted ourselves to diffeomorphism invariant states in Dph and D S ∩ Dph . While Lemma 4.2 is insensitive to this restriction (see its proof in Appendix ˜ l))} ˜ D) Corollary 4.1 has to be reformulated. The reason is that now the states {ρ(L( used to derive Eq. (4.8) cannot be chosen arbitrarily since there can exist a diffeomor˜ = l˜−1 . Then the diffeomorphism would induce an action on phism χ such that χ (l) ˜ l))} ˜ would have to be invariant with respect to the action. every DL˜ (l)˜ and every {ρ(L( ˜ l))} ˜ To ensure the invariance it is enough to require that the function ψ defining {ρ(L( satisfies ψ(−x) = eit ψ(x) for some real t (the map x → −x on R corresponds to the change of the orientation l˜ caused by χ ). Such a ψ, however, does not satisfy the second requirement of (4.7). What we can do in this situation is to replace that requirement by ¯ x2 ψ d x = 0 ψ∂ R
and show that the net 2 lim Tλ (φˆ S, f,λ ), ρλ
(4.14)
λ
is also divergent. Hence we have 2 ∈A ˇ on D S ∩ Dph . Instead Corollary 4.3. There is no natural way to evaluate the φˇ S, f there is a net of operators {Tλ (φˆ 2 )}, each of which can be evaluated on D S . S, f,λ
λ
4.3. Comparison with the compact case . At this point we are able to perceive some important differences between the quantum ‘toy theory’ and a quantum model in the compact case like e.g. LQG. Let us begin by analyzing the space of quantum states. In the compact case the space of (pure) quantum states is the Hilbert space H = L 2 (A, dµAL ), where A is the Ashtekar-Isham configuration space of generalized connections [19,28] and dµAL is the Ashtekar-Lewandowski measure [20] on A. The Hilbert space H can be obtained equivalently by means of the inductive limit (1.3). On the other hand the space D of quantum states of the ‘toy theory’ is defined as the projective limit (3.39) and is not a Hilbert space. Let us now emphasize another essential difference between the spaces D and H. Given a tame hoop group L˜ = λ, in the former case there exists a canonical projection πλ : D → Dλ , while in the latter one, there exists a canonical embedding pγ : Hγ → H, where γ is a graph. Since any projection is equivalent to a loss of some information, any state ρλ := πλ ρ belonging to Dλ should be considered only as an approximation of the state ρ ∈ D no matter how large and complicated the group L˜ is. On the other hand, every state vγ ∈ Hγ is naturally identified with a state v := pγ vγ ∈ H, i.e. it is
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just a state, no matter how trivial the graph γ is. Thus to use the projective techniques to build the space of quantum states means to construct the space by a sequence (or, more precisely, by a net) of approximations. Note moreover that every vector in H is a sum of at most countable cylindrical functions (like e.g. spin-networks—see [12,13,31])) and every cylindrical function depends on holonomies along a finite number of edges. Consequently, every vector can be seen as a function depending on at most a countable number of degrees of freedom represented by the holonomies. We can state this equivalently: every state depends on all the degrees of freedom but for most of them the dependence is trivial. This is, however, not true in the case of the quantum ‘toy theory’: given a loop l, ˜ is the tame group generevery state ρ ∈ D can be projected to ρλ , where λ = L(l) ated by l. Since ρλ is a density matrix on the Hilbert space Hλ defined as a space of L 2 functions on a non-compact space AL˜ (l) ∼ = R, it cannot be represented by | |, where is a constant function on AL˜ (l) . Thus ρλ does depend non-trivially on the degree of freedom defined by the holonomy along the loop l. Consequently, every state ρ depends non-trivially on all the (configuration) degrees of freedom which constitute a set of uncountable cardinality. This conclusion should not be surprising: in quantum mechanics we define the space of quantum states as a set of square-integrable functions on a non-compact configuration space and then each function depends non-trivially on all degrees of freedom. Here we just obtained an analogous result. Now let us focus our attention on Eq. (3.9) which (together with Eq. (3.8)) defines a flux operator. Given space CylL˜ , the flux operator φ S, f is defined as a finite sum of derivation operators such that each of them corresponds to a degree of freedom defined by the holonomy along a loop. Denote by φ S, f,l such a derivation operator associated with the loop l. Then φ S, f as an operator on the whole Cyl can be expressed as φ S, f,l , (4.15) φ S, f = L˜ (l)
where the sum runs through all tame groups generated by single loops. Thus the flux operator is given by an uncountable sum of operators corresponding to single degrees of freedom. In the compact case flux operators can also be expressed similarly as an uncountable sum (see e.g. [12,13]). We can see now why it is possible to define the action of flux operators on H (i.e. in the compact case), and why it is impossible to evaluate flux operators on D (i.e. in the non-compact case)—in the former case a state depends non-trivially only on at most countable number of degrees of freedom, so each time the flux operator acts on a state the uncountable sum in (4.15) reduces automatically to a countable one. In the latter case this is not true since the states in D do depend non-trivially on all the degrees of freedom. Thus in this case we are forced to reduce the sum by restricting ourselves to a finite number of operators {φ S, f,l }, obtaining as the result the net of the operators (4.11). The above discussion is summarized in Table 1. 4.4. Discretization of geometry. There is a common belief that quantum gravity may provide a discrete picture of the space-time instead of the continuous one assumed by general relativity. LQG partially supports this belief: quantum operators corresponding to basic geometric quantities like area and volume have discrete spectra [21,31,32], but still these operators are associated with regular subsets of the space-like manifold like e.g. bounded two-dimensional (analytic) submanifolds in the case of the area operator.
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Table 1. Cardinality of the set of degrees of freedom on which a state and a flux operator depend non-trivially compact case non-compact case
state countable uncountable
flux operator uncountable finite
In the quantum ‘toy theory’ the situation is quite different. We do not have, of course, any operators corresponding to classical area or volume since the classical ‘toy theory’ does not provide us with any metric. Therefore the only thing we can do is to analyze the quantum counterparts (4.11) of classical fluxes of the momentum field across surfaces in . However, Lemma 4.2 provides evidence that {tS, f,λ } do not define any non-trivial operator being a counterpart of the original flux operator φ S, f ∈ A. If this is really the case we are left with operators which do not correspond to the entire surface S underlying φ S, f since, given tS, f,λ , the surface S intersected originally by uncountably many loops is reduced to a finite number of points and to a finite number of loops passing through them. Thus in the ‘toy theory’ what seems to undergo the discretization is the surface as a set of points, while in the LQG case it is rather a feature of the surface (i.e. the area) which does not depend merely on the set of points constituting the surface. We can say then that in the case of the ‘toy theory’ the discretization appears to proceed at the more fundamental level than in the LQG case. Given surface S and smearing function f on it, one can ask which operator (4.11) should be considered as one representing the classical flux across S. The only answer we are able to give now refers to the interpretation of a state ρλ ∈ Dλ as an approximation of ρ = {ρλ } ∈ D. Recall that ρ depends essentially on the uncountable number of configuration degrees of freedom, so the whole information encoded in it is not available for us. We are then forced to use only a finite number of measuring instruments {h l1 , . . . , h l N } which results in doing an ordinary quantum mechanics on AL˜ ∼ = RN , where L˜ is generated by {l1 , . . . , l N }. Now, the choice of the instruments {h l1 , . . . , h l N } ˜ and consequently to the corresponding leads naturally to the operator φ S, f,λ with λ = L, discretization of the surface S. Thus the discretization in the quantum ‘toy theory’ seems to be observer-dependent—it depends on how (i.e. by means of which instruments) the observer perceives the physical reality. Taking into account the above conclusions we cannot exclude that if someone will manage to apply the quantization method presented in this paper to GR written in terms of the S L(2, C) connections then it will be not be possible to define a single area operator corresponding to a surface as a submanifold of —it may happen that non-compactness of S L(2, C) will force an analogous reduction of the surface to a finite set of points. This indicates that the change of the topology of the gauge group may result in a non-trivial change of the quantum theory which may force us to rethink some important geometric notions like the notion of a surface. In particular, it may be necessary to elaborate a new approach to the derivation of black hole entropy. 4.5. Diffeomorphism invariant states on B. In Subsect. 3.5.1 we constructed explicitly quite a plentiful set of diffeomorphism invariant states on the C ∗ -algebra B. This is in sharp contrast with the compact case where there is precisely one diffeomorphism invariant state [33] on the algebra of quantum observables which is the holonomy-flux ˆ hf [34]. Let us now point out the causes which are responsible for the differ∗-algebra A ence.
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ˆ hf does Note first that B contains exponentiated flux-like operators {tS, f,λ } while A just flux operators. To make the comparison more transparent let us first ‘de-exponentiate’ the operators {tS, f,λ }. Consider then an algebra of linear operators on Schwarz functions S N generated by all multipliers M N and derivative operators. Note that there ˜ we can use of the map IL (3.6) to pull is a natural ∗-involution on it. Given λ = L, ∗ (S ) ⊂ H . Denote this new back the algebra onto a ∗-algebra of operators acting on IL N λ ∗ ˆ algebra by Bλ . It is clear that Tλ (Aλ ) restricted to IL (S N ) is a subalgebra of Bλ . Given λ ≥ λ, taking into account Eq. (4.10) we define a map ∗ ˆ λ ) := id ⊗ u −1 ˆ λ ∈ Bλ , Bλ aˆ λ → πλλ (a λ λ a
where id is the identity operator on the Hilbert space H˜ λ λ given by the decomposition ∗ } (3.31) and u λ λ is defined by (3.32). Thus we obtained an inductive family {Bλ , πλλ whose inductive limit B can be thought of as generated by cylindrical functions Cyl and ˆ hf much more closely all flux-like operators {φˆ S, f,λ }. Thus the algebra B resembles A that B. It is clear that by applying the construction of diffeomorphism invariant states presented in Subsect. 3.5.1 we can obtain a large set of diffeomorphism invariant states on B . This means that the reasoning of [33] which leads to the uniqueness of a diffeomorˆ hf does not work in the case of B . There are at least three phism invariant state on A reasons for this: ˆ hf is defined as the invariance 1. In [33] diffeomorphism invariance of a state ω on A with respect to all semi-analytic diffeomorphisms on which form a larger group than analytic diffeomorphisms considered in this paper. 2. The crucial step of the proof of the uniqueness theorem of [33] shows that for every flux operator X S, f , ω( Xˆ ∗S, f Xˆ S, f ) = 0.
(4.16)
This result is obtained by observing that the expression ω( Xˆ ∗S, f Xˆ S, f ) defines a diffeomorphism invariant scalar product between functions on S and by showing that the invariance forces the product to be zero. This reasoning cannot be applied to B since the flux-like operators {φˆ S, f,λ } are not associated with surfaces but with finite subsets of . 3. In [33] the cylindrical functions are defined over graphs embedded in . In this case edges of a graph can be considered as compositions of edges of another one, i.e. we can transform a (smaller) graph into another (bigger) one by splitting its edges. In the case of B the cylindrical functions are defined over loops (as a special kind of graphs) which are almost analytic. As stated by Lemma 3.11 no almost analytic loop can be a composition of other such loops. This property is also a reason of plentitude of diffeomorphism invariant states on B . To see this assume that the loops are just piecewise analytic and consider edges {e0 , e1 , e2 } sharing the same beginning point and the same final one. Suppose also that there exists a diffeomorphism χ such that χ (ei ) = ei+1|mod 3 and define loops l0 := e1−1 ◦ e2 , l1 := e2−1 ◦ e0 , l2 := e0−1 ◦ e1 . Let ρ be a diffeomorphism invariant state constructed by the method presented in ˜ i )) (i = 1, 2, 3) is defined by a density Subsect. 3.5.1. Consequently, each ρ(L(l
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matrix ρ([li ]) on L 2 (R, d x). Since χ (li ) = li+1|mod 3 the matrix ρ([li ]) does not depend on i. Because l0−1 = l1 ◦ l2 for every bounded function ψ on R understood as a bounded operator on L 2 (R, d x), there holds the following condition: ρ([li ]) , κψ = ρ([li ]) ⊗ ρ([li ]) , ψ ,
(4.17)
where (κψ)(x) = ψ(−x) and ψ is a bounded function on R2 given by ( ψ)(x1 , x2 ) = ψ(x1 + x2 ). We thus see that if we allowed the loops to be compositions of other ones the number of diffeomorphism invariant states constructed in Subsect. 3.5.1 would be considerably reduced by the condition (4.17). However, this condition is not valid when the loops are almost analytic. Note also, that the set of almost analytic loops is not preserved by semi-analytic diffeomorphisms used in [33] and this is the main reason why in this paper we use only analytic ones. We conclude that (i) application of analytic diffeomorphisms, (ii) the discretization of the surfaces in the case of the flux-like operators and (iii) application of almost analytic loops make the diffeomorphism invariance much less restrictive than in the case considered in [33], and therefore the set of diffeomorphism invariant states on B , and consequently, on B is very large. Acknowledgement. I am very grateful to Prof. Jerzy Kijowski for the idea and discussions. I also would like the thank Prof. Abhay Ashtekar, Wojciech Kami´nski, Jerzy Lewandowski, Jorge Pullin and Thomas Thiemann for discussions and Jacek Jezierski, Kazimierz Napiórkowski and Wiesław Pusz for their help with the proof of Lemma 4.2. The author was partially supported by the Polish Ministerstwo Nauki i Szkolnictwa Wy˙zszego grant 1 P03B 075 29.
Appendix A. Physical Degrees of Freedom of the ‘Toy Theory’ Here we are going to construct an injective map from the set ]0, 1[×R into the class (2.9) of solutions of the constraints such that 1. given two distinct pairs (α, β), (α , β ) ∈]0, 1[×R there is no gauge transformation which maps the fields corresponding to (α, β) to ones corresponding to (α , β ); 2. given α ∈]0, 1[, the fields depend continuously on β ∈ R. Assume that = R3 . Consider the cylindrical coordinates (ρ, φ, z) on it and a smooth function on of the form b(ρ, z) = a(ρ) sin(ρ) exp(−z 2 ), where a(ρ) is a non-negative smooth function on [0, ∞[ such that it is zero on [0, 2π ] and is strictly positive on ]2π, ∞[. We assume moreover that for large ρ’s the function behaves like a Schwarz function, e.g. like exp(−ρ 2 ). Note that the function b(ρ, z) distinguishes infinite ‘rings’ in such that on each ring the function is either (strictly) positive or negative.
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Consider now a smooth non-negative function ξn,m (n = 1, 2, . . ., m = 0, 1, . . . , 9) on [0, ∞] × R such that (i) its support is a ball of the center at (2π + π2 n, m) and the radius 1 and (ii) the maximum value of ξn,m is 1 and is reached at precisely one point. Given α ∈]0, 1[, let αn be the n th digit of the decimal expansion of α. Define ζα := 1 −
αn ∞
ξn,m .
n=1 m=1 αn =0
Evidently, ζα is smooth, non-negative and not greater than 1. Note now that the function f α (ρ, φ, z) := b(ρ, z)ζα (ρ, z) encodes the full information about the number α in its zero set: in the n th ‘ring’ there are precisely αn pairwise distinct circles on which the value of the function is equal to 0. Define on a connection one-form z Aα,i d x i := [ f α (ρ, φ, z ) dz ] dρ 0
and a vector density i := β ˜ i jk Fα, jk , E˜ α,β
(A.1)
where β is a real number, and Fα,i j d x i ∧ d x j = f α (ρ, φ, z) dz ∧ dρ i ,A is the curvature form of Aα . It is easy to check that ( E˜ α,β α, j ) is a solution of all the constraints (see (2.10) and (2.11)). i ,A i Consider now two pairs ( E˜ α,β α, j ) and ( E˜ α ,β , Aα , j ) and assume that there exist a diffeomorphism χ and a Yang-Mills gauge transformation such that they map the former pair to the latter one. This means in particular that
(χ −1 )∗ Fα = Fα . This equation implies that the zero set of Fα is mapped bijectively onto the zero set of Fα . However, it is easy to realize that the zero set of Fα transformed by any diffeomorphism still encodes the full information about α. Hence α = α and Fα = Fα . It follows from (A.1) that β = β . This means that i , Aα, j ) (α, β) → ( E˜ α,β
is an injective map from ]0, 1[×R into the set of the gauge orbits contained in the physical i ,A phase space. Note moreover that, given α, the fields ( E˜ α,β α, j ) depend continuously on β. Therefore we can say that β parameterizes the degree of freedom corresponding to a fixed α. Since the interval ]0, 1[ is an uncountable set the ‘toy theory’ possesses an uncountable number of physical degrees of freedom.
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B. Dependence of πλλ on the Choice of Aλ λ Here we are going to show that the projection πλλ : Dλ → Dλ depends essentially on the choice of Aλ λ in the decomposition (3.13). Recall that (3.13) generates the decomposition of the Hilbert space Hλ into the tensor product H˜ λ λ ⊗ Hλ λ and πλλ is defined by evaluating the partial trace with respect to H˜ λ λ and by identifying the resulting density matrix on Hλ λ with one on Hλ (to identify the matrices we use a natural unitary map between the two Hilbert spaces). In fact, this is the partial trace which is sensitive to the choice of Aλ λ . We will show this by presenting a very simple example. Let R2 (R) play the role of AL˜ (AL˜ ) and let the projection R2 (x, y) → pr(x, y) := x ∈ R play the role of pr L˜ L˜ . Clearly, ker pr is the y-axis Y of R2 . Given Hilbert spaces L 2 (R2 , d xd y) and L 2 (R, d x), we want to find a projection π which corresponds to the projection pr and maps density matrices on L 2 (R2 , d xd y) to ones on L 2 (R, d x). Let us choose a subspace Ra of R2 spanned by a vector (1, a) as the space Aλ λ . Then the map R x → θa (x) := (x, ax) ∈ Ra ⊂ R2 satisfies pr ◦ θa = id (and hence is a counterpart of the map θλ λ introduced in Subsect. 3.2.1). Note that Y as ker pr and Ra give us a decomposition of R2 which is a counterpart of (3.13). Now we are going to decompose the Hilbert space L 2 (R2 , d yd x). Let (x , y ) be coordinates on R2 given by x = x . y = y − ax Evidently, the condition x = 0 describes the axis Y while y = 0 does Ra . Consequently, x restricted to Ra is a coordinate on it while y restricted to Y is a coordinate on the axis. It is straightforward to check that x (θa (x)) = x,
(B.1)
where x is meant to be the coordinate on Ra . Moreover, d yd x = dy d x and d x = θa∗ d x, where d x (d x) is understood here as a measure on Ra (R). Hence we get the desired decomposition of the Hilbert space L 2 (R2 , d yd x), L 2 (R2 , d yd x) = L 2 (Y, dy ) ⊗ L 2 (Ra , d x ). Consider a density matrix ρ on L 2 (R2 , d xd y). As it was said above the projection π should be defined by evaluating the partial trace tr Y of ρ with respect to L 2 (Y, dy ) and then by mapping the resulting density matrix tr Y ρ on L 2 (Ra , d x ) onto a density
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matrix πρ on L 2 (R, d x). To evaluate the partial trace tr Y we represent ρ by its integral kernel ρ(x, ˜ y˜ , x, y) = ρ(x˜ , y˜ + a x˜ , x , y + ax ). Then the kernel of tr Y ρ is given by ρ(x˜ , y + a x˜ , x , y + ax ) dy (tr Y ρ)(x˜ , x ) = R = ρ(x˜ , y , x , y + a(x − x˜ )) dy . R
(B.2)
We see then that tr Y ρ depends essentially on the choice of Ra . This dependence does not change after passing from tr Y ρ to πρ. It is clear that θa∗ defines the natural unitary map from L 2 (Ra , d x ) onto L 2 (R, d x), hence the passage from tr Y ρ to πρ is given by this map. Applying (B.1) we obtain the formula describing the kernel of πρ, (πρ)(x, ˜ x) = ρ(x, ˜ y , x, y + a(x − x)) ˜ dy , R
which still depends on the choice of Ra . C. Proof of Lemma 3.13 Lemma 3.13 follows immediately from: Lemma C.1. Suppose that a tame group L˜ generated by the set (l1 , . . . , l N ) of independent loops is a subgroup of a tame group L˜ generated by the set (l1 , . . . , l N ) of independent loops. Then (i) L˜ = L˜ if and only if N = N and (ii) L˜ is a proper subgroup of L˜ if and only if N > N . Proof. Let L˜ and L˜ be the tame groups of hoops occurring in Lemma C.1 generated respectively by sets L = {l1 , . . . , l N } and L = {l1 , . . . , l N } of independent loops. Applying a construction described in [20] we can get a set K = {k1 , . . . , kn } of independent loops such that: 1. Every loop belonging to L ∪ L is a composition of loops belonging to K. 2. Given a loop l ∈ L ∪ L, there exists a loop ki ∈ K such that (a) the loop l can be decomposed as l = k1 ◦ ki ±1 ◦ k2 , where k1 , k2 are loops built from the ones in K except ki . Without loss of generality we will assume that the orientation of the loop l is such that l = k1 ◦ ki ◦ k2 . (b) If the loop l ∈ L (L) then it is the only loop in L (L) in whose decomposition the loop ki appears. Now, decompose each loop l ∈ L ∪ L in terms of the loops {k1 , . . . , kn }. Thus we obtain the following decomposition of hoops: l˜I = l˜I =
n i=1 n i=1
(k˜i ) M I i := (k˜1 ) M I 1 ◦ . . . ◦ (k˜n ) M I n , (C.1) (k˜i )
MI i
:= (k˜1 )
MI 1
◦ . . . ◦ (k˜n )
MI n
,
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where every component of the matrices M = (M I i ) and M = (M I i ) belongs to Z. Clearly, the matrix M (M) has n columns and N (N ) rows. The properties of the decomposition of the loops in L (L) in terms of the loops in K imply that the columns of M (M) can be ordered in such a way that the first N (N ) columns of the matrix form the unit (N × N )-matrix ((N × N )-matrix) [15]. This means that the rank of M (M) is maximal and equal to N (N ). Since L˜ is a subgroup of L˜ there exists a matrix Q = (Q II ) of N rows and N columns such that every component of it belongs to Z and
l˜I =
N
(l˜I ) Q I I .
(C.2)
I =1
Taking into account Eqs. (C.1) the above equation can be rewritten as
n N n (k˜i ) M I i = ( (k˜i ) M I i ) Q I I . I =1
i=1
(C.3)
i=1
Because the loops in K are independent it is possible to find R-connections {A1 , . . . , An } on such that the holonomy h ki (A j ) = δi j . Hence the computation of the holonomies of the connections along the loops occurring in (C.3) give us by virtue of (3.2),
MI i =
N
Q I I M I i .
I =1
Since the rank of Q cannot be greater than N we have immediately N ≥ N.
(C.4)
˜ Then the transformation (C.2) is invertible which means that Assume now, that L˜ = L. the matrix Q is invertible, hence N = N. Suppose now that the above equality holds. Then properly chosen columns of the matrix M form a unit (N × N )-matrix. This means that the corresponding columns of the matrix M constitute the matrix Q −1 and that the components of Q −1 belong to Z. Hence
N
−1
(l˜I ) Q I I = l˜I ,
I =1
˜ Thus we have proved the statement (i) of the lemma. This which means that L˜ = L. statement together with (C.4) imply the statement (ii).
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D. Proof of Lemma 4.2 Let us fix a surface S and a non-zero function f on it. Then φ S, f = 0. In order to prove Statement 1 of the lemma note that for every λ ≥ λ, ˆ ||tS, f,λ || = ||id ⊗ u −1 λ λ exp( it Tλ (φ S, f,λ ) )||,
(D.1)
where the equality comes from Eq. (4.10) and the norm on the r.h.s. of the formula is the norm on Bλ . Recall that Hλ ∼ = L 2 (R N , d x), where the isomorphism between the Hilbert spaces is given by the map (3.6). Using the isomorphism we can convince ourselves that the operator on the r.h.s. of (D.1) corresponds to a translation operator Tx (x ∈ R N ), (Tx ψ)(x ) := ψ(x + x),
on L 2 (R N , d x). Consider now two operators Tx and Tx such that x = x and a function ψ ∈ L 2 (R N , d x) of compact support such that the support is contained in a ball of radius not greater than ||x − x ||/2. Then we have (Tx − Tx )ψ|(Tx − Tx )ψ = 2ψ|ψ , which means that ||Tx − Tx || ≥ 2.
(D.2)
Let L˜ = λ be an arbitrary tame hoop group. There exists a loop l such that l˜ ∈ L˜ and ˜ Taking φ S, f h l = 0. Let L˜ = λ ≥ λ be a tame group generated by hoops in L˜ and l. t t into account Eq. (4.10) we conclude that the operators S, f,λ and S, f,λ correspond to, respectively, Tx and Tx such that x = x . Now (D.2) gives us immediately ||tS, f,λ − tS, f,λ || ≥ 2. In other words, the net {tS, f,λ } is not convergent in the uniform topology on B. To prove Statement 2 note that if tS, f,λ is the unit of B then the evaluation of the operator on any state gives 1 as the result. On the other hand, there exists λ0 such that for every λ ≥ λ0 the operator tS, f,λ is not the unit and corresponds to a translation operator Tx on L 2 (R N , d x) ∼ = Hλ such that x = 0. Then for ρ = {ρλ } we have ˜ tS, f,λ , ρ = Tx , ρ , where ρ˜ is a density matrix on L 2 (R N , d x) corresponding to ρλ . There exists an orthonormal basis {ψn } of L 2 (R N , d x) and real non-negative numbers {ρ˜n } such that ρ˜ = ρ˜n |ψn ψn | and ρ˜n = 1. n
n
Given n, the Schwarz inequality gives us |ψn |Tx ψn | ≤ ||ψn || ||Tx ψn ||. Suppose now that both sides of the above formula are equal. This, however, can happen only if Tx ψn = αψn . Since Tx is unitary |α| has to be equal to 1, which means that
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ψ n ψn is a periodic function with the period x = 0 and thereby ψn cannot belong to L 2 (R N , d x). By virtue of this contradiction |ψn |Tx ψn | < ||ψn || ||Tx ψn || = 1 and |tS, f,λ , ρ | = |Tx , ρ | ˜ =|
ρ˜n ψn |Tx ψn | ≤
n
ρ˜n |ψn |Tx ψn | < 1.
n
Let finally justify Statement 3. Consider the state ρ = {ρλ } occurring in Eq. (4.13). ˜ where L˜ is generated by the independent loops {l1 , . . . , l N } we have For λ = L, tS, f,λ , ρ = exp( it Tλ (φˆ S, f,λ ) ), ρλ = exp( it TL˜ (l1 ) (φˆ S, f,L˜ (l1 ) ) ), ρL˜ (l1 ) . . . exp( it TL˜ (l ) (φˆ S, f,L˜ (l ) ) ), ρL˜ (l ) N
= t
S, f,L˜ (l1 )
, ρ . . . t
S, f,L˜ (l N )
, ρ .
N
N
(D.3)
There exists an uncountable set {l} of loops such that (i) they generate pairwise distinct ˜ tame groups {L(l)} and (ii) for every loop φ S, f h l = 0—the last property means that t ˜ n )) of pairwise is not the unit of B. Consequently, there exists a sequence (L(l S, f,L˜ (l) distinct groups such that for every n, |t
S, f,L˜ (l N )
, ρ | ≤ δ < 1.
Let λn = L˜ n be a tame group generated by the loops {l1 , . . . , ln }. Then it follows from (D.3) that for every λ ≥ λn , |tS, f,λ , ρ | ≤ δ n . This implies (4.13) immediately. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
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Commun. Math. Phys. 289, 383–400 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0719-7
Communications in
Mathematical Physics
Complete Einstein Metrics are Geodesically Rigid Volodymyr Kiosak1 , Vladimir S. Matveev2 1 Institute of Physics and Mathematics, K. Ushynsky South Ukraine Pedagogical University,
Odessa, Ukraine. E-mail: [email protected]
2 Institute of Mathematics, FSU Jena, 07737 Jena, Germany. E-mail: [email protected]
Received: 26 June 2008 / Accepted: 15 October 2008 Published online: 24 January 2009 – © Springer-Verlag 2009
Abstract: We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric g of nonconstant curvature is geodesically rigid: if any other complete metric g¯ has the same (unparametrized) geodesics with g, then the Levi-Civita connections of g and g¯ coincide. 1. Introduction 1.1. Definitions and results. Let (M n , g) be a connected Riemannian (= g is positive definite) or pseudo-Riemannian manifold of dimension n ≥ 3. We say that a metric g¯ on M n is geodesically equivalent to g, if every geodesic of g is a (reparametrized) geodesic of g. ¯ We say that they are affine equivalent, if their Levi-Civita connections coincide. We say that g is Einstein, if Ri j = Rn · gi j , where Ri j is the Ricci tensor of the metric g, and R := Ri j g i j is the scalar curvature. Our main result is Theorem 1. Let g and g¯ be complete geodesically equivalent metrics on a connected manifold M n , n ≥ 3. If g is Einstein, then at least one of the following possibilities holds: • they are affine equivalent, or • for certain constants c, c¯ ∈ R\{0} the metrics c · g and c¯ · g¯ are Riemannian metrics of curvature 1 (and, in particular, the manifolds (M n , c · g) and (M n , c¯ · g) ¯ are finite quotients of the standard sphere with the standard metric). For dimension ≥ 5, the assumption that the metrics are complete is important: if one of them is not complete, one can construct counterexamples (essentially due to [16,42]). For dimensions 3 and 4, (a natural modification of) Theorem 1 is true also locally: Theorem 2. Let g and g¯ be geodesically equivalent metrics on a connected 3- or 4-dimensional manifold M. If g is Einstein, then at least one of the following possibilities holds:
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• the metrics are affine equivalent, or • the metrics g and g¯ have constant curvature. Remark 1. In dimensions 3 and 4, Einstein metrics admitting nontrivial affine equivalent ones are completely understood [47,49]. Theorem 2 was announced in [25,44], with the extended sketch of the proof. The proof from [25,44] is very complicated: they prolonged (= covariantly differentiated) the basic equations (8) 6 times, and used the condition that the metric is Einstein at every stage of the prolongation. A partial case of Theorem 2 is also proved in [20]. Our proof of Theorem 2 is a relatively easy Linear Algebra (inspired by [9,14]) combined with a certain statement which is a relatively easy generalization of a certain result of Levi-Civita. Remark 2. Theorem 1 is also true in dimension 2 provided the scalar curvature of g is constant. Without this additional assumption Theorem 1 is evidently wrong, since every 2-dimensional metric satisfies Ri j = R2 · gi j . 1.2. History and motivation. The first examples of geodesically equivalent metrics are due to Lagrange [24]. He observed that the radial projection f (x, y, z) = − xz , − yz , −1 takes geodesics of the half-sphere S 2 := {(x, y, z) ∈ R3 : x 2 + y 2 + z 2 = 1, z < 0} to the geodesics of the plane E 2 := {(x, y, z) ∈ R3 : z = −1}, see the left-hand side of Fig. 1, since the geodesics of both metrics are intersection of the 2-plane containing the point (0, 0, 0) with the surface. Later, Beltrami [5] generalized the example for the metrics of constant negative curvature, and for the pseudo-Riemannian metrics of constant curvature. In the example of Lagrange, he replaced the half sphere by the half of one of the hyperboloids H±2 := {(x, y, z) ∈ R3 : x 2 + y 2 − z 2 = ±1}, with the restriction of the Lorentz metrics d x 2 + dy 2 − dz 2 to it. Then, the geodesics of the metric are also intersections of the 2-planes containing the point (0, 0, 0) with the surface, and, therefore, the stereographic projection sends it to the straight lines of the appropriate plane, see the right-hand side of Fig. 1 with the (half of the) hyperboloid H−2 . Though the examples of the Lagrange and Beltrami are two-dimensional, one can easily generalize them for every dimension. One of the possibilities in Theorem 1 is geodesically equivalent metrics of constant positive Riemannian curvature on closed manifold. Examples of such metrics are also
X
0
X f(X)
f(X) 0
Fig. 1. Surfaces of constant curvature are (locally) geodesically equivalent
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due to Beltrami [4]; we describe their natural multi-dimensional generalization. Consider the sphere def
2 S n = {(x1 , x2 , ..., xn+1 ) ∈ Rn+1 : x12 + x22 + ... + xn+1 = 1}
with the metric g which is the restriction of the Euclidean metric to the sphere. Next, A(v) consider the mapping a : S n → S n given by a : v → A(v) , where A is an arbitrary n+1 non-degenerate linear transformation of R . The mapping is clearly a diffeomorphism taking geodesics to geodesics. Indeed, the geodesics of g are great circles (the intersections of 2-planes that go through the origin with the sphere). Since A is linear, it takes planes to planes. Since the normalization w w → w takes punctured planes to their intersections with the sphere, the mapping a takes great circles to great circles. Thus, the pullback a ∗ g is geodesically equivalent to g. Evidently, if A is not proportional to an orthogonal transformation, a ∗ g is not affine equivalent to g. The success of general relativity suggested (see for example the popular paper [57]) to look for geodesically equivalent Einstein metrics. In particular, the classical textbook [15] has a chapter on geodesic equivalence. In our paper, in the proof of Corollary 1, we will use the following classical result of Weyl [56]: he proved that two conformally and geodesically equivalent metric are proportional with a constant coefficient of the proportionality. Later, geodesic equivalence of Einstein metrics was studied by many geometers and physicists (a simple search in mathscinet gives about 50 papers and few books). In particular, Petrov [46] proved that 4-dimensional Ricci-flat metrics of Lorenz signature can not be geodesically equivalent, unless they are affine equivalent. It is one of the results for which he obtained the Lenin prize (the most important scientific award of the Soviet Union) in 1972. He also explicitly asked [47, Problem 5 on p. 355] whether the result remains true in other dimensions. As we will prove in Lemma 3, the assumption that the second metric is Einstein is not important, since it is automatically fulfilled. By Theorem 2, the result of Petrov remains true for 4-dimensional metrics of other signatures. As we already mentioned in Sect. 1.1, the counterexamples independently constructed by Mikes [42] and Formella [16] show, that the result of Petrov fails in higher dimensions (so one indeed needs certain additional assumptions, for example the assumption that the metrics are complete as in Theorem 1, which is a standard assumption in problems motivated by physics.) Recent references include Barnes [3], Hall and Lonie [17,21], Hall [18,19]. They in particular studied the existence of projective transformations of Ricci-flat, Einstein, and FRW metrics, which is a stronger condition than the existence of geodesically equivalent metrics. Indeed, projective transformation of g allows to construct g¯ geodesically equivalent to g. Moreover, if g is Einstein, then g¯ is automatically Einstein as well, which essentially simplifies all formulas. One can find more historical details in the surveys [2,7,13,43], and in the introductions to the papers [33,34,38–41]. 2. Proof of Theorem 1 2.1. Schema of the proof. In Sect. 2.2 we list standard facts from theory of geodesically equivalent metrics, and introduce notation we will use through the paper. Most of these
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facts can be found in the book of Sinjukov [52], but unfortunately they are spread over the text, and it is not clear under which assumption they are true (Sinjukov always assumes real-analyticity, but actually needs smoothness). All the facts could be obtained by relatively simple tensor calculations; we will indicate how. The main result of Sect. 2.3 are Corollaries 3, 4.In Sect. 2.4 we explain that the ODE along geodesics given by Corollary 4 (that controls the reparametrization that makes g-geodesics from g-geodesics) ¯ can not have solutions such that they satisfy the condition that both metrics are complete provided that the Einstein metric g is pseudo-Riemannian, or Riemannian of nonpositive scalar curvature. Corollary 3 will be used in Sect. 2.5: we will see that combining Corollary 3 with a nontrivial result of Tanno [54] immediately gives Theorem 1 under the additional assumption that the metric is Riemannian of positive scalar curvature.
2.2. Standard formulas we will use . We work in tensor notations with the background metric g. That means we sum with respect to repeating indexes, use g for raising and lowing indexes (unless we explicitly mention), and use the Levi-Civita connection of g for covariant differentiation. As it was known already to Levi-Civita [26], two connections = ijk and ¯ = ¯ ijk have the same unparameterized geodesics, if and only if their difference is a pure trace: there exists a (0, 1)-tensor φ such that ¯ ijk = ijk + δki φ j + δ ij φk .
(1)
The reparameterization of the geodesics for and ¯ connected by (1) is done accord¯ the curve γ (τ (t)) is ing to the following rule: for a parametrized geodesic γ (τ ) of , a parametrized geodesic of , if and only if the parameter transformation τ (t) satisfies the following ODE: dτ 1 d α . (2) log φα γ˙ = 2 dt dt (We denote by γ˙ the velocity vector of γ with respect to the parameter t, and assume summation with respect to the repeating index α.) If and ¯ related by (1) are Levi-Cevita connections of metrics g and g, ¯ then one can find explicitly (following Levi-Civita [26]) a function φ on the manifold such that its differential φ,i coincides with the covector φi : indeed, contracting (1) with respect α = α + (n + 1)φ . From the other side, for the Levi-Civita to i and j, we obtain ¯ αi i αi α = 1 ∂ log(|det (g)|) . Thus, connection of a metric g we have αk 2 ∂ xk φi =
det(g) ¯ ∂ 1 = φ,i log 2(n + 1) ∂ xi det(g)
(3)
for the function φ : M → R given by
det(g) ¯ 1 . φ := log 2(n + 1) det(g)
In particular, the derivative of φi is symmetric, i.e., φi, j = φ j,i .
(4)
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The formula (1) implies that two metrics g and g¯ are geodesically equivalent if and only if for a certain φi (which is, as we explained above, the differential of φ given by (4)) we have g¯i j,k − 2g¯i j φk − g¯ik φ j − g¯ jk φi = 0,
(5)
where “comma” denotes the covariant derivative with respect to the connection . In¯ and deed, the left-hand side of this equation is the covariant derivative with respect to , vanishes if and only if ¯ is the Levi-Civita connection for g. ¯ Eq. (5) can be linearized by a clever substitution: consider ai j and λi given by ai j = e2φ g¯ αβ gαi gβ j ,
(6)
λi = −e2φ φα g¯ αβ gβi ,
(7)
where g¯ αβ is the tensor dual to g¯αβ : g¯ αi g¯α j = δ ij . It is an easy exercise to show that the following linear equations on the symmetric (0, 2)−tensor ai j and (0, 1)−tensor λi are equivalent to (5), ai j,k = λi g jk + λ j gik .
(8)
Remark 3. For dimension 2, the substitution (6, 7) was already known to R. Liouville [27] and Dini [12], see [10, Sect. 2.4] for details and a conceptual explanation. For arbitrary dimension, the substitution (6,7) and Eq. (8) are due to Sinjukov [52]. The background geometry is explained in [14]. Note that it is possible to find a function λ such that its differential is precisely the (0, 1)−tensor λi : indeed, multiplying (8) by g i j and summing with respect to repeating indexes i, j we obtain (g i j ai j ),k = 2λk . Thus, λi is the differential of the function λ :=
1 αβ g aαβ . 2
(9)
In particular, the covariant derivative of λi is symmetric: λi, j = λ j,i . Integrability conditions for Eq. (8) (we substitute the derivatives of ai j given by (8) α , which is true for every (0, 2)−tensor in the formula ai j,lk − ai j,kl = aiα R αjkl + aα j Rikl ai j ) were first obtained by Solodovnikov [53] and are α aiα R αjkl + aα j Rikl = λl,i g jk + λl, j gik − λk,i g jl − λk, j gil .
(10)
For further use let us recall the fact which can also be obtained by simple calculations: the Ricci-tensors of connections related by (1) are connected by the formula R¯ i j = Ri j − (n − 1)(φi, j − φi φ j ), ¯ where Ri j is the Ricci-tensor of and R¯ i j is the Ricci-tensor of .
(11)
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2.3. Local results. Within the whole paper we work on a smooth manifold of dimension n ≥ 3. Lemma 1. (Folklore) Let ai j be a solution of (8) for the metric g. Then, it commutes with the Ricci-tensor: aiα Rα j = a αj Riα .
(12)
Proof. Consider Eq (10). We are “cycling” the equation with respect to i, k, l: we sum it with itself after renaming the indexes according to (i → k → l → i) and with itself after renaming the indexes according to (i → l → k → i).The first term at the left-hand α + R α + R α = 0, side of the equation disappears because of the Bianchi equality Rikl kli lik the right-hand side vanishes completely, and we obtain aαi R αjkl + aαk R αjli + aαl R αjik = 0.
(13)
Multiplying with g jk , using the symmetries of the curvature tensor, and summing over the repeating indexes we obtain aαi Rlα − aαl Riα = 0 implying the claim, Lemma 2. Suppose the curvature tensor of the metric g satisfies Riαjk,α = 0. Then, for every solution ai j of (8) such that λi = 0 at a point p ∈ M n , in a sufficiently small neighborhood U ( p) of p we have 1
2
3
4
λk, j =c gk j + c Rk j + c ak j + c a αj Rαk ,
(14)
1 2 3 4
where the coefficients c, c, c, c are given by the formulas 1
c =
β
−λα aβα ξ β R + 2λλα,β β ξ α + aβα Rα − 4λα, α 4n
1 2 ; c = λα aβα ξ β ; 4
1 1 3 4 c = − λα,β β ξ α ; c = − aβα Rαβ , 4 4 where ξ is an arbitrary vector field such that λi ξ i = 1. Remark 4. The assumptions of the lemma are automatically fulfilled for Einstein spaces. Indeed, the second Bianchi identity for the curvature tensor is h h Rihjk,l + Rikl, j + Ril j,k = 0.
Contracting with respect to h and l, we obtain α α Riαjk,α + Rikα, j + Riα j,k = 0. −Rik, j
Ri j,k
If the metric is Einstein, then the second and the third components of the equation vanish, and we obtainRiαjk,α = 0. Moreover, we see that actually the condition Rik, j − Ri j,k = 0 is a necessary and sufficient condition for Riαjk,α = 0.
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Remark 5. The tensor Rik, j − Ri j,k is called a projective Yano tensor, and plays an important role in the theory of geodesically equivalent metrics; in particular, it is projectively invariant in dimension 2 [10,27], and is an essential part of the so-called tractor approach for the investigation of geodesically equivalent metrics [14]. Proof of Lemma 2. Consider the solution ai j of Eq. (8). Let us take the covariant derivative of Eq. (10) (the index of differentiation is “m”), and replace the covariant derivative of a by formula (8). We obtain α α λα R αjkl gim + λi Rm jkl + aαi R αjkl,m + λα Rikl g jm + λ j Rmikl + aα j Rikl,m = λl,im g jk + λl, jm gik − λk,im g jl − λk, jm gil .
(15)
We multiply with glm , sum with respect to repeating indexes l, m, and use Riαjk,α = 0. We obtain: α α α α λα Rik j + λα R jki − λi R jk − λ j Rik = λi,α g jk + λ j,α gik − λk,i j − λk, ji .
(16)
We now skew-symmetrise Eq. (16) with respect to k, j to obtain α α α 4λα Rik j = λ j,α gik − λk,α gi j − λk Ri j + λ j Rik .
(17)
Let us now rename the indexes i → k → j → α in (17), multiply the result by aiα , use the symmetries of the curvature tensor and sum over the repeating index α. We obtain β
4aiα Rα jkβ λβ = 4aiα Rk jα λβ
= aiα λα,β β gk j − λk,α α gi j − λ j,β β gkα − λ j Rαk + λα R jk = aiα λα,β β gk j − λ j,β β aki − λ j aiα Rαk + λα aiα Rk j .
(18)
Now we multiply Eq. (10) by λl and sum over the repeating index l. We see that the first component of the result is precisely the left-hand side of (18); we replace it bythe right-hand side of (18). We obtain
0 = aiα λα,β β − 4λα λα,i gk j − λ j,β β aki + λ j −aiα Rαk + 4λk,i + λα aiα Rk j + a αj λα,β β − 4λα λα, j gki − λi,β β ak j + λi −a αj Rαk + 4λk, j + λα a αj Rki . (19) We now skew-symmetrise (18) with respect to k, j, rename k ←→ i, and add the result to (19). After dividing by 2 for cosmetic reasons, and using that by Lemma 1 the tensor aiα Rαk is symmetric with respect to i, k, we obtain
α ai λα,β β − 4λα λα,i gk j + λα aiα Rk j − λi,β β ak j + λi −a αj Rαk + 4λk, j = 0. (20) We multiply (20) by g k j and sum over the repeating indexes k, j. We obtain (after dividing by n) α R β + 4λ α −a α, α
α β R 2λ λi,β β − λi = 0, (21) ai λα,β β − 4λα λα,i = − λα aiα + n n n
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where R := Rαβ g αβ
α is the scalar curvature of g. Substituting the expression for ai λα,β β − 4λα λα,i from (21) in (20), we obtain 0=
R β 2λ Rk j − gk j + λi,β gk j − a k j n n α β
−aβ Rα + 4λα, α α gk j + a j Rαk − 4λk, j . −λi n
λα aiα
(22)
Since λi = 0 at a point p, then λi ξ i = 1 for a certain vector field ξ in a sufficiently small neighborhood U ( p). Contracting Eq. (22) with this ξ i , we obtain R α i i β 2λ gk j − a k j 0 = λα ai ξ Rk j − gk j + ξ λi,β n n α β
−aβ Rα + 4λα, α α −λi (23) gk j + a j Rαk − 4λk, j . n We see that λ j,k is a linear combination of a αj Rαk , g jk , R jk and ak j as we want. The coefficients in the linear combination are as in the formula below.
4λk, j = aαk R αj +
β
−λα aβα ξ β R +2λλα,β β ξ α + aβα Rα − 4λα α g jk + λα aβα ξ β R jk − λα,β β ξ α ak j . n
Corollary 1. Assume g is an Einstein metric. Let ai j be a solution of (8). Assume λi = 0 at a point p. Then, in a sufficiently small neighborhood of p, λi, j is a linear combination of gi j and ai j : λi, j = µgi j + K ai j , R where the coefficients K := − n(n−1) and µ :=
(24)
λα, α −2K λ . n
Proof. By assumption, in a small neighborhood of p we have λi = 0; this implies that ai j is not proportional to gi j , because by the result of Weyl [56] if two metrics are geodesically and conformally equivalent, then they are proportional (with a constant coefficient of proportionality). As we explained in Remark 4, the assumptions of Lemma 2 are fulfilled if the metric is Einstein. Moreover, if the metric is Einstein, then the second term of the right-hand side of (14) is proportional to g, and the last term is proportional to a implying that λi, j is a linear combination of gi j and ai j . We need to calculate the coefficients of the linear combination. Substituting the condition that the metric is Einstein in (17), we obtain where
α λα Rik j = τ j gik − τk gi j , 1 R τi := λi,α α + λi . 4 n
(25) (26)
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Contracting Eq. (25) with g i j we obtain (n − 1)τ j = − Rn λ j implying τj = −
R λj. n(n − 1)
(27)
Now, since the metric is Einstein, the first bracket in the sum (22) is zero, and the term β β aβα Rα equals Rn δα aβα = 2 Rn λ, so the formula (22) reads
−2λ Rn + 4λα, α R β 2λ λi,β gk j − ak j − λi gk j + ak j − 4λk, j = 0. (28) n n n Combining (26) and (27), we obtain
R λi . λi,β β = 4k − n
(29)
Substituting this in (28), we obtain
−2λ Rn + 4λα, α 2λ R R gk j − ak j λi − gk j + ak j − 4λk, j λi = 0. (30) 4k − n n n n Since by assumption λi = 0, we obtain (24),
Remark 6. Assume g is an Einstein metric. Let ai j be a solution of (8). Then, (31) λα Yiαjk = 0, R δ hj gik − δkh gi j is the so-called concircular curvature of where Yihjk := Rihjk − n(n−1) g introduced by Yano [58]. Proof. Substituting (27) in (25), we obtain the claim,
Corollary 2. Assume g is an Einstein metric. Let ai j be a solution of (8). Consider λ α −2K λ R and the function µ := α, n . Then, the function µ satisfies the K := − n(n−1) equation µ,i = 2K λi .
(32)
Remark 7. In particular, under the assumptions of Corollary 2, for a certain const ∈ R, the function λ + const is an eigenfunction of the laplacian of g. Proof of Corollary 2. If λ is constant in a neighborhood of a point, Eq. (32) is automatically fulfilled. Below we will assume that λ is not constant. Differentiating the definition of µ and multiplying by n for cosmetic reasons, we obtain nµ,i = 2λα, α i − 2K λi .
(33)
By definition of curvature we have λi, jk − λi,k j = λα Riαjk . Contracting this with g i j , and using Ri j = Rn gi j , we obtain R λα, α k − λk,α α = − λk . n
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The formula (29) gives us λk,α α , whose substitution gives R λα, α k = −2 + 4K λk . n 2R Substituting this in (33), we obtain µ,i = − n(n−1) λi = 2K λi .
Corollary 3. Let g and g¯ be geodesically equivalent metrics, assume g is an Einstein metric. Then, the function λ given by (9) satisfies
(34) λ,i jk − K · 2λ,k gi j + λ, j gik + λ,i g jk = 0, R . where K := − n(n−1)
Proof. If λ is constant in a neighborhood of p, the equation is automatically fulfilled. Then, it is sufficient to prove Corollary 3 at points p such that λi ( p) = 0. Covariantly differentiating (24), we obtain λi, jk = µ,k gi j + K ai j,k . Substituting µ,k by (32), and ai j,k by (8), we obtain the claim, Lemma 3. Let g and g¯ be geodesically equivalent. Assume g is Einstein, and assume that λi = 0 at a point p. Then, the restriction of g¯ to a sufficiently small neighborhood U ( p) is Einstein as well. Moreover, the following formula holds (at every point of U ( p)), φi, j − φi φ j =
R R¯ gi j − g¯i j , n(n − 1) n(n − 1)
(35)
where R¯ is the scalar curvature of the metric g. ¯ Remark 8. The first statement of the lemma easily follows from certain formulas obtained in [42]. In dimension 4, under additional assumptions (R = 0 and Lorentz signature), the first statement was proved in [20]. Proof of Lemma 3. We covariantly differentiate (7) (the index of differentiation is “j”); then we substitute the expression (5) for g¯i j,k to obtain λi, j = −2e2φ φ j φα g¯ αβ gβi − e2φ φα, j g¯ αβ gβi + e2φ φα g¯ αγ g¯γ l, j g¯ lβ gβi = −e2φ φα, j g¯ αβ gβi + e2φ φα φγ g¯ αγ g¯i j + e2φ φ j φl g¯ lβ gβi ,
(36)
where g¯ αβ is the tensor dual to g¯αβ . We now substitute λi, j from (24), use that ai j is given by (6), and divide by e2φ for cosmetic reasons to obtain e−2φ µgi j + K g¯ αβ gα j gβi = −φα, j g¯ αβ gβi + φα φγ g¯ αγ g¯i j + φ j φl g¯ lβ gβi .
(37)
Multiplying with g iξ g¯ξ k , we obtain φk, j − φk φ j = (φα φβ g¯ αβ − e−2φ µ)g¯k j − K gk j . αβ
φ φ g¯ −e Let us now show that the coefficient K¯ := − α β n−1 (38) in (11), and using Ri j = Rn gi j , we obtain
−2φ µ
(38)
is constant. Substituting
R R R¯ i j = gi j − gi j − (φα φβ g¯ αβ − e−2φ µ)g¯i j . n n
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We see that R¯ i j is proportional to g¯i j . Then, g¯ is an Einstein metric; in particular, φ φ g¯ αβ −e−2φ µ R¯ K¯ := − α β n−1 is a constant equal to − n(n−1) , and (38) gives us the formula K¯ g¯i j = K gi j + φi, j − φi φ j , which is evidently equivalent to (35),
(39)
Corollary 4. Let g and g¯ be geodesically equivalent metrics, assume g is an Einstein ... ˙ φ¨ and φ metric. Consider a (parametrized) geodesic γ of the metric g, and denote by φ, the first, second and third derivatives of the function φ given by (4) along the geodesic. Then, along the geodesic, the following ordinary differential equation holds: ... ˙ 3, φ = 4K g(γ˙ , γ˙ )φ˙ + 6φ˙ φ¨ − 4(φ) (40) where g(γ˙ , γ˙ ) := gi j γ˙ i γ˙ j . Proof. If φi ≡ 0 in a neighborhood U , the equation is automatically fulfilled. Then, it is sufficient to prove Corollary 4 assuming φ is not constant. The formula (35) is evidently equivalent to (39), which is evidently equivalent to φi, j = K¯ g¯i j − K gi j + φi φ j .
(41)
Taking covariant derivative of (41), we obtain φi, jk = K¯ g¯i j,k + 2φi,k φ j + 2φ j,k φi .
(42)
Substituting the expression for g¯i j,k from (5), and substituting K¯ g¯i j given by (39), we obtain φi, jk = K¯ (2g¯i j φk + g¯ik φ j + g¯ jk φi ) + 2φi,k φ j + 2φ j,k φi = K (2gi j φk + gik φ j + g jk φi ) + 2(φk φi, j + φi φ j,k + φ j φk,i ) − 4φi φ j φk .
(43)
γ˙ i γ˙ j γ˙ k
Contracting with and using that φi is the differential of the function (4) we obtain the desired ODE (40). Corollary 5. Let g¯ (on a connected M n≥3 ) be geodesically equivalent to an Einstein metric g, but is not affine equivalent to g. Then, the restrictions of g and g¯ to any neighborhood are also not affine equivalent. Remark 9. The assumption that g is Einstein is important: Levi-Civita’s description of geodesically equivalent metrics [26] immediately gives counterexamples. Proof of Corollary 5. We consider the function φ given by (4). Suppose φi = 0 at a point p. Consider a geodesic γ (t) such that γ (0) = p, γ˙ α (0)φα ( p) = 0. Note that almost every geodesic with γ (0) = p satisfies γ˙ α (0)φα ( p) = 0. ˙ By Corollary 4, the function φ(t) = φ(γ (t)) (whose t−derivative is φ(t) = γ˙ i (t) φi (γ (t))) satisfies Eq. (40) along the geodesic. Clearly, every constant is a solution of ˙ the equation. Since φ(0) = 0, by uniqueness of the solutions of ODE, the restriction ˙ = 0 φ(t) to every open interval can not be constant. Hence, the subset of t such that φ(t) ˙ is everywhere dense. Since as we mentioned above, φ(0) = 0 for almost every geodesic γ (with γ (0) = p), we have that for every point p0 of every geodesic passing through p k→∞
there exists a sequence pk ∈ M n such that φi ( pk ) = 0 and such that pk −→ p0 . Since every point can be reached from the point p by a sequence of geodesics, we have that φi = 0 at every point of an open everywhere dense subset of M n .
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2.4. Proof of Theorem 1 for Riemannian metrics of nonpositive scalar curvature, and for pseudo-Riemannian metrics. Assume the metric g on a connected M n≥3 is Einstein and is either Riemannian (i.e., positive definite) with nonpositive scalar curvature, or there exist light-like vectors (i.e., for no constant c = 0 the metric c · g is Riemannian). Let g¯ be geodesically equivalent to g. Assume both metrics are complete. Our goal is to show that φ given by (4) is constant, because in view of (1) this implies that the metrics are affine equivalent. Consider a parameterized geodesic γ (t) of g. If the metric g is pseudo-Riemannian, we additionally assume that γ is a light-like geodesic i.e., γ˙ i γ˙ j gi j = 0. Since the metrics are geodesically equivalent, for a certain function τ : R → R the curve γ (τ ) is a geodesic of g. ¯ Since the metrics are complete, the reparameterization τ (t) is a diffeod morphism τ : R → R. Without loss of generality we can think that τ˙ := dt τ is positive, otherwise we replace t by −t. Then, Eq. (2) along the geodesic reads φ(t) =
1 log(τ˙ (t)) + const0 . 2
(44)
Now let us consider Eq. (40). Substituting 1 φ(t) = − log( p(t)) + const0 2
(45)
in it (since τ˙ > 0, the substitution is global), we obtain ... p = 4K g(γ˙ , γ˙ ) p. ˙
(46)
Since the length of the tangent vector is preserved along a geodesic, g(γ˙ , γ˙ ), and therefore 4K g(γ˙ , γ˙ ) is a constant. The assumptions above imply that this constant is nonnegative. Indeed, if γ˙ is a light-like vector, this constant is zero, since γ is an light-like geodesic. If the metric is Riemannian of nonpositive curvature, g(γ˙ , γ˙ ) ≥ 0, and K ≥ 0, so their product is nonnegative. Eq. (46) can be solved. We will first consider the case K g(γ˙ , γ˙ ) = 0. In this case, the solution of (46) is p(t) = C2 t 2 + C1 t + C0 . Combining (45) with (44), we see that τ˙ = C t 2 +C1 t+C . Then 2
1
0
τ (t) =
t
t0
C2
ξ2
dξ + const. + C1 ξ + C0
(47)
We see that if the polynomial C2 t 2 + C1 t + C0 has real roots (which is always the case if C2 = 0, C1 = 0), then the integral diverges (goes to infinity in finite time). If the polynomial has no real roots, but C2 = 0, the function τ is bounded. Thus, the only possibility for τ to be a diffeomorphism is C2 = C3 = 0, implying τ (t) = C10 t + const1 , implying τ˙ = C10 , implying φ is constant along the geodesic. Now, let us consider the case K g(γ˙ , γ˙ ) > 0. In this case, the general solution of Eq. (46) is √ √ C + C+ e2 K g(γ˙ ,γ˙ )t + C− e−2 K g(γ˙ ,γ˙ )t . (48) Then, the function τ satisfies the ODE τ˙ = implying
1
C + C+
√ e2 K g(γ˙ ,γ˙ )t
+ C− e−2
√
K g(γ˙ ,γ˙ )t
Complete Einstein Metrics are Geodesically Rigid
τ (t) =
t t0
C + C+
e2
√
395
dξ K g(γ˙ ,γ˙ )ξ
+ C− e−2
√
K g(γ˙ ,γ˙ )ξ
+ const.
(49)
If one of the constants C+ , C− is not zero, the integral (49) is bounded from one side, or diverges (goes to infinity in finite time). Thus, the only possibility for τ to be a diffeomorphism of R on itself is C+ = C− = 0. Finally, φ is a constant along the geodesic γ . Since every point of a connected manifold can be reached by a sequence of light-like geodesics in the pseudo-Riemannian case, or by a sequence of geodesics in the Riemannian case, φ is a constant, so that φi ≡ 0, and the metrics are affine equivalent by (1), Remark 10. A similar idea was used by Couty [11] in an investigation of projective transformations of Einstein manifolds, and by Shen [51] in an investigation of Finsler Einstein geodesically equivalent metrics. 2.5. Proof of Theorem 1 for Riemannian metrics of positive scalar curvature. We assume that g is a complete Einstein Riemannian metric of positive scalar curvature on a connected manifold (we do not need that the second metric is complete). Then, by Corollary 3, λ is a solution of (34). If the metrics are not affine equivalent, λ is not identically constant. Eqs. (34) was studied by Obata and Tanno in [45,54] in a completely different geometrical context. They proved (actually, Tanno [54], because the proof of Obata [45] has a mistake) that a complete Riemannian g such that there exists a nonconstant function λ satisfying (34) must have a constant positive curvature. Applying this result in our situation, we obtain the claim, 3. Proof of Theorem 2 It is sufficient to prove Theorem 2 in a neighborhood of a point p such that λi given by (7) does not vanish. Indeed, by Corollary 5, either such points are everywhere dense, or the metrics are affine equivalent. We will first formulate two simple lemmas from Linear Algebra, then prove a simple Lemma 6 which generalizes a certain result of Levi-Civita [26], and then obtain Theorem 2 as an easy corollary. 3.1. Two simple lemmas from Linear Algebra. We say that the vector v i lies in kernel of the tensor Z i jkl , if v i Z i jkl = 0. Lemma 4. Assume the tensor Z i jkl on R4 has the following symmetries: Z i jkl = Z kli j , Z i jkl = −Z jikl , g ik
and satisfies Z i jkl = 0. Suppose the vector in the kernel of Z i jkl . Then, Z = 0.
vi
such that g(v, v) :=
(50) v i v j gi j
= 0 lies
Remark 11. The assumption g(v, v) = 0 is important: one immediately constructs a counterexample. The dimension is also important: the claim fails for dimensions ≥ 5. Proof of Lemma 4. is an easy exercise and will be left to the reader. We recommend to consider a basis such that the first vector is v and the metric is given by the matrix ⎛ ⎞ ε1 ⎜ ε2 ⎟ , ⎝ ε3 ⎠ ε4
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where all εi = 0. Then, the conditions v i Z i jkl = 0 and Z i jkl g ik = 0 are a system of homogeneous linear equations on the components of Z which admit only trivial solution implying the claim. Lemma 5. Let a and Z be n × n matrices over C such that Z is skew-symmetric and such that their product a Z is symmetric. Let the geometric multiplicity of the eigenvalue ρ ∈ C of the matrix a be 1. Then, every vector v from the generalized eigenspace of ρ lies in the kernel of the matrix Z . (Recall that geometric multiplicity of ρ is the dimension of the kernel of (a − ρ · 1), and the generalized eigenspace of ρ is the kernel of (a − ρ · 1)n .) The proof of Lemma 5 is an easy exercise in linear algebra and will be left to the reader. We recommend to consider the basis such that the matrix a is in Jordan form, and then to calculate the matrix a Z . One immediately sees that it is block diagonal, and that if the eigenspace is one dimensional then the corresponding block is trivial, Corollary 6. Suppose Z i jkl is skew-symmetric with respect to indexes i, j. Suppose aiα Z α jkl + a αj Z αikl = 0
(51)
for a (1,1)-tensor a satisfying a αj gαi = aiα gα j , where (the metric) g is a symmetric nondegenerate (0, 2)-tensor. We assume that all components of Z , g, and a are real. Suppose there exists a (possibly, complex) eigenvalue ρ with geometric multiplicity 1. Then, there exists a vector v such that gi j v i v j = 0 lying in the kernel of Z . Proof. The condition a αj Z αi + aiα Z α j = 0 precisely means that the matrix a Z is symmetric. We see that this condition is the condition (51) with “forgotten” indexes k and l. Then, by Lemma 5, every vector v from the sum of the generalized eigenspaces of ρ and of its complex-conjugate ρ¯ lies in a kernel of Z . Since the generalized eigenspaces of ρ and of ρ¯ are orthogonal to all other generalized eigenspaces because of the condition a αj gαi = aiα gα j , and because the direct sum of all generalized eigenspaces coincides with the whole vector space, the sum of the generalized eigenspaces of ρ and of ρ¯ contains a (real) vector v such that gi j v i v j = 0, 3.2. If all eigenspaces are more than one-dimensional, the metrics are affine equivalent. Lemma 6. If geometric multiplicity of every eigenvalue of the solution ai j of (8) is at least two, then the function λ given by (9) is constant. Remark 12. For Riemannian metrics, the statement is due to Levi-Civita [26]. The proof for the pseudo-Riemannian case is essentially the same; the additional difficulties are due to possible Jordan blocks. In a certain form, it appears in [1]. Proof of Lemma 6. We prove the lemma assuming every Jordan-Block of a ij is at most 3-dimensional; this is sufficient for our four-dimensional goals. The proof for arbitrary dimensions of Jordan blocks can be done by induction. Let ρ be an eigenvalue of a ij ; let u i be an eigenvector corresponding to ρ. In a small neighborhood of almost every point, ρ a smooth (possibly, complex-valued) function. We will show that the differential ρ,i is proportional to u i . If the eigenspace of ρ is more than one-dimensional, this will imply that ρ,i is constant. This implies that if all
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eigenspaces are more than one-dimensional, the trace of a ij is constant implying the metrics are affine equivalent. Let u be an eigenvector corresponding to ρ, i.e., u α aiα = ρu i .
(52)
We take the covariant derivative and use (8). We obtain u α, j aiα + u α λα gi j + λi u j = ρ, j u i + ρu i, j .
(53)
We multiply (53) with u i and sum over i, to obtain (using (52)) 2λα u α u j = u α u α ρ, j .
(54)
We see that if u α u α = 0 (which is in particular always the case when the Jordan block corresponding to ρ is 1-dimensional), we are done. Suppose the Jordan block corresponding to ρ is more than 1-dimensional, i.e., there exists vi such that vα aiα = ρvi + u i .
(55)
Then, u i is automatically a light-like vector: indeed, multiplying (55) by u i , summing over i, and using (52), we obtain u α u α = 0.
(56)
u α,i u α = 0.
(57)
Differentiating (56), we obtain
Substituting (56) in (54), we obtain λα u α = 0. Differentiating (55) and using (8), we obtain vα, j aiα + vα λα gi j + λi v j = ρ, j vi + ρvi, j + u i, j .
(58)
Multiplying (58) by u i and summing over i, we obtain vα λα u j = vα u α ρ, j .
(59)
We see that if vα u α = 0, (which is in particular always the case when the Jordan block corresponding to ρ is 2-dimensional), we are done. Suppose the Jordan block corresponding to ρ is precisely 3-dimensional, i.e., there exists wi such that wα u α = 0
(60)
wα aiα = ρwi + vi .
(61)
and such that
We multiply (61) with v i and sum over i, to obtain wα u α = vα v α .
(62)
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We multiply (61) with u i and sum over i, to obtain u α v α = 0.
(63)
Differentiating (63), we obtain u α,i v α = −u α vα,i . Moreover, combining (63) with (59), we obtain λα obtain
(64) vα
= 0. Differentiating (61), we
wα, j aiα + wα λα gi j + λi w j = ρ, j wi + ρwi, j + vi, j . Contracting this with u i , we obtain (64)
wα λα u j = wα u α ρ, j + u α vα, j = wα u α ρ, j − u α,i v α . We multiply (58) with
vi
(65)
and sum over i to obtain vα, j u α = vα v α ρ, j + u α, j v α .
(66)
Using (64), we obtain (62)
2vα, j u α = vα v α ρ, j = wα u α ρ, j .
(67)
λα u
α Combining (67) with (65), we obtain 2wα j = 3u α w ρ, j . Combining this with (60), we obtain that the differential ρ,i is proportional to the eigenvector u i . If the eigenspace of ρ is more than one-dimensional, this implies that ρ,i = 0,
3.3. Proof of Theorem 2. If the dimension is 3, Theorem 2 follows from the well-known fact that every Einstein 3-manifold has constant curvature. We assume that g is an Einstein metric on M 4 . Let g¯ be geodesically equivalent to g. We consider the solution ai j of (8) given by (6). Assume that the corresponding λi = 0 at p. We will show that in a small neighborhood of p the metric g has constant curvature implying the metrics g¯ and gˆ have constant curvature as well by the Beltrami Theorem (see for example [38], or the original papers [4] and [50]). α = 0, where Substituting Eq. (24) in (10), we obtain aiα Z αjkl + aα j Z ikl Z ijkl = R ijkl − K · (δli g jk − δki g jl ).
(68)
We see that by construction the tensor Z i jkl has the symmetries (50). Since g is Einstein, the tensor Z i jkl satisfies Z i jkl g ik = 0. By Lemma 6, at almost every point there exists an eigenvalue of a ij with geometric multiplicity one. Then, by Corollary 6, there exists a vector v i such that g(v, v) = 0 and such that v i lies in the kernel of Z . By Lemma 5, the tensor Z ≡ 0 implying in view of (68) the claim, Acknowledgement. The results were obtained because Gary Gibbons asked the second author to check whether certain explicitly given Einstein metrics admit geodesic equivalence (these metrics admit integrals quadratic in velocities, and geodesic equivalence could lay behind the existence of such integrals, see [22,23,28–32], [35–37,49]). There exists an algorithmic method to understand whether a explicitly given metric admits a nontrivial geodesic equivalence (assuming we can explicitly differentiate components of the metrics, and perform algebraic operations). Unfortunately, the method is highly computational, and applying it to the metrics suggested by Gibbons, which are given by quite complicated formulas, resulted so huge output, that we could not convince even ourself that everything is correct. Therefore, we started to look for a theory that could simplify the calculations, and solved the problem in the whole generality.
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We thank Gary Gibbons for his question, and the referee for stylistic and grammatical corrections. The second author thanks Oxford, Cambridge, and Loughborough Universities, and MSRI for hospitality, and R. Bryant, A. Bolsinov, M. Eastwood, and G. Hall for useful discussions. Both authors were partially supported by Deutsche Forschungsgemeinschaft (Priority Program 1154 — Global Differential Geometry), and by FSU Jena.
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Commun. Math. Phys. 289, 401–433 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0665-4
Communications in
Mathematical Physics
Comparison Theory and Smooth Minimal C -Dynamics Andrew S. Toms Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario, Canada M3J 1P3. E-mail: [email protected] Received: 25 June 2008 / Accepted: 12 July 2008 Published online: 21 November 2008 – © Springer-Verlag 2008
Abstract: We prove that the C∗ -algebra of a minimal diffeomorphism satisfies Blackadar’s Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C∗ -algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C∗ -algebras with the same K-theory and tracial state space, providing a C∗ -algebraic analogue of McDuff’s uncountable family of II1 factors. We prove in passing that the range of the radius of comparison is exhausted by simple C∗ -algebras. 1. Introduction The comparison theory of projections is fundamental to the theory of von Neumann algebras, and is the basis for the type classification of factors. For a general C∗ -algebra this theory is vastly more complicated, but no less central. Blackadar opined in [1] that “the most important general structure question concerning simple C∗ -algebras is the extent to which the Murray-von Neumann comparison theory for factors is valid in arbitrary simple C∗ -algebras.” In this article we answer Blackadar’s question for the C∗ -algebras associated to smooth minimal dynamical systems, among others, and give several applications. This research was supported in part by an NSERC Discovery Grant.
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Tellingly, Blackadar’s quote makes no mention of projections. A C∗ -algebra may have few or no projections, in which case their comparison theory says little about the structure of the algebra. The appropriate replacement for projections is positive elements, along with a notion of comparison for the latter which generalises Murray-von Neumann comparison for projections. This idea was first introduced by Cuntz in [9] with a view to studying dimension functions on simple C∗ -algebras. His comparison relation is conveniently encoded in what is now known as the Cuntz semigroup, a positively ordered Abelian monoid whose construction is analogous to that of the Murray-von Neumann semigroup. When the natural partial order on this semigroup is governed by traces, then we say that the C∗ -algebra has strict comparison of positive elements (see Subsec. 2.2 for a precise definition); this property, first introduced in [1], is also known as Blackadar’s Fundamental Comparability Property for positive elements. It is the best available analogue among simple C∗ -algebras for the comparison theory of projections in a factor, and a powerful regularity property necessary for the confirmation of G. A. Elliott’s K-theoretic rigidity conjecture (see [11 and 27]). Its connection with the comparison theory of projections in a von Neumann algebra is quite explicit: if a unital simple stably finite C∗ -algebra A has strict comparison of positive elements, then Cuntz comparison for those positive elements with zero in their spectrum is synonymous with Murray-von Neumann comparison of the corresponding support projections in the bidual; the remaining positive elements have support projections which are contained in A, and Cuntz comparison for these elements reduces to Murray-von Neumann comparison of their support projections in A, as opposed to A∗∗ . Our main result applies to a class of C∗ -algebras which contains properly the ∗ C -algebras associated to minimal diffeomorphisms. Recall that a C∗ -algebra is subhomogeneous if there is a uniform bound on the dimensions of its irreducible representations, and approximately subhomogeneous (ASH) if it is the limit of a direct system of subhomogeneous C∗ -algebras. There are no known examples of simple separable amenable stably finite C∗ -algebras which are not ASH. Every unital separable ASH algebra is the limit of a direct sequence of recursive subhomogeneous C∗ -algebras, a particularly tractable kind of subhomogeneous C∗ -algebra ([19]). Theorem 1.1. Let (Ai , φi ) be a direct sequence of recursive subhomogeneous C∗ - algebras with slow dimension growth. Suppose that the limit algebra A is unital and simple. It follows that A has strict comparison of positive elements. We note that the hypothesis of slow dimension growth is necessary, as was shown by Villadsen in [31]. The relationship between Theorem 1.1 and the C∗ -algebras of minimal dynamical systems is derived from the following theorem: Theorem 1.2 ([17]). Let M be a compact smooth connected manifold, and let h : M → M be a minimal diffeomorphism. It follows that the transformation group C∗ -algebra C∗ (M, Z, h) is a unital simple direct limit of recursive subhomogeneous C∗ -algebras with slow dimension growth (indeed, no dimension growth). K-theoretic considerations show the class of C∗ -algebras covered by Theorem 1.1 to be considerably larger than the class covered by Theorem 1.2. Let us describe briefly the applications of our main result. In a C∗ -algebra A of stable rank one, the Cuntz semigroup can be identified with the semigroup of isomorphism classes of countably generated Hilbert A-modules—addition corresponds to the direct sum, and the partial order is given by inclusion of modules ([7]). It is also known that positive elements a, b ∈ A are approximately unitarily equivalent if and only if the canonical maps from C0 (0, 1] into A induced by a and b agree at the level of the Cuntz
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semigroup ([5]). Thus, to the extent that one knows the structure of the Cuntz semigroup, one also knows what the isomorphism classes of Hilbert A-modules and the closures of unitary orbits of positive operators look like. If A is in addition unital, simple, exact, and has strict comparison of positive elements, then its Cuntz semigroup can be described in terms of K-theory and traces (see [4, Theorem 2.6]), and the Ciuperca-Elliott classification of orbits of positive operators extends to self-adjoint elements. Thus, for the algebras of Theorem 1.1, under the additional assumption of stable rank one, we have a description of the countably generated Hilbert A-modules and of the closures of unitary orbits of self-adjoints in terms of K-theory and traces. (In fact, this description also captures the inclusion relation for the said modules, and the structure of their direct sums.) This result applies to the C∗ -algebras of minimal diffeomorphisms as these were shown to have stable rank one by N. C. Phillips ([23]). Our classification is quite practical, as the K-theory of these algebras is accessible through the Pimsner-Voiculescu sequence and their traces have a nice description as the invariant measures on the manifold M. The classification of Hilbert modules obtained is analogous to the classification of W∗ -modules over a II1 factor. (See [4] and Subsects. 5.4 and 5.5.) Finally, we note that Jacob has recently obtained a description of the natural metric on the space of unitary orbits of self-adjoint elements in a unital simple ASH algebra under certain assumptions, one of which is strict comparison. This gives another application of Theorem 1.1 ([15]). It was shown in [3, Theorem 6.4] that if the structure theorem for the Cuntz semigroup alluded to above holds for A, then the lower semicontinuous dimension functions on A are weakly dense in the space of all dimension functions on A, confirming a conjecture of Blackadar and Handelman from the early 1980s. This conjecture therefore holds for the algebras of Theorem 1.1. (See Subsects. 5.2 and 5.3.) If A is a unital stably finite C∗ -algebra, then one can define a nonnegative real-valued invariant called the radius of comparison which measures the extent to which the order structure on the Cuntz semigroup of A is determined by (quasi-)traces. This invariant has proved useful in the matter of distinguishing simple separable amenable C∗ -algebras both in general ([29]) and in the particular case of minimal C∗ -dynamical systems ([12]). The proof of Theorem 1.1 follows from a sharp upper bound that we obtain for the radius of comparison of a recursive subhomogeneous C∗ -algebra. This bound generalises and improves substantially upon our earlier bound for homogeneous C∗ -algebras ([30]). In addtion to being crucial for the proof of Theorem 1.1, this bound has other applications. We use it to prove that the range of the radius of comparison is exhausted by simple C∗ -algebras, and that there are uncountably many non-Morita-equivalent simple separable amenable C∗ -algebras which all have the same K-theory and tracial state space (Theorem 5.11). This last result is proved using approximately homogenenous (AH) algebras of unbounded dimension growth, and so may be viewed as a strong converse to the Elliott-Gong-Li classification of simple AH algebras with no dimension growth ([10]). It can also be viewed as a C∗ -algebraic analogue of McDuff’s uncountable family of pairwise non-isomorphic II1 factors ([18]). (See Subsect. 5.1 and 5.6.) W. Winter has recently announced a proof of Z-stability for a class of C∗ -algebras which includes unital simple direct limits of recursive subhomogeneous C∗ -algebras with no dimension growth, leading to an alternative proof of Theorem 1.1 under the stronger hypothesis of no dimension growth. Those working on G. A. Elliott’s classification program for separable amenable C∗ -algebras suspect that the conditions of slow dimension growth and no dimension growth are equivalent, but this problem remains open even for AH C∗ -algebras. Gong has shown that no dimension growth and a strengthened version
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of slow dimension growth are equivalent for unital simple AH algebras, an already difficult result (see [13]). The paper is organised as follows: Section 2 collects our basic definitions and preparatory results; Section 3 establishes a relative comparison theorem in the Cuntz semigroup of a commutative C∗ -algebra; Section 4 applies the said comparison theorem to obtain sharp bounds on the radius of comparison of a recursive subhomogeneous algebra; Section 5 describes our applications in detail. 2. Preliminaries 2.1. The Cuntz semigroup. Let A be a C ∗ -algebra, and let Mn (A) denote the n × n matrices whose entries are elements of A. If A = C, then we may simply write Mn . Let M∞ (A) denote the algebraic limit of the direct system (Mn (A), φn ), where φn : Mn (A) → Mn+1 (A) is given by a0 . a → 00 Let M∞ (A)+ (resp. Mn (A)+ ) denote the positive elements in M∞ (A) (resp. Mn (A)). Given a, b ∈ M∞ (A)+ , we say that a is Cuntz subequivalent to b (written a b) if there is a sequence (vn )∞ n=1 of elements of M∞ (A) such that n→∞
||vn bvn∗ − a|| −→ 0. We say that a and b are Cuntz equivalent (written a ∼ b) if a b and b a. This relation is an equivalence relation, and we write a for the equivalence class of a. The set W (A) := M∞ (A)+ / ∼ becomes a positively ordered Abelian monoid when equipped with the operation a + b = a ⊕ b and the partial order a ≤ b ⇔ a b. In the sequel, we refer to this object as the Cuntz semigroup of A. (It was originally introduced by Cuntz in [9].) The Grothendieck enveloping group of W (A) is denoted by K0∗ (A). Given a ∈ M∞ (A)+ and > 0, we denote by (a − )+ the element of C ∗ (a) corresponding (via the functional calculus) to the function f (t) = max{0, t − }, t ∈ σ (a). (Here σ (a) denotes the spectrum of a.) The proposition below collects some facts about Cuntz subequivalence due to Kirchberg and Rørdam. Proposition 2.1 ([16,26]). Let A be a C ∗ -algebra, and a, b ∈ A+ . (i) (a − )+ a for every > 0. (ii) The following are equivalent: (a) a b; (b) for all > 0, (a − )+ b; (c) for all > 0, there exists δ > 0 such that (a − )+ (b − δ)+ . (iii) If > 0 and ||a − b|| < , then (a − )+ b.
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2.2. Dimension functions and strict comparison. Now suppose that A is unital and stably finite, and denote by QT(A) the space of normalised 2-quasitraces on A (v. [2, Def. II.1.1]). Let S(W (A)) denote the set of additive and order preserving maps d : W (A) → R+ having the property that d(1 A ) = 1. Such maps are called states. Given τ ∈ QT(A), one may define a map dτ : M∞ (A)+ → R+ by dτ (a) = lim τ (a 1/n ). n→∞
(1)
This map is lower semicontinous, and depends only on the Cuntz equivalence class of a. It moreover has the following properties: (i) if a b, then dτ (a) ≤ dτ (b); (ii) if a and b are mutually orthogonal, then dτ (a + b) = dτ (a) + dτ (b); (iii) dτ ((a − )+ ) dτ (a) as → 0. Thus, dτ defines a state on W (A). Such states are called lower semicontinuous dimension functions, and the set of them is denoted LDF(A). QT(A) is a simplex ([2, Theorem II.4.4]), and the map from QT(A) to LDF(A) defined by (1) is bijective and affine ([2, Theorem II.2.2]). A dimension function on A is a state on K0∗ (A), assuming that the latter has been equipped with the usual order coming from the Grothendieck map. The set of dimension functions is denoted DF(A). LDF(A) is a (generally proper) face of DF(A). If A has the property that a b whenever d(a) < d(b) for every d ∈ LDF(A), then we say that A has strict comparison of positive elements. 2.3. Preparatory results. We now recall and improve upon some results that will be required in the sequel. Definition 2.2 (cf. Definition 3.4 of [30]). Let X be a compact Hausdorff space, and let a ∈ Mn (C(X )) be positive with (lower semicontinuous) rank function f : X → Z+ taking values in {n 1 , . . . , n k }, n 1 < n 2 < · · · < n k . Set Fi,a := {x ∈ X | f (x) = n i }. We say that a is well supported if, for each 1 ≤ i ≤ k, there is a projection pi ∈ Mn (C(Fi,a )) such that lim a(x)1/r = pi (x), ∀x ∈ Fi,a ,
r →∞
and pi (x) ≤ p j (x) whenever x ∈ Fi,a ∩ F j,a and i ≤ j. Theorem 2.3 (cf. Theorem 3.9 of [30]). Let X be a compact Hausdorff space, and let a ∈ Mn (C(X ))+ and > 0 be given. It follows that there is a˜ ∈ Mn (C(X ))+ with the following properties: (i) a˜ ≤ a; (ii) ||a − a|| ˜ < ; (iii) a˜ is well supported. Remark 2.4. In the statement of Theorem 3.9 of [30], X is required to be a finite simplicial complex, but this is only to ensure that some further conclusions about the approximant a˜ can be drawn. The proof of this theorem, followed verbatim, also proves Theorem 2.3—one simply ignores all statements which concern the simplicial structure of X . An alternative proof can be found in [17].
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For our purposes, we require a different and in some ways strengthened version of Theorem 2.3. It says that the well-supported approximant a˜ can be obtained as a cut-down of a, at the possible expense of condition (i). Lemma 2.5. Let X , a, and be as in the statement of Theorem 2.3. Suppose further that a has norm at most one. It follows that there is a positive element h of Mn (C(X )) of norm at most one such that the following statements hold: (i) ||hah − a|| < ; (ii) ||ha − a|| < /2 and ||ah − a|| < /2; (iii) hah is well-supported. Proof. Apply Theorem 2.3 to a with the tolerance /4 to obtain the approximant a. ˜ This approximant can be described as follows (the details can be found in the proof of Theorem 3.9, [30], which is constructive). At every x ∈ X there are mutually orthogonal positive elements a1 (x), . . . , ak (x) of Mn (C) such that a(x) = a1 (x) ⊕ a2 (x) ⊕ · · · ⊕ ak (x). Note that k varies with x, and that we make no claims about the continuity of the ai s. Our approximant then has the form a(x) ˜ = λ1 a1 (x) ⊕ λ2 a2 (x) ⊕ · · · ⊕ λk ak (x), where λi ∈ [0, 1]. We also have that ||ai (x)|| < /4 whenever λi = 1, and that there is an η > 0, independent of x, such that the spectrum of ai (x) is contained in [η, 1] whenever λi = 1. Let f : [0, 1] → [0, 1] be the continuous map given by t/η, t ≤ η f (t) = . 1, t > η Set h(x) = f (a(x)) ˜ = f (λ1 a1 (x)) ⊕ f (λ2 a2 (x)) ⊕ · · · ⊕ f (λk ak (x)), and note that h : X → Mn (C) is indeed a positive element of Mn (C(X )) since a˜ is. Let us first verify that ||ha − a|| < /2; the proof that ||ah − a|| < /2 is similar. For every x ∈ X we have h(x)a(x) − a(x) =
k
( f (λi ai (x))ai (x) − ai (x)) .
i=1
If λi = 1, then f (λi ai (x)) = pi (x), where pi (x) is the support projection of ai (x) in Mn (C). Thus, f (λi ai (x))ai (x) − ai (x) = pi (x)ai (x) − ai (x) = ai (x) − ai (x) = 0. Otherwise, ||ai (x)|| < /4 and || f (λi ai (x))|| ≤ 1, whence || f (λi ai (x))ai (x) − ai (x)|| < /4 + /4 = /2.
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We have shown that || f (λi ai (x))ai (x) − ai (x)|| < /2 for each i ∈ {1, . . . , k}, so that ||ha − a|| < /2, proving (ii). For (i), we have hah − a = hah − ha + ha − a ≤ h · ah − a + ha − a < /2 + /2 = . To complete the proof, we must show that hah is well-supported. The property of being well-supported depends only on the support projection of hah(x) as x ranges over X . It will thus suffice for us to show that the support projection of hah(x) is the same as that of a(x), ˜ since a˜ is well-supported. If λi is zero, then so is f (λi ai (x))ai (x) f (λi ai (x)), whence both it and λi ai (x) have the same support projection, namely, zero. If λi = 0, then f (λi ai (x))ai (x) f (λi ai (x)) is the image of λi ai (x) under the map t → f (t)(t/λi ) f (t). This map is nonzero on (0, 1], and it follows that λi ai (x) and f (λi ai (x))ai (x) f (λi ai (x)) again have the same support projection. Since these statements hold for each i ∈ {1, . . . , k}, we conclude that the support projections of a(x) ˜ and hah(x) agree for each x ∈ X . Proposition 2.6 (Proposition 4.2 (1) of [22]). Let X be a compact Hausdorff space of finite covering dimension d, and let E ⊂ X be closed. Let p, q ∈ Mn (C(X )) be projections with the property that 1 rank(q(x)) + (d − 1) ≤ rank( p(x)), ∀x ∈ X. 2 Let s0 ∈ Mn (C(E)) be such that s0∗ s0 = q| E and s0 s0∗ ≤ p| E . It follows that there is s ∈ Mn (C(X )) such that s ∗ s = q, ss ∗ ≤ p, and s0 = s| E . We record a corollary of Proposition 2.6 for use in the sequel. Corollary 2.7. Let X be a compact Hausdorff space of covering dimension d ∈ N, and let E 1 , . . . , E k be a cover of X by closed sets. Let p ∈ Mn (C(X )) and qi ∈ Mn (C(E i )) be projections of constant rank for each i ∈ {1, . . . , k}. Set n i = rank(qi ), and assume that n 1 < n 2 < · · · < n k . Assume that qi (x) ≤ q j (x) whenever x ∈ E i ∩ E j and i ≤ j. Finally, suppose that n i − rank( p) ≥ (1/2)(d − 1) for every i. The following statements hold: (i) there is a partial isometry w ∈ Mn (C(X )) such that w ∗ w = p and qi (x), ∀x ∈ X ; (ww ∗ )(x) ≤ {i | x∈E i }
(ii) if Y ⊆ X is closed, p|Y corresponds to a trivial vector bundle, and qi (y), ∀y ∈ Y, p(y) ≤ {i | y∈E i }
then p|Y can be extended to a projection p˜ on X which also corresponds to a trivial vector bundle and satisfies qi (x), ∀x ∈ X. p(x) ˜ ≤ {i | x∈E i }
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Proof.
(i) The rank inequality hypothesis and the stability properties of vector bundles imply that there is a partial isometry w1 ∈ Mn (C(E 1 )) such that w1∗ w1 = p and w1 w1∗ ≤ q1 . Since q1 (x) ≤ q j (x) whenever x ∈ E 1 ∩ E j , we have q j (x), ∀x ∈ E 1 . (w1 w1∗ )(x) ≤ { j | x∈E j }
Suppose now that we have found a partial isometry wi ∈ Mn (C(E 1 ∪ · · · ∪ E i )) such that wi∗ wi = p and (wi wi∗ )(x) ≤ q j (x), ∀x ∈ E 1 ∪ · · · ∪ E i . (2) { j | x∈E j }
We may now apply Proposition 2.6 with X = E i+1 , E = E i+1 ∩ (E 1 ∪ · · · ∪ E i ), and s0 = wi | Ei+1 ∩(E 1 ∪···∪Ei ) to extend wi to a partial isometry wi+1 ∈ Mn (C(E 1 ∪ · · · ∪ E i+1 )) which satisfies (2) with i + 1 in place of i. Continuing inductively yields the desired result. (ii) We will explain how to extend p|Y to p˜ defined on Y ∪ E 1 . The desired result then follows from iteration of this procedure. Let r be a projection over Y ∪ E 1 which corresponds to a trivial bundle and has rank equal to rank( p). Since r |Y and p|Y are Murray-von Neumann equivalent, we may use Proposition 2.6 to find a partial isometry s defined over Y ∪ E 1 with the property that s ∗ s = r and ss ∗ |Y = p. Thus, without loss of generality, we may assume that r |Y = p|Y . Now r |Y ∩E 1 corresponds to a trivial vector bundle and is subordinate to q1 |Y ∩E 1 . We may view this subordination as the statement that r |Y ∩E 1 is a partial isometry such that r |Y ∩E 1 = (r |Y ∩E 1 )∗ (r |Y ∩E 1 ) = (r |Y ∩E 1 )(r |Y ∩E 1 )∗ ≤ q1 |Y ∩E 1 . By Proposition 2.6, we may extend r |Y ∩E 1 to a partial isometry v defined on E 1 with the property that r | E 1 = v ∗ v ∼ vv ∗ ≤ q1 . We can extend v to Y ∪ E 1 by setting v(x) = r (x) for each x ∈ Y \E 1 . Now set p˜ = vv ∗ , so that p, ˜ by virtue of being equivalent to r , must correspond to a trivial vector bundle. It follows that for every x ∈ Y, v(x)v(x)∗ = (r |Y )(x) = ( p|Y )(x), and for every x ∈ E 1 p(x) ˜ = v(x)v(x)∗ ≤ q1 (x). The proof of the next lemma is contained in the proof of [30, Prop. 3.7]. Lemma 2.8. Let X be a compact Hausdorff space, and let a, b ∈ Mn (C(X )) be positive. Suppose that there is a non-negative integer k such that rank(a(x)) + k ≤ rank(b(x)), ∀x ∈ X. It follows that for every > 0 there is δ > 0 with the property that rank((a − )+ (x)) + k ≤ rank((b − δ)+ (x)), ∀x ∈ X.
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3. A Relative Comparison Result in Mn (C(X)) The goal of this section is to prove the following lemma. Lemma 3.1. There is a natural number N such that the following statement holds: Let X be a compact metrisable Hausdorff space of finite covering dimension d, and let Y ⊆ X be closed. Let a, b ∈ Mn (C(X )) be positive and, for a given tolerance 1 > > 0, satisfy (i) ||a(x) − b(x)|| < for each x ∈ Y , and (ii) rank(a(x)) + (d − 1)/2 ≤ rank(b(x)) for each x ∈ / Y. It follows that there are positive elements c, d and a unitary element u in M4n (C(X )) whose restrictions to Y are all equal to 1 ∈ M4n (C(Y )), and which, upon viewing a and b as elements of the upper left n × n corner of M4n (C(X )), satisfy the inequality √ ||(duc)b(duc)∗ − a|| < N . The proof of Lemma 3.1 proceeds in several steps. Lemma 3.2. Let X be a compact metrisable Hausdorff space, and let Y be a closed subset of X . Suppose that we have positive elements a, b ∈ Mn (C(X )), a tolerance > 0, and a natural number k satisfying (i) ||a|Y − b|Y || < , and (ii) rank(a(x)) + k ≤ rank(b(x) for each x ∈ / Y. It follows that there are a positive element a˜ ∈ Mn (C(X )) and open neighbourhoods U1 ⊆ U2 of Y with the following properties: (a) (b) (c) (d)
||a − a|| ˜ < 4; U1 ⊆ U2 ; a(x) ˜ = (b(x) − 2)+ for every x ∈ U2 \U1 ; rank(a(x)) ˜ + k ≤ rank(b(x)) for each x ∈ X \U1 .
Proof. By the continuity of a and b we can find open neighbourhoods U1 ⊆ U2 ⊆ U3 of Y such that U1 ⊆ U2 , U2 ⊆ U3 , and ||a|U3 − b|U3 || < (3/2). Let f : X → [0, 1] be a continuous map which is equal to zero on Y ∪ (X \U3 ) and equal to one on U2 \U1 . As a first approximation to our desired element a, ˜ we define a1 (x) = (1 − f (x))a(x) + f (x)b(x). We then have ||a1 |U3 − b|U3 || < 2 and ||a − a1 || < 2. Now find a continuous function g : X → [0, 1] which is zero on Y , and equal to one on X \U1 . Set a(x) ˜ = (a1 (x) − 2g(x))+ . Thus, conclusions (a) and (b) are satisfied. For each x ∈ U2 \U1 we have f (x) = g(x) = 1, so that a1 (x) = b(x) and a(x) ˜ = (b(x) − 2)+ . This establishes part (c) of the conclusion. For part (d) of the conclusion we treat two cases. For x ∈ U3 \U1 we have the estimate ||a1 (x) − a(x)|| < 2 and the fact that a(x) ˜ = (a1 (x) − 2)+ . Proposition 2.1 (iii) then implies that a(x) ˜ a(x), whence rank(a(x)) ˜ ≤ rank(a(x)) ≤ rank(b(x)) − k. For x ∈ X \U3 we have a1 (x) = a(x) and a(x) ˜ = (a1 (x) − 2)+ . Thus, a(x) ˜ a(x) and we proceed as before. We can now make our first reduction.
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Lemma 3.3. In order to prove Lemma 3.1, it will suffice to prove the following statement, hereafter referred to as (S): Let X be a compact metrisable Hausdorff space of covering dimension d ∈ N, and let Y ⊆ X be closed. Let 1 > > 0 be given. Suppose that a, ˆ bˆ ∈ Mn (C(X ))+ have the following properties: (i) (ii) (iii)
ˆ || < for some open set U ⊇ Y ; ||(aˆ − b)| U ˆb| X \U is well-supported; there are an open set V ⊇ U and γ > 0 such that ˆ a(x) ˆ = (b(x) − γ )+ , ∀x ∈ V \U ;
(iv) ˆ rank(a(x)) ˆ + (d − 1)/2 ≤ rank(b(x)), ∀x ∈ X \U. It follows that there are positive elements c, ˆ dˆ and a unitary element v in M4n (C(X )) whose restrictions to U are all equal to 1 ∈ M4n (C(U )), and which, upon viewing aˆ and bˆ as elements of the upper left n × n corner of M4n (C(X )), satisfy the inequality √ ˆ c) ˆ dv ˆ c) ||(dv ˆ b( ˆ ∗ − a|| ˆ < 4 . Proof. Let a and b be as in the hypotheses of Lemma 3.1. One can immediately find an open set U ⊇ Y such that ||a(x) − b(x)|| ≤ 0 < for every x ∈ U . By Lemma 2.8, there is a δ > 0 such that rank(a(x) − )+ ) + (d − 1)/2 ≤ rank(b(x) − δ)+ , ∀x ∈ X. Set η = min{ − 0 , δ}. Fix an open set W ⊇ Y such that W ⊆ U . Apply Lemma 2.5 to b| X \W with the tolerance η to produce a positive element hˆ ∈ Mn (C(X \W )) with the properties listed in the conclusion of that lemma. Fix a continuous map f : X → [0, 1] which is equal to one on W and equal to zero on X \U . Set ˆ ∀x ∈ X, h(x) = f (x)1Mn + (1 − f (x))h(x), ˆ and b(x) = h(x)b(x)h(x). For each x ∈ X \U , we have f (x) = 0. It follows that ˆb| X \U = (h| ˆ X \U )(b| X \U )(h| ˆ X \U ), whence, by part (i) of the conclusion of Lemma 2.5, ˆb| X \U is well-supported. We have ||hbh − b|| = sup ||h(x)b(x)h(x) − b(x)|| x∈X
ˆ ˆ = sup ||[ f (x)1 + (1 − f (x))h(x))]b(x)[( f (x)1 + (1 − f (x))h(x))] − b(x)|| < η, x∈X
where the last inequality follows from part (ii) of the conclusion of Lemma 2.5. Since ˆ || < . The inequality η ≤ δ implies that ||bˆ − b|| < δ. η ≤ − 0 , we have ||a|U − b| U Combining this fact with part (iii) of Proposition 2.1 yields ˆ rank((a(x) − )+ ) + (d − 1)/2 ≤ rank(b(x) − δ)+ ) ≤ rank(b(x)), ∀x ∈ X.
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ˆ (a − )+ , and 2 substituted for b, a, and , We will now apply Lemma 3.2 with b, respectively. Note that by shrinking U and W above, we may assume that they will serve as the sets U2 and U1 of Lemma 3.2, respectively. Form the approximant a˜ to (a − )+ provided in the conclusion of Lemma 3.2, and set aˆ = a. ˜ Note that aˆ − (a − )+ < 8. We have ˆ || ≤ ||(aˆ − (a − )+ )| || + ||((a − )+ − b)| ˆ || ||(aˆ − b)| U U U < 2(4) + 2 = 10 and ˆ a(x) ˜ = (b(x) − 4)+ , ∀x ∈ X. Our aˆ and bˆ now satisfy the hypotheses of statement (S) with 10 and 4 substituted ˆ and v be as in the conclusion of statement (S). Set for and γ , respectively. Let c, ˆ d, ˆ u = v, d = d, and c = ch. ˆ It follows that ˆ c)(hbh)( ˆ c) ˆ dv ˆ ∗ − a|| ||(duc)b(duc)∗ − a|| = ||(dv ∗ ˆ c) ˆ dv ˆ c) ˆ + ||aˆ − a|| ≤ ||((dv ˆ b( ˆ − a|| √ < 40 + aˆ − (a − )+ + a − (a − )+ √ √ < 40 + 9 < 49 . This shows that if (S) holds, then Lemma 3.1 holds (with N = 49).
The next lemma constructs the unitary u of Lemma 3.1. Lemma 3.4. Let X be a compact metrisable Hausdorff space of covering dimension d ∈ N, and let a, b ∈ Mn (C(X )) be well-supported positive elements with the property that 1 rank(a(x)) + (d − 1) ≤ rank((b − )+ (x)) 2 for some > 0 and every x ∈ X . Suppose further that a(y) ≤ (b(y) − )+ for each y in the closure of an open subset Y of X , and that a and b have norm at most one. For each k ∈ {0, 1, . . . , n}, set E k = {x ∈ Z | rank(a(x)) = k}; Fk = {x ∈ Z | rank(b(x)) = k}. For each x ∈ E k , let pk (x) be the support projection of a(x); for each x ∈ Fk let qk (x) be the support projection of b(x). Since a and b are well-supported, the continuous projection-valued maps x → pk (x) and x → qk (x) can be extended to E k and Fk , respectively. We also denote these extended maps by pk and qk . View Mn (C(X )) as the upper-left n × n corner of M4n (C(X )), and let Z ⊆ Y be closed. It follows that there is a unitary u ∈ M4n (C(X )) with the following properties: (i) u(z) = 14n ∈ M4n (C) for each z ∈ Z ; (ii) q j (x), ∀x ∈ E k , ∀k ∈ {0, . . . , n}; (u ∗ pk u)(x) ≤ { j | x∈F j }
(iii) u is homotopic to the unit of M4n (C(X )).
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A. S. Toms
Proof. Step 1. For each y ∈ Y , set v(y) = 1n . Let us verify conclusion (ii) above with v in place of u for each x ∈ Y ∩ E k . For each y ∈ Y we have a(y) ≤ b(y), and so n
n
a(y)1/2 ≤ b(y)1/2 , ∀n ∈ N. It follows that pk (y) ≤ q j (y) for each y ∈ Y ∩ E k ∩ F j , and so (v ∗ pk v)(y) = pk (y) ≤ q j (y) { j | y∈F j }
for each y ∈ Y ∩ E k and k ∈ {1, . . . , n}. It remains to prove that the inequality above holds when y ∈ Y ∩ E k . Set r (x) = χ[/2,1] (b(x)) for each x ∈ X , so that r (x) dominates the support projection of (b − )+ at x. It follows that pk (x) ≤ r (x) for each x ∈ E k . In fact, pk (x) ≤ r (x) ≤ q j (x) (3) { j | x∈F j }
for each k ∈ {0, . . . , n} and x ∈ E k , where the second inequality follows from the fact that q j (x) { j | x∈F j }
is the support projection of b(x) for each x ∈ E k . It will suffice to prove that the first inequality holds for y ∈ Y ∩ E k . It is well known that r (x) is an upper semicontinuous projection-valued map from X into Mn (C(X )). Fix y ∈ Y ∩ E k , and let (yn ) be a sequence in Y ∩ E k converging to y. Since pk (yn ) ≤ r (yn ) for each n ∈ N we have ( pk (yn )ξ |ξ ) ≤ (r (yn )ξ |ξ ), ∀ξ ∈ Cn , n ∈ N. Now ( pk (y)ξ |ξ ) = lim ( pk (yn )ξ |ξ ) ≤ lim sup(r (yn )ξ |ξ ) ≤ (r (y)ξ |ξ ), ∀ξ ∈ Cn , n→∞
n→∞
where the last inequaltiy follows from the upper semicontinuity of r . It follows that pk (y) ≤ r (y), as required. Step 2. We will construct partial isometries vk ∈ Mn (C(E k \Y )) with the following properties: (a) (vk∗ pk vk )(x) ≤
q j (x), ∀x ∈ E k \Y, ∀k ∈ {1, . . . , n};
{ j | x∈F j \Y }
(b) the vk s are compatible in the sense that for each x ∈ E i ∩ E j \Y with i ≤ j, (vi∗ pi vi )(x) = (v ∗j pi v j )(x); (c) for each x ∈ E k ∩ ∂Y , vk (x) = pk (x) = v(x) pk (x).
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In the third step of the proof, we will extend the v from Step 1 and the vk ’s above to produce the unitary u required by the lemma. We will prove the existence of the required vk ’s by induction on the number of rank values taken by a. Let us first address the case where a has constant rank equal to k0 . In this case E k0 = E k0 = X , and a is Cuntz equivalent to the projection pk0 ∈ Mn (C(X )). We set vk0 (y) = pk0 (y) for each y ∈ ∂Y , thus satisfying requirements (a) and (c) for these y. (Note that condition (b) is met trivially in the present case.) Let j1 < j2 < · · · < jl be the indices for which F ji = ∅. The existence of the required partial isometry extending the definition of vk0 on ∂Y now follows from repeated application of Proposition 2.6: one substitutes pk0 and q ji for q and p, respectively, in the hypotheses of said proposition. Now let us suppose that we have found partial isometries v0 , . . . , vk satisfying (a), (b), and (c) above. We must construct vk+1 , assuming k < n. We will first construct vk+1 on the boundary E k+1 ∩ (E 1 ∪ E 2 ∪ · · · ∪ E k ) ∩ Y c . For x ∈ E k+1 ∩ E k ∩ Y c , we have (vk∗ pk vk )(x) ≤
q j (x).
{ j | x∈F j \Y }
From (3) on E k+1 we also have that the rank of the right-hand side exceeds that of the left-hand side by at least 1 rank( pk+1 (x) − pk (x)) + (d − 1). 2 Working over E k+1 ∩ E k ∩ Y c , we have that ( pk+1 − pk ) is Murray-von Neumann equilvalent to a projection f k which is orthogonal to vk∗ pk vk and satisfies f k (x) ≤ q j (x). { j | x∈F j \Y }
(This follows from part (i) of Corollary 2.7.) Let wk be a partial isometry defined over E k+1 ∩ E k ∩ Y c such that wk∗ ( pk+1 − pk )wk = f k , and set vk+1 = vk + wk . With this definition we have ∗ (vk+1 pk+1 vk+1 )(x) ≤ q j (x), { j | x∈F j \Y }
and ∗ (vk∗ pk vk )(x) = (vk+1 pk vk+1 )(x)
for each x ∈ E k+1 ∩ E k ∩ Y c . Let us now show how to extend vk+1 one step further, to E k+1 ∩ (E k ∪ E k−1 ) ∩ Y c ; its successive extensions to the various E k+1 ∩ (E k ∪ · · · ∪ E k− j ) ∩ Y c , j ∈ {1, . . . , k − 1}, are similar, and the details are omitted.
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A. S. Toms
In this paragraph we work over the set E k+1 ∩ (E k ∪ E k−1 ) ∩ Y c . We will suppose that this set contains E k+1 ∩ E k ∩ Y c strictly, for there is otherwise no extension of vk+1 to be made. Over (E k+1 ∩ E k ∩ Y c ) ∩ E k−1 , we set wk−1 = vk+1 ( pk+1 − pk ). Thus, wk−1 is a partial isometry carrying (the restriction of) pk+1 − pk−1 to a subprojection of ⎛ ⎞ def ⎜ ⎟ ∗ Q(x) = ⎝ q j (x)⎠ − (vk−1 pk−1 vk−1 )(x), x ∈ (E k+1 ∩ E k ∩ Y c ) ∩ E k−1 . { j | x∈F j \Y }
We moreover have the rank inequality ∗ ∗ [rank(Q(x)) − rank((vk−1 pk−1 vk−1 )(x))] − rank((wk−1 wk−1 )(x)) ≥
1 (d − 1). 2
Applying part (i) of Corollary 2.7, we extend wk−1 to a partial isometry defined on all ∗ ) ≤ Q(x). Finally, set of E k+1 ∩ E k−1 ∩ Y c which has the property that (wk−1 wk−1 vk+1 = vk−1 + wk−1 on this set. It is straightforward to check that vk+1 has the required properties. Iterating the arguments above, we have an appropriate definition of vk+1 on E k+1 ∩ (E 1 ∪ E 2 ∪ · · · ∪ E k ) ∩ Y c . To extend the definition of vk+1 from the set above to all of E k+1 ∩ Y c , we simply apply part (i) of Corollary 2.7. Step 3. Set H−1 = Y and Hk = E k \Y , so that H−1 , . . . , Hn is a closed cover of X . For each k ∈ {−1, 0, . . . , n} we have a partial isometry vk ∈ Mn (C(Hk )) from Steps 1 and 2 (assuming that v−1 = v = 1). Let rk denote the source projection of vk . Notice that rk agrees with pk off Y . In this final step of our proof, we will construct the required unitary u in a manner which extends the vk : (u| Hk )rk = vk . Suppose that we have found a partial isometry wk ∈ M2n (C(H−1 ∪ · · · ∪ Hk )) with source projection equal to 1n (i.e., the unit of the upper-left n × n corner) and satisfying (wk | H j )r j = v j for each j ∈ {0, . . . , k}. Let us show that k can be replaced with k + 1, and that wk+1 may moreover be chosen to be an extension of wk . Over (H−1 ∪· · ·∪ Hk )∩ Hk+1 , wk carries the projection 1n −rk+1 into a subprojection ∗ . The rank of the latter projection exceeds that of the former by at least of 12n − vk+1 vk+1 (d −1)/2, and so the partial isometry wk (1n −rk+1 ) defined over (H−1 ∪· · ·∪ Hk )∩ Hk+1 can be extended to a partial isometry wk+1 defined over Hk+1 which carries 1n − rk+1 ∗ (cf. Proposition 2.6). Setting w into a subprojection of 12n − vk+1 vk+1 k+1 = vk+1 + wk+1 on Hk+1 and wk+1 = wk otherwise gives the desired extension. Iterating this extension process yields a partial isometry w ∈ M2n (C(X )) with source projection 1n satisfying (w| Hk )rk = vk . To complete the proof, it will suffice to find a unitary u ∈ M4n (C(X )) which is homotopic to the identity (for conclusion (iii)), satisfies u1n = w (for conclusion (ii)), and is equal to 1 ∈ M4n (C) over Z (for conclusion (i)). We will find a unitary s satisfying (ii) and (iii), and then modify it to obtain u. The complement of 1n in M2n (C(X )) is Murray-von Neumann equivalent to the complement of ww∗ , as both projections have the same K0 -class and are of rank at least (d − 1)/2. Let w be a partial isometry implementing this equivalence. It follows that w + w ∈ M2n (C(X )) is unitary. Setting s = (w + w ) ⊕ (w + w )∗ yields our precursor to the required unitary u ∈ M4n (C(X ))—the K1 -class of s is zero, so it is homotopic to 14n
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by virtue of its rank ([24, Theorem 10.12]). The unitary s|Y ∈ M4n (C(Y )) has the form 1n ⊕˜s , where s˜ is a 3n×3n unitary homotopic to the identity. (This follows from two facts: the K1 -class of 1n ⊕ s˜ is zero, and the natural map ι : U(M3n (C(Y )) → U(M4n (C(Y )) given by x → 1n ⊕ x is injective by [24, Theorem 10.12].) Let H : Y × [0, 1] → U(M3n (C)) be a homotopy such that H (y, 0) = s˜ (y) and H (y, 1) = 13n ∈ M3n (C). Let h : Y → [0, 1] be a continuous map equal to one on Z and equal to zero on ∂Y . Finally, define s(x), x∈ /Y u(x) = . 1n ⊕ H (x, f (x)), x ∈ Y The unitary u is clearly homotopic to s, and so satisfies conclusion (iii). Conclusion (i) holds for u by construction, and conclusion (ii) holds since u1n = s1n = w. Lemma 3.5. The statement (S) (cf. Lemma 3.3) holds. Proof. Step 1. To avoid cumbersome notation, we use a, b, c, and d in place of their “hatted” versions in the hypotheses and conclusion of (S). We will first find the unitary v and the positive elements c and d required by the conclusion of (S) with two failings: c and d are not necessarily equal to 1 ∈ M4n (C) at each point of U , and the estimate √ (cvd)b(cvd)∗ − a < 4 only holds on X \U . Both of these failings will be attributable to c and d alone, and will be repaired later in Steps 2. and 3. By combining the hypotheses (i) and (iii) of (S), we may, after perhaps shrinking the set V , assume that γ < . With this choice of V we also have that hypothesis (i) holds with V in place of U . We will also weaken hypothesis (iii) to an inequality. This has two advantages. First, by replacing a with (a − δ)+ for some small δ > 0, we can assume that (iv) holds with b replaced by (b − η)+ for some γ > η > 0. Second, we can assume that a| X \U is well-supported by using the following procedure: let W ⊇ Y be an open set whose closure is contained in U ; replace a with a suitably close approximant a˜ on X \W , as provided by Lemma 2.5; choose a continuous map f : X → [0, 1] which is equal to one on W and equal to zero on X \U ; replace the original a with the positive element equal to f (x)a(x) + (1 − f (x))a(x) ˜ at each x ∈ X . Let us summarise our assumptions: (i) ||(a − b)|V || < for some open set V ⊇ Y ; (ii) b| X \U and a| X \U are well-supported (and U ⊆ V ); (iii) there is 0 < γ < such that a(x) ≤ (b(x) − η)+ , ∀x ∈ V \U ; (iv) rank(a(x)) + (d − 1)/2 ≤ rank((b − η)+ (x)), ∀x ∈ X \U. Set Z = X \U and W = V \U . For each k ∈ {0, 1, . . . , n}, set E k = {x ∈ Z | rank(a(x)) = k}; Fk = {x ∈ Z | rank(b(x)) = k}.
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A. S. Toms
For each x ∈ E k , let pk (x) be the support projection of a(x). Similarly, define qk (x) to be the support projection of b(x) for each x ∈ Fk . Since (the restrictions of) a and b are well-supported on Z , the continuous projection-valued maps x → pk (x) and x → qk (x) can be extended to E k and Fk , respectively. We also denote these extended maps by pk and qk . Let V 1 be an open subset of X such that U ⊆ V 1 ⊆ V 1 ⊆ V , and set V1 = V 1 ∩ Z . Apply Lemma 3.4 with b| Z , a| Z , Z , W , V1 , and η substituted for the variables b, a, X , Y , Z , and in the hypotheses of the lemma, respectively. Let u be the unitary in M4n (C(Z )) provided by the conclusion of said lemma. Define v ∈ M4n (C(X )) to be the unitary which is equal to u on Z and equal to 1 ∈ M4n (C) at each point of U . This v will serve as the unitary required in the conclusion of (S). We will simply use v in place of v| Z whenever it is clear that we are working over Z . From conclusion (ii) of Lemma 3.4 we have ⎡ ⎤ ⎢ ⎥ pk (x) ≤ v(x) ⎣ q j (x)⎦ v(x)∗ , ∀x ∈ E k , ∀k ∈ {0, . . . , n}. (4) { j | x∈F j }
For each δ > 0 let f δ , gδ : [0, 1] → [0, 1] be given by the formulas ⎧ t ∈ [0, δ/2] ⎨ 0, f δ (t) = (2t − δ)/δ, t ∈ (δ/2, δ) , ⎩ 1, t ∈ [δ, 1] and
gδ (t) =
0, t ∈ [0, δ/2] . f δ (t)/t, t ∈ (δ/2, 1]
Note that f δ (t) and gδ (t) are continuous, and that tgδ (t) = f δ (t). Consider the following product in M4n (C(Z )): √ √ √ √ ( av gδ (b))b( av gδ (b))∗ = ( av gδ (b))b( gδ (b)v ∗ a). As δ → 0 we have [ gδ (b)b gδ (b)](x) = f δ (b)(x) →
(5)
q j (x), ∀x ∈ Z .
{ j | x∈F j }
√
√
Thus, by (4), [v gδ (b)b gδ (b)v ∗ ](x) converges to a projection which dominates the support projection of a(x). It follows that the product (5), evaluated at x ∈ Z , converges to a(x) as δ → 0. We will prove that this convergence is uniform in norm on Z . If δ < κ, then f δ (b) ≥ f κ (b). It follows that √ √ √ √ av f δ (b)v ∗ a ≥ av f κ (b)v ∗ a. (6) Since b ≤ 1, we have
√ √ √ √ av f δ (b)v ∗ a ≤ avv ∗ a = a,
and similarly for f κ (b). Combining this with (6) yields √ √ √ √ 0 ≤ a − av f δ (b)v ∗ a ≤ a − av f κ (b)v ∗ a.
Comparison Theory and Smooth Minimal C∗ -Dynamics
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417
√ √ √ √ a − av f δ (b)v ∗ a ≤ a − av f κ (b)v ∗ a .
(7)
Let (δn ) be a sequence of strictly positive tolerances converging to zero. By (7), √ √ √ √ [a − av gδn (b)b gδn (b)v ∗ a](x) = [a − av f δn (b)v ∗ a](x) is a monotone decreasing sequence converging to zero for each x ∈ Z . By Dini’s Theorem, this sequence converges uniformly to zero on Z . For the remainder of the proof we fix > δ > 0 with the property that √ √ (8) a − av gδ (b)b gδ (b)v ∗ a < . √ √ Extend a and gδ (b) to positive elements c and d in M4n (C(X )), respectively. This choice of c and d completes Step 1. Step 2. We must now modify our choice of c and d to address their failings, outlined at the beginning of Step 1. This modification will be made in three smaller steps. In a slight abuse of notation, we will use c and d to denote the successive modifications of the present c and d. For each x ∈ W we have b(x) − a(x) ≥ 0 and ||b(x) − a(x)|| < . It is a straightforward exercise to show that √ b(x) − a(x) < . Choose a continuous map f 1 : Z √→ [0, 1] which is equal √ to one on Z \W and equal to zero√on V1 . √ Set a1 (x) = f 1 (x) a(x) + (1 − f 1 (x)) b(x) for each x ∈ Z , and set s = v gδ (b)b gδ (b)v ∗ for brevity. Note that ||s|| ≤ 1. Now for each x ∈ W we have ||[a sa ](x)|| √1 1 √ √ √ √ √ = [ a + (1 − f 1 )( b − a)](x)s(x)[ a + (1 − f 1 )( b − a)](x) √ √ = [ as a](x) + r (x) , √ where ||r (x)|| < 2 + . We revise our definition of c by setting it equal to a1 on X \U and extending it in an arbitrary fashion to a positive element of M4n (C(X )). Combining this new definition of c with (8) above we have the estimate √ [(cvd)b(cvd)∗ ](x) − a(x) < 2( + ), ∀x ∈ X \U. (9) Choose an open subset V2 of Z such that U ⊆ V2 ⊆ V2 ⊆ V1 , and a continuous map f 2 : Z → [0, √1] equal to zero √ on Z \V1 and equal to one on V2 . For each x ∈ V1 we have c(x) = b(x), d(x) = gδ (b), and v(x) = 1, whence [(cvd)b(cvd)∗ ](x) − a(x) = b(x)2 gδ (b)(x) − a(x) (10) = ||b(x) f δ (b)(x) − a(x)|| (11) ≤ ||b(x) f δ (b)(x) − b(x)|| + ||b(x) − a(x)|| < 2. (12) For each s ∈ [0, 1] define 2ts/δ, h s (t) = (2t − δ)[(1 − s)/(2 − δ)] + s,
t ∈ [0, δ/2] . t ∈ (δ/2, 1]
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A. S. Toms
It straightforward to verify that h s (t) is a homotopy of maps such that 2t/δ, t ∈ [0, δ/2] h 0 (t) = t; h 1 (t) = . 1, t ∈ (δ/2, 1] Set
gδ,s (t) =
0, t ∈ [0, δ/2] . f δ (t)/ h s (t), t ∈ (δ/2, 1]
With these definitions we have h s (t)gδ,s (t) = f δ (t), ∀s, t ∈ [0, 1]. For each x ∈ V1 , we adjust our definitions of c(x) and d(x) as follows: c(x) = h f2 (x) (b(x)); d(x) = gδ, f 2 (x) (b(x)). Since f 2 (x) = 0 on ∂ V1 , the definitions of c(x) and d(x) are not altered on ∂ V1 . Thus, our modified versions of c and d are still positive elements of M4n (C(Z )), and the estimate (9) still holds on Z \V1 . For x ∈ V1 we have [(cvd)b(cvd)∗ ](x) − a(x) = h f (x) (b(x))gδ, f (x) (b(x))b(x) − a(x) (13) 2 2 = || f δ (b(x))b(x) − a(x)|| < 2, (14) where the last inequality follows from (10) above. Thus, (9) continues to hold with our new definitions of c and d. Choose an open subset V3 of Z such that U ⊆ V3 ⊆ V3 ⊆ V2 , and a continuous map f 3 : Z → [0, 1] equal to zero on Z \V2 and equal to one on V3 . For each s ∈ [0, 1] define continuous maps rs , ws : [0, 1] → [0, 1] by rs (t) = max s, h 1 (t) ; ws (t) = max s, gδ,1 (t) . Thus, rs and ws define homotopies of self-maps of [0, 1] such that r0 = h 1 , w0 = gδ,1 , and r1 = w1 = 1. For each x ∈ V2 we adjust our definitions of c(x) and d(x) as follows: c(x) = r f3 (x) (b(x)); d(x) = w f3 (x) (b(x)). Since f 3 = 0 on ∂ V2 , the definitions of c(x) and d(x) are not altered on ∂ V2 . Thus, our modified versions of c and d are still positive elements of M4n (C(Z )), and the estimate (9) still holds on Z \V2 . For x ∈ V2 we have [(cvd)b(cvd)∗ ](x) − a(x) = r f (x) (b(x))w f (x) (b(x))b(x) − a(x) < 2 3 3 by a functional calculus argument similar to (13) above—one need only observe that f δ (t) ≤ rs (t)ws (t) ≤ 1, ∀s, t ∈ [0, 1]. Thus, (9) continues to hold with our new definitions of c and d. Moreover, we have c(x) = d(x) = 1 ∈ M4n (C) for each x ∈ V3 . We may thus extend our definitions of c and d to all of X by setting c(x) = d(x) = 1 ∈ M4n (C) for every x ∈ U ∪ V3 . With this final definition of c and d, we see that [(cvd)b(cvd)∗ ](x) − a(x) = ||b(x) − a(x)|| < , ∀x ∈ U ∪ V3 . We conclude that the estimate (9) holds on all of X , whence (S) holds. With (S) in hand, we have completed the proof of Lemma 3.1.
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4. A Comparison Theorem for Recursive Subhomogeneous C∗-Algebras 4.1. Background and notation. Let us recall some of the terminology and results from [22]. Definition 4.1. A recursive subhomogeneous algebra (RSH algebra) is given by the following recursive definition: (i) If X is a compact Hausdorff space and n ∈ N, then Mn (C(X )) is a recursive subhomogeneous algebra. (ii) If A is a recursive subhomogeneousalgebra, X is a compact Hausdorff space, X (0) ⊆ X is closed, φ : A → Mk (C(X (0) )) is a unital ∗-homomorphism, and ρ : Mk (C(X )) → Mk (C(X (0) )) is the restriction homomorphism, then the pullback A ⊕Mk (C(X (0) )) Mk (C(X )) = {(a, f ) ∈ A ⊕ Mk (C(X )) | φ(a) = ρ( f )} is a recursive subhomogeneous algebra. It is clear from the definition above that a C∗ -algebra R is an RSH algebra if and only if it can be written in the form R = · · · C0 ⊕C (0) C1 ⊕C (0) C2 · · · ⊕C (0) Cl , (15) 1
2
l
with Ck = Mn(k) (C(X k )) for compact Hausdorff spaces X k and integers n(k), with Ck(0) = Mn(k) (C(X k(0) )) for compact subsets X k(0) ⊆ X (possibly empty), and where the maps Ck → Ck(0) are always the restriction maps. We refer to the expression in (15) as a decomposition for R. Decompositions for RSH algebras are not unique. Associated with the decomposition (15) are: (i) its length l; (ii) its k th stage algebra Rk = · · · C0 ⊕C (0) C1 ⊕C (0) C2 · · · ⊕C (0) Ck ; 1
2
k
(iii) its base spaces X 0 , X 1 , . . . , X l and total space lk=0 X k ; (iv) its matrix sizes n(0), n(1), . . . , n(l) and matrix size function m : X → N given by m(x) = n(k) when x ∈ X k (this is called the matrix size of R at x); (v) its minimum matrix size mink n(k) and maximum matrix size maxk n(k); (vi) its topological dimension dim(X ) and topological dimension function d : X → N ∪ {0} given by d(x) = dim(X k ) when x ∈ X k ; (vii) its standard representation σ R : R → ⊕lk=0 Mn(k) (C(X k )) defined to be the obvious inclusion; (viii) the evaluation maps evx : R → Mn(k) for x ∈ X k , defined to be the composition of evaluation at x on ⊕lk=0 Mn(k) (C(X k )) and σ R . Remark 4.2. If R is separable, then the X k can be taken to be metrisable ([22, Prop. 2.13]). If R has no irreducible representations of dimension less than or equal to N , then we may assume that n(k) > N . It is clear from the construction of Rk+1 as a pullback of Rk and Ck+1 that there is a canonical surjective ∗-homomorphism λk : Rk+1 → Rk . By composing several such, one has also a canonical surjective ∗-homomorphism from R j to Rk for any j > k. Abusing notation slightly, we denote these maps by λk as well.
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Remark 4.3. The C∗ -algebra Mm (R) ∼ = R ⊗ Mm (C) is an RSH algebra in a canonical way: Ck and Ck(0) are replaced with Ck ⊗ Mm (C) and Ck(0) ⊗ Mm (C), respectively, and (0) are replaced with the amplifications the clutching maps φk : Rk → Ck+1 (0)
φk ⊗ idm : Ck ⊗ Mm (C) → Ck+1 ⊗ Mm (C). From here on we assume that Mm (R) is equipped with this canonical decomposition whenever R is given with a decomposition. We will abuse notation by using φk to denote both the original clutching map in the given decomposition for R and its amplified versions.
4.2. A comparison theorem. Lemma 4.4. Let X be a compact metrisable Hausdorff space, and Y a closed subset of X . If a ∈ Mn (C(Y )) is positive, then a can be extended to a˜ ∈ Mn (C(X )) with the property that a(x) ˜ is invertible for every x ∈ X \Y . If u = v ⊕ v ∗ for a unitary v ∈ Mn (C(Y )), then u can be extended to a unitary u˜ ∈ M2n (C(X )). Proof. By the semiprojectivity of the C∗ -algebras they generate, both a and u can be extended to the closure of an open neighbourhood V of Y . We will also denote these extensions by a and u. Fix a continuous map f : X → [0, 1] which is equal to zero on Y , equal to one on X \V , and nonzero at every x ∈ X \Y . Define a(x) + f (x)(||a|| − a(x)), x ∈ V a(x) ˜ = . ||a||, x ∈ X \V Clearly, a˜ belongs to Mn (C(X )) and extends a. It follows that for each x ∈ X \Y , either a(x) ˜ = ||a|| ∈ GLn (C), or a(x) ˜ = a(x) + f (x)(||a|| − a(x)) = f (x)||a|| + (1 − f (x))a(x) ≥ f (x)||a|| > 0. In the latter case we conclude that the rank of a(x) ˜ is n, whence a(x) ˜ ∈ GLn (C) as desired. Now let us turn to u. We have u|V \Y = v|V \Y ⊕ v ∗ |V \Y ∼h 1 ∈ M2n C V \Y by the Whitehead Lemma, where ∼h denotes homotopy within the unitary group. Let H (x, t) : V \Y × [0, 1] → U(M2n (C)) be an implementing homotopy, with H (x, 0) = u|V \Y and H (x, 1) = 1. Define ⎧ x ∈Y ⎨ u(x), u(x) ˜ = H (x, f (x)), x ∈ V \Y . ⎩ 1, x ∈ X \V It is straightforward to check that u˜ is a unitary in M2n (C(X )), and u˜ extends u by definition.
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Lemma 4.5. Let A be a separable RSH algebra with a fixed decomposition as above. Let a, b ∈ A be positive, and suppose that ||λk (b − a)|| < inside the k th stage algebra Ak , k < l. Suppose further that (0)
rank(a(x)) + (d(x) − 1)/2 ≤ rank(b(x)), ∀x ∈ X j \X j , j > k. A as the It follows that there are m ∈ N and v ∈ Mm (A) such that, upon considering √ upper-left 1 × 1 corner of Mm (A) we have ||λk+1 (vbv ∗ − a)|| < N for the constant N of Lemma 3.1 and (0)
rank(a(x)) + (d(x) − 1)/2 ≤ rank((vbv ∗ )(x)), ∀x ∈ X j \X j , j > k + 1. (0)
Proof. Let φk : Ak → Ck+1 be the k th clutching map. Our hypotheses imply that (0) (0) φk (b), φk (a) ∈ Ck+1 = Mn(k+1) (C(X k+1 ) satisfy ||φk (b) − φk (a)|| < . Apply Lemma (0) 3.1 with φk (a), φk (b), X k+1 , X k+1 , and in place of a, b, X, Y , and , respectively. The conclusion of Lemma 3.1 provides us with positive elements c, d and a unitary element u in M4n(k+1) (C(X k+1 )) such that √ (i) ||(cud)φk (b)(cud)∗ − φk (a)|| < N , and (0) (ii) c(x) = d(x) = u(x) = 1 ∈ M4n(k+1) (C) for every x ∈ X k+1 . Using (ii) we extend c, d, and u to M4 (Ak+1 ) (keeping the same notation) by setting λk (c) = λk (d) = λk (u) = 1 ∈ M4 (Ak ). Set vk+1 = cud ∈ M4 (Ak+1 ). We claim that
√ ∗ − λk+1 (a)|| < N . ||vk+1 λk+1 (b)vk+1
∗ −λ It will suffice to prove that the image of vk+1 λk+1 (b)vk+1 k+1 (a) under the standard representation
σM4 (Ak+1 ) : M4 (Ak+1 ) →
k+1
M4n( j) (C(X j ))
j=0
√ is of norm at most N . This in turn need only be checked in each of the direct summands of the codomain. In the summand ⊕kj=0 M4n( j) (C(X j )) the desired estimate follows from two facts: σM4 (Ak+1 ) (vk+1 ) is equal to the unit of said summand (see (ii) above), √ and the images of a and b in this summand are at distance strictly less than < N . In the summand M4n(k+1) (C(X k+1 )) the desired estimate follows from (i) above. If m ≥ 4, then any v ∈ Mm (A) which, upon viewing M4 (A) as the upper-left 4 × 4 corner of Mm (A), √ has the property that λk+1 (v) = vk+1 will at least satisfy ||λk+1 (vbv ∗ − a)|| < N . It remains, then, to find such a v, while ensuring that (0)
rank(a(x)) + (d(x) − 1)/2 ≤ rank((vbv ∗ )(x)), ∀x ∈ X j \X j , j > k + 1. If k +1 = l, then there is nothing to prove. Suppose that k +1 < l. Let us first construct an element vk+2 of M8 (Ak+2 ) with the following properties: λk+1 (vk+2 ) = vk+1 , and (0)
∗ rank(a(x)) + (d(x) − 1)/2 ≤ rank((vk+2 bvk+2 )(x)), ∀x ∈ X k+2 \X k+2 .
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Define ck+1 = c ⊕ 0, dk+1 = d ⊕ 0, and u k+1 = u ⊕ u ∗ . Use Lemma 4.4 to extend φk+1 (ck+1 ), φk+1 (dk+1 ), and φk+1 (u k+1 ) to positive elements c˜k+2 , d˜k+2 and a unitary element u˜ k+2 , respectively, in M8n(k+2) (C(X k+2 )), all of which are invertible (0) . Consider M8 (Ak+2 ) as a subalgebra of ⊕k+2 at every x ∈ X k+2 \X k+2 j=0 M8n( j) (C(X j )) via its standard representation, and define ck+2 to be equal to ck+1 in the first k + 1 summands, and equal to c˜k+2 in the last summand; define dk+2 and u k+2 similarly. Setting vk+2 = ck+2 u k+2 dk+2 we have that λk+1 (vk+2 ) = = = = =
λk+1 (ck+2 u k+2 dk+2 ) ck+1 u k+1 dk+1 (c ⊕ 0)(u ⊕ u ∗ )(d ⊕ 0) cud ⊕ 0 vk+1 .
(0) , we have Moreover, for each x ∈ X k+2 \X k+2
vk+2 (x) = c˜k+2 (x)u˜ k+2 (x)d˜k+2 (x) ∈ GL8n(k+2) (C). It follows that (0) , rank((vbv ∗ )(x)) = rank(b(x)) ≥ (d(x) − 1)/2 + rank(a(x)), ∀x ∈ X k+2 \X k+2
as required. If k + 2 = l then we set v = vk+2 to complete the proof. Otherwise, we repeat the arguments in the paragraph above using ck+2 , dk+2 , and u k+2 in place of c, d, and u, respectively, to obtain vk+3 ∈ M82 (Ak+3 ) such that λk+2 (vk+3 ) = vk+2 and (0) ∗ rank(a(x)) + (d(x) − 1)/2 ≤ rank((vk+3 bvk+3 )(x)), ∀x ∈ X k+3 \X k+3 .
Continuing this process until we arrive at vk+(l−k) = vl and setting v = vl yields the lemma in full. Theorem 4.6. Let A be a separable RSH algebra with a fixed decomposition as above. Let a, b ∈ A be positive, and suppose that (0)
rank(a(x)) + (d(x) − 1)/2 ≤ rank(b(x)), ∀x ∈ X k \X k , k ∈ {0, 1, . . . , l}. It follows that a b. Proof. We view A as the upper-left 1 × 1 corner of Mm (A), and adopt the standard notation for the decompositions of A and Mm (A). Let > 0 be given; we must find m ∈ N and v ∈ Mm (A) such that ||vbv ∗ − a|| < . Let l√be the length of the fixed decomposition for A. Given δ0 > 0, we define δk = N δk−1 for each k ∈ {1, . . . , l}, where N is the constant of Lemma 3.1. It follows that 1/2k
δk = δ0
k−1
j
N 1/2 .
j=0
Assume that δ0 has been chosen so that δl < .
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Apply Lemma 3.1 with λ0 (a), λ0 (b), X 0 , and ∅, in place of a, b, X, and Y . Since Y is empty, we can arrange to have any value of appear in the conclusion of Lemma 3.1. We choose = δ02 /N 2 , so that the norm estimate in the conclusion of Lemma 3.1
is strictly less than N δ02 /N 2 = δ0 . Let c0 , d0 , and u 0 denote the positive elements and the unitary element, respectively, of M4n(0) (C(X 0 )) produced by Lemma 3.1. Apply the arguments of the second-to-last paragraph in the proof of Lemma 4.5 with c0 , d0 , and u 0 in place of c, d, and u, respectively, to produce an element v0 of M32 (A) such that ||λ0 (v0 bv0∗ − a)|| < δ0 , and (0)
rank(a(x)) + (d(x) − 1)/2 ≤ rank((v0 bv0∗ )(x)), ∀x ∈ X j \X j , j > 0. Suppose that we have found m k ∈ N and vk ∈ Mm k (A) such that ||λk (vk bvk∗ − a)|| < δk and (0)
rank(a(x)) + (d(x) − 1)/2 ≤ rank((vk bvk∗ )(x)), ∀x ∈ X j \X j , j > k. ∗ − An application of Lemma 4.5 yields vk+1 ∈ M8m k (A) such that ||λk+1 (vk+1 vk bvk∗ vk+1 √ a)|| < N δk = δk+1 and (0)
∗ )(x)), ∀x ∈ X j \X j , j > k + 1. rank(a(x)) + (d(x) − 1)/2 ≤ rank((vk+1 vk bvk∗ vk+1
Starting with v0 , we use the fact above to find, successively, v1 , . . . , vl . With v = vl vl−1 · · · v0 we have ||vbv ∗ − a|| < δl < , as desired.
5. Applications 5.1. The radius of comparison and strict comparison. Let A be a unital stably finite C∗ -algebra, and let a, b ∈ M∞ (A) be positive. We say that A has r -strict comparison if a b whenever d(a) + r < d(b), ∀d ∈ LDF(A). The radius of comparison of A, denoted by rc(A), is defined to be the infimum of the set {r ∈ R+ | A has r − strict comparison} whenever this set is nonempty; if the set is empty then we set rc(A) = ∞ ([28]). The condition rc(A) = 0 is equivalent to A having strict comparison (see Subsect. 2.2). The radius of comparison should be thought of as the ratio of the topological dimension of A to its matricial size, despite the fact that both may be infinite. It has been useful in distinguishing C∗ -algebras which are not K-theoretically rigid in the sense of G. A. Elliott ([12,29]). Here we give sharp upper bounds on the radius of comparison of a recursive subhomogeneous algebra. These improve significantly upon the upper bounds established in the homogeneous case by [30, Theorem 3.15].
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Theorem 5.1. Let A be a separable RSH algebra with a fixed decomposition of length l and matrix sizes n(0), . . . , n(l). It follows that rc(A) ≤ max
0≤k≤l
dim(X k ) − 1 . 2n(k)
Proof. Use r to denote the upper bound in the statement of the theorem, and suppose that we are given a, b ∈ M∞ (A)+ such that dτ (a) + r < dτ (b) for every τ ∈ T(A). (0) Associated to each x ∈ X k \X k , 0 ≤ k ≤ l, is an extreme point of T(A), denoted by τx , obtained by composing evx with the normalised trace on Mn(k) . For any a ∈ M∞ (A)+ we have dτx (a) = [rank(evx (a))]/n(k), and so rank(evx (a)) rank(evx (b)) rank(evx (a)) dim(X k ) − 1 + ≤ +r < . n(k) 2n(k) n(k) n(k) Multiplying through by n(k) we have rank(a(x)) +
dim(X k ) − 1 < rank(b(x)) 2
(0)
for every x ∈ X k \X k and k ∈ {0, . . . , l}, whence a b by Theorem 4.6, as desired. Specialising to the homogeneous case we have the following corollary. Corollary 5.2. Let X be a compact metrisable Hausdorff space of covering dimension d ∈ N, and p ∈ C(X ) ⊗ K a projection. It follows that rc( p(C(X ) ⊗ K) p) ≤
d −1 . 2rank( p)
Proof. The algebra p(C(X ) ⊗ K) p admits a recursive subhomogeneous decomposition in which every matrix size is equal to rank( p) and each X k has covering dimension at most d. (This decomposition comes from the fact that p corresponds to a vector bundle of finite type—see Sect. 2 of [22].) The Corollary now follows from Theorem 5.1. Corollary 5.2 improves upon [30, Theorem 3.15], or rather, the upper bound on the radius of comparison that can be derived from it: the latter result leads to an upper bound of (9d)/rank( p). The bound achieved here is sharp, as can by seen from [28, Theorem 6.6]. The property of strict comparison is a powerful regularity property with agreeable consequences. We will see some examples of this in Subsects. 5.2, 5.3, and 5.4; a fuller treatment of this topic can be found in [11]. Theorem 5.3. Let (Ai , φi ) be a unital direct sequence of recursive subhomogeneous algebras with slow dimension growth. If A = limi→∞ (Ai , φi ) is simple, then A has strict comparison of positive elements. Proof. Let us first show that lim inf i→∞ rc(Ai ) = 0. We assume that each Ai is equipped i X i,k denote the total space of Ai , di : Yi → with a fixed decomposition. Let Yi = lk=0 {0} ∪ N its topological dimension function, and n i (0), . . . , n i (li ) its matrix sizes. From [23, Def. 1.1], (Ai , φi ) has slow dimension growth if the following statement holds: for
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every i ∈ N, projection p ∈ M∞ (Ai ), and N ∈ N, there exists j0 > i such that for every j ≥ j0 and y ∈ Yi we have ev y (φi, j ( p)) = 0 or rank(ev y (φi, j ( p))) ≥ N d j (y); (0)
if p = 1 Ai , then only the latter statement can hold. If y ∈ X j,k \X j,k , then rank(ev y (φi, j (1 Ai ))) = rank(ev y (1 A j )) = n j (k) ≥ N dim(X j,k ). It now follows from Theorem 5.1 that lim inf i→∞ rc(Ai ) = 0. Theorem 4.5 of [30] would give us strict comparison for A if only each φi were injective. The origin of this injectivity hypothesis lies in [30, Lemma 4.4]—the proof of [30, Theorem 4.5] only uses injectivity of the φi in its appeal to this lemma. Thus, we must drop injectivity from the assumptions of [30, Lemma 4.4]; we must prove the following claim: Claim. Let B be the limit of an inductive sequence (Bi , ψi ) of C∗ -algebras, and let a, b ∈ M∞ (B) be positive. If ψi,∞ (a) ψi,∞ (b), then for every > 0 there is a j > i such that (ψi, j (a) − )+ ψi, j (b). Proof of Claim. If will suffice to prove the claim for a, b ∈ B. By assumption, there is a sequence (vk ) in B such that vk bvk∗ → a. We may assume that the vk lie in the dense local C∗ -algebra ∪i ψi,∞ (Bi ) (see the proof of [30, Lemma 4.4]). In fact, by compressing our inductive sequence, we may as well assume that vk = φk,∞ (wk ) for some wk ∈ Bk . The statement that vk bvk∗ → a can now amounts to n→∞
ψk,∞ (wk ψi,k (b)wk∗ − ψi,k (a)) −→ 0. Fix k0 large enough that the left-hand side above is < . Since ψk0 , j (x) → ψk0 ,∞ (x) for any x ∈ Ak0 we may find j > i such that ψk0 , j (wk0 ψi,k0 (b)wk∗0 − ψi,k0 (a)) < . Setting r j = ψk0 , j (wk0 ) and appealing to part (iii) of Proposition 2.1 we have (ψi, j (a) − )+ r j ψi, j (b)r ∗j ψi, j (b), as desired. This proves the claim, and hence the theorem. We collect an improvement of [30, Theorem 4.5] as a corollary. Corollary 5.4. Let A be the limit of an inductive sequence of stably finite C∗ -algebras (Ai , φi ), with each Ai and φi unital. Suppose that A is simple, and that lim inf rc(Ai ) = 0. i→∞
It follows that A has strict comparison of positive elements. Proof. Follow the proof of [30, Theorem 4.5] but use the claim in the proof of Theorem 5.3 instead of [30, Lemma 4.4]. Corollary 5.5. Let M be a compact smooth connected manifold and h : M → M a minimal diffeomorphism. It follows that the transformation group C∗ -algebra C∗ (M, Z, h) has strict comparison of positive elements. Proof. By the main result of [17], C∗ (M, Z, h) can be written as the limit of an inductive sequence of recursive subhomogeneous C∗ -algebras with slow dimension growth. Apply Theorem 5.3.
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5.2. The structure of the Cuntz semigroup. The Cuntz semigroup is a sensitive invariant in the matter of distinguishing simple separable amenable C∗ -algebras, and has recently received considerable attention (see [3–5,7,8,11,27], and [30]). It is, however, very difficult to compute in general—see [27, Lemma 5.1]. This situation improves dramatically in the case of simple C∗ -algebras with strict comparison of positive elements. Let A be a unital, simple, exact, stably finite C∗ -algebra. In this case we may write W (A) = V (A) W (A)+ (as sets), where V (A) denotes the semigroup of Murray-von Neumann equivalence classes of projections in M∞ (A)—here interpreted as those Cuntz equivalence classes represented by a projection—and W (A)+ denotes the subsemigroup of W (A) consisting of Cuntz classes represented by positive elements having zero as an accumulation point of their spectrum (cf. [21]). Let LAff b (T(A))++ denote the set of lower semicontinuous, affine, bounded, strictly positive functions on the tracial state space of A, and define a map ι : W (A) → LAff b (T(A))++ by ι(a)(τ ) = dτ (a). We endow the set V (A) LAff b (T(A))++ with an Abelian binary operation +W which restricts to the usual semigroup operation in each component and is given by x +W f = ι(x) + f for x ∈ V (A) and f ∈ LAff b (T(A))++ . We also define a partial order ≤W on this set which restricts to the usual partial orders in each component and satisfies (i) x ≤W f if and only if ι(x) < f , and (ii) x ≥W f if and only if ι(x) ≥ f . Theorem 5.6 ([3,7]). Let A be a simple, unital, exact, and stably finite C∗ -algebra with strict comparison of positive elements. It follows that the map idι
V (A) W (A)+ −→ V (A) LAff b (T(A))++ is a semigroup order embedding. If A is infinite-dimensional and monotracial, then the embedding of Theorem 5.6 is an isomorphism. We suspect that the monotracial assumption is unneccessary. Theorem 5.6 applies to ASH algebras as in Theorem 5.3, and so to the minimal diffeomorphism C∗ -algebras C∗ (M, Z, h) considered above. 5.3. A conjecture of Blackadar-Handelman. Blackadar and Handelman conjectured in 1982 that the lower semicontinuous dimension functions on a C∗ -algebra should be dense in the set of all dimension functions. This conjecture was proved for C∗ -algebras as in Theorem 5.6 in [3, Theorem 6.4]. Thus, we have the following result. Theorem 5.7. Let A be a C∗ -algebra as in Theorem 5.6 (in particular, A could be the C∗ -algebra of a minimal diffeomorphism). It follows that the lower semicontinuous dimension functions on A are weakly dense in the set of all dimension functions on A.
5.4. Classifying Hilbert modules. In [7], Coward, Elliott, and Ivanescu gave a new presentation of the Cuntz semigroup. Given a C∗ -algebra A, they considered positive elements in A ⊗K (as opposed to M∞ (A), as we have done—the difference is ultimately
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immaterial). If A is separable, then the hereditary subalgebras of A⊗K are singly generated, and any two generators of a fixed hereditary subalgebra are Cuntz equivalent. Thus, Cuntz equivalence factors through the passage from a positive element to the hereditary subalgebra it generates. These hereditary subalgebras are in one-to-one correspondence with countably generated Hilbert A-modules, and in [7] the notion of Cuntz equivalence, considered as a relation on hereditary subalgebras, is translated into a relation on Hilbert modules. Thus, we may speak of Cuntz equivalence between countably generated Hilbert A-modules. Theorem 5.8 ([7]). Let A be a C∗ -algebra of stable rank one. It follows that countably generated Hilbert A-modules X and Y are Cuntz equivalent if and only if they are isomorphic. Corollary 5.9. Let A be as in Theorem 5.3. Suppose further that A has stable rank one. (In particular, A could by the C∗ -algebra of a minimal diffeomorphism, as these have stable rank one by the main result of [23].) It follows that countably generated Hilbert A-modules X and Y are isomorphic if and only if they are Cuntz equivalent. If X and Y as in Corollary 5.9 are finitely generated and projective, then they are Cuntz equivalent if and only if the projections in A ⊗ K which generate them as closed right ideals have the same K0 -class. Otherwise, X has associated to it an affine function on the tracial state space of A: one extends the map ι of Subsect. 5.2 to have domain A ⊗ K, applies it to any positive element of A ⊗ K which generates X as a closed right ideal. This function determines non-finitely generated X up to isomorphism. This classification of Hilbert A-modules is analogous to the classification of W∗ -modules over a II1 factor. We refer the reader to Sect. 3 of [4] for further details. 5.5. Classifying self-adjoints. We say that self-adjoint elements a and b in a unital C∗ -algebra A are approximately unitarily equivalent if there is a sequence (u n )∞ n=1 of unitaries in A such that u n au ∗n → b. For a ∈ A+ we let φa : C∗ (a, 1) → A denote the canonical embedding. Denote by Ell(a) the following pair of induced maps: K0 (φa ) : K0 (C∗ (a, 1)) → K0 (A); φa : T(A) → T(C∗ (a, 1)). Theorem 5.10 ([4]). Let A be a unital simple exact C∗ -algebra of stable rank one and strict comparison (in particular, A could have stable rank one and satisfy the hypotheses of Theorem 5.3). If a, b ∈ A+ , then a and b are approximately unitarily equivalent if and only if σ (a) = σ (b) and Ell(a) = Ell(b). 5.6. The range of the radius of comparison, with applications. The classification theory of operator algebras is a rich field. It was begun by Murray and von Neumann with their type classification of factors in the 1930s, and has been active ever since. In the presence of certain regularising assumptions, the theory is well-behaved. For instance, there is a complete classification of injective factors with separable predual (due to Connes and Haagerup—see [6 and 14]), and a similarly successful classification program for simple C∗ -algebras upon replacing injectivity and separability of the predual with amenability and norm-separability, respectively (see [11 and 25] Without these regularising assumptions, the theory is fractious, but nonetheless interesting. One of the landmarks on this side of the theory is McDuff’s construction of
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uncountably many non-isomorphic factors of type II1 ([18]). (More recently there is Popa’s work on II1 factors with Betti numbers invariants—see [20].) One might view McDuff’s result as saying that there are uncountably many non-isomorphic factors which all have the same naive invariant, namely, the mere fact that they are II1 factors. (Connes proved that there is only one injective II1 factor with separable predual.) Here we prove an analogue of McDuff’s theorem for simple, separable, amenable C∗ -algebras, where the corresponding naive invariant consists of Banach algebra K-theory and positive traces. We even obtain a somewhat stronger result, replacing non-isomorphism with non-Morita-equivalence. In passing we prove that the range of the radius of comparison is exhausted by simple C∗ -algebras, a result which represents the first exact calculations of the radius of comparison for any simple C∗ -algebra. Recall that the Elliott invariant of a C∗ -algebra A is the 4-tuple Ell(A) := (K0 A, K0 A+ , A ), K1 A, T+ A, ρ A , (16) where the K-groups are the Banach algebra ones, K0 A+ is the image of the Murrayvon Neumann semigroup V(A) under the Grothendieck map, A is the subset of K0 A corresponding to projections in A, T+ A is the space of positive tracial linear functionals on A, and ρ A is the natural pairing of T+ A and K0 A given by evaluating a trace at a K0 -class. Theorem 5.11. There is a family {A(r ) }r ∈R+ \{0} of simple, separable, amenable C∗ -algebras such that rc(Ar ) = r and Ell(Ar ) ∼ = Ell(As ) for every s, r ∈ R+ \{0}. In particular, Ar As whenever r = s. If As and Ar are Morita equivalent, then s/r ∈ Q. Proof. The general framework for the construction of A(r ) follows [31]. Find sequences of natural numbers (n i ) and (li ) and a natural number m 0 with the following properties: (i) n i → ∞; (ii) n0 n1n2 · · · ni i→∞ · −→ r ; 2m 0 (n 1 + l1 )(n 2 + l2 ) · · · (n i + li ) (iii) li = 0 for infinitely many i; (iv) every natural number divides some m i := m 0 (n 1 + l1 )(n 2 + l2 ) · · · (n i + li ). j
Set X 1 = [0, 1]n 0 and set X i+1 = (X i )n i+1 . Let πi : X i+1 → X i , 1 ≤ j ≤ n i+1 be the co-ordinate projections. Let Ai be the homogeneous C∗ -algebra Mm i (C(X i )), and let φi : Ai → Ai+1 be the ∗-homomorphism given by ! φi ( f )(x) = diag f ◦ πi1 (x), . . . , f ◦ πin i+1 (x), a(xi1 ), . . . , a(xili ) , ∀x ∈ X i+1 , where xi1 , . . . , xili ∈ X i are to be specified. Set A(r ) = limi→∞ (Ai , φi ), and define φi, j := φ j−1 ◦ · · · ◦ φi . Let φi,∞ : Ai → A be the canonical map. We note that the xi1 , . . . , xili ∈ X i may be chosen to ensure that A is simple (cf. [31]); we assume that they have been so chosen, whence A(r ) is unital, simple, separable, and amenable.
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By Theorem 5.1, we have lim rc(Ai ) = lim
i→∞
i→∞ 2m 0 (n 1
n0 n1 · · · ni − 1 = r. + l1 )(n 2 + l2 ) · · · (n i + li )
Since the construction of A(r ) is the same as that of [29, Theorem 4.1], we conclude that rc(A(r ) ) ≤ r by [29, Prop. 3.3]. Let η > 0 be given. We will exhibit positive elements a, b ∈ M∞ (A(r ) ) with the property that dτ (a) + r − η < dτ (b), ∀τ ∈ T(A(r ) ), and yet a b in W (A(r ) ). This will show that rc(A(r ) ) ≥ r − η for every η > 0, whence rc(A(r ) ) = r , as desired. Choose i large enough that dim(X i )/2 − 1 > r − η/4. mi It follows from [28, Theorem 6.6] that there are a, b ∈ M∞ (Ai )+ such that a b in W (Ai ) and yet dτ (a) + r − η < dτ (b), ∀τ ∈ T(Ai ). Assumption (ii) above ensures that n i = 0, whence each φi is injective. We may thus identify a and b with their images in A(r ) so that dτ (a) + r − η < dτ (b), ∀τ ∈ T(A(r ) ). We need only prove that a b in W (A(r ) ). The technique for proving this is an adaptation of Villadsen’s Chern class obstruction argument from [31]. With Ni := n 0 n 1 · · · n i , we have Ai = Mm i (C([0, 1] Ni )). The element b of M∞ (Ai ) has the following properties: there is a closed subset Y of [0, 1] Ni homeomorphic to S2k , Ni − 2 ≤ 2k ≤ Ni , such that the restriction of b to Y is a projection of rank k corresponding to the k-dimensional Bott bundle ξ over S2k ; and the rank of b is at most 2k over any point in X i = [0, 1] Ni . The element a has constant rank—it is a projection corresponding to a trivial line bundle over X i —and need only have normalised rank strictly less than 3η/4. By increasing i, and hence m i , if necessary, we may assume that the normalised rank of a is at least η/2. This leads to η! η η! dτ (a) > = 2r ≥ dτ (b) , ∀τ ∈ T(Ai ). (17) 2 4r 4r The map φi, j : Ai → A j has the form
! k l φi, j ( f ) = diag f ◦ πi,1 j (x), . . . , f ◦ πi,i,j j (x), f (xi1 ), . . . , f (xi i, j ) , ∀x ∈ X j , where ki, j = n i+1 n i+2 · · · n j and li, j = m j /m i − ki, j . Following [31], we have that φi, j (a) is a projection of rank rank(a)m j /m i corresponding to the trivial vector bundle k
θrank(a)m j /m i , while the restriction of φi, j (b) to Y ki, j ⊆ X i i, j = X j is of the form ξ ×ki, j ⊕ f b , where f b is a constant positive element of rank at most 2kli, j . If p is the image of 1 ∈ Ai under the eigenvalue maps of φi, j which are co-ordinate projections,
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then pφi, j (b) p = ξ ×ki, j . Let x ∈ A j . Restricting to Y ki, j (and using the same notation for the restriction of x) we have x(ξ ×ki, j ⊕ f b )x ∗ − θrank(a)m j /m i = [x( p ⊕ f b )](ξ ×ki, j ⊕ θrank( f b ) )[x( p ⊕ f b )]∗ − θrank(a)m j /m i . 1/2
1/2
If we can show that θrank(a)m j /m i is not Murray-von Neumann equivalent to a subprojection of ξ ×ki, j ⊕ θrank( fb ) , then we will have that the last quantity above is ≥ 1/2 (cf. [30, Lemma 2.1]). It will then follow that for every j > i and every x ∈ A j , xφi, j (b)x ∗ − φi, j (a) ≥ 1/2; in particular, a b, as desired. By a straightforward adaptation of [31, Lemma 2.1] (using the fact that the top Chern class of ξ is not zero), θrank(a)m j /m i will fail to be equivalent to a subprojection of ξ ×ki, j ⊕ θrank( f b ) if rank(a)m j /m i > rank( f b ). We have rank( f b ) − rank(a) ·
mj mj ≤ 2kli, j − rank(a) · mi mi mj mj ≤ Ni − ki, j − rank(a) · mi mi mj = (Ni − rank(a)) · − n0n1 · · · n j , mi
so it will be enough to prove that n 0 n 1 · · · n j > (Ni − rank(a)) ·
mj . mi
Rearranging and using the definitions of m i and Ni we must show that (n i+1 + li+1 ) · · · (n j + l j ) · (1 − rank(a)/Ni ) < 1. n i+1 · · · n j Now rank(a) > (η/2)m i , so the right hand side above is less than (n i+1 + li+1 ) · · · (n j + l j ) mi η . · 1− n i+1 · · · n j 2Ni
(18)
The sequence (m i η)(2Ni ) is convergent to a nonzero limit, so for some γ > 0, for all i sufficiently large, the expression in (18) is strictly less than (n i+1 + li+1 ) · · · (n j + l j ) · (1 − γ ). n i+1 · · · n j
(19)
Increasing i if necessary we may assume that (n i+1 + li+1 ) · · · (n j + l j ) 1 < , n i+1 · · · n j 1−γ whence the expression in (19) is strictly less than one, as required. This completes the proof that rc(A(r ) = r .
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Since each natural number divides some m i and each X i is contractible, we have K0 (A(r ) ) ∼ = Q, with the usual order structure and order unit. The contractibility of X i also implies that K1 (Ai ) = 0 for every i, whence K1 (A(r ) ) = 0, too. The pairing ρ between traces and K0 is determined uniquely since there is only one state on K0 (A(r ) ). In order to complete the proof that Ell(A(r ) ) ∼ = Ell(A(s) ) for every r, s ∈ R+ \{0}, we (r ) (s) ∼ must prove that T(A ) = T(A ). Recall that the tracial state space of Mk (C(X )) is homeomorphic to the space P(X ) (r ) (s) of regular positive Borel probability measures on X . Let (Ai , φi ) and (Ai , ψi ) be inductive sequences as above, with simple limits A(r ) and A(s) , respectively. We have (r ) (s) Spec(Ai ) = [0, 1] Ni and Spec(Ai ) = [0, 1] Mi . Using the superscript to denote the map induced on traces by a ∗-homomorphism, we have T(A(r ) ) ∼ = lim (P([0, 1] Ni , φi ); T(A(s) ) ∼ = lim (P([0, 1] Mi , ψi ). ←−
←−
We require sequences (γi ) and (δi ) of continuous affine maps making the triangles in the diagram P([0,O 1] N1 ) o
P([0,O 1] N2 ) o P([0,O 1] N3 ) o u · · · p φ2 pp φ3 uu p p u pp ppp γ3 γ1 γ2 p p uu p p p δ1 p δ2 uu δ3 p p u p p wp wp zu ··· P([0, 1] M2 ) o P([0, 1] M3 ) o P([0, 1] M1 ) o
φ1
ψ1
ψ2
ψ3
(20) commute ever more closely on ever larger finite sets as i → ∞. We will in fact be able to arrange for near-commutation on the entire source space in each triangle. Let µ be a probability measure on X N , and K a subset of {1, . . . , N }. We use µ K to denote the measure on X |K | defined by integrating out those co-ordinates of X N not N /N contained in K . Straightforward calculation shows that upon viewing X i+1 as X i i+1 i we have ⎡ ⎤ /Ni Ni+1 N n i+1 li+1 i ⎣ φi (µ) = µ{l} ⎦ + λi , n i+1 + li+1 Ni+1 n i+1 + li+1 l=1
where λi is a convex combination of finitely many point masses. A similar statement holds for ψi . Since li+1 /(n i+1 + li+1 ) is negligible for large i, we may in fact assume that
φi (µ) =
Ni Ni+1
/Ni Ni+1
µ{l} ; ψi (µ) =
l=1
Mi Mi+1
/Mi Mi+1
µ{k}
k=1
for the purposes of our intertwining argument. We may also assume, by compressing our sequences if necessary, that N1 " M1 " N2 " M2 " · · · . Define γ1 (µ) =
1 M1 /N1
M 1 /N1 l=1
µ{(l−1)N1 +1,...,l N1 } .
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Now set Bk = {(k − 1)M1 + 1, . . . , k M1 } for each 1 ≤ k ≤ N2 /M1 , and Dt = {(t − 1)N1 , . . . , t N1 } for each 1 ≤ t ≤ N2 /N1 . Define 1 δ1 (µ) = N2 /M1
N 2 /M1
σk∗ (µ Bk ),
k=1
where σk∗ is the map induced on measures by the homeomorphism σk : Bk → Bk defined by the following property: if j is the first co-ordinate of Bk contained in a Dt which is itself contained in Bk , then σk is the permutation which subtracts j − 1(mod|Bk |) from each co-ordinate. (The idea is that σk moves all of the Dt ’s contained in Bk “to the beginning”.) Let L be the number of Dt ’s which are contained in some Bk . Since N1 " M1 " N2 , we have that (N2 − N1 L)/N2 is (arbitrarily) small. Now γ1 ◦ δ1 (µ) =
1 L
µ Dt ,
{t | Dt ⊆Bk , for some k}
while φ1 (µ)
N 2 /N1 1 = µ Dk . N2 /N1 k=1
The difference [(γ1 ◦ δ1 ) − φ1 ](µ) is a measure of total mass at most 2(N2 − N1 L)/N2 , and so the first triangle from the diagram (20) commutes to within this tolerance on all of P([0, 1] N2 ). The subsequent γi ’s and δi ’s are defined in a manner analogous to our definition of δ1 , and this leads to the desired intertwining. We conclude that Ell(A(r ) ) ∼ = Ell(A(s) ), as desired. It remains to prove that if A(r ) and A(s) are Morita equivalent, then r/s ∈ Q. Suppose that they are so. By the Brown-Green-Rieffel Theorem, A(r ) and A(s) are stably isomorphic, and so there are projections p, q ∈ A(r ) ⊗ K such that A(r ) ∼ = p(A(r ) ⊗ K) p and (r ) ⊗ K)q. Since K (A(r ) ⊗ K) = K (A(r ) ) = Q, there are natural numbers A(s) ∼ q(A = 0 0 n and m such that n[ p] = m[q] in K0 . It is proved in [31] that the construction used to arrive at A(r ) and A(s) always produces C∗ -algebras of stable rank one, whence A(r ) ⊗ K n has stable rank one. Thus, ⊕i=1 p and ⊕mj=1 q are Murray-von Neumann equivalent, and n n Mn (A(r ) ) ∼ p)(A(r ) ⊗ K)(⊕i=1 p) = (⊕i=1 m m ∼ q)(A(r ) ⊗ K)(⊕i=1 q) ∼ = (⊕i=1 = Mm (A(s) ).
By [28, Prop. 6.2 (ii)] we have ! ! r/n = rc Mn (A(r ) ) = rc Mm (A(s) ) = s/m, whence r/s ∈ Q, as required.
Acknowledgements. Part of this work was carried out at the Fields Institute during its Thematic Program on Operator Algebras in the fall of 2007. We are grateful to that institution for its support. We would also like to thank N. P. Brown and N. C. Phillips for several helpful conversations.
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References 1. Blackadar, B.: Comparison theory for simple C∗ -algebras, Operator Algebras and Applications, Vol. 1, 21–54, London Math. Soc. Lecture Note Ser., 135, Cambridge Univ. Press, Cambridge, 1988, pp. 21–54 2. Blackadar, B., Handelman, D.: Dimension Functions and Traces on C ∗ -algebras. J. Funct. Anal. 45, 297– 340 (1982) 3. Brown, N.P., Perera, F., Toms, A.S.: The Cuntz semigroup, the Elliott conjecture, and dimension functions on C∗ -algebras, J. Reine Angew. Math., to appear 4. Brown, N.P., Toms, A.S.: Three applications of the Cuntz semigroup. Int. Math. Res. Not., Article ID rnm068, (2007), 14 pages 5. Ciuperca, A., Elliott, G.A.: A remark on invariants for C∗ -algebras of stable rank one. Int. Math. Res. Not., to appear 6. Connes, A.: Classification of injective factors. Cases I I1 , I I∞ , I I Iλ , λ = 1. Ann. of Math. (2) 104, 73–115 (1976) 7. Coward, K., Elliott, G.A., Ivanescu, C.: The Cuntz semigroup as an invariant for C∗ -algebras. J. Reine Angew. Math., to appear 8. Ciuperca, A., Santiago, L., and Robert, L.: The Cuntz semigroup of ideals and quotients, and a generalized Kasparov Stabilization Theorem, J. Op. Th., to appear 9. Cuntz, J.: Dimension Functions on Simple C ∗ -algebras. Math. Ann. 233, 145–153 (1978) 10. Elliott, G.A., Gong, G., Li, L.: On the classification of simple inductive limit C ∗ -algebras, II: The isomorphism theorem. Invent. Math. 168, 249–320 (2007) 11. Elliott, G.A., Toms, A.S.: Regularity properties in the classification program for separable amenable C ∗ -algebras. Bull. Amer. Math. Soc. 45, 229–245 (2008) 12. Giol, J., Kerr, D.: Subshifts and perforation. Preprint, 2007 13. Gong, G.: On the classification of simple inductive limit C ∗ -algebras I. The reduction theorem. Doc. Math. 7, 255–461 (2002) 14. Haagerup, U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type I I I1 . Acta. Math. 158, 95–148 (1987) 15. Jacob, B.: A remark on the distance between unitary orbits in ASH algebras with unique trace. Preprint, 2008 16. Kirchberg, E., Rørdam, M.: Non-simple purely infinite C ∗ -algebras. Amer. J. Math. 122, 637–666 (2000) 17. Lin, Q., Phillips, N.C.: The structure of C∗ -algebras of minimal diffeomorphisms. In preparation 18. McDuff, D.: Uncountably many II1 factors. Ann. of Math. (2) 90, 372–377 (1969) 19. Ng, P.W., Winter, W.: A note on Subhomogeneous C ∗ -algebras. C. R. Math. Acad. Sci. Soc. R. Can. 28, 91–96 (2006) 20. Popa, S.: On a class of II1 factors with Betti numbers invariants. Ann. of Math. (2) 163, 809–899 (2006) 21. Perera, F., Toms, A.S.: Recasting the Elliott conjecture. Math. Ann. 338, 669–702 (2007) 22. Phillips, N.C.: Recursive subhomogeneous algebras. Trans. Amer. Math. Soc. 359, 4595–4623 (2007) 23. Phillips, N.C.: Cancellation and stable rank for direct limits of recursive subhomogeneous algebras. Trans. Amer. Math. Soc. 359, 4625–4652 (2007) 24. Rieffel, M.A.: Dimension and stable rank in the K-theory of C ∗ -algebras. Proc. London Math. Soc. (3) 46, 301–333 (1983) 25. Rørdam M (2002) Classification of Nuclear C ∗ -Algebras, Encyclopaedia of Mathematical Sciences 126, Berlin-Heidelberg: Springer-Verlag, 2002 26. Rørdam, M.: The stable and the real rank of Z-absorbing C ∗ -algebras. Int. J. Math. 15, 1065–1084 (2004) 27. Toms, A.S.: On the classification problem for nuclear C ∗ -algebras. Ann. of Math. (2) 167, 1059– 1074 (2008) 28. Toms, A.S.: Flat dimension growth for C ∗ -algebras. J. Funct. Anal. 238, 678–708 (2006) 29. Toms, A.S.: An infinite family of non-isomorphic C∗ -algebras with identical K-theory. Trans. Amer. Math. Soc., to appear 30. Toms, A.S.: Stability in the Cuntz semigroup of a commutative C ∗ -algebra. Proc. London Math. Soc. (3) 96, 1–25 (2008) 31. Villadsen, J.: Simple C ∗ -algebras with perforation. J. Funct. Anal. 154, 110–116 (1998) Communicated by Y. Kawahigashi
Commun. Math. Phys. 289, 435–482 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0716-x
Communications in
Mathematical Physics
Noncommutative Independence from the Braid Group B∞ Rolf Gohm1 , Claus Köstler2 1 Institute of Mathematics and Physics, Aberystwyth University, Aberystwyth,
SY23 3BZ, UK. E-mail: [email protected]
2 Department of Mathematics, Computer Science and Statistics, St. Lawrence University,
Canton, NY 13617, USA. E-mail: [email protected] Received: 25 June 2008 / Accepted: 19 September 2008 Published online: 4 February 2009 – © Springer-Verlag 2009
Abstract: We introduce ‘braidability’ as a new symmetry for infinite sequences of noncommutative random variables related to representations of the braid group B∞ . It provides an extension of exchangeability which is tied to the symmetric group S∞ . Our key result is that braidability implies spreadability and thus conditional independence, according to the noncommutative extended de Finetti theorem [Kös08]. This endows the braid groups Bn with a new intrinsic (quantum) probabilistic interpretation. We underline this interpretation by a braided extension of the Hewitt-Savage Zero-One Law. Furthermore we use the concept of product representations of endomorphisms [Goh04] with respect to certain Galois type towers of fixed point algebras to show that braidability produces triangular towers of commuting squares and noncommutative Bernoulli shifts. As a specific case we study the left regular representation of B∞ and the irreducible subfactor with infinite Jones index in the non-hyperfinite I I1 -factor L(B∞ ) related to it. Our investigations reveal a new presentation of the braid group B∞ , the ‘square root of 1/2 free generator presentation’ F∞ . These new generators give rise to braidability while the squares of them yield a free family. Hence our results provide another facet of the strong connection between subfactors and free probability theory [GJS07]; and we speculate about braidability as an extension of (amalgamated) freeness on the combinatorial level. Contents Introduction and Main Results . . . . . . . . . . . . . . . . . . . . 1. Distributional Symmetries . . . . . . . . . . . . . . . . . . . . 2. Random Variables Generated by the Braid Group B∞ . . . . . 3. Endomorphisms Generated by the Braid Group B∞ . . . . . . 4. Another Braid Group Presentation, k-Shifts and Braid Handles 5. An Application to the Group von Neumann Algebra L(B∞ ) . . 6. Some Concrete Examples . . . . . . . . . . . . . . . . . . . .
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436 440 444 448 454 461 470
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A. Appendix: Operator Algebraic Noncommutative Probability . . . . . . . . . 474 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 Introduction and Main Results The braid groups Bn were introduced by Artin in [Art25] where it is shown that, for n ≥ 2, Bn is presented by n − 1 generators σ1 , . . . , σn−1 satisfying the relations σi σ j σi = σ j σi σ j σi σ j = σ j σi
if | i − j | = 1; if | i − j | > 1.
(B1) (B2)
One has the inclusions B2 ⊂ B3 ⊂ · · · ⊂ B∞ , where B∞ denotes the inductive limit. For notational convenience, σ0 will denote the unit element in B∞ and B1 is the subgroup σ0 . The Artin generator σi and its inverse σi−1 will be presented as geometric braids according to Fig. 0.1. Due to their rich algebraic and topological properties, braid groups are a key structure in mathematics and their better understanding is crucial for many applications, for example entanglement in quantum information theory [KL07]. Of special interest for us will be that the braid group Bn is an extension of the symmetric group Sn and contains the free group Fn−1 as a subgroup [Bir75,BB05]. Aside from their algebraic and topological properties, Sn and Fn have also intrinsic probabilistic interpretations which are connected to independence structures. This was revealed for the symmetric groups Sn already in the 1930s by the celebrated work of de Finetti on exchangeability [Fin31]. Here the groups Sn are represented by automorphisms of the underlying probability space and we will generalize this idea to braid groups in this paper. In the case of the free groups Fn a breakthrough result had to wait until the 1980s when Voiculescu discovered freeness during his investigations of free group von Neumann algebras [Voi85] and soon after established its intimate connection to random matrix theory [Voi91]. Even more directly, the Fn ’s serve as noncommutative models for the underlying probability spaces themselves. On the other hand, it is known that braid groups carry a probabilistic interpretation through quantum symmetries [EK98], in particular quantum groups [Maj95,FSS03]. Their statistical and entropy properties are examined in [DN97,DN98,VNB00,NV05, MM07] and a physical interpretation in terms of quantum coin tosses is given in [KM98]. Moreover, a probabilistic facet of braid groups is apparent in subfactor theory, for example from Markov traces or commuting squares [Jon83,Pop83b,GHJ89,Pop90,Jon91, Pop93,JS97]. In this paper we take a new way towards a probabilistic interpretation of braid groups and look at braided structures from the perspective of distributional symmetries and invariance principles [Ald85,Kal05]. The guiding idea is that, as representations of S∞ are connected to exchangeability of infinite random sequences, the representations of B∞ should be connected to a new symmetry called ‘braidability’ such that a noncommutative notion of conditional independence appears. Two important pillars for the realization
Fig. 0.1. Artin generators σi (left) and σi−1 (right)
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437
of this idea are the noncommutative extended de Finetti theorem obtained by one of the authors [Kös08] and product representations of endomorphisms studied by the other author [Goh04]. Our main result is that ‘braidability’ of infinite random sequences implies conditional independence and provides an interesting symmetry between exchangeability and spreadability; of course these notions are meant in a noncommutative sense. At first sight this might be surprising for a probabilist, since Ryll-Nardzewski showed that exchangeability and spreadability are equivalent for infinite random sequences [RN57]. But we will see that this equivalence fails in the noncommutative realm. Hence we need to consider noncommutative random variables and stochastic processes and, because topological and analytical arguments are vital together with algebraic ones, we use a framework of von Neumann algebras. In particular this allows a rich theory of conditioning and independence. To get more directly to the heart of the matter without many preliminaries, we have collected in Appendix A what we need about an operator algebraic noncommutative probability theory; and the reader may find it necessary to consult this appendix from time to time. Note however that for a first reading it is fine to concentrate on tracial states and to avoid the Tomita-Takesaki theory; in fact most of our examples in this paper are tracial. As an additional help for the reader and to streamline the flow of arguments around our main results some important information is demoted to the level of remarks. These remarks serve a number of purposes, from bringing together the different areas touched upon in this paper to providing background information for readers interested in future developments or open problems. In the following we comment on the most significant issues and describe briefly the contents of the paper. Throughout this paper a probability space (A, ϕ) consists of a von Neumann algebra A with separable predual and a faithful normal state ϕ on A. A random variable ι from (A0 , ϕ0 ) to (A, ϕ) is an injective *-homomorphism ι : A0 → A such that ϕ0 = ϕ ◦ ι and ι(A0 ) embeds as a ϕ-conditioned von Neumann subalgebra of A (see Definition A.1). A random sequence I is an infinite sequence of (identically distributed) random variables ι ≡ (ιn )n∈N0 from (A0 , ϕ0 ) to (A, ϕ). We may assume A0 = ι0 (A0 ) ⊂ A and ϕ0 = ϕ|A0 whenever it is convenient. If we restrict or enlarge (where possible) the domain A0 of the random variables ι to another von Neumann algebra C0 (with ϕ-conditioned embedding), then to simplify notation we will just write IC0 instead of I . Our notion of (noncommutative) conditional independence actually emerges from the noncommutative extended de Finetti theorem [Kös08]. Consider the random sequence IC0 which generates the von Neumann subalgebras C I with I ⊂ N0 and the tail algebra C tail : C I := ιi (C0 ), C tail := ιk (C0 ). i∈I
n∈N0 k≥n
We say that IC0 is (full/order) C tail -independent if the family (C I ) I ⊂N0 is (full/order) C tail -independent (in the sense of Definition A.6). Note that we do not require C tail to be contained in C I since this would be far too restrictive in the context of conditioning and distributional symmetries. We next introduce several distributional symmetries on an intuitive level (see Sect. 1 for equivalent definitions which are less intuitive but more efficient in proofs). Given the two random sequences I and I˜ with random variables (ιn )n≥0 resp. (˜ιn )n≥0 from (A0 , ϕ0 ) to (A, ϕ), we write
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R. Gohm, C. Köstler distr
(ι0 , ι1 , ι2 , . . .) = (˜ι0 , ι˜1 , ι˜2 , . . .) if all their multilinear functionals coincide: ϕ ιi(1) (a1 )ιi(2) (a2 ) · · · ιi(n) (an ) = ϕ ι˜i(1) (a1 )˜ιi(2) (a2 ) · · · ι˜i(n) (an ) for all n-tuples i : {1, 2, . . . , n} → N0 , (a1 , . . . , an ) ∈ An0 and n ∈ N. Now a random sequence I is said to be exchangeable if its multilinear functionals are invariant under permutations: distr
(ι0 , ι1 , ι2 , . . .) = (ιπ(0) , ιπ(1) , ιπ(2) , . . .) for any finite permutation π ∈ S∞ of N0 . We say that the random sequence I is spreadable if every subsequence has the same multilinear functionals: distr
(ι0 , ι1 , ι2 , . . .) = (ιn 0 , ιn 1 , ιn 2 , . . .) for any subsequence (n 0 , n 1 , n 2 , . . .) of (0, 1, 2, . . .). Finally, I is stationary if the multilinear functionals are shift-invariant: distr
(ι0 , ι1 , ι2 , . . .) = (ιk , ιk+1 , ιk+2 , . . .) for all k ∈ N. The key definition of ‘braidability’ is motivated from the characterization of exchangeability in Theorem 1.9. Definition 0.1. A random sequence I consisting of random variables (ιn )n∈N0 from (A0 , ϕ0 ) to (A, ϕ) is said to be braidable if there exists a representation of the braid group, ρ : B∞ → Aut(A, ϕ), such that the properties (PR) and (L) are satisfied: ιn = ρ(σn σn−1 · · · σ1 )ι0 for all n ≥ 1;
ι0 = ρ(σn )ι0 if n ≥ 2.
(PR)
(L)
Here the σi ’s are the Artin generators and Aut(A, ϕ) denotes the ϕ-preserving automorphisms of A. Throughout this paper we will make use of fixed point algebras of the braid group representation ρ. We denote by Aρ(σn ) the fixed point algebra of ρ(σn ) in A, and by Aρ(B∞ ) the fixed point algebra of ρ(B∞ ). Given (A, ϕ) and a braid group representation ρ : B∞ → Aut(A, ϕ) then a ϕ-conditioned von Neumann subalgebra C0 of A with the localization property ρ C0 ⊂ A0 := Aρ(σn ) n≥2
induces canonically a braidable random sequence IC0 by ι0 = id |C0 and (PR). The ρ maximal choice is C0 = A0 . Our main result is a refinement of the noncommutative extended de Finetti theorem by inserting braidability into the scheme of distributional symmetries.
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Theorem 0.2. Consider the following assertions for the random sequence IC0 : (a) (b) (c) (d) (do )
IC0 IC0 IC0 IC0 IC0
is exchangeable; is braidable; is spreadable; is stationary and full C tail -independent; is stationary and order C tail -independent.
Then we have the implications: (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (do ). Here (a) ⇒ (c) ⇒ (d) ⇒ (do ) is the extended de Finetti theorem [Kös08] which we review in Sect. 1 as Theorem 1.7. We are left to prove (a) ⇒ (b) ⇒ (c). The first implication is obtained below as a consequence of Theorem 1.9, the second is shown in Theorem 2.2. A fine point of Theorem 0.2 is that braidability is intermediate to distributional symmetries which are of purely probabilistic nature. Our main result also makes it clear that spreadability is not tied to the symmetric group in the noncommutative realm since it can be obtained from the much wider context of braid groups. In fact we do not exclude the possibility of an even wider context where spreadability can be produced. An auxiliary result in the braid group context is that we can identify the tail algebra as the fixed point algebra of the representation. This presents a braided extension of the Hewitt-Savage Zero-One Law. See Theorem 2.5. In the last part of Sect. 2 we define fixed point algebras for certain subgroups in a Galois type manner and apply the noncommutative de Finetti theorem to obtain commuting squares. This sets the stage for the next step. Our stochastic processes are stationary and this allows, as an additional tool, to introduce the time shift endomorphism. This is done in Sect. 3. Within the tower of the fixed point algebras this endomorphism can be written as an infinite product of automorphisms. It has been investigated by one of us (see [Goh04]) how such product representations can be used to study operator theoretic and probabilistic structures, and the present setting is a particularly neat example for that. In Appendix A we develop some refinements of this theory which put the setting of this paper into a wider context. In particular, from Theorem A.12 it becomes clear why we think of the tower of fixed point algebras as a ‘Galois type’ structure and that it is unavoidable to choose exactly this tower to express the probabilistic structures associated to braids. In fact, if we have commuting squares for the braids as in Sect. 2, then the associated tower is necessarily the tower of fixed point algebras. See Theorem A.12 and Corollary A.13. The results of Sect. 2 can be extended to obtain triangular towers of commuting squares from which we deduce that the time shift is a noncommutative Bernoulli shift (see Definition A.7). Theorem 0.3. The limit α := lim ρ(σ1 σ2 · · · σn−1 σn ) n→∞
exists on ρ
A∞ :=
Aρ(σk )
n∈N k≥n ρ
in the pointwise strong operator topology and defines an endomorphism of A∞ such that α n ◦ ι0 = ιn . Then α restricted to n∈N0 α n (C0 ∨ Aρ(B∞ ) ) is a full Bernoulli shift
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Fig. 0.2. Braid diagrams of γ1 , γ2 and γ3 (left to right)
over Aρ(B∞ ) with generator B0 := C0 ∨ Aρ(B∞ ) . Further we have Aα = B tail = Aρ,tail = Aρ(B∞ ) . Here Aα is the fixed point algebra of α and B tail resp. Aρ,tail are the tail algebras for ρ ρ ρ B0 , α(B0 ), α 2 (B0 ) . . . resp. A0 , α(A0 ), α 2 (A0 ) . . . This is proved after Theorem 3.9. By modifying the product representations we also obtain examples of stationary random sequences which are order independent but not spreadable. In Sect. 4 we give a new presentation for the braid groups Bn in terms of the generators {γi | 1 ≤ i ≤ n − 1} subject to the relations γl γl−1 (γl−2 · · · γk )γl = γl−1 (γl−2 · · · γk )γl γl−1
for 0 < k < l < n.
We shall see that the first three γi ’s are depicted as the geometric braids shown in Fig. 0.2. This new presentation may be regarded to be intermediate between the Artin presentation [Art25] and the Birman-Ko-Lee presentation [BKL98]. We name it the ‘square root of free generator’ presentation, since the squares {γi2 | 1 ≤ i ≤ n − 1} generate the free group Fn−1 . This is followed by the discussion of various shifts on B∞ which are all cocycle perturbations of the shift in the Artin generators but which can be better understood by considering the new generators. We apply this to characterize relative conjugacy classes in B∞ . These results are a preparation for Sect. 5 in which we study in detail the left regular representation of B∞ and some of the corresponding stochastic processes. The group von Neumann algebra is a non-hyperfinite I I1 -factor and we can also control the fixed point algebras occurring in our theory. Using the Artin generators we get a random sequence which is conditionally independent but not spreadable. On the other hand the square roots of free generators turn out to be spreadable and we speculate about a braided extension of free probability theory which is suggested by this picture. Section 6 discusses a few other examples of braid group representations such as the Gaussian representation, Hecke algebras and R-matrices. While these examples are well known it is interesting to reinterpret their properties in the context of braidability and our general theory. Of course the reader may take her favorite braid group representation and investigate what our theory is able to tell about it. For example it was tempting at this point to go straight into the braid group representations in Jones’ subfactor theory. But we only give a short hint in this direction because we felt that this is a topic on its own which is better postponed to a consecutive paper. 1. Distributional Symmetries A noncommutative version of de Finetti’s theorem was obtained by one of us [Kös08]. We report here some of the obtained results, as far as they are needed for the present paper.
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Distributional symmetries of random objects lead to deep structural results in probability theory and the reader is referred to Kallenberg’s monograph [Kal05] for a recent account on this classical subject. Here we are interested to study some of these basic symmetries in the context of noncommutative random objects; and we will constrain ourselves to infinite sequences of noncommutative random variables (in the sense introduced above). We start with the introduction of some equivalence relations to prepare the definition of stationarity, spreadability and exchangeability in the broad sense of distributional symmetries. Notation 1.1. The group S∞ is the inductive limit of the symmetric groups Sn , n ≥ 2, where Sn is generated on N0 by the transpositions πi : (i − 1, i) → (i, i − 1) with 1 ≤ i < n. We write π0 for the identity of S∞ . By [n] we denote the ordered set {1, 2, . . . , n}. The symmetric group S∞ is presented by the transpositions (πi )i∈N , subject to the relations πi π j πi = π j πi π j if | i − j | = 1;
(B1)
πi π j = π j πi if | i − j | > 1;
(B2)
πi2 = π0 for all i ∈ N.
(S)
Definition 1.2. Let i, j : [n] → N0 be two n-tuples. (i) i and j are translation equivalent, in symbols: i ∼θ j, if there exists k ∈ N0 such that i = θk ◦ j
or
θ k ◦ i = j.
Here denotes θ the right translation m → m + 1 on N0 . (ii) i and j are order equivalent, in symbols: i ∼o j, if there exists a permutation π ∈ S∞ such that i=π ◦j
and
π |j([n]) is order preserving.
(iii) i and j are symmetric equivalent, in symbols: i ∼π j, if there exists a permutation π ∈ S∞ such that i = π ◦ j. We have the implications (i ∼θ j) ⇒ (i ∼o j) ⇒ (i ∼π j). Remark 1.3. Order equivalence was introduced in the context of noncommutative probability in [KS07]. Our present formulation is equivalent to that given in [KS07]. For the notation of mixed higher moments of random variables, it is convenient to use Speicher’s notation of multilinear maps.
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Notation 1.4. Let ι ≡ (ιi )i∈N0 : (M0 , ψ0 ) → (M, ψ) be given. We put, for i : [n] → N0 , a = (a1 , . . . , an ) ∈ Mn0 and n ∈ N, ι[i; a] := ιi(1) (a1 )ιi(2) (a2 ) · · · ιi(n) (an ), ψι [i; a] := ψ (ι[i; a]) .
(1.1) (1.2)
Next we define the distributional symmetries in terms of the mixed moments of a sequence of random variables. Definition 1.5. A sequence of random variables ι ≡ (ιi )i∈N0 : (M0 , ψ0 ) → (M, ψ), also called (noncommutative) random sequence, is (i) exchangeable if, for any n ∈ N, ψι [i; · ] = ψι [j; · ] whenever i ∼π j; (ii) spreadable if, for any n ∈ N, ψι [i; · ] = ψι [j; · ] whenever i ∼o j; (iii) stationary if, for any n ∈ N, ψι [i; · ] = ψι [j; · ] whenever i ∼θ j. It is obvious from Definition 1.2 that we have the implications (i) ⇒ (ii) ⇒ (iii). Remark 1.6. ‘Spreadable’ is also called ‘contractable’ in the literature, for example in [Kal05], and is also in close contact with ‘subsymmetric’ in Banach space theory [JPX07]. Here the first notion is more suitable, since ‘contractable’ in the context of distributional symmetries should not be confused with the notion of a contraction in the context of operator theory. Roughly speaking, de Finetti’s celebrated theorem states that exchangeable infinite commutative random sequences are mixed i.i.d. Inspired by [Kal05], a noncommutative dual version of this result has been obtained by one of the authors. Theorem 1.7 ([Kös08]). Let I be a random sequence with (identically distributed) random variables ι ≡ (ιi )i∈N0 : (A0 , ϕ0 ) → (A, ϕ) and tail algebra Atail :=
ιk (A0 ).
n≥0 k≥n
Consider the following conditions: (a) I (b) I (c) I (co ) I
is exchangeable; is spreadable; is stationary and full Atail -independent; is stationary and order Atail -independent.
Then we have the implications (a) ⇒ (b) ⇒ (c) ⇒ (co ). See Definitions A.4 and A.6 for our concept of independence. If the von Neumann algebras considered are commutative, then one finds a dual version of the extended de Finetti theorem stated in [Kal05, Theorem 1.1]. Note that the implication (a) ⇒ (b) is obvious from Definition 1.5, so is (c) ⇒ (co ) from Definition A.6. The implication (b) ⇒ (c) is established by means from noncommutative ergodic theory. For the proof and a more-in-depth discussion of this result the reader is referred to [Kös08]. The following characterization of exchangeability motivates our notion of braidability as introduced in Definition 0.1. In fact, we have exactly the definition of braidability if in Theorem 1.9(b) the symmetric group S∞ is replaced by the braid group B∞ .
Noncommutative Independence from the Braid Group B∞
Definition 1.8. A random sequence I is minimal if A =
443
n≥0 ιn (A0 ).
Theorem 1.9. The following are equivalent for a minimal random sequence I : (a) I is exchangeable; (b) There exists a representation of the symmetric group, ρ : S∞ → Aut(A, ϕ), such that the properties (PR) and (L) are satisfied: ιn = ρ(πn πn−1 · · · π1 )ι0 for all n ≥ 1; ι0 = ρ(πn )ι0 if n ≥ 2.
(PR)
(L)
Proof. ‘(a) ⇒ (b)’: The minimality of the random sequence ensures that the monomials ι[i; a] with n-tuples i : [n] → N0 and a ∈ (A0 )n , n ∈ N, are a weak* total set (see Notation 1.4). By exchangeability for every π ∈ S∞ , ϕι [i; a] = ϕι [π ◦ i; a], hence ρ(π ) : ι[i; a] → ι[π ◦ i; a] is well defined and extends to an element of Aut(A, ϕ). Then ρ : S∞ → Aut(A, ϕ) is a representation and properties (PR) and (L) are easily verified. ‘(b) ⇒ (a)’: Because S∞ is generated by the π j , it is enough to prove from (b) that for all j ∈ N, a ∈ (A0 )n , ϕι [i; a] = ϕι [π j ◦ i; a]. As shown in Lemma 2.1 (in the more general situation of braidability) ρ(π j )ιk = ιk if k∈ / { j − 1, j}. Further ρ(π j )ι j−1 = ι j (by (PR)) and ρ(π j )ι j = ρ(π j )−1 ι j = ι j−1 (by (S): π 2j = π0 ). Summarizing, for all k ∈ N0 , ρ(π j )ιk = ιπ j (k) . Using this together with ρ(π j ) ∈ Aut(A, ϕ) we obtain ϕι [i; a] = ϕ(ιi1 (a1 ) · · · ιin (an )) = ϕ(ρ(π j )ιi1 (a1 ) · · · ρ(π j )ιin (an )) = ϕι [π j ◦ i; a], which is what we wanted to prove.
We give the proof of the first implication stated in our main result. Proof of Theorem 0.2 (a) ⇒(b). We need to show that exchangeability implies braidability. Comparing the formulations of Theorem 1.9(b) and Definition 0.1, this is accom plished by the canonical epimorphism: B∞ → S∞ satisfying σi = πi for all i ∈ N. Finally, we will need the following noncommutative generalization of the Kolmogorov Zero-One Law. Theorem 1.10 ([Kös08]). Suppose the random sequence I is order N -independent with N ⊂ Atail . Then we have N = Atail . In particular, an order C-independent random sequence has a trivial tail algebra. We will make use of this result within the proof of a ‘braided’ noncommutative version of the Hewitt-Savage Zero-One Law (see Theorem 2.4).
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2. Random Variables Generated by the Braid Group B∞ This section is devoted to the construction of spreadable random sequences from braid group representations and the study of some of their properties. Our results give an application for the noncommutative version of de Finetti’s theorem, Theorem 1.7. Moreover this section provides the proof of Theorem 0.2. Throughout this section, let ρ : B∞ → Aut(A, ϕ) be a given representation on the probability space (A, ϕ). For the construction of random sequences we are interested in the subgroups Bn,∞ := σk | n ≤ k < ∞ of B∞ and the corresponding fixed point algebras Aρ(Bn,∞ ) := {x ∈ A | ρ(σ )(x) = x for all σ ∈ Bn,∞ }. These algebras provide us with a tower of von Neumann algebras:
ρ
Aρ(B∞ ) = Aρ(B1,∞ ) ⊂ Aρ(B2,∞ ) ⊂ · · · ⊂ Aρ(Bn,∞ ) ⊂ · · · ⊂ A∞ :=
Aρ(Bn,∞ ) .
n∈N
For short, we write as
ρ An
:= Aρ(Bn+2,∞ ) for n ∈ N so that the above tower can be written ρ
ρ
ρ
ρ
ρ
Aρ(B∞ ) = A−1 ⊂ A0 ⊂ A1 ⊂ · · · ⊂ An−2 ⊂ · · · ⊂ A∞ . These fixed point algebras give us a framework for the following construction of spreadable random sequences. We need some preparation. Lemma 2.1. Consider the braidable random sequence IAρ and let m, n ∈ N0 . If 0 n∈ / {m, m + 1}, then we have ρ(σn )ιm = ιm . Proof. For n = 0 this is trivial because ρ(σ0 ) is the identity. We observe that ρ A0 = Aρ(B2,∞ ) ⊂ Aρ(σn ) for n ≥ 2. Because ιm = ρ(σm · · · σ1 σ0 )|Aρ , it is suffi0 cient to show that (σm σm−1 · · · σ1 σ0 )σn if n > m + 1 σn (σm σm−1 · · · σ1 σ0 ) = . (σm σm−1 · · · σ1 σ0 )σn+1 if 0 = n < m This is obvious for n > m + 1 by (B2). The remaining case 0 = n < m follows from (B1) and (B2): σn (σm σm−1 · · · σ1 σ0 ) = σm σm−1 · · · σn+2 σn σn+1 σn σn−1 · · · σ1 σ0 = σm σm−1 · · · σn+2 σn+1 σn σn+1 σn−1 · · · σ1 σ0 = (σm σm−1 · · · σ1 σ0 )σn+1 . As discussed after Definition 0.1 it is useful to introduce some flexibility here by conρ sidering ϕ-conditioned subalgebras C0 ⊂ A0 .
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Theorem 2.2. A braidable random sequence IC0 is spreadable. ρ
Proof. Clearly it is enough to give the proof for the maximal case C0 = A0 . Using the multi-linear maps ρ
ϕι [i; · ] : (A0 )n → C (see Notation 1.4), we need to show that, for any n ∈ N, we have ϕι [i; · ] = ϕι [j; · ] whenever i ∼o j. Note that i ∼o j if and only if there exists a finite sequence of orderequivalent n-tuples (ik )k=1,...,K : [n] → N0 satisfying the following conditions: (i) i1 = i and i K = j; (ii) for each k ∈ {1, . . . , K − 1}, there exists a nonempty subset A ⊂ [n] such that ik |[n]\A = ik+1 |[n]\A , and such that ik (A) = {l} and ik+1 (A) = {l } with |l −l | ≤ 1 for some l, l ∈ N0 . Thus it is sufficient to prove that ϕι [ik ; · ] = ϕι [ik+1 ; · ]. Let ik and ik+1 be two n-tuples meeting the above conditions for some set A and nonnegative integers l and l . The case l = l is trivial. It is sufficient to consider the case l = l + 1 (otherwise reverse the order). We note l + 1 ∈ / ik ([n]) for later purposes. Since ϕ = ϕ ◦ ρ(σl+1 ), we obtain that, ρ for some n-tuple a = (a1 , . . . , an ) ∈ (A0 )n , ϕι [ik ; a] = ϕ ιik (1) (a1 ) · · · ιik (n) (an ) = ϕ ρ(σl+1 )ιik (1) (a1 ) · · · ρ(σl+1 )ιik (n) (an ) . We consider each factor ρ(σl+1 )ιik ( j) (a j ) separately, for fixed j ∈ [n]. Since l + 1 ∈ / ik ([n]), one of the following two cases occurs: Case l + 1 ∈ / {ik ( j), ik ( j) + 1}: We conclude j ∈ / A and ρ(σl+1 )ιik ( j) = ιik ( j) = ιik+1 ( j) with Lemma 2.1. Case l + 1 = ik ( j) + 1: We infer j ∈ A and ρ(σl+1 )ιik ( j) = ρ(σl+1 )ιl = ιl+1 = ιik ( j)+1 = ιik+1 ( j) from the definition of the random variable ιl and the relation between ik and ik+1 . Altogether, we conclude that ρ(σl+1 )ιik (1) (a1 ) · · · ρ(σl+1 )ιik (n) (an ) = ιik+1 (1) (a1 ) · · · ιik+1 (n) (an ) and thus ϕι [ik ; a] = ϕι [ik+1 ; a]. Now a finite induction on k ∈ {1, . . . , K } shows that ϕι [i; a] = ϕι [i1 ; a] = · · · = ϕι [i K ; a] = ϕι [j; a]. Remark 2.3. The proof actually shows that for order equivalent tuples i ∼o j there always ρ exists a braid τ ∈ B∞ such that ρ(τ )(ι[i; a]) = ι[j; a] for all a ∈ (A0 )n . Note however that while it is always possible to construct a representation of S∞ from an exchangeable sequence it is not clear at the present state which additional probabilistic conditions would allow us to construct braid group representations and braidability from spreadable sequences. We are now in the position to apply the noncommutative extended de Finetti theorem 1.7. Theorem 2.4. A braidable random sequence IC0 is stationary and full C tail -independent. Proof. The random sequence IC0 is spreadable by Theorem 2.2. Thus its stationarity and full C tail -independence follows directly from the implication (b) ⇒ (c) of Theorem 1.7.
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Another immediate implication of Theorem 2.2 and the noncommutative version of de Finetti’s Theorem, Theorem 1.7, is a noncommutative generalized version of the famous Hewitt-Savage Zero-One Law. More precisely, in the context of exchangeable commutative infinite random sequences and representations of the symmetric group S∞ , the tail algebra of the random sequence is identified as the fixed point algebra of S∞ (see [Kal05], for example). Now the Hewitt-Savage Zero-One Law states that these two algebras are trivial if the random sequence is (order) C-independent. With Theorem 2.2 at our disposal, the tail algebra C tail := ιk (C0 ), n≥0 k≥n
is identified in the much broader context of braid group representations and we obtain a ‘braided’ extension of the Hewitt-Savage Zero-One Law. Theorem 2.5. A braidable random sequence IC0 satisfies C tail ⊂ Aρ(B∞ ) . ρ
Suppose Aρ(B∞ ) ⊂ C0 (⊂ A0 ). Then we have an equality C tail = Aρ(B∞ ) . In particular, these two algebras are trivial if the random sequence IC0 is order C-independent. ρ
Note that the assumption Aρ(B∞ ) ⊂ C0 is superfluous for the maximal choice C0 = A0 . tail ρ(B∞ ) it suffices to show that C tail ⊂ Aρ(σl ) for any l ∈ N. Proof. To prove C ⊂A tail But C ⊂ k>l ιk (C0 ), hence the assertion follows from ρ(σl )ιk = ιk for all k > l, by Lemma 2.1. Now assume Aρ(B∞ ) ⊂ C0 . We verify Aρ(B∞ ) ⊂ C tail . Indeed, since Aρ(B∞ ) ⊂ ρ C0 ⊂ Aρ(B2,∞ ) = A0 , we have Aρ(B∞ ) = ιk (Aρ(B∞ ) ) ⊂ ιk (C0 ) for all k. This implies ρ( B ) A ∞ ⊂ n≥0 k≥n ιk (C0 ) = C tail . Finally, we conclude C C tail = Aρ(B∞ ) from the C-independence of ι by applying Theorem 1.10.
Remark 2.6. The assumptions do not require the global ρ(B2 )-invariance of the von Neumann algebra ι0 (C0 ) ∨ ι1 (C0 ) = C0 ∨ ρ(σ1 )(C0 ). This invariance property is automatic if the representation ρ is a representation of the symmetric group S∞ , or in other words, if we have ρ(σn2 ) = id for all n ∈ N: ρ(σ1 ) (C0 ∨ ρ(σ1 )(C0 )) = ρ(σ1 )(C0 ) ∨ ρ(σ12 )(C0 ) = ρ(σ1 )(C0 ) ∨ C0 . Our next result states that from braid group representations we can produce commuting squares (see Appendix A) in the tower of fixed point algebras. Theorem 2.7. Assume that the probability space (A, ϕ) is equipped with the represenρ tation ρ : B∞ → Aut(A, ϕ) and let An−1 := Aρ(Bn+1,∞ ) , the fixed point algebra of ρ(Bn+1,∞ ) (with n ∈ N0 ). Then ρ
ρ
ρ
A−1 ⊂ A0 ⊂ A1 ⊂ · · · ⊂ A
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447
is a tower of von Neumann algebras such that, for all n ∈ N0 , ρ(σn+1 ) restricts to an ρ automorphism of An+1 and ρ
ρ
ρ(σn+1 )(An ) ⊂ An+1 ∪ ∪ ρ ρ An−1 ⊂ An is a commuting square. ρ
Proof. The global invariance of An+1 under the action of ρ(σn+1 ) is concluded from ρ An+1 = Aρ(Bn+3,∞ ) and relation (B2). The existence of the conditional expectations needed to define a commuting square follows along the lines sketched in Appendix A. ρ ρ ρ We claim for n = 0 the order A−1 -independence of A0 and ρ(σ1 )A0 . Indeed, the ρ ρ ρ ρ corresponding random sequence IAρ enjoys ι0 (A0 ) = A0 and ι1 (A0 ) = ρ(σ1 )A0 . 0 ρ ρ Thus we deduce from Theorem 2.4 combined with Theorem 2.5 that A0 and ρ(σ1 )A0 ρ are A−1 -independent. This establishes our claim. To handle the general case we consider, for n ≥ 0 fixed, the ‘n-shifted’ representation ρn of B∞ , defined by the multiplicative extension of ρ(σn+k ) if k > 0 ρn (σk ) := . ρ(σ0 ) = id if k = 0 ρ
ρ
ρ
ρ
n = An−1 . Now we obtain the An−1 -independence of An Thus we have Aρn (B∞ ) = A−1 ρ and ρ(σn+1 )(An ) along the same lines of arguments as before, based on the ‘n-shifted’ ρ (n) (n) random sequence ι(n) ≡ (ιk )k≥0 : (An , ϕ|Aρn ) → (A, ϕ) with ιk = ρn (σk σk−1 · · · σ1 σ0 ).
Remark 2.8. The proof of the above theorem uses the fact that from a given braid group representation ρ one can easily produce ‘n-shifted’ representations ρn := ρ ◦ shn , where the shift sh : B∞ → B∞ sends the generator σi to σi+1 . Note that this endomorphism sh is injective (see [Deh00, Lemma 3.3]). The prospect of passing to a shifted representation is of interest when the fixed point algebra Aρ(B2,∞ ) turns out to be trivial or too small for the required task, and a fixed point algebra Aρ(Bn+2,∞ ) , with n ∈ N, fulfills the requirements. This idea is used in Sect. 5, Theorem 5.9. Remark 2.9. The spreadable random sequence ι from Theorem 2.2 is induced by positive braids of the form σn σn−1 · · · σ1 for n ∈ N. Using the group automorphism inv : B∞ → B∞ which sends the generator σi to σi−1 for all i ∈ N and given the representation ρ, we obtain the representation ρ inv := ρ ◦ inv. Thus the random variables −1 −1 −1 −1 ιinv n = ρ(σn σn−1 · · · σ1 σ0 )|Aρ , n ∈ N0 , 0
define another spreadable random sequence, since Aρ (B2,∞ ) = Aρ(B2,∞ ) and consequently Theorem 2.2 applies. The random sequences ι and ιinv have the same tail algebra. inv This is easily concluded from Theorem 2.5 and Aρ (B∞ ) = Aρ(B∞ ) . More generally, inv we have Aρ (Bn,∞ ) = Aρ(Bn,∞ ) for all n ∈ N. Thus the commuting squares constructed in Theorem 2.7 from starting with the representation ρ inv are just those coming from the representation ρ, but now with ρ(σn−1 ) in the upper left corner, instead of ρ(σn ). inv
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3. Endomorphisms Generated by the Braid Group B∞ Suppose (A, ϕ) is equipped with the representation ρ : B∞ → Aut(A, ϕ). Then the ρ fixed point algebras An−2 := Aρ(Bn,∞ ) with n ≥ 1 and Bn,∞ = σn , σn+1 , . . . provide a tower ρ
ρ
ρ
ρ
Aρ(B∞ ) = A−1 ⊂ A0 ⊂ A1 ⊂ · · · ⊂ A∞ ⊂ A, ρ ρ,alg ρ where A∞ denotes the weak closure of A∞ := k Ak . Definition 3.1. The representation ρ : B∞ → (A, ϕ) has the generating property if ρ
A = A∞ . If a representation of B∞ has the generating property then A is also generated by the fixed point algebras Aρ(σn ) . This generating property is not always fulfilled from the outset (see Proposition 3.3), but we may always restrict a representation to a generating one. Proposition 3.2. The representation ρ : B∞ → Aut(A, ϕ) restricts to the generρ ρ ρ ρ ating representation ρ res : B∞ → Aut(A∞ , ϕ∞ ) such that ρ(σi )(A∞ ) ⊂ A∞ and E Aρ∞ E Aρ(σi ) = E Aρ(σi ) E Aρ∞ (for all i ∈ N). ρ ρ ρ Proof. We study the action of ρ(σi ) on An and A∞ . Since An = k≥n Aρ(σk+2 ) , it ρ ρ holds that ρ(σi )(An ) ⊂ An for i ∈ {1, . . . , n} ∪ {n + 2, n + 3, . . .}. If i = n + 1, then ρ ρ ρ ρ ρ An ⊂ An+1 implies ρ(σn+1 )An ⊂ ρ(σn+1 )An+1 ⊂ An+1 . From this we conclude that ρ ρ ρ ρ ρ(σi )An ⊂ An+1 and therefore ρ(σi )(A∞ ) ⊂ A∞ for all i ∈ N. A similar argument ρ ρ ρ ensures the inclusion ρ(σi−1 )(A∞ ) ⊂ A∞ for all i ∈ N. Consequently, A∞ is globally invariant under the action of ρ(B∞ ) and the representation ρ : B∞ → Aut(A, ϕ) ρ ρ restricts to the representation ρ res : B∞ → Aut(A∞ , ϕ∞ ) which, by construction, has the generating property. That E Aρ∞ and E Aρ(σi ) commute is concluded by routine arguments from E Aρ∞ ρ(σi )E Aρ∞ = ρ(σi )E Aρ∞ , and thus E Aρ∞ ρ(σi ) = ρ(σi )E Aρ∞ , and by an application of the mean ergodic theorem (as in [Kös08, Theorem 8.3], for example): E Aρ(σi ) = lim
N →∞
N −1 1
ρ(σin ). N n=0
Here the limit is taken in the pointwise strong operator topology.
The following proposition gives a method to construct new braid group representations from a given (simpler) one. In this way we can find many interesting examples with and without the generating property. Proposition 3.3. Given the representation ρ : B∞ → Aut(A, ϕ) suppose γ ∈ Aut(A, ϕ) is an automorphism commuting with all ρ(σi )’s. Then the multiplicative extension of γρ(σi ) if i > 0 ργ (σi ) := id if i = 0 defines another representation of B∞ in Aut(A, ϕ) such that:
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(i) the restriction ργres of ργ to the fixed point algebra Aγ has the generating property if ρ has the generating property, and ργres coincides with the restriction of ρ to Aγ ; (ii) the representation ργ does not have the generating property if all ρ(σi )’s are N -periodic but γ N = id for some N ∈ N. Proof. An elementary calculation shows that ργ satisfies the braid relations ργ (σi )ργ (σ j )ργ (σi ) = ργ (σ j )ργ (σi )ργ (σ j ) for |i − j| = 1; ργ (σi )ργ (σ j ) = ργ (σ j )ργ (σi ) for |i − j| > 1. Clearly, ϕ ◦ ργ (σ ) = ϕ. So ργ is a representation from B∞ into Aut(A, ϕ). (i) Since γ commutes with all ρ(σi )’s and ργ (σi )’s, both representations ρ and ργ restrict to Aγ . An elementary calculation shows that these two restrictions coincide; and ρ res
ρ
we denote them both by ργres . We show next that Ak γ = Ak ∩ Aγ . For this purpose let ρ E k and E γ be the ϕ-preserving conditional expectations from A onto Ak = Aρ(Bk+2,∞ ) γ resp. A . Since all ρ(σi )’s and γ commute, we conclude that γ E k = E k γ for all k. But this entails E k E γ = E γ E k by an application of the mean ergodic theorem (similar as for ρ res
ρ
Proposition 3.2). Consequently, Ak γ = Ak ∩ Aγ . Finally, the generating property of ρ implies that limk→∞ E k = id (in the pointwise sot-sense). Thus limk→∞ E γ E k = E γ . So ργres has the generating property. (ii) We infer from the N -periodicity of the ρ(σi )’s that ργ (σi ) N = γ N . Consequently, Aργ (σi ) ⊂ Aργ (σi ) = Aγ N
N
and, passing to the intersections of the fixed point algebras Aργ (σi ) , ρ
Ak γ = Aργ (Bk+2,∞ ) ⊂ Aγ . N
ρ
Altogether this gives the inclusions A∞γ ⊂ Aγ ating property if γ is not N -periodic.
N
⊂ A. So ργ does not have the gener-
Let us consider some concrete examples. Example 3.4. Period N = 1 means that ρ is trivial, i.e. ρ(σi ) = id for all i. Now any non-trivial γ ∈ Aut(A, ϕ) gives a representation ργ without the generating property. The simplest example is a (classical) probability space with two points, each with probability ργ 1 2 2 , on which γ acts by interchanging the two points. Here we find A∞ C = C = A. Example 3.5. The case of period N = 2 covers the representations of S∞ in Aut(A, ϕ) and to obtain a non-generating representation one just needs to find a non-idempotent γ ∈ Aut(A, ϕ) which commutes with all ρ(σi )’s. Interesting examples come from infinite tensor products N M2 with product states. The canonical tensor product flips on neighboring factors provide us with a state-preserving representation ρ of S∞ , and thus of B∞ with period N = 2. Now implement the automorphism γ as a Xerox action, in other words: as the infinite tensor product γ = N γ0 , where γ0 is a state-preserving automorphism of M2 . It is easy to check that γ commutes with all ρ(σi )’s. Since γ 2 = id if and only if γ02 = id, we have plenty of choices for γ0 such that ργ does not have the generating property. On the other hand ρ itself clearly has the generating property and by Proposition 3.3(i) this is inherited by the restriction ργres to the fixed point algebra of γ .
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Example 3.6. In a non-tracial situation we always have a non-trivial modular automorphism group commuting with ρ (see Appendix A) which gives further possibilities to apply Proposition 3.3. From now on we will assume that the representation ρ has the generating property: ρ
A = A∞ . This allows us to define an endomorphism α in the following way. Due to the fixed point ρ,alg properties of the tower with respect to the ρ(σk )’s and the weak denseness of A∞ , it is easily verified that α := sot- lim ρ(σ1 σ2 · · · σn )
(PR-0)
n→∞
exists pointwise in A and defines an adapted endomorphism α of A with a product representation. This is discussed in detail in Appendix A, see Definition A.3. In particular (for k, n ∈ N0 ) ρ(σk )(Aρn ) = Aρn
(k ≤ n),
(PR-1)
ρ(σk )|Aρn = id |Aρn (k ≥ n + 2).
(PR-2)
Next we address how the endomorphism α relates to the spreadable (and thus stationary) random sequence ι ≡ (ιn )n∈N0 from Theorem 2.2. For this purpose we need an elementary result on the Artin generators of the braid group. Lemma 3.7. If σ1 , . . . , σm are Artin generators of the braid group Bm+1 then σ1 σ2 · · · σm−1 σm σm−1 · · · σ2 σ1 = σm σm−1 · · · σ2 σ1 σ2 · · · σm−1 σm . In words: Pyramids up and down are the same. Proof. For m = 2 this is (B1). The general case follows by induction.
Proposition 3.8. The endomorphism α for (A, ϕ), given by α = lim ρ(σ1 σ2 · · · σn ), n→∞
ρ
ρ
and the random sequence ι ≡ (ιn )n∈N0 : (A0 , ϕ0 ) → (A, ϕ), given by ιn = ρ(σn σn−1 · · · σ1 σ0 )|Aρ , 0
are related by α n |Aρ = ιn 0
ρ
for all n ∈ N0 . (Above we have put ϕ0 = ϕ|Aρ .) 0
(3.1)
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Proof. This is trivial for n = 0, since ι0 = ρ(σ0 ) and σ0 is the identity in B∞ . Using ρ induction and Lemma 3.7, we get for x ∈ A0 = Aρ(B2,∞ ) , α n+1 (x) = = = = =
αα n (x) = α ιn (x) ρ(σ1 σ2 · · · σn+1 )ρ(σn σn−1 · · · σ0 )(x) ρ(σn+1 σn · · · σ2 σ1 σ2 · · · σn σn+1 )(x) ρ(σn+1 σn · · · σ1 σ0 )(x) ιn+1 (x).
We can interpret Proposition 3.8 by saying that α implements the time evolution of the stationary process associated to the random sequence ι. Note that, even with the generating property of the representation, the minimal part ρ ρ α n (A0 ) = ιn (A0 ) n∈N0
n∈N0
may be strictly contained in A. Theorem 3.9. Assume that the probability space (A, ϕ) is equipped with the generating ρ representation ρ : B∞ → Aut(A, ϕ) and let An−1 := Aρ(Bn+1,∞ ) , with n ∈ N0 . Then one obtains a triangular tower of inclusions such that each cell forms a commuting square: ρ
ρ
ρ
ρ
ρ
A−1 ⊂ A0 ⊂ A1 ⊂ A2 ⊂ A3 ⊂ · · · ⊂ A ∪ ∪ ∪ ∪ ∪ ρ ρ ρ ρ A−1 ⊂ α(A0 ) ⊂ α(A1 ) ⊂ α(A2 ) ⊂ · · · ⊂ α(A) ∪ ∪ ∪ ∪ ρ ρ ρ A−1 ⊂ α 2 (A0 ) ⊂ α 2 (A1 ) ⊂ · · · ⊂ α 2 (A) ∪ ∪ ∪ .. .. .. . . . Proof of Theorem 3.9. All inclusions stated in the triangular tower follow from the ρ ρ adaptedness property α(An−1 ) ⊂ An (for all n) of the endomorphism α, which is an immediate consequence of (PR-1) and (PR-2) (see also Appendix A). We are left to prove that all its cells are commuting squares. We know already from Theorem 2.7 that, for any n ≥ 1, ρ
ρ
An−1 ⊂ An ∪ ∪ ρ ρ An−2 ⊂ ρ(σn )(An−1 ) is a commuting square. Introducing the automorphism γn := ρ(σ1 · · · σn ), ρ
ρ
γn−1 (An−1 ) ⊂ γn−1 (An ) ∪ ∪ ρ ρ γn−1 (An−2 ) ⊂ γn−1 ρ(σn )(An−1 )
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is obviously a commuting square. Let us consider counterclockwise the corners of this diagram, starting with the lower left corner. One readily verifies: ρ
ρ
(PR-0) & (PR-2)
⇒
γn−1 (An−2 ) = α(An−2 );
(PR-0) & (PR-2)
⇒
γn−1 ρ(σn ) (An−1 ) = α(An−1 );
(PR-1) (PR-1)
⇒ ⇒
γn−1 (Aρn ) = γn ρ(σn )−1 (Aρn ) = γn (Aρn ) = Aρn ; ρ ρ γn−1 (An−1 ) = An−1 .
ρ
ρ
Summarizing this corner discussion, we have shown that, for any n ≥ 1, ρ
ρ
An−1 ⊂ An ∪ ∪ ρ ρ α(An−2 ) ⊂ α(An−1 ) is a commuting square. Inductively acting with α on the corners of such a commuting square, we conclude further for any k, n ∈ N: ρ
ρ
α k−1 (An−1 ) ⊂ α k−1 (An ) ∪ ∪ ρ ρ α k (An−2 ) ⊂ α k (An−1 ) is a commuting square. But this is the general form of a cell in the triangular tower of inclusions. Cells involving the column on the right are also commuting squares by the generating property of the representation. ρ
Proof of Theorem 0.3. Here we consider a ϕ-conditioned subalgebra C0 of A0 . If ρ A−1 = Aρ(B∞ ) is contained in C0 then, with the results of Theorem 2.7 or 3.9, we ρ ρ can apply Theorem A.10 to obtain a Bernoulli shift over A−1 with generator C0 . If A−1 ρ is not contained in C0 then we have to use B0 = C0 ∨ A−1 as the generator. Now in Aα = B tail = Aρ,tail = Aρ(B∞ ) the first two equalities follow from Theorem A.10 while the last equality is Theorem 2.5. Remark 3.10. Given a braid group representation Theorem 3.9 shows that we find commuting squares in the tower of fixed point algebras and, with Theorem A.10 and Theorem 0.3, a corresponding Bernoulli shift. In Theorem A.12 we show that we can reconstruct the tower of fixed point algebras if we are given the commuting squares. This is what we mean when we think of this tower as a ‘Galois type’ tower. Next we produce a more general family of examples for adapted endomorphisms with product representations. Corollary 3.11. Under the assumptions of Theorem 3.9, let a sequence ε = (εk )k∈N ∈ {1, −1}N be given. Then αε (x) := sot- lim ρ(σ1ε1 σ2ε2 · · · σnεn )(x) n→∞
(PR-0’)
defines an endomorphism for (A, ϕ) such that, for all k, n ≥ 0, ρ
ρ(σkεk )(Aρn ) = An (k ≤ n);
(PR-1’)
ρ(σkεk )|Aρn = id |Aρn (k ≥ n + 2).
(PR-2’)
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Moreover, one obtains a family of triangular towers of commuting squares, indexed by the sequence ε: ρ
A−1 ⊂
ρ
ρ
ρ
ρ
A0 ⊂ A1 ⊂ A2 ⊂ A3 ⊂ · · · ⊂ A ∪ ∪ ∪ ∪ ∪ ρ ρ ρ ρ αε (A−1 ) ⊂ αε (A0 ) ⊂ αε (A1 ) ⊂ αε (A2 ) ⊂ · · · ⊂ αε (A) ∪ ∪ ∪ ∪ .. .. .. .. . . . .
Proof. We can argue in the same way as for Theorem 3.9. In particular, to prove that each cell forms a commuting square, we take advantage of the following observation: ρ
ρ
An An−1 ⊂ ∪ ∪ ρ ρ An−2 ⊂ ρ(σn )(An−1 ) is a commuting square if and only if ρ
ρ
An−1 ⊂ An ∪ ∪ ρ ρ An−2 ⊂ ρ(σn−1 )(An−1 ) is a commuting square.
Corollary 3.11 provides a rich source of stationary order Aρ(B∞ ) -independent random sequences (by Theorem A.10) which come from braid group representations and in general are no longer spreadable. Thus one obtains an interesting class of random sequences for which the implication (co ) ⇒ (b) in the noncommutative de Finetti theorem, Theorem 1.7, fails to be true. Remark 3.12. Equation (3.1) can be expressed in terms of fundamental braids. This suggests that Garside structures may be relevant for possible generalizations of braidability. We close this section with some information on this connection for the interested reader. For a solution of the word and conjugacy problem for braid groups Garside introduced in [Gar69] the fundamental braid ∆n := (σ1 σ2 · · · σn−1 )(σ1 σ2 · · · σn−2 ) · · · (σ1 σ2 )(σ1 ) in Bn . Since ∆2n generates the center Z (Bn ) of Bn , the fundamental braid ∆n is also called the ‘square root’ of the center. In a recent new approach to the word and conjugacy problem, Birman et. al. introduced in [BKL98] the fundamental braid δn := σn−1 σn−2 · · · σ1 , which satisfies δnn = ∆2n and thus can be thought of to be the ‘n th root’ of the center. Since ρ(∆n )|Aρ = α n−1 |Aρ and ρ(δn )|Aρ = ιn−1 , 0
0
0
the relationship between the random sequence ι and the endomorphism α, α n−1 |Aρ = ιn−1 , 0
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reads in terms of the fundamental braids ∆n and δn as ρ(∆n )|Aρ = ρ(δn )|Aρ . 0
0
Furthermore, we have ρ(∆2n )ρ inv (δn )|Aρ = ρ(δn )|Aρ , 0
(3.2)
0
which defines also a spreadable random sequence according to Remark 2.9. Indeed, it follows from the definition of ∆n and Lemma 2.1 that ρ(∆2n )|Aρ = ρ(∆n )ρ(δn )|Aρ = ρ(σ1 σ2 · · · σn−1 )ρ(δn )|Aρ . 0
Next we multiply with
0
ρ inv (δ
n)
0
= ρ(inv(δn )) from the left. We note that
−1 −1 σn−2 · · · σ1−1 inv(δn ) = σn−1
and use that ∆2n generates the center of Bn . This gives (3.2). We also conclude from the above discussion that the random sequences ι and ιinv ≡ (ιinv n )n∈N0 (from Remark 2.9) are connected according to ρ(∆2n+1 )ιinv n = ιn for all n ∈ N. Finally, we point out to the reader that the Artin presentation and the Birman-Ko-Lee presentation are presently the only known presentations of Bn which possess a Garside structure [Bir08,Deh08]. 4. Another Braid Group Presentation, k-Shifts and Braid Handles We begin by providing a presentation for braid groups in terms of generators γi which will play a distinguished role within our investigations of spreadability for random sequences coming from braid group representations. This presentation may be regarded as being ‘intermediate’ between the Artin presentation [Art25] and the Birman-Ko-Lee presentation [BKL98]. Theorem 4.1 (Square root of free generators presentation). The braid group Bn (for n ≥ 3) is presented by the generators {γi | 1 ≤ i ≤ n −1} subject to the defining relations γl γl−1 (γl−2 γl−3 · · · γk+1 γk )γl = γl−1 (γl−2 γl−3 · · · γk+1 γk )γl γl−1
(EB)
for 0 < k < l < n. It will become apparent in the proof that the two sets of generators {γi } and {σi } are related by the formulas (see also Fig. 4.1): −1 γk = (σ1 · · · σk−1 )σk (σk−1 · · · σ1−1 );
(4.1)
−1 (γ1−1 · · · γk−1 )γk (γk−1 · · · γ1 ).
(4.2)
σk =
The geometrical picture will be further discussed later. We note that the elements 2 γ12 , . . . , γn−1 generate the free group Fn−1 (cf. [Bir75, Sect. 1.4] and [Jon91, Lect. 5]). This connection to the free group motivated us to call this presentation as titled in Theorem 4.1. Similarly, we will say that the γi ’s are square roots of free generators (see also Figs. 4.2 and 4.3).
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Fig. 4.1. Braid diagrams of γ3 = σ1 σ2 σ3 σ2−1 σ1−1 (left) and γ32 = σ1 σ2 σ32 σ2−1 σ1−1 (right)
Fig. 4.2. Braid relation γ2 γ1 γ2 = γ1 γ2 γ1
Fig. 4.3. Braid relation γ3 γ2 γ1 γ3 = γ2 γ1 γ3 γ2
Proof. We start with the relations (B1) and (B2) for the Artin generators σi and suppose that the γ j ’s are defined according to (4.1). Then a straightforward computation yields −1 −1 γl γl−1 γl−2 γl−3 · · · γk+1 γk = σ1 σ2 · · · σl−1 σl σk−1 σk−2 · · · σ2−1 σ1−1 .
We multiply the equation above with γl from the right and see that the left-hand side of (EB) equals −1 −1 σl−2 · · · σ1−1 ). (σ1 σ2 · · · σl−1 σl )(σk σk+1 σk+2 · · · σl−2 σl−1 σl )(σl−1
(4.3)
Similarly, we obtain for the right-hand side of (EB) the expression −1 −1 (σ1 σ2 · · · σl−1 )(σk σk+1 σk+2 · · · σl−2 σl−1 σl )(σl−2 σl−3 · · · σ1−1 ).
Thus the formulas (4.3) and (4.4) are equal if and only if −1 σl (σk σk+1 σk+2 · · · σl−2 σl−1 σl )σl−1 = (σk σk+1 σk+2 · · · σl−2 σl−1 σl ).
(4.4)
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Indeed, an application of (B1) and (B2) to the left-hand side of this equation shows −1 −1 σl (σk σk+1 σk+2 · · · σl−2 σl−1 σl )σl−1 = (σk σk+1 σk+2 · · · σl−2 σl σl−1 σl )σl−1 −1 = (σk σk+1 σk+2 · · · σl−2 σl−1 σl σl−1 )σl−1 .
Thus the relations (B1) and (B2) for the σi ’s imply the relations (EB) for the γi s. Conversely, suppose the group G is generated by γ1 , γ2 , · · · γn−1 subject to the relations (EB). We denote by γ0 the identity of G and show that, as an intermediate step, the ‘new’ generators −1 σi := γi−1 γi γi−1 , 0 < i < n,
σj = σ j σi for |i − j| > 1 satisfy the relations (B1) and (B2). We begin with proving σi −1 and assume without loss of generality i + 1 < j. For this purpose, multiply (EB) by γl−1 from the left side and by γl−1 from the right side to obtain −1 γl γl−1 )(γl−2 γl−3 · · · γk+1 γk ) = (γl−2 γl−3 · · · γk+1 γk )(γl γl−1 γl−1 ). (γl−1
The relations (EB) include the braid relations γl γl−1 γl = γl−1 γl γl−1 , which entails −1 σl = γl−1 γl γl−1 = γl γl−1 γl−1 .
(4.5)
This establishes the following commutation relations: σl σl (γl−2 γl−3 · · · γk+1 γk ) = (γl−2 γl−3 · · · γk+1 γk )
for 0 < k + 1 < l.
(4.6)
σi with It follows that σl commutes with all γi with i ≤ l − 2 and hence also with all i ≤ l − 2. This establishes the relation (B2) for the σi ’s. We are left to prove σl σk σl = σk σl σk for |k − l| = 1 and assume without loss of generality that l = k + 1. We have already shown that σk+1 and γk−1 commute. Thus, also using (4.5), −1 −1 σk σk+1 σk = (γk−1 γk γk−1 ) σk+1 (γk−1 γk γk−1 ) −1 −1 = (γk−1 γk ) σk+1 (γk γk−1 ) = (γk−1 γk )γk−1 γk+1 γk (γk γk−1 ) −1 = γk−1 γk+1 γk2 γk−1 .
On the other hand, −1 σk σk+1 = σk+1 (γk−1 γk γk−1 ) σk+1 σk+1 −1 = γk−1 σk+1 γk σk+1 γk−1 −1 = γk−1 (γk−1 γk+1 γk )γk (γk−1 γk+1 γk )γk−1 −1 −1 = γk−1 γk (γk+1 γk γk+1 )γk γk−1 −1 −1 = γk−1 γk (γk γk+1 γk )γk γk−1 −1 = γk−1 γk+1 γk2 γk−1 .
Altogether we have shown that the relations (EB) for the γi ’s imply the relations (B1) and (B2) for the σ j ’s. Finally, we note that the commutation relations (4.6) imply −1 −1 γl−2 ) σl (γl−2 γl−3 · · · γ2 γ1 ). σl = (γ1−1 γ2−1 · · · γl−3
This establishes that the σk ’s defined by (4.2) satisfy the braid relations (B1) and (B2).
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Remark 4.2. The relations (EB) are ‘extended’ versions of the Artin braid relation γl γl−1 γl = γl−1 γl γl−1. On the other hand, the generators γl are a subset of {σs,t | 1 ≤ s < t ≤ n}, the generators of the Birman-Ko-Lee presentation (cf. [BKL98]). Following the notation of [BB05], these generators are of the form −1 −1 −1 σs,t = (σt−1 σt−2 · · · σs+1 )σs (σs+1 · · · σt−2 σt−1 )
and γl = σ1,l+1 is easily verified in the geometric picture. Remark 4.3. We are indebted to Patrick Dehornoy [Deh08] for the observation that the ‘square root of free generators presentation’ is closely related to the Sergiescu presentations associated to planar graphs [Ser93]. In fact, the relations (EB) for γ1 , . . . , γn−1 are also satisfied for the n − 1 generators associated to a star-shaped graph with n − 1 edges. But then we conclude from Theorem 4.1 together with the Hopfian property of Bn (see [MKS76], Sect. 3.7 and 6.5) that γ1 , . . . , γn−1 satisfy all relations associated to a star-shaped graph with n − 1 edges. This shows that there is much more symmetry involved than originally stated, for example we have γ j γk γ j = γk γ j γk , 1 ≤ j, k ≤ n − 1, γl γk γ j γl = γk γ j γl γk = γ j γl γk γ j , 1 ≤ j < k < l ≤ n − 1, etc. It is a nice exercise to give direct algebraic proofs of such relations from (EB). We would also like to thank Joan Birman [Bir08] who pointed out to us the more recent work of Han and Ko on positive braid group presentations from linearly spanned graphs which contains another alternative (minimal) collection of relations which is equivalent to (EB) (see Lemma 3.3 and its proof in [HK02] for further details). Remark 4.4. The group Bn is also presented by the set of generators {γ˜i | 1 ≤ i ≤ n − 1} subject to the defining relations γ˜l (γ˜k γ˜k+1 · · · γ˜l−2 )γ˜l−1 γ˜l = γ˜l−1 γ˜l (γ˜k γ˜k+1 · · · γ˜l−2 )γ˜l−1
(EB)
for 0 < k < l < n. This is immediate from Proposition 4.1 if one lets γ˜i := inv(γi−1 ) or if one notes that the transformation τ → inv(τ −1 ) reverses the ordering of the letters of the word τ (given in terms of the σi±1 ’s). The Garside fundamental braid ∆n (see Remark 3.12) has a simple form in terms of the γk ’s: ∆n = (γn−1 · · · γ1 )(γn−2 · · · γ1 ) · · · (γ2 γ1 )γ1 . Also the fundamental braid δn takes a simple form, but now in the terms of the generators γ˜i : δn = γ˜n−1 γ˜n−2 · · · γ˜2 γ˜1 .
may be A more detailed investigation of these presentations with relations (EB) or (EB) of interest on its own for the word and conjugacy problem in braid groups.
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Fig. 4.4. Braid diagram of the action τ → sh1 (τ ) = σ1 sh(τ )σ1−1
We continue with the introduction of several closely related shifts on B∞ and relate them to the geometric operation of inserting ‘handles’ into braid diagrams. This geometric approach turns out to be fruitful for results on orbits generated by the action of a shift on braids, as they are needed in Sect. 5. For further material on braid handles, the reader is referred to [Deh97] and [Deh00, Chap. III]. Definition 4.5. The shift sh is given by the endomorphism on B∞ defined by sh(σn ) = σn+1 for all n ∈ N. The m-shift shm on B∞ , with fixed m ∈ N, is given by the endomorphism −1 shm (τ ) := σm σm−1 · · · σ1 sh(τ )σ1−1 · · · σm−1 σm−1 .
Lemma 4.6. The endomorphisms sh and shm on B∞ are injective for all m ∈ N. Moreover there exists, for every τ ∈ B∞ , some n ∈ N such that −1 −1 −1 −1 shm (τ ) = (σm+1 σm+2 · · · σn−1 σn ) τ (σn σn−1 · · · σm+2 σm+1 ).
(4.7)
Proof. A geometric proof for the injectivity of sh is given in [Deh00, Chap. I, Lemma 3.3]. The endomorphism shm is the composition of an injective endomorphism and automorphisms, and thus injective. We observe that, using the braid relations (B1) and (B2), −1 −1 −1 σi+1 = σ1−1 σ2−1 · · · σi−1 σi σi+1 σi (σi+1 σi σi−1 · · · σ2 σ1 ) = sh(σi ). Now let τ ∈ B∞ be given. Then there exists some n ∈ N such that τ ∈ Bn and −1 −1 σn )τ (σn σn−1 · · · σ2 σ1 ). sh(τ ) = (σ1−1 σ2−1 · · · σn−1
This proves (4.7), since shm (τ ) = σm · · · σ1 sh(τ )σ1−1 · · · σm−1 .
It is advantageous at this point to use the picture of geometric braids (cf. [Deh00, Sect. I.1]) and to introduce the notion of braid handles to visualize the action of the shifts sh and shm . Here the action of sh on some initial braid τ ∈ B∞ corresponds to inserting a new strand on the left of the initial braid diagram. Since sh1 (τ ) = σ1 sh(τ )σ1−1 , we see that the action of sh1 corresponds to inserting a new strand between the first and second strand of the braid diagram for τ and above the other strands. This looks like an upper handle, as illustrated in Fig. 4.4. The new strand is not entangled with the strands of the initial braid τ . In particular, σ1 → sh1 (σ1 ) = σ1 σ2 σ1−1 (see Fig. 4.5) and sh1 (σi ) = σi+1 for i ≥ 2. Thus the action of sh1 coincides with the action of sh on σi for i ≥ 2. Generalizing the above discussion and motivated by [Deh00, Def. 3.3], we introduce:
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Fig. 4.5. Braid diagram of the action σ1 → sh1 (σ1 ) = σ1 σ2 σ1−1
Fig. 4.6. Braid diagram of the action τ → sh3 (τ ) = σ3 σ2 σ1 sh(τ )σ1−1 σ2−1 σ3−1
Definition 4.7. An upper m-handle is a braid of the form shm (τ ) with m ≥ 1 and τ ∈ B∞ . We discuss the geometric interpretation of the shift shm for m ≥ 1. The action of shm on the σi ’s can easily be identified to be ⎧ ⎪ if i < m; ⎨σi −1 shm (σi ) = σi σi+1 σi if i = m; ⎪ ⎩σ if i > m. i+1 The action of shm turns the initial braid τ into an upper m-handle and geometrically corresponds to inserting an upper strand between the m th and (m + 1)th strand of the initial braid diagram (see Fig. 4.6 for m = 3). Further simplification occurs if we replace the generators σ1 , σ2 , . . . by the square roots of free generators γ1 , γ2 , . . . and express the initial braid τ in terms of the γi±1 . Lemma 4.8.
shm (γi ) =
γi γi+1
if i < m; if i ≥ m.
In particular, for all n ∈ N, sh1 (γn ) = γn+1 . We see that as soon as we work in the square root of free generators presentation the 1-shift is nothing but the shift in the generators while the m-shifts for m > 1 are partial shifts. The upper m-handle shm (τ ) can be expressed in terms of the generators γi±1 without the appearance of γm±1 . Now we provide results on relative conjugacy classes which will be needed in Sect. 5. Definition 4.9. The total width tw(τ ) of a braid τ ∈ B∞ is defined as the function tw : B∞ → N0 , where tw(τ ) is the minimal number n ∈ N0 such that τ ∈ Bn+1 .
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In other words: if tw(τ ) = n, then we can express the braid τ as a word in the symbols ±1 ±1 , σn±1 , but not as a word in σ0 , σ1±1 , . . . , σn−1 . Note that tw(τ ) = 0 σ0 , σ1±1 , . . . , σn−1 if and only if τ = σ0 . Remark 4.10. The total width tw(τ ) should not be confused with the width of a nontrivial braid τ introduced in [Deh00, Def. 3.16]. The latter is the number of strands ‘really involved in τ ’ which is less or equal tw(τ ). For example, σ6 σ7−1 σ92 has the width (9 − 6 + 2) = 5, but the total width 9. Proposition 4.11. Suppose σ0 = τ ∈ B∞ and let k, l ∈ N0 . Then we have: tw(sh1 (τ )) = tw(τ ) + 1. It particular, the set {(sh1 )k (τ ) | k ∈ N0 } is infinite. If tw(τ ) ≥ m then these assertions remain valid for shm instead of sh1 . Proof. Suppose σ0 = τ ∈ B∞ with total width tw(τ ) = n. We have tw(sh1 (τ )) ≤ n + 1. In the following we will use geometric arguments to prove that tw(sh1 (τ )) = n + 1. We have already seen that the action of sh1 turns the initial braid τ into an upper 1-handle (see Fig. 4.4). Suppose now that, using the braid relations (B1) and (B2), it is possible to write the upper 1-handle sh1 (τ ) as a word in σ1±1 , . . . , σn±1 only. Then it is geometrically clear, because the new upper strand is not entangled, that we can use the same operations ±1 , contrary to our assumption about to write the initial braid τ as a word in σ1±1 , . . . , σn−1 tw(τ ). Hence tw(sh1 (τ )) = n + 1. This proves the proposition for sh1 . The general case can be done similarly. The condition n = tw(τ ) ≥ m is needed to ensure that the new upper strand is inserted between the n + 1 strands used to model Bn+1 , so that the geometric argument still works. Proposition 4.12. Let Cm (τ ) := {wτ w −1 | w ∈ Bm+1,∞ } denote the relative conjugacy class of τ ∈ B∞ for some m ≥ 1. Then we have: τ ∈ Bm
⇐⇒ Cm (τ ) = {τ } ⇐⇒ Cm (τ ) is finite.
Proof. The implication ‘⇒’ is immediate from the braid relation (B2). Conversely, let m ≥ 1 be fixed and suppose τ ∈ B∞ with total width tw(τ ) ≥ m. In other words, we assume that the braid τ is not contained in Bm . Now consider the shift shm and note that by Lemma 4.6 always −1 −1 −1 shm (•) = σm+1 σm+2 · · · σn−1 σn+1 • σn+1 σn · · · σm+2 σm+1
with some n and hence {(shm )k (τ ) | k ∈ N0 } ⊂ Cm (τ ). We conclude with Proposition 4.11 that Cm (τ ) is infinite. Lemma 4.13. B∞ is an ICC group. Moreover, the inclusion B2,∞ ⊂ B∞ has infinite group index [B∞ : B2,∞ ]. Proof. The first assertion is immediate from Proposition 4.12 for m = 1. For the second assertion note that the cosets (σ1 )n B2,∞ are all different from each other for all n ∈ N. This follows geometrically or from Theorem 5.6(vi) below which implies that they are pairwise orthogonal with respect to the trace of the group algebra.
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5. An Application to the Group von Neumann Algebra L(B∞) This section is devoted to the construction of noncommutative random sequences from the left regular representation of the braid group B∞ . This will bring us in contact with free probability which has been introduced by Voiculescu for the study of free group von Neumann algebras. The group von Neumann algebra L(B∞ ) is generated by the left-regular representation {L σ | σ ∈ B∞ } of B∞ on the Hilbert space 2 (B∞ ), where L σ f (σ ) := f (σ −1 σ ). Let δσ ∈ 2 (B∞ ) be the function
δσ (σ ) =
1 if σ = σ 0 otherwise.
Then the complex linear extension of tr ∞ (L σ ) := δσ0 , L σ δσ0 defines the normal faithful tracial state tr ∞ on L(B∞ ). We denote its restriction to L(Bn ) by tr n . Definition 5.1. The inclusion N ⊂ M of two von Neumann algebras is said to be irreducible if the relative commutant is trivial, i.e. N ∩ M C. Theorem 5.2. The inclusion L(B2,∞ ) ⊂ L(B∞ ) is irreducible. Proof. In analogy with the standard arguments for ICC groups (cf. [Tak03a, V.7]), to prove irreducibility of the inclusion L(B2,∞ ) ⊂ L(B∞ ) it is sufficient to show that the relative conjugacy class C1 (τ ) := {wτ w −1 | w ∈ B2,∞ } is infinite for every τ ∈ B∞ with τ = σ0 . But this follows from Proposition 4.12. We collect some auxiliary results on L(B∞ ) which are of interest on its own. Since we could not find proofs in the literature, we provide them here for the convenience of the reader. The second author is indebted to Benoit Collins and Thierry Giordano for stimulating discussions on Property (Γ ) resp. Property (T) for braid groups. Corollary 5.3. (i) L(B∞ ) is a non-hyperfinite I I1 -factor; (ii) L(B∞ ) has Property (Γ ); (iii) L(B∞ ) does not have Kazhdan Property (T); (iv) L(B2,∞ ) ⊂ L(B∞ ) is a subfactor inclusion with infinite Jones index. In particular, L(B∞ ) and L(Fn ) are non-isomorphic (2 ≤ n ≤ ∞). Proof. (i) The factoriality of L(B∞ ) is immediate from Theorem 5.2 and the definition of irreducibility. We are left to prove the non-hyperfiniteness. B3 and hence B∞ contains F2 as a subgroup (cf. [Jon91, Lect. 5]). Since any subgroup of an amenable discrete group is amenable ([Tak03c, XIII, Ex. 4.4(iii)]) and F2 is non-amenable ([Tak03c, XIII, Ex. 4.4(v)]), we conclude that the group B∞ is non-amenable. But this implies the non-hyperfiniteness of L(B∞ ) (see [Tak03c, XIII, Theorem 4.10]). (ii) Let [x, y] := x y − yx. We conclude from the braid relation (B2) that
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lim τ [L σk , x)]∗ [L σk , x)] = 0
k→∞
for any x ∈ alg{L σ | σ ∈ B∞ }. This extends to x ∈ L(B∞ ) by a density argument. Thus (L σk )k is a non-trivial central sequence. (iii) B∞ does not have Property (T). This can be deduced along the hints of [BdlHV08, Ex. 1.8.14], compare also [GdlH91]. But an ICC group has Property (T) if and only if its group von Neumann algebra does so [Con82,CJ85]. Thus L(B∞ ) does not have Property (T). (iv) Since sh(B∞ ) = B2,∞ , we conclude the factoriality of L(B2,∞ ) from the injectivity of sh (see Lemma 4.6). The infinity of the Jones index is clear from Lemma 4.13 and [B∞ : B2,∞ ] = [L(B∞ ) : L(B2,∞ )] (see also [Tak03c, Ex. 2.4]). Finally, L(B∞ ) and L(Fn ) are not isomorphic since L(B∞ ) has a non-trivial central sequence and thus cannot be full (see [Tak03c, Theorem 3.8]). But the free group factors L(Fn ) are full for 2 ≤ n ≤ ∞ (see [Tak03c, Theorem 3.9]). Remark 5.4. L(Bn ) is not a factor for 2 ≤ n < ∞, since Z (Bn ), the center of the group Bn , is non-trivial. We are now going to identify the relative commutants L(Bn,∞ ) ∩ L(B∞ ) for n ≥ 2. Note that the case n = 2 is already covered by Proposition 4.11 and Theorem 5.2. Theorem 5.5. We have L(Bn+1,∞ ) ∩ L(B∞ ) = L(Bn ) for all n ≥ 1. Proof. The inclusion L(Bn ) ⊂ L(Bn+1,∞ ) is clear from the braid relations (B2). For the converse inclusion, we conclude in analogy to arguments for ICC groups (cf. [Tak03a, V.7]). Let x ∈ L(B∞ ) with x = τ ∈B∞ x(τ )L τ , where x(τ ) are scalars such that τ ∈B∞ |x(τ )|2 < ∞. We have x ∈ L(Bn+1,∞ ) if and only if the coefficients x(τ ) are constant on the relative conjugacy class Cn (τ ). Thus the square summability of the non-zero coefficients x(σ ) implies that Cn (τ ) is finite for every non-zero coefficient from this that τ ∈ Bn by Proposition 4.12. This shows that x(τ ). We conclude x ∈ L(Bn+1,∞ ) implies x ∈ L(Bn ). We turn our attention to braid group representations on the probability space (L(B∞ ), tr ∞ ). Here we are interested in considering the representation ρ : B∞ → Aut(L(B∞ )), defined by ρ(σ ) := Ad(L σ ) := L σ • L ∗σ with σ ∈ B∞ . We note that tr ∞ is automatically ρ(σ )-invariant and thus ρ(σ ) ∈ Aut(L(B∞ ), tr ∞ ). Furthermore, the representation ρ has the generating property (see Definition 3.1): L(Bn+1 ) = L(B∞ ). L(B∞ ) ⊃ (L(B∞ ))ρ(Bn+2,∞ ) ⊃ n≥0
n≥0
At this point we have verified all assumptions of Theorem 3.9. We next identify the fixed point algebras as they appear in Theorem 3.9. We have from Theorem 5.2 that (L(B∞ ))ρ(B∞ ) = Z(L(B∞ )) C, and from Theorem 5.5 that, for all n ≥ 0, (L(B∞ ))ρ(Bn+2,∞ ) = L(Bn+2,∞ ) ∩ L(B∞ ) = L(Bn+1 ).
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Furthermore, a straightforward computation shows that the action of the endomorphism α (from Theorem 3.9) comes from the shift sh on B∞ ; more precisely: α(L τ ) = L sh(τ ) . Let us summarize the discussion above. Theorem 5.6. Consider the probability space (L(B∞ ), tr ∞ ), equipped with the representation ρ : B∞ → Aut(L(B∞ ), tr ∞ ) given by τ → ρ(τ ) := Ad L τ . We arrive at the following conclusions: (i) ρ has the generating property; (ii) (L(B∞ ))ρ(B∞ ) C; (iii) (L(B∞ ))ρ(Bn+2,∞ ) = L(Bn+1 ) for all n ≥ 0; (iv) the map α(x) := sot- lim ρ(σ1 σ2 · · · σn )(x) n→∞
is an endomorphism for (L(B∞ ), tr ∞ ) such that α(L τ ) = L sh(τ ) , with τ ∈ B∞ ; (v) Each cell of the triangular tower of inclusions is a commuting square: C ⊂ C ⊂ L(B2 ) ⊂ L(B3 ) ⊂ L(B4 ) ⊂ · · · ⊂ L(B∞ ) ∪ ∪ ∪ ∪ ∪ C ⊂ C ⊂ α(L(B2 )) ⊂ α(L(B3 )) ⊂ · · · ⊂ α(L(B∞ )) ∪ ∪ ∪ ∪ C ⊂ C ⊂ α 2 (L(B2 )) ⊂ · · · ⊂ α 2 (L(B∞ )) ∪ ∪ ∪ .. .. .. . . . (α)
(vi) The maps ιn sequence
:= α n | L(B2 ) define a stationary and full C-independent random ι(α) ≡ (ι(α) n )n∈N0 : (L(B2 ), tr 2 ) → (L(B∞ ), tr ∞ ) .
In other words, α is a full Bernoulli shift over C with generator L(B2 ) (see Definition A.7). (vii) The random sequence ι(α) is not spreadable. Remark 5.7. For all n ∈ N0 , ιn (L σ1 ) = L shn (σ1 ) = L σn+1 . Hence the left regular representation of the Artin generators gives us a full C-independent sequence which is not spreadable, and thus also not braidable by Theorem 0.2. The random sequence ι(α) shows that the class of stationary and conditionally full independent random sequences is strictly larger than the class of spreadable random sequences in our setting of the noncommutative de Finetti theorem, Theorem 1.7.
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Proof. (i) to (iv) are shown above. (v) is immediate from (ii), (iii) and Theorem 3.9. We are left to prove (vi) and (vii). The random sequence ι(α) is stationary, since it is induced by the endomorphism α. We show first that ι(α) is order C-independent. For this purpose let I = {n 1 , . . . , m 1 } and J = {n 2 , . . . , m 2 } be ‘intervals’ in N0 with I < J , or more explicitly: n1 ≤ m 1 < n 2 ≤ m 2 . Indeed, I < J implies the order C-independence of k∈I α k (L(B2 )) and l k l∈J α (L(B2 )) by the following arguments. We have α (L σ1 ) = L shk (σ1 ) = L σk+1 , from which we conclude α k (L(B2 )) ⊂ L(Bm 1 +2 ) k∈I
and
αl (L(B2 )) ⊂ α m 1 +1 (L(B∞ )) .
l∈J
Looking at the triangular tower, this clearly implies the order C-independence. Moreover, this entails the order C-independence of the random sequence ι(α) . We still need to show that order C-independence upgrades to full C-independence, valid whenever I ∩ J = ∅ (see Definition A.6). We will prove this by induction. Let A I :=
N
A Ik
with A Ik =
k=1
A J :=
N
α i (L(B2 )),
i∈Ik
A Jl
with A Jl =
l=1
α j (L(B2 )),
j∈Jl
where the non-empty finite ‘intervals’ {Ik }k=1,...,N and {Jl }l=1,...,N satisfy I1 < J1 < I2 < J2 < · · · < I N < J N . Since α comes from the symbolic shift sh on the Artin generators σi (see (iv)), we know from the assumptions on the ‘intervals’ and the braid relation (B2) that A Ik and A Ik commute for k = k . So do A Jl and A Jl for l = l . We conclude from this by a simple induction on k and l that A I (k) = A Ik · A I (k−1)
sot
for I (k) :=
k
Ik ,
k =1
A J (l) = A J (l−1) · A Jl
sot
for J (l) =
l
Jl .
l =1
By linearity and sot-density arguments, it is sufficient to consider elements x := x (N ) ∈ A I and y := y (N ) ∈ A J of a product form which is inductively defined by x (k) := xk x (k−1)
with xk ∈ A Ik and x (k−1) ∈ A I (k−1) ,
y (l) := y (l−1) yl
with yl ∈ A Jl and y (l−1) ∈ A J (l−1) .
This puts us into the position to use order C-independence in the next calculation:
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tr ∞ (x (k) y (k) ) = tr ∞ xk x (k−1) y (k−1) yk = tr ∞ xk x (k−1) y (k−1) tr ∞ (yk ) = tr ∞ (xk ) tr ∞ x (k−1) y (k−1) tr ∞ (yk ).
We iterate this factorization and then, after everything is factorized, we undo it for x and y separately. This gives
tr ∞ (x y) =
N
tr ∞ (xk )
k=1
N
tr ∞ (yl ) = tr ∞ (x) tr ∞ (y).
l=1
Finally, the factorization properties on sot-total sets of A I and A J extend linearly to A I and A J by approximation. Doing so the proof of full C-independence of the random sequence ι(α) is completed. We next verify that ιn (L σ1 ) = L shn (σ1 ) = L σn+1 . Indeed, this is obvious from (iv) and the definition of the shift sh. We are left to prove (vii), the non-spreadability of ι(α) . For this purpose consider the two words
w1 := σ1 σ2 σ1 σ2−1 σ1−1 σ2−1 and w2 := σ1 σ3 σ1 σ3−1 σ1−1 σ3−1 .
We note that these two words have order-equivalent index tuples
(1, 2, 1, 2, 1, 2) and (1, 3, 1, 3, 1, 3).
Since the braid relations (B1) and (B2) imply
tr ∞ (L w1 ) = tr ∞ (L σ0 ) = 1 and tr ∞ (L w2 ) = tr ∞ (L σ
−1 1 σ3
we conclude that ι(α) is not spreadable.
) = 0,
Actually there is an abundance of non-spreadable random sequences which are still order C-independent.
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Corollary 5.8. Under the assumptions of Theorem 5.6, let the sequence ε = (εk )k∈N ∈ {1, −1}N be given. Then αε (x) := sot- lim ρ(σ1ε1 σ2ε2 · · · σnεn )(x) n→∞
defines an endomorphism for (L(B∞ ), tr ∞ ) such that one obtains a family of triangular towers of commuting squares, indexed by the sequence ε: C ⊂ C ⊂ L(B2 ) ⊂ L(B3 ) ⊂ L(B4 ) ⊂ · · · ⊂ L(B∞ ) ∪ ∪ ∪ ∪ ∪ C ⊂ C ⊂ αε (L(B2 )) ⊂ αε (L(B3 )) ⊂ · · · ⊂ αε (L(B∞ )) ∪ ∪ ∪ ∪ .. .. .. .. . . . . ε) In particular, ι(α := αεn | L(B2 ) defines a family, indexed by ε, of stationary and order n C-independent random sequences ε) ι(αε ) ≡ (ι(α n )n∈N0 : (L(B2 ), tr 2 ) → (L(B∞ ), tr ∞ ) .
In other words, α is an ordered Bernoulli shift over C with generator L(B2 ) (see Definition A.7). Proof. Combine Theorem 5.6 and Corollary 3.11.
Let us now replace sh by sh1 and the Artin generators by the square roots of free generators. We can do this systematically in the following way. Recall that the isomorphism inv : B∞ → B∞ sends the generator σi to σi−1 , for example: inv(σ23 σ3−1 σ4 ) = −1 −1 σ2−3 σ3 σ4−1 and inv(σ1 σ2 · · · σn−1 σn ) = σ1−1 σ2−1 · · · σn−1 σn . Theorem 5.9. Consider the probability space (L(B∞ ), tr ∞ ), equipped with the 1-shifted inverse representation ρ1inv : B∞ → Aut(L(B∞ ), tr ∞ ) given by τ → ρ1inv (τ ) := Ad L sh(inv(τ )) . Then we conclude: (i) ρ1inv has the generating property; inv (ii) (L(B∞ ))ρ1 (B∞ ) C; inv (iii) (L(B∞ ))ρ1 (Bn+2,∞ ) = L(Bn+2 ) for all n ≥ 0; (iv) The map β(x) := sot- lim ρ1inv (σ1 σ2 · · · σn )(x) n→∞
= sot- lim ρ(σ2−1 σ3−1 · · · σn−1 )(x) n→∞
is an endomorphism for (L(B∞ ), tr ∞ ) such that β(L τ ) = L sh1 (τ ) , with τ ∈ B∞ ; (v) Each cell of the triangular tower of inclusions is a commuting square: C ⊂ L(B2 ) ⊂ L(B3 ) ⊂ L(B4 ) ⊂ · · · ⊂ L(B∞ ) ∪ ∪ ∪ ∪ C ⊂ β(L(B2 )) ⊂ β(L(B3 )) ⊂ · · · ⊂ β(L(B∞ )) ∪ ∪ ∪ C ⊂ β 2 (L(B2 )) ⊂ · · · ⊂ β 2 (L(B∞ )) ∪ ∪ .. .. . .
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(β)
(vi) The maps ιn := β n | L(B2 ) define a braidable, spreadable and full C-independent random sequence ι(β) ≡ (ι(β) n )n∈N0 : (L(B2 ), tr 2 ) → (L(B∞ ), tr ∞ ). In particular, β is a full Bernoulli shift over C with generator L(B2 ) (see Definition A.7). (vii) The random sequence ι(β) is not exchangeable. Remark 5.10. For all n ∈ N0 , ι(β) n (L γ1 ) = L shn1 (γ1 ) = L γn+1 . Hence we have the very remarkable fact that the roots of free generators form a spreadable random sequence and behave better in this respect than the usual Artin generators. Proof. It is easy to see that the state tr ∞ on L(B) is ρ1inv (τ )-invariant for every τ ∈ B∞ . Thus we have a representation ρ1inv : B∞ → Aut(L(B∞ , tr ∞ )). (i) The generating property of ρ1inv follows from inv inv L(B∞ ) ⊃ (L(B∞ ))ρ1 (Bn+2,∞ ) = (L(B∞ ))ρ (Bn+3,∞ ) n≥0
=
n≥0
(L(B∞ ))ρ(Bn+3,∞ ) ⊃
n≥0
L(Bn+2 ) = L(B∞ ).
n≥0
At this point we have verified all assumptions of Theorem 3.9. Since inv (L(B∞ ))ρ1 (Bn+2,∞ ) = L(B∞ )ρ(Bn+3,∞ ) ,
(ii) and (iii) are directly concluded from Theorem 5.2 and 5.5. (iv) The action of the endomorphism β on L τ is identified by Lemma 4.6 as β(L τ ) = L σ −1 σ −1 ···σn−1 L τ L σn ···σ3 σ2 = L sh1 (τ ) 2
3
(with n sufficiently large). (v) is immediate from (ii) and Theorem 3.9. (vi) Braidability of ι(β) follows by definition (with the representation ρ1inv ) and spreadability is then a consequence of Theorem 2.2. Using (iv) and sh1 (γi ) = γi+1 (Lemma 4.8) (β) the image of L γ1 under ιn is identified to be γn+1 . (vii) We need to show the non-exchangeability of ι(β) . From (EB) (see Proposition 4.1) we have γ3 γ2 γ1 γ3 = γ2 γ1 γ3 γ2 , but if we interchange the subscripts 1 and 2 then γ3 γ 1 γ 2 γ 3 = γ1 γ 2 γ 3 γ 1 . In fact, if in the geometric picture we represent the two words by four-strands braids (see Fig. 5.1) and then remove the third and fourth strand to obtain braids in B2 , then the left hand side yields the identity σ0 but the right-hand side yields σ12 . It follows that tr 4 (L γ3 L γ2 L γ1 L γ3 L γ −1 L γ −1 L γ −1 L γ −1 ) = 1 2
3
1
2
= 0 = tr 4 (L γ3 L γ1 L γ2 L γ3 L γ −1 L γ −1 L γ −1 L γ −1 ), 1
and hence
ι(β)
is not exchangeable.
3
2
1
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Fig. 5.1. Four-strand braids γ3 γ1 γ2 γ3 (left) and γ1 γ2 γ3 γ1 (right)
1/2
Definition 5.11. Let 0 < r < t ≤ ∞. We denote by Fr, t the subgroup of B∞ generated by the square root of free generators {γs | r ≤ s ≤ t} and by Fr, t the subgroup of B∞ with the free generators {γs2 | r ≤ s ≤ t}. 1/2
1/2
Clearly Fr, t ⊂ Fr, t . We have F1, n = Bn+1 and F1, n Fn , where Fn denotes the 1/2
1/2
group in n free generators, and hence shk1 (Fs,t ) = Fs+k,t+k , as well as shk1 (Fs,t ) = Fs+k,t+k . In the rest of this section we assume that the reader is familiar with some notions in free probability as they have been introduced by Voiculescu. As a reminder see for example [VDN92]. Corollary 5.12. The square root of free generator presentation γi |i ∈ N of B∞ gives rise to the system of Haar unitaries {L γi | i ∈ N} such that 1/2
L(Fr, t ) =
{L γs | r ≤ s ≤ t},
and such that one has the following triangular tower of commuting squares: 1/2
1/2
1/2
1/2
1/2
C ⊂ L(F1, 1 ) ⊂ L(F1, 2 ) ⊂ L(F1, 3 ) ⊂ L(F1, 4 ) ⊂ · · · ⊂ L(F1, ∞ ) ∪ ∪ ∪ ∪ ∪ 1/2 1/2 1/2 1/2 C ⊂ L(F2, 2 ) ⊂ L(F2, 3 ) ⊂ L(F2, 4 ) ⊂ · · · ⊂ L(F2, ∞ ) ∪ ∪ ∪ ∪ 1/2 1/2 1/2 C ⊂ L(F3, 3 ) ⊂ L(F3, 4 ) ⊂ · · · ⊂ L(F3, ∞ ) ∪ ∪ ∪ .. .. .. . . . The squared family {L 2γi | i ∈ N} is a free system of Haar unitaries in the sense of Voiculescu whose generated triangular tower is a restriction of the above triangular tower:
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C ⊂ L(F1, 1 ) ⊂ L(F1, 2 ) ⊂ L(F1, 3 ) ⊂ L(F1, 4 ) ⊂ · · · ⊂ L(F1, ∞ ) ∪ ∪ ∪ ∪ ∪ C ⊂ L(F2, 2 ) ⊂ L(F2, 3 ) ⊂ L(F2, 4 ) ⊂ · · · ⊂ L(F2, ∞ ) ∪ ∪ ∪ ∪ C ⊂ L(F3, 3 ) ⊂ L(F3, 4 ) ⊂ · · · ⊂ L(F3, ∞ ) ∪ ∪ ∪ .. .. .. . . . Moreover, each cell in this tower forms a commuting square C⊂D ∪ ∪, A⊂ B such that B and C are freely independent with amalgamation over A. Proof. It is immediate from the definition of the left regular representation that L γi is a Haar unitary, i.e., tr ∞ (L nγi ) = 0 for all n ∈ Z\{0}. From Theorem 5.9 we get the 1/2
commuting squares for the L(Fs, t )’s and, by restriction, also for the L(Fs, t )’s. This is independence in the sense of Definition A.4, but for the squared family more is true: It is a basic result of free probability theory that they are (amalgamated) free. See [VDN92, Ex. 2.5.8 and 3.8.3]. It is easily seen that freeness with amalgamation implies conditional independence in the sense of Definition A.4. The converse fails to be true since our notion of independence is more general. But it would be of interest to determine combinatorial formulas for the mixed moments of square root random sequences such that these formulas extend the combinatorics of noncrossing partitions from free probability. 1/2 The close connection between the square root presentation F∞ and the free group F∞ invites to ask whether there exists a deeper parallel between objects considered in free probability theory and the appropriately chosen ‘square root objects’ in a ‘braided probability theory’. We are going to illustrate this in an example. Since L γ1 and L γ 2 are Haar unitaries, the selfadjoint operators L γi + L ∗γi and L γ 2 + L ∗γ 2 1
i
i
have both the arcsine law on the interval [−2, 2] as spectral distribution (see [NS06, Lect. 1]). But this can be taken one step further: According to Theorem 0.2 we may consider any von Neumann subalgebra C0 of 1/2 L(F1, 2 ) and the restriction of the random sequence to C0 will again be a spreadable random sequence. Among the interesting choices is of course 1/2
C0
:=
{L γ1 + L ∗γ1 }. 1/2
This gives the ‘random sequence’ (β n (C0 ))n ⊂ L(B∞ ) which can be understood as a braided counterpart of a free sequence. If we put C0 := {(L γ1 + L ∗γ1 )2 } and observe that (L γ1 + L ∗γ1 )2 = 2 + L γ 2 + L ∗γ 2 , so the ‘squared random sequence’ (β n (C0 ))n ⊂ L(F∞ ) 1
is free.
1
Conjecture 5.13. There exists a braided extension of free probability.
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Such an extension should, of course, be related to some ‘braided independence’ as a specific form of C-independence (in the sense of Definition A.4), but it must necessarily lie beyond independence with universality rules (in the sense of Speicher) which is completely classified and leaves no room for such an idea, see [Spe97,BGS02,NS06]). At the moment we have no definite formulation of such a theory but we suggest that there are interesting concrete problems on the way. For example, in view of Corollary 5.12, it is intriguing to ask whether the combinatorics of free probability theory can be appropriately extended to a combinatorics of free square root presentations. A promising starting point for such investigations are random walks on free square root presentations, in parallel to Kesten’s work on symmetric random walks on groups [Kes59]. For example Kesten determined for the symmetric random walk on F2 the spectral distribution of the free Laplacian ∆free =
1 L 2 + L ∗γ 2 + L γ 2 + L ∗γ 2 ) 2 1 2 2 γ1
with respect to the trace tr 3 . It is determined by the moment generating function ∞
tr 3 (∆nfree )z n
n=0
√ 2 1 − 12z 2 − 1 = 1 − 16z 2
which can effectively be determined with freeness (see [NS06, Lect. 4]). The combinatorics involved for determining the n th moment or the moment generating function of the corresponding braided Laplacian ∆braid =
1 L γ1 + L ∗γ1 + L γ2 + L ∗γ2 ) 2
amounts to answer the following question on 3-strand braids: Consider all words w of length n written in the alphabet of 4 letters γ1±1 , γ2±1 . How many words w among the 4n -words describe the trivial 3-strand braid? An answer to this question immediately gives the spectral measure of ∆braid in terms of moments. Related explicit calculations for randomly growing braids on three strands are contained in [MM07] and involve random Garside normal forms for the Artin presentation of B3 . Unfortunately this approach does not generalize for Bn with n ≥ 4. It would be of interest to investigate if the square root of free generator presentation, which has no Garside structure for n ≥ 4 [Bir08,Deh08] but coincides for n = 3 with the Artin presentation, gives an alternative approach to such problems (see also Sect. 4). Further background information and additional structures may come from the closeness of our approach to subfactor theory and the recent progress on the connection between subfactors, large random matrices and free probability theory (see [GJS07] and references therein). 6. Some Concrete Examples In the following we discuss a few concrete examples which are well known but which can be looked upon from a new point of view by integrating them into our theory of spreadable random sequences from braid group representations. It appears that this strategy allows to simplify some arguments. Of course there are many other examples.
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Example 6.1 (Gaussian Representations). Choose 2 ≤ p ∈ N and ω :=
ex p(2π i/ p) if p is odd; ex p(π i/ p) if p is even.
Then consider unitaries (ei )i∈N0 satisfying p
ei = 1 ei e j =
ω2 e
for all i; j ei
whenever i < j.
A pair ei , e j with i < j can be realized in (and generates) the ( p × p)−matrices and, taking the weak closure with respect to the trace tr, the sequence (ei )i∈N0 generates the hyperfinite I I1 -factor. This is the noncommutative probability space in this class of examples. ∗ e (for all i ∈ N) then If we now define (vi )i∈N by vi := ωei−1 i p
vi = 1; vi vi+1 = ω2 vi+1 vi ; vi v j = v j vi if |i − j| > 1. 2
We remark that k → ωk vik is well defined for k mod p and so the following sums can always be interpreted as sums over the cyclic group Z p . Then a direct computation shows that (u i )i∈N defined by p−1 1 k2 k u i := √ ω vi p k=0
are unitary and satisfy the braid relations (B1) and (B2). Hence we obtain a unitary representation of B∞ , called the Gaussian representation in [Jon91, Subsect. 5.8] because of the Gaussian sums in related computations. As usual, let us consider the endomorphism α = lim Ad(u 1 u 2 · · · u n ). n→∞
This endomorphism has already been studied in [Rup95] as an example of a noncommutative Bernoulli shift. Let us summarize what we can say about it from the point of view of our theory. From the relations ei v j =
ω2 v j ei if j = i or j = i + 1, v j ei otherwise,
it is readily checked that this is a product representation for α (satisfying (PR-1), (PR-2) of Definition A.3) with respect to the tower (Mn )∞ n=0 with Mn generated by e0 , . . . , en .
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More precisely we have α(ei ) = ei+1 for all i ∈ N0 . In fact, 1 (k 2 −k 2 ) k u 1 e0 u ∗1 = ω v1 e0 v1−k p k,k
1 (k 2 −k 2 ) −2k = ω ω e0 v1(k−k ) p k,k
−2 1 2k(−1) = ω e0 v1 ω p
−1
=ω
k
e0 v1 = e1 , etc.
Iterated application of α to M0 produces a braidable sequence which by Theorem 0.2 is spreadable. For the sequence e0 , e1 , e2 , . . . this follows also more directly from the fact that order preserving transformations preserve the commutation relations for the en . Proposition 6.2. The endomorphism α is a full Bernoulli shift (in the sense of Definition A.7) over M−1 = Mα = Mtail = MB∞ = C with generator p−1
M0 = lin{e0k }k=0 C p . Further we have for all n ≥ −1, Mn = M ∩ {u k : k ≥ n + 2} . Proof. We conclude by Theorem 0.3 that α is a full Bernoulli shift over the fixed point algebra MB∞ with generator M0 . We can also check, by direct computation, the commuting squares assumptions of Theorem A.10 with M−1 = C. Now everything follows from Theorem A.10 together with Theorem A.12 for Mn . Note that for p = 2 we have a Clifford algebra with anticommuting en ’s and we can check that the sequence e0 , e1 , e2 , . . . is exchangeable. This is no longer the case for p > 2. For example tr(e1 e2 e1∗ e2∗ ) = ω2 = ω−2 = tr(e2 e1 e2∗ e1∗ ). Further results about more general product representations with respect to this tower can be found in [Goh01]. Example 6.3. (Hecke algebras) Recall from [Jon94, Ex. 3.1]: The Hecke algebra over C with parameter q ∈ C is the unital algebra with generators g0 , g1 , . . . and relations gn2 = (q − 1)gn + q, gm gn = gn gm if | n − m |≥ 2, gn gn+1 gn = gn+1 gn gn+1 . Then if q is a root of unity it is possible to define an involution and a trace such that the gn are unitary, and the tower (Mn )∞ n=0 with Mn generated by g0 , . . . , gn is embedded into the hyperfinite I I1 -factor M with its trace. In [Jon94] the commuting square
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473
assumptions required for our Theorem A.10 are checked for the Ad(gn )’s with M−1 = C. As in the previous example we conclude that α := lim Ad(g1 · · · gn ) n→∞
defines a full Bernoulli shift over C with generator M0 , that Mα = Mtail = MB∞ = C by Theorem 0.3, and that Mn = M ∩ {gk : k ≥ n + 2} by Theorem A.12. Such a situation occurs in particular in Jones’ subfactor theory [Jon83] if the index belongs to the discrete range. Example 6.4 (R-matrices). Now take the tower M0 := M p ⊂ M1 := M p ⊗ M p ⊂ M2 := M p ⊗ M p ⊗ M p ⊂ · · · ⊂ M, where M p denotes the ( p × p)-matrices and the embeddings are given by X → X ⊗ 1, all sitting inside the hyperfinite I I1 -factor M with trace tr. By an R-matrix we mean an element R˘ of M p ⊗ M p satisfying the (constant quantum) Yang-Baxter equation (YBE) R˘ 12 R˘ 23 R˘ 12 = R˘ 23 R˘ 12 R˘ 23 , where we use a leg notation, i.e., the subscripts describe the embedding of R˘ into a triple tensor product of M p ’s. See [Jon91] for the role of R-matrices in providing interesting ˘ where P is the flip operator, we braid group representations. Note that with R = P R, get another familiar form of the YBE, R12 R13 R23 = R23 R13 R12 , which plays an important role in the theory of quantum groups. For an overview see [Kas95,Maj95]. ˘ If R˘ is a unitary R-matrix then evidently (u n )∞ n=1 with u n := Rn−1,n satisfy the braid relations and provide us with a product representation for an adapted endomorphism α := limn→∞ Ad(u 1 · · · u n ). By Theorem 0.3 it is (or restricts to) a full Bernoulli shift over Mα = Mtail = MB∞ with generator M0 ∨ MB∞ which produces a braidable random sequence by iterated applications to M0 . For example we can take p = 2 and (with respect to a basis δ0 ⊗ δ0 , δ0 ⊗ δ1 , δ1 ⊗ δ0 , δ1 ⊗ δ1 ) ⎛ ⎞ 1000 ⎜0 0 1 0 ⎟ R˘ = ⎝ 0 1 0 0⎠ 000ω with |ω| = 1. YBE is easily checked directly. By computing the first commuting square in Theorem A.10 we find that M−1 = C. This full Bernoulli shift over C with generator M0 is called the ω-shift in [Rup95]. The corresponding subfactors have been investigated in [KSV96]. Note that these examples include the usual tensor shift (ω = 1) and the CAR-shift (ω = −1) in its Jordan-Wigner form [BR81].
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Example 6.5. (R-matrices, non-homogeneous case) Slightly varying the construction in Example 6.4 provides us with very elementary examples of non-spreadable ordered Bernoulli shifts over C. Define ⎞ ⎛ 100 0 ⎜0 0 1 0 ⎟ , un = ⎝ 010 0⎠ 0 0 0 ωn embedded at positions n − 1 and n as before, and again α := lim Ad(u 1 · · · u n ). If ωn n→∞
depends on n then α continues to be an ordered Bernoulli shift over C (by Theorem A.10) but may fail to produce spreadable random sequences. Take for example ω1 = 1 and ω2 = −1, i.e., we mix the tensor shift and the CAR-shift. Then for ! 01 A0 x = 10 we find: α(x) = Ad(u 1 )(x) = 1 ⊗ x, ! 1 0 ⊗ 1 ⊗ x, α 2 (x) = Ad(u 1 u 2 )(x) = 0 −1 [x α(x)]2 = [x ⊗ x]2 = 1, " ! #2 2 0 −1 [x α 2 (x)] = ⊗ 1 ⊗ x = −1. 1 0 Therefore tr(x α(x) x α(x)) = tr(x α 2 (x) x α 2 (x)), which shows that already the mixed moments of 4th order are not invariant under order equivalence. A. Operator Algebraic Noncommutative Probability Here we present in a condensed form some well known and some less well known constructions of noncommutative probability theory in the framework of von Neumann algebras. This provides a general context for the results of this paper. Similar settings are considered in [Küm85,Küm88,HKK04,AD06,JP08,Kös08]. The last part about connections between product representations and Bernoulli shifts (A.10–A.12) is new in this form. Definition A.1. A (noncommutative) probability space (M, ψ) consists of a von Neumann algebra M with separable predual and a faithful normal state ψ on M. A ψ-conditioned probability space (M, ψ, M0 ) consists of a probability space (M, ψ) and a von Neumann subalgebra M0 of M such that the ψ-preserving conditional expectation E M0 from M onto M0 exists. As an abbreviation we also say in this case that M0 is a ψ-conditioned subalgebra. Note that probability spaces always come with a standard representation on a separable Hilbert space via GNS construction for ψ. By Takesaki’s theorem (see ψ [Tak03b]), the ψ-preserving conditional expectation E M0 exists if and only if σt (M0 ) = M0 for all t ∈ R, where σ ψ is the modular automorphism group associated to (M, ψ). Thus the existence of such a conditional expectation is automatic if ψ is a trace.
Noncommutative Independence from the Braid Group B∞
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Lemma A.2. A ψ-preserving automorphism α always commutes with the modular automorphism group σ ψ . Proof. The modular automorphism group σ ψ is uniquely characterized by the KMSψ condition with respect to ψ, see [Tak03b]. Hence it is enough to show that α −1 σt α also satisfies the KMS-condition with respect to ψ. Using ψ ◦ α = ψ this is easily done. We denote the set of ψ-preserving automorphisms α of M by Aut(M, ψ). Combined with Takesaki’s theorem we conclude that the following subalgebras are always ψ-conditioned: − fixed point algebras of a ψ-preserving automorphism (or of a set of such automorphisms) − the image of a ψ-conditioned algebra under a ψ-preserving automorphism − algebras generated by ψ-conditioned algebras. By combining these items all the subalgebras in this paper turn out to be ψ-conditioned. We also encounter a special type of (non-surjective) ψ-preserving endomorphisms. In the probability space (M, ψ) consider a tower M−1 ⊂ M0 ⊂ M1 ⊂ M2 ⊂ · · · of ψ-conditioned subalgebras such that M = n≥−1 Mn , and a family of automorphisms (αk )k≥1 ⊂ Aut(M, ψ) satisfying (for all n ≥ −1) αk (Mn ) = Mn if k ≤ n,
(PR-1)
αk |Mn = id |Mn if k ≥ n + 2.
(PR-2)
Note that M−1 is pointwise fixed by all αk , by (PR-2). Definition A.3. Given (PR-1) and (PR-2) then α = lim α1 α2 α3 · · · αn , n→∞
(PR-0)
(in the pointwise strong operator topology) defines a ψ-preserving endomorphism of M which we call an adapted endomorphism with product representation. So PR stands for product representation. The existence of the limit is easily deduced from (PR-1) and (PR-2), in fact for n ∈ N and x ∈ Mn−1 , α(x) = α1 · · · αn (x), a finite product. From the limit formula it also follows that α commutes with the modular automorphism group σ ψ , so that the corresponding remarks above about automorphisms apply here as well. Another immediate consequence from the axioms is that for all n ∈ N, α(Mn−1 ) ⊂ Mn . Recall that a (noncommutative) random variable is an injective *-homomorphism ι into a probability space, i.e. ι : A → M, see [AFL82]. We also write ι : (A, ψ0 ) → (M, ψ), where ψ0 = ψ ◦ ι. For us A is a von Neumann algebra and we include the
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property that ι(A) is ψ-conditioned (see Definition A.1) into our concept of random variables. Generalizing terminology from classical probability we may say that for an adapted endomorphism α the random variables ι1 := α |M0 : ι2 := α 2 |M0 : ··· ιn := α n |M0 :
M0 → M1 ⊂ M, M0 → M2 ⊂ M, M0 → Mn ⊂ M
are adapted to the filtration M0 ⊂ M1 ⊂ M2 ⊂ · · · and α is the time evolution of a stationary process. This explains our terminology. Product representations with (PR-1) and (PR-2) give us a better grasp on adaptedness from an operator theoretic point of view. This idea has been introduced and examined in [Goh04, Chap. 3], where more comments on the general philosophy and additional details can be found. In fact, the definition in [Goh04, Chap. 3] is a bit more general but Definition A.3 above seems to be more handy and, as the results in this paper show, it contains a lot of interesting examples. An important class of product representations arises in the following way: Assume that there exists a sequence of unitaries (u n )n∈N ⊂ Mψ , where Mψ is the centralizer, such that α = lim Ad(u 1 u 2 · · · u n ),
(PR-0u)
u n ∈ Mn ,
(PR-1u)
u n+2 ∈ Mn
(PR-2u)
n→∞
for all n. Here Mn denotes the commutant and the limit is taken in the pointwise sotsense. It is easy to see that this provides us with an endomorphism α which is adapted with product representation. One of the most important procedures in classical probability is conditioning, and the framework of ψ-conditioned probability spaces allows us to do that in a natural way also in a noncommutative setting. We use a concept of conditional independence as introduced by Köstler in [Kös08], where a more detailed discussion is given. Definition A.4. Let (M, ψ) be a probability space with three ψ-conditioned von Neumann subalgebras N , N1 and N2 . Then N1 and N2 are said to be N -independent or conditionally independent if E N (x y) = E N (x)E N (y) for all x ∈ N1 ∨ N and y ∈ N2 ∨ N . Note that N1 and N2 are (N = C)-independent if and only if ψ(x y) = ψ(x)ψ(y) for all x ∈ N1 and y ∈ N2 and in this case we recover the definition of Kümmerer [Küm88]. One of the attractive features of Definition A.4 from the point of view of operator algebras is the connection to commuting squares.
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Lemma A.5. Suppose N ⊂ N1 ∩ N2 ⊂ M. Then the following are equivalent: (i) N1 and N2 are N -independent. (ii) The following is a commuting square: N2 ⊂ M ∪ ∪ N ⊂ N1 The notion of commuting squares has been introduced by Popa [Pop83a,Pop83b,PP86]. There are many equivalent formulations: (iii) E N1 (N2 ) = N , (iv) E N1 E N2 = E N , (v) E N1 E N2 = E N2 E N1 and N = N1 ∩ N2 , etc. Note in particular that the equality N = N1 ∩ N2 is automatic in this situation. For further discussion see [GHJ89], Prop. 4.2.1. The proof of the equivalences above is given in the tracial case there but it generalizes immediately to ψ-conditioned probability spaces. Definition A.6. A family of ψ-conditioned subalgebras (Mn )n∈N0 of a probability space (M, ψ) is said to be (CI) full N -independent or conditionally full independent, if the von Neumann sub algebras i∈I Mi and j∈J M j are N -independent for all I, J ⊂ N0 with I ∩ J = ∅. (CIo ) order N -independent or conditionally order independent, if i∈I Mi and j∈J M j are N -independent for all I, J ⊂ N0 with I < J or I > J . (Here the order relation I < J means i < j for all i ∈ I and j ∈ J .) Clearly, (CI) implies (CIo ); but the converse is open in the generality of our setting. We will deliberately drop the attributes ‘full’ or ‘order’ if we want to address conditional independence on an informal level or if it is clear from the context. All these notions of independence translate to random variables by saying that random variables are independent if this is true for their ranges. This notion of conditional independence includes classical, tensor and free independence, including their amalgamated variants. Moreover, it goes beyond noncommutative independences with universality rules [Spe97,BGS02,NS06]. It applies to all examples of generalized or noncommutative Gaussian random variables, as long as they respect the properties of a white noise functor [Küm96,GM02] and generate von Neumann algebras equipped with a faithful normal state (given by the vacuum vector of the underlying deformed Fock space), as they appear in [BS91,BS94,BKS97,GM02,Kró02]. We refer to [HKK04,Kös08] for a more detailed treatment of conditional independence and for further examples coming from quantum probability. Now we are in a position to generalize many classical concepts related to independence to a noncommutative setting. Definition A.7. Let B0 be a ψ-conditioned subalgebra of (M, ψ) and α a ψ-preserving endomorphism. Further let N ⊂ B0 be a ψ-conditioned subalgebra which is pointwise fixed by α (i.e. N ⊂ Mα ). Then β defined as the restriction of α to B := n∈N0 α n (B0 ) is called a (full/ordered) Bernoulli shift over N with generator B0 if (β n (B0 ))n∈N0 is (full/order) N -independent.
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A trivial example is α = id with N = B0 = M, but in more interesting examples the generator is usually small, often finite dimensional. Remark A.8. The above definition of a Bernoulli shift is equivalent to that given in [Kös08]. This relies on the fact that a Bernoulli shift β automatically commutes with the modular automorphism group: As required by the definition of independence, the ranges β n (B0 ) are ψ-conditioned and so is B. Consequently, the modular automorphism ψ group σt restricts from M to B. But the ranges β n (B0 ) are ψ-conditioned if and only ψ if β commutes with this restriction of σt . Lemma A.9. If β is a (full/ordered) Bernoulli shift over N then N = B β = B tail , where B β is the fixed point algebra of β and B tail := N is uniquely determined by the endomorphism β. Proof. See [Kös08, Cor. 6.9].
n∈N
k≥n
β k (B0 ). In particular
Noncommutative (ordered) Bernoulli shifts with N = C have been introduced by Kümmerer [Küm88] and further developed by Rupp in [Rup95] where the connection to commuting squares is recognized and schemes similar to the following Theorem A.10 are considered. See also [HKK04 and Kös08]. How can we construct noncommutative Bernoulli shifts? It turns out that product representations as introduced in A.3 are a powerful tool for this task in the framework of conditional order independence. Theorem A.10. In (M, ψ) let (Mn )n≥−1 be a tower of ψ-conditioned subalgebras such that M = n≥−1 Mn and let α be an adapted endomorphism with product representation by factors (αn )n∈N , as in Definition A.3. Suppose that for all n ∈ N, Mn−1 ⊂ Mn ∪ ∪ Mn−2 ⊂ αn (Mn−1 ) is a commuting square. Then one obtains a triangular tower of inclusions such that all cells form commuting squares: M−1 ⊂ M0 ⊂ M1 ⊂ M2 ⊂ M3 ⊂ · · · ⊂ M ∪ ∪ ∪ ∪ ∪ M−1 ⊂ α(M0 ) ⊂ α(M1 ) ⊂ α(M2 ) ⊂ · · · ⊂ α(M) ∪ ∪ ∪ ∪ . M−1 ⊂ α 2 (M0 ) ⊂ α 2 (M1 ) ⊂ · · · ⊂ α 2 (M) ∪ ∪ ∪ .. .. .. . . . If B0 is a ψ-conditioned subalgebra such that M−1 ⊂ B0 ⊂ M0 then β defined as the restriction of α to B := n∈N0 α n (B0 ) is an ordered Bernoulli shift over N with generator B0 , where N is given by N = M−1 = Mα = Mtail = B β = B tail .
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Proof. The proof of Theorem 3.9 is written up in a way that easily transfers to the present setting and yields the triangular tower of commuting squares given above. Now suppose I < J , say i ≤ n < j for all i ∈ I and j ∈ J . By adaptedness i∈I β i (B0 ) ⊂ Mn and j n+1 (M). Inspection of the triangular tower of commuting squares j∈J β (B0 ) ⊂ α shows that these algebras are order M−1 -independent. This proves that β is an ordered Bernoulli shift over M−1 , and hence N = M−1 . The equalities N = B β = B tail follow from Lemma A.9. The fixed point algebra Mα and the tail algebra Mtail cannot be strictly bigger than M−1 because it is readily checked from (PR-1), (PR-2) and the commuting squares that α maps the ψ-orthogonal complement of Mk−1 in Mk into the ψ-orthogonal complement of Mk in Mk+1 , for all k ∈ N0 . But this means that the isometry on the GNS Hilbert space induced by α is a shift operator on the orthogonal complement of M−1 and hence there can be no fixed point or tail of α outside M−1 . The last assertion of Theorem A.10 provides us with a fixed point characterization which is very useful in applications. Suppose we want to interpret an endomorphism β as a Bernoulli shift with a generator B0 , but we have not yet identified N . We know from Lemma A.9 that N equals B β = B tail ; but how can one effectively determine B β or B tail in applications? As soon as we succeed to realize β as the restriction of a shift α in the way of Theorem A.10, i.e. with M−1 ⊂ B0 ⊂ M0 , then N = B β = B tail must be equal to the left lower corner M−1 of the first commuting square. In general this commuting square is easier to access in applications than tail or fixed points of β. So, besides a nice general structure, Theorem A.10 provides a convenient way to identify N . We demonstrate this idea for some examples in Sect. 6. An additional strong point of Theorem A.10 is that it identifies M−1 , the first algebra in the given tower, as the fixed point algebra of the endomorphism α = limm→∞ α1 α2 α3 · · · αm . Dropping the first factors in this product representation, we can also identify the other algebras in the tower as fixed point algebras of certain partial shifts: Corollary A.11. With assumptions as in Theorem A.10, for all n ≥ −1 the algebra Mn is equal to the fixed point algebra of αn+2,∞ := lim αn+2 αn+3 αn+4 · · · αn+m m→∞
and equal to the tail algebra for the random sequence produced by αn+2,∞ from Mn+1 as a range of the time 0-random variable. (Compare with Sect. 2.) Proof. Apply Theorem A.10 for αn+2,∞ and the tower Mn ⊂ Mn+1 ⊂ · · ·.
Corollary A.11 shows that, given the commuting squares in Theorem A.10, the whole tower M−1 ⊂ M0 ⊂ · · · is determined in terms of fixed point algebras by the factors (αn )n∈N of the product representation. The following consequence for braid group representations can be used to determine these fixed point algebras explicitly, compare with Sect. 6. Theorem A.12. Let the setting be as in Theorem A.10. In addition assume that for all k ∈ N we have αk = ρ(σk ) for a representation ρ : B∞ → Aut(M, ψ). Then for all n ≥ −1, Mn = Mρn , ρ
using the notation Mn = Mρ(Bn+2,∞ ) introduced in Sect. 2. As a special case, if ρ(σk ) = Ad u k for all k ≥ n +2, then Mn is equal to the relative commutant M∩{u k | k ≥ n +2} .
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Proof. Combine Corollary A.11 with Theorem 2.5 for the shifted representation.
In other words, under the given assumptions the tower we started from is automatically the tower of fixed point algebras. Note that this is a kind of converse to Theorem 2.7 where we proved that starting from a braid group representation we can always construct commuting squares for the tower of fixed point algebras and establish the situation obtained in Theorem A.12 from another direction. This is why we think of this structure as a kind of Galois type tower (compare with Remark 3.10). Let us finally rewrite Theorem A.12 to obtain a more explicit form of the assumptions. Corollary A.13. Given a representation ρ : B∞→ Aut(M, ψ) and a tower (Mn )n≥−1 of ψ-conditioned subalgebras such that M = n≥−1 Mn . Suppose further that, for all n ≥ −1, we have Mn ⊂ Mρn and we have a commuting square Mn+1 ⊂ Mn+2 ∪ ∪ . Mn ⊂ ρ(σn+2 )(Mn+1 ) Then for all n ≥ −1 Mn = Mρn . Proof. Note that the corresponding endomorphism α = limn→∞ ρ(σ1 · · · σn ) has alρ ready been defined and studied in Sect. 3. The assumption Mn ⊂ Mn for all n ≥ −1 α gives M−1 ⊂ M for n = −1 and (PR-2) for n ∈ N0 . Now (PR-2) together with the fact that α j = ρ(σ j ) and αk = ρ(σk ) commute for | j − k| ≥ 2 imply (PR-1). Hence we have verified all assumptions of Theorem A.12. Acknowledgement. The authors are grateful to the anonymous referee for several comments and suggestions helping us to improve the clearness of our presentation.
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Communicated by Y. Kawahigashi
Commun. Math. Phys. 289, 483–527 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0816-2
Communications in
Mathematical Physics
A Local Families Index Formula for ∂ -Operators on Punctured Riemann Surfaces Pierre Albin1, , Frédéric Rochon2, 1 Department of Mathematics, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, USA. E-mail: [email protected] 2 Department of Mathematics, University of Toronto, 100 st. George Street, Toronto, Ontario, MSS 363, Canada. E-mail: [email protected] Received: 25 June 2008 / Accepted: 6 January 2009 Published online: 26 April 2009 – © Springer-Verlag 2009
Abstract: Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of ∂-operators on the Teichmüller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space Mg,n in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Hyperbolic Cusp Operators . . . . . . . . . . . . . . . . . . . . . . 2. The Boundary Compactification of a Riemann Surface with Puncture 3. The ∂-Operator as a Dirac-Type hc-Operator . . . . . . . . . . . . . 4. The Teichmüller Space and the Teichmüller Universal Curve . . . . . 5. The Canonical Connection on the Universal Teichmüller Curve . . . 6. A Local Formula for the Family Index . . . . . . . . . . . . . . . . 7. The Spectral hc-Zeta Determinant . . . . . . . . . . . . . . . . . . . 8. The Curvature of the Quillen Connection . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The first author was partially supported by a NSF postdoctoral fellowship.
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The second author was supported by a postdoctoral fellowship of the Fonds québécois de la recherche
sur la nature et les technologies.
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Introduction Let X be a smooth even dimensional oriented compact manifold with boundary ∂ X = ∅. Assume that the boundary is the total space of a fibration Z
∂X Y
(1)
φ
where Y and Z are compact oriented manifolds, Y being the base and Z being a typical fibre. Let x ∈ C ∞ (X ) be a boundary defining function for X and c : ∂ X × [0, )x → N ⊂ X
(2)
a corresponding collar neighborhood of ∂ X in X . Let ghc be a metric on X \∂ X which takes the form c∗ ghc =
dx + φ ∗ gY + x 2 g Z x2
(3)
in the collar neighborhood (2), where g Z is a metric for the vertical tangent bundle of (1) and gY is a metric on the base Y which is lifted to ∂ X using a choice of connection for the fibration (1). Such a metric is called a product fibred hyperbolic cusp metric (product d-metric in the terminology of [31]). If the manifolds X , Y and Z are spin, one can construct a Dirac operator associated to the metric ghc . More generally, one can consider a Dirac type operator D constructed using the metric ghc and a Clifford module E → X with a choice of Clifford connection. In his thesis [31], Vaillant studied the index and the spectral theory of such operators. To do so, he introduced the conformally related operator x D and defined the vertical family by D V := x D|∂ X , which is a family of operators on ∂ X parametrized by the base Y and acting on each fibre of (1). Assuming that the rank of ker D V → Y is constant so that it is a vector bundle over Y (constant rank assumption), Vaillant also introduced a horizontal operator D H : C ∞ (Y ; ker D V ) → C ∞ (Y ; ker D V )
(4)
which governs the continuous spectrum of D with bands of continuous spectrum starting at the eigenvalues of D H and going out at infinity. In particular, the operator D is Fredholm if and only if D H is invertible. In that case, Vaillant was able to obtain a formula for its index using heat kernel techniques and Getzler’s rescaling along the lines of [22], 1 hc ) Ch(F E/Shc ) − gY ) ind(D) = η(D V ) − η(D H ), (5) A(R A(R 2 X Y the first term being the usual Atiyah-Singer integral, η(D V ) being the Bismut-Cheeger eta form of the vertical family and η(D H ) being the eta invariant of D H . In [2], the authors, inspired by the work of Melrose and Piazza in [24 and 25], generalized the formula of Vaillant to families of Dirac type operators. Via the use of Fredholm perturbations, a notion intimately related to spectral sections, it was also possible to study situations where the constant rank assumption is not satisfied, allowing among other things to generalize the index theorem of Leichtnam, Mazzeo and Piazza [20].
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The present paper, which is a sequel to [2], intends to put into use the index formula of [2] to study the following fundamental example arising in Teichmüller theory. Assume that 2g + n ≥ 3 and let Tg,n be the Teichmüller space of Riemann surfaces of genus g with n punctures. Let p : Tg,n → Tg,n be the Teichmüller universal curve whose fibre above [] ∈ Tg,n is the corresponding Riemann surface of genus g with n punctures. i, j i, j Let Tv Tg,n → Tg,n be the (i, j) vertical tangent bundle and let v be its dual. In 1,0 particular, K v := v restricts on each fibre to the corresponding canonical line bundle K := 1,0 . For each ∈ Z, one can associate a family of ∂-operators ∂ : C ∞ (Tg,n ; K v ) → C ∞ (Tg,n ; 0,1 v ⊗ Kv )
(6)
acting fibre by fibre on p : Tg,n → Tg,n and parametrized by the base Tg,n . By the uniformization theorem for Riemann surfaces, each fibre of p : Tg,n → Tg,n comes equipped with a hyperbolic metric g . Compactifying each fibre by a compact Riemann surface with boundary, these metrics can be seen as product hyperbolic cusp metrics, the fibration structure on the boundary being the collapsing map onto a point. With these metrics, the family ∂ can be interpreted as a family of Dirac-type operators associated to a family of product hyperbolic cusp metrics. Using the criterion of Vaillant [31], one can check that each member of the family is Fredholm. The formula of [2] therefore applies. As described in [34], the fibration p : Tg,n → Tg,n is equipped with a canonical connection. This allows one to interpret the formula of [2] at the level of forms. In general, the eta forms involved in this formula are quite hard to compute. However, in this specific example, an explicit computation is possible using a result of Zhang [36], the vertical family being defined on a circle fibration. The main result of this paper, Theorem 1, gives the following local family index formula: 1 n Ch(Ind(∂ )) = Ch(Tv− (Tg,n )) Td(Tv Tg,n ) + sign( − ) 2 2 Tg,n /Tg,n N n ∞ 2 ∂AtD −(At )2 1 1 1 D ei − − e d Str − dt, (7) √ ei ∂t 2 tanh 2π −1 0 2 i=1 where AtD is the Bismut superconnection and N is the number operator in Tg,n . To define the form ei , let Li → Tg,n be the complex line bundle which at [] ∈ Tg,n is given by the restriction of K v at the i th puncture (marked point) of := p −1 ([]). Then ei is the Chern form of Li as defined by Wolpert [35]. Since the Teichmüller space Tg,n is contractible, formula (7) only contains cohomological information in its degree zero part. However, since it is local and each of its terms is invariant under the action of the Teichmüller modular group Mod g,n , formula (7) also holds on the moduli space Mg,n = Tg,n / Mod g,n in the sense of orbifolds, where the fibration p : Tg,n → Tg,n is replaced by the forgetful map πn+1 : Mg,n+1 → Mg,n and where it acquires a topological meaning in higher degrees (see Corollary 6.7). For instance, on the moduli space Mg,n , the Chern form ei represents the Miller class ψi = c1 (Li ), while the first term on the left-hand side of (7) represents a linear combination of the Mumford-Morita classes j+1
κ j = (πn+1 )∗ (c1 (ψn+1 )),
j ∈ N0 .
(8)
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This formula could be thought of as a local version of the Grothendieck-Riemann-Roch theorem applied to the forgetful map πn+1 : Mg,n+1 → Mg,n and a certain sheaf on Mg,n+1 depending on (when = 0, it is the sheaf of sections of the trivial line bundle). When = 0 or = 1, our formula agrees modulo boundary terms with the one obtained by Bini [6] using the Grothendieck-Riemann-Roch theorem. Our results should be compared with the result of Takhtajan and Zograf [30] and Wolpert [35], who gave the two form part of (7) by interpreting it as the first Chern form of the corresponding determinant line bundle equipped with the Quillen connection. As in the compact case, the definition of the Quillen connection makes use of the determinant of the Laplacian. However the presence of cusps induces continuous spectrum for the Laplacian and the usual definition of its determinant via zeta-regularization is necessarily more delicate. Takhtajan and Zograf sidestepped this issue by defining the determinant in terms of the Selberg zeta function, in analogy with the compact case [13,29]. The precise description of the heat kernel in [31] allows us to proceed along the lines of [12,22,28] and extend the zeta function definition to this context via renormalization. Unlike previous efforts (see, e.g., [14,15 and 26]) this definition does not make use of the hyperbolic structure of the underlying manifold and works more generally for the metrics considered in [2,31]. Furthermore we show that, for hyperbolic metrics on surfaces with cusps, the resulting zeta-regularized determinant coincides with that defined using the Selberg zeta function up to a universal constant (see Theorem 2 and Corollary 7.5)
α,g,n Z (), ≥ 2; det ( ) = (9) (1), = 0, 1; α,g,n Z when ≥ 0, where α,g,n is a constant only depending on , g and n. With this determinant and thanks to the fact ker ∂ is a holomorphic vector bundle on Tg,n , the construction of the Quillen connection and the computation of the curvature are essentially as in [5,8] with only minor changes. In this way, we recover the index formula of [30] (see also [32 and 35]), √ −1 Q 2 1 (∇ ) = Ch T − (Tg,n /Tg,n ) · Td T Tg,n /Tg,n 2π 2πi Tg,n /Tg,n [2]
n ei , − 12 i=1
see Theorem 3 and Corollary 8.5 below. Our approach and the one of Takhtajan and Zograf [30] use substantially the fact that the dimension of the kernel of the family ∂ does not jump, so that these kernels fit together into a vector bundle on the Teichmüller space. More generally, one can ask if the work of Bismut, Gillet and Soulé [9–11] for the determinant of ∂-operators arising on (compact) Kähler fibrations could be adapted to non-compact situations in order to deal with examples where the rank of the kernel jumps. The paper is organized as follows. In Sect. 1, we review the definition and main properties of hyperbolic cusp operators. In Sect. 2, we explain the passage from a punctured Riemann surface to a compact Riemann surface with boundary. In Sect. 3, we describe how the ∂-operator on a punctured Riemann surface can be seen as a Dirac type hyperbolic cusp operator. We also check that Vaillant’s formula (5) agrees with the
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Riemann-Roch theorem in this case. In Sect. 4 and Sect. 5, we make a quick review of Teichmüller theory from our perspective. We then obtain our main result in Sect. 6 by computing the eta forms appearing in the family index formula of [2]. We also compare our formula with the Grothendieck-Riemann-Roch theorem. In Sect. 7, we study the determinant of various Laplacians on Riemann surfaces of finite area and relate them to Selberg’s zeta function following [12]. Finally, in Sect. 8, we adapt the standard computation of the curvature of the Quillen connection to our context and compare our result with those of Takhtajan-Zograf [30], Weng [32] and Wolpert [35]. 1. Hyperbolic Cusp Operators Let X be a smooth compact manifold with boundary ∂ X = ∅. Let x ∈ C ∞ (X ) be a boundary defining function, that is, x is a positive function in the interior vanishing on the boundary such that its differential d x is nowhere zero on ∂ X . For > 0 sufficiently small, there is an induced collar neighborhood of ∂ X in X , c : ∂ X × [0, )x → N := { p ∈ X | x( p) < } ⊂ X.
(1.1)
Consider a Riemannian metric ghc in the interior X \∂ X taking the form c∗ ghc =
dx2 + x 2 π L∗ g∂ X x2
(1.2)
in the collar neighborhood (1.1), where g∂ X is a Riemannian metric on ∂ X and π L : ∂ X ×[0, )x → ∂ X is the projection on the left factor. Such a metric is called a product hyperbolic cusp metric (or product d-metric in the terminology of Vaillant [31]). This is a complete metric on the interior of X , hence the boundary ∂ X is at infinity. Notice however that the volume of X is finite with respect to the metric ghc . Following the philosophy of Melrose, one can get operators that are adapted to this geometry at infinity by considering the space of hyperbolic cusp vector fields Vhc (X ), that is, the space of smooth vector fields on X with length uniformly bounded with respect to the metric ghc , Vhc (X ) := {ξ ∈ C ∞ (X ; T X ) | ∃ c > 0 such that ghc (ξ( p), ξ( p)) < c ∀ p ∈ X \∂ X }.
(1.3)
If z = (z 1 , . . . , z n−1 ) are local coordinates on ∂ X , then in the collar neighborhood (1.1), a hyperbolic cusp vector field ξ takes the form ∂ bi ∂ + , ∂x x ∂z i n−1
ξ = ax
(1.4)
i=1
where a, b1 , . . . , bn−1 are smooth functions on X . It is possible to define a vector bundle X on X in such a way that its space of smooth sections is canonically identified with hyperbolic cusp vector fields, hc T
C ∞ (X ; hc T X ) = Vhc (X ).
(1.5)
In the interior X \∂ X , the vector bundle hc T X is isomorphic to the tangent bundle T X . This identification does not extend to an isomorphism on the boundary of X . The metric ghc naturally induces a metric on hc T X which is also well-defined on the boundary.
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A quick check indicates that Vhc (X ) is not closed under the Lie bracket. To define higher order hyperbolic cusp operators, it is convenient to consider the conformally related metric gcu :=
1 ghc . x2
(1.6)
The metric gcu is called a product cusp metric. One can consider the corresponding cusp vector fields Vcu (X ) := xVhc (X ) = {ξ ∈ C ∞ (X ; T X ) | ∃ c > 0 such that gcu (ξ( p), ξ( p)) < c ∀ p ∈ X \∂ X }.
(1.7)
Alternatively, one can define cusp vector fields by Vcu (X ) := {ξ ∈ C ∞ (X ; T X ) | ξ x ∈ x 2 C ∞ (X )},
(1.8)
which makes it clear that the definition only depends on the choice of boundary defining function x and not on the choice of metric gcu . There is also an associated vector bundle cu T X over X whose space of smooth sections is canonically identified with the space of cusp vector fields, C ∞ (X ; cu T X ) = Vcu (X ).
(1.9)
In the collar neighborhood (1.1), a cusp vector field ξ has to be of the form ∂ ∂ + bi ∂x ∂z i n−1
ξ = ax 2
(1.10)
i=1
with a, b1 , . . . , bn−1 ∈ C ∞ (X ). As opposed to Vhc (X ), the space Vcu (X ) is closed under the Lie bracket, so that it is naturally a Lie algebra. Its corresponding universal enveloping algebra is the space Diff ∗cu (X ) of cusp differential operators. In the collar neighborhood (1.1), a cusp differential operator of order k, P ∈ Diff kcu (X ), takes the form l α ∂ 2 ∂ P= pl,α x , pl,α ∈ C ∞ (X ). (1.11) ∂x ∂z l+|α|≤k
More generally, Mazzeo and Melrose in [21] defined the space of cusp pseudodifferk (X ). These operators are closed under composition, ential operators of order k, cu k l k+l (X ) ◦ cu (X ) ⊂ cu (X ). cu
(1.12)
There is a corresponding cusp Sobolev space of order m ∈ N0 , m m (X ) := { f ∈ L 2gcu (X ) | P f ∈ L 2gcu (X ) ∀ P ∈ cu (X )}. Hcu
(1.13)
m (X ) by some power x k of the boundOne can also consider its weighted version x k Hcu m (X ) then defines a ary defining function. A cusp pseudodifferential operator P ∈ cu bounded linear map l l−m P : x k Hcu (X ) → x k Hcu (X ).
(1.14)
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One interesting feature of the cusp operators is that if a cusp pseudodifferential P ∈ m (X ) is invertible as a bounded linear map (1.14), then its inverse is given by a cusp operator of order −m. Generalizing the relation Vhc (X ) = x1 Vcu (X ), one can define the space of hyperbolic cusp pseudodifferential operators of order m by m m hc (X ) := x −m cu (X ).
(1.15)
m (X ) naturally induces a bounded linear map A hyperbolic cusp operator P ∈ hc l l−m P : x k Hcu (X ) → x k−m Hcu (X ).
(1.16)
So far we have considered operators acting on functions on X , but if E → X and F → X are complex vector bundles on X , it is no more difficult to define the space of ∗ (X ; E, F) acting from sections of E to sections of F. hyperbolic cusp operators hc In [21], Mazzeo and Melrose gave a very elegant criterion to determine when a cusp operator is Fredholm. They first introduced a notion of principal symbol adapted to the geometry at infinity, that is, involving the cosphere bundle S ∗ (cu T X ) of cu T X , k σk : cu (X ; E, F) → C ∞ (S ∗ (cu T X ); hom(π ∗ E, π ∗ F)),
(1.17)
k (X ; E, F) where π : S ∗cu T X → X is the bundle projection. A cusp operator A ∈ cu is said to be elliptic if its principal symbol σk (A) is invertible. In that case, by a standard k (X ; F, E) such that construction, one can obtain a parametrix B ∈ cu −∞ B A − Id E ∈ cu (X ; E),
−∞ AB − Id F ∈ cu (X ; F).
(1.18)
−∞ (X ; E) are not compact in general, this does not insure However, since elements of cu that the operator A is Fredholm. One needs some extra decay at infinity for the error term to be compact. Precisely, the subset of compact operators in −∞ (X ; E) is given by x −∞ (X ; E). It is possible to insure the error term is in that subset provided A is ‘invertible at infinity’. This condition is determined by the normal operator map k k N : cu (X ; E, F) → sus (∂ X ; E, F),
(1.19)
k (∂ X ; E, F) is the space of suspended operators of order k introduced by where sus Melrose in [23]. These are operators on ∂ X × R which are translation invariant in the R direction. Essentially, the normal operator N (A) of A is its asymptotically translation invariant part at infinity. The criterion of Mazzeo and Melrose can now be stated as follows. k (X ; E, F) is Fredholm Proposition 1.1. (Mazzeo-Melrose). A cusp operator A ∈ cu if and only if it is elliptic and its normal operator N (A) is invertible.
For hyperbolic cusp operators, the situation is much more delicate. For simplicity, 1 (X ; E, F). let us restrict to a first order hyperbolic cusp differential operator ðhc ∈ hc 1 Then xðhc ∈ cu (X ; E, F) is a cusp operator and we can use Proposition 1.1 to determine whether or not xðhc is Fredholm. If it is Fredholm, then it is not hard to see that ðhc is Fredholm as well. In fact, in that case, the spectrum of ðhc is then necessarily discrete −1 (X ; F, E) is a compact operator. since its parametrix in xcu However, even if xðhc is not Fredholm, it is still possible for ðhc to be Fredholm. Define the vertical family of ðhc to be V := (xðhc )|∂ X ∈ 1 (∂ X ; E, F). ðhc
(1.20)
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When ðhc is a self-adjoint Dirac type operator with E = F a Clifford bundle, the vertical V is invertible if and only if the normal operator N (xð ) is invertible. In his family ðhc hc thesis [31], Vaillant gave the following criterion to determine if a Dirac-type self-adjoint operator ðhc is Fredholm. The vertical family does not have to be invertible, but if it is H acting on the finite dimensional vector space not, Vaillant defined another operator ðhc V and called it the horizontal family. If denotes the projection from K := ker ðhc 0 2 L (∂ X ; E) onto K, then the horizontal family is defined by extending an element ξ ∈ K into the interior to an element ξ ∈ C ∞ (X ; E) and then applying ðhc and 0 , H ðhc ξ ∂ X . ξ := 0 ðhc
(1.21)
In his thesis [31], Vaillant gave the following criterion: 1 Proposition 1.2 (Vaillant [31], Sect.3). A Dirac type self-adjoint operator ðhc ∈ hc H (X ; E) is Fredholm if and only if ðhc is invertible. Moreover, the continuous spectrum H with bands of continuous spectrum starting at the eigenvalues of ðhc is governed by ðhc H of ðhc and going to infinity.
2. The Boundary Compactification of a Riemann Surface with Puncture Let be a Riemann surface of type (g, n), that is, = \ {x1 , . . . , xn }, where is a compact Riemann surface of genus g and x1 , . . . , xn are pairwise distinct points on . We will assume that 2g +n ≥ 3. The surface is a compactification of . An alternative way of compactifying the Riemann surface is to consider the radial blow up b of at the points {x1 , . . . , xn } with blow-down map β : b → .
(2.1)
This gives a compactification of in which each puncture is replaced by a circular boundary. The Riemann surface with boundary b also comes equipped with a natural choice of boundary defining function ρ ∈ C ∞ (b ) as we will see. This choice is dictated by the uniformization theorem for Riemann surfaces. Recall that, by the uniformization theorem, there is a canonical hyperbolic metric g on obtained by taking the unique metric of constant scalar curvature equal to −1 in the conformal class defined by the complex structure of . Consider the upper-half plane H = {x + i y ∈ C | y > 0}
(2.2)
equipped with the Poincaré metric gH :=
d x 2 + dy 2 . y2
(2.3)
Let ∞ be the discrete Abelian group generated by the parabolic isometry z → z + 1. The horn is the quotient H := ∞ \H.
(2.4)
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Via the change of variable r = 1y , one sees that the horn is isometric to (0, +∞)r × R/Z equipped with the metric dr + r 2d x 2. r2
(2.5)
A cusp end is a subspace of H of the form (0, a] × R/Z. Near a puncture xi of , the geometry of (, g ) is modeled on a cusp end. That is, around each puncture xi , there exists a neighborhood Ni ⊂ and an isometry ϕi : Ni → Ci with a cusp end Ci = (0,
1 yi
(2.6)
] × R/Z. Each cusp end has a natural compactification 1 C i = 0, × R/Z, (2.7) yi ri
where the coordinate ri can be seen as a boundary defining function for the boundary {0} × R/Z ⊂ C i . This boundary defining function can in fact be defined intrinsically in terms of the hyperbolic metric (2.5). Indeed, we define a horocycle to be an embedded circle in a cusp end which is perpendicular to all geodesics emanating from the cusp. This definition is formulated purely in terms of the metric. On the other hand, as one can check, the horocycles are precisely given by the level sets of the function ri . Moreover, the value of the function ri on a horocycle γ = {u} × R/Z is also determined by the hyperbolic metric. It is the area of the smaller cusp end (0, u) × R/Z, namely u ri (u, v) = area((0, u) × R/Z) = dr d x = u. (2.8) 0
R/Z
Thus, intuitively, the boundary defining function ri is the ‘area function’ for the cusp end Ci . The compactification C i induces a corresponding compactification N i via the isometry (2.6), and thus a compactification hc of into a compact surface with boundary naturally diffeomorphic to b . To get a global boundary defining function, choose a smooth non-decreasing function χ ∈ C ∞ ([0, +∞)) such that
x, if 0 ≤ x ≤ 21 , (2.9) χ (x) := 1, if x ≥ 1, and consider χ (x) := χ ( x ) for 0 < < min{ y11 , . . . ,
1 yn }.
On each (compactified)
cusp end C i , consider the function χ (ri ). Then the function
ϕi∗ (χ ◦ ri )(σ ), if σ ∈ N i , i ∈ {1, . . . , n}, ρ, (σ ) := , otherwise,
(2.10)
is a boundary defining function for ∂hc in hc . Since the choice of the number is not of primary importance, we will usually denote the function ρ, simply by ρ . With respect to this boundary defining function, the hyperbolic metric g is a product hyperbolic metric. That is, in the coordinates (x, ρ ) on Ni , it is of the form g = near the boundary.
2 dρ 2 ρ
2 + ρ dx2
(2.11)
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3. The ∂-Operator as a Dirac-Type hc-Operator Let K := 1,0 denote the canonical line bundle on . This line bundle and all of its tensor powers K have natural holomorphic structures. In particular, for each ∈ Z, there is a well-defined ∂ operator ∂ : C ∞ (; K ) → C ∞ (; 0,1 ⊗ K ),
(3.1)
where 0,1 → is the bundle of (0, 1)-forms on . In a cusp end Ci , where the canonical line bundle is trivialized by the holomorphic section dz, it takes the form 1 ∂ ∂ ∂ = (d x − idy) +i dz ∂z 2 ∂x ∂y 1 dr 1 ∂ 2 ∂ − ir , r= , = (d x + i 2 ) r 2 ∂x ∂r y dr 1 ∂ ∂ 1 rdx + i − ir . (3.2) = 2 r r ∂x ∂r Thus, near the boundary ∂hc , the ∂-operator is of the form 1 ∂ ∂ 1 dρ ρ d x + i − iρ . ∂= 2 ρ ρ ∂ x ∂ρ
(3.3)
Since ρ1 ∂∂x − iρ ∂ρ∂ is a hc-operator and 21 (ρ d x + i dρ ρ ) is naturally a section of hc T ∗ ⊗ C, we see that the ∂ -operator naturally extends to give a hc-operator R
∂ : C ∞ (hc ; hc K ) →
1 ∞ hc C (hc ; hc 0,1 ⊗ K ), ρ
(3.4)
hc hc hc ∗ where hc 0,1 is the complex conjugate of K and K ⊂ T ⊗R C is such that it is identified with K in the interior of hc and it is trivialized by the section ρ dz = ρ d x − i dρ ρ near each connected component of the boundary. The metric g induces
hc hc 0,1 a Hermitian metric on K and 0,1 , as well as on K and . We denote by H,i hc hc the Hilbert space of square integrable sections of K ⊗ ( 0,1 )i with respect to the natural scalar product f 1 , f 2 H,i := f 1 (σ ), f 2 (σ )g dg (σ ), (3.5) hc
where dg is the natural extension of the volume form of g on hc . The operator ∂ is Fredholm. To see this, recall (see for instance Proposition 3.67 in [5]) that √ ∗ D := 2(∂ + ∂ ) (3.6) is a Dirac type operator induced by the Chern connection on K with Clifford action on ν ∈ C ∞ (hc ; hc ) given by √ c( f )ν = 2(ε( f 0,1 ) − ι( f 1,0 ))ν, f ∈ C ∞ (hc ; hc ), (3.7)
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where ε( f 0,1 ) denotes exterior multiplication by f 0,1 . The operator D is formally self-adjoint. The vertical family DV of D is given by c(du)
∂ ∂u
(3.8)
hc acting on C ∞ (R/Z; hc K ⊕ hc 0,1 ⊗ K ) on each circular boundary component of hc , ∂ where u = −x is such that { ∂u σ } is an oriented orthonormal basis of Tσ ∂hc for each σ ∈ ∂hc . In particular, K = ker(DV ) is a complex vector space of dimension 2n. By Proposition 1.2, we need to show that the horizontal family DH : K → K is invertible.
Proposition 3.1. On each circular boundary component of hc , the horizontal family is given by 1 H ic(du). D = − 2
Proof. The bundle on which D acts is C ⊕ 0,1 ⊗ K . Choose a spin structure on and let S be the corresponding spinor bundle. It is wellknown (see for instance [19]) that, seen as a complex line bundle, S is a square root of the canonical line bundle K so that S ⊗C S = K . Moreover, we have also that C ⊕ 0,1 ∼ = S ⊗R S ∗ . Thus the operator D acts on S ⊗R S ∗ ⊗C K , which means that D is a Dirac operator twisted by the bundle S ∗ ⊗C K . As a bundle with connection, the bundle S ∗ ⊗ K certainly does not have a product structure near the boundary since it has non-zero curvature. Thus, according to Proposition 3.15, p.44 in [31], the horizontal family DH at each cusp is given by
∂ −i Rc ∂u
= −i
1 ∂ − c , 2 ∂u
where i Rdg is the curvature of the complex vector bundle S ∗ ⊗C K (cf. (3.15)). Collecting the contributions at each cusp end, we get the desired result. This gives the following corollary.
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Corollary 3.2. The operators D =
√
2
∗
0 ∂ ∂ 0
∗
, ∂ , and ∂
are Fredholm. Notice that Proposition 3.1 is also consistent with the well-known fact that the band of 2 continuous spectrum of the Hodge Laplacian D2 starts at 21 − and goes to infinity. In his thesis [31], Vaillant obtained a general formula for the index of a Dirac type operator on a fibred hyperbolic cusp operator. For the index of the operator ∂ , this formula is given by the usual Atiyah-Singer integrand together with two corrections coming from the boundary, namely the eta invariants associated to the vertical family of ∂ and the horizontal family DH , 1 1 ind(∂ ) = Ch(hc K ) Td(hc K −1 ) − η(DV ) − η(DH ). (3.9) 2 2 hc The eta invariant of the vertical family is easily seen to be zero. This is because modulo standard identifications, η(DV ) corresponds to n times the eta invariant of the self-adjoint operator ∂ 1 ∂ =i : C ∞ (R/Z) → C ∞ (R/Z). i ∂x ∂u But the spectrum of
1 ∂ i ∂x
η(
(3.10)
is 2π Z and its eta functional
1 ∂ , s) = 2π k|2π k|−s , Re s >> 0 i ∂x
(3.11)
k=0
is identically zero. Thus its spectral asymmetry or eta invariant, which is the value at s = 0 of the analytic continuation of η(i ∂∂x , s), is zero. The corresponding eta invariant η(DV ) = nη(i ∂∂x ) therefore vanishes. For the computation of the spectral asymmetry of DH , there is no regularization involved since DH is just an endomorphism of a finite dimensional vector space. From Proposition 3.1, we compute directly (see [2, (4.14)]) that 1 H . (3.12) η(D ) = n sign − 2 The index is therefore given by 1 n ind(∂ ) = − . Ch(hc K ) Td(hc K −1 ) + sign hc 2 2
(3.13)
The integral is also easy to compute. Let denote the curvature of hc T 1,0 . Then the integrand is given by i i 2π hc hc 1,0 − 2π Ch( K ) Td( T ) = e i 1 − e− 2π 1 i = 1+ − . (3.14) 2 2π
Local Families Index for ∂-Operators
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By a standard computation (see for instance p.77 in [16]), we know that i κ 1 = dg = − dg , 2π 2π 2π
(3.15)
where κ = −1 is the Gaussian curvature of g . By the Gauss-Bonnet theorem applied to , we get that 1 1 κ n ind(∂ ) = − dg + sign − 2 2 2 2π hc n 1 1 − χ () + sign − = 2 2 2 n 1 1 − (2 − 2g − n) + sign − . (3.16) = 2 2 2 This gives the following formula. Proposition 3.3. The index of ∂ is given by
(2 − 1)(g − 1) + n, ≤ 0, ind(∂ ) = (2 − 1)(g − 1) + ( − 1)n, > 0. In fact, using the Riemann-Roch theorem on the compact Riemann surface , it is also possible to compute explicitly the dimension of the kernel and the cokernel of ∂ (cf. p. 404 in [30]). By definition, an element of f ∈ ker ∂ is a holomorphic section of K , so in each cusp end N j , it has a Laurent series expansion ∞
f (z) =
( j)
ak e2πikz (dz) .
(3.17)
k=−∞
When > 0, this expansion has to be of the form f (z) =
∞
( j)
ak e2πikz (dz)
(3.18)
k=1
in order for f to be an element of H,0 . Such an f is said to be a cusp form of weight (2, 0). When ≤ 0, we can also have a constant coefficient in the series, f (z) =
∞
( j)
ak e2πikz (dz) .
(3.19)
k=0
When = 0, using the coordinate ζ := e2πi z near each puncture x j in , we see that such a f naturally extends to give a holomorphic function on . It is therefore constant, so that dimC ker ∂ 0 = 1. When ≥ 1, the section f takes the form f (z) =
∞ k=1
( j)
ak ζ k
dζ 2πiζ
(3.20)
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in the coordinate ζ near the puncture x j . Thus, it naturally extends to a meromorphic
section of K → with poles of order not exceeding − 1 at each puncture x1 , . . . , xn and holomorphic elsewhere. Conversely, such a meromorphic section corresponds to an element of ker ∂ . We can thus compute dimC ker ∂ by applying the Riemann-Roch theorem on to the line bundle
LD ⊗ K ,
(3.21)
where L D is the holomorphic line bundle associated to the divisor n D= ( − 1)xi on .
(3.22)
i=1
This gives
dimC ker ∂ = h 0 (L D ⊗ K )
= h 0 (K ⊗ (L D ⊗ K )−1 ) + deg(L D ⊗ K ) − g + 1
= h 0 (K ⊗ (L D ⊗ K )−1 ) + n( − 1) + (2 − 1)(g − 1), (3.23) where h 0 (L) denotes the dimension of the space of holomorphic sections of the holomorphic line bundle L. Now we compute that deg(K ⊗ L −1 D ⊗K
−
) = −( − 1)(2g + n − 2).
(3.24)
When = 1, K ⊗ (L D ⊗ K )−1 is the trivial line bundle, so h 0 (K ⊗ (L D ⊗ K )−1 ) = 1 − in this case. When > 1, deg(K ⊗ L −1 D ⊗ K ) < 0 since we assume that 2g + n ≥ 3, − and therefore h 0 (K ⊗ L −1 D ⊗ K ) = 0. Finally, when < 0, elements of ker ∂ correspond to holomorphic sections of K with zeros of degree at least − at each puncture. These in turn correspond to the holomorphic sections of a holomorphic line bundle of negative degree (since 2g + n ≥ 3), so that ker ∂ = 0 in that case. Hence, we see that the dimension of the kernel of ∂ is given by ⎧ 0, ⎪ ⎨ 1, dim ker ∂ = ⎪ ⎩ g, (2 − 1)(g − 1) + n( − 1),
< 0, = 0, =1 l ≥ 2.
(3.25)
Comparing with the index (3.16), we also get that ⎧ −(2 − 1)(g − 1) − n, ⎪ ⎨ ∗ g, dim ker ∂ = 1, ⎪ ⎩ 0,
< 0, = 0, =1 l ≥ 2.
(3.26)
These formulas are consistent with Kodaira-Serre duality, which asserts in this case that ∗ ker ∂ ∼ = ker ∂ 1− .
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497
4. The Teichmüller Space and the Teichmüller Universal Curve So far we have assumed that the complex structure on was fixed. By changing the complex structure, one can get instead a family of ∂ operators. The universal case is obtained by considering all at once the moduli space of all complex structures on a surface of type (g, n), two complex structures being identified whenever there is a conformal transformation between them homotopic to the identity. It is called the Teichmüller space of Riemann surfaces of genus g with n punctures and is denoted Tg,n . It is a complex manifold of complex dimension 3g − 3 + n which can be identified with an open set of C3g−3+n . The Teichmüller space Tg,n comes together with a universal bundle, the universal Teichmüller curve Tg,n with bundle projection p : Tg,n → Tg,n
(4.1)
p −1 ([])
and fibre the Riemann surface of type (g, n) corresponding to the point i, j [] ∈ Tg,n . Denote by Tv Tg,n → Tg,n the vertical (i, j) tangent bundle of the fibrai, j tion (4.1) for i, j ∈ {0, 1}. On each fibre := p −1 ([]), the restriction of Tv Tg,n is i, j i, j canonically identified with T i, j . Denote by v → Tg,n the dual of Tv . On each fibre we also have a ∂-operator. These operators fit together to give a family of operators 1 0,1 1,0 ∂ ∈ ρ −1 cu (Tg,n /Tg,n ; (1,0 v ) , v ⊗ (v ) ),
(4.2)
where ρ is an appropriate boundary defining function (whose precise definition we postpone to (5.16)). Each element of the family is a Fredholm operator so that we have a family index in K 0 (Tg,n ), ind(∂ ) ∈ K 0 (Tg,n ).
(4.3)
Since the Teichmüller space is contractible, this families index really only encodes the numerical index of any member of the family under the identification K 0 (Tg,n ) ∼ = K 0 (pt) ∼ = Z. Still, it is possible to exhibit an explicit representative of the K -class ind(∂ ) ∈ K 0 (Tg,n ), providing in this way a local description of the family index. This is because, according to (3.25) and (3.26), the dimensions of the kernel and the cokernel of elements of the family ∂ are always the same (they only depend on , g and n, not on the complex structure). This means that ∗
ker ∂ → Tg,n and ker ∂ → Tg,n
(4.4)
form complex vector bundles on Tg,n and the family index of ∂ can then be expressed as the virtual difference of these two vector bundles, ∗
ind ∂ = [ker ∂ ] − [ker ∂ ] ∈ K 0 (Tg,n ).
(4.5)
In fact, as we will recall in a moment, these vector bundles both come equipped with a natural connection. We can therefore express their respective Chern characters at the level of forms. This provides a local description of the Chern character of the family index ∗
Ch(ind ∂ ) := Ch(ker ∂ ) − Ch(ker ∂ ) ∈ C ∞ (Tg,n , ev (Tg,n )).
(4.6)
On the Teichmüller space itself, this local description of the index does not contain more cohomological information than (3.16). However, the local descriptions (4.5) and (4.6) are invariant under the action of the Teichmüller modular group Mod g,n . This means that these local descriptions descend to the moduli space Tg,n / Mod g,n (in the sense of orbifolds), which typically has a non-trivial topology as well as singularities.
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5. The Canonical Connection on the Universal Teichmüller Curve The fibration p : Tg,n → Tg,n comes together with a canonical connection P. To describe this connection, one possible approach is to describe Riemann surfaces as certain quotients of the upper half-plane H. If is a Riemann surface of genus g with n punctures, then it can be represented as a quotient \H of the upper half-plane by the action of a torsion-free finitely generated Fuchsian group . The group ⊂ PSL(2, R) is of type (g, n), which is to say it is generated by 2g hyperbolic transformations A1 , B1 , . . . , A g , Bg and n parabolic transformations S1 , . . . , Sn satisfying −1 −1 −1 the single relation A1 B1 A−1 1 B1 · · · A g Bg A g Bg S1 · · · Sn = 1. Since H is simply connected, in fact contractible, it is the universal cover of under the quotient map H → \H. From this perspective, the canonical hyperbolic metric g associated to the (conformal structure of the) complex structure is precisely the metric on \H induced from the Poincaré metric gH :=
d x 2 + dy 2 on H. y2
(5.1)
The punctures of then correspond to the image of the fixed points z 1 , . . . , z n in R ∪ {∞} of the parabolic transformations S1 , . . . , Sn under the quotient map H → \H. Let i be the cyclic subgroup of generated by the parabolic transformation Si for i = 1, . . . , n. It can be identified with the cyclic group ∞ by choosing σi ∈ PSL(2, R) such that σi ∞ = z i , so that 1 ±1 −1 σi Si σi = , σi−1 i σi = ∞ . (5.2) 0 1 0,1 m On , sections of (1,0 ) ⊗ (( ) ) correspond to automorphic forms of weight (2, 2m) with respect to the group , that is, functions f : H → C such that m
f (γ z)γ (z) γ (z) = f (z) ∀z ∈ H, ∀γ ∈ .
(5.3)
For instance, the natural Kähler metric associated to the hyperbolic metric g , seen as 0,1 a section of 1,0 ⊗ , corresponds to the automorphic form of weight (2, 2) 1 on H. y2
(5.4)
In the correspondence between Riemann surfaces and quotients of H, a change of complex structure corresponds to a change of the Fuchsian group . This provides a canonical identification between the Teichmüller space Tg,n of Riemann surfaces of type (g, n) and the Teichmüller space of Fuchsian groups of type (g, n). Under this identification, the ∗ tangent space of Tg,n at [] can be identified with the subspace −1,1 () = ker ∂ −1 ⊂ H−1,1 of harmonic Beltrami differentials. Each element of µ ∈ −1,1 () has the form µ = y 2 ϕ for a unique ϕ ∈ ker ∂ 2 , so that dimC −1,1 () = 3g − 3 + n. In particular, an element of −1,1 () decays exponentially fast as one approaches a puncture (using ∗ T the coordinates of (2.7)). The (holomorphic) cotangent space T[] g,n can be identified −1,1 () via the pairing with ker ∂ 2 on , this space being naturally dual to (µ, ϕ) := µϕ, µ ∈ −1,1 (), ϕ ∈ ker ∂ 2 . (5.5)
Local Families Index for ∂-Operators
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To get complex coordinates on Tg,n we can use the fact that to every µ ∈ −1,1 () satisfying µ L ∞ = sup |µ(z)| < 1,
(5.6)
z∈
one can associate a unique diffeomorphism f µ : H → H satisfying the Beltrami equation ∂f µ ∂f µ =µ ∂z ∂z
(5.7)
and fixing the points 0, 1, ∞, where µ in (5.7) is seen as an automorphic form of weight (−2, 2) on H. From this solution, one gets a new Fuchsian group by considering µ := f µ ( f µ )−1 , that is, a new complex structure by considering the Riemann surface µ := µ \H. The diffeomorphism f µ also naturally descends to the quotient \H to give a diffeomorphism f µ : \H → µ \H.
(5.8)
Now, if one chooses a basis µ1 , . . . , µ3g−3+n of −1,1 () and sets µ = ε1 µ1 + · · · + ε3g−3+n µ3g−3+n , then the correspondence (ε1 , . . . , ε3g−3+n ) → [ µ ] defines complex coordinates in a neighborhood of [] ∈ Tg,n called Bers coordinates. In the overlapping of neighborhoods of two points [] and [ µ ], the Bers coordinates transform complex analytically (see for instance p. 409 in [30]), defining on Tg,n a complex structure. The Bers coordinates provide a local trivialization of the fibration p : Tg,n → Tg,n of the universal Teichmüller curve, in fact, of its universal cover, the Bers fibre space BFg,n (see p. 138 in [34]). If U ⊂ Tg,n is the open set where the Bers coordinates (ε1 , . . . , ε3g−g+n ) associated to [] are defined, then this trivialization is given by the commutative diagram ν / −1 U × J p (U) JJJ pr JJJ1 JJJ p J% U
(5.9)
where pr 1 is the projection on the first factor and ν is given by ν(µ, σ ) = f µ (σ ) ∈ p −1 ([ µ ]), where f µ denotes the map (5.8). This local trivialization also induces a lift of T[] Tg,n to T Tg,n p−1 ([]) , namely (see p. 142 in [34]), a vector µ ∈ T[] Tg,n has a canonical lift pr1∗ µ ∈ T (U × )|{[]}× , ∗ and therefore a canonical lift ν∗ (pr 1 µ) ∈ T Tg,n p−1 ([]) . More generally, introducing Bers coordinates at each [] ∈ Tg,n , we can get in this way a canonical horizontal lift of T Tg,n to T Tg,n . In other words, associated to the fibration p : Tg,n → Tg,n , there is a canonical connection P, that is, P ⊂ T Tg,n is a distribution of hyperplanes such that p∗ : Pz → T p(z) Tg,n
(5.10)
is an isomorphism for every z ∈ Tg,n . It is also possible to define a covariant derivative 0,1 m ∞ ∗ ∗ 1,0 0,1 m ∇ P : C ∞ (Tg,n ; (1,0 v ) ⊗ (v ) ) → C (Tg,n ; p (Tg,n ) ⊗ (v ) ⊗ (v ) ). (5.11)
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P. Albin, F. Rochon
0,1 m This allows one to differentiate sections of (1,0 v ) ⊗ (v ) with respect to vectors on the base Tg,n . At [] ∈ Tg,n , the differentiation can be described by using the Bers coordinates associated to T[] Tg,n ∼ = −1,1 () with the local trivialization (5.9) of 0,1 m p : Tg,n → Tg,n near []. In this trivialization, a section ω of (1,0 v ) ⊗ (v ) corre∗ 0,1 m sponds to a section ω of (pr ∗2 1,0 ) ⊗ (pr 2 ) on U × , where pr 2 : U × → is the projection on the second factor. Precisely, in terms of automorphic forms of weight (2, 2m), we have that µ µ m ∂f ∂f
ω(ε, σ ) = ω ◦ f µ , (5.12) ∂z ∂z
where µ = ε1 µ1 + · · · + ε3g−3+n µ3g−3+n . On = p −1 ([]) ⊂ Tg,n , there is a canonical 1,0 0,1 m 0,1 m identification between (1,0 v ) ⊗ (v ) and ( ) ⊗ ( ) . Under this identification, the covariant derivative of ω takes the form (cf. p. 409 in [30]), ∂ P ∇ ∂ ω =
ω(ε, σ ) . (5.13) ∂εi ∂εi p −1 ([]) ε=0 An important example is given by the family of fibrewise hyperbolic area forms dg , which as was shown in [1] gives a parallel section of 1,1 v with respect to the connection P, ∇ P dg = 0. This corresponds to the fact that the automorphic form of weight (2, 2) y12 is parallel with respect to the connection P. However, notice that this does not imply the family of hyperbolic metrics g , [] ∈ Tg,n is parallel with respect to P as a section of Tv∗ Tg,n ⊗ Tv∗ Tg,n . In fact, they cannot be parallel with respect to any connection, since otherwise this would mean that these metrics are all isometric, a contradiction since essentially by definition of the Teichmüller space, these metrics are not even conformal to one another. It is also possible to define the covariant derivative of families of operators using the connection P. If Aε : H,m ( µ ) → H ,m ( µ ) is such a family in the trivialization (5.9) given by the Bers coordinates, then the covariant derivative of Aε at [] is given by ∂ P ε µ ∗ ε µ∗ −1 ∇∂ A = (f ) A (f ) , ∂ε ∂εi i [] ε=0 ∂ ∇ P∂ Aε = ( f µ )∗ Aε ( f µ∗ )−1 . (5.14) ∂ε ∂ε i
[]
ε=0
i
For example, the covariant derivatives of ∂ and (2.6) in [30])
∗ ∂
at [] are given by (see formula
∗
∇µP ∂ = µ∂ +1 u, ∇µP ∂ = 0, ∗
∗
∇µP ∂ = 0, ∇µP ∂ = µ∂ −1 u −1 , where u :=
1 y2
0,1 is seen as a section of 1,0 ⊗ .
(5.15)
Local Families Index for ∂-Operators
501
As we have seen, each Riemann surface of type (g, n) has a boundary compactification hc constructed using the metric g . These compactifications fit together to give a fibrewise boundary compactification hc Tg,n of the universal Teichmüller curve. In terms of the local trivializations of (5.9), this is because the solution f µ to the Beltrami equation (5.7) is real analytic (see for instance Proposition 4.6.2 in [18]), it maps the fixed points of to the fixed points of µ and, seen as a map f µ : → µ , it is asymptotically holomorphic as one approaches any puncture of . Since the canonical connection P is obtained by using Bers coordinates and infinitesimal deformations induced by the solutions of the Beltrami equation (5.7), we see that it also naturally lifts to provide a canonical connection hc P to the fibration hc
p : hc Tg,n → Tg,n .
To get a natural boundary defining function for hc Tg,n , we use the construction of (2.10) in each fibre. This definition depends on the choice of a number > 0 which has to be chosen so that each cusp end Ni in a given surface has area strictly greater than . To get a global definition hc Tg,n , we should replace the number by a smooth function a : Tg,n → R+ such that in a given fibre := p −1 ([]), the area of each cusp end Ni is strictly greater than a([]). We can then define our global defining function on hc Tg,n to be ρ(σ ) = ρ,a([]) (σ ) for σ ∈ := p −1 ([]), [] ∈ Tg,n ,
(5.16)
where ρ, : hc → R is defined in (2.10) for the Riemann surface and a choice of small > 0. 6. A Local Formula for the Family Index hc The family of operators ∂ ∈ 1 (Tg,n /Tg,n ; hc K v , hc 0,1 v ⊗ K v ) is a particular example of the families of φ-hc operators considered in [2]. When we apply this local index theorem to our family ∂ with the canonical connection hc P for the fibration hc p : hc T g,n → Tg,n , we get the family version of (3.9), Ch(Tv− (Tg,n )) Td(Tv Tg,n ) − η(DV ) Ch(Ind(∂ )) =
Tg,n /Tg,n
− η(DH ) −
1 √
2π −1
N
2
∞
d
Str 0
∂AtD ∂t
e
−(AtD )2
dt,
(6.1)
where the eta invariants of the vertical and horizontal families are replaced by the corresponding eta forms of Bismut and Cheeger [7] (with non-standard Z2 grading for D H ). This is an equality at the level of forms. Notice that in [2] the first term is expressed in form. However, thanks to Theorem 5.5 in [34] and its reformulation in terms of the A Eq. 5.3 of [34], the fibration p : Tg,n → Tg,n is Kähler fibration (see [10] for a defintion) so that it is possible to rewrite the first term using the Todd form instead. In the last term, AtD is the rescaled Bismut superconnection while N is the number operator in Tg,n , that is, the action of N on forms of degree k on Tg,n is multiplication by k. Remark 6.1. In this paper, our convention for the Chern character differs from that of [5]. This is why we need to include these extra factors of 2πi in the last term. In principle, the eta forms would also require such factors, so really, by an eta form, we mean N (2πi)− 2 times the eta form of Bismut and Cheeger (cf. Eq. 4.101 in [7]).
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When we take the degree zero part of (6.1), we get back the numerical index (3.9) by evaluating it at a given point [] ∈ Tg,n . In fact, as we have seen, the degree zero part of η(DV ) is identically zero, while the degree zero part of η(DH ) is n2 sign( − 21 ). However, the higher degree components of η(DH ) vanish identically at the level of forms as we will see in a moment. Let ∂Tg,n := ρ −1 (0) be the union of the boundaries of the fibres of hc p : hc Tg,n → Tg,n . The map hc p induces a fibration structure ∂ p : ∂Tg,n → Tg,n
(6.2)
with typical fibre the disjoint union of n circles. In fact, the manifold ∂Tg,n has precisely n components, ∂Tg,n =
n
∂i Tg,n
(6.3)
i=1
with ∂i Tg,n the component associated to the i th cusp. There is a corresponding fibration structure ∂ pi : ∂i Tg,n → Tg,n . Recall that the vertical family DV decomposes as 0 DV,− V D = DV,+ 0
(6.4)
(6.5)
with respect to the Z2 grading of the Clifford bundle. Lemma 6.2. The Chern form of ker DV,+ → Tg,n vanishes in positive degrees, Ch(ker DV,+ )[2k] = 0, k ∈ N. V be the vertical family of the i th component ∂ T Proof. Let D,i i g,n of ∂Tg,n . Via the identification
−c(
dρ ): ρ
hc
hc hc 0,1 v ⊗ K v −→ K v
given by Clifford multiplication, the operator DV,+ can be identified with 1 ∇ ∂ = i∇ ∂ : C ∞ (R/Z; hc K v ) → C ∞ (R/Z; hc K v ), ∂u i ∂x
(6.6)
∂ where u = −x is such that ∂u is an oriented orthonormal basis of Tσ (∂i Tg,n /Tg,n ) V,+ for each σ ∈ ∂i Tg,n . Thus, ker D,i → Tg,n defines a complex line bundle over the Teichmüller space Tg,n and
ker DV,+ =
n i=1
V,+ ker D,i .
(6.7)
Local Families Index for ∂-Operators
503
For the corresponding Chern characters, this gives Ch(ker DV,+ ) =
n
V,+ Ch(ker D,i ).
(6.8)
i=1 V,+ )[2] = 0 for To prove the lemma, it therefore suffices to show that Ch(ker D,i
V,+ by a parallel i ∈ {1, . . . , n}. This will be true provided we can trivialize ker D,i
V,+ with (6.6), a choice of trivializing section is section. From the identification of D,i given by taking dρ s,i : [] → ρd x − i ∈ hc K v , (6.9) (∂hc )i ρ (∂hc )i
p −1 ([]) and ρ
where = is the boundary defining function of (5.16). Notice that the section (6.9) is completely determined by the canonical family of hyperbolic metrics gTg,n /Tg,n . Conversely, for = 1, the section s1,i completely determines the asymptotic behavior of gTg,n /Tg,n as one approaches the i th puncture. If the family of metrics gTg,n /Tg,n were parallel with respect to the canonical connection P, we could conclude immediately that the section s,i is parallel. This is not the case, but at least the family of metrics gTg,n /Tg,n is asymptotically parallel as one approaches a puncture. Indeed, from (5.15), we see that the parallel transport (along a path on Tg,n ) defined by the canonical connection P is asymptotically holomorphic as one approaches a puncture. This is because the Beltrami differential µ in (5.7) vanishes exponentially fast as one approaches a puncture (using the coordinates of (2.7)). Thus, parallel transport is asymptotically a conformal transformation for the family of metrics gTg,n /Tg,n . Since ∇ P dgTg,n /Tg,n = 0,
(6.10)
this means that the parallel transport defined by the connection P is asymptotically an isometry as one approaches a puncture. That is, ∇ P gTg,n /Tg,n is asymptotically zero as one approaches a puncture. In particular, this implies that for each i ∈ {1, . . . , n}, the section s,i of (6.9) is parallel with respect to the connection P. Together with the boundary defining function ρ, the family of metric gTg,n /Tg,n induces a natural family of metrics gi for each fibre of the fibration (6.4) in such a way that each fibre becomes isometric to the circle S1 := R/Z of length 1 (cf. [35]). With these identifications, we get a natural action of S1 on each fibre, giving (6.4) the structure of a principal S1 -bundle. By construction, the family of metrics gi is S1 -equivariant with respect to the S1 action. The canonical connection hc P naturally induces a connection Pi on (6.4). Lemma 6.3. The family of metrics gi is parallel with respect to the connection Pi , that is, the connection Pi is unitary with respect to the metric gi . In particular, on the i th cir∂ cular boundary component, the vector field ∂u is parallel with respect to the connection Pi . Proof. By the proof of Lemma 6.2, the family of metrics gTg,n /Tg,n is asymptotically parallel as one approaches a cusp, from which the result follows. We can now show that the eta form of DH vanishes in positive degrees.
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Lemma 6.4. For each k ∈ N, the degree 2k part of the form η(DH ) vanishes identically, η(DH )[2k] = 0, k > 0.
Proof. Since DH is just an endomorphism of ker DV , we see from Proposition 3.1, Lemma 6.3 and the definition of the eta form that (see [2, (4.12)]) η(DH )
1 1 Ch(ker DV,+ ). = sign − 2 2
The result then follows from Lemma 6.2.
(6.11)
On the other hand, the eta form of the vertical family gives a contribution in higher degrees. In fact, since the geometry of the boundary fibration is very special, it is possible to compute the eta form explicitly. With respect to the decomposition (6.3), the vertical family DV admits a corresponding decomposition DV =
n
V D,i ,
(6.12)
i=1 V is a family of self-adjoint Dirac operators on the fibration (6.4). In terms of where D,i this decomposition, the eta form of DV can be expressed as
η(DV ) =
n
V η(D,i ).
(6.13)
i=1
By (5.15) (see also the proof of Lemma 8.1), the family of Dirac operators D is asymptotically parallel with respect to the canonical connection P as one approaches a cusp. V is parallel with respect to the connecThis means that each of the vertical families D,i tion Pi on (6.4). This fact, together with the fact the family of metric gi is parallel with respect to the connection Pi and is equivariant with respect to the circle action, means that we can apply the result of Zhang (Theorem 1.7 in [36]) to get an explicit formula V ). for the eta form η(D,i V is given by Proposition 6.5 (Zhang, [36], Theorem 1.7). The eta form of D,i V η(D,i )=
1 1 ei − , e 2 tanh 2 i
√
−1 where ei := 2π i is the curvature form of the circle bundle ∂ pi : ∂i Tg,n → Tg,n with connection Pi and curvature i , the Lie algebra of S1 being identified with iR.
Remark 6.6. Notice in particular that this implies that the eta form is zero in degree 2k for k = 0 modulo 2. Moreover, it is a closed form, an unusual feature for an eta form.
Local Families Index for ∂-Operators
505
Before stating our main theorem, let us give an alternate description of the Chern form ei . Namely, to the circle bundle (6.4) with connection Pi and family of metrics (2π )gi , we can associate in a canonical way a complex line bundle Li → Tg,n equipped Li with a Hermitian metric h i and √a unitary connection ∇ in such a way that the curvature form of Li is precisely (−2π −1)ei . The line bundle Li is such that its unit circle bundle with induced metric and connection is precisely the circle bundle (6.4) with family of metrics 2πgi and connection Pi . Thinking of a fibre := p −1 ([]) as a punctured Riemann surface = − {x1 , . . . , xn },
(6.14)
one can also define the line bundle Li by
∗ , [] ∈ Tg,n . Li,[] := (Tx1,0 ) = K x i i
(6.15)
Moreover, from this perspective, the Hermitian metric h i and the unitary connection ∇ Li are easily seen to be the same as the one introduced by Wolpert [35]. Thus, the form ei corresponds to the Chern form c1 ( can,i ) of Corollary 7 in [35]. Now, combining (6.1) with Lemma 6.4 and Proposition 6.5, we obtain the following formula. Theorem 1. The local family index of the family of operators √ 1 hc D+ := 2 ∂ ∈ ρ −1 cu (Tg,n /Tg,n ; hc K v , hc 0,1 v ⊗ Kv ) associated to the Teichmüller universal curve p : Tg,n → Tg,n and its canonical connection P is given by 1 n + − − Ch(ind ker D ) = Ch(Tv (Tg,n )) Td(Tv Tg,n ) + sign 2 2 Tg,n /Tg,n t N ∞ n 2 ∂A D −(At )2 1 1 1 D − − e d Str − dt, √ ei ∂t 2 tanh e2i 2π −1 0 i=1
(6.16) where AtD is the rescaled Bismut superconnection associated to the family D , ei is the canonical Chern form of the (holomorphic) cotangent bundle along the i th cusp Li → Tg,n and N is the number operator on Tg,n . As in [30], each of the terms in our formula is invariant under the action of the Teichmüller modular group Mod g,n . Thus, formula (6.16) also holds on the moduli space Mg,n := Tg,n / Mod g,n in the sense of orbifolds with the fibration p : Tg,n → Tg,n replaced by the forgetful map πn+1 : Mg,n+1 → Mg,n . In fact, on the moduli space Mg,n , the formula acquires a topological meaning in higher degrees. To see this, define T g,n to be the space obtained from Tg,n by filling each puncture of each fibre by a marked point. There is still a fibration p : T g,n → Tg,n , but now with fibres being compact Riemann surfaces of genus g with n marked points. Let K v → Tg,n denote the corresponding vertical canonical line bundle (the dual of the vertical (1, 0) th tangent bundle). Let Di ⊂ T g,n be the divisor associated nto the i marked points and let L D be the line bundle associated to the divisor D := i=1 Di . Then, by analogy with
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P. Albin, F. Rochon
the discussion in Sect. 3, we see that the family index of ∂ is the same as the family index of the family of ∂-operators −1 ∞ 0,1 ∂ : C ∞ (Tg,n ; K v ⊗ L −1 D ) → C (Tg,n ; v ⊗ K v ⊗ L D )
(6.17)
for > 0 and ∂ : C ∞ (Tg,n ; K v ⊗ L D ) → C ∞ (Tg,n ; 0,1 v ⊗ K v ⊗ L D)
(6.18)
for ≤ 0. On the fibration πn+1 : Mg,n+1 → Mg,n , this corresponds to the following situation. Let ωπn+1 be the relative dualizing sheaf of this fibration, that is, the sheaf of sections of K v . Let ωπn+1 (D) be the logarithmic variant of ωπn+1 , which means that the local sections of ωπn+1 (D) are sections of ωπn+1 with possibly simple poles at the first n marked points. Then the line bundle K v ⊗ L D on Tg,n corresponds to the sheaf ωπn+1 (D) on Mg,n+1 . Going back to the formula of Theorem 1, we see that the form ei then represents the Miller class ψi = c1 (Li ). On the other hand, since the Miller class ψn+1 on Mg,n+1 is given by ψn+1 = c1 (ωπn+1 (D)) (see for instance p. 254 in [33]), the first term in the righthand side of (6.16) can be seen to represent a linear combination of the Mumford-Morita classes j+1 j+1 κ j := (πn+1 )∗ (ψn+1 ) = (πn+1 )∗ (en+1 ) , j ∈ N0 , (6.19) where en+1 is the Chern form of the vertical canonical line bundle K v ∼ = K v ⊗ L D . The precise formula involves the Bernouilli numbers Bm and the Bernouilli polynomials Bm (), which are defined by the following identities: x xm = , B m ex − 1 m! m≥0
ex x xm = . B () m ex − 1 m!
(6.20)
m≥0
Thus, the first term in (6.16) is seen to represent the cohomology class ψn+1 e ψn+1 κm−1 (πn+1 )∗ = . Bm () ψ n+1 e −1 m!
(6.21)
m≥1
On the moduli space, Theorem 1 therefore gives the following local formula (in the sense of orbifolds). Corollary 6.7. In the sense of orbifolds, the Chern character of the index of the family ∂ associated to the forgetful map πn+1 Mg,n+1 → Mg,n is given at the form level by Bm () 1 n km−1 + sign − Ch(ker D+ ) − Ch(ker D− ) = m! 2 2 m≥1 N ∞ n 2 ∂AtD −(At )2 1 1 1 D − e − d Str − dt, √ ei ∂t 2 tanh e2i 2π −1 0 i=1
(6.22) m+1 ) (πn+1 )∗ (en+1
where km = and ei are canonical form representatives of the Morita-Mumford-Miller classes κm and ψi .
Local Families Index for ∂-Operators
507
If Mg,n denote the Deligne-Mumford compactification of the moduli space Mg,n , then Theorem 1 can be intuitively interpreted as a local version of the GrothendieckRiemann-Roch theorem applied to the morphism πn+1 : Mg,n+1 → Mg,n and the sheaf
ωπn+1 (D)−1 ⊗ ωπn+1 , > 0,
ω := (6.23) ωπn+1 (D) , ≤ 0. In this context, the Grothendieck-Riemann-Roch theorem was first studied and used by Mumford [27] in the case n = 0 with formula given by Ch((π1 )∗ ωπ 1 ) =
Bm () κm−1 + (terms coming from ∂Mg ). m!
m≥1
When n > 0, a Grothendieck-Riemann-Roch formula was obtained for the sheaf ωπ n+1 by Bini [6], Ch((πn+1 )∗ ωπ n+1 ) =
Bm ()
κm−1 + (terms coming from ∂Mg,n ), m!
(6.24)
m≥1
where κm := (πn+1 )∗ c1 (ωπn+1 )m+1 . When = 0 and = 1, it makes sense to compare our formula with the one of Bini. In that case, using the relation κm = κm +
n
ψim
i=1
proved by Arbarello and Cornalba [4] together with the identity x 2 tanh
x 2
=
x x + , ex − 1 2
we can easily check that, as expected, our formula agrees with the interior contribution of (6.24). 7. The Spectral hc-Zeta Determinant On any geometrically finite hyperbolic surface = \H, the Selberg’s zeta function is defined for Re (s) > 1 to be ∞ Z (s) = 1 − e−(s+k)(γ ) ,
(7.1)
{γ } k=0
where the outer product goes over conjugacy classes of primitive hyperbolic elements of and (γ ) is the length of the corresponding closed geodesic. On closed hyperbolic surfaces, a well-known result of D’Hoker and Phong [13] says that the determinant of the Laplacian , acting on sections of K can be expressed in terms of special values of the Selberg’s Zeta function, det( , ) = Z ()e−c−1 χ () , ≥ 2, (1)e−c0 χ () , = 0, 1, det ( , ) = Z
(7.2)
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P. Albin, F. Rochon
where c :=
1 2 (2 − 2m − 1) log(2 − m) − + 2 1
0≤m<− 2
1 log 2π + 2ζ Riem (−1). + + 2
(7.3)
Shortly after, it was shown by Sarnak [29] that for the geometric Laplacian with nonnegative spectrum , −χ () 2 det ( + s (s − 1)) (s) 2 = e E−s(s−1) , (7.4) (2π )s Z (s) (s) where E = − 41 − 21 log 2π + 2ζ Riem (−1), 2 is the Barnes double Gamma function. As indicated in [29], the formula of D’Hoker and Phong can be recovered relatively easily from (7.4). On a Riemann surface with cusps, the Selberg Zeta function as defined above still makes sense. However, since the Laplacian has a continuous spectrum, the definition of its determinant is more subtle. It was studied by Efrat [14,15] and by Müller [26] using scattering theory to understand the contribution from the continuous spectrum. In this paper, we use renormalized integrals to extend the usual definition of the determinant via zeta-regularization to these manifolds, with the advantage that this does not require the metric to have constant curvature. We then use the analysis of [12] to show that, on hyperbolic surfaces, our definition satisfies (7.4) with the right-hand-side replaced by a meromorphic function depending only on the genus and the number of punctures, an important feature for our purposes.
7.1. The determinant of the Laplacian. To relate the determinant with the Selberg Zeta function and get an analog of formula (7.4), it is convenient to work first with the (positive) geometric Laplacian instead of the ∂-Laplacian. Recall that the two are the same modulo a multiplicative constant, ∂ =
1 . 2
Following [22, Sect. 9.5] and [17, Sect. 3], we define the zeta function of using the renormalized trace (see e.g., [2]) ∞ dt 1 ζ (z) := . t z R Tr e−t − Pker (z) 0 t Since is Fredholm, zero is spectrally isolated and the integrand decays exponentially for large times. Thus the integral defines a holomorphic function for Re(z) large enough. The small-times asymptotics of the integrand (whose existence follows from the construction of the heat kernel in [31]) allow us to extend the function meromorphically to the whole complex plane. We denote the meromorphic extension by the same symbol and define log det := −ζ (0) .
Local Families Index for ∂-Operators
509
We can find a more explicit expression for the zeta function by subtracting the first few terms in the expansion of the heat kernel at t = 0. The form of this expansion can be deduced for arbitrary φ-hc operators of Laplace-type from Vaillant’s construction (see the Appendix of [3] for such an approach), but for the case at hand the expansion is well-known (see, e.g., (2.3) in [26]) R
Tr(e−t ) =
√ log t a− 1 a−1 + a− 1 √ + √ 2 + a0 + O( t) as t → 0+ . 2 t t t
(7.5)
1
√ t + a 1 t − 2 + a0 and choosing any C > 0, we Thus, writing f 0 (t) = a−1 t −1 + a− 1 log −2 t 2 have the expression
ζ (z) =
C dt 1 t z R Tr e−t − f 0 (t) (z) 0 t ∞ dt 1 + t z R Tr e−t − dim ker − ( ) (z) C t 1
C z− 2 a− 1 Cz + (a0 − dim ker − ( )) + 2 (z + 1) z − 21 (z) 1
a− 1 C z− 2 log C 1 C z−1 a−1 2 . + − + 1 1 (z) (z − 1) (z) z−2 (z − 2 )2 Differentiating and setting z = 0, we get ζ
(0) =
dt Tr e−t − f 0 (t) t 0 ∞ dt R + Tr e−t − dim ker − ( ) t C C
R
1
+ (log C + γe ) (a0 − dim ker − ( )) − 2C − 2 a− 1 2
−C
−1
a−1 + a− 1 C 2
− 21
(−4 − 2 log C)
(7.6)
1 1 1 z=0 = 0, ∂z z=0 (z) = 1 and ∂z z=0 (z+1) = γe is Euler’s by using the fact that (z) gamma constant. More generally, and to connect with (7.4), we can consider the determinant of with its spectrum shifted by a complex number w, that is, the determinant of + w. Just as before we have ∞ dt 1 ζ (z; w) = t z R Tr e−t − f 0 (t) e−tw (z) 0 t 1 1
a 1 z − − log w z − a0 −2 log 2 2 + z + z− 21 w (z) w a− 1 z − 1 a−1 2 2 + z−1 (z − 1)−1 , + (7.7) 1 w w z− 2 (z)
510
P. Albin, F. Rochon
where the function log (z) is defined to be ∞ dt log (z) := t z e−t log t t 0
(7.8)
for Re z > 0. Since it satisfies the recurrence relation log (z + 1) = zlog (z) + (z), it has a meromorphic continuation to the whole complex plane with poles at −N0 = 0, −1, −2 . . .. In particular, it has no pole at z = − 21 . Taking the derivative of ζ (z; w) with respect to z and setting z = 0, we get ζ (0; w) = − log det ( + w) ∞ dt R = Tr e−t − f 0 (t) e−tw t 0 √ √ −a0 log w − 2 πa− 1 w + a−1 w (−1 + log w) 2 √ 1 1 + a− 1 w log (− ) − log w(− ) . 2 2 2
(7.9)
7.2. Relation with the Selberg Zeta function. To relate the determinant with the Selberg Zeta function, we will follow [12] and use a description of the Selberg Zeta function in terms of the resolvent of the Laplacian. Given a hyperbolic surface of genus g with n cusps, we denote by R (s) := ( + s(s − 1))−1
(7.10)
the resolvent of the geometric Laplacian with respect to the hyperbolic metric. The Schwartz kernel of R (s) is singular along the diagonal. A natural way to remove this singular part is to subtract the resolvent of the model hyperbolic space RH (s) = ( H + s(s − 1))−1 .
(7.11)
This resolvent has Schwartz kernel defined on H×H. Hence, thinking of as the quotient \H of the hyperbolic half-plane by some appropriate discrete subgroup ⊂ SL(2, R), there is a natural lift of the Schwartz kernel G (s; z, w) of R (s) to H×H. Since locally R (s) and RH (s) have the same full symbol (being a parametrix for H + s(s − 1)), they will have the same singularities along the diagonal. This means the function ϕ (s; z) := (2s − 1) [G (s; z, w) − G H (s; z, w)]w=z
(7.12)
will be smooth in z ∈ H and meromorphic in s. Because of the SL(2, R) invariance of H , the Schwartz kernel G H will also be SL(2, R) invariant when restricted to the diagonal in H×H. This means in particular that the function ϕ (s; z) will be -invariant and so will descend to give a function on = \H. Thus, we can define the function R φ (s) := ϕ (s; z)dg (z), (7.13)
Local Families Index for ∂-Operators
511
where dg is the volume form associated to the hyperbolic metric, and the integral is renormalized using ρ , the boundary defining function of (2.10). This function has a meromorphic continuation to the complex plane with possible poles at 21 − N0 /2. As a particular example, we can consider the Horn H := ∞ \H of (2.4). The end obtained as y → +∞ is a cusp end, so we can pick a boundary defining function as usual there, but the other end when y → 0+ is not a cusp end, but a funnel, and for that end, one should take a boundary defining function which is given by y near y = 0. With this choice, we can make sense of the function φ H (s). The following proposition is due to Borthwick, Judge, and Perry [12]. Proposition 7.1 ([12], Prop. 4.3). Let be a Riemann surface of genus g and with n cusps. Then (s) Z = φ (s) − nφ H (s). Z (s)
The function φ H (s) can be computed explicitly [12, Prop. 2.4], 1 1 + φ H (s) = − log 2 − s + , 2 2s − 1
(7.14)
(z) . Thus, to relate the logarithmic derivative of where (z) is the digamma function (z) Z (s) with the determinant we need to understand the function φ (s) in terms of the heat kernel instead of the resolvent.
Lemma 7.2. In the sense of distributions, we have that ∞ ( + s)−1 = K (t)e−st dt, for Re s > 0. 0
Proof. Let f ∈ Cc∞ () be a test function. Then by definition of the heat kernel, we have that ∂t K (t) f + K (t) f = 0,
K (0) f = f.
Using integration by part, this implies that ∞ −∞ −st K (t) f e dt = −∂t K (t) f e−st dt 0 0 ∞ −st ∞ −s K (t) f e−st dt = −K (t) f e 0 0 ∞ K (t) f e−st dt, (7.15) = f −s 0
which shows that
( + s)
∞ 0
K (t) f e−st dt
= f.
512
P. Albin, F. Rochon
This means that
∞
f → 0
K (t) f e−st dt
is a right inverse for ( + s). The same computation shows that it is a left inverse since K (t) f = K (t) f by uniqueness of the solution for the heat equation.
From this lemma we conclude that ∞ ϕ (s; z) = (K (t, z, z) − K H (t; z, z)) e−s(s−1)t dt. 2s − 1 0
(7.16)
Since the left-hand side is smooth, the right-hand side is smooth as well, which means that K (t; z, z) and K H (t; z, z) have the same term of order t −1 in their asymptotic expansions as t 0. Integrating (7.16) in z and taking the finite part, we get φ (s) = 2s − 1
R
∞ 0
(K (t; z, z) − K H (t; z, z)) e−ts(s−1) dtdg (z).
(7.17)
The order of integration can be interchanged, since for Re(w) 1,
∞
x w (K (t; z, z) − K H (t; z, z)) e−ts(s−1) dtdg (z) 0 ∞ x w (K (t; z, z) − K H (t; z, z)) e−ts(s−1) dtdg (z). =
0
(7.18)
Hence, we get that φ (s) = 2s − 1
∞ 0
R
Tr(K (t)) − R Tr (K H (t)) e−ts(s−1) dt,
(7.19)
where R
Tr (K H (t)) := FP
K H (t; z, z)dg (z).
We recall that K H (t) itself does not descend to × , but its restriction to the diagonal in H × H does descend to the diagonal in × . Indeed, it is well-known that because the hyperbolic metric on H is SL(2, R)-invariant, K H (t; z, z) is constant in z, say equal to kH (t)dgH (z) for some function of t. Since the surfaces we are studying have finite area, we have R
Tr(K H (t)) = kH (t) area().
Local Families Index for ∂-Operators
513
Corollary 7.3. The function kH (t) has an asymptotic expansion given by kH (t) ∼ k−1 t −1 + o(t −1 ) as t 0. Therefore, if is a Riemann surface of genus g with n cusps, then the regularized trace of its heat kernel has the asymptotic expansion R
1
Tr(K (t)) ∼ k−1 area()t −1 + O(t − 2 log t) as t 0.
Proof. Consider the case where has no cusp. Then it is well-known that R Tr(K (t)) = Tr(K (t)) has asymptotic expansion of the form Tr(K (t)) ∼ αt −1 + β + O(t) as t 0, where α and β are some constants. By formula (7.19), of order t −1 as t 0, hence we get that k−1 :=
R
Tr(K H (t)) has the same term
1 α = area() 4π
is such that kH (t) ∼ k−1 t −1 + o(t −1 ) as t 0. When we consider a surface with cusps, R Tr(K ) will have the same term of order t −1 as R Tr (K H (t)) as t 0, hence R
1
Tr(K (t)) ∼ k−1 area()t −1 + O(t − 2 log t), as t 0.
Consider the functional D (s) := log det ( + s (s − 1)) = −ζ (0; s(s − 1)). If we differentiate with respect s and use formula (7.9), we find that ∞ (s) 1 D R = Tr e−t − a−1 t −1 e−ts(s−1) dt 2s − 1 D (s) 0 +a−1 log (s(s − 1)) .
(7.20)
(7.21)
Combining formula (7.21) and (7.19) and using Corollary 7.3, we get that (s) D (s) Z 1 − = 2s − 1 D (s) Z (s) ∞ R Tr(K (t)) − a−1 t −1 − R Tr(K (t)) + R Tr (K H (t)) e−ts(s−1) dt 0
n φ H (s) − k−1 area() log (s(s − 1)) + 2s − 1 ∞ R Tr(K H (t)) − k−1 area()t −1 e−ts(s−1) dt = 0
+ k−1 area() log (s(s − 1)) +
n φ H (s), 2s − 1
(7.22)
514
P. Albin, F. Rochon
so that
(s) D Z (s) n − = φ H (s) D (s) Z (s) 2s − 1 ∞ kH (t) − k−1 t −1 e−ts(s−1) dt + k−1 log (s(s − 1)) . + area()
1 2s − 1
(7.23)
0
In particular, for fixed g and n, we see that the right hand side does not depend on the since the area is given by −2π χ () by the Gauss-Bonnet theorem, a quantity that only depends on g and n. From (7.14), we see that 1 1 d φ H (s) = −s log 2 − log (s + ) + log(2s − 1) . (7.24) ds 2 2 Then, according to (7.23) and (7.4), there exists a constant C such that n −χ () √ 2 D (s) (s) 2s − 1 2 = C e E − s(s − 1) (2π )s , Z (s) (s) 2s (s + 21 )
(7.25)
where E := − 41 − 21 log 2π + 2ζ (−1). As in [29], the constant C can be determined by the asymptotic expansion of the logarithm of both sides of (7.25) as s approaches infinity. For the left side, it is clear from (7.1) that log Z (s) has a trivial asymptotic expansion as s → +∞. Thus, from (7.9), we conclude that the asymptotic behavior of the logarithm of the left side is given by √ D (s) = a0 log s(s − 1) + 2 πa− 1 s(s − 1) log 2 Z (s) √ 1 − a− 1 s(s − 1) log (− ) + 2 π log(s(s − 1)) 2 2 (7.26) − a−1 s(s − 1) (−1 + log(s(s − 1))) + o(1) as s → +∞. On the other hand, if we set √ √ 2s − 1 2 = , Z cu (s) = 1 2s (s + 2 ) 2s s − 21 (s − 21 )
(7.27)
we see using Stirling’s formula that its logarithm has the following asymptotic behavior: 1 1 log (Z cu (s)) = − log(2π ) + (1 − log 2) s − 2 2 1 1 log s − + o(1) (7.28) − s− 2 2 as s → +∞. Since
1 s(s − 1) = s − + o(1), 2 1 1 log s − + o(1), s(s − 1) log (s(s − 1)) = 2 s − 2 2
(7.29) (7.30)
Local Families Index for ∂-Operators
515
as s → +∞, we can rewrite (7.26) as √ 1 1 D (s) = a0 log s(s − 1) + 2 πa− 1 − s− a− 1 log (− ) log 2 2 Z (s) 2 2 √ 1 1 log s − −4 π a− 1 s − 2 2 2 (7.31) −a−1 s(s − 1) (−1 + log(s(s − 1))) + o(1) as s → +∞. Now, from [29]1 , we have also that 2 (s)2 (2π )s = log e E−s(s−1) (s) 1 1 s(s − 1) − log s(s − 1) + s(s − 1) − log s(s − 1) + o(1) 6 2 2
(7.32)
as s → +∞. This asymptotic behavior only involves terms of the form log (s(s − 1)) and s(s −1) log (s(s − 1)). Thus, in (7.31), the terms involving a− 1 and a− 1 counterbalance 2 2 the asymptotic behavior of (7.28) while the terms involving a−1 and a0 counterbalance the asymptotic behavior of (7.32). Comparing (7.31) with (7.32), we find a−1 = g − 1 = −
χ () χ () , a0 = . 2 6
(7.33)
Comparing (7.31) with (7.28), we also get
a− 1 2
n n = √ , a− 1 = √ 2 4 π 2 π
log (− 21 ) 1 − log 2 + . √ 4 π
Now, recall that in [29], the constant E is chosen so that 2 E−s(s−1) 2 (s) s log e (2π ) = (s) −a−1 s(s − 1) (−1 + log (s(s − 1))) + a0 log (s(s − 1)) .
(7.34)
(7.35)
This means the constant C has to be chosen to compensate the constant term of (7.28), that is, n log C = − log 2π 2
⇒
n
C = (2π )− 2 .
(7.36)
This gives the following result. Theorem 2. For a Riemann surface of genus g with n cusps satisfying 2g − 2 + n > 0 and equipped with the hyperbolic metric, we have −χ () 2 2 (s) e E−s(s−1) (s) (2π )s det ( + s(s − 1)) = Z (s) n . 2s π(s − 21 )(s − 21 ) 1 In (2.19) of [29], the coefficient of log s(s − 1) is 1 , but it is supposed to be 1 . 24 6
516
P. Albin, F. Rochon
As a consequence, we see that the ratio det ( + s(s − 1)) Z (s) is a meromorphic function in s which only depends on the genus g and the number of cusps n. This means that, up to a multiplicative constant depending only on g and n, the (1). The formula of Theorem 2 can also be expressed determinant of is given by Z in terms of the ∂-Laplacian ∂ =
1 . 2
For the heat kernel of the ∂-Lapacian, we have the following short time asymptotic expansion, t R Tr e−t ∂ = R Tr e− 2 √ √ √ 2 a− 1 2a− 1 − 2 a− 1 log 2 √ 2a−1 2 2 + √ 2 log t + = + a0 + O( t) √ t t t (7.37) as t → 0+ , where a−1 , a− 1 , a− 1 , a0 are the coefficients in (7.5). From formula (7.7), 2 2 we conclude that s(s − 1) ζ ∂ z; = 2z ζ (z; s(s − 1)) . 2 Hence, ζ ∂
s(s − 1) 0; = (log 2) ζ (0; s(s − 1)) + ζ (0; s(s − 1)) , 2
which means that s(s − 1) = 2−ζ (0;s(s−1)) det ( + s(s − 1)) . det ∂ + 2 Now, ζ (0; s(s − 1)) can be computed explicitly from (7.7) and (7.33), ζ (0; s(s − 1)) = a0 − a−1 s(s − 1) 1 s(s − 1) + . = χ () 6 2
(7.38)
From Theorem 2, we get the following formula. Corollary 7.4. For a Riemann surface of genus g with n cusps satisfying 2g−2+n > 0 and equipped with the hyperbolic metric, we have 1 s(s−1) −χ () (s)2 s 6 + 2 e E−s(s−1) 2 2 (2π ) (s) s(s − 1) det ∂ + = Z (s) . n 2 1 1 s 2 π(s − 2 )(s − 2 )
Local Families Index for ∂-Operators
517
7.3. The determinant of for ≥ 1. As indicated in [29], it is possible to express the determinant of in terms of Selberg Zeta function by using Corollary 7.4. This is because the spectrum of is essentially given by a shifted version of the spectrum of ∗ 0 = ∂ ∂. Recall first that these various Laplacians are related by the recurrence relation (see2 for instance (1.3) in [30]) ∗
∗
∂ u = ∂ u ( −1 + − 1) ,
(7.39)
0,1 where u := y12 is seen as a section of 1,0 ⊗ on . Taking the formal adjoint of (7.39), we get
u ∗ ∂ = ( −1 + − 1) u ∗ ∂ ,
(7.40)
−1 ⊗ (0,1 )−1 . where u ∗ is the conjugate u of u seen as a section of (1,0 ) −1
∗
Since the operator ∂ is Fredholm, it has a well-defined parametrix ∂ : (ker ∂ )⊥ → (ker ∂ )⊥ . Applying this parametrix to both sides of (7.40), we get −1 = u ∗ ∂ ( −1 + − 1) u ∗ ∂ .
(7.41)
In the compact case, this directly implies that det ( ) = det( −1 + − 1)
(7.42)
for ≥ 2 since ∂ is surjective in that case. When = 1, the operator ∂ 1 is not surjective, but the cokernel of u ∗ ∂ , ∗
ker ∂ 1 u = ker ∂ 0 = ker 0 is precisely the kernel of 0 , so that we have in that case det ( 1 ) = det ( 0 ). Using (7.41) once more and (3.25), we have on the other hand that for k > 0,
g−1 det ( 0 + k) , = 2; (k) det ( −1 + k) = (k)(2−1)(g−1) det ( −2 + k + − 2) , ≥ 3. Applying this recursively, we get ⎧ = 1, ⎨ det ( 0 ), = 2, det ( ) = det ( 0 + 1) , ⎩ δ det ( + ( − 1)) , ≥ 3, ,g 0
(7.43)
(7.44)
where δ,g is a number depending only on and g. In the non-compact case, one has to be more careful since the regularized trace does not necessarily vanish on a commutator. Taking this into account, the analog of (7.41) in the non-compact case is det ( ) = D,n det ( −1 + − 1) 2 We have − 1 instead of −1 in [30] since we use the convention |dz|2 = 2. 2
(7.45)
518
P. Albin, F. Rochon
with
−log D,n
d 1 = dz (z)
∞
t 0
zR
∗
Tr [(u ∂ )
−1 −t −1
e
∗
, u ∂ ] e
− t (−1) 2
dt t
z=0
(7.46) regularizing as in (7.6). Although the term D,n might be hard to compute, what is clear is that it only depends on and the number n of cusps. This is because the regularized trace of a commutator [A, B] ‘localizes’ near the boundary in the sense that it only depends on the Taylor expansion of the integral kernels at the boundary of the diagonal. Recall that to construct the heat kernel (see [2,31]) we start with a ‘parametrix’ for the heat equation which solves a model equation at the cusp. The solution of this model equation is then used iteratively to construct the Taylor expansion of the heat kernel as we approach the cusp, before finally solving away the remaining error in the interior. The upshot is that, since all cusps have isometric neighborhoods, the term D,n only depends on and n as required. Thus using recursively (7.45) and applying Corollary 7.4, we get the following. Corollary 7.5. For a Riemann surface of genus g ≥ 2 with n cusp, we have
α,g,n Z (), ≥ 2, det ( ) = (1), = 0, 1, α,g,n Z where each constant α,g,n > 0 only depends on , g and n. A similar statement holds of the determinant of D− D+ = 2 . 8. The Curvature of the Quillen Connection Recall that the determinant bundle of the family of operators ∂ is by definition −1 λ := det ind ∂ = max ker ∂ ⊗ max coker ∂
(8.1)
where ∈ Z and max denotes the maximal exterior power of a vector space. The definition is particularly simple in this case because ker ∂ l is a vector bundle over Tg,n . The L 2 -norm on ker ∂ defines a canonical metric on λ , the L 2 -metric, denoted · . An alternative metric which is more interesting geometrically is the Quillen metric, − 1 · Q := det D− D+ 2 · .
(8.2)
Following the discussion of Sect. 9.7 in [5], we will associate to · Q a compatible connection called the Quillen connection. In order to do that, consider over Tg,n the Z2 -graded bundle (8.3) E = E+ ⊕ E− , E+ := ,0 Tg,n /Tg,n , E− := ,1 Tg,n /Tg,n . Let also π∗ E → Tg,n be the Fréchet bundle whose fiber at [] ∈ Tg,n is 1 π∗ E,[] := C˙∞ ; El | ⊗ | | 2 ,
(8.4)
Local Families Index for ∂-Operators
519
1 where | | is the density bundle on and C˙∞ ; El | ⊗ | | 2 is the space of 1
smooth sections of El | ⊗ | | 2 with rapid decay at infinity. The family of Dirac type operators √ ∗ 2 ∂ + ∂ ,
D :=
D+ =
√ 2∂ ,
D− =
√
∗
2∂ ,
(8.5)
acts from π∗ E to π∗ E . One of the reasons that motivates the introduction of the fibre density bundle in the definition of π∗ E is that in this way the canonical connection on π : Tg,n → Tg,n induces a connection on π∗ E , denoted ∇ π∗ E , which is automatically compatible with the metric of π∗ E (cf. Prop. 9.13 in [5]). Notice also that the density bundle | | is canonically trivialized by the section |dg | so that D acts on π∗ E in a natural way. To the family of Dirac type operators D , we can associate a superconnection A := D + ∇ π∗ E
(8.6)
As := s 2 D + ∇ π∗ E .
(8.7)
and its rescaled version 1
For s ∈ R+ , we can define two differential forms α± ∈ A(Tg,n )(s), α± (s)
∂As −(As )2 e := Tr π∗ E ± ∂s 1 R −(As )2 ± D e , Tr = 1 π∗ E 2s 2 R
(8.8)
by taking the trace with respect to E+ and E− respectively. The 0-form component of
(A2 )s is s D2 , while its 1-form component is s 2 [∇ π∗ El , D ]. On the other hand, the s 2 1-form component of e−(A ) is given by 1
s 2 e−(A )
[1]
1
0 1
= (−s) 1
= −s 2
0
e−(1−σ )s D s − 2 [∇ π∗ E , D ]e−σ s D dσ 2
1
2
e−(1−σ )s D [∇ π∗ E , D ]e−σ s D dσ. 2
2
(8.9)
The following observation will turn out to be very useful. Lemma 8.1. The Schwartz kernel of [∇ π∗ E , D± ] vanishes to all order at the front face. In particular, for P ∈ −∞ (Tg,n /Tg,n ; E ), R
STr [[∇ π∗ E , Dl± ], P] = 0.
520
P. Albin, F. Rochon
Proof. Let [] ∈ Tg,n be given. If µ ∈ −1,1 () is a harmonic Beltrami differential, let f µ : H → H be the unique diffeomorphism satisfying the Beltrami equation ∂f µ ∂f µ =µ ∂z ∂z and fixing the points 0, 1, ∞. In particular, since µ is a cusp form, it decreases rapidly as z → ∞. This means that f µ is asymptotically holomorphic as z → ∞. From the definition of the canonical connection on π : Tg,n → Tg,n , this means that the Schwartz kernel [∇ π∗ E , D+ ](z, z ) decreases quickly as z and z approaches a cusp in . For √ ∗ ∗ D− = 2∂ , the same is true, but since ∂ = −u −1 ∂u − , we also need to use the fact that u = y12 is parallel with respect to the canonical connection on π : Tg,n → Tg,n . Now, we know that R STr [[∇ π∗ E , D± ], P] depends linearly on the asymptotic expansions of [∇ π∗ E , D± ] and P at the corner of ×. The asymptoptic expansion of [∇ π∗ E , D± ] being trivial, the result follows. Alternatively, the result also follows directly from the explicit formulas (5.15). With this lemma, the discussion of Sect. 9.7 in [5] applies almost directly to our context.
Lemma 8.2. The one form component of the differential forms α+ (s) satisfies α+ (s)[1] = α− (s)[1] and has an asymptotic expansion of the form
α+ (s)[1] ∼
∞
k
s 2 (ak + bk log s)
−N
as s → 0+ . Proof. The asymptotic expansion as s → 0+ follows from the construction of the heat kernel by Vaillant [31], its generalization in [2] and an application of the pushforward theorem for manifolds with corners. From (8.9), we have that
α+ (s)[1]
1 = − R Tr π∗ E+ 2 =−
1R STr π∗ E 2
D
D−
1
e
−(1−σ )s D2
0
0
1
[∇
π∗ E
, D ]e
−σ s D2
dσ
2 2 e−(1−σ )s D [∇ π∗ El , D+ ]e−σ s D dσ . (8.10)
Local Families Index for ∂-Operators
521
Taking the complex conjugate and using the fact that (D+ )∗ = D− and that ∇ π∗ E is a unitary connection, we have that α+ (s)[1]
= = =
=
1 1R 2 2 − −σ s D π E −(1−σ )s D + ∗ − Tr π∗ E+ e [D , ∇ ]e D dσ 2 0 1 1R 2 2 STr π∗ E e−(1−σ )s D [D− , ∇ π∗ E ]e−σ s D D+ dσ 2 0 1 1R 2 2 STr π∗ E D+ e−(1−σ )s D [D− , ∇ π∗ E ]e−σ s D dσ 2 0 1 1R − −(1−σ )s D2 π∗ E −σ s D2 + STr π∗ E e [D , ∇ ]e dσ, D + 2 0 1 1R − + −(1−σ )s D2 π∗ E −σ s D2 Tr π∗ E − D e [D , ∇ ]e dσ + 0 − 2 0
= α− (s)[1] ,
(8.11)
where Lemma 8.1 was used in the line before the last one. We would like to consider the one-forms β± (z) := 2
∞ 0
t z α± (t)[1] dt.
(8.12)
By Lemma 8.2, this integral is holomorphic for Re z >> 0 and admits a meromorphic extension to the whole complex plane. Thus, in this sense, β+ (z) and β− (z) are well-defined meromorphic families of one-forms. We are interested in their finite part at z = 0. More precisely, we would like to consider the one-forms β± :=
d 1 ± β (z) , dz (z) z=0
(8.13)
where the evaluation at zero means that we take the finite part at z = 0. More generally, we will use the notation
∞
γ dt :=
0
whenever the integral
∞ 0
d 1 dz (z)
∞
t z γ (t)dt
0
z=0
t z−1 γ (t)dt varies meromorphically in z. Thus, in this notation, βl±
∞
=2 0
α± (t)[1] dt.
Lemma 8.3. Seen as a function on Tg,n , the differential of ζ (0; D− D+ ) is given by dζ (0; D− D+ ) = − β+ + β− .
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P. Albin, F. Rochon
Proof. Using Duhamel’s formula, we have that dζ (0; D− D+ ) is given by ∞ 1 t −(t−s)D − D + π∗ E − + −s D − D + R [∇ ds Tr π∗ E+ − e , D D ]e dt = t 0 0 1 ∞ − + − + R Tr π∗ E+ − e−(1−s)t D D [∇ π∗ E , D− D+ ]e−st D D ds dt. (8.14) 0
0
On the other hand, we have ∞ + β = 2 α+ (t)[1] dt 0
=− =−
∞ 0 ∞ 0
while β−
=−
∞
0 ∞
=
∞
=
R
R
∞
0 ∞
= 0
Tr π∗ E+
e D+
e
STr π∗ E
1
e
D− [∇ π∗ E ,
−(1−s)t D2
[∇
π∗ E
,
2 D+ ]e−st D ds
2 D− ]e−st D ds
dt,
(8.15)
dt
2 2 e−(1−s)t D [∇ π∗ E , D− ]e−st D ds dt
D+ ,
STr π∗ E
1
−(1−s)t D2
0
R
0
1
R
−(1−s)t D2
e 0 1
D+
STr π∗ E
2 2 e−(1−s)t D [∇ π∗ E , D+ ]e−st D ds dt
0
Tr π∗ E −
STr π∗ E
1 0
1
R
D−
0
+
R
Tr π∗ E+
0
R
1 0
[∇
π∗ E
,
2 D− ]D+ e−st D ds
dt
e−(1−s)t D [∇ π∗ E , D− ]e−st D ds
−(1−s)t D2
2
[∇
π∗ E
0
,
2
2 D− ]D+ e−st D ds
dt
dt + 0, (8.16)
using Lemma 8.1 in the last step. The result then follows by combining (8.14), (8.15) and (8.16) and using the formula [∇ π∗ E , D− D+ ] = [∇ π∗ E , D− ]D+ − D− [∇ π∗ E , D+ ]. π∗ E+
If P0 : → ker ∂ denotes the orthogonal projection onto the kernel of ∂ , then the connection ∇ ker ∂ = P0 ∇ π∗ E P0 +
(8.17)
is compatible with the L 2 -metric. It is holomorphic, so that ∇ ker ∂ is the Chern connection of ker ∂ with respect to the L 2 -metric · . It defines a connection on det ∂ , ∇ det ∂ , which is the Chern connection of det ∂ with respect to the L 2 -metric. We define the Quillen connection on det ∂ to be the connection given by ∇ Q := ∇ det ∂ + β+ .
(8.18)
Local Families Index for ∂-Operators
523
Proposition 8.4. The Quillen connection is the Chern connection of det ∂ with respect to the Quillen metric · Q . Proof. We need to check that ∇ Q is holomorphic and is compatible with the Quillen metric. To see that it is holomorphic, it suffices to check that β+ is a (1, 0)-form. Since √ D+ = 2∂ is a family of operators that varies holomorphically on Tg,n , the form [∇ π∗ E , D+ ] has to be a (1, 0)-form (cf. with (5.15)). Directly from the definition of β+ , we thus see it has to be a (1, 0)-form. To see that ∇ Q is compatible with the Quillen metric, notice that in general, a connection which is compatible with the Quillen metric is of the form 1 ∇ det ∂ l − dζ 0; D− D+ + ω, 2
(8.19)
where ω is any imaginary one-form. The result then follows by noticing that, taking ω=
β+ −β+ 2
and using Lemma 8.2 and Lemma 8.3, we get the Quillen connection.
We can now compute the curvature of the Quillen connection. Theorem 3. The curvature of the Quillen connection is given by √ −1 Q 2 − (∇ ) = Ch T (Tg,n /Tg,n ) · Td T Tg,n /Tg,n 2π Tg,n /Tg,n
[2]
−
n i=1
ei . 12
Proof. With respect to the connection ∇ det ∂ , we have √ −1 det ∂ l 2 ∇ = Ch ∇ ker ∂ l . [2] 2π
(8.20)
But by definition, since ∇ π∗ E = A[1] for A the Bismut superconnection, (cf. Prop. ker ∂ l is the connection used in Theorem 1. Thus, 10.16 we have by (8.17) that ∇ in [5]), is given by formula (6.16), so that Ch ∇ ker ∂ [2]
√ −1 det ∂ 2 (∇ ) = Ch T − (Tg,n /Tg,n ) · Td T Tg,n /Tg,n 2π Tg,n /Tg,n [2] ∞ n 1 ei ∂At −A2t R − e − STr dt. (8.21) √ d 12 2π −1 0 ∂t [1] i=1
On the other hand, from the definition of the Quillen connection, we have 2 2 ∇ Q = ∇ det ∂ l + dβ+ .
(8.22)
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P. Albin, F. Rochon
From Lemma 8.3, d(β+ + β− ) = 0, hence 2 2 1 ∇ Q = ∇ det ∂ l + d β+ − β− . 2
(8.23)
But using the fact the Bismut superconnection A is given by D + ∇ π∗ E up to terms of degree 2, we have ∞ ∞ + 1 + ∂At −A2t R β − β− = e α (t)[1] − α− (t)[1] dt = STr dt 2 ∂t 0 0 [1] ∞ ∂At −A2t R = STr dt. (8.24) e ∂t 0 [1] 2 In the last step, we have used the fact R STr ∂∂tAt e−At is integrable in t, so that there is no need to regularize. Combining (8.21), (8.23) and (8.24), the result follows. We should compare our result with the local index formula of Takhtajan and Zograf [30], (∇ Q )2 =
62 − 6 + 1 1 ωWP − ωTZ , 12π 2 9
(8.25)
where ωW P is the Weil-Peterson Kähler form on Tg,n and ωTZ is the Kähler form on Tg,n defined by Takhtajan and Zograf in terms of the cusp ends of the fibres of p : Tg,n → Tg,n . A well-known result of Wolpert [34] (see also p. 424 in [30]) shows that − Ch T (Tg,n /Tg,n ) · Td T Tg,n /Tg,n Tg,n /Tg,n
=
62
[2]
− 6 + 1 ωWP . 12π 2
(8.26)
Thus, comparing Theorem 3 with (8.25), we get the following relation. Corollary 8.5. (Weng [32], Wolpert [35]). For ≥ 0 and n > 0, we have η(DV )[2] =
n 1 ei = ωTZ . 12 9 i=1
The fact the Takhtajan-Zograf Kähler form is a rational multiple of the curvature of a Hermitian line bundle was first obtained by Weng [32] using Arakelov theory. This was later improved and finalized by Wolpert [35], who obtained more generally that ei = 43 ωTZ,i (ωTZ,i is defined in (8.31) below) via a natural intrinsic way to define metrics on the line bundles Li . For completeness, let us recall how the Takhtajan-Zograf Kähler form ωT Z is defined. Given a fibre of p : Tg,n → Tg,n , identify it with a quotient of the upper half-plane, ∼ = \H, where is the corresponding Fuchsian group of type (g, n). Let 1 , . . . , n be the list of non-conjugate parabolic subgroup of as in (5.2) so that σi−1 i σi = ∞
Local Families Index for ∂-Operators
525
for i ∈ {1, . . . , n}. The Eisenstein-Mass series E i (z, s) associated to the i th cusp of the group is defined for Re s > 1 by the formula E i (z, s) :=
Im(σi−1 γ z)s .
(8.27)
γ ∈i \
The Eisenstein-Mass series naturally descends to define a function on the quotient = \H. Recall that under the identification of T[] Tg,n with the space of harmonic Beltrami differentials −1,1 (), the Weil-Peterson Kähler metric is defined by µ, νWP :=
µνdg =
µ, ν K −1 ⊗0,1 dg
(8.28)
for µ, ν ∈ T[] Tg,n with corresponding Kähler form given by √
−1 µ, νWP . 2
ωWP (µ, ν) =
(8.29)
To define their Kähler metric, Takhtajan and Zograf considered instead µ, νi =
µν E i (·, 2)dg , i = 1, . . . n.
(8.30)
Each of these scalar products gives rise to a Kähler metric on Tg,n with corresponding Kähler form ωTZ,i (µ, ν) =
√ −1 µ, νi , i = 1, . . . , n. 2
(8.31)
The sum of these metrics is the Takhtajan-Zograf Kähler metric µ, νTZ :=
n µ, νi
(8.32)
i=1
with corresponding Kähler form given by √ ωTZ (µ, ν) =
−1 µ, νTZ . 2
(8.33)
We know from Corollary 8.5 that the eta form η(DV )[2] is the Kähler form of a Kähler metric. This is consistent with Theorem 1 asserting that the eta form η(DV ) is closed. Acknowledgement We would like to thank Leon Takhtajan and Peter Zograf for explaining to us their results. We are also grateful to Rafe Mazzeo, Richard Melrose, Gabriele Mondello and Sergiu Moroianu for helpful conversations.
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References 1. Ahlfors, L.: Some remarks on Teichmüller’s space of Riemann surfaces. Ann. Math. 74, 171–191 (1961) 2. Albin, P., Rochon, F.: Family index for manifolds with hyperbolic cusp singularities. IMRN 2009, 625–697 (2009) 3. Albin, P., Rochon, F.: Some index formulae on the moduli space of stable parabolic vector bundles. preprint, available at http://arxiv.org/abs/0812.2223v1[math.DG], 2008 4. Arbarello, E., Cornalba, M.: Combinatorial and algebro geometric cohomology classes on the moduli space of curves. J. Alg. Geom. 5, 705–749 (1996) 5. Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Berlin: Springer-Verlag, 1992 6. Bini, G.: Generalized Hodge classes on the moduli space of curve. Cont. Alg. Geom. 44(2), 559–565 (2003) 7. Bismut, J.M., Cheeger, J.: η-invariants and their adiabatic limits. J. Amer. Math. Soc. 2(1), 33–70 (1989) 8. Bismut, J.M., Freed, D.S.: The analysis of elliptic families. I. metric and connections on the determinant bundles. Commun. Math. Phys. 106, 159–176 (1986) 9. Bismut, J.M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. I. Bott-Chern forms and analytic torsion. Commun. Math. Phys. 115, 49–78 (1988) 10. Bismut, J.M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. II. Direct images and Bott-Chern forms. Commun. Math. Phys. 115, 79–126 (1988) 11. Bismut, J.M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. III. Quillen metrics on holomorphic determinants. Commun. Math. Phys. 115, 301–351 (1988) 12. Borthwick, D., Judge, C., Perry, P.A.: Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces. Comment. Math. Helv. 80(3), 483–515 (2005) 13. D’Hoker, E., Phong, D.H.: On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys. 104, 537–545 (1986) 14. Efrat, I.: Determinants of Laplacians on surfaces of finite volume. Commun. Math. Phys. 119, 443–451 (1988) 15. Efrat, I.: Erratum: Determinants of Laplacians on surfaces of finite volume. Commun. Math. Phys. 138, 607 (1991) 16. Griffiths, P., Harris, J.: Principle of Algebraic Geometry. New York: Wiley, 1994 17. Hassell, A.: Analytic surgery and analytic torsion. Commun. Anal. Geom. 6(2), 255–289 (1998) 18. Hubbard, J.H.: Teichmüller Theory and Applications to Geometry, Topology, and Dynamics Volume I: Teichmüller Theory. Ithaca NY: Matrix Editions, 2006 19. Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton, NJ: Princeton Univ. Press, 1989 20. Leichtnam, E., Mazzeo, R., Piazza, P.: The index of Dirac operators on manifolds with fibered boundaries. Bull. Belgian Math. Soc. 13(5), 845–855 (2006) 21. Mazzeo, R., Melrose, R.B.: Pseudodifferential operators on manifolds with fibred boundaries. Asian J. Math. 2(4), 833–866 (1998) 22. Melrose, R.B.: The Atiyah-Patodi-Singer Index Theorem. Research notes in mathematics, Wellesley, MA: A.K. Peters, 1993 23. Melrose, R.B.: The eta invariant and families of pseudodifferential operators. Math. Res. Lett. 2(5), 541–561 (1995) 24. Melrose, R.B., Piazza, P.: Families of Dirac operators, boundaries and the b-calculus. J. Diff. Geom. 46(1), 99–180 (1997) 25. Melrose, R.B., Piazza, P.: An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary. J. Diff. Geom. 46(2), 287–334 (1997) 26. Müller, W.: Spectral geometry and scattering theory for certain complete surfaces of finite volume. Invent. Math. 109, 265–305 (1992) 27. Mumford, D.: Towards an Enumerative Geometry of the Moduli Space of Curves. Arithmetic and Geometry Part II, M. Artin, J. Tate, eds., Bosel-Boston: Birkhäuser 1983, pp. 271–328 28. Piazza, P.: Determinant bundles, manifolds with boundary and surgery. Commun. Math. Phys. 178(3), 597–626 (1996) 29. Sarnak, P.: Determinants of Laplacians. Commun. Math. Phys. 110(1), 113–120 (1987) 30. Takhtajan, L.A., Zograf, P.G.: A local index theorem for families of ∂-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces. Commun. Math. Phys. 137(2), 399–426 (1991) 31. Vaillant, B.: Index and spectral theory for manifolds with generalized fibred cusps. Ph.D. dissertation, Bonner Math. Schriften 344, Univ. Bonn., Mathematisches Institut, Bonn (2001), available online at http://arXiv.org/abs/math/0102072v1[math.DG], 2001 32. Weng, L.: -admissible theory II. Deligne pairings over moduli spaces of punctured Riemann surfaces. Math. Ann. 320, 239–283 (2001) 33. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv. Diff. Geo. 1, 243–310 (1991)
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34. Wolpert, S.: Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85, 119–145 (1986) 35. Wolpert, S.: Cusps and the family hyperbolic metric. Duke Math. J. 138(3), 423–443 (2007) 36. Zhang, W.: Circle bundles, adiabatic limits of η-invariants and Rokhlin congruences. Ann. Inst. Fourier 44(1), 249–270 (1994) Communicated by L. Takhtajan
Commun. Math. Phys. 289, 529–559 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0697-9
Communications in
Mathematical Physics
The Deformation Quantizations of the Hyperbolic Plane P. Bieliavsky1 , S. Detournay2 , Ph. Spindel3 1 Géométrie et Physique mathématique, Université Catholique de Louvain,
2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium. E-mail: [email protected] 2 INFN, Sezione di Milano, Via Celoria, 16, 20133 Milano, Italy. E-mail: [email protected] 3 Mécanique et Gravitation, Université de Mons-Hainaut, 20 Place du Parc, 7000 Mons, Belgium. E-mail: [email protected] Received: 30 June 2008 / Accepted: 22 July 2008 Published online: 8 January 2009 – © Springer-Verlag 2008
Abstract: We describe the space of (all) invariant, both formal and non-formal, deformation quantizations on the hyperbolic plane D as solutions of the evolution of a second order hyperbolic differential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a unified method producing every quantization of D, and provides, in the 2-dimensional context, an exact solution to Weinstein’s WKB quantization program within geometric terms. The construction reveals the existence of a metric of Lorentz signature canonically attached (or ‘dual’) to the geometry of the hyperbolic plane through the quantization process. 1. Introduction 1.1. Motivations. The idea of the formal deformation quantization program [1], initiated by Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer, is to generalize the Weyl product to an arbitrary symplectic (or Poisson) manifold. In this context, the framework of quantum mechanics is the same as classical mechanics, observables are the same, and quantization arises as a deformation of the algebra of functions on the manifold, from a commutative to a non-commutative one. A formal star product (or deformation quantization) of a symplectic manifold (M, ω) is an associative C[[]]-bilinear product: : C ∞ (M)[[]] × C ∞ (M)[[]] −→ C ∞ (M)[[]] : (u, v) → u v such that, for u, v ∈
C ∞ (M),
(1)
the formal series uv =
involves bi-differential operators Ck : following properties:
∞
k Ck (u, v)
k=0 C ∞ (M)
(2)
× C ∞ (M) → C ∞ (M) satisfying the
530
P. Bieliavsky, S. Detournay, Ph. Spindel
(i) C0 ( u , v ) = u v, (ii) C1 (u, v) − C1 (v, u) = 2i{ u , v }, where { , } denotes the Poisson bracket on C ∞ (M) associated to ω. Two such star products i (i = 1, 2) on the same symplectic manifold M are called equivalent if there exists a formal series of the form T = I +
∞
k Tk ,
(3)
k=1
where the Tk ’s are differential operators on C ∞ (M) such that for all u, v ∈ C ∞ (M), one has u 2 v = T ( T −1 u 1 T −1 v );
(4)
the latter expression will be shortened by 2 = T (1 ). The first existence proofs were given in the 80’s independently by Dewilde-Lecomte, Fedosov and Omori-Maeda-Yoshioka [2–4]. Equivalence classes of star products on a symplectic manifold are in 1-1 correspondence with the space of formal series with coefficients in the second de Rham cohomology space of M [5,6]. The most important example of star product is the so-called Moyal star product 0 on the plane R2 endowed with its standard symplectic structure : ∞
(i /2)k i1 j1 · · · ik jk ∂i1 ···ik u∂ j1 ··· jk v, u 0 v := u.v + i {u, v} + 2 k!
(5)
k=2
=
∞ k (i /2)k k! k− p p p k− p ∂a ∂ u ∂a ∂ v, (−1) p k! p!(k − p)!
(6)
p=0
k=0
where j
ik k j = −δi ,
{u, v} = i j ∂i u∂ j v := ∂a u∂ v − ∂ u∂a v.
(7)
In the context of formal deformation quantization, one does not worry about the convergence of the series (2). In particular, a formal star product does not, in general, underlie any operator algebraic or spectral theory. However, situations exist where star products occur as asymptotic expansions of operator calculi. The best known example is provided by the Weyl product which has the following integral representation: 2 0 (u W v)(x) = (π )−2n u(y)v(z) ei S (x,y,z) dydz, (8) R2n ×R2n
where S 0 (x, y, z) = (x, y) + (y, z) + (z, x),
(9)
denoting the standard symplectic two-form on R2n . The product represented by (8) enjoys an important property: it is internal on the Schwartz space, the space of rapidly decreasing functions, on R2n [7]. Thus, the product of two (Schwartz) functions is again a (Schwartz) function—rather than a formal power series as in the formal context. Such a situation will be referred to as non-formal deformation quantization. Note that Moyal’s star product (5) can be defined as a formal asymptotic expansion of Weyl’s product.
Deformation Quantizations of the Hyperbolic Plane
531
In this paper, we will be interested in invariant deformation quantizations. When a group L acts on M by symplectomorphisms, a star product is said to be L- invariant if for all g ∈ L, u, v ∈ C ∞ (M): g (u v) = g u g v.
(10)
When L preserves a symplectic connection ∇ on M: L ⊂ Aff(∇) ∩ Symp(ω),
(11)
then, Fedosov’s construction yields an L-invariant star product. The notion of equivalence of L-invariant star products is the same as above except that each differential operator Tk is required to commute with the action of L. The set of L-equivariant equivalence classes of L-invariant star products is in this case parametrized by the space 2 (M)L [[]] of series with coefficients in the classes of L-invariant 2-forms on M HdR [8]. Although the study of deformation quantization admitting a given symmetry is natural, one may mention important specific situations where the symmetric situation is relevant. Firstly, the observable algebra of a conformal field theory defines an algebra of vertex operators. Thus any finite dimensional limit of such a theory must contain some remanent of this structure. In particular if the limit procedure is equivariant the remanent will also be symmetric. For instance, in the seminal work of Seiberg and Witten [9] it is shown that , in the limit of a large B-field on a flat brane, the limit of the vertex operator algebra is an associative invariant product on the space of functions on the brane that has the symmetry of the initial brane and fields configuration. Accordingly it becomes natural to inquire about all the associative composition laws of functions compatible with a given symmetry group (especially for non-compact symmetries). Actually, let us emphasize that our considerations go beyond string theory, but concern any Kac-Moody invariant theory. Moreover, it is known since the seminal works [9,10] that noncommutative geometry (and deformation theory) has intricate links with string theory. A manifestation of this statement stems from the fact that, in flat space-time, the worldvolume of a D-brane, on which open strings end, is deformed into a noncommutative manifold in the presence of a B-field. In particular, in the limit in which the massive open string modes decouple, the operator product expansion of open string tachyon vertex operators is governed by the Moyal-Weyl product[9,11]. This can be schematically written as M
ei P.X (τ )ei Q.X (τ ) ∼ (ei P.X ei Q.X )(τ ), where X represents the string coordinates on the brane, P and Q the momenta of the corresponding string states, τ and τ parameM
terize the worldsheet boundary, and where the classical limit f g → f.g is recovered for B → 0. It is not clear how this generalizes to an arbitrary curved string background supporting D-brane configurations. Nevertherless, some particular examples have been tackled in the literature. The SU(2) WZW model for instance supports symmetric D-branes wrapping S 2 spheres in the SU (2) group manifold. It has been shown that in an appropriate limit the worldvolume of these branes are deformed into fuzzy spheres [12–15]. The latter fuzzy spheres can be related to the Berezin/geometric quantization of S 2 hence to star product theory (see [16] and references therein). The appearance of noncommutative structures was also pointed out in relation with other backgrounds, like the Melvin Universe [17–23] or the Nappi-Witten plane wave [19,24], see e.g. [25] for a recent review. Another model that has attracted much attention in recent years is the SL(2, R) WZW model, describing string propagation on Ad S3 (and its euclidian counterpart, the H3+ model), see e.g. [26 and 27]. This model has played, and still
532
P. Bieliavsky, S. Detournay, Ph. Spindel
plays an important role in the understanding of string theory in non-compact and curved space-times. It appears as ubiquitous when dealing with black holes in string theory and is also particularly important in connection with the AdS/CFT correspondence. In this perspective, much effort has been devoted in studying the D-brane configurations this model can support (see e.g. [28]). The most simple and symmetric D-branes in Ad S3 turn out to wrap Ad S2 and H2 spaces in the SL(2, R) group manifold. The question one could then ask is: what is the low energy effective dynamics of open string modes ending on such branes? By analogy with the flat case, it seems not unreasonable to expect a field theory defined on a noncommutative deformation of the D-branes’ worldvolume. This deformation would have to enjoy some properties, namely to respect the symmetries of the model, just like the Moyal product and the fuzzy sphere construction do in their respective cases. In the present paper, we will be concerned with the specific situation of formal and non-formal deformation quantizations of the hyperbolic plane. In this particular case, a third motivation relies on the relevance of the hyperbolic plane in the study of Riemann surfaces through the uniformization theorem (establishing the hyperbolic plane D as the metric universal covering space of every constant curvature hyperbolic surface). Therefore, invariant deformations of the Poincaré disk could constitute a step towards a spectral theory of non-commutative Riemann surfaces. 1.2. What is done in the present work. We now summarize what is done in the present article as well as the method used. Given the geometric data of an affine space (i.e. a manifold endowed with an affine connection), one defines the notion of contracted space as a pair constituted by the same manifold but endowed with a connection whose Riemann’s curvature tensor appears as the initial one but where some components were ‘contracted’ to zero. One then starts from the standard intuitive idea that every geometric theory (such as deformation quantization for instance) formulated at the level of the initial space turns into a simpler theory after the contraction process. In order to describe the theory at the initial level, one may try to describe the contracted theory first and then apply to the latter an operator that reverses the contraction process. This is exactly what is done here regarding deformation quantization of the hyperbolic plane. We observe that the hyperbolic plane admits a unique contraction into a generic co-adjoint orbit of the Poincaré group in dimension 1+1. The set of all Poincaré invariant deformation quantizations (formal and non-formal) of the latter contracted orbits was earlier bijectively parametrized by an open subset of the algebra of pseudo-differential operators of the line [29,30]. The correspondence as well as the composition products were given there in a totally explicit manner. Both geometries (i.e. affine connections) on the hyperbolic and contracted hyperbolic planes, although very different, share however a common symmetry realized by a simply transitive action of the affine group of the real line: S ‘ax + b’. From earlier results at the formal level, one knows that, up to a redefinition of the deformation parameter, two S-invariant star products are equivalent to each other under a convolution operator by a (formal) distribution u on the Lie group S [8]. In the situation where one of them is SL2 (R)-invariant and the other one is Poincaré-invariant, the distribution u is shown to solve a second order canonical hyperbolic differential evolution equation. Solving the latter evolution problem therefore reverses the contraction process, allowing to recover the set of invariant star products (formal or not) on the hyperbolic plane from the set of contracted ones on the above mentioned Poincaré orbit. It is worth to point out that the Lorentz metric underlying the above Dalembertian, that realizes per se the “de-contraction” process, is canonically
Deformation Quantizations of the Hyperbolic Plane
533
attached to the geometry of the (quantum) hyperbolic plane. Indeed, the latter metric actually does not depend on any choice made. To our knowledge, the relevance of the above metric within hyperbolic geometry is new. The physical meaning of this canonical quantity has still to be clarified. It turns out that the method of separation of variables applies to the above-mentioned evolution equation yielding a space of solutions under a totally explicit form. Every solution u can be realized as a superposition of specific modes u s given in terms of Bessel functions: u = u(s) ˜ u s ds. R
These modal solutions u s provide non-formal invariant deformation quantizations of the hyperbolic plane. In particular, one of them (u 0 ) corresponds to a deformational version of Unterberger ’s Bessel calculus on the hyperbolic plane [31,32]. From explicitness, one also deduces a geometric solution of Weinstein’s WKB quantization program. The latter program proposes the study of invariant star products on symplectic symmetric spaces (see Sec. 2) expressed as an oscillatory integral, analogous to the oscillatory integral formulation (8) of Weyl’s composition. In [33], A. Weinstein suggested a beautiful geometrical interpretation of the asymptotics of the phase S = S(x, y, z) occurring in the oscillatory kernel in terms of the area of a geodesic triangle admitting points x, y and z as midpoints of its geodesic edges. In Sec. (3.7), we illustrate this by establishing an exact formula for the kernel in terms of a specific geometrical quantity of three points hereafter denoted by Scan , in accord with Weinstein’s asymptotics. 1.3. Comparison with other approaches. Noncommutative hyperbolic surfaces have mostly been considered within the context of operator (C -) algebras. In the seminal work [34] by Klimek and Lesniewski , Berezin-Toeplitz quantization is used to produce a ‘Strict Deformation Quantization’ —in the sense of M. Rieffel— of the hyperbolic plane D, i.e. a field of C -algebras over a real interval that deform C0 (D) in the direction of the SU (1, 1)-invariant Poisson structure1 . Klimek and Lesniewski gave in [36] a important extension of the work on the unit disc to higher genus Riemann surfaces g := g \D. In the latter and in [37], Berezin-Toeplitz quantization (or generalizations of it) is performed at the level of a universal covering exact Kähler space and then “quotiented down” to the compact base. This “covering” technique yields an alternative to geometric quantization of the base that avoids the pathology of the latter of only producing finite dimensional quantum observable C -algebras. Much less explicitly, but in a sense more directly from the operator algebraic viewpoint, Natsume and Nest in [38] produce a strict deformation quantization of g where each fiber is naturally Morita equivalent to a twisted group C -algebra C ( g , σ ), where σ, denotes a specific group 2-cocycle. Analogously, in their beautiful work [39], Carey et al. design within a geometrical context a variant of Connes ‘kernel algebra’ —called imprimitivity algebra— that they show to be Morita equivalent to the relevant twisted group C -algebra. At this stage, it seems relevant to point out that the crucial geometric ingredient in our construction is constituted by what we call the “admissible three-point function” 1 In [35], Cahen et al. essentially describe a procedure that naturally associates to the BerezinToeplitz quantization of a Kaehler manifold a formal -product in an analogous way a standard stationary phase expansion method yields Moyal’s -product from the non-formal Weyl quantization.
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Scan (see Sec. 2.3 ) which is intimately related with the Kubo cyclic cocycle described in Sec. 6 of Carey et al.’s work. Indeed, in the contracted case (M) Scan simply coincides with the symplectic area of the double triangle while in the Kähler case (D) they share the same asymptotic behaviour on “small” triangles. Therefore, it is very tempting to view the Kubo cocycle as the (first order approximation of the) pull back under the inverse of the double triangle map of our function Scan . The imprimitivity algebra (or alternatively Connes’ kernel algebra) would therefore appear as some kind of “resolution” of the operator algebra naturally attached to our framework2 . Nevertheless, not only from the methodological point of view but also regarding the results, the present work differs in several aspects from the above-mentioned previous works. Indeed, apart from the fact that it explicitly refers to Weyl-Moyal algebra, one cannot say that our construction belongs to the operator algebraic framework in the sense that it does not rely on any particular Hilbert representation. Also, the question addressed here concerns more the ‘space of all quantizations’ of D rather than the analysis of a particular one such as Berezin-Toeplitz within the non-formal context or Fedosov’s within the formal one. Of course, the fact that every such quantization is realized by intertwining the MoyalWeyl algebra under a convolution operator whose kernel is a solution of a second order linear PDE allows to define SU (1, 1)-equivariant operator calculi simply by structure transportation. However, the latter important aspect of our construction has not been investigated in the present article except, somewhat indirectly, in Sec. 3.7, where we reproduce Unterberger’s symbolic calculus. In particular the question of characterizing, within the context of the present construction, positive quantizations such as BerezinToeplitz or more generally coherent state quantizations will be addressed in a subsequent work. 2. Symplectic Symmetric Spaces 2.1. Definitions and elementary properties. Everything in this subsection is entirely standard and can be found e.g. in [40] and references therein. A symplectic symmetric space is a triple (M, ω, s), where M is a connected smooth manifold endowed with a non-degenerate two-form ω and where s : M × M → M : (x, y) → s(x, y) =: sx y is a smooth map such that for all points x in M, the partial map: sx : M → M is an involutive diffeomorphism of M (i.e. sx2 = id M ) which preserves the two-form ω (i.e. sx ω = ω) and which admits x as an isolated fixed point. One furthermore requires the following property: s x s y s x = ss x y to hold for any pair of points (x, y) in M × M. In this situation, the space M is endowed with a preferred affine connection. Indeed, for every triple of tangent vector fields X, Y and Z on M, the following formula: ωx ( ∇ X Y , Z ) :=
1 X x .ω( Y + sx Y , Z ) 2
2 Somewhat similarly as Klimek-Lesniewski ’s construction is related to the imprimitivity algebra in [39].
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defines3 a torsion-free affine connection ∇ on M that enjoys the properties of being preserved by every symmetry sx as well as being compatible with ω in the sense that: ∇ω = 0. This last fact implies in particular that ω is closed, turning it into a symplectic form on M. An important class of symplectic symmetric spaces is constituted by the non-compact Hermitian symmetric spaces. Such a space is a coset space M = G/K of a non-compact simple Lie group G by a maximal compact subgroup K that admits a non-discrete center Z (K ), the first example being the hyperbolic plane D := SL2 (R)/ SO(2). The compactness of K implies in particular that a Hermitian symmetric space admits a G-invariant Riemannian metric for which the connection ∇ is the Levi-Civita connection. However, in general a symplectic symmetric space needs not to be Riemannian, even not pseudo-Riemannian, in the sense that there is in general no metric tensor g on M such that ∇g = 0. In this sense a symplectic symmetric space is a purely symplectic object. In the problem we are concerned with in the present work, such a “non-metric” symplectic symmetric space will play a central role. Two symplectic symmetric spaces (Mi , s (i) , ω(i) )(i = 1, 2) are called isomorphic is there exists a diffeomorphism ϕ : M1 → M2 that is symplectic, i.e. such that ϕ ω(2) = (1) (2) ω(1) and that intertwines the symmetries: ϕsx ϕ −1 = sϕ(x) for all x in M (1) . Given a symplectic symmetric space (M, ω, s), its automorphism group Aut(M, ω, s) therefore turns out to be the intersection of the diffeomorphism group of affine transformations of (M, ∇) with the symplectic group of (M, ω): Aut(M, ω, s) = Aff(∇) ∩ Symp(ω).
(12)
The latter group is therefore a (finite dimensional) Lie group of transformations of M. One then shows that since it contains the symmetries {sx }x∈M its action on M is transitive, turning (M, ω) into a homogeneous symplectic space. In particular, every symplectic symmetric space is a coset space. Up to isomorphism, the list of homogeneous spaces underlying simply connected 2-dimensional symplectic symmetric spaces is the following: 1. the flat plane : R2 , 2. the hyperbolic plane : D := S L 2 (R)/S O(2), 3. the universal covering space of the anti-de Sitter surface : 2 := S L 2 AdS (R)/S O(1, 1), 4. the sphere : S 2 := S O(3)/S O(2), 5. the universal covering space of the Galileo coset : S O(2) × R2 /R, 6. the Poincaré coset : M := S O(1, 1) × R2 /R. Items 5 and 6 provide the first examples of non-metric symplectic symmetric spaces. An old classical result, independently due to Kirillov and Kostant [41,42], asserts that every simply connected homogeneous symplectic space is isomorphic to the universal covering space of a co-adjoint orbit of some Lie group. In our situation of symplectic symmetric spaces, this can be easily visualized. Indeed, fixing a base point o ∈ M the conjugation σ˜ : Aut(M, ω, s) → Aut(M, ω, s) : g → σ˜ (g) := so gso
(13)
3 See [40] and Appendix A for an explicit computation in the case of generic co-adjoint orbits of the
Poincaré (1,1) group.
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defines an involutive automorphism of the group Aut(M, ω, s). Its differential at the unit element therefore yields an involutive automorphism at the Lie algebra level: σ := σ˜ e : aut(M, ω, s) → aut(M, ω, s),
(14)
where aut(M, ω, s) denotes the Lie algebra of the automorphism group. The latter Lie algebra therefore decomposes into a direct sum of (±1)-eigenspaces for σ : aut(M, ω, s) = k ⊕ p.
(15)
Note that the differential at e of the coset projection: π : Aut(M, ω, s) → M, when restricted to the (−1)-eigensubspace p, provides a linear isomorphism with the tangent space at o: πe |p : p−→T ˜ o (M).
(16)
The pull back of the symplectic structure at o therefore defines a symplectic bilinear two-form on p: := (πe |p) ωo .
(17)
When extended by zero to the entire aut(M, ω, s), the element is easily seen to be a Chevalley 2-cocycle in the sense that: ([X, Y ], Z ) + ([Z , X ], Y ) + ([Y, Z ], X ) = 0.
(18)
The 2-cocycle needs in general not to be exact at the level of the automorphism algebra aut(M, ω, s). That is, there does not always exist a linear form ξo on aut(M, ω, s) such that (X, Y ) = ξo [X, Y ] .
(19)
Note nevertheless that the only 2-dimensional non-exact case corresponds to the flat plane R2 . We now adopt the notation G := Aut(M, ω, s); g := aut(M, ω, s).
(20)
Denoting by K the stabilizer subgroup of G of the base point o, one gets the G-equivariant identification G/K −→ M : g K → go,
(21)
where the symmetry map reads sgK (g K ) = g σ˜ (g −1 g ) K .
(22)
It turns out that the 2-cocycle is exact if and only if the action of G on (M, ω) is Hamiltonian in the sense that, endowing C ∞ (M) with the Lie algebra structure defined by the Poisson bracket associated to the symplectic structure ω, there exists a Lie algebra map λ : g −→ C ∞ (M) : X → λ X
(23)
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that satisfies the following property: {λ X , f }(x) =
d |t=0 f (exp(−t X )x) =: X x . f dt
(24)
for all functions f ∈ C ∞ (M). Note that the map λ is then necessarily G-equivariant in the sense that λAd(g)X (x) = λ X (gx)
(25)
for all g ∈ G, X ∈ g and x ∈ M. When Hamiltonian, one may choose: ξo (X ) := λ X (o).
(26)
In that case, denoting by O the co-adjoint orbit of the element ξo ∈ mapping:
g ,
J : M −→ O ⊂ g : x → [J (x) : g → R : X → λ X (x)]
the moment (27)
realizes a G-equivariant covering from M onto the co-adjoint orbit O. Note that the co-adjoint orbit O is itself endowed with a canonical symplectic structure ωO defined at the level of the fundamental vector fields for the co-adjoint action by ωξO (X , Y ) := ξ [X, Y ]
(ξ ∈ O).
(28)
With respect to the latter structure, the moment map J : M → O is symplectic. When non-exact, a passage to a central extension of g yields an entirely similar situation. Indeed, the transvection group (generated by all the sx ◦ s y and forming a normal subgroup of Aut(M) ) does not act in a strongly Hamiltonian manner on M. However, one may consider the (non-split) central extension 0 → RZ → g˜ := g ⊕ RZ → g → 0 defined by [X, Y ]∼ := [X, Y ] ⊕ (X, Y )Z , mimicking the passage from R2n to the Heisenberg algebra in the flat situation. In this new set-up, M may now be realized as a co-adjoint orbit of the extended group and the above results remain valid under essentially the same form [40]. To close this subsection, we observe that when the co-adjoint orbit O is simply connected the moment mapping J : M → O is then necessarily a global G-equivariant symplectomorphism. This turns out to be the case in items 1, 2, 4 and 6 in the above list of 2-dimensional spaces. 2.2. Group type symplectic symmetric surfaces, curvature contractions. A symplectic symmetric space (M, ω, s) is said to be of group type if there exists in its automorphism group G a Lie subgroup S that acts on M in a simply transitive manner, i.e. in a way that for all x in M there is one and only one element g in S with x = go. In that case, one has a S-equivariant diffeomorphism: S → M : g → go.
(29)
The symplectic structure ω on M then pulls back to the group manifold S as a leftinvariant symplectic structure ωS . Also, the symmetry at o corresponds to a symplectic involution M : S → S
(30)
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that encodes at the level of S the whole structure of symmetric space of (M, s): the symmetry at a point g ∈ S is given by: sgM (g ) := g. M (g −1 .g ).
(31)
A quick look at the above list in the 2-dimensional case leads us to the observation that: Up to isomorphism, there are two and only two non-flat symplectic symmetric affine geometries that are of group type: the hyperbolic plane D and the Poincaré orbit M. The corresponding automorphism subgroups are in fact both isomorphic to the 2-dimensional affine group S that we realize as S := {(a, )} with the group law:
(a, ).(a , ) := ( a + a , e−2a + ).
(32)
The unit element is then e = (0, 0) and the inverse map is given by (a, )−1 = (−a, −e2a ). Within this setting (see Appendix A), the affine structures are encoded by the maps : M (a, ) := ( −a , − ); and D
(a, ) :=
1 2 −a − log(1 + ) , − . 2
(33)
(34)
Regarding the symplectic structures, one observes that the constant 2-form da ∧ d is invariant under the left-action. Therefore, for every k ∈ R+0 , the symplectic structure √ ω(k) := k da ∧ d (35) induces the following symplectic symmetric surfaces: (S, s D , ω(k) ) and (S, s M , ω(k) ). The first one is isomorphic to the hyperbolic plane D with curvature −1 k . In the second one, the parameter k is indifferent in the sense that for all k, one has the isomorphism: (S, s M , ω(k) ) (S, s M , ω(1) ). 2.3. Admissible functions on symplectic symmetric surfaces. In [33], Alan Weinstein conjectures the relevance of a certain three-point function, here denoted SW , as the essential constituent of the phase of an oscillatory kernel defining an invariant star product on the hyperbolic plane D or more generally on any (reasonable) symplectic symmetric space O = G/K . Essentially, when three points x, y and z in O are close enough to one another, Weinstein function SW = SW (x, y, z) is defined as the symplectic area of the geodesic (defined by the Loos connection) triangle, in O, admitting points x, y, z as mid-points of its geodesic edges. Additionally to its invariance under the symmetries, the three-point function SW (when defined) has been shown in [44] to satisfy the following so-called admissibility condition (36). A three-point function S ∈ C ∞ (O × O × O, R) is called admissible if it is invariant under the diagonal action of the symmetries on O × O × O, is totally skewsymmetric, and if it satisfies the following property: S(x, y, z) = −S(x, sx y, z).
(36)
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539
Fig. 1. Contraction of the Kähler orbit into a non-metric hyperbolic cylinder
As it will appear, the above condition is in fact the crucial one regarding star-products. In the case of a symplectic symmetric surface, every such (regular) admissible function turns out to coincide with an odd function of a canonical admissible function. The latter function, denoted hereafter Scan , is defined in terms of the co-adjoint orbit realization of the symplectic symmetric surface at hand. We now explain how to define it. Locally, every geodesic line starting at the base point o := K and ending at x can be realized as the (parametrized) orbit of o by a one-parameter subgroup exp(t X ) of G with X ∈ p (see e.g. [43]). An invariant totally skewsymmetric smooth function S ∈ C ∞ (O3 ) will be called regular admissible if S(o, x, y) = S(o, x, exp(t X ).y), for all t ∈ R and y ∈ O. Observe that regular admissibility implies admissibility as a consequence of the following classical identity: exp(X ) = sExp
1 o( 2
X)
◦ so
(X ∈ p),
(37)
where Expo denotes the exponential mapping at point o with respect to the canonical connection. The inclusion p ⊂ g induces a linear projection : g → p . The dual space p naturally carries a (constant) symplectic 2-form here denoted again by . Identifying points with vectors in p , given two points ξ and η, the quantity (ξ, η) therefore represents the flat symplectic area of the (flat) Euclidean triangle in p admitting points 0, ξ and η as vertices. By transversality, the restriction of the projection to the co-adjoint orbit O ⊂ g is locally (around o) a diffeomorphism. Denoting by x2 the unique point in a small neighbourhood of o such that s 2x o = x, the following formula: Scan (x, y, z) := ( (s x2 (y)) , (s x2 (z)) ) defines a (local) regular admissible function on O. In the case where the surface O is of group type, the function Scan is globally defined and smooth. Moreover every regular admissible function is an odd function of Scan . The proof essentially relies in the fact that the -projected exp(t X )-orbits in O exactly coincide with straight lines in p . Indeed, we first observe that the diffeomorphism establishes a bijection between the exp(t X )-orbits (X ∈ p) in O and the straight lines in p . Indeed, for x ∈ O and X ∈ p, one has < Ad (exp(t X ))x − x, X >= 0, where Ad denotes the co-adjoint action, which means that the x-translated exp (t X )-orbit of x lies in the plane in g orthodual to X ∈ p. This plane is generated by the kernel k of the projection : g → p and an element X ⊥ of p orthodual to X . In particular, it projects onto the line directed by X ⊥ . Now consider the two-point function κ(x, y) := S(o, x, y) on M induced by the data of an admissible function S on M. This function corresponds to a two-point function κ 0 on p via the diffeomorphism . By admissibility and the above observation, one has κ 0 (ξ, η) = κ 0 (ξ, η + tξ ) for all t ∈ R,
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Fig. 2. The canonical three-point phase obtained from projecting the orbit
Fig. 3. The double geodesic triangle
which is precisely the property of admissibility for a two-point function with respect to the flat structure on p . The rest then follows from Proposition 3.3 in [44]. It is remarkable that, in the case of Poincaré orbit M, the equation sx s y sz t = t admits a unique solution t[x,y,z] for all data of three points x, y and z in M. Moreover, the ‘double triangle’ mapping4 : : M × M × M → M × M × M : (x, y, z) → (t[x,y,z] , sz t[x,y,z] , s y sz t[x,y,z] ) is a global diffeomorphism whose jacobian determinant equals Jac (x0 , x1 , x2 ) = 16 cosh(2(a0 − a1 )) cosh(2(a1 − a2 )) cosh(2(a2 − a0 )). This is a straightforward computation based on the following formulas for the symmetries: M s(a,) (a , ) = ( 2a − a , 2 cosh(2(a − a )) − ),
as well as for the mid-point map: m : M × M → M : (x, y) → m(x, y) =
1 1 (ax +a y ), (x + y )sech(ax −a y ) 2 2
(38) , (39)
4 Within A. Weinstein terminology.
Deformation Quantizations of the Hyperbolic Plane
541
defined by the relation M x = y. sm(x,y)
One has −1 (x, y, z) = (m(x, y), m(y, z), m(z, x)); and a computation yields Jac −1 (x0 , x1 , x2 ) =
1 sech(a0 − a1 )sech(a1 − a2 )sech(a2 − a0 ). 16
One then obtains the announced formula by using the relation: Jac = ( Jac −1 )−1 . At last, the canonical admissible three-point function, in this particular situation of M, exactly coincides with Weinstein’s function: ( M)
( M) Scan = Symplectic Area of = SW .
In coordinates, one has ( M) (x0 , x1 , x2 ) = sinh(2(a0 − a1 )) 2 + sinh(2(a2 − a0 )) 1 + sinh(2(a1 − a2 )) 0 . Scan
In the hyperbolic plane case, however, the situation is not as nice. Indeed, a bit of reflection leads to the fact that Weinstein’s function is not well-defined for every triple of (D ) points x, y and z in D. In particular, since by construction Scan is smooth it must differ (D ) from SW . Nevertheless, as proven above, they should locally be odd functions of each other. Precisely, one has the relation: (D )
SW
(D ) = 2 arcsin(Scan ).
In terms of the Kähler potential ζ of the hyperbolic plane realized as the Poincaré unit disc, one has (D ) (0, z, w) = 4 ζ (z) ζ (w) Im(zw). Scan
In coordinates, one has (D ) Scan (0, x0 , x1 ) = 1 sinh 2a0 − 0 sinh 2a1 +
0 1 (0 e2a0 − 1 e2a1 ). 2k
(40)
As we will see in the sequel, the 3-point kernel defining the associative deformation product in the case of the hyperbolic plane turns out to be expressed as a special function (of one real variable) of the function Scan only (no approximation). While in the solvable contracted case, the latter coincides (up to a co-boundary) with the phase of the quantization kernel. This canonical function therefore is a unifying notion for both contracted and un-contracted situations.
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3. Deformation Quantization 3.1. The space of star products on the Poincaré orbit M := S O(1, 1) × R2 /R. It is proven in [30] that every Poincaré invariant star product on M is realized as a formal asymptotic expansion in powers of of an oscillatory integral expression in terms of the geometrical quantities defined above: i (M) P(a0 − a1 )P(a2 − a0 ) 1 u ,P v (x0 ) = 2 Jac (x0 , x1 , x2 ) e Scan (x0 ,x1 ,x2 ) M× M P(a1 − a2 ) × u(x1 ) v(x2 ) dx1 dx2 ; where dx denotes the Liouville symplectic measure on M and where P is an essentially arbitrary nowhere vanishing, complex-valued one variable function, possibly depending smoothly in the real deformation parameter . The function P therefore is the only degree of freedom (see [30] for details). We now briefly recall how the above result is obtained. The key point relies in the fact that the global Darboux coordinate system (a, ) on S enjoys a property of compatibility with the hyperbolic action of G := S L 2 (R) on D = S. In the case L is a Lie group, with Lie algebra l, that acts on a symplectic manifold (M, ω) in a Hamiltonian manner, a (not necessarily L-invariant) star product on (M, ω) is called l-covariant if, denoting by λ : l −→ C ∞ (M) : X → λ X
(41)
the associated (dual) moment mapping, the following equalities hold: i i [ λ X , λY ] := (λ X λY − λY λ X ) = { λ X , λY } = λ[X,Y ]
(42)
for all X, Y ∈ l. In this situation, one has a representation of l on C ∞ (M)[[]] by derivations of the star product : ρ : l −→ Der(), i [ λ X , u ] . ρ(X ) u :=
(43) (44)
It turns out that in coordinates (a, ) the Moyal product (5) is sl2 (R)-covariant with respect to the hyperbolic action of G := S L 2 (R) on the hyperbolic plane D = S [40]. Precisely, presenting the Lie algebra g := sl2 (R) as generated over R by H, E and F satisfying: [H, E] = 2E, [H, F] = −2F, [E, F] = H, the moment map associated with the action of G on D = G/K = S reads: √ √ √ k −2a k 2a e e (1 + 2 ). λH = k ; λE = ; λF = − 2 2
(45)
(46)
The associated fundamental vector fields are given by: H = −∂a ; E = −e−2a ∂ ; F = e2a (∂a − (k + 2 )∂ ).
(47)
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543
At last, the representation of g by derivations of 0 admits the expression: ρ(H ) = −∂a , e−2a sin(∂ ), ρ(E) = − 2a 2 2 sin(∂ ) sin(∂ )∂a + cos(∂ )∂a − (k + ) . ρ(F) = e 4 We now consider the partial Fourier transform in the -variable ∞ F(ϕ)(a, ζ ) = e−iζ ϕ(a, )d, −∞
(48) (49) (50)
(51)
and denote by S˜ := {(a, ζ )} the space where the Fourier transformed F(g) is defined. ˜ Defining the following one-parameter family of diffeomorphisms of S: sinh( ζ ) φ(a, ζ ) = a, , (52) ˜ (resp. S(S)) on and denoting by S˜ (resp. S) the space of Schwartz test functions S(S) S˜ = {(a, ζ )} (resp. on S(S) of S = {(a, )}), one observes the following inclusions: φ S˜ ⊂ S˜ and S˜ ⊂ (φ−1 ) S˜ ⊂ S˜ .
(53)
Therefore, every data of (reasonable) one parameter smooth family of invertible functions P = P(b) yields an operator on the Schwartz space S(S) of S = {(a, )}: T −1 : S(S) −→ S(S) defined as T −1 ϕ(a0 , 0 ) =
1 2π
−i
eiζ 0 P(ζ ) e sinh(ζ ) ϕ(a0 , ) d dζ.
(54)
(55)
More generally, denoting by M f the pointwise multiplication operator by f , the operator: T : S(S) −→ S (S)
(56)
T := F −1 ◦ (φ−1 ) ◦ M 1 ◦ F P
(57)
defined as
is a left-inverse of T −1 . Intertwining Moyal-Weyl’s product by T yields the above product as ,P = T (0). Observe that the latter closes on the range space E := T S yielding a (non-formal) one parameter family of associative function algebras: (E, ,P ). Asymptotic expansions of these non-formal products produce genuine Poincaré-invariant formal star products on M [45]. For generic P the above oscillatory integral product defines, for all real value of , an associative product law on some function space (as opposed to formal power series space) on M, the most remarkable case being probably the one where P is pure phase. In the latter case, the above product formula extends (when = 0) to the space L 2 (M) of
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square integrable functions as a Poincaré invariant Hilbert associative algebra. The star product there is strongly closed: for all u and v in L 2 (M), u ,P v belongs to L 1 (M), and one has: u ,P v = u v. M
M
The contracted situation is therefore, to some extent, relatively well understood. 3.2. sl2 (R)- triplets of derivations and an unexpected Lorentzian structure. Let us now consider any Poincaré invariant formal star product on the contracted plane M. Denote by Der() its algebra of derivations. Note that from formal equivalence with Moyal, every derivation is interior. Consider in Der() any element D with the property that D, H and E form an sl2 (R)-triplet of -derivations. The crucial point, which the entirety of the present paper relies on, resides now in the following totally unexpected fact: Intertwining the derivation D by the partial Fourier transform yields a second order hyperbolic differential operator := F ◦ D ◦ F −1 whose principal symbol defines a Lorentzian metric which does not depend on any particular choice made (of , sl2 (R)triplet derivation algebra and D). The latter Lorentzian metric is therefore a new object that is canonically associated with the hyperbolic plane, its canonical contraction and their quantization. To prove the above assertion we first observe that the above covariance property implies that the operator D0 := T ◦ ρ(F) ◦ T −1
(58)
is a derivation of := T (0 ). Now the particular choice of P(ζ ) ≡ 1
(59)
yields the following expression for := F ◦ D0 ◦ F −1 , a second order differential operator: 2 2a ζ ∂a2 + ζ (1 + 2 ζ 2 )∂ζ2 + (1 + 2 ζ 2 )∂a ∂ζ + 2 ζ ∂a + (2 + 32 ζ 2 )∂ζ (a,ζ ) = ie 4 −ζ (k − 2 ) . (60) ˜ Note the occurrence of the Lorentzian metric on S: 2 [g ] := e (1 + ζ ) ij
2a
2
2
ζ 4 (1+2 ζ 2 )
1
1 ζ
.
(61)
Now, consider an arbitrary Poincaré invariant star product on M. Note that, denoting by s the Lie algebra of S, the above Moyal-covariance property yields a s-quantum moment for , i.e. a linear map sl2 (R) → C ∞ (M)[[]]; X → X such that 1 1 2 [ X , Y ] = [X,Y ] and V = 2 [ V , . ] for all V ∈ s. Fix D0 in Der() with the same property as D and set D =: D0 + D1 :=: D0 + [λ1 , . ] with λ1 ∈ C ∞ (M)[−1 , ]]
Deformation Quantizations of the Hyperbolic Plane
545
(one knows from the equivalence with Moyal that Der() is interior). The triplet condition yields the following conditions: [E , D1 ] = 0 and [H , D1 ] = −2D1 . This implies 21 [ E , λ1 ] = E .λ1 = c E and H .λ1 = −2λ1 + c H , where c E and c H are formal constants. From the expressions (47), one gets λ1 = −c E e2a + const. i.e. D = D0 + c [e2a , . ] , where c ∈ C[−1, ]]. We now argue that the operator F D1 F −1 := F ◦ c [e2a , . ] ◦ F −1 is differential and at most first order. Indeed, starting with D0 := T −1 ρ(F)T, where = T (0 ), one observes by looking at the expression (46) of the classical moment that D exactly corresponds to λ F affected by a translation in ): (a, ) → (a, + c)5 (and a re-definition of k). From expression (50), one deduces that T D1 T −1 is a linear combination of cos(∂ ), and sin(∂ ). The latter correspond under the T equivalence to multiplication and vector fields operators. To complete the argument, we end by observing (see [30]) that one passes from one Poincaré invariant star product to by an equivalence of the form F −1 ◦ M Q ◦ F, where M Q denotes the multiplication operator by a one variable (formal) function Q = Q(ξ ) independent of s. The latter has the effect of a simple gauge transformation affecting only by lower order terms. 3.3. The de-contraction procedure: evolution of the Dalembert operator. Let us now consider an invariant formal star product on the hyperbolic plane D. This star product is in particular S-invariant. One knows from [8], that the set of S-equivariant equivalence classes of S-invariant star products is in one-to-one correspondence with the set of formal series with coefficients in the S-invariant second de Rham cohomology space. In the pres2 S ent two-dimensional situation, the latter space is simply R[[]] since Hde Rham (D) is generated by the (non trivial) class of the invariant symplectic structure (or area form). From [46], one may therefore pass from one equivalence class of star products to another by re-defining the deformation parameter. In particular, up to a change of parameter, our star product can be obtained by intertwining a given Poincaré invariant star product on the contracted plane M through a formal equivalence U that commutes with the left action of S (identifying D, M and S). The equivalence U between and = U () must therefore be a convolution operator by a (formal) distribution u ∈ D (S) on S, i.e. of the form: (U f )(x) = u(y −1 x) f (y) d L y, (62) S
where d L y denotes a left-invariant Haar measure on S (remark that it coincides with the Liouville area form on S = D = M). Now, on the one hand, the element D := D F := U −1 ◦ F ◦ U is a derivation of that generates together with E and H a sl2 (R)-triplet derivation algebra. On the other hand the above subsection provides us with an explicit expression for D. The identity D ◦ U −1 = U −1 ◦ F
(63)
may then be interpreted as an equation that must be satisfied by the intertwiner U (or rather U −1 ). Denoting by v the distribution on S defining the kernel of U −1 , for any test function f , the latter corresponds to −1 L (64) Dx v(y x) f (y) d y = v(y −1 x) (Fy f ) d L y, 5 Note that this symplectic transformation leaves Moyal’ star product invariant.
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which yields, since F is a symplectic vector field: Dx [v(y −1 x)] f (y)d L y = − Fy [v(y −1 x)] f (y)d L y.
(65)
Since the vector field Fy and the operator Dx commute, the last equation leads to the following evolution equation for D: Dx [v(y −1 x)] = −Fy [v(y −1 x)].
(66)
In particular, we have shown that: Every S L 2 (R)-invariant star product on the hyperbolic plane D can be obtained by intertwining an arbitrary Poincaré invariant star product on the contracted hyperbolic plane M through a left invariant convolution operator U −1 on S whose associated kernel v ∈ D (S) is solution of the following problem: − z˜ W (x, z˜ ) = −Fx W (x, z˜ ), with
W ( x , (b, ζ ) ) :=
∞ −∞
e−iζ v(x −1 .(b, )) d;
(67)
(68)
˜ (x ∈ S , z˜ = (b, ζ ) ∈ S). We end this subsection by observing that the inverse map S −→ S : x → x −1
(69)
induces a duality between the space of kernels v of the above inverted intertwiners U −1 and that of kernels u defining the direct intertwiners U . To see this, we first observe that given a Poincaré invariant star product with associated trace form Tr , there exists a Poincaré equivariant equivalence: D (S)[[]] −→ D (S)[[]] : ϕ → ϕ such that every S-commuting intertwiner U may be expressed as
U (ϕ)(x) = Tr L x −1 u ∨ ϕ ;
(70)
(71)
where ϕ ∨ (x) := ϕ(x −1 ). The latter being obvious for any strongly closed star product [47] (with, in this case, ϕ = ϕ), one gets it for every star product by use of an equivalence with a strongly closed one. Right composition by Poincaré equivariant equivalences obviously preserves the space of S-commuting intertwiners. Hence, every S-commuting intertwiner U may also be expressed as
U (ϕ)(x) = Tr L x −1 u ∨ ϕ , (72) for some u ∈ D (S)[[]]. Therefore, the equation U D = F U admits the expression:
(73) Tr L x −1 u ∨ Dϕ = Tr Fx (L x −1 u ∨ ) ϕ . The element D being a derivation of , one has:
Tr L x −1 u ∨ Dϕ = −Tr D(L x −1 u ∨ ) ϕ ,
(74)
Deformation Quantizations of the Hyperbolic Plane
547
which yields: D(L x −1 u ∨ ) = −Fx (L x −1 u ∨ ).
(75)
Now, from the particular form of Poincaré equivariant equivalence as power series in the left-invariant E˜ with constant coefficient6 [30]: ck k E˜ k , (76) τ˜ := I + k≥1
one observes that u ∨ = τ˜ (u ∨ ) = (τ (u))∨ , with τ := I +
ck k E k .
(77)
(78)
k≥1
3.4. Finding solutions of the Dalembertian evolution by variable separation. For convenience, we fix a particular choice of closed star product on M associated, as above, with the function P(ζ ) = P closed (ζ ) := cosh(ζ ). (79) We denote the corresponding intertwiner by T closed . We then note that, from the above discussion, the analogue of Eq. (67) for the direct operator U in this case can be rewritten under the following form:
with
− |¯z W (x, z¯ ) = Fx W (x, z¯ ),
(80)
W (x, z¯ ) = (φ∗ )−1 ◦ F ◦ T closed u(z −1 x), |z
(81)
and where the operator is given by (60). Observe that from (81) the function W (x, z¯ ) = W (a, n, b, r ) takes the form W (a, n, b, r ) = (1 + 2 r 2 )1/4 q e−ir nq F(q, r ), with q = e2(a−b) and
F(q, r ) =
1 ei q r y u( ln q, y) dy, 2
(82)
(83)
and where (a, n) are the coordinates of x, (b, ) those of z, and, (b, r ) those of z¯ , r being the variable conjugated to through the Fourier transformation F which only affects the second z-coordinate. The right-hand side of Eq. (80) is simpler:
(84) Fx W (x, z¯ ) = e2 a q 2 q n ∂q − (k + n 2 )∂n W (q, n, r ),
6 E˜ := d x exp(t E) x dt 0
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P. Bieliavsky, S. Detournay, Ph. Spindel
k being for the operator Fx the corresponding of k for the operator F in ρ1 (F)|z (see Eqs. (47)), so finally Eq. (80) simplifies to the following equation for F(q, r ):
−r 2 q 2 (1 + 2 r 2 )∂q2 − r (1 + 2 r 2 )2 ∂r2 + 2 q (1 + 2 r 2 )2 ∂r ∂q −2 2 r 2 (1 + 2 r 2 )∂r + V F(q, r ) = 0, (85) with V = V (q, r ) := r (−
2 (2 + 2 r 2 ) + (1 + 2 r 2 )(k − k q 2 )). 4
(86)
The problem amounts thus to solve the corresponding equation for F(q, r ), from which we could find u by inverting (83). We set G(q, r ) = q 1/2 (1 + r 2 2 )1/4 F(q, r ),
(87)
1 1 U = q 2 sinh( ζ ), V = q 2 cosh( ζ ), 2 2
(88)
and use the variables:
with r =:
1 sinh(ζ ).
(89)
The equation then becomes ˜ ∂U ∂V + Q(U, V ) G(U, V ) = 0,
(90)
with UV 4 2 ˜ Q(U, V) = − 2 2 ( − k + k (U 2 − V 2 )2 ). (U − V 2 )2 4 Note that here : U 2 − V 2 = −q, hence U 2 − V 2 < 0. Setting U 2 = V 2 = 21 (t + x) and G(q, r ) = H (t, x), one then gets x 2 (∂t2 − ∂x2 ) −
1 2 2 − k + k [ x ] H (t, x) = 0. 2 4
(91) 1 2 (t
− x),
(92)
Applying a method of separation of variables to the latter yields solutions as superpositions of the following modes: F,s (q, r ) = h s (q)(1 + 2 r 2 )−1/4 eisq
√
1+2 r 2
,
(93)
with √ 2 2 h s (x) = A J k [ s + k / x] + B Y k [ s 2 + k /2 x], √
(94)
Deformation Quantizations of the Hyperbolic Plane
549
where Jµ and Yµ denote Bessel functions of the first and second kinds (see ref. [48]). From (83) e2a 2a (95) u(a, l) = e−i r l e F(e2a , r ) dr, 2π thus (up to constant factors) u ,s (a, l) = e2a h s (e2a )
Hs (a, r ) e−i b l e dr, 2a
(96)
with Hs (a, r ) = (1 + 2 r 2 )−1/4 eis
√
1+2 r 2 e2a
.
(97)
We may use a similar technique to determine the kernel of the inverse operator U −1 . By writing −1 v(z −1 x) f (z)dz, (98) [U f ](x) = one finds:
with
x¯ M(x, ¯ z) = −Fz M(x, ¯ z)
(99)
v P (z −1 x). M(x, ¯ z) = (φ∗ )−1 ◦ F ◦ T P |x
(100)
Solutions to (99) can be deduced by using a slight generalization of what we have done here above. We finally get, √ 2 2 −2a v,s (a, l) = e−2a h s (e−2a ) (1 + 2 r 2 )−1/4 eis 1+ r e e−ibl db. (101)
3.5. Normalizations and asymptotic expansions. Observe that the change of variables (88) becomes singular when vanishes, while Eq. (85) degenerates into (−r ∂r2 + 2 q ∂r ∂q + r (k − k q 2 ))F(q, r ) = 0.
(102)
The general solution of this equation can be expressed as a superposition of modes σ q r2 − i 1 F0,σ = C √ ei 2 e 2 σ q
k q +k q
.
(103)
Let us notice that these modes can be obtained from special combinations of modes (94) and a rescaling of the wave-number s into s = σ/2 .7 This rescaling is dictated by the necessity of maintaining an r dependence in the limit → 0 of the modes (94): √ σ s 1 + 2 r 2 ≈ s + s 2 r 2 = 2 + σ r 2 , (104) 7 For illustrative purpose, we limit ourselves to s > 0.
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P. Bieliavsky, S. Detournay, Ph. Spindel
while the oscillating factor can be controlled by considering the asymptotic behavior of the Hankel functions ([48], Sec. 8.41)
√ σ 2 k 2 −i σ 2q i k π − 2iσ qk +k q (2) 2 √ e H k( + q) ≈ e e . (105) 4 2 πσq Thus by choosing the coefficients A and B in the linear combination (94) as : √ 1 π σ −i k π e 2 A = −i B = C 2
(106)
we obtain modes whose limit for = 0 is well defined. This explains the results of our computations below. Note that, given an operator U , it is not straightforward to extract the inverse operator U −1 since one has to determine how to superpose these different modes. In Sec. 3.7 however, we will discuss a particular example where the superposition is known. The operators U we consider hereafter will correspond to (96), (94) with B = 0. This special choice is motivated by the bad behavior of the Y -Bessel function near the origin. We have: 1 √ (Us f )(a, l) = C(s) dr dn d x J k ( s 2 + k /2 x)(1 + 2 r 2 )− 4 R 2 ×R +
×e
√ i s x 1+2 r 2 i r lx −i r n
e
e
f (a −
1 ln x, n). 2
(107)
A necessary condition for Us to define a star product is that (Us 1) = 1. This leads to the condition8 √
k √ √ k k − i s[] 2 C(s) = . √ 2π k + i s[]
(108)
Let us notice that we allow an dependence in s; we don’t assume it a priori to be constant but leave it as a free (regular) function of the deformation parameter. On the other hand, the requirement Us f −→ f → 0
(109)
to get the right classical limit further imposes k = k.
(110)
Let us now turn to the asymptotic expansion of the operator (107). To this end we introduce the following Fourier-like transforms: 1 −2a f˜− (k, p) = (111) f (a, l)e−ike e−i pl e−2a da dl, 2π 2 −2a f (a, l) = f˜− (k, p)eike ei pl dkdp, (112) 8 A useful relation in this derivation (see ref.[48], Sec. 13.2, Eq. [8]) is :
√ β −λ [ √β 2 −κ 2 +iκ]λ . β 2 −κ 2
∞ (iκ−)ζ Jλ (β ζ ) dζ = 0 e
Deformation Quantizations of the Hyperbolic Plane
551
from which one obtains √ (Us f )(a, l) = k
(1 + 2
p 2 )−1/4
k
+ 2 (s 2 () − κ 2 ) + i √
(k + 2 (s 2 () − κ 2 ))1/2
κ
κ = r e−2a + p l + s()
k
k + i s()
× f˜− (r, p) dp dr, where
√
(113) √
1 + 2 r 2 .
(114)
Setting X = r e−2a + p l, the integrand can be Taylor expanded around = 0 as √
k
(1+2
p 2 )−1/4
(k + 2 (s 2 ()−κ 2 ))1/2
k +2 (s 2 ()−κ 2 )+i κ √ k +i s()
√
k
= ei X P(X, p 2 ), (115)
where (115) defines the power series P(X, p 2 ), which, at first (non-trivial) order9 , reads as: (1 + i s) 2 (1 − 2 i s) s i 3 + X+ X + X + O(4 ). (116) P(X, p 2 ) = 1 + 2 − p 2 4 k 2k 6k The important point in the previous expression is the structure of the expansion, and in particular the occurrence of the factor ei X . Using this result, the expansion of the operator U follows: (Us f )(a, l) ≈ P(X, p 2 )ei X f˜− (r, p) dp dr = P[−i∂σ , −∂α2 ] eiσ X eiαp f˜(k, p)− dp dr σ =1,α=0 1 = P[−i∂σ , −∂α2 ] f (a − ln σ, σ l + α) . (117) 2 σ =1,α=0 It therefore appears that the product so-defined deforms the pointwise product on the hyperbolic plane in the direction of the SL(2, R)-invariant Poisson bracket. A similar computation can be performed starting from the inverse operator. One has (with x = e2(b−a) ) 1 D(s) (Vs f )(a, l) = dr dn d x J √k ( s 2 + k /2 x)(1 + 2 r 2 )− 4 2 R 2 ×R + √ 1 2 2 (118) ×ei s x 1+ r eir (n x−l) f (a + ln x, n). 2 The requirement Vs 1 = 1, imposes √
k √ √ k k − i s() 2 D(s) = √ π k + i s()
(119)
9 If we assume s to be constant, instead of a function of , this expansion involves only even powers of .
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P. Bieliavsky, S. Detournay, Ph. Spindel
while the right classical limit implies 10 k = k,
(120)
in accordance with Eq. 110. Using again the expression of the Fourier transform of the Bessel function but the Fourier transform 1 2a f˜+ (k, p) = f (a, l)e−ike e−i pl e2a da dl, (121) 2π 2 +2a f (a, l) = f˜+ (k, p)eike ei pl dkdp, (122) one is led to √ k k + 2 (s 2 () − ρ 2 ) + i ρ (1 + 2 p 2 )−1/4 ei pl ei q n (Vs f )(a, l) = √ 2π (k + 2 (s 2 () − ρ 2 ))1/2 k + i s() × f˜+ (r, q) dp dr dn dq, where
√
k
(123)
ρ = r e2a − p n + s() 1 + 2 p 2 .
(124)
Setting Y = r e2a − p n, we may write √
k
(1+2
p 2 )−1/4
(k +2 (s 2 ()−ρ 2 ))1/2
k
+ 2 (s 2 ()−ρ 2 )+i √
ρ
√
k
k + i s()
= eiY P(Y, p 2 ), (125)
where P has the same expression as in (115). (Let us remark that this holds only because we use the strongly closed S-invariant star product.) One successively gets 1 (Vs f )(a, l) = dp dr dn dq ei p l ei q n P(Y, p 2 )eiY f˜+ (r, q) 2π 1 = dp dr dn dq P[−i∂σ , −∂l2 ]eiσ Y ei p l ei q n f˜+ (k, q) σ =1 2π 1 1 = dp dn P[−i∂σ , −∂l2 ] f (a + ln σ, n)ei p (l−n σ ) 2π 2 σ =1 1 l 1 = P[−i∂σ , −∂l2 ] f (a + ln σ, ) . (126) σ 2 σ σ =1 To end this section, let us remark that Eq. 96 and Eq. 101 together with the normalisation conditions (108,110) provide explicit expressions of the intertwiners between Sl(2, R)-invariant star products on the hyperbolic plane D and a specific strongly closed Poincaré-invariant star product on the contracted plane M. On the formal side, Eqs. (117,126) provide generating functions for asymptotic expansions up to any order. 10 A useful relation in that case is (see ref.[48], Sec. 13.2, Eq. (7)) :
√
[ β 2 −κ 2 +iκ]λ . λ βλ
∞ e(iκ−)ζ Jλ (β ζ ) dζ = 0 ζ
Deformation Quantizations of the Hyperbolic Plane
553
2 3.6. The case k = 4 and Zagier type deformations. We now consider the case of our closed star product := T closed (0) together with a particular derivation D ∈ Der()
defined as setting k = 4 in the expression (50) and then intertwining by T closed following (58). We then proceed as in Subsect. 3.4, but use the following variables rather than (88): 1 2 1 (127) ξ = q 2 sinh( ζ ), η = q 2 cosh( ζ ). 2 2 Note that in contrast with the coordinates (88), the above ones are not singular in the 1 1 limit → 0. Indeed, they become ξ0 = q 2 ζ and η0 = q 2 in the limit. Within these normalisations, the Eq. (90) then becomes: 2
∂ξ ∂η G = 4k ηξ G;
(128)
G(q, r ) = q 1/2 (1 + r 2 2 )1/4 F(q, r ).
(129)
with, as before, Note that from (83), the classical limit → 0 of the latter is prescribed to equal Fcl := δ0 (a) ⊗ 1r . (130) One way to achieve this requirement is to allow k to depend on in such a way that the condition (110) remains satisfied lim k () = 0.
→0
(131)
More naturally, this corresponds to considering the -independent wave equation in the case k = 0: ∂ξ ∂η G = 0, which trivially admits the ξ -independent solution: √
1 √ 2 a 2 G = G(η) = G e 1 + 1 + 2 r 2 ; 2
(132)
(133)
with G(ea ) := ea δ0 (a). The above discussion leads us to the particular solution: 1 √ √ 2a ∞ 2 2 2 2e ir e2a 1 + 1 + r u (a, ) := e √ 4π −∞ 1 + 2 r 2 √ 1 1 2 2 × δ0 a + log( (1 + 1 + r ) ) dr ; 2 2 that admits the correct limit: u 0 (a, ) = δe (a, ).
(134)
(135)
(136)
The associated convolution operator U produces a deformation quantization that is invariant under the infinitesimal action of sl2 (R) on an open S-orbit in R2 (the group S acting by special linear transformations). The corresponding underlying geometry here is flat (k = 0). In particular, this class of solutions reproduces star products of the same type as the one considered by Connes and Moscovici in [49], primarily constructed by Zagier as a (sl)2 (R)-invariant deformation of the algebra of modular forms [50].
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P. Bieliavsky, S. Detournay, Ph. Spindel
3.7. Unterberger type solutions and Weinstein’s asymptotics. In [31,32] an associative SL(2, R) invariant composition law, though not a star product, has been derived in a totally different context by A. and J. Unterberger, for the composition of symbols in the so-called Bessel calculus. We are going to show that a slight modification of their formula yields one of the simplest products in the family we have found. As a preamble, let us notice that in their works, these authors also make use of an interwiner operator linking an SL(2, R)- invariant composition law with an AN -invariant star product, but not the strongly closed one we have adopted. Instead of this one, the star product they use may be expressed in our framework as the one obtained from (55) with special weight function P(ξ ) = cosh(ξ ), as proven in [51]. The convolution kernel of an intertwiner between an SL(2, R)-invariant star product # and a general AN -invariant product, itself built from the Moyal star product twisted by an operator TP is obtained from the operator U we constructed in Sect. 3.4 by: UP = U ◦ (T closed )−1 ◦ TP .
(137)
In particular, if TP is of the form of Eq. (55), the kernel of UP is then given as a superposition of modes, φ(s )u P (138) u P (a, l) = ,s (a, l) ds , with 2a 2a uP ,s (a, l) = e h s (e )
1
(1+2 r 2 )− 2 P(arcsinh(r )) eis
√
1+2 r 2 e2a irle2a
e
dr,
(139)
where h s (x) ∝ J√k/[ s 2 + k/2 x].
An analog computation yields the kernel of the inverse operator UP−1 again as a superposition of modes, each of them being given by √ 2 2 −2a P −1 (arcsinh( r ))eis 1+ r e e−irl dr. (140) vP,s (a, l) = e−2a h s (e−2a )
Let us plug B = 0 in (94), and normalize the modes accordingly to Eqs. (108, 119) with k = k (see Eqs. (110, 120). The operator U considered in [31,32] then corresponds to the value s = 0 in (96). This obviously amounts to pick up φ(s ) = δ(s ) in Eq. (138). By taking P( ξ ) = cosh( ξ ), and denoting by UU the corresponding operator, one finds √ √ k 1 (141) (UU f )(a, l) = d x J √k ( k x/) f (a − ln x, l x). 2 The inverse operator, UU−1 corresponds to the kernel resulting from the superposition v,s (a, l) = ψ(s)U,s (a, l)ds (142) of modes √
U,s (a, l) =
k 2π
√
k
− i s()
√ k + i s()
√ k 2
J √k ( s 2 + k /2 x)
√
ei s x 1+ r e−irl dr (1 + 2 r 2 ) (143) 2 2
Deformation Quantizations of the Hyperbolic Plane
555
that is obtained by using ψ(s) = (i − s) δ (s)
(144)
in the superposition of modes like those appearing in Eq. (126), normalized according to Eq. (119). Explicitly, one gets √ √ l k 1 −1 (145) (UU f )(a, l) = d x J √k ( k x/) f (a + ln x, ). 2 x Apart from the -dependent pre-factors, these operators are exactly the ones found in [31], see Eqs. (4.16) and (4.7). Of course, the asymptotic expansion allows to build U −1 perturbatively, as an expansion in . For illustrative purpose let us mention, that for a fixed value of s we obtain at fourth order: i 1 Us−1 = Vr − 1 − i 2 r ( + i r ) ∂r Vr + O(5 ) 4π 4 r =−s 2 2 i 2 2 2 (1 + + i r )δ[r + s] + (1 − i r + π r )δ (r + s) Vr dr = 16 π 2 4π 4 +O(5 ).
(146)
By a long but straightforward computation, the above explicit expressions of UU and UU−1 yield (for k = 1) the following integral formula for the invariant star product on the hyperbolic plane D: 1 u v (x) = K (Scan (x, y, z)) u(y) v(z) dy dz; 16π 3 4 D×D where dy denotes the Liouville measure on D and where ∞ i s 2 J 1 (s/) e s ds. K ( ) := 0
Using tabulated Laplace transforms, one first computes ∞ [ p + (1 + p 2 )1/2 ]−µ Fµ ( p) = e− pt Jµ (t)dt = , (1 + p 2 )1/2 0
(147)
(148)
with p = ε − i , ε being a small positive real part necessary to ensure convergence, and one finally gets µ √ 1 ei 2 f W ( ) if 2 < 1, 2 1− Fµ (−i ) = π ( 2 − 1)−1/2 [| | + ( 2 − 1)1/2 ]−µ ei 2 (µ+1)sign( ) if 2 > 1; (149) where f W = 2 arcsin is the odd function that links Scan to SW . Note that, when 2 < 1, the phase of Fµ is precisely given by f W ( ), while when 2 > 1, it is constant up to the sign of . The kernel K 1/ is now simply described by K 1/( ) = −
d2 F1/(−i ). d 2
(150)
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P. Bieliavsky, S. Detournay, Ph. Spindel
The phase S1/ of the kernel K 1/ can then be determined from (150). For 2 < 1, one gets √ arcsin( ) 3 1 − 2 S1/( ) = . − arctan 1 − 2 − (1 + 22 ) 2
(151)
In other words, for 2 < 1, which corresponds to triples of points in D for which Weinstein’s SW = 2 arcsin( ) is well defined, the above kernel K can be expressed under the WKB oscillatory form. The expression of the corresponding phase is then the following: S =
3 sin SW SW − arctan[ ]. 2 2 1 − 4 + (1 + 22 ) cos SW
(152)
In particular, one has the following asymptotics: S ∼
SW D ∼ Scan , 2
agreeing with A. Weinstein’s picture in [33]. Acknowledgements. P.B. and Ph.S. acknowledge partial support from the IISN-Belgium (convention 4.4511.06). P.B. acknowledges partial support from the IAP grant ‘NOSY’ delivered by the Belgian Federal Government. S.D. and Ph.S. warmly thank the IHES for its hospitality and the exceptional working conditions provided to its guests. At last we thank the referee for precious suggestions and comments.
A. Complements A.1. On D and M. 1. Group type structures. To obtain the expressions (33) and (34), we proceed as follows. Consider for instance the hyperbolic plane D = S L 2 (R)/S O(2) =: G/K . Consider the following usual presentation of g := sl2 (R) as generated over R by the elements E, F, H with table [E, F] = H , [H, E] = 2E and [H, F] = −2F. Then one may set k := R.(E − F), a := R.H ⊂ p =: k⊥ and ln := R.E. The group S may then be realized as the connected Lie subgroup of G admitting s := a ⊕ n as Lie algebra. Within these notations, the Iwasawa decomposition G = SK yields the above-mentioned identification S = D and the following global coordinate system: s −→ S : (a, ) := a H + E → exp(a H ) exp(E).
(153)
In this simple group context, the involution σ˜ of G is nothing else than the Cartan involution associated with the data of K . The symmetry at the base point K of G/K therefore reads s K (g K ) := σ˜ (g)K which for a ∈ A := exp(a) and n ∈ N := exp(n) corresponds to σ˜ (an)K = a−1 σ˜ (n)K . Observing that σ (E) = −F, a small computation then yields exp(F) exp(− 21 log(1+2 )H ) exp(−E) ∈ K , hence the above expression of D . The case of the Poincaré orbit is similar, details can be found in [44].
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A.2. Co-adjoint orbits of iso(1, 1): the Poincaré plane. Let us emphasize that when the Lie group under consideration is not semi-simple (e.g. the Poincaré group I so(1, 1), in opposition to the SL(2, R) group), only the co-adjoint orbits make sense a priori in the framework of Subsect. (2.1). An elementary calculation shows that on I so(1, 1) the co-adjoint orbits consist generically of hyperbolic cylinder sheets, otherwise four planes and a line of fixed point remain. If we denote by b, e0 and e1 a basis of generators of iso(1, 1) that obey the commutation relations: [b, e0 ] = e1 , [b, e1 ] = e0 , [e0 , e1 ] = 0,
(154)
and by β, ε0 and ε1 a basis of the dual space iso∗ (1, 1), the co-adjoint orbits are v β + cosh(α) ε0 − sinh(α) ε1 , the generic ones: x = k k = 0, v ∈ R, a ∈ R, v β − sinh(α) ε0 + cosh(α) ε1 (155) + the four null ones: x = v β ± α(ε0 ± ε1 ), v ∈ R, α ∈ R0 , (156) the pointlike ones: x = vβ, v = C te ∈ R. (157) The first ones are coset of the Poincaré group by the subgroup of translations in time (or in space); the second ones are coset obtained by dividing by light-like translation; both are topologically R2 , α and v providing coordinates on them. The fundamental vector fields, on a generic orbit, are given by (denoting a point x by its coordinates α and v): ∗ b(α,v) = ∂α , e0∗ (α,v) = − sinh α ∂v , e1∗ (α,v) = cosh α ∂v ,
∗ b(α,v)
= ∂α ,
e0∗ (α,v)
= cosh α ∂v ,
e1∗ (α,v)
(158)
= − sinh α ∂v ,
(159)
and the symplectic form is given by: ω(α,v) = dα ∧ dv.
(160)
In terms of the (a, l) = (α/2, v)-coordinates (32) used throughout this paper, these fundamental vector fields would be re-expressed as H ∗ = −∂a , E ∗ = −e−2a ∂l = −(e0∗ (a,l) + e1∗ (a,l) ) and F ∗ = −k e2a ∂l = ±k(e0∗ (a,l) − e1∗ (a,l) ). An affine, torsion free, connection will be I so(1, 1) invariant if its coefficients verify the two sets of equations: α α
βγ = γβ ,
ξ
µ
(161)
µ α α α ∂µ βγ −∂µ ξ α βγ +∂β ξ µ µγ +∂γ ξ µ βµ +∂βγ ξ α = 0,
∗
ξ =β ,
e0∗ ,
e1∗. (162)
These equations imply that in (v, α) coordinates, only two connection coefficients are non vanishing, depending on two constants A and B: v α
α,α = −v + A, α,α = B.
(163)
If we moreover require the connection to be symplectic (∇ω = 0), this implies that only v = −v + A is non zero.
α,α Finally imposing that the symmetry transformations (38) preserve the connection imposes that A = 0. On a symplectic manifold (M, ω, s) such a connection: torsionless, preserving the two-form ω, and invariant with respect to the symmetries s is unique. It constitutes the so-called Loos connection ([40]), intrinsically defined by
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ωx (∇ X Y, Z ) =
1 X x ωx (Y + sx Y, Z ), 2
(164)
or in terms of the symmetry expressed in coordinates x ρ as x ρ [s P (Q)] = sρ (x µ [P], x ν [Q]): 1 ∂ 2 sρ (x µ [P], x ν [Q]) ρ µ
σ τ (x [P]) = . 2 ∂ x σ [Q]∂ x τ [Q] Q=P The geodesic differential equations are a¨ = 0, ¨ − 4 a˙ 2 = 0
(165)
from which we infer immediately the equations of the affine geodesic curves (in terms of an affine parameter s, starting from the point of coordinates (a0 , 0 ) with tangent vector, in natural components, ( p0 , q0 ): a = p0 s + a0 , = 0 cosh(2 p0 s) + q0
sinh(2 p0 s) . 2 p0
(166)
From these we may recover the symmetry (38) and mid-point (39) equations; but let us emphasize that these are defined directly in terms of the group action (32) and the involution (33). References 1. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization I and II. Ann. Phys. 111, 61–151 (1978) 2. De Wilde, M., Lecomte, P.: Existence of star-products and of formal deformations of the Poisson Lie algebra of arbitrary symplectic manifolds. Lett. Math. Phys. 7(6), 487–496 (1983) 3. Fedosov, B.V.: Formal quantization. In: Some problems in modern mathematics and their applications to problems in mathematical physics (Russian), Moscow: Moskov. Fiz.-Tekhn. Inst., 1985., pp. 129–136 4. Omori, H., Maeda, Y., Yoshioka, A.: Weyl manifolds and deformation quantization. Adv. Math. 85(2), 224–255 (1991) 5. Bertelson, M., Cahen, M., Gutt, S.: Equivalence of star products. Geometry and physics. Classical Quantum Gravity 14(1A), A93–A107 (1997) 6. Nest, R., Tsygan, R.: Algebraic index theorem. Commun. Math. Phys. 172, 223–262 (1995) 7. Hansen, F.: Quantum mechanics in phase space. Rep. Math. Phys. 19(3), 361–381 (1984) 8. Bertelson, M., Bieliavsky, P., Gutt, S.: Parametrizing equivalence classes of invariant star products. Lett. Math. Phys. 46(4), 339–345 (1998) 9. Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 09, 032 (1999) 10. Connes, A., Douglas, M.R., Schwarz, A.S.: Noncommutative geometry and matrix theory: Compactification on tori. JHEP 9802, 003 (1998) 11. Schomerus, V.: D-branes and deformation quantization. JHEP 9906, 030 (1999) 12. Alekseev, A.Y., Recknagel, A., Schomerus, V.: Brane dynamics in background fluxes and non-commutative geometry. JHEP 0005, 010 (2000) 13. Alekseev, A.Y., Recknagel, A., Schomerus, V.: Open strings and non-commutative geometry of branes on group manifolds. Mod. Phys. Lett. A16, 325–336 (2001) 14. Alekseev, A.Y., Recknagel, A., Schomerus, V.: ‘Non-commutative world-volume geometries: Branes on SU(2) and fuzzy spheres. JHEP 9909, 023 (1999) 15. Schomerus, V.: Lectures on branes in curved backgrounds. Class. Quant. Grav. 19, 5781–5847 (2002) 16. Bieliavsky, P., Jego, C., Troost, J.: Nucl. Phys. B782, (2007) 94–133 17. Hashimoto, A., Thomas, K.: Non-commutative gauge theory on d-branes in Melvin universes. JHEP 0601, 083 (2006) 18. Hashimoto, A., Sethi, S.: Holography and string dynamics in time-dependent backgrounds. Phys. Rev. Lett. 89, 261601 (2002)
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19. Halliday, S., Szabo, R.J.: Noncommutative field theory on homogeneous gravitational waves. J. Phys. A39, 5189–5226 (2006) 20. Behr, W., Sykora, A.: Construction of gauge theories on curved noncommutative spacetime. Nucl. Phys. B 698, 473–502 (2004) 21. Cai, R.-G., Lu, J.-X., Ohta, N.: Ncos and d-branes in time-dependent backgrounds. Phys. Lett. B551, 178–186 (2003) 22. Cai, R.-G., Ohta, N.: Holography and d3-branes in Melvin universes. Phys. Rev. D73, 106009 (2006) 23. Cai, R.-G., Ohta, N.: On the thermodynamics of large N non-commutative super Yang-Mills theory. Phys. Rev. D61, 124012 (2000) 24. Halliday, S., Szabo, R.J.: Isometric embeddings and noncommutative branes in homogeneous gravitational waves. Class. Quant. Grav. 22, 1945–1990 (2005) 25. Szabo, R.J.: Symmetry, gravity and noncommutativity. Class. Quant. Grav. 23, R199–R242 (2006) 26. Maldacena, J.M., Ooguri, H.: Strings in AdS(3) and SL(2,R) WZW model. I. J. Math. Phys. 42, 2929–2960 (2001) 27. Teschner, J.: On structure constants and fusion rules in the SL(2,C)/SU(2) WZNW model. Nucl. Phys. B546, 390–422 (1999) 28. Bachas, C., Petropoulos, M.: JHEP 0102, (2001) 025 29. Bieliavsky, P., Detournay, S., Spindel, P., Rooman, M.: Star products on extended massive non-rotating BTZ black holes. JHEP 06, 031 (2004) 30. Bieliavsky, P.: Non-formal deformation quantizations of solvable Ricci-type symplectic symmetric spaces. J. Phys.: Conf. Ser. 103, 012001 (2008) 31. Unterberger, A. et J.: Quantification et analyse pseudo-différentielle. Ann. Scient. Ec. Norm. Sup. 4E série t. 21, 133–158 (1988) 32. Unterberger, A. et J.: La série discrète de SL(2,R) et les opérateurs pseudo-différentiels sur une demidroite. Ann. Scient. Ec. Norm. Sup. 4E série t.17, 83–116 (1984) 33. Weinstein, A.: Traces and triangles in symmetric symplectic spaces. Contemp. Math. 179, 262–270 (1994) 34. Klimek, S., Lesniewski, A.: Quantum Riemann surfaces. I. The unit disc. Commun. Math. Phys. 146(1), 103–122 (1992) 35. Cahen, M., Gutt, S., Rawnsley, J.: Quantization of Kähler manifolds. II. Trans. Amer. Math. Soc. 337(1), 73–98 (1993) 36. Klimek, S., Lesniewski, A.: Quantum Riemann surfaces. II. The discrete series. Lett. Math. Phys. 24(2), 125–139 (1992) 37. Hawkins, E.: Quantization of multiply connected manifolds. Commun. Math. Phys. 255(3), 513–575 (2005) 38. Natsume, T., Nest, R.: Topological approach to quantum surfaces. Commun. Math. Phys. 202(1), 65–87 (1999) 39. Carey, A., Hannabuss, K., Mathai, V., McCann, P.: Quantum Hall effect on the hyperbolic plane. Commun. Math. Phys. 190(3), 629–673 (1998) 40. Bieliavsky, P.: Espace symétriques symplectiques. PhD. thesis, ULB, 1995 41. Kirillov, A.A.: Elementy teorii predstavleni˘ı. (Russian) [Elements of the theory of representations] Moscow: Izdat. “Nauka”, 1972 42. Kostant, B.: Quantization and unitary representations. In: Lectures in modern analysis and applications, III, Lecture Notes in Math., vol. 170, Berlin: Springer, 1970, pp. 87–208 43. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Londen-NewYork: Academic Press, 1978 44. Bieliavsky, P.: Strict quantization of solvable symmetric spaces. J. Sympl. Geom. 1(2), 269–320 (2002) 45. Bieliavsky, P., Bonneau, Ph., Maeda, Y.: Universal deformation formulae, symplectic Lie groups and symmetric spaces. Pacific J. Math. 230(1), 41–57 (2007) 46. Bieliavsky, P., Bonneau, Ph.: On the geometry of the characteristic class of a star product on a symplectic manifold. Rev. Math. Phys. 15(2), 199–215 (2003) 47. Connes, A., Flato, M., Sternheimer, D.: Closed star products and cyclic cohomology. Lett. Math. Phys. 24(1), 1–12 (1992) 48. Watson, G.N.: A treatise on the Theory of Bessel Function. Second edition, Cambridge: Cambridge Univ. Press, 1966 49. Connes, A., Moscovici, H.: Rankin-Cohen brackets and the Hopf algebra of transverse geometry. Mosc. Math. J. 4(1), 111–130 (2004) 50. Zagier, D.: Modular forms and differential operators. In: K. G. Ramanathan memorial issue, Proc. Indian Acad. Sci. Math. Sci. 104(1), 57–75 (1994) 51. Bieliavsky, P., Massar, M.: Oscillatory integral formulae for left-invariant star products on a class of Lie groups. Lett. Math. Phys. 58, 115–128 (2001) Communicated by A. Connes
Commun. Math. Phys. 289, 561–577 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0694-z
Communications in
Mathematical Physics
Feynman Graphs, Rooted Trees, and Ringel-Hall Algebras Kobi Kremnizer1 , Matt Szczesny2 1 Department of Mathematics, University of Chicago, Chicago, IL, USA.
E-mail: [email protected]
2 Department of Mathematics, Boston University, Boston, MA, USA.
E-mail: [email protected] Received: 4 July 2008 / Accepted: 19 August 2008 Published online: 18 December 2008 – © Springer-Verlag 2008
Abstract: We construct symmetric monoidal categories LRF, LFG of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of LRF, LFG, HLRF , HLF G are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman diagrams. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras. 1. Introduction The Connes-Kreimer Hopf algebras on rooted trees and Feynman graphs HT , H F G , introduced in [2,6], describe the algebraic structure of the BPHZ algorithm in the renormalization of perturbative quantum field theories. If we let T denote the set of (nonplanar) rooted trees, and Q{T} the Q–vector space spanned by these, then as an algebra, HT = Sym(Q{T}), and the coalgebra structure is given by the coproduct PC (T ) ⊗ RC (T ), (T ) = C admissible cut
where PC (T ) is the forest of branches resulting from the cut C, and RC (T ) is the root component remaining “above” the cut (see [2] for a more detailed definition). H F G is defined analogously, with Feynman graphs in place of rooted trees. More precisely, given a perturbative QFT, and denoting by Q{} the vector space spanned by the one-piece irreducible graphs of the theory (1 PI graphs), H F G = Sym(Q{}) as an algebra. Its coalgebra structure is given by γ ⊗ /γ , () = γ ∈
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where the sum is over all (not necessarily connected) subgraphs of , and /γ denotes the graph obtained from by shrinking each connected component of γ to a point. HT and H F G are graded commutative Hopf algebras, and so by the Milnor-Moore theorem, their graded duals H∗T and H∗F G are isomorphic to the universal enveloping algebras U(nT ), U(n F G ) of the Lie algebras nT , n F G of their primitive elements. We refer to nT and n F G as the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs respectively. In this paper, we present a categorification of the Hopf algebras U(nT ), U(n F G ), by showing that they arise naturally as the Ringel-Hall algebras of certain categories LRF, LFG of labeled rooted forests and Feynman graphs respectively. We briefly recall the notion of Ringel-Hall algebra. Given an abelian category A (typically linear over a finite field Fq ) satisfying the finiteness properties | Hom(M, N )| < ∞ and | Ext 1 (M, N )| < ∞ (such an abelian category is called finitary), the Ringel-Hall algebra of A, HA , is the Q–vector space Q{[M]} spanned by the isomorphism classes [M] ∈ A. It becomes an associative algebra under the product [M] × [N ] =
L g M,N
aM aN
[L],
L where g M,N is the number of short exact sequences
0→M →L→N →0 and a M = | Aut(M)|. HA also possesses a coproduct (see sect. 2), which in good cases makes it a co-commutative Hopf algebra, isomorphic to the universal enveloping algebra U(nA ) of the Lie algebra of its primitive elements nA , called the Ringel-Hall Lie algebra of A. The categories LRF and LFG are not abelian, or even additive, but possess all the necessary properties to define the corresponding Ringel-Hall Hopf algebras. These are enumerated in sect. 4. We prove that nT ∼ = nLRF and n F G ∼ = nLF G . This paper is organized as follows. In sect. 2 we review the notion of Ringel-Hall algebra of a finitary abelian category. Section 3 introduces some terminology relating to rooted trees and forests, and recalls the Connes-Kreimer Lie algebra on rooted trees. In sect. 4 we construct the category LRF of labeled rooted forests, and describe some of its properties. The following section (5) applies the notion of Ringel-Hall algebra to LRF to obtain U(nT ). Finally, in sects. 6, 7 we construct the category LFG in an analogous manner, and show that its Ringel-Hall algebra is isomorphic to U(n F G ). 2. Ringel-Hall Algebras Associated to Finitary Abelian Categories In this section, we briefly recall the construction of the Ringel-Hall algebra associated to a finitary abelian category. The notion was introduced in [9], and our treatment borrows heavily from [10], where we refer the reader for details and proofs. Recall that a small abelian category A is called finitary if: i) For any two objects M, N ∈ Ob(A) we have |H om(M, N )| < ∞,
(2.1)
ii) For any two objects M, N ∈ Ob(A) we have |E xt (M, N )| < ∞.
(2.2)
1
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For M, N , L ∈ Ob(A), let G LM,N denote the set of all exact sequences of the form 0 → M → L → N → 0. L By (i), G LM,N is a finite set. Let g M,N = |G LM,N | and a M = |Aut (M)|. As a Q–vector space,
HA :=
Q{[M]},
[M]∈I (A)
where I (A) denotes the set of isomorphism classes of objects in A, and [M] the isomorphism class of the object M. HA is an associative algebra with respect to the product
[M] × [N ] =
[L]∈I (A)
L g M,N
aM aN
[L]
(2.3)
which is finite by property (ii) of A. This product clearly counts the number of extensions of N by M up to isomorphism. A more geometric way of expressing this product is as follows. Let F(A) denote the vector space of Q–valued functions on A supported on finitely many isomorphism classes, i.e. F(A) := { f : I (A) → Q||supp( f )| < ∞}, and let Fl2 (M) denote the space parametrizing flags of length two in A, 0 = M0 ⊂ M1 ⊂ M2 = M,
Mi ∈ A
(note that this is the same as a short exact sequence of objects in A). F(A) is equipped with a convolution product: for f, g ∈ F(A), let f × g(M) :=
Fl2 (M)
f (M1 /M0 )g(M2 /M1 ) .
(2.4)
Identifying the symbol [M] with the characteristic functions δ M of the isomorphism class of M ∈ A, we see that the product δ M × δ N corresponds to [M] × [N ], and so F(A) = HA as associative algebras. In this formulation, the algebra F(A) possesses a natural coproduct : F(A) → F(A) ⊗ F(A), ( f )(M, N ) := f (M ⊕ N ),
(2.5)
which in “good” cases endows it with the structure of a co-commutative bialgebra. The primitive elements of F(A) are those functions supported on indecomposable elements of A, and form a Lie algebra nA . F(A) can be naturally identified with the universal enveloping algebra U (nA ).
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3. Rooted Trees and Forests Let T denote the set of rooted trees. An element T ∈ T is a tree (finite, one-dimensional contractible simplicial complex), with a distinguished vertex r (T ), called the root of T . Let V (T ) and E(T ) denote the set of vertices and edges of T , and let |T | = #V (T ). A labeling of a tree T by a set S is a bijection S → V (T ). In what follows, we will frequently consider trees labeled by subsets S ⊂ N of the natural numbers. For example, 2
6
1
9
2
3 are labeled rooted trees, with the vertex pictured at the top. Let LT denote the set of rooted trees labeled by subsets of N, and for T ∈ LT, let lab(T ) ⊂ N denote the set of labels (which is canonically identified with V (T )). A labeled rooted forest F is a set of labeled rooted trees, i.e. F := {T1 , T2 , . . . , Tn },
Ti ∈ LT .
An admissible cut of a labeled tree T is a subset C(T ) ⊂ E(T ) such that at most one member of C(T ) is encountered along any path joining a leaf to the root. Removing the edges in an admissible cut divides T into a forest PC (T ) and a rooted tree RC (T ), where the latter is the component containing the root. A simple cut is an admissible cut consisting of a single edge. For example, if T :=
4 = 7
3 =
1
2
5
6
and the cut edges are indicated with “=”, then PC (T ) =
7
2
1
5
and RC (T ) = 4 3 6
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We also allow the empty and full cuts Cnull , C f ull , where (PCnull (T ), RCnull (T )) = (∅, T ) and (PC f ull (T ), RC f ull (T )) = (T, ∅) respectively. The latter is considered simple. More generally, given a labeled forest F := {T1 , · · · , Tn }, an admissible cut on F is an n–tuple {C1 , C2 , · · · , Cn }, where Ci is an admissible cut of Ti , and PC (F) := {PC1 (T1 ), . . . , PCn (Tn )} and RC (F) := {RC1 (T1 ), . . . , RCn (Tn )}. Definition 3.1. By a subforest of a labeled rooted forest F we mean a forest of the form G = PC (F), for an admissible cut C. We write G ⊂ F. Given a labeled rooted forest F and two admissible cuts C1 , C2 , we write C1 < C2 if the cut edges of C2 occur closer to the root than those of C1 along any path joining a leaf to the root. Similarly, we write C1 ≤ C2 if the cut edges of C2 occur at those of C1 or closer to the root. The relation ≤ defines a partial order on cuts. We also define (1) the cut C1 ∪ C2 by the property that Ci ≤ C1 ∪ C2 , i = 1, 2, and if Ci ≤ D for some cut D, then C1 ∪ C2 ≤ D, (2) the cut C1 ∩ C2 by the property that C1 ∩ C2 ≤ Ci , and if D ≤ Ci for some cut D, then D ≤ C1 ∩ C2 . In other words, C1 ∪ C2 involves cutting the edge closer to the root, and C1 ∩ C2 the farther one. Note that both operations ∪, ∩ are associative and commutative. The following two observations will be important below: Remark 1. If G = PC (F) is a subforest of F, and C is an admissible cut on F, then C induces a unique admissible cut on G. It is the restriction of the cut C ∩ C to G. Remark 2. If G = PC (F) is a subforest of F, then there is a bijection between subforests of RC (F) and subforests H of F such that G ⊂ H ⊂ F. Both correspond to cuts C on F such that C ≤ C . 3.1. The Connes-Kreimer Lie algebra on rooted trees. In this section, we recall the definition of the Connes-Kreimer Lie algebra on rooted trees nT (see [2]). As a vector space, nT = Q{T}, i.e. the span of unlabeled rooted trees. On nT , we have a pre-Lie product “∗”, given, for T1 , T2 ∈ T by a(T1 , T2 ; T )T, T1 ∗ T2 = T ∈T
where a(T1 , T2 ; T ) := |{e ∈ E(T )|PCe (T ) = T1 , RCe (T ) = T2 }| and Ce denotes the cut severing the edge e. The Lie bracket on nT is given by [T1 , T2 ] := T1 ∗ T2 − T2 ∗ T1 .
(3.1)
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Thus, for example if T1 :=
and T2 :=
,
then [T1 , T2 ] =
−
−3
4. The Category LRF of Labeled Rooted Forests Labeled rooted forests can be made into a category LRF as follows. Let Ob(LRF) = { labeled forests } ∪ {∅}, where ∅ denotes the empty forest, which plays the role of zero object. Definition 4.1. We say that two labeled rooted trees T1 and T2 are isomorphic, and write f : T1 ∼ = T2 if there exists a root and incidence-preserving bijection f : lab(T1 ) → lab(T2 ). Two forests F1 := {T1 , . . . , Tn } and F2 := {U1 , . . . , Un } are isomorphic if there exists a permutation σ ∈ Sn such that f : Ti ∼ = Uσ (i) for i = 1, . . . , n. We write f : F1 ∼ = F2 . If F1 , F2 ∈ LRF, we now define Hom(F1 , F2 ) := {(C1 , C2 , f )|Ci is an admissible cut of Fi , f : RC1 (F1 ) ∼ = PC2 (F2 )} and the image of (C1 , C2 , f ), I m(C1 , C2 , f ) (or I m( f ) if C1 , C2 are understood) to be the subtree PC2 (F2 ) of F2 . For F ∈ LRF, the morphism (Cnull , C f ull , id) is the identity morphism. Example. If F1 := 2 1 3
F2 :=
6 5
7
8 4
6 9
2
then a morphism is given by the triple (C1 , C2 , f ) where: • C1 is the full cut on the first tree in F1, and on the second severs the edge joining 8 to 4. • C2 is the cut on F2 which severs the edge joining 7 to 6. • f : RC1 (F1 ) ∼ = PC2 (F2 ) is defined by f (5) = 9, f (6) = 6 , f (8) = 2. Remark 3. Note that (C1 , C2 , f ), where f (5) = 2, f (6) = 6, f (8) = 9 is also a morphism.
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The composition of morphisms Hom(F1 , F2 ) × Hom(F2 , F3 ) → Hom(F1 , F3 ) is defined as follows. Suppose that (C1 , C2 , f ) ∈ Hom(F1 , F2 ), and (D2 , D3 , g) ∈ Hom(F2 , F3 ). By Remark 1, the cut D2 on F2 induces a cut on the subforest PC2 (F2 ) ∼ = RC1 (F1 ), which by Remark 2 corresponds to a subforest PE 1 (F1 ) of F1 containing PC1 (F1 ). The image g ◦ f (R E 1 (F1 )) ⊂ F3 is a subforest PE 3 (F3 ). We define the composition (D2 , D3 , g) ◦ (C1 , C2 , f ) to be (E 1 , E 3 , g ◦ f ). It is easy to see that this composition is associative. We thus obtain: Theorem 4.1. With the above definitions of Ob(LRF) and Hom, LRF forms a category. LRF has among other, the following properties: (1) Given labeled forests F1 , F2 we denote their disjoint union by F1 ⊕ F2 . The disjoint union of forests equips LRF with a symmetric monoidal structure. (2) The empty forest {∅} is an initial, terminal, and therefore null object. (3) Every morphism (C1 , C2 , f ) : F1 → F2
(4.1)
possesses a kernel (Cnull , C1 , id) : PC (F1 ) → F1 , where Cnull denotes the empty cut, and id the identity map id : PC1 (F1 ) = RCnull (PC1 (F1 )) ∼ = PC1 (F1 ). (4) Similarly, every morphism 4.1 possesses a cokernel (C2 , C f ull , id) : F2 → RC2 (F2 ), where id is the identity map RC2 (F2 ) ∼ = RC2 (F2 ) = PC f ull (RC2 (F2 )). We will frequently use the notation F2 /F1 for coker ((C1 , C2 , f )). Note. Properties 3 and 4 imply that the notion of exact sequence makes sense in LRF. (5) All monomorphisms are of the form (Cnull , C1 , f ) : PC1 (F1 ) → F1 , where f is an automorphism of PC1 (F1 ). Once the image subforest PC1 (F1 ) is fixed, all monomorphisms with that image form a torsor over Aut(PC1 ), and there are therefore | Aut PC1 (F1 )| of them. All epimorphisms are of the form (C2 , C f ull , g) : F2 → RC2 (F2 ), where g is an automorphism of RC2 (F2 ).The epimorphisms with fixed kernel subforest PC2 (F2 ) form a torsor over Aut(RC2 (F2 )), and so there are | Aut(RC2 (F2 ))| of them.
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(6) Sequences of the form ∅ → PC (F)
(Cnull ,C,id)
−→
F
(C,C f ull ,id)
−→
RC (F) → ∅
(4.2)
are exact, and it follows from the last property that all other short exact sequences arise by composing with automorphisms of PC (F) and RC (F) on the left and right respectively. (7) By Remark 2, given a forest F and an admissible cut C, there is a bijection between sub-objects F of F containing PC (F), i.e. chains PC (F) ⊂ F ⊂ F, and subobjects of RC (F). (8) Hom(F1 , F2 ) and Ext n (F1 , F2 ) are finite sets, where Ext n (F1 , F2 ) is understood in the sense of Yoneda Ext’s, as equivalence classes of exact sequences of length n. (9) We may define the Grothendieck group of LRF, K (LRF), as
K (LRF) =
Z[M]/ ∼ ,
[M]∈I(LRF )
where the equivalence relation ∼ is defined by: [M] ∼ [N ] iff there exists a short exact sequence 0 → M → L → N → 0. It is then easy to see that K (LRF) = Z, as every forest is a direct sum of trees, and each tree is an extension of the one-vertex tree. 5. The Ringel-Hall Algebra of LRF We proceed to define the Ringel-Hall algebra of the category LRF as in the case of finitary abelian categories. Let I (LRF) denote the isomorphism classes of objects in LRF, and let HLRF := { f : I (LRF) → Q||supp( f )| < ∞}, i.e. the space of Q–valued functions supported on finitely many isomorphism classes in LRF. We equip HLRF with the convolution product f × g(F) =
f (G)g(F/G),
(5.1)
G⊂F
where the notation F/G is used as explained in Property 4 of LRF. It is clear that this sum is finite, as any object in LRF possesses finitely many sub-objects. The proof of the following theorem is essentially identical to that in the case of finitary abelian categories in [10]. We include it for the sake of completeness. Theorem 5.1. The multiplication × in (5.1) is associative.
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Proof. Suppose f, g, h ∈ HLRF , and F ∈ I (LRF). We have f (G)(g × h)(F/G) ( f × (g × h))(F) = G⊂F
=
f (G)g(H )h((F/G)/H )
G⊂F,H ⊂F/G
=
f (G)g(H/G)h(F/H ),
G⊂H ⊂F
(( f × g) × h)(F) =
( f × g)(K )h(F/K )
K ⊂F
=
f (L)g(K /L)h(F/K )
K ⊂F,L⊂K
=
f (L)g(K /L)h(F/K ),
L⊂K ⊂F
where the equality between the second and third lines follows from Property 7 which yields a bijection between the sets {H ⊂ F/G}
↔
{G ⊂ H ⊂ F}
satisfying H = H/G and (F/G)/H = F/H. Remark 4. The only properties of the category LRF that are needed to establish the associativity of the product 5.1 are 3, 4, 6 Property 6 of LRF implies that if g FK1 ,F2 is the number of short exact sequences of the form ∅ → F1 → K → F2 → ∅
(5.2)
and h KF1 ,F2 is the number of sub-objects L ⊂ K such that L ∼ = F1 and K /L ∼ = F2 , then h KF1 ,F2 =
g FK1 ,F2 | Aut(F1 )|| Aut(F2 )|
.
Thus, if δ F1 , δ F2 ∈ HLRF are the characteristic functions of the isomorphism classes of F1 , F2 , we have δ F1 × δ F2 =
K ∈I (LRF )
h KF1 ,F2 δ K =
K ∈I (LRF )
g FK1 ,F2 | Aut(F1 )|| Aut(F2 )|
δK .
The algebra HLRF is graded by K (LRF) = Z, which coincides with the grading by the number of vertices in a forest. We introduce a coproduct on HLRF , as in the case of a finitary abelian category, by : HLRF → HLRF ⊗ HLRF , ( f )(F, G) = f (F ⊕ G).
(5.3)
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Theorem 5.2. HLRF is a co-commutative Hopf algebra isomorphic to U(nT ). Proof. is co-commutative since F⊕G = G⊕F, and the uniqueness of decompositions into indecomposable objects (labeled trees) in LRF yields coassociativity. It is very easy to check that is compatible with ×. HLRF is therefore a graded connected bialgebra, and thus a Hopf algebra. The Milnor-Moore theorem implies that HLRF is isomorphic to the universal enveloping algebra of its primitive elements nLRF , which for the coproduct are exactly the indecomposable elements of LRF—the characteristic functions supported on the isomorphism class of a single labeled tree. It remains to verify that for δT1 , δT2 ∈ nLRF , the bracket [δT1 , δT2 ]× := δT1 × δT2 − δT2 × δT1 coincides with the Lie bracket 3.1 under the map j : T → HLRF , j (T ) = δT (this makes sense since any two labelings of a tree are isomorphic). Extending j linearly, it is easy to see that for unlabeled rooted trees T1 , T2 , j (T1 T2 ) = δT1 × δT2 − δT1 ⊕T2 , which implies that j (T1 T2 − T2 T1 ) = δT1 × δT2 − δT2 × δT1 . This proves the result.
Remark 5. The Hopf algebra HLRF is canonically isomorphic to the Grossman-Larson Hopf algebra (see [4]—the fact that the Grossman-Larson Hopf algebra is isomorphic to U (nT ) was first proved in [8], with certain inaccuracies corrected in [5]). For instance, for the forest F = {T1 , . . . , Tn } ∈ I (LRF), (δ F ) = δ FJ ⊗ δ F[n]\J , J ⊂[n]={1,...,n}
where if J = { j1 , . . . , jk } ⊂ [n], FJ := {T j1 , . . . , T jk } ∈ I (LRF). The GrossmanLarson product involves a summation over all subtree attachments which easily are seen to correspond to the enumeration of all exact sequences 5.2. 6. Feynman Graphs In this section we show how to equip Feynman graphs with the structure of a category LFG possessing properties completely parallel to those of LRF, in such a way that HLF G ∼ = U(n F G ), where n F G is the Connes-Kreimer Lie algebra on Feynman graphs. We thus arrive at an interpretation of the latter as the Ringel-Hall Lie algebra of LFG. Our treatment of the combinatorics of graphs is taken from [11]. In order to not get bogged down in notation, we focus on the special case of φ 3 theory (the case of trivalent graphs with only one edge-type). The results of this section extend to the general case in a completely straightforward manner. We begin with a series of definitions.
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Definition 6.1. A graph consists of a set H = H () of half-edges, a set V = V () of vertices, a set of vertex-half edge adjacency relations (⊂ V × H ), and a set of half edge-half edge adjacency relations (⊂ H × H ), with the requirements that each half edge is adjacent to at most one other half edge and to exactly one vertex. Note that graphs may not be connected. Half edges which are not adjacent to another half edge are called external edges, and denoted E x = E x() ⊂ E = E(). Pairs of adjacent half edges are called internal edges, and denoted I nt (). Definition 6.2. A half edge S–labeled graph, (labeled graph for short), is a triple (, S, ρ), where is a graph, S is a set such that |S| = |H |, and ρ : H → S is a bijection. S will usually be obvious from context. Definition 6.3. (1) A Feynman graph is a graph where each vertex is incident to exactly three half-edges, and each connected component has 2 or 3 external edges. We denote the set of Feynman graphs by F G. (2) Similarly, we can define a labeled Feynman graph. We denote the set of labeled Feynman graphs by L F G. (3) A graph (or a labeled graph) is 1-particle irreducible (1 PI) if it is connected, and remains connected under the removal of an arbitrary internal edge. Example. The graph eg is a 1 PI Feynman graph with two external edges. We have
labeled each vertex and edge, and each half-edge can be thought of as labeled by a pair (v, e), where v is a vertex, and e is an edge incident to v. Definition 6.4. Given a Feynman graph , a subgraph γ is a Feynman graph such that V (γ ) ⊂ V (), H (γ ) ⊂ H (), and such that if v ∈ V (γ ), and (v, e) ∈ V () × H (), then e ∈ H (γ ) (i.e. the subgraph has to contain all half-edges incident to its vertices). We also insist that dimQ (H1 (γ , Q)) > 0 (i.e. that a subgraph contain at least one loop). We write γ ⊂ . The same definition applies to labeled graphs. Example. We define a subgraph γeg ⊂ eg as follows. Let V (γeg ) = {v3, v4}, E(γeg ) := {all half-edges incident to v3, v4}, i.e. E(γeg ) = {(v4, e8), (v3, e7), (v3, e9), (v3, e10), (v4, e9), (v4, e10)},
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and all incidences inherited from eg . We proceed to define the contraction of subgraphs of Feynman graphs. Definition 6.5. Let be a Feynman graph, and γ ⊂ a connected subgraph. The quotient graph /γ is defined as follows. If γ has 3 external edges, then /γ is the Feynman graph with (1) V (/γ ) set the vertex set of with all vertices of γ removed, and a new trivalent vertex v added. (2) H (/γ ) the half edge set of , with all half edges corresponding to internal half edges of γ removed. (3) All adjacencies inherited from , and the external half edges of γ joined to v. If γ has 2 external edges, then /γ is the Feynman graph with (1) V (/γ ) is V () with all the vertices of γ removed. (2) H (/γ ) is H () with all half edges of γ removed. (3) All adjacencies inherited from , as well as the adjacency of the external half-edges of γ . Finally, if γ ⊂ is an arbitrary (not necessarily connected) Feynman subgraph, then /γ is defined to be the Feynman graph obtained by performing successive quotients by each connected component. Note that the order of collapsing does not matter. Example. With eg , γeg as above, eg /γeg is:
Remark 6. If γ1 , γ2 ⊂ are subgraphs of a (labeled) Feynman graph, then γ1 ∩ γ2 is a Feynman subgraph of γi (and of course also ). Remark 7. If γ ⊂ is a subgraph of a (labeled) Feynman graph, then there is a bijection between subgraphs of /γ , and subgraphs γ of such that γ ⊂ γ ⊂ . 6.1. The Connes-Kreimer Lie algebra on Feynman graphs. In order to define the Connes-Kreimer Lie algebra structure on Feynman graphs, we must first introduce the notion of inserting a graph into another graph. Let Q{L F G} denote the vector space spanned by labeled Feynman graphs.
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Definition 6.6. Let 1 , 2 ∈ L F G. If 1 has three external edges, v ∈ V (2 ), and f : E x(1 ) → H (v) is a bijection (where H (v) are the labeled half-edges incident to the vertex v), then let 2 ◦v, f 1 be the labeled Feynman graph such that • V (2 ◦v, f 1 ) = V (2 ) ∪ V (1 )\v. • H (2 ◦v, f 1 ) = H (1 ) ∪ f H (2 ) - i.e. the unions of the half-edges of each graph, with the identifications induced by f . • The adjacencies induced from those of 1 and 2 . If 1 has two external edges, {e1 , e2 } ∈ I nt (2 ) ⊂ H × H , and f is a bijection between E x(1 ) and {e1 , e2 } (there are two of these), then 2 ◦e, f 1 is the labeled Feynman graph such that • V (2 ◦e, f 1 ) = V (1 ) ∪ V (2 ). • H (2 ◦e, f 1 ) = H (1 ) ∪ H (2 ). • The adjacency induced by f as well as those induced from 1 and 2 . Let n L F G denote the Q–vector space spanned by unlabeled Feynman graphs. Given a labeled Feynman graph , denote by the corresponding unlabeled Feynman graph. Thus, n L F G = Q{L F G}/ ∼ , where ∼ iff = . We now equip n L F G with the pre-Lie product “”, defined by 1 2 := 2 ◦v, f 1 v∈V (2 ), f :E x(1 )→H (v)
if 1 has three external edges, and 1 2 :=
2 ◦v, f 1
e∈I nt (2 ), f :E x(1 )→{e1 ,e2 }
if 1 has two external edges, and extended linearly (in the above formulas, we first choose an arbitrary labeling of the Feynman graphs). Finally, we can define the Lie bracket on n L F G by [1 , 2 ] := 1 2 − 2 1 .
(6.1)
Example. Suppose
(6.2)
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then
n L F G has an alternative presentation as follows (see [3]). For (unlabeled) Feynman graphs 1 , 2 , let a(1 , 2 , ) := |{ subgraphs γ ⊂ |γ ∼ = 1 , /γ = 2 }|. n L F G can now be equipped with the pre-Lie structure 1 #2 := a(1 , 2 , ),
(6.3)
where the sum is taken over all unlabeled Feynman graphs. We thus obtain “another” Lie bracket on n L F G : [1 , 2 ] := 1 #2 − 2 #1 .
(6.4)
The two structures (6.1) and (6.4) are shown in [3] to be isomorphic via the map → | Aut()|. 7. The Category LF G of Labeled Feynman Graphs Labeled Feynman graphs form a category LFG as follows. Let Ob(LFG) = { labeled Feynman graphs } ∪ {∅}, where ∅ denotes the empty Feynman graph, which plays the role of zero object. Note that objects of LFG may have several connected components. Definition 7.1. We say that two labeled Feynman graphs 1 and 2 are isomorphic if there exist bijections f V : V (1 ) → V (2 ), f H : H (1 ) → H (2 ) which induce bijections on all incidences. We write f : 1 ∼ = 2 . If 1 , 2 ∈ LFG, we now define Hom(1 , 2 ) := {(γ1 , γ2 , f )|γi is a subgraph of i , f : 1 /γ1 ∼ = γ2 }. For ∈ LFG, (∅, , id) is the identity map in Hom(, ). The composition of morphisms in LFG Hom(1 , 2 ) × Hom(2 , 3 ) → Hom(1 , 3 ) is defined as follows. Suppose that (γ1 , γ2 , f ) ∈ Hom(1 , 2 ), and (τ2 , τ3 , g) ∈ Hom(2 , 3 ). By Remark 6, the subgraph τ2 on 2 induces a subgraph τ2 ∩γ2 of γ2 ⊂ 2 , which by Remark 7 corresponds to a subgraph of ξ of 1 containing γ1 . The image g ◦ f (ξ ) ⊂ 3 is a subgraph ρ ⊂ τ3 . We define the composition (τ2 , τ3 , g) ◦ (γ1 , γ2 , f ) to be (ξ, ρ, g ◦ f ). It is easy to see that this composition is associative. We thus obtain:
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Theorem 7.1. With the above definitions of Ob(LFG) and Hom, LFG forms a category. LFG has a list of properties completely analogous to LRF: (1) Given labeled Feynman graphs 1 , 2 we denote their disjoint union by 1 ⊕ 2 . This operation equips LFG with a symmetric monoidal structure. (2) The empty graph {∅} is an initial, terminal, and therefore null object. (3) Every morphism (γ1 , γ2 , f ) : 1 → 2
(7.1)
possesses a kernel (∅, γ1 , id) : γ1 → 1 where id is the identity map id : γ1 /∅ ∼ = γ1 . (4) Similarly, every morphism (7.1) possesses a cokernel (γ2 , 2 /γ2 , id) : 2 → 2 /γ2 , where id is the identity map id : 2 /γ2 → 2 /γ2 , We will frequently use the notation 2 / 1 for coker ((γ1 , γ2 , f )). Note. Properties 3 and 4 imply that the notion of exact sequence makes sense in LFG. (5) All monomorphisms are of the form (∅, γ1 , f ) : γ1 → γ1 , where f is an automorphism of γ1 ⊂ 1 . and all epimorphisms are of the form (γ2 , 2 /γ2 , g) : 2 → 2 /γ2 , where g is an automorphism of 2 /γ2 . (6) Sequences of the form (∅,γ ,id)
∅ → γ −→
(γ ,/γ ,id)
−→
/γ → ∅
(7.2)
are exact, and all other short exact sequences arise by composing with automorphisms of γ and /γ on the left and right respectively. (7) By Remark 7, given a labeled Feynman graph and γ ⊂ , there is a bijection between sub-objects γ of containing γ , i.e. chains γ ⊂ γ ⊂ , and sub-objects of /γ . (8) Hom(1 , 2 ) and Ext n (1 , 2 ) are finite sets. (9) We may define the Grothendieck group of LFG, K (LFG), as K (LFG) = Z[M]/ ∼, [M]∈I(LF G )
where the equivalence relation ∼ is defined by: [M] ∼ [N ] iff there exists a short exact sequence 0 → M → L → N → 0. K (LFG) = Z[P], where P is the set of isomorphism classes of primitive Feynman graphs, which are those Feynman diagrams not containing any proper subgraphs. This follows since every Feynman graph is a union of connected ones, and each connected component is obtained by repeated insertions of primitive graphs.
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We may now proceed exactly as in sect. 5 to define the Ringel-Hall algebra of LFG. We define: HLF G := { f : I (LFG) → Q||supp( f )| < ∞}. We may equip HLF G with the product (5.1) and coproduct (5.3). Since the category LFG satisfies the conditions in 4, the proof of associativity goes through as in Theorem 5.1. Finally, the argument of Theorem 5.2 establishes the following result: Theorem 7.2. HLF G is a Hopf algebra isomorphic to U (n L F G ). 8. Further Directions Once Lie algebras of Feynman graphs are thought of categorically as Ringel-Hall Lie algebras, interesting and natural questions emerge. In particular, combining the results of this paper with those of [1] yields an interesting link between perturbative quantum field theory and the geometry of irregular connections on P1 and their Stokes phenomena. Very briefly, in [1], the authors study isomonodromic families of irregular connections with values in the Ringel-Hall Lie algebra of a finitary Abelian category A, nA . The connections are parametrized by families of stability conditions on the category A. Recall that a stability condition on an abelian category A is a homomorphism of abelian groups Z : K (A) → C (where K (A) is the Grothendieck group of A ), such that Z (K >0 (A)) ⊂ H, where K >0 (A) is the positive cone generated by the classes of nonzero objects, and H is the upper half plane. Thus, a stability condition is a choice of Z (M) ∈ H for each non-zero object M ∈ H, such that Z is additive across exact sequences. Given Z , each object M has a well-defined phase, φ(M) =
1 arg(Z ) ∈ (0, 1), π
and M is said to be semi-stable with respect to Z if every non-zero sub-object N ⊂ M satisfies φ(N ) ≤ φ(M). As Z changes, and passes through “walls”, the collection of semi-stable objects changes. Stability conditions certainly make sense for categories LRF, LFG, even though they are not abelian. For LFG, whose Grothendieck group is generated by primitive Feynman graphs (see Property 9 of LFG), a stability condition Z assigns a phase to each primitive graph. For a given Z , the semi-stable graphs will be allowed to contain certain primitive subgraphs, and not others (additively). Changing Z amounts to changing the allowed subgraphs. These, and other connections with the results in [1] are explored in [7]. Acknowledgement. M.S. would like to thank Dirk Kreimer and Valerio Toledano-Laredo for valuable conversations and for helpful suggestions.
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References 1. Bridgeland, T., Toledano-Laredo, V.: Stability conditions and Stokes factors. http://arXiv.org/abs/0801. 3974vI[math.AG], 2008 2. Connes, A., Kreimer, D.: Hopf algebras, renormalization, and noncommuative geometry. Commun. Math. Phys. 199, 203–242 (1998) 3. Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210(1), 249–273 (2000) 4. Grossman, R., Larson, R.G.: Hopf-algebraic structure of families of trees. J. Algebra 126, 184–210 (1989) 5. Hoffman, M.: Combinatorics of rooted trees and Hopf algebras. Trans. AMS 355, 3795–3811 (2003) 6. Kreimer, D.: On the Hopf algebra structure of perturbative quantum field theory. Adv. Theor. Math. Phys. 2, 303–334 (1998) 7. Kremnizer, K., Szczesny, M.: Feynman diagrams, connections, and Stokes factors. In preparation 8. Panaite, F.: Relating the Connes-Kreimer and Grossman-Larson Hopf algebras built on rooted trees. Lett. Math. Phys. 51, 211–219 (2000) 9. Ringel, C.: Hall algebras. In: Topics in algebra, Part 1 (Warsaw, 1988), Banach Center Publ., 26, Part 1, Warsaw: PWN, 1990, pp. 433–447 10. Schiffmann, O.: Lectures on Hall algebras. http://arXiv.org/list/math.RT/0611617, 2006 11. Yeats, K.: Growth estimates for Dyson-Schwinger equations. Ph.D. thesis. Boston University, 2008 Communicated by A. Connes
Commun. Math. Phys. 289, 579–596 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0775-7
Communications in
Mathematical Physics
Black Hole Formation from a Complete Regular Past Mihalis Dafermos Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom. E-mail: [email protected] Received: 7 July 2008 / Accepted: 17 July 2008 Published online: 17 March 2009 – © Springer-Verlag 2009
Abstract: An open problem in general relativity has been to construct an asymptotically flat solution to a reasonable Einstein-matter system containing a black hole and yet causally geodesically complete to the past, containing no white holes. We construct such a solution in this paper–in fact a family of such solutions, stable in a suitable sense–where matter is described by a self-gravitating scalar field. 1. Introduction The problem of gravitational collapse is typically formulated as the study of the future maximal evolution of complete asymptotically flat Cauchy data for an appropriate Einstein-matter system. The question of identifying physically admissible initial data, however, is best characterized by properties of their past evolution. In particular, it seems reasonable to restrict to Cauchy data whose past evolution is regular. Unfortunately, however, with the exception of certain classical results for dust [19], current theorems on the evolution of asymptotically flat Cauchy data ensuring a regular past [9,10,22] also ensure a regular future; this is due to the fact that these theorems depend on smallness in function-space norms that do not distinguish past from future.1 As a consequence, even a single example of a solution to a reasonable Einstein-matter system with a regular asymptotically flat past and a singular future has thus far been lacking. In the present paper, we shall prove Theorem 1. There exist smooth asymptotically flat initial data for the coupled Einsteinscalar field equations (1)–(3) such that the following holds. Let (M, g, φ) denote the maximal development. Then 1. (M, g) is causally geodesically complete to the past. 1 One can compare with the cosmological case, where choice of gauge based on the expansion of the universe can be used to prove future completeness theorems in cases where the past is known to be singular. See for instance [1,21].
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2. The future of past null infinity is the entire M. 3. (M, g) is future-causally geodesically incomplete, and there exist future incomplete causal geodesics along which the curvature blows up. 4. (M, g) has a complete future null infinity whose past is bounded by a regular event horizon. The above theorem thus constructs solutions which terminate to the future in a singularity hidden in a black hole, but have a complete past without white holes. In fact, a large class of such solutions will be here constructed. The solutions are spherically symmetric and their Penrose diagram2 is as follows: singularity
Fu Cauchy surface
st Pa
Past Timelike Infinity
Nu l
lI
nf
in
ity
r=0
ev
ty
i in
f In
en th or iz
ll Nu
on
re tu
BLACK HOLE
The singular boundary will be acausal except for a (possibly empty) null component emerging from the center.3 The curvature will blow up towards the remaining part of the singular boundary, which in particular will be non-empty.4 Moreover, the above Penrose diagram will be stable to a certain class of perturbations. Although the solutions we construct will admit complete asymptotically flat Cauchy surfaces, they will not be constructed a priori as developments of such data. Rather, the examples of this paper will be constructed by smoothly pasting together the solutions to three distinct characteristic initial value problems. In particular, the issue of formulating a criterion on Cauchy data ensuring a singular future yet regular past is sidestepped in this paper. Replacing the Cauchy problem with several distinct characteristic initial value problems allows one to completely separate the problem of the future from the past. The future is determined by data prescribed on a future-outgoing cone (see A in the diagram below), whereas the past is determined by data on a past-outgoing cone (see B). 2 These diagrams are defined to be the image of a time-orientation preserving conformal representation of the quotient manifold Q = M/S O(3) to a bounded subset of 2-dimensional Minkowski space, with certain conventional names for various subsets of the induced boundary. See the Appendix of [17] for a detailed discussion in this context. 3 The case where such a component is non-empty has been depicted above. By passing to the class of BV solutions introduced in [7], Christodoulou has shown [5] that for generic solutions, such a component is not present. We shall not pursue this further here, however. 4 In fact, the spacetime can be shown to be C 0 -inextendible beyond this part of the boundary.
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To complete the spacetime, however, one has to construct (and understand) the solution “between” the two cones (see ):
Α
r=0
Γ
Β
Since in spherical symmetry, the equations reduce to a hyperbolic system in 2 spacetime dimensions, the latter problem–let us call this problem –is also a wellposed characteristic initial value problem. Conceptually, this is perhaps most easily understood by replacing the two-dimensional metric on the quotient spacetime Q by its negative, thus interchanging the notions of space and time.5 The data for the initial value problem B can be taken to be trivial, but alternatively, from Christodoulou [7,9], one can prescribe an arbitrary scalar field sufficiently small in an appropriate norm, and this will also guarantee a past complete development. On the other hand, Christodoulou has given in [8] a criterion on initial data ensuring that for the initial value problem A, a black hole forms in the future. The important feature, from the point of view of this paper, is that this criterion can be applied to data which are not trapped initially, and for which the mass aspect function µ and the scalar field multiplied by the area radius r φ can be arbitrarily small in the supremum norm. Theorem 1 thus reduces to completely understanding the evolution of problem . It should be noted that the global features of this problem are quite different from the more standard characteristic initial value problems A and B; in particular, there are no a priori global energy bounds. The main analytical result of this paper is that leads to a “complete” wedge in an asymptotically flat spacetime, given sufficiently small initial data. The requisite smallness, however, is asymmetric in the two initial characteristics; on the future outgoing characteristic, only pointwise smallness of µ and r φ are required. In particular, the requirements for the data for are compatible with the data necessary in A to produce a black hole. In the next section, the Einstein-scalar field equations will be presented under spherical symmetry. The initial value problem is studied in Sect. 3. From this, the theorem of this paper is easily obtained in Sect. 4 along the considerations outlined above. The stability of the geometry of these spacetimes is briefly noted in Sect. 5.
2. The Einstein-Scalar Field Equations under Spherical Symmetry For a detailed discussion of the role of the Einstein-scalar field system in the study of the problem of gravitational collapse, the reader may refer to [13]. In rationalized units, the equations take the form 5 Estimates in the direction depicted by have many applications in the study of 2-dimensional hyperbolic problems. See [4,11,12,17].
582
M. Dafermos
1 Rgµν = 2Tµν , 2 g µν φ;µν = 0, 1 = φ;µ φ;ν − gµν φ ;α φ;α , 2
Rµν −
Tµν
(1) (2) (3)
where gµν is a Lorentzian metric defined on a 4-dimensional manifold M and φ is a scalar function (the scalar field). Recall that spherical symmetry is the assumption that S O(3) acts by isometry on the spacetime (M, g) and preserves φ. Our spacetimes will be maximal developments of asymptotically flat spherically symmetric Cauchy data (, g, ¯ K ) with one end. The quotient /S O(3) will then inherit the structure of a 1-dimensional manifold with boundary, and thus Q = M/S O(3) inherits the structure of a time-oriented Lorentzian manifold with boundary with metric g, ˆ such that Q can be covered by a system of global null coordinates u, v. The metric g on M can be expressed as gµν d x µ d x ν = π ∗ gˆ + r 2 γS2 = −2 dudv + r 2 γS2 , where π : M → Q denotes the standard projection, r : Q → R is given by 1 Area(π −1 ( p)), and where γS2 is the standard metric on the unit 2-sphere, r ( p) = 4π to be denoted henceforth simply by γ . The function r vanishes precisely on the boundary of Q. Since the scalar field is constant on group orbits, it descends to a function on Q which may again be denoted simply as φ. Equations (1)–(3) reduce to the following second order system on Q: 1 1 ∂u ∂v r = − ∂u r ∂v r − 2 , r 4r 1 2 −2 −2 ∂u ∂v log = r + r ∂u r ∂v r − ∂u φ∂v φ, 4 ∂u ∂v φ = −r −1 ∂u φ∂v r − r −1 ∂u φ∂u r, ∂u (−2 ∂u r ) = −r −2 (∂u φ)2 , ∂v (−2 ∂v r ) = −r −2 (∂v φ)2 .
(4) (5) (6) (7) (8)
3. The Characteristic Initial Value Problem In this section, we will study a characteristic initial value problem posed in the direction indicated by in the Introduction. This will be the main new analytic element in the proof of Theorem 1 in the next section.
Black Hole Formation from a Complete Regular Past
583
3.1. A local existence theorem. (See [16] for further developments of the propositions below.) Proposition 3.1.1. Let , r , φ be functions defined on X = [−d, 0] × {0} ∪ {0} × [0, d]. Let k ≥ 0, and assume r > 0 is C k+2 (u) on [−d, 0] × {0} and C k+2 (v) on {0} × [0, d], assume that and φ are C k+1 (u) on [−d, 0] × {0} and C k+1 (v) on {0} × [0, d]. Suppose that Eqs. (7), (8) hold initially on [−d, 0] × {0} and {0} × [0, d], respectively. Let ||n,u denote the C n (u) norm of on [−d, 0] × {0}, ||n,v the C n (v) norm on {0} × [0, d], etc. Define N = sup{||1,u , ||1,v , |−1 |0 , |r |2,u , |r |2,v , |r |−1 0 , |φ|1,u , |φ|1,v }. Then there exists a δ, depending only on N , and a C k+2 function (unique among C 2 functions) r and C k+1 functions (unique among C 1 functions) and φ, satisfying Eqs. (4)–(8) in [−δ ∗ , 0] × [0, δ ∗ ], where δ ∗ = min{d, δ}, such that the restriction of these functions to [−d, 0] × {0} ∪ {0} × [0, d] is as prescribed. The proposition is illustrated in the figure below: The additional regularity assumed ( 0, d ) X ( 0, 0 )
( −d, 0 )
on r is natural in view of the constraint equations (7), (8). The expressions ∂u φ, ∂u have additional regularity in v, etc. In particular, in the case k = 0, the statement of the above proposition implies that Eqs. (5), (6) hold pointwise–in particular the mixed null second derivatives exist–despite the fact that and φ are only assumed C 1 . Proof. The above theorem can be proven with straightforward pointwise estimates for (4)–(6) obtained via integration along characteristics. Differentiating Equations (7)–(8), one then shows that these propagate. We now proceed to globalize the above. We have Proposition 3.1.2. Let , r , φ be functions defined on (−∞, −R] × {R} ∪ {−R} × [R, ∞). Let k ≥ 0, and assume r > 0 is C k+2 (u) on (−∞, R] × {R} and C k+2 (v) on {−R} × [R, ∞), assume that and φ are C k+1 (u) on (−∞, R] × {R} and C k+1 (v) on {−R} × [R, ∞). Suppose that Eqs. (7), (8) hold initially on (−∞, R] × {R} and {−R} × [R, ∞), respectively. Consider the initial value problem described above. There exists a unique subset ∅ = D ⊂ (−∞, −R] × [R, ∞), open in the subspace topology of the latter set, and a unique set of sufficiently regular functions r , φ, and defined on D, such that
584
M. Dafermos
1. The functions r , φ, restrict to the prescribed values on ((−∞, −R] × {R} ∪ {−R} ×[R, ∞)) ∩ D. 2. The functions r , φ, satisfy (4)–(8). 3. If (u, v) ∈ D, then (u, ˜ v) ˜ ∈ D, for all u˜ ≥ u, v˜ ≤ v. ˜ and sufficiently regular r˜ , , ˜ φ, ˜ satisfying properties 1, 2, and 3, 4. Given any other D, ˜ ˜ ˜ ˜ it follows that D ⊂ D, and r˜ = r , = , and φ = φ on D. Let ∂D denote the boundary of D in the topology of (−∞, −R] × [R, ∞). Let us denote by J−− ( p) the past of p with respect to the Lorentzian metric +dudv on R1+1 , time-oriented so that the vector ∂v − ∂u is future directed.6 Assumption 3 is then simply that D is a past set with respect to this metric. Proposition 3.1.3. Let D be as above, and suppose that p ∈ D ∪ ∂D is such that (J−− ( p) \ { p}) ∩ (D ∪ ∂D) ⊂ D. Then either p ∈ D, or
sup
J−− ( p)∩D
|| + |−1 | + |r | + |r −1 | + |∂u φ| = ∞.
(9)
Proof. The proof of Proposition 3.1.2 follows easily from set theoretic arguments and Proposition 3.1.1. The improved extension principle of Proposition 3.1.3 follows again from Proposition 3.1.1 and straightforward pointwise estimates applied to the evolution system (4)–(6), which can be viewed as essentially linear if (9) does not hold7 , yielding estimates for all first derivatives. At the last step, one obtains control of ∂v2 r , etc., from the constraints (7)–(8).
3.2. A reformulation of the equations. To understand the global behavior of the system (4)–(8), the following reformulation of the equations is useful. Define the Hawking mass
m=
r r (1 − |∇r |2gˆ ) = (1 + 4−2 ∂v r ∂u r ). 2 2
(10)
Introduce the first order derivatives ∂u r = ν, ∂v r = λ, r ∂u φ = ζ , r ∂v φ = θ , and the so-called mass ratio function µ = 2m r . Note that, if ν = 0, λ = 0, then it follows from
1 − µ = −4−2 λν
(11)
6 Let our convention be that p ∈ J − ( p). − 7 This takes advantage of the null structure. In fact, the ∂ φ term could also be easily eliminated from (9). u
Black Hole Formation from a Complete Regular Past
585
that 1 − µ = 0. For such points, we obtain the system ∂u r = ν, ∂v r = λ, 2ν m , ∂u λ = λ 1 − µ r2 2λ m ∂v ν = ν , 1 − µ r2 2 ζ 1 ∂u m = (1 − µ) ν, 2 ν 2 θ 1 ∂v m = (1 − µ) λ, 2 λ ζλ ∂u θ = − , r θν ∂v ζ = − . r From the above we also derive φ 2λν ∂v (φν + ζ ) = 2 m, r 1−µ φ 2λν ∂u (φλ + θ ) = 2 m. r 1−µ
(12) (13) (14) (15) (16) (17) (18) (19)
(20) (21)
We shall see that the r −2 factor on the right hand side above allows us to show that the quantities φλ + ζ , φλ + θ have better decay in r than φ, θ , or ζ ; this fact plays an important role in our argument. We also easily obtain from the above equations 1 ζ 2 λ λ ∂u = (22) ν 1−µ r ν 1−µ and ∂v
1 ν = 1−µ r
2 θ ν . λ λ 1−µ
(23)
3.3. The initial data. For the time being, fix constants R > 0, C > 0, M > 0, and m 0 ≥ 0. The initial characteristic segments will be u = −R and v = R. At (−R, R), we prescribe r (−R, R) = R, m(−R, R) = m 0 . On u = −R, we prescribe the functions of vr (−R, v) : [R, ∞) → R, φ(−R, v) : [R, ∞) → R so as to satisfy ∂v r = λ = 1, |r φ| ≤ C, m ≤ M,
(24) (25) (26)
586
M. Dafermos
where m is deduced by solving (17). The function r is of course smooth under this prescription, in fact, r = v. For now, let us assume only that φ is C 2 , although later we shall require it to be smooth. On v = R, we prescribe functions r (u, R) : (−∞, −R] → R, and φ(u, R) : (−∞, −R] → R, so as to satisfy ∂u r = ν = −1, |∂u (r φ)| = |ζ + νφ| ≤ C|u|−2 , m ≤ M,
(27) (28) (29)
where m is deduced by solving (16). Again, for now, let us require only that φ is C 2 . Such a prescription is easily seen to determine initial data (, r, φ) on segments (−∞, R) × {R} ∪ {R} × [R, ∞) satisfying (7)-(8). We may thus apply Proposition 3.1.2 to obtain D. 3.4. Global bounds. Proposition 3.4.1. For the intial value problem described above, ν < 0 and λ > 0 throughout D. Proof. By continuity, the above inequalities indeed hold in a neighborhood in D of initial data. Thus, assuming the proposition is false, there exists a (u, v) such that either ν(u, v) = 0 or λ(u, v) = 0, but for which ν < 0, λ < 0 in J−− (u, v)\{(u, v)}: Let us first
( − R, R )
( u, v )
assume that λ(u, v) = 0. Note that from (11) we have that 1−µ ≥ 0 in J−− (u, v)∩D, and 1 − µ > 0, where ν < 0, λ > 0. Integrating now (14) on [u, ˜ −R] × v, for u < u˜ < −R, as we can since ν < 0, λ > 0, we obtain λ(u, ˜ v) = λ(−R, v)e ≥ λ(−R, v) = 1.
uˆ
2ν m −R 1−µ r 2 d u˜
By continuity, λ(u, v) ≥ 1. But this contradicts the assumption λ(u, v) = 0. In an entirely similar fashion, integrating (23), the assumption ν(u, v) = 0 leads to a contradiction. The proposition is thus proven.
Black Hole Formation from a Complete Regular Past
587
In view of the above proposition, ν = 0, λ = 0 and thus 1 − µ > 0, and we may henceforth apply Eqs. (12)–(23) in all of D. Proposition 3.4.2. In D, r m ν λ 0
≥ R, ≥ m0, ≤ −1, ≥ 1, < 1 − µ ≤ 1.
(30) (31) (32) (33)
Proof. The proof is immediate from the previous proposition and the signs in Eqs. (12)– (17) and (22)–(23). We now proceed to the main result of this section: Proposition 3.4.3. Fix α > 1. Consider an initial value problem of the type described before, where8 2 4α 2 4α M 1 + 16α 4 M 32α 4 M 6α 2 C 2 , , R > max , , . α−1 α log α M(α − 1) 1 − α Then D = (−∞, −R] × [R, ∞), and the following estimates hold: |r φ| ≤ αC, |νφ + ζ | ≤ (1 + α 4 M)C|u|−2 , −1 ≥ ν ≥ −α, 1 ≤ λ ≤ α, 1 − µ ≥ (2α)−1 .
(34) (35) (36) (37) (38)
˜ −2 , |λφ + θ |(R, v) ≤ Cv
(39)
|λφ + θ |(R, v) ≤ (C˜ + α 4 MC)v −2
(40)
Moreover, if in addition
then
holds as well. Proof. Consider the region R defined to be the set of all p such that the inequalities: |r φ| < 2αC, |νφ + ζ | < 2αC|u| ν > −2α,
(41) − 23
,
8 We have here not attempted to give a scale-invariant condition.
(42) (43)
588
M. Dafermos
λ < 2α, 1 − µ > (2α)−1 , m < 2α M,
(44) (45) (46)
hold for all q ∈ J−− ( p) ∩ D. With the exception of (42) and (45), the above inequalities hold on the initial data segments regardless of the value of R. Inequality (42) holds since R > 1, and (45) holds since R>
4M . 1 − α −1
By Proposition 3.1.2 and continuity, R is non-empty; moreover, it is easily seen to be open in the topology of D. We shall show first that in R, (41)–(46) hold with α replacing 2α. Let (u, v) thus be in R. It follows by continuity that (41)–(46) hold in J−− (u, v), where, however, the < sign is replaced by the ≤ sign. We estimate r φ(u, v) as follows. Integrating the equation ∂u (r φ) = νφ + ζ, we obtain from (25) and (42), |r φ(u, v))| ≤ |r φ(−R, v)| +
−R
|νφ + ζ |d u˜
u 1
≤ C + 4Cα R − 2 1
= C(1 + 4R − 2 α) < αC 2 since R > 4α(α − 1)−1 . On the other hand, integrating (20) and applying (28), (41), (43), (45), (46), we estimate v |φ| 2λ|ν| |νφ + ζ |(u, v) ≤ |νφ + ζ |(u, R) + md v˜ 2 R r 1−µ r (u,v) |φ| 2|ν| mdr ≤ C|u|−2 + r2 1 − µ R r (u,v) dr −2 4 ≤ C|u| + 32Cα M r3 R ≤ C|u|−2 + 16Cα 4 Mr (u, R)−2 = [C + 16Cα 4 M]|u|−2 1
3
≤ C R − 2 [1 + 16α 4 M]|u|− 2 3
< αC|u|− 2 , where for the last inequality we use that R > α −2 (1 + 16α 4 M)2 .
Black Hole Formation from a Complete Regular Past
589
To estimate ν(u, v), we integrate (15), applying (27), (43), (45), (46), to obtain ν(u, v) ≥ ν(u, R)e ≥ −e
v
32α 4 M
≥ −e32α > −α,
2 λ|ν| R r 2 1−µ md v˜
r (u,v) R
dr r2
4 M R −1
since R > 32α 4 M(log α)−1 . Similarly, from (14) we estimate λ(u, v), applying (24), (44), (45), (46): λ(u, v) ≤ λ(−R, v)e ≤e
32α 4 M
≤ e32α < α.
−R u
2 λ|ν| md u˜ r 2 1−µ
−R
dr r (u,v) r 2
4 M R −1
For m, from (16) we compute
u 11−µ m(u, v) = m(−R, v) + ζ 2 d u˜ 2 ν −R u 11−µ ≤M+ [|ζ + νφ| + |νφ|]2 d u˜ 2 ν −R u 1−µ
(ζ + νφ)2 + ν 2 φ 2 d u˜ ≤M+ ν −R −R −R dr 2 2 −3 2 2 ≤ M + 4α C |u| ˜ d u˜ + 4α C 2 u r (u,v) r
≤ M + 2α 2 C 2 R −2 + 4α 2 C 2 R −1 < α M,
where for the last inequality, we use R > max 6α 2 C 2 M −1 (α − 1)−1 , 1 . ≥ 1 − 4αRM > α −1 , since Finally from this, we obtain 1 − µ = 1 − 2m r −1 −1 R > 4α M(1 − α ) . Since9 J−− (R) ∩ D ⊂ J−− (R) ∩ D = R, we have just shown that for all (u, v) ∈ R, the inequalities (41)–(46) hold throughout − J−g (u, v) with α replacing 2α. Thus (u, v) ∈ R, so R = R. Since D is connected, and R is open, it follows that R = D. 9 Here closure refers to the topology of D.
590
M. Dafermos
The above estimates and Proposition 3.1.3 imply now that ∂D = ∅, and thus D = R = (−∞, −R] × [R, ∞). To complete the proof of the proposition, it remains to show (35), and, under the additional assumption (39), (40). Integrating (20) and applying (28), (34), (36), (38), (40), we obtain v |φ| 2λ|ν| |νφ + ζ |(u, v) ≤ |νφ + ζ |(u, R) + md v˜ 2 R r 1−µ r (u,v) |φ| 2|ν| ≤ C|u|−2 + mdr r2 1 − µ R r (u,v) dr −2 4 ≤ C|u| + 2Cα M r3 R ≤ C|u|−2 + Cα 4 Mr (u, R)−2 = (1 + α 4 M)C|u|−2 . This gives (35). On the other hand, assuming (39) and integrating (21), we obtain
−R
|λφ + θ |(u, v) ≤ |λφ + θ |(−R, v) +
u
|φ| 2λ|ν| md u˜ r2 1 − µ
r (u,v)
|φ| 2λ ˜ ≤ Cv mdr + r2 1 − µ R r (u,v) dr −2 4 ˜ ≤ Cv + 2Cα M r3 R −2 4 −2 ˜ ≤ Cv + Cα Mr (−R, v) ˜ = (C + Cα 4 M)v −2 . −2
This completes the proof.
4. Proof of the Theorem In this section, we shall combine Proposition 3.4.3 with previous results of Christodoulou to give the proof of Theorem 1. Fix α, C and M, and let R be as in Proposition 3.4.3. On the ray v = R, u ≤ 0, prescribe m(0, R) = 0, r = −u, and an arbitrary C 2 function φ(u) satisfying
0
−∞
|∂u2 (uφ)|du < ∗ ,
|∂u (uφ)|(u, R) < C|u|−2 ,
(47) (48)
Black Hole Formation from a Complete Regular Past
591
for u ≥ R, and |Rφ(−R, R)| < C, 0 1 2 2 −∞ 2 (∂u φ) u < M.
(49)
If ∗ > 0 is sufficiently small, then, by Theorem 6.1 of [7], (47) implies that this initial value problem has a past-causally geodesically complete asymptotically flat development:
r=0
characteristic initial segment
Pa
st
Nu
ll
In
fin
ity
Past Timelike Infinity
When written as a first order system on Q, the boundary conditions imposed on the axis will be r = 0, −u + v = R. Moreover, if φ is C ∞ , then this development is also C ∞ . Let us assume that φ above is indeed chosen C ∞ in what follows. From the above spacetime, consider the ray u = −R, and extend it off the shaded region to all positive values of v:
r=0
r=R
Where v ≥ R, prescribe r =v
(50)
and denote by C0 = |Rφ(−R, R)|. Now, choose some r1 satisfying R < r1 < R + 1, and, setting m(−R, R) = m 0 , choose some m 1 satisfying m 0 < m 1 < M. Define the function
592
M. Dafermos
I (V ) = min
C − C0 2
2
1 M − m1 , . (V − r1 )2 2R 2 (V − r1 )
Fixing for the time being a V > r1 , from the definition of I , it is clear by a partition of unity argument that one can construct a large class of functions φ : −R × [0, ∞) → R, coinciding with the φ induced from the solution already obtained in {−R} × [0, R], and such that: 1. The function φ, when viewed as a function of r on [0, ∞), is C ∞ . 2. The inequality (∂v φ)2 (u, v) > I (V ),
(51)
|r φ|(−R, v) < C, m(−R, v) < M,
(52) (53)
holds for v ∈ [r1 , V ]. 3. The inequalities
hold for all v ≥ R, where m is defined by integrating (17). We will denote the class of functions satisfying properties 1, 2, and 3 above as FV . Define the quantities η(V ) = min V −1 (m(−R, V ) − m 1 ) φ∈FV
and δ(V ) = V r1−1 − 1. It follows that η(V ) = min V −1 φ∈FV
V
r1
≥ min R V 2
φ∈FV
−1
1 21−µ θ dv 2 λ V
r1
1 (∂v φ)2 dv 2
1 ≥ R 2 V −1 (V − r1 )I (V ) 2 1 ≥ R 2 (R + 1)−1 (V − r1 )I (V ). 2 Thus for small enough V − r1 we have η(V ) ≥
M − m1 . 4(R + 1)
Recall the function E(y) defined in [8] by 1 y log + 5 − y . E(y) = (1 + y)2 2y
(54)
Black Hole Formation from a Complete Regular Past
593
Note that E(y) → 0 as y → 0. Choose V − r1 small enough so that E(δ) <
M − m1 . 4(R + 1)
(55)
Inequalities (54) and (55) together imply η > E(δ).
(56)
The choice of φ and r determines initial data for future evolution as in [8]. From Theorem 5.1 of [8], property (56) ensures that initial data so determined for φ in FV has an incomplete future development; moreover, its conformal structure is as depicted below: (See [8] for details, in particular for the curvature blow-up and inextendibility Future Null Infinity
singularity
BLACK HOLE
r=V r = r1
r=0
r=R
Pa
st
Nu l
lI
nf
in i
ty
Past Timelike Infinity
properties of the necessarily non-empty part of the singular boundary not containing a null segment emanating from the center.) On the other hand, (48), (49), (52), and (53) imply that the rays u = −R, and v = R, and the functions r , φ, and m defined on them determine smooth initial data satisfying the conditions of Proposition 3.4.3. Applying this proposition one obtains D. Pasting D to the two regions already obtained, one forms a spacetime as depicted in the conformal diagram of the Introduction. The resulting spacetime can easily be seen to be smooth: Define new null coordinates on the regions so that the normalizations agree on v = R, u = −R, say by requiring r as a function of u, v, resp., to agree.10 At the point p ∈ Q, where the three regions intersect, by construction, all null derivatives of φ, , r up to first order match at p, whereas tangential null derivates up to any order match along v = R and u = −R. Differentiating the system (4)–(8) it follows that all null derivatives up to any order must match at p, and integrating, it follows that all null derivatives up to any order must match along v = R, and u = −R. The smoothness of the spacetime follows. The spacetime is clearly globally hyperbolic, with Cauchy surface for instance u = −v. This hypersurface will in general be boosted, however. To find a good asymptotically flat slice, let us change coordinates in D by normalizing ν = −1 at (u, ∞) and 10 This is why property 1 defining F imposes regularity with respect to r . V
594
M. Dafermos
λ = 1 at (−∞, v). This is clearly possible in view of the estimates of Sect. 3.4, which will still hold in these new coordinates with slightly different constants. One easily sees now that the slice u + v = 0 is asymptotically flat. We leave the details to the reader. Note that the completeness of future null infinity follows from [8]. See also [15]. We have thus shown statements 2, 3 and 4 of Theorem 1. It remains to show statement 1. Let α(s) = (u(s), v(s), x 1 (s), x 2 (s)) be a past directed causal geodesic emanating from the Cauchy surface u + v = 0. (We may revert to our original coordinates.) Since we know that the solution of problem B is past geodesically complete, it suffices to consider the case where (u(0), v(0)) ∈ D with u (0) ≤ 0. We have that u u (s) = −uu (u (s))2 − uAB (x A ) (s)(x B ) (s).
Suppose that (u(s), v(s)) ∈ D for some s > 0. It follows that u(s)
u
u(s)
u
u(s)
u
u(s)
u
u(s)
u
u (s) = u (0)e− u(0) uu (u,v(u))du s u(s) u − (u,v(u))du − uAB (x A ) (˜s )(x B ) (˜s )e u(˜s ) uu d s˜ 0
= u (0)e− u(0) uu (u,v(u))du u(s) u 1 s −4 u C D 4 − (u,v(u))du − r AB γ r γC D (x C ) (˜s )(x D ) (˜s )e u(˜s ) uu d s˜ 2 0
= u (0)e− u(0) uu (u,v(u))du u(s) u 1 s −4 u AB − u(˜ s ) uu (u,v(u))du d s − r AB γ X e ˜ 2 0 = u (0)e− u(0) uu (u,v(u))du s u(s) u − (u,v(u))du − 2−2 r −3 λX e u(˜s ) uu d s˜ 0
= u (0)e− u(0) uu (u,v(u))du s u(s) u 1 − µ −3 − u(˜ s ) uu (u,v(u))du d s r Xe − ˜, 2(−ν) 0
where X denotes the conserved angular momentum of the geodesic, and we have used u = ∂ log 2 and thus Eq. (5) can be the relation uAB = −2−2 r λγ AB . We have uu u rewritten λν −3 u r m − 2∂u φ∂v φ. = −4 ∂v uu 1−µ In what follows, let C denote a generic constant. In view of the fact that −R u ≤ sup log(2 (−R, R)−2 (u, R)) ≤ C < ∞, (u, R)du uu −∞
u≤−R
we have that u(s) 4λν −3 u |2∂ ≤ C + + r (u, v(u))du m φ∂ φ| du dv u v 1 − µ u(0) uu D ≤C
Black Hole Formation from a Complete Regular Past
595
by our bounds from Sect. 3.4. We have thus a uniform bound |u (s)| ≤ C(|u (0)| + X )
(57)
for as long as (u(s), v(s)) remains in D. It follows that either α(s) exists for all s > 0 and remains in D11 , or there exists a finite time s , where v(s ) = R, and α thus enters the region of spacetime described by problem B. The affine completeness of α then follows from the completeness of the past evolution of problem B. 5. Stability The strict inequalities in (48), (49), (52), and (53) immediately indicate one notion according to which our solutions are stable in the spherically symmetric category. Alternatively, considering a Cauchy surface S intersecting (−R, R), it follows that given any solution of the kind we have constructed, sufficiently small12 perturbations supported on S in r < R lead to a Cauchy development with a similar conformal diagram. For on the backwards characteristic, (47), (48), and (49) hold on account of Cauchy stability (this takes care of r ≤ R) and the domain of dependence theorem (for r ≥ R), thus guaranteeing the past completeness. On the other hand, Cauchy stability implies that η > E(δ) for these purturbations, as this refers to a compact set of the development. Thus, the formation of a black-hole in the future is also a stable feature, in this sense. Another point is worth mentioning. As the formation of a black hole depends only on the solution on −R × [r1 , V ], we can replace the part of the solution for v ≥ V with the solution of yet another characteristic initial value problem, where initial data are given on v = V and on past null infinity for v ≥ V . In particular, one can provide solutions with the conformal diagram indicated in the Introduction, for which there is no incoming radiation near spacelike infinity, i.e. for which m is constant along past null infinity in a neighborhood of spacelike infinity. On the other hand, by the results of [4], if past null infinity is complete and m is constant throughout, then either m = 0, in which case the solution is Minkowski space, or else it contains a white hole. Acknowledgement. The importance of identifying asymptotically flat initial data whose future development is singular but whose past development is regular has been stressed by Sergio Dain and Alan Rendall. I thank them for several very useful discussions. The author is supported in part by the NSF grant DMS-0302748.
References 1. Andréasson, H., Rein, G., Rendall, A.D.: On the Einstein-Vlasov system with hyperbolic symmetry. Math. Proc. Cambridge Philos. Soc. 134(3), 529–549 (2003) 2. Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Commum. Math. Phys. 14, 329–335 (1969) 3. Christodoulou, D.: The global initial value problem in general relativity. In: Proceedings of the Ninth Marcel Grossman Meeting. (Rome 2000), River Edge, NJ: World Scientific, 2002, pp. 44–54 4. Christodoulou, D.: A mathematical theory of gravitational collapse. Commum. Math. Phys. 109(4), 613–647 (1987) 5. Christodoulou, D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. of Math. 149(1), 183–217 (1999) 6. Christodoulou, D.: On the global initial value problem and the issue of singularities. Class. Quant. Grav. 16(12A), A23–A35 (1999) 11 In the case where α is timelike this can in fact be excluded by deducing a lower bound for v (s). 12 For instance, in the C ∞ topology, if we consider only C ∞ solutions.
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7. Christodoulou, D.: Bounded variation solutions of the spherically symmetric Einstein-scalar field equations. Comm. Pure Appl. Math 46(8), 1131–1220 (1992) 8. Christodoulou, D.: The formation of black holes and singularities in spherically symmetric gravitational collapse. Comm. Pure Appl. Math. 44(3), 339–373 (1991) 9. Christodoulou, D.: The problem of a self-gravitating scalar field. Commun. Math. Phys. 105(3), 337–361 (1986) 10. Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of the Minkowski Space. Princeton Mathemetical Series 41, Princeton, NJ: Princeton Univ. press, 1993 11. Christodoulou, D., Tahvildar-Zadeh, A.S.: On the regularity of spherically symmetric wave maps. Comm. Pure Appl. Math 46(7), 1041–1091 (1993) 12. Dafermos, M.: On “time-periodic” black-hole solutions to certain spherically symmetric Einstein-matter systems. Commum Math. Phys. 238(3), 411–427 (2003) 13. Dafermos, M.: Stability and instability of the Cauchy horizon for the spherically-symmetric EinsteinMaxwell-scalar field equations. Ann. Math. 158, 875–928 (2003) 14. Dafermos, M.: The interior of charged black holes and the problem of uniqueness in general relativity. Comm. Pure Appl. Math. 58, 445–504 (2005) 15. Dafermos, M.: Spherically symmetric spacetimes with a trapped surface. Class. Quant. Grav. 22(11), 2221–2232 (2005) 16. Dafermos, M.: On naked singularities and the collapse of self-gravitating Higgs fields. Adv. Theor. Math. Phys. 9, 575–591 (2005) 17. Dafermos, M., Rodnianski, I.: A proof of Price’s law for the collapse of a self-gravitating scalar field. Invent. Math. 162, 381–457 (2005) 18. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-time Cambridge Monographs on Mathematical Physics, No. 1, London-New york: Cambridge University Press, 1973 19. Oppenheimer, J.R., Snyder, H.: On continued gravitational contraction. Phys. Rev. Lett. 56, 455–459 (1939) 20. Penrose, R.: Gravitational collapse and space-time singularities. Phys. Rev. Lett. 14, 57–59 (1965) 21. Rein, G.: On future geodesic completeness for the Einstein-Vlasov system with hyperbolic symmetry Math. Proc. Cambridge Philos. Soc. 137(1), 237–244 (2004) 22. Rein, G., Rendall, A.: Global existence of solutions of the spherically symmetric Vlasov-Einstein system with small initial data. Commum. Math. Phys. 150, 561–583 (1992) Communicated by G.W. Gibbons
Commun. Math. Phys. 289, 597–652 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0787-3
Communications in
Mathematical Physics
Local Asymptotic Normality for Finite Dimensional Quantum Systems Jonas Kahn1 , M˘ad˘alin Gu¸ta˘ 2 1 Département de Mathématiques, Université Paris-Sud 11, Bât 425,
91405 Orsay Cedex, France
2 School of Mathematical Sciences, University of Nottingham, University Park,
NG7 2RD, Nottingham, UK. E-mail: [email protected] Received: 7 July 2008 / Accepted: 16 January 2009 Published online: 28 March 2009 – © Springer-Verlag 2009
Abstract: Previous results on local asymptotic normality (LAN) for qubits [16,19] are extended to quantum systems of arbitrary finite dimension d. LAN means that the quantum statistical model consisting of n identically prepared d-dimensional systems with joint state ρ ⊗n converges as n → ∞ to a statistical model consisting of classical and quantum Gaussian variables with fixed and known covariance matrix, and unknown means related to the parameters of the density matrix ρ. Remarkably, the limit model splits into a product of a classical Gaussian with mean equal to the diagonal parameters, and independent harmonic oscillators prepared in thermal equilibrium states displaced by an amount proportional to the off-diagonal elements. As in the qubits case [16], LAN is the main ingredient in devising a general two step adaptive procedure for the optimal estimation of completely unknown d-dimensional quantum states. This measurement strategy shall be described in a forthcoming paper [18]. 1. Introduction Quantum statistics deals with problems of statistical inference arising in quantum mechanics. The first significant results in this area appeared in the seventies and tackled issues such as quantum Cramér-Rao bounds for unbiased estimators, optimal estimation for families of states possessing a group symmetry, estimation of Gaussian states, optimal discrimination between non-commuting states. It is impossible to list all contributions but the following references may give the flavour of these developments [7,8,27,28,47,48]. The more recent theoretical advances [2,3,6,23,24,35] are closely related to the rapid development of quantum information and quantum engineering, and are often accompanied by practical implementations [1,20,39,40]. An important topic in quantum statistics is that of optimal estimation of an unknown state using the results of measurements performed on n identically prepared quantum systems [4,5,9,13,14,25,26,30,32,45]. In the case of two dimensional systems, or qubits, the problem has been solved explicitly in the Bayesian set-up, in the particular case
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of an invariant prior and figure of merit based on the fidelity distance between states [5]. However the method used there does not work for more general priors, loss functions, or higher dimensions. In the pointwise approach, Hayashi and Matsumoto [26] showed that the Holevo bound [28] for the variance of locally unbiased estimators can be achieved asymptotically, and provided a sequence of measurements with this property. Their results, building on earlier work [21,22], indicate for the first time the emergence of a Gaussian limit in the problem of optimal state estimation for qubits. The extension to d-dimensional case is analysed by Matsumoto in [33]. In [16,19] we performed a detailed analysis of this phenomenon (again for qubits), and showed that we deal with the quantum generalization of an important concept in mathematical statistics called local asymptotic normality. As a corollary, we devised a two steps adaptive measurement strategy for state estimation which is asymptotically optimal for a large class of loss functions and priors, and could be practically implemented using continuous-time measurements. In ‘classical statistics’, the idea of approximating a sequence of statistical models by a family of Gaussian distributions appeared in [46], and was fully developed by Le Cam [31] who coined the term “local asymptotic normality”. Among the many applications we mention its role in asymptotic optimality theory and in proving the asymptotic normality of certain estimators such as the maximum likelihood estimator. The aim of this paper is to extend the results of [16,19] to systems of arbitrary dimension d < ∞, and thus provide the main tool for solving the open problem of optimal state estimation for d-dimensional quantum systems [18]. Before stating the main result of the paper we shall explain briefly the meaning of local asymptotic normality for two dimensional systems [16,19]. We are given n qubits identically prepared in an unknown state ρ. Asymptotic normality means that for large n we can encode the statistical information contained in the state ρ ⊗n into a Gaussian model consisting of a classical random variable with distribution N (u, I −1 ), and a quantum harmonic oscillator prepared in a (Gaussian) displaced thermal state φζ . The term local refers to how ρ is related to the parameters θ = (u, ζ ), as explained below. For a more precise formulation let us parametrise the qubit states by their Bloch → → → → vectors ρ(− r ) = 21 (1 + − r − σ ), where − σ = (σx , σ y , σz ) are the Pauli matrices. The − → neighbourhood of the state ρ0 with r0 = (0, 0, 2µ − 1) and 1/2 < µ < 1, is a threedimensional ball parametrised by the deviation u ∈ R of diagonal elements and ζ ∈ C of the off-diagonal ones, µ+u ζ∗ , θ = (u, ζ ) ∈ R × C. (1.1) ρθ = ζ 1−µ−u Note that ρ0 is to be considered fixed and known but otherwise arbitrary, and can be taken to be diagonal without any loss of generality. Consider now n identically prepared √ qubits whose individual states are in a neighbourhood of ρ0 of size 1/ n, so that their ⊗n for some unknown θ . We would like to understand the joint state is ρθn := ρθ/√n structure of the family (statistical experiment) Qn := {ρθn : θ ≤ C},
(1.2)
as a whole, more precisely what is its asymptotic behavior as n → ∞ ? For this we consider a quantum harmonic oscillator with position and momentum operators satisfying the commutation relations [Q, P] = i1. We denote by {|k , k ≥ 0} the eigenbasis of the number operator and define the thermal equilibrium state
Local Asymptotic Normality for Finite Dimensional Quantum Systems
φ = (1 − e−β )
∞
e−kβ |k k|,
k=0
e−β =
599
1−µ , µ
which has centered Gaussian distributions for both Q and P with variance 1/(4µ − 2) > 1/2. We define a family of displaced thermal equilibrium states φ ζ := D ζ (φ) := W (ζ / 2µ − 1) φ W (ζ / 2µ − 1)∗ , (1.3) where W (ζ ) := exp(ζ a ∗ − ζ¯ a) is the unitary displacement operator with ζ ∈ C. Additionally we consider a classical Gaussian shift model consisting of the family of normal distributions N (u, µ(1 − µ)) with unknown center u and fixed known variance. The classical-quantum statistical experiment to which we alluded above is defined by the family of densities R := {φ θ := N (u, µ(1 − µ)) ⊗ φ ζ : θ ≤ C},
(1.4)
where the unknown parameters θ = (u, ζ ) ∈ R × C are the same as those of Qn . Theorem 1.1 [16,19]. Let Qn be the quantum statistical experiment (1.2) and let R be the classical-quantum experiment (1.4). Then for each n there exist quantum channels (normalized completely positive maps) Tn : M (C2 )⊗n → L 1 (R) ⊗ T (L 2 (R)), Sn : L 1 (R) ⊗ T (L 2 (R)) → M (C2 )⊗n , with T (L 2 (R)) the trace-class operators, such that
lim sup φθ − Tn ρθn 1 = 0, n→∞ θ≤C
lim
sup ρθn − Sn (φθ ) 1 = 0,
n→∞ θ≤C
for an arbitrary constant C > 0. The norm of trace class operators is τ 1 := Tr(|τ |). The theorem shows that from a statistical point of view the joint qubits states are asymptotically indistinguishable from the limit Gaussian system. At the first sight one might object that the local nature of the result prevents us from drawing any conclusions for the original model of a completely unknown state ρ. However this is not a limitation, but reflects the correct normalisation of the parameters with n → ∞. Indeed as n grows we have more information √ about the state which can be pinned down to a region of size slightly larger than 1/ n by performing rough measurements on a small proportion of the systems. After this ‘localisation’ step, we can use more sophisticated techniques to better estimate the state within the local neighbourhood of the first step estimator, and it is here where we use the local asymptotic normality result. Indeed, since locally the states are uniformly close to displaced Gaussian states we can pull back the optimal (heterodyne) measurement for estimating the latter to get an asymptotically optimal measurement for the former. Based on this insight we have proposed a realistic measurement set-up for this purpose using an atom-field interaction and continuous measurements in the field [16].
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This paper deals with the extension of the previous result to d-dimensional systems. Like in the two-dimensional case we parametrise the neighbourhood of a fixed (diagonal) state ρ0 by a vector u ∈ Rd−1 of diagonal parameters and d(d − 1)/2 complex parameters ζ = (ζ j,k : j < k), √ one for each off-diagonal matrix element (cf. (4.2) and (4.4)). We consider the same 1/ n−scaling and look at the family Qn =
⊗n ρθ/√n : θ = ( u , ζ ) ∈ n ⊂ Rd−1 ⊗ Cd(d−1)/2 ,
where n is a ball of local parameters whose size is allowed to grow slowly with n. As in the 2-dimensional case, the limit model is the product of a classical statistical model depending on the parameters u and a quantum model depending on ζ . Moreover the quantum part splits into a tensor product of displaced thermal states of quantum oscillators, one for each off-diagonal matrix element ζ j,k with j < k. Thus φ θ = N ( u , Iρ−1 )⊗ 0
ζ
φ j,kj,k ,
θ = ( u , ζ ).
j
Here, Iρ0 is the Fisher information matrix of the multinomial model with parameters ζ
(µ1 , . . . , µd ) described in Example 3.4, and φ j,kj,k is the displaced thermal equilibrium state defined in (4.15) with inverse temperature β = ln(µ j /µk ). Theorem 4.3 is the main result of the paper and shows the convergence of Qn to the Gaussian model Rn = φ θ : θ ∈ n ⊂ Rd−1 ⊗ Cd(d−1)/2 , in the spirit of Theorem 1.1. On the technical side, the uniform convergence holds over local neighbourhoods n which are allowed to grow with n rather than being fixed balls. This is essential for constructing the two stage optimal measurement: first localise within a neighbourhood n , and then apply the optimal Gaussian measurement. The details of this construction are similar to the two dimensional case and will be given in a subsequent paper [18]. Despite the similarity to the two dimensional case, the proof of the d-dimensional result has additional features which may be responsible for the fact that the optimal estimation problem has remained unsolved until now. The proof is based on the following observations: • the n systems space (Cd )⊗n decomposes into a direct sum of irreducible representations of SU (d), each representation being labelled by a Young diagram λ (cf. Theorem 4.1); ⊗n√ • the joint state ρθ/ has the block diagonal form (4.8), the block weights λ → pλθ,n n depend only on the diagonal parameters u and are closely related to the multinomial distribution of Example 3.4. This classical statistical model converges to the (d − 1)dimensional Gaussian shift model N ( u , Iρ−1 ); 0 • there exists an isometry Vλ mapping basis vectors |m, λ of the irreducible representation Hλ almost into number vectors |m of the multimode Fock space, where m = {m j,k : j < k} is the collection of eigenvalues of the number operators for all oscillators.
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• given a typical λ, the conditional block-state ρλθ,n can be mapped with Vλ into a mulζ
timode state which is close (in trace norm) to the Gaussian product state ⊗ j
The first item is the well known Weyl duality which is extensively used in quantum statistics for i.i.d. states. The probability distribution of the second point has also been analysed in the context of large deviations [30] for the estimation of the state eigenvalues. The third point shows that the basis |m, λ is almost orthogonal for indices m which are not too big. This basis is obtained by projecting tensors of the form f a := f a(1) ⊗ · · · ⊗ f a(n) onto a subspace of (Cd )⊗n which is isomorphic to Hλ (cf. Theorem 5.2). Let us place the indices {a(i) : i = 1 . . . n} in the boxes of the diagram λ along rows, starting from the left end of the first row, to obtain a tableau ta . It turns out that we only need to consider f a for which ta is a semistandard tableau (nondecreasing along rows, increasing along columns). Then the label m := {m i, j : j > i} is the collection of integers m i, j equal to the number of j s on the row i, and is in one to one correspondence with a. The following is an example of such semistandard tableau 1 1 1 1 1 1 1 1 2 2 2 3 3
tm = 2 2 2 2 2 3 3 3 3
,
with m 1,2 = 3, m 1,3 = 2, m 2,3 = 1.
The relatively large number of i’s in the row i is intentional, since it turns out that the ‘relevant’ vectors, i.e. those carrying the states ρλθ,n , have indices m i, j small compared with the length of the rows (λi ≈ nµi for typical representations λ). More precisely, in Sect. 7.2 we prove the following quasi-orthogonality result which allows us to carry the block states over to the oscillator space: if m = l and |l| ≤ |m| ≤ n η , then | m, λ|l, λ | = O(n (9η−2)|m−l|/12 ) −−−→ 0 n→∞
for η < 2/9.
The proof of the fourth point involves a detailed analysis of the state ρλθ,n through its coefficients in the basis |m, λ of Hλ . When θ = 0 the state is diagonal and its coeffi cients approach uniformly those of the multidimensional thermal state φ 0 = ⊗ j
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In Sect. 5 we introduce the basis |m, λ and the isometry Vλ allowing us to define the channels Tn and Sn connecting the two statistical models. In Sect. 6 we break the proof of the main theorem into manageable lemmas, essentially by using triangle inequalities. Each lemma deals with a different aspect of the convergence and has an interest in its own. Finally, the technical proofs are collected in Sect. 7. Notably, Subsect. 7.1 and Lemma 7.1 contain the combinatorial substance of the paper. Our investigation relies on the theory of representations of SU (d). We refer to [11, 12,15] for proofs of standard results and more details. As in the two-dimensional case [16], local asymptotic normality provides a two stage adaptive measurement strategy which is asymptotically optimal for both Bayesian and pointwise viewpoints, and for a large range of ‘distances’ on the state space [18]. Our result is derived under the assumption that the density matrix ρ0 is strictly positive and has no degenerate eigenvalues. Although we believe that a limit experiment exists for all states, the limit model may have a different structure. For example in the case d = 2, when µ1 = µ2 = 1/2 the limit is a completely classical Gaussian shift experiment as it can be gleaned from the quantum Central Limit Theorem [37] and from the alternative version of local asymptotic normality based on ‘weak convergence’ [17]. More drastically, when the state ρ0 is pure (but we do not have this information), the classical Gaussian shift describing the eigenvalue parameter is replaced by a Poisson experiment and the correct scaling is 1/n. A detailed analysis of these remaining cases will be pursued in a separate publication. Throughout, we will use the following symbols: ϕ, ψ for states, φ, ρ for density matrices, T, S, M for channels (randomisations), E, P, Q for statistical models (experiments), θ, ζ, u for parameters, α, β, γ , δ, for positive constants, λ for Young diagrams. 2. Classical and Quantum Statistical Experiments In this section we introduce some basic notions from classical statistics with the aim of defining the Le Cam distance between statistical models and local asymptotic normality. In parallel, we shall define the quantum analogues and point out their relevance in quantum statistics. The reader may find the conceptual framework helpful in understanding the quantum version of the result. Let X be a random variable with values in the measure space (X , X ), and let us assume that its probability distribution P belongs to some family {Pθ : θ ∈ } where the parameter θ is unknown. Statistical inference deals with the question of how to use the available data X in order to draw conclusions about some property of θ . We shall call the family E := {Pθ : θ ∈ },
(2.1)
a statistical experiment or statistical model over (X , X ) [31]. In quantum statistics the data is replaced by a quantum system prepared in a state ϕ which belongs to a family {ϕθ : θ ∈ } of states over an algebra of observables. In order to make a statistical inference about θ one first has to measure the system, and then apply statistical techniques to draw conclusions from the data consisting of the measurement outcomes. An important difference with the classical case is that the experimenter has the possibility to choose the measurement set-up M, and each set-up (M) (M) will lead to a different classical model {Pθ : θ ∈ }, where Pθ is the distribution of outcomes when performing the measurement M on the system prepared in state ϕθ .
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The guiding idea of this paper is to investigate the structure of the family of quantum states Q := {ϕθ : θ ∈ }, which will be called a quantum statistical experiment. We shall show that in an important asymptotic set-up, namely that of a large number of identically prepared systems, the joint state can be approximated by a multidimensional quantum Gaussian state, for all possible preparations of the individual systems. This will bring a drastic simplification in the problem of optimal estimation for d-dimensional quantum systems, which can then be solved in the asymptotic framework [18]. 2.1. Classical and quantum randomisations. Any statistical decision (e.g. estimator, test) can be seen as data processing using a Markov kernel. Suppose we are given a random variable X taking values in (X , X ) and we want to produce a ‘decision’ y ∈ Y based on the data X . The space Y may be for example the parameter space in the case of estimation, or just the set {0, 1} in the case of testing between two hypotheses. For every value x ∈ X we choose y randomly with probability distribution given by K x (dy). Assuming that K : X × Y → [0, 1] is measurable with respect to x for all fixed A ∈ Y , we can regard K as a map from probability distributions over (X , X ) to probability distributions over (Y, Y ) with K (P)(A) = K x (A)P(d x), A ∈ Y . (2.2) A statistic S : X → Y is a particular example of such a procedure, where K x is simply the delta measure at S(x). Besides statistical decisions, there is another important reason why one would like to apply such treatment to the data, namely to summarize it in a more convenient and informative way for future purposes as illustrated in the following simple example. Consider n independent identically distributed random variables X 1 , . . . , X n with values in {0, 1} and distribution Pθ := (1 − θ, θ ) with θ ∈ := (0, 1). The associated statistical experiment is En := {Pθn : θ ∈ }. n It is easy to see that X¯ n = n1 i=1 X i is an unbiased estimator of θ and moreover it is a sufficient statistic for En , i.e. the conditional distribution Pθn (·| X¯ n = x) ¯ does not depend on θ ! In other words the dependence on θ of the total sample (X 1 , X 2 , . . . , X n ) is completely captured by the statistic X¯ n which can be used as such for any statistical decision problem concerning En . If we denote by P¯θn the distribution of X¯ n then the experiment E¯n = { P¯θn : θ ∈ }, is statistically equivalent to En . The Markov kernel K defined in (2.2) maps the experiment E of Eq. (2.1) into the experiment F := {Q θ : θ ∈ }, over (Y, Y ) with Q θ = K (Pθ ). For mathematical convenience it is useful to represent such transformations in terms of linear maps between linear spaces. Definition 2.1. A positive linear map T∗ : L 1 (X , X , P) → L 1 (Y, Y , Q)
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is called a stochastic operator or transition if T∗ (g)1 = g1 for every g ∈ L 1+ (X ). A positive linear map T : L ∞ (Y, Y , Q) → L ∞ (X , X , P) is called a Markov operator if T 1 = 1, and if for any f n ↓ 0 in L ∞ (Y) we have T f n ↓ 0. A pair (T∗ , T ) as above is called a dual pair if f T (g)d P = T∗ ( f )gd Q, for all f ∈ L 1 (X , X , P) and g ∈ L ∞ (Y, Y , Q). It is a theorem that for any stochastic operator T∗ there exists a unique dual Markov operator T and vice versa. What is the relation between Markov operators and Markov kernels ? Roughly speaking, any Markov kernel defines a Markov operator when we restrict to families of dominated probability measures. Let us assume that all distributions Pθ of the experiment E defined in (2.1) are absolutely continuous with respect to a fixed probability distribution P, such that there exist densities pθ := d Pθ /d P : X → R+ . Such an experiment is called dominated and in concrete situations this condition is usually satisfied. Let K x (dy) be a Markov kernel (2.2) such that Q θ = K (Pθ ), then we define associated Markov operator (T ( f ))(x) := f (y)k x (dy) and have Q θ = Pθ ◦ T, ∀θ.
(2.3)
When the probability distributions of two experiments are related to each other as in (2.3), we say that F is a randomisation of E. From the duality between T and T∗ we obtain an equivalent characterization in terms of the stochastic operator T∗ : L 1 (X , X , P) → L 1 (Y, Y , Q) such that T∗ (d Pθ /d P) = d Q θ /d Q, ∀θ . The concept of randomisation is weaker than that of Markov kernel transformation, but under the additional condition that (Y, Y ) is locally compact space with countable base and Borel σ -field, it can be shown that any randomisation can be implemented by a Markov kernel [41]. What is the analogue of randomisations in the quantum case ? In the language of operator algebras L ∞ (X , X , P) is a commutative von Neumann algebra and L 1 (X , X , P) is the space of (densities of) normal linear functionals on it. The stochastic operator T∗ is the classical version of quantum channel, i.e. a completely positive normalized (tracepreserving) map T∗ : A∗ → B∗ , where A∗ , B∗ are the spaces of normal states on the von Neumann algebra A and respectively B. Any normal state ϕ on A has a density ρ with respect to the trace such that ϕ(A) = Tr(ρ A) for all A ∈ A. The dual of T∗ is T : B → A, which is a unital completely positive map and has the property that T∗ (ϕ)(b) = ϕ(T (b)) for all b ∈ B and ϕ ∈ A∗ . We interpret such quantum channels as possible physical transformations from input to output states.
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A particular class of channels is that of measurements. In this case the input is the state of a quantum system described by an algebra A, and the output is a probability distribution over the space of outcomes (X , X ). Any measurement is described by a positive linear map M : L ∞ (X , X , P) → A, which is completely specified by the image of characteristic functions of measurable sets, also called positive operator valued measure (POVM). The corresponding channel acting on states is a positive map M∗ : A∗ → L 1 (X , X , P) given by M∗ (ϕ)(A) = ϕ(M(A)) = Tr(ρ M(A)), where ρ is the density matrix of ϕ. By applying the channel M to the quantum statistical experiment consisting of the family of states Q = {ϕθ : θ ∈ } on A we obtain a classical statistical experiment Q M := {M∗ (ϕθ ) : θ ∈ }, over the outcomes space (X , X ). As in the classical case, quantum channels can be seen as ways to compare quantum experiments. The first steps in this direction were made by Petz [34,36,38] who developed the theory of quantum sufficiency dealing with the problem of characterizing when a sub-algebra of observables contains the same statistical information about a family of states, as the original algebra. More generally, two experiments Q := {A, ϕθ : θ ∈ } and R := {B, ψθ : θ ∈ } are called statistically equivalent if there exist channels T : A → B and S : B → A such that ψθ ◦ T = ϕθ and ϕθ ◦ S = ψθ ∀θ. As consequence, for any measurement M : L ∞ (X , X , P) → A there exists a measurement T ◦ M : L ∞ (X , X , P) → B such that the resulting classical experiments coincide Q M = RT ◦M . Thus for any statistical problem, and any procedure concerning the experiment Q there exists a procedure for R with the same risk (average error), and vice versa. 2.2. The Le Cam distance and its statistical meaning. We have seen that two experiments are statistically equivalent when they can be transformed into each other by means of quantum channels. When this cannot be done exactly, we would like to have a measure of how close the two experiments are when we allow any channel transformation. We define the deficiency of R with respect to Q as δ(R, Q) = inf sup ϕθ − ψθ ◦ T , T
θ
(2.4)
where the infimum is taken over all channels T : A → B. The norm distance between two states on A is defined as ϕ1 − ϕ2 := sup{|ϕ1 (a) − ϕ2 (a)| : a ∈ A, a ≤ 1}, and for A = B(H) it is equal to ρ1 − ρ2 1 := Tr(|ρ1 − ρ2 |), where ρi is the density matrix of the state ϕi . When δ(R, Q) = 0 we say that R is more informative than
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Q. Note that δ(R, Q) is not symmetric but satisfies a triangle inequality of the form δ(R, Q) + δ(Q, T ) ≥ δ(R, T ). By symmetrizing we obtain a proper distance over the space of equivalence classes of experiments, called Le Cam’s distance [31], (Q, R) := max (δ(Q, R) , δ(R, Q)) .
(2.5)
What is the statistical meaning of the Le Cam distance ? We shall show that if δ(R, Q) ≤ , then for any statistical decision problem with loss function between 0 and 1, any measurement procedure for Q can be matched by a measurement procedure for R whose risk will be at most larger than the previous one. A decision problem is specified by a decision space (X , X ) and a loss function Wθ : X → [0, 1] for each θ ∈ . We are given a quantum system prepared in the state ϕθ ∈ A∗ with unknown parameter θ ∈ and would like to perform a measurement with outcomes in X such that the expected value of the loss function Wθ is small. Let M : L ∞ (X , X , P) → A, be such a measurement, and Pθ(M) = ϕθ ◦ M, then the risk at θ is (M) Wθ (x)Pθ (d x). R(M, θ ) := X
Since the point θ is unknown one would like to obtain a small risk over all possible realizations Rmax (M) = sup R(M, θ ). θ∈
The minimax risk is then Rminmax := inf M Rmax (M). Coming back to the experiments Q and R we shall compare their achievable risks for a given decision problem as above. Consider the measurement N : L ∞ (X , X , P) → B given by N = T ◦ M, where T : A → B is the channel which achieves the infimum in (2.4). Then R(N , θ ) = Wθ (x)Pθ(N ) (d x) = ψθ (T ◦ M(Wθ )) X
≤ ψθ ◦ T − ϕθ + ϕθ (M(Wθ )) ≤ δ(R, Q) + R(M, θ ), where we have used the fact that 0 ≤ Wθ ≤ 1. Lemma 2.2. For every achievable risk R(M, θ ) for Q there exists a measurement N : L ∞ (X , X , P) → B for R such that R(N , θ ) ≤ R(M, θ ) + δ(R, Q). As a consequence, Rminmax (R) ≤ Rminmax (Q) + δ(R, Q). A similar inequality holds in the Bayesian framework where one optimises the average risk with respect to a prior distribution π over : Rπ (M) = R(M, θ )π(dθ ).
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3. Local Asymptotic Normality in Statistics In this section we describe the notion of local asymptotic normality and its significance in statistics [31,41–43]. Suppose that we observe X 1 , . . . , X n , where X i take values in a measurable space (X , X ) and are independent, identically distributed with distribution Pθ indexed by a parameter θ belonging to an open subset ⊂ Rm . The full sample is a single observation from the product Pθn of n copies of Pθ on the sample space (n , n ). Local asymptotic normality means that for large n such statistical experiments can be approximated by Gaussian experiments after √a suitable reparametrisation. Let θ0 be a fixed point and define a local parameter u = n(θ − θ0 ) characterizing points in a small neighbourhood of θ0 , and rewrite Pθn as Pθn +u/√n seen as a distribution depending on 0 the parameter u. Local asymptotic normality means that for large n the experiments m Pθn +u/√n : u ∈ Rm and N (u, Iθ−1 , ) : u ∈ R 0 0
have the same statistical properties when the models θ → Pθ are sufficiently ‘smooth’. The point of this result is that while the original experiment may be difficult to analyse, the limit one is a tractable Gaussian shift experiment in which we observe a single sample . Here, from the normal distribution with unknown mean u and fixed variance matrix Iθ−1 0 Iθ0 i j = Eθ0 θ0 ,i θ0 , j is the Fisher information matrix at θ0 , with θ,i := ∂ log pθ /∂θi the score function and pθ is the density of Pθ with respect to a reference probability distribution P. There exist two formulations of the result depending on the notion of convergence which one uses. In this paper we only discuss the strong version based on convergence with respect to the Le Cam distance, and we refer to [43] for another formulation using the so-called weak convergence (convergence in distribution of finite dimensional marginals of the likelihood ratio process), and to [17] for its generalization to quantum statistical experiments. Before formulating the theorem, we explain what sufficiently smooth means. The least restrictive condition is that pθ is differentiable in quadratic mean, i.e. there exists a measurable function θ : X → R such that 1/2 1/2 1/2 2 d P = 0. pθ+u − pθ − u t θ pθ lim u→0
Note that θ must still be interpreted as a score function since under some regularity 1/2 1/2 conditions we have ∂ pθ /∂θi = 21 (∂ log pθ /∂θi ) pθ . Theorem 3.1. Let E := {Pθ : θ ∈ } be a statistical experiment with ⊂ Rd and Pθ P such that the map θ → pθ is differentiable in quadratic mean. Define En = {Pθn +u/√n : u ≤ C}, F = {N (u, Iθ−1 ) : u ≤ C}, 0 0
with Iθ0 the Fisher information matrix of E at point θ0 , and C a positive constant. Then (En , F) → 0. In other words, there exist sequences of randomisations Tn and Sn such that: lim
sup Tn (Pθn +u/√n ) − N (u, Iθ−1 ) = 0, 0
lim
sup Pθn +u/√n − Sn (N (u, Iθ−1 )) = 0. 0
n→∞ u≤C
n→∞ u≤C
0
0
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Remark 3.2. Note that the statement of the theorem is not of Central Limit type which typically involves convergence in distribution to a Gaussian distribution at a single point θ0 . Local asymptotic normality states that the convergence is uniform in a 1/sqr tnneighbourhood of θ0 , and moreover the variance of the limit Gaussian is fixed whereas the variance obtained from the Central Limit Theorem depends on the point θ . Additionally, the randomisation transforming the data (X 1 , . . . , X n ) into the Gaussian variable √ is the same for all θ = θ0 + u/ n and thus does not require a priori the knowledge of θ . Remark 3.3. Local asymptotic normality is the basis of many important results in asymptotic optimality theory and explains the asymptotic normality of certain estimators such as the maximum likelihood estimator. The quantum version introduced in the next section plays a similar role for the case of the quantum statistical model. An asymptotically optimal estimation strategy based on local asymptotic normality was derived in [16] for two-dimensional systems. Example 3.4. Let Pµ = (µ1 , . . . , µd ) be a probability distribution with unknown param satisfying µi > 0 and i≤d−1 µi < 1. The Fisher eters (µ1 , . . . , µd−1 ) ∈ Rd−1 + information at a point µ is I (µ)i j =
d−1
µk (δik µi−1 · δ jk µ−1 j )+(1 −
k=1
d−1
µl )−1 = δi j µi−1 +(1 −
l=1
d−1
µl )−1 ,
(3.1)
l=1
and its inverse is V (µ)i j := [I (µ)−1 ]i j = δi j µi − µi µ j .
(3.2)
Pµn
The multinomial experiment consisting of n i.i.d. date with distribution Pµ converges locally to F := (N (u, V (µ)) : u ∈ Rd−1 , u ≤ C). The experiment F will appear again in Theorem 4.3, as the classical part of the limit Gaussian shift experiment. 4. Local Asymptotic Normality in Quantum Statistics In this section we present the main result of the paper. Local asymptotic normality for d-dimensional quantum systems means roughly the following: the sequence Qn of experiments consisting of joint states ρ ⊗n of n identical quantum systems prepared independently in the same state ρ, converges to a limit experiment R which is a quantumclassical Gaussian model involving displaced thermal equilibrium states of d(d − 1)/2 oscillators and a (d − 1)-dimensional classical Gaussian shift √ model. As in the classical case, the result has a local nature reflecting the 1/ n rate of convergence of state estimation. A neighbourhood of a fixed diagonal state ρ0 = Diag(µ1 , . . . , µd ) is parametrised by (changes in the) diagonal parameters u ∈ Rd−1 and off-diagonal parameters ζ ∈ Cd(d−1)/2 . The latter can be implemented by small unitary rotations. The limit Gaussian model has a classical part N ( u , V (µ)) with fixed known variance ζ j,k ζ j,k V (µ), and a quantum part ⊗ j
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the proof. However as we shall see in Sect. 4.5, the limit experiment can be formulated in a ‘coordinate-free’ way in terms of quasifree states on a CC R-algebra. Although it is not needed in the main theorem, we include this formulation linking our result to the Quantum Central Limit Theorem. We stress again that local asymptotic normality is not a consequence of the Central Limit Theorem, indeed the latter is not even an ingredient in the proof but gives an indication as to what is the limit state when all parameters are zero. 4.1. The n-tuple of d-dimensional systems. As explained in Sect. 3 for the classical case, our theory will be local in nature, so we shall be interested in a (shrinking) neighbourhood of an arbitrary but fixed faithful state with density given by the diagonal matrix ρ0 = Diag(µ1 , µ2 , . . . , µd ) with µ1 > µ2 > · · · > µd > 0.
(4.1)
A sufficiently small neighbourhood of ρ0 in the state space can be parametrised by θ := ( u , ζ ) as follows: ⎡ ⎤ ∗ ∗ µ1 + u 1 ζ1,2 ... ζ1,d .. ⎢ ⎥ .. ⎢ ζ1,2 µ2 + u 2 ⎥ . . ⎢ ⎥, u i ∈ R, ζ j,k ∈ C. (4.2) ρ˜θ := ⎢ . ⎥ . . .. .. ∗ ⎣ .. ⎦ ζd−1,d d−1 ζ1,d . . . ζd−1,d µd − i=1 ui Indeed, note that if θ is small enough then ρ˜θ is a density matrix. Let δ := inf 1≤i≤d µi − µi+1 , with µd+1 = 0, be the minimal separation between the eigenvalues. Throughout the paper we restrict to states satisfying (4.1), for which the minimal separation δ is strictly positive. √ In the first order in θ/ δ, the family ρ˜θ is obtained by first perturbing the diagonal elements of ρ0 with u and then performing a small unitary transformation with ⎡ ⎛ ⎞⎤ Re(ζ j,k )T j,k + Im(ζ j,k )Tk, j ⎠⎦ , U (ζ ) := exp ⎣i ⎝ (4.3) √ µ j − µk 1≤ j
where T j,k are generators of the Lie algebra of SU (d) defined in (7.2). The advantage of the latter parametrisation is that we can fully exploit the machinery of irreducible group representations. For this reason, in all subsequent computations we shall work with the ‘unitary’ family d−1 ρθ :=U (ζ )Diag µ1 + u 1 , µ2 + u 2 , . . . , µd − u i U ∗ (ζ ), u i ∈ R, ζ j,k ∈ C. i=1
(4.4) but we keep in mind the relationship with (4.2). √ As in the classical case, the parameter θ will be scaled by the factor 1/ n meaning that we zoom in around ρ0 with the rate equal to the typical estimation rate based on n ⊗n√ samples. Let ρ θ,n := ρθ/ and let Qn be the sequence of statistical experiments n " ! Qn := ρ θ,n : θ ∈ n ,
(4.5)
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consisting of n systems, each one prepared in a state ρθ/√n situated in a local neigh− → → u , ζ ) belongs to a neighbourhood of bourhood of ρ . The local parameter θ = (− 0
n
the origin of Rd−1 × Cd(d−1)/2 which is allowed to grow slowly with n in a way that will be made precise later. One of the principal tools in our result is the representation theory of the special unitary group SU (d). Due to lack of space we shall not include any proofs and refer to [11,12,15] for details. In particular we shall be working with the well known tensor representation which will be analysed in increasing depth across the following sections. The space (Cd )⊗n carries two commuting group representations: that of SU (d) given by πn (U ) : f 1 ⊗ · · · ⊗ f n → U f 1 ⊗ · · · ⊗ U f n , U ∈ SU (d),
(4.6)
and that of the permutation group S(n) given by π˜ d (τ ) : f 1 ⊗ · · · ⊗ f n → f τ −1 (1) ⊗ · · · ⊗ f τ −1 (n) , τ ∈ S(n).
(4.7)
Since the two group representations commute with each other, the representation space decomposes into a direct sum of tensor products of irreducible representations. It turns out that the irreducible representations of SU (d) and S(n) are indexed by Young diagrams with d rows for the former and n boxes for the latter. A Young diagram is defined by a tuple of ordered integers λ = (λ1 ≥ λ2 · · · ≥ λk ) with λi the number of boxes on row i (see Fig. 1). As we shall see later this pictorial representation will be very useful in understanding the structure of the irreducible representations (Hλ , πλ ) of SU (d). The following theorem called Schur-Weyl duality shows that the only tensor products appearing in the above mentioned direct sum are those of irreducible representations indexed by the same λ, and in particular the algebras generated by πn (u) and respectively π˜ d (τ ) are each other’s commutant! Theorem 4.1. Let πn and π˜ d be the representations of SU (d) and respectively S(n) on (Cd )⊗n . Then the representation space decomposes into a direct sum of tensor products of irreducible representations of SU (d) and S(n) indexed by Young diagrams with d lines and n boxes: # (Cd )⊗n ∼ Hλ ⊗ Kλ , = πn ≡
#
λ
πλ ⊗ 1Kλ , π˜ d ≡
λ
#
1Hλ ⊗ π˜ λ .
λ
⊗n√ In particular ρ θ,n = ρθ/ and π˜ d (τ ) commute for all τ . Hence we have the block n diagonal form for the joint states # θ,n θ,n 1Kλ ρ θ,n = pλ ρλ ⊗ , (4.8) Mn (λ) λ
Fig. 1. Young diagram with λ = (5, 3, 3, 2)
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611
where Mn (λ) is the dimension of Kλ , pλθ,n is a probability distribution over the Young diagrams, and ρλθ,n is a density matrix on Hλ . From (4.4) and the Schur-Weyl duality, we get the expression of the block states √ √ ρλθ,n = Uλ (ζ / n) ρλu,0,n Uλ (ζ / n)∗ . (4.9) We interpret the decomposition (4.8) as follows: by doing a ‘which block’ measurement we obtain information about θ through the probability density pλθ,n . In fact it is easy to see that pλθ,n does not depend on ζ , so it only gives information about the diagonal parameters u . Later on we shall see that the model p θ,n has the same limit as the classical multinomial model described in Example 3.4. Once this information has been obtained, one still possesses a conditional quantum state ρλθ,n . It turns out that this state carries information about the rotation parameters ζ , and we shall show that the statistical model described by the conditional state converges to a ‘purely quantum’ Gaussian shift experiment. 4.2. Displaced thermal equilibrium states of a harmonic oscillator. The ground state of a quantum harmonic oscillator or the laser state of a monochromatic light pulse are well known examples of quantum Gaussian states. Both physical systems are described by the same algebra of observables generated by the canonical ‘position’ and ‘momentum’ observables Q and P satisfying the Heisenberg commutation relation: QP − PQ = i1. These observables can be represented on the Hilbert space L 2 (R) as (Q f )(x) = x f (x), (P f )(x) = −i
df (x), dx
f ∈ L 2 (R).
(4.10)
The space L 2 (R) has a special orthonormal basis {|0 , |1 , . . . } with the vector |m given by Hm (x)e−x
2 /2
√ /( π 2m m!)1/2 ,
where Hm are the Hermite polynomials [10]. These are the eigenvectors of the number operator N := 21 (Q2 + P2 − 1) counting the number of ‘excitations’ of the oscillator or the number of photons in the case of the light beam, such that N |m √ = m |m . ∗ The creation √and annihilation operators a := (Q − iP)/ 2 and respectively a := (Q + iP)/ 2, satisfy [a, a∗ ] = 1 and act as ‘ladder’ operators on the number basis: √ √ a |m = m |m − 1 , a∗ |m = m + 1 |m + 1 . It can be easily checked that both Q and P have Gaussian distribution with respect to the vacuum state |0 . In fact they are ‘jointly Gaussian’ 1 0| exp(iuQ + ivP)| |0 = exp − (u 2 + v 2 ) . 4 We shall often use the complex form of the unitary Weyl operators √ W (z) := exp(za∗ − z¯ a) = exp(i p0 Q − iq0 P), z = (q0 + i p0 )/ 2 ∈ C,
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J. Kahn, M. Gu¸ta˘
which satisfy the Weyl relations W (z)∗ W (z )W (z) = exp 2iIm(¯z z) W (z ). The coherent (vector) states |z are obtained by displacing the vacuum state with Weyl operators |z := W (z) |0 = exp(−|z|2 /2)
∞ zm √ |m . m! m=0
(4.11)
They are Gaussian states √ with the same variance as the vacuum, and means z| Q |z = √ 2Re(z) and z| P |z = 2Im(z): 1 2 z| W (z ) |z = exp − |z − z | + 2iIm(¯z z) . 2 Besides, coherent states, an important role in our discussion will be played by the thermal equilibrium states. For every β > 0 we define the Gaussian state |z|2 . (4.12) ϕβ (W (z)) = exp − 2 tanh(β/2) Its density matrix consists of a mixture of k-photon states with geometrical weights φβ = (1 − e−β )
∞
e−kβ |k k| ,
(4.13)
k=0
and can also be obtained by ‘smearing’ the coherent states with a Gaussian kernel: eβ − 1 (4.14) exp −(eβ − 1)|z|2 |z z| dz. φβ = π C The thermal equilibrium states can be shifted in ‘phase space’ by means of displacement operations D z which act by adjoining with unitaries W (z), i.e. D z (·) := Ad[W (z)](·) = W (z)∗ · W (z). The result is a Gaussian state ϕβz with the same variance as ϕβ and the same means as |z z|: |z|2 z + 2iIm(¯z z) , ϕβ (W (z )) := exp − 2 tanh(β/2) φβz := D z (φβ ) := W (z)∗ φβ W (z). (4.15) 4.3. The multimode Fock space and the limit Gaussian shift experiment. We now consider d(d − 1)/2 commuting harmonic oscillators, with a joint state consisting of independent Gaussian states. Let us define the multimode Fock space,
F := L 2 (R), 1≤ j
in which we identify the number basis
$ % ! " $m j,k , m = m j,k ∈ N : j < k . |m = j
(4.16)
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613
For each of the oscillators we define the thermal equilibrium state φ j,k := φβ j,k , where β j,k = ln(µ j /µk ), and {µ1 , . . . , µd } are the eigenvalues of the density matrix ρ0 (cf. (4.1)). We now use the Weyl operators to displace these states by an amount proportional to the off-diagonal elements ζ j,k of ρ θ (cf. (4.2) and (4.4)) ∗ ζ j,k ζ j,k ζ j,k . φ j,k W φ j,k := W √ √ 2 µ j − µk 2 µ j − µk
Next we define the joint state ϕ ζ of the oscillators with density matrix
ζ j,k φ j,k ∈ T1 (F), φζ =
(4.17)
j
where T1 (F) is the space of trace-class operators on F. The states φ ζ form the quantum part of the limit Gaussian experiment. The classical part is identical to the (d − 1)-dimensional Gaussian shift model N ( u , V (µ)) of Example 3.4, where µ = {µ1 , . . . , µd }. Definition 4.2. On the algebra L ∞ (Rd−1 )⊗B(F) we define normal state ϕ θ with density
φ θ := N ( u , V (µ)) ⊗ φ ζ ∈ L 1 (Rd−1 ) ⊗ T1 (F),
(4.18)
where N ( u , V (µ)) is the Gaussian density of Example 3.4. The quantum-classical Gaussian experiment R is defined by R = {φ θ : θ = ( u , ζ ) ∈ Rd−1 × Cd(d−1)/2 }. 4.4. The main theorem. We are now ready to formulate the main result of the paper. In view of subsequent application to optimal state estimation, it is essential to consider (slowly) growing domains of the local parameters. For given β, γ > 0 we define n,β,γ = (ζ , u ) : ζ ∞ ≤ n β , u ∞ ≤ n γ . Recall that δ is the separation between the eigenvalues of ρ0 given by Eq. (4.1). Though we use parametrisation (4.4) for density matrices ρθ , recall that in the first order this is approximated by ρ˜θ defined in (4.2). In fact it can be shown that the same theorem holds for the latter parametrisation. Theorem 4.3. Let δ > 0, let β < 1/9 and γ < 1/4. Let the quantum experiments " " ! ! Qn = ρ θ,n : θ ∈ n,β,γ , Rn = φ θ : θ ∈ n,β,γ ,
⊗n√ is the state on M (Cd )⊗n given by Eq. (4.4), and φ θ is given by where ρ θ,n = ρθ/ n (4.18). Then, there exist channels (completely positive, normalised maps) Tn : M(Cd )⊗n → L 1 (Rd−1 ) ⊗ T1 (F), Sn : L 1 (Rd−1 ) ⊗ T1 (F) → M(Cd )⊗n
(4.19) (4.20)
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J. Kahn, M. Gu¸ta˘
with T1 (F) the space of trace-class operators on F, such that & & sup &φ θ − Tn (ρ θ,n )&1 = O(n −κ ),
(4.21)
& & sup & Sn (φ θ ) − ρ θ,n &1 = O(n −κ ),
(4.22)
θ∈n,β,γ θ∈n,β,γ
where κ > 0 depends only on δ, β and γ . In particular we have lim (Qn , Rn ) = 0,
n→∞
where (·, ·) is the Le Cam distance defined in (2.5). In other words, we get polynomial speed of convergence of the approximation, which is enough to build two-step evaluation strategies in the finite experiments globally asymptotically equivalent to strategies in the limit experiment [18]. The main steps of the proof are given in a sequence of lemmas in Sect. 6 assembled into Theorem 6.7. The bound (4.22) follows easily from (4.21) as shown in Sect. 6.2. 4.5. The relation between LAN and CLT. One way to think of local asymptotic normality is the following: we would like to understand the asymptotic behaviour of the collective (fluctuation) observables (4.25) with respect to a whole neighbourhood of the state ρ, how the limit distribution changes as we change the reference state ρ ⊗n . The quantum Central Limit Theorem [37] describes the asymptotic behaviour of the same observables with respect to a fixed state, and is one of the ingredients in the proof of a different version of LAN based on weak convergence [17]. However, in the case of strong convergence, which is the object of this paper, CLT does not play any role since we are interested in convergence in norm rather than in distribution, and uniformly over a range of parameters. The purpose of the section is to derive a ‘coordinate free’ version of the limit Gaussian experiment using the Central Limit Theorem and the notion of symmetric logarithmic derivative. The reader interested in the proof of the main theorem may skip the following pages and continue with Sect. 5. Let ρ be the density matrix of a fixed faithful state on M(Cd ). To ρ we associate an algebra of canonical commutation relations carrying a Gaussian state ϕ. The Quantum Central Limit Theorem [37] says that ϕ is the limit distribution of certain multi-particle observables with respect to product states ρ ⊗n . Let (A, B)ρ := Tr(ρ A ◦ B), where A ◦ B :=
AB + B A , 2
be a positive inner product on the real linear space of selfadjoint operators M(Cd )sa . We define the Hilbert space with inner product (·, ·)ρ , L 2 (ρ) := {A ∈ M(Cd )sa : Tr(Aρ) = 0}. Let σ be the symplectic form on L 2 (ρ), σ (A, B) :=
i Tr(ρ [A, B]). 2
Local Asymptotic Normality for Finite Dimensional Quantum Systems
615
The C ∗ -algebra of canonical commutation relations CC R(L 2 (ρ), σ ) is generated by the Weyl operators W (A) satisfying the relations W (A)∗ = W (−A), W (A)W (B) = W (A + B) exp(−iσ (A, B)),
A, B ∈ L 2 (ρ).
On CC R(L 2 (ρ), σ ) we define the Gaussian (quasifree) state 1 ϕ(W (A)) := exp − A2ρ , A2ρ = (A, A)ρ . 2
(4.23)
The state ϕ is regular, i.e. there exists a representation (π, H) of the algebra CC R(L 2 (ρ), σ ) such that the one parameter family t → π(W (t A)) is weakly continuous and ϕ extends to a normal state on the von Neumann algebra generated by π(CC R(L 2 (ρ), σ )). This means that there exist selfadjoint ‘field operators’ B(A) such that π(W (t A)) = exp(it B(A)), and there exists a density matrix φπ ∈ T1 (H) such that ϕ(W (A)) = Tr (exp(i B(A))φπ ) ,
A ∈ L 2 (ρ).
The representation (π, H) can be obtained through the GNS construction, or by ‘diagonalising’ the CCR algebra as we shall see in a moment. From (4.23) we deduce that the distribution of B(A) with respect to ϕ is a centred normal distribution with variance A2ρ . From the Weyl relations it follows that the fields satisfy the following canonical commutation relations [B(A), B(C)] ' = 2iσ (A, C)1. Consider now the tensor product nk=1 M(Cd ) which is generated by elements of the form A(k) = 1 ⊗ · · · ⊗ A ⊗ · · · ⊗ 1,
(4.24)
⊗n with A acting on the k th position of the Hilbert space tensor product Cd . We are interested in the asymptotics as n → ∞ of the joint distribution under the state ρ ⊗n , of ‘fluctuation’ elements of the form n 1 (k) Fn (A) := √ A . n
(4.25)
k=1
Theorem 4.4 [Quantum CLT]. Let A1 , . . . , As ∈ L 2 (ρ). Then the following holds: s s ( ( Fn (Al ) =ϕ lim Tr ρ ⊗n (B(Al )) , n→∞
l=1
lim Tr ρ
n→∞
⊗n
s ( l=1
l=1
exp(i Fn (Al ))
=ϕ
s (
W (Al ) .
l=1
Although the algebra CC R(L 2 (ρ), σ ) may look rather abstract, its structure can be easily understood by ‘diagonalising’ it. Let us assume that ρ is a diagonal matrix ρ0 = Diag(µ1 , . . . , µd ). The Hilbert space L 2 (ρ0 ) decomposes as direct sum of orthogonal subspaces Hρ0 ⊕ Hρ⊥0 , where Hρ0 := Lin{A : [A, ρ0 ] = 0, Tr(Aρ0 ) = 0}, and Hρ⊥0 = Lin{T j,k , j = k}, (4.26) with T j,k the generators of the su(d) algebra defined in (7.2).
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The elements W (A) with A ∈ Hρ0 generate the center of the algebra which is isomorphic to the algebra of bounded continuous functions Cb (Rd−1 ). Explicitly, we identify the coordinates in Rd−1 with the basis {di = −µ1 + E i,i : i = 1, . . . d − 1} of Hρ0 , (see (7.2) for the definition of E i,i ). Then the covariance matrix for the basis vectors is (di , d j )ρ0 = Tr(ρ0 di d j ) = δi, j µi − µi µ j = [V (µ)]i, j , where Vµ is the covariance matrix (3.2). Moreover ) t j,k := T j,k / 2(µ j − µk ),
j = k,
(4.27)
form an orthogonal and symplectic basis of Hρ⊥0 , i.e. σ (t j,k , tk, j ) = −1/2,
j < k, and σ (t j,k , tl,m ) = 0 for { j, k} = {l, m},
which means that {t j,k , tk, j } generate isomorphic algebras of quantum harmonic oscillator which we denote by CC R(C). From t j,k 2ρ0 = Tr(ρ0 t 2j,k ) =
µ j + µk 2(µ j − µk )
and (4.12) we conclude that each of the oscillators is prepared independently in the thermal equilibrium state ϕ j,k = ϕβ j,k with β j,k = ln(µ j /µk ). Based on the discussion of Sects. 4.2 and 4.3 we can choose H := L 2 (Rd−1 )⊗F and define the regular representation π of CC R(L 2 (ρ0 ), σ ) on this space in a straightforward way and its von Neumann completion is L ∞ (Rd−1 ) ⊗ B(F). The state ϕ decomposes as
ϕ∼ ϕ j,k , (4.28) = N (0, Vµ ) ⊗ j
0), defined in (4.18). which is precisely the state ϕ θ for θ = ( u , ζ ) = (0, Remark. The family of states ϕ θ of the experiment R is obtained by shifting ϕ 0 with the help of symmetric logarithmic derivatives. Since this falls outside the scope of this paper we refer the reader to [26] for more details. 5. Explicit Form of the Channels and First Steps of the Proof 5.1. Second look at the irreducible representations of SU (d). Before explaining the steps involved in the proof, let us take a closer look at the block states (4.9). Recall that we have the decomposition of Theorem 4.1 over Young diagrams with n boxes and # θ,n 1Kλ ρ θ,n = . ρλ ⊗ Mn (λ) λ
Let { f 1 , . . . , f d } be the eigenvectors of ρ0 , i.e. the standard basis vectors of Cd . Then the eigenvectors of ρ0⊗n = ρ 0,n are tensor products
and the eigenvalues
f a := f a(1) ⊗ f a(2) ⊗ · · · ⊗ f a(n) ,
* k
λa(k) do not depend on the order of the vectors in the product.
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617
5.1.1. Projecting onto a copy of Hλ . Our aim is to ‘project’ to an irreducible representation Hλ and obtain an explicit expression for the eigenvectors of the block components ρλθ,n . Such a projection is not unique, in fact for any rank one operator |v u| ∈ B(Kλ ) with u|v = 1 we can define a (not necessarily orthogonal) projection y = y 2 on a copy of Hλ , yλ (u, v) := 1Hλ ⊗ |v u| : (Cd )⊗n → Hλ ⊗ |v . However the action of yλ (u, v) on basis vectors f a depends on a particular identification between (Cd )⊗n and the direct sum in Theorem 4.1. Therefore we need a direct way of defining such a projection and the key observation is that yλ (u, v) is a minimal projection in the algebra Alg(π˜ d (τ ) : τ ∈ S(n)), i.e. it cannot be decomposed into a sum of nonzero projections, and vice-versa any minimal projection is of this form. The following recipe (given without proof) shows how to construct minimal projections in the S(n) group algebra. We recall that the group ∗ -algebra A(S(n)) is the linear space spanned by the group elements endowed with a product stemming from the group product a(τ )τ, b = b() ⇒ ab = a(τ )b()τ a= τ ∈S(n)
=
σ ∈S(n)
⎛ ⎝
∈S(n)
τ,∈S(n)
⎞
a(σ s −1 )b(s)⎠ σ,
s∈S(n)
and with adjoint a ∗ = τ ∈S(n) a(τ )τ −1 . Let λ be a Young diagram with n boxes and consider the (standard) Young tableau t in which the boxes are filled with the numbers {1, . . . , n} in increasing order from left to right along rows, starting with the top row and down columns, as shown in the left-side tableau of Fig. 2. Define the group algebra elements Pλ = σ, Q λ = sgn(τ )τ, σ ∈Rλ
τ ∈Cλ
where Rλ is the S(n)-subgroup of permutation leaving the rows of t invariant, and Cλ is the subgroup of permutations leaving the columns of t invariant. Note that Pλ and Q λ are self-adjoint elements of the S(n) group algebra satisfying Pλ Pλ = |Rλ |Pλ = (
d (
λi !)Pλ ,
Q λ Q λ = |C(λ)|Q λ = (
i=1
d (
i λi −λi+1 )Q λ .
i=1
The Young symmetriser is defined as Yλ := Q λ Pλ .
Fig. 2. Left: a standard Young tableaux. Right: a semi-standard Young tableau for d = 3
(5.1)
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J. Kahn, M. Gu¸ta˘
Theorem 5.1. Up to a scalar normalising factor, the Young symmetriser Yλ is minimal projection in A(S(n)) and yλ := qλ pλ = π˜ d (Q λ )π˜ d (Pλ ) projects onto a copy of Hλ ⊂ (Cd )⊗n . The action of the Young symmetriser yλ on basis vectors f a ∈ (Cd )⊗n follows easily from the definition of Yλ . For each f a we fill the boxes of λ with the indices a(k) going along rows from left to right, starting with the top row and finishing with the bottom one. For example, if λ = and f a = f 2 ⊗ f 2 ⊗ f 1 ⊗ f 2 ⊗ f 1 then ta = 22 21 1 . S(n) has an obvious action on the set of tableaux by permuting the content of the boxes which are numbered from 1 to n in the standard way as in Fig. 2. The action of the Young symmetriser yλ = qλ pλ on f a is deduced from the action on the tableau ta : one first symmetrises with respect to components which are in the same row, and then antisymmetrises with respect to components in the same column. For example if λ = then yλ ( f 2 ⊗ f 1 ⊗ f 3 ) = f 2 ⊗ f 1 ⊗ f 3 + f 1 ⊗ f 2 ⊗ f 3 − f 3 ⊗ f 1 ⊗ f 2 − f 3 ⊗ f 2 ⊗ f 1 . 5.1.2. Finding a basis in Hλ . By the previous theorem the vectors yλ f a span Hλ , but are not linearly independent. We show now how to select a basis (subset of linearly independent vectors spanning Hλ ). A semistandard Young tableau is a diagram filled with numbers in {1, . . . , d} such that the entries are non-decreasing along rows from left to right and increasing along columns from top to bottom, as in the right-side of Fig. 2. Theorem 5.2. The vectors yλ f a for which ta is a semistandard Young tableau form a (non-orthogonal) basis of the irreducible representation (πλ , Hλ ). Since the values in the rows are nondecreasing, there is a one-to-one correspondence between Young tableaux ta and vectors m = (m i, j )1≤i< j≤d , where m i, j is the number of j’s appearing in line i of the Young tableau ta . Note that we need only consider m i, j for j > i, as there is no j in line i if j < i (the columns are increasing), 1 1 2 3 3 then and the number of i in line i is λi − dj=i+1 m i, j . For example, if ta = 23 3 m = {m 1,2 = 1, m 1,3 = 2, m 2,3 = 1}. By a slight abuse of notation we shall denote the corresponding vectors by yλ f m and the normalised vectors |m, λ := N (m, λ)yλ f m ,
(5.2)
where N (m, λ) = 1/yλ f m . This constant is in general not easy to compute but we shall describe its asymptotic properties in Sect. 7.3. Using (5.1) we have yλ f a |yλ f b = qλ pλ f a |qλ pλ f b = pλ f a |qλ2 pλ f b = (
d (
i λi −λi+1 ) pλ f a |yλ f b .
i=1
(5.3) In order to get further simplifications, we examine some special vector states, that we shall call by analogy with the Fock spaces finite-dimensional coherent states. The first is the special vector |0, λ , the highest weight vector of the representation (πλ , Hλ ), which later on will play the role of the finite-dimensional vacuum. This vector,
Local Asymptotic Normality for Finite Dimensional Quantum Systems
619
as we have seen, corresponds to the semi-standard Young tableau where all the entries in row i are i. An immediate consequence is that pλ | f 0 = (
d (
λi !)| f 0 .
(5.4)
i=1
Moreover f 0 |qλ f 0 = 1 since any column permutation produces a vector orthogonal to f 0 . Thus the normalised vector is: 1 yλ | f 0 . √ λi −λi+1 i=1 λi ! i
|0, λ = *d
(5.5)
The finite-dimensional coherent states are defined as πλ (U )|0λ for U ∈ SU (d). From *d λi !)U |0λ , thus [ pλ , πλ (U )] = 0 and (5.4), we get pλ πλ (U )|0λ = ( i=1 + , d ,( yλ f m |πλ (U )|0, λ = - i λi −λi+1 pλ f m |qλ πλ (U ) f 0 . (5.6) i=1
The latter expression holds for any linear combination of f m on the left-hand side, in particular πλ (V ) f 0 for another unitary operator V . In Lemma 7.1, we shall examine asymptotics of (5.6) for specific sequences of unitaries U when n → ∞. One of the main tools will be formula (7.6). The following expressions of the dimensions of Kλ and Hλ are given without proof: Let gl,m be the hook length of the box (l, m), defined as one plus the number of boxes under plus the number of boxes to the right. For example the diagram (5, 3, 3) has the hook lengths :
7 6 5 2 1 4 3 2 3 2 1
.
The dimension Mn (λ) of Kλ is Mn (λ) = *
n! l=1...d m=1...λl
gl,m
,
and can be rewritten in the following form which is more adapted to our needs: ( λl − λk + k − l n . Mn (λ) = λl + k − l λ1 , . . . , λ d
(5.7)
l=1...d k=l+1...d
To summarise, we have defined a non-orthonormal basis {|m, λ } of Hλ such that |m, λ are eigenvectors of ρ 0, u ,n for all λ, with eigenvalues: u ,n m i, j d d ( ( µj u ,n u ,n λi 0, |m, λ = (µi ) , (5.8) m, λ|ρ u ,n i=1 j=i+1 µi √ √ where µiu ,n = µi + u i / n for 1 ≤ i ≤ (d − 1) and µud ,n = µd − ( i u i )/ n. The next step is to take into account the action of the unitary U (ζ ). We define the automorphism
ζ ,n : M((Cd )⊗n ) → M((Cd )⊗n ),
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J. Kahn, M. Gu¸ta˘
by √ √ τ → ζ ,n (τ ) = Ad[U (ζ , n)](τ ) := U (ζ / n)⊗n τ U ∗ (ζ / n)⊗n .
(5.9)
Then we have ρ ζ , u ,n = ζ ,n (ρ 0, u ,n ). By Theorem 4.1 and using the decomposition (4.8), we get the blockwise action on irreducible components # ζ ,n ζ ,n (ρ ⊗n ) = λ (ρλ ) ⊗ 1Kλ , λ
ζ ,n
where λ
= Ad[Uλ (ζ , n)]. In particular we have ζ , u ,n
ρλ
ζ ,n
= λ (ρλ0, u ,n ).
(5.10)
With these notations, we can set about building the channels Tn . 5.2. Description of Tn . We look for channels Tn : M((Cd )⊗n ) → L 1 (Rd−1 ) ⊗ T1 (F) of the form: Tn : ρ θ,n −→
λ
pλθ,n τλn ⊗ Vλ ρλθ,n Vλ∗ .
(5.11)
Here, Vλ is an isometry from Hλ to F, i.e. Vλ∗ Vλ = 1Hλ . On the classical side, τλn is a probability law on Rd−1 . We may view τ n as a Markov kernel (2.2) from the set of diagrams λ to Rd−1 . The channel Tn can be described by the following sequence of operations. We first perform a ‘which block’ measurement over the irreducible representations and get a result λ. Then, on the one hand, we apply a classical randomisation to λ, and on the other hand we apply a channel depending on our result λ to the conditional state ρλ . The underlying ideas are the following: 1) The probability distribution pλθ,n is essentially a multinomial depending only on u , as it can be deduced from (5.8) and (5.7). As we have seen in Example 3.4, this converges (in Le Cam sense) to a classical Gaussian shift experiment. Here, in order to obtain the strong norm convergence we need to smooth the discrete distribution into a continuous one with respect to the Lebesgue measure. We choose a particular smoothing distribution that will ensure the uniform L 1 convergence to the Gaussian model (Lemma 6.1). Definition 5.3. Let τλn be the probability density on Rd−1 defined for all λ such that λi = n, by: τλn (dx) = τλn (x)dx = dx n (d−1)/2 χ (Aλ,n ),
(5.12)
where Aλ,n = {x ∈ Rd−1 : |n 1/2 xi + nµi − λi | ≤ 1/2, 1 ≤ i ≤ d − 1}. We further denote bλθ,n = pλθ,n τλn , depending on θ only through u .
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2) For the quantum part, we map the ‘finite-dimensional vacuum’ |0, λ to the Fock space vacuum |0 , and the basis vectors |m, λ of Hλ ‘near’ the basis vectors |m of the Fock space F (cf. Definitions (5.2) and respectively (4.16)). Here we need to tackle the problem that {|m, λ } is not an orthonormal basis but only becomes so asymptotically. The following lemma provides the isometry Vλ appearing in (5.11). Lemma 5.4. Let η < 2/9. Suppose that λi − λi+1 ≥ δn for all 1 ≤ i ≤ d, with the convention λd+1 = 0. Then for n > n 0 (η, δ, d) there exists an isometry Vλ : Hλ → F such that, V |0, λ = |0 and for 0 < |m| ≤ n η , m| Vλ = )
1 ˜ (9η−2)/12 /δ 1/3 1 + (Cn)
m, λ| ,
˜ where C˜ = C(η, d) is a constant. More precisely, n 0 can be taken of the form (C(d)/δ 2 )1/(1−3η) . Proof. See Sect. 7.2. The main tool is Lemma 7.3.
For Young diagrams which do not satisfy the assumption of the previous lemma, the isometry Vλ can be defined arbitrarily. The reason is that those blocks have vanishing collective weight and can be neglected altogether (cf. Lemma 6.2). From this operational description we conclude that Tn is a proper channel since τ n is a Markov kernel and Vλ is an isometry. We then want to prove that Tλ (ρλ0, u ,n ) is close ζ ,n
to φ 0 and that the finite-dimensional operations λ have almost the same action as the displacement operators D ζ of the Fock space, cf. (4.15). Finite-dimensional coherent states and formula (4.14) will be the stepping stone to those results. 6. Main Steps of the Proof 6.1. Why Tn does the work. We shall break (4.21) in small manageable pieces. The result and brief explanatory remarks, repeating those in the derivation, are given from (6.3) on. We introduce first a few shorthand notations: the restriction of Tn to the block λ is Tλ : ρλθ,n → Vλ ρλθ,n Vλ∗ , so that Tn : ρ θ,n →
λ
Tλ∗
pλθ,n τλn ⊗ Tλ (ρλθ,n ) =
Vλ∗ φVλ .
: φ → We also define We expand (5.11) as θ,n Tn (ρ θ,n ) = bλ ⊗ φλθ,n
and note that
λ
= N ( u , Vµ ) ⊗ φ ζ − (N ( u , Vµ ) −
Tλ∗ Tλ
λ
λ
bλθ,n ⊗ φλθ,n .
= IdHλ .
bλθ,n ) ⊗ φ ζ −
λ
bλθ,n ⊗ φ ζ − φλθ,n .
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J. Kahn, M. Gu¸ta˘
Proving (4.21) then amounts to proving θ,n θ,n u , Vµ ) − bλ ) ⊗ φ ζ + bλ ⊗ (φ ζ − φλθ,n )1 ≤ Cn −/δ . sup (N ( θ∈n,
λ
λ
We now use the triangle inequality to upper bound this norm by a sum of “elementary” terms to be treated separately in the following sections: θ,n θ,n bλ ) ⊗ φ ζ + bλ ⊗ (φ ζ − φλθ,n )1 (N ( u , Vµ ) − λ
≤ (N ( u , Vµ ) −
λ
ζ
λ
bλθ,n ) ⊗ φ ζ 1
≤ φ 1 (N ( u , Vµ ) −
λ
bλθ,n )1
+
λ
+
λ
bλθ,n ⊗ (φ ζ − φλθ,n )1
bλθ,n 1 (φ ζ − φλθ,n )1 .
u , Vµ )1 = φλθ,n = 1, we have φ ζ − φλθ,n 1 ≤ 2. Similarly Since φ ζ 1 = N ( θ,n θ,n θ,n λ bλ 1 = 1 because bλ 1 = pλ . We split the sum over λ in two parts, one for which it is expected that φ ζ − φλθ,n 1 is small, and the other on which the sum of all bλθ,n 1 is small. Specifically, define the set of typical Young diagrams, n,α := {λ : |λi − nµi | ≤ n α , 1 ≤ i ≤ d}, for α > 1/2,
(6.1)
then
Tn (ρ θ,n ) − N ( u , Vµ ) ⊗ φ ζ θ,n bλ 1 + sup φ ζ − φλθ,n 1 + 2 bλθ,n 1 . ≤ N ( u , Vµ ) − λ∈n,α
λ
(6.2)
λ∈n,α
The first term corresponds to the convergence of the classical experiment in the Le Cam sense. If the second term is small, then on n,α , the (purely quantum) family ρλθ,n is near the family φ ζ . The last term is small due to the concentration of pλθ,n around the representations with shape λi = nµi . In other words, the only representations that matter are those in n,α . The hardest term to dominate (notice that the two others are classical) is the second. We transform it until we reach tractable fragments:
φ ζ − φλθ,n 1 = φ ζ − Tλ (ρλθ,n )1 ζ ,n
= D ζ (φ 0 ) − [Tλ λ Tλ∗ ](Tλ (ρλ0, u ,n ))1
ζ ,n
= D ζ (φ 0 ) − D ζ (Tλ (ρλ0, u ,n )) + D ζ (Tλ (ρλ0, u ,n )) − [Tλ λ Tλ∗ ](Tλ (ρλ0, u ,n ))1
ζ ,n
ζ ,n
≤ D ζ (φ 0 ) − D ζ (Tλ (ρλ0, u ,n ))1 + [D ζ − Tλ λ Tλ∗ ](Tλ (ρλ0, u ,n ) − φ 0 )1 ζ ,n
+[D ζ − Tλ λ Tλ∗ ](φ 0 )1
≤ 3Tλ (ρλ0, u ,n ) − φ 0 1 + [D ζ − Tλ λ Tλ∗ ](φ 0 )1 , where in the last inequality we have used the fact that the displacement operators are isometries.
Local Asymptotic Normality for Finite Dimensional Quantum Systems
623
Note that the first term does not depend on ζ and the second term is small if the ζ ,n
displacement operators λ and D ζ have ‘similar action’ on an appropriate domain. Using the integral formula (4.14) for gaussian states φβ and the fact that φ 0 is a tensor product of such states (cf. (4.18)), we bound the second term by ζ ,n
[D ζ − Tλ λ Tλ∗ ](φ 0 )1 ≤
ζ ,n
f ( z )[D ζ − Tλ λ Tλ∗ ](| z z |)1 d z ,
Cd(d−1)/2
where ( µi − µ j µi − µ j 2 f ( z ) = exp − |z i, j | , πµj µj i< j
and | z z | = D z (|0 0|) is the multimode coherent state, so ζ ,n
ζ ,n
[D ζ − Tλ λ Tλ∗ ](| z z |) = [D ζ D z − Tλ λ Tλ∗ D z ](|0 0|). Now, f is a probability density, and the norm in the integrand is dominated by two. By splitting the integral we obtain ζ ,n
[D ζ − Tλ λ Tλ∗ ](φ 0 )1 ≤ 2
f ( z )d z + sup [D ζ D z
z >n β
z ≤n β
ζ ,n
−Tλ λ Tλ∗ D z ](|0 0|)1 . By adding and subtracting additional terms ζ ,n
ζ + z ,n
D ζ D z − Tλ λ Tλ∗ D z = D ζ + z − Tλ λ ζ + z ,n
+Tλ λ
ζ ,n
Tλ∗ ζ ,n
Tλ∗ − Tλ λ λ z ,n Tλ∗ ζ ,n
+Tλ λ λ z ,n Tλ∗ − Tλ λ Tλ∗ D z , we deduce that
ζ ,n
ζ + z ,n
[D ζ − Tλ λ Tλ∗ ](| z z |)1 ≤ [D ζ + z − Tλ λ ζ + z ,n
+[λ
Tλ∗ ](|0 0|)1
ζ ,n
− λ λ z ,n ](|0, λ 0, λ|)1
+[λ z ,n Tλ∗ − Tλ∗ D z ](|0 0|)1 ,
where the last two terms on the right side have been simplified using properties of ζ ,n
Tλ , Tλ∗ , λ . Notice that the first and third norms are essentially the same and the three ζ
terms are small if the action of λ is mapped into that of the displacement operators D ζ .
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J. Kahn, M. Gu¸ta˘
Putting all this together, our ‘expanded’ form for (4.21) is
u , Vµ )1 sup Tn (ρ θ,n ) − φ ζ ⊗ N (
θ∈n,β,γ
≤
u , Vµ ) − sup N (
θ∈n,β,γ
+ 2 sup
θ∈n,β,γ λ∈ n,α
+ 3 sup
λ
(6.3)
bλθ,n 1
(6.4)
bλθ,n 1
(6.5)
sup φ 0 − Tλ (ρλ0, u ,n )1
(6.6)
θ∈n,β,γ λ∈n,α
ζ + z ,n
+ sup
sup
sup [D ζ + z − Tλ λ
+ sup
sup
sup [D z − Tλ λ z ,n Tλ∗ ](|0 0|)1
sup
sup [λ
z ≤n β θ∈n,β,γ λ∈n,α z ≤n β
Tλ∗ ](|0 0|)1
θ∈n,β,γ λ∈n,α
+ sup
ζ + z ,n
z ≤n β θ∈n,β,γ λ∈n,α
ζ ,n
− λ λ z ,n ](|0, λ 0, λ|)1
(6.7) (6.8) (6.9)
+2
z ≥n β
f ( z )d z .
(6.10)
The last Gaussian tail term is less than C exp(−δn 2β ), where C depends only on the dimension d. Under the hypothesis n 2β > 2/δ, this can be bounded again by O(n −2β ). The following lemmas provide upper bounds for each of the terms. Before each lemma we remind the reader what is the significance of the bound. The proofs are gathered in Sect. 7. The classical part of the channel is a Markov kernel τ (see Definition 5.3) mapping the ‘which block’ distribution pλθ,n into the density bλθ,n on Rd−1 which approaches uniformly the gaussian shift experiment (6.4). Recall that bλθ,n depends only on u and not on ζ , so that we have the same parameter set for the two classical experiments. Lemma 6.1. With the above definitions, for any , we have θ,n sup N ( u , Vµ ) − bλ 1 = O(n −1/4+ /δ, n −1/2+γ /δ). θ∈n,β,γ
λ
The next lemma deals with (6.5) by showing concentration around Young diagrams λ in the ‘typical subset’ (6.1). This allows we to restrict to this set of diagrams in further estimates. Lemma 6.2. Let α − γ − 1/2 > 0. Then, with the above definitions we have 2 sup bλθ,n 1 = O n d exp(−n 2α−1 /2) , θ∈n,β,γ λ∈ n,α
with the O(·) term converging to zero.
The term (6.6) shows that when the rotation parameter is zero, the block states ρλ0, u ,n are essentially thermal equilibrium states, as one would expect from the quantum Central Limit Theorem 4.4. However the convergence here is in norm rather than in distribution, and uniform over the various parameters.
Local Asymptotic Normality for Finite Dimensional Quantum Systems
625
Lemma 6.3. Let 0 < η < 2/9. With the above definitions, we have sup
sup φ 0 − Tλ (ρλ0, u ,n )1 = O(n −1/2+γ +η /δ, n (9η−2)/24 /δ 1/6 , exp(−δn η )).
θ∈n,β,γ λ∈n,α
The terms (6.7) and (6.8) show that the ‘finite dimensional coherent states’ obtained by performing small rotations on the ‘finite-dimensional vacuum’ are uniformly close to their infinite dimensional counterparts, thus justifying the coherent state terminology. Lemma 6.4. Let > 0 be such that 2β + ≤ η < 2/9. Then, sup
sup
z ≤n β
ξ ≤n −1/2+2β /δ
sup
ζ + z ,ξ ,n
sup [D ζ + z − Tλ λ
θ∈n,β,γ λ∈n,α
Tλ∗ ](|0 0|)1 = R(n)
with R(n)2 = O(n (9η−2)/12 δ −1/3 , n −1+2β+η δ −1 , n −1/2+3β+2 δ −3/2 , n −1+α+2β δ −1 , n −1+α+η δ −1 , n −1+3η δ −1 , n −β ). (6.11) For estimating the terms (6.7, 6.8), the case when ξ = 0 is sufficient. This more general form is useful for the proof of Lemma 6.5. The unitary operation is defined as ζ ,ξ,n := Ad[Uλ (ζ , ξ, n)] with U (ζ , ξ, n)) the general SU (d) element of (7.1). λ
Finally (6.9) shows that the ‘finite-dimensional’ displacement operators multiply as the corresponding displacement operators when acting on the vacuum. Lemma 6.5. With the above definitions, under the same hypotheses as in Lemma 6.4, we have sup
sup
ζ + z ,n
sup [λ
z ≤n β θ∈n,β,γ λ∈n,α
ζ ,n
− λ λ z ,n ](|0, λ 0, λ|)1 = R(n)
with R(n) given by Eq. (6.11). From the last three lemmas, together with the bound on the remainder integral (6.10) we obtain the following lemma which can be plugged into the bound (6.2): Lemma 6.6. With the above notations under the same hypotheses as in Lemma 6.4, we have sup
sup φ ζ − φλθ,n = R(n) + O(n −1/2+γ +η /δ)
θ∈n,β,γ λ∈n,α
with R(n) given by Eq. (6.11). Gathering all these results and using the inequalities α − γ − 1/2 > 0, 2β + ≤ η < 2/9 we get the following relations between the error terms: n −1/2+β+η/2 /δ 1/2 = o(n −1/2+3η/2 /δ 1/2 ) and n −1/2+α/2+β /δ 1/2 = o(n −1/2+α/2+η/2 /δ 1/2 ). This yields the next theorem which provides the bound (4.21).
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Theorem 6.7. For any δ > 0, 0 < γ < 1/4, > 0, 1/2 + γ < α < 1, η < 2/9, 0 < β < (η − )/2, the sequence of channels Tn satisfies & & sup &Tn (ρ θ,n ) − φ &1 = O(n −1/4+3β/2+ δ −3/2 + n −1/2+α/2+η/2 δ −1/2 θ∈n,β,γ
+ n −1/2+3η/2 δ −1/2 + n −β/2 + n −1/2+γ +η /δ + n (9η−2)/24 /δ 1/6 + exp(−δn η )).
(6.12)
For any given 0 < δ < 1, β < 1/9 and γ < 1/4, we can choose α, η, satisfying the above conditions, such that the right side is of order O(n −κ ), with κ > 0 depending on β, γ , δ. 6.2. Definition of Sn and proof of its efficiency. The channel Sn is essentially the inverse of Tn and as we shall see, (4.22) can be deduced from (4.21). On the classical side we need a Markov kernel completing the equivalence between the family pλu ,n and N ( u , Vµ ). Let σ n be defined by σ n : x ∈ Rd−1 → δλx ,
(6.13)
where λx is the Young diagram such that d1 λi = n, and |n 1/2 xi + nµi − λi | < 1/2, for 2 ≤ i ≤ d. No such diagram exists, we set λx to any admissible value, for example (n, 0, . . . , 0). Notice that with (5.12), σ n ◦ τ n ◦ σ n = σ n . Moreover any probability on the λ such that d1 λi = n is in the image of σ n , so that σ n ◦ τ n ( p θ,n ) = p θ,n . Lemma 6.8. With the above definitions, for any , we have & & & & sup &σ n N ( u , Vµ ) − p u ,n & = O(n −1/2+ /δ, n −1/4+γ /δ). u ≤n γ
1
Proof. See the end of Sect. 7.5.
The channel Sn is given by the following sequence of operations acting on the two spaces of the product L 1 (Rd−1 ) ⊗ T1 (F) . Given a sample from the probability distribution N ( u , Vµ ), we use the Markov kernel σ n to produce a Young diagram λ. Conditional on λ we send the quantum part through the channel 1Kλ Sλ : φ → S˜λ (φ) ⊗ Mn (λ) with S˜λ : φ → Tλ∗ φ + (1 − Tr(Tλ∗ (φ)))|0, λ 0, λ|. The second term is rather arbitrary and ensures that S˜λ is trace preserving map. What is important is that for any density operator ρλ on the block λ, the operator S˜λ reverts the action of Tλ : S˜λ Tλ (ρλ ) = Tλ∗ Tλ (ρλ ) + (1 − Tr(Tλ∗ Tλ (ρλ )))|0, λ 0, λ| = ρλ + (1 − Tr(ρλ ))|0, λ 0, λ| = ρλ .
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627
Now
u , Vµ ) ⊗ φ ζ ) = Sn (N (
# 1Kλ , [σ n N ( u , Vµ )](λ) S˜λ (φ ζ ) ⊗ Mn (λ) λ
and with the notation σ n Nλu := [σ n N ( u , Vµ ))](λ) and qλu ,n := min(σ n Nλu , pλu ,n ) we have # u ,n Sn (φ ζ ⊗ N ( qλ ( S˜λ (φ ζ ) − ρλθ,n ) + (σ n Nλu − qλu ,n ) S˜λ (φ ζ ) u , Vµ )) − ρ θ,n = λ
1Kλ . − ( pλu ,n − qλu ,n )ρλθ,n ⊗ Mn (λ) Taking L 1 norms, and using that all φ’s and ρ’s have trace 1 and that channels (such as S˜λ ) are trace preserving, we get the bound: u ,n u , Vµ )) − ρ θ,n 1 ≤ qλ ( S˜λ (φ ζ ) − ρλθ,n )1 + |σ Nλu − pλu ,n | Sn (φ ζ ⊗ N ( ≤2
λ∈n,α
≤2
λ∈n,α
λ
qλu ,n
+ sup S˜λ (φ λ∈n,α
ζ
) − ρλθ,n 1
λ
+ σ N ( u , Vµ ) − p n
u ,n
1
qλu ,n + sup φ ζ − Tλ (ρλθ,n )1 + σ n N ( u , Vµ ) − p u ,n 1 . λ∈n,α
Now the first term is smaller than the remainder term of the gaussian outside a ball whose radius is n α . Hence this term is going to zero faster than any polynomial, independently on δ and u for u ≤ n γ . The second term is treated in Lemma 6.6 (recalling that θ,n θ,n φλ = Tλ (ρλ )), and the third term is treated in Lemma 6.8. This ends the proof of (4.22). 7. Technical Proofs 7.1. Combinatorial and representation theoretical tools. Here we continue the analysis of the SU (d) irreducible representations (πλ , Hλ ) started in Sect. 5.1. The purpose of this section is to provide good estimates of quantities of the type m, λ | πλ (U ) | l, λ which will be needed in the proofs of Lemmas 7.3 and 6.4. We shall use the following form of a general SU (d) element and the shorthand notations: ⎡ ⎛ ⎞⎤ d−1 Re(ζ j,k )T j,k + Im(ζ j,k )Tk, j ⎠⎦ , U (ζ , ξ ) := exp ⎣i ⎝ ξi Hi + √ µ j − µk i=1 1≤ j
(7.2)
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We first introduce some new notations and remind the reader about the already existing ones: 1) We write l(c) for the length of the column c in the Young diagram λ. There are then λi − λi+1 columns such that l(c) = i. An alternative definition is l(c) = inf{i : λi ≥ c}. 2) Recall that we denote by f a the basis vectors f a(1) ⊗ · · · ⊗ f a(n) , and to each vector we associate a Young tableau ta , where the indices a(i) fill the boxes of a diagram λ in a particular way. We denote by tac the column c of ta , i.e. the function tac : {1, . . . , l(c)} → {1, . . . , d} that associates to the row number r the value of the entry of that Young tableau in column c, row r . For example, if ta = 22 21 1 we get the values: ta1 (1) = 2, ta1 (2) = 2, ta2 (1) = 2, ta2 (2) = 1, ta3 (1) = 1. We shall often be interested in the image tac ({1, . . . , l(c)}) as an unordered set, or compare tac to Idc , the identity function on the integers {1, . . . , l(c)}. 3) Recall also that Hλ is spanned by the vectors yλ f a for which ta is a semistandard Young tableau, and yλ = qλ pλ is the Young symmetriser (cf. Theorem 5.2). If ta is semistandard then we can use the alternative notation f m for f a since a is in one-to-one correspondence with m = {m i, j : 1 ≤ i < j ≤ d}, where m i, j is the number of j’s in the row i of ta . The normalised vectors are |m, λ := yλ f m /yλ f m . 4) Let Oλ (m) be the orbit of f m under the subgroup Rλ of row permutations. This consists of vectors f b which have exactly m i, j boxes with j in row i, and the rest are i. In particular, row i has no entries smaller than i. Since the action of permutations is transitive, we have pλ f m =
σ ∈Rλ
f a◦σ =
f b ∈Oλ (m)
#Rλ fb . #Oλ (m)
(7.3)
5) Since we antisymmetrize with qλ , we are only interested in the ta (not necessarily semistandard) which do not have two equal entries in the same column. Such tableaux ta (or vectors f a ) shall be called admissible and their set is denoted V. 6) For any f a ∈ Oλ (m) we define ( f a ) := |m| − #{1 ≤ c ≤ λ1 : tac = Idc }, . and denote by V (m) the set of vectors f a ∈ Oλ (m) V with ( f a ) = . Then we have / 0 Oλ (m) V= V (m). ∈N
Note that ( f a ) ≥ 0 and is zero if and only if each column tac is either Idc or of the form tac (r ) = jδr =i + r δr =i for some i ≤ l(c) < j. A tac of this form will be called an (i, j)-substitution. The following ‘algorithm’ shows how to build all the possible f a ∈ V (m), thus enabling us to estimate the size of V (m).
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Algorithm. Let (m, λ) be fixed but otherwise arbitrary. In order to generate a particular admissible f a ∈ Oλ (m) we need to select the m i, j boxes on row i which are filled with j, for all i < j. The rest of the boxes are filled automatically with i’s. The constraint is that no column should have two boxes filled with the same number. Generating a diagram can be described intuitively as follows. We start with the ‘vacuum’ vector (tableau) f 0 := f m=0 (row i is filled exclusively with i’s), and with a set of |m| bricks containing m i, j identical bricks labelled (i, j), for each pair i < j. To change the content of a box from i into j we place an (i, j)-brick in that box. This procedure is repeated until all bricks have been used, each box being modified at most once. At this stage each column c may contain several bricks placed in the appropriate boxes, so that its configuration is uniquely defined by the set of bricks κ which shall be called a column-modifier. For example if κ = {(i, j), ( f, l)}, then the column has entries ⎧ ⎨ j if k = i; tac (k) = l if k = f ; ⎩ k otherwise. Note that a column-modifier is not an arbitrary collection of bricks but one that can be used to produce a column with different entries. In the previous example, if i < f this means either ( j = f and j, l > l(c)) or ( j = f and l > l(c)). The elementary onebrick column-modifier denoted κ(i, j) can only be used in a column with i ≤ l(c) < j, otherwise the entry j would appear twice. Now, since the length of a column is at most d and all entries must be different, there are less than d! different types of column-modifiers. Another important remark is that a column-modifier always increases the value of the modified cells, so that in this case tac ({1, . . . , l(c)}) = {1, . . . , l(c)}. Alternatively to the above scenario where the bricks are inserted sequentially, we can first cluster them into |m| − column-modifiers, and then apply each column-modifier to a particular column. A given collection of column-modifiers is uniquely determined by {m κ : κ}, where m κ is the multiplicity of κ. This procedure is detailed in the following 3 stages: I. Choose bricks among our |m|. As we have d(d − 1)/2 different types of bricks (recall that i > j), and we do not distinguish between identical bricks, there are at most [d(d − 1)/2] possibilities. For = 0, we have only one choice. II. Consider the remaining bricks as a set of elementary column-modifiers. Starting from these, we sequentially add each of the bricks selected in the first stage, to one of these elementary column-modifiers to form non-elementary ones. At each step we have at most d! different types of column modifiers to which we can attach the new brick. Note that we do not distinguish between column modifiers of the same type, but rather consider them as an unordered set. Hence, we have less than (d!) possibilities. If = 0 there is only one possibility. Note that at the end of Stage II at least max{0, |m| − 2 } of the column-modifiers are elementary, and that m κ(i, j) ≤ m i, j . III. Apply the column-modifiers to the columns of f 0 , so that no two modifiers are applied to the same column and the resulting f a ∈ Oλ (m) is admissible. By construction ( f a ) = and all admissible tableaux can be generated in this way. For counting the number of possibilities for the third stage we apply the column modifiers sequentially, but *since some of them may be identical we need to divide by the combinatorial factor κ m κ !, where m κ is the number of column modifiers of type κ.
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We distinguish between elementary column modifiers of type κ(i, j) and composite ones. There are less than n possibilities of inserting a composite column-modifier κ. An elementary one of type κ(i, j) can only be inserted in a column with at least i rows, and since the resulting vector has to be admissible, the column cannot contain another j, so its length is smaller than j. There are λi − λ j such columns. Hence the number of possibilities at stage three of the algorithm is upper bounded by ( κ=κ(i, j )
n m κ ( (λi − λ j )m κ(i, j) · . mκ ! m κ(i, j) !
(7.4)
i< j
When = 0, for each elementary column modifier κ(i, j) the number of available columns is at least (λi − λ j − |m|)+ := max{0, λi − λ j − |m|}. Thus we have the following lower bound: i, j ( (λi − λ j − |m|)m + . m i, j !
(7.5)
i< j
Note that the upper bound (7.4) depends on the set of multiplicities {m κ }. We now return to our list of notations and definitions: 7) To each column of ta we associated a column modifier which completely determines its content. If m aκ is the number of columns with column-modifer κ, we collect all multiplicities in E := {m aκ : κ}. In particular is a function of E, m aκ . ( f a ) = |m| − κ
Vectors for which ( f a ) = 0 have the same multiplicity set E 0 , where m κ(i, j) = m i, j for all i < j and the.other m κ = 0. Similarly to V (m), we denote by V E (m) the set of tableaux in Oλ (m) V with E( f a ) = E, in particular 0 V E (m). V (m) = E: (E)=
8) To each column c of ta we associate two disjoint sets: the added entries {tac (1), . . . , tac (l(c))} \ {1, . . . , l(c)} and the deleted entries {1, . . . , l(c)} \ {tac (1), . . . , tac (l(c))}. This data is placed into a single set by attaching a ± sign to each entry, indicating if it is added or deleted. It is easy to verify that if ta is admissible, the set of added and deleted entries is uniquely determined by the column-modifer κ associated to c, and hence shall be denoted by S(κ). For example S(κ(i, j)) = {(i, −), ( j, +)} and for κ ={(i, j), ( j, k)} we have S(κ) = {(i, −), (k, +)}. We define the multiplicities m aS = κ:S(κ)=S m aκ and F( f a ) := {m aS : S} . To summarise, we have defined the maps f a −→ E( f a ) −→ F( f a ). We now state our estimates. The first point of the following lemma is an exact formula serving as the main tool to prove some of the bounds below. Lemma 7.1. 1. For any unitary operator U ∈ M(Cd ), for any basis vectors f a and f b , we have * c c f a |qλ U ⊗n f b = 1≤c≤λ1 det(U ta ,tb ), (7.6)
Local Asymptotic Normality for Finite Dimensional Quantum Systems
631
where U ta ,tb is the l(c) × l(c) minor of U given by [U ta ,tb ]i, j = Utac (i),tbc ( j) . Under the assumptions ⎫ |m| ≤ n η , λ ∈ n,α ⎪ ⎪ ⎬ inf i |µi − µi+1 | ≥ δ, µd ≥ δ β −1/2+2β ζ 1 ≤ Cn , ξ 1 ≤ n /δ, ⎪ −1 1/(1−α) ⎪ ⎭ β ≤ 1/2 n > 2δ c c
c c
(7.7)
we have the following estimates with remainder terms uniform in the eigenvalues µ• : 2. The number of admissible f a ∈ Oλ (m) with ( f a ) = 0 is ( (λi − λ j )m i, j (1 + O(n −1+2η /δ)). (7.8) #V 0 (m) = m i, j ! j>i
3. Let E := {m κ : κ} with (E) = . The number of admissible f a ∈ Oλ (m) with E( f a ) = E is bounded by: ( (λi − λ j )m κ(i, j) . (7.9) #V E (m) ≤ n − + i< j (m i, j −m κ(i, j) ) m κ(i, j) ! j>i
4. The number of admissible f a ∈ Oλ (m) with ( f a ) = is bounded by: ( (λi − λ j )m i, j , #V (m) ≤ C n − δ −2 |m|2 m i, j !
(7.10)
j>i
for a constant C = C(d). a b 5. Let f a ∈ V (l), and consider V (m) ⊂ Oλ (m) for some fixed b . Then: 0 if b = |m| − |l| + a | f a |qλ f b | ≤ , (7.11) b (C|m|) otherwise b f b ∈V (m)
with C = C(d). 6. If f a ∈ V 0 (m), then f a |qλ
f b = 1.
(7.12)
f b ∈Oλ (m)
7. If f a ∈ V 0 (m) so that its set of elementary column-modifiers is E 0 = {m κ(i, j) = m i, j }, then m i, j ζ 22 ( ζi, j ⊗n r (n), f a |qλ U (ζ , ξ , n) f 0 = exp iφ − √ √ 2 n µi − µ j i< j
(7.13) with the phase and error factor φ=
d−1 √ n (µi − µi+1 )ξi , i=1
r (n) = 1 + O n −1+2β+η δ −1 , n −1/2+2β δ −1 , n −1+2β+α δ −1 .
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J. Kahn, M. Gu¸ta˘
8. If f a ∈ V E (m), so that its set of column-modifiers is E = {m κ : κ} and (E) = , then $ $ $ $ $ f a |qλ U (ζ , ξ , n)⊗n f 0 $ − + i< j (m i, j −m κ(i, j) ) m κ(i, j) ( ζ 22 ζi, j Cζ ≤ exp − r (n), √ √ √ 2 n µi − µ j nδ i< j
(7.14) with C = C(d) a constant and r (n) as in point 7 above. 9. Under the further hypotheses that z ≤ n β , m i, j ≤ 2|ζi, j + z i, j |n β+ for some > 0, we have: f a |qλ U (ζ + z , ξ , n) f 0 f a ∈Oλ (m)
√ ( (ζi, j + z i, j )( n √µi − µ j ) m i, j r (n), m i, j !
ζ + z 22 = exp iφ − 2
i< j
(7.15) with r (n) = 1
+ O n −1+2β+η δ −1 , n −1+2β+α δ −1 , n −1+2η δ −1 , n −1+α+η δ −1 , δ −3/2 n −1/2+3β+2 .
10. Under the further hypotheses that |l| ≤ |m| and n 1−3η > 2C/δ 2 , where C = C(d), f a |qλ f b | | f a ∈Oλ (l)
f b ∈Oλ (m) |m|−|l|
≤ (C|m|)
a ( (λi − λ j )li, j C|l|2 |m| min (l,m) li, j ! nδ 2
(7.16)
i< j
with a (l, m) ≥ min
11. We have f a ∈Oλ (m)
f a |qλ
(|l − m| + 3|l| − 3|m|)+ . 6
fb =
*
f b ∈Oλ (m)
m i, j
i< j
(λi −λ j ) m i, j !
1 + O(n 3η−1 /δ) .
Proof of (7.6). We first express f a |U ⊗n f b as a product of matrix entries of U : ( ( f a |U ⊗n f b = f tac (r ) |U f tbc (r ) 1≤c≤λ1 1≤r ≤l(c)
=
(
(
1≤c≤λ1 1≤r ≤l(c)
Utac (r ),tbc (r ) .
(7.17)
(7.18)
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633
Since the subgroup of column permutations Cλ is the product of the permutation groups of each column, each σ ∈ Cλ is σ = s1 . . . sλ1 with sc a permutation of column c which transforms tbc (r ) into tbc (sc (r )). Then ( ( (σ ) Utac (r ),tbc (sc (r )) f a |qλ U ⊗n f b = f a |U ⊗n qλ f b = σ ∈Cλ
=
(
1≤c≤λ1 1≤r ≤l(c)
1≤c≤λ1 sc ∈Sc
=
(
(
(sc )
1≤r ≤l(c)
Utac (r ),tbc (sc (r ))
det(U ta ,tb ). c c
1≤c≤λ1
Proof of (7.8). The number of admissible f a such that ( f a ) = 0 is given by the products of the possibilities at each stage of the algorithm. For the first two stages, there is 0 is the number of possibilities at the exactly one possibility when = 0. Hence #V* third stage. Here the upper bound (7.4) reads: as j>i (λi − λ j )m i, j /m i, j !. On the other hand, we may use (7.5) as a lower bound, recalling that λi − λ j ≥ δn/2 and |m| ≤ n η (cf. (7.7)). This yields the result (7.8). Proof of (7.9). The number of f a in V E is given by the third stage of the algorithm (the two first stages yield a particular E). We then obtain (7.9) by applying (7.4) and neglecting the m κ ! factors, while noticing that κ m κ = |m| − . Proof of (7.10). The set V is the union of all V E with (E) = . Now the first two stages of the algorithm imply that there are at most C different E with the latter property, with C = C(d). m κ(i, j) ≥ |m| − 2 , Now we use (7.9) to upper-bound V E as follows. Since * * we may write κ m κ(i, j) ! ≥ i< j m i, j ! supi< j m i,−2 . Moreover λi − λ j ≥ δn/2. By j putting together we obtain #V E ≤ n − δ −2 |m|2
( (λi − λ j )m i, j , ∀E with (E) = . m i, j ! j>i
Multiplying by the number of possible E yields the result. Proof of (7.11). We are applying (7.6) with U = 1. Since both f a and f b are products of basis vectors, the scalar product f a | qλ f b is equal to −1 or 1 if tac ([1, l(c)]) = tbc ([1, l(c)]) for all columns, and 0 otherwise. Here we denote by tac ([1, l(c)]) the set of entries {tac (1), . . . , tac (l(c))}. Now, since a modified column cannot satisfy tac ([1, l(c)]) = [1, l(c)] (and the same for b), the vectors f a and f b are orthogonal unless they have the same number of modified columns. Finally, that number is |l| − a for f a and |m| − b for f b . This yields the first line of (7.11). We now concentrate on the case when b = |m| − |l| + a . Since | f a | qλ f b | ≤ 1, we can bound the sum of scalar products by the number of non-zero inner products. The question is how many diagrams f b have the same content (seen as an unordered set) in each column as f a : tac ([1, l(c)]) = tbc ([1, l(c)]), or equivalently S(κac ) = S(κbc ).
634
J. Kahn, M. Gu¸ta˘
For building the relevant f b , we can follow the algorithm with the further condition that, at stage three, all the column-modifiers are applied in such a way that the unordered column content is identical to that of f a . b The first two stages of the algorithm are the same so they yield a C factor. We now have a collection {m κ } of column modifiers which have to be placed so that they match the column content of f a . For each S we identify the column modifiers κ1 , . . . , κr (S) such that S(κi ) = S for all 1 ≤ i ≤ r (S). The total number of such objects is m S := i≤r (S) m κi and the number of ways in which they can be inserted to produce distinct diagrams is mS . m κ1 . . . m κr (S) Recall that the number of elementary column-modifiers i< j m κ(i, j) is at least |m| − 2 b . Moreover, each elementary column-modifier κ(i, j) corresponds to a different S(κ(i, j)) = {(i, −), ( j, +)}. Thus |m| − 2 b ≤ m κ(i, j) ≤ max m κ . i< j
Since
mS =
S
we obtain
S
κ:S(κ)=S
m κ = |m| − b ,
κ
m S − max m κ ≤ b . S
κ:S(κ)=S
This implies ( S
mS m κ1 . . . m κr (S)
≤ |m| . b
Multiplying by the C of the first stages, we get (7.11). b
Proof of (7.12). As shown above the only non-zero contributions come from f b ∈ V 0 ⊂ Oλ (m). Since b = 0, the constant from the two first stages of the algorithm is 1, m S = m i, j = m κ(i, j) for all S corresponding to an elementary column-modifier, and 0 otherwise. So the combinatorial factor is again one: we do not have any choice in our placement of column-modifiers. In other words, the only f b such that f a | qλ f b = 0 is f a . Finally, f a | qλ f a = 1. Proof of (7.13). From (7.6) we deduce ( c c det(U ta ,Id ), U = U (ζ , ξ , n). f a |qλ U (ζ , ξ , n)⊗n f 0 = 1≤c≤λ1
We shall use the Taylor expansion of the unitary U (ζ , ξ , n) to estimate the above determinants.
Local Asymptotic Normality for Finite Dimensional Quantum Systems
635
Entry-wise, for all 1 ≤ i ≤ d on the first line, and all 1 ≤ i < j ≤ d on the second and third lines: Ui,i (ζ , ξ , n) = 1 + i
ξi δi=d − ξi−1 δi=1 1 |ζi, j |2 − √ 2n |µi − µ j | n j=i
+ O(ζ n δ , ζ ξ n −1 δ −1/2 , ξ 2 n −1 ); ζi,∗ j 1 Ui, j (ζ , ξ , n) = − √ √ + O(ζ 2 n −1 δ −1 , ζ ξ n −1 δ −1/2 ); n µi − µ j ζi, j 1 U j,i (ζ , ξ , n) = √ √ + O(ζ 2 n −1 δ −1 , ζ ξ n −1 δ −1/2 ). µ n i − µj 3 −3/2 −3/2
If ζ = O(n β ), ξ ≤ n −1/2+2β /δ, and β < 1/2, the remainder terms are O(n −3/2+3β δ −3/2 ) for the first line and O(n −1+2β δ −1 ) for the last two lines. Therefore, when our parameters are in this range, we can give precise enough evaluations of the determinants. The idea is to find the dominating terms in the expansion of the determinant ( det A = (σ )Ai,σ (i) . σ
i
Note that we can use the above Taylor expansions inside the determinant since the number of terms in the product is at most d. Since f a ∈ V 0 , all tac are either Idc , or an (i, j)-substitution. If tac = Idc , the summands with more than two non-diagonal terms are of the same order * as the remainder term, so that only the identity and the transpositions count in σ i Ai,σ (i) . Let l = l(c), then υ(l) := det(U Id
c ,Idc
ξl 1 (ζ , ξ , n)) = 1 + i √ − 2n n
1≤i≤l l+1≤ j≤d
|ζi, j |2 + O(n −3/2+3β δ −3/2 ). µi − µ j
Note that for l = d, we get the usual determinant of U (ζ , ξ , n) which is 1. Consider now the case tac = Idc . Since tac (r ) ≥ r for all r , there √ exists a whole column c c of U ta ,Id whose entries are smaller in modulus than O(ζ / nδ) = O(n −1/2+β δ −1 ). In particular if tac is an (i, j)-substitution, then the only summand that is of this order comes from the identity. So that ζi, j c c υ(i, j) := det(U ta ,Id (ζ , ξ , n)) = √ √ + O(n −1+2β δ −1 ). n µi − µ j
(7.19)
Note that this approximation does not depend on l(c), but only on i and j. We now put together the estimated determinants in the product (7.6). For each i < j there are m i, j columns of the type (i, j)-substitution. Out of the λl − λl+1 columns of length l = l(c) there are λl − λl+1 − Rl of the type Idc , with 0 ≤ Rl ≤ |m|. Hence: f a |qλ U (ζ , ξ , n)⊗n f 0 =
d ( l=1
(υ(l)))λl −λl+1
( 1≤i< j≤d
(υ(i, j))m i, j
d (
(υ(l))−Rl .
l=1
(7.20)
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J. Kahn, M. Gu¸ta˘
Now υ(l) = 1 + O(n −1+2β δ −1 ) and Rl ≤ |m| ≤ n η , so the last product is 1 + O(n −1+2β+η δ −1 ). Similarly, since λ ∈ n,α we have λl − λl+1 = n(µl − µl+1 ) + O(n α ), and we can use Lemma 7.2 given at the end of this section to estimate the first product as follows: ⎞ ⎛ d (
υ(l)λl −λl+1 =
l=1
d ( l=1
⎜ 1 exp ⎜ ⎝iφl − 2
= exp iφ −
|ζi, j |2
1≤i≤l l+1≤ j≤d
ζ 22 2
µl − µl+1 ⎟ ⎟ r (n) µi − µ j ⎠
r (n),
with r˜ (n) = 1 + O(n −1+α+2β δ −1 , n −1/2+2β δ −1 ), √ φl = δl=d n(µl − µl+1 )ξl , φ=
d−1 √ n (µl − µl+1 )ξl . l=1
We now turn our attention to the middle product on the right side of (7.20), m i, j ζi, j m i, j 1 + O n −1+2β+η δ −1 , = √ √ υ(i, j) n µi − µ j where we have used that |m| ≤ n η . Inserting into (7.20) yields (7.13). Note that f a |qλ U (ζ , ξ , n)⊗n f 0 = 0 if there exist i < j such that ζi, j = 0 and m i, j = 0 . Proof of (7.14). We may write, much like in (7.20), f a |qλ U (ζ , ξ , n)⊗n f 0 =
d (
(υ(l)))λl −λl+1
(
(υ(κ))m κ
κ
l=1
d ( (υ(l))−Rl , l=1
where 0 ≤ Rl ≤ |m| − and υ(κ) is the determinant of the minor of U corresponding to having applied the column-modifier κ. We can further split the column-modifers into elementary ones κ(i, j) and non-elementary ones κ . Then f a |qλ U (ζ , ξ , n)⊗n f 0 can be written as d ( l=1
(υ(l)))λl −λl+1
( i< j
(υ(i, j))m κ(i, j)
d ( l=1
(υ(l))−Rl
(
m υ(κ ) κ . κ
The first three products on the right side can be treated as above. For the fourth product we give a rough upper bound based on the following observation. If the entries in the column have been modified in an admissible way, then tac (i) = j > l(c) for some i, so √ that |υ(κ)| ≤ Cζ / nδ for any κ, with some constant C = C(d).
Local Asymptotic Normality for Finite Dimensional Quantum Systems
Thus by using the previous point $ $ $ $ $ f a |qλ U (ζ , ξ , n)⊗n f 0 $ ζ 22 Cζ ≤ exp − √ 2 nδ
κ
(
mκ
i< j
|ζi, j | √ √ n µi − µ j
637
m κ(i, j)
r (n).
(7.21)
We obtain (7.14) by noting that the number of non-elementary modifiers is m κ = − + (m i, j − m κ(i, j) ). κ
i< j
can bring Proof of (7.15). Note that only admissible vectors in Oλ (m) . : non-zero contributions. We shall split the sum into sub-sums using Oλ (m) V = E V E (m), and 0 compare each sub-sum against the benchmark V 0 = V E . β From the bounds on ζ and z we obtain ζ + z = O(n ), so we can apply the previous points with ζ + z instead of ζ . Using (7.8) and (7.13) and recalling that λ ∈ n,α , we get: f a |qλ U (ζ + z , ξ , n)⊗n f 0 f a ∈V 0
( (ζi, j + z i, j )√n √µi − µ j m i, j 2 r (n) = exp iφ − ζ + z 2 /2 m i, j ! i< j
with error factor r (n) = 1 + O n −1+2β+η δ −1 , n −1/2+2β δ −1 , n −1+2β+α δ −1 , n −1+2η δ −1 , n −1+α+η δ −1 . For E = E 0 we combine (7.14) and (7.9) to obtain f a |qλ U (ζ + z , ξ , n) f 0 | · | f a |qλ U (ζ + z , ξ , n) f 0 |−1 | f a ∈V 0
f a ∈V E
− ( λi − λ j m κ(i, j) −m i, j m i, j ! ζ + z ≤n √ n m κ(i, j) ! δn i< j √ m κ(i, j) −m i, j ( δn|ζi, j + z i, j | × r (n) √ √ ζ + z n µi − µ j i< j √ ( |ζi, j + z i, j | µi − µ j − (1/2+β) − /2 ≤ O(n )δ m i, j ζ + z i< j:m i, j =0 ≤ O (2δ −3/2 n −1/2+3β+2 ) , −
with O(·) uniform in . In the second inequality we used m −m m i, j !/m κ(i, j) ! ≤ m i, i,j j κ(i, j) , (m κ(i, j) − m i, j ) ≥ −2 , i< j
m κ(i, j) −m i, j
λ ∈ n,α ,
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and in the third inequality we used m i, j ≤ 2|ζi, j + z i, j |n
β+
,
√ |ζi, j + z i, j | µi − µ j ≤ 1. m i, j ζ + z
Furthermore, for a given , there are at most C different E such that (E) = , corresponding to the possible choices in the first two stages of the algorithm, where C = C(d). Hence, if n is large enough, so that 2Cδ −3/2 n −1/2+3β+2 < 1, we have: f a |qλ U (ζ + z, ξ , n) f 0 = f a |qλ U (ζ + z, ξ , n) f 0
f a ∈Oλ (m)
f a ∈V
= 1 + O(δ −3/2 n −1/2+3β+2 ) exp iφ − ζ + z22 /2 m i, j √ √ ( ζ + z) ( n µ − µ ) ( i, j i j r (n) × m i, j ! i< j m i, j √ √ ( (ζ + z)i, j ( n µi − µ j ) r2 (n), = exp iφ − ζ + z22 /2 m i, j ! i< j
where the sum over was bounded using a geometric series and r2 (n) = 1
+O n −1+2β+η δ −1 , n −1+α+β δ −1 , n −1+2η δ −1 , n −1+α+η δ −1 , δ −3/2 n −1/2+3β+2 .
This is exactly (7.15).
Proof of (7.16). We choose a and b satisfying the condition b − a = |m| − |l| under which the inner products in (7.11) are non-zero. By multiplying (7.10) and (7.11), we see that: a ( (λi − λ j )li, j C|l|2 b | f a |qλ f b | ≤ (C|m|) li, j ! nδ 2 b a f a ∈V
(l)
i< j
f b ∈V (m)
= (C|m|)
|m|−|l|
a ( (λi − λ j )li, j C|l|2 |m| . li, j ! nδ 2
i< j
(7.22) It remains to sum up the upper bounds over all relevant pairs ( a , b ). If n 1−3η > 2C/δ 2 , the dominating term in the sum of bounds is that corresponding to the smallest possible a . The question is, what is the smallest possible value of a leading to non-zero inner products? A necessary condition for f a not to be orthogonal to f b is that for each set S of suppressed and added values, the two vectors have the same multiplicities m aS = m bS . The following argument provides a lower bound for ( f a ) + ( f b ). The idea is to count the minimum number of ‘horizontal box shuffling’ operations necessary in order to transform a Young tableau ta ∈ Oλ (m) into the tableau ta . Since |m| ≤ n η and λd ≥ δn + O(n α ), the tableau ta can be chosen to have at most one modified box per
Local Asymptotic Normality for Finite Dimensional Quantum Systems
639
column (thus ( f a ) = 0), and such that each of the modified columns of ta are also modified in ta . We also choose tb in a similar fashion. Now at each step we horizontally move one elementary column modifier κ(i, j) of ta (or tb ) into an already modified column, with the aim of constructing ta (or tb ). Each such operation increases ( f a ) + ( f b ) by one. On the other hand the opera tion has the following effect on the m aS (or m bS ): the multiplicities m {(i,−),( j,+)} and m S0 decrease by one, and m S0 +{(i,−),( j,+)} increases by one. Here S0 is the signature of the a b column to which the box (i, j) is moved. Hence the distance S |m S − m S | decreases by at most three. Since initially this quantity was equal to i< j |li, j − m i, j |, we need at least i< j |li, j − m i, j |/3 such operations before reaching our goal m aS = m bS . This means that ( f a ) + ( f b ) ≥ |l − m|/3. Together with b − a = |m| − |l|, this result yields a ≥ (|l − m| + 3|l| − 3|m|)/6. Moreover a is non-negative. Replacing in the above equation yields (7.16). Proof of (7.18). Since l = m, Eqs. (7.8) and (7.12) prove that
the bound (7.22) is saturated when a = 0, up to the error factor 1 + O(n −1+2η /δ) . Hence the remainder term due to the other consist in a geometric series with factor
C|m|3 nδ 2
= O(n 1−3η /δ 2 ).
Above we used the following lemma whose proof can be found in [29]: Lemma 7.2. If xn = O(n 1/2− ), then x n n 1+ = exp(xn )(1 + O(n − )). n 7.2. Proof of Lemma 5.4 and non-orthogonality issues. Lemma 7.3. Let Young tableaux with diagram λ and (m, λ) and (l, λ) be semistandard define |m| := i< j m i j and |l − m| := i< j |li, j − m i j |. If m i, j − m j,i = li, j − l j,i j>i
j
j>i
j
for some 1 ≤ i ≤ d, then m, λ|l, λ = 0. Otherwise, we derive an upper bound under the following conditions. We assume that λi − λi+1 > δn for all 1 ≤ i ≤ d − 1 and λd > δn, for some δ > 0. Furthermore we assume |l| ≤ |m| ≤ n η for some η < 1/3 and that Cn 3η−1 /δ 2 < 1, where C = C(d) is a constant. Then: | m, λ|l, λ | ≤ (C n)−η(|m|−|l|)/4 (C n)(9η−2)|m−l|/12 δ (|m|−|l|)/2−|m−l|/3 ×(1 + O(n −1+3η /δ)), (7.23) where C = C (d, η) and the constant in the remainder term depends only on d. The right side is of order less than n (9η−2)|m−l|/12 and converges to zero for η < 2/9 when n → ∞.
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Proof. We know that |m, λ is a linear combination of n-tensor product vectors in which the basis vector f i appears exactly λi − j>i m i, j + ji m i, j + ji li, j + j
(7.24)
We use the fact that qλ is a projection, up to a constant factor (cf. (5.1),(5.3)), and erase the qλ at the left of each scalar product, and we decompose pλ f m and pλ f l on orbits as in (7.3). Since the multiplicity of the elements in the orbits are the same in numerator and denominator, we end up with: m, λ|l, λ = fa ∈Oλ (m)
fa ∈Oλ (m) f a |qλ fb ∈Oλ (l) f b f a |qλ f ∈Oλ (m) f a · fb ∈Oλ (l) f b |qλ f a
b ∈Oλ (l)
f b
.
(7.25) We use (7.18) for the denominator: f a |qλ f a ∈Oλ (m)
=
( 1≤i< j≤d
f a ∈Oλ (m)
f a ·
f b ∈Oλ (l)
f b |qλ
f b
f b ∈Oλ (l)
(λi − λ j )(m i, j +li, j )/2 (1 + O(n 3η−1 /δ))), m i, j ! li, j !
and the numerator is bounded as in (7.16). Then, under the assumption |m| ≥ |l| we have ; 3 min ( |m|−|l| C|m| (li, j −m i, j )/2 m i, j ! | m, λ|l, λ | ≤ (C|m|) · · (λi − λ j ) δ2 n li, j ! i< j × 1 + O(n 3η−1 /δ) , where min = ((|l − m| + 3|l| − 3|m|)/6) ∧ 0. The factorials can be bounded as ; ( m i, j ! ≤ |m| (m i, j −li, j )+ /2 = |m|(|m−l|+|m|−|l|)/4 . li, j ! i< j
Since |m| ≤ n η and Cn 3η−1 /δ 2 < 1, we have 3η−1 (|l−m|+3|l|−3|m|)/6 C|m|3 min Cn ≤ . 2 δ n δ2 Since λi − λ j > nδ we have ( (λi − λ j )(li, j −m i, j )/2 ≤ (nδ)(|l|−|m|)/2 . 1≤i< j≤d
Local Asymptotic Normality for Finite Dimensional Quantum Systems
641
The constant C = C(d) can be replaced by another constant C = C (d, η) such that all powers of n appear in the form (C n)γ . Putting the bounds together we get | m, λ|l, λ | ≤ δ (|m|−|l|)/2−|m−l|/3 (C n)−η(|m|−|l|)/4 (C n)(9η−2)|m−l|/12 ×(1 + O(n −1+3η /δ)). Corollary 7.4. Let η < 2/9 and let (m, λ) be such that |m| ≤ n η . Assume as in Lemma 7.3 that λi − λi+1 > δn for all 1 ≤ i ≤ d − 1 and λd > δn, for some δ > 0, and that Cn 3η−1 /δ 2 < 1, where C = C(d) is a constant. Then there exists a constant C = C (d, η) such that
| m, λ|l, λ | ≤ (C n)(9η−2)/12 δ −1/3 .
(7.26)
|l|≤n η l=m
Proof. Recall that the bound (7.23) is given for |m| ≥ |l|. If on the contrary |l| > |m|, we must change all the |m − l| into |l − m|, so that these terms are always positive. Now, they are always in exponents of values less than one. We shall therefore neglect all those terms. Hence the expression on the left side of (7.26) is bounded from above by 2
k N (k) (C n)(9η−2)/12 δ −1/3 ,
k≥1
where N (k) is the number of l’s for which |m − l| = k. Since there are d(d − 1)/2 pairs 1 ≤ i < j ≤ d, there are at most (k + 1)d(d−1)/2 different choices for the values {|li, j − m i, j | : i < j} satisfying |li, j − m i, j | = k. Moreover, there are 2d(d−1)/2 sign choices which fix l = {li, j } completely. Thus N (k) ≤ (2(k + 1))d(d−1)/2 ≤ ck for some constant c which can be incorporated in the geometric series starting at k = 1, hence the desired estimate. We use this quasi-orthogonality to build an isometry Vλ : Hλ → F which maps the relevant finite-dimensional vectors |m, λ ‘close’ to their Fock counterparts |m . This is the aim of Lemma 5.4. Lemma 7.5. Let A be a contraction (i.e. A∗ A ≤ 1) from a finite space H to an infinite space K. Then there is an R : H → K such that A + R is an isometry and Range(A) ⊥ Range(R). As a consequence, for any unit vector f , we have R f 2 = 1 − A f 2 . Proof. As K is infinite-dimensional, we may consider a subspace H of K, orthogonal to Range(A), and the same dimension as H, so that we can find an isomorphism I from √ H to H . We then take R = I 1 − A∗ A. Proof of Lemma 5.4. Let Aλ : Hλ → F be defined by Aλ :=
1 1 + (Cn)(9η−2)/12 /δ 1/3
|l|≤n η
|l l, λ| .
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Then, A∗λ Aλ =
1 1 + (Cn)(9η−2)/12 /δ 1/3
|l|≤n η
|l, λ l, λ| ≤ 1Hλ .
where the last inequality follows from Corollary 7.4 and the following argument. It is enough to show that all eigenvalues of A∗λ Aλ are smaller than 1. Let m cm |m, λ be an eigenvector of A∗λ Aλ , and a the corresponding eigenvalue. Then by the linear independence of |m, λ we get that for each l, 1 l, λ|m, λ cm = acl . (9η−2)/12 1/3 1 + (Cn) /δ η |m|≤n
If l0 is an index for which |cl | is maximum, then by taking absolute values on both sides we obtain 1 | l, λ|m, λ | ≤ 1. a≤ (9η−2)/12 1/3 1 + (Cn) /δ η |m|≤n
Now we may apply Lemma 7.5, and find an Rλ such that Aλ + Rλ is an isometry, and Range(Rλ ) ⊥ Range(A), so that m| Rλ = 0. We define Vλ := Aλ + Rλ . Then m| Vλ = m| (Aλ + Rλ ) = m| Aλ 1 m| |l l, λ| = 1 + (Cn)(9η−2)/12 /δ 1/3 |l|≤n η =
1 1 + (Cn)(9η−2)/12 /δ 1/3
m, λ| .
Recall By Lemma 7.3 we have m, λ|l, λ = 0 if m i = li for some i, where m i in the total number of i in m (cf. (7.38)). In particular, |0, λ is orthogonal on all other basis vectors. This means that we can choose the isometry Vλ to satisfy Vλ |0, λ = |0 , and such that the relation above holds for all 0 < |m| ≤ n η . 7.3. Proof of Lemma 6.4 on mapping rotations into displacements. We first recall a few definitions and notations. We denote by D z the displacement operation (super-operator) acting on observables in the multimode Fock space F as D z (W ( y )) := Ad[W ( z )] (W ( y )) = e2iσ ( y , z ) W ( z + y ),
y , z ∈ Cd(d−1)/2 .
The operation acts as displacement on coherent states, in particular D ζ + z (|0 0|) = |ζ + z ζ + z |. ⊗n we have the action (cf. (7.1)) Similarly, on the finite dimensional space Cd
√ √ √ √ ζ ,ξ ,n (A) = Ad[U (ζ , ξ , n)](A) := U (ζ / n, ξ / n)⊗n A U ∗ (ζ / n, ξ / n)⊗n , ζ ,ξ ,n
whose restriction to the block λ is λ
= Ad[Uλ (ζ , ξ , n)].
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643
The isometric embedding Tλ (·) := Vλ · Vλ∗ and its ‘adjoint’ Tλ∗ (·) := Vλ∗ · Vλ satisfy ζ + z ,ξ ,n
Tλ λ
Tλ∗ (|0 0|) = Vλ |ζ + z , ξ , λ ζ + z , ξ , λ|Vλ∗ ,
where |ζ + z , ξ , λ := Uλ (ζ + z , ξ , n) |0, λ are the ‘finite dimensional coherent states’. According to Lemma 5.4, the coordinates of Vλ |ζ + z , ξ , λ in the Fock basis are described by: m, λ|Uλ (ζ + z , ξ , n)|0, λ (1 + O(n (9η−2)/12 δ −1/3 )) if |m|≤n η ; m|Vλ |ζ + z , ξ , λ = something not important if |m| > n η . (7.27) Using the relation | f f | − | f f |1 = 2 1 − | f | f |2 , which holds for unital vectors f, f , the statement of the lemma is equivalent to $ $ $ $ sup sup sup sup 1 − $ z + ζ |Vλ |ζ + z , ξ , λ $ = R(n)2 , (7.28) z ≤n β ζ ∈n,β ξ ≤n −1/2+2β/δ λ∈n,α
with R(n) the original remainder term. We shall prove formula (7.28) by decomposing these vectors in the Fock basis, that is ζ + z |Vλ |ζ + z , ξ , λ = ζ + z |m m|Vλ |ζ + z , ξ , λ . (7.29) m
The estimates are based on the following observations. 1) The coherent states have significant coefficients ζ + z |m only for ‘small’ m’s, i.e. those in the set M := {m : m i, j ≤ |(ζ + z )i, j |2 n , i < j}.
(7.30)
In particular, since 2β + < η we have M ⊂ {m : |m| ≤ n η }. 2) The coefficients m|Vλ |ζ + z , ξ , λ are uniformly close to exp(iφ) ζ + z |m , where φ is a fixed real phase, in particular uniformly over m ∈ M. 3) If am and bm are the two sets of coefficients, such that m |am |2 = m |bm |2 = 1, then $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ 1−$ am bm $ ≤ 1 − $ am bm $ + $ am bm $ $ $m $ $ $ $ m∈M m∈ /M $ $ $ $ $ $ ≤ 2 1−$ (7.31) am bm $ . $ $ m∈M
The precise statement in point 1) is | ζ + z |m |2 ≤ d 2 n −β . m∈M
(7.32)
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J. Kahn, M. Gu¸ta˘
Indeed, the inner products can be written as a product over the (i, j) oscillators and we have the bound
| ζ + z |m |2 ≤
m∈M
exp(−xi, j )
i< j
k>xi, j
n
xikj k!
, xi, j = |(ζ + z )i, j |2 . β
Each of the terms in the sum is a tail of Poisson distribution and is bounded by n −n if xi, j ≥ 1 and by n −β if xi, j < 1. We turn now to point 2). From the third line of (7.27) we get yλ f m |yλ U (ζ + z , ξ , n)| f 0 (1 + O(n (9η−2)/12 δ −1/3 )) m|Vλ Uλ (ζ + z , ξ , n)|0, λ = √ √ yλ f 0 |yλ f 0 yλ f m |yλ f m pλ f m |qλ U (ζ + z , ξ , n) f 0 = (1 + O(n (9η−2)/12 δ −1/3 )), √ pλ f m |qλ pλ f m where we have used (5.3) and (5.6). We recall that Oλ (m) is the orbit in (Cd )⊗n of f m under Rλ and that we have the decomposition pλ f m =
f a ∈Oλ (m)
#Rλ fa . #Oλ (m)
Then, by employing formulas (7.15) and (7.18), we can write m|Vλ Uλ (ζ + z , ξ , n)|0, λ z , ξ , n) f 0 f a ∈Oλ (m) f a |qλ U (ζ + ) = (1 + O(n (9η−2)/12 δ −1/3 ) f |q f f a , f b ∈Oλ (m) a λ b =e
iφ−ζ + z 22 /2
m ( (ζ + z )i, i,j j n(µi − µ j ) m i, j /2 r (n). λi − λ j m i, j ! i≤ j
(7.33)
The corresponding remainder term is r (n) = 1 + O n (9η−2)/12 δ −1/3 , n −1+2β+η δ −1 , n −1/2+3β+2 δ −3/2 , n −1+α+2β δ −1 , n −1+α+η δ −1 , n −1+3η δ −1 and the phase is: φ=
d−1 √ n (µi − µi+1 )ξi . i=1
Since λ ∈ n,α and the eigenvalues are separated by δ we have O(n α−1+η /δ)
n(µi −µ j ) m i, j /2 λi −λ j
1+ and the error can be absorbed in r (n). In conclusion, for m satisfying (7.30), we have: m|Vλ U (ζ + z , ξ , n)|0, λ = exp(iφ) m|ζ + z r (n).
=
Local Asymptotic Normality for Finite Dimensional Quantum Systems
645
Inserting this result into (7.29), and using (7.31) and (7.32), we get ⎛ ⎞ $ $ $ $ 1 − $ z + ζ |Vλ U (ζ + z , ξ , n)|0, λ $ = O ⎝1 − r (n), | m|ζ + z |2 ⎠ = R2 (n), m∈M
with
R2 (n) = O n (9η−2)/12 δ −1/3 , n −1+2β+η δ −1 , n −1/2+3β+2 δ −3/2 , n −1+α+2β δ −1 , n −1+α+η δ −1 , n −1+3η δ −1 , n −β .
Through expression (7.28), noticing that R2 (n) = R(n)2 , we see that we have proved the lemma. ⊗n√ 7.4. Proof of Lemma 6.2 on typical Young diagrams. Recall that the state ρ θ,n := ρθ/ n has the decomposition over ‘blocks’ λ given by (4.8). The probability distribution over ζ , u ,n
Young diagrams pλ
depends only on the diagonal parameters u and is given by
ζ , u ,n pλ
=
cnλ
d d ( ( (µiu ,n )λi m∈λ i=1
j=i+1
µu j ,n
m i, j
µiu ,n
,
with cnλ =
* ( d λl ! dk=l+1 (λl − λk + k − l) n . (λl + d − l)! λ1 , λ2 , . . . , λ d
l=1
The above formula can be understood as follows. By invariance under rotations we can ⊗n take ζ = 0 and the state is diagonal in the standard basis Cd formed by the vector *d u ,n m i f a . Each eigenprojector carries a weight i=1 (µ ) , where m i is the multiplicity of the vector f i in the tensor product f a . Thus, we only need to add all multiplicities over vectors that are ‘inside’ the block λ. Since the irreducible representation has basis f m labelled by semistandard Young tableaux, we get a factor u ,n m i, j d d d ( ( ( µj u ,n m i u ,n λi (µi ) = (µi ) . u ,n i=1 i=1 j=i+1 µi The additional factor cnλ is the dimension of Kλ , on which the state is proportional to the identity. √ √ Recall that µiu ,n = µi + u i / n for 1 ≤ i ≤ (d − 1) and µud ,n = µd − ( i u i )/ n. u ≤ n γ . Moreover m i, j ≤ n If δ ≥ 2dn α−1 ≥ 2dn γ −1/2 , then µu j ,n /µiu ,n ≤ 1 for all 2
for all (i, j), so the total number of m’s is smaller than n d . Thus u ,n m i, j ( µj 2 (µu ,n )λi ≤ nd . u ,n µi m i< j
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J. Kahn, M. Gu¸ta˘
On the other hand m = 0 is always in the set of possible m, so that m i, j ( µu j ,n ≥ 1. u ,n m i< j µi One can easily verify that 1≥
d ( λl ! l=1
*d
k=l+1 (λl
− λk + k − l) 1 ≥ . 2 (λl + d − l)! (n + d)d
The remaining factor is the multinomial law. We now show that this is the dominating part. Let us write (Y1 , . . . , Yd ) for the multinomial random variable. Then we have 2x 2 u ,n . (7.34) P[|Yi − nµi | ≥ x] ≤ 2 exp − n Indeed each Yi is a sum of independent Bernoulli variables X 1 , . . . , X n with P(X k = 1) = µiu ,n and P(X k = 0) = 1 − µiu ,n , and by Hoeffding’s inequality [44], n 2x 2 . X k − E[X k ]| ≥ x] ≤ 2 exp − P[| n k=1
By definition, for any λ ∈ / n,α there exists an i such that |λi − nµi | ≥ n α , which u ,n α implies |λi − nµi | ≥ n − dn γ +1/2 . With n α−γ −1/2 > 2d, the upper bound is simply n α /2 and we have λ∈n,α /
bλθ,n 1
= P[λ ∈ n,α ] ≤ n
d2
d
P[|Yi − nµiu ,n | ≥ n α /2]
i=1
≤ 2dn
d2
exp(−n 2α−1 /2).
7.5. Proof of Lemma 6.1 and Lemma 6.8 on classical LAN. We shall use multinomials as an intermediate step. Recalling that bλθ,n = pλθ,n τλn , we can write: θ,n N ( u , Vµ ) − bλ 1 ≤ p θ,n − M n u ,n u , Vµ ) u ,n 1 + N ( λ
−
µ1 ,...,µd
M n u ,n
λ
where M n u ,n
µ1 ,...,µud ,n
µ1 ,...,µud ,n
(λ)τλn 1 ,
(7.35)
is the d-multinomial with coefficients µiu ,n .
Concisely, what we really prove in this lemma is the asymptotic equivalence of the following classical experiments, together with an explicit rate: u ≤ nγ , Pn = p u ,n , γ , Mn = M n u ,n , u ≤ n u ,n µ1 ,...,µd
" Gn = N ( u , Vµ ), u ≤ nγ . !
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The equivalence of Mn and Gn is well known [31] and that of Pn and Mn can be treated similarly to the d = 2 case [16]. Complete details of the calculations leading to the desired rate of convergence for Lemma 6.1 can be found in [29]. From here, proving Lemma 6.8 (that is the inverse direction) is easy enough. Indeed, recall that σ n τ n p θ,n = p θ,n and that σ n is a contraction. Then
σ n N ( u , Vµ ) − p ζ , u ,n 1 = σ n N ( u , Vµ ) − σ n τ n p ζ , u ,n 1
≤ N ( u , Vµ ) − τ n p ζ , u ,n 1 .
Thus we have the same speed and conditions as those of Lemma 6.1.
7.6. Proof of Lemma 6.3 on convergence to the thermal equilibrium state. We recall that the state φ on CC R(L 2 (ρ), σ ) was defined in (4.28) and is the product of a classical Gaussian distribution and d(d − 1)/2 Gaussian states φi, j of quantum harmonic oscillators, one for each pair i < j. φi, j are thermal equilibrium states with inverse temperature ' β = ln(µi /µ j ) (cf. (4.14)). The joint state φ 0 := i< j φi, j is then displaced to obtain
φ ζ but Lemma 6.3 is only concerned with φ 0 . It is well known that thermal equilibrium states are diagonal in the number basis and in our case ( µi − µ j µ j m i, j φ0 = |m m|. (7.36) µi µi d(d−1)/2 m∈N
i< j
As shown in (5.8), a similar formula holds for the finite dimensional block states ρλ0, u ,n : m, λ|ρλ0, u ,n |m, λ
=
Cλu
u ,n d ( µj i< j
µiu ,n
m i, j
,
(7.37)
√ where Cλu is a normalisation constant, µiu ,n = µi + u i / n for 1 ≤ i ≤ (d − 1) and √ µud ,n = µd − ( i u i )/ n.
However there is a caveat: although |m, λ are eigenvectors of ρλ0, u ,n , they are not orthogonal to each other so we cannot directly use |m, λ m, λ| as eigenprojectors in the spectral decomposition. However, Lemma 5.4 gives us an estimate of the error that we incur by doing just that. Note first that the eigenvalues of ρλ0, u ,n are labelled by the total multiplicities m i of the index i in the semistandard Young tableaux: m i := λi − m i, j + m j,i . (7.38) j>i
j
By Lemma 7.3 we have that m, λ|l, λ = 0 if |m| = |l|. This allows us to split Hλ into a direct sum of orthogonal subspaces ⊥ := Lin{|l, λ : |l| > n η }, Hλ,η := Lin{|m, λ : |m| ≤ n η }, and Hλ,η
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and similarly for the Fock space F = Fη ⊕ Fη⊥ . Note the hidden dependence on n in the
definition of the subspaces. Asymptotically, the state ρλ0, u ,n and φ 0 concentrate on the ‘low excitations’ spaces Hλ,η and Fη with corresponding orthogonal projections Pλ,η and Pη , respectively. More precisely,
⊥ 0, ⊥ ρλ u ,n Pλ,η ) Tλ (ρλ0, u ,n ) − φ 0 1 = Tλ (Pλ,η ρλ0, u ,n Pλ,η ) − Pη φ 0 Pη 1 + Tλ (Pλ,η
−Pη⊥ φ 0 Pη⊥ 1
≤ 2Tλ (Pλ,η ρλ0, u ,n Pλ,η ) − Pη φ 0 Pη 1 + 2Pη⊥ φ 0 Pη⊥ 1 . (7.39)
From Definition 4.17 of φ 0 and that of thermal states (4.13) we see that the second term η on the right side of order max j
=
Cλu
d (
(µiu ,n )m i −λi P({m i }).
{m i } i=1
As in Lemma 5.4 we have that for |m| ≤ n η , P({m i }) =
1 1 + Cn (9η−2)/12 δ −1/3
|m, λ m, λ| + E({m i }),
m:{m i }
where the sum runs over those m with total multiplicities {m i }. The (positive) remainder has trace norm Tr(E({m i })) = O(n (9η−2)/12 δ −1/3 ) · dim(H({m i })). By summing over all {m i } satisfying |m| ≤ n η we get Pλ,η ρλ0, u ,n Pλ,η
=
1
ρ˜ 0, u ,n 1 + Cn (9η−2)/12 δ −1/3 λ
+ Cλu
d (
(µiu ,n )m i −λi E({m i }),
{m i } i=1
where ρ˜λ0, u ,n is the approximate state
ρ˜λ0, u ,n := Cλu
d (
(µiu ,n )m i −λi
{m i } i=1
|m, λ m, λ|.
m:{m i }
The error term has trace norm of the order O(n (9η−2)/12 δ −1/3 ) · Cλu
d (
(µiu ,n )m i −λi dim(H({m i })) = O(n (9η−2)/12 δ −1/3 ),
{m i } i=1
where we have used the normalisation of the block state ρλ0, u ,n .
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In conclusion
Pλ,η ρλ0, u ,n Pλ,η − ρ˜λ0, u ,n 1 = O(n (9η−2)/12 δ −1/3 ).
(7.40)
The next step is to show that the block states ρ˜λ0, u ,n are mapped by Tλ close to Pη φ 0 Pη . Using (5.8), we can write m i, j ( µu j ,n u ,n 0, u Tλ (ρ˜λ ) = Cλ Tλ (|m, λ m, λ|). (7.41) u ,n m∈λ i< j µi If n α−1 ≤ δ/2 and α > 1/2 > η, we know that all m such that |m| ≤ n η ‘fit into’ λ. Since µiu ,n = µi + O(n −1/2+γ ), when |m| ≤ n η ,
µu j ,n
m i, j
=
µiu ,n
µj µi
m i, j
(1 + O(n −1/2+γ +η /δ)).
For the normalisation constant we can write: m i, j ( µu j ,n ( µu j ,n u −1 + (Cλ ) = u ,n u ,n |m|≤n η i< j µi m∈λ:|m|≥n η i< j µi
(7.42)
m i, j
.
η
If 2dn γ −1/2 < δ/2 then the second part is less than n d (1 − δ/2)n which is negligible compared to the other error terms. Hence: ( µ j m i, j (Cλu )−1 = (1 + O(n −1/2+γ +η /δ)) µ i |m|≤n η i< j ( µ j m i, j = (1 + O(n −1/2+γ +η /δ) µ i m∈Nd(d−1)/2 i< j ( µi (1 + O(n −1/2+γ +η /δ)). (7.43) = µi − µ j 2
i< j
We then recall that for unit vectors, we have | f f | − | f f |1 = 2 1 − | f | f |2 . So that, using Lemma 5.4, we get that for |m| ≤ n η , Tλ (|m, λ m, λ|) − |m m|1 = Vλ |m, λ m, λ|Vλ∗ − |m m|1 = O(n (9η−2)/24 /δ 1/6 ).
(7.44)
Putting the estimates (7.42), (7.43), (7.44) back into formula (7.41), we obtain Tλ (ρ˜λ0, u ,n ), so that ( µi − µ j µ j m i, j Tλ (ρ˜λ0, u ,n ) = |m m| µi µi η |m|≤n i< j
+O(n −1/2+γ +η /δ, n (9η−2)/24 /δ 1/6 ). Comparing with (7.36), and using (7.39) and (7.40) we get the desired result.
(7.45)
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7.7. Proof of Lemma 6.5 on local linearity of SU (d). The key is to notice that, as we are dealing with a group, there is an r such that n)U (ζ , 0, n)U ( z , 0, n) = U (−ζ − z , 0, n)U (ζ , 0, n)U ( z , 0, n) U −1 (ζ + z , 0, = U ( r , s , n), and similarly for the operation . We shall prove below that under the condition that both ζ and z are smaller than n β , then r + s = O(n −1/2+2β /δ). Let us call this the domination hypothesis for further reference. Now, as the actions are unitary, we may rewrite the norm in Lemma as 6.5: ζ + z ,n
A = [λ
−(ζ + z ),n
= λ
ζ ,n
− λ λ z ,n ](|0, λ 0, λ|)1 ζ + z ,n
[λ
ζ ,n
− λ λ z ,n ](|0, λ 0, λ|)1
= [Id − rλ , s ,n ](|0, λ 0, λ|)1 .
As Tλ is an isometry, we may also let it act on the left and Tλ∗ on the right and get: A = Tλ (|0, λ 0, λ|) − Tλ rλ , s ,n Tλ∗ (|0 0|)1 ≤ |0 0| − | r r | 1 + | r r | − Tλ rλ , s ,n Tλ∗ (|0 0|)1 + Tλ (|0, λ 0, λ|) − |0 0| 1 . −1/2+2β /δ, hence By the domination hypothesis, the norm of r is smaller r |0 = than n −1+4β 2 1 − O(n /δ ). Using | f f | − | f f |1 = 2 1 − | f | f |2 we get that the first term on the right side of the inequality is O(n −1/2+2β /δ). Notice that this is dominated by R(n) given in Eq. (6.11) since η > 2β. For the second term, we apply Lemma 6.4, with z = 0. By the domination hypothesis, s ≤ n −1/2+2β /δ, so we may apply Lemma 6.4, and the remainder is given by R(n) in Eq. (6.11). The last term is O(n (9η−2)/24 /δ 1/6 ) as shown in (7.44) which is dominated by R(n). We finish the proof of the lemma, and simultaneously that of Theorem 4.3, by proving the domination hypothesis. Recall that an arbitrary element in SU (d) can be written in the exponential form ⎡ ⎛ ⎞⎤ d−1 Re(r j,k )T j,k + Im(r j,k )Tk, j ⎠⎦ , U ( r , s ) := exp ⎣i ⎝ si Hi + √ µ j − µk i=1
1≤ j
where ( r , s ) ∈ Cd(d−1)/2 × Rd−1 , and Ti, j , Hi are the generators of SU (d) defined in In general, the map ( (7.2). A special case of this is U ( r ) := U ( r , 0). r , s ) → U ( r , s ) is not injective but becomes so if we restrict to a small enough neighbourhood C of the origin (0, 0) ∈ Cd(d−1)/2 × Rd−1 . On this neighbourhood it makes sense to define the inverse as a sort of ‘logarithm’ log U ( r , s ) := ( r , s ), which is a C ∞ function. d(d−1)/2 By continuity of the product, if x , y ∈ C are small enough, then U (− x− √ √ y )U ( x )U ( y ) ∈ C. Since ζ + z / n ≤ n β−1/2 /δ, we can apply this to x = ζ / n, y = 1 √ z / n for n > (C/δ) 1/2−β with the constant C depending only on the dimension, and get √ √ √ √ √ √ √ ( r / n, s / n) = f (ζ / n, z / n) := log U (−(ζ + z )/ n)U (ζ / n)U ( z / n) .
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Since f is a C ∞ function we can expand in Taylor series and it is easy to show that 0) = (0, 0), the first order partial derivatives are zero as well, and the second order f (0, derivatives are uniformly bounded in a neighbourhood of the origin. Thus we get √ ζi, j 2 z i, j 2 , = O(n −1/2+2β /δ). r = n O n(µi − µ j ) n(µi − µ j ) Acknowledgements. We thank Richard Gill for suggesting the research topic and for many useful discussions. J. Kahn acknowledges the support of the French Agence Nationale de la Recherche (ANR), under grant StatQuant (JC07 205763) “Quantum Statistics”. M. Gu¸ta˘ ’s research is supported by the Engineering and Physical Sciences Research Council (EPSRC).
References 1. Armen, M.A., Au, J.K., Stockton, J.K., Doherty, A.C., Mabuchi, H.: Adaptive Homodyne Measurement of Optical Phase. Phys. Rev. Lett. 89, 133602 (2002) 2. Artiles, L., Gill, R., Gu¸ta˘ , M.: An invitation to quantum tomography. J. Royal Statist. Soc. B (Methodological) 67, 109–134 (2005) 3. Audenaert, K.M.R., Nussbaum, M., Szkola, A., Verstraete, F.: Asymptotic Error Rates in Quantum Hypothesis Testing. Commun. Math. Phys. 279, 251–283 (2008) 4. Bagan, E., Baig, M., Munoz-Tapia, R.: Optimal Scheme for Estimating a Pure Qubit State via Local Measurements. Phys. Rev. Lett. 89, 277904 (2002) 5. Bagan, E., Ballester, M.A., Gill, R.D., Monras, A., Munõz-Tapia, R.: Optimal full estimation of qubit mixed states. Phys. Rev. A 73, 032301 (2006) 6. Barndorff-Nielsen, O.E., Gill, R., Jupp, P.E.: On quantum statistical inference (with discussion). J. R. Statist. Soc. B 65, 775–816 (2003) 7. Belavkin, V.P.: Optimal multiple quantum statistical hypothesis testing. Stochastics 1, 315–345 (1975) 8. Belavkin, V.P.: Generalized Heisenberg uncertainty relations, and efficient measurements in quantum systems. Theor. Math. Phys. 26, 213–222 (1976) 9. Cirac, J.I., Ekert, A.K., Macchiavello, C.: Optimal Purification of Single Qubits. Phys. Rev. Lett. 82, 4344 (1999) 10. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. vol. II. New York-Toronto-London: McGraw-Hill, 1953 11. Fulton, W.: Young Tableaux, with Applications to Representation Theory and Geometry. Cambridge: Cambridge University Press, 1997 12. Fulton, W., Harris, J.: Representation Theory, A First Course. New York: Springer Verlag, 1991 13. Gill, R.D.: Asymptotic information bounds in quantum statistics. In: Quantum Stochatics and Information: Statstics Filtering and Control, Belavin, V.P., Guta, M. (eds.), River Edge, NJ: World Scientific, 2008 14. Gill, R.D., Massar, S.: State estimation for large ensembles. Phys. Rev. A 61, 042312 (2000) 15. Goodman, R., Wallach, N.R.: Representations and Invariants of the Classical Groups. Cambridge: Cambridge University Press, 1998 16. Gu¸ta˘ , M., Janssens, B., Kahn, J.: Optimal estimation of qubit states with continuous time measurements. Commun. Math. Phys. 277, 127–160 (2008) 17. Gu¸ta˘ , M., Jençová, A.: Local asymptotic normality in quantum statistics. Commun. Math. Phys. 276, 341 –379 (2007) 18. Gu¸ta˘ , M., Kahn, J.: Optimal estimation of d-dimensional quantum states. In preparation 19. Gu¸ta˘ , M., Kahn, J.: Local asymptotic normality for qubit states. Phys. Rev. A 73, 052108 (2006) 20. Hannemann, T., Reiss, D., Balzer, C., Neuhauser, W., Toschek, P.E., Wunderlich, C.: Self-learning estimation of quantum states. Phys. Rev. A 65, 050303 (2002) 21. Hayashi, M.: Presentations at MaPhySto and QUANTOP Workshop on Quantum Measurements and Quantum Stochastics, Aarhus, 2003, and Special Week on Quantum Statistics, Isaac Newton Institute for Mathematical Sciences, Cambridge, 2004 22. Hayashi, M.: Quantum estimation and the quantum central limit theorem. Bulletin of the Mathematical Society of Japan 55, 368–391 (2003) (in Japanese; Translated into English in quant-ph/0608198)
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23. Hayashi, M., (ed.): Asymptotic Theory of Quantum Statistical Inference: Selected Papers, River Edge, NJ: World Scientific, 2005 24. Hayashi, M.: Quantum Information. Berlin-Heidelberg: Springer-Verlag, 2006 25. Hayashi, M., Matsumoto, K.: Statistical Model with Measurement Degree of Freedom and Quantum Physics. In: Hayashi, M. ed., Asymptotic theory of quantum statistical inference: selected papers, River Edge, NJ: World Scientific, 2005, pp. 162–170 (English translation of a paper in Japanese published in Surikaiseki Kenkyusho Kokyuroku, vol. 35, pp. 7689–7727, 2002) 26. Hayashi, M., Matsumoto, K.: Asymptotic performance of optimal state estimation in quantum two level system. http://arxiv.org/abs/quant-ph/0411073v2, 2006 27. Helstrom, C.W.: Quantum detection and estimation theory. J. Stat. Phys. 1, 231–252 (1969) 28. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. Amsterdam: North-Holland, 1982 29. Kahn, J.: Quantum Local Asymptotic Normality and other Questions in Quantum Statistics. Ph.D. thesis, University of Leiden, 2008 30. Keyl, M., Werner, R.F.: Estimating the spectrum of a density operator. Phys. Rev. A 64, 052311 (2001) 31. Le Cam, L.: Asymptotic Methods in Statistical Decision Theory. New York: Springer Verlag, 1986 32. Massar, S., Popescu, S.: Optimal extraction of information from finite quantum ensembles. Phys. Rev. Lett. 74, 1259–1263 (1995) 33. Matsumoto, K.: Unpublished manuscript 34. Ohya, M., Petz, D.: Quantum Entropy and its Use. Berlin-Heidelberg: Springer Verlag, 2004 ˇ 35. Paris, M.G.A., Rehᡠcek, J. (eds.): Quantum State Estimation, Lecture Notes in Phys. vol. 649, New York: Springer, 2004 36. Petz, D.: Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Commun. Math. Phys. 105, 123–131 (1986) 37. Petz, D.: An Invitation to the Algebra of Canonical Commutation Relations. Leuven: Leuven University Press, 1990 38. Petz, D., Jencova, A.: Sufficiency in quantum statistical inference. Commun. Math. Phys. 263, 259– 276 (2006) 39. Schiller, S., Breitenbach, G., Pereira, S.F., Müller, T., Mlynek, J.: Quantum statistics of the squeezed vacuum by measurement of the density matrix in the number state representation. Phys. Rev. Lett. 77, 2933–2936 (1996) 40. Smith, G.A., Silberfarb, A., Deutsch, I.H., Jessen, P.S.: Efficient Quantum-State Estimation by Continuous Weak Measurement and Dynamical Control. Phys. Rev. Lett. 97, 180403 (2006) 41. Strasser, H.: Mathematical Theory of Statistics. Berlin-New York: De Gruyter, 1985 42. Torgersen, E.: Comparison of Statistical Experiments. Cambridge: Cambridge University Press, 1991 43. van der Vaart, A.: Asymptotic Statistics. Cambridge: Cambridge University Press, 1998 44. van der Vaart, A., Wellner, J.A.: Weak Convergence and Empirical Processes. New York: Springer, 1996 45. Vidal, G., Latorre, J.I., Pascual, P., Tarrach, R.: Optimal minimal measurements of mixed states. Phys. Rev. A 60, 126 (1999) 46. Wald, A.: Statistical Decision Functions. New York: John Wiley & Sons, 1950 47. Yuen, H., Kennedy, R., Lax, M.: Optimum testing of multiple hypotheses in quantum detection theory. IEEE Trans. Inform. Theory 21, 125–134 (1975) 48. Yuen, H.P., Lax, M.: Multiple-parameter quantum estimation and measurement of non-selfadjoint observables. IEEE Trans. Inform. Theory 19, 740 (1973) Communicated by M. B. Ruskai
Commun. Math. Phys. 289, 653–662 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0752-1
Communications in
Mathematical Physics
KPZ in One Dimensional Random Geometry of Multiplicative Cascades Itai Benjamini1,2 , Oded Schramm2 1 Department of Mathematics, Weizmann Institute of Science,
Rehovot 76100, Israel. E-mail: [email protected]
2 Microsoft Research, Redmond, WA, USA
Received: 8 July 2008 / Accepted: 7 November 2008 Published online: 26 February 2009 – © Springer-Verlag 2009
Abstract: We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity [KPZ88] appears. This note was inspired by the recent work of Duplantier and Sheffield [DS08] proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension.
1. Introduction A highlight of quantum gravity in the physics literature is the mysterious KPZ formula of Knizhnik, Polyakov and Zamolodchikov [Pol87,KPZ88] relating the dimensions of fractals in the random geometry to the corresponding dimension in Euclidean geometry. More specifically, the KPZ formula is − 0 =
(1 − ) , k+2
(1.1)
where 2 − 2 is the dimension of a set in quantum gravity and 2 − 2 0 is the dimension of the corresponding set in Euclidean geometry.1 1 The parameter k comes up since it is presumed that there is essentially one free parameter in the construction of quantum gravity. This parameter is intimately related to the central charge, and the various variants of quantum gravity are believed to arise by weighting a uniform measure with the partition function of a statistical physics model.
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Recently Duplantier and Sheffield [DS08] were able to prove an expected box count dimension version of the KPZ formula in Liouville gravity. In their setup, they avoid the very difficult issue of defining the random metric2 , and instead define a random measure. Multiplicative cascades is a well studied object and defines naturally a random metric ρ on [0, 1]. (The definition will be recalled in Sect. 2.) The metric space ([0, 1], ρ) is a path metric space, which in this case just means that it is isometric with some interval [0, ] with its Euclidean metric |x − y|. The length = ρ(0, 1) of [0, 1] in the metric ρ is in general random. In this note we prove a formula relating the Hausdorff dimensions of sets K ⊂ [0, 1] with respect to the random metric ρ on the one hand and with respect to the Euclidean metric on the other hand. The KPZ relation appears precisely when the random variables defining the cascade are exponentials of Gaussians. Interestingly, this is hinted at in the introduction of [OW00]. One goal of the note is to establish in this simple one-dimensional setup a Hausdorff dimension version of the KPZ relation. We also generalize the discussion to the nonGaussian setting, mainly to gain perspective on the underpinnings of the KPZ relation. Early versions of multiplicative cascades were introduced by Kolmogorov already in 1941 [Kol91] and were developed by Yaglom [Yag66] and by Mandelbrot [Man74]. Many fundamental properties of multiplicative cascades were first proved in the remarkable paper [KP76] by Kahane and Peyrière. For further background and references regarding the long history of multiplicative cascades see e.g., [OW00]. In Sect. 2, we describe the basic setup of multiplicative cascades and define the random metric ρ. Our main result is stated and proved in Sect. 3. An appendix follows in which we prove some essentially known necessary background facts about multiplicative cascades which we need to use. In some cases, the proofs in the appendix are simpler than the proofs that we were able to find in the literature, and in other cases, the results are slightly stronger. 2. Setup We now describe our setup. Let In denote the set of dyadic subintervals of [0, 1] of length 2−n ; namely, In := k 2−n , (k + 1) 2−n : n ∈ N, k ∈ {0, 1, . . . , 2n − 1} . Then each interval in In has precisely two subintervals in In+1 , its left half and its right half. Also set I = n∈N In . Let W be some positive random variable with mean 1, and let W I , I ∈ I, be an independent collection of random variables, each of which has the distribution of W . We now define inductively a sequence of random measures on [0, 1]. Let µ0 denote Lebesgue measure on [0, 1], and let µ1 := W[0,1] µ0 . Let µ2 denote the measure which agrees with W[0,1/2] µ1 on [0, 1/2] and agrees with W[1/2,1] µ1 on [1/2, 1]. Inductively, define µn+1 as the measure that on every I ∈ In agrees with W I µn . Alternatively, µn := wn µ0 ,
where
wn (x) :=
n−1
W I j (x) ,
j=0
and I j (x) denotes the interval I ∈ I j that contains x (and if there is more than one, the one whose maximum is x, say). We will need the following general result regarding multiplicative cascades. 2 We use the term “metric” to mean a “distance function”, not a Riemannian metric tensor.
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655
Theorem 2.1. The weak limit µ := limn→∞ µn exists almost surely. Moreover, if E W log2 W < 1 then µ[0, 1] > 0 a.s. and µ has no atoms a.s. This theorem is entirely or almost entirely proved in [KP76], but we present in the appendix a different perhaps simpler proof. Since µ[0, s] is a positive martingale for every s ∈ [0, 1], thefirst claim is very easy to verify. In [KP76] they show that µ[0, 1] > 0 a.s. if and only if E W log2 W < 1. The claim about the non-existence of atoms would follow from the last remark in [KP76], but we were unable to verify the justification of that remark (though it does follow under additional moment assumptions). Henceforth, we will be assuming that E W log2 W < 1. (2.1) By the result of [KP76] mentioned above, this assumption is necessary for the limit µ to be nonzero. On [0, 1], define the random metric ρ by ρ(x, y) := µ[x, y],
for all 0 ≤ x ≤ y ≤ 1.
If F(x) := µ[0, x], then ρ is just the pullback of the Euclidean metric on [0, F(1)] under F. We also set := µ[0, 1]. (2.2) n := µn [0, 1], Clearly, E n = 1 and E ≤ 1. In fact, E = 1 by [KP76], but we do not need this result. 3. Hausdorff Dimension Theorem 3.1. Suppose that E W −s < ∞
for all s ∈ [0, 1)
(3.1)
(in addition to the standard assumptions E W log2 W < 1, E W = 1, and P W > 0 = 1.) Let K ⊂ [0, 1] be some (deterministic) nonempty set, let ζ0 denote its Hausdorff dimension with respect to the Euclidean metric, and let ζ denote its Hausdorff dimension with respect to the random metric ρ. Then a.s. ζ is the unique solution of the equation 2 ζ0 =
2ζ E Wζ
(3.2)
in [0, 1]. As our proof shows, the assumption (3.1) may be significantly relaxed. See Theorems 3.4 and 3.5. Now consider the case in which log W is a Gaussian random variable. Since E[W ] = 1, this implies W = exp(σ Y − σ 2 /2), where Y is a standard Gaussian of zero mean and unit variance and σ ≥ 0. The assumption (2.1) is then equivalent to the requirement σ 2 < log 4. In this case, the moments E W s are easily evaluated and (3.2) gives
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ζ0 − ζ =
σ2 ζ (1 − ζ ), log 4
(3.3)
in agreement with (1.1). 3 For comparison, suppose instead that W = 1 ± σ , each with probability 1/2. Then (2.1) becomes |σ | < 1 and (3.2) transforms to 2 ζ0 =
2ζ 1 ζ 2 (1 − σ )
+ 21 (1 + σ )ζ
,
or ζ0 = 1 + ζ − log2 (1 − σ )ζ + (1 + σ )ζ . We now proceed to prove Theorem 3.1. Define φ(s) := s − log2 E W s . Then (3.2) reads ζ0 = φ(ζ ). The following lemma implies the uniqueness of the ζ satisfying (3.2). Lemma 3.2. The function φ is continuous, strictly monotone increasing in [0, 1] and maps [0, 1] onto [0, 1]. Proof. Set ψ(s) := E (W/2)s . Continuity of ψ follows from the dominated convergence theorem and convexity of ψ is immediate by the convexity of (W/2)s in s. Since ψ is convex and 1 1 ψ (1−) = E (W/2) log(W/2) = E W log W − log 2 < 0, 2 2 it is strictly monotone decreasing in [0, 1]. The lemma follows since φ = − log2 ψ, φ(0) = 0 and φ(1) = 1. The following simple lemma can serve to motivate Theorem 3.1, and is also important in its proof. Lemma 3.3. Let x, y ∈ [0, 1], and let s ∈ (0, 1]. Then E ρ(x, y)s ≤ 8 |x − y|φ(s) . Proof. Let [a, b] ∈ In . Then by the construction of ρ and the independence of the different variables W I , I ∈ I, n E ρ(a, b)s = 2−ns E W s E s = |a − b|φ(s) E s . 3 In (1.1), 2 (1 − ) is the dimension and similarly for . The factor of 2 comes from the fact that the 0 ambient space is two dimensional, and that the dimension is defined in terms of the measure, not the distance function. The transition from 0 to 1 − 0 is a passage from the dimension to the co-dimension, and does not change the form of (1.1).
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s Now, Jensen’s inequality gives E s ≤ E ≤ 1. Note that if |y − x| ∈ (2−n−1 , 2−n ], then the interval joining x and y can be covered by two consecutive intervals in In , say [a, b] and [b, c]. Then E ρ(x, y)s ≤ E (ρ(a, b) + ρ(b, c))s ≤ E (2 ρ(a, b))s + (2 ρ(b, c))s = 21+s E ρ(a, b)s ≤ 21+s |a − b|φ(s) ≤ 21+s+φ(s) |x − y|φ(s) . The lemma follows, since φ(s) ≤ 1 by Lemma 3.2.
Theorem 3.4. Let K , ζ0 and ζ be as in Theorem 3.1. Then a.s. φ(ζ ) ≤ ζ0 . It is worth pointing out that we are not assuming (3.1) here. Proof. Let s ∈ [0, 1] and assume that t := φ(s) > ζ0 . We now show that s ≥ ζ a.s. Let > 0. Then there is a covering of K by at most countably many intervals [xi , yi ] such that i |xi − yi |t < . By Lemma 3.3, we have
ρ(xi , yi )s ≤ 8 |xi − yi |t ≤ 8 . E i
i
√ we have a covering of K By Markov’s inequality, with probability at least 1s − √ with balls whose radii in the ρ metric satisfy ri ≤ 8 . Thus s ≥ ζ a.s. Hence ζ ≤ inf φ −1 (ζ0 , 1]. By Lemma 3.2, the theorem follows. Theorem 3.5. Let K , ζ0 and ζ be as in Theorem 3.1. Then a.s. ζ ≥ sup s ∈ (0, 1) : φ(s) < ζ0 , E[W −s ] < ∞ . Proof. Suppose that s ∈ (0, 1) satisfies t := φ(s) < ζ0 and E[W −s ] < ∞. We need to prove that ζ ≥ s. Since E[W s ] is convex in s and equals to 1 at s = 0, 1, we have t ≥ s ≥ 0. Since ζ0 > t, by Frostman’s lemma [Mat95, Chap. 8] there is a Borel probability measure ν0 supported on K such that
dν0 (x) dν0 (y) Et (ν0 ) := < ∞. (3.4) |x − y|t Set a := E[W s ], Z := W s /a and Z I := W Is /a. Define Z I j (x) , νn := f n ν0 . f n (x) := j
Since for every a ∈ [0, 1] the sequence νn [0, a] is a non-negative martingale, it easily follows that the weak limit ν := limn→∞ νn exists. (See, e.g., the proof of Theorem 2.1 in the appendix.) Then the support of ν is contained in the support of ν0 and therefore in K . Define ρn (x, y) := ρ(x, y) ∨ µ (In (x)) ∨ µ (In (y)).
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In order to estimate the expectation of
Es (νn ; ρn ) :=
dνn (x) dνn (y) ρn (x, y)s
(3.5)
we fix x, y ∈ [0, 1], and estimate E f n (x) f n (y) ρn (x, y)−s . Let k be the smallest integer such that [x, y] contains some interval of Ik . Then |x − y| < 4 · 2−k .
(3.6)
Let J ∈ Ik satisfy J ⊂ [x, y], and let J ∈ Ik−1 satisfy J ⊃ J . Then x ∈ J or y ∈ J . By symmetry, assume x ∈ J . Let G denote the σ -field W I j (x) , W I j (y) : j < n. Assume first that k ≤ n. Then
E ρ(x, y)
−s
k−1 G ≤ E µ(J )−s G = 2ks E −s W I−s . j (x) j=0
By Lemma A.3 in the appendix and our assumption that E W −s −s < ∞. Using (3.6), we therefore obtain E
< ∞, we have
k−1 W I−s , E ρ(x, y)−s G ≤ O(1) |x − y|−s j (x) j=0
where the implied constant may depend on s and the law of W . Now, f n (x) = a −n
n−1
W Isj (x) ,
j=0
and we have a similar expression for f n (y). Thus, n−1 n−1 E f n (x) f n (y) ρ(x, y)−s G ≤ O(1) a −2n |x − y|−s W Isj (x) W Isj (y) . j=k
j=0
Note that for j ≥ k we have I j (x) = I j (y). Taking expectations and using the definition of a yields E f n (x) f n (y) ρn (x, y)−s ≤ E f n (x) f n (y) ρ(x, y)−s (3.6)
≤ O(1) |x − y|−s a −k ≤ O(1) |x − y|−s+log2 a = O(1) |x − y|−t . Now, if k > n, we have instead n−1 W I−s . E ρn (x, y)−s G ≤ O(1) E µ (In (x))−s G ≤ O(1) 2sn j (x) j=0
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The above argument therefore gives in this case, E f n (x) f n (y) ρn (x, y)−s ≤ O(1) 2ns a −n ≤ O(1) |x − y|−t . Thus, we have E f n (x) f n (y) ρn (x, y)−s ≤ O(1) |x − y|−t for every x, y ∈ [0, 1]. Integrating this with respect to dν0 (x) × dν0 (y) and applying Fubini, one obtains (3.7) E Es (νn ; ρn ) ≤ O(1) Et (ν0 ). Since ρn (x, y) ≤ holds for x, y ∈ [0, 1], this estimate gives E (νn [0, 1])2 −s ≤ O(1) Et (ν0 ) . Now Hölder’s inequality comes into play: 1/(1+s) s/(1+s) E (νn [0, 1])2/(1+s) ≤ E (νn [0, 1])2 −s E ≤ O(1) Et (ν0 )1/(1+s) . + s) Thus, the martingale sequence νn [0, 1] is uniformly bounded in L p with p = 2/(1 > 1. It follows by the corresponding martingale convergence theorem that E ν[0, 1] = ν0 [0, 1] = 1, and in particular, ν[0, 1] > 0 with positive probability. The event ν[0, 1] > 0 is clearly independent of σ -field generated by any finite number of the ran dom variables W I , and therefore has probability 0 or 1, and in this case, P ν[0, 1] > 0 = 1. Since a.s. ρ is continuous, ρn → ρ uniformly as n → ∞ and νn → ν weakly, we have a.s. (3.7)
Es (ν; ρ) ≤ lim inf Es (νn ; ρn ) < ∞ . n→∞
The proof is now completed by appealing to Frostman’s criterion [Mat95, Chap. 8], since ν[0, 1] > 0 a.s. Proof of Theorem 3.1. The theorem follows immediately from Lemma 3.2 and Theorems 3.4 and 3.5. A. Some Multiplicative Cascades Background Lemma A.1. Our standing assumption E W log2 W < 1 implies that > 0 a.s. Proof. Set a := E W log2 W < 1. We first prove that > 0 with positive probability. Since n := µn [0, 1] is a positive martingale, we have a.s. convergence n → . The proof will come out of a recurrence relation for the sequence bn := E n log2 n . Let n and n have the law of n and be independent and independent from W . Then the law of n+1 is the same as the law of W (n + n )/2. Thus, bn+1 = E (W/2) (n + n ) log2 W + E (W/2) (n + n ) log2 (n + n ) + E (W/2) (n + n ) log2 (1/2) . We now use independence, E n = E W = 1, the symmetry between n and n , and the definition of a, and simplify the above to bn+1 = a + E n log2 (n + n ) − 1.
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Thus, bn+1 − bn = a − 1 + E n log2 (n + n ) − E n log2 n = a − 1 + E n log2 (n + n )/n . Since log2 (1 + x/n ) is concave as a function of x, and since n is independent from n , Jensen’s inequality applied to the above gives bn+1 − bn ≤ a − 1 + E n log2 (1 + E[n ]/n ) (A.1) = a − 1 + E n log2 (1 + 1/n ) . Because inf{bn : n ∈ N} ≥ inf{x log2 x : x > 0} > −∞ and a < 1, the set S := {n ∈ N : bn+1 − bn > (a − 1)/2} is infinite. For n ∈ S, (A.1) implies E g(n ) > (1 − a)/2 , where g(x) := x log2 (1 + 1/x) . Note that c := sup{g(x) : x > 0} < ∞. For all > 0 and n ∈ S, we have (1 − a)/2 < E g(n ) ≤ c P n ≥ + sup {g(x) : 0 < x < } . By taking sufficiently small, we can make sure that the last summand is at most (1 − a)/4. Then, P n > > (1 − a)/(4 c). This proves that n does not tend to zero in probability, and hence does not tend to zero a.s. Thus P > 0 > 0. Set := 2 µ[0, 1/2]/W[0,1] and := 2 µ[1/2, 1]/W[0,1] . Then , and W[0,1] are independent, and each of and has the law of = (W[0,1] /2) ( + ). But = 0 if and only if = 0 = , since W > 0 a.s. Thus, 2 P = 0 = P = 0, = 0 = P = 0 . Since P > 0 > 0, we conclude that > 0 a.s., which completes the proof. Proof of Theorem 2.1. Since E µn [0, 1] = 1, the sequence of measures µn is tight in the space of Borel measures on [0, 1] with the topology of weak convergence. Thus, some subsequence converges to a limit µ. For every rational r ∈ [0, 1] ∩ Q the sequence µn [0, r ] is a positive martingale, and hence the limit f (r ) := limn→∞ µn [0, r ] exists for all r ∈ Q∩[0, 1] a.s. It is immediate to verify that µ[0, x) = sup { f (r ) : r ∈ Q ∩ [0, x)} and µ[0, 1] = f (1). This implies that the subsequential limit µ is unique, and hence is the weak limit of the entire sequence µn . Thus, the theorem follows from Lemma A.1 and the next lemma. Lemma A.2. A.s. µ has no atoms. Proof. Let Z have the distribution of the µ-measure of the largest atom in [0, 1]. Clearly, E Z ≤ 1. Let Z 1 and Z 2 have the law of Z with W, Z 1 and Z 2 independent. Then W max(Z 1 , Z 2 )/2 has the law of Z . Therefore, E Z 1 + Z 2 /2 = E Z = E W max(Z 1 , Z 2 )/2 = E max(Z 1 , Z 2 ) /2 . Since Z 1 + Z 2 ≥ max(Z 1 , Z 2 ), and the expectations are the same, it follows that they are equal a.s. Thus, on the event Z 1 > 0, we have Z 2 = 0 a.s. Since Z 1 and Z 2 are independent, we get Z 1 = 0 a.s., as required.
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Lemma A.3. Suppose that E W −r < ∞ for some constant r > 0. Then also
(A.2)
E −r < ∞ .
A very slightly weaker form of this lemma can be found in [Liu01], where the assumption (A.2) is the same, but the conclusion is that E −s < ∞ for all s ∈ [0, r ). However, the setup in [Liu01] is more general. Proof. Let 1 := 2 µ[0, 1/2]/W and 2 := 2 µ[1/2, 1]/W . By construction 1 , 2 and W are independent and each of 1 and 2 has the law of . Moreover = W (1 + 2 )/2 .
(A.3)
Assume, for a moment, that there is some δ > 0 such that E −δ < ∞ .
(A.4)
By the means inequality and (A.3), we have ≥ W 1 2 .
(A.5)
Now independence gives for every s > 0, −s/2 −s/2 2 E 1 = E W −s E −s/2 . E −s ≤ E W −s E 1 (A.6) Let denote the set of s ∈ [0, ∞) such that E −s < ∞. By (A.4), we have δ ∈ S. Since S−s is a convex E W function of s, the set S must be an interval. Moreover, [0, δ] ⊂ S. Similarly, E W −s < ∞ for every s ∈ [0, r ]. Now, (A.6) shows that [0, r ] ∩ (2 S) ⊂ S, where 2 S = {2 s : s ∈ S}. This implies that [0, r ] ⊂ S, as needed. It remains to prove (A.4). From (A.2) we know that lim x −r P W < x = 0 . (A.7) x0
By (A.3), for every b, x > 0, we have that if < b x/2 then W < x or 1 + 2 < b. Thus, P < b x/2 ≤ P W < x + P 1 + 2 < b (A.8) ≤ P W < x + P 1 < b, 2 < b 2 = P W < x +P
j −1
for j ∈ N. Then j+1 = 2 2j . Let b0 > 0 satisfy P < b0 ≤ 0 .
(Theorem 2.1 implies the existence of such a b0 .) Set x j := sup x : P W < x ≤ 2j .
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Then our assumption (A.7) implies that x j ≥ j define
for all but finitely many j. Inductively
b j+1 = b j x j /2 . Then using induction, the relation j+1 = 2 2j and the estimate (A.8) give P < bj ≤ j .
(A.9)
The definition of b j gives b j = 2− j b0
j−1
x k ≥ C 2− j
k=0
j−1
2/r
k
,
k=0 2/r
where C > 0 is the product of b0 and the finitely many xk that satisfy xk < k . Taking into account the definition of j , we get b j ≥ C 2− j−2( j−1+2
j )/r
.
Now, ∞ E −δ ≤ b0−δ + b−δ j+1 P b j+1 ≤ < b j . j=0 5δ2 /r , while For all but finitely many j, the above estimate on b j gives b−δ j+1 ≤ 2 (A.9) j P b j+1 ≤ < b j ≤ P < b j ≤ j = 2−2 −1 . Thus, we have (A.4) for every δ ∈ (0, r/5), and the proof is thus complete. j
Acknowledgements. We are obliged to Scott Sheffield for numerous discussions explaining his insights to us. Yuval Peres and Ed Waymire have been very helpful in enlightening us with regards to the published work on multiplicative cascades.
References [DS08] [Kol91] [KP76] [KPZ88] [Liu01] [Man74] [Mat95] [OW00] [Pol87] [Yag66]
Duplantier, B., Sheffield, S.: In preparation, 2008 Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Translated from the Russian by V. Levin, Turbulence and stochastic processes: Kolmogorov’sideas 50 years on, Proc. Roy. Soc. London Ser. A 434(1890), 9–13 (1991) Kahane, J.-P., Peyrière, J.: Sur certaines martingales de benoit mandelbrot. Adv. Math. 22(2), 131–145 (1976) Knizhnik, V.G., Polyakov, A.M., Zamolodchikov, A.B.: Fractal structure of 2d-quantum gravity. Mod. Phys. Lett. A 3(8), 819–826 (1988) Liu, Q.: Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stoch. Proc. Appl. 95(1), 83–107 (2001) Mandelbrot, B.: Intermittent turbulence in self similar cascades: divergence of high moments and dimension of carrier. J. Fluid Mech. 62, 331–333 (1974) Mattila, P.: Geometry of sets and measures in Euclidean spaces. Volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 1995 Ossiander, M., Waymire, E.C.: Statistical estimation for multiplicative cascades. Ann. Stat. 28(6), 1533–1560 (2000) Polyakov, A.M.: Quantum gravity in two dimensions. Mod. Phys. Lett. A 2(11), 893–898 (1987) Yaglom, A.M.: Effect of fluctuations in energy dissipation rate on the form of turbulence characteristics in the inertial subrange. Dokl. Akad. Nauk SSSR 166, 49–52 (1966)
Communicated by M. Aizenman
Commun. Math. Phys. 289, 663–699 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0750-3
Communications in
Mathematical Physics
A Construction of Einstein-Weyl Spaces via LeBrun-Mason Type Twistor Correspondence Fuminori Nakata Department of Mathematics, Graduate School of Science and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro, 152-8551, Japan. E-mail: [email protected] Received: 9 July 2008 / Accepted: 4 December 2008 Published online: 3 March 2009 – © Springer-Verlag 2009
Abstract: We construct infinitely many Einstein-Weyl structures on S 2 × R of signature (− + +) which is sufficiently close to the model case of constant curvature, and on which the space-like geodesics are all closed. Such a structure is obtained as a parameter space of a family of holomorphic disks which is associated to a small perturbation of the diagonal of CP1 × CP1 . The geometry of constructed Einstein-Weyl spaces is well understood from the configuration of holomorphic disks. We also review Einstein-Weyl structures and their properties in the former half of this article. 1. Introduction Twistor type correspondences for the following structures are known (see [6]): (T1) projective structures on complex 2-manifolds, (T2) self-dual conformal structures on complex 4-manifolds, and (T3) Einstein-Weyl structures on complex 3-manifolds. (T2) is the original twistor theory introduced by R. Penrose [15]. (T3) is called Hitchin correspondence or minitwistor correspondence. There has been much progress on these twistor theories; more detailed or concrete investigation [13,14], real objects and reduction theory [1,4,5,7,16], relation with the theory of integrable systems [2,3], and so on. The geometric structures considered in these papers are either complex or real slices of complex objects, hence they are all analytic. On the other hand, the real indefinite case, for example, admits non-analytic solutions. Recently, C. LeBrun and L. J. Mason developed another type of twistor theory by which we can also analyse such non-analytic solutions [9,10] (see also [11,12]). The structures investigated by LeBrun and Mason are This work is partially supported by Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science.
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(LM1) Zoll projective structures on S 2 or S 2 /Z2 , and (LM2) self-dual conformal structures of signature (++−−) on S 2 ×S 2 or (S 2 ×S 2 )/Z2 . Here, a projective structure is called Zoll if and only if all the maximal geodesics are closed. Notice that (LM1) and (LM2) are the real objects corresponding to (T1) and (T2) respectively. There are several remarkable points for LeBrun-Mason theory. First, the twistor space is given as a pair (Z , N ) of a complex manifold Z and a totally real submanifold N in Z . The “twistor lines”, also known as the “nonlinear gravitons”, are given by holomorphic disks on Z with boundaries lying on N while in Penrose’s case or Hitchin’s case the twistor lines are embedded CP1 . Second, the structures (LM1) and (LM2) are obtained from a small perturbation of N in Z . By this reason, we have only been able to deal with the objects which are sufficiently close to the model case up to now. Lastly, the corresponding geometry satisfies a global condition, for example, Zoll condition in (LM1) case. In light of this research, in this article, we investigate another possibility, the LeBrunMason type correspondence for Einstein-Weyl structures. We now review the definitions and then we state the conjecture and the main theorem. Let X be a real (or complex) manifold. Definition 1.1. Let [g] be the conformal class of a definite or an indefinite metric g (or holomorphic bilinear metric for the complex case) on X , and ∇ be a (holomorphic) connection on T X . The pair ([g], ∇) is called a Weyl structure on X if there exists a (holomorphic) 1-form a on X such that ∇g = a ⊗ g.
(1.1)
Definition 1.2. A Weyl structure ([g], ∇) is called Einstein-Weyl if the symmetrized Ricci tensor R(i j) = 21 (Ri j + R ji ) is proportional to the metric tensor gi j , that is, if we can write R(i j) = gi j
(1.2)
using a function which depends on the choice of g ∈ [g]. Let [g] be an indefinite conformal structure on a real manifold X . A tangent vector v on X is called time-like if g(v, v) < 0, space-like if g(v, v) > 0 and light-like or null if g(v, v) = 0. We introduce the following global condition. Definition 1.3. An indefinite Weyl structure ([g], ∇) is called space-like Zoll if and only if every maximal space-like geodesic is closed. Now we state the conjecture for the LeBrun-Mason type correspondence for Einstein-Weyl structures. Conjecture 1.4. There is a natural one-to-one correspondence between • equivalence classes of space-like Zoll Einstein-Weyl structures on S 2 × R; and • equivalence classes of totally real embeddings ι : CP1 → CP1 × CP1 , at least in a neighborhood of the standard objects. Here the standard embedding CP1 → CP1 ×CP1 is given by ζ → (ζ, ζ¯ −1 ) using the inhomogeneous coordinate ζ of CP1 . The standard Einstein-Weyl structure is explained in Sect. 5. Before we state the main theorem, we define the following notion.
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Definition 1.5. Let Z be a complex manifold and D ⊂ Z be a holomorphic disk with boundary embedded in Z . Let v ∈ T p Z be a non-zero tangent vector at p ∈ ∂D. Then v is said to be adapted to D (denoted by v D) if and only if v ∈ T p ∂D and v has the same orientation as the orientation of ∂D which is induced from the complex orientation of D. The main theorem (Theorem 1.6) gives half of the correspondence in the above conjecture; from the embedding ι to the Einstein-Weyl space. We also claim that the geometry of the constructed Einstein-Weyl space is characterized by the holomorphic disks in the following way. Theorem 1.6. Let N be the image of any embedding of CP1 into Z = CP1 × CP1 which is C 2k+5 close to the standard one. Then there is a unique family of holomorphic disks {Dx }x∈S 2 ×R such that each boundary ∂Dx lies on N , and that the parameter space M = S 2 × R has a unique C k indefinite Einstein-Weyl structure ([g], ∇) satisfying the following properties. 1. For each p ∈ N , S p = {x ∈ M | p ∈ ∂Dx } is a maximal connected null surface on M and every null surface can be written in this form. 2. For each p ∈ Z \ N , C p = {x ∈ M | p ∈ Dx } is a maximal connected time-like geodesic and every time-like geodesic on M can be written in this form. 3. For each p ∈ N and non zero v ∈ T p N , C p,v = {x ∈ M | p ∈ ∂Dx , v Dx } is a maximal connected null geodesic on M and every null geodesic on M can be written in this form. 4. For each distinguished p, q ∈ N , C p,q = {x ∈ M | p, q ∈ ∂Dx } is a connected closed space-like geodesic on M and every space-like geodesic on M can be written in this form. In particular, this Einstein-Weyl structure is space-like Zoll. The organization of this paper is as follows. We first review projective structures in Sect. 2. Next, we study complex, definite or indefinite Einstein-Weyl spaces in Sect. 3. We prove that, in each case, the Einstein-Weyl condition can be translated to an integrability condition for certain distributions. Applying this method, we review the proof of the Hitchin correspondence in Sect. 4. In Sect. 5, the model case of the LeBrun-Mason type correspondence is explained. The standard Einstein-Weyl space is obtained as a double cover of a real slice of Hitchin’s example. We also study some properties of this model case. From Sect. 6, we deal with the perturbation of the model case. In Sect. 6, we prove that, for a small perturbation of the real submanifold N , there is a unique family of holomorphic disks with boundaries lying on N . This family has similar properties to the model case, especially for the double fibration, which is studied in Sect. 7. Finally in Sect. 8, we prove that there is a unique Einstein-Weyl structure on the parameter space of the constructed family of holomorphic disks. We also prove that the geometry of the Einstein-Weyl space is characterized by the holomorphic disks as in Theorem 1.6. 2. Projective Structures In this section, we review projective structures. Let X be a real smooth n-manifold and x i (i = 1, . . . , n) be a local coordinate on X . The following argument also works well in the complex case by considering x i as a complex coordinate, and using holomorphic functions instead of smooth functions.
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Definition 2.1. Two connections ∇ and ∇ on the tangent bundle T X are called projectively equivalent if their geodesics coincide without considering parameterizations. A projectively equivalent class [∇] is called a projective structure on X . Let ∇ and ∇ be connections on T X , and let ijk and ijk be their Christoffel symbols respectively, that is, ∇∂k ∂ j = ijk ∂i and so on, where we denote ∂i = ∂∂x i . Notice that ∇ is torsion-free if and only if ijk = ki j . The following proposition is readily checked (see [6]). Proposition 2.2. Suppose that both ∇ and ∇ are torsion-free, then they are projectively equivalent if and only if there exist functions f i (i = 1, . . . , n) on X such that the following condition holds: 1 i ijk = jk + (δ ij f k + δki f j ). 2
(2.1)
In the complex case, we can prove the following. Proposition 2.3. Let X be a complex n-manifold, and F be a holomorphic family of holomorphic curves on X . Suppose that, for each non-zero tangent vector v ∈ T X , there is a unique member of F which tangents to v. Then there is a unique projective structure [∇] on X so that F coincides to the family of geodesics. Proof. Let p : T X \ 0 X → X and π : T X \ 0 X → P(T X ) be the projections, where 0 X is the zero section and P(T X ) is the projectivization of T X . We use a local coordinate (x i ) on X , and let (y i ) be the fiber coordinate on T X with respect to the frame ∂∂x i . First we consider a curve c : (−ε, ε) → X given by c(t) = (x i (t)). We also write c for the i image of c. Then the natural lift c˜ : (−ε, ε) → T X is given by c(t) ˜ = x i (t); ddtx (t) . We obtain the velocity vector field of c, ˜ and this vector field uniquely extends to the vector field along p −1 (c) of the form v = yi
∂ ∂ + G i (x, y) i ∂ xi ∂y
(2.2)
so that G i satisfies G i (x, ay) = a 2 G i (x, y) for each non-zero a ∈ C. Notice that v descends to a line distribution on π(c) ˜ ⊂ P(T X ) by π∗ . This is the tangent distribution of the lift of c on P(T X ), hence it does not depend on the parametrization of c. Now we go back to the statement. Since the statement is local, we can assume P(T X ) = X × CPn−1 . Let CPn−1 = ∪Wα be an affine open cover. By the assumption, a foliation F˜ on P(T X ) is defined so that each leaf of F˜ is the natural lift of a curve in F. We notice the curves in F of which the lift intersects with X × Wα . Taking a parametrization of them, we obtain a holomorphic vector field vα = y i
∂ ∂ + G iα (x, y) i ∂ xi ∂y
on π −1 (X × Wα ) by the above construction. In this way, we have obtained the vector fields {vα }. Since vα and vβ descend to the same line distribution on X × Wα ∩ X × Wβ , we can write vα − vβ = f αβ (x, y)y i ∂∂y i
on π −1 (X × Wα ) ∩ π −1 (X × Wβ ), where f αβ is a holomorphic function satisfying
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f αβ (x, ay) = a f αβ (x, y) for each non-zero a ∈ C. Since H1 (Pn−1 , O(1)) = 0, we can take {vα } so that f αβ = 0 by changing the parametrizations. Hence we obtain a vector field on the whole of T X \ 0 X of the form (2.2). Then G i must be a degree-two polynomial for y, so we obtain a torsion-free connection ∇ so that G i (y) = ijk y j y k . For this ∇, each curve of F is a geodesic by construction. Here ∇ is determined up to projective equivalence since the ambiguity of taking v remains.
3. Einstein-Weyl Structures In this section, we study the basic properties of 3-dimensional Einstein-Weyl structures. We will prove that the Einstein-Weyl condition is equivalent to the integrability condition of certain distributions. We consider the complex, definite, and indefinite cases separately. Complex case. Let X be a complex 3-manifold and ([g], ∇) be a Weyl structure on X . We pick a g ∈ [g], however, the statements do not depend on the choice of g. We denote TC X = T X ⊗ C = T 1,0 X ⊕ T 0,1 X and TC∗ X = T ∗ X ⊗ C = T ∗ 1,0 X ⊕ T ∗ 0,1 X. Notice that g induces complex bilinear metrics on T 1,0 X, T 0,1 X, T ∗ 1,0 X and T ∗ 0,1 X which we also denote g. Definition 3.1. For each x ∈ X , a complex two-dimensional subspace V ⊂ Tx1,0 X is called a null plane if the restriction of g on V degenerates. The following property is easily checked. Lemma 3.2. If v ∈ Tx1,0 X is null, then v ⊥ is a null plane. Conversely, every null plane is written as v ⊥ for some null vector v. Notice that v ⊥ = ker v ∗ for every v ∈ Tx1,0 X , where v ∗ = g(v, ·) ∈ Tx∗ 1,0 X , and that v is null if and only if v ∗ is null. Let N (T ∗ 1,0 X ) be the null cotangent vectors, and Z = P(N (T ∗ 1,0 X )) be its complex projectivization. Notice that each point u ∈ Z corresponds to the null plane Vu = ker λ, where λ ∈ N (T ∗ 1,0 X ) is the cotangent vector satisfying u = [λ]. We can define a complex 2-plane distribution D ⊂ T 1,0 Z so that Du ⊂ Tu1,0 Z is the horizontal lift of the null plane Vu with respect to ∇. Notice that the horizontal lift is well-defined since N (T ∗ 1,0 X ) is parallel to ∇ because of the compatibility condition (1.1). Proposition 3.3. Let X be a complex 3-manifold. A Weyl structure ([g], ∇) with torsionfree ∇ on X is Einstein-Weyl if and only if the induced distribution D on Z is integrable, in other words, involutive. Proof. Let {e1 , e2 , e3 } be an orthonormal complex local frame on T 1,0 X with respect to g ∈ [g], and {e1 , e2 , e3 } be the dual frame on T ∗ 1,0 X . Let ω = (ωij ) be the connection form of ∇ with respect to {ei }, and let K ij = K ijkl ek ∧ el be its curvature form. Then from the compatibility condition (1.1), we obtain the following symmetry for K : K ijkl = Aijkl + δ ij Bkl , j
Aijkl = −Aijlk = −Aikl and Bkl = −Blk .
(3.1)
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Since the frame is orthonormal, the Einstein-Weyl equations are R(12) = R(23) = R(31) = 0 and R(11) = R(22) = R(33) , and this is equivalent to A1213 + A1312 = A2321 + A2123 = A3132 + A3231 = 0 and A1212 = A2323 = A3131 . (3.2) Now let N = N (T ∗ 1,0 X )\0 X , and π : N → Z be the projection where 0 X is the zero section. Then D is integrable if and only if the pull-back π ∗ D is integrable. Here π ∗ D ⊂ T 1,0 N is the complex 3-plane distribution defined by π ∗ D = {v ∈ T N | π∗ (v) ∈ D}. ˜ ⊂ T 1,0 N which is defined in a On the other hand, there is a 2-plane distribution D similar way to D, that is, Du is the horizontal lift of the null plane Vu . These distributions ˜ ⊕ ϒ, where are related by π ∗ D = D ϒ=
λi
∂ ∂λi
(3.3)
is the Euler differential. Now we define several 1-forms on N by θ=
λi ei , θi = dλi −
j
λ j ωi
and τi j = λi θ j − λ j θi .
(3.4)
˜ = {v ∈ T N | θ (v) = θi (v) = 0 (∀i) } and π ∗ D = {v ∈ T N | θ (v) = τi j (v) = Then D 0 (∀i, j) }. Hence D is integrable if and only if the 1-forms {θ, τi j } on N are involutive. Notice that τ23 /λ1 = τ31 /λ2 = τ12 /λ3 , hence τi j are proportional to each other. Let us prove that D is integrable if and only if (3.2) holds. First, we claim that dθ ≡ 0 mod θ, τi j always holds. Indeed, since θ1 /λ1 ≡ θ2 /λ2 ≡ θ3 /λ3 , we have
θi ∧ ei ≡
θ1 ∧θ ≡0 λ1
mod θ, τi j .
On the other hand, we have the torsion-free condition: dei + dθ =
dλi ∧ ei +
λi dei =
θi ∧ ei +
ωij ∧ e j = 0. Then
λi (dei + ωij ∧ e j ) ≡ 0
Next, a direct calculation shows that j j dτ12 ≡ − λ1 λ j K 2 + λ2 λ j K 1
mod θ, τi j .
mod τ12 ,
and we can check that dτ12 ≡ 0 holds if and only if 0 = λ3 − A2323 λ21 − A3131 λ22 − A1212 λ23 + (A3132 + A3231 )λ1 λ2 + (A2321 + A2123 )λ3 λ1 + (A1213 + A1312 )λ2 λ3
(3.5)
for every (λi ) satisfying λi2 = 0. Hence D is integrable if and only if the Einstein-Weyl equation (3.2) holds.
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The distribution D can be explicitly described in the following way. As in the above proof, let us take a local orthonormal frame {e1 , e2 , e3 } on an open set U ⊂ X . From the compatibility condition (1.1), the connection form ω of ∇ is written ⎛ ⎞ φ η21 η31 ⎜ ⎟ j ω = ⎝η12 φ η32 ⎠ , with ηi = −ηij . (3.6) η13 We can write
η23
λi2 = 0 λi ei and Z|U = [λ1 : λ2 : λ3 ] λi2 = 0 .
N (T ∗ 1,0 X )|U =
Then we obtain
φ
τ23 = λ2 dλ3 − λ3 dλ2 + λ1 λ1 η32 + λ2 η13 + λ3 η21 .
(3.7)
∼
Let U × CP1 → Z|U be a trivialization given by (x, ζ ) −→ [i(1 + ζ 2 ) : 1 − ζ 2 : 2ζ ],
(3.8)
where ζ ∈ C ∪ {∞} is a inhomogeneous coordinate. The horizontal lift v˜ of v ∈ Tx U at (x, ζ ) ∈ Z|U is 2 1 η32 + iη31 ∂ 1 2 η3 − iη3 − iζ η2 + ζ (3.9) v˜ = v + (v) . 2 2 ∂ζ For (x, ζ ) ∈ Z|U , the corresponding null plane on Tx1,0 X is spanned by m1 (ζ ) = ie1 + e2 + ζ e3 and m2 (z) = ζ (−ie1 + e2 ) − e3 .
(3.10)
˜ 1 (ζ )x and m ˜ 2 (ζ )x . Therefore the Einstein-Weyl condition Hence D(x,ζ ) is spanned by m ˜ 1, m ˜ 2 ] ∈ D. Proposition 3.3 could also be is equivalent to the involutive condition [m proved in this way. However, it is rather easier to check the integrability condition for π ∗ D as we did. Definite case. Let X be a real 3-manifold and ([g], ∇) be a definite Weyl structure, that is, a Weyl structure on X with positive definite [g]. In this case, we can define complex null planes on TC X . If we put Z = P(N (TC∗ X )), then we can define the complex 2-plane distribution D ⊂ TC Z in the same manner as the complex case by using the horizontal lift defined by (3.9). The complex conjugation TC∗ X → TC∗ X induces a fixed-point-free involution σ : Z → Z which is fiber-wise antiholomorphic. Notice that D satisfies σ ∗ D = D. We also define a complex 3-plane distribution E ⊂ TC Z by E = D ⊕ V 0,1 , where V 0,1 ⊂ TC Z is the (0, 1)-tangent vectors on the fiber of : Z → X . Here, we also obtain σ ∗ E = E. Proposition 3.4. Let ([g], ∇) be a definite Weyl structure on a 3-manifold X . Let : Z → X be the CP1 -bundle and E be the distribution on Z constructed above. Then there is a unique continuous distribution L of real lines on Z which satisfies L ⊗ C = E ∩ E on Z. Moreover the projection (C) of each integral curve C of L is a geodesic.
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Proof. If we take a real local frame {ei }, then we can describe the situations in a similar ˜ 1, m ˜ 2 and E = Span m ˜ 2 , ∂∂ζ¯ . Since ˜ 1, m way to (3.6) to (3.10). Then D = Span m E + E = TC Z, L exists uniquely by a dimension counting argument. Now let us define
l = ζ¯ m1 + m2 = 2(Im ζ )e1 + 2(Re ζ )e2 + (|ζ |2 − 1)e3 . Notice that l is real. We can take a unique function γ on Z so that ˜1+m ˜2+γ l † := ζ¯ m
∂ ∂ ζ¯
is real. Then we obtain L = Span l † . Let p : E → D be the natural projection, then ˜ 2 . By construction, the image of an integral curve ˜1+m p(L) = Span l˜ , where l˜ = ζ¯ m of p(L) by is a geodesic. Pulling back to E by p, we obtain the statement.
Proposition 3.5. Let X be a real 3-manifold, and ([g], ∇) be a definite Weyl structure on X with torsion-free ∇. Then ([g], ∇) is Einstein-Weyl if and only if E is integrable, in other words, involutive. Proof. The distribution E is integrable, if and only if π ∗ E is integrable, where π : N = N (TC∗ X )\0 X → Z. If we take an orthonormal frame field {e1 , e2 , e3 } of TC X , and if we use the complex fiber coordinate {λi } for TC∗ X , then we can define 1-forms θ , θi , τi j on N by (3.4). In this case, we obtain π ∗ E = π ∗ D + π ∗ V 0,1 , and π ∗ E = {v ∈ T ∗ N | θ (v) = τi j (v) = 0 (∀i, j)}. Hence E is integrable if and only if θ, τi j is involutive. By similar arguments, this occurs if and only if ([g], ∇) is Einstein-Weyl.
Remark 3.6. Locally speaking, E/L defines an almost complex structure on the space of geodesics on X . Proposition 3.5 means that this almost complex structure is integrable if and only if ([g], ∇) is Einstein-Weyl (see also [14]). Indefinite case. Let X be a real 3-manifold and ([g], ∇) be a Weyl structure on X for which the conformal structure [g] has signature (− + +). Let {e1 , e2 , e3 } be a local frame field on T X such that ⎛ ⎞ −1 1 ⎠. (3.11) (gi j ) = (g(ei , e j )) = ⎝ 1 A non-zero tangent vector v ∈ T X is called time-like, space-like or null when g(v, v) is negative, positive, or zero respectively. The following properties are easily checked. Lemma 3.7. 1. For each space-like vector v, there are just two real null planes which contain v. 2. Each time-like vector is transverse to every real null plane. Similar to the definite case, we define N (TC∗ X ), the space of complex null cotangent vectors, and Z = P(N (TC∗ X )), the space of complex null planes. In the indefinite case, we can also define N (T ∗ X ), the space of real null cotangent vectors, and ZR = P(N (T ∗ X )), the space of real null planes. There is a natural embedding ZR → Z. The complex conjugation TC∗ X → TC∗ X induces an involution σ : Z → Z which is fiber-wise antiholomorphic and for which the fixed point set coincides with ZR .
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Let us describe the situation explicitly using the above frame {ei } and its dual {ei }. From the compatibility condition (1.1), the connection form ω of ∇ is written: ⎛
φ ⎜ 1 ω = ⎝η2 η31
η21 φ −η32
⎞ η31 ⎟ η32 ⎠ . φ
(3.12)
We can write N (TC∗ X )|U =
λi ei − λ21 + λ22 + λ23 = 0
and Z|U = [λ1 : λ2 : λ3 ] | − λ21 + λ22 + λ23 = 0 .
(3.13)
∼
Let U × CP1 → ZR |U be a trivialization over an open set U ⊂ X such that (x, ζ ) −→ (1 + ζ 2 )e1 + (1 − ζ 2 )e2 + 2ζ e3 .
(3.14)
Here ZR corresponds to {(x, ζ ) ∈ U × CP1 | ζ ∈ R ∪ {∞}}. The horizontal lift v˜ of v ∈ Tx U at (x, ζ ) ∈ ZR |U is v˜ = v +
η32 + η31 η2 − η31 − ζ η21 + ζ 2 3 2 2
(v)
∂ . ∂ζ
(3.15)
If we define m1 (ζ ) = −e1 + e2 + ζ e3 and m2 (ζ ) = ζ (e1 + e2 ) − e3 ,
(3.16)
then m1 (ζ ) and m2 (ζ ) span the null plane corresponding to (x, ζ ) ∈ ZR . Define the real ˜ i are the vector ˜ 1, m ˜ 2 , where m 2-plane distribution DR ⊂ T ZR so that DR = Span m ˜ i (x,ζ ) is the horizontal lift of m(ζ )x . fields on ZR such that m ˜ i meromorphically We can extend m on Z, and define the complex 2-plane distribu ˜ 2 . We also define a complex 3-plane distribution E ˜ 1, m tion D ⊂ TC Z by D = Span m by E = D ⊕ V 0,1 , where V 0,1 ⊂ TC Z is (0, 1)-tangent vectors. Then we obtain σ ∗ D = D, σ ∗ E = E, DR ⊗ C = D|ZR and DR = D ∩ T ZR = E ∩ T ZR .
Proposition 3.8. Let ([g], ∇) be an indefinite Weyl structure on a 3-manifold X . Let : Z → X be the CP1 -bundle and E be the distribution on Z constructed above. Then there is a unique continuous distribution L of real lines on Z which satisfies L ⊗ C = E ∩ E on Z\ZR and L ⊂ DR on ZR . Moreover each integral curve C of L is contained in either Z\ZR or ZR , and the projection (C) is time-like geodesic if C ⊂ Z\ZR , and null-geodesic if C ⊂ ZR .
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Proof. Let us define a real vector field l on X by l = m1 − ζ¯ m2 = −(1 + |ζ |2 )e1 + (1 − |ζ |2 )e2 + (ζ + ζ¯ )e3 .
(3.17)
Notice that l is time-like if Im ζ = 0, and null if Im ζ = 0. We can take a unique function γ on Z so that ˜ 1 − ζ¯ m ˜2+γ l† = m
∂ ∂ ζ¯
˜ 1 − ζ¯ m ˜ 2 is real on ZR , γ = 0 and l † = l˜ on ZR . If we put L = l † , is real. Since l˜ = m then we obtain L ⊗ C = E ∩ E on Z\ZR and L ⊂ DR on ZR . L is unique since E + E¯ = TC Z on Z\ZR . The remaining statements are proved in a similar way to the definite case (Proposition 3.4).
Proposition 3.9. Let X be a real 3-manifold, and ([g], ∇) be an indefinite Weyl structure on X with torsion-free ∇. Then the following conditions are equivalent: • ([g], ∇) is Einstein-Weyl, • the real distribution DR is integrable, • the complex distribution E is integrable. Proof. If we put ∂ ∂ ∂ + λ2 + λ3 , ∂λ1 ∂λ2 ∂λ3 = λ1 θ2 + λ2 θ1 , τ13 = λ1 θ3 + λ3 θ1 and τ23 = λ2 θ3 − λ3 θ2
ϒ = −λ1 τ12
instead of (3.3) and (3.4), then the situation is parallel to the complex or definite case.
A direct calculation shows
τ23 = λ2 dλ3 − λ3 dλ2 − λ1 λ1 η32 + λ2 η31 − λ3 η21 .
(3.18)
Equation (3.18) will be used in Sect. 8. ˜1 ⊕m ˜ 2 locally, hence c1 (D) = c1 ( m ˜ 1 )+c1 ( m ˜2 )= Remark 3.10. We can write D = m −2 along each CP1 -fiber of : Z → X . Since c1 (V 0,1 ) = −2, we also obtain c1 (E) = −4 along each fiber. 4. Hitchin Correspondence In this section, we recall the twistor correspondence for complex Einstein-Weyl structures introduced by Hitchin [6]. Let Z be a complex 2-manifold and Y be a non-singular rational curve on Z with the normal bundle NY /Z ∼ = O(2). Let X be the space of twistor lines, that is, the rational curves which are obtained by small deformation of Y in Z . By Kodaira’s theorem, X has a natural structure of a 3-dimensional complex manifold, and its tangent space at x ∈ X is identified with the space of sections of the normal bundle NYx /Z , where Yx is the twistor line corresponding to x. Proposition 4.1. There is a unique Einstein-Weyl structure on X such that
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• each non-null geodesic on X corresponds to a one-parameter family of twistor lines on Z passing through two fixed points, and • each null geodesic on X corresponds to a one-parameter family of twistor lines each of which passes through a fixed point and is tangent to a fixed non-zero vector there. Proof. We have NYx /Z ∼ = O(2) for each x ∈ X since Yx is a small deformation of Y . We have Tx X ∼ = (Yx , NYx /Z ) by definition. Each holomorphic section of NYx /Z O(2) corresponds to a degree-two polynomial s(ζ ) = aζ 2 + bζ + c, where ζ is the inhomogeneous coordinate on Yx . We can define the conformal structure [g] so that a tangent vector in Tx X is null if and only if the corresponding polynomial s(ζ ) has double roots, that is, when b2 − 4ac = 0. If we fix two, possibly infinitely near, points in Z , then the twistor lines passing through these points make a one-parameter family. This family corresponds to a holomorphic curve on X . Let F be the family of such holomorphic curves. Then, by Proposition 2.3, we obtain a unique projective structure [∇] on X such that F coincides with the geodesics. Now, we prove that there is a unique torsion-free ∇ ∈ [∇] such that ([g], ∇) defines a Weyl structure. For this purpose, we first fix an arbitrary torsion-free ∇ ∈ [∇], and check that the second fundamental form on each null surface with respect to ∇ vanishes. For each point p ∈ Z , the two-parameter family of twistor lines passing through p corresponds to a null surface S on X . Notice that S is totally geodesic and naturally foliated by null geodesics each of which corresponds to a tangent line at p. Let N = T X | S /T S be the normal bundle of S. The second fundamental form II : T S ⊗ T S → N is defined by v ⊗ w → [∇v w] N , where the value does not depend on how we extend w. Take a frame field {e1 , e2 , e3 } on T X | S so that e1 is null and T S = e1 , e2 . Then the metric tensor is ⎛ ⎞ 0 0 ∗ g = (gi j ) = ⎝0 ∗ ∗⎠ . ∗ ∗ ∗ Since ∇ is torsion-free, ∇e1 e2 −∇e2 e1 = [e1 , e2 ] ∈ T S, so g(∇e1 e2 , e1 ) = g(∇e2 e1 , e1 ). Since g13 = 0, we obtain 3 3 12 = 21 .
(4.1)
On the other hand, since S is totally geodesic, we obtain 3 3 0 = g(∇ξ ξ, e1 ) = ξ 1 ξ 2 g13 12 + 21 3 + 3 = 0, and comfor every tangent vector ξ = ξ 1 e1 + ξ 2 e2 on S. So we obtain 12 21 3 3 bining with (4.1), we obtain 12 = 21 = 0. Hence g(∇ξ η, e1 ) = 0 for every vector field ξ and η on S, and this means II = 0 on S. Next we claim that there are functions ai , bi (i = 1, 2, 3) on X such that
1 1 (∇g)i jk = ai g jk + b j gik + bk gi j . 2 2
(4.2)
Since II = 0 for every null surface, we obtain ∇η g(ξ, ξ ) = 0
(4.3)
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for every null vector ξ and every vector η satisfying g(η, ξ ) = 0. Let us fix a local frame {ei } on X . If we put ξ = ξ i ei , η = ηi ei (i = 1, 2, 3) and ϕi jk = ∇ei (e j , ek ), then (4.3) is written (ϕi jk ξ j ξ k )ηi = 0.
(4.4)
Since ξ runs over all null vectors, (ξ i ) moves the conic C = [ξ 1 : ξ 2 : ξ 3 ] ∈ CP2 ξ i ξ j gi j = 0 . For fixed ξ , (ηi ) moves the line
L(ξ ) = { [η1 : η2 : η3 ] ∈ CP2 ηi (ξ j gi j ) = 0}.
Since (4.4) holds for every [ηi ] ∈ L(ξ ), we can take a function b(ξ ) satisfying ϕi jk ξ j ξ k = b(ξ )ξ j gi j for every ξ ∈ C and i = 1, 2, 3. Then we can take b(ξ ) to be a degree-one polynomial. Actually, since ξ j gi j (i = 1, 2, 3) do not all vanish at the same time, b(ξ ) = (ϕi jk ξ j ξ k )/(ξ j gi j ) defines a holomorphic section of O(1) over CP2 . If we put b(ξ ) = bk ξ k , then we obtain (ϕi jk − bk gi j )ξ j ξ k = 0 for i = 1, 2, 3. Here the bk (k = 1, 2, 3) are functions on X . Since these equations hold for every ξ ∈ C, there are functions ai on X such that (ϕi jk − bk gi j )X j X k = ai (g jk X j X k ) for every (X j ) ∈ C3 and i = 1, 2, 3. Noticing the symmetry, we obtain (4.2). Finally, if we define a new connection ∇˜ by 1 1 ˜ ijk = ijk + b j + bk , 2 2
(4.5)
then ∇˜ ∈ [∇] and ∇˜ satisfies ˜ i jk = (ai − bi )g jk , (∇g) ˜ is Einstein-Weyl, since the which means ∇˜ is compatible with [g]. Moreover, ([g], ∇) integrable condition in Proposition 3.3 is automatically satisfied by construction. Notice that such a connection is unique since the compatibility condition is not satisfied for any other torsion-free connection in [∇].
Remark 4.2. Let X = {(x, p) ∈ X × Z | p ∈ Yx }, then we obtain the double fibration
f
X ← X → Z , where and f are the projections. Each u ∈ X defines a null plane at (u) ∈ X as a tangent plane of the null surface corresponding to f(u) ∈ Z . Hence we obtain a natural map X → Z = P(N (TC∗ 1,0 X )) which is in fact biholomorphic. Identifying X with Z, we obtain D = ker{f∗ : TC1,0 X → TC1,0 Z }.
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Hitchin introduced two examples of Einstein-Weyl spaces, each of which is obtained from a complex twistor space [6]. The twistor space of one of them is Z = [z 0 : z 1 : z 2 : z 3 ] ∈ CP3 | z 12 + z 22 + z 33 = 0 . In this case, the twistor lines are the plane sections, and the corresponding Einstein-Weyl space is flat. In the other case, the twistor space is Z = [z 0 : z 1 : z 2 : z 3 ] ∈ CP3 | z 02 + z 12 + z 22 + z 33 = 0 . (4.6) In this case, the twistor lines are also the plane sections, and the corresponding EinsteinWeyl space is constant curvature space. We study the latter example in more detail in the next section. 5. The Standard Case In this section, the standard model of LeBrun-Mason type correspondence is explained. We start from Hitchin’s example (4.6), and construct the model case as a real slice of it (see also [14]). If we change the coordinate, (4.6) can be written {[z i ] ∈ CP3 | z 0 z 3 = z 1 z 2 } which coincides with the image of the Segre embedding CP1 × CP1 → CP3 , ([u 0 : u 1 ], [v0 : v1 ]) −→ [u 0 v0 : u 0 v1 : u 1 v0 : u 1 v1 ]. So we usually identify Z with CP1 × CP1 . Since the twistor lines are the plane sections, the twistor lines are parametrized by X = CP∗3 . We introduce a homogeneous coordinate [ξ i ] ∈ CP∗3 so that [ξ i ] corresponds to the plane {[z i ] ∈ CP3 | ξ i z i = 0}. Let X sing = [ξ i ] ∈ CP∗3 ξ 0 ξ 3 = ξ 1 ξ 2 be the set of planes tangent to Z . If [ξ i ] ∈ X sing , then the plane section degenerates to two lines 1 CP × [−ξ 1 : ξ 0 ] ∪ [−ξ 2 : ξ 0 ] × CP1 intersecting at the tangent point. We call such a plane section a singular twistor line on Z . Since Proposition 4.1 does not work on X sing , the Einstein-Weyl structure is defined only on X \X sing . Next we introduce real structures, that is, antiholomorphic involution on Z . There are several ways to introduce such a structure. For example, if we take the fixed-point-free involution σ : ([u 0 : u 1 ], [v0 : v1 ]) −→ ([u¯ 1 : u¯ 0 ], [v¯1 : −v¯0 ]), then σ extends to an involution on CP3 by [z 0 : z 1 : z 2 : z 3 ] −→ [¯z 3 : −¯z 2 : −¯z 1 : z¯ 0 ]. Then we also obtain an antiholomorphic involution on X . Let X R be its fixed point set. Since X R ∩ X sing is empty, we obtain a real Einstein-Weyl structure on the whole of
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XR ∼ = RP3 as a real slice of the complex Einstein-Weyl structure on X \X sing . This is nothing but the definite Einstein-Weyl structure induced from the standard constant curvature metric on RP3 . Our main interest is, however, in the indefinite case. Let σ : ([u 0 : u 1 ], [v0 : v1 ]) −→ ([v¯1 : v¯0 ], [u¯ 1 : u¯ 0 ]), be another involution on Z for which the fixed point set is denoted by Z R . The involution σ extends to an involution on CP3 by [z 0 : z 1 : z 2 : z 3 ] −→ [¯z 3 : z¯ 1 : z¯ 2 : z¯ 0 ]. Then we also obtain an involution on X . Let X R be its fixed point set. In this case, X R,sing = X R ∩ X sing is nonempty. Let (η1 , η2 ) = (u 0 /u 1 , v0 /v1 ) be a coordinate on Z = CP1 × CP1 , and let us write τ (η) for η¯ −1 . Then σ (η1 , η2 ) = (τ (η2 ), τ (η1 )) and Z R = {(η, τ (η)) | η ∈ CP1 }. In this coordinate, each non-singular twistor line l is written as a graph of some Möbius transform f : CP1 → CP1 , that is, l = {(η, f (η)) | η ∈ CP1 }. The twistor line l is σ -invariant if and only if τ ( f (η)) = f −1 (τ (η)), and in this case we can write f (η) =
Aη − B ¯ −C Bη
for some (A, B, C) ∈ R × C × R satisfying |B|2 − AC = 0. The intersection l ∩ Z R is nonempty if |B|2 − AC > 0, and is empty if |B|2 − AC < 0. In the non-singular case, the parameters (A, B, C) can be normalized so that |B|2 − AC = ±1. Since (A, B, C) and (−A, −B, −C) defines the same Möbius transform, we obtain X R \X R,sing ∼ = H H , where H = (A, B, C) ∈ R × C × R |B|2 − AC = 1 / ± and H = (A, B, C) ∈ R × C × R |B|2 − AC = −1 / ± . We obtain an indefinite Einstein-Weyl structure on H and a definite Einstein-Weyl structure on H as a real slice of X \X sing . The conformal structures are the class of g = |d B|2 − d AdC, which is indefinite on H and definite on H . ∼ If we identify CP1 → Z R by ω → ω, ω¯ −1 , then the intersection of Z R with the twistor line corresponding to [A, B, C] ∈ H is the circle ¯ +C =0 . ω ∈ CP1 A|ω|2 − B ω¯ − Bω (5.1) Hence H is naturally identified with the set of circles on CP1 , and its double cover H˜ = (A, B, C) ∈ R × C × R |B|2 − AC = 1 ∼ = S2 × R is identified with the set of oriented circles on CP1 . Since each circle divides the twistor line into two holomorphic disks, H˜ is identified with the set of holomorphic disks in Z with boundaries lying on Z R .
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There is a natural action of PSL(2, C) on H , H and H˜ defined in the following way. Each φ ∈ PSL(2, C) = Aut(CP1 ) induces an automorphism on Z by φ∗ : (η1 , η2 ) → (φ(η1 ), τ φτ (η2 )).
(5.2)
The automorphism φ∗ maps each σ -invariant twistor line to another σ -invariant twistor line. Since φ∗ preserves Z R , φ∗ preserves H and H . Obviously this action lifts to an automorphism on H˜ , and we will see later that this action on H˜ is transitive. Now we introduce an explicit description of the holomorphic disks corresponding to H˜ . Let M ∼ = CP1 × R = U1 ∪ U2 , where the Ui = {(λi , t) ∈ C × R} are patched by −1 λ2 = λ1 . Let : X+ → M be the disk bundle X+ = (U1 × D) ∪ (U2 × D), (λ1 , t; z 1 ) ∼ (λ2 , t; z 2 ) ⇐⇒ λ2 = λ−1 1 , z2 =
λ¯ 1 z1, λ1
where D = {z ∈ C | |z| ≤ 1}. We denote XR = (U1 × ∂D) ∪ (U2 × ∂D), and notice that XR is a circle bundle with c1 (XR ) = 2 along each fiber of . Let us define a smooth map f : X+ → Z by z 1 + r λ1 r z 1 − λ1 , −λ¯ 1 z 1 + r r λ¯ 1 z 1 + 1 λ¯ 2 z 2 + r r λ¯ 2 z 2 − λ and U2 × D (λ2 , t; z 2 ) −→ , , −z 2 + r λ2 r z 2 + λ2 U1 × D (λ1 , t; z 1 ) −→
where r = et . In this way, we have obtained the following double fibration: || || | | | ~|
M
X+ A AA f AA AA
(5.3)
Z
We use the coordinate λ ∈ C ∪ {∞} = CP1 satisfying λ = λ1 on U1 , and we define D(λ,t) = f ◦ −1 (λ, t). Then {D(λ,t) }(λ,t)∈M gives the family of holomorphic disks which coincides with the family corresponding to H˜ above. Hence naturally M ∼ = H˜ . Notice that we made our construction in such a way that the center of D(λ,t) , that is, the point given by z = 0, lies on Q = (λ, −λ) ∈ Z λ ∈ CP1 which is a twistor line on Z corresponding to [1, 0, 1] ∈ H . We have already defined a PSL(2, C)-action on M = H˜ by (5.2). For each element φ ∈ PSU(2) ⊂ PSL(2, C), we can check that φ∗ (D(λ,t) ) = D(φ(λ),t) . Since PSU(2) acts transitively on CP1 , PSU(2) acts transitively on CP1 × {t} ⊂ M for each t ∈ R. On the other hand, −t e ∈ PSL(2, C) (5.4) φ= et
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gives the automorphism φ∗ which maps the disk D(0,1) to D(0,2t) . Hence the action of PSL(2, C) on M = H˜ is transitive. Let S(T Z R ) = (T Z R \0 Z R )/R+ be the circle bundle on Z R , where 0 Z R is the zero section and R+ is positive real numbers acting on T Z R by scalar multiplication. On XR , we can take a nowhere vanishing vertical vector field v, that is, ∗ (v) = 0, so that the orientation matches the complex orientation of the fiber of : X+ → M. Since f∗ (v) does not vanish anywhere, we can define a smooth map ˜f : XR → S(T Z R ) by u → [f∗ (vu )]. Then we obtain the following diagram: XR B BB BB B f BB!
f˜
ZR
/ S(T Z R ) vv v vv v v v {v
Proposition 5.1. Let St = CP1 × {t} ⊂ M, and let ft and ˜ft be the restriction of f and ˜f on −1 (St ) respectively. Then, for each t ∈ R, 1. ft : (X+ \XR )| St → Z \Z R is diffeomorphic, 2. ˜ft : XR | St → S(T Z R ) is diffeomorphic, and 3. ft : XR | St → Z R is an S 1 -fibration such that each fiber is transverse to the vertical distribution of : XR → M. In particular, {D(λ,t) }λ∈CP1 gives a foliation on Z \Z R for each t ∈ R. Remark 5.2. Notice that, from 2 above, the following holds: for each t ∈ R, p ∈ Z R and non-zero v ∈ T p Z R , there is a unique x ∈ St such that p ∈ ∂ Dx and v Dx (see Definition 1.5). Proof of Proposition 5.1. We can assume t = 0 by changing the parameter t ∈ R by the automorphism of type (5.4). When t = 0, we can interpret the situation as a geometry on S 2 in the following 2 ∼ xi = 1} and p : CP1 → S 2 be the stereographic way. Let S 2 = {(x1 , x2 , x3 ) ∈ R3 | projection, p : λ −→
2 Re λ 2 Im λ 1 − |λ|2 , , , . 1 + |λ|2 1 + |λ|2 1 + |λ|2 ∼
We identify Z with S 2 × S 2 by the diffeomorphism Z → S 2 × S 2 : (η1 , η2 ) → (p(η1 ), p ◦ τ (η2 )), where τ (η) = η¯ −1 . Notice that Z R corresponds to the diagonal in this identification. Recall that D(λ,0) is the image of D → Z : z −→ (η1 , η2 ) =
z+λ z−λ , . ¯ ¯ −λz + 1 λz + 1
Then ∂ D(λ,0) ⊂ Z R corresponds to the big circle on the diagonal S 2 ⊂ S 2 × S 2 cut out by the plane 2(Re λ)x1 + 2(Im λ)x2 + (1 − |λ|2 )x3 = 0.
(5.5)
Hence we obtain a one-to-one correspondence between λ ∈ CP1 and the oriented big circle p(∂ D(λ,0) ), where the orientation is induced from the natural orientation of p(D(λ,0) ). Moreover, we claim that the following conditions are equivalent:
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(A1) (η1 , η2 ) ∈ Z lies on D(λ,0) , (A2) the oriented big circle p(∂ D(λ,0) ) winds anti-clockwise around p(η1 ), and this big circle coincides with the set of points on S 2 which have the same distance from p(η1 ) and p ◦ τ (η2 ) with respect to the standard Riemannian metric on S 2 . Indeed, if (η1 , η2 ) ∈ D(λ,0) , then the point p(η1 ) + p ◦ τ (η2 ) ∈ R3 lies on the plane (5.5), hence the big circle p(∂ D(λ,0) ) satisfies (A2). The converse is easy. In particular, the following conditions are equivalent: (B1) (η1 , η2 ) ∈ Z R lies on ∂ D(λ,0) , (B2) the big circle p(∂ D(λ,0) ) passes through p(η1 ) = p ◦ τ (η2 ). The statement follows directly from this interpretation. Actually, for each p = (η1 , η2 ) ∈ Z \Z R , the big circle satisfying (A2) exists uniquely, hence 1 holds. For each p = (η1 , η2 ) ∈ Z R , S(T p Z R ) corresponds to the oriented big circles satisfying (B2), hence 2 and 3 follow.
The geometry on M is characterized by the double fibration (5.3) in the following way: Proposition 5.3. 1. For each p ∈ Z R , S p = {x ∈ M | p ∈ ∂ Dx } = ◦ f−1 ( p) is a maximal connected null surface on M and every null surface can be written in this form. 2. For each p ∈ Z \Z R , C p = {x ∈ M | p ∈ Dx } = ◦ f−1 ( p) is a maximal connected time-like geodesic on M and every time-like geodesic can be written in this form. 3. For each p ∈ Z R and each non-zero v ∈ T p Z R , C p,v = {x ∈ M | p ∈ ∂ Dx , v Dx } = ◦ ˜f−1 ([v]) is a maximal connected null geodesic on M and every null geodesic can be written in this form. 4. For each distinguished p, q ∈ Z R , C p,q = {x ∈ M | p, q ∈ ∂ Dx } = S p ∩ Sq is a closed connected space-like geodesic on M and every space-like geodesic can be written in this form. Proof. Since {∂ D(λ,t) } is the set of oriented circles of the form (5.1), we obtain • S p S 1 × R for each p ∈ Z R , • C p,v R for each p ∈ Z R and non zero vector v ∈ T p Z R , • C p,q S 1 for each distinguished p, q ∈ Z R . Since S p is a real slice of a complex null surface, it is a real null surface. Moreover, it is a maximal connected null surface since S p is closed in M. Hence 1 holds. In a similar way, we can see that C p,v is a maximal connected real null geodesic, so 3 holds. C p,q is also a maximal connected real non-null geodesic. Notice that C p,q is contained in the null surface S p . Since a null plane never contains time-like vectors, C p,q is a space-like geodesic (see Lemma 3.7). Hence 4 holds. Now we check 2. Let p ∈ Z \Z R and notice that every σ -invariant twistor line passing through p also passes through σ ( p). So C p is a real slice of the complex geodesic corresponding to the two points p and σ ( p). Hence C p is a geodesic. From Proposition 5.1, we obtain C p R which is closed in M. Hence C p is a maximal connected geodesic. To see that C p is a time-like geodesic, it is enough to check that C p is transversal to every null plane at each point (see Lemma 3.7). Notice that, if we fix three points on Z , there is at most one twistor line containing them. Hence C p ∩ Sq = {x ∈ M | p, σ ( p), q ∈ Dx } is at most one point for each q ∈ Z R . Thus C p is time-like.
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In particular, we obtain the following. Corollary 5.4. The indefinite Einstein-Weyl structure on M constructed above is spacelike Zoll. Let X = X+ ∪XR X− be a CP1 bundle over M, where X− = X+ is the copy of X+ with fiber-wise opposite complex structure. On the other hand, we have a CP1 -bundle Z on M equipped with the distributions DR , E, L and so on. Then, similar to Remark 4.2, ∼ there is a natural identification X → Z such that • for each p ∈ • for each p ∈ • for each p ∈ in XR .
Z R , f−1 ( p) corresponds to an integral surface of DR , Z \Z R , f−1 ( p) corresponds to an integral curve of L in X \XR , and Z R and [v] ∈ S(T p Z R ), ˜f−1 ([v]) corresponds to an integral curve of L
Hence the following holds: • DR = E ∩ T XR = ker{f∗ : T XR → T Z R } on XR , • L = ker{f∗ : T X → T Z } on X+ \XR , and • L = ker{˜f∗ : T XR → S(T Z R )} on XR . Recall that we denote St = CP1 × {t} and let us denote Xt = −1 (St ), where : X → M is the projection. Let Et = E ∩ TC Xt for each t. Then, since L ∩ T Xt = 0, we obtain E = (L ⊗ C) ⊕ Et . From E ∩ E = L ⊗ C and E ⊕ E = T X , we obtain Et ⊕ Et = T Xt . Moreover, since E is integrable, Et is also integrable. Hence Et defines a complex structure on Xt . Now we claim that ft : (X+ \XR )| St → Z \ZR is holomorphic with respect to the above complex structure. Consider the complex Einstein-Weyl space MC = X \X sing defined at the beginning of this section, and let ZC = P(N (T ∗ 1,0 MC )). Then we obtain fC
the double fibration MC ← ZC → Z , where fC is holomorphic. On the other hand, there is natural embedding i t : (X+ \XR )| St → ZC which is holomorphic since it preserves the distributions. Since ft = fC ◦ i t , ft is holomorphic on (X+ \XR )| St . 0,1 Z ) ⊂ f−1 (T 0,1 Z ) on X \X . From the above argument, we obtain Et = (ft )−1 + R ∗ (T ∗ 0,1 Z ) on X \X . Since L ⊗ C = ker f∗ there, we obtain E = (L ⊗ C) ⊕ Et ⊂ f−1 + R ∗ (T 1,0 Z ). Since E + E = T X and E ∩ E = L ⊗ C, we Then we also have E ⊂ f−1 C + ∗ (T 0,1 Z ) on X \X . obtain E = f−1 + R ∗ (T In this way, we have proved the following: Proposition 5.5. Identifying X = X+ ∪ X− with Z, 1. 2. 3. 4.
0,1 Z ) on X where f : T X → T Z , E = f−1 + ∗ C + C ∗ (T DR = E ∩ T XR = ker{f∗ : T XR → T Z R } on XR , L = ker{f∗ : T X+ → T Z } on X+ \XR , and L = ker{˜f∗ : T XR → S(T Z R )} on XR .
It is convenient to consider the compactification of M and X+ . Let I = [−∞, ∞] be the natural compactification of R. If we put Mˆ = CP1 × I , then we obtain a natural ˆ ˆ embedding ι : M → M. Next, let : X+ → M × Z be the embedding defined by (u) = ι ◦ (u), f(u) . Let us define Xˆ+ and XˆR as the closure of (X+ ) and (XR )
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in Mˆ × Z . Then we obtain the double fibration (Xˆ+ , XˆRK) KKK fˆ w w ˆ ww KKK w KKK ww w % {w (Z , Z R ) Mˆ
(5.6)
where ˆ and ˆf are the projections. ˆ but a marked CP1 Notice that ˆ −1 (x) is no longer a disk for x = (λ, ±∞) ∈ ∂ M, −1 ˆ for which the marking point is ˆ (x) ∩ XR . We denote these marked CP1 by D(λ,∞) = ˆ −1 (λ, −∞) = {λ} × CP1 and D(λ,−∞) = ˆ −1 (λ, ∞) = CP1 × {−λ},
(5.7)
where D(λ,∞) is marked at (λ, λ¯ −1 ) and D(λ,−∞) is marked at (−λ¯ −1 , −λ). Recall the definitions of C p and C p,v introduced in Proposition 5.3. We define Cˆ p and Cˆ p,v as the compactification of C p and C p,v in Mˆ respectively. Then the following properties are easily checked. Proposition 5.6. 1. For each p ∈ Z \Z R , XˆR |Cˆ is homeomorphic to S 2 and the restricp tion ˆf : XˆR |Cˆ → Z R is a homeomorphism. In particular, {∂ Dx }x∈Cp gives a foliation p on Z R \{2 points}. 2. For each p ∈ Z R and non-zero v ∈ T p Z R , XˆR |Cˆ is homeomorphic to S 2 and p,v the restriction ˆf : XˆR | ˆ → Z R is surjective. Moreover, this is one-to-one on Cp,v
the complement of the curve ˆf−1 ( p), hence {(∂ Dx \{ p})}x∈Cp,v gives a foliation on Z R \{ p}. Remark 5.7. For distinguished points p, q ∈ Z R CP1 , there are two families of circles called “Apollonian circles”. One of them is the family of the circles passing through p, q, which corresponds to the space-like geodesic C p,q . The other family gives a foliation on CP1 \{ p, q}, which corresponds to a time-like geodesic and the foliation coincides with the one given in 1 of Proposition 5.6. The Case 2 of Proposition 5.6 corresponds to the degenerate case. 6. Perturbation of Holomorphic Disks We now investigate the deformation of holomorphic disks. For a complex manifold A and its submanifold B, we use the term holomorphic disk on (A, B) for a continuous map (D, ∂D) → (A, B) which is holomorphic on the interior of D = {z ∈ C | |z| ≤ 1}. As in the previous section, we put Z = CP1 × CP1 and Z R = {(η, η¯ −1 ) | η ∈ CP1 }. We have the family of holomorphic disks {D(λ,t) } defined from the double fibration (5.3), and we call each D(λ,t) a standard disk. In this section, we treat a small perturbation N of Z R , and prove that there is a natural (S 2 × R)-family of holomorphic disks on (Z , N ) each of which is close to some standard disk. From the general theory by LeBrun [8], one can show that there exists a real three-parameter family of holomorphic disks on
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(Z , N ) near each standard disk. We, however, use the method given in [9] so that we can consider the holomorphic disks in more detail. First of all, we recall the C k -topology of the space of deformations of Z R in Z . A small perturbation N of Z R can be written −1 N = η, φ(η) η ∈ CP1 using an automorphism φ : CP1 → CP1 which is sufficiently close to the identity map. Let {Ai } be finitely many compact subsets and {Bi } be open subsets on CP1 with complex coordinates ηi , which satisfy Ai ⊂ Bi , φ(Ai ) ⊂ Bi and ∪i Ai = CP1 . Then φ is identified with a combination of functions (h i )i , where h i ∈ C k (Ai , C) is defined by φ(ηi ) = ηi + h i (ηi ) for each i. The C k -topology of the set of deformations of Z R in Z is defined by the norm φC k = sup h i C k (Ai ) , i
where h i C k (Ai ) is the supremum on Ai of absolute values of all partial derivatives of h i for which the order is less than or equal to k. In particular, let A ∈ CP1 be a compact subset contained in a coordinated open subset of CP1 , which we denote B, and suppose φ(A) ⊂ B, then hC k (A) is sufficiently small if φ is sufficiently close to the identity where φ(η) = η + h(η). Lemma 6.1. Fix a standard holomorphic disk D = D(λ,t) . If N ⊂ Z is the image of any embedding CP1 → Z which is sufficiently close to the standard one in the C k+l -topology with k, l ≥ 1, then there is a real three-parameter C l -family of holomorphic disks on (Z , N ) each of which is L 2k close to D. Proof. Since there is a transitive action of PSL(2, C) on the standard disks, we can assume (λ, t) = (0, 0), that is, D = {(z, z) ∈ Z | z ∈ D}, where D = {z ∈ C | |z| ≤ 1}. If we put A = {η ∈ C | 21 ≤ |η| ≤ 2}, then N can be written −1 η , η + h(η) ∈ Zη∈ A near ∂ D using a function h ∈ C k+l (A) for which the C k+l -norm is sufficiently small. Then a small perturbation of ∂ D is given as the image of −1 ¯ S 1 → N : θ → ei(θ+u(θ)) , e−i(θ+u(θ)) + h¯ ei(θ+u(θ)) , ¯ ¯ ) for u(θ ) and h(η) where u is a C-valued function on S 1 = R/2π Z. Here we write u(θ 2 2 1 k+l 1 for h(η). Then we define the maps Fi : L k (S , C) × C (A, C) → L k (S , C) by [F1 (u, h)](θ ) = ei(θ+u(θ)) −1 ¯ and [F2 (u, h)](θ ) = e−i(θ+u(θ)) + h¯ ei(θ+u(θ)) .
(6.1)
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For a given h, we want to choose u ∈ L 2k (S 1 , C) so that [Fi (u, h)](θ ) extends holomorphically to {|z| < 0} for z = eiθ . Taking the derivation Fi , we obtain ˙ [F1∗(0,0) (u, ˙ h)](θ ) = ieiθ u(θ ˙ ) ¯˙ iθ ). ˙ ¯˙ ) − e2iθ h(e ˙ h)](θ ) = ieiθ u(θ and [F2∗(0,0) (u,
(6.2)
Now, we introduce some bounded operators (see [9]). Set 2 ilθ 2 L ↓ = al e al ∈ C, |al | < ∞ l<0 l<0 and L 2k ↓ = al eilθ al ∈ C, l 2k |al |2 < ∞ = L 2k (S 1 , C) ∩ L 2↓, l<0
l<0
and define : L 2k (S 1 , C) → L 2k ↓ by ∞ ilθ al eilθ = al e . l=−∞
l<0
Similarly let us define p : L 2k (S 1 , C) → C by p
∞
al eilθ = a0 .
l=−∞
Then, for k, l ≥ 1, we define a C l -map F : L 2k (S 1 , C) × C k+l (A, C) −→ L 2k ↓ ×L 2k ↓ ×C k+l (A, C) × C × C × C given by F = ( ◦ F1 ) × ( ◦ F2 ) × L × (p ◦ F1 ) × (p ◦ F2 ) × x, where L : L 2k (S 1 , C) × C k+l (A, C) −→ C k+l (A, C) is the factor projection, and x : L 2k (S 1 , C) × C k+l (A, C) −→ C is given by x(u, h) =
1 2π
2π
u(θ )dθ,
0
in other words, x(u, h) = p(u). The map F is C l since , L, p and x are all bounded linear operators, and its derivative is given by F∗ = ( ◦ F1∗ ) × ( ◦ F2∗ ) × L × (p ◦ F1∗ ) × (p ◦ F2∗ ) × x.
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In particular, if we write u(θ ˙ )= ⎡ ⎢ i ⎢ ⎢ u˙ ⎢ F∗(0,0) ˙ = ⎢ h ⎢ ⎢ ⎣
n
u n einθ , then we obtain
⎤ inθ n<0 iu n−1 e ¯˙ iθ ) ⎥ inθ − e2iθ h(e ⎥ n<0 u¯ 1−n e h˙ iu −1 ¯˙ iθ ) i u¯ 1 − p e2iθ h(e u0
⎥ ⎥ ⎥. ⎥ ⎥ ⎦
Since F∗(0,0) has a bounded inverse, the Banach-space inverse function theorem tells us that there is an open neighborhood U of (0, 0) ∈ L 2k (S 1 , C) × C k+l (A) and open neighborhood V of 0 ∈ L 2k ↓ ×L 2k ↓ ×C k+l (A, C) × C × C × C such that F|U : U → V is a C l -diffeomorphism. Hence, for a given h, we obtain a complex three-parameter C l -family of holomorphic disks defined from (u, h) = F−1 (0, 0, h, α1 , α2 , β), where α1 , α2 , β are small complex numbers. It contains, however, real three-dimensional ambiguity which comes from the disk automorphism. To kill this ambiguity, it is enough to use the inverse of (0, 0, h, α, −α, iβ) ∈ L 2k ↓ ×L 2k ↓ ×C k+l (A, C) × C × C × C,
(6.3)
in F for (α, β) ∈ C × R which is sufficiently close to (0, 0). Now the statement follows since hC k+l (A) is sufficiently small if N is sufficiently close to Z R .
Remark 6.2. 1. Let D be any holomorphic disk on (Z , N ) constructed as in the above lemma. Then D intersects with N only on the boundary ∂D. Actually, let D → Z : −1 z → (ϕ1 (z), ϕ2 (z)) be the map corresponding to D and denote N = {(η , φ(η) ) | −1 η ∈ CP1 }. Notice that η → φ(η) maps ϕ1 (∂D) to ϕ2 (∂D) and maps the interior of ϕ1 (D) to the outside of ϕ2 (D). Suppose that there is an interior point z ∈ D −1 such that ϕ2 (z) = φ(ϕ1 (z)) . Then ϕ1 (z) is contained in the interior of ϕ1 (D), and −1 φ(ϕ1 (z)) is outside of ϕ2 (D). This is a contradiction. 2. We can take V so that V = V1 × V2 × W × V1 × V2 × V3 with W = h ∈ C k+l (A, C) hC k+l (A) < ε0 ,
(6.4)
where Vi ⊂ L 2k ↓ and Vi ⊂ C are small open sets and ε0 > 0 is a constant. This notation is used in the following arguments. Next we want to prove that, if N is sufficiently close to Z R , then the method of Lemma 6.1 works for all standard disks at once. Then we need a uniform estimate of the deformation N of Z R among all standard disks. In LeBrun-Mason’s case [9,10], the parameter spaces of holomorphic disks are compact and homogeneous, so the uniform estimate is automatically deduced from the local estimate. In our case, however, the parameter space is a non-compact space S 2 × R, so we need more detailed arguments. For this purpose, it is enough to show that the deformations of the disks are “tame”, as in the following lemma, on the neighborhood of the boundary of the parameter space.
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Lemma 6.3. Let {D(λ,t) } be the standard disks. Suppose N ⊂ Z is sufficiently close to Z R in the C k+l -topology. Then a three-parameter family of holomorphic disks on (Z , N ) near D(λ,t) always exists for each (λ, t) ∈ CP1 × R with t $ 0. Proof. It is enough to consider the case λ = 0. We fix a small constant c > 0 and let Bc = {z ∈ C | |z| < c}. Notice that the compact subset Bc × CP1 ⊂ Z contains all holomorphic disks of the form D(0,t) if et > 2c−1 . We can write −1 (6.5) N ∩ (Bc × CP1 ) = η, (η + h(η)) η ∈ Bc using h ∈ C k+l (Bc , C). We claim that if hC k+l (Bc ) <
ε√0 , 4 2
then a three-parameter
family of holomorphic disks on (Z , N ) near D(0,t) exists for all et > 2c−1 . Here ε0 is the constant defined in (6.4). Now we show that it is enough to prove the case when h(0) = 0 and hC k+l (Bc ) < ε√0 . In the general case, if we change the coordinate (η1 , η¯ 2−1 ) ∈ Z to (ξ1 , ξ¯2−1 ) by the 2 2 relation ξ1 = η1 , ξ2 = η2 + h(0), then we can write N ∩ (Bc × CP1 ) =
−1 ξ, (ξ + g(ξ )) ξ ∈ Bc
using g(ξ ) = h(ξ ) − h(0). Here we obtain gC k+l (Bc ) <
ε√0 , 2 2
because
ε0 sup |g(ξ )| < sup |h(ξ )| + |h(0)| < √ 2 2 ξ ∈Bc ξ ∈Bc ε0 and sup |Dg(ξ )| = sup |Dh(ξ )| < √ , 4 2 ξ ∈Bc ξ ∈Bc where D is any partial derivative of degree less than or equal to k + l. Hence, if we replace g with h, we can assume h(0) = 0 and hC k+l (Bc ) < ε√0 . 2 2
From now on we write r for et . A small perturbation of ∂ D(0,t) is given as the image of −1 ¯ , + h¯ r −1 ei(θ+u(θ)) S 1 → N : θ → r −1 ei(θ+u(θ)) , r −1 e−i(θ+u(θ)) where u is a C-valued function on S 1 . r −1 r Let A = {z ∈ C | 2 ≤ |z| ≤ 2r −1 } and A = A1 , then Ar is a compact subset of Bc if r > 2c−1 . We define the maps Fri : L 2k (S 1 , C) × C k+l (Ar , C) → L 2k (S 1 , C) by [Fr1 (u, h)](θ ) = r −1 ei(θ+u(θ)) −1 ¯ and [Fr2 (u, h)](θ ) = r −1 e−i(θ+u(θ)) + h¯ r −1 ei(θ+u(θ)) . Putting h r (z) = r h(r −1 z), we obtain [Fr1 (u, h)](θ ) = r −1 [F1 (u, h r )](θ ) and [Fr2 (u, h)](θ ) = r [F2 (u, h r )](θ ), (6.6)
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where Fi is the map given by (6.1). Notice that the map ρ r : h → h r gives an isomorphism of Banach spaces C k+l (Ar , C) → C k+l (A, C). In a similar way to how we defined F in Lemma 6.1, we define Fr : L 2k (S 1 , C) × C k+l (Ar , C) −→ L 2k ↓ ×L 2k ↓ ×C k+l (Ar , C) × C × C × C given by Fr = ( ◦ Fr1 ) × ( ◦ Fr2 ) × L × (p ◦ Fr1 ) × (p ◦ Fr2 ) × x, where L is the projection. Then we can relate Fr to F in the following way. Let m(r ) be multiplication of r on L 2k ↓ or C, then we obtain the following commutative diagram L 2k (S 1 , C)×C k+l (Ar , C)
Fr
id×ρ r
L 2k (S 1 , C) × C k+l (A, C)
/ L 2↓ ×L 2↓ ×C k+l (Ar , C) × C×C×C k k
(6.7)
r
F
/ L 2↓ ×L 2↓ ×C k+l (A, C) × C × C × C k k
where r = m(r ) × m(r −1 ) × ρ r × m(r ) × m(r −1 ) × id. Notice that the vertical arrows in the above diagram are isomorphisms, and that the restriction F|U : U → V gives a C l -diffeomorphism from the proof of Lemma 6.1. Hence the restriction Fr : (id × ρ r )−1 (U) −→ ( r )−1 (V) is a C l -diffeomorphism. If we take V to be the product as in (6.4), then ( r )−1 (V) = r −1 V1 × r V2 × (ρ r )−1 (W) × r −1 V1 × r V2 × V3 . We want to show that h| Ar ∈ (ρ r )−1 (W), or equivalently h r C k+l (A) < ε0 , for all r > 2c−1 . Let x, y be the real coordinate such that η = x +i y, and let D = ∂ m /∂ x j ∂ y m− j be a derivation of degree m ≤ l + k, then we obtain Dh r (η) = r 1−m Dh(r −1 η). Hence ε0 sup |Dh r (η)| ≤ r 1−m sup |Dh(r −1 η)| ≤ r 1−m sup |Dh(ζ )| < √ r 1−m < ε0 , 2 2 η∈A η∈A ζ ∈Ar if m ≥ 1. For m = 0, notice that 1 1 1 dh ∂h ∂h (tη) dt ≤ (tη) |x|dt + (tη) |y|dt |h(η)| ≤ dt ∂x ∂y 0 0 0 ε0 ε0 < √ (|x| + |y|) < |η|, 2 2 2 hence we obtain sup |h r (η)| = r sup |h(r −1 η)| = r sup |Dh(ζ )| <
η∈A
η∈A
ζ ∈Ar
r ε0 sup |ζ | = ε0 . 2 ζ ∈Ar
In this way, we have obtained h r (η)C k+l (A) < ε0 for all r > 2c−1 .
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Remark 6.4. Lemma 6.3 also holds for t & 0. Exchange the role of factors of Z = CP1 × CP1 and replace t with −t to prove this case. From Lemmas 6.1 and 6.3, we obtain the following statement. Proposition 6.5. If N ⊂ Z is the image of any embedding CP1 → Z which is sufficiently close to the standard one in the C k+l -topology with k, l ≥ 1, then there is a C l family of holomorphic disks on (Z , N ), each of which is L 2k close to some standard disk on (Z , Z R ). We will strengthen this statement in Proposition 7.3. 7. The Double Fibration In this section, we investigate some properties for the family of holomorphic disks constructed in Sect. 6. We continue to use the notation F, Fi , U, V and so on. For each h ∈ C k+l (A, C), we define C l -maps h , Fih : U → L 2k (S 1 , C) so that h (α, β), h = F−1 (0, 0, h, α, −α, iβ), F1h (α, β)(eiθ ) = F1 h (α, β), h (θ ) = exp i θ + h (α, β)(θ ) , and F2h (α, β)(eiθ ) = F2 h (α, β), h (θ ), where U ⊂ C × R is a small open neighborhood of (0, 0) depending on h. By definition, the functions Fih (α, β)(z) extend to holomorphic functions on D = {|z| ≤ 1}, and satisfy F1h (α, β)(0) = α and F2h (α, β)(0) = −α. If we expand h (α, β)k eikθ , (7.1) h (α, β)(θ ) = k
then we obtain h (α, β)0 = iβ by definition. Notice that we can also define the derivatives ∗h and Fih∗ which satisfy h ˙ 0 = F−1 ˙ ˙ β), ˙ −α, ˙ i β), ∗ (α, ∗ (0, 0, 0, α, ˙ iθ ) = F1 ∗ h (α, ˙ 0 (θ ) = i F1h (eiθ ) ∗h (α, ˙ ), F1h∗ (α, ˙ β)(e ˙ β), ˙ β)(θ h iθ h ˙ (α, ˙ ˙ 0 (θ ), F2 ∗ (α, ˙ β)(e ) = F2 ∗ ˙ β), h h ˙ ˙ ˙ 0 = i β. ˙ F1 ∗ (α, ˙ β)(0) = α, ˙ F2 ∗ (α, ˙ β)(0) = −α˙ and ∗h (α, ˙ β) Let N ⊂ Z be the image of any embedding CP1 → Z which satisfies ProposiN tion 6.5. Let us denote by B(α,β) the holomorphic disk on (Z , N ) which corresponds to the element (0, 0, h, α, −α, iβ) ∈ V in the notation of the proof of Lemma 6.1. Then h N (7.2) = F1 (α, β)(z), F2h (α, β)(z) ∈ Z z ∈ D , B(α,β) N and {B(α,β) }(α,β)∈U gives a three-parameter family of holomorphic disks, each of which N 2 is L k -close to the standard disk D(0,0) . Notice that B(α,β) passes through (α, −α) when N z = 0, hence, for fixed α, {B(α,β) }β defines a one-parameter family of holomorphic disks which pass through (α, −α). In the standard case, the following statement holds.
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R Proposition 7.1. B(α,β) coincides with the standard disk D(α,β) .
Proof. Since the disk Z
R B(α,β) =
F10 (α, β)(z), F20 (α, β)(z) ∈ Z | z ∈ D
coincides with one of the standard disks near D(0,0) , there is a unique element (λ, t) ∈ CP1 × R near (0, 0) such that eiθ + et λ F10 (α, β)(eiθ ) = exp i θ + 0 (α, β)(θ ) = . −λ¯ eiθ + et
(7.3)
Then we obtain α = F 0 (α, β)(0) = λ. On the other hand, taking the derivative of (7.3), we obtain ˙ )= i 0∗ (α, ˙ β)(θ
(λ˙ + λt˙)e−(iθ−t) eiθ−t λ¯˙ − t˙ + . 1 + λe−(iθ−t) 1 − λ¯ eiθ−t
If we expand the right hand side and compare the constant terms, then we find ˙ 0 = i t˙. i β˙ = 0∗ (α, ˙ β) On the other hand, it is easy to see that t = β when α = 0. Hence (λ, t) = (α, β) for each (α, β) ∈ U .
Let M be the parameter space of the family of holomorphic disks on (Z , N ) constructed in Proposition 6.5. Then M has the natural structure of a real 3-manifold and we can take a coordinate system on M in the following way. For each (λ, t) ∈ CP1 × R, choose an element T = T (λ, t) ∈ PSL(2, C) such that T∗ (D(λ,t) ) = D(0,0) , where of (0, 0) {D(λ,t) } are the standard disks. Let U T ⊂ C × R be an open neighborhood T (N )
T (N )
∗ ∗ such that B(α,β) is defined for all (α, β) ∈ U T . Then T∗−1 B(α,β)
(α,β)∈U T
gives the
family of holomorphic disks on (Z , N ) each of which is close to D(λ,t) , and {U T (λ,t) } gives a coordinate system on M. Using the above coordinates, we prove the following lemma. Lemma 7.2. Suppose N ⊂ Z is sufficiently close to Z R so that Proposition 6.5 holds, and consider the constructed family of holomorphic disks on (Z , N ). Then, for each q = (α, −α) ∈ Z , there is an R-family of holomorphic disks each of which passes through q. Moreover there is a natural compactification of this family and the boundary points ±∞ correspond to marked CP1 . Proof. We can assume α = 0. Take any t so that |t| is sufficiently small, and consider the standard disk D(0,t) . If we define T ∈ PSL(2, C) by # t $ e2 T = , t e− 2 T∗ (N ) gives then T∗ (η1 , η2 ) = (et η1 , e−t η2 ) and T∗ (D(0,t) ) = D(0,0) . Now T∗−1 B(0,β ) β ∈V
a one-parameter family of holomorphic disks on (Z , N ) each of which is close to D(0,t) and pass through (0, 0). Here V is the set {β ∈ R | (0, β ) ∈ U T }.
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689 T (N )
∗ Since |t| is small, there is an open set V ⊂ V such that T∗−1 B(0,β ) is sufficiently close to D(0,0) for all β ∈ V . Hence, for each β ∈ V , there is a unique (α, β) such that
T (N )
N ∗ T∗−1 B(0,β ) = B(α,β) .
(7.4)
Now N and T∗ (N ) can be written locally as −1 −1 T η∈A , η ∈ A and T N : η, η + h(η) η + h (N ) : η, (η) ∗ (7.5) using a C k+l -function h which is defined on a neighborhood of A = {z ∈ C | 21 ≤ |z| ≤ 2}. Here we write h T to mean T hT −1 . Then (7.4) is equivalent to e−t F1h (0, β )(z) = F1h (α, β)(z) T
on z ∈ D.
Evaluating for z = 0, we obtain α = 0. Moreover, this is also equivalent to it + h (0, β )(θ ) = h (α, β)(θ ) T
on θ ∈ S 1 .
Comparing the constant terms for eiθ , we obtain β = β + t. Hence (7.4) is equivalent N to (α, β) = (0, β + t). So the one-parameter family {B(0,β) }(0,β)∈U extends to β ∈ R (0, β) ∈ U or (0, β − t) ∈ U T T (N )
∗ N by putting B(0,β) = T∗−1 B(0,β−t) . In this way, we can define the one-parameter family N {B(0,β) } for all β ∈ R. The statement of the compactification is obtained from Lemma 6.3 and its proof. Indeed, in the notation of (6.5), if we take the limit t → ∞, the holomorphic disk −1 parametrized by t degenerates to {0} × CP1 marked at (0, h(0) ). As we explained in 1 Remark 6.4, we also obtain another marked CP by taking the limit t → −∞.
Now the following statement is easily proved. Proposition 7.3. If N ⊂ Z is the image of any embedding CP1 → Z which is sufficiently close to the standard one in the C k+l -topology with k, l ≥ 1, then there is a C l family of holomorphic disks on (Z , N ) parametrized by S 2 × R which satisfies the following properties: • each disk is L 2k -close to some standard disk, • there is a natural compactification of the family such that the compactified family is parameterized by S 2 × I , and each boundary point on S 2 × I corresponds to a marked CP1 embedded in (Z , N ), where I = [−∞, ∞] is the compactification of R. Proof. Let Q = {(λ, −λ) ∈ Z | λ ∈ CP1 }. For each q ∈ Q, there is an R-family of holomorphic disks constructed in Lemma 7.2. Since this family varies continuously, we obtain the family of holomorphic disks parametrized by Q ×R S 2 ×R. The statement for the compactification is obvious from Lemma 7.2.
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For each (λ, t) ∈ CP1 × R, we define T (N )
∗ D(λ,t) = T∗−1 B(0,0) ,
where T = T (λ, t) ∈ PSL(2, C) is an element which satisfies T∗ (D(λ,t) ) = D(0,0) . Then we obtain the continuous map j : CP1 × R → M : (λ, t) → D(λ,t) . Moreover, we can prove that j is an isomorphism in the following way. For each constructed holomorphic T∗ (N ) disk D on (Z , N ), we can choose (λ, t) and T = T (λ, t) so that D = T∗−1 B(0,β) . Here λ is uniquely defined so that the center of D is (λ, −λ). Then D = D(0,β+t) from Lemma 7.2 and its proof, so j is surjective. The injectivity and the continuity of j −1 is also deduced from the above procedure of choosing (λ, t), hence j is isomorphism. Let us construct the double fibration. Let U ⊂ CP1 × R be a sufficiently small neighborhood of (0, 0). For each (λ, t) ∈ U , we define T = T (λ, t) ∈ PSL(2, C) by 1 1 −et λ , T = % t t ¯ e− 2 1 + |λ|2 λ e then we obtain T∗ (D(λ,t) ) = D(0,0) . Introducing C k+l -functions h and h T similar to those in (7.5), we define a map f : U × D → Z by T T f(λ, t; z) = T∗−1 F1h (z), F2h (z) . Then f is C l for (λ, t) and C k−1 for z, and we obtain D(λ,t) = {f(λ, t; z) ∈ Z | z ∈ D}. Constructing a similar map for each neighborhood of CP1 × R, and patching them, we obtain the double fibration (X+ , XRK) o KKK o o o o KKfK o o KKK o o woo % (Z , N ) M CP1 × R
(7.6)
where is a disk bundle. By construction, X+ is the same disk bundle as the standard case. In particular, we obtain c1 (XR ) = 2 along each fiber of and that is C ∞ . Lemma 7.4. Let N ⊂ Z be the image of any embedding CP1 → Z which is sufficiently close to the standard one in the C k+l -topology with k, l ≥ 1, and consider the double fibration (7.6). Then f ∗ (v) = 0 for each non-zero vector v ∈ T XR such that ∗ (v) = 0. Proof. For each (u, h) ∈ L 2k (S 1 , C) × C k+l (A, C), we have d d i(θ+u(θ)) [F1 (u, h)](θ ) = e = ei(θ+u(θ)) (i + iu (θ )), dθ dθ so this does not vanish if u L 2 is sufficiently small. Hence, by shrinking U and V 1 smaller if needed, the statement holds for v ∈ ker ∗ over U ⊂ M, where U is the neighborhood introduced above. Now, recall the diagram (6.7) in the proof of Lemma 6.3. Notice that the L 2k (S 1 , C) component does not change by the vertical arrow, so we can estimate u ∈ L 2k (S 1 , C) d uniformly so that dθ [Fr1 (u, h)](θ ) does not vanish for all r . Hence the statement holds for all v ∈ ker ∗ .
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691
By Lemma 7.4, we can define the lift ˜f of f by ˜f : XR → S(T N ) : u → [f∗ (vu )]. Here v is a nowhere vanishing vertical vector field, that is, ∗ (v) = 0, for which the orientation matches the complex orientation of the fiber of : X+ → M. The next proposition is the perturbed version of Proposition 5.1. Proposition 7.5. Let N ⊂ Z be the image of any embedding CP1 → Z which is sufficiently close to the standard one in the C k+l -topology with l ≥ 1, k ≥ 2. Consider the double fibration (7.6), let St = CP1 × {t} ⊂ M, and let ft and ˜ft be the restriction of f and ˜f on −1 (St ) respectively. Then, for each t ∈ R, 1. ft : (X+ \XR )| St → Z \N is diffeomorphic, 2. ˜ft : XR | St → S(T N ) is diffeomorphic, 3. ft : XR | St → N is an S 1 -fibration such that each fiber is transverse to the vertical distribution of : XR → M. In particular, {D(λ,t) }λ∈CP1 gives a foliation on Z \N for each t ∈ R. Remark 7.6. From 2 above, it follows that: for each t ∈ R, p ∈ N and non zero v ∈ T p N , there is a unique x ∈ St such that p ∈ ∂Dx and v Dx . Proof of Proposition 7.5. Since St is compact and f is C 1 -close to the standard case, we can assume the derivation of ft to be non-zero everywhere by shrinking W smaller if required. Here W is the open set defined in Remark 6.2. Notice that we can define W so that this property holds for all t ∈ R at once by Lemma 6.3 and its proof. Thus ft gives a proper local diffeomorphism on (X+ \XR )| St , and this is actually a diffeomorphism since ft is close to the standard case. By a similar argument for the lift ˜f : XR | St → S(T N ), we obtain property 2. If there are x ∈ St and p ∈ N such that −1 (x) and f−1 t ( p) are not transversal at u ∈ XR , then (ft )∗ (vu ) = 0. This contradicts Lemma 7.4, hence 3 holds.
From Proposition 7.3, we obtain the natural compactification of and f which gives the following double fibration: (Xˆ+ , XˆRJ) JJJ fˆ w w ˆ ww JJJ w JJJ w w % {ww (Z , N ) Mˆ
(7.7)
which is studied in Sect. 8. In the last part of this section, we prove the following technical lemma which enables us to prove the non-degeneracy of the induced conformal structure. Let us denote C p = ◦ f−1 ( p) = {x ∈ M | p ∈ Dx } for each p ∈ Z \N , then C p is an embedded R in M from Proposition 7.5. Notice that C p is a closed subset in M since it connects two ˆ boundaries of M. Lemma 7.7. Let x ∈ M, then there are two points p1 , p2 ∈ Dx \∂Dx such that C p1 and C p2 intersect transversally at x. Proof. We can assume x = (0, 0), and we use the local coordinate (α, β) ∈ U around ˙ ∈ C×R ∼ ˙ β) x. Each tangent vector on T(0,0) M is given by (α, = T(0,0) (C × R). Notice ˙ that the tangent vector (α, ˙ β) ∈ T(0,0) M induces the vector field ˙ ˙ ˙ β)(z), F2 ∗ (α, ˙ β)(z) F1 ∗ (α,
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F. Nakata
along D(0,0) . Here we identified C×C with the tangent vectors on each point of C×C ⊂ ˙ and F2 ∗ (α, ˙ are holomorphic functions on D and their zeros coincide ˙ β) ˙ β) Z . F1 ∗ (α, since ˙ iθ ) = ieiθ ∗ (α, ˙ ) F1 ∗ (α, ˙ β)(e ˙ β)(θ ˙ iθ ) = ieiθ ∗ (α, ˙ ) and F2 ∗ (α, ˙ β)(e ˙ β)(θ ˙ by (6.2). If β˙ = 0, then F1 ∗ (0, β)(z) is not a zero function since F∗ is bijective, and ˙ ˙ ∈ T(0,0) M is tangent to C(0,0) F1 ∗ (0, β)(0) = 0 by definition. This means that (0, β) since the one-parameter family of holomorphic disks fixing (0, 0) ∈ D ⊂ Z is unique ˙ ˙ and this family corresponds to the vector field F1 ∗ (0, β)(z), F2 ∗ (0, β)(z) along D. Now consider the vector field ˙ ˙ F1 ∗ (t α, ˙ β)(z), F2 ∗ (t α, ˙ β)(z) ˙ is a ˙ β) for t ∈ [0, 1] and non-zero α˙ ∈ C with sufficiently small |α|. ˙ Then F1 ∗ (t α, non-zero holomorphic function on D for all t, and its zeros vary continuously depending ˙ 1 ) = 0 but z 1 cannot be 0 on t. Hence there exists a z 1 ∈ D near 0 such thatF1 ∗ (α, ˙ β)(z ˙ because F1 ∗ (α, ˙ β)(0) = α˙ = 0. If we put p2 = F1 (0, 0)(z 1 ), F2 (0, 0)(z 1 ) ∈ D(0,0) , ˙ ∈ T(0,0) M is tangent to C p2 . Hence p1 = (0, 0) and p2 satisfies then we find that (α, ˙ β) the statement.
8. Construction of Einstein-Weyl spaces In this section, we construct an Einstein-Weyl structure on the parameter space of the family of holomorphic disks on (Z , N ) constructed in the previous sections. The following proposition is critical. Proposition 8.1. Let M be a smooth connected 3-manifold and let : X → M be a smooth CP1 -bundle. Let ρ : X → X be an involution which commutes with , and is fiber-wise anti-holomorphic. Suppose ρ has a fixed-point set Xρ which is an S 1 -bundle over M, and which disconnects X into two closed 2-disk bundles X± with common boundary Xρ . Let D ⊂ TC X be a distribution of complex 3-planes which satisfies the following properties: • • • • •
ρ∗ D = D, the restriction of D to X+ is C k , k ≥ 1 and involutive, D + D = TC X on X \Xρ , D ∩ ker ∗ is the (0, 1) tangent space of the CP1 fibers of , the restriction of D to a fiber of X has c1 = −4 with respect to the complex orientation, and • the map X → P(T M) : z → ∗ (D ∩ D)z is not constant along each fiber of .
Then M admits a unique C k−1 indefinite Einstein-Weyl structure ([g], ∇) such that the null-surfaces are the projections via of the integral manifolds of real 2-plane distribution D ∩ T Xρ on Xρ . Proof. Let V 0,1 be the (0, 1) tangent space of the fibers, then = D/V 0,1 is a rank two vector bundle on X . We can define a continuous map ψ : X → Gr2 (TC X ) by
Constructing Einstein-Weyl Spaces via LeBrun-Mason Twistor Correspondence
693
z → ∗ (D|z ) which makes the following diagrams commute: X? ?? ?? ?? ?
ψ
X
/ Gr2 (TC X ) uu uu u u uu zu u
X
ψ
X
/ Gr2 (TC X )
(8.1)
c
ψ
/ Gr2 (TC X )
Using the involutiveness of D, we can prove that ψ is fiber-wise holomorphic by a similar argument to that in [9,10]. Let P : Gr2 (TC X ) −→ P(∧2 TC X ) ∼ = P(TC∗ X ) be the natural isomorphism. Then we obtain the fiber-wise holomorphic map ψˆ = P ◦ ψ : X → P(TC∗ X ). By definition, we obtain ψˆ ∗ O(−1) = ∧2 . On the other hand, since c1 (V 0,1 ) = −2 and c1 (D) = −4 on any fiber of , we have c1 (∧2 ) = c1 () = −2. Hence ψˆ is fiber-wise degree 2. ˆ either a non-degenerate conic or a For each fiber, there are only two possibilities for ψ; ramified double cover of a projective line CP1 ⊂ CP2 . The latter is, however, removable. Indeed, any line CP1 ⊂ CP2 corresponds to the planes in C3 containing a fixed line. Notice that, for each z ∈ X \XR , ∗ (D ∩ D)z = ∗ (D|z ) ∩ ∗ (D|z ) = ∗ (D|z ) ∩ ∗ (D|ρ(z) ) is independent on z if the image of −1 (x) under ψˆ is a line. This contradicts the hypothesis. Now we define a conformal structure [g]. Let U ⊂ M be an open set and let U × ∼ CP1 → X |U be a trivialization on U . Let ζ be an inhomogeneous coordinate on CP1 such that ρ(x, ζ ) = (x, ζ¯ ). Then we can choose a C k frame field {e1 , e2 , e3 } on T M|U so that & ' ˆ ψ(x, ζ ) = (1 + ζ 2 )e1 + (1 − ζ 2 )e2 + 2ζ e3 , (8.2) where {ei } is the dual frame. Define an indefinite metric g on U so that g(ei , e j ) is given by (3.11). Here, the frame {ei } is uniquely defined by (8.2) up to scalar multiplication, and the coordinate change of ζ causes an S O(1, 2) action on the frame {ei }. Hence the ˆ So we can obtain an indefinite conformal conformal structure [g] is well-defined by ψ. structure [g] on M. Next we prove that a unique torsion-free connection ∇ on T M is induced, and ([g], ∇) gives an Einstein-Weyl structure on M. We also prove that D agrees with the distribution E defined in Sect. 3. We fix an indefinite metric g ∈ [g], and take a local frame field {e1 , e2 , e3 } of T M on an open set U ⊂ M as above. It is enough to construct ∇ on U . Notice that (8.2) ∼ gives a natural identification X → Z = P(N (TC∗ M)) on U . If we define the maps mi : U × C → T M for i = 1, 2 by m1 = −e1 + e2 + ζ e3 and m2 = ζ (e1 + e2 ) − e3 ,
(8.3)
then we obtain ∗ (D|(x,ζ ) ) = Span m1 , m2 (see (3.16)). ˜ i be the vector fields on U × C ⊂ U × CP1 X |U such that m ˜ i ∈ D and the Let m ˜ i are written in the following form: m ˜ 1 = m1 + α m
∂ ∂ζ
˜ 2 = m2 + β and m
∂ , ∂ζ
(8.4)
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where α and β are functions on X . Then α and β are uniquely defined and C k . Moreover, α and β are holomorphic for ζ , since ∂ ∂α ∂ ˜1 = ≡0 mod D, ,m ¯ ∂ζ ∂ ζ¯ ∂ζ and so on. ∂ By a similar argument for ζ −1 mi on {(x, ζ ) ∈ U ×CP1 | ζ = 0}, we find that ζ −1 α ∂ζ
∂ and ζ −1 β ∂ζ extends to holomorphic vector fields on {ζ = 0}, hence we can write
∂ , ∂ζ ∂ ˜ 2 = m2 + (β0 + β1 ζ + β2 ζ 2 + β3 ζ 3 ) , and m ∂ζ ˜ 1 = m1 + (α0 + α1 ζ + α2 ζ 2 + α3 ζ 3 ) m
(8.5)
where αi and βi are C k functions on U . Recall that the compatibility condition ∇g = a ⊗g holds if and only if the connection form ω of ∇ is written ⎛ ⎞ φ η21 η31 ⎜ ⎟ ω = (ωij ) = ⎝η21 (8.6) φ η32 ⎠ 1 2 η3 −η3 φ with respect to the frame {ei } (see (3.12)). For each vector v ∈ T U , the horizontal lift ˜ i (x,ζ ) is the v˜ with respect to the connection defined from (8.6) is given by (3.15). If m horizontal lift of mi (ζ )x , then ηij must be 2 1 1 η32 = η3,0 + f e1 , η31 = η3,0 + f e2 , η21 = η2,0 − f e3 ,
(8.7)
where f is an unknown function on U and α0 + α2 + β1 + β3 1 −α0 − α2 + β1 + β3 2 e + e + (−α3 − β0 )e3, 2 2 α0 − α2 + β1 − β3 1 −α0 + α2 + β1 − β3 2 e + e + (α3 − β0 )e3, (8.8) = 2 2 −α1 + α3 + β0 − β2 1 α1 + α3 − β0 − β2 2 e + e . = 2 2
2 = η3,0 1 η3,0 1 and η2,0
We claim that there is a unique pair ( f, φ) such that the connection (8.6) is torsionfree, that is, ω satisfies dei + ωij e j = 0. (8.9) First, we fix a connection for which the connection form is ⎛ 1 1 ⎞ 0 η2,0 η3,0 1 2 ⎠ 0 η3,0 ω0 = (ωij,0 ) = ⎝η2,0 . 1 2 η3,0 −η3,0 0
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Let λi be the fiber coordinate on TC∗ X with respect to {ei }. We consider the distribution π ∗ D on N = N (TC∗ M)\0 M , where π : N → Z X is the projection. We define 1-forms θ, θi,0 , τi j,0 on N (see (3.4)) by j θ= λi ei , θi,0 = dλi − λ j ωi,0 , τi j,0 = λi θ j,0 − λ j θi,0 . If we simply write τ = τ23,0 , then we have (see (3.18)) 2 1 1 τ = λ2 dλ3 − λ3 dλ2 − λ1 λ1 η3,0 . + λ2 η3,0 − λ3 η2,0 Similar to the proofs of Propositions 3.3 or 3.5, we obtain π ∗ D = {v ∈ T N | θ (v) = τi j,0 (θ ) = 0}. Hence the 1-forms {θ, τi j } are involutive. Since θi,0 ∧ ei ≡ 0 mod θ, τi j , we obtain dθ ≡ µ mod θ, τi j , where 1 1 1 2 + λ3 η3,0 ) ∧ e1 + (λ1 η2,0 − λ3 η3,0 ) ∧ e2 µ = (λ2 η2,0 1 2 +(λ1 η3,0 + λ2 η3,0 ) ∧ e3 + λi dei .
(8.10)
Then we can write µ = µ23 e2 ∧ e3 + µ31 e3 ∧ e1 + µ12 e1 ∧ e2 ,
(8.11)
where the µi j = µi jl λl are linear in λ. Notice that the µi jl are C k−1 functions because θ is C k . Since dθ ≡ 0 mod θ, τi j , there are 1-forms 1 and 2 such that µ=
1
∧τ +
2
∧ θ.
(8.12)
The 1-form 1 is, however, zero since µ does not contain dλi . Hence we obtain µ∧θ = 0, and this is equivalent to −µ231 = µ312 = µ123 , µ122 + µ313 = 0, µ233 + µ121 = 0 and µ311 + µ232 = 0.
(8.13)
Thus, if we put f = 21 µ123 and φ = µ313 e1 + µ121 e2 + µ232 e3 , then µ = −φ ∧ θ + f − λ1 e2 ∧ e3 + λ2 e3 ∧ e1 + λ3 e1 ∧ e2 . Here f and φ are C k−1 . Comparing the coefficients of λi with (8.10), we obtain 1 1 − f e3 ) ∧ e2 + (η3,0 + f e1 ) ∧ e3 = 0, de1 + φ ∧ e1 + (η2,0 1 2 de2 + (η2,0 − f e3 ) ∧ e1 + φ ∧ e2 + (η3,0 + f e1 ) ∧ e3 = 0, 1 2 and de3 + (η3,0 + f e1 ) ∧ e1 − (η3,0 + f e1 ) ∧ e2 + φ ∧ e3 = 0.
This is nothing but the torsion-free condition for the connection defined from f and φ above. Since ( f, φ) is uniquely defined, we have obtained the unique torsion-free C k−1 connection ∇. For this ∇, the distribution E on Z X agrees with D by construction. Hence ([g], ∇) is Einstein-Weyl from Proposition 3.9. The remaining condition is deduced from the fact that D ∩ T Xρ corresponds to DR .
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Remark 8.2. In the statement of Proposition 8.1, the last hypothesis • ∗ (D ∩ D)z is not constant along the fiber is not removable. Actually, ∗ (D ∩ D)z can be constant when the metric degenerates so that the light cone degenerates to a line, which occurs as a limit of an indefinite metric. Proposition 8.3. Let N be any embedding of CP1 into Z = CP1 × CP1 which is C 2k+5 close to the standard one. Let {Dx }x∈S 2 ×R be the constructed family of closed holomorphic disks on (Z , N ). Then a C k indefinite Einstein-Weyl structure ([g], ∇) is naturally induced on M = S 2 × R. Proof. We apply Proposition 7.3 by putting k + 3 instead of k and l = k + 2. Let
f
M ← X+ → Z be the constructed double fibration (the diagram (7.6)), then f is C k+2 in this case. Let X− be a copy of X+ and let X = X+ ∪ X − be the CP1 bundle over X which is obtained by identifying the boundaries ∂X+ and ∂X− where X − is a copy of X− with fiber-wise opposite complex structure. Let ρ : X → X be the involution which interchanges X+ and X− . 0,1 Z ) on X . Let f∗ : TC X → TC Z be the differential of f. We define D = f−1 + ∗ (T ∂ Then, along XR = ∂X+ , D is spanned by ∂ ζ¯ and the distribution of real planes tangent to the fibers of f : XR → N . So we can extend D to the whole of X so that D = ρ ∗ D on XR . Let us check the hypotheses in Proposition 8.1:
• ρ∗ D = D follows from the construction. • D is C k+1 on X+ \XR since f∗ is C k+1 , and D is involutive since T 0,1 Z is involutive. 0,1 Z ) + f−1 (T 1,0 Z ) = f−1 (T Z ) = T X on X \X since f is • D + D = f−1 + C C + R ∗ (T ∗ ∗ surjective. • For each fiber −1 (x) = X+ |x , the restriction fx : X+ |x → Z of f is a holomorphic 0,1 Z ) = V 0,1 . embedding. Hence D ∩ ker ∗ = (fx )−1 ∗ (T 0 • D is C -close to the D of the standard case, so c1 (D) = −4 on each fiber of . • For each x ∈ M, there are p, q ∈ Dx such that C p and Cq intersects transversally at −1 −1 (x), then we obtain x (Lemma 7.7). If we put z = f−1 x ( p) = f ( p) ∩ (Tx C p ) ⊗ C = ∗ (TC z f−1 ( p)) = ∗ (ker f∗ )z = ∗ (D ∩ D)z . Similarly (Tx Cq ) ⊗ C = ∗ (D ∩ D)z for z = f−1 x (q). Hence ∗ (D ∩ D) is not constant. Thus all the hypotheses in Proposition 8.1 are fulfilled, so we obtain the unique C k indefinite Einstein-Weyl structure on M.
Recall that we obtained a lift ˜f : XR → S(T N ) of f : XR → N in Sect. 7. Proposition 8.4. Identifying X with Z, 1. 2. 3. 4.
0,1 Z ) on X where f : T X → T Z , E = f−1 + ∗ C + C ∗ (T DR = E ∩ T XR = ker{f∗ : T XR → T N } on XR , L = ker{f∗ : T X+ → T Z } on X+ \XR , and L = ker{˜f∗ : T XR → S(T N )} on XR .
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Proof. 1 and 2 follow from Propositions 8.1 and 8.3 and their proofs. We also have 1,0 Z ), so L ⊗ C = E ∩ E = ker f : T X → T Z . Hence 3 follows. E = f−1 ∗ C + C ∗ (T ∼ Let us prove 4. Let U × CP1 → X |U be a trivialization on U such that ρ(x, ζ ) = 1 (x, ζ¯ ). Notice that X± |U = √ {(x, ζ ) ∈ U × CP | ± Im ζ ≥ 0}. Let us denote ζ = ξ + −1η using a real coordinate (ξ, η). We fix a point √ (x0 , ξ0 ) ∈ XR |U and let c(s) be a curve defined by Iε → −1 (x) : s → (x0 , ξ0 + −1s), where Iε = (−ε, ε) is a small interval. Now, we define a map : Iε × Iε → X : (s, t) → (s, t) so that (s, 0) = c(s) and ∗ ( ∂t∂ ) = l † , where l † is a ρ-invariant real vector field such that L = Span l † . ∂ Let ! be the image of , and let ν = ( ∂s ) which is a tangent vector field along ! † ∂ on ! ∩ XR . Indeed, we such that T ! = Span l , ν . Moreover, ν is proportional to ∂η have ρ ◦ (s, t) = (−s, t) by definition, so ρ∗ ν = −ν. Hence ν is “pure imaginary" ∂ using a real-valued function a on XR . Taking ε on XR , that is, we can write ν = a ∂η ∂ ∂ ) = ∂η . small, we can assume a is a positive function since ν(x0 ,ξ0 ) = c∗ ( ∂s † † Since {l , ν} is involutive, there are functions A, B on ! such that [l , ν] = Al † + Bν. Let ϕ be a positive function on ! such that l † ϕ = −B, then [l † , ϕν] = ϕ Al † . We define ∂ a positive function ψ on ! ∩ XR by ϕν = ψ ∂η . Now, f : X+ → Z = CP1 × CP1 is described as f(x, ζ ) = (F1 (x, ζ ), F2 (x, ζ )) in the neighborhood of (x0 , ξ0 ) using functions Fi which are holomorphic on ζ . Let p1 : Z → CP1 be the first projection. Then its restriction p1 : N → CP1 is diffeomor1 phism. Hence, identifying N with CP by p1 , f : XR → N is described by F1 . Since L = Span l † = ker f∗ on X+ \XR , we have l † Fi =0 on X+ . Then
l † (ϕν Fi ) = [l † , ϕν]Fi + ϕν(l † Fi ) = 0, Fi and so l † ψ ∂∂η = 0 on ! ∩ XR .
∂ Fi ∂ξ
√ Fi = − −1 ∂∂η for i = 1, 2. Thus we
∂ Fi =0 l† ψ ∂ξ
(8.14)
Since the Fi are holomorphic for ζ , we have have obtained
on ! ∩ XR for i = 1, 2. Since ˜f(x, ξ ) = ∂∂ξF1 (x, ξ ) by definition, and since ψ is a positive function, (8.14) means ˜f∗ (l † ) = 0. From 2 of Proposition 7.5, the fiber of ˜f is at
most one-dimensional, hence L = ker{˜f∗ : T XR → S(T N )} on XR . Proposition 8.5. The Einstein-Weyl structure ([g], ∇) constructed in Proposition 8.3 satisfies the following properties: 1. For each p ∈ N , S p = {x ∈ M | p ∈ ∂Dx } is a connected maximal null surface on M and every null surface can be written in this form. 2. For each p ∈ Z \N , C p = {x ∈ M | p ∈ Dx } is a connected maximal time-like geodesic and every time-like geodesic on M can be written in this form. 3. For each p ∈ N and non-zero v ∈ T p N , C p,v = {x ∈ M | p ∈ ∂Dx , v Dx } is a connected maximal null geodesic on M and every null geodesic on M can be written in this form.
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Proof. From Proposition 8.4 and the properties of DR and L, we obtain • S p = ◦ f−1 ( p) is a null surface for each p ∈ N , • C p = ◦ f−1 ( p) is a time-like geodesic for each p ∈ Z \N , • C p,v = ◦ ˜f−1 ([v]) is a null geodesic for each p ∈ N and non-zero v ∈ T p N . Moreover from Proposition 7.5, • S p S 1 × R for each p ∈ N , • C p R for each p ∈ Z \N , • C p,v R for each p ∈ N and non-zero v ∈ T p N , and they are all closed in M. Hence the statement follows.
Recall the compactification of the double fibration given by (7.7). Let Cˆ p and Cˆ p,v be the compactification of C p and C p,v in Xˆ+ respectively. Proposition 8.6. 1. For each p ∈ Z \N , XˆR |Cˆ is homeomorphic to S 2 and the restricp tion ˆf : XˆR |Cˆ → N is a homeomorphism. In particular, {∂Dx }x∈Cp gives a foliation p on N \{2 points}. 2. For each p ∈ N and non-zero v ∈ T p N , XˆR |Cˆ is homeomorphic to S 2 and the p,v restriction ˆf : XˆR | ˆ → N is surjective. Moreover, this is one-to-one on the comCp,v
plement of the curve ˆf−1 ( p), hence {(∂Dx \{ p})}x∈Cp,v gives a foliation on N \{ p}. Proof. Let p ∈ Z \N , then XR |Cp is an S 1 -bundle over C p R. Since XˆR |Cˆ is the comp
pactification of XR |Cp with extra two points, it is isomorphic to S 2 . Since f is C 0 -close to the f of the standard case, ˆf : XˆR | ˆ → N is a degree one map. Cp
Let f∗ : T (XR |Cp ) → T Z R be the differential. We claim that ker f∗ = 0 everywhere. Indeed, if there exists a non-zero w ∈ Tz (XR |Cp ) such that f∗ (w) = 0, then w ∈ Dz and ∗ (w) = 0. Then ∗ (w) must be null with respect to the constructed conformal structure. On the other hand ∗ (w) tangents to C p , so this is time-like. This is a contradiction. Hence ˆf : XˆR |Cˆ → N is locally homeomorphic degree one map, that is, it is a p homeomorphism. Next, let p ∈ N . By a similar argument, XˆR |Cˆ S 2 and ˆf : XˆR |Cˆ → N is p,v p,v degree one, hence surjective. We claim that ker{f∗ : T (XR |Cp,v ) → T N } = 0 on z ∈ XR |Cp,v \f−1 ( p) . Indeed, if there exists non-zero w ∈ Tz (XR |Cp,v ) such that f∗ (w) = 0, then ∗ (w) is non-zero and null. Notice that ∗ (w) is tangent to the null surface Sf(z) . On the other hand, ∗ (w) is tangent to C p,v ⊂ S p . Since f(z) = p, Sf(z) and S p are different null surfaces, hence T (z) Sf(z) and T (z) S p are different null planes at (z). Then ∗ (w) ∈ T (z) Sf(z) ∩ T (z) S p must be a space-like vector which is a contradiction. Hence the statement follows.
Proposition 8.7. Let ([g], ∇) be the Einstein-Weyl structure constructed in Proposition 8.3. Then, for each distinguished p, q ∈ N , C p,q = {x ∈ M | p, q ∈ ∂Dx } is a connected closed space-like geodesic on M and every space-like geodesic on M can be written in this form. In particular, this Einstein-Weyl structure is space-like Zoll.
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Proof. Since C p,q is the intersection of the null surfaces S p and Sq , this is either empty or a space-like geodesic. We claim that C p,q is not empty and is homeomorphic to S 1 . For each non-zero v ∈ T p N , there is a unique x ∈ C p,v such that q ∈ ∂Dx , since {(∂Dx \{ p})}x∈Cp,v foliates N \{ p} by 2 of Proposition 8.6. Then x ∈ C p,q , so C p,q is not empty. Moreover there is a one-to-one continuous map S(T p N ) → C p,q , so C p,q S 1 .
The main theorem (Theorem 1.6) follows from Propositions 8.3, 8.5 and 8.7. References 1. Calderbank, D.M.J.: Selfdual 4-manifolds, projective surfaces, and the Dunajski-West construction. http:// arxiv.org/abs/math.DG/0606754, 2006 2. Dunajski, M.: A class of Einstein-Weyl spaces associated to an integrable system of hydrodynamic type. J. Geom. Phys. 51(1), 126–137 (2004) 3. Dunajski, M., Mason, L.J., Tod, P.: Einstein-Weyl geometry, the dKP equation and twistor theory. J. Geom. Phys. 37(1–2), 63–93 (2001) 4. Dunajski, M., West, S.: Anti-self-dual conformal structures with null Killing vectors from projective structures. Commum. Math. Phys. 272(1), 85–118 (2007) 5. Dunajski, M., West, S.: Anti-self-dual conformal structures in neutral signature. http://arxiv.org/abs/ math/0610280V4[math.DG], 2008, to appear in Recent Developments in pseudo In: Riemannian geometry, ESI-Series on Math and Physics 6. Hitchin, N.J.: Complex manifolds and Einstein’s equations. In: Twistor Geometry and Non-Linear Systems, Lecture Notes in Mathematics, Vol. 970, 1982 7. Jones, P.E., Tod, K.P.: Minitwistor spaces and Einstein-Weyl spaces. Class. Quant. Grav. 2, 565–577 (1985) 8. LeBrun, C.: Twistors, Holomorphic Disks, and Riemann Surfaces with Boundary. In: Perspectives in Riemannian geometry, CRM Proc. Lecture Notes, 40, Providence, RI: Amer. Math. Soc. 2006, pp. 209–221 9. LeBrun, C., Mason, L.J.: Zoll Manifolds and complex surfaces. J. Diff. Geom. 61, 453–535 (2002) 10. LeBrun, C., Mason, L.J.: Nonlinear Gravitons, Null Geodesics, and Holomorphic Disks. Duke Math. J. 136(2), 205–273 (2007) 11. Nakata, F.: Singular self-dual Zollfrei metrics and twistor correspondence. J. Geom. Phys. 57(6), 1477–1498 (2007) 12. Nakata, F.: Self-dual Zollfrei conformal structures with α-surface foliation. J. Geom. Phys. 57(10), 2077–2097 (2007) 13. Pedersen, H.: Einstein-Weyl spaces and (1,n)-curves in the quadric surface. Ann. Global Anal. Geom. 4(1), 89–120 (1986) 14. Pedersen, H., Tod, K.P.: Three-dimensional Einstein-Weyl geometry. Adv. Math. 97, 74–109 (1993) 15. Penrose, R.: Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7, 31–52 (1976) 16. Tod, K.P.: Compact 3-dimensional Einstein-Weyl structures. J. London Math. Soc (2) 45, 341–351 (1992) Communicated by G.W. Gibbons
Commun. Math. Phys. 289, 701–723 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0721-0
Communications in
Mathematical Physics
Eigenfunctions in a Two-Particle Anderson Tight Binding Model Victor Chulaevsky1 , Yuri Suhov2 1 Département de Mathématiques et d’Informatique, Université de Reims,
Moulin de la Housse, B.P. 1039, 51687 Reims Cedex 2, France. E-mail: [email protected]
2 Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK Received: 10 July 2008 / Accepted: 14 October 2008 Published online: 13 January 2009 – © Springer-Verlag 2008
Abstract: We establish the phenomenon of Anderson localisation for a quantum twoparticle system on a lattice Zd with short-range interaction and in presence of an IID external potential with sufficiently regular marginal distribution. 1. The Two-Particle Tight Binding Model. Decay of Green’s Functions and Localisation This paper focuses on a two-particle Anderson tight binding model on lattice Zd with interaction. Our goal is three-fold. First, we establish a theorem deducing exponential localisation from a property of decay of Green’s functions in the two-particle model (Theorem 1.2 below). Second, we outline the so-called multi-scale analysis (MSA) scheme for the two-particle model. Finally, we perform the initial and the inductive steps of two-particle MSA and therefore establish the phenomenon of Anderson localisation in a two-particle model for a large disorder. See our main result, Theorem 1.1. d d We consider the Hilbertspace of the two-particle system 2 (Z × Z ). The Hamil(2) (2) = HU,V,g (ω) is a lattice Schrödinger operator of the form tonian H H 0 + U + g(V1 + V2 ), acting on functions φ ∈ 2 (Zd × Zd ), given by H (2) φ(x) = H 0 φ(x) + (U (x) + gW (x; ω))φ(x) = φ(y) + [U (x) + gW (x; ω)] φ(x),
(1.1)
y∈Zd ×Zd : y−x=1
W (x; ω) = V (x1 ; ω) + V (x2 ; ω), x = (x1 , x2 ) ∈ Zd × Zd . Here and such as x,y, u etc. for points in Zd × Zd . Next, boldface letters below we use (1) (d) (1) (d) and y j = y j , . . . , y j stand for the coordinate vectors of xj = xj , . . . , xj particles in Zd , j = 1, 2, and · is the sup-norm: for v = (v1 , v2 ) ∈ Rd × Rd : v = max v j , j=1,2
(1.2.1)
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where
v = max v(i) , i=1,...,d
for v = (v(1) , . . . , v(d) ) ∈ Rd .
(1.2.2)
We will consider the distance on Rd × Rd and Rd generated by the norm · . Throughout this paper, the random external potential V (x; ω), x ∈ Zd , is assumed to be real IID, with a common distribution function FV on R. Of course, the random variables W (x; ω) form an array with dependencies (which is the main source of difficulties in spectral analysis of multi-particle quantum systems in a random environment). A popular assumption is that FV has a probability density function (PDF). The condition on FV guaranteeing the validity of all results presented in this paper is as follows: FV has a PDF pV which is bounded and has a compact support.
(1.3)
This will allow us to use, in Sect. 3, some results on single-particle localisation proved in [A94] with the help of the fractional-moment method (FMM), an alternative of the MSA for single-particle models; see [AM,ASFH]). We note that a number of important facts proven or used here remain true under considerably weaker assumptions on FV . For example, Wegner-type bounds (3.5), (3.6) hold under the condition that for some δ > 0 and all > 0, sup (FV (a + ) − FV (a)) ≤ δ ;
a∈R
see [CS1]. Moreover, we stress that with the help of a technically more elaborate argument it is possible to obtain the main result of this paper (Theorem 1.1 below) under an assumption weaker than (1.3). We would also like to note that the IID property of {V (x; ω), x ∈ Zd } can be relaxed. See our forthcoming manuscript [CS2]. Parameter g ∈ R is traditionally called the coupling, or amplitude, constant. The interaction potential U is assumed to satisfy the following properties: (i) U is a bounded real function Zd × Zd → R symmetric under the permutation of variables: U (x) = U (σ x), where σ x = (x2 , x1 ) for x = (x1 , x2 ), x1 , x2 ∈ Zd .
(1.4)
U (x) = 0, if x1 − x2 > r0 .
(1.5)
(ii) U obeys
Here r0 ∈ [1, +∞) is a given value (the interaction range). Let P stand for the joint probability distribution of RVs {V (x; ω), x ∈ Zd }. The main assertion of this paper is Theorem 1.1. Consider the two-particle random Hamiltonian H (2) (ω) given by (1.1). Suppose that U satisfies conditions (1.4) and (1.5), and the random potential {V (x; ω), x ∈ Zd } is IID, with a marginal distribution function FV obeying (1.3). Then there exists g ∗ ∈ (0, +∞) such that for any g with |g| ≥ g ∗ , with P-probability one, the spectrum of operator H (2) (ω) is pure point. Furthermore, there exists a nonrandom constant m + = m + (g) > 0 (the effective mass) such that all eigenfunctions Ψ j (x; ω) of H (2) (ω) admit an exponential bound: |Ψ j (x; ω)| ≤ C j (ω) e−m + x .
(1.6)
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The assertion of Theorem 1.1 can also be stated in the form where ∀ given m ∗ > 0, ∃ g∗ = g∗ (m ∗ ) ∈ (0, +∞) such that ∀ g with |g| ≥ g∗ , the eigenfunctions Ψ j (x; ω) of H (2) (ω) admit exponential bound (1.6) with effective mass m + ≥ m ∗ . The conditions of Theorem 1.1 are assumed throughout the paper. As was said earlier, the proof of Theorem 1.1 uses mainly MSA, in its two-particle version. Most of the = H (2) (ω) time we will work with finite-volume approximation operators H (2) Λ L (u) Λ L (u) given by (2)
= H (2) Λ L (u) + Dirichlet boundary conditions H Λ L (u) and acting on vectors φ ∈ CΛ L (u) by (2) φ(x) = φ(y) + [U (x) + gW (x; ω)] φ(x), H Λ L (u) y∈Λ L (u): y−x=1
(1.7)
(1.8)
x = (x1 , x2 ) ∈ Λ L (u), with W (x) as in (1.1). Here and below, Λ L (u) stands for the ‘two-particle lattice box’ (d) d (a box, for short) of size 2L around u = (u 1 , u 2 ), where u j = (u(1) j , . . . , uj ) ∈ Z : 2 d (i) d d Λ L (u) = × × [u(i) . (1.9) − L , u + L] ∩ Z × Z j j j=1 i=1
(2)
Denoting by |Λ L (u)| the cardinality of Λ L (u), H is a Hermitian operator in the Λ L (u) Hilbert space 2 (Λ L (u)) of dimension |Λ L (u)|. In fact, the approximation (1.7) can be used for any finite subset Λ ⊂ Zd × Zd of (2) cardinality |Λ| producing Hermitian operator H in 2 (Λ). Λ (2) admit the permutation symmetry. Hamiltonian H (2) and its approximants H Λ d d Namely, let S be the unitary operator in 2 (Z × Z ) induced by map σ : Sφ(x) = φ(σ x).
(1.10)
(with natural embeddings CΛ , Cσ Λ ⊂ Then S −1 H (2) S = H (2) and S −1 H S = H σΛ Λ d d 2 (Z ×Z )). This implies, in particular, that for any finite Λ ⊂ Zd ×Zd , the eigenvalues (2) (2) are identical. This fact is accounted for in the course of the of operators H and H σΛ Λ presentation. Like its single-particle counterpart, the two-particle MSA scheme involves a number of technical parameters playing roles similar to those in the paper [DK]. In this and the following section we make use of some of these parameters (to begin with, see Theorem 1.2). More precisely, given a positive number α > 1 and starting with L 0 > 0 large enough and m 0 > 0, define an increasing sequence L k : (2)
(2)
L k = L α0 , k ≥ 1, k
(1.11)
and a decreasing positive sequence m k (depending on a positive number γ ): mk = m0
k j=1
−1/2
1 − γ Lk
, k ≥ 1.
(1.12)
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We will also use in Theorem 1.2 parameter p; our assumptions on α, γ and p in this theorem will be that p > αd > 1, γ ≥ 40.
(1.13)
Note that sequence m k is indeed positive, and the limit lim m k ≥ m 0 /2 when L 0 is k→∞
sufficiently large. We will also assume that L 0 > r0 . The single-particle MSA scheme was used in [DK] to check, for IID potentials, decay properties of the Green’s functions (GFs). In this paper we adopt a similar strategy. For the two-particle model, the GFs in a box Λ L (u) are defined by: −1 (2) (2) G (E; x, y) = H −E δx , δy , x, y ∈ Λ L (u), (1.14) Λ L (u) Λ L (u) where δx (v) = 1 (v = x) is the lattice Dirac delta-function (considered as a vector in CΛ L (u) ). Following [DK], we introduce Definition 1. Fix E ∈ R and m > 0. A two-particle box Λ L (u) is said to be (E, m)(2) (E; u, u ) defined by (1.14) for non-singular (in short: (E, m)-NS) if the GFs G Λ L (u) (2) from (1.8) satisfy the Hamiltonian H Λ L (u) (1.15) max G (2) (E; u, y) ≤ e−m L . Λ (u) L y∈∂ Λ L (u) Otherwise, it is called (E, m)-singular (or (E, m)-S). Here ∂Λ L (u) stands for the interior boundary (or briefly, of box Λ L (u): it is formed by points y ∈ Λ L (u) the boundary) such that ∃ a site v ∈ Zd × Zd \Λ L (u) with y − v = 1. A similar concept can be introduced for any set Λ ⊂ Zd × Zd , for which we use the same notation ∂Λ. The first step in the proof of Theorem 1.1 is Theorem 1.2 below. More precisely, Theorem 1.2 deduces exponential localisation from a postulated property of decay of two-particle GFs. The proof of Theorem 1.2 given in Sect. 2 follows that of its singleparticle counterpart from earlier works (see Sect. 1 in [FMSS] and Theorem 2.3 from [DK]). Nevertheless, this theorem is an important part of our method (as in [FMSS] and [DK]); having established Theorem 1.2 one can attempt to prove two-particle localisation (2) by analysing only GFs G (E; u, y). Λ L (u) It is convenient to introduce the following Definition 2. A pair of two-particle boxes Λ L (u), Λ L (v) is called R-distant (R-D, for short) if min {u − v, σ u − v} > 8R.
(1.16)
Here, σ was defined in (1.4). Theorem 1.2. Let I ⊆ R be an interval. Assume that for some m 0 > 0 and L 0 > 1, lim m k ≥ m 0 /2, and for any k ≥ 0 the following properties hold:
k→∞
(DS.k, I)
∀ u, v ∈ Zd × Zd such that Λ L k (u) and Λ L k (v) are 8L k -D,
−2 p P ∀ E ∈ I : Λ L k (u) or Λ L k (v) is (m k , E)−NS ≥ 1 − L k . (1.17)
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Here L k and m k are defined in (1.11), (1.12), and σ by (1.4), with p, α and γ satisfying (1.13). Then, with probability one, the spectrum of operator H (2) (ω) in I is pure point. Furthermore, there exists a constant m + ≥ m 0 /2 such that all eigenfunctions Ψ j (x; ω) of H (2) (ω) with eigenvalues E j (ω) ∈ I decay exponentially fast at infinity, with the effective mass m + : |Ψ j (x; ω)| ≤ C j (ω) e−m + x .
(1.18)
We stress that it is the property (DS.k, I) encapsulating decay of the GFs which enables the two-particle MSA scheme to work. (Here and below, DS stands for ‘double singularity’.) Clearly, Theorem 1.1 would be proved, once the validity of property (DS.k, I) had been established for I = R and for all k ≥ 0. However, an important remark is that, to deduce Theorem 1.1, we actually need to check the conditions of Theorem 1.2 for an arbitrary interval I ⊂ R of unit length (but of course with a fixed sequence of values m k and L k from (1.11), (1.12)). In fact, by covering the whole spectral line R by a countable family of such intervals, we will get that the whole spectrum of H (2) is pure point with P-probability one, with a ‘universal’ effective mass m + > 0. We will therefore focus on establishing property (DS.k, I) for an arbitrary unit interval I and all k ≥ 0; this is done in Sects. 3–5 below. Nevertheless, many details of the presentation in Sects. 3–5 do not require the assumption that the length of I is 1; we will choose appropriate conditions on an ad hoc basis. 2. Proof of Theorem 1.2 It is well-known (see, e.g., [B,S]) that almost every energy E with respect to the spectral measure of H (2) is a generalised eigenvalue of H (2) , i.e., solutions Ψ of the equation H (2) Ψ = EΨ are polynomially bounded. Therefore, it suffices to prove that the generalised eigenfunctions of H (2) decay exponentially with P-probability one. Let E ∈ I be a generalised eigenvalue of Hamiltonian H (2) from Eq. (1.1), and Ψ be a corresponding generalised eigenfunction. Following [DK], we will prove that ∀ ρ ∈ (0, 1) : lim sup ln x→∞
|Ψ (x; ω)| ≤ − ρ m, x
(2.1)
where m > 0 is the constant from the statement of Theorem 1.1. Given u ∈ Zd × Zd and an integer k = 0, 1, 2, . . . , set R(u) = σ u − u, bk (u) = 1 + R(u)L −1 k , Mk (u) = Λ L k (u) ∪ σ Λ L k (u); (2.2) cf. (1.4). Note that ∀ u ∈ Zd × Zd , ∀ k ≥ 1 : Mk (u) ⊂ Λbk L k (u), and lim bk (u) = 1.
(2.3)
Ak+1 (u) = Λbk+1 L k+1 (u) \ Λbk L k (u)
(2.4)
k→∞
Now set
and define the event Ωk (u) = {∃ E ∈ I and x ∈ Ak+1 (u) : Λ L k (x) and Λ L k (u) are (m, E) − S}. (2.5)
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Observe that, owing to the definition of M L k+1 (u) (see (2.2)), if x ∈ Ak+1 (u), then
dist Λ L k (x), Λ L k (u) ∪ σ Λ L k (u) ≥ 8L k . Thus, by the hypothesis of the theorem, P { Ωk (u) } ≤
(2bk+1 L k+1 + 1)2d 2p Lk
≤
(2bk+1 + 1)2d 2 p−2α
Lk
.
(2.6)
Since p > α and by virtue (2.3), the series ∞
P { Ωk (u) } < ∞.
(2.7)
k=0
Consider the event Ω<∞ (u) = {∀ u ∈ Zd × Zd , Ωk (u) occurs finitely many times}.
(2.8)
Then, owing to (2.7) and the Borel–Cantelli lemma, P { Ω<∞ } = 1. So, it suffices to pick a potential sample {V (y; ω), y ∈ Zd } with ω ∈ Ω<∞ and prove exponential decay of any generalised eigenfunction Ψ of operator H (2) , with the respective eigenvalue E ∈ I , for the specified ω ∈ Ω<∞ . From now on, the argument in the proof of the theorem becomes deterministic, and we omit symbol ω, except for the places where its presence is instructive. Since Ψ is polynomially bounded, there exist C, t ∈ (0, +∞) such that ∀ x ∈ Zd × Zd , |Ψ (x)| ≤ C (1 + x)t .
(2.9)
d d Further, since Ψ is not zero, ∃u ∈ Z × Z such that Ψ (u) = 0. For any identically then the values of Ψ inside Λ L k (u) can be recovered given k, if E ∈ spec H (2) Λ L k (u) from its boundary values, in particular, (2) 1{v−v =1} G (E; u, v) Ψ (v ). (2.10) Ψ (u) = Λ L k (u) v∈∂ Λ L k (u) v ∈∂ + Λ L k (u)
Here and below, ∂ + Λ L k (u) denotes the exterior boundary of box Λ L k (u) formed by the points y ∈ ∂ Zd × Zd \Λ L k (u) , i.e., the points y ∈ Zd × Zd \Λ L k (u) with dist (y, Λ L k (u)) = 1. Suppose that a two-particle box Λ L k (u) is (m, E)-NS for an infinite number of values k (i.e. for arbitrarily big values of k). Then, by Definition 1.1, (2) −m L k G , Λ L k (u) (E; u, v) ≤ e yielding, by (2.9) and (2.10), that |Ψ (u)| ≤ 4d(2L k + 1)2d−1 e−m L k C(1 + u + L k )t → 0, as k → ∞.
(2.11)
This would mean that, in fact, Ψ (u) = 0, which contradicts the above assumption. Therefore, ∃ an integer k1 = k1 (ω, E, u) < ∞ such that ∀ k ≥ k1 box Λ L k (u) is
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(m, E)-S. At the same time, owing to the choice of ω ∈ Ω<∞ , ∃ k2 = k2 (ω, u) such that ∀ k ≥ k2 , the event Ωk (u) does not occur. Then ∀ k ≥ max {k1 , k2 } : ∀ x ∈ Ak+1 (u), box Λ L k (x) is (m, E)-NS . Next, for a given ρ ∈ (0, 1) and b >
1+ρ 1−ρ ,
(2.12)
define
Ak+1 (u) = Λ2b/(1+ρ)L k (u)\Λ2/(1−ρ)L k (u) ⊂ Ak+1 (u). Naturally, the above inclusion is valid for sufficiently large L k+1 , or, with L 0 > 1 being fixed, for sufficiently large values of k, which we will assume in this argument. For any x ∈ Ak+1 (u), we have that dist(x, ∂ Ak+1 (u)) ≥ ρ x − u. Furthermore, if x − u ≥ L 0 /(1 − ρ), then ∃ k ≥ 0 such that x ∈ Ak+1 (u). }, {k box Λ For any k ≥ max , k (u) must be (m, E)-NS, and therefore, 1 2 L k E ∈ spec H (2) . Hence, Eq. (2.10) holds, with x instead of u: Λ L k (u) Ψ (x) =
1(v − v = 1)G
v∈∂ Λ L k (x) v ∈∂ + Λ L k (x)
(2)
Λ L k (x)
(E; u, v) Ψ (v ). (2.13)
Further, by virtue of non-singularity of Λ L k (u), ∃ v1 ∈ ∂ + Λ L k (u) such that |Ψ (x)| ≤ 4d(2L k + 1)2d−1 e−m L k |Ψ (v1 )|.
(2.14)
In fact, it suffices to apply bound (1.15) and pick, in the RHS of (2.13), a point v incident (2) to v providing the maximal absolute value of the GF G (E; u, v). Λ L k (x) Next, pick a value ρ ∈ (0, 1) and write it as a product ρ = ρρ , where ρ, ρ ∈ (0, 1). Further, pick any b > 8 + 1 + ρ/(1 − ρ). We can iterate bound (2.14) at least ((L k + 1)−1 ρx − u times, obtaining the following inequality: (L k +1)−1 ρx−u |Ψ (x)| ≤ 4d(2L k + 1)2d−1 e−m L k C (1 + u + bL k+1 )t . (2.15) This provides an exponential decay rate of Ψ arbitrarily close to ρ. Namely, ∃ an integer k3 ≥ max {k1 , k2 } such that ∀ k ≥ k3 , if x − u ≥ L k /(1 − ρ) then
|Ψ (x)| ≤ e−ρρ mx−u . Therefore, lim sup x→∞
1 ln |Ψ (x)| ≤ −ρρ m. x
Equation (2.1) then follows. This completes the proof of Theorem 1.2.
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3. The Two-Particle MSA Scheme. Non-Interactive Pairs of Singular Boxes In view of Theorem 1.2, our aim is to check property (DS.k, I) in Eq. (1.17). We now outline the two-particle MSA which is used for this purpose. It bears many features borrowed from its single-particle counterpart. In both single- and two-particle versions, the MSA scheme is an elaborate induction dealing with GFs G (2) and involving several Λ L k (u) mutually related parameters; some of them have been used in Sects. 1 and 2. Here we give the complete list, following specifications (1.11), (1.12) of sequences L k and m k (these specifications are assumed for the rest of the paper). • Parameter α ∈ (1, 2) determines the rate of growth of sequence L k in Eq. (1.11). • Constant γ > 0 determines sequence m k in Eq. (1.12). • Parameter p > 0 controls the power of the decay of probability of double singularity; see property (DS.k, I) in (1.17). • Parameters m 0 > 0 (the initial mass) and L 0 > 1 (the initial length) define the initial step of the induction. These values are related to the threshold g ∗ ∈ (0, ∞) for the coupling constant g in Theorem 1.1, roughly, by the constraint m 0 L 0 ∼ ln g ∗ ; see the proof of the initial inductive step in Theorem 3.1 below. In addition, to complete the inductive step, L 0 should be large enough: L 0 ≥ L ∗ ; see Theorem 3.2. • Parameters m k > 0 and L k > 1 (the mass and the length at step k) are chosen to follow Eqs. (1.11) and (1.12) since it allows us to check Eq. (1.17) with substantial use of the single-patricle MSA scheme. Next, • Parameter β ∈ (0, 1) controls the important property of tolerated resonances; see Eq. (3.2). • Parameter q > 0 is responsible for decay of probability of non-tolerated resonances. Thresholds L ∗ and g ∗ are functionally described as L ∗ = L ∗ (d, β, α, m 0 , p, q) and g ∗ = g ∗ (d, β, α, m 0 , p, q). An initial insight into the values of these thresholds is provided by writing L ∗ = max [L ∗0 , L ∗1 ] and g ∗ = max [g0∗ , g1∗ ]. Here L ∗0 and g0∗ are (rather explicitly) determined from Eq. (3.1) and Theorem 3.1 whereas L ∗1 and g1∗ are encrypted into Theorem 3.2. A further insight into the values of L ∗ and g ∗ is provided in the course of the presentation below. The initial mass m 0 > 0 can be chosen at will (but of course the choice of m 0 affects that of L ∗ and g ∗ ). To start with, we assume L 0 ≥ L ∗0 , where L ∗0 is large enough, so that ∞ j=1
−1/2
1−γLj
≥
1 . 2
(3.1)
Technically, it is convenient for us to run the two-particle MSA scheme under the following conditions on parameter values: p > 12d + 9, q > 4 p + 12d, β = 1/2, α = 3/2, γ = 40.
(3.2)
We will assume Eqs. (3.1), (3.2) for the rest of the paper, although we will use symbols α and β to make analogies with [DK] more fulfilling. We will also use results of the single-particle MSA formulated and proved in [DK]. The property of decay (with high probability) of GFs of the single-particle Anderson tight binding model is proved, in particular, under assumption of large disorder: |g| ≥ g˜ > 0, where the threshold g˜ is defined in terms of the single-particle Hamiltonian. We always assume, directly or indirectly, that the two-particle threshold g ∗ , introduced in this paper,
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satisfies g ∗ ≥ g, ˜ so that for all g with |g| ≥ g ∗ , all results of [DK] for the single-particle model are valid. In order to avoid confusion, we will denote by p and q parameters analogous to p and q but related to the single-particle model. It is worth mentioning that, according to the results of [DK], one can choose p and q arbitrarily large, provided that |g| is sufficiently large. Therefore, we can also assume that p and q are as large as required for our arguments, provided that g˜ is sufficiently large and |g| ≥ g ∗ ≥ g. ˜
(3.3)
The initial step of the two-particle MSA scheme consists in establishing properties (S.0) and (SS.0); see Eqs. (3.7) and (3.8). The inductive step of the two-particle MSA consists in deducing property (SS.k + 1) from property (SS.k); again see Eq. (3.8). These properties are equivalent to properties (DS.0, I), (DS.k + 1, I) and (DS.k, I), respectively, figuring in Theorem 1.2 (in the form of (SS. · ) they are slightly more convenient to deal with). Both the initial and the inductive step are done with the assistance of properties (W1) and/or (W2) (Wegner-type bounds, see Eqs. (3.5) and 3.6) below) which should be established independently. In our context, i.e. for a two-particle system, properties (W1) and (W2) have been proved in [CS1]. Definition 3. Given E ∈ R, v ∈ Zd × Zd and L > 1, we call the box Λ L (v) E-resonant (briefly: E-R) if the spectrum of the Hamiltonian H (2) satisfies Λ L (v) β (2) < e−L . dist E, spec H (3.4) Λ L (v) Given an L 0 > 1, introduce the following properties (W1) and (W2) of Hamiltonians (2) H , l ≥ L .0 . Λl (W1) (W2)
∀ l ≥ L 0 , box Λl (x) and E ∈ R: P { Λl (x) is E-R } < l −q , ∀ l ≥ L 0 and 8l-D boxes Λ (x) and Λ (y), P { ∃ E ∈ R : both Λl (x) and Λl (y) are E−R } < l −q .
(3.5) (3.6)
Lemma 3.1. (cf. [CS1].) Under the above assumptions on {V (x; ω)} and U (see (1.3)(1.5)), properties W1, W2 hold true. Further, let I ⊆ R be an interval. Given m 0 > 0 and L 0 > 1, consider property (S.0) :
p (S.0) ∀ x ∈ Zd , P ∃ E ∈ I : Λ L 0 (x) is (E, m 0 )−S < L −2 (3.7) 0 . Next, for interval I ⊆ R and values L k and m k , k ≥ 0, as in (1.11) and (1.12), we introduce property (SS.k): (SS.k)
∀ L k -D boxes Λ L k (x) and Λ L k (y):
−2 p P ∃ E ∈ I : both Λ L k (x), Λ L k (y) are (E, m k )−S < L k .
(3.8)
The initial MSA step is summarised in Theorem 3.1. ∀ given m 0 and L 0 > 0 and ∀ bounded interval I ⊂ R, there exists g0∗ = g0∗ (m 0 , L 0 , |I |) ∈ (0, +∞) such that for |g| ≥ g0∗ , properties (S.0) and (SS.0) hold true.
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Proof of Theorem 3.1. Obviously, property (S.0) implies (SS.0), so we focus on the former. Property (S.0) is established along the lines of [DK]; see [DK], Prop. A.1.2. Without loss of generality, we can assume that g > 0. Let E 0 ∈ R be the middle point of I and 2η be its length: I = (E 0 − η, E 0 + η). Note that if ∀ x = (x1 , x2 ) ∈ Λ L 0 (u) we have |W (x) − E 0 | ≥ 4d + 2η + em 0 L 0 , then ∀ E ∈ [E 0 − η, E 0 + η] G Λ L
0 (u)
(E) ≤ e−m 0 L 0 .
Next, with c0 = c0 (d, η, m 0 , L 0 ) := 4d + 2η + em 0 L 0 , observe that
P ∃ x ∈ Λ L 0 (u) : |W (x) − E 0 | ≤ c0 (x1 ; ω) + V (x2 ; ω)] − [E 0 − U (x)]| ≤ c0 = P ∃ x ∈ Λ L 0 (u) : |g[V ≤ Λ L 0 (u) max P V (x1 ; ω) + V (x2 ; ω) − g −1 [E 0 − U (x)] ≤ c0 g −1 . x∈Λ L 0 (u) For x = (x1 , x2 ) with x1 = x2 , random variables V (x1 ; ·) and V (x2 ; ·) are independent and have a common bounded PDF pV of compact support. The sum V (x1 ; ·) + V (x2 ; ·) has a bounded PDF pV ∗ pV , the convolution of pV with itself. Thus, for x = (x1 , x2 ) with x1 = x2 , P V (x1 ; ω) + V (x2 ; ω) − g −1 [E 0 − U (x)] ≤ c0 g −1 ≤ c0 (max pV ∗ pV ) g −1 . For x = (x1 , x1 ), we have V (x1 ; ω) + V (x2 ; ω) = 2V (x1 ; ω), so that
P V (x1 ; ω) + V (x2 ; ω) − g −1 [E 0 − U (x)] ≤ c0 g −1
= P V (x1 ; ω) − (2g)−1 (E 0 − U (x)) ≤ c0 · (2g)−1 ≤ c0 (max pV ) g −1 . We see that in both cases P { |W (x) − E 0 | ≤ c0 } → 0 as g → ∞, uniformly in x. Property (S.0) then follows. To complete the inductive MSA step, we will prove Theorem 3.2. ∀ given m 0 > 0, there exist g1∗ ∈ (0, +∞) and L ∗1 ∈ (0, +∞) such that the following statement holds. Suppose that |g| ≥ g1∗ and L 0 ≥ L ∗1 . Then, ∀ k = 0, 1, . . . and ∀ interval I ⊆ R, property (SS.k) implies (SS.k + 1) . The proof of Theorem 3.2 occupies the rest of the paper. Before we proceed with the proof, let us repeat that the property (DS.k, I) (or, equivalently, (SS.k)), for ∀ k ≥ 0 and ∀ unit interval I ⊂ R, follows directly from Theorems 3.1 and 3.2. Proof of Theorem 3.2. To deduce property (SS.k + 1) from (SS.k), we introduce Definition 4. Consider the following subset in Zd × Zd : Dr0 = {x = (x1 , x2 ) ∈ Zd × Zd : x1 − x2 ≤ r0 }.
(3.9)
A two-particle box Λ L (u) is called interactive when Λ L (u)∩Dr0 = ∅, and non-interactive if Λ L (u)∩Dr0 = ∅. For a non-interactive box Λ L (u), the interaction potential U (x)=0, ∀x ∈ Λ L (u). For brevity, we use the terms I-box and NI-box, respectively.
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The procedure of deducing property (SS.k + 1) from (SS.k) is done here separately for the following three cases: (I) Both Λ L k+1 (x) and Λ L k+1 (y) are NI-boxes. (II) Both Λ L k+1 (x) and Λ L k+1 (y) are I-boxes. (III) One of the boxes is I, while the other is NI. In the remaining part of this section we consider case (I). Cases (II) and (III) are treated in Sects. 4 and 5, respectively. We repeat that all cases require the use of property (W1) and/or (W2). The plan for the rest of Sect. 3 is as follows. We aim to derive property (SS.k + 1) for a pair of non-interactive L k+1 -D boxes Λ L k+1 (x), Λ L k+1 (y), and we are allowed to assume property (SS.k) for every pair of L k -D boxes Λ L k ( x), Λ L k ( y), where x, y, x, d d y ∈ Z × Z . In fact, we are able to establish property (SS.k + 1) for non-interactive L k+1 -D boxes Λ L k+1 (x), Λ L k+1 (y) directly, without referring to (SS.k). (In Cases (II) and (III) such a reference is needed.) An important part of our argument is a single-particle result stated as Theorem 3.3. Let Λ L k+1 (u) be an NI-box, where u = (u 1 , u 2 ). We represent it as the Cartesian product Λ L k+1 (u) = Λ L k+1 (u 1 ) × Λ L k+1 (u 2 ). Here and below, for given > 1 and v = (v(1) , . . . , v(d) ) ∈ Rd : d (i) (i) Λ (v) := × v j − , v j + ∩ Zd .
(3.10)
(3.11)
i=1
We call sets Λ (v) single-particle boxes; as before, |Λ (v)| denotes the cardinality of Λ (v). The boundary ∂Λ (v) is also defined in a similar fashion: it is formed by the points y ∈ Λ (v) for which ∃y ∈ Zd \ Λ (v) with y − y ≤ 1. (2) Since the potential U vanishes on Λ L k+1 (u), the Hamiltonian H takes the Λ L k+1 (u) form (2) H φ(x) = φ(y) + g V (x j ; ω)φ(x), (3.12) Λ L k+1 (u) (u): y∈Λ L j=1,2 k+1 y−x=1
x = (x1 , x2 ) ∈ Λ L k+1 (u), or, algebraically, (2)
(1)
(1)
H = H1;Λ L (u 1 ) ⊗ I + I ⊗ H2;Λ L (u 2 ) . Λ L k+1 (u) k+1 k+1 (1)
(3.13)
Here H j;Λ L (u j ) is the single-particle Hamiltonian acting on variable x j ∈ Λ L k+1 (u j ), k+1 j = 1, 2: (1) H j;Λ L (u j ) ϕ (x j ) = ϕ(y j ) + gV (x j ; ω)ϕ(x j ), k+1 (3.14) (u ): y j ∈Λ L k+1 j y j −x j =1
and I is the identity operator on the complementary variable.
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Let ψ j;s (x) be the eigenvectors of operators H j;Λ L
k+1 (u j )
and E j;s be their eigenvalues,
can be represented as s = 1, . . . |Λ L k+1 (u j )|. Then the eigenvectors Ψs1 ,s2 of H (2) Λ L k+1 (u) tensor products: Ψs1 ,s2 (x) = ψ1;s1 (x1 )ψ2;s2 (x2 ), (2)
are written as sums: while the eigenvalues E s1 ,s2 of H Λ L k+1 (u) E s1 ,s2 = E 1;s1 + E 2;s2 , with s1 = 1, . . . , |Λ L k+1 (u 1 )|, s2 = 1, . . . , |Λ L k+1 (u 2 )|. We make use of the following definition: Definition 5. Fix m > 0 and a positive integer . Given v ∈ Zd , consider the single(1) particle Hamiltonian HΛ (v) in Λ (v) acting on vectors ϕ ∈ CΛ (v) : (1) HΛ (v) ϕ (x) = ϕ(y) + gV (x; ω)ϕ(x), x ∈ Λ (u). (3.15) y∈Λ (u): y−x=1
Let ψs (x) be the normalised eigenvectors and E s the corresponding eigenvalues of (1) -non-tunnelling ( m -NT, for short), HΛ (v) . We say that a single-particle box Λ (v) is m if (1) m . (3.16) max max |ψs (v)ψs (y)| : E s ∈ spec HΛ (v) ≤ e− y∈∂Λ (v)
Otherwise we call it m -tunnelling ( m -T ). A two-particle box Λ (v) is called m -non-tunnelling if both of its projections Π1 Λ (v) and Π2 Λ (v) are m -non-tunnelling. In the future, the eigenvectors of finite-volume Hamiltonians appearing in arguments and calculations, will be assumed normalised. Remark. Observe that (i) property m -NT implies m -NT for any m ∈ [0, m ]. Next, (ii) properties m -T and m -NT refer only to single-particle Hamiltonians. As we will see later, in our two-particle MSA inductive procedure, we can use the (2m 0 )-NT property while working with boxes Λ L k (x), ∀ k ≥ 0. The following statement gives a formal description of a property of NI two-particle boxes which will be referred to as property (NIRoNS) (‘non-interactive boxes are resonant or non-singular’). As we said earlier, property (NIRoNS) is established for all k ≥ 0, by combining known results from the single-particle localisation theory, established via MSA or the FMM. It is worth mentioning that a property close to (NIRoNS) was formulated in [FMSS], Proposition in Sect. 6, p.43. However, the context here is different. Lemma 3.2. Consider a pair of single-particle boxes Λ L k (u j ), j = 1, 2, where u 1 − u 2 > L k + r0 . Given m > 0, assume that Λ L k+1 (u 1 ) and Λ L k+1 (u 2 ) are m -NT. Next, assume that the two-particle non-interactive box Λ (u) = Λ (u ) × Λ L k (u 2 ) is L L 1 k k β −1 2d (1) E-NR. If L k L k + ln (2L k + 1) < 1, then Λ L k (u) is m -NS with −1+β 2d . (3.17) m (1) = m 1 − Lk − L −1 k ln (2L k + 1) −1+β
2d In particular, if L −1 ≤ Lk k ln (2L k + 1)
−1+β
, then m (1) ≥ m (1 − 2L k
).
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Proof of Lemma 3.2. By definition of the GFs, G (2) (u, y; E) = Λ L k (u)
|Λ L k (u 1 )| |Λ L k (u 2 )|
s1 =1
s2 =1
ψ1;s1 (u 1 )ψ¯ 1;s1 (y1 )ψ2;s2 (u 2 )ψ¯ 2;s2 (y2 ) . E − E 1;s1 + E 2;s2 (3.18)
Here, as before, E j;s and ψ j;s , s = 1, . . . , |Λ L k (u j )|, j = 1, 2, are the eigenvalues and (1) . the corresponding eigenvectors of H j;Λ L k (u j ) Since Λ L k (u) is E-NR, the absolute values |E − (E 1;s1 + E 2;s2 )| of the denominators β
in (3.18) are bounded from below by e−L k . The sum of numerators can be bounded as follows. First, note that if u−y = L k , then either u 1 − y1 = L k , or u 2 − y2 = L k . Without loss of generality, suppose that u 2 − y2 = L k , then ψ1;s1 (u 1 )ψ¯ 1;s1 (y1 )ψ2;s2 (u 2 )ψ¯ 2;s2 (y2 ) s1 ,s2 ψ1;s (u 1 )ψ¯ 1;s (y1 ) ¯ ≤ 1 1 s2 ψ2;s2 (u 2 )ψ2;s2 (y2 ) s1 m = e− m L k (2L + 1)2d , ≤ |Λ L k (u 1 )| · 1 · |Λ L k (u 2 )| e− k
owing to the hypothesis of non-tunnelling. Finally, we obtain (2) β 2d L k − m Lk m (1) L k G ≤ e− . Λ L k (u) (u, y; E) ≤ (2L k + 1) e This yields Lemma 3.2.
Now introduce the following property of single-particle Hamiltonians HΛ(1) (v) : (NT. k, s) P
single-particle box Λ L k (v) is (2m 0 )−NT ≥ 1 − L −s k , (3.19)
where s > 0. Lemma 3.2 implies the following Lemma 3.3. Assume Property (W2). Suppose that ∀ k ≥ 0, the single-particle Hamil(1) tonians HΛ L (v) satisfy (NT. k, s) with s ≥ q: k
−q P Λ L k (v) is (2m 0 )−NT ≥ 1 − L k .
(3.20)
Suppose also that −1+β
2d ≤ L0 L −1 0 ln (2L 0 + 1)
≤
1 . 4
Then, ∀ interval I ⊆ R, ∀ k ≥ 0 and ∀ pair of non-interactive L k -D two-particle boxes Λ L k (x) and Λ L k (y),
−q (3.21) P ∃ E ∈ I : Λ L k (x) and Λ L k (y) are (E, m k )−S ≤ 5L k .
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Proof of Lemma 3.3. By virtue of Lemma 3.1,
P ∃ E ∈ I : Λ L k (x) and Λ L k (y) are (E, m k )−S
≤ P Λ L k (x) is 2m k −T + P Λ L k (y) is 2m k −T
+P ∃ E ∈ I : Λ L k (x) and Λ L k (y) are E−R −q
≤ 2 · 2L −s k + Lk
−q
≤ 5L k .
The validity of (3.20) is guaranteed by (1)
Theorem 3.3. Consider single-particle Hamiltonians HΛ L (v) , v ∈ Zd , k = 0, 1, . . .. k Then ∃ g2∗ , L ∗2 ∈ (0, +∞) such that when |g| ≥ g2∗ and L 0 ≥ L ∗2 , the following bound holds true for all k ≥ 0:
p − 2(1 + α)d . P Λ L k (v) is (2m 0 )−NT ≥ 1 − L −s k , s = α
(3.22)
In other words, Theorem 3.3 asserts Property (NT. k, s) with s = [ p − 2(1 + α)d] α. So, it suffices to assume that p ≥ αq + 2(1 + α)d. Since, as we observed before, p= p (g) → ∞ as |g| → ∞, the latter inequality holds for |g| large enough. As was said in Sect. 1, the reader can find, in the forthcoming manuscript [CS2], a proof of Theorem 3.3 based on an adaptation of MSA techniques from [DK] and valid under the IID assumption and condition (1.3). However, a stronger estimate was proved in [A94], with the help of the FMM, under condition (1.2) (in fact, the assumptions on the external potential V (x; ω), x ∈ Zd , adopted in [A94] are more general than IID, and, according to [AW08], [AW] they can be further relaxed. Namely, bound (1.6) from [A94] implies that
m Lk P Λ L k (v) is (2m 0 )−NT ≥ 1 − e− , (3.23) where m =m (g) → ∞ as |g| → ∞. We also recall that, for one-dimensional singleparticle models, exponential bounds of probability of exponential decay of eigen-functions in finite volumes were obtained in [GMP] (for Schrödinger operators on R) and in [KS] (for lattice Schrödinger operators on Z). We thus come to the following conclusion. Theorem 3.4. ∀ given interval I ⊆ R and k = 0, 1, . . . , Property (SS.k) holds for all pairs of L k -D non-interactive boxes Λ L k (x), Λ L k (y). Summarising the above argument: the validity of Property (SS.k + 1) for a pair of two-particle NI-boxes did not require us to assume (SS.k). However, in the course of deriving (SS.k + 1) for NI-boxes we used property (3.20) for single-particle boxes, as well as the Wegner-type property (W2). This completes the analysis of the Case (I), where both boxes Λ L k+1 (x) and Λ L k+1 (y) are NI. For future use, we also give Lemma 3.4. Consider a two-particle box Λ L k+1 (u). Let M(Λ L k+1 (u); E) be the maximal number of (E, m k )-S, pair-wise L k -D NI-boxes Λ L k (u( j) ) ⊂ Λ L k+1 (u). The following property holds:
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−2 p −2 p 2d(1+α) P ∃E ∈ I : M(Λ L k+1 (u); E) ≥ 2 ≤ L k · 5L k < L 4dα · Lk . k (3.24) Proof of Lemma 3.4. The number of possible pairs of centres u(1) , u(2) is bounded by 2d (2L k+1 + 1)2d ≤ (2L αk + 1)2d ≤ (L α+1 = L 2d(α+1) , k ) k
while for a given pair of centres one can apply Theorem 3.3.
4. Interactive Pairs of Singular Boxes Speaking informally, Case (II) corresponds to a two-particle system with ‘confinement’: in both boxes Λ L k+1 (x) and Λ L k+1 (y), particles are at a distance ≤ 2L k + r0 from each other, and form a ‘compound quantum object’ which can be considered as a ‘single particle’ subject to a random external potential. It is not entirely surprising, then, that such a compound object should feature localisation properties resembling those from the single-particle theory. The reader may see that the analysis needed to cover Case (II) is rather similar to that in [DK]. It relies essentially upon Properties (W1) and (W2) (Wegner-type estimates). However, it is worth mentioning that the derivation of estimates (W1) and (W2) required new ideas due to strong dependencies in the random potential gV (x1 ; ω) + gV (x2 ; ω). As was said before, these dependencies do not decay as x1 − x2 → ∞. Our proofs given in [CS1] are based on Stollmann’s lemma (cf. [St1,St2]) rather than on the original ideas of Wegner. The main outcome in Case (II) is Theorem 4.1 placed at the end of this section. Before we proceed further, let us state a geometric assertion (see Lemma 4.1 below) which we prove in Sect. 6. Given a two-particle box Λ L (u), with u = (u 1 , u 2 ), and (1) (d) u j = (u j , . . . , u j ) ∈ Zd , set Π Λ L (u) = Π1 Λ L (u) ∪ Π2 Λ L (u) ⊂ Zd .
(4.1)
Here Π1 Λ L (u) and Π2 Λ L (u) denote the projections of Λ L (u) to the first and the second factor in Zd × Zd : d (i) (i) Π j Λ L (u) = × [u j − L , u j + L] ∩ Zd , j = 1, 2; i=1
cf. (1.9). In other words, Π j Λ L (u) describes a ‘supporting domain’ of the single-particle external potential {V (x), x ∈ Zd } contributing to the potential field W (x), x ∈ Λ L (u). Lemma 4.1. Let be two interactive 8L-D boxes Λ L (u ) and
L > r0 and consider Λ L (u ), with dist Λ L (u ), Λ L (u ) > 8L. Then Π Λ L (u ) ∩ Π Λ L (u ) = ∅.
(4.2)
Lemma 4.1 is used in the proof of Lemma 4.2 which, in turn, is important in establishing Theorem 4.1. Actually, it is a natural complement to Lemma 2.2 in [CS1]. Let I ⊆ R be an interval. Consider the following assertion: (IS.k) :
∀ pair of interactive L k -D boxes Λ L k (x) and Λ L k (y):
−2 p P ∃ E ∈ I : both Λ L k (x), Λ L k (y) are (E, m k )-S ≤ L k .
(4.3)
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Lemma 4.2. Given k ≥ 0, assume that Property (IS.k) holds true. Consider a box Λ L k+1 (u) and let N (Λ L k+1 (u); E) be the maximal number of (E, m k )-S, pair-wise L k -D I-boxes Λ L k (u( j) ) ⊂ Λ L k+1 (u). Then ∀ n ≥ 1,
−2np 2n(1+dα) · Lk . (4.4) P ∃ E ∈ I : N (Λ L k+1 (u); E) ≥ 2n ≤ L k Proof of Lemma 4.2. Suppose ∃ I-boxes Λ L k (u(1) ), . . ., Λ L k (u(2n) ) ⊂ Λ L k+1 (u) such that any two of them are L k -D, i.e., are at the distance > 8L k . By virtue of Lemma 4.1, it is readily seen that (a) ∀ pair Λ L k (u (2i−1) ), Λ L k (u (2i) ), the respective (random) operators H (2) (2i−1) (ω) Λ L k (u ) (2) and H (ω) are independent, and so are their spectra and GFs. Λ L k (u(2i) ) (b) Moreover, the pairs of operators, (2) (2) H (ω), H (ω) , i = 1, . . . , n, (4.5) Λ L k (u(2i−1) ) Λ L k (u(2i) ) form an independent family. Indeed, operator H (2) (i) , with i ∈ {1, . . . , 2n}, is measurable relative to the sigmaΛ L k (u ) algebra Bi generated by random variables {V (x), x ∈ Π Λ L k (u(i) )}, with Π Λ L k (u(i) ) = Π1 Λ L k (u(i) ) ∪ Π2 Λ L k (u(i) ) ⊂ Zd . Now, by Lemma 4.2, the sets Π Λ L k (u(i) ), i ∈ {1, . . . , 2n}, are pairwise disjoint, so that all sigma-algebras Bi , i ∈ {1, . . . , 2n}, are independent. Remark. This property formalises the observation made in the beginning of this section: a pair of particles corresponding to an interactive box of size 2L k forms a “compound quantum object” of size < 8L k , and their analysis is quite similar to that from the single-particle MSA. Thus, any collection ofevents A1 , . . . , An−1 related to the corresponding pairs (2) (2) , i = 1, . . . , n, also form an independent family. H ,H Λ L k (u(2i−1) ) Λ L k (u(2i) ) Now, for i = 1, . . . , n − 1, set Ai = ∃ E ∈ I : both Λ L k (u(2i−1) ), Λ L k (u(2 j+2) ) are (E, m k )-S . (4.6) Then, by virtue of (IS.k),
−2 p P A j ≤ L k , 0 ≤ j ≤ n − 1, and by virtue of independence of events A0 , . . . , An−1 , we obtain ⎧ ⎫ ⎨n−1 ⎬ n−1 −2 p n Aj = P A j ≤ Lk . P ⎩ ⎭ j=0
(4.7)
(4.8)
j=0
To complete the proof, note that the total number of different families of 2n boxes Λ L k ⊂ Λ L k+1 (u) with required properties is bounded from above by 2n 2n 1 1 2n(1+dα) 2(L k + r0 + 1)L dk+1 4L k L dk+1 ≤ ≤ Lk , (2n)! (2n)!
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since their centres must belong to the subset D L k +r0 ∩ Λ L k+1 (u). Here D L k +r0 = (x1 , x2 ) ∈ Zd × Zd : x1 − x2 ≤ L k + r0 is a ‘layer’ of width 2(L k + r0 ) adjoint to the set D := {x = (x, x), x ∈ Zd }, the diagonal in Zd × Zd . Recall also that r0 < L 0 ≤ L k , k ≥ 0, by our assumption. This yields Lemma 4.2. Lemma 4.3. Let K (Λ L k+1 (u) ; E) be the maximal number of (E, m k )-S, pair-wise L k -D boxes Λ L k (u( j) ) ⊂ Λ L k+1 (u) (interactive or non-interactive). Then ∀ n ≥ 1,
−2 p −2np 2n(1+dα) P ∃E ∈ I : K (Λ L k+1 (u) ; E) ≥ 2n + 2 ≤ L 4dα · Lk + Lk · Lk . k
(4.9)
Proof of Lemma 4.3. Assume that K (Λ L k+1 (u) ; E) ≥ 2n + 2. Let M(Λ L k+1 (u) ; E) be as in Lemma 3.4 and N (Λ L k+1 (u) ; E) as in Lemma 4.2. Obviously, K (Λ L k+1 (u) ; E) ≤ M(Λ L k+1 (u) ; E) + N (Λ L k+1 (u) ; E). Then either M(Λ L k+1 (u) ; E) ≥ 2 or N (Λ L k+1 (u) ; E) ≥ 2n. Therefore,
P ∃E ∈ I : K (Λ L k+1 (u) ; E) ≥ 2n + 2
≤ P ∃E ∈ I : M(Λ L k+1 (u) ; E) ≥ 2
+ P ∃E ∈ I : N (Λ L k+1 (u) ; E) ≥ 2n −2 p
≤ L 4dα · Lk k by virtue of (3.24) and (4.4)
−2np
+ L 2n(1+dα) · Lk k
,
An elementary calculation now gives rise to the following Corollary 4.1. Under assumptions of Lemma 4.3, with n ≥ 4, p ≥ 12d +9, p ≥ 3 p+3d, α = 3/2, for L 0 ≥ 2 large enough, we have
−2 p−1 P ∃E ∈ I : K (Λ L k+1 (u) ; E) ≥ 2n + 2 ≤ L k+1 .
(4.10)
Remark. Our lower bounds on values of n, p and p are not sharp. Definition 4.1. A box Λ L k+1 (v) is called (E, J )-completely non-resonant ((E, J )-CNR in brief), if the following properties are fulfilled: (i) Λ L k+1 (v) is E-NR; (ii) all boxes of the form Λ j (L k +1) (y) ⊂ Λ L k+1 (v), y ∈ Λ L k+1 (v), j = 1, . . . , J , are E-NR. As follows from Definition 4.1 and Property (W.2), we have
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Lemma 4.4. Let = Λ L k+1 (u ), = Λ L k+1 (u ) be two L k+1 -D boxes. Then, for L 0 > (J + 1)2 ,
P ∀ E ∈ I : either Λ L k+1 (u ) or Λ L k+1 (u ) is (E, J )−CNR (4.11) −(qα −1 −2α) −(q −4) > 1 − L k+1 , q := q/α. ≥ 1 − (J + 1)2 L k+1 The statement of Lemma 4.5 below is a simple reformulation of Lemma 4.2 from [DK], adapted to our notations. Indeed, the reader familiar with the proof given in [DK] can see that the structure of the external potential is irrelevant to this completely deterministic statement. So it applies directly to our model with potential U (x) + gW (x). For that reason, the proof of Lemma 4.5 is omitted. Lemma 4.5. Fix an odd positive integer J and suppose that the following properties are fulfilled: (i) Λ L k+1 (v) is (E, J )-CNR, and (ii) K (Λ L k+1 (u) ; E) ≤ J . Then for sufficiently large L 0 , box Λ L k+1 (v) is (E, m k+1 )-NS with 5J + 6 > m 0 /2 > 0. (4.12) m k+1 ≥ m k 1 − (2L k )1/2 Remark. In [DK], it is also assumed that α < (J + 1)(d + 1/2). In our case, this is automatically satisfied with α = 3/2 and J ≥ 1. In particular, with J = 9, we obtain 51 40 > m k 1 − 1/2 , m k+1 ≥ m k 1 − (4.13) 1/2 (2L k ) Lk which explains our assumption (3.1) and the recursive definition (1.13) with γ = 40. Now comes a statement which extends Lemma 4.1 from [DK] to pairs of two-particle L k -D I-boxes. Theorem 4.1. ∀ given interval I ⊆ R, there exists L ∗3 ∈ (0, +∞) such that if L 0 ≥ L ∗0 , then, ∀ k ≥ 0, Property (IS.k) in (4.1) implies (IS.k + 1) . Proof of Theorem 4.1. Let x, y ∈ Zd × Zd and assume that Λ L k+1 (x) and Λ L k+1 (y) are L k -D I-boxes. Consider the following two events:
B = ∃ E ∈ I : both Λ L k+1 (x) and Λ L k+1 (y) are (E, m k+1 )-S , and, for a given odd integer J ,
Σ = ∃ E ∈ I : neither Λ L k+1 (x) nor Λ L k+1 (y) is (E, J )−CNR . By virtue of Lemma 4.4, we have, with L 0 large enough (L 0 ≥ J + 1)2 ) and α = 3/2: −(q −4)
P { Σ } < L k+1 Further,
, q := q/α.
(4.14)
P { B } = P { B ∩ Σ } + P B ∩ Σc ≤ P { Σ } + P B ∩ Σc , −q +4
and we know that P { Σ } ≤ L k+1 . So, it suffices now to estimate P { B ∩ Σ c }. Within the event B ∩ Σ c , for any E ∈ I , one of the boxes Λ L k+1 (x), Λ L k+1 (y) must be (E, J )
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719
-CNR. Without loss of generality, assume that for some E ∈ I , Λ L k+1 (x) is (E, J )-CNR and (E, m k+1 )-S. By Lemma 4.5, for such value of E, K (Λ L k+1 (x); E) ≥ J + 1. We see that
B ∩ Σ c ⊂ ∃E ∈ I : K (Λ L k+1 (x); E) ≥ J + 1 and, therefore, by Lemma 4.3, with the same values of parameters as in Corollary 4.1, as before:
−2 p P B ∩ Σ c ≤ P ∃E ∈ I : K (Λ L k+1 (X ); E) ≥ J + 1 ≤ L k . (4.15) In what follows we consider J = 9 although it will be convenient to use symbol J , in particular, to stress analogies with [DK]. 5. Mixed Pairs of Singular Two-Particle Boxes It remains to derive the property (SS.k + 1) in Case (III), i.e., for mixed pairs of twoparticle boxes (where one is I and the other NI). Here we use several properties which have been established earlier in this paper for all scale lengths, namely, (W1), (W2), (NT. k, s) with s ≥ q, (NIRoNS), and the inductive assumption (IS.k + 1) which we have already derived from (IS.k) in Sect. 4. A natural counterpart of Theorem 4.1 for mixed pairs of boxes is the following Theorem 5.1. ∀ given interval I ⊆ R, there exists a constant L ∗4 ∈ (0, +∞) with the following property. Assume that L 0 ≥ L ∗4 and, for a given k ≥ 0, the property (SS.k) holds: (i) ∀ pair of L k -D NI-boxes Λ L k ( x), Λ L k ( y), and (ii) ∀ pair of L k -D I-boxes Λ L k ( x), Λ L k ( y). Let Λ L k+1 (x), Λ L k+1 (y) be a pair of L k+1 -D boxes, where Λ L k+1 (x) is I and Λ L k+1 (y) NI. Then
−2 p P ∃ E ∈ I : both Λ L k+1 (x), Λ L k+1 (y) are (E, m k+1 )−S ≤ L k+1 . (5.1) Before starting a formal proof we give an informal description of our strategy. 1. We are going to list several situations which may give rise to singularity of a mixed pair Λ L k+1 (x) (an I-box), Λ L k+1 (y) (an NI-box). Next, we show that each situation is covered by an event of (negligibly) small probability. Finally, we show that if neither of these events occurs, the pair of boxes in question cannot be (E, m k+1 )-S. 2. Given a pair of an I-box and an NI-box, which are (E, m)-S for some (and the same) E, we note first that, owing to (NIRoNS), with high probability, the NI-box has to be E-R. If it is not, we count such an event as an unlikely situation which may give rise to simultaneous singularity of the pair in question. 3. Assuming that the NI-box Λ L k+1 (y) is E-R, we apply the Wegner-type estimate (W2) and conclude that, with high probability, neither the I-box Λ L k+1 (x), nor any of its sub-boxes of size 2L k is E-R. Again, the presence of ‘unwanted’ E-R boxes is considered as an unlikely situation. Otherwise, we conclude that Λ L k+1 (x) is (E, J )CNR. 4. Focusing on the I-box Λ L k+1 (x), we use properties (W2) and (IS.k) to prove that, with high probability, it contains a limited number of distant sub-boxes of size 2L k which are (E, m k )-S. Specifically, it is unlikely that Λ L k+1 (x) contains at least two L k -D NI-sub-boxes of size 2L k (by (NIRoNS) and (W2)); it is also unlikely that it contains at least (J − 1) L k -D I-sub-boxes of size 2L k , by virtue of (IS.k).
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5. Finally, if a two-particle box of width 2L k+1 is both (E, J )-CNR and contains at most (J − 2) + (2 − 1) = J − 1 distant sub-boxes, it must be (E, m k+1 )-NS, which is a possibility outside the event in Eq. (5.1). So, the sum of probabilities of the above-mentioned events gives an upper bound for the probability of simultaneous singularity of the given mixed pair of boxes. (2)
Proof of Theorem 5.1. Recall that the Hamiltonian H is decomposed as in Λ L k+1 (y) Eqs. (3.12), (3.13). Consider the following three events:
B = ∃ E ∈ I : both Λ L k+1 (x), Λ L k+1 (y) are (E, m k+1 )-S ,
T = either Λ L k+1 (y1 ) or Λ L k+1 (y2 ) is (2m 0 )-T , and
Σ = ∃ E ∈ I : neither Λ L k+1 (x) nor Λ L k+1 (y) is (E, J )-CNR . Event B is the one figuring in the bound (5.1), and we are interested in estimating its probability. Recall that by virtue of (3.22), we have P { T } ≤ L −s k+1 , where s =
p − 5d p − 2(1 + α)d = , α α
(5.2)
while for event Σ we have again, by virtue of Lemma 4.4 and inequality (4.13), with our choice of parameters J and L 0 (J = 9 and L 0 large enough), −q+2
P { Σ } ≤ L k+1 .
(5.3)
Further, P { B } = P { B ∩ T } + P { B ∩ Tc } c ≤ P { T } + P { B ∩ Tc } ≤ L −s k+1 + P { B ∩ T }. Now, we estimate P { B ∩ Tc }: P { B ∩ Tc } = P { B ∩ Tc ∩ Σ } + P { B ∩ Tc ∩ Σ c } −q+2 ≤ P { Σ } + P { B ∩ Tc ∩ Σ c } ≤ L k+1 + P { B ∩ Tc ∩ Σ c }. So, it suffices to estimate P { B ∩ Tc ∩ Σ c }. Within the event B ∩ Tc ∩ Σ c , one of the boxes Λ L k+1 (x), Λ L k+1 (y) is E-NR. It cannot be the NI-box Λ L k+1 (y). Indeed, by Corollary 4.1, had box Λ L k+1 (y) been both E-NR and (2m 0 )-NT, it would have been (E, m k+1 )-NS, which is not allowed within the event B. Thus, the I-box Λ L k+1 (x) must be E-NR, but (E, m k+1 )-S: B ∩ Tc ∩ Σ c ⊂ {∃ E ∈ I : Λ L k+1 (x) is (E, m k+1 )-S
and
E-NR}.
However, applying Lemma 4.5, we see that {∃ E ∈ I : Λ L k+1 (x) is (E, m k+1 )-S and E-NR} ⊂ {∃ E ∈ I : K (Λ L k+1 (x); E) ≥ J + 1}.
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Therefore, with the same values of parameters as in Corollary 4.1,
P { B ∩ Tc ∩ Σ c } ≤ P ∃ E ∈ I : K (Λ L k+1 (x); E) ≥ 2n + 2 −2 p
≤ 2L −1 k+1 L k+1 .
(5.4)
Finally, we get, with q := q/α, P { B } ≤ P { B ∩ T } + P { Σ } + P { B ∩ Tc ∩ Σ c } −q +4
≤ L −s k+1 + L k+1
−2 p
(5.5)
1 , 2L −1 k+1 ≤ . 3
(5.6)
−2 p
+ 2L −1 k+1 L k+1 ≤ L k+1 ,
if we can guarantee that −s+2 p max L k+1 ,
−q +4+2 p
L k+1
The bound (5.6) follows from our assumptions, provided that L 0 is large enough and s − 2p =
p − 5d − 2 p > 1, q − 2 p − 4 > 1. α
This completes the proof of Theorem 5.1.
(5.7)
Therefore, Theorem 3.2 is proven. In turn, this completes the proof of Theorem 1.1. 6. Proof of Lemma 4.1 Recall that we deal with boxes Λ := Λ L (u ) and Λ := Λ L (u ) such that two-particle (i) dist Λ , Λ > 8L and (ii) Λ ∩ Dr0 = ∅ = Λ ∩ Dr0 . Recall that we denote by D the diagonal in Zd × Zd : D = {x = (x, x), x ∈ Zd }. Then property (ii) implies that Λ L+r0 (u ) ∩ D = ∅, Λ L+r0 (u ) ∩ D = ∅,
(6.1)
so that ∃ x = ( x , x ) ∈ Λ L+r0 (u ) and ∃ x = ( x , x ) ∈ Λ L+r0 (u ). Next, observe that dist(Λ L+r0 (u ), Λ L+r0 (u )) ≥ dist(Λ L (u ), Λ L (u )) − 2r0 > 8L − 2r0 > 0, (6.2) owing to the assumption L > r0 , and therefore, x − x = x1 − x1 = x2 − x2 > 8L − 2r0 ,
(6.3)
since x , x ∈ D. Further, for arbitrary points x ∈ Λ , x ∈ Λ , and any j ∈ {1, 2}, we can write the triangle inequality as follows: dist( x j , x j ) ≤ dist( x j , x j ) + dist(x j , x j ) + dist(x j , x j ) or, equivalently, dist(x j , x j ) ≥ dist( x j , x j ) − dist( x j , x j ) − dist(x j , x j ) > 8L − 2r0 − (2L + 2r0 ) − (2L + 2r0 ) = 4L − 6r0 ≥ 2L > 0,
(6.4)
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since dist( x j , x j ) ≤ diam(Λ L+r0 ) = 2L + 2r0 , x j ). We see that, for j = 1, 2, and the same upper bound holds for dist(x j , dist(Π j Λ , Π j Λ ) > 2L > 0,
(6.5)
so that Π1 Λ ∩ Π1 Λ = ∅, Π2 Λ ∩ Π2 Λ = ∅. Finally, to reach the same conclusion for Π1 Λ ∩ Π2 Λ and Π2 Λ ∩ Π1 Λ , it suffices to replace Λ by σ Λ and to use the definition of L-D boxes: min( dist(Λ , Λ ),
dist(σ Λ , Λ )) > 8L .
Indeed, we have Π1 (σ Λ ) = Π2 Λ , Π2 (σ Λ ) = Π1 Λ , so that an analogue of inequality (6.5) for boxes σ Λ and Λ reads dist(Π j (σ Λ ), Π j Λ ) > 2L > 0,
(6.6)
dist(Π2 Λ , Π1 Λ ) > 2L > 0, dist(Π1 Λ , Π2 Λ ) > 2L > 0,
(6.7)
yielding
so that Π2 Λ ∩ Π1 Λ = ∅, Π1 Λ ∩ Π2 Λ = ∅. Now we see that Π1 Λ ∪ Π2 Λ ∩ Π1 Λ ∪ Π2 Λ = ∅. This completes the proof of Lemma 4.1.
(6.8)
Acknowledgements. We thank the referees for numerous suggestions improving the quality of the paper. VC thanks The Isaac Newton Institute and DPMMS, University of Cambridge, for hospitality during visits in 2003, 2004, 2007 and 2008. YS thanks the Département de Mathématiques, Université de Reims for hospitality during visits in 2003 and 2006–2008, in particular, for a Visiting Professorship in the Spring of 2003. YS thanks IHES, Bures-sur-Yvette, and STP, Dublin Institute for Advanced Studies, for hospitality during visits in 2003–2007. YS thanks the Departments of Mathematics of Penn State University and of UC Davis, for hospitality during Visiting Professorships in the Spring of 2004, Fall of 2005 and Winter of 2008. YS thanks the Department of Physics, Princeton University and the Department of Mathematics of UC Irvine, for hospitality during visits in the Spring of 2008. YS acknowledges the support provided by the ESF Research Programme RDSES towards research trips in 2003–2006.
References [A94] [AW08] [AM] [ASFH] [AW] [B]
Aizenman, M.: Localization at weak disorder: some elementary bounds. Rev. Math. Phys. 6, 1162–1184 (1994) Aizenman, M., Warzel, S.: Localization bounds for multiparticle systems. Commun. Math. Phys., arXiv: math-ph/0809.3436, 2008 Aizenman, M., Molchanov, S.A.: Localization at large disorder and extreme energies: an elementary derivation. Commun. Math. Phys. 157, 245–278 (1993) Aizenman, M., Schenker, J.H., Friedrich, R.M., Hundertmark, D.: Finite-volume fractionalmoment criteria for anderson localization. Commun. Math. Phys. 224, 219–253 (2001) Aizenman, M., Warzel, S.: Lecture Notes on Random Schrödinger Operators, in preparation Berezanskii, J.M.: Expansion in eigenfunctions of self-adjoint operators. Transl. Math. Monographs 17. Providence, R.I.: Amer. Math. Soc. 1968
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[CS1] [CS2] [DK] [FS] [FMSS] [GMP] [KS] [S] [St1] [St2]
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Chulaevsky, V., Suhov, Y.: Wegner bounds for a two-particle tight binding model. Commun. Math. Phys. 283, 479–489 (2008) Chulaevsky, V., Suhov, Y.: Eigenfunctions in a two-particle Anderson tight binding model. In preparation von Dreifus, H., Klein, A.: A new proof of localization in the anderson tight binding model. Commun. Math. Phys. 124, 285–299 (1989) Fröhlich, J., Spencer, T.: Absence of diffusion in the anderson tight binding model for large disorder or low energy. Commun. Math. Phys. 88, 151–184 (1983) Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: A constructive proof of localization in anderson tight binding model. Commun. Math. Phys. 101, 21–46 (1985) Goldsheid, I.Ya., Molchanov, S.A., Pastur, L.A.: A pure point spectrum of the one dimensional schrödinger operator. Funct. Anal. Appl. 11, 1–10 (1977) Kunz, H., Souillard, B.: Sur le spectre des opérateurs aux différences finies aléatoires. Commun. Math. Phys. 78, 2011–246 (1980) Simon, B.: Schrödinger semigrooups. Bull. Amer. Math. Soc. 7, 447 (1983) Stollmann, P.: Wegner estimates and localization for continuous anderson models with some singular distributions. Arch. Math. 75, 307–311 (2000) Stollmann, P.: Caught by disorder. Basel-Boston: Birkhäuser, 2001
Communicated by B. Simon
Commun. Math. Phys. 289, 725–764 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0781-9
Communications in
Mathematical Physics
The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model Jian Ding1, , Eyal Lubetzky2 , Yuval Peres2, 1 Department of Statistics, UC Berkeley, Berkeley, CA 94720, USA.
E-mail: [email protected]
2 Microsoft Research, One Microsoft Way, Redmond, WA 98052-6399, USA.
E-mail: [email protected]; [email protected] Received: 11 July 2008 / Accepted: 8 December 2008 Published online: 1 April 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for 1 any fixed β < 1 there is cutoff at time 2(1−β) n log n with a window of order n, whereas the mixing-time at the critical temperature β = 1 is (n 3/2 ). It is natural to ask how the mixing-time transitions from (n log n) to (n 3/2 ) and finally to exp ((n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point βc = 1. In particular, √ we find a scaling window of order 1/ n around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ 2 n → ∞ with n, the mixing-time has order (n/δ) log(δ 2 n), and exhibits cutoff with constant 21 and window size n/δ. In the critical window, β = 1 ± δ, where δ 2 n is O(1), there is no cutoff, and the mixing-time has order n 3/2 . At low temperature, β = 1 + δ for δ > 0 with δ 2 n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order nδ exp ( 43 + o(1))δ 2 n .
1. Introduction The Ising Model on a finite graph G = (V, E) with parameter β ≥ 0 and no external magnetic field is defined as follows. Its set of possible configurations is = {1, −1}V , where each configuration σ ∈ assigns positive or negative spins to the vertices of the graph. The probability that the system is at a given configuration σ is given by the Gibbs distribution Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.
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⎛ ⎞ 1 exp ⎝β µG (σ ) = σ (x)σ (y)⎠, Z (β) x y∈E
where Z (β) (the partition function) serves as a normalizing constant. The parameter β represents the inverse temperature: the higher β is (the lower the temperature is), the more µG favors configurations where neighboring spins are aligned. At the extreme case β = 0 (infinite temperature), the spins are totally independent and µG is uniform over . The Curie-Weiss model corresponds to the case where the underlying geometry is the complete graph on n vertices. The study of this model (see, e.g., [7–9,11]) is motivated by the fact that its behavior approximates that of the Ising model on high-dimensional tori. It is convenient in this case to re-scale the parameter β, so that the stationary measure µn satisfies β σ (x)σ (y) . (1.1) µn (σ ) ∝ exp n x
1 |φ(σ ) − ψ(σ )|. 2 σ ∈
The (worst-case) total-variation distance of (X t ) to stationarity at time t is dn (t) := max Pσ (X t ∈ ·) − µn TV , σ ∈
where Pσ denotes the probability given that X 0 = σ . The total-variation mixing-time of (X t ), denoted by tmix (ε) for 0 < ε < 1, is defined to be tmix (ε) := min {t : dn (t) ≤ ε} . A related notion is the spectral-gap of the chain, gap := 1 − λ, where λ is the largest absolute-value of all nontrivial eigenvalues of the transition kernel. (n) Consider an infinite family of chains (X t ), each with its corresponding worst-dis(n) (n) tance from stationarity dn (t), its tmix , etc. We say that (X t ) exhibits cutoff
mixing-times (n)
iff for some sequence wn = o tmix ( 41 ) we have the following: for any 0 < ε < 1 there exists some cε > 0, such that (n)
(n)
tmix (ε) − tmix (1 − ε) ≤ cε wn
for all n.
(1.2)
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Fig. 1. Illustration of √ the mixing time evolution as a function of the inverse-temperature β, with a scaling window of order 1/ n around the critical point. We write δ = |β − 1| and let ζ be the unique positive root tanh(βx)−x of g(x) := 1−x tanh(βx) . Cutoff only occurs at high temperature
That is, there is a sharp transition in the convergence of the given chains to equilibrium (n) at time (1 + o(1))tmix ( 41 ). In this case, the sequence wn is called a cutoff window, and (n) the sequence tmix ( 41 ) is called a cutoff point. It is well known that for any fixed β > 1, the Glauber dynamics (X t ) mixes in exponential time (cf., e.g., [10]), whereas for any fixed β < 1 (high temperature) the mixing time has order n log n (see [2] and also [3]). Recently, Levin, Luczak and Peres [11] established that the mixing-time at the critical point β = 1 has order n 3/2 , and 1 that for fixed 0 < β < 1 there is cutoff at time 2(1−β) n log n with window n. It is therefore natural to ask how the phase transition between these states occurs around the 1 critical βc = 1: abrupt mixing at time ( 2(1−β) + o(1))n log n changes to a mixing-time 3/2 of (n ) steps, and finally to exponentially slow mixing. In this work, we determine this phase transition, and characterize the mixing-time of the dynamics as a function of the parameter β, as it approaches its critical value βc = 1 both from below √and from above. The scaling window around the critical temperature βc has order 1/ n, as formulated by the following theorems, and illustrated in Fig. 1. Theorem 1. (Subcritical regime) Let δ = δ(n) > 0 be such that δ 2 n → ∞ with n. The Glauber dynamics for the mean-field Ising model with parameter β = 1 − δ exhibits cutoff at time 21 (n/δ) log(δ 2 n) with window size n/δ. In addition, the spectral gap of the dynamics in this regime is (1 + o(1))δ/n, where the o(1)-term tends to 0 as n → ∞. √ Theorem 2. (Critical window) Let δ = δ(n) satisfy δ = O(1/ n). The mixing time of the Glauber dynamics for the mean-field Ising model with parameter β = 1 ± δ has order n 3/2 , and does not exhibit cutoff. In addition, the spectral gap of the dynamics in this regime has order n −3/2 .
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0.008
=0.95 =1.01 0.006
=1.05 =1.1
0.004
0.002
−1.0
− 0.5
0.5
1.0
Fig. 2. The stationary distribution of the normalized magnetization chain (average of all spins) for the dynamics on n = 500 vertices. The center of mass at high temperatures (see β = 0.95) is at 0. Low temperatures feature two centers of mass at ±ζ (where ζ is the unique positive solution of tanh(βx) = x), leading to the exponential mixing time
Theorem 3. (Supercritical regime) Let δ = δ(n) > 0 be such that δ 2 n → ∞ with n. The mixing-time of the Glauber dynamics for the mean-field Ising model with parameter β = 1 + δ does not exhibit cutoff, and has order ζ n n 1 + g(x) texp (n) := exp dx , log δ 2 0 1 − g(x) where g(x) := (tanh(βx) − x) / (1 − x tanh(βx)), and ζ is the unique positive root of g. In particular, in the special case δ → 0, the order of the mixing time is nδ exp(( 43 + o(1)) δ 2 n), where the o(1)-term tends to 0 as n → ∞. In addition, the spectral gap of the dynamics in this regime has order 1/texp (n). As we further explain in Sect. 2, the key element in the proofs of the above theorems is understanding the behavior of the sum of all spins (known as the magnetization chain) at different temperatures. This function of the dynamics turns out to be an ergodic Markov chain as well, and namely a birth-and-death chain (a one-dimensional chain, where only moves between neighboring positions are permitted). In fact, the reason for the exponential mixing at low-temperature is essentially that this magnetization chain has two centers of mass, ±ζ n (where ζ is as defined in Theorem 3), with an exponential commute time between them. Fig. 2 demonstrates how the single center of mass around 0 that this chain (rescaled) has at high near-critical temperature proceeds to split into two symmetric centers of mass that drift further and further apart as the temperature decreases. In light of this, a natural question that rises is whether the above mentioned bottleneck between the two centers of mass at ±ζ n is the only reason for the exponential mixing-time at low temperatures. Indeed, as shown in [11] for the strictly supercritical
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regime, β > 1 fixed, if one restricts the Glauber dynamics to non-negative magnetization (known as the censored dynamics), the mixing-time becomes (n log n) just like in the subcritical regime. Formally, the censored dynamics is defined as follows: at each step, a new state σ is generated according to the original rule of the Glauber dynamics, and if a negative magnetization is reached (S(σ ) < 0) then σ is replaced by −σ . Interestingly, this simple modification suffices to boost the mixing-time back to order n log n, just as in the high temperature case, and thus raises the question of whether the symmetry between the high temperature regime and the low temperature censored regime applies also to the existence of cutoff. In a companion √ paper [5], we strengthen the result of [11] by showing that the scaling window of 1/ n exists also for the censored low temperature case, beyond which cutoff indeed occurs (yet at a different location than in the symmetric high temperature point). Theorem 4. Let δ > 0 be such that δ 2 n → ∞ arbitrarily slowly with n. Then the censored Glauber dynamics for the mean field Ising model with parameter β = 1 + δ has a cutoff at 1 n 1 + log(δ 2 n) tn = 2 2(ζ 2 β/δ − 1) δ with a window of order n/δ. In the special case of the dynamics started from the all-plus configuration, the cutoff constant is [2(ζ 2 β/δ − 1)]−1 (the order of the cutoff point and the window size remain the same). Theorem 5. Let δ > 0 be such that δ 2 n → ∞ arbitrarily slowly with n. Then the censored Glauber dynamics for the mean field Ising model with parameter β = 1 + δ has a spectral gap of order δ/n. Recalling√Theorem 1, the above confirms that there is a symmetric scaling window of order 1/ n around the critical temperature, beyond which there is cutoff both at high and at low temperatures, with the same order of mixing-time (yet with a different constant), cutoff window and spectral gap. The rest of this paper is organized as follows. Section 2 contains a brief outline of the proofs of the main theorems. Several preliminary facts on the Curie-Weiss model and on one-dimensional chains appear in Sect. 3. Sections 4, 5 and 6 address the high temperature regime (Theorem 1), critical temperature regime (Theorem 2) and low temperature regime (Theorem 3) respectively. 2. Outline of Proof In what follows, we present a sketch of the main ideas and arguments used in the proofs of the main theorems. We note that the analysis of the critical window relies on arguments similar to those used for the subcritical and supercritical regimes. Namely, to obtain the order of the mixing-time in Theorem 2 (critical window), we study the magnetization chain using the arguments that appear in the proof of Theorem 1 (high temperature regime). It is then straightforward to show that the mixing-time of the entire Glauber dynamics has the very same order. In turn, the spectral-gap in the critical window is obtained using arguments similar to those used in the proof of Theorem 3 (low temperature regime). In light of this, the following sketch will focus on the two non-critical temperature regimes.
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2.1. High temperature regime. Upper bound for mixing. As mentioned above, a key element in the proof is the analysis of the normalized magnetization chain, (St ), which is the average spin in the system. That is, for a given configuration σ , we define S(σ ) to be n1 i σ (i), and it is easy to verify that this function of the dynamics is an irreducible and aperiodic Markov chain. Clearly, a necessary condition for the mixing of the dynamics is the mixing of its magnetization, but interestingly, in our case the converse essentially holds as well. For instance, as we later explain, in the special case where the starting state is the all-plus configuration, by symmetry these two chains have precisely the same total variation distance from equilibrium at any given time. In order to determine the behavior of the chain (St ), we first keep track of its expected value along the Glauber dynamics. To simplify the sketch of the argument, suppose that our starting configuration is somewhere near the all-plus configuration. √ In this case, one can show that ESt is monotone decreasing in t, and drops to order 1/δn precisely at the cutoff point. Moreover, if we allow the dynamics to perform another (n/δ) steps (our cutoff window), then the magnetization will hit 0 (or n1 , depending on the parity of n) with probability arbitrarily close to 1. At that point, we essentially achieve the mixing of the magnetization chain. It remains to extend the mixing of the magnetization chain to the mixing of the entire Glauber dynamics. Roughly, keeping in mind the above comment on the symmetric case of the all-plus starting configuration, one can apply a similar argument to an arbitrary starting configuration σ , by separately treating the set of spins which were initially positive and those which were initially negative. Indeed, it was shown in [11] that the following holds for β < 1 fixed (strictly subcritical regime). After a “burn-in” period of order n steps, the magnetization typically becomes not too biased. Next, if one runs two instances of the dynamics, from two such starting configurations (where the magnetization is not too biased), then by the time it takes their magnetization chains to coalesce, the entire configurations become relatively similar. This was established by a so-called Two Coordinate Chain analysis, where the two coordinates correspond to the current sum of spins along the set of sites which were initially either positive or negative respectively. By extending the above Two Coordinate Chain Theorem to the case of β = 1 − δ, where δ = δ(n) satisfies δ 2 n → ∞, and combining it with second moment arguments and some additional ideas, we were able to show that the above behavior holds throughout this mildly subcritical regime. The burn-in time required for the typical magnetization to become “balanced” now has order n/δ, and so does the time it takes the full dynamics of two chains to coalesce once their magnetization chains have coalesced. Thus, these two periods are conveniently absorbed in our cutoff window, making the cutoff of the magnetization chain the dominant factor in the mixing of the entire Glauber dynamics. Lower bound for mixing. While the above mentioned Two Coordinate Chain analysis was required in order to show that the entire Glauber dynamics mixes fairly quickly once its magnetization chain reaches equilibrium, the converse is immediate. Thus, we will deduce the lower bound on the mixing time of the dynamics solely from its magnetization chain. The upper bound in this regime relied on an analysis of the first and second moments of the magnetization chain, however this approach is too coarse to provide a precise lower bound for the cutoff. We therefore resort to establishing an upper bound on the third
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moment of the magnetization chain, using which we are able to fine-tune our analysis of how its first moment changes along time. Examining the state of the system order n/δ steps before the alleged cutoff point, using concentration inequalities, we show that the magnetization chain is typically substantially far from 0. Recalling Fig. 2, this implies a lower bound on the total variation distance of the magnetization chain to stationarity, as required. Spectral gap analysis. In the previous arguments, we stated that the magnetization chain essentially dominates the mixing-time of the entire dynamics. An even stronger statement holds for the spectral gap: the Glauber dynamics and its magnetization chain have precisely the same spectral gap, and it is in both cases attained by the second largest eigenvalue. We therefore turn to establish the spectral gap of (St ). The lower bound follows directly from the contraction properties of the chain in this regime. To obtain a matching upper bound, we use the Dirichlet representation for the spectral gap, combined with an appropriate bound on the fourth moment of the magnetization chain.
2.2. Low temperature regime. Exponential mixing. As mentioned above, the exponential mixing in this regime follows directly from the behavior of the magnetization chain, which has a bottleneck between ±ζ . To show this, we analyze the effective resistance between these two centers of mass, and obtain the precise order of the commute time between them. Additional arguments show that the mixing time of the entire Glauber dynamics in this regime has the same order. Spectral gap analysis. In the above mentioned proof of the exponential mixing, we establish that the commute time of the magnetization chain between 0 and ζ has the same order as the hitting time from 1 to 0. We can therefore apply a recent result of [6] for general birth-and-death chains, which implies that in this case the inverse of the spectral-gap (known as the relaxation-time) and the mixing-time must have the same order. 3. Preliminaries 3.1. The magnetization chain. The normalized magnetization of a configuration σ ∈ , denoted by S(σ ), is defined as S(σ ) :=
n 1 σ (i). n i=1
Suppose that the current state of the Glauber dynamics is σ , and that site i has been selected to have its spin updated. By definition, the probability of updating this site to a positive spin is given by p + (S(σ ) − σ (i)/n), where p + (s) :=
eβs 1 + tanh(βs) . = eβs + e−βs 2
(3.1)
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Similarly, the probability of updating the spin of site i to a negative one is given by p − (S(σ ) − σ (i)/n), where p − (s) :=
e−βs 1 − tanh(βs) = . eβs + e−βs 2
(3.2)
It follows that the (normalized) magnetization of the Glauber dynamics at each step is a Markov chain, (St ), with the following transition kernel: ⎧ 1+s − −1 ⎪ if s = s − n2 , ⎨ 2 p (s − n ) + −1 PM (s, s ) = 1−s (3.3) if s = s + n2 , 2 p (s + n ) ⎪ ⎩1 − 1+s p − (s − n −1 ) − 1−s p + (s + n −1 ) if s = s. 2 2 An immediate important property that the above reveals is the symmetry of St : the distribution of (St+1 | St = s) is precisely that of (−St+1 | St = −s). As evident from the above transition rules, the behavior of the Hyperbolic tangent will be useful in many arguments. This is illustrated in the following simple calculation, showing that the minimum over the holding probabilities of the magnetization chain is nearly 21 . Indeed, since the derivative of tanh(x) is bounded away from 0 and 1 for all x ∈ [0, β] and any β = O(1), the Mean Value Theorem gives 1−s 2 )= (1 + tanh(βs)) + O(n −1 ), n 4 1+s 2 (3.4) PM (s, s − ) = (1 − tanh(βs)) + O(n −1 ), n 4 1 PM (s, s) = (1 + s tanh(βs)) − O(n −1 ). 2 Therefore, the holding probability in state s is at least 21 − O n1 . In fact, since tanh(x) is monotone increasing, PM (s, s) ≤ 21 + 21 s tanh(βs) for all s, hence these probabilities are also bounded from above by 21 (1 + tanh(β)) < 1. Using the above fact, the next lemma will provide an upper bound for the coalescence time of two magnetization chains, St and S˜t , in terms of the hitting time τ0 , defined as τ0 := min{t : |St | ≤ n −1 }. PM (s, s +
Lemma 3.1. Let (St ) and ( S˜t ) denote two magnetization chains, started from two arbitrary states. Then for any ε > 0, there exists some cε > 0, such that the following holds: if T > 0 satisfies P1 (τ0 ≥ T ) < ε, then St and S˜t can be coupled in a way such that they coalesce within at most cε T steps with probability at least 1 − ε. Proof. Assume without loss of generality that | S˜0 | < |S0 |, and by symmetry, that σ = |S0 | ≥ 0. Define τ := min t : |St | ≤ | S˜t | + n2 . Recalling the definition of τ0 , clearly we must have τ < τ0 . Next, since the holding probability of St at any state s is bounded away from 0 and 1 for large n (by the discussion preceding the lemma), there clearly exists a constant 0 < b < 1 such that
P St+1 = S˜t+1 |St − S˜t | ≤ n2 > b > 0
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1 (for instance, one may choose b = 10 (1 − tanh(β)) for a sufficiently large n). It there˜ fore follows that |Sτ +1 | = | Sτ +1 | with probability at least b. Condition on this event. We claim that in this case, the coalescence of (St ) and ( S˜t ) (rather than just their absolute values) occurs at some t ≤ τ0 +1 with probability at least b. The case Sτ +1 = S˜τ +1 is immediate, and it remains to deal with the case Sτ +1 = − S˜τ +1 . Let us couple (St ) and ( S˜t ) so that the property St = − S˜t is maintained henceforth. Thus, at time t = τ0 we obtain |St − S˜t | = 2|St | ≤ n2 , and with probability b this yields St+1 = S˜t+1 . Clearly, our assumption on T and the fact that 0 ≤ σ ≤ 1 together give
Pσ (τ0 ≥ T ) ≤ P1 (τ0 ≥ T ) < ε. Thus, with probability at least (1 − ε)b2 , the coalescence time of (St ) and ( S˜t ) is at most T . Repeating this experiment a sufficiently large number of times then completes the proof. In order to establish cutoff for the magnetization chain (St ), we will need to carefully track its moments along the Glauber dynamics. By definition (see (3.3)), the behavior of these moments is governed by the Hyperbolic tangent function, as demonstrated by the following useful form for the conditional expectation of St+1 given St (see also [11, (2.13)]), E [St+1 | St = s] = s + n2 PM s, s + n2 + s PM (s, s) + s − n2 PM s, s − n2 = (1 − n −1 )s + ϕ(s) − ψ(s),
(3.5)
where 1 2n s ψ(s) = ψ(s, β, n) := 2n ϕ(s) = ϕ(s, β, n) :=
tanh β(s + n −1 ) + tanh β(s − n −1 ) ,
tanh β(s + n −1 ) − tanh β(s − n −1 ) .
3.2. From magnetization equilibrium to full mixing. The motivation for studying the magnetization chain is that its mixing essentially dominates the full mixing of the Glauber dynamics. This is demonstrated by the next straightforward lemma (see also [11, Lemma 3.4]), which shows that in the special case where the starting point is the all-plus configuration, the mixing of the magnetization is precisely equivalent to that of the entire dynamics. Lemma 3.2. Let (X t ) be an instance of the Glauber dynamics for the mean field Ising model starting from the all-plus configuration, namely, σ0 = 1, and let St = S(X t ) be its magnetization chain. Then P1 (X t ∈ ·) − µn TV = P1 (St ∈ ·) − πn TV , where πn is the stationary distribution of the magnetization chain.
(3.6)
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Proof. For any s ∈ {−1, −1 + n2 , . . . , 1 − n2 , 1}, let s := {σ ∈ : S(σ ) = s}. Since by symmetry, both µn (· | s ) and P1 (X t ∈ · | St = s) are uniformly distributed over s , the following holds: 1 |P1 (X t = σ ) − µn (σ )| P1 (X t ∈ ·) − µn TV = 2 s σ ∈s 1 P1 (St = s) µn (s ) − = | | 2 s |s | s σ ∈s
= P1 (St ∈ ·) − πn TV . In the general case where the Glauber dynamics starts from an arbitrary configuration σ0 , though the above equivalence (3.6) no longer holds, the magnetization still dominates the full mixing of the dynamics in the following sense. The full coalescence of two instances of the dynamics occurs within order n log n steps once the magnetization chains have coalesced. Lemma 3.3. ([11, Lemma 2.9]) Let σ, σ˜ ∈ be such that S(σ ) = S(σ˜ ). For a coupling (X t , X˜ t ), define the coupling time τ X, X˜ := min{t ≥ 0 : X t = X˜ t }. Then for a sufficiently large c0 > 0 there exists a coupling (X t , X˜ t ) of the Glauber dynamics with initial states X 0 = σ and X˜ 0 = σ˜ such that
lim sup Pσ,σ˜ τ X, X˜ > c0 n log n = 0. n→∞
Though Lemma 3.3 holds for any temperature, it will only prove useful in the critical and low temperature regimes. At high temperature, using more delicate arguments, we will establish full mixing within order of nδ steps once the magnetization chains have coalesced. That is, the extra steps required to achieve full mixing, once the magnetization chain cutoff had occurred, are absorbed in the cutoff window. Thus, in this regime, the entire dynamics has cutoff precisely when its magnetization chain does (with the same window). 3.3. Contraction and one-dimensional Markov chains. We say that a Markov chain, assuming values in R, is contracting, if the expected distance between two chains after a single step decreases by some factor bounded away from 0. As we later show, the magnetization chain is contracting at high temperatures, a fact which will have several useful consequences. One example of this is the following straightforward lemma of [11], which provides a bound on the variance of the chain. Here and throughout the paper, the notation Pz , Ez and Var z will denote the probability, expectation and variance respectively given that the starting state is z. Lemma 3.4. ([11, Lemma 2.6]) Let (Z t ) be a Markov chain taking values in R and with transition matrix P. Suppose that there is some 0 < ρ < 1 such that for all pairs of starting states (z, z˜ ), Ez [Z t ] − Ez˜ [Z t ] ≤ ρ t |z − z˜ |. (3.7) Then vt := supz 0 Var z 0 (Z t ) satisfies vt ≤ v1 min t, 1/ 1 − ρ 2 .
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Remark. By following the original proof of the above lemma, one can readily extend it to the case ρ ≥ 1 and get the following bound:
vt ≤ v1 · ρ 2t min t, 1/ ρ 2 − 1 . (3.8) This bound will prove to be effective for reasonably small values of t in the critical window, where although the magnetization chain is not contracting, ρ is only slightly larger than 1. Another useful property of the magnetization chain in the high temperature regime is its drift towards 0. As we later show, in this regime, for any s > 0 we have E [St+1 |St = s] < s, and with probability bounded below by a constant we have St+1 < St . We thus refer to the following lemma of [12]: Lemma 3.5. ([12, Chap. 18]) Let (Wt )t≥0 be a non-negative supermartingale and τ be a stopping time such that (i) W0 = k, (ii) Wt+1 − Wt ≤ B, (iii) Var(Wt+1 | Ft ) > σ 2 > 0 on the event τ > t. If u > 4B 2 /(3σ 2 ), then Pk (τ > u) ≤
4k √ . σ u
This lemma, together with the above mentioned properties of (St ), yields the following immediate corollary: Corollary 3.6. ([11, Lemma 2.5]) Let β ≤ 1, and suppose that n is even. There exists a constant c such that, for all s and for all u, t ≥ 0, cn|s| P( |Su | > 0, . . . , |Su+t | > 0 | Su = s) ≤ √ . t
(3.9)
Finally, our analysis of the spectral gap of the magnetization chain will require several results concerning birth-and-death chains from [6]. In what follows and throughout the paper, the relaxation-time of a chain, trel , is defined to be gap−1 , where gap denotes its spectral-gap. We say that a chain is b-lazy if all its holding probabilities are at least b, or simply lazy for the useful case of b = 21 . Finally, given an ergodic birth-and-death chain on X = {0, 1, . . . , n} with stationary distribution π , the quantile state Q(α), for 0 < α < 1, is defined to be the smallest i ∈ X such that π({0, . . . , i}) ≥ α. Lemma 3.7. ([6, Lemma 2.9]) Let X (t) be a lazy irreducible birth-and-death chain on 1 {0, 1, . . . , n}, and suppose that for some 0 < ε < 16 we have trel < ε4 · E0 τ Q(1−ε) . Then for any fixed ε ≤ α < β ≤ 1 − ε: 3 E Q(α) τ Q(β) ≤ t · E0 τ Q( 1 ) . (3.10) 2 2ε rel Lemma 3.8. ([6, Lemma 2.3]) For any fixed 0 < ε < 1 and lazy irreducible birth-anddeath chain X , the following holds for any t: P t (0, ·) − π TV ≤ P0 (τ Q(1−ε) > t) + ε,
(3.11)
P t (k, ·) − π TV ≤ Pk (max{τ Q(ε) , τ Q(1−ε) } > t) + 2ε.
(3.12)
and for all k ∈ ,
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Remark. As argued in [6] (see Theorem 3.1 and its proof), the above two lemmas also hold for the case where the birth-and-death chain is not lazy but rather b-lazy for some constant b > 0. The formulation for this more general case incurs a cost of a slightly different constant in (3.10), and replacing t with t/C (for some constant C) in (3.11) and (3.12). As we already established (recall (3.4)), the magnetization chain is indeed b-lazy for any constant b < 21 and a sufficiently large n. 3.4. Monotone coupling. A useful tool throughout our arguments is the monotone coupling of two instances of the Glauber dynamics (X t ) and ( X˜ t ), which maintains a coordinate-wise inequality between the corresponding configurations. That is, given two configurations σ ≥ σ˜ (i.e., σ (i) ≥ σ˜ (i) for all i), it is possible to generate the next two states σ and σ˜ by updating the same site in both, in a manner that ensures that σ ≥ σ˜ . More precisely, we draw a random variable I uniformly over {1, 2, . . . , n} and
independently draw another random variable
U uniformly over [0, 1]. To generate σ ) , otherwise σ (I ) = −1. We from σ , we update site I to +1 if U ≤ p + S(σ ) − σ (I n
perform an analogous process in order to generate σ˜ from σ˜ , using the same I and U as before. The monotonicity of the function p + guarantees that σ ≥ σ˜ , and by repeating this process, we obtain a coupling of the two instances of the Glauber dynamics that always maintains monotonicity. Clearly, the above coupling induces a monotone coupling for the two corresponding magnetization chains. We say that a birth-and-death chain with a transition kernel P and a state-space X = {0, 1, . . . , n} is monotone if P(i, i + 1) + P(i + 1, i) ≤ 1 for every i < n. It is easy to verify that this condition is equivalent to the existence of a monotone coupling, and that for such a chain, if f : X → R is a monotone increasing (decreasing) function then so is P f (see, e.g., [6, Lemma 4.1]).
3.5. The spectral gap of the dynamics and its magnetization chain. To analyze the spectral gap of the Glauber dynamics, we establish the following lemma which reduces this problem to determining the spectral-gap of the one-dimensional magnetization chain. Its proof relies on increasing eigenfunctions, following the ideas of [13]. Proposition 3.9. The Glauber dynamics for the mean-field Ising model and its onedimensional magnetization chain have the same spectral gap. Furthermore, both gaps are attained by the largest nontrivial eigenvalue. Proof. We will first show that the one-dimensional magnetization chain has an increasing eigenfunction, corresponding to the second eigenvalue. Recalling that St assumes values in X := {−1, −1+ n2 , . . . , 1− n2 , 1}, let M denote its transition matrix, and let π denote its stationary distribution. Let 1 = θ0 ≥ θ1 ≥ · · · ≥ θn be the n + 1 eigenvalues of M, corresponding to the eigenfunctions f 0 ≡ 1, f 1 , . . . , f n . Define θ = max{θ1 , |θn |}, and notice that, as St is aperiodic and irreducible, 0 < θ < 1. Furthermore, by the existence of the monotone coupling for St and the discussion in the previous subsection, whenever a function f : X → R is increasing so is M f . Define f : I → R by f := f 1 + f n + K 1, where 1 is the identity function and K > 0 is sufficiently large to ensure that f is monotone increasing (e.g., K = n2 f 1 + f n L ∞ easily suffices). Notice that, by symmetry of St , π(x) = π(−x) for all x ∈ X , and in particular x∈X xπ(x) = 0, that is to say, 1, f 0 L 2 (π ) = 0. Recalling that for all
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i = j we have f i , f j L 2 (π ) = 0, it follows that for some q1 , . . . , qn ∈ R we have n f = i=1 qi f i with q1 = 0 and qn = 0, and thus n
m θ −1 M f = qi (θi /θ )m f i . i=1
Next, define
g=
q1 f 1 if θ = θ1 , 0 otherwise
and h =
and notice that
2m f = g + h, and lim θ −1 M m→∞
qn f n if θ = −θn , 0 otherwise
lim
m→∞
θ −1 M
2m+1
f = g − h.
As stated above, M m f is increasing for all m, and thus so are the two limits g + h and g − h above, as well as their sum. We deduce that g is an increasing function, and next claim that g ≡ 0. Indeed, if g ≡ 0 then both h and −h are increasing functions, hence necessarily h ≡ 0 as well; this would imply that q1 = qn = 0, thus contradicting our construction of f . We deduce that g is an increasing eigenfunction corresponding to θ1 = θ , and next wish to show that it is strictly increasing. Recall that for all x ∈ X , 2 2 2 2 g x− + M(x, x)g(x) + M x, x + g x+ . (Mg)(x) = M x, x − n n n n Therefore, if for some x ∈ X we had g(x − n2 ) = g(x) ≥ 0, the fact that g is increasing would imply that θ1 g(x) = (Mg)(x) ≥ g(x) ≥ 0, and analogously, if g(x) = g(x + n2 ) ≤ 0 we could write θ1 g(x) = (Mg)(x) ≤ g(x) ≤ 0. In either case, since 0 < θ1 < 1 (recall that θ1 = θ ) this would in turn lead to g(x) = 0. By inductively substituting this fact in the above equation for (Mg)(x), we would immediately get g ≡ 0, a contradiction. Let 1 = λ0 ≥ λ1 ≥ · · · ≥ λ||−1 denote the eigenvalues of the Glauber dynamics, and let λ := max{λ1 , |λ2n −1 |}. We translate g into a function G : → R in the obvious manner: n 1 G(σ ) := g(S(σ )) = g σ (i) . n i=1
One can verify that G is indeed an eigenfunction of the Glauber dynamics corresponding to the eigenvalue θ1 , and clearly G is strictly increasing with respect to the coordinatewise partial order on . At this point, we refer to the following lemma of [13]: Lemma 3.10. ([13, Lemma 4]) Let P be the transition matrix of the Glauber dynamics, and let λ1 be its second largest eigenvalue. If P has a strictly increasing eigenfunction f , then f corresponds to λ1 .
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The above lemma immediately implies that G corresponds to the second eigenvalue of Glauber dynamics, which we denote by λ1 , and thus λ1 = θ1 . It remains to show that λ = λ1 . To see this, first recall that all the holding probabilities of St are bounded away from 0, and the same applies to the entire Glauber dynamics by definition (the magnetization remains the same if and only if the configuration remains the same). Therefore, both θn and λ2n −1 are bounded away from −1, and it remains to show that gap = o(1) for the Glauber dynamics (and hence also for its magnetization chain). To see this, suppose P is the transition kernel of the Glauber dynamics, and recall the Dirichlet representation for the second eigenvalue of a reversible chain (see [12, Lemma 13.7], and also [1, Chap. 3]): E( f ) 1 − λ1 = min : f ≡ 0, Eµn ( f ) = 0 , (3.13) f, f µn where Eµn ( f ) denotes 1, f µn , and
"2 1 ! f (σ ) − f (σ ) µn (σ )P(σ, σ ). 2
σ,σ ∈ n By considering the sum of spins, h(σ ) = i=1 σ (i), we get E(h) ≤ 2, and since the spins are positively correlated, Var µn i σ (i) ≥ n. It follows that E( f ) = (I − P) f, f µn =
1 − λ1 ≤ 2/n, and thus gap = 1 − λ1 = 1 − θ1 for both the Glauber dynamics and its magnetization chain, as required. 4. High Temperature Regime In this section we prove Theorem 1. Subsect. 4.1 establishes the cutoff of the magnetization chain, which immediately provides a lower bound on the mixing time of the entire dynamics. The matching upper bound, which completes the proof of cutoff for the Glauber dynamics, is given in Subsect. 4.2. The spectral gap analysis appears in Subsect. 4.3. Unless stated otherwise, assume throughout this section that β = 1 − δ, where δ 2 n → ∞. 4.1. Cutoff for the magnetization chain. Clearly, the mixing of the Glauber dynamics ensures the mixing of its magnetization. Interestingly, the converse is also essentially true, as the mixing of the magnetization turns out to be the most significant part in the mixing of the full Glauber dynamics. We thus wish to prove the following cutoff result: Theorem 4.1. Let β = 1 − δ, where δ > 0 satisfies δ 2 n → ∞. Then the corresponding magnetization chain (St ) exhibits cutoff at time 21 · nδ log(δ 2 n) with a window of order n/δ. Notice that Lemma 3.2 then gives the following corollary for the special case where the initial state of the dynamics is the all-plus configuration: Corollary 4.2. Let δ = δ(n) > 0 be such that δ 2 n → ∞ with n, and let (X t ) denote the Glauber dynamics for the mean-field Ising model with parameter β = 1 − δ, started from the all-plus configuration. Then (X t ) exhibits cutoff at time 21 (n/δ) log(δ 2 n) with window size n/δ.
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4.1.1. Upper bound. Our goal in this subsection is to show the following: 1 n n 2 lim lim sup dn · log(δ n) + γ = 0, γ →∞ n→∞ 2 δ δ
(4.1)
where dn (·) is with respect to the magnetization chain (St ) and its stationary distribution. This will be obtained using an upper bound on the coalescence time of two instances of the magnetization chain. Given the properties of its stationary distribution (see Fig. 2), we will mainly be interested in the time it takes this chain to hit near 0. The following theorem provides an upper bound for that hitting time. Theorem 4.3. For 0 < β < 1 + O(n −1/2 ), consider the magnetization chain started from some arbitrary state s0 , and let τ0 = min{t : |St | ≤ n −1 }. Write β = 1 − δ, and for γ > 0 define ⎧ n n ⎪ log(δ 2 n) + (γ + 3) δ 2 n → ∞, ⎨ 2δ δ tn (γ ) =
(4.2)
# ⎪ ⎩ 200 + 6γ 1 + 6 δ 2 n n 3/2 δ 2 n = O(1). √ Then there exists some c > 0 such that Ps0 (τ0 > tn (γ )) ≤ c/ γ . Proof. For any t ≥ 1, define:
! " st := Es0 |St |1{τ0 >t} .
Suppose s > 0. Recalling (3.5) and bearing in mind the concavity of the Hyperbolic tangent and the fact that ψ(s) ≥ 0, we obtain that 1 E(St+1 St = s) ≤ s + (tanh(βs) − s) . n Using symmetry for the case s < 0, we can then deduce that " ! 1 E |St+1 | St ≤ |St | + (tanh(β|St |) − |St |) for any t < τ0 . n
(4.3)
Hence, combining the concavity of the Hyperbolic tangent together with Jensen’s inequality yields 1 1 st+1 ≤ 1 − st + tanh(βst ). (4.4) n n Since the Taylor expansion of tanh(x) is tanh(x) = x −
x 3 2x 5 17x 7 + − + O(x 9 ), 3 15 315
(4.5)
3
we have tanh(x) ≤ x − x5 for 0 ≤ x ≤ 1, giving 1 1 1 1 (st )3 st + tanh(βst ) ≤ 1 − st + βst − st+1 ≤ 1 − n n n n 5n = st −
δ (st )3 st − . n 5n
(4.6)
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J. Ding, E. Lubetzky, Y. Peres
For some 1 < a ≤ 2 to be defined later, set bi = a −i /4, and u i = min{t : st ≤ bi }. Notice that st is decreasing in t by (4.6), thus for every t ∈ [u i , u i+1 ] we have bi /a = bi+1 ≤ st ≤ bi . It follows that st+1 ≤ st − and u i+1 − u i ≤
b3 δ bi · − i3 , n a 5a n
bi3 δ bi a−1 5(a − 1)a 2 n . bi / + 3 ≤ a n a 5a n 5δa 2 + bi2
(4.7)
For the case δ 2 n → ∞, define:
√ i 0 = min{i : bi ≤ 1/ δn}.
The following holds: i0 i0 5(a − 1)a 2 n 5(a − 1)a 2 n 5(a − 1)a 2 n (u i+1 − u i ) ≤ ≤ + 5δa 2 5δa 2 + bi2 bi2 i=1 i=1 i≤i i≤i
≤
0
0
bi2 >δ
bi2 ≤δ
a−1 n 5na 2 + · log(δ 2 n), δ(a + 1) 2 log a δ
where in the last inequality we used the fact that the series {bi−2 } is a geometric series with a ratio a 2 , and that, as bi2 ≥ 1/(δn) for all i ≤ i 0 , the number of summands such √ that bi2 ≤ δ is at most loga ( δ 2 n). Therefore, choosing a = 1 + n −1 , we deduce that: i0 1 5 n n + O(n −1 ) + + O(n −1 ) log(δ 2 n) (u i+1 − u i ) ≤ 2 δ 2 δ i=1
≤3
n n + log(δ 2 n), δ 2δ
(4.8)
where the last inequality holds for any sufficiently large n. Combining the above inequality and the definition of i 0 , we deduce that √ ! " (4.9) ( stn (0) = ) Es0 |Stn (0) |1{τ0 >tn (0)} ≤ 1/ δn. Thus, by Corollary 3.6 (after taking expectation), for some fixed c > 0, √ P(τ0 > tn (γ )) ≤ c/ γ . For the case δ 2 n = O(1), choose a = 2, that is, bi = 2−(i+2) , and define # i 1 = min{i : bi ≤ n −1/4 ∨ 5 |δ|}.
(4.10)
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741
1 2 Substituting a = 2 in (4.7), while noting that δ > − 25 bi for all i < i 1 , gives i1
i1
i1 20n n n (u i+1 − u i ) ≤ ≤ 100 ≤ 200 2 2 2 20δ + bi b bi1 i=1 i=1 i=1 i n ≤ 200n 3/2 , ≤ 200n 3/2 ∧ 8 |δ|
where the last inequality in the first line incorporated the geometric sum over {bi−2 }. By (4.10),
# bi1 ≤ n −1/4 ∨ 5 |δ| ≤ n −1/4 1 + 5(δ 2 n)1/4 , and as in the subcritical case, we now combine the above results with an application of Corollary 3.6 (after taking expectation), and deduce that for some absolute constant c > 0, √ P(τ0 > tn (γ )) ≤ c/ γ , as required.
Apart from drifting toward 0, and as we had previously mentioned, the magnetization chain at high temperatures is in fact contracting; this is a special case of the following lemma. Lemma 4.4. Let (St ) and ( S˜t ) be the corresponding magnetization chains of two instances of the Glauber dynamics for some β = 1 − δ (where δ is not necessarily positive), and put Dt := St − S˜t . The following then holds: δ |Dt | E[Dt+1 − Dt | Dt ] ≤ − Dt + 2 + O(n −4 ). n n Proof. By definition (recall (3.5)), we have
(4.11)
E[Dt+1 − Dt | Dt ] = E[St+1 − St + S˜t − S˜t+1 | Dt ] S˜t − St = + ϕ(St ) − ϕ( S˜t ) − ψ(St ) − ψ( S˜t ) . n The Mean Value Theorem implies that β ϕ(St ) − ϕ( S˜t ) ≤ (St − S˜t ), n and applying Taylor expansions on tanh(x) around β St and β S˜t , we deduce that 1 St S˜t ψ(St ) − ψ( S˜t ) = 2 . + O − n4 n cosh2 (β St ) n 2 cosh2 (β S˜t ) Since the derivative of the function x/ cosh2 (βx) is bounded by 1, another application of the Mean Value Theorem gives |S − S˜ | 1 t t . + O ψ(St ) − ψ( S˜t ) ≤ n2 n4 Altogether, we obtain (4.11), as required.
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Indeed, the above lemma ensures that in the high temperature regime, β = 1 − δ, where δ > 0, the magnetization chain is contracting: " ! δ E |Dt+1 | Dt ≤ 1 − (4.12) |Dt | for any sufficiently large n. 2n We are now ready to prove that hitting near 0 essentially ensures the mixing of the magnetization. Lemma 4.5. Let β = 1 − δ for δ > 0 with δ 2 n → ∞, (X t ) and ( X˜ t ) be two instances of the dynamics started from arbitrary states σ0 and σ˜ 0 respectively, and (St ) and ( S˜t ) be their corresponding magnetization chains. Let τmag denote the coalescence time τmag := min{t : St = S˜t }, and tn (γ ) be as defined in Theorem 4.3. Then there exists some constant c > 0 such that √ P τmag > tn (3γ ) ≤ c/ γ for all γ > 0. (4.13) Proof. Set T = tn (γ ). We claim that the following holds for large n: 2 2 |ESt | ≤ √ and |E S˜t | ≤ √ for all t ≥ T . δn δn
(4.14)
To see this, first consider the case where n is even. The above inequality then follows directly from (4.9) and the decreasing property of st (see (4.6)), combined with the fact that E0 St = 0 (and thus √ ESt = 0 for all t ≥ τ0 ). In fact, in case n is even, |ESt | and |E S˜t | are both at most 1/ δn for all t ≥ T . For the case of n odd (where there is no 0 state for the magnetization chain, and τ0 is the hitting time to ± n1 ), a simple way to show that (4.14) holds is to bound |E 1 St |. By definition, PM ( n1 , n1 ) ≥ PM ( n1 , − n1 ) (see (3.3)). n Combined with the symmetry of the positive and negative parts of the magnetization t ( 1 , k ) ≥ P t ( 1 , − k ) for any odd k > 0 chain, one can then verify by induction that PM M n n n n and any t. Therefore, by symmetry as well as the fact that Es0 St ≤ s0 for positive s0 , we conclude that |E 1 St | is decreasing with t, and thus is bounded by n1 . This implies that n (4.14) holds for odd n as well. Combining (4.14) with the Cauchy-Schwartz inequality we obtain that for any t ≥ T , $ $ 4 4 ˜ ˜ E|St − St | ≤ E|St | + E| St | ≤ Var(St ) + + Var( S˜t ) + . δn δn Now, combining Lemma 3.4 and Lemma 4.4 (and in particular, (4.12)), we deduce that 4 Var St ≤ δn , and plugging this into the above inequality gives 10 E|St − S˜t | ≤ √ δn
for any t ≥ T.
We next wish to show that within 2γ n/δ additional steps, St and S˜t coalesce with prob√ ability at least 1 − c/ γ for some constant c > 0. Consider time T , and let Dt := St − S˜t . Recall that we have already established that √ EDT ≤ 10/ δn, (4.15)
Mixing Time Evolution of Glauber Dynamics
743
and assume without loss of generality that DT > 0. We now run the magnetization chains St and S˜t independently for T ≤ t ≤ τ1 , where τ1 := min t ≥ T : Dt ∈ {0, − n2 } , and let Ft be the σ -field generated by these two chains up to time t. By Lemma 4.4, we deduce that for sufficiently large values of n, if Dt > 0 then E[Dt+1 − Dt | Ft ] ≤ −
δ Dt ≤ 0, 2n
(4.16)
and Dt is a supermartingale with respect to Ft . Hence, so is Wt := DT +t ·
n 1{τ >t} , 2 1
and it is easy to verify that Wt satisfies the conditions of Lemma 3.5 (recall the upper bound on the holding probability of the magnetization chain, as well as the fact that at most one spin is updated at any given step). Therefore, for some constant c > 0, P (τ1 > tn (2γ ) | DT ) = P(W0 > 0, W1 > 0, . . . , Wtn (2γ )−T > 0 | DT ) cn DT . ≤ √ γ n/δ Taking expectation and plugging in (4.15), we get that for some constant c , c
P (τ1 > tn (2γ )) ≤ √ . γ
(4.17)
From time τ1 and onward, we couple St and S˜t using a monotone coupling, thus Dt becomes a non-negative supermartingale with Dτ1 ≤ n2 . By (4.16), E [Dt+1 − Dt | Ft ] ≤ −
δ for τ1 ≤ t < τmag , n2
and therefore, the Optional Stopping Theorem for non-negative supermartingales implies that, for some constant c
, E(τmag − τ1 ) c
P τmag − τ1 ≥ n/δ ≤ ≤ . n/δ γ Combining (4.17) and (4.18) we deduce that for some constant c, c P τmag > tn (3γ ) ≤ √ , γ completing the proof.
(4.18)
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4.1.2. Lower bound. We need to prove that the following statement holds for the distance of the magnetization at time t from stationarity: lim lim inf dn
γ →∞ n→∞
1 n n · log(δ 2 n) − γ 2 δ δ
= 1.
(4.19)
The idea is to show that, at time 21 · nδ log(δ 2 n)−γ nδ , the expected magnetization remains large. Standard concentration inequalities will then imply that the magnetization will typically be significantly far from 0, unlike its stationary distribution. To this end, we shall first analyze the third moment of the magnetization chain. Recalling the transition rule (3.3) of St under the notations (3.1),(3.2), p + (s) =
1 + tanh(βs) , 2
p − (s) =
1 − tanh(βs) , 2
the following holds: 3 E St+1 | St = s
1+s − 2 3 1−s + 2 3 −1 −1 = + p (s − n ) s − p (s + n ) s + 2 n 2 n 1+s − 1 − s p (s − n −1 ) − p + (s + n −1 ) s 3 + 1− 2 2
6s 2 1
· −2s + tanh β(s − n −1 ) + tanh β(s + n −1 ) = s3 +
n 4
s c2 + s tanh β(s − n −1 ) − tanh β(s + n −1 + c1 2 + 3 . n n
(4.20)
As tanh(x) ≤ x for x ≥ 0, for every s > 0 we get s 3s 2
c2 3 −2s + β(s − n −1 ) + β(s + n −1 ) + c1 2 + 3 E St+1 | St = s ≤ s 3 + 2n n n δ 3 c1 c2 3 = s − 3 s + 2s + 3. (4.21) n n n If s = 0, the above also holds, since in that case |St+1 |3 ≤ (2/n)3 . Finally, by symmetry, 3 | = −S 3 given S = s is the same as that of S 3 if s < 0 then the distribution of |St+1 t t+1 t+1 given St = |s|, and altogether we get: δ c1 c2 E |St+1 |3 | St = s ≤ |s|3 − 3 |s|3 + 2 |s| + 3 . n n n We deduce that δ c1 c2 E|St+1 |3 ≤ E |St |3 − 3 |St |3 + 2 |St | + 3 n n n 3δ c c 1 2 E|St |3 + 2 E|St | + 3 . ≤ 1− n n n
(4.22)
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Note that the following statement holds for the first moment of St : Es0 [|St |] ≤ (Es0 St )2 + Var s0 (St ) $ 16 4 δ t ≤ (st )2 + |s0 | + √ . ≤ 1− δn n δn Hence,
δ t 3δ c1 2 c2 3 Es0 |St | + 2 1 − Es0 |St+1 | ≤ 1 − |s0 | + √ + 3 2 n n n n δn n
δ2 c c 4 1 = η3 Es0 |St |3 + ηt 2 |s0 | + √ + 2 2 , n n n 2 δn 3
where η = 1−δ/n, and the extra error term involving c2 absorbs the change of coefficient of Es0 |St |3 and also the 1/n 3 term. Iterating, we obtain t t c1
c2 δ 2 3 j 3 3t 3 t c1 2j η + η Es0 |St+1 | ≤ η |s0 | + η 2 |s0 | √ + 2 n n n 2 δn j=0 j=0 c1
c2 δ 2 |s0 | 1 3t 3 t c1 ≤ η |s0 | + η 2 · + √ + 2 · 2 2 n 1−η n 1 − η3 n δn ≤ η3t |s0 |3 + ηt
c1
c2 δ c1 + |s0 | + . δn (δn)3/2 n
(4.23)
the Define Z t := |St |η−t , whence Z 0 = |S0 | = |s0 |. Recalling (3.5), and combining Taylor expansion of tanh(x) given in (4.5) with the fact that |ψ(s)| = O s/n 2 , we get that for s > 0, " ! s s3 − 2. E |St+1 | St = s ≥ ηs − 2n n By symmetry, an analogous statement holds for s < 0, and altogether we obtain that " ! |St | |St |3 − 2 . E |St+1 | St ≥ η|St | − 2n n
(4.24)
Remark. Note that (4.24) in fact holds for any temperature, having followed from the basic definition of the transition rule of (St ), rather than from any special properties that this chain may have in the high temperature regime. Rearranging the terms and multiplying by η−(t+1) , we obtain that for any sufficiently large n, & % 1 2 E 1 − 2 Z t − Z t+1 | St ≤ η−t |St |3 , n n where we used the fact that η−1 ≤ 2. Taking expectation and plugging in (4.23), we deduce that % & c1
c2 δ 2 1 c1 η2t |s0 |3 + |s0 |+η−t . (4.25) Es0 1− 2 Z t − Z t+1 ≤ + n n δn (δn)3/2 n
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Set t=
n log(δ 2 n) − γ n/δ, 2δ
and notice that when n is sufficiently large, 1 − multiplying (4.25) by (1 −
2 −(t+1) ) n2
2 n2
−(t+1)
≤ 2 for any t ≤ t. Therefore,
and summing over gives:
c1
c2 δ 2|s0 |3 c1 2η−t + |s0 | − 2Es0 Z t ≤ + t 2 |s0 | + n(1 − η2 ) δn n(1 − η) (δn)3/2 n c1
c2 δ 2|s0 |3 c1 log(δ 2 n) c2 ≤ + |s0 | + 3/2 + 3/2 + √ δ 2δ 2 n δ n δn n 3 √ 2|s0 | + o( δ + |s0 |), = δ where√the last inequality follows from the assumption δ 2 n → ∞. We now select s0 = δ/3, which gives √
√ √ δ/3 − 2Es0 Z t ≤ 2 δ/27 + o( δ),
and for a sufficiently large n we get Es0 Z t ≥
√
δ/9.
Recalling the definition of Z t , and using the well known fact that (1−x) ≥ exp(−x/(1− x)) for 0 < x < 1, we get that for a sufficiently large n, √ eγ /2 Es0 |St | ≥ ηt δ/9 ≥ √ =: L . 10 δn
(4.26)
Lemma 3.4 implies that max{Var s0 (St ), Var µn ( S˜t )} ≤ 16/δn. Therefore, recalling that Eµn S˜t = 0, Chebyshev’s inequality gives 16/(δn) = ce−γ , Ps0 (|St | ≤ L/2) ≤ Ps0 (|St | − Es0 |St | ≥ L/2) ≤ L 2 /4 16/(δn) = ce−γ . Pµn (| S˜t | ≥ L/2) ≤ L 2 /4 " ! Hence, letting π denote the stationary distribution of St , and A L denote the set − L2 , L2 , we obtain that Ps0 (St ∈ ·) − π TV ≥ π(A L ) − Ps0 (|St | ∈ A L ) ≥ 1 − 2ce−γ , which immediately gives (4.19).
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4.2. Full mixing of the Glauber dynamics. In order to boost the mixing of the magnetization into the full mixing of the configurations, we will need the following result, which was implicitly proved in [11, Sects. 3.3, 3.4] using a Two Coordinate Chain analysis. Although the authors of [11] were considering the case of 0 < β < 1 fixed, one can follow the same arguments and extend this result to any β < 1. Following is this generalization of their result: Theorem 4.6. ([11]) Let (X t ) be an instance of the Glauber dynamics and µn the stationary distribution of the dynamics. Suppose X 0 is supported by 0 := {σ ∈ : |S(σ )| ≤ 1/2}. For any σ0 ∈ 0 and σ˜ ∈ , we consider the dynamics (X t ) starting from σ0 and an additional Glauber dynamics ( X˜ t ) starting from σ˜ , and define: τmag := min{t : S(X t ) = S( X˜ t )}, U (σ ) := |{i : σ (i) = σ0 (i) = 1}| , V (σ ) := |{i : σ (i) = σ0 (i) = −1}| , := {σ : min{U (σ ), U (σ0 ) − U (σ ), V (σ ), V (σ0 ) − V (σ ))} ≥ n/20} , R(t) := U (X t ) − U ( X˜ t ) , t2 H1 (t) := {τmag ≤ t}, H2 (t1 , t2 ) := ∩i=t {X i ∈ ∧ X˜ i ∈ }. 1
Then for any possible coupling of X t and X˜ t , the following holds: max Pσ0 (X r2 ∈ ·) − µn TV ≤ max Pσ0 ,σ˜ (H1 (r1 )) σ0 ∈0
σ0 ∈0 σ˜ ∈
# αc1 · + Pσ0 ,σ˜ (Rr1 > α n/δ) + Pσ0 ,σ˜ (H2 (r1 , r2 )) + √ r2 − r1
$ & n , δ
(4.27)
where r1 < r2 and α > 0. The rest of this subsection will be devoted to establishing a series of properties satisfied by the magnetization throughout the mildly subcritical case, in order to ultimately apply the above theorem. First, we shall show that any instance of the Glauber dynamics concentrates on 0 once it performs an initial burn-in period of n/δ steps. It suffices to show this for the dynamics started from s0 = 1: to see this, consider a monotone-coupling of the dynamics (X t ) starting from an arbitrary configuration, together with two additional instances of the dynamics, (X t+ ) starting from s0 = 1 (from above) and (X t− ) starting from s0 = −1 (from below). By definition of the monotone-coupling, the chains (X t+ ) and (X t− ) “trap” the chain (X t ), and by symmetry it indeed remains to show that P1 (|St0 | ≤ 1/2) = 1 − o(1), where t0 = n/δ. " ! Recalling (4.4), we have st+1 ≤ (1 − nδ )st , where st = E |St |1{τ0 >t} , thus ! " E1 |St0 |1{τ0 >t0 } ≤ e−1 .
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Adding this to the fact that E1 St0 1{τ0 ≤ t0 } = 0, which follows immediately from symmetry, we conclude that E1 St0 ≤
e−1 . Next, applying Lemma 3.4 to our case and noting that (3.7) holds for ρ = 1 − Var(St ) ≤ ν1
1 − n tanh( βn ) ≤ 1 − nδ , we conclude that
1 n
n ≤ δ
2 4 16 n = for all t. n δ δn
Hence, Chebyshev’s inequality gives that |St0 | ≤ 1/2 with high probability. We may therefore assume henceforth that our initial configuration already belongs to some good state σ0 ∈ 0 . Next, set: T := tn (γ ), r0 := tn (2γ ), r1 := tn (3γ ), r2 := tn (4γ ). We will next bound the terms in the righthand side of (4.27) in order. First, recall that Lemma 4.5 already provided us with a bound on the probability of H1 (r1 ), by stating there for constant c > 0, c P(τmag > r1 ) ≤ √ . γ
(4.28)
Our next task is to provide an upper bound on Rr1 , and namely, to show that it typi√ cally has order at most n/δ. In order to obtain such a bound, we will analyze the sum of the spins over the set B := {i : σ0 (i) = 1}. Define Mt (B) :=
1 X t (i), 2 i∈B
and consider the monotone-coupling of (X t ) with the chains (X t+ ) and (X t− ), starting from the all-plus and all-minus positions respectively, such that X t− ≤ X t ≤ X t+ . By defining Mt+ and Mt− accordingly, we get that E(Mt (B))2 ≤ E(Mt+ (B))2 + E(Mt− (B))2 = 2E(Mt+ (B))2 . By (4.14), we immediately get that for t ≥ T , |EMt+ (B)| ≤ nδ . We will next bound
the variance of Mt+ (B), by considering the following two cases:
(i) If every pair of spins of X t+ is positively correlated (since X 0+ is the all-plus configuration, by symmetry, the covariances of each pair of spins is the same), then we can infer that ⎛ ⎞ 4n 1 n2 Var(Mt+ (B)) ≤ Var ⎝ Var S(X t+ ) ≤ . X t+ (i)⎠ = 2 4 δ i∈[n]
(ii) Otherwise, every pair of spins of X t+ is negatively correlated, and it follows that 1 + n + Var(Mt (B)) ≤ Var X t (i) ≤ . 2 4 i∈B
Mixing Time Evolution of Glauber Dynamics
749
Altogether, we conclude that for all t ≥ T , E|Mt (B)| ≤ E (Mt (B))2 ≤ 2 Var(Mt+ (B) + 2 (EMt (B))2 $ $ 8n 2n n + ≤8 . ≤ δ δ δ
(4.29)
This immediately implies that $ n , ERr1 = E|Mr1 (B) − M˜ r1 (B)| ≤ E|Mr1 (B)| + E| M˜ r1 (B)| ≤ 16 δ and an application of Markov’s inequality now gives $ 16 n P(Rr1 ≥ α )≤ . δ α
(4.30)
It remains to bound the probability of H2 (r1 , r2 ). Define: Y := 1{|Mt (B)| > n/64}, r1 ≤t≤r2
and notice that P
r2 '
{|Mt (B)| ≥ n/32} ≤ P(Y > n/64) ≤
t=r1
c0 E[Y ] . n
Recall that the second inequality of (4.29) actually gives E|Mt (B)|2 ≤ standard second moment argument gives 1 P(|Mt (B)| > n/64) = O . δn
5n δ .
Hence, a
Altogether, Eσ0 Y = O(1/δ 2 ) and r 2 ' 1 {|Mt (B)| ≥ n/32} = O . Pσ0 δ2 n t=r 1
Applying an analogous argument to the chain ( X˜ t ), we obtain that r 2 ' 1 , | M˜ t (B)| ≥ n/32 Pσ˜ =O δ2 n t=r 1
and combining the last two inequalities, we conclude that
1 Pσ0 ,σ˜ H2 (r1 , r2 ) = O . δ2 n
(4.31)
Finally, we have established all the properties needed in order to apply Theorem 4.6. At the cost of a negligible number of burn-in steps, the state of (X t ) with high probability belongs to 0 . We may thus plug in (4.28), (4.30) and (4.31) into Theorem 4.6, choosing √ α = γ , to obtain (4.1).
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4.3. Spectral gap analysis. By Proposition 3.9, it suffices to determine the spectral gap of the magnetization chain. The lower bound will follow from the next lemma of [4] (see also [12, Theorem 13.1]) along with the contraction properties of the magnetization chain. Lemma 4.7. ([4]) Suppose is a metric space with distance ρ. Let P be a transition matrix for a Markov chain, not necessarily reversible. Suppose there exists a constant θ < 1 and for each x, y ∈ , there is a coupling (X 1 , Y1 ) of P(x, ·) and P(y, ·) satisfying Ex,y (ρ(X 1 , Y1 )) ≤ θρ(x, y). If λ is an eigenvalue of P different from 1, then |λ| ≤ θ . In particular, the spectral gap satisfies gap ≥ 1 − θ . Recalling (4.12), the monotone coupling of St and S˜t implies that δ δ ˜ |s − s˜ | . Es,˜s S1 − S1 ≤ 1 − + o n n Therefore, Lemma 4.7 ensures that gap ≥ (1 + o(1)) nδ . It remains to show a matching upper bound on gap, the spectral gap of the magnetization chain. Let M be the transition kernel of this chain, and π be its stationary distribution. Similar to our final argument in Proposition 3.9 (recall (3.13)), we apply the Dirichlet representation for the spectral gap (as given in [12, Lemma 13.7]) with respect to the function f being the identity map 1 on the space of normalized magnetization, we obtain that gap ≤
(I − M)1, 1π E(1) Eπ [E [St St+1 | St ]] = =1− , 1, 1π 1, 1π Eπ St2
(4.32)
where Eπ Stk is the k th moment of the stationary magnetization chain (St ). Recall (4.24) (where η = 1 − nδ ), and notice that the following slightly stronger inequality in fact holds: " ! |St | |St |3 E sign(St )St+1 St ≥ η|St | − − 2 2n n (to see this, one needs to apply the same argument that led to (4.24), then verify the special cases St ∈ {0, n1 }). It thus follows that " ! S2 S4 E St St+1 St ≥ ηSt2 − t − t2 , 2n n and plugging the above into (4.32) we get gap ≤
1 Eπ St4 δ 1 + · + . n 2n Eπ St2 n 2
(4.33)
In order to bound the second term in (4.33), we need to give an upper bound for the fourth moment in terms of the second moment. The next argument is similar to the one used earlier to bound the third moment of the magnetization chain (see (4.20)), and hence will be described in a more concise manner.
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751
For convenience, we use the abbreviations h + := tanh β(s + n −1 ) and h − := tanh β(s − n −1 ) . By definition (see (3.3)) the following then holds: 2 3 s −2s + h − + h + + sh − − h + n 6 2 + 2 s 2 + h + − h − − sh − + h + n 8 + 3 s 3 −2s + h − + h + + sh − − h + n 4 + 4 2 + h + − h − − sh − + h + n 4δ 4 12 2 16 ≤ 1− s + 2s + 4. n n n
4 | St = s] = s 4 + E[St+1
Now, taking expectation and letting the St be distributed according to π , we obtain that Eπ St4 ≤
3 4 Eπ St2 + 3 . δn δn
Recalling that, as the spins are positively correlated, Var π (St ) ≥ n1 , we get 4 Eπ St2 . Eπ St4 ≤ 3 + n nδ
(4.34)
Plugging (4.34) into (4.33), we conclude that gap ≤
δ n
1 δ 1+O = (1 + o(1)) . 2 δ n n
5. The Critical Window In this section we prove Theorem 2, which establishes that the critical window has a mixing-time of order n 3/2 without a cutoff, as well as a spectral-gap of order n −3/2 .
5.1. Upper bound. Let (X t ) denote the Glauber dynamics, started from an arbitrary configuration σ , and let ( X˜ t ) denote the dynamics started from the stationary distribution µn . As usual, let (St ) and ( S˜t ) denote the (normalized) magnetization chains of (X t ) and ( X˜ t ) respectively. Let ε >0. The case δ 2 n = O(1) of Theorem 4.3 implies that, for a sufficiently large γ > 0, Pσ τ0 ≥ γ n 3/2 < ε. Plugging this into Lemma 3.1, we deduce that there exists some cε > 0, such that the chains St and S˜t coalesce after at most cε n 3/2 steps with probability at least 1 − ε. At this point, Lemma 3.3 implies that (X t ) and ( X˜ t ) coalesce after at most O(n 3/2 ) + O(n log n) = O(n 3/2 ) additional steps with probability arbitrarily close to 1, as required.
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5.2. Lower bound. Throughout this argument, recall that δ is possibly negative, yet satisfies δ 2 n = O(1). By (4.22), δ c1 c2 3 3 3 E|St+1 | ≤ E |St | − 3 |St | + 2 |St | + 3 n n n 3δ c c 1 2 E|St |3 + 2 E|St | + 3 . ≤ 1− n n n Recalling Lemma 4.4, and plugging the fact that δ = O(n −1/2 ) in (4.11), the following holds. If St and S˜t are the magnetization chains corresponding to two instances of the Glauber dynamics, then for some constant c > 0 and any sufficiently large n, Es,˜s |S1 − S˜1 | ≤ (1 + cn −3/2 )|s − s˜ |.
(5.1)
Combining this with the extended form of Lemma 3.4, as given in (3.8), we deduce that if t ≤ εn 3/2 for some small fixed ε > 0, then Var s0 St ≤ 4t/n 2 . Therefore, √ δ t 2 t 2 . Es0 [|St |] ≤ |Es0 St | + Var s0 St ≤ 1 − |s0 | + n n Therefore, √ c1 t δ t 3δ c1 3 Es0 |St+1 | ≤ 1 − Es0 |St | + 2 1 − |s0 | + 3 n n n n √
t c c 1 ≤ η3 Es0 |St |3 + ηt 2 |s0 | + 1 3 , n n 3
where again η = 1 − δ/n. Iterating, we obtain Es0 |St+1 | ≤ η |s0 | 3
3t
3
√ t c1 t 3 j η + 3 η n j=0 j=0 √ c1 t η3t − 1 η2t−1 − 1 |s0 | + 3 · 3 · η2 − 1 n η −1 √ c1 t · 2t|s0 | + 3 · 3t, n
c1 + ηt 2 |s0 | n
c1 n2 c1 ≤ η3t |s0 |3 + ηt 2 n
≤ η3t |s0 |3 + ηt
t
2j
(5.2)
where the last inequality holds for sufficiently large n and t ≤ εn 3/2 with ε > 0 small enough (such a choice ensures that ηt will be suitably small). Define Z t := |St |η−t , whence Z 0 = |S0 | = |s0 |. Applying (4.24) (recall that it holds for any temperature) and using the fact that η−1 ≤ 2, we get E[Z t+1 | St ] ≥ Z t −
1 −t η |St |3 + O(1/n) , n
for n large enough, hence E[Z t − Z t+1 | St ] ≤
1 −t η |St |3 + O(1/n) . n
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Taking expectation and plugging in (4.23),
3/2 1 c2 t 2t 3 −t c2 t Es0 [Z t − Z t+1 ] ≤ + O(1/n) . η |s0 | + 2 |s0 | + η n n n3 Set t = n 3/2 /A4 for some large constant A such that we obtain that
1 2
(5.3)
≤ ηt ≤ 2. Summing over (5.3)
c2 1 − η2t |s0 |3 + t 2 3 |s0 | + 2η−t · t 5/2 /n 4 + O(t/n 2 ) 2 n(1 − η ) n c 2 √2 4 2 √ 2 ≤ 4 n|s0 |3 + 8 |s0 | + 10 e δ n/A n −1/4 + O(n −1/2 ). A A A
|s0 | − Es0 Z t ≤
We now select s0 = An −1/4 for some large constant A; this gives 2 2 √2 4 c2 An −1/4 − Es0 Z t ≤ + 7 + 10 e δ n/A n −1/4 + O(n −1/2 ). A A A Choosing A large enough to swallow the constant c2 as well as the term δ 2 n (using the fact that δ 2 n is bounded), we obtain that Es0 Z t ≥
1 −1/4 . An 2
Translating Z t back to |St |, we obtain Es0 |St | ≥ ηt ·
1 −1/4 √ −1/4 An ≥ An =: B, 2
(5.4)
provided that A is sufficiently large (once again, using the fact that ηt is bounded, this time from below). Since Var s0 (St ) ≤ 16t/n 2 =
16 −1/2 n , A4
(5.5)
the following concentration result on the stationary chain ( S˜t ) will complete the proof: Pµn (| S˜t | ≥ An −1/4 )} ≤ ε(A), and lim ε(A) = 0. A→∞
(5.6)
Indeed, combining the above two statements, Chebyshev’s inequality implies that Ps0 (St ∈ ·) − π TV ≥ π([−B/2, B/2]) − Ps0 (|St | ≤ B/2) √ 64 ≥ 1 − 5 − ε( A). A
(5.7)
It remains to prove (5.6). Since we are proving a lower bound √ for the mixing-time, it suffices to consider a sub-sequence of the δn -s such that δn n converges to some constant (possibly 0). The following result establishes the limiting stationary distribution of the magnetization chain in this case.
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√ Theorem 5.1. Suppose that limn→∞ δn n = α ∈ R. The following holds: 4 Sµn s2 s − α . → exp − n −1/4 12 2
(5.8)
Proof. We need the following theorem: Theorem 5.2. ([8, Theorem 3.9]) Let ρ denote some probability measure, and let Sn (ρ)= 1 n X j=1 j (ρ), where the {X j (ρ) : j ∈ [n]} have joint distribution n ( ) n 1 (x1 + · · · + xn )2 * exp dρ(x j ), Zn 2n j=1
and Z n is a normalization constant. Suppose that {ρn : n = 1, 2, . . .} are measures satisfying exp(x 2 /2)dρn → exp(x 2 /2)dρ. Suppose further that ρ has the following properties: (1) Pure: the function G ρ (s) :=
s2 − log 2
esx dρ(x)
has a unique global minimum. (2) Centered at m: let m denote the location of the above global minimum. (3) Strength δ and type k: the parameters k, δ > 0 are such that G ρ (s) = G ρ (m) + δ
(s − m)2k + o((s − m)2k ), (2k)!
where the o(·)-term tends to 0 as s → m. If, for some real numbers α1 , . . . , α2k−1 we have G (ρj) (m) = n
αj + o(n −1+ j/2k ), n 1− j/2k
j = 1, 2, . . . , 2k − 1, n → ∞,
then the following holds: Sn (ρn ) → 1{s=m} and
⎧ ⎪ ⎪ ⎪ ⎪ ⎨
α1 1 N − , −1 , if k = 1, δ δ ⎛ ⎞ Sn (ρn ) − m 2k−1 → , sj s2k ⎪ n −1/2k ⎪ ⎝ exp −δ − α j ⎠, if k ≥ 2, ⎪ ⎪ ⎩ (2k)! j! j=1
where δ −1 − 1 > 0 for k = 1.
(5.9)
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Let ρ denote the two-point uniform measure on {−1, 1}, and √ let ρn denote the twopoint uniform measure on {−βn , βn }. As |1 − βn | = δn = O(1/ n), the convergence requirement (5.9) of the measures ρn is clearly satisfied. We proceed to verify the properties of ρ:
s4 s6 s2 s2 − log esx dρ(x) = − log cosh(s) = − + O(s 8 ). G ρ (s) = 2 2 12 45 This implies √ that G ρ has a unique global minimum at m = 0, type k = 2 and strength δ = 2. As δn n → α, we deduce that the G ρn -s satisfy s2 − log cosh(βn s), 2 (3) G (1) ρn (0) = G ρn (0) = 0, G ρn (s) =
2α 2 −1/2 G (2) ). ρn (0) = 1 − βn = δn (2 − δn ) = √ + o(n n This completes the verification of the conditions of the theorem, and we obtain that 4 s2 s Sn (ρn ) . (5.10) → exp − − α n −1/4 12 2 Recalling that, if xi = ±1 is the i th spin,
⎛ β 1 exp ⎝ µn (x1 , . . . , xn ) = Z (β) n
⎞ xi x j ⎠ ,
(5.11)
1≤i< j≤n
clearly Sn (ρn ) has the same distribution as Sµn for any n. This completes the proof of the theorem. Remark. One can verify that the above analysis of the mixing time in the critical window holds also for the censored dynamics (where the magnetization is restricted to be non-negative, by flipping all spins whenever it becomes negative). Indeed, the upper bound immediately holds as the censored dynamics is a function of the original Glauber dynamics. For the lower bound, notice that our argument tracked the absolute value of the magnetization chain, and hence can readily be applied to the censored case as-well. 3/2 Altogether, the censored √ dynamics has a mixing time of order n in the critical window 1 ± δ for δ = O(1/ n). 5.3. Spectral gap analysis. The spectral gap bound in the critical temperature regime is obtained by combining the above analysis with results of [6] on birth-and-death chains. The lower bound on gap is a direct consequence of the fact that the mixing time has order n 3/2 , and that the inequality trel ≤ tmix 14 always holds. It remains to prove the matching bound tmix 41 = O(trel ). Suppose that this is false, that is, trel = o tmix 41 . Let A be some large constant, and let s0 = An −1/4 . Notice that the case δ 2 n = O(1) in Theorem 4.3 implies that E1 τ0 = O(n 3/2 ). Furthermore, by Theorem 5.2, there exists a strictly positive function of A, ε(A), such that lim A→∞ ε(A) = 0 and 1 ε(A) ≤ π(S ≥ s0 ) ≤ 2ε(A) 2
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for sufficiently large n. Applying Lemma 3.7 with α = π(S ≥ s0 ) and β = 21 gives Es0 τ0 = o(n 3/2 ). As in Subsect. 5.2, set t¯ = n 3/2 /A4 for some large constant A. Combining Lemma 3.8 with Markov’s inequality gives the following total variation bound for this birth-and-death chain: Ps0 (St¯ ∈ ·) − π TV ≤ 4ε(A) + o(1). However, the lower bound (5.7) obtained in Subsect. 5.2 implies that: √ Ps0 (St¯ ∈ ·) − π TV ≥ 1 − 4ε( A/2) − 64/A5 .
(5.12)
(5.13)
Choosing a sufficiently large constant A, (5.12) and (5.13) together lead to a contradiction for large n. We conclude that gap = O(n −3/2 ), completing the proof. Note that, as the condition gap · tmix ( 41 ) → ∞ is necessary for cutoff in any family of ergodic reversible finite Markov chains (see, e.g., [6]), we immediately deduce that there is no cutoff in this regime. Remark. It is worth noting that the order of the spectral gap at βc = 1 follows from a simpler argument. Indeed, in that case, the upper bound on gap can alternatively be derived from its Dirichlet representation, similar to the argument that appeared in the proof of Proposition 3.9 (where we substitute the identity function, i.e., the sum of spins in the Dirichlet form). For this argument, one needs a lower bound for the variance of the stationary magnetization. Such a bound is known for βc = 1 (see [9]), rather than throughout the critical window. 6. Low Temperature Regime In this section we prove Theorem 3, which establishes the order of the mixing time and the spectral gap in the super critical regime (where the mixing of the dynamics is exponentially slow and there is no cutoff).
6.1. Exponential mixing. Recall that the normalized magnetization chain St is a birthand-death chain on the space X = {−1, −1+ n2 , . . . , 1− n2 , 1}, and for simplicity, assume throughout the proof that n is even (this is convenient since in this case we can refer to the 0 state. Whenever n is odd, the same proof holds by letting n1 take the role of the 0 state). The following notation will be useful. We define X [a, b] := {x ∈ X : a ≤ x ≤ b}, and similarly define X (a, b), etc. accordingly. For all x ∈ X , let px , qx , h x denote the transition probabilities of St to the right, to the left and to itself from the state x, that is: 1 − x 1 + tanh(β(x + n −1 )) · , px := PM x, x + n2 = 2 2 1 + x 1 − tanh(β(x − n −1 )) qx := PM x, x − n2 = · , 2 2 h x := PM (x, x) = 1 − px − qx .
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By well known results on birth-and-death chains (see, e.g., [12]), the resistance r x and conductance cx of the edge (x, x +2/n), and the conductance c x of the self-loop of vertex x for x ∈ X [0, 1] are (the negative parts can be obtained immediately by symmetry) rx =
* y∈X (0,x]
qy , cx = py
* y∈X (0,x]
py hx , c x = (cx−2/n + cx ), qy px + qx
(6.1)
and the commute-time between x and y, C x,y for x < y (the minimal time it takes the chain, starting from x, to hit y then return to x) satisfies EC x,y = 2c S R(x ↔ y),
(6.2)
where c S :=
(cx + c x ) and
R(x ↔ y) :=
x∈X
rz .
z∈X [x,y)
Our first goal is to estimate the expected commute time between 0 and ζ . This is incorporated in the next lemma. Lemma 6.1. The expected commute time between 0 and ζ has order texp :=
ζ n n 1 + g(x) log exp d x, δ 2 0 1 − g(x)
(6.3)
where g(x) := (tanh(βx) − x) / (1 − x tanh(βx)). In particular, in the special case δ → 0 we have EC0,ζ = nδ exp ( 43 + o(1))δ 2 n , where the o(1)-term tends to 0 as n → ∞. Remark. If ζ ∈ X , instead we simply choose a state in X which is the nearest possible to ζ . For a sufficiently large n, such a negligible adjustment would keep our calculations and arguments intact. For the convenience of notation, let ζ denote the mentioned state in this case as well. To prove Lemma 6.1, we need the following two lemmas, which establish the order of the total conductance and effective resistance respectively. Lemma 6.2. The total conductance satisfies $ cS =
ζ n n 1 + g(x) exp dx . log δ 2 0 1 − g(x)
Lemma 6.3. The effective resistance between 0 and ζ satisfies # R(0 ↔ ζ ) = ( n/δ). Proof of Lemma 6.2. Notice that for any x ∈ X , the holding probability h x is uniformly bounded from below, and thus c x can be uniformly bounded from above by (cx +cx−2/n ). It therefore follows that c S = (c˜ S ), where c˜ S := x∈X cx , and it remains to determine
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c˜ S . We first locate the maximal edge conductance and determine its order, by means of classical analysis, py 1 − y 1 + tanh(β(y + n −1 )) log cx = · log = log qy 1 + y 1 − tanh(β(y − n −1 )) y∈X (0,x] y∈X (0,x] 1 + g(y) 1 + g(y) = log log + O(1/n) = + O(x). 1 − g(y) 1 − g(y) y∈X (0,x]
y∈X (0,x]
(6.4) Note that g(x) has a unique positive root at x = ζ , and satisfies g(x) > 0 for x ∈ (0, ζ ) and g(x) < 0 for x > ζ . Therefore, log cx ≤ log cζ + O(x) ≤ log cζ + O(1), thus we move to estimate cζ . As log cζ is simply a Riemann sum (after an appropriate rescaling), we deduce that
n ζ 1 + g(x) 1 + g(x) log cζ = + O(1) = d x + O(1), log log 1 − g(x) 2 0 1 − g(x) x∈X (0,ζ ]
and therefore ζ n 1 + g(x) cζ = exp dx , log 2 0 1 − g(x) cx = O(cζ ).
(6.5) (6.6)
Next, consider the ratio cx+2/n /cx ; whenever x ≤ ζ , g(x) ≥ 0, hence we have cx+2/n = cx Whenever
√1 δn
px+2/n qx+2/n
≤x ≤ζ−
≥
1+g(x) 1−g(x)
− O(1/n) ≥ 1 + 2g(x) − O(1/n).
√1 (using the Taylor expansions around 0 and around ζ ) δn √ ≥ 21 δ/n. Combining this with the fact that x tanh(βx) is √
we obtain that tanh(βx) − x always non-negative, we obtain that for any such x, 2g(x) ≥ δ/n. Therefore, setting $ $ $ 1 1 1 , ξ2 := ζ − , ξ3 := ζ + , (6.7) ξ1 := δn δn δn we get cx+2/n ≥1+ cx
$
δ − O(1/n) for any x ∈ X [ξ1 , ξ2 ]. n
(6.8)
Using the fact that δ 2 n → ∞, the sum of c√ x -s in the above range is at most the sum of a geometric series with a quotient 1/(1 + 21 δ/n) and an initial position cζ : $ n · cζ . cx ≤ 3 (6.9) δ x∈X [ξ1 ,ξ2 ]
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We now treat x ≥ ξ3 ; since √ g(ζ ) = 0 and g(x) is decreasing for any x ≥ ζ , then in particular whenever ζ + δ/n ≤ x ≤ 1 we have −1 = g(1) ≤ g(x) ≤ 0, and therefore px+2/n cx+2/n = ≤ 1 + g(x) + O(1/n). cx qx+2/n √ Furthermore, for any √ ζ + δ/n ≤ x ≤ 1 (using √ Taylor expansion around ζ ) we have tanh(βx) − x ≤ − δ/n, and hence g(x) ≤ − δ/n. We deduce that $ cx+2/n δ ≤1− + O(1/n) for any x ∈ X [ξ3 , 1], cx n and therefore
$ cx ≤ 2
x∈X [ξ3 ,1]
n · cζ . δ
(6.10)
Combining (6.9) and (6.10) together, and recalling (6.6), we obtain that $
# n c˜ S = cζ . cx ≤ 2 |X [0, ξ1 ]| + 5 n/δ + |X [ξ2 , ξ3 ]| cζ = O δ x∈X
Finally, consider x ∈ X [ξ2 , ξ3 ]; an argument similar to the ones above (i.e., perform Taylor expansion around ζ and bound the ratio of cx+2/n /cx ) shows that cx is of order cζ in this region. This implies that for some constant b > 0, $ n cζ , c˜ S ≥ cx ≥ b|X [ξ2 , ξ3 ]|cζ ≥ b (6.11) δ x∈X [ξ2 ,ξ3 ]
and altogether, plugging in (6.5), we get ζ $ n n 1 + g(x) c˜ S = exp dx . log δ 2 0 1 − g(x)
(6.12)
Proof of Lemma 6.3. Translating the conductances, as given in (6.8), to resistances, we get $ r x+2/n δ − O(1/n) for any x ∈ X [ξ1 , ξ2 ], ≤1− rx n and hence
# # r x ≤ rξ1 2 n/δ ≤ 2 n/δ,
x∈X [ξ1 ,ξ2 ]
where in the last inequality we used the fact that r x ≤ r x−2/n (≤ r0 = 1) for all x ∈ X [0, ζ ], which holds since qx ≤ px for such x. Altogether, we have the following upper bound: rx + rx + rx R(0 ↔ ζ ) = x∈X [0,ξ1 ]
≤ |X [0, ξ1 ]| + 2
x∈X [ξ1 ,ξ2 ]
$
x∈X [ξ2 ,ζ ]
# n + |X [ξ2 , ζ ]| ≤ 4 n/δ. δ
(6.13)
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For a lower bound, consider x ∈ X [0, ξ1 ]. Clearly, for any x ≤ g(x) =
tanh(βx−x) 1−x tanh(βx)
≤ 2δx, and hence
√1 δn
we have
r x+2/n 1 − g(x) + O(1/n) ≥ 1 − 5xδ ≥ exp(−6xδ), = rx 1 + g(x) yielding that
rξ1 ≥ exp −3δn · ξ12 ≥ e−3 . Altogether, # R(0 ↔ ζ ) ≥ e−3 |X [0, ξ1 ]| ≥ e−4 n/δ,
√ and combining this with (6.13) we deduce that R(0 ↔ ζ ) = ( n/δ).
Proof of Lemma 6.1. Plugging in the estimates for c˜ S and R(0 ↔ ζ ) in (6.2), we get ζ n n 1 + g(x) exp dx . (6.14) log EC0,ζ = δ 2 0 1 − g(x) This completes the proof of Lemma 6.1.
Note that by symmetry, the expected hitting time from ζ to −ζ is exactly the expected commute time between 0 and ζ . Hence, Eζ [τ−ζ ] = (texp ).
(6.15)
In order to show that the above hitting time is the leading order term in the mixingtime at low temperatures, we need the following lemma, which addresses the order of the hitting time from 1 to ζ . Lemma 6.4. The normalized magnetization chain St in the low temperature regimes satisfies E1 τζ = o(texp ). Proof. First consider the case where δ is bounded below by some constant. Notice that, as px ≤ qx for all x ≥ ζ , in this region St is a supermartingale. Therefore, Lemma 3.5 (or simply standard results on the simple random walk, which dominates our chain in this case) implies that E1 τζ = O(n 2 ). Combining this with the fact that texp ≥ exp(cn) for some constant c in this case, we immediately obtain that E1 τζ = o(texp ). Next, assume that δ = o(1). Note that in this case, the Taylor expansion tanh(βx) = βx − 13 (βx)3 + O((βx)5 ) implies that √ ζ = 3δ/β 3 − O((βζ ))5 = 3δ + O(δ 3/2 ). (6.16) Recalling that E[St+1 | St = s] ≤ s + n1 (tanh(βs) − s) (as s ≥ 0), Jensen’s inequality (using the concavity of the Hyperbolic tangent) gives E[St+1 − St ] = E(E[St+1 − St | St ]) ≤ ≤
1 (tanh(βESt ) − ESt ) . n
1 (E tanh(β St ) − ESt ) n (6.17)
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Further note that the function tanh(βs) has the following Taylor expansion around ζ (for some ξ between s and ζ ): tanh(βs) = ζ + β(1 − ζ 2 )(s − ζ ) + β 2 (−1 + ζ 2 )ζ (s − ζ )2 β3 tanh(4) (ξ ) + (−1 + 4ζ 2 − ζ 4 )(s − ζ )3 + (s − ζ )4 . 3 4!
(6.18)
Since tanh(4) (x) < 5 for any x ≥ 0, (6.18) implies that for a sufficiently large n the term − 13 (s − ζ )3 absorbs the last term in the expansion (6.18). Together with (6.16), we obtain that √ tanh(βs) ≤ ζ + β(1 − ζ 2 )(s − ζ ) + β 2 (−1 + ζ 2 ) δ(s − ζ )2 . Therefore, (6.17) follows: √ E[St+1 − St ] ≤ −
δ (ESt − ζ )2 . 2n
(6.19)
Set bi = 2−i , i 2 = min{i : bi <
√ δ} and u i = min{t : ESt − ζ < bi },
noting that this gives bi /2 ≤ ESt − ζ ≤ bi for any t ∈ [u i , u i+1 ]. It follows that u i+1 − u i ≤
bi /2 √ δ bi 2 2n ( 2 )
4n =√ , δbi
and hence i2 i=1
u i+1 − u i ≤
i:bi2 >δ
4n = O(n/δ), √ δbi
where we used the fact that the series {bi−1 } is geometric with ratio 2. We claim that this implies the required bound on E1 τζ . To see this, recall (6.19), according to which Wt := n(St − ζ )1{τζ >t} is a supermartingale with bounded increments, whose variance is uniformly bounded from below on the event τζ > t (as the holding probabilities of (St ) are uniformly bounded from above, see (3.4)). Moreover, the above argument gives √ EWt ≤ n δ for some t = O(n/δ). Thus, applying Lemma 3.5 and taking expectation, we deduce that E1 τζ = O(n/δ + δn 2 ) = O(δn 2 ), which in turn gives E1 τζ = o(texp ). Remark. With additional effort, we can establish that E0 τ±ζ = o(texp ) (for more details, see the companion paper [5]), where τ±ζ = min{t : |St | ≥ ζ }. By combining this with the of St symmetry and applying the geometric trial method, we can obtain the expected commute time between 0 and ζ : Eζ τ0 = ( 21 + o(1))EC0,ζ = (texp ), and therefore conclude that E1 τ0 = (texp ).
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6.1.1. Upper bound for mixing. Combining Lemma 6.4 and (6.15), we conclude that E1 τ−ζ = (texp ) and hence E1 τ0 = O(texp ). Together with Lemma 3.1, this implies that the magnetization chain will coalescence in O(texp ) steps with probability arbitrarily close to 1. At this point, Lemma 3.3 immediately gives that the Glauber dynamics achieves full mixing within O(n log n) additional steps. The following simple lemma thus completes the proof of the upper bound for the mixing time. Lemma 6.5. Let texp be as defined in Lemma 6.1. Then n log n = o(texp ). Proof. In case δ ≥ c > 0 for some constant c, we have texp ≥ n exp(c n) for some constant c > 0 and hence n log n = o(texp ). It remains to treat the case δ = o(1). Suppose first that δ = o(1) and δ ≥ cn −1/3 for some constant c > 0. In this case, we have texp = nδ exp ( 43 + o(1))δ 2 n and thus n exp( 21 n 1/3 ) = O(texp ), giving n log n = o(texp ). Finally, if δ = o(n −1/3 ), we can simply conclude that n 4/3 = O(texp ) and hence n log n = o(texp ). 6.1.2. Lower bound for mixing. The lower bound will follow from showing that the probability of hitting −ζ within εtexp steps is small, for some small ε > 0 to be chosen later. To this end, we need the following simple lemma: Lemma 6.6. Let X denote a Markov chain over some finite state space , y ∈ denote a target state, and T be an integer. Further let x ∈ denote the state with the smallest probability of hitting y after at most T steps, i.e., x minimizes Px (τ y ≤ T ). The following holds: Px (τ y ≤ T ) ≤
T . Ex τ y
Proof. Set p = Px (τ y ≤ T ). By definition, Pz (τ y ≤ T ) ≥ p for all z ∈ , hence the hitting time from x to y is stochastically dominated by a geometric random variable with success probability p, multiplied by T . That is, we have Ex τ y ≤ T / p, completing the proof. The final fact we would require is that the stationary probability of X [−1, −ζ ] is strictly positive. This is stated by the following lemma. Lemma 6.7. There exists some absolute constant 0 < Cπ < 1 such that Cπ ≤ π(X [ζ, 1]) ( = π(X [−1, −ζ ]) ). Proof. Repeating the derivation of (6.11), we can easily get ζ $ n n 1 + g(x)
exp dx . (cx + cx ) ≥ log cX [ζ,1] := δ 2 0 1 − g(x) x∈X [ζ,1]
Combining the above bound with (6.12), we conclude that there exists some Cπ > 0, such that π(X [ζ, 1]) ≥ Cπ . Plugging in the target state −ζ into Lemma 6.6, and recalling that the monotone-coupling implies that, for any T , the initial state s0 = 1 has the smallest probability (among
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all initial states) of hitting −ζ within T steps, we deduce that, for a sufficiently small ε > 0, P1 (τ−ζ ≤ εtexp ) ≤
1 Cπ . 2
This implies that 1 Cπ , texp = O tmix 2 which in turn gives texp = O tmix 41 . 6.2. Spectral gap analysis. The lower bound is straightforward (as the relaxation time is always at most the mixing time) and we turn to prove the upper bound. Note that, by Lemma 6.7, we have π(X [ζ, 1]) ≥ Cπ > 0. Suppose first that gap · tmix ( 41 ) → ∞. In this case, one can apply Lemma 3.7 onto the birth-and-death chain (St ), with a choice of α = π(X [ζ, 1]) and β = 1 − π(X [ζ, 1]) (recall that tmix ( 41 ) = (E1 τ−ζ )). It follows that Eζ τ−ζ = o E1 τ−ζ . However, as both quantities above should have the same order as tmix ( 41 ), this leads to a contradiction. We therefore have gap · tmix ( 41 ) = O(1), completing the proof of the upper bound. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Aldous, D., Fill, J.A.: Reversible markov chains and random walks on graphs, In preparation. http://www. stat.berkeley.edu/~aldous/RWG/book.html, 2002 2. Aizenman, M., Holley, R.: Rapid Convergence to Equilibrium of Stochastic Ising Models in the Dobrushin Shlosman regime, (Minneapolis, Minn., 1984), IMA Vol. Math. Appl., Vol. 8, New York: Springer, 1987, pp. 1–11 3. Bubley, R., Dyer, M.: Path coupling: A technique for proving rapid mixing in Markov chains. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS), 1997 pp. 223–231, DOI 10.1109/SFCS.1997.646111, 2002 4. Chen, M.-F.: Trilogy of couplings and general formulas for lower bound of spectral gap. In: Probability towards 2000, (New York, 1995), Lecture Notes in Statist., Vol. 128, New York: Springer, 1998, pp. 123–136 5. Ding, J., Lubetzky, E., Peres, Y.: Censored Glauber dynamics for the mean-field Ising model. http://arxiv. org/abs/0812.0633, 2008 6. Ding, J., Lubetzky, E., Peres, Y.: Total-variation cutoff in birth-and-death chains. Prob. Th. Rel. Fields, to appear, DOI 10.1007/s00440-008-0185-3, 2009 7. Ellis, R.S.: Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften, Vol. 271, New York: Springer-Verlag, 1985 8. Ellis, R.S., Newman, C.M.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44(2), 117–139 (1978)
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9. Ellis, R.S., Newman, C.M., Rosen, J.S.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. II. Conditioning, multiple phases, and metastability. Z. Wahrsch. Verw. Gebiete 51(2), 153–169 (1980) 10. Griffiths, R.B., Weng, C.-Y., Langer, J.S.: Relaxation Times for Metastable States in the Mean-Field Model of a Ferromagnet. Phys. Rev. 149, 301–305 (1966) 11. Levin, D.A., Luczak, M., Peres, Y.: Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probability Theory and Related Fields, to appear, DOI 10.1007/s00440008-0189-z, 2009 12. Levin, D.A., Peres, Y., Wilmer, E.: Markov chains and mixing times. Providence, RI: Amer. Math. Soc. 2009, available at http://www.uoregon.edu/~dlevin/MARKOV/, 2007 13. Nacu, S.: ¸ Glauber Dynamics on the Cycle is Monotone. Probab. Th. Rel. Fields 127, 177–185 (2003) Communicated by H. Spohn
Commun. Math. Phys. 289, 765–776 (2009) Digital Object Identifier (DOI) 10.1007/s00220-008-0717-9
Communications in
Mathematical Physics
Local Critical Perturbations of Unimodal Maps Alexander Blokh1, , Michał Misiurewicz2, 1 Department of Mathematics, University of Alabama in Birmingham, University Station,
Birmingham, AL 35294-2060, USA. E-mail: [email protected]
2 Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN
46202-3216, USA. E-mail: [email protected] Received: 16 July 2008 / Accepted: 17 September 2008 Published online: 13 January 2009 – © Springer-Verlag 2009
Abstract: We introduce a new complete metric in the space V2 of unimodal C 2 -maps of the interval, with two maps close if they are close in the C 2 -metric and differ only on a small interval containing their critical points. We identify all structurally stable maps in the sense of this metric. They are maps for which either (1) the trajectory of the critical point is attracted to a topologically attracting (at least from one side) periodic orbit, but never falls into this orbit, or (2) the critical point is mapped by some iterate to the interior of an interval consisting entirely of periodic points of the same (minimal) period. We verify the generalized Fatou conjecture for V2 and show that structurally stable maps form an open dense subset of V2 . 1. Introduction One of the central problems in dynamical systems theory is to investigate how the dynamics of a map changes if the map is slightly perturbed. The size of a perturbation is normally measured by the C r -distance between maps. The dynamics is said to change in an essential way if the new map is not conjugate to the original one. If a map f has a neighborhood U such that all maps in U are conjugate to f then f is said to be structurally stable. The definition of structural stability depends on the topology in the space of maps, but since the standard agreement is that maps are considered with C r -topology, this dependence is omitted from the definition. According to the generalized Fatou conjecture, structurally stable maps should form an open dense subset in the entire space and should have some nice properties (like hyperbolicity). The original Fatou conjecture goes back to Poincaré, Fatou and Andronov and was stated in [3] by Fatou for rational maps. For interval maps, density of Axiom A maps has been proved by Jakobson [4] in C 1 -topology, and more recently, in C r -topology (r = 2, 3, . . . , ∞, ω) by Kozlovski [5] for unimodal maps and by Kozlovski, Shen and van Strien [6] for multimodal maps. Partially supported by NSF grant DMS 0456748.
Partially supported by NSF grant DMS 0456526.
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Let us look at this situation from a different point of view. Suppose that maps g and h are close to a map f so that the C r -distances between g and f and between h and f are the same. Suppose however that the map g differs from f on the entire space while the map h differs from f only on a small open set. Then intuitively one should declare that h is closer to f than g. To reflect this intuitive feeling one needs to introduce another metric in the space of all C r -maps. Now, suppose that f is not structurally stable. Then there are maps with different dynamics which are very close to f in the C r -metric, and one can ask if these maps differ from f on big sets (like g) or on small sets (like h). Theoretically it might happen that to change the dynamics of a structurally unstable f one needs to introduce small changes everywhere, while the dynamics of f persists under small changes on small sets. If this were the case, it would reflect a version of stability of the dynamics of f. In our view, making a distinction between these two situations is natural and important. We would like to employ this point of view in the context of smooth unimodal maps. Despite apparent simplicity, these maps provide a variety of hard to study dynamical phenomena. The crucial role for the description of their dynamics is played by the behavior of the trajectory of the critical point, on which the dynamics depends almost completely. Therefore, if we want to achieve the maximal impact by slightly changing the map on a small set, it is natural to choose the set to be a small neighborhood of the critical point. In other words, one can expect that the critical point is the most sensitive to the changes of the map and makes the greatest impact upon the dynamics. This motivates us to study the question of which unimodal maps change their dynamics under small perturbations on small neighborhoods of their critical points, and which are stable with respect to such perturbations. An additional motivation is the fact that this was the main kind of perturbations used by Jakobson in [4] and that it works for Collet-Eckmann maps in higher smoothness (see [1]). We suggest a certain formalism which reflects the above considerations and may be helpful if we compare our approach to the standard one. For a closed interval I and a C r -function ϕ : I → R we will denote by ϕr,I the C r -sup-norm of ϕ; the corresponding metric is denoted by dr,I . We want some smoothness, so we will assume that r ≥ 1 and consider the space C r (I, R) of real-valued C r -functions on I, as well as a subspace C r (I, J ) of C r (I, R), consisting of the functions with values in an interval J. Without loss of generality, by a unimodal C r -map we mean a map f ∈ C r (K , K ) of a closed interval K = [a, b] into itself such that it has a unique critical point c = c f , f attains a local maximum at c f , and f (a) = f (b) = a (we will keep the notation K , a, b, c throughout the paper). Given an interval I, we denote its length by |I |. Normally this space of maps is endowed with the C r -metric and the corresponding topology. To implement the ideas described earlier, we introduce a finer metric. Let f be a unimodal C r -map and g be its perturbation.1 Then denote by Di( f, g) the open set of all points x such that f (x) = g(x). The idea is to measure both how big Di( f, g) is and how far it is from the critical points of f and g. To do so, let I ( f, g) be the smallest closed interval containing Di( f, g), c f and cg . Set ρr ( f, g) = |I ( f, g)| + f − gr,K . Then ρr ( f, g) ≥ f − gr,K . In the next section (Proposition 2.1) we will show that ρr is a complete metric.
1 We will use the word “perturbation” basically in 2 different meanings. The first one is the usual “process of perturbing.” The second one is as here: we start with a map f, perturb it, and get a map g which is a perturbation of f. However, when we speak of a “small perturbation,” we mean that g − f (not g) is small.
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Observe that the metric ρr is strictly finer than the C r -metric. For instance, if K =[0, 1] and f ∈ C r (K , K ) is unimodal then unimodal self-mappings of K defined as f ε = (1−ε) f converge to f in C r -sense, while ρr ( f ε , f ) > |K | = 1 for any ε. Finally, one can introduce topology analogous to the C ∞ -topology in the space of all unimodal C ∞ -maps by declaring that C ∞ -maps converge to a map if they converge to it in all ρr -metrics. Since the metric ρr reflects closeness of maps around their critical points and coincidence far away from the critical points, it serves our purpose and will be used from now on. To distinguish between ours and standard terminology we speak of (arbitrarily) c-small perturbations meaning (arbitrarily) small perturbations in ρr -metric, c-structurally (un)stable meaning structurally (un)stable in ρr -metric, c-close meaning close in ρr -metric, and so on (it will follow from the context what smoothness r we consider). Let us fix a closed interval K = [a, b]. Think of trivial ways in which a c-small perturbation of a unimodal map f is not topologically conjugate to f. One such way would be to have f (c f ) or f 2 (c f ) to be equal to an endpoint of K , a property that can be easily destroyed by such a perturbation. Unless f (a) = a, this reflects the choice of K , rather that the dynamics of f and its perturbations. However, we excluded such phenomena by requiring that f (a) = f (b) = a. Another trivial situation when a c-small perturbation produces a map which is not topologically conjugate to f occurs when the critical point of f is degenerate, that is, f (c f ) = 0. Then there are c-small perturbations g with c f as a critical point, but with g (c f ) > 0. Thus, g is not unimodal, so it cannot be conjugate to f. This basically means that unimodal maps with a degenerate critical point should be considered c-structurally unstable. However, the metric ρr that we are using, is not defined outside the space of unimodal maps. Fortunately, there is another point of view, that we will adopt here. Namely, we consider only perturbations that are unimodal. Note that if the critical point is non-degenerate, its C 2 -c-small perturbation is also unimodal. Hence, our decision will influence only classification of unimodal maps with the degenerate critical point. We will denote the space of unimodal C r -maps of K into itself by Vr and its subspace consisting of maps with the critical point non-degenerate (that is, with the second derivative at the critical point non-zero) by Ur . Those spaces are considered with the metric ρr . We mentioned already that the space Vr is complete. As we will prove in Lemma 6.1, Ur is its open dense subset. We are interested in describing all c-structurally stable maps in Vr and proving that c-structurally stable maps are dense in Vr . To do so we need a couple of definitions. An interval I is said to be periodic of period k ≥ 1 if f k (I ) ⊂ I and the interiors of f i (I ) are pairwise disjoint for i = 0, 1, 2, . . . , k − 1. If f ∈ Vr is a map such that either (1) the trajectory of c f is attracted to a topologically attracting (at least from one side) periodic orbit, but never falls into this orbit, or (2) c f is mapped by some iterate of f to the interior of an interval consisting entirely of periodic points of the same (minimal) period, then we call f stabilized. Moreover, in the case (1) we call f attractively stabilized, and in the case (2) neutrally stabilized. Observe that in both cases c f is mapped by some iterate of f to the interior of a periodic interval I of period k with f k | I monotone. Moreover, if x belongs to such interval then either x is periodic or the trajectory of x is attracted to a topologically attracting (at least from one side) periodic orbit. Denote the set of all stabilized maps in V2 by St2 . Our main result is the following theorem, in which we verify the generalized Fatou conjecture for V2 . Theorem 1.1. A map f ∈ V2 is c-structurally stable if and only if it is stabilized. Moreover, St2 is an open dense subset of V2 .
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cf
cf
Fig. 1.1. Attractively stabilized (left) and neutrally stabilized (right) maps
Let us compare this theorem with a theorem of Kozlovski in which he verifies the generalized Fatou conjecture for unimodal maps. Theorem 1.2 (Kozlovski). The set of real analytic unimodal maps whose critical points are attracted by attracting periodic orbits is dense in the set of all unimodal C r -maps in the C r -topology for r = 1, 2, . . . , ∞, ω. We use this theorem in an essential way when we consider infinitely renormalizable unimodal maps in Sect. 5. However, a major tool in [5] are analytic maps, which implies that in [5] perturbations inevitably differ from the original map on the entire interval except for finitely many points. Therefore the arguments from [5] are not directly applicable to V2 . On the other hand, if the question of density of structurally stable maps is considered for smaller spaces of maps (like, for example, analytic unimodal maps, space of real polynomials, etc.) then even the question of density in the sense of ρr is not always reasonable because then the space with ρr -metric may become discrete (as in the cases mentioned above). We deal with the problem in a step-by-step fashion. Section 2 contains preliminaries. In Sect. 3 we show that stabilized maps from V2 are c-structurally stable. In Sect. 4 we consider finitely renormalizable unimodal maps in U2 and prove that if they are not stabilized then they are not c-structurally stable. The decay of geometry established in [2,11] is crucial for our arguments. Then in Sect. 5 we obtain similar results for infinitely renormalizable maps in U2 (and even in U∞ ). Finally in Sect. 6 we deduce the results for V2 from the results for U2 . When the paper was ready for submission, we learned about Mike Todd’s thesis [13] in which, among others, a problem of stability of unimodal maps was considered. In the case of non-recurrent critical points the results of [13] are similar to ours (as we point out before Lemma 4.1, they are known). In the case of a recurrent critical point it is shown in [13] that if a unimodal map f with recurrent critical point c is analytic or such that ω(c) is minimal, then for any point x ∈ ω(c) arbitrarily small C k -perturbations of f supported on an arbitrary neighborhood of x may change the combinatorial type of f. Todd uses complex analytic tools based upon [5,7] and obtains results related to ours. However, [13] covers only a limited class of maps while we solve the problem for all C 2 -unimodal maps. 2. Preliminaries We start by proving the proposition promised in the Introduction.
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Proposition 2.1. The function ρr is a complete metric in Vr . Proof. Let us show first that |I ( f, g)| understood as the distance between f and g is a metric. It is clear that |I ( f, g))| is symmetric and reflexive. To prove the triangle inequality observe that I ( f, g) ∪ I (g, h) is a closed interval (it is the union of two closed intervals, each of which contains cg ) outside which we have f = g = h. Moreover, c f , cg , ch ∈ I ( f, g) ∪ I (g, h). Hence I ( f, h) ⊂ I ( f, g) ∪ I (g, h) and the triangle inequality follows. Thus, ρr as a sum of two metrics, is a metric. Let us now show that it is complete. Suppose that we have a Cauchy sequence (in the sense of ρr ) of unimodal maps f 1 , f 2 , . . . . Then it follows that there exists f ∈ C r (K , K ) such that f i → f ∈ C r (K , K ) in the sense of the C r -metric. Let Im = ∞ j=m I ( f j , f j+1 ). Since for any j the intervals I ( f j , f j+1 ) and I ( f j+1 , f j+2 ) intersect each other (both contain c f j+1 ), Im is a closed interval. Since we deal with a Cauchy sequence in the sense of ρr , for every ε > 0 there exists m such that |I ( f m , f i )| ≤ ε for every i ≥ m. Hence Im ⊂ [c fm − ε, c fm + ε] and |Im | → 0. Since Im ⊃ Im+1 . . . we see that ∞ m=1 Im = {c} for some point c. Then c = lim m→∞ c f m is a local maximum of f. Now, for every x = c there is a neighborhood U of x and an integer m such that f coincides with f m on U and c fm ∈ / U. Thus, c is the only critical point of f, so f is unimodal (note that in general unimodal C r -maps may converge in the C r -sense to a non-unimodal map, e.g., to a map with an interval on which it is constant). Therefore f ∈ Vr and |I ( f m , f )| → 0. This proves that ρr ( f m , f ) → 0.
For a unimodal map f : K → K set c = c f , c = c and for x = c define x as the unique point not equal to x with f (x ) = f (x). Observe that the map x → x is defined on the entire K because we assume that f (a) = f (b). Given a set A ⊂ K , set A = {x : x ∈ A}. In what follows any interval I ⊂ K such that I = I will be called symmetric. We will also use the notation c1 = f (c). In order to make a c-small perturbation of a unimodal map, we will be adding to this map a small “bump” concentrated close to the critical point. The following lemma describes this procedure. Lemma 2.2. Let an integer r ≥ 2, a map f ∈ Vr on an interval K , and a symmetric interval I ⊂ K be given. Then for every ε > 0 there is δ > 0 such that if z ∈ K is a point with 0 < |z − f (c)| < δ then there exists g ∈ Ur such that f and g coincide outside I, have the same critical point c, g(c) = z, and ρr ( f, g) < ε. Moreover, if f is of class C ∞ , then g can be chosen also of class C ∞ . Proof. Fix functions ψ+ , ψ− : R → R of class C ∞ , equal 0 outside the interval [−1, 1], (0) = 0, ψ (0), ψ (0) < 0 and such that ψ > 0 on (−1, 1), while with ψ+ (0) = ψ− + + − ψ− < 0 on (−1, 1). We will look for g of the form x −c g(x) = f (x) + α · ψ± β for some α, β > 0, where we choose ψ+ when z > f (c) and ψ− when z < f (c). Clearly, if f is of class C ∞ , then g is also of class C ∞ . We fix β so small that [c − β, c + β] ⊂ I and β < min(1, ε/4). We have α (i) x − c (i) (i) (2.1) |g (x) − f (x)| = i · ψ± β β
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for i = 0, 1, . . . , r, so ρr ( f, g) < ε/2 +
α ψ± r,[−1,1] . βr
(2.2)
If α is small enough then the right-hand side of (2.2) is smaller than ε. On the other hand, if δ > 0 is sufficiently small, we can choose the corresponding small α to satisfy g(c) = z and, by (2.2), to guarantee that ρr ( f, g) < ε. Thus, it remains to prove that for a sufficiently small α the map g is unimodal with the critical non-degenerate point c. (0) = 0, ψ (0) < 0 and f (c) ≤ 0, we get g (c) = 0 and g (x) < 0 for Since ψ± ± every x in a small neighborhood U = (y, z) of c with f (y) > 0 and f (z) < 0. Then in U , the function g(x) has positive derivative if x < c and negative derivative if x > c. Outside U, if α is sufficiently small, the sign of g is the same as the sign of f . Thus, c is the maximum of g and g is unimodal. This completes the proof.
Any perturbation from the above lemma (or the next one) will be called a bump perturbation. The next lemma uses the construction from Lemma 2.2 but includes some additional estimates and is proven only in C 2 . Lemma 2.3. Let f ∈ U2 and suppose that γ > 0 is given. Then there exists δ > 0 such 1| that for any interval T = [a, a ] with |T | ≤ δ and any point d ∈ K with |d−c | f (T )| ≤ δ there exists a unimodal map g ∈ U2 with the same critical point c such that g = f outside T, ρ2 ( f, g) ≤ γ and g(c) = d. Proof. We use the same construction as in Lemma 2.2, but only with ψ = ψ+ and admitting α < 0. Thus, we look for g of the form g(x) = f (x) + α · ψ x−c β for some α, β ∈ R, β > 0. According to (2.1) and (2.2), we have |α| ζ β2
(2.3)
|α| ζ, β2
(2.4)
|g (x) − f (x)| ≤ for all x ∈ K , and ρ2 ( f, g) ≤ 2β +
where ζ = ψ2,[−1,1] . Since f (c) = 0, there exists σ > 0 such that for any interval T = [a, a ], 2 | f (T )| ≤ σ min(c − a, a − c) , and there exists τ > 0 such that | f (x)| > τ for any x ∈ (c − τ, c + τ ). Set γ γ τ δ = min , , 1, τ, 4 2ζ σ σζ and if T = [a, a ] is an interval with |T | ≤ δ which we fix from now on, set β = min(c − a, a − c).
(2.5)
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Note that β ≤ |T | ≤ δ ≤ min(1, τ ) and | f (T )| ≤ σβ 2 .
(2.6)
1| Take a point d ∈ K with |d−c | f (T )| ≤ δ, and set α = d − c1 . Then g(c) = f (c) + α = d and (in view of (2.5) and (2.6)) γ |α| |d − c1 |ζ σ ≤ δζ σ ≤ min ,τ . (2.7) ζ ≤ β2 | f (T )| 2
To see that g is unimodal, observe that g = f outside (c − δ, c + δ) ⊂ (c − τ, c + τ ), and for x ∈ (c − τ, c + τ ) we have |g (x)| ≥ | f (x)| − | f (x) − g (x)| > τ − | f (x) − g (x)|, and this is positive by (2.3) and (2.7). Moreover, both f and g − f have a critical point at c, so c is also a critical point of g. Finally, by (2.4)–(2.7), ρ2 ( f, g) ≤ 2β + This completes the proof.
|α| γ ζ ≤ 2δ + ≤ γ . 2 β 2
If f is a smooth piecewise monotone map then an interval I is said to be a homterval if for every n there is no critical point of f n in its interior. Clearly, a point belongs to the interior of a homterval if and only if it is not approximated by precritical points. We need the following theorem, proved in [10]. We state it here in a form convenient for us. Theorem 2.4. Let f ∈ U2 . If J is a homterval then J is eventually mapped into a periodic homterval. In particular, if c is not periodic then the critical value c1 is approximated by precritical points if and only if none of the following happens: 1. c1 is mapped to the interior of a periodic homterval by some iterate of f, 2. f (c) = b and a is an endpoint of an invariant homterval. If there exists a periodic interval I containing c of period greater than 1 then f is said to be renormalizable. If there exists a number k such that a periodic interval containing c cannot be of period greater than k then f is said to be finitely renormalizable, otherwise f is said to be infinitely renormalizable. An interval T = [ p, q] ⊂ K such that c ∈ ( p, q) and all forward images of p, q miss ( p, q) is said to be nice (see [8]). We always assume that nice intervals are small and map off themselves. Fix a nice interval T. Consider a point x ∈ K . Then we can define the first entry map RT into int T as the first positive iterate of f (if any) which maps x into int T. Clearly, there are points at which RT is not defined. The set of all points on which RT is defined is the union DT of pairwise disjoint open intervals called domains of RT . There is one exceptional domain of RT , called central return domain of T and defined as follows. Suppose that c is recurrent. Then there exists the minimal positive integer m with f m (c) ∈ int T. We can choose an open interval J c1 such that f m−1 (J ) = T and f m−1 | J has no critical points. The interval f −1 (J ) ∩ T is then called the central return domain of T and is
; moreover, f maps T
in a 2-to-1 fashion onto J. Clearly, even though denoted by T T is not necessarily symmetric, its central return domain always is. Observe that the
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are mapped by f m into one endpoint of T. Otherwise a domain I of RT endpoints of T is mapped by RT diffeomorphically onto T. The properties of domains of RT in the case when T is nice are well known. For the sake of completeness we state them in the following lemma. It is partially proved in [9], p. 341, partially is obvious. Lemma 2.5. Let T c be a nice interval, and let Q be a domain of RT . Then for some m the map RT | Q coincides with f m | Q , the endpoints of Q are mapped by RT to the endpoint(s) of T and have the positive f -orbits disjoint from int T, and there are two possibilities: (1) RT | Q is monotone and RT (Q) = T (if Q is not a central domain of T ), or (2) RT | Q has only one critical point (if Q is a central domain of T ). Moreover, intervals Q, f (Q), . . . , f m (Q) = RT (Q) are pairwise disjoint, and also disjoint from T, except for the case when Q ⊂ T, in which case the only nonempty intersection may be between Q and f m (Q) = RT (Q). Finally we state a result which we rely upon considering the finitely renormalizable recurrent case in Sect. 4. It has been proven in [2,11], and establishes the decay of geometry for maps in U3 . Moreover, the same result is true in U2 , see [11,12]. To state it we need some definitions. Given intervals I, M with I ⊂ int M we set |I | |I | , , ε(M, I ) = max |M − | |M + | where M − and M + are components of M\I, and if ε(M, I ) < ε, we say that I is ε-inside M. Theorem 2.6. Let f ∈ U2 be a finitely renormalizable map with a recurrent and nonperiodic critical point c. Then for any ε > 0 there exists a nice interval I c such that the central return domain
I is ε-inside I. 3. Stabilized Maps and Their Structural Stability Here we prove the simple direction of Theorem 1.1 and show that stabilized maps are c-structurally stable. We also state a useful (although trivial) lemma dealing with maps whose critical points are periodic. Theorem 3.1. For r = 0, 1, . . . ∞, if f ∈ Str then it is c-structurally stable in Vr . Proof. If f ∈ Str then two cases are possible. First suppose that f is neutrally stabilized. Then for some minimal m the point f m (c f ) belongs to an open interval I consisting of periodic points of minimal period k. Choose pullbacks of I along c f , . . . , f m (c f ) and denote them I−1 , . . . , I−m with c f ∈ I−m . Choose a small neighborhood U ⊂ I−m of c f . If a perturbation is sufficiently c-small then it will not change the map outside U, and on U the new map g will act so close to f that g(U ) ⊂I−m+1 , . . . , g m (U ) ⊂ I. m Clearly, a homeomorphism which acts as the identity outside i=1 I−i = H and inside i i H simply maps intervals g (U ) onto the intervals f (U ) appropriately, conjugates g and f which shows that neutrally stabilized maps are c-structurally stable. A very similar argument shows that if a map is attractively stabilized then it is c-structurally stable too. We leave this case to the reader.
The next lemma deals with the case when the orbit of the critical point is periodic.
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Lemma 3.2. Suppose that f ∈ Vr (r = 1, 2, . . . , ∞) and the orbit of c f is periodic. Then f is not c-structurally stable in Vr , but arbitrarily c-close to f there are attractively stabilized maps. Proof. Let n be the period of c = c f . For any sufficiently c-small bump perturbation g of f the point c is a critical point of g and g n (c) = c, so g is not topologically conjugate to f. Moreover, if the perturbation is sufficiently c-small, the attracting periodic orbit of period n persists and attracts the g-trajectory of c. Thus, g is attractively stabilized.
4. Finitely Renormalizable Maps In this section we consider finitely renormalizable unimodal maps. The main result here is Theorem 4.3 in which we show that a finitely renormalizable f ∈ / St2 is not structurally stable. We begin with the non-recurrent case (this result is known, but we include it for completeness). Lemma 4.1. Let f ∈ Ur (r = 2, 3, . . . , ∞) be a map with the critical point c which is non-recurrent and none of the following happens: 1. c1 is mapped to the interior of a periodic homterval by some iterate of f, 2. f (c) = b and a is an endpoint of an invariant homterval. Then arbitrarily c-close to f in Ur there are maps with critical periodic points. Proof. By Theorem 2.4 there are precritical points arbitrarily close to c1 . Let us show that then there exists a neighborhood U of c such that arbitrarily close to c1 there are precritical points whose trajectories never enter U before they hit c. Consider two cases. First, it may happen that there is a small neighborhood of c which contains no precritical points. Then choose this neighborhood as U. Now, suppose that there are precritical points in any neighborhood of c. Since c is non-recurrent, we can choose a precritical point y < c so close to c that the orbit of c1 is disjoint from [y, y ]. Moreover, we can choose y so that no point inside (y, y ) = U is mapped into c before y. That is, the order (the time necessary to get to c) of every precritical point from (y, y ) is strictly higher than that of y (and of y ). Let V be a small neighborhood of c1 and let z be the precritical point of the smallest order m in V. Let us show that the trajectory of z never enters U before it hits c. Suppose that for some k < m we have f k (z) ∈ U. By the choice of U then f k (c1 ) ∈ / U and hence there exists a point t ∈ V such that f k (t) = y or f k (t) = y . By the choice of y it follows that t is precritical and of smaller order than z, a contradiction. Hence the trajectory of z misses U before it hits c. Now, for every bump perturbation g of f such that Di( f, g) ⊂ U, the point z found above is also precritical for g. By Lemma 2.2 we can choose g such that it is of the same class as f, differs from f on an arbitrarily small neighborhood W ⊂ U, and once W is fixed, g(c) = z (so c is periodic for g) and z is so close to c1 that g − f has as small C r -norm as we want.
We get the following corollary to the above lemma. Corollary 4.2. If f ∈ Ur \Str (r = 2, 3, . . . , ∞) and c is non-recurrent then f is not c-structurally stable in Vr .
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Proof. We may assume that f (c) = b as otherwise f is obviously not c-structurally stable. If the orbit of c is infinite then c is not mapped into the interior of a homterval because f ∈ / Str , and in view of Theorem 2.4 every infinite trajectory of a point in an interior of a homterval is attracted to a periodic orbit. Thus, in this case f is not c-structurally stable by Lemma 4.1. If the orbit of c is finite, then c is preperiodic. If k is its eventual period then again because f ∈ / Str , there exists an arbitrarily c-small bump perturbation g of f for which c is not preperiodic with the eventual period k (this explains the definition of a neutrally stabilized map).
Now we deal with the recurrent finitely renormalizable case. Theorem 4.3. If f ∈ U2 \St2 is finitely renormalizable then f is not c-structurally stable. Moreover, arbitrarily c-close to f there are maps g with periodic critical points. Proof. By Lemma 3.2 we may assume that c is not periodic. By Lemma 4.1 we may assume that c is recurrent. Suppose that γ > 0 is given. Now, choose δ applying Lemma 2.3 to f and γ . By Theorem 2.6 there exists an arbitrarily small nice interval I = [x, x ] such that its central return domain
I = [y, y ] is δ-inside I. It follows
)| < δ. that if I is sufficiently small then || ff ((II )| Now, by Lemma 2.3 there exists a map g with ρ2 (g, f ) < γ and g(c) = f (y). Since by Lemma 2.5 the f -orbit of f (y) avoids (x, x ), we see that the g-orbit of g(c) = f (y) avoids (x, x ) because g = f outside (x, x ). Since c is recurrent for f, it implies that f and g are not topologically conjugate. Consider a family of maps gt , 0 ≤ t ≤ 1, defined as gt (x) = t f + (1 − t)g. Clearly, all these maps are unimodal C 2 -maps, with the same critical point c, coinciding with g = f outside (x, x ). Since f and g are not topologically conjugate it follows that for some t the point c will be gt -periodic. By the choice of g we have ρ2 ( f, gt ) < γ which completes the proof.
5. Infinitely Renormalizable Maps The main result of this section is based upon a simple observation that for infinitely renormalizable maps the restriction of the appropriate iterate of the map onto a small periodic interval can be viewed as a globally defined map. We will need the following simple and well known lemma. Lemma 5.1. For closed intervals I and J, a positive integer r and a C r -function ψ : J → R, the map f → ψ ◦ f from C r (I, J ) to C r (I, R) is continuous. We will also need the following lemma about a specific extension of a function defined on an interval onto a greater interval. Lemma 5.2. Let r be a positive integer and let I, J, T be closed intervals such that I ⊂ int(J ) ⊂ J ⊂ int(T ). Then there exists a constant M = Mr (I, J ) such that the following holds. Given two functions f ∈ C r (T, R), h ∈ C r (I, R) there exists a function F ∈ C r (T, R) such that F| I = h| I , F|T \J = f |T \J and F − f r,T < Mh − f r,I . If f and h are of class C ∞ then F can be chosen of class C ∞ except perhaps at the endpoints of I. Proof. Consider the left endpoint p of I and the left component L of J \ int(I ). If t = h − f r,I then |(h − f )(k) ( p)| ≤ t for k = 0, 1, 2, . . . , r. For x ∈ L set h(x) = f (x) +
r
k=0
ak (x − p)k , where ak =
(h − f )(k) ( p) . k!
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This formula extends h to a function of class C r on I ∪ L and h − f r,I ∪L ≤ (r + 1) max(1, |L|r )t. We do the same construction with the right endpoint of I and the right component of J \ int(I ). In this way we get an extension of h to a C r -function on J with h − f r,J ≤ (r + 1) max(1, |J |r )h − f r,I . Now we fix a function ϕ : R → [0, 1] of class C ∞ , depending only on I and J, such that ϕ is 1 on I and 0 on a neighborhood of the closure of R\J. Finally, we define F = f + ϕ(h − f ). Clearly, F| I = h| I and F|T \J = f |T \J . By the formulas for the derivatives of a product of two functions and by the estimate for h − f r,J , there is a constant M, depending only on r, I, J, such that F − f r,T < Mh − f r,I . If f and h are of class C ∞ then F constructed as above is of class C ∞ except perhaps at the endpoints of I.
Now we can prove the main result of this section. A closed interval I will be called strongly periodic (of period n) if I, f (I ), . . . , f n−1 (I ) are pairwise disjoint while f n (I ) ⊂ I. Theorem 5.3. Assume that f ∈ Ur (r ∈ {2, . . . , ∞}) is infinitely renormalizable. Then f is not c-structurally stable. Moreover, arbitrarily c-close to f there is a map F for which the trajectory of its critical point is attracted to an attracting periodic orbit. Proof. Fix ε > 0. Since f is infinitely renormalizable and by Theorem 2.4, there exists a positive integer n and a closed strongly periodic interval I ⊂ K of length less than ε/2 and period n such that c = c f ∈ int I. Then there exists a closed interval J of length less than ε, containing I in its interior and disjoint from f k (I ), k = 1, 2, . . . , n − 1. Assume first that r is finite. By Theorem 1.2 there exists a C ∞ -map g : I → I arbitrarily C r -close to f n | I and such that its critical point is attracted to an attracting periodic orbit. Since f n−1 | f (I ) is a diffeomorphism and f | I = ( f n−1 | f (I ) )−1 ◦ f n | I , the map h : I → f (I ) defined as h = ( f n−1 | f (I ) )−1 ◦ g is arbitrarily C r -close to f | I by Lemma 5.1. By Lemma 5.2 we can extend h to a C r -map F on R so that F − f is supported on J and is arbitrarily C r -close to 0. In particular, the sign of F will be the same as the sign of f outside of I. Since additionally we know that F| I = h| I is unimodal, F is also unimodal. Since F = f on f k (I ), k = 1, 2, . . . , n − 1, we get F| I = f n−1 | f (I ) ◦ h| I = g, and therefore the critical point of F is attracted to an attracting cycle. If r = ∞ we first fix a finite s and apply the above construction for s instead of r to get a map Fs with ρs (Fs , f ) < 1/s. By Lemma 5.2, we may assume that Fs is of class C ∞ except perhaps at the endpoints of I. Then a standard construction (like in the second part of the proof of Lemma 5.2) produces a map G s of class C ∞ , arbitrarily close to Fs in the C s -topology and coinciding with Fs outside J (which is shorter than 1/s). By the definition of the C ∞ -topology the sequence (G s )∞ s=1 converges to f in this topology, which completes the proof for r = ∞.
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6. Proof of Theorem 1.1 First we show how to deal with the maps with degenerate critical points. Lemma 6.1. For every r ≥ 2 the set Ur is open and dense in Vr . Proof. Openness is clear, and density follows immediately from Lemma 2.2.
Now we can prove Theorem 1.1. One direction is immediate: if f ∈ St2 then f is c-structurally stable by Theorem 3.1. Assume now that f ∈ U2 is c-structurally stable. If f is finitely renormalizable, then it belongs to St2 by Theorem 4.3. If f is infinitely renormalizable, then by its c-structural stability and Theorem 5.3, it is conjugate to a map with the critical point attracted to an attracting periodic orbit. Thus, the same property has to hold for f. This prevents infinite renormalizability, so this case cannot occur. Hence, if f ∈ U2 is c-structurally stable then f ∈ St2 . Assume now that f ∈ V2 \U2 is c-structurally stable. By its c-structural stability and Lemma 6.1, f is conjugate to a c-structurally stable g ∈ U2 . As we proved, this g belongs to St2 . By the definition of St2 , any map conjugate to a map from St2 also belongs to St2 . Thus, f ∈ St2 . Finally, by the definition of c-structural stability, the set of all c-structurally stable elements of V2 is open in V2 . We will show that it is dense in V2 . By Lemma 6.1 it is enough to show that it is dense in U2 . However, this density follows immediately from Lemma 4.1, Theorem 4.3, and Theorem 5.3. This completes the proof of Theorem 1.1. References 1. Blokh, A., Misiurewicz, M.: Collet-Eckmann maps are unstable. Commun. Math. Phys. 191, 61–70 (1998) ´ atek, G.: Metric attractors for smooth unimodal maps. Ann. Math. 159, 2. Graczyk, J., Sands, D., Swi¸ 725–740 (2004) 3. Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. France, 48, 33–94, 208–314 (1920) 4. Jakobson, M.V.: Smooth mappings of the circle into itself (Russian). Mat. Sb. 85, 163–188 (1971) 5. Kozlovski, O.S.: Axiom A maps are dense in the space of unimodal maps in the C k topology. Ann. Math. 157, 1–43 (2003) 6. Kozlovski, O.S., Shen, W., van Strien, S.: Density of hyperbolicity in dimension one. Ann. Math. 166, 145–182 (2007) 7. Levin, G., van Strien, S.: Bounds for maps of an interval with one critical point of inflection type II. Invent. Math. 141, 399–465 (2000) 8. Martens, M.: Distortion results and invariant Cantor sets of unimodal maps. Ergodic Th. and Dyn. Syst. 14, 331–349 (1994) 9. de Melo, W., van Strien, S.: One-Dimensional Dynamics. Berlin: Springer Verlag, 1993 10. de Melo, W., van Strien, S.: A structure theorem in one-dimensional dynamics. Ann. Math. 129, 519–546 (1989) 11. Shen, W.: Decay of geometry for unimodal maps: An elementary proof. Ann. Math. 163, 383–404 (2006) 12. Shen, W.: Decay of geometry – the C 2 case. preprint, 2008 13. Todd, M.: One-dimensional dynamics: cross-ratios, negative Schwarzian and structural stability. PhD Thesis, University of Warwick, Department of Mathematics, 2003 Communicated by G. Gallavotti
Commun. Math. Phys. 289, 777–801 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0749-9
Communications in
Mathematical Physics
Equivariant Differential Characters and Symplectic Reduction Eugene Lerman , Anton Malkin Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL 61801, USA. E-mail: [email protected]; [email protected] Received: 22 July 2008 / Accepted: 13 November 2008 Published online: 26 February 2009 – © Springer-Verlag 2009
Abstract: We describe equivariant differential characters (classifying equivariant circle bundles with connections), their prequantization, and reduction. Contents 0.
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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 Prequantization . . . . . . . . . . . . . . . . . . . . . . . 0.2 Lie group actions . . . . . . . . . . . . . . . . . . . . . . 0.3 The stack of bundles and the stack of differential characters 0.4 Invariant connections . . . . . . . . . . . . . . . . . . . . 0.5 Symplectic reduction . . . . . . . . . . . . . . . . . . . . 0.6 General Lie groupoids . . . . . . . . . . . . . . . . . . . . 0.7 Structure of the paper . . . . . . . . . . . . . . . . . . . . 0.8 Previous results . . . . . . . . . . . . . . . . . . . . . . . 0.9 Higher degree versions . . . . . . . . . . . . . . . . . . . Circle Bundles with Connections and Prequantization . . . . . 1.1 Prequantization . . . . . . . . . . . . . . . . . . . . . . . 1.2 Equivariant S 1 -bundles on groupoids . . . . . . . . . . . . 1.3 Connections on equivariant S 1 -bundles . . . . . . . . . . . 1.4 Singular and de Rham cohomology of groupoids . . . . . . 1.5 Differential characters on groupoids . . . . . . . . . . . . 1.6 Equivariant Kostant theorem . . . . . . . . . . . . . . . . Invariant Connections and the Moment Map . . . . . . . . . . 2.1 The moment map . . . . . . . . . . . . . . . . . . . . . . 2.2 Invariant connections . . . . . . . . . . . . . . . . . . . . Supported in part by NSF grant DMS-0603892.
Supported in part by NSF grant DMS-0456714.
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2.3 Prequantization on global quotients . . . . . . . . . . Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 General reduction . . . . . . . . . . . . . . . . . . . 3.2 Reduction for action groupoids . . . . . . . . . . . . 4. Differential Characters as Characters . . . . . . . . . . . 4.1 Differential characters on manifolds and groupoids . 4.2 Special cases . . . . . . . . . . . . . . . . . . . . . 4.3 Invariant differential characters on an action groupoid 5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Example: classifying stack . . . . . . . . . . . . . . 5.2 Example: coadjoint orbits of compact Lie groups . . 5.3 Example: coadjoint orbits of SU (2) and S 1 -reduction References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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0. Introduction This paper is a continuation of [11]. The goal of the project is to understand geometric (pre)quantization and symplectic reduction in the context of stacks. There are two practical reasons for doing that. First, even if one is only interested in global Lie group actions on manifolds, symplectic quotients are generally orbifolds (i.e. DeligneMumford stacks). Second, the language of stacks allows one to work locally and thus to ˇ avoid messy Cech-type arguments.
0.1. Prequantization. The classic geometric prequantization theorem (due to Weil [16] and Kostant [10]) says that given a closed integral differential 2-form ω on a manifold M there exists a principal S 1 -bundle P → M with a connection A such that the curvature of A is equal to ω, and moreover, the set of isomorphism classes of such pairs (P, A) is a 1 principal homogeneous space of the group of flat S -bundles. Recall that a 2-form ω on a manifold M is integral if S ω ∈ Z for any closed smooth singular 2-chain S ∈ Z 2 (M). It would be preferable for prequantization procedure to produce unique output (an S 1 -bundle with connection). For example, this is clearly required to make sense of statements like “prequantization commutes with reduction” - see below. In order to have unique output of the prequantization one has to refine its input. One way to do it, due to Cheeger and Simons [5], is by using differential characters (an alternative approach is provided by Deligne cohomology - see [3]). A differential character of degree 2 (degree 1 in Cheeger-Simons grading) is a pair (ω, χ ), where ω ∈ 2 (M) is a closed 2-form and χ : Z 1 (M) → R/Z is a character of the group Z 1 (M) of smooth singular 1-cycles. This pair should satisfy the following compatibility condition: χ (∂ S) = ω mod Z, S
for any smooth singular 2-chain S ∈ C2 (M). One can show (cf. [5,9,11] for various versions of the proof) that differential characters classify isomorphism classes of principal S 1 -bundles with connections. The classifying bijection (whose inverse is the prequantization map) associates to a principal S 1 -bundle P with a connection A its differential character (ω, χ ), where ω is the curvature of A and χ is the holonomy of A (we identify S 1 with R/Z throughout the paper).
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0.2. Lie group actions. Let G be a Lie group, g the Lie algebra of G, g∗ the dual space of g, and <, > the natural pairing between g and g∗ . Suppose G acts on a manifold M and consider a G-equivariant principal S 1 -bundle P → M with an invariant connection A. That means we are given a lifting of the G-action to the total space P (commuting with the S 1 -action) and the connection form A ∈ 1 (P) is G-invariant. Then the curvature ω of A is also G-invariant and moreover there is a G-equivariant map µ : P → g∗ given by < µ, X >= i εx A, where ε X is the vector field on P generating the action of an element X ∈ g. The map µ is S 1 -invariant and hence descends to a map from M to g∗ which we also denote by µ. It follows from the definition of the curvature that i εx ω = − < dµ, X >. Hence µ is the moment map in the sense of equivariant symplectic geometry (see for example [8]). Note that if G-action is free and proper (so that the quotient M/G is a manifold) then a G-equivariant bundle P with an invariant connection A descends to M/G iff the moment map is identically equal to zero, in other words iff the forms A and ω are G-basic. Recall that a differential form is G-basic if it is G-invariant and G-horizontal (i.e. vanish on vector fields generating the action of g). Equivariant prequantization procedure should produce a G-equivariant principal S 1 -bundle with an invariant connection given a closed invariant integral 2-form ω on M. Obviously there are some obstructions. First of all there should exist a moment map µ such that i εx ω = − < dµ, X > for any X ∈ g. This means that the action is Hamiltonian. If G is connected and simply-connected then existence of the moment map ensures existence of an equivariant prequantization bundle with an invariant connection (cf. [10]). For a general G there are additional obstructions. The origin of these complications is that in the equivariant situation it is not enough to think of isomorphism classes of bundles – one needs to consider an actual bundle in order to lift the G-action to the total space. This becomes much more clear if one uses an equivalent definition of a o an equivariant bundle in terms of the action groupoid M o G × M, where the two p maps are the projection and the action. In this language an equivariant S 1 -bundle is a ∼ → p ∗ P of the two pull-backs of bundle P → M together with an isomorphism ϕ : a ∗ P − P to G × M. Note that we have to use a morphism in the category of bundles on G × M, so it is not enough to consider isomorphism classes of bundles. Therefore in order to understand equivariant prequantization, we have to upgrade the prequantization map to a functor from some category of differential characters to the category of principal S 1 -bundles with connections. 0.3. The stack of bundles and the stack of differential characters. Let DBS 1 (M) be the category of S 1 -bundles with connections on a manifold M. In [9] Hopkins and Singer introduced a category of differential characters DC 22 (M) and showed that it is equivalent to DBS 1 (M). As the name suggests, the isomorphism classes of objects of DC 22 (M) are in bijection with differential characters. We give the complete definition of the category DC 22 (M) in Subsect. 1.1. Let us just mention that an object of DC 22 (M) is a triple (c, h, ω), where c ∈ Z 2 (M, Z) is a smooth singular 2-cocycle, ω ∈ 2 (M) is a closed 2-form, and h ∈ C 1 (M, R) is a smooth singular 1-chain satisfying dh = c − ω. In the previous paper [11] we introduced a prequantization functor Preq : DC 22 (M) → DBS 1 (M), such that if Preq((c, h, ω)) is a bundle P with a connection A, then the Chern class of P is [c] ∈ H 2 (M, Z) and the curvature of A is ω. Moreover we showed (and it was
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crucial for our definition of Preq) that both DC 22 and DBS 1 are stacks over the category of differentiable manifolds and Preq is an equivalence of stacks. This allowed us to define geometric prequantization of arbitrary stacks over the category of manifolds, in particular prequantization of orbifolds and equivariant prequantization of manifolds with Lie group actions. For example, given a manifold M with an action of a Lie group G we have an equivalence functor Preq
DC 22 (M ⇔ G × M) DC 22 ([M/G]) −−→ DBS 1 ([M/G]) DBS 1 (M ⇔ G × M). Here [M/G] is the quotient stack. Objects of the category DBS 1 (M ⇔ G × M) are G-equivariant principal S 1 -bundles on M with G-basic connections. The category DC 22 (M ⇔ G × M) of equivariant differential characters is defined in Sect. 1. 0.4. Invariant connections. In the present paper we are interested in equivariant bundles with G-invariant (but not necessarily basic) connections. These form a category dBS 1inv (M ⇔ G × M). The corresponding category of differential characters is defined in Sect. 2 and is denoted by DC 2inv (M ⇔ G × M). In particular a G-invariant differential character (i.e. an isomorphism class of objects of the category DC 2inv (M ⇔ G × M)) is a triple (ω, µ, ), where ω ∈ 2 (M) is a 2-form, µ : M → g∗ is a (moment) map, and is a character Z 1 (M ⇔ G × M) → R/Z of the group Z 1 (M ⇔ G × M) of smooth singular 1-cycles on the action groupoid. We refer the reader to Sect. 4 for details, but let us give here two examples of subgroups of Z 1 (M ⇔ G × M). The first one is the group Z 1 (M) of 1-cycles on M. The restriction of to Z 1 (M) together with ω form a differential character on M corresponding to a bundle with connection (without equivariant structure). The second typical subgroup of Z 1 (M ⇔ G × M) is the stabilizer group (in G) of a point on M. The restriction of to this subgroup corresponds (under prequantization map) to the action of the stabilizer group on the fiber of the prequantization S 1 -bundle. Let us also note that the character is equivariant (in other words, belongs to DC 22 (M ⇔ G × M) DC 22 ([M/G]) iff ω is basic or, equivalently, iff µ vanishes identically. Here is the first result of the present paper stated both in terms of categories and isomorphism classes of objects (see Theorems 2.3.1 and 4.3.1 for more precise/complete statements) . Theorem 0.4.1. Suppose a Lie group G acts on a manifold M. • There is an equivalence of categories (prequantization functor) ∼
Preq : DC 2inv (M ⇔ G × M) − → dBS 1inv (M ⇔ G × M) from the category DC 2inv (M ⇔ G × M) of G-invariant differential characters to the category dBS 1inv (M ⇔ G × M) of G-equivariant principal S 1 -bundles with G-invariant connections. • Isomorphism classes of G-equivariant S 1 -bundles with G-invariant connections are in bijection with G-invariant differential characters, i.e. triples (ω, µ, ), where ω ∈ 2 (M) is a G-invariant 2-form (the curvature of the connection), µ : M → g∗ is a G-equivariant (moment) map, and is a G-invariant character Z 1 (M ⇔ G × M) → R/Z.
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0.5. Symplectic reduction. We continue with our setup of a Lie group G acting on a manifold M, a G-equivariant principal S 1 -bundle P → M with a G-invariant connection A ∈ 1 (P), the curvature ω ∈ 1 (M), and the moment map µ : M → g∗ . Assume zero is a regular value of the moment map. Then µ−1 (0) is a manifold and the restriction of the pair (P, A) to µ−1 (0) is a principal S 1 -bundle with a basic connection, hence a bundle with connection on the stack [µ−1 (0)/G]. On the other hand one can restrict the corresponding differential character (ω, µ, ) to µ−1 (0) and the restricted character descends to the quotient stack [µ−1 (0)/G]. In particular the form ω|µ−1 (0) descends to [µ−1 (0)/G], which is the point of symplectic reduction (to be honest, a deeper statement is about non-degeneracy of the reduction of a non-degenerate 2-form ω [13,14], but the non-degeneracy condition is not relevant for geometric (pre)quantization). Our second result is that the prequantization bijection between bundles with connections and differential characters commutes with reduction. Since we are interested in equivalences of categories let us change the point of view and consider all bundles/characters such that their moment map vanishes on a given G-stable submanifold N ⊂ M. We denote the corresponding subcategories by dBS 1inv,N (M ⇔ G × M) and DC 2inv,N (M ⇔ G × M). Then we have the following result (see Theorem 3.2.1). Theorem 0.5.1. Let a Lie group G act on a manifold M and i : N → M be an inclusion of a G-stable submanifold. The diagram DC 2inv (M ⇔ G × M) O ? DC 2inv,N (M ⇔ G × M) i∗ DC 22 (N ⇔ G × N ) DC 22 ([N /G])
Preq
Preq
Preq
Preq
/ dBS 1 (M ⇔ G × M) inv O ? (M ⇔ G × M) inv,N
/ dBS 1
i∗ / DBS 1 (N ⇔ G × N ) / DBS 1 ([N /G])
commutes up to natural transformations. Here the first row represents G-invariant characters/connections, the second row has moments vanishing on the submanifold N , the third is the restriction of the second to N , and the fourth is the descent of the third to the quotient (which makes sense because of the moment vanishing). 0.6. General Lie groupoids. The above discussion concerned bundles, connections, and differential characters, in the setup of a Lie group G acting on a manifold M. However it should be clear by now that definitions and results are best stated using the language of an action groupoid M ⇔ G × M. Hence it makes sense to work in the context of a general Lie groupoid M ⇔ , which we do most of the time in the present paper. Our other motivation for using general groupoids is in that they provide atlases for stacks (and in particular for manifolds). In other words, we try to describe prequantization/reduction locally. One should be careful however with stacky interpretations of groupoid results. Not every construction is atlas-independent (i.e. makes sense for the underlying stack). For example, basic connections descend to the quotient stack while invariant ones don’t (in general).
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0.7. Structure of the paper. Our goal is to describe equivariant bundles with invariant connections in terms of differential characters. We do this in two steps. First, in Sect. 1, we define a category dBS 1 (M ⇔ ) of equivariant principal S 1 -bundles with arbitrary connection. Roughly speaking, this category sits in between the category BS 1 (M ⇔ ) of equivariant bundles without connections and the category DBS 1 (M ⇔ ) of equivariant bundles with basic connections. Then in Theorem 1.5.1 we construct a (prequantizaion) equivalence ∼
Preq : DC 22−1 (M ⇔ ) − → dBS 1 (M ⇔ ), where DC 22−1 (M ⇔ ) is certain category of differential characters sitting between the category DC 21 (M ⇔ ) classifying equivariant principal S 1 -bundles and the category DC 22 (M ⇔ ) classifying equivariant bundles with basic connections. This is the main technical result of the paper. Our construction of the functor Preq has two important features. First, it is explicit enough to allow us to track invariance properties of the connections via the corresponding differential characters. More precisely, a connection on an equivariant bundle canonically determines a 1-form α ∈ 1 ( ) and this form is a part of the differential character (see Theorem 1.5.1 for details). Another property of the functor Preq is that it commutes (up to natural transformations) with pull-backs (in particular, restrictions). This implies the “prequantization commutes with reduction” theorem. On the second step, in Sect. 3, we translate invariance property of connections into conditions on the corresponding differential character and thus prove the first part of Theorem 0.4.1. We also discuss the moment map and invariant connections/curvatures on a general groupoid. In particular, we observe that the invariance condition only makes sense on the zero level of the moment map. So in the groupoid case the usual (global action) reduction procedure: consider invariant symplectic forms – restrict to the zero level of the moment map – take quotient, should be reversed as follows: restrict to the zero level of the moment map – consider invariant forms – take quotient (in fact, the quotient is a tautology in the stacky language). In Sect. 3 we discuss reduction and prove Theorem 0.5.1. In Sect. 4 we provide an explicit description of the actual differential characters on a groupoid (i.e. of the isomorphism classes of objects of DC 22−1 (M ⇔ )) and prove the second part of Theorem 0.4.1. Finally, Sect. 5 provides some examples of differential characters and reduction. 0.8. Previous results. There are several recent papers dealing with equivariant geometric prequantization and reduction. In the case of a global Lie group action Gomi [7] describes prequantization via equivariant Deligne cohomology. Lupercio and Uribe [12] describe equivariant differential characters for a finite group action (their construction is different from ours). Behrend and Xu [2] prove a version of Kostant’s prequantization theorem on proper groupoids (we explain how to derive a generalization of their result using differential characters in Subsect. 1.6). Besides being restricted to global quotients (except [2]) the above papers deal with isomorphism classes rather than stacks of bundles. As explained above we believe that it is impossible to properly understand geometric quantization on stacks without working with categories of bundles (i.e. with both objects and morphisms). Bos [4] deals with symplectic reduction in a Lie groupoid case. His approach (and in fact his problem) is different from ours – he studies certain internal symplectic structure/reduction.
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As always when talking about symplectic structures and groupoids one should not confuse symplectic structures on the manifolds of objects (our situation) and on arrows (“symplectic groupoids” of Weinstein [17]).
0.9. Higher degree versions. In this paper we are dealing with invariant differential characters of degree 2. Similar results hold for degree 3 characters (classifying S 1 -gerbes with connective structures). In particular equivariant string connections appear naturally in this approach. We wanted to keep this text free of obscure notions of higher stacks and their descent conditions and so postponed discussion of gerbes and their characters for another paper.
1. Circle Bundles with Connections and Prequantization 1.1. Prequantization. We recall the construction of the prequantization functor in [11]. Let M be a differentiable manifold and consider the category DBS 1 (M) of principal S 1 -bundles with connections on M, i.e. the objects of DBS 1 (M) are S 1 -bundles with connections and morphisms are smooth S 1 -equivariant maps preserving connections. In [11] we described a prequantization functor Preq : DC 22 (M) → DBS 1 (M), where DC 22 (M) is the category of differential characters defined below. We also showed that Preq is an equivalence of categories. Thus one can say that differential characters classify S 1 -bundles with connections. To define the category of differential characters (following Hopkins-Singer [9]) we start with a complex DCs• (M) of abelian groups: DCsn (M) = {(c, h, ω) | ω = 0 if n < s} ⊂ C n (M, Z) × C n−1 (M, R) × n (M), where C n (M, R) denotes the group of smooth singular cochains with coefficients in a ring R, n (M) denotes the group of differential forms, and s is a fixed positive integer (truncation degree). The differential in the complex DCs• (M) is given by d(c, h, ω) = (dc , ω − c − dh , dω).
(1.1.1)
The complex DCs• (M) is called the homotopy product of C • (M, Z) and •≥s (M) over C • (M, R). As a motivation for the differential (1.1.1) note that a closed cochain (cocycle) in DCsn (M) consists of a closed integral singular n-cochain c and a closed n-form ω, such that c and ω define the same cohomology class with values in R: ω − c = dh with a given h ∈ C n−1 (M, R). The group of Cheeger-Simons differential characters described in the Introduction is isomorphic to H 2 (DC2• (M)), cf. Sect. 4.1.
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The category DC 22 (M) is defined as follows: • Objects are 2-cocycles: (c, h, ω) ∈ DC22 (M), d(c, h, ω) = 0 • Morphisms are 1-cochains up to exact 1-cochains. The set of morphisms from (c, h, ω) to (c , h , ω ) is {(a, t, 0) ∈ DC21 (M) | d(a, t, 0) = (c , h , ω ) − (c, h, ω)} (a, t, 0) ∼ (a, t, 0) + d(m, 0, 0) for (m, 0, 0) ∈ DC20 (M)
.
• Composition of morphisms is the addition in DC21 (M). Note that given a smooth map ρ : M → N , one can define natural pull-back functors DC 22 (ρ) : DC 22 (N ) → DC 22 (M) and DBS 1 (ρ) : DBS 1 (N ) → DBS 1 (M). This means that DC 22 and DBS 1 form presheaves of groupoids over the category of manifolds, and it is shown in [11] that these presheaves are in fact stacks. We don’t use stacky language in the present paper but it is needed for the proof of the prequantization theorem below. Now we are ready to define the prequantization functor. Let DBS 1triv (M) denote the full subcategory of DBS 1 (M) consisting of trivial bundles with arbitrary connections and consider a functor DChtriv : DBS 1triv → DC 22 given by: • DChtriv ((M × S 1 , β + dθ )) = (0, β, dβ) on objects. Here β ∈ 1 (M), θ ∈ R/Z is the coordinate on S 1 , and β + dθ is a connection 1-form on the trivial bundle M × S 1 → M. • DChtriv ( f ) = [(d( f˜ − f ), − f˜, 0)] ∈ DC21 /d(DC20 ) on morphisms. Here we think of a morphism in DBS 1triv from (M × S 1 , β + dθ ) to (M × S 1 , β + dθ ) as a smooth function f : M → S 1 = R/Z such that d f = β − β and let f˜ ∈ C 0 (M, R) be a lift of f . One of the main results of [11] is: Theorem 1.1.1. There is unique (up to a natural transformation) equivalence of stacks DCh : DBS 1 → DC 22 , extending DChtriv . In other words, there is a family of equivalence functors (one for each manifold): DCh(M) : DBS 1 (M) → DC 22 (M) such that (i) DCh commutes (up to coherent natural transformations) with pull-backs, and (ii) on trivial bundles with connections DCh is equal to DChtriv . We call a quasi-inverse functor of DCh a prequantization functor and denote it Preq (it is defined uniquely up to a natural transformation). The idea behind this theorem is that any bundle becomes trivial after pull-back to a contractible open subset of M. A similar argument (cf. [11]) shows the category DC 21 (M) is equivalent to the category BS 1 of principal S 1 -bundles (without connections).
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1.2. Equivariant S 1 -bundles on groupoids. Consider a Lie groupoid M ⇔ . Its nerve is the simplicial manifold
M
oo
∂1 =t ∂0 =s
∂2
oo
o
∂0
∂3
o
o
2 oo
··· , ∂0
where
n = × M × M . . . × M . n
The maps ∂0 = s, ∂1 = t : → M are the source and the target maps of the groupoid respectively. We use notations (s, t) and (∂0 , ∂1 ) interchangeably throughout the paper. The maps ∂0 , ∂n : n → n−1 forget the first and the last -factors respectively, and finally, ∂i : n → n−1 for 1 ≤ i ≤ n −1 multiplies (composes) i th and (i +1)th factors. We omit face maps n−1 → n (in particular the unit map) in the above diagram. A principal S 1 -bundle P → M is -equivariant if one can lift it to a simplicial principal S 1 -bundle o Po πM
o Mo
∂1P ∂0P ∂1M ∂0M
o P1 oo π
o
oo
oo P2 oo
···
π 2
2
oo
oo
···
More precisely, such a lift provides an equivariant structure on the bundle. An equivalent definition of an equivariant structure on P is a bundle isomorphism ∼ → (∂0M )∗ P satisfying a cocycle condition on 2 . In the case of an ϕ : (∂1M )∗ P − action groupoid M ⇔ G × M an equivariant structure on a bundle P → M is equivalent to a lift of the G-action on M to an action on the total space P.
1.3. Connections on equivariant S 1 -bundles. Now suppose we are given a connection A on the bundle P → M. We think of A as a 1-form on P. The connection A is -basic (with respect to a given equivariant structure on P) if (∂1P )∗ A = (∂0P )∗ A. In the case of an action groupoid M ⇔ G × M a connection is basic if it is G-invariant and vanishes on vector fields generating the infinitesimal action of the Lie algebra g of G. Equivariant S 1 -bundles with basic connections form a category DBS 1 (M ⇔ ) with morphisms being equivariant bundle maps preserving connections. In fact (cf. [11]) this category (up to an equivalence) depends only on the stack [M/ ], not on the actual (atlas) groupoid M ⇔ . In other words, it is Morita-equivariant. In the present paper we are interested in arbitrary (not necessarily basic) connections on an equivariant S 1 -bundle. So we introduce a category dBS 1 (M ⇔ ) of
-equivariant bundles on M with arbitrary connections. Morphisms in dBS 1 (M ⇔ ) are equivariant bundle maps preserving connections. Note that DBS 1 (M ⇔ ) is a full subcategory of dBS 1 (M ⇔ ).
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1.4. Singular and de Rham cohomology of groupoids. Let us recall the definitions of singular and de Rham complexes on a groupoid M ⇔ . We refer the reader to [1] and [6] for details and examples of applications. The de Rham complex • (M ⇔ ) is the total complex of the following double complex (denoted by • ( • )) : .. .O
.. .O −d
d
1 (M) O
δ
/ 1 ( ) O
o Mo
/ ···
δ
/ ···
−d
d
0 (M)
δ
δ
∂1
/ 0 ( )
∂0
o
oo
∂2
···
∂0
where δ is the simplicial differential: δ = (−1)i ∂i∗ . Analogously, given a ring R, the smooth singular cochain complex C • (M ⇔ , R) is defined as the total complex of the double complex C • ( • , R). One can also define singular chains complex in a similar fashion (with arrows in the opposite direction). Smooth singular chains are dual to smooth singular cochains. The usual de Rham Theorem on manifolds implies (by a spectral sequence argument) the de Rham Theorem on groupoids: H n (C • (M ⇔ , R)) is isomorphic to H n (• (M ⇔ )). The isomorphism is provided by integrating forms over smooth chains. 1.5. Differential characters on groupoids. Similarly to singular and de Rham cochain complexes on a groupoid M ⇔ one can define the complex of differential characters DCs• (M ⇔ ) as the total complex of the double complex DCs• ( • ). It is shown in [11] that, provided s > 0, the total cohomology of DCs• ( • ) is isomorphic to the total cohomology of its truncated version .. .O
.. .O −d
d
DCs2 (M) O
δ
/ DC 2 ( ) sO
/ ···
δ
/ ···
−d
d DCs1 (M) d DCs0 (M)
δ
δ
/
DCs1 ( ) d DCs0 ( )
which we’ll use from now on. Translating complexes into categories, we get a category DC 2s (M ⇔ ) associated to the second total cohomology of the above (truncated) complex (2-cocycles are objects, 1-cochains modulo exact 1-cochains are morphisms). A standard stacky argument (cf. [11]) allows one to extend the prequantization functor (on manifolds) to an equivalence
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∼
Preq : DC 22 (M ⇔ ) − → DBS 1 (M ⇔ ) . Similarly DC 21 (M ⇔ ) is equivalent to the category BS 1 (M ⇔ ) of equivariant S 1 -bundles without connections. However presently we are interested in the intermediate category dBS 1 (M ⇔ ) of equivariant bundles with arbitrary connections. So we consider a double complex .. .O
.. .O −d
d
DC22 (M) O
δ
/ DC 2 ( ) 1O
δ
/ ···
δ
/ ···
−d
d DC21 (M) d DC20 (M)
δ
/
DC11 ( ) d DC10 ( )
Note that the truncation degree for differential forms is 2 in the first column and 1 in the second. This corresponds to the fact that equivariant structure on our bundles (i.e. isomorphism of the two pull-backs to ) does not necessarily preserve connection. Roughly speaking, we would like our characters to correspond to bundles with connections on M 2 (M ⇔ ) and bundles without connections on . We denote the above complex DC2−1 2 and the corresponding category of total degree-2 cochains DC 2−1 (M ⇔ ). The com2 (M ⇔ ) contains DC 2 (M ⇔ ) as a subcomplex, and the category plex DC2−1 2 DC 22−1 (M ⇔ ) contains DC 22 (M ⇔ ) as a full subcategory. Explicitly, a degree-2 2 (M ⇔ ) (i.e an object of DC 2 (M ⇔ )) looks like cochain in DC2−1 2−1 (c, h, ω)
•
•
[(b, f, α)]
where c ∈ C 2 (M, Z), b ∈ C 1 ( , Z), h ∈ C 1 (M, R), f ∈ C 0 ( , R), ω ∈ 2 (M), α ∈ 1 ( ), and [(b, f, α)] means an equivalence class with respect to the relation (b, f, α) ∼ (b + dn, f − n, α), n ∈ C 0 ( , Z). We write this cochain as ((c, h, ω), [(b, f, α)]). It belongs to DC22 (M ⇔ ) iff α = 0. 2 (M ⇔ ) (i.e. a morphism in DC 2 (M ⇔ )) looks A degree-1 cochain in DC2−1 2−1 like •
•
[(a, t, 0)]
•
with a ∈ C 1 (M, Z), t ∈ C 0 (M, R), and (a, t, 0) ∼ (a+dm, t −m, 0) for m ∈ C 0 (M, Z). The first result of the present paper is the following prequantization theorem on groupoids:
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Theorem 1.5.1. The prequantization functor ∼
Preq : DC 22 (M ⇔ ) − → DBS 1 (M ⇔ ) extends to an equivalence of categories ∼
Preq : DC 22−1 (M ⇔ ) − → dBS 1 (M ⇔ ) such that if π
Preq((c, h, ω), [(b, f, α]) = (P → M, A) then ω is the curvature of the connection A, while α is the difference of the two pull-backs of A to P1 considered as a 1-form on : π ∗ α = ∂1∗ A − ∂0∗ A. • (M ⇔ ) clasIn particular the second total cohomology group of the complex DC2−1 sifies equivariant principal S 1 -bundles on M ⇔ with arbitrary connections.
Proof. We give an explicit construction of the quasi-inverse functor DCh : dBS 1 (M ⇔ ) → DC 22−1 (M ⇔ ) of the extended prequantization functor Preq. Let us begin with objects. Given an equivariant bundle P → M with a connection A ∈ 1 (P) we apply the DCh-functor on M (cf. Subsect. 1.1.1) to get a cochain (c, h, ω) = DCh ((P, A)) ∈ DC22 (M) such that d(c, h, ω) = 0.
(1.5.1) ∼
Now the equivariant structure on P is a bundle isomorphism ∂1∗ P − → ∂0∗ P or a trivi∼ alization σ : × S 1 − → ∂1∗ P ⊗ (∂0∗ P)−1 . We put σ ∗ (∂1∗ A − ∂0∗ A) = α + dθ, where 1 α ∈ ( ). Applying the DCh-functor on to the isomorphism σ and using the explicit form of DCh for trivial bundles with connections (cf. Subsect. 1.1) we get (a class of) cochain DCh(σ ) = [(b, f, 0)] ∈ DC21 ( )/DC20 ( ) such that d(b, f, 0) = (∂1∗ (c, h, ω) − ∂0∗ (c, h, ω)) − (0, α, dα).
(1.5.2)
Note that [(b, f, 0)] encodes not only the trivialization σ but also natural transformations coming from the fact that the functor DCh commutes only weakly with pull-backs and products. In any case, (1.5.2) means that d(b, f, α) = δ(c, h, ω).
(1.5.3)
Since σ satisfies the cocycle condition on 2 and DCh commutes strongly with pullbacks on the level of morphisms, we get δ[(b, f, α)] = 0.
(1.5.4)
Equations (1.5.1), (1.5.3), (1.5.4), mean that ((c, h, ω), [(b, f, α)]) is a cocycle in DC 22−1 (M ⇔ ) and we put DCh((P, A, σ )) = ((c, h, ω), [(b, f, α)]).
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This completes the construction of DCh on objects of dBS 1 (M ⇔ ). Note that the differential form parts ω and α of the cocycle ((c, h, ω), [(b, f, α)]) correspond precisely to their description in the theorem. In particular A is basic iff α = 0. Hence the restriction of DCh to the subcategory DBS 1 ⊂ dBS 1 takes values in DC 22 ⊂ dDC 22−1 and coincides with DCh of [11]. Since morphisms in DC 22−1 (M ⇔ ) (resp. dBS 1 (M ⇔ )) are the same as morphisms in DC 22 (M) (resp. DBS 1 (M)) we just use the original functor DCh on M for morphisms. The above construction is clearly reversible (using Preq-functors on M and as quasi-inverses of DCh-functors). Hence the functor DCh : dBS 1 (M ⇔ ) → DC 22−1 (M ⇔ ) is an equivalence of categories and we denote the quasi-inverse functor Preq. Note that morphisms in the categories dBS 1 (M ⇔ ) and DC 22−1 (M ⇔ ) change neither the connection A nor differential form parts ω and α of the cocycle. Therefore the condition of the theorem is satisfied for any choice of the quasi-inverse functor.
1.6. Equivariant Kostant theorem. Recall that Kostant’s prequantization theorem on a manifold M states that, given a closed 2-form ω ∈ 2 (M) with integral periods, there exists a principal S 1 -bundle on M with a connection whose curvature is equal to ω. Moreover the set of such bundles with connections is in bijection with the cohomolgy H 1 (C • (M, R/Z)) of the complex of smooth singular cochains with values in R/Z. In this subsection we prove an equivariant version of the Kostant theorem on a Lie groupoid M ⇔ . Denote 20 (M ⇔ ) ⊂ 2 (M ⇔ ) = 2 (M) × 1 ( ) × 0 ( 2 ) the group of closed 2-forms on M ⇔ which are integral (i.e. have integral periods or, equivalently, represent images of integral cohomology classes in de Rham cohomology) and have vanishing 0 ( 2 )-component. Suppose = (ω, α, 0) ∈ 20 ⊂ 2 (M) × 1 ( ) × 0 ( 2 ). Since represents the image of an integral cohomology class in real smooth singular cohomology there exists an integral cochain (c, b, a) ∈ C 2 (M ⇔ , Z) = C 2 (M, Z) × C 1 ( , Z) × C 0 ( 2 , Z) and a real cochain (h, f ) ∈ C 1 (M ⇔ , R) = C 1 (M, R) × C 0 ( , R) such that dtot (h, f ) = (ω, α, 0) − (c, b, a), where dtot is the total differential in the double complex of smooth singular chains on M ⇔ . This implies that the projection map • η : H 2 (DC2−1 (M ⇔ )) → 20 (M ⇔ ), η ( [((c, h, ω), (b, f, α), (a, 0, 0))] ) = (ω, α, 0)
is surjective, and it is easy to see that the kernel of η is isomorphic (by the map [((c, h, 0), (b, f, 0), (a, 0, 0))] → [(h mod Z, f mod Z)] to the cohomology group H 1 (C • (M ⇔ , R/Z)) of smooth singular 1-cochains with values in R/Z. Hence we obtain a short exact sequence • 0 → H 1 (C • (M ⇔ , R/Z)) → H 2 (DC2−1 (M ⇔ )) → 20 (M ⇔ ) → 0. (1.6.1)
This sequence (in the manifold case) first appeared in the original Cheeger and Simons • (M ⇔ )) classifies equivariant principal S 1 -bundles on paper [5]. Since H 2 (DC2−1 M ⇔ with arbitrary connections (cf. Theorem 1.5.1), we can interpret (1.6.1) as follows:
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Theorem 1.6.1. Let = (ω, α, 0) ∈ 2 (M ⇔ ) = 2 (M) × 1 ( ) × 0 ( 2 ) be an integral closed 2-form on a Lie groupoid M ⇔ . Then there exists a -equivariant principal S 1 -bundle P → M with a connection A ∈ 1 (P) such that ω is the curvature of the connection A and π ∗ α = ∂1∗ A − ∂0∗ A. Moreover the set of isomorphism classes of such pairs (P, A) is in bijection with H 1 (C • (M ⇔ , R/Z)). For proper groupoids this theorem was proved (by a different argument) in [2]. A typical non-proper example is the classifying groupoid BR = ∗ ⇔ R of the additive group of real numbers. In this case an integral 2-form is given by = (0, µ d x, 0), where µ is a real number and x is the coordinate on R. The unique S 1 -bundle on the point ∗ is S 1 → ∗ and this bundle has unique connection A = dθ . An equivariant structure on this bundle is a group representation R → S 1 = R/Z. Now the above theorem assigns the representation x → (µx mod Z) to the form = (0, µ d x, 0). In this example H 1 (C • (∗ ⇔ R, R/Z)) vanishes since the groupoid is contractible. See Sect. 5.1 for a generalization of this example. 2. Invariant Connections and the Moment Map In this section we identify differential characters corresponding to invariant connections. 2.1. The moment map. We again think of an equivariant principal S 1 -bundle on a groupoid M ⇔ as a simplicial bundle oo o o ··· P1 oo P2 oo Po πM
o Mo
π
o
oo
π 2
2
oo
oo
···
Recall (see e.g. [15]) that the Lie algebroid P of the Lie groupoid P ⇔ P1 is a vector bundle over P whose fiber at p ∈ P is equal to Ker ds ⊂ T p P1 (here we consider P as a subset of P1 via the unit map). Another way to think of sections of P is as right-invariant vector fields in Ker ds ⊂ T P1 . The Lie algebroid P comes equipped with a bundle map ρ := dt : P → T P called the anchor. Let P∗ be the dual bundle of P. Given a connection A ∈ 1 (P) we define the moment map µ PA as a section of P∗ given by: µ PA (X ) = A(ρ(X )) for a section X of P. One can rewrite this definition as µ PA (X ) = A(dt (X )) = A(dt (X ) − ds(X )) = (dt ∗ (A) − ds ∗ (A))(X ) = (δ A)(X ).
(2.1.1)
Here δ is, as usual, the simplicial differential, and we think of X as a right-invariant section of Ker ds ⊂ T P1 and of µ PA (X ) as a right-invariant function on P1 , hence a function on P. Since a connection and all the structure maps of a simplicial principal S 1 -bundle are preserved under the action of S 1 , the moment map µ PA is a pullback under π M of a ∗ section µ M A of the dual bundle G of the Lie algebroid G of the base groupoid M ⇔ .
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P Without much risk of confusion we call both µ M A and µ A the moment maps and often drop the top index. By definition a connection A is horizontal iff µ A = 0. For example, the Lie algebroid of an action groupoid P ⇔ G × P is the trivial bundle P × g and the moment map is a map µ : P → g∗ given by < µ, X>= i ρ(X ) A, X ∈ g.
2.2. Invariant connections. We continue with the setup of the previous subsection. Our goal is to define what it means for a connection to be P1 -invariant. In the case of a global action P ⇔ G × P a connection A ∈ 1 (P) is G-invariant iff g ∗ A = A for any g ∈ G. In the groupoid language an element g ∈ G should be thought of as a constant section of the source map s, and the condition g ∗ A = A can be written as (t ∗ A − s ∗ A)|T = 0 or (δ A)|T = 0.
(2.2.1)
For a general (non-action) groupoid there is no natural choice of the section through a point p ∈ P1 . One can fix a local regular foliation of P1 by bisections (submanifolds transverse to both s and t) and say that A is -invariant if (2.2.1) holds. This condition depends on the choice of . Note that we have a decomposition T P1 = T ⊕ Ker ds.
(2.2.2)
Hence the two equations (2.2.1) and (2.1.1) completely determine δ A. In particular, if the moment map µ vanishes at some point p ∈ P1 then the condition (2.2.1) at the point p does not depend on the choice of . If µ vanishes identically then the choice of is irrelevant and the connection is invariant (for some ) iff it is basic (δ A = 0). In other words, as in the global quotient case, the connection is basic iff it is horizontal and invariant. However for a general groupoid the second condition is natural only if the first one is satisfied. 2.3. Prequantization on global quotients. We would like to characterize bundles with invariant connections via their differential characters. Let us restrict ourselves to a global quotient case M ⇔ G × M to avoid choice of a local foliation . Given an equivariant bundle P with an invariant connection A we apply the functor DCh (cf. Theorem 1.5.1) to obtain a character DCh ((P, A)) = ((c, h, ω), [(b, f, α)]) , where π ∗ α = δ A ∈ 1 (G × P). Then, according to the previous subsection, G-invariance of the connection A is equivalent to the following vanishing condition on the form α ∈ 1 (G × M): iv α = 0 for any vector field v ∈ T M ⊂ T (G × M) = T G ⊕ T M. This means that α = <µ, dg g −1 >,
(2.3.1)
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where dg g −1 is the right-invariant Maurer-Cartan form on G (pulled back to G × M) and µ : M → g∗ is the moment map (pulled back to G × M). Putting everything together we obtain the following description of equivariant bundles with invariant connections. Theorem 2.3.1. Suppose a Lie group G acts on a manifold M. Then the prequantization functor ∼
Preq : DC 22−1 (M ⇔ G × M) − → dBS 1 (M ⇔ G × M) restricts to an equivalence ∼
Preq : DC 2inv (M ⇔ G × M) − → dBS 1inv (M ⇔ G × M), where dBS 1inv (M ⇔ G × M) is the category of G-equivariant bundles with G-invariant connections and DC 2inv (M ⇔ G × M) is the category of differential characters of the form ((c, h, ω), [(b, f, <µ, dg g −1 >)]). To summarize we have two towers of full subcategories related by prequantization/ character equivalence: DC 22−1 (M ⇔ G × M) O
Preq
/ dBS 1 (M ⇔ G × M) O
? DC 2inv (M ⇔ G × M) O
Preq
? / dBS 1 (M ⇔ G × M) inv O
? DC 22 (M ⇔ G × M)
Preq
? / DBS 1 (M ⇔ G × M)
The bottom row consists of equivariant bundles with basic connections and characters with vanishing α-component; the middle row bundles have invariant connections and the characters have α =< µ, dg g −1 >; finally the top row does not involve any conditions on the connection or α-component of the character. 3. Reduction The idea of reduction is that after restriction to the zero level of the moment map an invariant connection becomes basic and thus defines a connection on a bundle on the quotient stack. From our point of view it is more natural to consider categories of bundles rather than a single bundle. Hence we start with the “zero level set” and consider all bundles with the moment map vanishing on that manifold. 3.1. General reduction. Let M ⇔ be a Lie groupoid, N a submanifold of M such that s −1 N = t −1 N , i : N → M the inclusion map, and N ⇔ N the corresponding full subgroupoid of M ⇔ (i.e. N = ×s Mi N = ×t Mi N ). We denote by dBS 1N (M ⇔ ) the full subcategory of dBS 1 (M ⇔ ) consisting of equivariant principal S 1 -bundles with connections (P, A) such that the restriction
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i ∗ (P, A) of (P, A) to N is an equivariant bundle with a basic connection (i.e. such that s ∗ i ∗ A = t ∗ i ∗ A). As explained in the previous section, this condition can be split into two: (3.1.i) the moment map of A vanishes on N : µ A | N = 0; (3.1.ii) i ∗ A is N -invariant (with respect to any bisection). Similarly we define a full subcategory DC 2N (M ⇔ ) of DC 22−1 (M ⇔ ) consisting of differential characters ((c, h, ω), [(b, f, α)]) such that α| = 0. N It follows from Theorem 1.5.1 that the prequantization functor ∼
Preq : DC 22−1 (M ⇔ ) − → dBS 1 (M ⇔ ) restricts to an equivalence ∼
Preq : DC 2N (M ⇔ ) − → dBS 1N (M ⇔ ). The restriction i ∗ dBS 1N (M ⇔ ) of the category dBS 1N (M ⇔ ) to the submanifold N is equivalent to the category DBS 1 (N ⇔ | N ) of equivariant bundles with basic connections on N , and similarly i ∗ DC 2N (M ⇔ ) DC 22 (N ⇔ N ). Hence we obtain the following result: Theorem 3.1.1. Prequantization commutes with reduction, i.e. the following diagram commutes up to natural transformations: DC 22−1 (M ⇔ ) O ? DC 2N (M ⇔ )
Preq
Preq
i∗
DC 22 (N ⇔ N ) DC 22 ([N/ N ])
Preq
Preq
/ dBS 1 (M ⇔ ) O ? / dBS 1 (M ⇔ ) N i∗ / DBS 1 (N ⇔ ) N / DBS 1 ([N/ ]) N
Note that prequantization functors in the above diagram are defined differently for each row (the top two rows are constructed in the present paper while the bottom two are from [11]). The meaning of the theorem is that these definitions are compatible. 3.2. Reduction for action groupoids. On an action groupoid M ⇔ G × M the reduction can be performed in a more familiar way: first consider subcategories dBS 1inv (M ⇔ G × M), DC 2inv (M ⇔ G × M) of invariant connection/characters and then further subcategories dBS 1inv,N (M ⇔ G × M), DC 2inv,N (M ⇔ G×M) of invariant connections/characters with the moment map vanishing on a given G-stable submanifold N . The categories dBS 1inv,N (M ⇔ G × M), DC 2inv,N (M ⇔ G × M) are in general
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proper subcategories of dBS 1N (M ⇔ G×M), DC 2N (M ⇔ G×M) introduced in the previous subsection since in the latter ones we impose the invariance condition only on the submanifold N . This version of reduction also commutes with prequantization. Theorem 3.2.1. The diagram DC 22−1 (M ⇔ G × M) O
Preq
? DC 2inv (M ⇔ G × M) O
Preq
? DC 2inv,N (M ⇔ G × M)
Preq
i∗ DC 22 (N ⇔ G × N ) DC 22 ([N/G])
Preq
Preq
/ dBS 1 (M ⇔ G × M) O ? / dBS 1 (M ⇔ G × M) inv O ? (M ⇔ G × M) inv,N
/ dBS 1
(3.2.1)
i∗ / DBS 1 (N ⇔ G × N ) / DBS 1 ([N/G])
commutes up to natural transformations. The proof is again a simple application of Theorem 1.5.1 (more precisely, of the explicit relation between δ A and the α-part of a differential character). 4. Differential Characters as Characters In this section we provide an explicit description of the actual differential characters on groupoids, i.e. of the isomorphism classes of objects of DC 22−1 (M ⇔ ) or, in other • (M ⇔ ). words, of the second total cohomology group of the complex DC2−1 4.1. Differential characters on manifolds and groupoids. Let us first recall the description of Cheeger-Simons differential characters in the case of a manifold M (cf. [5,9]). We are interested in the cohomology group H 2 (DC2• (M) =
{(c, h, ω) ∈ C 2 (M, Z) × C 1 (M, R) × 2 (M) | dc = 0, ω − c − dh = 0, dω = 0} . (c, h, ω) ∼ (c + da, h − a − dt, ω), (a, t) ∈ C 1 (M, Z) × C 0 (M, R)
It is easy to see that the equivalence (cohomology) class of (c, h, ω) is completely determined by ω and the values of h mod Z on smooth 1-cycles. To make a precise statement let us introduce the group of differential characters on M. A differential character is a pair (ω, χ ), where ω ∈ 2 (M), dω = 0, and χ is a character Z 1 (M) → R/Z of the group of smooth 1-cycles Z 1 (M). This pair should satisfy the following condition: (4.1.1) χ (∂ S) = ω mod Z S
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for any smooth 2-chain S ∈ C2 (M, Z). The map [(c, h, ω)] → (ω, h| Z 1 (M) mod Z) provides an isomorphism between H 2 (DC2• (M)) and the group of differential characters. Condition (4.1.1) ensures that c = ω − dh is an integral cochain. Let us turn to the groupoid case. We would like to have an explicit description of • (M ⇔ ). An element of the second total cohomology group of the complex DC2−1 • 2 H (DC2−1 (M ⇔ )) is a closed cochain ((c, h, ω), [(b, f, α)]) up to addition of the total differential of [(a, t, 0)], see 1.5 for notation. The cochain being closed means d(c, h, ω) = 0, δ(c, h, ω) = d[(b, f, α)], δ[(b, f, α)] = 0.
(4.1.2) (4.1.3) (4.1.4)
Similar to the manifold case c and b are determined by h, f , ω and α: b = α − d f − δh, c = ω − dh.
(4.1.5) (4.1.6)
Turning to differential form components, we put = (ω, α, 0) ∈ 2 (M) ⊕ 1 ( ) ⊕ 0 ( 2 ) = 2 (M ⇔ ).
(4.1.7)
Then is a closed 2-form on M ⇔ with vanishing 0 ( 2 )-component. Finally the pair (h, f ) is determined up to coboundary (total differential of [(a, t, 0)]) by its values mod Z on smooth 1-cycles Z 1 (M ⇔ ). Recall that a cycle (generator of Z 1 ) on a groupoid looks like this (see for example [1]):
O
g1 ? _ _ _/ ?? ??γ2 γ1 ?? ?
gn
g2
|| | | || }|| γ3
Here γi : [0, 1] → M are smooth paths in M and gi ∈ are arrows connecting end-points of these paths. This is a particular example of a smooth singular 1-chain on and so it makes sense to evaluate the cochain (h, f ) on this chain: the groupoid h(γi ) + f (gi ) ∈ R. Taking values mod Z we get a character = (h, f )| Z 1 (M⇔ ) mod Z
:
Z 1 (M ⇔ ) → R/Z.
The pair (, ) uniquely determines the cohomology class of ((c, h, ω), [(b, f, α)]). Conversely, a pair ∈ 2 (M ⇔ ), : Z 1 (M ⇔ ) → R/Z corresponds to a coho• (M ⇔ )) iff it satisfies the following conditions: mology class in H 2 (DC2−1 (4.1.i) is a closed 2-form on (M ⇔ ) with vanishing 0 ( 2 )-component. This ensures that the third components of the relations (4.1.2)–(4.1.4) hold true.
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(4.1.ii) (g1 g2 g3 ) = 0 for g1 , g2 , g3 ∈ such that g1 g2 g3 = id p , p ∈ M. Note that such a triple g1 , g2 , g3 is a cycle in Z 1 (M ⇔ ): g p __? _ _ 1 _ _ _/ ? ? g3 ? g2
This condition ensures that δ f = 0. (4.1.iii) For any smooth path η : [0, 1] → one has (∂η) = α mod Z, η
where ∂η is the total boundary of η, i.e. the following cycle in Z 1 (M ⇔ ): η(0) O _ _ _/ −s(η)
o_ _ _
t (η)
η(1)−1
This condition ensures that b(η) defined by (4.1.5) belongs to Z. (4.1.iv) For any smooth 2-chain S on M one has (∂ S) = ω mod Z. S
Note that ∂ S is a 1-cycle on M but we think of it as an element of Z 1 (M ⇔ ) O
ED @A S BC
∂S
This condition ensures that c(S) defined by (4.1.6) belongs to Z. Let us summarize the above discussion as the following version of Theorem 1.5.1. Theorem 4.1.1. Isomorphism classes of equivariant S 1 -bundles with (not necessarily basic) connections on a groupoid M ⇔ are in bijection with differential characters, i.e. pairs (, ), where ∈ 2 (M ⇔ ) and is a character Z 1 (M ⇔ ) → R/Z, satisfying conditions (4.1.i)–(4.1.iv) above. Characters (, ) with vanishing 1 ( )-component of correspond to equivariant bundles with basic connections, i.e. bundles with connections on the quotient stack. Remark. Note that (| M , | Z 1 (M) ) defines a differential character on M corresponding to an S 1 -bundle with connection on M. The above theorem says that equivariant structures on this bundle are in 1-1 correspondence with equivariant extensions of its character.
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4.2. Special cases. Because of conditions (4.1.ii)–(4.1.iv) the character is determined by its value on the following classes of 1-cycles for each point p ∈ M: (1) generators of the fundamental group of M with the base point p, (2) representatives of the connected components of the inertia group I p = s −1 p ∩ t −1 p, and (3) cycles of the form pj Y
_
e
) ,
where as before the solid line is a path in M and the broken line is an arrow in . Moreover it is enough to consider one base point p in each connected (via sequences of paths in M and arrows in ) component of M. This description of , though economical, is not very natural because homotopy of paths/points involves integration of ω and/or α. However in some cases it can clarify (and simplify) things substantially. Here are some examples: (4.2.i) If = 0 then is just a homomorphism of the product of the (suitably defined) fundamental groups of connected components of the stack [M/ ] into R/Z, i.e. a local system on [M/ ]. (4.2.ii) If the groupoid is source-connected (e.g. it is the action groupoid of a connected group action) then is determined by its values on Z 1 (M) (or just on generators of fundamental groups of connected components of M). (4.2.iii) For any point p ∈ M the character restricts to a homomorphism I p → R/Z, where I p is the inertia group at p. Moreover this homomorphism is determined by its values on representatives of the connected components of I p . Note however that, unless the quotient of I p by its identity component I pe has a section or the form α vanishes, does not lift to a homomorphism I p/I pe → R/Z. (4.2.iv) In the case of a finite group G acting on a point, a differential character is just a homomorphism G → R/Z. 4.3. Invariant differential characters on an action groupoid. In the case of an action groupoid M ⇔ G × M we have a notion of invariant connections (cf. Subsect. 2.3) which fit between arbitrary and basic ones. We would like to describe the corresponding differential characters. Invariance of a connection corresponds to vanishing of the form α ∈ 1 (G × M) on constant sections of the source map (projection) s : G × M → M. Because of relation (4.1.iii) this vanishing condition is equivalent to vanishing of the character on cycles of the type g O _ _ _/ γ
g◦γ
_ _ _/ g
Here γ : [0, 1] → M is a path in M, and we denote by the same letter g an element of G, the corresponding constant section of s, and the corresponding diffeomorphism of M. Taking into account (4.1.ii) we see that invariance of the connection means that the character is invariant with respect to the natural action of g ∈ G on Z 1 (M ⇔ M × G). Note that assuming (4.1.i), (4.1.ii), and (4.1.iv), it is enough to require (4.1.iii) on homotopy representatives of paths in G × M and any path in G × M is homotopic to a composition of a constant section of s and a 1-parametric subgroup in G acting on a point in M. The above invariance condition takes care of constant sections of s, so let us
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consider a 1-parametric subgroup et X , X ∈ g. Given a point p ∈ M the relation (4.1.ii) applied to the path (et X , p) ⊂ G × M reads et X
( p j Y
_
e−X
) e X p) =< µ( p), X >
e
mod Z.
Here the solid arrow represents the path et X p, t ∈ [0, 1], in the manifold M, the broken arrow the element (et X , p) of G × M, and µ : M → g∗ is the moment map. This discussion together with Subsect. 2.3, leads us to the following description of the invariant characters for an action groupoid. Theorem 4.3.1. Isomorphism classes of equivariant S 1 -bundles with invariant connections on an action groupoid M ⇔ G × M are in bijection with invariant differential characters, i.e. triples (ω, µ, ), where ω ∈ 2 (M) is a 2-form, µ : M → g∗ is a (moment) map, and is a character Z 1 (M ⇔ G × M) → R/Z. This triple should satisfy the following conditions: (4.3.i) ω and are G-invariant, µ is G-equivariant. (4.3.ii) i ε X ω = − < dµ, X >, where X ∈ g and ε X is the action vector field of X . (4.3.iii) (ω, | Z 1 (M) ) is an (invariant) differential character on M, i.e. (∂ S) =
ω mod Z S
for any S ∈ C2 (M). (4.3.iv) One has et X
( p j Y
_
e−X
e
) e X p) =< µ( p), X >
mod Z
for any p ∈ M, X ∈ g. (4.3.v) restricts to a character p : I p → R/Z of the stabilizer (inertia) group I p for each point p ∈ M. The connection is basic iff µ ≡ 0. Here we rewrote the condition that the form ω, < µ, dg g −1 >, 0 ∈ 2 (M ⇔ G × M) is closed in a more familiar form (see for example [8]): (1) ω is closed and G-invariant, (2) µ : M → g∗ is G-equivariant, and (3) i ε X ω = − < dµ, X >. As for a general groupoid, one only needs to specify the character on some elements of Z 1 (M ⇔ G × M). However a natural choice of these generators depends on the particular situation. 5. Examples In this section we sketch several examples of differential characters and reduction. We leave the details to an interested reader.
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5.1. Example: classifying stack. Consider a Lie group G acting on a point ∗. A differential character on the groupoid ∗ ⇔ G × ∗ is a homomorphism : G → R/Z together with an element µ ∈ g∗ , such that d = µ. Such a pair (, µ) is determined by the character which has to be smooth. So the differential character on ∗ ⇔ G × ∗ is the same as a smooth character of G.
5.2. Example: coadjoint orbits of compact Lie groups. Let M be a coadjoint orbit of a compact connected Lie group G: M = G · λ for λ ∈ g∗ . If we assume the moment map µ : M → g∗ is the inclusion, then the rest of a differential character is uniquely determined by Theorem 4.3.1 as follows. Let τ : G → M be given by τ (g) = g · λ. The condition (4.3.ii) implies τ ∗ ω = d < λ, g −1 dg >, where g −1 dg is the left-invariant Maurer-Cartan form. This equality completely determines ω since τ is a surjective submersion. Now we turn to the character . Note that is G-invariant and G is compact. Therefore is determined by its values on G-invariant cycles in Z 1 (G × M) containing λ. Moreover, G being connected, we can assume these cycles to be closed paths et X · λ ⊂ M, 0 ≤ t ≤ 1,
X ∈ g,
produced by actions of 1-parametric subgroups of G on λ. For such cycles Eq. (4.3.iv) implies ({et X · λ}) =< λ, X > + (e X ) mod Z. Note that e X belongs to the stabilizer Iλ of λ in G (in other words to the inertia group at λ); we denote iλ the Lie algebra of Iλ . It remains to determine the values of the character | Iλ : Iλ → R/Z. Because of (4.3.iv) we have d| Iλ = λ|iλ and Iλ being connected this determines | Iλ , completing construction of the differential character (ω, µ, ). Of course there are still conditions to be checked for (ω, µ, ) to be a differential character. For example, λ|iλ should actually lift to a character of the stabilizer group Iλ . In fact (we don’t prove it in the present paper), this is the only condition (for existence of a differential character on the coadjoint orbit of λ ∈ g∗ ). One can say that the differential character on an orbit is induced from a smooth character of the stabilizer group of a point on the orbit. Let us remark that usually the integrality condition is stated in terms of the form ω on M and then additional conditions on λ are imposed to ensure that the Lie algebra action on the prequantization bundle lifts to an action of the group. For example, if G is simply connected then the differential character exists iff ω is integral. Our point of view is that it is more natural to emphasise the character rather than the form ω. As an extreme example, if G = T is a torus (compact connected abelian group) then each coadjoint orbit is a point λ ∈ t∗ , hence has unique (integral) 2-form ω (= 0). However only those points for which λ is a differential of a character of T are prequantizable according to the above discussion. In fact the set of prequantizable orbits in t∗ is the lattice of characters of T . Moreover, since an orbit is a point, the differential character is just a smooth character of T .
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5.3. Example: coadjoint orbits of SU (2) and S 1 -reduction. Let S 2 be a sphere about the origin in R3 . We think of S 2 as a coadjoint orbit of SU (2) in su∗2 ≈ R3 . In particular, SU (2) acts on S 2 by rotations (via the 2-fold covering map SU (2) → S O(3, R)). It is easy to see that given an integral 2-form ω ∈ 2 (S 2 ) one has a unique SU (2)invariant differential character (ω, µ, ) on S 2 . For example, let us compute the restriction χ : {±1} → R/Z of the character to the center {±1} ⊂ SU (2). Here we think of the center as a subgroup of the inertia group I p at a point p ∈ S 2 (note that is SU (2)-invariant and hence χ does not depend on the point p). Let X be (unique up to sign) element of the Lie algebra su2 such that e X = −1 ∈ SU (2), < µ( p), X >= 0, and γ (t) = et X p, 0 < t < 1, is a big circle on S 2 bounding a half-sphere B ⊂ S 2 . Then we have 1 (4.3.iii) X (4.3.iv) χ (−1) = (e ) = (∂ B) = ω= ω mod Z. 2 S2 B In fact, we did not have to assume that < µ( p), X >= 0. For an arbitrary X ∈ su2 such that e X = −1, one has (4.3.iv) (4.3.iii) ω− < µ( p), X > χ (−1) = (e X ) = (∂ B)− < µ( p), X > = B 1 = ω mod Z, 2 S2 where B is either of the two parts of S 2 bounded by the circle γ (t) = et X p, 0 < t < 1, and the last equality follows from the Archimedes (or the Duistermaat-Heckman) Theorem. Let us now turn to the reduction procedure with respect to the action of a torus T ≈ S 1 ⊂ SU (2). We denote by h the restriction of the moment map µ to the Lie algebra of T (so if T is the group of rotations about the vertical axis, then h is the height function on S 2 up to scale). According to Sect. 3 the reduction is the following sequence of functors (cf. (3.2.1)):
DC 2inv S 2 ⇔ SU (2) × S 2 → DC 2inv S 2 ⇔ T × S 2
∼ → DC 22 h −1 (0) ⇔ T ×h −1 (0) − → DC 22 ( p ⇔ {±1} × p). Here p is a point on the circle h −1 (0), the first arrow is the restriction, the second is the actual reduction (i.e. restriction to the zero level of the moment map), and the last (restriction) isomorphism is due to the fact that DC 22 is a stack. The category DC 22 ( p ⇔ {±1} × p) is just the category of characters (or 1-dim representations) of the group {±1}. On the other hand, an isomorphism class of an object (c, h, ω) of DC 2inv S 2 ⇔ T × S 2 is determined by ω. Since the reduction is just the restriction of differential characters we see that it takes ω to the character χ : {±1} → R/Z given, as above, by 1 χ (−1) = ω mod Z . 2 S2 One of our main motivations for developing the theory of invariant differential characters was to provide a context for the above formula. It corresponds to the fact that an irreducible representation of SU (2) has a T -invariant vector iff its highest weight is even.
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References 1. Behrend, K.: Cohomology of stacks. In: Intersection theory and moduli, ICTP Lect. Notes, XIX, Abdus Salam Int. Center. Theo. Phys., pp. 249–294 (electronic) (2004) 2. Behrend, K., Xu, P.: Differentiable Stacks and Gerbes, available at http://arXiv.org/abs/math.DG/0605694 v1 [math.DG], 2006 3. Be˘ılinson, A.: Higher regulators and values of L-functions, Current problems in mathematics, Vol. 24, Moscow: Akad Nauk SSR Vsesoyuz, pp. 181–238 (Russian), 1984 4. Bos, R.: Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids. Int. J. Geom. Meth. Mod. Phys. 4(3), 389–436 (2007) 5. Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and topology, LNM 1167 Berlin-Heidelberg, New York: Springer-Verlag, pp. 50–80 (1985) 6. Dupont, J.: Curvature and characteristic classes. Lecture Notes in Mathematics, Vol. 640, Berlin:Springer-Verlag, 1978 7. Gomi, K.: Equivariant smooth Deligne cohomology. Osaka J. Math. 42(2), 309–337 (2005) 8. Guillemin, V., Sternberg, S.: Supersymmetry and equivariant de Rham theory. Berlin-Heidelberg, New York: Springer, 1999 9. Hopkins, M., Singer, I.: Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70(3), 329–452 (2005) 10. Kostant, B.: Quantization and unitary representations. I. Prequantization. Lectures in modern analysis and applications, III, LNM 170 Berlin-Heidelberg, New York: Springer, pp. 87–208, 1970 11. Lerman, E., Malkin, A.: Differential characters as stacks and prequantization. J. Gökova Geom. Topol. 2, 14–39 (2008) 12. Lupercio, E., Uribe, B.: Differential characters on orbifolds and string connections. I. Global quotients. In: Gromov-Witten theory of spin curves and orbifolds, Contemp. Math., Vol. 403 Providence, RI: Amer. Math. Soc., pp. 127–142 (2006) 13. Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1), 121–130 (1974) 14. Meyer, K.: Symmetries and integrals in mechanics. In: Dynamical systems, New York: Academic Press, pp. 259–272 (1973) 15. Moerdijk, I., Mrcun J.: Introduction to foliations and Lie groupoids. Cambridge Studies in Advanced Mathematics, Vol. 91, Cambridge: Cambridge University Press, 2003 16. Weil, A.: Sur les théorèmes de de Rham. Comment. Math. Helv. 26, 119–145 (French), (1952) 17. Weinstein, A.: Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. (N.S.) 16(1), 101–104 (1987) Communicated by S. Zelditch
Commun. Math. Phys. 289, 803–824 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0807-3
Communications in
Mathematical Physics
On the Well-Posedness of the Spatially Homogeneous Boltzmann Equation with a Moderate Angular Singularity Nicolas Fournier1 , Clément Mouhot2 1 Centre de Mathématiques, Faculté de Sciences et Technologies,
Université Paris 12, 61, avenue du Général de Gaulle, 94010 Créteil Cedex, France. E-mail: [email protected] 2 Ceremade, Université Paris IX-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris, France. E-mail: [email protected] Received: 10 March 2007 / Accepted: 20 February 2009 Published online: 19 May 2009 – © Springer-Verlag 2009
Abstract: We prove an inequality on the Kantorovich-Rubinstein distance–which can be seen as a particular case of a Wasserstein metric–between two solutions of the spatially homogeneous Boltzmann equation without angular cutoff, but with a moderate angular singularity. Our method is in the spirit of [7]. We deduce some well-posedness and stability results in the physically relevant cases of hard and moderately soft potentials. In the case of hard potentials, we relax the regularity assumption of [6], but we need stronger assumptions on the tail of the distribution (namely some exponential decay). We thus obtain the first uniqueness result for measure initial data. In the case of moderately soft potentials, we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy (for very moderately soft potentials), plus sometimes an additionnal moment condition. We thus improve significantly on all previous results, where weighted Sobolev spaces were involved.
Contents 1. Introduction . . . . . . . . . 2. Main Results . . . . . . . . . 3. The General Inequality . . . . 4. Application to Hard Potentials 5. Application to Soft Potentials References . . . . . . . . . . . . .
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1. Introduction 1.1. The Boltzmann equation. We consider a spatially homogeneous gas in dimension d ≥ 2 modeled by the Boltzmann equation. Therefore the time-dependent density
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f = f t (v) of particles with velocity v ∈ Rd solves ∂t f t (v) = dv∗ dσ B(|v − v∗ |, θ ) f t (v ) f t (v∗ ) − f t (v) f t (v∗ ) , Rd
Sd−1
(1.1)
where v =
|v − v∗ | v + v∗ |v − v∗ | v + v∗ + σ, v∗ = − σ 2 2 2 2
∗) and θ is the so-called deviation angle defined by cos θ = (v−v |v−v∗ | · σ . The collision kernel B = B(|v − v∗ |, θ ) = B(|v − v∗ |, θ ) is given by physics and is related to the microscopic interaction between particles. In dimension d = 3 it is related to the probabilistic cross-section Bˆ of the distribution of possible outgoing velocities v and v∗ arising from a collision with two particles with velocities v and v∗ , by the formula ˆ We refer to the review papers of Desvillettes [5] and Villani [18] for B = |v − v∗ | B. more details. Conservation of mass, momentum and kinetic energy hold at least formally for solutions to (1.1), that is for all t ≥ 0, f t (v) ϕ(v) dv = f 0 (v) ϕ(v) dv, ϕ = 1, v, |v|2 ,
Rd
Rd
and we classically may assume without loss of generality that Rd f 0 (v) dv = 1. 1.2. Assumptions on the collision kernel. We shall assume that the collision kernel takes the form B(|v − v∗ |, θ ) sind−2 θ = (|v − v∗ |) β(dθ )
(A1)
for some function : R+ → R+ and some nonnegative measure β on (0, π ]. In the case of an interaction potential V (s) = 1/r s in dimension d = 3, with s ∈ (2, ∞), one has (z) = cst z γ , β(θ ) ∼ cst θ −1−ν , with γ =
2 s−5 , ν= . s−1 s−1
(1.3)
One classically names hard potentials the case when γ ∈ (0, 1) (i.e., s > 5 in dimension d = 3), Maxwellian molecules the case when γ = 0 (i.e., s = 5 in dimension d = 3), and soft potentials the case when γ ∈ (−d, 0) (i.e., s ∈ (2, 5) in dimension d = 3). Let us emphasize that 0+ β(dθ ) = +∞, which expresses the affluence of grazing collisions, but in any case, π θ 2 β(dθ ) < +∞. (1.4) 0
In this paper we shall deal with a moderate angular singularity, that is we shall assume that the collision kernel satisfies π κ1 = θ β(dθ ) < +∞, (A2) 0
which corresponds to s ∈ (3, ∞) in (1.3)).
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We will also assume that behaves as a power function, namely that for some γ ∈ (−d, 1], there exists some constant C such that for all z, z˜ ∈ R+ , (z) ≤ C z γ ; |(z) − (˜z )| ≤ C |z γ − z˜ γ |.
(A3(γ ))
Sometimes, we will need a lowerbound: there exists c > 0 such that for all z ∈ R+ , (z) ≥ c z γ .
(A4(γ ))
In the case of hard potentials, we will also sometimes use an additional technical assumption in order to obtain the propagation of some exponential moments: β(dθ ) = b(cos θ ) dθ, where b is nondecreasing, convex and C 1 on [−1, 1).
(A5)
In the case of moderately soft potentials, we will sometimes use β(dθ ) = β(θ ) dθ with β(θ ) ∼θ→0 cst θ −1−ν
(A6(ν))
for some positive constant. In practise, all these assumptions are met when one deals with the interaction potential V (s) = 1/r s in dimension d = 3, with s ∈ (3, ∞). 1.3. Goals, existing results and difficulties. We study in this paper the well-posedness of the spatially homogeneous Boltzmann equation for singular collision kernel as introduced above. In particular we focus on the questions of uniqueness and stability with respect to the initial condition which were open, for collision kernel with angular cutoff, until the two recent papers [6,7] (except in the special case of Maxwell molecules, see below). π In the case of a collision kernel with angular cutoff, that is when 0 β(dθ ) < +∞, there are some optimal existence and uniqueness results: Mischler-Wennberg [13] in the space of L 1 non-negative functions with finite non-increasing kinetic energy (for counter-examples of spurious solutions with increasing kinetic energy, see [20] in the hard spheres case, and [11] in the case of hard potentials with or without angular cutoff), Lu-Mouhot [10] in the space of non-negative measures with finite non-increasing kinetic energy. However, the case of collision kernels without cutoff is much more difficult. At the same time it is crucial from the physical viewpoint since it corresponds to the fundamental class of the interactions deriving from the inverse power-law between particles. This difficulty is not surprising, since there is a difference of nature in the collision process between the two cases: on each compact time interval, each particle collides with infinitely (resp. finitely) many others in the case without (resp. with) cutoff. Until recently, the only uniqueness result obtained for the non-cutoff collision kernel was concerning Maxwellian molecules, studied successively by Tanaka [15], HorowitzKarandikar [9], Toscani-Villani [16]: it was proved in [16] that uniqueness holds for the Boltzmann equation as soon as is constant and (1.4) is met, for any initial (measure) datum with finite mass and energy, that is Rd (1 + |v|2 ) f 0 (dv) < +∞. There have been recently two papers in the case where β is non-cutoff and is not constant. The case where is bounded (together with additional regularity assumptions) was treated in Fournier [7], for essentially any initial (measure) datum such that Rd (1 + |v|) f 0 (dv) < ∞. More realistic collision kernels have been treated by Desvillettes-Mouhot [6] (including the physical important cases of hard and moderately soft potentials without cutoff), for initial data in some weighted W 1,1 spaces.
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In the present paper, we extend and improve the method of [7]: • it can deal with the physical collision kernels corresponding to hard and moderately soft potentials, as in [6]: in dimension d = 3 we obtain well-posedeness for interaction potentials 1/r s with s ∈ (3, ∞), • the proof is simplified as compared to [7]: it is shorter, allows measure initial conditions (for technical reasons, we had to consider only functions in [7]), and it does not refer anymore to probabilistic arguments. Finally let us compare our results with those in [6], when applied to the case of an interaction potential V (s) = 1/r s in dimension d = 3. • Our result is much better in the case of moderately soft potentials (s ∈ (3, 5)). Indeed, we assume only that the initial condition f 0 has finite mass, energy and entropy (plus, if s ∈ (3, 3.48), a moment condition Rd |v|q f 0 (v)dv < ∞ for q large enough). All these conditions, together with f 0 ∈ L p (Rd ) ∩ W 1,1 (Rd , (1 + |v|)2 dv) (for some p > 1 depending on the collision rate) were assumed in [6]. • Our result is different in the case of hard potentials (s ∈ (5, ∞)). We allow any meaγ sure initial f 0 condition such that for some ε > 0, Rd eε|v| f 0 (dv) < ∞, where γ = (s − 5)/(s − 1). In [6], the case where f 0 ∈ W 1,1 (Rd , (1 + |v|)2 dv) was treated. We thus assume much less regularity, but much more localization. Let us remark that our result is quasi-optimal when s ∈ (3.48, 5), since the finiteness of entropy and energy is physically very reasonable. It might be possible to relax the entropy condition, but it is not clear: one reasonably has to assume a small regularity on f 0 to get the uniqueness, since the collision rate involves |v − v∗ |γ with γ < 0, and we remark that |v − v∗ |γ f 0 (dv) f 0 (dv∗ ) is infinite when f 0 contains, e.g., Dirac measures. Let us emphasize that, as in [6,7], we are only able to prove well-posedness in the case of a moderate angular singularity (Assumption (A2)). To our knowledge, there is no uniqueness result under the general assumption (1.4), except for Maxwellian molecules (see [16]). 1.4. Notation. Let us denote by Lip(Rd ) the set of globally Lipschitz functions ϕ : Rd → R, and by Lip1 (Rd ) the set of functions ϕ ∈ Lip(Rd ) such that ϕ Lip(Rd ) = sup v =v˜
|ϕ(v) − ϕ(v)| ˜ ≤ 1. |v − v| ˜
Let also L p (Rd ) denote the Lebesgue space of measurable functions f such that 1/ p f L p (Rd ) := f p dv < +∞. Rd
Let P(Rd ) be the set of probability measures on Rd , and d d P1 (R ) = f ∈ P(R ), m 1 ( f ) < ∞ with m 1 ( f ) := |v| f (dv). Rd
We denote by L ∞ ([0, T ], P1 (Rd )) the set of measurable families ( f t )t∈[0,T ] of probability measures on Rd such that sup m 1 ( f t ) < +∞,
[0,T ]
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and by L ∞ ([0, T ], P1 (Rd )) ∩ L 1 ([0, T ], L p (Rd )) the set of measurable families ( f t )t∈[0,T ] of probability measures on Rd such that
T
sup m 1 ( f t ) < +∞,
[0,T ]
0
f t L p (Rd ) dt < +∞.
For v, v∗ ∈ Rd , and σ ∈ Sd−1 , we write v = v (v, v∗ , σ ) =
v + v∗ |v − v∗ | + σ, 2 2
and we write σ = (cos θ, sin θ ξ )
with ξ ∈ Sd−2 , θ ∈ [0, π ],
in some orthonormal basis of Rd with first vector (v − v∗ )/|v − v∗ |. Finally we denote x ∧ y = min{x, y} and x+ = max{x, 0}, and for some set E we write 11 E the usual indicator function of E. 2. Main Results Let us define the notion of weak (measure) solutions we shall use. Definition 2.1. Let B be a collision kernel which satisfies (A1-A2). A family f = ( f t )t∈[0,T ] ∈ L ∞ ([0, T ], P1 (Rd )) is a weak solution to (1.1) if
T
dt
Rd
0
f t (dv)
Rd
f t (dv∗ ) (|v − v∗ |) |v − v∗ | < +∞,
(2.1)
and if for any ϕ ∈ Lip(Rd ), and any t ∈ [0, T ], d ϕ(v) f t (dv) = f t (dv) f t (dv∗ ) A[ϕ](v, v∗ ), dt Rd Rd Rd
(2.2)
where A[ϕ](v, v∗ ) = (|v − v∗ |)
π 0
β(dθ )
ξ ∈Sd−2
ϕ(v ) − ϕ(v) dξ.
(2.3)
Note that for any σ ∈ Sd−1 ,
|v − v| = |v − v∗ |
1 − cos θ θ ≤ |v − v∗ |, 2 2
(2.4)
so that thanks to Assumption (A2), (2.1) ensures that all the terms in (2.2) are welldefined.
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Let us now introduce the distance on P1 (Rd ) we shall use. For g, g˜ ∈ P1 (Rd ), let H(g, g) ˜ be the set of probability measures on Rd × Rd with first marginal g and second marginal g. ˜ We then set
d1 (g, g) ˜ = inf |v − v| ˜ G(dv, d v), ˜ G ∈ H(g, g) ˜ R d ×R d
= min |v − v| ˜ G(dv, d v), ˜ G ∈ H(g, g) ˜ R d ×R d
ϕ(v) g(dv) − g(dv) ˜ , ϕ ∈ Lip1 (Rd ) . (2.5) = sup Rd
This distance is the Kantorovitch-Rubinstein distance, and can be viewed as a particular Wasserstein distance. We refer to Villani [19, Sect. 7] for more details on this distance, and for proofs that the equalities in (2.5) hold. Our main result is the following inequality, which will be applied in the sequel to hard and soft potentials separately. Theorem 2.2. Let B be a collision kernel which satisfies (A1-A2). Let us consider two weak solutions f, f˜ to (1.1) lying in L ∞ ([0, T ], P1 (Rd )), and satisfying T
f t (dv) f t (dv∗ ) + f˜t (dv) f˜t (dv∗ ) (1 + |v|) (|v − v∗ |) < +∞. dt 0
R d ×R d
(2.6)
For s ∈ [0, T ], let Rs ∈ H( f s , f˜s ) be such that d1 ( f s , f˜s ) = |v − v| ˜ Rs (dv, d v). ˜ R d ×R d
Then for all t ∈ [0, T ], |Sd−2 | t ˜ ˜ d1 ( f t , f t ) ≤ d1 ( f 0 , f 0 ) + κ1 ds Rs (dv, d v) ˜ Rs (dv∗ , d v˜∗ ) 2 0 R d ×R d R d ×R d ˜ × 8 ((|v − v∗ |) ∧ (v˜ − v˜∗ )) |v − v| + ((|v − v∗ |) − (v˜ − v˜∗ ))+ |v − v∗ | + ((|v˜ − v˜∗ |) − (v − v∗ ))+ |v˜ − v˜∗ | . (2.7) The meaning of this inequality can be understood by means of probabilistic arguments, see [7] for details. Consider however two infinite particle systems, whose velocity distributions are f and f˜ respectively. The main ideas are that the first term on the righthand side expresses an increase of the optimal coupling due to simultaneous collisions (in both systems), whose rate is (optimally) the minimum between the two rates. Next, the second and third terms explain that the optimal coupling also increases due to a difference between the rates of collision in the two systems. Note that these two last terms equal zero in the case of Maxwellian molecules. We now give the application of our inequality to the study of hard potentials. Corollary 2.3. Let B be a collision kernel which satisfies (A1-A2), and (A3)(γ ) for some γ ∈ (0, 1].
Well-Posedness of Some Singular Boltzmann Equations
809
(i) Let ε > 0 be fixed. There exists a constant K ε > 0 such that for any pair of weak solutions ( f t )t∈[0,T ] , ( f˜t )t∈[0,T ] to (1.1), lying in L ∞ ([0, T ], P1 (Rd )) and satisfying
γ C T, f + f˜, ε := sup (2.8) eε|v| f t + f˜t (dv) < +∞, [0,T ] Rd
there holds for all t ∈ [0, T ]: d1 ( f t , f˜t ) ≤ d1 ( f 0 , g0 ) t + K ε C T, f + f˜, ε d1 ( f s , f˜s ) 1 + log d1 ( f s , f˜s ) ds. 0
(Rd ),
(ii) As a consequence for any f 0 ∈ P1 there exists at most one weak solution f ∈ L ∞ ([0, T ], P1 (Rd )) to (1.1) starting from f 0 and such that C(T, f, ε) < +∞. (iii) Let us now give an existence and uniqueness result, assuming (here only) additionally (A4)(γ ) and (A5). Consider f 0 ∈ P1 (Rd ) such that, for some ε0 > 0, K > 0, we have γ eε0 |v| f 0 (dv) ≤ K < +∞. (2.9) Rd
Then there exists a unique weak solution ( f t )t∈[0,∞) ∈ L 1loc ([0, ∞), P1 (Rd )) starting from f 0 . Furthermore, there exist ε1 > 0 and K¯ > 0, depending only on ε0 , K , B, such that for all T > 0, C(T, f, ε1 ) ≤ K¯ . (iv) Finally let us give a result on the dependence according to the initial datum. Consider a family ( f n )n≥1 , f ∞ of weak solutions to (1.1) such that, for some ε > 0, T > 0, we have sup C(T, f ∞ + f n , ε) < +∞. n≥1
Then lim d1 ( f 0n , f 0∞ ) = 0
n→∞
⇒
lim sup d1 ( f tn , f t∞ ) = 0.
n→∞ [0,T ]
Let us recall that this result applies in particular to hard potentials in dimension d = 3 (that is inverse power-law potentials with s > 5). In [6], under very similar conditions on the collision kernel, a well-posedness and stability result was obtained in the space L ∞ ([0, T ], W 1,1 (Rd , (1 + |v|2 ) dv)). We thus relax the regularity assumption, but we require more moments. We finally apply our inequality to the study of soft potentials. Corollary 2.4. Let B be a collision kernel which satisfies (A1-A2), and (A3)(γ ) for some γ ∈ (−d, 0). (i) Let p ∈ (d/(d + γ ), ∞] be fixed. There exists a constant K p > 0 such that for any pair of weak solutions ( f t )t∈[0,T ] , ( f˜t )t∈[0,T ] to (1.1) on [0, T ], lying in L ∞ ([0, T ], P1 (Rd )) ∩ L 1 ([0, T ], L p (Rd )), there holds ∀ t ∈ [0, T ],
K d1 ( f t , f˜t ) ≤ d1 ( f 0 , g0 ) e p
C(t, f, p)+C(t, f˜, p)+t
,
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N. Fournier, C. Mouhot
where ∀ t ∈ [0, T ],
t
C(t, f, p) = 0
f s L p (Rd ) ds.
Uniqueness and stability thus hold in L ∞ ([0, T ], P1 (Rd )) ∩ L 1 ([0, T ], L p (Rd )). (ii) Let p ∈ (d/(d + γ ), ∞]. For any initial condition f 0 ∈ P1 (Rd ) ∩ L p (Rd ), local existence and uniqueness hold, that is there exists T∗ = T∗ f 0 L p (Rd ) , B > 0 such that there exists a unique weak solution ( f t )t∈[0,T∗ ) to (1.1) which furthermore belongs to d p d [0, T L∞ ), P (R ) ∩ L (R ) . ∗ 1 loc (iii) Assume now furthermore that γ ∈ (−1, 0), (A4)(γ ), and (A6)(ν) for some ν ∈ (−γ , 1). Consider an initial datum f 0 ∈ P1 (Rd ) with finite energy and entropy, that is f 0 (v)(|v|2 + | log f 0 (v)|)dv < ∞. (2.10) Rd
Assume also that for some q > γ 2 /(ν + γ ), f 0 ∈ L 1 (Rd , |v|q dv). Then there exists a unique weak solution ( f t )t∈[0,∞) to (1.1), which furthermore belongs to d 1 d q 2 1 p d [0, ∞), P [0, ∞), L (R ) ∩ L (R , (|v| + |v| ) dv) ∩ L (R ) L∞ 1 loc loc for some (explicit) p ∈ (d/(d + γ ), d/(d − ν)). Let us recall that point (iii) applies, in dimension d = 3, to the case of moderately soft potentials, that is inverse power-law potentials with s ∈ (3, 5). In such a case, one has γ = (s − √ 5)/(s − 1) and ν = 2/(s − 1) ∈ (−γ , 1). We observe that for s ∈ (s0 , 5), with s0 = 2 5 − 1 3.472, the choice q = 2 is possible, so that our conditions reduce to the finiteness of entropy and energy. On the contrary, for s > 3 close to 3, q has to be chosen very large, e.g., for s = 3.01, we have to take q 200. A similar result was obtained in [6, Theorem 1.3], assuming that f 0 ∈ L p (Rd ) ∩ 1 L (Rd , |v|q dv) ∩ W 1,1 (Rd , (1 + |v|2 ) dv), with p > d/(d + γ ) and q > γ 2 /(ν + γ ). We thus relax a large part of these conditions. The rest of the paper is dedicated to the proof of these results: we establish Theorem 2.2 in Sect. 3. Applications to hard and soft potentials are studied in Sects. 4 and 5 respectively. 3. The General Inequality As a preliminary step, we shall parameterize precisely the post-collisional velocities. We follow here the approach of [8], which was strongly inspired by Tanaka [15], and we extend it to any dimension d ≥ 2.
Well-Posedness of Some Singular Boltzmann Equations
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The first step is to define a parameterization of the sphere orthogonal to some given vector X ∈ Rd . This parameterization shall not be smooth of course. We identify in the sequel S0 = {−1, +1}. For X ∈ Rd \{0}, we set S X to be the symmetry with respect to the hyperplane X ⊥ H X = ed − |X | (where ed = (0, . . . , 0, 1)) if ed = X/|X |, and S X = Id otherwise. We set C X = U ∈ Rd ; |U | = |X | and U, X = 0 . Then we parameterize C X by Sd−2 as follows: we set ∀ ξ = (ξ1 , ..., ξd−1 ) ∈ Sd−2 , (ξ ) = (ξ1 , . . . , ξd−1 , 0) ∈ Sd−1 ⊂ Rd and (X, ξ ) = |X | S X ( (ξ )). It is easy to check that for a given X , the map ξ ∈ Sd−2 → (X, ξ ) is a bijection onto C X and is a unitary parameterization. Therefore, for ξ ∈ Sd−2 , θ ∈ [0, π ], and X, v, v∗ ∈ Rd , one may write v = v (v, v∗ , θ, ξ ) = v +
cos θ − 1 sin θ (v − v∗ ) + (v − v∗ , ξ ) , 2 2
and for all ϕ ∈ Lip(Rd ), recalling (2.3) A[ϕ](v, v∗ ) = (|v − v∗ |)
π 0
β(dθ )
Sd−2
dξ ϕ v (v, v∗ , θ, ξ ) − ϕ(v) .
A problem of this parameterization is its lack of smoothness. To overcome this difficulty, we shall prove the following fine version of a lemma due to Tanaka [15], whose proof may be found in [8, Lemma 2.6] in dimension 3. Lemma 3.1. There exists a measurable map ξ0 : Rd × Rd × Sd−2 → Sd−2 such that for any X, Y ∈ Rd \ {0}, the map ξ → ξ0 (X, Y, ξ ) is a bijection with jacobian 1 from Sd−2 into itself (when d ≥ 3), and ∀ ξ ∈ Sd−2 , |(X, ξ ) − (Y, ξ0 (X, Y, ξ ))| ≤ 3 |X − Y |.
(3.1)
This implies that for all v, v∗ , v, ˜ v˜∗ ∈ Rd , all θ ∈ [0, π ], all ξ ∈ Sd−2 , we have v (v, v∗ , θ, ξ ) − v (v, ˜ v˜∗ , θ, ξ0 (v − v∗ , v˜ − v˜∗ , ξ ) ≤ |v − v| ˜ + 2 θ (|v − v| ˜ + |v∗ − v˜∗ |) .
(3.2)
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N. Fournier, C. Mouhot
Proof of Lemma 3.1. The case d = 2 is trivial, therefore we assume d ≥ 3. Let us consider X, Y ∈ Rd \ {0}. If X/|X | = Y/|Y |, it is enough to choose ξ0 (X, Y, ξ ) = ξ . Indeed in this case S X = SY so that |(X, ξ ) − (Y, ξ )| = ||X | − |Y || ≤ |X − Y |. Now assume that X/|X | = Y/|Y |. Then let us define R X,Y to be the axial rotation of Rd transforming X/|X | into Y/|Y | around a line perpendicular to the plane determined by X and Y . Let us then define ξ0 by the identity (Y, ξ0 (X, Y, ξ )) =
|Y | R X,Y ((X, ξ )) ∈ CY . |X |
For any X, Y ∈ Rd \ {0}, the application ξ → ξ0 (X, Y, ξ ) is the restriction to Sd−2 of the following orthogonal linear transformation on Rd−1 : O X,Y (Z ) = −1 ◦ SY ◦ R X,Y ◦ S X ◦ (Z ).
∀ Z ∈ Rd−1 ,
Therefore it has unit jacobian. Finally let us check the control (3.1): |Y | |(X, ξ ) − (Y, ξ0 (X, Y, ξ ))| = (X, ξ ) − R X,Y (X, ξ ) |X | |Y | |Y | + ≤ (X, ξ ) 1 − (X, ξ ) − R X,Y (X, ξ ) |X | |X | Y X ≤ 3 |X − Y |. ≤ |X − Y | + |Y | − |Y | |X | Since the transformation ξ0 (X, Y, ·) has unit jacobian, one may finally rewrite (2.3), for all ϕ ∈ Lip(Rd ), all X, Y ∈ Rd (which may depend on v, v∗ , θ ), as A[ϕ](v, v∗ ) = (|v − v∗ |)
π
β(dθ )
0
Sd−2
dξ ϕ v (v, v∗ , θ, ξ0 (X, Y, ξ )) − ϕ(v) . (3.3)
We may finally give the Proof of Theorem 2.2. We denote
ϕ
h t :=
Rd
ϕ(v)
f t − f˜t (dv)
for ϕ ∈ Lip1 (Rd ), t ∈ [0, T ]. We also set h t = d1 ( f t , f˜t ), and we recall that ht =
|v − v| ˜ Rt (dv, d v) ˜ =
R d ×R d
sup ϕ ∈ Lip1 (Rd )
ϕ
ht .
Well-Posedness of Some Singular Boltzmann Equations
813
Step 1. Let us thus consider ϕ ∈ Lip1 (Rd ). Using (2.2), that Rt ∈ H( f t , f˜t ) and (3.3), we immediately obtain, using the map ξ0 built in Lemma 3.1, d ϕ ht = f t (dv) f t (dv∗ ) A[ϕ](v, v∗ ) − ˜ f t (d v˜∗ ) A[ϕ](v, ˜ v˜∗ ) f˜t (d v) d d dt R d ×R d R ×R Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) (A[ϕ](v, v∗ ) − A[ϕ](v, ˜ v˜∗ )) = R d ×R d R d ×R d π = Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) β(dθ ) dξ R d ×R d R d ×R d Sd−2 0 (3.4) × (|v − v∗ |) ϕ v (v, v∗ , θ, ξ ) − ϕ(v) ˜ v˜∗ , θ, ξ0 (v − v∗ , v˜ − v˜∗ , ξ ) − ϕ(v) ˜ . −(|v˜ − v˜∗ |) ϕ v (v, We now use the shortened notation v = v (v, v∗ , θ, ξ ) and v˜ = v (v, ˜ v˜∗ , θ, ξ0 (v − v∗ , v˜ − v˜∗ , ξ )). Noting that for all x, y ∈ R, x = x ∧ y + (x − y)+ , we easily deduce from (3.4) that π d ϕ h = Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) β(dθ ) dξ dt t 0 R d ×R d R d ×R d Sd−2 ˜ × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) × ϕ(v ) − ϕ(v˜ ) − ϕ(v) + ϕ(v) + (|v − v∗ |) − (|v˜ − v˜∗ |) + × ϕ(v ) − ϕ(v) + (|v˜ − v˜∗ |) − (|v − v∗ |) + × ϕ(v) ˜ − ϕ(v˜ ) ϕ
ϕ
ϕ
=: I1 (t) + I2 (t) + I3 (t), where the last equality stands for a definition. Using that ϕ ∈ Lip1 (Rd ), (2.4), and (A2), we get |Sd−2 | ϕ ϕ I2 (t) + I3 (t) ≤ κ1 Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) 2 R d ×R d R d ×R d × (|v − v∗ |) − (|v˜ − v˜∗ |) + |v − v∗ | + (|v˜ − v˜∗ |) − (|v − v∗ |) + |v˜ − v˜∗ | . Next, using again that ϕ ∈ Lip1 (Rd ), we get that for all ε ∈ (0, π ), ε ϕ I1 (t) ≤ Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) β(dθ ) dξ R d ×R d R d ×R d Sd−2 0 ˜ × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) × |v − v| + |v˜ − v| π + Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) β(dθ ) dξ R d ×R d R d ×R d Sd−2 ε ˜ × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) × |v − v˜ | − |v − v| π + Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) β(dθ ) dξ R d ×R d R d ×R d Sd−2 ε ˜ − (ϕ(v) − ϕ(v)) ˜ × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) × |v − v| ϕ
ϕ
ϕ
=: J1 (t, ε) + J2 (t, ε) + J3 (t, ε),
814
N. Fournier, C. Mouhot ϕ
where the last equality stands for a definition. First for J2 (t, ε), using (3.2) and (A2), we immediately get, by symmetry, that π ϕ d−2 θ β(dθ ) |S | Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) J2 (t) ≤ 2 R d ×R d R d ×R d ε ˜ + |v∗ − v˜∗ | × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) |v − v| Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) ≤ 4 κ1 |Sd−2 | R d ×R d R d ×R d ˜ × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) |v − v|. Next, setting αε = |S
d−2
|
ε
θ β(dθ ),
0
it is not hard to obtain, using (2.4), the fact that Rt ∈ H( f t , f˜t ) and a symmetry argument, that αε ϕ J1 (t, ε) ≤ Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) 2 R d ×R d R d ×R d ˜ + |v∗ | + |v˜∗ |) × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) (|v| + |v| f t (dv) f t (dv∗ ) (|v − v∗ |) |v| ≤ αε Rd Rd + αε ˜ ˜ f˜t (d v) f˜t (d v˜∗ ) (|v˜ − v˜∗ |) |v| Rd Rd ≤ C t, f, f˜ αε , where the constant C t, f, f˜ belongs to L 1 ([0, T ]) due to (2.6). ϕ
Finally for J3 (t, ε) we notice that the integrand is nonnegative (since ϕ ∈ Lip1 (Rd )) and does not depend on θ, ϕ. Hence, denoting π Sε := |Sd−2 | β(dθ ) < +∞, ε
we have, for any A > 0, ϕ
ϕ
ϕ
J3 (t, ε) ≤ K 1 (t, ε, A) + K 2 (t, ε, A), where
ϕ K 1 (t, ε, A) = A Sε Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) R d ×R d R d ×R d × |v − v| ˜ − (ϕ(v) − ϕ(v)) ˜ , ϕ Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ), K 2 (t, ε, A) = Sε R d ×R d R d ×R d ˜ × (|v − v∗ |) ∧ (|v˜ − v˜∗ |) 11{(|v−v∗ |)∧(|v− ˜ v˜∗ |)>A} |v − v|.
Well-Posedness of Some Singular Boltzmann Equations
815
Using that Rt ∈ H( f t , f˜t ), and that it achieves the Wasserstein distance, we get
ϕ ϕ K 1 (t, ε, A) = A Sε d1 ( f t , f˜t ) − h t . Next, we obtain ϕ K 2 (t, ε,
A) ≤ Sε
Rd
f t (dv)
Rd
f t (dv∗ ) |v| (|v − v∗ |) 11{(|v−v∗ |)>A}
˜ ˜ (|v˜ − v˜∗ |) 11{(|v− f˜t (d v) f˜t (d v˜∗ ) |v| ˜ v˜∗ |)>A} Rd ≤ Sε C A t, f, f˜ . +Sε
Rd
Due to (2.6), we observe that lim
A→∞ 0
T
C A t, f, f˜ dt = 0.
Step 2. Gathering all the previous estimates, we observe that for any ϕ ∈ Lip1 (Rd ), t ∈ [0, T ], ε > 0, A > 0, we have d ϕ ϕ h t ≤ Ht + ε,A (t) + A Sε h t − h t , dt where
and
(3.5)
ε,A (t) := αε C t, f, f˜ + Sε C A t, f, f˜ , |Sd−2 | Rt (dv, d v) ˜ Rt (dv∗ , d v˜∗ ) 2 R d ×R d R d ×R d ˜ × 8 (|v − v∗ |) ∧ (v˜ − v˜∗ ) |v − v| + (|v − v∗ |) − (v˜ − v˜∗ ) + |v − v∗ | + (|v˜ − v˜∗ |) − (v − v∗ ) + |v˜ − v˜∗ | .
Ht := κ1
ϕ
Recall that h t = supϕ∈Lip (Rd ) h t , and that our aim is to prove that 1 t ht ≤ h0 + Hs ds.
(3.6)
0
We immediately deduce from (3.5) that t t ϕ A Sε t ϕ A Sε s ht e Hs + ε,A (s) ds + A Sε ≤ h0 + e h s e A Sε s ds. 0
0
Then we take the supremum over ϕ ∈ Lip1 Lemma which states that
(Rd )
u t ≤ gt + a
and we use the generalized Gronwall
t
u s ds 0
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N. Fournier, C. Mouhot
implies that
t
u t ≤ g0 eat +
ea (t−s)
0
dgs ds, ds
which yields ht e
A Sε t
≤ h0 e
A Sε t
+e
t
A Sε t
0
Hs + ε,A (s) ds,
so that for all t ∈ [0, T ], ht ≤ h0 +
t
T
Hs ds + 0
0
ε,A (t) dt.
This inequality holding for any ε > 0, A > 0, we easily conclude that (3.6) holds, since lim
T
A→∞ 0
ε,A (t) dt = αε
T
C t, f, f˜ dt
0
with
T
C t, f, f˜ dt < +∞
0
and
αε = |S
d−2
due to (A2).
| 0
ε
θ β(dθ ) −−→ 0 ε→0
4. Application to Hard Potentials 4.1. Propagation of exponential moments. We first prove a lemma on the propagation (and appearance) of exponential moment, which is a variant of results first obtained in [3,2] (and also developed in [14,12]). Lemma 4.1. Let B be a collision kernel satisfying Assumptions (A1-A2-A5) and (A3)-(A4)(γ ) for some γ ∈ (0, 1]. Let f 0 ∈ P1 (Rd ). (i) Assume that for some ε0 > 0, some s ∈ (0, 2), s eε0 |v| f 0 (dv) ≤ Cε0 ,s < +∞. Rd
Then there exists ε1 > 0 and a constant C > 0, depending only on s, ε0 , Cε0 ,s , such that for any T > 0, any weak solution ( f t )t∈[0,T ] to (1.1) satisfies s sup eε1 |v| f t (dv) ≤ C < +∞. [0,T ] Rd
Well-Posedness of Some Singular Boltzmann Equations
817
(ii) Assume now only that e0 = Rd |v|2 f 0 (dv) < ∞. For any s ∈ (0, γ /2), any τ > 0, there exists ε > 0 and C > 0, depending only on s, τ , and an upperbound of e0 such that for any T > 0, any weak solution ( f t )t∈[0,T ] to (1.1) satisfies s sup eε|v| f t (dv) ≤ C < +∞. t∈[τ,T ] Rd
Proof of Lemma 4.1. We first recall that for any t ∈ [0, T ], |v|2 f t (dv) = |v|2 f 0 (dv) =: e0 , Rd
Rd
(4.1)
and we observe that for all v ∈ Rd , all t ≥ 0, since γ ∈ (0, 1] and since f t ∈ P1 (Rd ), γ /2 γ γ |v − v∗ | f t (dv∗ ) ≥ |v| − |v∗ |γ f t (dv∗ ) ≥ |v|γ − e0 . (4.2) Rd
Rd
Let us fix 0 < s < 2. We define for any p ∈ R+ , m p (t) := |v|sp f t (dv). Rd
Step 1. The evolution equation (2.2) yields dm p = (|v − v∗ |) K p (v, v∗ ) f t (dv) f t (dv∗ ), dt R d ×R d
(4.3)
where, using (A5) and a symmetry argument, sp 1 π K p (v, v∗ ) := |v | + |v∗ |sp − |v|sp − |v∗ |sp b(cos θ ) dθ dξ. 2 0 Sd−2 Let us split b = bηc + bηr for some η ∈ (0, π ) with bηc (cos θ ) = b(cos θ ) 11θ≥η + b(cos η) + b (cos η) (cos θ − cos η) 110≤θ≤η for θ ∈ (0, π ]. Due to (A5), we know that bηc ≤ b, so that bηr ≥ 0. We can split corc,η r,η respondingly K p = K p + K p . We also easily check that for each η ∈ (0, π ), bηc is convex, non-decreasing, and bounded on [−1, 1). We are thus in a position to apply [3, Corollary 1], which yields that for p > 2/s, sp/2 2 2 K c,η (v, v ) ≤ α (η) |v| + |v | − K (η) |v|sp + |v∗ |sp , ∗ p ∗ p
(4.4)
where (α p (η)) p is strictly decreasing and satisfies ∀ p > 2/s, 0 < α p <
C(η) sp + 1
(4.5)
for some constant C(η) depending on an upper bound of bηc , and some constant K (η) depending on a lower bound of the mass of βηc . Therefore K can be made uniform according to η as η → 0.
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For the other part of the collision kernel we use for instance [6, Lemma 2.1] and Assumption (A2) to deduce that (as soon as sp ≥ 2) sp sp K r,η (4.6) p (v, v∗ ) ≤ δ(η) |v| + |v∗ | with
π
δ(η) ≤ cst 0
θ bηr (cos θ ) dθ → 0
as η → 0, due to (A2). Combining (4.4,4.5,4.6) and fixing carefully η we thus find for all p > 2/s, sp/2 K p (v, v∗ ) ≤ α¯ p |v|2 + |v∗ |2 − K¯ |v|sp + |v∗ |sp for some constant K¯ > 0 and where (α¯ p ) p is strictly decreasing and satisfies, for some constant C¯ > 0, ∀ p > 2/s, 0 < α¯ p <
C¯ . sp + 1
We of course deduce that for p large enough, say p ≥ p0 > 2/s, sp/2 K p (v, v∗ ) ≤ α¯ p |v|2 + |v∗ |2 − |v|sp − |v∗ |sp − K¯ |v|sp + |v∗ |sp , (4.7) changing if necessary the value of K¯ > 0. We now insert (4.7) in (4.3). Using (A4)(γ ) and (4.2), we get, for p ≥ p0 , dm p γ /2 ≤ α¯ p Q p − K (m p+γ /s − e0 m p ) dt
(4.8)
for some new constant K > 0 and with sp/2 2 2 sp sp |v| + |v∗ | (|v − v∗ |) f t (dv) f t (dv∗ ). − |v| − |v∗ | Q p := R d ×R d
Step 2. Using (A3)(γ ) and following line by line the proof of [14, Lemma 4.7] from [14, Eq. (4.13)] which is the same as (4.8) here to [14, Eq. (4.19)] (this proof is itself essentially based on [3]), we obtain the following conclusion. Set k p = [sp/4 + 1/2] (here [·] stands for the integer part). Set also, with the usual Gamma function, mp z p := and Z p := max z (2k+γ )/s z p−2k/s , z 2k/s z p−2k/s+γ /s . k=1,..,k ( p + 1/2) p Then for some constants A > 0, A > 0, A > 0, for all p ≥ p0 , 1+ γ dz p ≤ A p γ /s−1/2 Z p − A p γ /s z p sp + A z p . dt
(4.9)
Step 3. Next, point (i) can be checked following the ideas of [12, Proposition 3.2] (for γ = 1) and using that classically, for any p ≥ 0, supt∈[0,∞) m p ≤ C p , for some constant depending only on B and m p (0), see e.g., [18, Theorem 1-(ii)] and [6, Lemma 2.1]. Step 4. Finally, point (ii) can be proved following line by line the proof of [14, Lemma 4.7].
Well-Posedness of Some Singular Boltzmann Equations
819
4.2. Proof of Corollary 2.3. We first recall the following variant of a classical lemma used by Yudovitch [21] in his Cauchy theorem for bidimensional incompressible nonviscious flow. See [4, Lemme 5.2.1, p. 89] for a proof. Lemma 4.2. Consider a nonnegative bounded function ρ on [0, T ], a real number a ∈ [0, ∞), and a strictly positive, continuous and non-decreasing function µ = µ(x) on (0, ∞). Assume furthermore that 1 dx = +∞, µ(x) 0 and that for all t ∈ [0, T ],
ρ(t) ≤ a +
t
µ(ρ(s)) ds.
0
Then (i) if a = 0, then ρ(t) = 0 for all t ∈ [0, T ]; (ii) if a > 0, then ∀ t ∈ [0, T ], m(a) − m(ρ(t)) ≤ t, where
1
m(x) = x
dy . µ(y)
We may now give the Proof of Corollary 2.3. We thus consider γ ∈ (0, 1], and we assume (A1)-(A2)-(A3)(γ ). We also consider some ε > 0 fixed. Step 1. Let us first prove point (i). Let us consider two weak solutions ( f t )t∈[0,T ] , ( f˜t )t∈[0,T ] to (1.1), lying in L ∞ ([0, T ], P1 (Rd )) and satisfying (2.8). We are in position to apply Theorem 2, since (A3)(γ ) and (2.8) clearly guarantee that (2.6) holds. We thus know that (2.7) holds. Using (A3)(γ ), simple computations show that ˜ ˜ ≤ C |v|γ + |v∗ |γ |v − v|, ((|v − v∗ |) ∧ (|v˜ − v˜∗ |)) |v − v| while (|v − v∗ |) − (|v˜ − v˜∗ |) + |v − v∗ | ≤ C |v − v∗ |γ − |v˜ − v˜∗ |γ |v − v∗ | ∧ |v˜ − v˜∗ | + |(|v − v∗ | − |v˜ − v˜∗ |)| ≤ Cγ (|v − v∗ | ∧ |v˜ − v˜∗ |)γ −1 |(|v − v∗ | − |v˜ − v˜∗ |)| (|v − v∗ | ∧ |v˜ − v˜∗ |) +C |v − v∗ |γ + |v˜ − v˜∗ |γ |v − v| ˜ + |v∗ − v˜∗ | ˜ + |v∗ − v˜∗ | . ˜ γ + |v˜∗ |γ |v − v| ≤ C(1 + γ ) |v|γ + |v∗ |γ + |v| We hence obtain by inserting these inequalities in (2.7) and using symmetry properties, that for some constant D > 0, t ˜ ˜ d1 ( f t , f t ) ≤ d1 ( f 0 , f 0 ) + D ds Rs (dv, d v) ˜ Rs (dv∗ , d v˜∗ ) R d ×R d R d ×R d 0 γ ˜ ˜ γ + |v˜∗ |γ |v − v|. × |v| + |v∗ |γ + |v|
820
N. Fournier, C. Mouhot
Recall now that Rs ∈ H( f s , f˜s ) achieves the Wasserstein distance. It is thus clear (recall that C(T, f + f˜, ε) was defined in (2.8)) that sup Rs (dv∗ , d v˜∗ ) |v∗ |γ + |v˜∗ |γ ≤ Aε C(T, f + f˜, ε) [0,T ] Rd ×Rd
for some constant Aε . We thus get
t d1 ( f s , f˜s ) ds d1 ( f t , f˜t ) ≤ d1 ( f 0 , f˜0 ) + D Aε C T, f + f˜, ε 0 t +D ˜ ds Rs (dv, d v) ˜ |v|γ + |v| ˜ γ |v − v|. 0
R d ×R d
Next, for any s ∈ [0, T ] and a > 0, we have Rs (dv, d v) ˜ |v|γ + |v| ˜ γ |v − v| ˜ ≤ 2 a γ d1 ( f s , f˜s ) R d ×R d ˜ 11{|v|>a} + 11{|v|>a} Rs (dv, d v) ˜ |v|γ + |v| ˜ γ |v| + |v| + ˜ R d ×R d
γ γ γ γ ˜γ ˜ Rs (dv, d v) ˜ e−εa /2 eε|v| + e−εa /2 eε|v| ≤ 2 a d1 ( f s , f s ) + L ε R d ×R d γ ≤ 2 a γ d1 ( f s , f˜s ) + L ε e−εa /2 C T, f + f˜, ε , for some constant L ε such that
ε|v|γ /2 γ γ ˜ γ /2 ˜γ ≤ L ε eε|v| + eε|v| . ˜ γ |v| + |v| + eε|v| ˜ e |v| + |v| Choosing a such that a γ = 2 log d1 ( f s , f˜s )/ε , we finally get 4 ˜ ≤ d1 ( f s , f˜s ) log d1 ( f s , f˜s ) Rs (dv, d v) ˜ |v|γ + |v| ˜ γ |v − v| ε R d ×R d + L ε C T, f + f˜, ε d1 ( f s , f˜s ). We finally obtain, setting K ε = D (Aε + L ε + 4/ε), that t ˜ ˜ ˜ d1 ( f s , f˜s ) 1 + log d1 ( f s , f˜s ) ds. d1 ( f t , f t ) ≤ d1 ( f 0 , f 0 ) + K ε C T, f + f , ε 0
Step 2. Points (ii) and (iv) are immediate consequences of point (i) and Lemma 4.2 applied with µ(x) = x (1 + | log x|). Step 3. Finally, we check point (iii). We thus assume (A1)-(A2)-(A3)(γ )-(A4)(γ )-(A5), and consider an initial condition f 0 ∈ P1 (Rd ) satisfying (2.9) for some ε0 > 0. Then we know from Lemma 4.1-(i) that any weak solution starting from f 0 satisfies (2.8) for some ε1 > 0. We thus deduce the uniqueness part from point (ii).
Well-Posedness of Some Singular Boltzmann Equations
821
Next, we approximate f 0 by a sequence of initial conditions f 0n with finite entropy satisfying (2.9) uniformly (in n), and such that d1 ( f 0 , f 0n ) tends to 0. Then, using for example the existence result of Villani [17, Theorem 1], we know that for each n, there exists a weak solution ( f tn )t≥0 to (1.1) starting from f 0n . Due to Lemma 4.1-(i), we deduce that there exists ε1 > 0 such that for all T > 0, supn C(T, f tn , ε1 ) < ∞. It is then not hard to deduce from point (i) and Lemma 4.2 that there exists ( f t )t≥0 such that for all T > 0, C(T, f, ε1 ) < ∞ and limn sup[0,T ] d1 ( f tn , f t ) = 0. An easy consequence is that ( f t )t≥0 is a weak solution to (1.1) starting from f 0 . 5. Application to Soft Potentials The application to soft potentials is easier, since we shall apply the standard Gronwall Lemma instead of that of Yudovitch. Proof of Corollary 2.4. We consider γ ∈ (−d, 0), and assume that (A1)-(A2)-(A3)(γ ). We observe at once that for α ∈ (−d, 0), and for q ∈ (d/(d + α), ∞], there exists a constant Cα,q such that for any g ∈ P1 (Rd ) ∩ L q (Rd ), any v ∈ Rd , g(v∗ ) |v − v∗ |α dv∗ ≤ g(v∗ ) |v − v∗ |α dv∗ + g(v∗ ) dv∗ Rd
|v∗ −v|<1
|v∗ −v|≥1
≤ Cα,q g L q (Rd ) + 1.
(5.1)
Step 1. We first prove point (i). Let thus p ∈ (d/(d + γ ), ∞]. We consider two solutions f, f˜ as in the statement. In order to apply Theorem 2.2, we have to check that (2.6) holds. But using (5.1), since p > d/(d + γ ), we get for t ∈ [0, T ], f t (dv) f t (dv∗ ) (1 + |v|) |v − v∗ |γ Rd Rd ≤ f t (dv) (1 + |v|) sup f t (dv∗ ) |v − v∗ |γ Rd
≤
Rd
v∈Rd
Rd
f t (dv) (1 + |v|) Cγ , p f t L p (Rd ) + 1 .
The same estimate holds for f˜, and therefore we conclude that the estimate (2.6) holds using that f and f˜ belong to L ∞ ([0, T ], P1 (Rd )) ∩ L 1 ([0, T ], L p (Rd )). Hence we deduce that (2.7) holds. Simple computations using (A3)(γ ) show that ˜ ≤ C|v − v∗ |γ |v − v|, ˜ ((|v − v∗ |) ∧ (|v˜ − v˜∗ |)) |v − v| while ((|v − v∗ |) − (|v˜ − v˜∗ |))+ |v − v∗ | ≤ C |v − v∗ |γ − |v˜ − v˜∗ |γ (|v − v∗ | ∧ |v˜ − v˜∗ | + |(|v − v∗ | − |v˜ − v˜∗ |)|) ≤ C|γ | (|v − v∗ | ∧ |v˜ − v˜∗ |)γ −1 |(|v − v∗ | − |v˜ − v˜∗ |)| (|v − v∗ | ∧ |v˜ − v˜∗ |) +C |v − v∗ |γ ∨ |v˜ − v˜∗ |γ |(|v − v∗ | − |v˜ − v˜∗ |)| ≤ C(1 + |γ |) |v − v∗ |γ + |v˜ − v˜∗ |γ (|v − v| ˜ + |v∗ − v˜∗ |).
822
N. Fournier, C. Mouhot
Inserting these inequalities in (2.7) and using a symmetry argument, we obtain that for some constant D > 0, t d1 ( f t , f˜t ) ≤ d1 ( f 0 , f˜0 ) + D ds Rs (dv, d v) ˜ Rs (dv∗ , d v˜∗ ) R d ×R d R d ×R d 0 ˜ × |v − v∗ |γ + |v˜ − v˜∗ |γ |v − v|. Recall now that Rs ∈ H( f s , f˜s ) and achieves the Wasserstein distance. Hence, Rs (dv∗ , d v˜∗ ) |v − v∗ |γ + |v˜ − v˜∗ |γ sup v,v˜
R d ×R d
≤ sup v
Rd
f t (dv∗ ) |v − v∗ |γ + sup v˜
Rd
f˜t (d v˜∗ ) |v˜ − v˜∗ |γ
≤ Cγ , p f t L p (Rd ) + Cγ , p f˜t L p (Rd ) + 2
≤ Cγ , p f t L p (Rd ) + f˜t L p (Rd ) + 2, where we used (5.1). Since Rs (dv, d v) ˜ |v R d ×R d
− v| ˜ = d1 ( f s , f˜s )
we obtain finally, choosing K p := D (Cγ , p + 2),
t d1 ( f t , f˜t ) ≤ d1 ( f 0 , f˜0 ) + K p f t L p (Rd ) + f˜t L p (Rd ) + 1 d1 ( f s , f˜s ) ds. 0
The Gronwall Lemma then allows us to conclude the proof. Step 2. We now check point (ii). We only have to prove the existence of solutions, since uniqueness follows from point (i). Using some results of Villani [17, Theorems 1 and 3], we know that for γ ∈ (−d, 0), for any f 0 ∈ P1 (Rd ) such that f 0 (v) |v|2 + | log f 0 (v)| dv < +∞, there exists a weak solution f ∈ L ∞ ([0, ∞), (1 + |v|2 ) dv) to (1.1) starting from f 0 . Then the existence result of point (ii) follows immediately from point (i) together with the following a priori estimates, which guarantee that if f 0 ∈ P1 (Rd ) ∩ L p (Rd ), then this bound propagates locally (in time): first there exists C = C(B) such that (see [6, Proposition 3.2] and its proof) for any p ∈ (d/(d + γ ), ∞], any weak solution to (1.1) satisfies d f t L p ≤ C 1 + f t 2L p (Rd ) , dt so that for 0 ≤ t < T∗ :=
1 C (π/2
− arctan || f 0 || L p ), we have
f t L p ≤ tan (arctan f 0 L p + C t). Next, we easily check, using (2.2), (2.4) and (A2) that |Sd−2 | d |v| f t (dv) ≤ κ1 f t (dv) f t (dv∗ ) |v − v∗ |1+γ . dt Rd 2 Rd Rd
(5.2)
Well-Posedness of Some Singular Boltzmann Equations
823
If 1 + γ ≥ 0, we immediately conclude, since |v − v∗ |1+γ ≤ 1 + |v| + |v∗ |, that d |Sd−2 | 1+2 |v| f t (dv) ≤ κ1 |v| f t (dv) , dt Rd 2 Rd so that for t ≥ 0, we have d−2 |v| f t (dv) ≤ eκ1 |S |t Rd
Rd
|v| f 0 (dv) + 1 .
If 1 + γ ≤ 0, we use (5.1) (with α = 1 + γ and q = p, which is valid since p > d/(d + γ ) > d/(d + α)), and we deduce that |Sd−2 | d C1+γ , p f t L p (Rd ) + 1 =: A f t L p (Rd ) + A , |v| f t (dv) ≤ κ1 dt Rd 2 so that for 0 ≤ t < T∗ , we have, recalling (5.2), t |v| f t (dv) ≤ |v| f 0 (dv) + A tan (arctan f 0 L p + C s) ds + A t. Rd
Rd
0
Step 3. We now assume additionally that γ ∈ (−1, 0), (A4)(γ ), and (A6)(ν) for some ν ∈ (−γ , 1). We consider an initial datum f 0 with finite energy and entropy (2.10), and such that for some q > q0 = γ 2 /(ν + γ ), f 0 ∈ L 1 (Rd , |v|q dv). Applying the result of Villani [17, Theorem 1], we know that there exists a weak solution ( f t )t∈[0,∞) to (1.1). To conclude the proof, it suffices to apply point (i), and to check that for any weak solution ( f t )t∈[0,∞) to (1.1) starting from f 0 , ∞ ([0, ∞), L 1 (Rd , (|v|2 + |v|q )dv)), (a) f ∈ L loc 1 ([0, ∞), L p (Rd )). (b) there exists p > p0 := d/(d + γ ) such that f ∈ L loc
Point (a) follows from a straightforward application of (2.2), using (A1)-(A2)-(A3)(γ ) and that γ ∈ (−1, 0), and concluding with the Gronwall Lemma. To check point (b), we follow the line of [6, Proposition 3.3] (see (3.2), (3.3) and (3.4) in [6]), which was relying on exploiting the entropy production and its regularization property obtained by Alexandre-Desvillettes-Villani-Wennberg [1]. Exactly as in [6, (3.2)], we get that for any α > 0, T (1 + |v|)γ −α f t (v) L d/(d−ν) (Rd ) dt ≤ C (1 + T ) (5.3) 0
for any α > 0, any T > 0, and some constant C > 0 (depending on α). Using point (a), we also know that for all T > 0, C T := sup (1 + |v|)q f t (v) L 1 (Rd ) < ∞. [0,T ]
(5.4)
By interpolation between estimates (5.3) and (5.4), we see that for all T > 0, for another constant C T depending on T , T f L p (Rd ) dt ≤ C T (5.5) 0
824
N. Fournier, C. Mouhot
for any 1 < p < d/(d − ν) as soon as, for instance p−1 q > (α − γ ) . (5.6) 1 − p(d − ν)/d Since p0 = d/(d + γ ) < d/(d − ν) (because γ > −ν) and since by assumption, γ2 p0 − 1 = , 1 − p0 (d − ν)/d γ +ν we clearly have (5.6) when choosing α > 0 small enough and p > p0 close enough. This concludes the proof of point (b). q > q0 = (−γ )
References 1. Alexandre, R., Desvillettes, L., Villani, C., Wennberg, B.: Entropy dissipation and long-range interactions. Arch. Rat. Mech. Anal. 152(4), 327–355 (2000) 2. Bobylev, A.V.: Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88(5-6), 1183–1214 (1997) 3. Bobylev, A.V., Gamba, I.M., Panferov, V.A.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116(5-6), 1651–1682 (2004) 4. Chemin, J.Y.: Fluides parfaits incompressibles. Astérisque, no. 230 (1995) 5. Desvillettes, L.: Boltzmann’s Kernel and the spatially homogeneous Boltzmann equation. Rivista di Matematica dell’Universita di Parma, 6(4) 1–22 (special issue) (2001) 6. Desvillettes, L., Mouhot, C.: Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. Arch. Rat. Mech. Anal. (2009, to appear). http://arXiv. org/abs/math.AP/0606307 7. Fournier, N.: Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff. J. Stat. Phys. 125(4), 927–946 (2006) 8. Fournier, N., Méléard, S.: A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comp. 71(238), 583–604 (2002) 9. Horowitz, J., Karandikar, R.L.: Martingale problems associated with the Boltzmann equation. Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989), Progr. Probab., 18, Boston, MA: Birkh¨auser Boston, 1990, pp. 75–122 10. Lu, X., Mouhot, C.: About measures solutions of the spatially homogeneous Boltzmann equation. Work in progress 11. Lu, X., Wennberg, B.: Solutions with increasing energy for the spatially homogeneous Boltzmann equation. Nonlinear Anal. Real World Appl. 3(2), 243–258 (2002) 12. Mischler, S., Mouhot, C., Rodriguez Ricard, M.: Cooling process for inelastic Boltzmann equations for hard spheres. I. The Cauchy problem. J. Stat. Phys. 124(2-4), 655–702 (2006) 13. Mischler, S., Wennberg, B.: On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(4), 467–501 (1999) 14. Mouhot, C.: Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials. Commun. Math. Phys. 261(3), 629–672 (2006) 15. Tanaka, H.: Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrscheinlichkeitstheorie Und Verw. Gebiete 46(1), 67–105 (1978–1979) 16. Toscani, G., Villani, C.: Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94(3-4), 619–637 (1999) 17. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rat. Mech. Anal. 143(3), 273–307 (1998) 18. Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, Amsterdam: North-Holland, 2002, pp. 71–305 19. Villani, C.: Topics in optimal transportation. Graduate Studies in Mathematics 58. Providence, RI: Amer. Math. Soc., 2003 20. Wennberg, B.: An example of nonuniqueness for solutions to the homogeneous Boltzmann equation. J. Statist. Phys. 95(1-2), 469–477 (1999) 21. Yudovich, V.: Non stationary flow of an ideal incompressible liquid. Zhurn. Vych. Mat 3, 1032–1066, (1963) (in Russian) Communicated by P. Constantin
Commun. Math. Phys. 289, 825–840 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0808-2
Communications in
Mathematical Physics
TYZ Expansion for the Kepler Manifold Todor Gramchev, Andrea Loi Dipartimento di Matematica e Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy. E-mail: [email protected]; [email protected] Received: 8 November 2007 / Accepted: 7 February 2009 Published online: 29 April 2009 – © Springer-Verlag 2009
Abstract: The main goal of the paper is to address the issue of the existence of Kempf’s distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic expansion for the Kepler manifold - an important example of non-compact manifold. Motivated by the recent results for compact manifolds we construct Kempf’s distortion function and derive a precise TYZ asymptotic expansion for the Kepler manifold. We get an exact formula: finite asymptotic expansion of n −1 terms and exponentially small error terms uniformly with respect to the discrete quantization parameter m → ∞ ( = m −1 → 0 standing for Planck’s constant and |x| → ∞, x ∈ Cn ). Moreover, the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the Kepler manifold. We show that our estimates are sharp by analyzing the nonharmonic behaviour of Tm for m → +∞. The arguments of the proofs combine geometrical methods, quantization tools and functional analytic techniques for investigating asymptotic expansions in the framework of analytic-Gevrey spaces. 1. Introduction and Statements of the Main Results Let g be a Kähler metric on an n–dimensional complex manifold M. Assume that g is polarized with respect to a holomorphic line bundle L over M, i.e. c1 (L) = [ω], where ω is the Kähler form associated to g and c1 (L) denotes the first Chern class of L. Let m ≥ 1 be a non-negative integer and let h m be a Hermitian metric on L m = L ⊗m such that its Ricci curvature Ric(h m ) = mω. Here Ric(h m ) is the two–form on M whose local expression is given by i Ric(h m ) = − ∂ ∂¯ log h m (σ (x), σ (x)), 2
(1.1)
The first author was supported in part by the project PRIN (Cofin) n. 2006019457 with M.I.U.R., Italy. The second author was supported in part by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”.
826
T. Gramchev, A. Loi
for a trivializing holomorphic section σ : U → L m \{0}. In the quantum mechanics terminology L m is called the quantum line bundle, the pair (L m , h m ) is called a geometric quantization of the Kähler manifold (M, mω) and = m −1 plays the role of Planck’s constant (see e.g. [2]). Consider the separable complex Hilbert space Hm consisting of global holomorphic sections s of L m such that ωn h m (s(x), s(x)) s, sm = < ∞. n! M Let x ∈ M and let q ∈ L m \{0} be a fixed point of the fiber over x. If one evaluates s ∈ Hm at x, one gets a multiple δq (s) of q, i.e. s(x) = δq (s)q. The map δq : Hm → C is a continuous linear functional [9]. Hence from Riesz’s theorem, there exists a unique eqm ∈ H such that δq (s) = s, eqm m , ∀s ∈ Hm , i.e. s(x) = s, eqm m q.
(1.2)
m = c−1 em , ∀c ∈ C∗ . The holomorphic section em ∈ H is called the It follows that ecq m q q coherent state relative to the point q. Thus, one can define a smooth function on M,
Tm (x) = h m (q, q) eqm 2 , eqm 2 = eqm , eqm ,
(1.3)
where q ∈ L m \{0} is any point on the fiber of x. If s j , j = 0, . . . dm , (dm + 1 = dim Hm ≤ ∞) form an orthonormal basis for (Hm , ·, ·m ) then one can easily verify that Tm (x) =
dm
h m (s j (x), s j (x)).
(1.4)
j=0
Notice that when M is compact Hm = H 0 (L m ), where H 0 (L m ) denotes the space of global holomorphic sections of L m . Hence in this case dm < ∞ and (1.4) is a finite sum. The function Tm has appeared in the literature under different names. The earliest one was probably the η-function of J. Rawnsley [42] (later renamed to function in [9]), defined for arbitrary Kähler manifolds, followed by the distortion function of Kempf [27] and Ji [26], for the special case of Abelian varieties and of Zhang [48] for complex projective varieties. The metrics for which Tm is constant were called critical in [48] and balanced in [17] (see also [3,32 and 33]). If Tm are constants for all sufficiently large m then the geometric quantization (L m , h m ) associated to the Kähler manifold (M, g) is called regular. Regular quantization plays a prominent role in the theory of quantization by deformation of Kähler manifolds developed in [9]. Fix m ≥ 1. Under the hypothesis that for each point x ∈ M there exists s ∈ Hm non-vanishing at x, one can give a geometric interpretation of Tm as follows. Consider the holomorphic map of M into the complex projective space CP dm : ϕm : M → CP dm : x → [s0 (x) : · · · : sdm (x)].
(1.5)
i ϕm∗ (ω F S ) = mω + ∂ ∂¯ log Tm , 2
(1.6)
One can prove that
TYZ Expansion for the Kepler Manifold
827
where ω F S is the Fubini–Study form on CP dm , namely the form which in homogeneous m coordinates [Z 0 , . . . , Z dm ] reads as ω F S = 2i ∂ ∂¯ log dj=0 |Z j |2 . Clearly (1.6) leads to ϕm∗ (ω F S ) i ¯ −ω = ∂ ∂ log Tm , m 2m
(1.7)
therefore the term Em (x) :=
i ¯ ∂ ∂ log Tm , 2m
(1.8)
turns out to play a role of the “error” of the approximation of ω (resp. g) by ϕm∗ (ω F S )/m (resp. ϕm∗ (g F S )/m). Observe that by (1.6), if there exists m such that mg is a balanced metric, or more generally if Tm is harmonic, then Em (x) is identically zero and hence mg is projectively induced via the coherent states map ϕm (see [2,17 and 18] for more details on the link between balanced metrics and quantization of Kähler manifolds). Recall that a Kähler metric g on a complex manifold M is projectively induced if there exists a Kähler (i.e. a holomorphic and isometric) immersion ψ : M → CP N , N ≤ ∞, such that ψ ∗ (g F S ) = g. Projectively induced Kähler metrics enjoy important geometrical properties and were extensively studied in [8] (see also the beginning of Sect. 4 below). Not all Kähler metrics are balanced or projectively induced. Nevertheless, when M is compact, G. Tian [46] and W. Ruan [43] solved a conjecture posed by Yau by proving ϕ ∗ (ω ) that the sequence of metrics m m F S C ∞ -converges to ω. In other words, any polarized metric on compact complex manifold is the C ∞ -limit of (normalized) projectively induced Kähler metrics. Zelditch [47] generalized the Tian–Ruan theorem by proving a complete asymptotic expansion in the C ∞ category, namely Tm (x) ∼
∞
a j (x)m n− j ,
(1.9)
j=0
where a j , j = 0, 1, . . ., are smooth coefficients with a0 (x) = 1, and for any nonnegative integers r, k the following estimates hold: ||Tm (x) −
k
a j (x)m n− j ||C r ≤ Ck,r m n−k−1 ,
(1.10)
j=0
where Ck,r are constant depending on k, r and on the Kähler form ω and || · ||C r denotes the C r norm in local coordinates. (Notice that similar asymptotic expansion were used in [9–12,35 and 36]) to construct the star product on Kähler manifolds.) Later on, Lu [34], by means of Tian’s peak section method, proved that each of the coefficients a j (x) in (1.9) is a polynomial of the curvature and its covariant derivatives at x of the metric g. Such polynomials can be found by finitely many algebraic operations. Furthermore a1 (x) = 21 ρ, where ρ is the scalar curvature of the polarized metric g (see also [30 and 31] for the computations of the coefficients a j ’s through Calabi’s diastasis function). The expansion (1.9) is called the TYZ (Tian–Yau–Zelditch) expansion and is a key ingredient in the investigations of balanced metrics [17]. The aim of the present paper is to address the problem of TYZ expansions for noncompact manifolds (see also the recent paper [22]). Our motivation is two-folded.
828
T. Gramchev, A. Loi
First, we are inspired by the investigations of geometric quantization problems, in particular, by the recent works of M. Engliš [20,21,23] (see also the fundamental paper of L. Boutet de Monvel and J. Sjöstrand [7]), where analytical tools have been applied in order to extend Berezin’s quantization method (cf. [4,5]) to non-homogeneous complex domains on Cn (see also [37–39] and the references therein for the study of coherent states and relations to geometric quantization). Secondly, it is a purely geometrical question in the framework of the quantization theory of its own interest. We choose as a noncompact manifold the Kepler manifold (X, ω), namely the cotangent bundle of the n-dimensional sphere minus its zero section endowed with the standard symplectic form ω (see [45 and 41]). We summarize the main novelties of our work. First, we compute explicitly Kempf’s distortion function Tm (x) for the Kepler manifold (X, ω). Secondly, based on this computation we find an analogue of the results of S. Zelditch and Z. Lu for (X, ω). More precisely, building upon the explicit representation of Tm as an action of “singular derivatives” and using precise estimates of nonlinear compositions in functional spaces, we show that the TYZ expansion for the Kepler manifold has two remarkable features: • the TYZ expansion is finite. More precisely, it consists of n − 1 terms, (n − 2)(n − 1) n−1 2ak n−k m + m + Rm (|x|), 2|x| |x|k n−2
Tm (x) = m n +
k=2
where ak , k ≥ 2 can be computed explicitly by recursive formulas, • the remainder term has an exponentially small decay O(e−c|x|m ) as m → ∞ uniformly with respect to |x| ≥ δ > 0. We stress that these constructions on non compact manifolds might be of some interest from the numerical point of view. Our approach should be compared with the recent quantization numerical results of S. Donaldson [19], obtained by projectively induced metrics on compact manifolds. We stress that the approach for other noncompact manifolds in [7,20,21 and 23]) provides good approximations of the Bergman kernel via Fourier integral operators. For the Kepler manifold we rely on an explicit computation. We also demonstrate uniform analytic–Gevrey regularity estimates for Tm keeping the exponential decay for m → ∞, |x| → ∞, which resemble the simultaneous analytic–Gevrey estimates and exponential decay for solitary waves via the use of global analytic–Gevrey pseudodifferential operators in Rn (cf. [13,14] and the references therein). We mention also the recent works [15,16], where exponentially small error terms in the framework of Gevrey spaces appear in the study of oscillatory integrals with non–Morse functions and divergent normal forms. Observe that as for the compact case our expansion shows that g (the metric g associated to the Kepler manifold (M, ω)) is the C ∞ -limit of (suitable normalized) projectively induced Kähler metrics, namely limm→∞ m1 ϕm∗ (g F S ) = g, where ϕm : X → CP ∞ is the coherent states map. A geometric construction is proposed showing that our estimates are sharp. Indeed, we show that g is not projectively induced, i.e. there cannot exist any Kähler immersion of (X, ω) into a finite or infinite dimensional complex projective space. The arguments use Calabi’s tools which provide necessary and sufficient conditions for a Kähler metric to be projectively induced. The paper is organized as follows. We propose an explicit construction of Kempf’s distortion function Tm for the Kepler manifold (X, ω) in Sect. 2. In Sect. 3 we derive an
TYZ Expansion for the Kepler Manifold
829
exact TYZ asymptotic expansion when m → ∞. In Sect. 4 we prove (see Theorem 4.4) that our estimate is sharp. 2. Kempf’s Distortion Function for the Kepler Manifold The (regularized) Kepler manifold [45] is (may be identified with) the 2n-dimensional symplectic manifold (X, ω), where X = T ∗ S n \0 is the cotangent bundle of the n-dimensional sphere minus its zero section endowed with the standard symplectic form ω. This may further be identified with X = {(e, x) ∈ Rn+1 × Rn+1 | e · e = 1, x · e = 0, x = 0}. Here · denotes the standard scalar product on Rn+1 . J. Souriau [45] showed that the Kepler manifold admits a natural complex structure. He proved, by introducing z = x |x|e + ix = |x|(e + is) ∈ Cn+1 , s = |x| ∈ S n , that X is diffeomorphic to the isotropic cone 2 = 0, z = 0} ⊂ Cn+1 , C = {z ∈ Cn+1 | z · z = z 12 + · · · + z n+1
and hence X inherits the complex structure of C via this diffeomorphism. Later on J. Rawnsley [42] observed that the symplectic form ω is indeed a Kähler form with respect to this complex structure and can be written (up to a factor) as ω=
i ¯ ∂ ∂|x|. 2
(2.11)
We denote by g the Kähler metric metric induced by ω. Remark 2.1. Although the Kepler manifold is not complete (cf. Remark 4.1) and hence not homogeneous, its isometry group is very large, being equal to S 1 × O(n + 1) (see the appendix in [41]). The radial symmetries of the metric has the great advantage that its diastasis function and Kempf’s distortion function can be computed explicitly as function of the polar radius in Cn . Another interesting Kähler metric gG on T ∗ S n using a Grauert tube function ρ has been studied by Guillemin and Stenzel [24,25], Lempert and Szöke [28,29] and Patrizio and √ Wong [40]). The metric gG is uniquely determined by a Kähler potential ρnsuch that ρ satisfies the Monge–Ampère equation and the metric induced by gG on S (viewed as the zero section of T ∗ S n ) equals the round metric on S n . We can show, taking advantage of the conic structure of the Kepler manifold, the singularity at the zero section and the radial symmetry of g, that there is no smooth map f : N G \S n → N K \S n such that f ∗ (gG ) = g, where N K (resp. N G ) is an arbitrary open neighborhood of S n ⊂ T ∗ S n . Moreover, since ω is exact, it is trivially integral and hence there exists a holomorphic line bundle L over X such that c1 (L) = [ω]. For n ≥ 3, X is simply-connected, so L m is holomorphically trivial (L m = X × C) and we can identify H 0 (L m ) with the set of holomorphic functions on X . Furthermore, we can define a Hermitian metric h m on L m = X × C by h m (σ (z), σ (z)) = e−m|x| ,
(2.12)
where σ : X → X × C, is the global holomorphic section such that σ (z) = (z, 1). It follows by (1.1) above that the pair (L m , h m ) is a geometric quantization of the Kepler
830
T. Gramchev, A. Loi
manifold (X, ω). Then the Hilbert space Hm consists of the set of holomorphic functions f of X such that ωn (z) i ¯ n | f (z)|2 e−m|x| dµ(z) < ∞, dµ(z) = .
f 2m := = ( ∂ ∂|x|) n! 2 X Notice that in this case Tm (z) = e−m|x| K (m) (z, z), where K (m) (z, z) is the reproducing Kernel for the Hilbert space Hm . At p. 412 in [42] J. Rawnsley explicitly computed K (z, z) = K (1) (z, z) (the reproducing kernel for H = H1 ) and hence the corresponding Kempf’s distortion function, which in our notations is read: T1 (z) = e−|x| K(z, z) = 2n−1 e−|x|
∞ ( j + n − 2)! |x|2 j , (2 j + n − 2)! j!
2|x|2 = z · z¯ . (2.13)
j=0
Next, we compute Kempf’s distortion functions Tm (z), m ∈ Z+ . The change of variable mz = w yields 2
f m = | f (w/m)|2 e−| Im w| m −n dµ(w). X
Consequently, the operator T : T f (w) := m −n/2 f (w/m)
(2.14)
is a unitary isomorphism from Hm onto H. Denote by K(m) (w, z) ≡ Km z (w) the reproducing kernel of Hm , m ≥ 1 (and write K(w, z) ≡ K z (w) if m = 1). We have, on the one hand, (m) f (z) = f, K(m) z m = T f, T Kz
for any f ∈ Hm , while, on the other hand, n
n
f (z) = m 2 T f (mz) = T f, m 2 Kmz . n/2 T −1 K (w) = m n K (mw), i.e., = m 2 Kmz , and K(m) Thus T K(m) mz mz z z (w) = m n
K(m) (w, z) = m n K(mw, mz). Substituting this into Rawnsley’s formula (2.13), we thus get Tm (z) = e−m|x| K(m) (z, z) = 2n−1 m n e−m|x|
∞ ( j + n − 2)! (m|x|)2 j . (2 j + n − 2)! j!
(2.15)
j=0
Remark 2.1. From (2.15) one sees that Tm (x) = m n T1 (mx). In the compact case the relationship between Tm and T1 is in general unknown. This is also true in the Bargman Fock case. In fact in [44] one can find a general result which explains this kind of relation in the case of Bergman metrics.
TYZ Expansion for the Kepler Manifold
831
Notice that the growth of Tm (z) as m → ∞ is not clear from representation (2.15). The following proposition gives us important analytic information about Tm as m → ∞. Proposition 2.2. Kempf’s distortion function for the Kepler manifold can be written in the following two forms: Tm (z) = 2m n e−m|x|
∞
(1 + τ j )
j=0
(m|x|)2 j , (2 j)!
(2.16)
with τj = 1 −
( j + 1) · · · ( j + n − 2) −→ 0 ( j + 1/2) · · · ( j + (n − 2)/2)
for j → ∞,
and Tm (z) = 2m n e−ξm
1 ∂ ξm ∂ξm
n−2
ξmn−2
eξm + (−1)n−2 e−ξm + Q(ξm ) 2
, (2.17)
where ξm = m|x|, Q(ξm ) is a polynomial of degree ≤ n − 4 in the variable ξm . Proof. From (2.15) one gets Tm (z) = e
−m|x|
(m)
K
(z, z) = 2
n−1
n −m|x|
m e
∞ ( j + n − 2)! (m|x|)2 j (2 j + n − 2)! j! j=0
∞ ( j + n − 2)!(2 j)! (m|x|)2 j n−1 n −m|x| =2 m e j!(2 j + n − 2)! (2 j)! j=0
= 2m n e−m|x|
∞ j=0
= 2m n e−m|x|
∞
(m|x|)2 j ( j + 1) · · · ( j + n − 2) ( j + 1/2) · · · ( j + (n − 2)/2) (2 j)! (1 + τ j )
j=0
(m|x|)2 j . (2 j)!
(2.18)
In order to prove (2.17) set 2 ym = m|x| = ξm , ym , ξm ∈ R\{0}.
Then, since
∂ ∂ym
=
1 ∂ 2ξm ∂ξm
one gets
Tm (z) = 2n−1 m n e−ξm
∞ j ( j + n − 2)! ym (2 j + n − 2)! j! j=0
=2
n−1
n −ξm
m e
= 2m n e−ξm
∂ ∂ym
1 ∂ ξm ∂ξm
n−2 ∞ j=0
n−2
⎡
j+n−2
ym (2 j + n − 2)!
⎣ξmn−2
∞ j=0
⎤ 2 j+n−2 ξm ⎦. (2 j + n − 2)!
832
T. Gramchev, A. Loi
If n is even then Tm (z) = 2m n e−ξm
1 ∂ ξm ∂ξm
n−2 [ξmn−2 (cosh ξm − P(ξm ))],
with n−4
2j 2 ξm , P(ξm ) = (2 j)! j=0
while n odd leads to Tm (z) = 2m n e−ξm
1 ∂ ξm ∂ξm
n−2 [ξmn−2 (sinh ξm − R(ξm ))],
where n−5
2 j+1 2 ξm . R(ξm ) = (2 j + 1)! j=0
Hence we get (2.17).
3. TYZ Expansion for the Kepler Manifold The key ingredient to find the TYZ expansion of Tm is (2.17). Clearly we have Tm (z) = 2m n F(m|x|), where F(y) = e−y
1 d y dy
y n−2 e + (−1)n−2 e−y y n−2 + Q(y) , y ∈ R. 2
(3.19)
(3.20)
The explicit representation (3.19)–(3.20) of Tm (z) for the Kepler manifold has a remarkable feature, namely, it is defined by a generating function F(y) depending on one variable. Note that in fact Tm (z) is independent of the base variables e ∈ S n . We show the first main result for the TYZ expansion for the Kepler manifold. Theorem 3.1. Let F satisfy (3.20). Then the following representation holds: F(y) =
n−2 bj j=0
yj
+ (y) + (y),
(3.21)
where (y) = e−2y
n−2 pj , yj
(3.22)
j=0
(y) = e−y
n−2 rj , yj j=0
(3.23)
TYZ Expansion for the Kepler Manifold
833
and the constants a j , p j , r j are written explicitly. The functions (y), (y) and therefore, F(y) as well, can be extended to meromorphic functions in C. In particular, by (3.19) and (3.21) we get Tm (z) =
n−2
a j (x)m n− j + 2m n (m|x|) + 2m n (m|x|), m ∈ N,
(3.24)
j=0
where a j (x) =
2b j , |x| j
j = 0, 1, . . . , n − 2,
(3.25)
and a0 (x) = 1, (n − 2)(n − 1) a1 (x) = . 2|x|
(3.26) (3.27)
Moreover, there exists an absolute constant C0 > 0 such that for every δ ∈]0, 1], sup |Dxα m (x)|, ≤ C0α+1
|x|≥δ
α! −mδ/2 e δα
(3.28)
for all m ∈ N, where = , . Therefore, we have the following estimates ⎛ ⎞ n−2 α! |Dxα ⎝Tm − a j (x)m n− j ⎠ | ≤ C0α+1 α e−mδ/2 δ
(3.29)
j=0
for all |x| ≥ δ, α ∈ Zn+ . Proof. We recall the well known Faà di Bruno type formula for the derivative of g ◦ ϕ, namely, for a given α ∈ N we have Dtα (g(ϕ(t))) = Dyα (g(ϕ(y)))|y=t = =
α g ( j) (ϕ(t)) j=1 α j=1
j! g ( j) (ϕ(t)) j!
Dxα (ϕ(x) − ϕ(t)) j |x=t α1 +···+α j =α α1 ≥1,··· ,α j ≥1
(3.30)
α! ϕ (α1 ) (t) . . . ϕ (α j ) (t), α1 ! . . . α j !
(3.31)
where ϕ (k) (t) stands for Dtk ϕ(t). y −1 Dy into Dt via the singular change of the variable y = y(t) = √ Next, we straighten 2 2t, t = t (y) = y /2. Therefore, setting √ G(t) = F( 2t), t > 0,
F(y) = G
y2 2
, y > 0,
(3.32)
834
T. Gramchev, A. Loi
we get by (3.20), F(y) = G(t) = e
√ − 2t
d dt
n−2
(2t)(n−2)/2
√
e
2t
√
+ (−1)n−2 e− 2
2t
√ + Q( 2t) . (3.33)
The next assertion is instrumental in the proof. Lemma 3.2. Let N ∈ N, c ∈ R, and r > 0. Then 1 d N r cy ψ Nc,r (y) := e−y (y e ) y dy √ d N √ √ c,r c,r − 2t ((2t)r/2 ec 2t ) = ψ N ( 2t) =: ϕ N (t) = e dt
(3.34)
has the following representation: −(1−c) ϕ c,r N (t) = e
√
2t
(2t)(r −N )/2
N s=0
κs , (2t)s/2
(3.35)
i.e. ψ Nc,r (y) = e−(1−c)y yr −N
N κs s=0
ys
,
(3.36)
where
⎛ ⎞ N −−1 N r c N −s N ⎝ − q ⎠ 2 N −r/2 (−1)+s−N κs = (N − s)! 2 =N −s
×
1 +···+ N −s =
q=0
N −s −1 1 −1 1 1 ! − q1 . . . − q N −s 1 ! . . . N −s ! q 2 2 q N −s
1
1 ≥1,··· , N −s ≥1
(3.37) for s = 0, . . . , N − 1 and κ N = 2 N −r/2
N −1 q=0
r ( − q). 2
(3.38)
Proof. By Faà di Bruno type formula (3.31) we derive N N √ √ d N r/2 c 2t DtN − (t r/2 )Dt (ec 2t ) r,c (t) = (t e ) = N dt =0 ⎛ ⎞ N −−1 N √ √ c j 2 j/2 r N ⎝ = DtN (t r/2 )ec 2t + ( − q)⎠ t r/2−N + ec 2t 2 j! ×
1 +···+ j =
1 ≥1,..., j ≥1
=1
q=0
! D 1 (t 1/2 ) · · · Dt j (t 1/2 ) 1 ! · · · j ! t
j=1
(3.39)
TYZ Expansion for the Kepler Manifold
with the convention 1 2
µ
Dt (t 1/2 ) =
−1
q=0 ...
835
= 1. Since
1 1 (2µ − 3)!! 1/2−µ t − 1 ... − µ + 1 t 1/2−µ = (−1)µ−1 2 2 2µ (3.40)
for all positive integers µ, with (−1)!! := 1, (2µ − 3)!! := 1 · · · (2µ − 3) if µ ≥ 2, combining (3.39) and (3.40), we obtain 1 +···+ j =
! t j/2− Dt1 (t 1/2 ) · · · Dt j (t 1/2 ) = (−1)− j , j 1 ! · · · j ! 2
(3.41)
1 ≥1,..., j ≥1
with
, j :=
! (21 − 3)!! · · · (2 j − 3)!!. 1 ! · · · j !
1 +...+ j =
(3.42)
1 ≥1,..., j ≥1
We note that , = !, −1 !. ,−1 = 2
(3.43) (3.44)
Therefore, by (3.39)–(3.41), r,c N (t)
=
N −1 q=0
r 2
−q
t r/2−N ec
√
2t
⎛ ⎞ N −−1 N r N ⎝ − q ⎠ 2 N −−r/2 (2t)r/2−N + + 2 =1
q=0
cj
×
j=1
j!
(−1)− j , j (2t) j/2−
√ r/2−N /2 c 2t
= (2t)
e
2 N −r/2
N √ r/2−N /2 c 2t
+(2t)
e
j=1
⎛ ×⎝
N −−1 q=0
⎞
N −1 r
q=0 2 (2t) N /2
−q
N N cj j!(2t)(N − j)/2 = j
r ( − q)⎠ 2 N −−r/2 (−1)− j , j 2
836
T. Gramchev, A. Loi
2 N −r/2
√ r/2−N /2 c 2t
= (2t)
e
+(2t)r/2−N /2 ec ⎛ ×⎝
√
s=0 N −−1 q=0
= (2t)r/2−N /2 ec
q=0 2 (2t) N /2
N −1
2t
⎞
N −1 r
−q
N c N −s N (N − s)!(2t)s/2 =N −s
r − q ⎠ 2 N −−r/2 (−1)+s−N ,N −s 2
√
2t
N s=0
κs , (2t)s/2
(3.45)
where κs is defined by ⎛ ⎞ N −−1 N c N −s N ⎝ r κs := ( − q)⎠ 2 N −−r/2 (−1)+s−N ,N −s . (N − s)! 2 =N −s
q=0
In view of the definition of , j with the convention ,0 = 1, it is equivalent to (3.37), (3.38). This ends the proof of the lemma. We conclude the proof of the theorem by applying the previous lemma for z = m|x| and obtain the value of as = κs /2 by setting c = 1, r = N = (n − 2); ps = (−1)n−2 κs /2 by setting c = −1, r = N = n − 2, and rs = qn−2−s
n−2
(2n − 2 − s − 2),
=1
provided Q(z) =
n−3
qjz j.
j=0
Remark 3.2. In view of (3.24), we have (n − 2)(n − 1) n−1 2bk n−k m + m + Rm (|x|), 2|x| |x|k n−2
Tm (z) = m n +
(3.46)
k=2
with Rm (x) being exponentially small e−c|x|m away from the origin x = 0. Remark 3.3. The novelty of the theorem above is twofold. First, our TYZ type expansion is finite, i.e., a j = 0 for j ≥ n − 1 (compare (1.10)). Secondly, the remainder is exponentially small. Moreover, the coefficients a j can be computed explicitly. We also mention that our approach allows us to investigate the asymptotic behaviour of the obstruction term Em (z) =
n+1 j,=1
j,
Em (z)dz j ∧ d z¯
TYZ Expansion for the Kepler Manifold
837
in (1.8) and to prove that the coefficients decay polynomially of the type m −2 . More precisely, for some C > 0, they behave like C (1 + o(1)) m 2 |z|3
m → ∞,
(3.47)
uniformly for |z| away from the origin in Cn . In fact, we are able to show an abstract theorem for the asymptotic behaviour of obstruction terms similar to (1.8) on conic manifolds of Kepler type. The proof is based on a suitable choice of global singular coordinates parametrizing the Kepler manifold and the use of implicit function theorem arguments. Consequently, by (1.6), the metric g associated to ω can be approximated by suitable normalized projectively induced Kähler metrics with an error of the type m −2 , m → ∞. The details are to be found in another work. 4. Proof that our Estimate is Sharp As a consequence of Theorem 3.1 the Kähler form g on the Kepler manifold X is the C ∞ -limit of suitable normalized projectively induced Kähler metrics, namely 1 ∗ ϕ (g F S ) = g, m→∞ m m lim
where ϕm : X → CP ∞ is the coherent states map. In this section we show that g is not projectively induced (via any map) and then that our estimate in Theorem 3.1 is sharp. We need to recall briefly some results about Calabi’s diastasis function referring the reader to [8] for details and further results. Let M be a complex manifold endowed with a real analytic Kähler metric g. Then, in g a neighborhood of every point p ∈ M, one can introduce a special Kähler potential D p (diastasis, cf. [8]) for the Kähler form ω associated to g. Recall that a Kähler potential ¯ is an analytic function defined in a neighborhood of a point p such that ω = 2i ∂∂ . A Kähler potential is not unique: it is defined up to addition to the real part of a holomorphic function. By duplicating the variables z and z¯ , a potential can be complex ˜ defined in a neighborhood U of the diagonal conanalytically continued to a function ¯ ¯ taining ( p, p) ¯ ∈ M × M (here M denotes the manifold conjugated to M). The diastasis g function is the Kähler potential D p around p defined by ˜ ˜ p, p) ˜ p, q) ˜ D gp (q) = (q, q) ¯ + ( ¯ − ( ¯ − (q, p). ¯ g
Observe that the diastasis does not depend on the potential chosen, D p (q) is symg metric in p and q and D p ( p) = 0. The diastasis function is the key tool for studying the Kähler immersions of a Kähler manifold into another Kähler manifold as expressed by the following lemma. Lemma 4.1. (Calabi [8]) Let (M, g) be a Kähler manifold which admits a Kähler immersion ϕ : (M, g) → (S, G) into a real analytic Kähler manifold (S, G). Then g is real g G analytic. Let D p : U → R and Dϕ( p) : V → R be the diastasis functions of (M, g) and G ) = D g on ϕ −1 (V ) ∩ U . (S, G) around p and ϕ( p), respectively. Then ϕ −1 (Dϕ( p p)
838
T. Gramchev, A. Loi
When (S, G) is the N -dimensional complex projective space S = CP N equipped with the Fubini–Study metric G = g F S , one can show that for all p ∈ CP N the diasg tasis function D pF S around p is globally defined except in the cut locus H p of p where gF S
it blows up. Moreover, e−D p is globally defined (and smooth) on CP N (see [8] for details). Then, by Lemma 4.1 one immediately gets the following: Lemma 4.2. Let g be a projectively induced Kähler metric on a complex manifold M. g Then, e−D p is globally defined on all M. Corollary 4.3. Let g∗ be the Kähler metric on C∗ whose associated Kähler form is given ¯ by ω∗ = 2i ∂ ∂|η|, η = x + iy. Then g∗ is not projectively induced. Proof. Fix any point α ∈ C∗ . A globally defined Kähler potential for the Kähler metric g∗ around α is given by (η) = |η| and Calabi’s diastasis function around α reads: Dαg∗ : U → R, η → |η| + |α| − ηα¯ − ηα, ¯ where U ⊂ C∗ is a suitable simply-connected open subset of C∗ around α (as a maximal g domain of definition of Dα∗ one can take U = C∗ \L, where L is any half-line starting g 2 from the origin of C = R such that α ∈ / L). Neither the function Dα∗ nor the function g∗ e−Dα can be extended to all C∗ . Hence we are done by Lemma 4.2. We are now in the position to prove that our estimate is sharp. Theorem 4.4. Let g be the Kähler metric on the Kepler manifold X whose associated Kähler form is given by (2.11). Then g is not projectively induced. Proof. First observe that the map j : (C∗ , g∗ ) → (X, g)
(4.48) j ∗ (g)
defined by j (η) = (η, iη, 0, . . . , 0) is a Kähler immersion satisfying = g∗ , with g∗ as in Corollary 4.3. Assume by contradiction that g is projectively induced, namely there exists N ≤ ∞ and a Kähler immersion ϕ : (X, g) → (CP N , g F S ). Then the map ϕ ◦ j : (C∗ , g∗ ) → (CP N , g F S ) would be a Kähler immersion contradicting Corollary 4.3. Remark 4.1. From the proof of the previous theorem one can easily see that the metric g of the Kepler manifold X is not complete. Indeed, if it were complete the same would 2 2 √ +dy on C∗ = R2 \{0} since the map (4.48) is totally be true for the metric g∗ = 41 dx 2 2 x +y
geodesic. On the other hand the length of the geodesic segment {(t, 0) |0 < t ≤ 1 } from 1 the origin to (1, 0) is finite since it is given by: 14 0 √1x dx = 18 < ∞ and thus g∗ is not complete. Acknowledgements. The authors are indebted to the anonymous referees for their critical remarks and suggestions which led to improvements of the revised version of the paper. The authors thank Simon Donaldson for the encouragement to study TYZ type asymptotic expansions for noncompact manifolds and Miroslav Engliš for useful discussions on reproducing kernels of complex domains and for pointing out the idea of using an isomorphism (2.14) for computing Tm from T1 . Thanks are also due to Antonio J. Di Scala, Ivailo Mladenov, Vassil Tsanov and Cornelis Van Der Mee for useful discussions and for providing references related to the subject of the paper.
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References 1. Ali, S.T., Engliš, M.: Quantization methods: a guide for physicists and analysts. Rev. Math. Phys. 17(4), 391–490 (2005) 2. Arezzo, C., Loi, A.: Quantization of Kähler manifolds and the asymptotic expansion of Tian–Yau– Zelditch. J. Geom. Phys. 47, 87–99 (2003) 3. Arezzo, C., Loi, A.: Moment maps, scalar curvature and quantization of Kähler manifolds. Commun. Math. Phys. 246, 543–549 (2004) 4. Berezin, F.A.: Quantization. Izv. Akad. Nauk SSSR Ser. Mat. 38, 1116–1175 (1974) (Russian) 5. Berezin, F.A.: Quantization in complex symmetric spaces. Izv. Akad. Nauk SSSR Ser. Mat. 39(2), 363– 402, 472 (1975) (Russian) 6. Berezin, F.A., Shubin, M.A.: The Schrödinger Equation. Translated from the 1983 Russian edition by Yu. Rajabov, D.A. Le˘ıtes, N.A. Sakharova and revised by Shubin. With contributions by G. L. Litvinov and Le˘ıtes. Mathematics and its Applications (Soviet Series), 66. Dordrecht: Kluwer Academic Publishers Group, 1991 7. Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szeg˝o, Journées: Équations aux Dérivées Partielles de Rennes (1975), Astérisque 34-35, 123–164 (1976) 8. Calabi, E.: Isometric imbeddings of complex manifolds. Ann. of Math. 58, 1–23 (1953) 9. Cahen, M., Gutt, S., Rawnsley, J.H.: Quantization of Kähler manifolds I: Geometric interpretation of Berezin’s quantization. JGP 7, 45–62 (1990) 10. Cahen, M., Gutt, S., Rawnsley, J.H.: Quantization of Kähler manifolds II. Trans. Amer. Math. Soc. 337, 73–98 (1993) 11. Cahen, M., Gutt, S., Rawnsley, J.H.: Quantization of Kähler manifolds III. Lett. Math. Phys. 30, 291–305 (1994) 12. Cahen, M., Gutt, S., Rawnsley, J.H.: Quantization of Kähler manifolds IV. Lett. Math. Phys. 34, 159–168 (1995) 13. Cappiello, M., Gramchev, T., Rodino, L.: Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients. J. Func. Anal. 237, 634–654 (2006) 14. Cappiello, M., Gramchev, T., Rodino, L.: Semilinear pseudo-differential equations and travelling waves. In: Pseudo-differential operators: partial differential equations and time-frequency analysis, Fields Inst. Commun. 52, Providence, RI: Amer. Math. Soc., 2007, pp. 213–238 15. Cardin, F., Gramchev, T., Lovison, A.: Exponential estimates for oscillatory integrals with degenerate phase functions. Nonlinearity 21(3), 409–433 (2008) 16. Carletti, T.: Exponentially long time stability for non-linearizable analytic germs of (Cn , 0). Ann Inst. Fourier (Grenoble) 54, 989–1004 (2004) 17. Donaldson, S.: Scalar curvature and projective embeddings. I. J. Diff. Geom. 59, 479–522 (2001) 18. Donaldson, S.: Scalar curvature and projective embeddings. II. Q. J. Math. 56, 345–356 (2005) 19. Donaldson, S.: Some numerical results in complex differential geometry. http://arXiv.org/abs/math.DG/ 0512625, 2005 20. Engliš, M.: Berezin Quantization and reproducing Kernels on complex domains. Trans. Amer. Math. Soc. 348, 411–479 (1996) 21. Engliš, M.: A Forelli–Rudin contruction and asymptotics of weighted Bergman kernels. J. Func. Anal. 177, 257–281 (2000) 22. Feng, R.: Szasz analytic functions and noncompact toric varieties. http://arXiv.org/abs/0809.2436v3 [math.DG], 2008 23. Engliš, M.: The asymptotics of a Laplace integral on a Kähler manifold. J. Reine Angew. Math. 528, 1–39 (2000) 24. Guillemin, V., Stenzel, M.: Grauert tubes and the homogeneous Monge-Ampère equation. J. Diff. Geom. 34, 561–570 (1991) 25. Guillemin, V., Stenzel, M.: Grauert tubes and the homogeneous Monge-Ampère equation. II. J. Diff. Geom. 35, 627–641 (1992) 26. Ji, S.: Inequality for distortion function of invertible sheaves on Abelian varieties. Duke Math. J. 58, 657–667 (1989) 27. Kempf, G.R.: Metric on invertible sheaves on abelian varieties. Topics in algebraic geometry (Guanajuato), 1989 28. Lempert, L., Sz˝oke, R.: Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290, 689–712 (1991) 29. Lempert, L., Sz˝oke, R.: The tangent bundle of an almost complex manifold. Canad. Math. Bull. 44, 70–79 (2001) 30. Loi, A.: The Tian–Yau–Zelditch asymptotic expansion for real analytic Kähler metrics. Int. J. of Geom. Methods Mod. Phys. 1, 253–263 (2004)
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31. Loi, A.: A Laplace integral, the T-Y-Z expansion and Berezin’s transform on a Kaehler manifold. Int. J. of Geom. Methods Mod. Phys. 2, 359–371 (2005) 32. Loi, A.: Balanced metrics on Cn . J. Geom. Phys. 57, 1115–1123 (2007) 33. Loi, A.: Regular quantizations and covering maps. Geom. Dedicata 123, 73–78 (2006) 34. Lu, Z.: On the lower terms of the asymptotic expansion of Tian–Yau–Zelditch. Amer. J. Math. 122, 235–273 (2000) 35. Moreno, C., Ortega-Navarro, P.: ∗-products on D 1 (C), S 2 and related spectral analysis. Lett. Math. Phys. 7, 181–193 (1983) 36. Moreno, C.: Star-products on some Kähler manifolds. Lett. Math. Phys. 11, 361–372 (1986) 37. Odzijewicz, A.: On reproducing kernels and quantization of states. Commun. Math. Phys. 114, 577–597 (1988) 38. Odzijewicz, A.: Coherent states and geometric quantization. Commun. Math. Phys. 150, 385–413 (1992) 39. Mladenov, I.M., Tsanov, V.V.: Reduction in stages and complete quantization of the MIC-Kepler problem. J. Phys. A 32, 3779–3791 (1999) 40. Patrizio, G., Wong, P.M.: Stein manifolds with compact symmetric centers. Math. Ann. 289(3), 355–382 (1991) 41. Rawnsley, J.H.: A nonunitary pairing of polarizations for the Kepler problem. Trans. Amer. Math. Soc. 250, 167–180 (1979) 42. Rawnsley, J.H.: Coherent states and Kähler manifolds. Quart. J. Math. Oxford 28(2), 403–415 (1977) 43. Ruan, W.D.: Canonical coordinates and Bergmann metrics. Comm. in Anal. Geom. 6(2), 589–631 (1998) 44. Shiffman, B., Tate, T., Zelditch, S.: Harmonic analysis on toric varieties, Explorations in complex and Riemannian geometry. Contemp. Math. 332, Providence, RI: Amer. Math. Soc., 2003, pp. 267–286 45. Souriau, J.M.: Sur la varie’te’ de Kepler. Symposia Mathematica XIV, London-New York: Academic Press, 1974 46. Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Diff. Geom. 32, 99–130 (1990) 47. Zelditch, S.: Szegö Kernels and a Theorem of Tian. Internat. Math. Res. Notices 6, 317–331 (1998) 48. Zhang, S.: Heights and reductions of semi-stable varieties. Comp. Math. 104, 77–105 (1996) Communicated by S. Zelditch
Commun. Math. Phys. 289, 841–861 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0843-z
Communications in
Mathematical Physics
Stability of a Composite Wave of Two Viscous Shock Waves for the Full Compressible Navier-Stokes Equation Feimin Huang 1, , Akitaka Matsumura 2, 1 Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing, China.
E-mail: [email protected]
2 Department of Pure and Applied Mathematics, Graduate School of Information
Science and Technology, Osaka University, Suita, Osaka, Japan. E-mail: [email protected] Received: 21 February 2008 / Accepted: 20 April 2009 Published online: 2 June 2009 – © Springer-Verlag 2009
Abstract: In this paper we investigate the asymptotic stability of a composite wave consisting of two viscous shock waves for the full compressible Navier-Stokes equation. By introducing a new linear diffusion wave special to this case, we successfully prove that if the strengths of the viscous shock waves are suitably small with same order and also the initial perturbations which are not necessarily of zero integral are suitably small, the unique global solution in time to the full compressible Navier-Stokes equation exists and asymptotically tends toward the corresponding composite wave whose shifts (in space) of two viscous shock waves are uniquely determined by the initial perturbations. We then apply the idea to study a half space problem for the full compressible Navier-Stokes equation and obtain a similar result. 1. Introduction In this paper, we consider the 1-dimensional compressible Navier-Stokes equation in Lagrangian coordinates: ⎧ v − u x = 0, ⎪ ⎪ t ux ⎨ u t + px = µ( )x , (1.1) v ⎪ ⎪ uu x u2 θx ⎩ )x , (e + )t + ( pu)x = (κ + µ 2 v v where x ∈ R, t > 0, and v(x, t) > 0, u(x, t), θ (x, t) > 0, e(x, t) > 0 and p(x, t) are the specific volume, fluid velocity, internal energy, absolute temperature, and pressure respectively, while µ > 0 and κ > 0 denote the viscosity and heat conduction Research is supported in part by NSFC Grant No. 10471138, NSFC-NSAF Grant No. 10676037 and 973
project of China, Grant No. 2006CB805902, in part by Japan Society for the Promotion of Science, the Invitation Fellowship for Research in Japan (Short-Term). Research is supported in part by Grant-in-Aid for Scientific Research (B) 19340037, Japan.
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coefficients respectively. Here we study the ideal and polytropic fluids so that p and e are given by the state equations p=
Rθ R , e= θ + const., v γ −1
(1.2)
where γ > 1 is the adiabatic exponent and R > 0 is the gas constant. In the present paper, we study the Cauchy problem for (1.1)-(1.2) with the initial data (v, u, θ )(x, 0) = (v0 , u 0 , θ0 )(x),
lim (v0 , u 0 , θ0 ) = (v± , u ± , θ± ),
x→±∞
(1.3)
where v± > 0, θ± > 0 and u ± ∈ R are given constants. In the perfect fluids, i.e., κ = µ = 0, (1.1) becomes the well-known compressible Euler system: ⎧ v − u x = 0, ⎪ ⎨ t u t + px = 0, (1.4) 2 ⎪ ⎩ (e + u ) + ( pu) = 0, t x 2 which is one of the most important nonlinear strictly hyperbolic systems of conservation laws. The basic waves for the system (1.4) are dilation invariant solutions: shock waves, rarefaction waves, and contact discontinuities, and the linear combinations of these basic waves, called Riemann solutions, govern both local and large time asymptotic behavior of general solutions to the inviscid Euler system (1.4) ([1,9,21]). Since the inviscid system (1.4) is an idealization when the dissipative effects are neglected, thus it is of great importance to study the large time asymptotic behavior of solutions to the corresponding viscous systems, such as (1.1), toward the viscous versions of these basic waves. Indeed, there have been great interest and intensive studies in this respect since 1985, starting with studies on the nonlinear stability of viscous shock profiles by Goodman [2] and Matsumura-Nishihara [16]. Deeper understanding has been achieved on the asymptotic stability of each wave pattern, which has been shown to be nonlinearly stable for quite general perturbation for the compressible Navier-Stokes system (1.1) and the more general system of viscous strictly hyperbolic conservation laws, and new phenomena have been discovered and new techniques, such as weighted characteristic energy methods, uniform approximate Green’s functions and Evans function approach, have been developed based on the intrinsic properties of the underlying waves, see [3–8,10–14,18,20,22–24] and the references therein. However, the important progress above is only established for the single wave pattern, i.e., viscous shock wave, viscous rarefaction wave or viscous contact wave. Few results are known for the composite wave, which consists of at least two wave patterns, due to various difficulties. In this paper, we study the asymptotic stability of a composite wave consisting of 1-viscous shock wave and 3-viscous shock wave under general initial perturbations for the full Navier-Stokes system (1.1). More precisely, let (V1 , U1 , 1 )(x, t) be the 1-viscous shock wave connecting the left state (v− , u − , θ− ) with an intermediate state (vm , u m , θm ) and (V3 , U3 , 3 )(x, t) be the 3-viscous shock wave connecting (vm , u m , θm ) with the right state (v+ , u + , θ+ ), where the viscous shock waves are given in (2.3) and (2.5) and v± , vm , u ± , u m and θ± , θm are given constants. We denote a composite wave consisting of the two viscous shock waves (Vi , Ui , i ), i = 1, 3 by (v, ¯ u, ¯ θ¯ )(x, t) = (V1 + V3 − vm , U1 + U3 − u m , 1 + 3 − θm ). The main purpose of this paper is to investigate the nonlinear stability of the above composite wave (v, ¯ u, ¯ θ¯ ) under small general initial perturbations without zero integral assumption for the full
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compressible Navier-Stokes system (1.1). It is worth pointing out here that the stability of the above composite wave with zero integral perturbation is straightforward by a similar method of [7]. The main novelty of this paper is to overcome the difficulty arising from the general initial perturbation by introducing a new linear diffusion wave. The above problem is exactly motivated by our study on an initial boundary value problem on the half space for the system (1.1), i.e., ⎧ ⎪ ⎪ vt − u x = 0, u ⎪ x ⎪ ⎪ u t + px = µ( )x , ⎪ ⎪ v ⎪ ⎪ ⎨ uu x u2 θx )x , x > 0, t > 0, (e + )t + ( pu)x = (κ + µ (1.5) 2 v v ⎪ ⎪ ⎪ u(x, 0) = 0, θx (x, 0) = 0, x > 0, ⎪ ⎪ ⎪ (v, u, θ )(x, 0) = (v0 , u 0 , θ0 )(x), x > 0, ⎪ ⎪ ⎪ ⎩ lim (v , u , θ )(x) = (v , u , θ ), v > 0, u < 0, θ > 0, x→+∞
0
0
0
+
+
+
+
+
+
where the initial data are supposed to satisfy the boundary condition as a compatibility condition. In the 2 × 2 isentropic case, i.e., ⎧ vt − u x = 0, ⎪ ⎪ ⎪ ux −γ ⎪ ⎪ u ⎨ t + (v )x = µ( )x , x > 0, t > 0, v (1.6) u(x, 0) = 0, x > 0, ⎪ ⎪ ⎪ (v, u)(x, 0) = (v , u )(x), x > 0, 0 0 ⎪ ⎪ ⎩ lim (v0 , u 0 )(x) = (v+ , u + ), v+ > 0, u + < 0, x→+∞
it is shown by Matsumura-Mei [15] that the solutions tend to the 2-viscous shock wave connecting (vb , 0) with (v+ , u + ) as t goes to infinity, where the constant vb is uniquely determined by the Rankine-Hugoniot (RH) condition ([21]) so that (vb , 0) is located on the 2-shock curve passing through (v+ , u + ), under an additional assumption that the viscous shock wave is initially far away from the boundary. Later, Matsumura-Nishihara [19] introduced a different approach to study (1.6) by extending the half space problem (1.6) into an initial value problem by setting v(x, 0) = v(−x, 0), u(x, 0) = −u(−x, 0) when x < 0. The above assumption of [15], that the viscous shock wave is initially far away from the boundary, is removed. It is naturally expected that the solutions of (1.5) also time-asymptotically tend to a 3-viscous shock wave connecting (vb , 0, θb ) and (v+ , u + , θ+ ), with the constants vb and θb being uniquely determined by the RH condition so that (vb , 0, θb ) is located on the 3-shock curve passing through (v+ , u + , θ+ ) in the same way as in 2 × 2 case. In the spirit of [19], we define for x < 0, (v, θ )(x, 0) = (v, θ )(−x, 0), u(x, 0) = −u(−x, 0),
(1.7)
which satisfies the boundary condition (u, θx )(x, 0) = 0. Notice that the transformation (1.7) is invariant to the system (1.1), thus we successfully change the half space problem (1.5) into a special case of the initial value problem (1.1)-(1.3) with (v− , u − , θ− ) = (v+ , −u + , θ+ ) and (vm , u m , θm ) = (vb , 0, θb ). Roughly speaking, our main results are as follows: 1. The composite wave (v, ¯ u, ¯ θ¯ ) = (V1 + V3 − vm , U1 + U3 − u m , 1 + 3 − θm ) for the Cauchy problem (1.1)-(1.3) is asymptotically stable with shifts in space if the strengths of viscous shock waves are suitably small with the same order and also the initial perturbations are suitably small. The precise statement is given in Theorem 1 of Sect. 2.
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2. For the initial boundary value problem (1.5), there exists a 3-viscous shock wave connecting (vb , 0, θb ) and (v+ , u + , θ+ ), where the constants vb and θb are uniquely determined by the RH condition associated with the 3-shock curve, and this 3-viscous shock wave is asymptotically stable with a shift in space if the strength of the shock wave and the initial perturbations are suitably small. The precise statement is given in Theorem 2 of Sect. 2. The rest of the paper will be arranged as follows. In Sect. 2, a linear diffusion wave is introduced and main results are stated. In Sect. 3, the compressible Navier-Stokes system is reformulated to an integrated system and the a priori estimates are derived. In the end of this section, the main theorems are proved. Notations. Throughout this paper, generic positive constants are denoted by c and C without confusion. For function spaces, L p (), 1 ≤ p ≤ ∞ denotes the usual Lebesgue space on ⊂ R = (−∞, ∞) with its norm given by f L p () :=
| f (x)| p d x
1
p
, 1 ≤ p < ∞, f L ∞ () := ess.sup | f (x)|.
W k, p () denotes the k th order Sobolev space with its norm f W k, p () :=
k
j
p
∂x f L p ()
1
p
, 1 ≤ p < ∞.
j=0
And if p = 2, we note H k () = W k,2 (), · = · L 2 () and · k = · W k,2 () for simplicity. The domain will be often abbreviated without confusion. 2. Linear Diffusion Wave and Main Results Before stating the main results, we recall the Riemann problem for the compressible Euler equation (1.4) with the Riemann initial data (v− , u − , θ− ), x < 0, (v, u, θ )(x, 0) = (2.1) (v+ , u + , θ+ ), x > 0. √ It is known that the system (1.4) has three eigenvalues: λ1 = − γ p/v < 0, λ2 = 0, λ3 = − λ1 > 0, where the second characteristic field is linear degenerate and the others are genuinely nonlinear. In the present paper, we focus our attention on the situation where the Riemann solution of (1.4),(2.1) consists of two shock waves (and three constant states), that is, there exists an intermediate state (vm , u m , θm ) such that (v− , u − , θ− ) connects with (vm , u m , θm ) by the 1-shock wave with the shock speed s1 < 0 and (vm , u m , θm ) connects with (v+ , u + , θ+ ) by the 3-shock wave with the shock speed s3 > 0. Here the shock speeds s1 and s3 are constants determined by the RH condition and satisfy entropy conditions λ1 (v− , u − , θ− ) > s1 > λ1 (vm , u m , θm ), λ3 (vm , u m , θm ) > s3 > λ3 (v+ , u + , θ+ ). By the standard arguments (e.g. [21]), for each (v− , u − , θ− ) we can see our situation takes place provided (v+ , u + , θ+ ) is located on a curved surface in a neighborhood of
Composite Wave of Two Viscous Shock Waves
845
(v− , u − , θ− ). In what follows, the neighborhood of (v− , u − , θ− ) is denoted by − . To describe the strengths of the shock waves for later use, we set δ1 = |vm − v− | + |u m − u − | + |θm − θ− |, δ3 = |vm − v+ | + |u m − u + | + |θm − θ+ | and δ = min(δ1 , δ3 ). When we choose |(v+ − v− , u + − u − , θ+ − θ− )| small in our situation for the fixed (v− , u − , θ− ), we note that it holds δ1 + δ3 ≤ C|(v+ − v− , u + − u − , θ+ − θ− )|,
(2.2)
where C is a positive constant depending only on (v− , u − , θ− ). Then, if it also holds δ1 + δ3 ≤ Cδ, δ1 + δ3 → 0
(2.3)
for a positive constant C, we call the strengths of the shock waves “small with same order”. In what follows, we always assume (2.3). Next we recall the definitions of viscous shock waves of (1.1) which correspond to the above shock waves. We see that the 1-viscous shock wave which corresponds to the 1-shock wave is a traveling wave solution of (1.1) with the formula (V1 , U1 , 1 )(x −s1 t) which is determined by ⎧ −s1 V1 − U1 = 0, ⎪ ⎪ ⎪ U ⎪ ⎪ ⎨ −s1 U1 + p(V1 , 1 ) = µ( V11 ) , U2
−s1 (e(V1 , 1 ) + 21 ) + ( p(V1 , 1 )U1 ) = (κ V11 + µ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (V1 , U1 , 1 )(−∞) = (v− , u − , θ− ), (V1 , U1 , 1 )(+∞) = (vm , u m , θm ) where = µ
d dξ ,
U1 U1 V1 ) ,
(2.4)
ξ = x − s1 t. Integrating (2.4) with respect to x, we have
U1,x R1 = P1 + s12 V1 − b1 , b1 = p− + s12 v− = pm + s12 vm , P1 = . V1 V1
(2.5)
Similarly, the 3-viscous shock wave (V3 , U3 , 3 )(x − s3 t) is defined by ⎧ −s3 V3 − U3 = 0, ⎪ ⎪ ⎪ ⎪ −s U + p(V , ) = µ( U3 ) , ⎪ 3 3 ⎨ 3 3 V3 U2
−s3 (e(V3 , 3 ) + 23 ) + ( p(V3 , 3 )U3 ) = (κ V33 + µ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (V3 , U3 , 3 )(−∞) = (vm , u m , θm ), (V3 , U3 , 3 )(+∞) = (v+ , u + , θ+ ) where = µ
d dη ,
U3 U3 V3 ) ,
(2.6)
η = x − s3 t and
U3,x R3 = P3 + s32 V3 − b3 , b3 = p+ + s32 v+ = pm + s32 vm , P3 = . V3 V3
By the results in [7], we have
(2.7)
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Proposition 2.1. For each (v− , u − , θ− ), suppose (v+ , u + , θ+ ) ∈ − and the Riemann solution of (1.4),(2.1) consists of two shock waves. Also, assume 1 < γ ≤ 2. Then there exist positive constants c and C which depend only on (v− , u − , θ− ) such that, for i = 1, 3, |(V1 − vm , U1 − u m , 1 − θm )(x, t)| ≤ Cδ1 e−cδ1 |x−s1 t| , x > s1 t, t ≥ 0, |(V3 − vm , U3 − u m , 3 − θm )(x, t)| ≤ Cδ3 e−cδ3 |x−s3 t| , x < s3 t, t ≥ 0, Ui,x (x, t) < 0, |(Vi,x , Ui,x , i,x )(x, t)| ≤ Cδi2 e−cδi |x−si t| , x ∈ R, t ≥ 0, |i,x (x, t)| ≤ C|Vi,x (x, t)|, x ∈ R, t ≥ 0, |si2 − γvpmm | ≤ Cδi .
(2.8)
Remark 2.1. The condition 1 < γ ≤ 2 is due to [7]. Based on the same reason, we also assume the same condition in Theorems 1 and 2 below. This condition might be removed by more careful and complicated estimates. But here we only consider the simple case, i.e. 1 < γ ≤ 2. Let E = θ +
γ −1 2 2R u
¯ with and m(x, t) = t (v, u, E), m(x, ¯ t) = t (v, ¯ u, ¯ E),
v¯ = V1 (x − s1 t) + V3 (x − s3 t) − vm , u¯ = U1 (x − s1 t) + U3 (x − s3 t) − u m , −1 2 E¯ = E 1 (x − s1 t) + E 3 (x − s3 t) − E m , E m = θm + γ2R um .
(2.9)
Since
∞ the present paper is concerned with a general initial perturbation, the integral ¯ 0) d x is in general not zero. To use the anti-derivative technique, we −∞ (m − m)(x,
∞ need to find an ansatz m˜ such that −∞ m(x, t) − m(x, ˜ t) d x = 0; meanwhile |m˜ − m| ¯ tends to zero as t → ∞. This idea was first applied in [10] to study the asymptotic stability of a single viscous shock wave for general viscous conservation laws with uniformly artificial viscosities by introducing linear or nonlinear diffusion waves. In this paper, we will borrow the idea of [10] to construct the desired ansatz m˜ by introducing a new linear diffusion wave. In the following, we shall show how to construct the linear diffusion wave with details. Let ⎛ ⎞ 0 −1 0 p R ⎠ 0 A(v, u, θ ) = ⎝ − v v γ −1 0 0 R p be the Jacobi matrix of the flux t (−u, p, γ −1 R pu) with respect to (v, u, θ ). Then the second right eigenvector of the matrix A(v, u, θ ) at the point (vm , u m , θm ) is pm . r2 = t 1, 0, R Furthermore, a direct computation yields that the three vectors r1 = t (vm − v− , u m −u − , E m − E − ), r2 and r3 = t (v+ −vm , u + −u m , E + − E m ) are linearly independent in R 3 if δ1 + δ3 is small. So if the initial mass m(x, 0) − m(x, ¯ 0) d x is not zero, we can distribute the initial mass along the three independent directions r1 , r2 and r3 , that is, (m(x, 0) − m(x, ¯ 0)) d x =
3
αi ri ,
(2.10)
i=1
where αi , i = 1, 2, 3, are constants uniquely determined by the initial data. The excessive mass α1r1 in the first characteristic field can be eliminated by the translated 1-viscous
Composite Wave of Two Viscous Shock Waves
847
shock wave with a shift α1 , i.e., V1 (x − s1 t + α1 ). Similarly, we can eliminate α3r3 by replacing V3 (x − s3 t) by V3 (x − s3 t + α3 ). So the remaining problem is how to remove the excessive mass in the second characteristic field, i.e., α2 r2 . For this, we look for our ˜ as in the form ansatz m˜ = t (v, ˜ u, ˜ E) v˜ = V1 (x − s1 t + α1 ) + V3 (x − s3 t + α3 ) − vm + (x, t), u˜ = U1 (x − s1 t + α1 ) + U3 (x − s3 t + α3 ) − u m + f (x, t), E˜ = E 1 (x − s1 t + α1 ) + E 3 (x − s3 t + α3 ) − E m + pm + g(x, t).
(2.11)
R
Here we expect t (, 0, pRm ) = r2 is a basic approximation of the diffusion wave associated with the second characteristic field which tends to zero as t → +∞ and carries excessive mass α2 r2 , and so expect it holds ∞ C|α2 | − cx 2 (x, t) d x = α2 . (2.12) |(x, t)| ≤ √ e 1+t , 1+t −∞ We also expect that f (x, t) and g(x, t) are higher order correction terms compared with (x, t) which behave like derivatives of the diffusion wave, i.e., | f (x, t)| + |g(x, t)| ≤ and do not carry any mass, i.e., ∞ f (x, t) d x = −∞
∞
−∞
C|α2 | − cx 2 e 1+t , 1+t
(2.13)
g(x, t) d x = 0.
(2.14)
˜ well approximates a solution of the system Finally, we should expect that m˜ = t (v, ˜ u, ˜ E) (1.1) as in the form ⎧ v˜t − u˜ x = 0, ⎪ ⎪ ⎪ ⎪ u˜ x ⎪ ⎨ u˜ t + p˜ x = µ + R˜ 1,x , v˜ x (2.15) ⎪ µ 2 ˜x ⎪ u ˜ θ ⎪ ⎪ u˜ u˜ x + R˜ 2,x , + ( p˜ u) ˜ x =κ + ⎪ ⎩ e˜ + 2 v˜ v˜ x t x
where the error terms R˜ i (i = 1, 2) should decay fast enough for a priori estimates we will establish later (see (3.22)). Under the assumptions (2.2) and (2.12), by Proposition 2.1, we have the estimates for wave interactions for small δ and α2 as in [10 and 22], |V1 − vm ||V3 − vm | ≤ Cδ1 δ3 (e−cδ1 (|x|+t)+cδ1 |α1 | + e−cδ3 (|x|+t)+cδ3 |α3 | ) ≤ Cδ 2 e−cδ(|x|+t) ,
(2.16)
and for i = 1, 3, |Vi − vm | ≤ Cδi e−cδi (|x|+t)+cδi |αi |
|α2 |
cx 2
1 2
e− 1+t + Cδi |α2 |e−c(|x|+t) ,
(1 + t) |α2 | − cx 2 ≤ C|α2 |δ e−cδ(|x|+t) + C e 1+t + C(δ + |α2 |)e−c(|x|+t) , (2.17) 3 (1 + t) 2 3 2
848
F. Huang, A. Matsumura
where we used the fact that δ1 α1 and δ3 α3 are uniformly bounded by (2.10) for small δ1 and δ3 as long as the initial perturbation stays bounded. By the estimates (2,16) and (2.17), we expect R˜ i (i = 1, 2) satisfy | R˜ i |, | R˜ i,x | ≤ C(δ 2 + |α2 |δ 2 )e−cδ(|x|+t) + C
|α2 |
3
(1 + t)
cx 2
3 2
e− 1+t + C(δ + |α2 |)e−c(|x|+t) , (2.18)
where and in what follows the constants c and C are uniformly positive constants with respect to small δ and α2 . It is remarked here that we do not rely on any arguments of [10 or 22] to get (2.18) in the end, but see it by inspection. In the following, we denote A ≈ B when 3
|A − B| ≤ C(δ 2 + |α2 |δ 2 )e−cδ(|x|+t) + C
|α2 | (1 + t)
cx 2
3 2
e− 1+t + C(δ + |α2 |)e−c(|x|+t) holds. (2.19)
To obtain (2.15) and (2.18), we are now trying to choose the functions (, f, g) which satisfy (2.12)-(2.14). Here we emphasize that the estimates below to get (2.18) do not rely on any arguments of [10 and 22]. If we substitute (2.11) into (1.1)1 and use (2.4) and (2.6), we easily observe that t − f x = 0.
(2.20)
In view of (2.12),(2.13),(2.16) and (2.17), we obtain γ −1 pm γ −1 2 u˜ ≈ 1 + 3 − θm + +g− u m f. θ˜ = E˜ − 2R R R Hence we choose g =
γ −1 R um
(2.21)
f so that
pm γ −1 2 u˜ ≈ 1 + 3 − θm + . θ˜ = E˜ − 2R R In the same way, we have ⎧ R θ˜ ⎪ ⎪ p˜ = ≈ P1 + P3 − pm , ⎪ ⎪ ⎪ v˜ ⎪ ⎪ ⎨ p˜ u˜ ≈ P1 U1 + P3 U3 − pm u m + pm f (x, t), U1,x U3,x θ˜x 1,x 3,x u˜ x pm ⎪ ≈ ≈ + , + + x , ⎪ ⎪ v˜ V1 V3 v˜ V1 V3 Rvm ⎪ ⎪ ⎪ ⎪ ⎩ u˜ u˜ x ≈ U1 U1,x + U3 U3,x . v˜ V1 V3
(2.22)
(2.23)
Substituting (2.11), (2.22)-(2.23) into (1.1)3 and using (2.3),(2.5), we have γ −1 γ − 1 κ pm pm t + pm f x = x x + εx , ε ≈ 0. R R R Rvm
(2.24)
Then (2.20) yields t =
(γ − 1)κ x x + εx , ε ≈ 0. γ Rvm
(2.25)
Composite Wave of Two Viscous Shock Waves
849
Therefore we choose the function by x2 α2 (γ − 1)κ (x, t) = √ e− 4a(1+t) , a = >0 γ Rvm 4πa(1 + t)
(2.26)
as the unique solution of t = ax x , |t=−1 = α2 δ(x),
∞
−∞
(x, t) d x = α2 ,
(2.27)
and also choose f and g by f (x, t) = ax (x, t), g(x, t) =
γ −1 u m ax (x, t). R
(2.28)
It is easy to see (, f, g) defined above satisfy (2.12)-(2.14), and so (2.17),(2.22),(2.23), (2.15) and (2.18) are justified. Now we return to (2.11). Substituting (, f, g) defined as in (2.26)-(2.28) into (2.11), we have v˜ = V1 (x − s1 t + α1 ) + V3 (x − s3 t + α3 ) − vm + , u˜ = U1 (x − s1 t + α1 ) + U3 (x − s3 t + α3 ) − u m + ax , E˜ = E 1 (x − s1 t + α1 ) + E 3 (x − s3 t + α3 ) − E m + pm + R
γ −1 R u m ax .
(2.29)
Then it follows from (2.10) that
∞
(m(x, 0) − m(x, ˜ 0)) d x ∞ = (m(x, 0) − m(x, ¯ 0)) d x +
−∞
=
−∞ 3
αi ri +
i=1
i =2
∞
−∞
t
∞ −∞
(m(x, ¯ 0) − m(x, ˜ 0)) d x
(Vi (x) − Vi (x + αi ), Ui (x) − Ui (x + αi ),
(2.30) E i (x) − E i (x + αi )) d x − α2 r2 = 0,
∞ ˜ t) is the desired ansatz. where we have used the fact that −∞ d x = α2 . Thus m(x, Remark 2.2. Since the L 2 norms of the error terms R˜ 1 and R˜ 2 are integrable with respect to time t by (2.18), it is good enough to deduce the desired a priori estimates shown in Sect. 3. It is worth recalling that in the study of a single viscous shock wave for viscous conservation laws with artificial viscosities, to derive good error terms like R˜ 1 and R˜ 2 , besides the nonlinear or linear diffusion waves of Liu [10], Szepessy and Xin [22] introduced additional coupled linear diffusion waves which do not carry any mass. Here our linear diffusion wave is somehow like that of Liu [10] and ax is like the coupled diffusion wave of Szepessy and Xin [22]. Furthermore, in this particular system of compressible Navier-Stokes equations, we get an explicit formula of the coupled diffusion wave, i.e., ax , which is a higher order correction corresponding to the linear diffusion wave .
850
F. Huang, A. Matsumura
Remark 2.3. The approach used here is essentially based on the property that a characteristic variable associated with the second characteristic field is decoupled from other characteristic ones up to second order at the hyperbolic level. Therefore, this approach could be extended to the arguments on the stability of viscous shock patterns of general hyperbolic-parabolic system for which all characteristic fields in which a shock is not present have the similar properties of decoupling. This fact was kindly pointed out by the anonymous referee in the referee process. Here the authors thank the referee for his/her deep insight. Under the above preparations, we are now at the stage to state our main results. We first fix any (v− , u − , θ− ), and assume that (v+ , u + , θ+ ) ∈ − and the Riemann solution of (1.4),(2.1) consists of two shock waves. Then the composite wave (v, ¯ u, ¯ θ¯ )(x, t) alluded above is well defined, and we assume the initial data (1.3) satisfy ¯ 0)) ∈ H 1 ∩ L 1 . (v0 − v(·, ¯ 0), u 0 − u(·, ¯ 0), θ0 − θ(·, (2.31) ˜ in (2.27) is well defined and the shifts α1 Under the assumption (2.31), m˜ = (v, ˜ u, ˜ E)
and α3 are uniquely determined by (2.10) so that (m(x, 0) − m(x, ˜ 0)) d x = 0. Thus, if we set ∞ ∞ 0 (x) = − (v0 (y) − v(y, ˜ 0)) dy, 0 (x) = − (u 0 (y) − u(y, ˜ 0)) dy, W¯ 0 (x) = −
x
x ∞
x
((e0 +
u 20 2
)(y) − (e˜ +
u˜ 2 2
(2.32)
)(y, 0)) dy,
we can assume the initial data further satisfy (0 , 0 , W¯ 0 ) ∈ L 2 ,
(2.33) 2 ¯ where it is noted that (2.31) and (2.33) imply (0 , 0 , W0 ) ∈ H . To state the results precisely, we also set v¯α1 ,α3 (x, t) = V1 (x − s1 t + α1 ) + V3 (x − s3 t + α3 ) − vm , u¯ α1 ,α3 (x, t) = U1 (x − s1 t + α1 ) + U3 (x − s3 t + α3 ) − u m , θ¯α1 ,α3 (x, t) = 1 (x − s1 t + α1 ) + 3 (x − s3 t + α3 ) − θm ,
(2.34)
where we note (v, ¯ u, ¯ θ¯ )0,0 = (v, ¯ u, ¯ θ¯ ). Our first main theorem is Theorem 1. For each (v− , u − , θ− ), assume that (v+ , u + , θ+ ) ∈ − and the Riemann solution of (1.4),(2.1) consists of two shock waves whose strengths satisfy (2.3). Further assume that the initial data satisfy (2.31) and (2,33), and also 1 < γ ≤ 2. Then there exist positive constants δ0 and 0 , such that if |(v+ − v− , u + − u − , θ+ − θ− )| ≤ δ0 and ¯ 0)) H 1 ∩L 1 + (0 , 0 , W¯ 0 ) ≤ 0 , (v0 − v(·, ¯ 0), u 0 − u(·, ¯ 0), θ0 − θ(·, then the Cauchy problem (1.1)-(1.3) of the compressible Navier-Stokes system admits a unique global solution in time (v, u, θ ) satisfying (v − v, ˜ u − u, ˜ θ − θ˜ ) ∈ C([0, +∞); H 1 ), v − v˜ ∈ L 2 (0, ∞; H 1 ), (u − u, ˜ θ − θ˜ ) ∈ L 2 (0, ∞; H 2 ), where (v, ˜ u, ˜ θ˜ ) is defined by (2.29), and the shifts α1 , α3 in (v, ˜ u, ˜ θ˜ ) are defined by (2.10). Furthermore, it holds sup |(v − v¯α1 ,α3 , u − u¯ α1 ,α3 , θ − θ¯α1 ,α3 )(x, t)| → 0, t → ∞. x∈R
Composite Wave of Two Viscous Shock Waves
851
Remark 2.4. Let us make a comment on the condition (2.33) for (0 , 0 , W¯ 0 ). It is easy to see that a sufficient condition which implies (2.33) is given by ¯ 0))| ≤ β(1 + |x|)−α , x ∈ R |(v0 (x) − v(x, ¯ 0), u 0 (x) − u(x, ¯ 0), θ0 (x) − θ(x, for some constants α > 3/2 and β > 0, and the smallness of (0 , 0 , W¯ 0 ) in the assumption of Theorem 1 is clearly implied by assuming β small. Remark 2.5. Our result is the first one concerning the asymptotic stability of viscous shock waves to physical system for general initial perturbations obtained by an elementary energy method. However, as we pointed out in Remark 2.3, this approach works because the second characteristic field is hyperbolically decoupled, and so does not yield asymptotic stability of the single viscous shock wave. For the initial boundary value problem (1.5), it is known that if u + is negative and suitably small, the Riemann problem (1.4),(2.1) with (v− , u − , θ− ) = (v+ , −u + , θ+ ) admits the unique solution consisting of the two shock waves where the intermediate state (vm , u m , θm ) is given in the form (vb , 0, θb ). Then the viscous shock wave (V3 , U3 , 3 ) (x − s3 t) which corresponds to the 3-shock wave and connects the state (vb , 0, θb ) to (v+ , u + , θ+ ) is well defined and expected to be the asymptotic state of the solution of (1.5). As for the initial data, we assume (v0 − v+ , u 0 − u + , θ0 − θ+ ) ∈ H 1 (R+ ) ∩ L 1 (R+ ), u 0 (0) = 0.
(2.35)
Under the conditions of (2.35), extending the data in the whole space by (1.7), we see ˜ in (2.29) is well defined and the shifts α1 and α3 are uniquely deterthat m˜ = (v, ˜ u, ˜ E) mined as in the above Cauchy problem, where we notice α1 = −α3 due to the symmetry of the data. Then we also define (0 , 0 , W¯ 0 )(x) by (2.32) for x ∈ R+ , and assume (0 , 0 , W¯ 0 ) ∈ L 2 (R+ ).
(2.36)
Our second main theorem is Theorem 2. Assume u + < 0 and 1 < γ ≤ 2. Further assume that the initial data satisfy (2.35) and (2.36). Then there exist positive constants δ0 and 0 , such that if |u + | ≤ δ0 and the initial data (v0 , u 0 , θ0 ) satisfies (v0 − v+ , u 0 − u + , θ0 − θ+ ) H 1 (R+ )∩L 1 (R+ ) + (0 , 0 , W¯ 0 ) L 2 (R+ ) ≤ 0 , then the initial boundary value problem (1.5) of the compressible Navier-Stokes system admits a unique global solution in time (v, u, θ ) satisfying (v − v, ˜ u − u, ˜ θ − θ˜ ) ∈ C([0, +∞); H 1 (R+ )), v − v˜ ∈ L 2 (0, +∞; H 1 (R+ )), (u − u, ˜ θ − θ˜ ) ∈ L 2 (0, +∞; H 2 (R+ )), where (v, ˜ u, ˜ θ˜ ) is defined by (2.29) for the extended initial data by (1.7), and the shifts α1 , α3 = −α1 in (v, ˜ u, ˜ θ˜ ) are defined by (2.10). Furthermore, sup |(v, u, θ )(x, t) − (V3 , U3 , 3 )(x − s3 t + α3 )| → 0, t → ∞ holds. x>0
852
F. Huang, A. Matsumura
3. Reformulation of the Problem and a Priori Estimates Following the arguments in [7], we first reformulate the Cauchy problem for (1.1) to one for an integrated system in terms of the perturbation from (v, ˜ u, ˜ θ˜ ). Since the systems ˜ both (1.1) for (v, u, θ ) and (2.15) for (v, ˜ u, ˜ θ ) are of conserved form, we can expect that (2.28) implies (m(x, t) − m(x, ˜ t)) d x = 0 for any t > 0, and so expect that the functions ∞ ∞ (x, t) = − (v(y, t) − v(y, ˜ t)) dy, (x, t) = − (u(y, t) − u(y, ˜ t)) dy, W¯ (x, t) = −
x
x ∞
((e +
x
u2 2
)(y, t) − (e˜ +
u˜ 2 2
(3.1)
)(y, t)) dy
can be well defined in some Sobolev space for all t > 0 and consistent with (2.32). Especially, by (2.33), we also expect (, , W¯ ) ∈ C([0, ∞); H 2 ). From (3.1), we naturally look for the solution of (1.1) in the form v − v˜ = x , u − u˜ = x ,
1 R (θ − θ˜ ) + |x |2 + u ˜ x = W¯ x . γ −1 2
(3.2)
Subtracting (2.15) from the system (1.1) and integrating the resulting system, we have the following integrated system for (φ, ψ, W¯ ): ⎧ t − x = 0, ⎪ ⎪ ⎨ µ µ t + p − p˜ = u x − u˜ x − R˜ 1 , (3.3) v v˜ ⎪ ⎪ ⎩ W¯ t + pu − p˜ u˜ = κ θx − κ θ˜x + µ uu x − µ u˜ u˜ x − R˜ 2 , v v˜ v v˜ with the initial data (φ, ψ, W¯ )(0) = (φ0 , ψ0 , W¯ 0 ) ∈ H 2 . Instead of the variable W¯ , which is the anti-derivative of the total energy, it is more convenient to introduce another variable related to the absolute temperature W =
γ −1 ¯ (W − u), ˜ R
which replaces (3.2) by v − v˜ = x , u − u˜ = x , θ − θ˜ = Wx −
γ −1 R
1 2 − u˜ x . 2 x
(3.4)
Then we rewrite the system (3.2) in terms of (φ, ψ, W ) as in the form ⎧ t − x = 0, ⎪ ⎪ ⎪ ⎪ µu˜ x p˜ R γ −1 µ ⎪ ⎪ − 2 x + Wx + u˜ x = x x + J1 − R˜ 1 , ⎪ t − ⎨ v˜ v˜ v˜ v˜ v˜ R µ u ˜ (γ − 1)κ κ θ˜x x ⎪ Wt + p˜ − x + u˜ t − (u˜ x )x + x ⎪ ⎪ ⎪ γ −1 v˜ v˜ R v v˜ ⎪ ⎪ κ ⎪ ⎩ = Wx x + J2 − R˜ 3 , v˜
(3.5)
Composite Wave of Two Viscous Shock Waves
853
where (v, u, θ ) is given by (3.4), R˜ 3 = R˜ 2 − u˜ R˜ 1 and γ −1 2 µ µ p˜ R 2 ˜ x + u˜ x x − x x x − p − p˜ + x − (θ − θ ) , J1 = 2v˜ v v˜ v v˜ v˜ v˜ µu˜ x µu x (γ − 1)κ κx (θ − θ˜ )x J2 = ( p˜ − p)x + − x − x x x − . v v˜ R v˜ v v˜
(3.6)
It is noted that for small (, , W )2 , J1 = O(1)(2x + x2 + Wx2 + |u˜ x |||2 + |x x ||x |), ˜ x ||x | + |x x ||x |). J2 = O(1)(2x + x2 + Wx2 + |u˜ x |||2 + |(θ − θ)
(3.7)
From now on, we focus on the Cauchy problem to the reformulated system (3.5) with the initial data (, , W )(0) = (0 , 0 , W0 ) ∈ H 2 ,
(3.8)
¯ ˜ where W0 = γ −1 0 ), and look for the global solution (, , W ) ∈ R ( W0 − u(0) C([0, ∞); H 2 ) under the smallness conditions on (0 , 0 , W0 ) and δ + |α2 |. Since the local existence is well known (e.g., see [7]), we omit it here for brevity. Following the arguments in [7], in order to prove the global existence of the small solution, we only need to establish an a priori estimate which assures to close the following a priori assumption: N (T ) := sup (, , W )(t)2 ≤ ε0 ,
(3.9)
t∈[0,T ]
where [0, T ] is a time interval on which the solution is supposed to exist, and ε0 is a positive small constant which is eventually determined by smallness of the initial data and the strength of shock waves. To state the a priori estimate precisely, we define the solution set X (I ) for any interval I ⊂ R by X (I ) = {(, , W ) ∈ C(I ; H 2 ) ; (x , Wx ) ∈ L 2 (I ; H 2 ) }. Then our desired a priori estimate which assures to close the assumption (3.9) and also to imply the asymptotic behavior of the solution is Proposition 3.1 (A Priori Estimate). For each (v− , u − , θ− ), there exist positive constants δ0 , ε0 and C such that, if δ + |α2 | ≤ δ0 and (, , W ) ∈ X ([0, T ]) is a solution of (3.5) for some positive T and N (T ) ≤ ε0 , then (, , W ) satisfies for t ∈ [0, T ], t (x 21 + (x , Wx )22 ) dτ (, , W )(t)22 + t + 0
0
∞
1
−∞
(|U1,x | + |U3,x |)( 2 + W 2 ) d xdτ ≤ C((0 , 0 , W0 )22 + δ02 ). (3.10)
To show Proposition 3.1, for given (v− , u − , θ− ), we first choose positive constants δ¯ and ε¯ by Proposition 2.1 and Sobolev’s lemma so that δ + |α2 | ≤ δ¯ and N (T ) ≤ ε¯ imply v, v, ˜ θ and θ˜ are uniformly positive on [0, T ]; more concretely, inf v˜ ≥ x,t
v− , 4
inf v ≥ x,t
θ− v− , inf θ˜ ≥ , x,t 2 4
inf θ ≥ x,t
θ− , 2
(3.11)
854
F. Huang, A. Matsumura
which assure all the coefficients in (3.5) are non-singular. Then the proof of Proposition 3.1 is given by the following series of lemmas, where we always assume that (, , W ) ∈ X ([0, T ]) is a solution of (3.5), δ + |α2 | ≤ δ0 ≤ δ¯ and N (T ) ≤ ε0 ≤ ε¯ . In all the lemmas, we use the notations γ −1 1 2 − u˜ x , (3.12) φ = x , ψ = x , ζ = Wx − R 2 x which exactly correspond to v − v, ˜ u − u˜ and θ − θ˜ by (3.4). We also notice that in the proofs the energy estimates are derived for smooth enough (, , W ) since the same estimates can be justified for (, , W ) ∈ X ([0, T ]) by standard arguments on mollifiers, and so we omit this process for simplicity. Then, the first lemma is Lemma 3.1. If δ0 and ε0 are suitably small, it holds that (, , W )(t) + 2
t + 0
t
(x , Wx )2 dτ
0 ∞
−∞
(|U1,x | + |U3,x |)( 2 + W 2 ) d xdτ 1
≤ C(0 , 0 , W0 )2 + C(N (T ) + δ02 )
t 0
1
(x 2 + ζx 2 + x x 2 ) dτ + Cδ02 . (3.13)
Proof. When δ0 is small, we know inf( p˜ − µvu˜˜ x ) > 0 due to Proposition 2.1. Let L = ( p˜ − µvu˜˜ x )−1 . Then, multiplying (3.5)1 by , (3.5)2 by vL, ˜ (3.5)3 by R L 2 W respectively and adding all the resultant equations, we have E 1,t + E 2 + E 3 + E 4 = −vL ˜ R˜ 1 + vL ˜ J1 − R˜ 3 R L 2 W + R L 2 J2 W + (· · · )x , (3.14) with E1 =
1 R2 2 2 + L 2 W 2 + vL , ˜ 2 (γ − 1)
1 ˜ t + (γ − 1)L u˜ x , A = − (vL) 2 R2 Rκ 2 Rκ 2 2 (3.15) L Lt W 2 + L L Wx , W Wx + E3 = − γ −1 v˜ v˜ x 2 L (γ − 1)κ 2 2 u˜ x L Wx + (γ − 1)κ u˜ x W E 4 = R(L u˜ t − L x )W + v˜ v˜ x κ + R L 2 θ˜x x W, v v˜
E 2 = A 2 + µL x x + µLx2 ,
here and in the sequel the notation (· · · )x represents the term in the conservative form so that it vanishes after integration. Since it has no effect on the energy estimates, we do
Composite Wave of Two Viscous Shock Waves
855
not write them out in details for simplicity. We estimate E i , i = 2, 3, 4, term by term. From (2.22) and (2.23), we have Lt ≈ −
(P1 −
µU1,x V1 )t
µU3,x V3 )t µU − V33,x )2
+ (P3 −
(P1 + P3 − pm −
µU1,x V1
≈−
(P1 − (P1 −
µU1,x V1 )t µU1,x 2 V1 )
−
(P3 − (P3 −
µU3,x V3 )t µU3,x 2 V3 )
(3.16) = L 1,t + L 3,t , L x ≈ L 1,x + L 3,x , µU
where L i = (Pi − Vii,x )−1 = (bi − si2 Vi )−1 , i = 1, 3, which, together with (2.15), implies 1 1 A ≈ − (V1 L 1 )t + (γ −1)L 1 U1,x +(− (V3 L 3 )t +(γ − 1)L 3 U3,x ) =: A1 + A3 . 2 2 (3.17) We will treat all the errors arising from the relation “≈” later. We further obtain from (2.5), (2.7) and Proposition 2.1, that for i = 1, 3, 1 1 1 si Vi,x L i + si Vi L i2 si2 Vi,x + (γ − 1)L i Ui,x = − Ui,x L i2 (bi − 2(γ − 1)L i−1 ) 2 2 2 ≥ c|Ui,x |((3 − γ ) pm − Cδi ). (3.18)
Ai =
Therefore, if δ0 is small, we get the following inequality for E 2 with the help of the Cauchy inequality and Proposition 2.1, E 2 ≥ c{(|U1,x | + |U3,x |) 2 + x2 }.
(3.19)
For E 3 , we only need to estimate the first term. Since − L 1,t − L 3,t = s1 L 21 s12 V1,x + s3 L 23 s32 V3,x = |U1,x |s12 L 21 + |U3,x |s32 L 23 ,
(3.20)
using the Cauchy inequality again, we have (
Rκ 2 Rκ 2 2 L )x W Wx ≤ L Wx + C(|U1,x |2 + |U3,x |2 + 2x )W 2 . v˜ 2v˜
Note that 2x ≈ 0, we get E 3 ≥ c{(|U1,x | + |U3,x |)W 2 + Wx2 }.
(3.21)
On the other hand, we note that L 2 u˜ t − L x = L 2 (u˜ t + p˜ x − (
µu˜ x )x ) = −L 2 R˜ 1,x ≈ 0. v˜
(3.22)
Thus the Cauchy inequality and Proposition 2.1 give 1
1
E 4 ≤ Cδ 2 (|U1,x | + |U3,x |)( 2 + W 2 ) + Cδ 2 (Wx2 + 2x ).
(3.23)
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We now estimate the terms J1 and J2 W . We only consider J2 W because J1 can be treated in the same way. From (3.7), we have |J2 W | ≤ C N (T )(2x + x2 + Wx2 + ζx2 + x2x ) + Cδ(|U1,x | + |U3,x |) 2 .
(3.24)
Finally, we estimate all the error terms arising from the relation “≈” and also the terms R˜ 1 and R˜ 3 W . It turns out that all the integrals of these terms on (0, t) × R can be bounded by t 1 2 (x , Wx )2 dτ ). (3.25) Cδ0 (1 + 0
˜ In fact, for example, the typical terms like RW x and R˜ ( R˜ ≈ 0) are estimated as t ∞ t ∞ t ˜ ˜ | R|(|W Wx 2 dτ + C | R||| d xdτ x | + ||) d xdτ ≤ Cδ0 0
−∞
and t
0
0
−∞
t ∞ 3 ˜ | R||| d xdτ ≤ C (δ 2 + |α2 |δ 2 )e−cδ(|x|+τ ) || d xdτ 0 −∞ 0 −∞ t ∞ t ∞ |α2 | − cx 2 1+τ || d xdτ + C e (δ + |α2 |)e−c(|x|+τ ) || d xdτ +C 3 0 −∞ (1 + τ ) 2 0 −∞ t t 3 |α2 | −cδτ 2 (δ + |α2 |δ)e dτ + C dτ + Cδ0 ≤C 5 0 0 (1 + τ ) 4 ∞
1
1
≤ C(δ 2 + |α2 |) + C|α2 | + Cδ0 ≤ Cδ02 . Integrating (3.14) on (0, t) × R, using (3.15), (3.19), (3.21), (3.23) and (3.24)-(3.25), and choosing δ0 and N (T ) suitably small, we complete the proof of Lemma 3.1. Lemma 3.2. If δ0 and ε0 are suitably small, it follows that t 2 2 (t)1 + (, W )(t) + (x , x , Wx )2 dτ 0 t ∞ + (|U1,x | + |U3,x |)( 2 + W 2 ) d xdτ 0 −∞ t 1 1 2 2 2 ζx 2 + x x 2 dτ + Cδ02 . ≤ C0 1 + C(0 , W0 ) + C(N (T ) + δ0 ) 0
(3.26) Proof. From (3.5)2 , we have µ 1 R γ −1 xt − t + x = Wx + u˜ x + R˜ 1 − J1 . v˜ vL ˜ v˜ v˜
(3.27)
Multiplying (3.27) by x yields µ µ 1 2 R γ −1 2x − x = Wx + u˜ x + R˜ 1 − J1 x . 2x − x t + 2v˜ 2v˜ t vL ˜ v˜ v˜ t (3.28)
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Since x t = (x )t − (t )x + x2 ,
(3.29)
we obtain, if δ0 is small, µ 2 1 x − x d x)t + 2x d x ( 2v˜ 2vL ˜ ≤ C (x2 + Wx2 )d x + C J12 d x + Cδ0 (|U1,x | + |U3,x |) 2 d x + R˜ 12 d x. (3.30) The Cauchy inequality and (3.7) yield 2 2 2 J1 d x ≤ C N (T )(x , x , Wx ) + C N (T )x x +Cδ0 (|U1,x |+|U3,x |) 2 d x. (3.31) From (2.18), it is easy to see
t 0
∞ −∞
R˜ 12 d xdτ ≤ Cδ0 .
(3.32)
Multiplying (3.10) by a large constant C and using (3.30)-(3.32) and choosing N (T ) and α2 suitably small, we complete the proof of Lemma 3.2.
Lemma 3.3. If δ0 and ε0 are suitably small, it follows that t 2 (φ, ψ, ζ )(t) + (ψx , ζx )2 dτ 0 t 2 ≤ C(φ0 , ψ0 , ζ0 ) + C(N (T ) + δ0 ) (φx 2 + ψx x 2 + ζx x 2 ) dτ 0 t t ∞ +C (x , x , Wx )2 dτ + C (|U1,x | + |U3,x |) 2 d xdτ + Cδ0 . 0
−∞
0
(3.33) Proof. Applying ∂x to the system (3.2), we have ⎧ φt − ψx = 0, ⎪ ⎪ µ ⎪ p˜ R p˜ R µu˜ x φ µ ⎪ ⎪ ⎪ φ ζ − + − φ + ζ − ψ + = ψx x − R˜ 1,x + Q 1 , ψ x ⎪ ⎨ t v˜ x v˜ x v˜ x v˜ x v˜ x v˜ 2 x v˜ κ (3.34) κ θ˜x φ R µ 2 2µ ⎪ ⎪ ζ u ˜ u ˜ + pψ ˜ + ( p − p) ˜ u ˜ + φ − ψ − ζ + t x x x x x ⎪ ⎪ γ −1 v v˜ x v v˜ x v˜ 2 ⎪ ⎪ x ⎪ ⎩ = κζx x + u˜ R˜ 1,x − R˜ 2x + Q 2 , v˜
where
Q1 =
µ µ ψx φ − u˜ x φ 2 − 2 v v˜ v v˜
= O(1)(φ + ζ 2
2
R p˜ ˜ p − p˜ + φ − (θ − θ) v˜ v˜
+ |φx ||φ| + |ψx ||φx | + ζx2
+ |ψx x ||φ|),
x
(3.35)
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F. Huang, A. Matsumura
µψx2 + Q 2 = ( p˜ − p)ψx + v = O(1)(φ + ζ 2
2
κ θ˜x 2 φ v v˜ 2
+ |φ||φx | + ψx2
− x
κφζx v v˜
x
+ |φx ||ζx | + |ζx x ||φ|).
(3.36)
Here we note that the system (3.34) is one directly obtained by subtracting (1.1) by (2.15) and setting (v − v, ˜ u − u, ˜ θ − θ˜ ) = (φ, ψ, ζ ). Multiplying (3.34)1 by φ, (3.34)2 v˜ R by p˜ ψ, (3.34)3 by p˜ 2 ζ , and using the relation (3.12), we can have Lemma 3.3 by the same way as in Lemma 3.1. Lemma 3.4. If δ0 and ε0 are suitably small, it follows that t (φx , ψx , ζx )2 dτ φ(t)21 + (ψ, ζ )(t)2 + 0 t 2 ≤ Cφ0 1 + C(ψ0 , ζ0 )2 + C(N (T ) + δ0 ) (ψx x 2 + ζx x 2 ) dτ 0 t t ∞ +C (x , x , Wx )2 dτ + C (|U1,x | + |U3,x |) 2 d xdτ 0
0
+ Cδ0 .
−∞
(3.37)
Proof. We rewrite Eq. (3.34)2 as µ µ p˜ R p˜ R µu˜ x φ φxt −ψt + φx = ζx − φ+ ζ− ψx + + R˜ 1,x − Q 1 . v˜ v˜ v˜ v˜ x v˜ x v˜ x v˜ 2 x (3.38) Multiply (3.38) by φx and estimate the resulting formula, where we use the fact that the term |φx Q 1 | can be estimated as t ∞ t C t 2 |φx Q 1 | d xdτ ≤ φx dτ + Q 1 2 dτ ( > 0) (3.39) 0 0 −∞ 0 and
t
t
Q 1 dτ ≤ C N (T ) (ψ2 + ζ 2 + (φx , ψx , ζx )2 + (ψx x , ζx x )2 ) dτ 0 0 t t ∞ ≤ C N (T )( ((x , Wx )2 + C (|U1,x | + |U3,x |) 2 d xdτ 0 0 −∞ t (3.40) + ((φx , ψx , ζx )2 + (ψx x , ζx x )2 ) dτ ) + Cδ0 . 2
0
In fact, for an example, the worst term ψx2 φx2 in Q 1 2 above is estimated by t t ∞ |φx |2 |ψx |2 d xdτ ≤ ψx ψx x φx 2 dτ 0 −∞ 0 t (3.41) ≤ C N (T ) (ψx x 2 + φx 2 ) dτ. 0
Then the estimate (3.37) is given by combining the resulting estimate with Lemma 3.3 in the same way as Lemma 3.2. Thus the proof of Lemma 3.4 is completed.
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Lemma 3.5. If δ0 and ε0 are suitably small, it follows that (ψx , ζx )(t)2 +
t
(ψx x , ζx x )2 dτ t t 2 2 ≤ C(ψx , ζx )(0) + C (ψx , ψx , ζx ) dτ + C (x , x , Wx )2 dτ 0 0 t ∞ +C (|U1,x | + |U3,x |) 2 d xdτ + Cδ0 . (3.42) 0
0
−∞
Proof. Multiplying (3.34)2 by −ψx x , (3.34)3 by −ζx x and using the Cauchy inequality, we have
t
(ψx , ζx )(t)2 +
(ψx x , ζx x ) dτ t t ≤ C(ψx , ζx )(0)2 + C (φx , ψx , ζx ) dτ + C (x , x , Wx )2 dτ + Cδ0 0 0 t ∞ t ∞ +C (|U1,x | + |U3,x |) 2 d xdτ + (|Q 1 ψx x | + |Q 2 ζx x |) d xdτ. 0
−∞
0
0
−∞
(3.43) Since the terms |Q 1 x x |, |Q 2 ζx x | can be estimated in the same way as (3.39)-(3.41), we have (3.42) from (3.43). Thus we complete the proof of Lemma 3.5. Proof of Proposition 3.1. Noting the relation (3.12), then combining Lemmas 3.1- 3.5 and choosing δ0 and N (T ) suitably small, it is straightforward to have Proposition 3.1.
Proof of Theorem 1. Once Proposition 3.1 is proved, combining with the local existence theorem, we can prove that the initial value problem (3.5), (3.8) has the global solution in time (, , W ) ∈ X ([0, ∞)) for small δ0 and (0 , 0 , W0 )2 , and the estimate (3.10) holds for all t ∈ [0, ∞). Furthermore, this estimate (3.10) and the system (3.5) imply that
∞
(x , x , Wx )(t) dt + 2
0
0
d (x , x , Wx )(t)2 dt < ∞, dt
∞
which together with Sobolev’s lemma easily leads to the asymptotic behavior of the solution sup |(, , W )(x, t)| + sup |(x , x , Wx )(x, t)| → 0, t → ∞. x∈R
x∈R
Then, setting v = v˜ + x , u = u˜ + x , θ = θ˜ + Wx −
γ −1 R
1 2 − u˜ x , 2 x
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and noting that the smallness assumptions on |(v+ − v− , u + − u − , θ+ − θ− )| and initial disturbance in Theorem 1 imply the smallness of δ0 and (0 , 0 , W0 )2 , Theorem 1 is proved by Proposition 3.1. Proof of Theorem 2. As we explained in Sect. 1, due to the transformation (1.7), the initial boundary value problem (1.5) can be reduced to a special case of the initial value problem of (1.1)-(1.3) with (v− , u − , θ− ) = (v+ , −u + , θ+ ). For this problem, we first fix (v+ , θ+ ) and take u + (< 0) small. Then, it is proved that the corresponding Riemann problem (1.4)(2.1) admits the solution consisting of two shock waves and the strengths of the shock waves are small with same order. Then Theorem 2 can be directly implied by Theorem 1 as a special case.
References 1. Courant, R., Friedrichs, K.O.: Supersonic Flows and Shock Waves. New York: Wiley-Interscience, 1948 2. Goodman, J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95(4), 325–344 (1986) 3. Huang, F.M., Matsumura, A., Shi, X.: On the stability of contact discontinuity for compressible Navier-Stokes equations with free boundary. Osaka J. Math. 41(1), 193–210 (2004) 4. Huang, F.M., Matsumura, A.: Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations. Arch. Rat. Mech. Anal. 179, 55–77 (2005) 5. Huang, F.M., Xin, Z.P., Yang, T.: Contact discontinuity with general perturbation for gas motion. Adv. Math. 219(4), 1246–1297 (2008) 6. Huang, F.M., Zhao, H.J.: On the global stability of contact discontinuity for compressible Navier-Stokes equations. Rend. Sem. Mat. Univ. Padova 109, 283–305 (2003) 7. Kawashima, S., Matsumura, A.: Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101(1), 97–127 (1985) 8. Kawashima, S., Matsumura, A., Nishihara, K.: Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas. Proc. Japan Acad. 62, Ser. A, 249–252 (1986) 9. Liu, T.-P.: Linear and nonlinear large time behavior of general systems of hyperbolic conservation laws. Comm. Pure Appl. Math. 30, 767–796 (1977) 10. Liu, T.-P.: Nonlinear stability of shock waves for viscous conservation laws. Mem. Amer. Math. Soc. 56(329), 1–108 (1985) 11. Liu, T.-P.: Shock waves for compressible Navier-Stokes equations are stable. Comm. Pure Appl. Math. 39, 565–594 (1986) 12. Liu, T.-P.: Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50(11), 1113–1182 (1997) 13. Liu, T.-P., Xin, Z.P.: Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations. Commun. Math. Phys. 118(3), 451–465 (1988) 14. Liu, T.-P., Xin, Z.P.: Pointwise decay to contact discontinuities for systems of viscous conservation laws. Asian J. Math. 1, 34–84 (1997) 15. Matsumura, A., Mei, M.: Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary. Arch. Rat. Mech. Anal. 146, 1–22 (1999) 16. Matsumura, A., Nishihara, K.: On the stability of traveling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985) 17. Matsumura, A., Nishihara, K.: Asymptotics toward the rarefaction waves of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 3(1), 3–13 (1985) 18. Matsumura, A., Nishihara, K.: Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas. Commun. Math. Phys. 144, 325–335 (1992) 19. Matsumura, A., Nishihara, K.: Global Solutions for Nonlinear Differential Equations–Mathematical Analysis on Compressible Viscous Fluids (In Japanese). Nippon Hyoronsha, 2004 20. Nishihara, K., Yang, T., Zhao, H.-J.: Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations. SIAM J. Math. Anal. 35(6), 1561–1597 (2004) 21. Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Second Edition, New York: Springer-Verlag, 1994 22. Szepessy, A., Xin, Z.P.: Nonlinear stability of viscous shock waves. Arch. Rat. Mech. Anal. 122, 53–103 (1993)
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23. Xin, Z.P.: On nonlinear stability of contact discontinuities. In: Hyperbolic Problems: Theory, Numerics, Applications (Stony Brook, NY, 1994), River Edge, NJ: World Sci. Publishing, 1996, pp. 249–257 24. Zumbrun, K.: Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In: Handbook of Mathematical Fluid Dynamics, Vol. III, Amsterdam: North-Holland, 2004, pp. 311–533 Communicated by P. Constantin
Commun. Math. Phys. 289, 863–906 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0817-1
Communications in
Mathematical Physics
Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension Guido Gentile1 , Michela Procesi2 1 Dipartimento di Matematica, Università di Roma Tre, Roma,
I-00146, Italy. E-mail: [email protected]
2 Dipartimento di Matematica, Università di Napoli “Federico II”,
Napoli, I-80126, Italy. E-mail: [email protected] Received: 7 May 2008 / Accepted: 2 March 2009 Published online: 2 May 2009 – © Springer-Verlag 2009
Abstract: We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist. 1. Introduction 1.1. A brief survey of the literature. The problem of the existence of finite-dimensional tori, i.e. of quasi-periodic solutions, for infinite-dimensional systems, such as nonlinear PDEs, has been extensively studied in the literature. A particularly significant example is the nonlinear Schrödinger equation (NLS) ivt − v + µv = f (x, |v|2 ) v,
(1.1)
with periodic boundary conditions; here is the Laplacian on T D , µ > 0 is the “mass”, and the function f is real-analytic in a neighbourhood of the origin, where it vanishes. For instance f (x, |v|2 ) = |v|2 gives the cubic NLS, which is a widely studied model appearing in many branches of physics, such as the theory of Bose-Einstein condensation, plasma physics, nonlinear optics, wave propagation, theory of water waves [1]. Another physically interesting case is the NLS “with potential”, where the mass is substituted by a multiplicative potential V (x). However many results on quasi-periodic solutions are on simplified models, such as ivt − v + Mσ v = f (x, |v|2 ) v,
(1.2)
where Mσ is a “Fourier multiplier” i.e. a linear operator, depending on a finite number of free parameters σ , which commutes with the Laplacian. The free parameters (as many
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as the fundamental frequencies of the solution) are chosen in such a way as to impose suitable Diophantine conditions on the “frequencies” of the linearised equation, i.e. the eigenvalues of − + Mσ . Up to recent times, the only available results on quasi-periodic solutions for PDEs were confined to the case of one space dimension (D = 1). In this context the first results were obtained by Wayne, Kuksin and Pöschel [27,28,30,35], for the nonlinear Schrödinger equation (1.2) and the nonlinear wave equation (NLW) with Dirichlet boundary conditions, by using KAM techniques. Later on, Craig and Wayne in [16] proved similar result, for both Dirichlet and periodic boundary conditions, with a rather different method based on the Lyapunov-Schmidt decomposition and a Newton scheme. The case of periodic boundary conditions within the framework of KAM theory was then obtained by Chierchia and You [15]. When looking for periodic solutions, one can work directly on Eq. (1.1) and use the mass as a free parameter. However, if the mass vanishes, then the system becomes completely resonant, i.e. all the frequencies of the linearised equation are rationally dependent and there are infinitely many linear solutions with the same period. This makes the problem much harder – already in the case of periodic solutions. The completely resonant case was discussed by several authors, and theorems on the existence of periodic solutions for a large measure set of frequencies were obtained by Bourgain [10] for the NLW with periodic boundary conditions, by Gentile, Mastropietro and Procesi [22] and by Berti and Bolle [4] for the NLW with Dirichlet boundary conditions. In [23], we constructed, for the NLS with Dirichlet boundary conditions, periodic solutions which at leading order are wave packets. The existence of quasi-periodic solutions for the completely resonant NLW with periodic boundary conditions has been proved by Procesi [32] for a zero-measure set of two-dimensional rotation vectors, by Baldi and Berti [3] for a large measure set of two-dimensional rotation vectors, and by Yuan [36] for a large measure set of – at least three-dimensional – rotation vectors. Finding periodic and quasi-periodic solutions for PDEs in higher space dimensions (D > 1) is much harder than in the one-dimensional case, mainly due to the high degeneracy of the frequencies of the linearised equation. The first achievements in this direction were due to Bourgain, and concerned the existence of periodic solutions for NLW [8] and of periodic solutions (also quasi-periodic in D = 2) for the NLS [9]. The case of quasi-periodic solutions in arbitrary dimension was solved by Bourgain [11] for the NLW and the NLS with a Fourier multiplier as in (1.2). Bourgain’s method is based on a Nash-Moser algorithm. A proof of existence and stability of quasi-periodic solutions in high dimension was given by Geng and You, using KAM theory. Their result holds for a class of PDEs, with periodic boundary conditions and with nonlinearities which do not depend on the space variable. Both conditions are required in order to ensure a symmetry for the Hamiltonian which simplifies the problem in a remarkable way. Their class of PDEs includes the nonlinear beam equation (NLB) [19] and the NLS with a smoothing nonlinearity [20] ivt − v + Mσ v = ( f (|(v)|2 )(v)), where is a convolution operator. This equation has been studied in the mathematical literature (see for instance [31]), because it allows some simplifications, but it does not appear to be a physically interesting model. Their approach does not extend to the NLS with local nonlinearities like (1.2), mainly because it would require a “second Melnikov condition” at each iterative KAM step, and such a condition does not appear to be satisfied by the local NLS.
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865
Successively, Eliasson and Kuksin [17], by using KAM techniques, proved the existence and linear stability of quasi-periodic solutions for the NLS, with local nonlinearities. In their paper the main point is indeed to prove that one may impose a second Melnikov condition at each iterative KAM step. However, their result does not extend to other PDEs, because, in general (see for instance the case of the NLW in D > 1), it can be too hard to impose a second Melnikov condition – even on the unperturbed eigenvalues. In Eliasson and Kuksin’s paper only finite regularity (in the space variables) is found for the solutions. This is a drawback which does not arise in Bourgain’s approach [11], where an exponential decay of the Fourier coefficients is obtained. Again very recently, Berti and Bolle [7] proved the existence of periodic solutions for PDE systems with C k nonlinearities – all the other papers are in the analytic setting. They use a Nash-Moser algorithm suited for finitely differentiable nonlinearities, already employed in the one-dimensional case [5], and they find solutions belonging to suitable Sobolev classes. In [24] we studied the NLS with nonlocal smoothing nonlinearities, and we proved the existence of periodic solutions. In particular we discussed the completely resonant case, for which we obtained for D = 2 “wave packet” solutions similar to those found in [23] in the one-dimensional case. The main purpose of [24] was to extend the Lindstedt series method – based on renormalisation group ideas and originally introduced in [21] – to high dimensional PDEs in a simple nontrivial case, i.e. the nonlocal NLS. The proofs however strongly rely on the fact that the nonlinearity is nonlocal and does not cover the local NLS. In the present paper we prove an “abstract theorem” on the existence of Gevreysmooth periodic solutions for a wide class of PDEs with analytic nonlinearities satisfying some “abstract conditions”. This class of PDEs contains the local NLS as well as the NLB and the NLW; formal statements of both assumptions and results will be given in Sect. 2. Our approach is based on a standard Lyapunov-Schmidt decomposition – which separates the original PDEs into two equations, traditionally called the P and Q equations – combined with renormalised expansions à la Lindstedt to handle the small divisor problem. Although the general strategy, at least as far as the small divisor problem is concerned, is similar to [24], we wish to stress that working with local nonlinearities as in (1.1) makes the small divisor problem much more delicate and requires some substantial work to overcome the consequent difficulties. Moreover, not only the result in the present paper is more general, but the proofs are simpler and more compact. More details will be given along the proofs. As a first application of our abstract theorem we recover various known results discussed in the aforementioned literature. Then – this is the main original result in our paper – we apply the abstract theorem to the completely resonant (µ = 0) local NLS and NLB equations and prove the existence of “wave packet” solutions in the spirit of [23]. Again, proving existence of “wave packet” solutions for the local NLS is much more challenging than for the nonlocal case discussed in [24]. Indeed the proof is completely different – see Subsect. 1.3 for a comparison. In the following Subsect. 1.2 we provide an informal overview of the main hypotheses and of the Lindstedt series method. Then in Subsect. 1.3 we describe the main applications, with particular attention to the completely resonant cases. We conclude this section by mentioning that periodic solutions are also found in the literature for NLS on R D with an external confining potential; see for instance [13,25,34]. In principle, one could expect that confining potentials have an effect similar to that of imposing Dirichlet boundary conditions on a finite domain, but the setting is rather different: the potentials
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are taken to depend on a small length scale h, and in the limit h → 0 the considered NLS reduce to (1.1). Thus, the unperturbed equation is the same, but on a completely different domain (the full space). For this class of equations solitonic solutions, periodic in time and exponentially decaying in space, are constructed in the quoted references. 1.2. Informal presentation of the main result and techniques. To describe the main hypotheses, consider equations of the form D(ε)u(x) = ε f (u(x), u(x)), ¯
x = (t, x),
(1.3)
where (for instance) x ∈ T D+1 , ε is a small real parameter, D(ε) is a linear operator depending on ε, and f (u, u) ¯ is an analytic function, possibly depending also on x and ε, which is superlinear at u = 0. We shall see that our class of Eqs. (2.7) and (2.8) reduce to the form (1.3) after some rescaling. We require three properties on Eqs. (1.3), which we call Hypotheses 1 to 3 (see Sect. 2 for a precise formulation). Informally, the properties are the following: 1. D(ε) is diagonal in the Fourier basis with real eigenvalues δν (ε) which are smooth in both ν and ε and satisfy appropriate bounds on the derivatives. 2. The Q equation at ε = 0 (bifurcation equation) has a non-degenerate solution which is analytic in space and time. 3. For each ε the set of “singular” frequencies S(ε) := {ν ∈ Z D+1 : |δν (ε)| ≤ 1/2} is of the form S(ε) = ∪ j∈N j (ε) where the j (ε) are disjoint finite sets which are “well separated” and “not too big”. Properties 1 and 3 are assumptions on the linear part, while Property 2 is an assumption on the nonlinearity. In particular the solution of the bifurcation equation identifies the solution of the linearised PDE from which the solution of the full PDE branches off. Property 1 can probably be weakened to cover cases in which D(ε) is not diagonal in the Fourier basis but its eigenfunctions are still “well localised” with respect to the Fourier basis (namely the Fourier coefficients of the eigenfunctions have a uniform exponential decay). Property 2 is required to solve the bifurcation equation by the implicit function theorem. Also this hypothesis could be weakened; see for instance [6] for a discussion of weaker hypotheses in the case of the NLW in dimension D = 1. Property 3 was introduced by Bourgain to prove the existence of periodic solutions for the NLS in high dimension [9]. This hypothesis is essential for our proof; a similar, weaker hypothesis appears in [7]. Assuming Properties 1 to 3 we prove our main result, which is the Main Theorem in Sect. 2, by a “renormalised series expansion”. The proof of the theorem is performed through two steps, formally described by Propositions 1 and 2 in Subsect. 4.3. To illustrate our method, write Eqs. (1.3) in Fourier space, δν (ε) u ν = ε f ν ,
f ν = f ν ({u ν , u¯ ν }ν ∈Z D+1 ),
(1.4)
where u ν and f ν are the ν th Fourier coefficients of u(x) and f (u(x), u(x)), ¯ respectively. Then one studies (1.4): the presence of the small parameter ε suggests to look for a recursive solution, and hence to write the solution-to-be in the form of a series expansion in powers of ε. This is a very natural approach for any problem in perturbation theory, and leads to a graphical representation of the solution order by order in terms of trees; see the diagrammatic expansion described in Sect. 5.3 and, in particular, Definition 5.7. Thus, the solution can be given a meaning as a formal power series, provided an irrationality
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condition is assumed on the eigenvalues δν (ε); cf. Lemma 5.10 with M = L = 0 in Subsect. 5.3. On the other hand the convergence of the series fails to be proved, and is likely not to hold. This fact suggests to modify Eqs. (1.4) into new equations which have a solution in the form of an absolutely convergent series (which is not a power series in ε). First of all we introduce some notation by rewriting (1.4) as D(ε) U = η F,
(1.5)
where η = ε, U := {u ν }ν ∈Z D+1 , F := { f ν }ν ∈Z D+1 , and D(ε) = diag{δν (ε)}ν ∈Z D+1 . Then, we change (1.5) by considering ε and η as two independent parameters, and adding “corrections” which are linear in U , that is (D(ε) + M ) U = η F + L U,
(1.6)
where M = M (ε) and L = L(η, ε) are self-adjoint matrices (called “counterterms”) with the only restriction that Mν ,ν (ε) = L ν ,ν (η, ε) = 0 if ν, ν do not both belong to the same j (ε) for some j. For technical reasons, we shall write M (ε) = χ 1 (ε)M χ1 (ε), where the matrices χ 1 (ε) impose the restriction described above, and M is a matrix of free parameters. As a matter of fact, the description above does not mention some technical intricacies. For instance we shall need η to be a suitable fractional power of ε (depending on the leading order of the nonlinearity). Moreover for convenience we shall double Eqs. (1.6). Hence both M and L will carry two further indices; see (2.10) and (4.3). The reason why we introduce counterterms which are linear in U is that the terms which give problems in the naive power series expansion for (1.5) can be eliminated by adding a linear term to F (this is in the same spirit as Moser’s modifying term theorem [29] in KAM theory for finite-dimensional systems). In fact, the terms which may be an obstacle in proving the convergence of the formal power series are classified in a relatively simple way through the language of the diagrammatic expansion. Assume for the time being that M = 0. Then it is possible to choose L = L(η, ε) as a convergent power series in η in such a way that the formal power series for the solution of the modified Eq. (1.6) converges, provided the eigenvalues δν (ε) satisfy some Diophantine conditions. Of course, there remains the major problem that the modified Eq. (1.6) with M = 0 and L = 0 is not the original (1.5). Then we introduce M = 0, and show that for any M it is possible to determine L = L(η, ε, M) as a function of M in such a way that Eqs. (1.6) turn out to be solvable. The proof proceeds essentially in the same way as for M = 0, provided ε and M are in an appropriate Cantor set so that the eigenvalues of D(ε) + M satisfy some Diophantine conditions (cf. Definition 5.21 and Lemma 5.24 in Subsect. 5.4). In particular, in order to be able to impose such conditions we shall use in a decisive way the block structure of the matrix M . This first step is essentially the content of Proposition 1 in Subsect. 4.3. Of course only if η = ε and M = L(η, ε, M) the modified equation reduces to the original one. Thus, once the first step is accomplished, we are left with the problem of solving the compatibility equation L(η, ε, M) = M . This will be done by showing that, at the cost of further shrinking the set of allowed values for ε, one can choose M = M(ε) in such a way to solve the compatibility equation. This second step in the proof corresponds to Proposition 2 in Subsect. 4.3.
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1.3. Applications of the abstract theorem. In Subsect. 3.1 we consider the NLS, NLW and NLB equations in the non-resonant case (under a Diophantine condition on the mass) and recover the known results on the existence of periodic solutions. In Subsect. 3.2 – this represents the main novelty of this paper – we discuss cases in which the bifurcation equation is infinite-dimensional, such as the zero-mass NLS and NLB. In the resonant case the linearised equation has an infinite-dimensional space of periodic solutions with the same period, so that in principle we have at our disposal infinitely many linear solutions with the same period which may be extended to solutions of the nonlinear equation. Indeed we find a denumerable infinity of solutions with the same minimal period even in the presence of the nonlinearity. More precisely, we prove the existence of periodic solutions which at leading order involve an arbitrary finite number of harmonics not too far from each other, and which therefore can be described as distorted wave packets. Solutions of this kind are very natural in the case of completely resonant PDEs, where all harmonics are commensurate in the absence of the nonlinearity. An essential ingredient for the existence of such solutions is the particular form of the bifurcation equation: the proof strongly relies on the fact that the leading order of the nonlinearity is cubic and gauge-invariant. In this latter case the most challenging problem is proving Property 2 in Subsect. 1.2, i.e. the non-degeneracy of the solutions of the bifurcation equation. The problem of the existence of periodic and quasi-periodic solutions in completely resonant systems in higher dimension was already considered by Bourgain in [9], where he constructed quasi-periodic solutions with two frequencies, in D = 2, for the resonant NLS with periodic boundary conditions (in contrast with the non-resonant case, where the Fourier multiplier allows to find quasi-periodic solutions with any number of frequencies). However, the non-degeneracy problem is especially complicated in the case of the Dirichlet boundary condition, which we explicitly consider in this paper. To prove non-degeneracy, we require that the nonlinearity does not depend explicitly on the space variables – this is a sufficient condition. Moreover we use a combinatorial Lemma, proved in [24], and some results in algebraic number theory. In [24], we proved the existence and non-degeneracy of the “wave packet” solutions of the bifurcation equation for the nonlocal NLS, but only in the case D = 2. For higher dimension we required some additional condition, which in practice should be checked case-by-case. In fact, the problem was reduced to that of inverting a finite number of matrices of finite – but very big – dimensions, so that a computer-assisted check should be relied upon. Moreover even in D = 2, the proof of the non-degeneracy of the bifurcation equation given in [24] does not extend to the local NLS. The proof in this paper is completely different and much simpler, at the cost of requiring that the nonlinearity does not depend on the space variables. Moreover the result holds in any dimension D. 2. Formal Statement of the Main Result In this section, we give a rigorous description of the PDE systems we shall consider, and a formal statement of the results that we shall prove in the paper. Throughout the paper we shall call a function F(t, x), with x = (x1 , . . . , x D ) ∈ R D and t ∈ R, even [resp. odd] in x – or even [resp. odd] tout court – if it is even [resp. odd] in each of its arguments xi . Let S be the D dimensional square [0, π ] D , and let ∂S be its boundary. We consider for instance the following class of equations:
Periodic Solutions for a Class of Nonlinear PDEs in Higher Dimensions
(i∂t + P(−) + µ) v = f (x, v, v), ¯ v(t, x) = 0
(t, x) ∈ R × S, (t, x) ∈ R × ∂S,
869
(2.1)
where is the Laplacian operator, P : R+ → R+ is a strictly increasing convex C ∞ function with P(0) = 0, P(−) eim·x = P(|m|2 ) eim·x ( · denotes the scalar product in R D ), µ is a real parameter which – we can assume – belongs to some finite interval (0, µ0 ), with µ0 > 0, and x → f (x, v(t, x), v(t, ¯ x)) is an analytic function which is super-linear in v, v¯ and odd (in x) for odd v(t, x), i.e. ar,s (x) vr v¯ s , N ≥ 1, (2.2) f (x, v, v) ¯ = r,s∈N:r +s≥N +1
with ar,s (x) even for odd r + s and odd otherwise; notice that the leading order of the nonlinearity is N +1. We shall look for odd 2π -periodic solutions with periodic boundary conditions in [−π, π ] D . We require for f in (2.2) to be of the form f (x, v, v) ¯ =
∂ H (x, v, v) ¯ + g(x, v), ¯ ∂ v¯
We also consider the class of equations ∂tt + (P(−) + µ)2 v = f (x, v), v(t, x) = 0, and finally the wave equation (∂tt − + µ) v = f (x, v), v(t, x) = 0,
H (x, v, v) ¯ = H (x, v, v). ¯
(2.3)
(t, x) ∈ R × S, (t, x) ∈ R × ∂S,
(2.4)
(t, x) ∈ R × S, (t, x) ∈ R × ∂S,
(2.5)
where f (x, v) is of the form (2.2) with s identically zero and ar (x) := ar,0 (x) real (by parity ar (x) is even for odd r and odd for even r ). We shall consider also (2.1), (2.4) and (2.5) with periodic boundary conditions: in that case, we shall drop the condition for f to be odd. Solutions of the linearised equations are superpositions of oscillations, e.g. in case (2.1) they are of the form vm ei m t eim·x ,
m = P(|m|2 ) + µ; m∈Z D
and similar expressions hold for the other equations. For all these classes of equations we prove a.e. in µ the existence of small periodic solutions with frequency ω close to a given linear frequency ω0 = m and in an appropriate Cantor set of positive measure. For concreteness we shall focus on the linear oscillation with m =√(1, 1, . . . , 1), which yields the frequency ω0 = P(D) + µ for (2.1) and (2.4) and ω0 = P(D) + µ for (2.5); of course, the analysis could be easily extended to any other harmonics. For P(x) = x and µ = 0 the system becomes completely resonant: in this case all the harmonics are commensurate with each other. We shall concentrate on the NLS and NLB, and shall prove that there exist periodic solutions which look like perturbations of wave packets, i.e. of superpositions of linear oscillations peaked around given harmonics.
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We introduce a smallness parameter by rescaling v(t, x) = ε1/N u(ωt, x), ε > 0,
(2.6)
with ω = P(D) + µ − ε for (2.1) and (2.4) and ω2 = P(D) + µ − ε for (2.5). We shall formulate our results in a more abstract context, by considering the following classes of equations with Dirichlet boundary conditions: (I) (II)
D(ε) u = ε f (x, u, u, ¯ ε1/N ), u(t, x) = 0, D(ε) u = ε f (x, u, ε1/N ), u(t, x) = 0,
(t, x) ∈ T × S, (t, x) ∈ T × ∂S, (t, x) ∈ T × S, (t, x) ∈ T × ∂S,
(2.7a) (2.7b)
where T := R/2π Z and D(ε) is a linear (possibly integro-)differential wave-like operator with constant coefficients depending on a (fixed once and for all) real parameter ω0 and on the parameter ε. We can treat the case of periodic boundary conditions in the same way: (I) (II)
D(ε) u = ε f (x, u, u, ¯ ε1/N ), D(ε) u = ε f (x, u, ε1/N ),
(t, x) ∈ T × T D , (t, x) ∈ T × T D ,
(2.8a) (2.8b)
with the same meaning of the symbols as in (2.7). In Case (I) we assume that f (x, u, u, ¯ ε1/N ) is a rescaling of a function f (x, u, u) ¯ defined as in (2.2) and satisfying (2.3). In Case (II) we suppose D(ε) real and f real for real u, so that it is natural to look for real solutions u = u. ¯ For ν ∈ Z D+1 set ν = (ν0 , m), with ν0 ∈ Z and m = (ν1 , . . . , ν D ) ∈ Z D and |ν| = |ν0 | + |m| = |ν0 | + |ν1 | + . . . + |ν D |. For x = (t, x) = (t, x1 , . . . , x D ) ∈ R D+1 , set ν ·x = ν0 t +m ·x = ν0 t +ν1 x1 +. . .+ν D x D . Set also Z+ = {0}∪N and Z∗D+1 = Z D+1 \{0}. Finally denote by δ(i, j) the Kronecker delta, i.e. δ(i, j) = 1 if i = j and δ(i, j) = 0 otherwise. Given a finite set A we denote by |A| the cardinality of the set. Throughout the paper, for z ∈ C we denote by z the complex conjugate of z. Since all the results of the paper are local (that is, they concern small amplitude solutions), we shall always assume that the hypotheses below are satisfied for all ε sufficiently small. Hypothesis 1 (Conditions on the linear part). 1. D(ε) is diagonal in the Fourier basis {eiν ·x }ν ∈Z D+1 with real eigenvalues δν (ε) which are C ∞ in both ν and ε. 2. For all ν ∈ Z∗D+1 , one has either δν (0) = 0 or |δν (0)| ≥ γ0 |ν|−τ0 , for suitable constants γ0 , τ0 > 0. 3. For all ν ∈ Z∗D+1 one has |∂ε δν (ε)| < c2 |ν|c0 and, if |δν (ε)| < 1/2, one has |∂ε δν (ε)| > c1 |ν|c0 as well, for suitable ε-independent constants c0 , c1 , c2 > 0. 4. For all ν ∈ Z∗D+1 such that |δν (ε)| < 1/2 one has |∂ε ∂ν δν (ε)| ≤ c3 |ν|c0 −1 , for a suitable ε-independent constant c3 > 0. 5. In Case (I) we require that if for some ε and for some ν 1 , ν 2 ∈ Z D+1 one has |δν 1 (ε)|, |δν 2 (ε)| < 1/2, then |ν 1 − ν 2 | ≤ |ν 1 + ν 2 |.
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We now pass to the equation for the Fourier coefficients. We write u(t, x) =
u ν eiν ·x ,
(2.9)
ν ∈Z D+1 + − and introduce the coefficients u ± ν by setting u ν := u ν and u ν := u ν . Analogously we define f ν = f ν ({u}, η) := [ f (x, u, u, ¯ η)]ν = [ar,s (x)u r u¯ s ]ν r,s∈N:r +s=N +1
+ ηr +s−N −1 [ar,s (x)u r u¯ s ]ν , r,s∈N:r +s>N +1
, [·]ν denotes the Fourier coefficient with label ν, and we set where {u} = {u σν }σν ∈=± Z D+1 f ν+ := f ν and f ν− := f ν . Naturally f νσ depends also on the Fourier coefficients of the + := ar,s,m and functions ar,s (x), which we denote by ar,s,m , with m ∈ Z D ; we set ar,s,m − ar,s,m := ar,s,m . Then in Fourier space Eqs. (2.7) and (2.8) give δν (ε) u σν = ε f νσ ({u}, ε1/N ), ν ∈ Z D+1 ,
(2.10)
and in the case of Dirichlet boundary conditions we shall require u σν = −u σSi (ν ) for all i = 1, . . . , D, where Si (ν) is the linear operator that changes the sign of the i th component of ν. Remark 2.1. The reality condition on H in (2.3) reads − + (s + 1) as+1,r −1,m = r ar,s,−m .
(2.11)
Moreover, by the analyticity assumption on the nonlinearity, one has |ar,s,m | ≤ Ar1+s e−A2 |m| for suitable positive constants A1 and A2 independent of r and s. Remark 2.2. We have doubled our equations by considering separately the equations for u +ν and u − ν – which clearly must satisfy a compatibility condition. In Case (II) one can + work only on u +ν , since u − ν = u −ν . In other examples it may be possible to reduce to solutions with u ν real for all ν ∈ Z D+1 , but we found it more convenient to introduce the doubled equations in order to deal with the general case. Following the standard Lyapunov-Schmidt decomposition scheme we split Z D+1 into two subsets called P and Q and treat the equations separately. By definition we call Q the set of those ν ∈ Z D+1 such that δν (0) = 0; then we define P = Z D+1 \Q. Equations (2.10) restricted to the P and Q subset are called respectively the P and Q equations. Hypothesis 2 (Conditions on the Q equation). 1. For all ν ∈ Q one has λν (ε) := ε−1 δν (ε) ≥ c > 0, where c is ε-independent.
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2. The Q equation at ε = 0, λν (0) u σν = f νσ ({u σ }, 0), ν ∈ Q, has a non-trivial non-degenerate solution (0) q (0) (t, x) = u ν eiν ·x , ν ∈Q
where non-degenerate means that the matrix
Jνσ,σ ,ν = λν (0) δ(ν, ν ) δ(σ, σ ) −
∂ f νσ
∂u σν
({q (0) }, 0)
(0) −λ0 |ν −ν | , is invertible. Moreover one has |u ν | ≤ 0 e−λ0 |ν | and (J −1 )σ,σ ν ,ν ≤ 0 e for suitable constants 0 and λ0 . Remark 2.3. The solution of the bifurcation equation, i.e. of the Q equation at ε = 0, could be assumed to be only Gevrey-smooth. Note also that, even when Q is infinitedimensional, the number of non-zero Fourier components of q (0) (t, x) can be finite. The non-degeneracy condition in Hypothesis 2 is required in order to apply implicit function arguments. In principle the assumption can be weakened; see for instance [6]. However, to find optimal conditions is a very difficult task, already in the finitedimensional case; see for instance [14], where the case of hyperbolic lower-dimensional tori with one normal frequency is investigated for finite-dimensional quasi-integrable systems. Definition 2.4 (The sets S(ε), S, and R). Let ε0 be a fixed positive constant. For ε ∈ [0, ε0 ] we set S(ε) := {ν ∈ P : |δν (ε)| < 1/2} and S = ∪ε∈[0,ε0 ] S(ε). Finally we call R the subset P\S. Remark 2.5. Note that ν ∈ R means that |δν (ε)| ≥ 1/2 for all ε ∈ [0, ε0 ]. The following definitions appear (in a slightly different form) in the papers by Bourgain. We shall use the formulation proposed by Berti and Bolle in [7], in terms of equivalence classes, because it will turn out to be very convenient. Definition 2.6 (The equivalence relation ∼). Let β and C2 be two fixed positive constants. We say that two vectors ν, ν ∈ S(ε) are equivalent, and we write ν ∼ ν , if the following happens: one has |δν (ε)|, |δν (ε)| < 1/2 and there exists a sequence {ν 1 , . . . , ν K } in S(ε), with ν 1 = ν and ν K = ν , such that δν (ε) < 1 , k 2
|ν k − ν k+1 | ≤
C2 (|ν k | + |ν k+1 |)β , 2
k = 1, . . . , K − 1.
Denote by j (ε), j ∈ N, the equivalence classes with respect to ∼. Remark 2.7. The equivalence relation ∼ induces a partition of S(ε) into disjoint sets { j (ε)} j∈N . Note also that, if ν, ν ∈ j (ε), then it is not possible that for some ε one has ν ∈ j1 (ε ) and ν ∈ j2 (ε ) with j1 = j2 .
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Hypothesis 3 (Conditions on the set S(ε): separation properties). There exist four ε-independent positive constants α, β, C1 , C2 , with α small enough and β ≤ α, such that the sets j (ε) constructed according to Definition 2.6 satisfy | j (ε)| ≤ C1 p αj (ε), where p j (ε) = minν ∈ j (ε) |ν|, for all j ∈ N. Remark 2.8. The condition that α be small will be essential in the following. On the contrary we could also allow β > α and this would also simplify the forthcoming analysis. However we prefer to consider directly the more relevant case β ≤ α because this is the case which arises in all applications. The relation between α and β is dictated by the explicit application one has in mind. On the contrary, it would be interesting to look for optimal bounds on the constant α in Hypothesis 3. Lemma 2.9. Hypothesis 3 implies the following properties: β C2 1. dist( j (ε), j (ε)) ≥ p j (ε) + p j (ε) ∀ j, j ∈ N such that j = j , 2 α+β 2. diam( j (ε)) ≤ C1 C2 p j (ε) ∀ j ∈ N, 3.
max |ν| ≤ 2 p j (ε)
ν ∈ j (ε)
∀ j ∈ N, α+β
and, furthermore, we can always assume that 2c0 −1 C1 C2 p j where the constants c1 and c3 are defined in Hypothesis 1.
≤ ζ p j , with ζ c3 < c1 /4,
Proof. Properties 1–3 follow immediately from Definition 2.6. Indeed, the bound | j (ε)| ≤ C1 p αj (ε), used in Definition 2.6, yields |ν| ≤ 2 p j (ε) for all ν ∈ j (ε). β
Then diam( j (ε)) ≤ C1 p αj C2 (4 p j (ε))/2 and, for ν ∈ j (ε) and ν ∈ j (ε) with j = j , one has |ν − ν | ≥ C2 (|ν| + |ν |)β /2 ≥ C2 ( p j (ε) + p j (ε))β /2. Remark 2.10. The sets j (ε) are locally constant, in the sense that for almost all ε¯ ∈ [0, ε0 ] there exists an interval I containing ε¯ such that j (ε) = j (¯ε ) for all ε ∈ I. Definition 2.11. Given f : T D+1 → C define the norm | f |κ := sup | f ν | eκ|ν |
1/2
ν ∈Z D+1
with
f (x) =
eiν ·x f ν .
(2.12)
ν ∈Z D+1
We can now state our main result. Main Theorem. Consider a PDE in the class described by (2.7) and (2.8), such that the Hypotheses 1, 2 and 3 hold. Then there exist two positive constants ε0 and κ, a Cantor set E ⊂ [0, ε0 ], and a function u(t, x) = u(t, x; ε) with the following properties: 1. u(t, x; ε) is 2π -periodic in time, Gevrey-smooth both in time and in space, and C 1 in ε ∈ [0, ε0 ]; 2. for all ε ∈ [0, ε0 ] one has u(t, x; ε) − q (0) (t, x)κ ≤ Cε, where q (0) is defined in Hypothesis 2; 3. u(t, x; ε) solves the PDE for ε ∈ E; 4. the set E has density 1 at ε = 0. The result above provides only Gevrey regularity. In [24] we could prove analyticity of the periodic solutions of the non-local NLS only in the non-resonant case. We leave as an open problem whether the solution in the Main Theorem is analytic in time and/or space.
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3. Applications 3.1. Non-resonant equations. Let us prove that Eqs. (2.1), (2.4), and (2.5) – in particular the NLS, the NLB and the NLW – comply with all Hypotheses 1 to 3 in Sect. 2 and therefore admit a periodic solution by the Main Theorem. 3.1.1. The NLS equation. Theorem 3.1. Consider the nonlinear Schrödinger equation in dimension D ¯ i∂t v − v + µ v = f (x, v, v), with Dirichlet boundary conditions on the square [0, π ] D , where µ ∈ (0, µ0 ) ⊂ R and f (x, v, v) ¯ = |v|2 v + O(|v|4 ), that is f is given according to (2.2) and (2.3), with N = 2, a2,1 = 1 and ar,s = 0 for r, s such that r + s = 3 and (r, s) = (2, 1). Then there exist a full measure set M ⊂ (0, µ0 ) and two positive constants ε0 and κ such that the following holds. For all µ ∈ M there exists a Cantor set E(µ) ⊂ [0, ε0 ], such that for all ε ∈ E(µ) the equation admits a solution v(t, x), which is 2π/ω-periodic in time and Gevrey-smooth both in time and in space, and such that √ v(t, x) − εq0 eiωt sin x1 . . . sin x D ≤ Cε, κ
ω = D + µ − ε,
D/2 4 |q0 | = . 3
The set E = E(µ) has density 1 at ε = 0. With the notations of Sect. 2 one has δν (ε) = −ωn + |m|2 + µ, with ω = ω0 − ε and ω0 = D + µ. Then it is easy to check that all items of Hypothesis 1 are satisfied provided µ is chosen in such a way that | − ω0 n + |m|2 | ≥ γ0 |n|−τ0 . This is possible for µ in a full measure√set; cf. Eq. (2.1) in [24]. Then Hypothesis 1 holds with c0 = c2 = c3 = 1 and c1 = 1/ 1 + 4ω0 . The subset Q is defined as Q := {(n, m) ∈ Z1+D : n = 1, |m i | = 1 ∀i = 1, . . . D}, and one can assume q0 to be real, so that, by the Dirichlet boundary conditions, Q is in fact one-dimensional, and u n,m = ±q0 for all (n, m) ∈ Q. The leading order of the Q equation is explicitly studied in [24], where it is proved that Hypothesis 2 is satisfied. Finally, Hypothesis 3 has been proven by Bourgain [9] (see also Appendix A6 in [24]). Of course, Theorem 3.1 refers to solutions with m = (1, 1, . . . , 1), but it easily extends to solutions which continue other harmonics of the linear equation; see comments in [24]. Also, the condition on the nonlinearity can be weakened. In general N can be any integer N > 1, and no other conditions must be assumed on the functions ar,s (x) beyond those mentioned after (2.2). In that case (for simplicity we consider the same solution of the linear equation as in Theorem 3.1), the leading order of the Q equation becomes q0 = sign(ε)A0 q0N (again by taking for simplicity’s sake q0 to be real), where A0 is a constant depending on the nonlinearity. If A0 is non-zero, this surely has a non-trivial non-degenerate solution q0 either for positive or negative values of ε. In general the non-degeneracy condition in item 2 of Hypothesis 2 has to be verified case by case by computing A0 .
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3.1.2. The NLW equation. Theorem 3.2. Consider the nonlinear wave equation in dimension D, ∂tt v − v + µ v = f (x, v), with Dirichlet boundary conditions on the square [0, π ] D , where µ ∈ (0, µ0 ) ⊂ R and f (x, v) = v 3 + O(v 4 ), that is f is given according to (2.2), with s = 0, N = 2, a3,0 = 1. Then there exist a full measure set M ⊂ (0, µ0 ) and two positive constants ε0 and κ such that the following holds. For all µ ∈ M there exists a Cantor set E(µ) ⊂ [0, ε0 ], such that for all ε ∈ E(µ) the equation admits a solution v(t, x), which is 2π/ω-periodic in time and Gevrey-smooth both in time and in space, and such that (D+1)/2 √ v(t, x)−q0 ε cos ωt sin x1 . . . sin x D ≤ Cε, ω = D+µ − ε, q0 = 4 . κ 3 The set E = E(µ) has density 1 at ε = 0. In that case one has δν (ε) = −ω2 n 2 + |m|2 + µ, with ω2 = ω02 − ε and ω02 = D 2 + µ. Once more, it is easy to check that Hypothesis 1 is satisfied provided µ is chosen in a full measure set, with c0 = c2 = c3 = 1 and c1 = 1/(1 + 4ω02 ). The subset Q is given by Q := {(n, m) ∈ Z1+D : n = ±1, |m i | = 1 ∀i = 1, . . . D}, and, if one chooses to look for solutions that are even in time, then Q is one-dimensional. The Q equation at ε = 0 can be discussed as in the case of the nonlinear Schrödinger equation. For instance for f as in the statement of Theorem 3.2 the non-degeneracy in item 2 of Hypothesis 2 can be explicitly verified. Again, the analysis easily extends to more general situations, under the assumption that the Q equation at ε = 0 admits a non-degenerate solution. For a fixed nonlinearity, this can be easily checked with a simple computation. Hypothesis 3 has been verified by Bourgain [8], under some strong conditions on ω. Recently the same separation estimates have been proved by Berti and Bolle [7], by only requiring that ω2 be Diophantine. 3.1.3. Other equations. The separation properties for the NLS equation imply similar separation also for the nonlinear beam (NLB) equation ∂tt v + ( + µ)2 v = f (x, v), and in that case we can also consider nonlinearities with one or two space derivatives. As in the previous cases one restricts µ to√ some full measure set, and Hypothesis 1 holds with c0 = c3 = 2, c2 = 1 and c1 = 1/ 1 + 2ω0 . This implies that the subset Q is one-dimensional, provided we look for real solutions which are even in time. The same kind of arguments holds for all equations of the form (2.1) and (2.4). The separation of the points (m, |m|2 ) in Z D+1 implies, by convexity, also the separation of (m, P(|m|2 )), with P(x) defined after (2.1).
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3.2. Completely resonant equations. Here we describe an application to completely resonant NLS and NLB equations, namely Eqs. (2.1) and (2.4) with P(x) = x and µ = 0, and with Dirichlet boundary conditions (the case of periodic boundary conditions is easier for fully resonant equations). Since the equation is completely resonant we need some assumption on the nonlinearity in order to comply with Hypothesis 2. We set f (x, v, v) ¯ = |v|2 v for the NLS and f (x, v) = v 3 for the NLB (the NLB falls in Case (II) and we look for real solutions), but our proofs extend easily to deal with higher order corrections which are odd and do not depend explicitly on the space variables. In the case of the NLS we say that the leading term of the nonlinearity is cubic and gauge-invariant.1 The validity of Hypothesis 1 can be discussed as in the non-resonant equations of Subsects. 3.1. The separation properties (Hypothesis 3) do not change in the presence of a mass term, and they have been already discussed in the non-resonant examples of Subsect. 3.1. Thus, we only need to prove the non-degeneracy of the solution of the Q equation. Since the nonlinearity does not depend explicitly on x we look for solutions such that u ν ∈ R. We follow closely [24], but we set ω0 = 1. This is done for purely notational reasons, and is due to the fact that a trivial rescaling of time allows us to put ω0 = 1. 3.2.1. The NLS equation. The subset Q is infinite-dimensional, i.e. Q := {(n, m) ∈ N × Z D : n = |m|2 }. We set u (n,m) = qm = am + O(ε1/2 ) for (n, m) ∈ Q and restrict our attention to the case qm ∈ R. At leading order, the Q equation is (cf. [24]) |m|2 am = am 1 am 2 am 3 . (3.1) m 1 ,m 2 ,m 3 m 1 +m 2 −m 3 =m m 1 −m 3 ,m 2 −m 3 =0
Note that in the case of [24], the left-hand side of (3.1) was |m|2+2s D −1 am , with √ sa free parameter; then (3.1) is recovered by setting s = 0 and rescaling by 1/ D the coefficients qm . By Lemma 17 of [24] – which holds for all values of s – for each N0 ≥ 1 there exist infinitely many finite sets M+ ⊂ Z+D with N0 elements such that Eq. (3.1) admits the solution (due to the Dirichlet boundary conditions we describe the solution in Z+D ) ⎧ 0, m ∈ Z+D \M+ ⎪ ⎪ ⎪ ⎛ ⎞ ⎨ am = 1 ⎝|m|2 − c1 ⎪ D+1 |m |2 ⎠, m ∈ M+ , ⎪ ⎪ − 3D ⎩ 2 m ∈M+
with c1 = 2 D+1 /(2 D+1 (N0 − 1) + 3 D ). The set M+ defines a matrix J on Z D such that Q m 1 am 2 am 3 − 2 am 1 am 2 Q m 3 , (3.2) (J Q)m = |m|2 − 2 m 1 ,m 2 ,m 3 m 1 +m 2 −m 3 =m m 1 −m 3 ,m 2 −m 3 =0
m 1 >m 2 ,m 3 m 1 +m 2 −m 3 =m m 1 −m 3 ,m 2 −m 3 =0
where m 1 > m 2 refers, say, to lexicographic ordering of Z D ; see in particular Eqs. (8.5) and (8.7) of [24]. 1 I.e. the equation up to the third order is invariant under the transformation v → veiα for any α ∈ R.
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Moreover we know (Lemma 18 of [24]) that the matrix J is block-diagonal with blocks of size depending only on N0 , D: we denote by K (N0 , D) the bound on such a size. Whatever the block structure, the matrix J has the form diag(|m|2 ) + 2T , where all the entries of T are linear combinations of terms qm i qm j with integer coefficients. If we multiply J by z := (2 D+1 − 3 D )(2 D+1 (N0 − 1) + 3 D ) – which is odd – we obtain a , where all the entries of T are integral linear combinations matrix J := diag(z|m|2 )+2T of the square roots of a finite number of integers. Let us call the prime factors of such integers p0 = 1, p1 , p2 , . . .. Definition 3.3 (The lattice Z1D ). Let Z1D := (1, 0, . . . , 0) + 2Z D be the affine lattice of D be its integer vectors such that the first component is odd and the others even. Let Z1,+ D D 2 intersection with Z+ . Of course, for all m ∈ Z1 one has |m| odd. Since we are working with odd nonlinearities which do not depend explicitly on the space variables we look for solutions such that u n,m = 0 if m ∈ / Z1D . Let 1, p1 , . . . , pk be prime numbers (as above), and let a1 , . . . ,√ a K be the set √ of all products of square roots of different numbers pi , i.e. a1 = 1, a2 = p1 , a3 = p1 p2 , etc. It is clear that the set of integral linear combinations of ai is a ring (of algebraic integers). We denote it by a. The following Lemma is a simple consequence of Galois theory [2]. For completeness, the proof is given in Appendix A. Lemma 3.4. The numbers ai are linearly independent over the rationals. Immediately we have the following corollary (I denotes the identity). Corollary 3.5. In a consider 2a, i.e. the set of linear combinations with even coefficients. • 2a is a proper ideal, and the quotient ring a/2a is thus a non-zero ring. • if a matrix M with entries in a is such that M − I has all entries in 2a, then M is invertible. with the entries The point of Corollary 3.5 is that the determinant of M = I + 2 M, in a, is 1 + 2α, with α ∈ a. Hence, by Lemma 3.4, 2α = ±1. of M D the matrix J defined by M is invertLemma 3.6. For all N0 and for all M+ ⊂ Z1,+ + ible. Its inverse is a block matrix with blocks of dimension depending only on N0 , D so that for some appropriate C one has (J −1 )m,m ≤ C if |m − m | ≤ K (N0 , D), while (J −1 )m,m = 0 otherwise.
Proof. We use Corollary 3.4, the fact that the matrix J has entries in a and the fact that D . z|m|2 is odd for all m ∈ Z1,+ Now, we can state our result on the completely resonant NLS. Theorem 3.7. Consider the nonlinear Schrödinger equation in dimension D, i∂t v − v = f (v, v), ¯ with Dirichlet boundary conditions on the square [0, π ] D , where f is given according to (2.2) and (2.3), with N = 2, a2,1 = 1, ar,s = 0 for r, s such that r + s = 3 and (r, s) = (2, 1), and ar,s (x) independent of x for r + s > 3 (so that in particular ar,s = 0 for even r + s). Then for any N0 ≥ 1 there exist sets M+ of N0 vectors in Z+D and real
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amplitudes {am }m∈M+ such that the following holds. There exist two positive constants ε0 and κ and a Cantor set E ⊂ [0, ε0 ], such that for all ε ∈ E the equation admits a solution v(t, x), which is 2π/ω-periodic in time and Gevrey-smooth both in time and in space, and such that, setting q0 (t, x) = (2i) D
2
am ei|m| t sin m 1 x1 . . . sin m D x D , ω = 1 − ε,
(3.3)
m∈M+
one has √ v(t, x) − εq0 (x, ωt) ≤ Cε. κ The set E has density 1 at ε = 0. 3.2.2. The beam equation. We set ω2 = ω02 − ε = 1 − ε (recall that we are assuming ω0 = 1 by a suitable time rescaling). The subset Q is given by Q := {(n, m) ∈ N × Z D : |n| = |m|2 }. We set u n,m = qm+ for n = |m|2 and u n,m = qm− for n = −|m|2 . We can require that qm+ = qm− ≡ qm for all m (we obtain a solution which is even in time). Since we look for real solutions, this implies that qm ∈ R if D is even and qm ∈ iR if D is odd. Since the nonlinearity does not depend explicitly on x, we can look for solutions u n,m such that m ∈ Z1D (see Definition 3.3). Finally the separation properties of the small divisors do not depend on the presence of the mass term, so that we only need to prove the existence and non-degeneracy of the solutions of the bifurcation equation. The Q equation at leading order is |m|4 am = (−1) D
am 1 am 2 am 3 ,
m 1 +m 2 +m 3 =m ±|m 1 |2 ±|m 2 |2 ±|m 3 |2 =±|m|2
where we have set |qm | = am + O(ε1/2 ). Lemma 3.8. The condition ±|m 1 |2 ± |m 2 |2 ± |m 3 |2 = ±|m|2 , for m i , m ∈ Z1D , is equivalent to m 1 + m 3 , m 2 + m 3 = 0. Proof. The condition |m 1 |2 +|m 2 |2 +|m 3 |2 = (m 1 +m 2 +m 3 )2 is equivalent to m 1 , m 2 + m 3 + m 2 , m 3 = 0, which is impossible since the left hand side is an odd integer. The same happens with the condition |m 1 |2 −|m 2 |2 −|m 3 |2 = (m 1 +m 2 +m 3 )2 . Thus, we are left with |m 1 |2 +|m 2 |2 −|m 3 |2 = (m 1 +m 2 +m 3 )2 , which implies m 1 +m 3 , m 2 +m 3 = 0.
Lemma 3.8 implies that the bifurcation equation, restricted to Z1D , is identical to that of a smoothing NLS with s = 2; cf. [24]. Indeed by recalling that qm = (−1) D q−m one has |m|4 am = am 1 am 2 am 3 . (3.4) m 1 +m 2 −m 3 =m m 1 −m 3 ,m 2 −m 3 =0
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Then we can repeat the arguments of the previous subsection. By Lemma 17 of [24] – which holds for all values of s – for each N0 ≥ 1 there exist infinitely many finite sets D with N elements such that Eq. (3.4) has the solution M+ ⊂ Z1,+ 0 ⎧ 0, m ∈ Z+D \M+ ⎪ ⎪ ⎪ ⎛ ⎞ ⎨ am = , 1 ⎝|m|4 − c1 ⎪ D+1 |m |4 ⎠, m ∈ M+ , ⎪ D ⎪ −3 ⎩ 2 m ∈M+
with c1 = 2 D+1 /(2 D+1 (N0 − 1) + 3 D ). The matrix J is defined as in (3.2), only with |m|4 on the diagonal. We know (Lemma 18 of [24] does not depend on the values of s) that the matrix J is block-diagonal with blocks of size bounded by K (N0 , D) (defined as in Subsect. 3.2.1). Whatever the block structure, the matrix J has the form diag(|m|4 ) + 2T , where all the entries of T are linear combinations of terms am i am j with integer coefficients. If we multiply J by z := (2 D+1 − 3 D )(2 D+1 (N0 − 1) + 3 D ) – which is odd – we obtain a matrix , where all the entries of T are linear combinations of the square J := diag(z|m|4 ) + 2T roots of a finite number of integers; finally z|m|4 is clearly odd and we can apply Lemma 3.4 to obtain the analogue of Lemma 3.6. Thus, a theorem analogous to Theorem 3.7 is obtained, with q0 (t, x) in (3.3) replaced with am cos |m|2 t sin m 1 x1 . . . sin m D x D , ω2 = 1 − ε. q0 (t, x) = 2 D+1 m∈M+
We leave the formulation to the reader. 4. Technical Set-up and Propositions 4.1. Renormalised P- Q equations. Group Eqs. (2.10) for ν ∈ S as a matrix equation. Setting σ σ =± U = {u σν }σν ∈=± S , V = {u ν }ν ∈R ,
D(ε) = diag {δν (ε)}σν ∈=± S,
Q = {u σν }σν ∈=± Q,
F = { f νσ }σν ∈=± S, (4.1)
the P equations spell D(ε) U = εF(U, V, Q, ε1/N ), u σν = εδν−1 (ε) f νσ (U, V, Q, ε1/N ),
ν ∈ R, σ = ±,
(4.2)
with a reordering of the arguments of the coefficients f νσ . Note that also the first line in (4.2) could be written for components as the second line (and it would look exactly like the second line); however we find more convenient the shortened writing for ν ∈ S. We shall proceed as follows. We introduce an appropriate “correction” to the left σ,σ =± hand side of (4.2). We shall consider self-adjoint matrices M (ε) := {Mνσ,σ ,ν (ε)}ν ,ν ∈S , such that, for each fixed ε, M (ε) is block-diagonal on the sets j (ε) (cf. Definition 2.6), namely Mνσ,σ ,ν (ε) = 0 can hold only if ν, ν ∈ j (ε) for some j. Note that in order to
σ,σ −σ ,−σ have u +ν = u − . ν we must require that Mν ,ν = Mν ,ν
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Definition 4.1 (The set G and the matrix χ 1 ). Call G = {1/4 > γ¯ > 0 : ||δν (0)| − γ¯ | ≥ γ¯0 /|ν|τ¯0 for all ν ∈ Z∗D+1 }, for suitable constants γ¯0 , τ¯0 > 0. For γ¯ ∈ G, we introduce the step function χ¯ 1 (x) such that χ¯ 1 (x) = 0 if |x| ≥ γ¯ and χ¯ 1 (x) = 1 if |x| < γ¯ , and set χ¯ 0 (x) = 1 − χ¯ 1 (x). We then introduce the (ε-dependent) diagonal matrices χ 1 = diag{χ¯ 1 (δν (ε))}σν ∈=± 0 = diag{χ¯ 0 (δν (ε))}σν ∈=± S and χ S. Remark 4.2. One has G = ∅. Moreover, for any interval U ⊂ (0, 1/4), the relative measure of the set U ∩ G tends to 1 as γ¯0 tends to 0, provided τ¯0 is large enough. Remark 4.3. Notice the difference with respect to [24], where the functions χ¯ i (x) were smooth. This new definition will allow us to strongly simplify notations and proofs in the forthcoming analysis. In particular one has χ 12 = χ 1 and χ 1 χ 0 = 0, with 0 the null matrix. Roughly speaking, this will allow us to invert matrices without mixing singular and regular frequencies. Definition 4.4 (Resonant sets). A set N = {ν 1 , . . . , ν m } ⊂ S is resonant if there exists ε ∈ [0, ε0 ] and j ∈ N such that ν 1 , . . . , ν m ∈ j (ε). A resonant set {ν 1 , ν 2 } with m = 2 will be called a resonant pair. Given a resonant set N = {ν 1 , . . . , ν m } we call CN the set of all ν ∈ S such that N ∪ {ν} is still a resonant set. Finally set C N (ε) := {ν ∈ CN : |δν (ε)| < γ¯ }, with γ¯ introduced in Definition 4.1. Define the renormalised P equation as (D(ε) + M ) U = η N F(U, V, Q, η) + L U, ν ∈ R, u σν = η N δν−1 (ε) f νσ (U, V, Q, η),
(4.3)
σ,σ =± σ,σ σ,σ =± where η is a real parameter, while M = {Mνσ,σ ,ν }ν ,ν ∈S and L = {L ν ,ν }ν ,ν ∈S are self-adjoint matrices with the properties:
σ,σ =± χ1 , where M = {Mνσ,σ 1. M = χ 1 M ,ν }ν ,ν ∈S ;
σ,σ 2. Mνσ,σ ,ν = L ν ,ν = 0 if {ν, ν } is not a resonant pair; −σ ,−σ −σ ,−σ 3. Mνσ,σ and L σ,σ . ,ν = Mν ,ν ν ,ν = L ν ,ν
Remark 4.5. Property 1 above implies that M has an ε-dependent block structure, which will be crucial in the convergence estimates. On the other hand we need to introduce free (i.e. ε-independent) parameters. Thus, we introduce the elements of the matrix M as ε-independent parameters, with the only restriction that they satisfy the ε-independent Properties 2 and 3. Eventually we shall manage to fix M as a function of the parameter ε, that is M = M(ε). Moreover an important property for the measure estimates will be that M(ε) depends smoothly on ε, at least in a large measure set. The renormalised Q equation is defined as σ u σν = (J −1 )σ,σ ν ,ν f ν (U, V, Q, η),
ν ∈ Q, σ = ±.
(4.4)
ν ∈Q σ =±
Remark 4.6. By looking at (4.3) and (4.4) it would be tempting to introduce 2×2 matrices Mν ,ν and (J −1 )ν ,ν instead of carrying along the subscripts σ, σ through all the equations. However, in the following, to introduce the diagrammatic expansion and check some symmetry properties, we shall have to write everything by components: hence it will be more convenient to keep also the σ labels, in order not to introduce too many notations.
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The parameter η and the counterterms L will have to satisfy eventually the identities (compatibility equation) η = ε1/N ,
M = L.
(4.5)
We proceed in the following way: first we solve the renormalised P and Q equations (4.3) and (4.4), then we impose the compatibility equation (4.5). 4.2. Matrix spaces. Here we introduce some notations and properties that we shall need in the following. Definition 4.7 (The Banach space Bκ ,ρ ). We consider the space of infinite-dimensional σ,σ =± σ,σ self-adjoint matrices {Mνσ,σ ,ν }ν ,ν ∈S such that Mν ,ν = 0 if {ν, ν } is not resonant. For ρ, κ > 0 we equip such a space with the norm κ|ν −ν |ρ |M|κ,ρ := sup sup Mνσ,σ , ,ν e ν ,ν ∈S σ,σ =±
so obtaining a Banach space that we call Bκ,ρ . For L a linear operator on Bκ,ρ define the operator norm |L|op = sup
M∈Bκ,ρ
|L M|κ,ρ . |M|κ,ρ
Definition 4.8 (Matrix norms). Let A be a d × d self-adjoint matrix, and denote with A(i, j) and λ(i) (A) its entries and its eigenvalues, respectively. We define the norms |A|∞ := max |A(i, j)|, 1≤i, j≤d
1 A := √ tr(A2 ), d
A2 := max |Ax|2 , |x|2 ≤1
where, given a vector x ∈ Rd , we denote by |x|2 its Euclidean norm. Lemma 4.9 Given a d × d self-adjoint matrix A, the following properties hold. 1. 2. 3. 4.
The norm A√ depends smoothly √ on the coefficients A(i, j). One has A/ d ≤ |A|∞ ≤ √ dA. One has max1≤i≤d |λ(i) (A)|/ d ≤ A ≤ max1≤i≤d |λ(i) (A)|. For invertible A one has ∂ A(i, j) A−1 (i , j ) = −A−1 (i , i) A−1 ( j, j ) and ∂ A(i, j) A = A(i, j)/dA. Here and henceforth we shall write A = D(ε) + M in (4.3).
Definition 4.10 (Small divisors). For ν ∈ S define Aν (ε) as the matrix with entries 2 χ¯ 1 (δν (ε)) Aσν 11 ,σ ,ν 2 such that ν 1 , ν 2 ∈ C ν (ε) and σ1 , σ2 = ±. If |δν (ε)| < γ¯ (cf. Definition 4.1), define also d ν (ε) := 2|C ν (ε)| and pν (ε) = min{|ν | : ν ∈ C ν (ε)}. For real positive ξ , define the small divisor −1 1 ν xν (ε) := ξ (A (ε))−1 , pν (ε) if A is invertible, and set xν (ε) = 0 if A is not invertible.
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Remark 4.11. Note that for ν ∈ j (ε) one has pν (ε) = p j (ε), dν (ε) ≤ 2| j (ε)|, and Aν (ε) = Aν (ε) for all ν ∈ C ν (ε). This shows that dν (ε), xν (ε) and pν (ε) are the same for all ν ∈ C ν (ε). Note also that, if ν ∈ j (ε) for some j ∈ N, then one has C ν (ε) = {ν ∈ j (ε) : |δν (ε)| < γ¯ }. Hypothesis 3 implies dν (ε) ≤ 2C1 pνα (ε). Definition 4.12 (The sets D0 , D1 (γ ), D2 (γ ), and D(γ )). We define D0 = {(ε, M) : ε ∈ [0, ε0 ], |M|κ ≤ C0 ε0 }, for a suitable positive constant C0 , and, for fixed τ, τ1 > 0 and γ < γ¯ , we set D1 (γ ) = {(ε, M) ∈ D0 : xν ≥ γ / pντ (ε) for all ν ∈ S}, D2 (γ ) = {(ε, M) ∈ D0 : ||δν (ε)| − γ¯ | ≥ γ /|ν|τ1 for all ν ∈ S}, and D(γ ) = D1 (γ ) ∩ D2 (γ ). Definition 4.13 (The sets I N (γ ) and I N (γ )). Given a resonant set N we define I N (γ ) := {ε ∈ [0, ε0 ] : ∃ν ∈ CN such that ||δν (ε)| − γ¯ | < γ |ν|−τ1 }, and set IN (γ ) := {(ε, M) ∈ D0 : ε ∈ I N (γ )}. 4.3. Main propositions. We state the propositions which represent our main technical results. The Main Theorem in Sect. 2 is an immediate consequence of Propositions 1 and 2 below. Proposition 1. There exist positive constants K 0 , K 1 , κ, ρ, η0 such that the following holds true. For (ε, M) ∈ D(γ ), there exists a matrix L(η, ε, M) ∈ Bκ,ρ , such that the following holds: 1. For each ε the matrix L(η, ε, M) is block-diagonal so as to satisfy L(η, ε, M) = χ 1 L(η, ε, M) χ1 . Moreover the L(η, ε, M) is analytic in η for |η| ≤ η0 , and uniformly bounded for (ε, M) ∈ D(γ ) as |L(η, ε, M)|κ,ρ ≤ |η| N K 0 . 2. There exists a uniquely determined solution u σν (η, M, ε) of Eqs. (4.3) and (4.4), which is analytic in η for |η| ≤ η0 , and such that for all ν ∈ Z D+1 and σ = ±, σ u (η, M, ε) ≤ |η| K 0 e−κ|ν |1/2 . ν
3. The matrix elements L σ,σ ν ,ν (η, ε, M) can be extended on the set D0 \I{ν ,ν } (γ ) to E σ,σ E σ,σ σ,σ 1 C functions L ν ,ν (η, ε, M), such that L ν ,ν (η, ε, M) = L ν ,ν (η, ε, M) for all (ε, M) ∈ D(2γ ). Moreover, for all (ε, M) ∈ D0 \I{ν ,ν } (γ ), the matrix elements L νE,νσ,σ (η, ε, M) satisfy the bounds ρ E σ,σ L ν ,ν (η, ε, M) ≤ e−κ|ν −ν | |η| N K 1 ,
ρ
ρ
−κ|ν −ν | |∂ε L νE,νσ,σ |η| N K 1 | pν |c0 , (η, ε, M)| ≤ e −κ|ν −ν | |∂η L νE,νσ,σ N |η| N −1 K 1 . (η, ε, M)| ≤ e
4. For all (ε, M) ∈ D0 \ ∪ I{ν ,ν } (γ ), where the union is taken over all the resonant pairs {ν, ν }, one has E E ≤ |η| N K 1 . ∂ ∂ L (η, ε, M) ≤ L (η, ε, M) M σ,σ Mν,ν op ν ∈S ν ∈Cν σ,σ =±
κ,ρ
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5. The functions u σν (η, ε, M) can be extended on the set D0 to C 1 functions u νE σ (η, ε, M), such that u νE σ (η, ε, M) = u σν (η, ε, M) for all (ε, M) ∈ D(2γ ) and 1/2 Eσ u ν (η, ε, M) ≤ |η| N K 1 e−κ|ν | , uniformly for (ε, M) ∈ D0 . Remark 4.14. In our analysis we choose M ∈ Bκ,ρ because eventually we obtain L ∈ Bκ,ρ , but – as the bound on the M-derivative in Item 4 of Proposition 1 suggests – we could also take M in a larger space, say B∞ with norm |M|∞ = supν ,ν ∈S supσ,σ =±
σ,σ |Mνσ,σ ,ν |. In Item 3 of Proposition 1 we need to work on the matrix elements L ν ,ν (η, ε, M) since the extensions hold for (ε, M) in the (ν, ν )-dependent sets D0 \I{ν ,ν } (γ ).
Once we have proved Proposition 1, we solve the compatibility equation (4.5) for the extended counterterms L E (ε1/N , ε, M), which are well defined provided we choose ε ≤ ε0 , with ε0 = η0N .
Proposition 2. There exist functions ε → (ε, Mνσ,σ ,ν (ε)) from [0, ε0 ] → D0 , with an appropriate choice of C0 in Definition 4.12, such that the following holds.
1. For ε ∈ I {ν ,ν } (γ ) one has Mνσ,σ ,ν (ε) = 0, while for ε ∈ [0, ε0 ]\I {ν ,ν } (γ ) the σ,σ 1 elements Mν ,ν (ε) are C , verify the equation
E σ,σ 1/N , ε, M(ε)), Mνσ,σ ,ν (ε) = L ν ,ν (ε
and satisfy the bounds ρ σ,σ Mν ,ν (ε) ≤ K 2 ε e−κ|ν −ν | ,
(4.6)
ρ σ,σ ∂ε Mν ,ν (ε) ≤ K 2 1 + εpνc0 (ε) e−κ|ν −ν | ,
for a suitable constant K 2 . 2. The functions u νE (ε) := u νE + (ε1/N , ε, M(ε)) are C 1 in [0, ε0 ]. 3. The set E(2γ ) := {ε ∈ [0, ε0 ] : (ε, M(ε)) ∈ D(2γ )} has density 1 at ε = 0, namely lim+
ε→0
meas(E(2γ ) ∩ (0, ε)) = 1. ε
4.4. Proof of the Main Theorem. By Items 1 and 2 in Proposition 1 for all (ε, M) ∈ D(γ ) we can find a matrix L(η, ε, M) so that there exists a unique solution u σν (η, ε, M) of (4.3) and (4.4) for all |η| ≤ η0 , for a suitable η0 , and for ε0 small enough. By Items 3 σ and 5 in Proposition 1 the matrix blocks L σ,σ ν ,ν (η, ε, M) and the solution u ν (η, ε, M)
Eσ can be extended to C 1 functions – denoted by L νE,νσ,σ (η, ε, M) and u ν (η, ε, M) – for all (ε, M) ∈ D0 \I{ν ,ν } (γ ) and for all (ε, M) ∈ D0 , respectively. Moreover σ,σ Eσ σ L νE,νσ,σ (η, ε, M) = L ν ,ν (η, ε, M) and u ν (η, ε, M) = u ν (η, ε, M) for all (ε, M) ∈ D(2γ ). Equation (4.3) coincides with our original (4.2) provided the compatibility equation (4.5) is satisfied. Now we fix ε0 < η0N so that L E (ε1/N , ε, M) and u νE σ (ε1/N , ε, M) are well defined for |ε| < ε0 . By Item 1 in Proposition 2, there exists a matrix M(ε) which
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satisfies the extended compatibility equation (4.6). Finally by Item 3 in Proposition 2 (we need Item 4 in Proposition 2 to prove it) the Cantor set E(2γ ) is well defined and of large relative measure. Set E = E(2γ ) and u(x; ε) = ν ∈Z D+1 u νE (ε)eiν ·x . The function u(t, x; ε) is C 1 in ε for ε ∈ [0, ε0 ] by Item 2 in Proposition 2. Moreover it is 2π -periodic and Gevrey-smooth, and satisfies the bound in Item 2, by Item 5 in Proposition 1. For all ε ∈ E the pair (ε, M(ε)) is by definition in D(2γ ), so that by Item 3 in Proposition 1 one has L ν ,ν (ε1/N , ε, M(ε)) = L νE,ν (ε1/N , ε, M(ε)) and by Item 5 in Proposition 1 one has u σν (ε1/N , ε, M(ε)) = u νE σ (ε1/N , ε, M(ε)), and hence u σν (ε1/N , ε, M(ε)) solves (4.3) for η = ε1/N . So, by Item 1 in Proposition 2, M(ε) solves the true compatibility equation (4.5) for all ε ∈ E. Then u(t, x; ε) is a true nontrivial solution of (4.3) and (4.4) in E. 5. Tree Expansion 5.1. Recursive equations. In this section we find a formal solution u σν , L of (4.3) and (4.4) as a power series on η; the solution u σν , L depends on the matrix M and it will be written in the form of a tree expansion. We shall introduce the trees in abstracto, by giving the rules how to construct them, that is, essentially, how to associate labels to unlabelled trees. Then, we shall show that both the solution u σν and the matrix L can be expressed in terms of labelled trees. Of course, the easiest way to see that the construction makes sense is to try to express u σν and L in terms of trees and then check that some constraints and relations must be imposed on the tree labels. We assume for u σν (η, ε, M) for all ν ∈ P and for the matrix L(η, ε, M) a formal series expansion in η: u σν (η, ε, M) =
∞
ηk u (k)σ ν ,
L(η, ε, M) =
k=N
∞
ηk L (k) ,
(5.1)
k=N
with the Ansatz that L (k)σ,σ = 0 if either χ¯ 1 (δν (ε))χ¯ 1 (δν (ε)) = 0 or the pair {ν, ν } is ν ,ν (k)σ not resonant, so that L = χ 1 L χ1 . We set also u ν = 0 for all k ≤ N and ν, ν ∈ P, (k)σ,σ and the same for L ν ,ν for ν, ν ∈ S. For ν ∈ Q we set ∞ + ηk u (k) (5.2) u σν (η, ε, M) = u (0) ν ν k=N (0)+
(0)
(0)−
(0)
with u ν = u ν and u ν = u ν (cf. Item 2 in Hypothesis 2 for notations). Again we set u (k)σ = 0 for 0 < k < N and ν ∈ Q. ν Inserting the series expansions (5.1) and (5.2) into (4.3) we obtain ⎧ (k−N )σ ⎪ fν (k)σ ⎪ ⎪ u = , ν ∈ R, σ = ±, ⎪ ν ⎪ ⎪ δν (ε) ⎪ (k)σ ⎪ (k)σ σ,σ ⎨u = (J −1 )ν ,ν f ν , ν ∈ Q, σ = ±, ν (5.3) ν ∈Q,σ =± ⎪ ⎪ ⎪ k−N ⎪ ⎪ ⎪ ⎪(D(ε) + M ) U (k) = F (k−N ) + L (r ) U (k−r ) . ⎪ ⎩ r =N
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5.2. Multiscale analysis. Definition 5.1 (The scale functions). Let χ be a non-increasing function C ∞ (R+ , [0, 1]), such that χ (x) = 0 if x ≥ 2γ and χ (x) = 1 if x ≤ γ , with γ given in Definition 4.12; moreover one has |∂x χ (x)| ≤ γ −1 for some positive constant . Let χh (x) = χ (2h x) − χ (2h+1 x) for h ≥ 0, and χ−1 (x) = 1 − χ (x). Remark 5.2. In contrast to the functions χ¯ i in Definition 4.1, the scale functions χh are smooth. Indeed, in this case smoothness is important, because we shall need to derivate such functions. On the other hand, the fact that the functions χ¯ i are sharp implies that the matrix A−1 has the same block structure as A. Recall that for each ε the matrix A = D(ε) + M is block diagonal with a diagonal part whose eigenvalues are larger than γ¯ > γ and a list of C1 pνα (ε) × C1 pνα (ε) blocks Aν containing small entries. In the following if Aν is invertible — i.e. if xν = 0 — we will denote the entries of (Aν )−1 by (A−1 )σ,σ ν ,ν even though it may be possible that the whole matrix A is not invertible. Definition 5.3 (Propagators). For ν, ν ∈ S, we define the propagators
(G i,h )σ,σ ν ,ν ⎧ −1 σ,σ ⎪ ⎨χh (xν (ε)) χ¯ 1 (δν (ε))χ¯ 1 (δν (ε))(A )ν ,ν , if i = 1 and χh (xν (ε)) = 0, = χ¯ 0 (δν (ε)) δν−1 (ε), if i = 0, ν = ν , σ = σ and h = −1, ⎪ ⎩ 0, otherwise.
In terms of the propagators we obtain A−1 =
∞
G i,h ,
(5.4)
i=0,1 h=−1
which provides the multiscale decomposition. Notice that if (A−1 )σ,σ ν ,ν = 0 then xν (ε) = xν (ε) (see Remark 4.11), so that the matrices G i,h are indeed self-adjoint. Remark 5.4. Only the propagator G 1,h can produce small divisors while the propagator G 0,−1 is diagonal and of order one. Hence, there exists a positive constant C such that we can bound the propagators as −ξ σ,σ G 0,−1 ≤ Cγ −1 , (5.5) (G 1,h )ν ,ν ≤ 2h Cγ −1 pν (ε) pνα (ε), ∞ where the condition dν (ε) ≤ 2C1 pνα (ε) – cf. Remark 4.11 – and Item 2 of Lemma 4.9 have been used. We write L (k) in (5.1) as 1 ,σ2 L (k)σ = ν 1 ,ν 2
∞ h=−1
1 ,σ2 χh (xν 1 (ε))L (k)σ h,ν 1 ,ν 2 ,
(5.6)
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(k)σ ,σ
for all resonant pairs {ν 1 , ν 2 }; we denote by L h the matrix with entries L h,ν 11,ν 22 . Finally we set U (k) =
∞
(k)
Ui,h ,
(5.7)
i=0,1 h=−1
so that (5.3) gives ⎧ (k)σ (k)σ ⎪ (J −1 )σ,σ , ν ∈ Q, σ = ±, ⎪u ν = ν ,ν f ν ⎪ ⎪ =± ⎪ σ ν ∈ Q ⎪ ⎪ ⎪ (k−N )σ ⎨ (k)σ fν uν = , ν ∈ R, σ = ±, (5.8) δν (ε) ⎪ ⎪ ⎪ ∞ k−N ⎪ (r ) (k−r ) ⎪ (k) ⎪ ⎪ L h U1,h 1 , i = 0, 1, h ≥ −1, Ui,h = G i,h F (k−N ) + δ(i, 1) G 1,h ⎪ ⎩ h 1 =−1 r =N
which are the recursive equations we want to study. 5.3. Diagrammatic rules. A connected graph G is a collection of points (vertices) and lines connecting all of them. We denote with V (G) and L(G) the set of nodes and the set of lines, respectively. A path between two nodes is the minimal subset of L(G) connecting the two nodes. A graph is planar if it can be drawn in a plane without graph lines crossing. Definition 5.5 (Trees). A tree is a planar graph G containing no closed loops. One can consider a tree G with a single special node v0 : this introduces a natural partial ordering on the set of lines and nodes, and one can imagine that each line carries an arrow pointing toward the node v0 . We can add an extra (oriented) line 0 exiting the special node v0 ; the added line 0 will be called the root line and the point it enters (which is not a node) will be called the root of the tree. In this way we obtain a rooted tree θ defined by V (θ ) = V (G) and L(θ ) = L(G) ∪ 0 . A labelled tree is a rooted tree θ together with a label function defined on the sets L(θ ) and V (θ ). We shall call equivalent two rooted trees which can be transformed into each other by continuously deforming the lines in the plane in such a way that the latter do not cross each other (i.e. without destroying the graph structure). We can extend the notion of equivalence also to labelled trees, simply by considering equivalent two labelled trees if they can be transformed into each other in such a way that also the labels match. Given two nodes v, w ∈ V (θ ), we say that v ≺ w if w is on the path connecting v to the root line. We can identify a line with the nodes it connects; given a line = (w, v) we say that enters w and exits (or comes out of) v, and we write = v . To help himself follow the diagrammatic construction, one can visualise the trees with the root to the left and the end-nodes to the right; in particular given a line (w, v) one has v ≺ w, with the endpoint w to the left of the endpoint v. Given two comparable lines and 1 , with 1 ≺ , we denote with P(1 , ) the path of lines connecting 1 to ; by definition the two lines and 1 do not belong to P(1 , ). We say that a node v is along the path P(1 , ) if at least one line entering or exiting v belongs to the path. If P(1 , ) = ∅ there is only one node v along the path (such that 1 enters v and exits v).
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Definition 5.6 (Lines and nodes). We call internal nodes the nodes such that there is at least one line entering them; we call internal lines the lines exiting the internal nodes. We call end-nodes the nodes which have no entering line. We denote with L(θ ), V0 (θ ) and E(θ ) the set of lines, internal nodes and end-nodes, respectively. Of course V (θ ) = V0 (θ ) ∪ E(θ ). As anticipated at the beginning of Subsect. 5.1, we first introduce the trees as abstract structures and prove thereafter that the quantities of interest can be expressed in terms of trees. For further details and to better understand the general strategy of the construction, we refer to the more pedagogical discussion in [24]. In fact, we first develop a naive but very natural tree expansion for the solution. Such an expansion would work for any choice of L if the solution were analytic in ε. However in our case we do not expect that analyticity holds, so we modify the expansion into a new one (renormalised expansion), where we fix L appropriately (see Definition 5.21) in such a way to eliminate the contributions which would cause the divergence of the series. We associate with the nodes (internal nodes and end-nodes) and lines of any tree θ some labels, according to the following rules. Definition 5.7 (Diagrammatic rules). Let θ be a tree. We associate with θ the following labels: 1. With each internal line ∈ L(θ ) one associates a label q, p or r . We say that is a p-line, a q-line or an r -line, respectively, and we call L q (θ ), L p (0) and L r (0) the set of internal lines ∈ L(θ ) which are q-lines, p-lines and r -lines, respectively. 2. With each line ∈ L(θ ) one associates the type label i = 0, 1 and the scale label h ∈ N ∪ {−1, 0}. 3. With each line ∈ L(θ ) except the root line 0 one associates a sign label σ = ±. 4. With each internal line ∈ L(θ ) one associates the momenta (ν , ν ) ∈ Z D+1 × Z D+1 . 5. With each line ∈ L(θ ) exiting an end-node one associates the momentum ν . 6. For each node v there are pv ≥ 0 entering lines. If pv = 0 then v ∈ E(θ ), if pv > 0 then either pv = 1 or pv ≥ N + 1 and v ∈ V0 (θ ), where N is introduced in (2.1). If L(v) is the set of lines entering v one has pv = |L(v)|. 7. With each end-node v ∈ E(θ ) one associates the mode label ν v ∈ Q, the order label kv = 0, and the sign label σv = ±. 8. With each internal node v ∈ V0 (θ ) one associates the mode label m v ∈ Z D , the order label kv ∈ N, and the sign label σv = ±, and one defines rv as the number of lines ∈ L(v) with σ = σv , and one sets sv = pv − rv . The following constraints and relations will be imposed on the labels. 9. Given an internal node v ∈ V0 (θ ), if pv = 1 let 1 be the line entering v and be the line exiting v. Then and 1 are both p-lines. Moreover one has i 1 = i = 1 and {ν , ν 1 } is a resonant pair. 10. If a line ∈ L(θ ) is not a p-line one sets i = 0. 11. If a line ∈ L(θ ) has i = 0, then h = −1. 12. Let ∈ L(θ ) be an internal line. If is a p-line with i = 0, then ν = ν . If is a p-line with i = 1, then {ν , ν } is a resonant pair. If is a q-line, then ν , ν ∈ Q. If is an r -line, then ν = ν ∈ R. 13. If exits an end-node v ∈ E(θ ), then one sets ν = ν v and σ = σv . 14. If two p-lines and have i = i = 1 and are such that {ν , ν , ν , ν } is a resonant set, then |h − h | ≤ 1.
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15. If is the line exiting v and 1 , · · · , pv are the lines entering v one has ν = (0, m v ) + σv (σ1 ν 1 + · · · + σ pv ν pv ) = (0, m v ) + σv σ ν , ∈L(v)
which represents a conservation rule for the momenta. 16. Given an internal node v ∈ V0 (θ ), if pv = 1 one has kv ≥ N , while if pv ≥ N one has kv = pv − 1. (0)σ 17. With each end-node v ∈ E(θ ) one associates the node factor ηv = u ν v v ; cf. Item 2 in Hypothesis 2 and (5.2) for notations. 18. Given an internal node v ∈ V0 (θ ), if pv > 1 one associates with v the node factor σ ηv = arσvv,sv ,m v , where ar,s,m satisfies Eq. (2.11), while if pv = 1 one associates with (kv )σv ,σ
v the node factor ηv = L h ,ν ,ν 1 , still to be defined (see Definition 5.21 below), 1 where and 1 are the lines exiting and entering v, respectively. 19. One associates with each line ∈ L(θ ) a line propagator g ∈ C with the following v rules. If is a p-line exiting the internal node v one sets g := (G i ,h )σν ,σ , if is ,ν an r -line one sets g := 1/δν (ε), if is a q-line exiting the internal node v one sets v g := (J −1 )σν ,σ , if exits an end-node one sets g = 1. ,ν 20. One defines the order of the tree θ as k(θ ) := kv , v∈V (θ)
the momentum of θ as the momentum ν of the root line , and the sign of θ as the sign σv0 of the node v0 which the root line exits. Remark 5.8. The “line propagators” defined in Item 19 are not to be confused with the “propagators” tout court introduced in Definition 5.3. In fact, the line propagator coincides with the propagator when the latter is defined, but there are lines with which no propagator is associated. (k)σ
(k)σ
Definition 5.9 (The sets of trees ν and ). We call ν the set of all the nonequivalent trees of order k, momentum ν and sign σ , defined according to the diagrammatic (k)σ rules of Definition 5.7. We call the sets of trees belonging to ν for some k ≥ 1, D+1 σ = ± and ν ∈ Z . The reason for introducing these sets of trees is summed in the following result. Lemma 5.10 For any given counterterm L ∈ Bκ,ρ such that L = χ 1 L χ1 , the coefficients (k)σ u ν can be written in terms of trees ⎛ ⎞⎛ ⎞ (k)σ ⎝ g ⎠ ⎝ ηv ⎠ . uν = (k)σ
θ∈ν
∈L(θ)
v∈V (θ)
Proof. The proof is easily obtained by standard arguments in Taylor series expansions. For instance, one can proceed by induction, using the diagrammatic rules and definitions given in this section; we refer to Lemma 3.6 of [24] for details. However, in general we cannot prove the convergence of the series (5.2) for arbitrary L. This compels us to change the expansion, in such a way that suitably fixing L all the dangerous contributions disappear from the expansion.
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5.4. Clusters and resonances. Definition 5.11 (Clusters). Given a tree θ ∈ (k)σ a cluster T on scale h is a connected ν maximal set of nodes and lines such that all the lines have a scale label ≤ h and at least one of them has scale h; we shall call h T = h the scale of the cluster. We shall denote by V (T ), V0 (T ) and E(T ) the set of nodes, internal nodes and the set of end-nodes, respectively, which are contained inside the cluster T , and with L(T ) the set of lines connecting them. Finally k(T ) = v∈V (T ) kv will be called the order of T . An inclusion relation is established between clusters, in such a way that the innermost clusters are the clusters with lowest scale, and so on. A cluster T can have an arbitrary number of lines entering it (entering lines), but only one or zero line coming out from it (exiting line or root line of the cluster); we shall denote the latter (when it exists) with T . Notice that, by definition, |V (T )| > 1 and all the entering and exiting lines have i = 1. Next we introduce the notion of resonances. The resonances identify the clusters which, if not eliminated, would produce a runaway accumulation of small divisors (and hence the divergence of the algorithm to construct the solution). The idea will be to choose the matrices L in such a way to eliminate (iteratively) the resonances. Definition 5.12 (Resonances). We call resonance on scale h a cluster T on scale h T = h such that 1. the cluster has only one entering line 1T and one exiting line T of scale h T ≥ h + 2, 2. one has that {ν , ν 1 } is a resonant pair and min{|ν 1 |, |ν |} ≥ 2(h−2)/τ , T
T
T
T
3. for all ∈ P(1T , T ) with i = 1 the pair {ν , ν 1 } is not resonant, T
4. for all ∈ L(T )\P(1T , T ) the pair {ν , ν 1 } is not resonant. T
The line T of a resonance will be called the root line of the resonance. (k)σ ,σ
Definition 5.13 (The sets of trees Rh,ν ,ν and R). For k ≥ N , h ≥ 1 and a resonant (k)σ σ
pair {ν, ν } such that min{|ν|, |ν |} ≥ 2(h−2)/τ , we define Rh,ν ,ν as the set of trees with (k)σ
the following differences with respect to ν
.
1. There is a single end-node, called e, with node factor ηe = 1 (but no label no labels ν e nor σe ). 2. The line e exiting e is a p-line. We associate with e the labels ν e = ν , σ = σ , and i e = 1 (but no labels ν nor h ), and the corresponding line propagator is ge = χ¯ 1 (δν (ε)). 3. The root line 0 is a p-line. We associate with 0 the labels i 0 = 1 and ν 0 = ν (but no labels ν 0 nor h 0 ), and the corresponding line propagator is g0 = χ¯ 1 (δν (ε)). Let v0 be the node which the line 0 exits: we set σv0 = σ . 4. One has max∈L(θ)\{0 ,e } h = h. 5. If ∈ P(e , 0 ) is such that {ν , ν } is resonant, then i = 0. 6. For ∈ / P(e , 0 ) one has that {ν , ν } is not a resonant pair. (k)σ σ
We call R the sets of trees belonging to Rh,ν ,ν for some k ≥ 1, h ≥ 1, σ, σ = ±, and ν, ν ∈ S such that {ν, ν } is resonant and min{|ν|, |ν |} ≥ 2(h−2)/τ .
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Definition 5.14 (Clusters for trees in R). Given a tree θ ∈ R, a cluster T on scale h T ≤ h is a connected maximal set of nodes v ∈ V (θ ) and lines ∈ L(θ )\{0 , e } such that all the lines have a scale label ≤ h T and at least one of them has scale h T .
Note that if θ ∈ R(k)σ,σ h,ν ,ν , then for any cluster T in θ one necessarily has h T ≤ h. Definition 5.15 (Resonances for trees in R). Given a tree θ ∈ R, a cluster T is a resonance if the four items of Definition 5.12 are satisfied. Remark 5.16. There is a one-to-one correspondence between resonances T of order k and scale h with ν 1 = ν , ν = ν, σv0 = σ , σ1 = σ (here v0 is the node which T T
(k)σ,σ
T
T
exits) and trees θ ∈ Rh,ν ,ν ; cf. [24], Sect. 3.4 and Fig. 7. (k)σ
(k)σ ,σ
Definition 5.17 (The sets of renormalised trees R,ν , R R,h,ν ,ν , R and R R ). We (k)σ,σ define the set of renormalised trees (k)σ R,ν and R R,h,ν ,ν as the set of trees defined as
(k)σ and R(k)σ,σ ν h,ν ,ν , respectively, but with no resonances and no nodes v with pv = 1. Analogously we define the sets R and R R .
In the following it will turn out to be convenient to introduce also the following set of trees. (k)σ ,σ
Definition 5.18 (The set of renormalised trees S R,h,ν ,ν and S R ). For k ≥ N , h ≥ 1 and
(k)σ,σ ν, ν ∈ S such that |ν | ≥ 2(h−2)/τ we define the set of renormalised trees S R,h, ν ,ν as the (k)σ,σ
set of trees with the following differences with respect to R R,h,ν ,ν (see Definition 5.13). Items 1 and 2 are unchanged. 3 One assigns to the line 0 the further label h 0 ≤ h, and requires |ν| ≥ 2(h 0 −2)/τ . 4 One has max∈L(θ)\{e } h = h Items 5 and 6 are unchanged. The set S R is defined analogously as R R .
(k)σ,σ Remark 5.19 Note that if θ ∈ R(k)σ,σ R,h,ν ,ν then Val(θ ) = Val(θ ) with θ ∈ S R,h,ν ,ν such that h 0 = h − 1. Thus, it is enough to study the set S R in order to obtain bounds for trees in R R .
Definition 5.20 (Tree values). For any tree or renormalised tree θ call ⎛ ⎞⎛ ⎞ Val(θ ) = ⎝ g ⎠ ⎝ ηv ⎠ ∈L(θ)
v∈V (θ)
the value of the tree θ . To make explicit the dependence of the tree value on ε and M, sometimes we shall write Val(θ ) = Val(θ ; ε, M).
Definition 5.21 (Counterterms). We define the node factors L (k)σ,σ h,ν ,ν (cf. item 21 in Definition 5.7) by setting (k)σ,σ Val(θ ), σ, σ = ±, (5.9) L h,ν ,ν = h
R,h ,ν,ν
for all k ≥ N , all h ≥ 1, and all resonant pairs {ν, ν }. The counterterms L are then expressed in terms of (5.9) through (5.1) and (5.6).
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(k)σ,σ
(k)−σ ,−σ
Lemma 5.22 For any tree θ ∈ R R,h,ν ,ν there exists a tree θ ∈ R R,h,ν ,ν Val(θ ) = Val(θ ).
such that
(k)σ,σ
Proof. Given a tree θ ∈ R R,h,ν ,ν , consider the path P = P(e , 0 ), and set P = {1 , · · · , N }, with 0 1 . . . N N +1 = e (if P = ∅, set N = 0 in the forthcoming discussion). For k = 0, . . . , N , denote by vk the node which the line k exits and by L 0 (vk ) the set L(vk )\{k+1 } (cf. Item 1 in Definition 5.7). (k)−σ ,−σ We construct a tree θ ∈ R R,h,ν ,ν in the following way: 1. We shift the sign labels down the path P and change their sign, so that σk → −σvk and σvk → −σk+1 for k = 0, . . . , N . In particular 0 acquires the label −σv0 , while e loses its label σe (which with the opposite sign becomes associated with the node v N ). 2. The end-node e becomes the root, and the root becomes the end-node e. In particular the line e becomes the root line, and the line 0 becomes the entering line, so that the arrows of all the lines ∈ P are reverted, while the ordering of all the lines and nodes outside P is not changed. 3. For all the lines ∈ P we exchange the labels ν , ν , so that ν k → ν k and ν k → ν k for k = 1, . . . , N , and we set ν e = ν and ν 0 = ν. 4. For all k = 0, . . . , N we replace m vk → −σvk σk+1 m vk . (k)−σ ,−σ
By construction, the tree θ belongs to R R,h,ν ,ν , and all line propagators and node factors of the lines and nodes, respectively, which do not belong to P remain the same. −σv ,−σ Moreover, the line propagator of each k ∈ P in θ is (G ik ,h k )ν ,νk k k
σ ,σv
= (G ik ,h k )ν k ,ν k , hence it does not change with respect to the line propagator of k
the corresponding line in θ . For each node vk , the conservation law ⎛ ⎞ ν k+1 = (0, −σvk σk+1 m vk ) − σk+1 ⎝−σvk ν k + σ ν ⎠
(5.10)
is assured by the conservation law (cf. Item 15 in Definition 5.7) ⎛ ⎞ σ ν ⎠ ν k = (0, m vk ) + σvk ⎝σk+1 ν k+1 +
(5.11)
∈L 0 (vk )
∈L 0 (vk )
for the corresponding node vk in θ : simply multiply (5.11) times σvk σk+1 in order to obtain (5.10). Finally we want to show that the product of the combinatorial factors times the node factors of the nodes v0 , . . . , v N do not change. Take a node v = vk , for k = 0, . . . , N , and call rv and sv the number of lines ∈ L 0 (v) with σ = σv and σ = −σv , respectively. Set σv = σ and σk+1 = σ . Consider first the case σ = σ . In that case in θ one has rv = rv + 1 and sv = sv , and the combinatorial factor contains a factor rv because there are rv lines entering v with σ = σ . In θ one has σv → −σ , rv → sv + 1, sv → rv and m v → −m v (because σ σ = 1). Moreover the corresponding combinatorial factor contains a factor (sv + 1) because there are sv + 1 lines entering v with σ = −σ . Therefore, taking
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into account also the combinatorics, the node factor associated with the node v in θ is (sv + 1)as−σ = rv arσv ,sv ,m v , i.e. the same as in θ , by the condition (2.11). v +1,rv −1,−m v Now, we pass to the case σ = −σ . In that case in θ one has rv = rv , sv = sv + 1. In θ one has the same values for rv , sv and σv , so that, by using also that −σ σ m v = m v in such a case, the node factors arσvv,sv ,m v do not change. Of course the combinatorial factors do not change either. In conclusion, one has Val(θ ) = Val(θ ), which yields the assertion. (k)
Remark 5.23. By Lemma 5.22 we have that the matrix L h is self-adjoint, and Definition 5.21 together with (5.6) implies that we can write (k)σ,σ
L ν ,ν
=
∞
C h (xν (ε))
h=−1
∞
Val(θ ), C h (x) =
(k)σ,σ θ∈R R,h,ν,ν
χh (x), σ = ±,
h =h+2
for all k ≥ N , all h ≥ 1, and all resonant pairs {ν, ν }. By construction xν (ε) = xν (ε) (k)σ,σ whenever L h,ν ,ν = 0, so that also L (k) is self-adjoint. Finally we have that L (k) = (k)σ,σ
χ 1 L (k) χ 1 (cf. the definition of the line propagators g0 and ge for trees θ ∈ R R,h,ν ,ν in Definition 5.13). Lemma 5.24. One has u (k)σ = ν
Val(θ ), σ = ±,
(5.12)
(k)σ θ∈ R,ν
for all k ≥ 1 and all ν ∈ Z D+1 . (k)σ
Proof. For any given counterterm L, the coefficients u ν tree values (k)σ uν = Val(θ ).
can be written as sums over
(k)σ
θ∈ν
This can be easily proved by induction, using the diagrammatic rules and definitions given in this section; we refer to Lemma 3.6 of [24] for details. Then, defining the counterterms according to Definition 5.21, all contributions arising from trees belonging to (k)σ (k)σ the set ν but not to the set R,ν cancel out exactly — see Lemma 3.13 of [24] for further details — and hence the assertion follows. 6. Bryuno Lemmas and Bounds Given a tree θ ∈ R , call Z(θ, γ ) the set of (ε, M) ∈ D0 such that for all ∈ L p (θ ) with i = 1 one has 2−h −1 γ ≤ |xν (ε)| ≤ 2−h +1 γ , h = −1, (6.1) |xν (ε)| ≥ γ , h = −1,
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and for all ∈ L p (θ ) one has |δν (ε)| ≤ γ¯ , |δν (ε)| ≤ γ¯ , i = 1, γ¯ ≤ |δν (ε)|, i = 0.
(6.2)
Here γ is the constant introduced in Definition 4.12. Define also D(θ, γ ) ⊂ D0 as the set of (ε, M) ∈ D0 such that for all ∈ L p (θ ) with i = 0 one has |δν (ε) ± γ¯ | ≥ γ /|ν |τ1 , while for all ∈ L p (θ ) with i = 1 one has xν (ε) ≥
γ , pντ (ε)
|δν (ε) ± γ¯ | ≥
γ ∀ν ∈ Cν ∪ Cν , |ν|τ1
(6.3)
for some τ, τ1 > 0. Note that the second condition in (6.3) does not depend on M. Analogously, given a tree θ ∈ S R , we call Z(θ, γ ) the set of (ε, M) ∈ D0 such that (6.1) holds for all ∈ L p (θ )\{e , 0 } with i = 1 and (6.2) holds for all ∈ L p (θ ), and γ ) the set of (ε, M) ∈ D0 such that (6.3) holds for all ∈ L p (θ )\{e , 0 } we call D(θ, with i = 1, while for all ∈ L p (θ ) with i = 0 one has |δν (ε) ± γ¯ | ≥ γ /|ν |τ1 . Remark 6.1. If (ε, M) ∈ Z(θ, γ ) then Val(θ ; ε, M) = 0, while (ε, M) ∈ D(θ, γ ) means that we can use the bounds (6.3) to estimate Val(θ ; ε, M). Analogous considerations hold for trees θ ∈ S R . Remark 6.2. If for some ε one has Val(θ ; ε, M) = 0 and for two comparable lines , ∈ L(θ ) the pair {ν , ν } is resonant, then all the set {ν , ν , ν , ν } is resonant. This motivates the condition in Item 14 in Definition 5.7. (k)σ,σ
Remark 6.3. If θ ∈ R R,h,ν ,ν is such that Val(θ ; ε, M) = 0, then ν, ν ∈ j (ε) for α+β
some j, so that pν (ε) = pν (ε) and |ν − ν | ≤ C1 C2 pν (ε) ≤ C1 C2 pν2α (ε). Moreover pν (ε) ≤ |ν|, |ν | ≤ 2 pν (ε). Such properties follow from Hypothesis 3 — cf. also Lemma 2.9.
Definition 6.4 (The quantity N h (θ )). Define Nh (θ ) as the number of lines ∈ L(θ ) with i = 1 and scale h ≥ h. Definition 6.5 (The quantity K (θ )). Define |m v | + |ν − ν | + |ν v |, K (θ ) = k(θ ) + v∈V0 (θ)
∈L q (θ)
v∈E(θ)
where k(θ ) is the order of θ . Lemma 6.6 There exists a constant B such that the following holds: 1. For all θ ∈ R and all lines ∈ L(θ ) one has |ν | ≤ B(K (θ ))1+4α . 2. If θ ∈ S R , for all lines ∈ L(θ )\(P(e , 0 ) ∪ {0 , e }) one has |ν | ≤ B(K (θ ))1+4α , while for all lines ∈ P(e , 0 ) ∪ {0 } one has |ν | ≤ B(|ν e | + K (θ ))1+4α . 3. Given a tree θ let , ∈ L(θ ) be two comparable lines, with ≺ , such that i = i = 1 and i = 0 for all the lines ∈ P(, ). If |ν − ν | ≥ B K (θ )1+4α , then one has Val(θ ) = 0 for all ε. 4. If θ ∈ S R , ∈ P(e , 0 ) ∪ {0 } and, moreover, i = 0 for all lines ∈ P(e , ), then |ν | ≤ |ν e | + B(K (θ ))1+4α .
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Proof. Let us consider first trees θ ∈ R . The proof is by induction on the order of the tree k = k(θ ). For k = 1 the bound is trivial. If the root line 0 is either a q-line or an r -line or a p-line with i 0 = 0, again the bound follows trivially from the inductive bound. If 0 is a p-line with i 0 = 1, call v0 the node such that 0 = v0 and θ1 , . . . , θs the subtrees with root in v0 . By the inductive hypothesis and Hypothesis 3 one obtains, for a suitable constant C and taking B large enough, 1+4α 2α(1+4α) + C |m v0 | + B(K (θ ) − 1 − |m v0 |) |ν | ≤ |m v0 | + B K (θ ) − 1 − |m v0 | ≤ B(K (θ ))1+4α , which proves the assertion for R in Item 1. As a byproduct also the bound for S R is obtained, as far as lines ∈ / P(e , 0 ) ∪ {0 , e } are concerned. The bound |ν | ≤ B(|ν e |+K (θ ))1+4α for the lines ∈ P(e , 0 )∪ {0 } can be proved similarly by induction. Thus, also Item 2 is proved. Given two comparable lines , such that i = 0 for all lines ∈ P(, ), then by momentum conservation one has min{|ν − ν |, |ν + ν |} ≤ B(K (θ ))1+4α in Case (I) and |ν − ν | ≤ B(K (θ ))1+4α in Case (II). This proves the bounds in Item 3 in Case (II) and in Item 4 for both Cases (I) and (II). In Case (I), if i = i = 1 and max{|δν (ε)|, |δν (ε)|} < 1/2, then |ν − ν | ≤ |ν + ν | by Item 5 in Hypothesis 1. On the other hand if i = i = 1 and max{|δν (ε)|, |δν (ε)|} ≥ 1/2, one has Val(θ ; ε, M) = 0. Hence Item 3 follows also in Case (I). The following bound will allow us to bound the tree values. This bound, and the analogous bound in Lemma 6.12, will be called a Bryuno Lemma, by analogy with the kind of bounds used in the Siegel-Bryuno arguments in the case of Siegel’s problem; see [12,18,33]. Lemma 6.7. Given a tree θ ∈ R such that D(θ, γ ) ∩ Z(θ, γ ) = ∅, for all h ≥ 1 one has Nh (θ ) ≤ max{0, c K (θ )2(2−h)β/2τ − 1}, where c is a suitable constant. Proof. Define E h := c−1 2(h−2)β/2τ . So, we have to prove that Nh (θ ) ≤ max{0, K (θ ) E h−1 − 1}. If a line is on scale h ≥ 0 then γ / pντ (ε) < xν (ε) ≤ 2−h+1 γ by (6.1) and (6.3). Hence B(K (θ ))2 ≥ B(K (θ ))1+4α ≥ |ν | ≥ pν (ε) > 2(h−1)/τ , by Lemma 6.6, so that K (θ )E h−1 ≥ cB −1/2 2(h−1)/2τ 2(2−h)β/2τ ≥ 2 for c suitably large. Therefore if a tree θ contains a line on scale h one has max{0, K (θ )E h−1 − 1} = K (θ )E h−1 − 1 ≥ 1. The bound Nh (θ ) ≤ max{0, K (θ )E h−1 − 1} will be proved by induction on the order of the tree. Let 0 be the root line of θ and call θ1 , . . . , θm the subtrees of θ whose root lines 1 , . . . , m are the lines on scale h i ≥ h −1 and i i = 1 which are the closest to 0 .
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If h 0 < h we can write Nh (θ ) = Nh (θ1 ) + · · · + Nh (θm ), and the bound follows by induction. If h 0 ≥ h then 1 , · · · , m are the entering lines of a cluster T with exiting line 0 ; in that case we have Nh (θ ) = 1+ Nh (θ1 )+. . .+ Nh (θm ). Again the bound follows by induction for m = 0 and m ≥ 2. The case m = 1 can be dealt with as follows. If {ν 0 , ν 1 } is a resonant pair, then either there exists a line ∈ P(1 , 0 ) with i = 1 such that {ν , ν 1 } is a resonant pair or there must be a line ∈ L(T )\P(1 , 0 ) with {ν , ν 1 } a resonant pair. In fact, the first case is not possible: indeed, also {ν 0 , ν } would be resonant (cf. Remark 6.2), so that |h − h 0 | ≤ 1 (cf. Item 14 in Definition 5.7), and hence the contradiction h − 2 ≥ h ≥ h 0 − 1 ≥ h − 1 would follow. In the second case, one has |ν | ≥ pν 1 (ε) > 2(h−2)/τ , hence if θ is the subtree with root line , then one has K (θ ) − K (θ1 ) > K (θ ) > 2E h , and the bound follows once more by the inductive hypothesis. If {ν 0 , ν 1 } is not a resonant pair, call ¯ the line along the path P(1 , 0 ) ∪ {1 } with i ¯ = 1 closest to 0 . Since i ¯ = 1 and by hypothesis h ¯ < h − 1 then {ν ¯, ν 0 } is not a resonant pair (see Item 14 in Definition 5.7). Call T˜ the set of nodes and lines preceding ¯ and define K (T ) = K (θ ) − K (θ1 ) and K (T˜ ) = K (θ ) − K (θ), ¯ 0 and following , ¯ Set also ν¯ = ν ¯ and ν 0 = ν . One has 2|¯ν − ν 0 | ≥ where θ¯ is the tree with root line . 0 β
C2 ( pν¯ (ε) + pν 0 (ε))β ≥ C2 pν 0 (ε) (see Lemma 2.9), so that by Lemma 6.6 one finds B(K (θ ) − K (θ1 )2 ≥ B(K (T˜ ))2 ≥ |¯ν − ν 0 | ≥
1 1 β C2 pν 0 (ε) ≥ C2 2(h−1)β/τ . 2 2
Hence (K (θ ) − K (θ1 ))E h−1 ≥ K (T )E h−1 ≥ K (T˜ )E h−1 ≥ 2, provided c is large enough. This proves the bound. Lemma 6.8. There exists positive constants ξ0 and D0 such that, if ξ > ξ0 in Definition 4.10, then for all trees θ ∈ R and for all (ε, M) ∈ D(θ, γ ) ∩ Z(θ, γ ) one has |Val(θ )| ≤ D0k e−κ K (θ)
−(ξ −ξ0 )
pν
(ε),
(6.4a)
∈L(θ) i =1
|∂ε Val(θ )| ≤ D0k e−κ K (θ)
−(ξ −ξ0 )
pν
(ε),
(6.4b)
∈L(θ) i =1
−(ξ −ξ ) 0 ∂ σ,σ Val(θ ) ≤ D k e−κ K (θ) pν (ε). 0 Mν,ν
ν ∈S ν ∈Cν σ,σ =±
(6.4c)
∈L(θ) i =1
(k)
Proof. The propagators are bounded according to (5.5), so that for all trees θ ∈ R,ν one has ⎛ ⎞⎛ ⎞ |Val(θ )| ≤ C k ⎝ e−A2 |m v | ⎠ ⎝ e−λ0 |ν −ν | ⎠ ⎛ ×⎝
v∈V0 (θ)
v∈E(θ)
⎞
∈L q (θ)
⎛
e−λ0 |ν v | ⎠ 2kh 0 ⎝
∞ h=h 0 +1
⎞ 2h Nh (θ) ⎠
∈L(θ) i =1
−ξ
pν (ε) pνa0 (ε),
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for arbitrary h 0 and for suitable constants C and a0 . For (ε, M) ∈ D(θ, γ ) ∩ Z(θ, γ ) one can bound Nh (θ ) through Lemma 6.7. Therefore, by choosing h 0 large enough the bound (6.4a) follows, provided ξ − a0 > 0 and κ is suitably chosen. When bounding ∂ε Val(θ ), one has to consider derivatives of the line propagators, i.e. ∂ε g . If is an r -line then |∂ε g | is bounded proportionally to |ν |c0 , whereas if is a p-line, then the derivative produces factors which admit bounds of the form −ξ
Cpνa1 (ε) 22h pνc0 (ε) pν (ε),
(6.5)
for suitable constants C and a1 ; see the proof of Lemma 4.2 in [24] for details (and use Item 3 in Hypothesis 1). The extra factor 2h can be taken into account by bounding the product of line propagators with 22h 0 k
∞
22h Nh (θ) .
h=h 0 +1
One can bound |ν | ≤ B(K (θ ))2 , and use part of the exponential decaying factors e−A2 |m v | , e−λ0 |ν −ν | , and e−λ0 |ν v | , to control the contribution v∈V0 (θ) |m v | + ∈L q (θ) |ν − ν | + v∈E(θ) |ν v | to K (θ ) (cf. Definition 6.5). Then, if ξ is large enough, so that ξ − a1 > 0 for all possible values of a1 in (6.5), the bound (6.4b) follows. Also the bound (6.4c) can be discussed in the same way. We refer again to [24] for the details. Remark 6.9. Note that for (ε, M) ∈ D(θ, γ ) the singularities of the functions χ¯ 1 are avoided, so that ∂ε χ¯ 1 (δν (ε)) = 0 for all ∈ L(θ ). Note also that the bound (6.4c) is not really needed in the following. Lemma 6.10. There are two positive constants B2 and B3 such that the following holds: β/2
1. Given a tree θ ∈ S R such that Val(θ ; ε, M) = 0, if K (θ ) ≤ B2 pν e (ε) then for all lines ∈ P(e , 0 ) one has i = 0. Moreover for all such lines , if {ν , ν e } is not a resonant pair, then one has |δν (ε)| ≥ 1/2. 2. Given a tree θ ∈ R R such that Val(θ ; ε, M) = 0, one has ν 0 − ν e ≤ B3 (K (θ ))1/ρ , with ρ depending on α and β. (k)σ,σ
Proof. Suppose that θ ∈ S R,h,ν ,ν and P(e , 0 ) contains lines with i = 1 and consequently with {ν , ν } not resonant (cf. Definition 5.18). Let ¯ be the one closest to e ; β β thus, one has |ν ¯ − ν | ≥ C3 (|ν ¯| + |ν |)β ≥ C3 pν (ε) = C3 pν (ε), so that we can apply β
Item 3 in Lemma 6.6 to obtain B(K (θ ))2 ≥ C pν (ε), for some positive constant C. This proves the first statement in Item 1. The proof of the second statement is identical, since |δν (ε)| < 1/2 implies that ν ∈ j1 (ε) for some j1 , so that if {ν , ν } is not a resonant β pair then ν ∈ / j1 (ε), and therefore |ν − ν | ≥ C3 pν (ε). α+β To prove Item 2, notice that |ν − ν | ≤ C1 C2 pν (ε) (cf. Remark 6.3). If K (θ ) > β/2 β/2 B2 pν (ε) then K (θ ) ≥ C|ν − ν |β/2(α+β) . If K (θ ) ≤ B2 pν (ε) then P(e , 0 ) has only lines with i = 0, so that by Item 3 in Lemma 6.6 one finds |ν − ν | ≤ B K (θ )2 .
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γ) ∩ Lemma 6.11. Given a tree θ ∈ S R such that D(θ, Z(θ, γ ) = ∅, if Nh (θ ) ≥ 1 for some h ≥ 1, then cK (θ )2(2−h)β/2τ ≥ 1, with c the same constant as in Lemma 6.7. (k)σ,σ
¯ Proof. Consider a tree θ ∈ S R,h, ¯ ν ,ν for some k ≥ 1, h ≥ 1, σ, σ = ± and ν, ν ∈ S
¯ such that |ν | ≥ 2(h−2)/τ . Assume Nh (θ ) ≥ 1 for some h¯ ≥ h ≥ 1. If there is a line ∈ L(θ ), which does not belong to P := P(e , 0 ), such that h ≥ h, then one can reason as at the beginning of the proof of Lemma 6.7 to obtain K (θ )E h−1 ≥ 2, with E h = c−1 2(h−2)β/2τ ≥ 1. Otherwise, there are lines ∈ P on scale h ≥ h, and hence such that i = 1 and, consequently, {ν , ν } is not a resonant pair. Let ¯ be the one closest to e among such β β lines; thus, one has |ν ¯ − ν | ≥ C3 pν (ε), so that one obtains B(K (θ ))2 ≥ C pν (ε) ≥ ¯
C2(h−2)β/τ , for some positive constant C. So, the desired bound follows once more.
γ) ∩ Lemma 6.12. Given a tree θ ∈ S R such that D(θ, Z(θ, γ ) = ∅, for all h ≥ 1 one has Nh (θ ) ≤ c K (θ )2(2−h)β/2τ , where c is the same constant as in Lemma 6.7.
(k)σ,σ ¯ Proof. Consider a tree θ ∈ S R, ¯ ν ,ν for some k ≥ 1, h ≥ 1, σ, σ = ± and ν, ν ∈ S h, ¯
such that |ν | ≥ 2(h−2)/τ . For k(θ ) = 1 one has Nh (θ ) ≤ 1, so that the bound follows from Lemma 6.11. For k(θ ) > 1 one can proceed as follows. Let 0 be the root line of θ and call θ1 , . . . , θm the subtrees of θ whose root lines 1 , . . . , m are the lines on scale h i ≥ h −1 and i i = 1 which are the closest to 0 . All the trees θi such that i ∈ / P(e , 0 ) belong to β/2 (k )± some R,iν i with ki < k. If K (θ ) ≥ B2 pν (ε) (cf. Lemma 6.10) it may be possible that (k ),σ ,σ
a line, say 1 , belongs to P(e , 0 ), so that Val(θ1 ) = g1 Val(θ1 ), with θ1 ∈ S R,h1 1 ,ν11 ,ν ¯ σ1 = ± and k1 < k. with h 1 ≤ h, If h 0 < h one has Nh (θ ) = Nh (θ1 ) + · · · + Nh (θm ), so that the bound Nh (θ ) ≤ K (θ )E h−1 follows by the inductive hypothesis. If h 0 ≥ h one has Nh (θ ) = 1 + Nh (θ1 ) + · · · + Nh (θm ). For m = 0 the bound can be obtained once more from Lemma 6.11, while for m ≥ 2 at least one tree, say θm , )± belongs to (k R,ν for some k and ν so that we can apply Lemma 6.7 and the inductive hypothesis to obtain Nh (θ ) ≤ 1 + (K (θ1 ) + · · · + K (θm−1 )) E h−1 + K (θm )E h−1 − 1 ≤ (K (θ1 ) + · · · + K (θm−1 )) E h−1 + K (θm )E h−1 ≤ K (θ )E h−1 ,
which yields the bound. / P(e , 0 ), again the Finally if m = 1 one has Nh (θ ) = 1 + Nh (θ1 ). Hence, if 1 ∈ bound follows from Lemma 6.7. If on the contrary 1 ∈ P(e , 0 ), one can adapt the discussion of the case m = 1 in the proof of Lemma 6.7.
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Lemma 6.13. There exist positive constants κ, ξ1 and D1 such that, if ξ > ξ1 in Defini γ) ∩ tion 4.10, then for all trees θ ∈ R R and for all (ε, M) ∈ D(θ, Z(θ, γ ), by setting ν = ν 0 and ν = ν e , one has ρ −(ξ −ξ1 ) |Val(θ )| ≤ D1k 2−h e−κ|ν −ν | pν (ε), (6.6a) ∈L(θ) i =1 ρ
|∂ε Val(θ )| ≤ D1k 2−h pνc0 (ε) e−κ|ν −ν | ν 1 ∈S ν 2 ∈Cν 1
−(ξ −ξ1 )
pν
(ε),
(6.6b)
∈L(θ) i =1
ρ −(ξ −ξ1 ) pν (ε), ∂ Mνσ1,ν,σ2 Val(θ ) ≤ D1k 2−h e−κ|ν −ν | 1 2
σ1 ,σ2 =±
(6.6c)
∈L(θ) i =1
with ρ as in Lemma 6.10. Proof. Set for simplicity P = P(e , 0 ) and (θ ) = |m v | + |ν − ν | + |ν v |, v∈V0 (θ)
⎛ (θ ) = ⎝
v∈V0 (θ)
∈L q (θ)
⎞⎛
e
A2 |m v |/8 ⎠ ⎝
v∈E(θ)
∈L q (θ)
⎞⎛
e
λ0 |ν −ν | ⎠ ⎝
⎞ e
λ0 |ν v | ⎠
.
v∈E(θ)
(k)σ,σ
If θ ∈ R R,h,ν ,ν for some k ≥ 1, h ≥ 1, σ, σ = ± and {ν, ν } resonant, then Nh (θ ) ≥ 1, so that K (θ ) = k+(θ ) > C2hβ/2τ , for some constant C, which imply 1 ≤ 2−h C k (θ ), for some constant C. This produces the extra factor 2−h . ρ By Item 2 in Lemma 6.10 one has (B3−1 |ν−ν |)ρ ≤ K (θ ), so that 1≤e−|ν −ν | C k (θ ), for some constant C. The factor (θ ) can be bounded by using part of the factors e−A2 |m v | , e−λ0 |ν v | , and e−λ0 |ν −ν | , associated with the nodes and with the q-lines. This proves the bound (6.6a), To prove the bound (6.6b) one has to take into account the further ε-derivative acting on the line propagator g , for some ∈ L(θ ). If the line does not belong to P then one can reason as in the proof of (6.4b) in Lemma 6.8. If ∈ P one has to distinguish between two cases. If there exists a line ¯ ∈ P such that i ¯ = 1, then β/2 K (θ ) > B2 pν (ε) by Item 1 in Lemma 6.10, so that, by Item 2 in Lemma 6.6, one has pν (ε) ≤ |ν | ≤ B(|ν e | + K (θ ))1+4α ≤ B(2 pν (ε) + K (θ ))1+4α ≤ C(K (θ ))4/β , for some constant C. If i = 0 for all lines ∈ P then, by Item 3 in Lemma 6.6, one has pν (ε) ≤ |ν | ≤ |ν e | + B(K (θ ))2 . Then Item 3 in Hypothesis 1 implies the bound (6.6b). To prove (6.6c) one has to study a sum of terms each containing a derivative ∂ Mνσ1,ν,σ2 g , 1 2
β/2
for some ∈ L(θ ). If ∈ P we distinguish between the two cases. If K (θ ) > B2 pν (ε), α+β α+β the sum over ν 1 , ν 2 has the limitations |ν 1 − ν 2 | ≤ C pν 1 (ε), |ν 1 − ν | ≤ C pν 1 (ε) and |ν | ≤ (|ν e | + B K (θ ))1+4α ≤ C(K (θ ))4/β , for some constant C: hence the sum over ν 1 , ν 2 produces a factor C(K (θ ))C for suitable constants C and C , and one has β/2 (K (θ ))C ≤ C k (θ ), for some constant C. If K (θ ) ≤ B2 pν (ε), then i = 0 for all lines ∈ P, so that the line propagators g do not depend on M. Finally if ∈ P then one
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has |ν | ≤ B(K (θ ))1+4α , so that the sum over ν 1 , ν 2 is bounded once more proportion ally to (K (θ ))C , for some constant C , and again one can bound (K (θ ))C ≤ C k (θ ), for some constant C. Remark 6.14. Both Lemmas 6.12 and 8.2 deal with the first derivatives of Val(θ ). One can easily extend the analysis so to include derivatives of arbitrary order, at the price of allowing larger constants ξ1 and D1 — and a factor pνc0 (ε) for any further ε-derivative. Therefore, one can prove that the function Val(θ ) is C r for any integer r , in particular for r = 1, which we shall need in the following — cf. in particular the forthcoming Lemma 7.2. 7. Proof of Proposition 1 Definition 7.1. (The extended tree values). Let the function χ−1 be as in Definition 5.1. Define ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎜ χ−1 (|xν (ε)| pντ (ε))⎟ Val E (θ ) = ⎜ ⎝ ⎠⎝
∈L(θ) ν ∈C{ν ,ν } i =1
∈L(θ) i =1
⎛
⎟ χ−1 (||δν (ε)| − γ¯ | |ν|τ1 )⎟ ⎠
⎞
⎜ ⎟ τ1 ⎟ ×⎜ χ (||δ (ε)| − γ ¯ | |ν | ) −1 ν ⎝ ⎠ Val(θ )
(7.1)
∈L p (θ) i =0
(k)
for θ ∈ R,ν , and
⎛
⎞
⎜ Val E (θ ) = ⎜ ⎝
∈L(θ)\{0 ,e } i =1
⎟ χ−1 (|xν (ε)| pντ (ε))⎟ ⎠ ⎞
⎛ ⎜ ×⎜ ⎝ ⎛ ⎜ ⎜ ×⎜ ⎝
∈L(θ)\{0 ,e } ν ∈C{ν ,ν } i =1
∈L p (θ) / C{ν,ν } i =0, ν ∈
⎟ χ−1 (||δν (ε)| − γ¯ | pντ1 (ε))⎟ ⎠ ⎞
⎟ ⎟ χ−1 (||δν (ε)| − γ¯ | |ν |τ1 )⎟ Val(θ ) ⎠
(7.2)
(k)
for θ ∈ R R,h,ν ,ν . We call Val E (θ ) the extended value of the tree θ . The following result proves Proposition 1. (k)σ,σ
Lemma 7.2. Given θ ∈ R R,h,ν ,ν , the function Val(θ ) can be extended to the function (7.1) defined and C 1 in D0 \I{ν ,ν } (γ ), such that, defining the “extended” counterterm according to Definition 5.21, with Val(θ ) replaced with Val E (θ ), the following L νE,νσ,σ holds:
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1. Possibly with different constants ξ1 and K 0 , Val E (θ ) satisfies for all (ε, M) ∈ D0 \I{ν ,ν } (γ ) the same bounds in Lemma 6.13 as Val(θ ) in D(γ ). 2. There exist constants ξ1 , K 1 , κ, ρ and η0 , such that, if ξ > ξ1 in Definition 4.10, satisfies, for all (ε, M) ∈ D0 \I{ν ,ν } (γ ) and |η| ≤ η0 , the bounds L νE,νσ,σ ρ ρ E σ,σ E σ,σ ∂ε L ν ,ν ≤ |η| N K 1 pνc0 e−κ|ν −ν | , L ν ,ν ≤ |η| N K 1 e−κ|ν −ν | , ρ E σ,σ ∂η L ν ,ν ≤ N |η| N −1 K 1 e−κ|ν −ν | , σ ,σ E σ,σ κ|ν −ν |ρ ≤ |η| N K 1 . ∂ Mν 1,ν 2 L ν ,ν e ν 1 ∈S ,σ1 =± ν 2 ∈Cν 1 ,σ2 =±
1 2
3. Val E (θ ) = Val(θ ) for (ε, M) ∈ D(2γ ) and Val E (θ ) = 0 for (ε, M) ∈ D0 \D(γ ). (k)σ
Analogously, given θ ∈ R,ν , the function Val(θ ) can be extended to the function (7.2) E (k)
defined and C 1 in D0 , such that, defining u ν as in Lemma 5.24 with Val(θ ) replaced with Val E (θ ), the following holds: 1. Possibly with different constants ξ1 and K 0 , Val E (θ ) satisfies for all (ε, M) ∈ D0 the same bounds in Lemma 6.8 as Val(θ ) in D(γ ). 2. There exist constants ξ1 , K 1 , κ and η0 such that, if ξ > ξ1 in Definition 4.10, u νE σ satisfies, for all (ε, M) ∈ D0 and |η| ≤ η0 , the bounds 1/2 E σ u ν ≤ |η| N K 1 e−κ|ν | for all ν ∈ Z D+1 . 3. Val E (θ ) = Val(θ ) for (ε, M) ∈ D(2γ ) and Val E (θ ) = 0 for (ε, M) ∈ D0 \D(γ ). (k)σ,σ
Proof. We shall consider explicitly the case of trees θ ∈ R R,h,ν ,ν . The case of trees (k)σ
θ ∈ R,ν can be discussed in the same way. Item 3 follows from the very definition. The bounds of Item 1 can be proved by reasoning as in Sect. 6, by taking into account the further derivatives which arise because of the compact support functions χ−1 in (7.2). On the other hand all such derivatives produce factors proportional to pνa2 (ε) for some constant a2 (again we refer to [24] for details); in particular we are using Item 2 in Hypothesis 1 to bound the derivatives of δν (ε) with respect to ε. Therefore by using Lemma 6.7 and possibly taking larger constants ξ1 and K 0 the bounds of Lemma 6.13 follow also for the extended function (7.2). Finally the bounds on L E in Item 2 come directly from the definition. Indeed, the counterterms L νE,νσ,σ are expressed in terms of the values Val(θ ) according to Remark 5.23, and the factor 2−h is used to perform the summation over the scale labels. Hence we have to control the sum over the trees. Let us fix ε. For each v ∈ E(θ ) the sum over |ν v | is controlled by using the exponential factors e−λ0 |ν v | . For each line ∈ L(θ ) the labels ν are fixed by the conservation rule of Item 12 in Definition 5.7, while the sum over ν gives a factor C1 pνα (ε) for the p-lines (see Item 2 in Hypothesis 3), and it is controlled by using the exponential factors e−λ0 |ν −ν | for the q-lines. The sums over i and h can be bounded by a factor 4. Finally the sum over all the unlabelled trees of order k is bounded by C k for some constant C. Thus, the bounds on L νE,ν are proved. Finally, the C 1 smoothness follows from Remark 6.14.
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901
8. Proof of Proposition 2 The following result proves Items 1 and 2 in Proposition 2. Here and henceforth we write L = L(η, ε, M) and L E = L E (η, ε, M), and we fix η = ε1/N .
Lemma 8.1. There exists constants ε0 > 0 such that there exist functions Mνσ,σ ,ν (ε) = σ,σ 1 Mν ,ν (ε) well defined and C for ε ∈ [0, ε0 ]\I {ν ,ν } (γ ), such that the “extended” compatibility equation
E σ,σ 1/N , ε, M(ε)) Mνσ,σ ,ν (ε) = L ν ,ν (ε
holds for all ε ∈ (0, ε0 )\I {ν ,ν } (γ ). The functions u νE σ (ε1/N , ε, M(ε)) are C 1 in [0, ε0 ].
Proof. By definition we set Mνσ,σ ,ν (ε) = 0 for all ε such that χ¯ 1 (δν (ε))χ¯ 1 (δν (ε)) = 0. σ,σ Consider the Banach space B of lists {Mν ,ν (ε)}, with {ν, ν } a resonant pair, such that
σ,σ 1 each Mνσ,σ ,ν (ε) is well defined and C in ε ∈ [0, ε0 ]\I {ν ,ν } (γ ), and Mν ,ν (ε) = 0 for ε ∈ I {ν ,ν } (γ ).
By definition {L νE1σ,ν12,σ2 (ε1/N , ε, {Mνσ,σ ,ν (ε)})} is well defined as a continuously dif-
1 ,σ2 ferentiable application from B in itself, since, for each tree θ ∈ R(k)σ R,h,ν 1 ,ν 2 , the value Val E (θ ) by definition smooths out to zero the value of each line propagator g in the corresponding intervals I {ν ,ν }(2γ )\I {ν ,ν } (γ ). Again by definition L E (0, 0, 0) = 0 and |∂ M L(0, 0, 0)|op = 0, so that we can apply the implicit function theorem. Analogously one discusses the smoothness of the functions u νE σ (ε1/N , ε, M(ε)).
Lemma 8.2. Let A = A(ε) be a self-adjoint matrix piecewise differentiable in the parameter ε. Then, if λ(i) (A) and φ (i) (A) denote the eigenvalues and the (normalised) eigenvectors of A, respectively, the following holds: 1. One has |λ(i) (A(ε))| ≤ A(ε)2 . 2. The eigenvalues λ(i) (A(ε)) are piecewise differentiable in ε. 3. One has |∂ε λ(i) (A(ε))| ≤ ∂ε A(ε)2 . Proof. See [26] for Items 1 and 2. Moreover, for each interval in which A is differentiable, let An be an analytic approximation of A in such an interval, with An → A as n → ∞: then the eigenvalues φ (i) (An ) are piecewise differentiable [26], and one has ∂ε λ(i) (An ) = ∂ε φ (i) , An φ (i) = λ(i) (An )∂ε φ (i) , φ (i) + φ (i) , ∂ε An φ (i) = φ (i) , ∂ε An φ (i) , which yields Item 3 when the limit n → ∞ is taken. For M ∈ Bκ,ρ we can write M = χ 1 M χ1 = j M j , where M j are block matrices, so that we can define M 2 = sup j M j 2 , with M j 2 given as in Definition 4.8. Lemma 8.3. For M ∈ Bκ,ρ one has M 2 ≤ Cε0 for some constant C depending on κ and ρ.
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Proof. If M ∈ Bκ,ρ then M = ρ
j
M j , with M j a block matrix with dimension d j
σ,σ = Mνσ,σ ,ν , for suitable ν, ν , σ, σ such that |Mν ,ν | ≤ for some constant D. Therefore
depending on j, and M j Dε0 e−κ|ν −ν |
(i, i )
M j 2= max M j x2 ≤ max 2 2 |x|2 ≤1
|x|2 ≤1
dj M j (i, i ) x(i ) M j (i, i ) x(i ) i,i ,i =1
1 M j (i, i ) M j (i, i ) x(i )2 + x(i )2 max 2 |x|2 ≤1 i,i ,i =1 ⎞2 ⎛ dj dj dj dj M j (i, i ) M j (i, i ) x(i )2 ≤ ⎝ M j (i, i )⎠ , ≤ max dj
≤
|x|2 ≤1
i =1
i=1
which yields the assertion.
i =1
i=1
Lemma 8.4. Let A, B be two self-adjoint d × d matrices. Then d (i) ( j) (i) ≤ (A + B) − λ (A) λ λ (B) j=1
for all i = 1, . . . , d. Proof. The result follows from Lidskii’s Lemma; cf. [26].
Define E1 = {ε ∈ [0, ε0 ] : xν (ε) ≥ 2γ / pντ (ε) ∀ν ∈ S} and E2 = {ε ∈ [0, ε0 ] : ||δν (ε)| − γ¯ | ≥ 2γ /|ν|τ1 ∀ν ∈ S}, and set E = E1 ∩ E2 . We can denote by λσν (A), with ν ∈ S and σ = ±, the eigenvalues of the block matrix A = D + M . If |δν (ε)| ≥ γ¯ , then λσν (ε) = δν (ε). Moreover for each ε ∈ [0, ε0 ] and each ν ∈ S such that |δν (ε)| < γ¯ , there exists a block Aν (ε) of the matrix A, of size dν (ε) ≤ 2C1 pνα (ε) such that λ± ν (A) depends only on the entries of such a block. This follows from Lemma 5.24 and Remark 5.4. Therefore we have to discard from [0, ε0 ] only values of ε such that |δν (ε)| < γ¯ for some ν ∈ S: for all such ν the matrix Aν (ε) is well defined, and one has λσν (A) = λσν (Aν (ε)). One has, by Item 3 in Lemma 4.9, xν (ε) ≥
1 ξ
min
pν (ε) i=1,...,dν (ε)
(i) ν λ (A (ε)) ≥
1 ξ
min min λσν (Aν (ε)) ,
pν (ε) ν ∈C ν (ε) σ =±
(8.1)
so that, by using that λσν (Aν (ε)) = λσν (Aν (ε)) = λσν (A) for all ν ∈ C ν (ε), we shall impose the conditions σ ν λ (A (ε)) ≥ γ2 , ν |ν|τ2
ν ∈ S,
σ = ±,
(8.2)
for suitable γ2 > 2γ . Thus, the conditions (8.2), together with the bound |ν| ≤ 2 pν (ε) (cf. Remark 6.3), will imply through (8.1) the bounds (6.3) for xν (ε).
Periodic Solutions for a Class of Nonlinear PDEs in Higher Dimensions
Define
γ2 Kσν = ε ∈ [0, ε0 ] : λσν (A) ≤ τ , |ν| 2
with τ2 = τ − ξ , so that we can estimate meas([0, ε0 ]\E1 ) ≤
ν ∈S σ =±
Moreover, by defining 2γ Hν ,σ = ε ∈ [0, ε0 ] : |δν (ε) − σ γ¯ | ≤ τ , |ν| 1 with τ1 to be determined, one has meas([0, ε0 ]\E2 ) ≤
ν ∈Z D+1
903
ν ∈ S,
σ = ±,
(8.3)
meas(Kσν ).
(8.4)
ν ∈ Cj,
(8.5)
j ∈ N, σ = ±,
meas(Hν ,σ ).
(8.6)
σ =±
Lemma 8.5. There exist constants w0 and w1 such that K± ν = ∅ for all ν such that |ν| ≤ w0 /ε0w1 . There exist constants y0 and y1 such that Hν ,± = ∅ for all ν such that y |ν| ≤ y0 /ε01 . Proof. We start by considering the sets Kσν for ν ∈ S and σ = ±. If |δν (ε)| < γ¯ one can write Aν (ε) = diag{δν (0), δν (0)}ν ∈C ν (ε) + B ν (ε), which defines the matrix B ν (ε) as B ν (ε) = diag{δν (ε) − δν (0)}σ=±
ν ∈C ν (ε)
+ M ν (ε),
where M ν (ε) is the block of M(ε) with entries Mνσ11,,σν 22 (ε) such that ν 1 , ν 2 ∈ C ν (ε). By Lemma 8.4, one has ν (ε) d σ (i) ν λ (A) − δν (0) ≤ λ (B (ε)) ≤ 2C1 p α (ε)B ν (ε)2 ,
ν
ν
ν ∈ S,
σ = ±,
i=1
(8.7) where we have used Remark 4.11 to bound dν (ε). One has |δν (0)| ≥ γ0 /|ν|τ0 ≥ γ0 /(2 pν (ε))τ0 by Item 2 in Hypothesis 1, whereas B ν (ε)2 ≤ c2 (2 pν (ε))c0 ε0 + M ν (ε)2 , by Items 1 and 2 in Hypothesis 1, and M ν (ε)2 ≤ M(ε)2 ≤ C0 ε0 by Lemma 8.3. Therefore (8.7) implies σ γ0 λ (A) ≥ − C pνc0 +1 (ε) ε0 , ν (2 pν (ε))τ0 for a suitable constant C, so that, by setting w1 = c0 + 1 + τ0 and choosing suitably the τ2 τ0 constants γ2 , τ and w0 , one has |λ± ν (A)| ≥ γ0 /2(2 pν (ε)) ≥ γ2 / pν (ε) for all ν such w1 that |ν| ≤ w0 /ε0 . For the sets Hν ,σ , one can reason in the same way, by using that γ¯ ∈ G (cf. Definition 4.1). Lemma 8.6. Let ξ > ξ1 and ε0 = η0N be fixed as in Lemma 7.2. There exist constants γ , τ and τ1 such that meas([0, ε0 ]\E) = o(ε0 ). Proof. First of all we have to discard from [0, ε0 ] the sets Hν ,σ . It is easy to see that one has
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meas(Hν ,σ ) ≤
2 2γ , τ 1 |ν| c1 |ν|c0
for some positive constant C, so that, by using the second assertion in Lemma 8.5, we find y (τ +c −D−1) meas(Hν ,σ ) ≤ meas(Hν ,σ ) ≤ Cε01 1 0 , ν ∈S σ =±1
ν ∈S y σ =±1 |ν |≥y0 /ε01
for some constant C, provided τ1 + c0 − D > 1, so that we shall require for τ1 to be such that τ1 + c0 − D > 1 and y1 (τ1 + c0 − D − 1) > 1. ν ν Next, we consider the sets K± ν . For all ν ∈ S consider A (ε) and write A (ε) = ν ν δν (ε)I + B (ε), which defines the matrix B (ε) as B ν (ε) = diag{δν (ε) − δν (ε)}σ =±
ν ∈C ν (ε)
+ M ν (ε),
with M ν (ε) defined as in the proof of Lemma 8.5. Then the eigenvalues of Aν (ε) are of the form λ(i) (Aν (ε)) = δν (ε) + λ(i) (B ν (ε)), so that for all ε ∈ [0, ε0 ]\I ν (γ ) one has (i) ν ∂ε λ (A ) ≥ |∂ε δν (ε)| − ∂ε B ν (ε)2 , where Item 3 in Lemma 8.2 has been used. One has |∂ε δν (ε)| ≥ c1 |ν|c0 , by Item 2 in Hypothesis 1, and ∂ε B ν (ε)2 ≤ maxν ∈C ν (ε) |∂ε (δν (ε) − δν (ε))| + ∂ε M ν (ε)2 ≤ ζ c3 pν (ε) pνc0 −1 (ε) + ε0 C pνc0 (ε), for a suitable constant C, as follows from Item 4 in Hypothesis 1, from Hypothesis 3 (see Lemma 2.9 for the definition of ζ ), from Lemma 7.2, and from Lemma 8.1. Hence we can bound |∂ε λ(i) (Aν )| ≥ c1 |ν 0 |c0 /2 for ε0 small enough. Therefore one has 2 2γ2 (α+β)(D+1) , (8.8) C|ν| meas(Kσν ) ≤ τ |ν| 2 c1 |ν|c0 for some constant C, where the last factor C|ν|(α+β)(D+1) arises for the following reason. The eigenvalues λσν (A) are differentiable in ε except for those values ε such that for some ν ∈ Cν one has |δν (ε)| = γ¯ and |δν (ε)| < γ¯ . Because of Item 3 in Hypothesis 1 all functions δν (ε) are monotone in ε as far as |δν (ε)| < 1/2, so that for each ν ∈ Cν the condition |δν (ε)| = γ¯ can occur at most twice. The number of ν ∈ Cν such that the conditions |δν (ε)| = γ¯ and |δν (ε)| < γ¯ can occur for some ε ∈ [0, ε0 ] is bounded by the volume of a sphere of centre ν and radius proportional to |ν|α+β (cf. Lemma 5.24). Hence C|ν|(α+β)(D+1) counts the number of intervals in [0, ε0 ]\I ν (γ ). Thus, (8.8) yields, by making use of the first assertion of Lemma 8.5, 8γ −τ2 −c0 w (τ +c −2α−D−1) meas(Kσν ) ≤ |ν| C|ν|2α ≤ Cε0 1 2 0 , c 1 D+1 σ =± ν ∈S
ν ∈Z w |ν |≥w0 /ε0 1
for some positive constant C, provided τ2 + c0 − 2α − D = τ + c0 − 2α − D − ξ > 1, w (τ +c −D−1) so that (8.4) implies that meas([0, ε0 ]\E1 ) ≤ Cε0 1 2 0 . Therefore, the assertion follows provided min{τ1 , τ2 − 2α} > D − c0 + 1, y1 (τ1 + c0 − D − 1) > 1 and w1 (τ2 + c0 − 2α − D − 1) > 1.
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905
A. Proof of Lemma 3.4 Lemma 3.4 is a consequence of the following elementary proposition in Galois theory. Proposition. If p1 , . . . , pk are distinct primes then the field √ √ √ F := Q[ p1 , p2 , . . . , pk ] √ obtained from the rational numbers Q by adding the k square roots pi has dimension √ 2k over Q with basis the elements i∈I pi as I varies on the 2k subsets of {1, 2, . . . , k}. The group of automorphisms2 of F which fix Q (i.e. the Galois group of F/Q) is an √ √ Abelian group generated by the automorphisms τi defined by τi ( p j ) = (−1)δ(i, j) p j . Proof. We prove by induction √ both statements. Let us assume the statements valid for √ √ √ p1 , . . . , pk−1 and let F := Q[ p1 , p2 , . . . , pk−1 ] so that F = F [ pk ]. We first √ √ prove that pk ∈ / F . Assume it to be false. Since√( pk )2 is integer, each element – √ say τ – of the Galois group of F /Q must either fix pk or transform it into − pk (by √ definition τ ( pk ) = τ ( p k )2 = pk ). Now any element b ∈ F is by induction uniquely expressed as √ aI pi , a I ∈ Q. b= I ⊂{1,2,...,k−1}
i∈I
If h of the numbers a I are non-zero, it is easily seen that b has 2h transforms√(changing the signs of each of the a I ) under the√Galois group of F√ . Therefore b = p k if and only if h = 1, that is one should have pk = m/n i∈I pi , I ⊂ {1, 2, . . . , k − 1} for m, n integers. This implies that pk n 2 = m 2 i∈I pi which is impossible by the unique factorisation of integers. This proves the first statement. To construct ..,k −1 √the Galois √ group of F/Q we extend the action of τi for i = 1, . √ by setting τi ( pk ) = pk . Finally we define the automorphism τk as τk ( p j ) = √ (−1)δ(k, j) p j for j = 1, . . . , k. Acknowledgement. We thank Claudio Procesi and Massimiliano Berti for useful discussions. The paper was written while M. Procesi was supported by the European Research Council under FP7.
References 1. Ablowitz, M., Prinari, B.: Nonlinear Schrodinger systems: continuous and discrete. Scholarpedia 3(8), 5561 (2008) 2. Artin, M.: Algebra. Englewood Cliffs, NJ: Prentice Hall, 1991 3. Baldi, P., Berti, M.: Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 17(3), 257–277 (2006) 4. Berti, M., Bolle, Ph.: Cantor families of periodic solutions for completely resonant nonlinear wave equations. Duke Math. J. 134(2), 359–419 (2006) 5. Berti, M., Bolle, Ph.: Cantor families of periodic solutions of wave equations with C k nonlinearities. Nonlinear Diff. Eqs. Appl. 15(1–2), 247–276 (2008) 6. Berti, M., Bolle, Ph.: Cantor families of periodic solutions for wave equations via a variational principle. Adv. Math. 217(4), 1671–1727 (2008) 7. Berti, M., Bolle, Ph.: Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions. Preprint, 2008 2 I.e. the linear transformations τ such that τ (uv) = τ (u)τ (v).
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8. Bourgain, J.: Construction of periodic solutions of nonlinear wave equations in higher dimension. Geom. Funct. Anal. 5, 629–639 (1995) 9. Bourgain, J.: Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations. Ann. of Math. 148(2), 363–439 (1998) 10. Bourgain, J.: Periodic solutions of nonlinear wave equations. In: Harmonic analysis and partial differential equations (Chicago, IL, 1996), Chicago Lectures in Mathematics, Chicago, IL: University Chicago Press, 1999, pp. 69–97 11. Bourgain, J.: Green’s Function Estimates for Lattice Schrödinger Operators and Applications. Ann. Math. Studies 158, Princeton, NJ: Princeton University Press, 2005 12. Bruno, A.D.: Analytic form of differential equations. I, II. Trudy Moskov. Mat. Obšˇc. 25, 119–262 (1971); ibid. 26, 199–239 (1972) 13. Buslaev, V.S., Perelman, G.S.: Nonlinear scattering: the states which are close to a soliton. J. Math. Sci. 77(3), 3161–3169 (1995) 14. Cheng, Ch.-Q.: Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems. Commun. Math. Phys. 177(3), 529–559 (1996) 15. Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211(2), 497–525 (2000) 16. Craig, W., Wayne, C.E.: Newton’s method and periodic solutions of nonlinear wave equations. Comm. Pure Appl. Math. 46, 1409–1498 (1993) 17. Eliasson, L.H., Kuksin, S.: KAM for non-linear Schrödinger equation. Ann. of Math., to appear, available at http://pjm.math.berkeley.edu/editorial/uploads/annals/accepted/080510-Eliasson/080510Eliasson-v2.pdf 18. Gallavotti, G.: Twistless KAM tori. Commun. Math. Phys. 164(1), 145–156 (1994) 19. Geng, J., You, J.: A KAM theorem for Hamiltonian partial differential equations in higher dimensional spaces. Commun. Math. Phys. 262(2), 343–372 (2006) 20. Geng, J., You, J.: KAM tori for higher dimensional beam equations with constant potentials. Nonlinearity 19(10), 2405–2423 (2006) 21. Gentile, G., Mastropietro, V.: Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method. J. Math. Pures Appl. (9) 83(8), 1019–1065 (2004) 22. Gentile, G., Mastropietro, V., Procesi, M.: Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions. Commun. Math. Phys. 256(2), 437–490 (2005) 23. Gentile, G., Procesi, M.: Conservation of resonant periodic solutio.ns for the one-dimensional nonlinear Schrödinger equation. Commun. Math. Phys. 262(3), 533–553 (2006) 24. Gentile, G., Procesi, M.: Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension. J. Diff. Eqs. 245(11), 3095–3544 (2008) 25. Gang, Zh., Sigal, I.M.: Relaxation of solitons in nonlinear Schrödinger equations with potential. Adv. Math. 216(2), 443–490 (2007) 26. Kato, T.: Perturbation Theory for Linear Operators. Berlin-New York: Springer-Verlag, 1976 27. Kuksin, S.B.: Nearly Integrable Infinite-Dimensional Hamiltonian Systems. Lecture Notes in Mathematics 1556, Berlin: Springer-Verlag, 1993 28. Kuksin, S.B., Pöschel, J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. of Math. 143(1), 149–179 (1996) 29. Moser, J.: Convergent series expansions for quasi-periodic motions. Math. Ann. 169, 136–176 (1967) 30. Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71(2), 269–296 (1996) 31. Pöschel, J.: On the construction of almost periodic solutions for a nonlinear Schrödinger equation. Erg. Th. Dynam. Syst. 22(5), 1537–1549 (2002) 32. Procesi, M.: Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D. Discrete Contin. Dyn. Syst. 13(3), 541–552 (2005) 33. Siegel, C.L.: Iteration of analytic functions. Ann. of Math. 43, 607–612 (1942) 34. Soffer, A., Weinstein, M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, 9–74 (1999) 35. Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990) 36. Yuan, X.: Quasi-periodic solutions of completely resonant nonlinear wave equations. J. Diff. Eqs. 230(1), 213–274 (2006) Communicated by G. Gallavotti
Commun. Math. Phys. 289, 907–923 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0818-0
Communications in
Mathematical Physics
Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra Daniel Lenz1 , Robert V. Moody2 1 Mathematisches Institut, Fakultät für Mathematik und Informatik,
Friedrich-Schiller-Universität Jena, D-07737 Jena, Germany. E-mail: [email protected] 2 Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W3P4, Canada. E-mail: [email protected] Received: 16 May 2008 / Accepted: 15 February 2009 Published online: 9 May 2009 – © Springer-Verlag 2009
Abstract: The paper investigates how correlations can completely specify a uniformly discrete point process. The setting is that of uniformly discrete point sets in real space for which the corresponding dynamical hull is ergodic. The first result is that all of the essential physical information in such a system is derivable from its n-point correlations, n = 2, 3, . . . . If the system is pure point diffractive an upper bound on the number of correlations required can be derived from the cycle structure of a graph formed from the dynamical and Bragg spectra. In particular, if the diffraction has no extinctions, then the 2 and 3 point correlations contain all the relevant information.
1. The Setting 1.1. Quasicrystals and dynamical systems. The defining feature of physical cyrstals and quasicrystals is the prominent appearance of Bragg peaks in their diffraction diagrams. Mathematically the diffraction is a positive measure and the Bragg peaks comprise the pure point component of this measure. In a ‘perfect’ crystal or quasicrystal, the diffraction should be entirely pure point, and that is the situation that we shall assume here. We shall simply refer to these as quasicrystals in the sequel. The diffraction does not, on its own, determine the internal structure of the quasicrystal that created it. However, the diffraction does determine the 2-point correlation, which is its Fourier transform. Under appropriate conditions (Assumptions A(i),(ii) below), knowledge of all the correlations (2-point, 3-point, etc.) does determine the internal structure (Thm. 1.4). The primary objective of this paper is to explore the details behind this in the case that the diffraction is a pure point measure. In this case, under fairly mild conditions, one does not at all need the entire set of correlations. In fact in the best situation, where there are no extinctions in the Bragg spectrum (this term is explained below), the 2- and 3-point correlations alone (in fact just the 3-point correlations) are enough to determine the structure (Thm. 2.8 and its Corollary).
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D. Lenz, R. V. Moody
Our approach here is to use a setting familiar from statistical mechanics and from the theory of tilings and long-range aperiodic order. Rather than deal with a single quasicrystal Λ, we work instead with translation invariant families of them. The intuition is that such a family, X , will consist of all those quasicrystals which are in some sense locally indistinguishable from one another, or which cannot be isolated from one another by the physical considerations at hand. As for the individual quasicrystals, we model these simply as uniformly discrete1 point sets in space, with the points representing the positions of the atoms. Since the dimension does not play any special role here, we work in general d-dimensional space Rd . Thus our main result deals with general ergodic uniformly discrete point processes. It is always the case that the dynamical spectrum is generated, as a group, by the diffraction spectrum. But here we prove that if it has pure point spectrum and the dynamical spectrum is expressible as a sum of finitely many copies of the diffraction spectrum then the point process is determined by its n-point-correlations for some finite n. In the context of aperiodic order this becomes particularly relevant as it gives a way of assessing the degree of ‘complexity’ of the long range order.
1.2. Background. It may be of interest to briefly discuss the reasons that such an elaborate formalism is relevant to what might seem a fairly straightforward exercise in spectral theory. Pure point diffraction from aperiodic structures was not predicted, either by mathematicians or crystallographers. When it did appear, both in tiling theory and experimentally in the discovery of aperiodic metallic alloys, the projection method was quickly utilized and it was generally believed that one could use standard techniques like the Poisson summation formula (applied to lattices in higher dimensions) to explain the diffraction. It was A. Hof who, in his much-cited papers [13,14], showed that diffraction in aperiodic structures is not business as usual. The Bragg spectrum of an aperiodic material is not lattice-like as in the periodic case, but is typically dense in Fourier space. The d problem is that for a countable aperiodic set of scatterers in R the Fourier transform νˆ of their Dirac comb ν = x∈ δx is not in general a measure. The sums involved diverge, even locally. This is in contrast to the lattice case. For this reason the theory of diffraction has developed by defining the diffraction as the Fourier transformation of the volume averaged autocorrelation. It is the ‘quadratic’ nature of autocorrelation which produces the necessary convergence, and this is now the standard approach to diffraction in the aperiodic case. Assume now that we are in the case of a countable and uniformly discrete set of scatterers. It was pointed out in [29] that a good way to study an aperiodic set was to follow ideas from statistical mechanics and form a compact space from its translation orbit, the so-called hull X . This is a dynamical system (with Rd as the acting group) and allows one to use spectral theory. S. Dworkin [11] then showed how the dynamical spectrum could be linked to the diffraction spectrum by using spectral measures. This linkage is now often called Dworkin’s argument. It was the use of hulls and Dworkin’s argument that first allowed rigorous proof of the pure pointedness of model sets (or cut and project sets). This is a good example of a situation where the result is seemingly clear from the Poisson summation formula, but on 1 A subset Λ of d-dimensional space Rd is uniformly discrete (or more specifically r -uniformly discrete) if for some r > 0 and for all x, y ∈ Λ with x = y, |x − y| ≥ r .
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closer inspection one is confronted with divergent sums with no obvious mathematical meaning. The first proof by A. Hof, and its full generalization to all model sets by M. Schlottmann [31], of the pure pointedness of diffraction from model sets uses the fact that the hull is measure-theoretically a compact Abelian group, and so pure point, followed by Dworkin’s argument using spectral measures. By the way, unlike the periodic case, the hull is topologically not a group, and indeed its highly subtle topology has been the focus of many mathematicians recently (see [5,15] for reviews and further discussion). Dworkin’s argument still left the precise connection between the diffraction and dynamical spectrum unresolved. Further developments on the hull and its connection to diffraction and to point processes were made in [5 and 12] where it is shown that under the assumption of ergodicity the autocorrelation of the points sets of X exist almost surely (in the sense of the invariant measure µ on X ) and almost surely are equal to the first moment of the Palm measure of X . In [8] this was extended to show that all the higher correlations of ∈ X exist almost surely and they completely determine µ. This fact is not generally true for point processes but here follows from the assumed uniform discreteness of the point sets under consideration. In the pure point case it comes pretty much for free from the spectral structure, as we see in the present paper. In [17] (see [2,12,22] as well) it is proved that the diffraction is pure point if and only if the dynamical spectrum is pure point. This is remarkable since we know that in general the diffraction can fail to see great chunks of the dynamical spectrum - even the pure point part of the spectrum. In [8] the diffraction/dynamics connection is made even more precise by showing that there is an isometric embedding of L 2 (Rd , ω) into L 2 (X, µ), where Rd is the Fourier space with its diffraction measure ω. This embedding allows one to see how the eigenfunctions transfer across. Precisely, each Bragg peak located at position k gives rise to an eigenfunction f k whose value at the point set ∈ X is almost surely 1 R→∞ vol C R
f k () = lim
e2πik·x .
x∈∩C R
We mention this, first because the convergence of this sum away from 0 is precisely what is called the Bombieri-Taylor conjecture in [13,14], (which was precisely that, until recently). There is a proof of convergence in the L 2 - sense in [8]. There is also a recent proof of the point-wise convergence of the limit within the context of uniform convergence in ergodic theorems in [18]. From the point of view of our present paper, it is these functions f k which lie at its heart. It is interesting to note that the way in which the isometric embedding connection between diffraction and dynamics is defined, the eigenfunctions are nowhere in sight. It is only in the L 2 -completion that the eigenfunctions appear, and even then the way in which they map (by the Bombieri-Taylor formula) is nothing like the original defining map, and has to be proved. Now we come to our present paper. Of course the underlying concern of much of diffraction theory is that the inverse problem (resolving structure from the diffraction) has no unique solution in general. The problem is that the embedding of diffraction into dynamics is not surjective. This problem is exacerbated in the aperiodic case. Our setting is an ergodic dynamical system of uniformly discrete point sets which are pure point diffractive almost surely. The main result of our paper is to show rather precisely the significant role that extinctions (places in the dynamical spectrum where there are no Bragg peaks) play in this ambiguity.
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In [24] D. Mermin made the remarkable suggestion that the second and third correlations should always determine the structure. The argument made there is quite simple – even trivial – except that it again deals with quantities facing the same problem originally pointed out in [13]; they are not convergent (see also §4). The author was well aware of the difficulties of his argument: he writes “Even granting that I have begged the question of when the density has a Fourier transform, and when the auto-correlation functions exist, this informal Fourier space argument that the identity of all second and third order correlations implies the identity of all higher order correlations is disarmingly trivial. I would very much like to learn of a comparably simple informal argument or an instructional counterexample in position space.” Our paper is a response to this question. If there are no extinctions, then the second and third correlations do suffice. If the extinctions are not too bad, we can at least get away with knowing only finitely many types of point correlations. As further discussed in Sect. 5 there are also recent results which show that in fact there are situations where one really does need higher moments than just the second and third to resolve aperiodic point sets (in fact, even model sets), [10]. Apart from its mathematical interest and the potential directions for further development, the results of our present paper seem to be physically relevant. The detailed atomic structures of quasicrystals are basically unknown in spite of over 20 years of work by theoreticians and experimentalists. Model sets are one of the primary modelling devices in the subject and diffraction is a fundamental tool. It is relevant to know the controlling influences on diffraction and to know how close diffraction, particularly in model sets, can come to determining the underlying structure. 1.3. Hulls. The basic objects of interest in this paper are pairs (X, µ), where X is a set of r -uniformly discrete point sets Λ of some real space Rd for some r > 0 and µ is a probability measure on X . The assumptions that we need to make on (X, µ) are listed in A(i), A(ii), A(iii) below. Throughout the paper C R , R > 0, denotes the open cube (−R/2, R/2)d ⊂ Rd . There is a uniform topology (called the local topology) on the set Dr (Rd ) of all r -uniformly discrete subsets of Rd . The uniformity is generated from the collection of all sets (entourages) of the form U (K , ), K ⊂ Rd being compact, and > 0, where U (K , ) = {(Λ, Λ ) ∈ Dr (Rd )×Dr (Rd ) : Λ ∩ K ⊂ Λ +C and Λ ∩ K ⊂ Λ+C }. (1) Thus Λ and Λ are ‘close’ if on some (‘large’) compact K and for some (‘small’) > 0, the points of Λ that are within K also lie in the -cubical neighbourhoods of the points of Λ , and vice-versa. It is relatively easy to see that Dr (Rd ) is compact in this topology [29] and that the translation action T : Tx (Λ) := x + Λ ∈ X, for all x ∈ Rd of Rd on it is continuous. An alternative description of this topology using the functions N f below can be found in [5]. We assume A(i) X is a closed translation invariant subset of Dr (Rd ); A(ii) µ is an ergodic probability Borel measure on X .
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A(i) obviously implies that X is compact and, together with A(ii), the pair (X, µ) along with the group action of Rd by translation is a dynamical system, both in the topological and measure theoretic senses. The assumption of ergodicity is that µ is a translation invariant measure and X cannot be decomposed into two measurable invariant subsets which each have positive measure. We can, if we wish, interpret µ as a measure on Dr (Rd ) whose support lies inside X . This makes it clear that µ is the actual relevant piece of data. The space X is only noted for convenience. The basic open neighbourhoods of Λ defined by the uniformity on X are of the form U (K , )[Λ] := {Λ ∈ X : (Λ, Λ ) ∈ U (K , )}. These consist of the point sets Λ that are sufficiently close to making the same pattern as Λ within the compact set K ⊂ Rd . The interpretation of µ(U (K , )[Λ]) is that it is the probability that a random element Λ of X will lie in U (K , )[Λ]. The measure µ thus gives the information about what patterns are possible and what their probabilities of occurrence are. Its support specifies which subsets of Dr (Rd ) are relevant. The ergodicity says that, when viewed from the origin, the translations of any element Λ from the support of µ will, almost surely, faithfully represent all possible local patterns with the correct frequencies. We take the attitude that this is all we can hope to know about our physical system, and thus our objective is to determine µ from other, physically observable, data. In our case this other data will consist of various correlations of the system (X, µ). 1.4. Diffraction and pure pointedness. Let S(Rd ) denote the Schwartz space of all complex-valued infinitely many times differentiable rapidly decreasing functions on the real space Rd . For each n = 1, 2, . . . the n + 1-point correlation of Λ ∈ Dr (Rd ) is the measure (n+1) on Rd × · · · × Rd (n-factors) defined by γΛ 1 R→∞ vol C R
γΛ(n+1) (F) = lim
F(−x + y1 , . . . , −x + yn )
x,y1 ,...,yn ∈Λ∩C R
for all F ∈ S(Rd × · · · × Rd ), if this limit exists. Theorem 1.1. [5,8,12]. Let (X, µ) satisfy A(i) and A(ii). Then (n)
(i) for µ almost every Λ ∈ X , all of the n-point correlations γΛ exist. Furthermore, they are almost surely independent of the point-set Λ chosen in X ; (ii) the 2-point correlation is almost surely Fourier transformable. The common n-point correlations are denoted simply as γ (n) , and even more simply (2) as γ for n = 2. The Fourier transform of γΛ is almost surely the Fourier transform γ of γ . Definition 1. γ is the diffraction of (X, µ). Starting with the work of Hof [13] the rigorous mathematical study of diffraction for aperiodic order has attracted quite some attention in recent years. We refer to [16,19,20] for recent surveys.
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A basic idea is that the correlations are, in principle, quantities that can be physically measured. Certainly measurement of the diffraction is standard, and hence its Fourier transform, the 2-point correlation, may be considered as known. There are reports of inference of higher correlations through fluctuation microscopy [33], though it is not clear that these correlations go beyond pair-pair correlations arising from squaring the 2-point correlation. The space L 2 (X, µ) of square integrable functions on X gets an Rd action through translation of functions: (Tt f )(x) = f (−t +x). A simple consequence of the translation invariance of µ is that this action is unitary for the basic inner product ·, · defined by f, g = f gdµ. X
As usual f ∈ with f = 0 is an eigenfunction of T (to the eigenvalue k ∈ Rd ) if Tt f = exp(−2πik · t) f for all t ∈ Rd . The fact that µ is assumed ergodic implies that the multiplicity of each eigenvalue is one. The dynamical system (X, µ) is said to be pure point if L 2 (X, µ) has a Hilbert basis of eigenfunctions. A key point is a theorem that relates L 2 (Rd , γ ) and L 2 (X, µ) and then relates the two concepts of pure pointedness. In order to discuss this further we need some more notation. For each f ∈ S(Rd ) let N f : X → C be defined by f (x). N f (Λ) = L 2 (X, µ)
x∈Λ
Lemma 1.2. The algebra generated by the N f , f ∈ S(Rd ), is dense in the algebra of continuous functions on X equipped with the supremum norm. In particular, it is dense in L 2 (X, µ). Proof. (See [2] for a similar argument.) Obviously, the algebra in question separates points, is closed under taking complex conjugates, and to each Λ ∈ X , there exists an f ∈ S(Rd ) with N f (Λ) = 0. Thus, the first statement follows by the Stone-Weierstrass theorem (see [30] for the version used here). The last statement is then clear. γ ) by Define an action U of Rd on L 2 (Rd , (Ut f )(x) = e−2πit·x f (x) γ ). for all t, x ∈ Rd , f ∈ L 2 (Rd , Theorem 1.3. Let (X, µ) satisfy A(i) and A(ii). Then the following holds. γ ) and there is a unique isometric (i) The set { f : f ∈ S(Rd )} is dense in L 2 (Rd , embedding θ : L 2 (Rd , γ ) −→ L 2 (X, µ) with θ ( f ) = N f for all f ∈ S(Rd ). This embedding intertwines U and T . (ii) γ is a pure point measure if and only if (X, µ) is pure point. (iii) For k ∈ Rd the equation 1 e2πik·x θ (1k ) = lim R→∞ vol C R x∈Λ∩C R
holds, where the limit is meant in the L 2 sense. Moreover, γ ({k}) = 0 if and only if θ (1k ) = 0. In this case, θ (1k ) is an eigenfunction of (X, µ) for the eigenvalue k.
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Remark 1. (a) The theorem may be found in the form stated here in [8]. It has a long history that includes [2,11,12,17,18]. In fact, (i) and (iii) are shown in [8], see [18] for extensions of (iii) as well. The statement (ii) is proven in various levels of generality in [2,11,12,17,22]. We will give an independent proof below in Corollary 2.2. (b) Under θ eigenfunctions go to eigenfunctions. However, the formula of part (i) is not applicable in (iii): the function 1{k} , which takes the value 1 at k and zero everywhere else, is not even remotely in S(Rd ). The limit stated here appears only after approximation by functions from S(Rd ). (c) As θ is an isometry, γ ({k}) = 0 is obviously equivalent to θ (1k ) = 0. γ ({k}) = 0 appearing in (iii) of the previous theorem The functions θ (1k ) for k with will play a crucial role in our considerations. We define f k := θ (1k ) whenever γ ({k}) = 0. Definition 2. Let E := {k ∈ Rd : k is an eigenvalue of (X, µ)}, S := {k ∈ Rd : γ ({k}) = 0}. E is the dynamical spectrum of (X, µ) and S is its Bragg spectrum. We note that the Bragg spectrum is sometimes also known as the Fourier-Bohr spectrum. For convenience of notation, we will write γ (k) instead of γ ({k}) in the remaining part of the paper. Let us briefly discuss the structure of E and S and their relationship. It is well known that E is a subgroup of Rd . In fact, the product of eigenfunctions is again an eigenfunction to the sum of the respective eigenvalues and the complex conjugate of an eigenfunction is an eigenfunction to the inverse of the corresponding eigenvalue. As shown in (iii) of Thm. 1.3, any Bragg peak k comes with a canoncial eigenfunction f k = θ (1k ). In particular, we have the inclusion S ⊂ E. In general, this inclusion is strict even in the pure point case. The limit formula for f k in Thm. 1.3 shows that f −k = f k
(2)
and that f 0 = 0. Thus, the statement on the eigenfunctions in (iii) of Thm. 1.3 shows that S satisfies 0 ∈ S, and S = −S. It is a fundamental fact that the canonical eigenfunction f k of the Bragg peak k can be related to the intensity of the Bragg peak. More precisely, note that due to the ergodicity the modulus of any eigenfunction is constant µ-almost everywhere. Thus, 1/2
1/2
| f k | = f k , f k L 2 (X,µ) = 1k , 1k L 2 (Rd , = γ (k)1/2 , γ) where the first equality holds µ-almost everywhere, see also [18] as well.
(3)
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D. Lenz, R. V. Moody
Our final assumption is: A(iii) γ is a pure point measure. Definition 3. A pair (X, µ) satisfying axioms A(i),(ii),(iii) is called a pure point ergodic uniformly discrete point process. In this pure point case, S generates E as a group, as shown in [2]. We will give an independent proof based on Thm. 1.3 in Corollary 2.2 below. Definition 4. The set X := E\S is called the set of extinctions of (X, µ). 1.5. Correlations and moments. Let (X, µ) satisfy A(i) and A(ii). For n = 1, 2, . . . , the nth-moment of µ is the measure µn defined on Rd × · · · × Rd (n-factors) by µn (h 1 , . . . , h n ) = Nh 1 . . . Nh n dµ X
S(Rd ).
for all h 1 , . . . , h n ∈ It is clear that these moment measures are invariant under simultaneous translation of all the variables. There is a standard procedure of eliminating this translation invariance resulting in the reduced moments µred n which are in one less variable. This works as follows: let g, h 1 , . . . , h n−1 ∈ S(Rd ) and let G be the function on (Rd )n whose value on (x, y1 , . . . , yn−1 ) is g(x)(Tx h 1 )(y1 ) · · · (Tx h n−1 )(yn−1 ). Then µn (G) = µn (g(Tx h 1 ) . . . (Tx h n−1 )) = g(x)dx µred n (h 1 , . . . , h n−1 ) , Rd
and this equation defines the reduced moments. Most importantly for our purposes, these reduced moments are also the correlations – see [7], Sect. 12.2 and [8], Sect. 7 2 . More precisely, the following holds. Theorem 1.4. Let (X, µ) satisfy A(i) and A(ii). (i) For each m ∈ N, µ is uniquely determined by its moments µn , n ≥ m. (ii) γ (n) = µred n , n = 2, 3, . . . . (iii) For n ≥ 2, µn is uniquely determined by µred n . (iv) The measure µ is uniquely determined by γ (n) , n = 2, 3, . . . . Proof. (i) For m = 1 the statement follows immediately from Lemma 1.2. Now, it suffices to show that the µn , n > m, determine µm . By Lemma 1.2, again, the constant function 1 can be approximated by elements of the algebra generated by the Nh , h ∈ S(Rd ). Thus, a product Nh 1 · · · Nh m = Nh 1 . . . Nh m .1 can be approximated by linear combinations of products of more than m functions in S(Rd ). Thus, µn , n > m, determine µm . The proof of (ii) can be found in [7], Prop. 12.2.V. The proof of (iii) can be found in [7], Sec. 10.4. Finally, (iv) is a direct consequence of (i), (ii) and (iii). Remark 2. The proof of (i) in the previous theorem does not require ergodicity. It only uses that the functions N f , f ∈ S(Rd ), are bounded and continuous on the compact X . The point of the previous theorem is that rather than correlations, we may instead look at corresponding moments. Our question becomes that of asking how many moment measures are required to pin down µ uniquely. 2 The reduced moments are also connected directly to Palm measures, a direction that is more fully explored in [8,12].
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2. Eigenfunctions and Cycles 2.1. The cycle function of (X, µ). Let (X, µ) be a pure point ergodic uniformly discrete point process. Thus, A(i),(ii),(iii) are valid. The elements of L 2 (Rd , γ ) are all the sums xk 1k , where |xk |2 γ (k) < ∞. k∈S
k∈S
As described in (i) and (iii) of Thm. 1.3 the map θ exhibits very different behaviour on functions h ∈ S(Rd ) and on functions 1k . This leads to two very different ways in which to write θ (L 2 (Rd , γ )). More precisely, both the linear span of the set of 1k , k ∈ S, and the set h, h ∈ S(Rd ), are dense in L 2 (Rd , γ ). As θ is an isometry, this gives γ )) = {Nh : h ∈ S(Rd )} = linear span { f k : k ∈ S}. θ (L 2 (Rd , Our next aim is to obtain similar statements for products. This requires some care as we will have to deal with products of infinite sums. The corresponding details are given in the next two lemmas. We will use repeatedly the elementary fact that {gm h} converges to gh in L 2 whenever {gm } is a sequence converging to g in L 2 and h is a bounded function. Lemma 2.1. Let n ∈ N and h 1 , . . . , h n ∈ S(Rd ) be given. Then, Nh 1 . . . Nh n = ... h1 (k1 ) . . . hn (kn ) f k1 . . . f kn , k1 ∈S
kn ∈S
where the sums exist in L 2 and are taken one after the other. In particular, ... Nh 1 . . . Nh n dµ = h1 (k1 ) . . . hn (kn )µ( f k1 . . . f kn ). k1 ∈S
kn ∈S
Proof. By (i) of Thm. 1.3 we have Nh j =
hj (k j ) f k j
k j ∈S
for each j. Therefore, ⎛ Nh 1 . . . Nh n = ⎝
k1 ∈S
⎞ h1 (k1 ) f k1 ⎠ Nh 2 . . . Nh n =
h1 (k1 ) f k1 Nh 2 . . . Nh n
k1 ∈S
and the first statement follows by induction. As µ is a finite measure, the last statement then follows easily. As a corollary of this lemma we obtain a new proof of the following known fact. Corollary 2.2. T has pure point spectrum, i.e. there exists a basis of L 2 (X, µ) consisting of eigenfunctions. Moreover, any eigenfunction is, up to a scalar factor, a finite product of functions f k , k ∈ S, and any eigenvalue is a sum of k ∈ S.
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Proof. By the previous lemma, any function of the form Nh 1 . . . Nh n with h j ∈ S(Rd ) can be approximated by linear combinations of products of the form f k1 . . . f kn , k j ∈ S. Lemma 1.2 then gives that functions of the form f k1 . . . f kn , n ∈ N, k j ∈ S, j = 1, . . . , n are total in L 2 (X, µ). As each function of the form f k1 . . . f kn is an eigenfunction to the eigenvalue k1 + · · · + kn the statement follows. (m)
Lemma 2.3. Let n ∈ N and k1 , . . . , kn ∈ S be given. For each j = 1, . . . , n, let {h j } be a sequence in S(Rd ) whose Fourier transforms converge to 1k j in L 2 (Rd , γ ). Then, f k1 . . . f kn = lim
lim . . . lim N
m 1 →∞ m 2 →∞
m n →∞
(m 1 )
h1
. . . Nh (m n ) , n
where the limits are taken in L 2 . In particular, µ( f k1 . . . f kn ) = lim
lim . . . lim µ(N
m 1 →∞ m 2 →∞
m n →∞
(m 1 )
h1
. . . Nh (m n ) ). n
Proof. The functions Nh (m) and the functions f k j , j = 1, . . . , n, m ∈ N are bounded. j
(m)
Thus, the convergence of the h j easily yields convergence of the products. (Note that the limits are taken one after the other). As µ is a finite measure the last statement then follows easily. We will now introduce a crucial object in our studies, namely the cycle function a. Notice that, by almost sure constancy of the modulus of the functions f k , 2 | f k1 . . . f kn | = f k1 . . . f kn f kn . . . f k1 dµ X
= f k1 , f k1 . . . f kn , f kn = γ (k1 ) . . . γ (kn ). If k1 + · · · + kn = 0 then f k1 . . . f kn is an eigenvector for 0, and hence is a multiple of the constant function 1 X . Thus, in this case, f k1 . . . f kn = a(k1 , . . . , kn ) γ (k1 )1/2 . . . γ (kn )1/2 1 X
(4)
for some a(k1 , . . . , kn ) ∈ U (1). Here, U (1) is the unit circle i.e. the set of all complex numbers of modulus one. For any k1 , . . . , kn ∈ S, we therefore obtain µ( f k1 . . . f kn ) = f k1 . . . f kn 1 X dµ = f k1 . . . f kn , 1 X
X a(k1 , . . . , kn ) γ (k1 )1/2 . . . γ (kn )1/2 if k1 + · · · + kn = 0; = (5) 0 if k1 + · · · + kn = 0 , since eigenfunctions for different eigenvalues are orthogonal. The next two results basically say that knowledge of the cycle function a determines the moments and vice versa. Proposition 2.4. Let n ∈ N be given. Then, the n th moment of µ is uniquely determined by γ and the quantities a(k1 , . . . , kn ) for k1 , . . . , kn ∈ S with k1 + · · · + kn = 0.
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Proof. This follows directly from Lemma 2.1 and (5).
Proposition 2.5. Let n ∈ N be given. Then, the values a(k1 , . . . , kn ) for k1 , . . . , kn ∈ S with k1 + · · · + kn = 0 are uniquely determined by γ and the n th moment of µ. Proof. This follows from Lemma 2.3 and (5).
It is convenient to introduce the Cayley graph G of E with respect to the set of generators S. Its vertices are the points of E and its edges are the pairs {k, l} of vertices whose differences k − l lie in S. Since S = −S, we may treat the edges as undirected. Any k = (k1 , . . . kn ) ∈ S n with k1 + · · · + kn = 0 leads to a cycle {0, k1 , k1 + k2 , . . . , k1 + · · · + kn−1 , k1 + · · · + kn = 0} in G. Thus the function a described above can be thought of as a function of the set Z of cycles of G. We shall call it the cycle function of (X, µ). 2.2. Properties of the cycle function. Given k, l ∈ Z , their concatenation kl := (k1 , . . . , kn , l1 , . . . , l p ) , is obviously also in Z . Proposition 2.6. The cycle function a has the following properties: for all k, l ∈ Z , a(k)a(l) = a(kl); a(0) = 1; for all k ∈ Z , a(k, −k) = 1; a(k1 , . . . , kn ) is independent of the order of the elements k1 , . . . , kn making up the cycle; (v) given any cycle k ∈ Z , then any pair {k, −k}, where k ∈ S, and also 0 can be inserted into or deleted from the symbols of k without affecting the value of the cycle function a.
(i) (ii) (iii) (iv)
Proof. (i) follows from (4). As for (ii), note that f 0 = a(0) γ (0)1/2 by (4). Since f 0 = θ (1{0} ) ≥ 0 from Thm. 1.3 (iii), γ (0) > 0, and a(0) ∈ U (1), we see that a(0) = 1. This proves (ii). Part (iii) follows from a(k, −k) γ (k) = f k f −k = f k f k = | f k |2 = γ (k). Items (iv) and (v) are trivial consequences of (4) and parts (i),(ii), and (iii).
Let Z n := {(k1 , . . . , kn ) ∈ Z }, Z 0 := {∅}, and Z ∞ = ∞ n=0 Z n . We introduce an equivalence relation on Z ∞ by transitive extension of the two rules: • k ∼ l if l is a permutation of the symbols of k. • k ∼ l if l can be obtained from k by inserting or removing pairs {k, −k}, k ∈ S, or by inserting or removing 0. Let Z := Z ∞ / ∼ .
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It is easy to see that k ∼ l , k ∼ l ⇒ kl ∼ k l , so multiplication descends from Z ∞ to Z. Indeed Z is an abelian group under this multiplication, with ∅∼ as the identity element. Of the various (k1 , . . . , kn ) that can represent a given element κ ∈ Z there is (at least) one of minimal length n. This minimal length is denoted by len(κ). Define Zn := {κ ∈ Z : len(κ) ≤ n}. We shall also write len(k) = len(κ) when k∼ = κ and call it the reduced length of k, Z=
∞
Zn ,
n=0
Zn Z p ⊂ Zn+ p
for all n, p.
Evidently the cycle function a determines a homomorphism, a, ˜ a˜ : Z −→ U (1) with a(κ) ˜ = a(k1 , . . . , kn ) if κ = (k1 , . . . , kn )∼ . It is clear from this that a˜ is known on Zq by its values on the sets Zn for n < q if Zn Z p . Zq = n+ p=q,0
2.3. Main results. Theorem 2.7. Let (X, µ) be a pure point uniformly discrete ergodic point process and suppose that · · + S = E. S + · n
Then Z is generated, as a group, by Z2n+1 . Proof. Let k = (k1 , . . . , k N ) be any cycle where N > 2n + 1. By assumption k1 + · · · + kn+1 ∈ E can be written in the form l1 + · · · + ln , where the li ∈ S. Then with j := (k1 , . . . , kn+1 ) and l = (l1 , . . . , ln ), k ∼ ((j)(−l)) (l((−j)k)) which is written it as the product of (j)(−l) and l((−j)k). These have reduced lengths at most 2n + 1 and N − 1 respectively, and this shows that Z N ⊂ Z2n+1 Z N −1 . The proof finishes by induction. Theorem 2.8. Let (X, µ) be a pure point uniformly discrete ergodic point process. If the dynamical spectrum is finitely generated by the diffraction spectrum, then µ is uniquely determined by a finite number of its moments µm . More precisely, if · · + S = E S + · n
then µ is uniquely determined by its moments µm , m = 2, . . . , 2n + 1.
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Proof. By (ii) of Thm. 1.4 the second moment determines the autocorrelation γ . By Proposition 2.5 the moments µm , m = 2, . . . , 2n + 1, then determine the function a on Zm for m = 2, . . . , 2n + 1. The previous theorem gives that the function a is then completely determined. Thus, by Proposition 2.4, all moments µm , m ≥ 2, are then determined. Now, the theorem follows by (i) of Thm. 1.4. Corollary 2.9. If there are no extinctions, so S = E, then µ is determined by its second and third correlations. Proof. This follows immediately from the previous theorem and parts (ii) and (iii) of Thm. 1.4. 3. Model Sets The theory of model sets is a good place to find examples of the types of point processes (X, µ) that we have been discussing. We start with a cut and project scheme ˜ with corresponding ‘torus’ T := (Rd × H )/ L. ˜ We suppose that H is a (Rd , H, L) complete metric space. The canonical mapping from the projected image L of L˜ in Rd to its projected image in H is denoted by (·) . We suppose also that we have a regular model set Λ = Λ(W ) := {x ∈ L : x ∈ W } given by some regular closed set W ⊂ H (i.e. W = W ◦ and W has boundary of measure 0). In this case, the hull X = X (Λ) of Λ is uniquely ergodic, all elements of X share a common autocorrelation γ , and the diffraction γ is pure point [31]. See [3,25] for basic material on model sets. We wish to consider the situation regarding the diffraction and extinctions. Let d , H , L˜ ◦ ) be the corresponding dual cut and project scheme, where L˜ ◦ is the dual (R d Rd is denoted by E. group of T, [25], Sec. 5. The natural projected image of L˜ ◦ in R We shall also use the -notation for the dual cut and project scheme. Since the projection has dense image, E is dense in H . of L˜ ◦ into H The diffraction of Λ is known [14,31] to be given by 2 γ (k) = |1 W (−k )| ,
(6)
for all k ∈ E. Here, the measure of the torus T is normalized to be one. The set of extinctions X is then the set of k ∈ E for which 1 W (−k ) vanishes, and the Bragg peaks make up the set S = E\X . We know that S generates E as a group and 0 ∈ S. Lemma 3.1. Suppose that X has no interior. Then S + S = E. Proof. We need to prove that every element of X is the sum of two elements of S. Let K be the kernel of ( ) . From the form of the diffraction (6), both S and X consist of unions of full cosets of K . Thus it suffices to show that every element of X is the sum of two elements of S . Let z ∈ X and suppose that z is not so expressible, i.e., (z − S ) ∩ S = ∅. Then z − S ⊂ X , so S ⊂ z − X . From this = E = S ∪ X = S ∪ X ⊂ (z − X ) ∪ X , H which is impossible by the Baire category theorem (since X has no interior).
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Remark 3. One should note that even in the same cut and project scheme, different windows can give rise to the same diffraction in (6), even though the windows are not translationally equivalent. An example of this, that derives from the covariogram problem, is discussed, along with references to covariogram literature, in [1]. This shows that the second moment alone is insufficient even to distinguish model sets from the same cut and project scheme. 4. Origins of the Problem in Questions of Symmetry The results in this paper have a number of points of contact with the ideas of D. Mermin [24] and subsequent works of R. Lifshitz, D. A. Rabson, and B. N. Fisher [23,26]. The starting point was a puzzle which arose almost immediately after the discovery of quasicrystals. The usual notions of symmetry from crystallography are not adequate in the theory of quasicrystals. First of all, translational symmetry is drastically diminished, often to the point of being non-existent. Second, even the finite symmetries are somewhat nebulous. ‘Icosahedrally symmetric’ quasicrystals, for example, have perfectly isosahedrally symmetric diffraction patterns, but they need not be literally icosahedrally symmetric in the sense that the structure is mapped precisely onto itself by the point symmetries of the icosahedral group. In this section we will discuss this question and how it can be resolved through the use of dynamical systems. Mermin’s solution to the symmetry problem was to realize it on the Fourier side of the picture rather than the physical side. Briefly, the idea was to express the density distribution ρ of the quasicrystal as a superposition of plane waves from its translation module (what we have called E above): ρ(r ) = ρ (k)e2πik·r , (7) k∈E
and then to remark that two densities ρ and ρ based on the same module of wave vectors are physically indistinguishable if their correlations of all orders are identical, something that should happen if ρ (k1 ) . . . ρ (kn ) = ρ (k1 ) . . . ρ (kn )
(8)
for all k1 , . . . kn ∈ E with k1 + · · · + kn = 0. Thus symmetry becomes symmetry in the sense of indistinguishable correlations. Now ρ and ρ will be indistinguishable if for some χ : E −→ R/Z we have (k) , ρ (k) = e2πiχ (k) ρ
(9)
and a symmetry g of E would appear as a symmetry of the physical system if, for all k ∈ E, ρ (g(k)) = e2πiχg (k) ρ (k) for some suitable χg : E → R/Z. Applying (8) at n = 3 in (9) already gives χ (k1 + k2 ) = χ (k1 ) + χ (k2 ) for all k1 , k2 ∈ E, which already provides all the information derivable from (8) for all other n. It was from this that Mermin concluded that 2- and 3-point correlations should determine everything.
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The main difficulty with this approach, and this is already made clear in [24], is to give any mathematical meaning to the expressions (7), except as tempered distributions. This issue is discussed in some detail by A. Hof in [14]. The measure ρ representing the distribution of density of the quasicrystal, say ρ = x∈Λ δx , is not in general Fourier transformable as a measure. If instead one treats it as a distribution then it is difficult to say what its Fourier transform looks like and particularly whether or not it is composed of a countable sum of weighted deltas at some (usually dense) subset of Rd . Certainly the formation of moments in this language would be a formidable task. The upshot of Hof’s study of diffraction was his approach to it using the Fourier transform of the autocorrelation (which is Fourier transformable as a measure), and this has been the basis of most subsequent mathematical work on diffraction. Thus [24] is better seen as a formal vision of how things should work out rather than a rigorous exposition of how they actually do, and in that sense it is prescient. In our approach the sums (7) appear in a form that is tamed by averaging, namely the expressions appearing in Thm. 1.3 (iii). Such an expression is non-zero only if k ∈ E is the position of a Bragg peak, i.e. there is no extinction there. In [24] it is claimed that extinctions can only occur due to symmetries, but the examples from model sets show that extinctions come from the Fourier transform of the window function and do not seem to be related specifically to symmetries. Furthermore the extinctions seem to be potential, but not absolute, obstructions to the correctness of the assertion about 2- and 3-point correlations, see §5 below. In hindsight one can see that the symmetry question amounts to symmetries of the correlations, hence of the moments, and ultimately of the measure µ itself. An isometry g of Rd gives rise to a mapping Λ → gΛ. This g is a symmetry of the point process if µ is g-invariant. If X is the support of µ, then this also entails that g(X ) ⊂ X . This is the idea of symmetry put forward by Radin in [27,28]. This symmetry can also be expressed by means of groupoids. In fact, there is a canonical groupoid structure associated to the dynamical system (X, Rd ). Associated with it is the groupoid of the transversal (see [5] and references therein). The point groups of symmetries of the system then act as isomorphisms of this transversal groupoid. The measure µ appears by giving a trace on the corresponding C ∗ -algebra. In this picture, invariance of the system under a symmetry means invariance of the trace under this symmetry. This invariance of the trace then amounts to invariance of the measure µ, which in turn implies invariance of diffraction and higher moments under the symmetry. 5. Final Comments There are a number of obvious questions that arise from this work. Foremost is the question of whether every cycle on Z arises from some sort of dynamical system of density distributions on Rd . In [21] we have taken a first look at this. This requires changing the setting from the study of uniformly discrete subsets of Rd to more general distributions of density arising by formalizing the main ideas of this current paper. There is also the interesting question of the real role of extinctions. Although we have seen that extinctions are an obstruction to our method of reconstructing the measure µ from its moments, we have not shown that this obstruction is one of principle rather than an artifact of our approach. Inspired by the present paper this topic has been taken up recently: In [9] it is shown that for model sets with real internal spaces, whether or not there are extinctions, the 2- and 3-point correlations determine the model set within the class of all model sets. On the other hand, as soon as the internal space is
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not purely real, model sets exist in which the extinctions force higher correlations to be used, [9]. The paper [10] offers examples of multi-atomic model sets in which the scattering strengths of the different point types are differently weighted and for which even the 2, 3, 4, 5-point correlations are insufficient to resolve them. Another question is the curious role of odd numbers in our main result. Are there general conditions under which even numbered higher correlations play the defining role? Finally it would be interesting to develop practical methods for actually constructing increasingly accurate atomic approximations to the densities using the increasing information from the 2-, 3-, . . . correlations. Acknowledgement. RVM gratefully acknowledges the support of this research by the Natural Sciences and Engineering Research Council of Canada. Part of the work was done while DL had a visiting position at Rice University. He would like to thank the Department of Mathematics there for its hospitality.
References 1. Baake, M., Grimm, U.: Homometric model sets and window covariograms. Z. Krist. 222, 54–58 (2007) 2. Baake, M., Lenz, D.: Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra. Ergod. Th. & Dyn. Syst. 24, 1867–1893 (2004) 3. Baake, M., Lenz, D., Moody, R.V.: Characterization of model sets by dynamical systems. Ergod. Th. & Dyn. Syst. 26, 1–42 (2006) 4. Baake, M., Moody, R.V. (eds.): Directions in mathematical quasicrystals, CRM Monogr. Ser., 13, Providence, RI: Amer. Math. Soc., 2000 5. Bellissard, J., Hermmann, D., Zarrouati, M.: Hull of Aperiodic Solids and Gap Labelling Theorems. in: [4], pp. 207–259 6. Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Berlin: Springer, 1975 7. Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer-Series in Statistics, Berlin-Heidelberg-NewYork: Springer, 1988 8. Deng, X., Moody, R.V.: Dworkin’s argument revisited: point processes, dynamics, diffraction, and correlations. J. Geom. Phys. 58, 506–541 (2008) 9. Deng, X., Moody, R.V.: How model sets can be determined by their two-point and three-point correlations. http://arxiv.org/abs/0901.4381v1[math-ph], 2009 10. Deng, X., Moody, R.V.: Weighted model sets and their higher point-correlations. arXiv:0904.4552 11. Dworkin, S.: Spectral theory and X-ray diffraction. J. Math. Phys. 34, 2965–2967 (1993) 12. Gouéré, J.-B.: Quasicrystals and almost periodicity. Commun. Math. Phys. 255, 651–681 (2005) 13. Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169, 25–43 (1995) 14. Hof, A.: Diffraction by aperiodic structures. In: The Mathematics of Long-Range Aperiodic Order ed. R. V. Moody, NATO-ASI Series C 489, Dordrecht: Kluwer, 1997, pp. 239–268 15. Kellendonk, J., Putnam, I.: Tilings, C ∗ -algebras, and K -theory. In: Directions in Mathematical Quasicrystals, CRM Monograph Series, Volume 13, Baake, M.B., Moody, R.V. eds., Providence, RI: Amer. Math. Soc, 2000, pp. 177–206 16. Lagarias, J.: Mathematical quasicrystals and the problem of diffraction. In: [4], pp. 61–93 17. Lee, J.-Y., Moody, R.V., Solomyak, B.: Pure point dynamical and diffraction spectra. Ann. H. Poincaré 3, 1003–1018 (2002) 18. Lenz, D.: Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287, 225–258 (2009) 19. Lenz, D.: Aperiodic order and pure point diffraction. Phil. Mag. 88, 2059–2071 (2008) (Special Issue: Quasicrystals: the silver jubilee) 20. Lenz, D.: Aperiodic order via dynamical systems: Diffraction theory for sets of finite local complexity. To appear in Contemp. Math., http://arxiv.org/abs/0712.1323v1[math.DS], 2007 21. Lenz, D., Moody, R.V.: Which distributions of matter have pure point diffraction? In preparation 22. Lenz, D., Strungaru, N.: Pure point spectrum for measure dynamical systems on locally compact Abelian groups. http://arxiv.org/abs/0704.2498v1[math-ph], 2007 23. Lifschitz, R.: The Symmetry of Quasiperiodic Crystals. Physica A 232, 633–647 (1996) 24. Mermin, D.: The symmetry of crystals. In: The Mathematics of Long-Range Aperiodic Order ed. R. V. Moody, NATO-ASI Series C 489, Dordrecht: Kluwer, 1997, pp. 377–401
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25. Moody, R.V.: Model sets and their duals. In: The Mathematics of Long-Range Aperiodic Order ed. R. V. Moody, NATO-ASI Series C 489, Dordrecht: Kluwer, 1997, pp. 239–268 26. Rabson, D., Fisher, B.: Fourier-Space Crystallography as Group Cohomology. Phys. Rev. B 65, 024201–024201 (2002) 27. Radin, C.: Symmetry and tilings. Notices Amer. Math. Soc. 42, 26–31 (1995) 28. Radin, C.: Miles of Tiles. Student Mathematical Library, Providence, RI: Amer. Math. Soc., 1999 29. Radin, C., Wolff, M.: Space tilings and local isomorphism. Geom. Ded. 42, 355–360 (1992) 30. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional analysis. Second edition. New York: Academic Press, Inc., 1980 31. Schlottmann, M.: Generalized model sets and dynamical systems. In: [4], pp. 143–159 32. Schwartz, L.: Théorie des Distributions. Paris: Hermann, 1966 33. Treacy, M.M., Gibson, J.M., Fan, L., Paterson, D.J., McNulty, I.: Fluctuation microscopy: a probe of medium range order. Rep. Prog. Phys. 68, 2899–2944 (2005) Communicated by G. Gallavotti
Commun. Math. Phys. 289, 925–955 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0823-3
Communications in
Mathematical Physics
A Mass for ALF Manifolds Vincent Minerbe Université Paris 6, Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75013 Paris, France. E-mail: [email protected] Received: 30 May 2008 / Accepted: 7 February 2009 Published online: 12 May 2009 – © Springer-Verlag 2009
Abstract: We prove a positive mass theorem on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length. Introduction In general relativity [ADM], one can define the mass of an asymptotically Euclidean spatial slice (M n , g) in a spacetime by the formula 1 ∂ j gi j − ∂ j gii ι∂i d x, lim (1) µg = ωn R−→∞ Sn−1 (R)
where Sn−1 (R) is the standard sphere with radius R in Rn and ωn is the volume of Sn−1 (1). The terminology “asymptotically Euclidean” means M minus a compact subset is diffeomorphic to Rn minus a ball and the metric g is asymptotic to the standard metric on Rn . The positive mass conjecture asserts this mass, if defined, is nonnegative when the scalar curvature Scalg is nonnegative, vanishing only when (M n , g) is isometric to Rn . It is a theorem in dimension 3 ≤ n ≤ 8 ([SY1,SY2]) and on spin manifolds of any dimension ([Wit]). J. Lohkamp [Loh] recently announced a proof of the positive mass theorem in all dimensions, without spin assumption. Moreover, R. Bartnik [Bart] proved the mass is a genuine Riemannian invariant (under sharp assumptions). The interested reader should certainly have a look at [LP,He1]. The mathematical interest of such a theorem is the rigidity result it involves: under a nonnegative curvature assumption, it asserts a model metric at infinity (the Euclidean metric, here) cannot be approached at any rate, the obstruction being precisely the mass. Positive mass theorems were proved in other settings. Asymptotically (real) hyperbolic manifolds were studied from this point of view in [MO,AD,CH], while asymptotically complex hyperbolic manifolds are treated in [He2,BH]. Motivated by theoretical
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physics considerations, Xianzhe Dai [Dai] pointed out a positive mass theorem for spin manifolds that are asymptotic to the product of a Euclidean space with a compact simply connected Calabi-Yau manifold. In this paper, we are interested in manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length. For instance, there are complete Ricci flat metrics on R2 ×Sn−2 that are asymptotic to the standard metric on Rn−1 × S1 ; in dimension four, an example is provided by the Euclidean Schwarzschild metric g=
dr 2 1−
2µ r
+ r 2 dω2 + 1 −
2µ r
dt 2 ,
where (r, ω, t) ∈ R∗+ ×Sn−1 ×S1 . Another kind of example, adapted to the Hopf fibration at infinity, is the hyperkähler Taub-NUT metric on R4 : g = 1+
2µ r
dr 2 + r 2 dω2 +
1 1+
2µ r
η2 ,
where η is the standard contact form on the three-spheres (lots of details will be given about these formulas in the text). In these expressions, the parameter µ is bound to be nonnegative (so as to get a complete metric) and it is interpreted as a mass by physicists. It is tempting to ask for a positive mass theorem involving this µ in these examples. Some motivation comes from the study of gravitational instantons, i.e. hyperkähler four-manifolds with decaying curvature at infinity (an example is provided by the TaubNUT metric). These manifolds arise in Euclidean quantum gravity and string theory, but also in gauge theory, so that it is of interest to try and understand their geometry. The main result of [Mi2] asserts that gravitational instantons with cubic volume growth are always asymptotic to circle fibrations over a Euclidean base (up to a finite group action) with an approximation rate of order r −1 , where r is the distance to some point. More generally, the result applies to a class of Ricci flat metrics, including Schwarzschild examples for instance. With this in mind, it seems interesting to wonder if there are non trivial examples of Ricci flat metrics g that are closer to their model at infinity, namely for instance: g = gR3 ×S1 + O(r −1− ), with > 0? The corresponding question with a nontrivial circle fibration at infinity can also be addressed. This paper will prove in particular that the answer is no, because of a positive mass theorem. To introduce the class of metrics we will work with, we first consider the total space X m+1 of a principal S1 -bundle π over Rm \Bm (the exterior of the unit ball in Rm , m ≥ 3). L times the Given a positive number L, we introduce the vector field T that is equal to 2π 1 infinitesimal generator of the S action and consider a “model” metric h on X , given by h = π ∗ gRm + η2 , where η is a connection one-form, namely a S1 invariant one-form on X such that η(T ) = 1. Observe the fibers have length L. For such a one-form η, one may write dη = π ∗ ω for some two-form ω on the base B and we will assume this “curvature” two-form ω decays at infinity, in that there is a positive number τ such that D i ω = O(r −τ −1−i ), 0 ≤ i ≤ 2.
(2)
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We will consider Riemannian manifolds (M m+1 , g) such that M minus a compact set is diffeomorphic to such an X and such that g is asymptotic to h in the following sense : ∇ h,i (g − h) = O(r −τ −i ), 0 ≤ i ≤ 3.
(3)
Such metrics g are known as “ALF metrics”. Our main result is a positive mass theorem, valid for any circle fibration at infinity under a nonnegativity assumption on the Ricci curvature. We will denote by B R the preimage by π of the ball of radius R on the base. Theorem 1. Let (M m+1 , g), m ≥ 3, be a complete oriented manifold with nonnegative Ricci curvature. We assume that, for some compact subset K , M\K is the total space of a circle fibration over Rm \Bm , which can be endowed with a model metric h such that (2) and (3) hold with τ > m−2 2 . Then the quantity defined by 1 B lim sup µG = − ∗h divh g + d Tr h g − 21 d g(T, T ) g ωm L R−→∞ ∂ BR
is a nonnegative Riemannian invariant and vanishes exactly when (M, g) is the standard Rm × S1 . We stress the fact that the assumption Ric ≥ 0 is too strong for the asymptotically Euclidean setting (for Bishop-Gromov the theorem immediately implies the manifold is isometric to Rn ), while it allows many interesting examples when the volume growth is slower. The assumption τ > m−2 2 is classical (cf. [Bart] for instance). Considering products of asymptotically Euclidean manifolds with S1 , one can rely on [DS] to see that this hypothesis is optimal (see also the remark before Theorem 4.3 in [Bart]). The assumption on the third derivative of g is only used to ensure the mass is a Riemannian invariant, namely does not depend on h. A cheap adaptation of our argument yields another positive mass theorem, valid for asymptotic trivial fibrations, with a spin assumption and a nonnegativity assumption on the scalar curvature. Theorem 2. Let (M m+1 , g), m ≥ 3, be a complete spin manifold with nonnegative scalar curvature. We assume there is a compact set K and a spin preserving diffeomorphism between M\K and Rm \Bm × S1 such that for some τ > m−2 2 : D i (g − gRm ×S1 ) = O(r −τ −i ), 0 ≤ i ≤ 3. Then the quantity defined by µgD
=−
1 ωm L
lim sup R−→∞
∗h 0 divh 0 g + d Tr h 0 g
∂ BR
is a nonnegative Riemannian invariant and vanishes exactly when (M, g) is the standard Rm × S1 . There is a definite link with Dai’s work [Dai]. In his paper, he points out he cannot include Schwarzschild-like metrics in the discussion because the spin structure on the circles at infinity is not trivial. Indeed, physicists even built examples of complete spin manifolds asymptotic to R3 × S1 , with nonnegative scalar curvature and with a
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negative mass ([BrH,CJ])! Here, we can “justify” the positivity of the mass of genuine Schwarzschild metrics, relying on the fact that their Ricci tensor vanishes (see Sect. 4). Moreover, we can cope with nontrivial fibrations at infinity, which is interesting in view of the Taub-NUT example. In this case, the “model” at infinity cannot be realized by a complete metric, so that the mass cannot vanish. It seems that it is the first positive theorem with such a feature. In these positive mass theorems, the base of the circle fibration at infinity is a Euclidean space. One might ask for a generalization to the case of Riemannian manifolds asymptotic to a circle fibration over a finite quotient of Rm (this occurs for the so-called Dk gravitational instantons). It fails. For instance, M. Atiyah and N. Hitchin [AH,Hit] introduced a famous hyperkähler (hence Ricci-flat) metric on the moduli space of two monopoles. Its asymptotic shape is that of (R3 ×S1 )/Z2 endowed with a metric of TaubNUT form, except that the mass parameter µ is bound to be negative! This problem with finite quotients was already noticed for Asymptotically Locally Euclidean manifolds [Dah,Le1,Nak]. The paper is organized as follows. In a first section, we introduce the class of metrics we are interested in. In a second section, we describe the analytical tools required for our arguments. We could have relied on Mazzeo-Melrose machinery; yet, we have chosen to include elementary proofs for everything we need. An advantage is we never use complete Taylor expansions at infinity; the theory is thus simpler, closer to what is usually done in the asymptotically Euclidean case and it does not require any familiarity with pseudo-differential calculus. We emphasize nonetheless that the paper [HHM] greatly inspired us, as well as the text of some lectures given by Frank Pacard [Pac]. In a third section, we prove the positive mass theorems. In the last section, we discuss examples and counter-examples. We will often use an Einstein summation convention without mentioning it: when an index is repeated, the expression should be summed over this index (with a range depending on the context). 1. Metrics Adapted to a Circle Fibration at Infinity Let us work on the total space X m+1 of a principal circle fibration π over B m := Rm \Bm , m ≥ 3. From a topological point of view, there are two families of such fibrations. In any dimension, we can consider the trivial circle fibration π0 over B = Rm \Bm . The total space is X = Rm \Bm × S1 . In dimension four (i.e. m = 3), the Hopf fibration S3 −→ S2 induces non trivial fibrations πk , k ∈ N∗ , where the total space X is the quotient of R4 \B4 by Zk ; the action of Zk is the complex scalar action of the k th unit root group on C2 = R4 . We wish to define a class of “model” metrics h on X . Their first feature is that the fibers of π will have constant length L (0 < L < +∞). It is therefore natural to introL duce the vertical vector field T that is equal to 2π times the infinitesimal generator of the 1 S -action; the flow of T goes around the fibers in time L. As a second feature, h should pullback the Euclidean metric on Rm as much as possible, i.e. on the orthogonal of the fibers. The way to do this is to pick a “connection” one-form η, namely a S1 -invariant one-form η such that η(T ) = 1. We define the corresponding model metric on X by h := π ∗ gRm + η2 = d x 2 + η2 , where xi := π ∗ xˇi , 1 ≤ i ≤ m, denote the lifts of the canonical coordinates xˇi on the base B ⊂ Rm . In addition, we will require some decay for the curvature dη of this
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connection. To express it, let us define r :=
x12 + · · · + xm2 and observe dη = π ∗ ω
for some two-form ω on the base B (dη is S1 -invariant and Cartan’s magic formula L T η = ιT dη + d(ιT η) implies ιT dη = 0). We will assume ω = O(r −τ −1 ) and Dω = O(r −τ −2 ) for some positive number τ (D is the flat connection on Rm ). Example 1. (Trivial fibration) In the trivial fibration case, we can choose a trivialization (x1 , . . . , xm , eit ) and put η = η0 := dt. Since dη0 = 0, any τ is convenient. Observe η0 defines foliations over r −1 (R) for any R, corresponding to the product foliation of Rm × S1 by circles. The model metric is the flat h 0 := d x 2 + dt 2 . Example 2. (Hopf fibration) In the Hopf fibration case, we define a connection form η on R4 \ {0} = C2 \ {0} by the formula η(z) := |z|−2 gR4 (T, .), where T is the infinitesimal generator of the scalar action of S1 ⊂ C∗ . Then τ = 1 is convenient. Note η is nothing but the standard contact form on S3 and h := d x 2 + η2 is the model at infinity for the Riemannian Taub-NUT metric (see Example 4.3 at the end of the paper). Let us describe a few features of such metrics. The coframe (d x1 , . . . , d xm , η) is obviously orthonormal. We let (X 1 , . . . , X m , T ) be the dual frame. Uniqueness of the Levi-Civita connection ensures that, for horizontal vector fields X , Y (namely η(X ) = η(Y ) = 0), π∗ ∇ Xh Y = Dπ∗ X π∗ Y . The computation of brackets of horizontal vector fields, together with T , contains the geometric information about the fibration. First, since η([X i , T ]) = −dη(X i , T ) = 0 and π∗ [X i , T ] = [π∗ X i , π∗ T ] = 0, T commutes with any vector field X i . Besides, with η([X i , X j ]) = −dη(X i , X j ) and π∗ [X i , X j ] = [π∗ X i , π∗ X j ] = 0, we obtain [X i , X j ] = −dη(X i , X j )T . Koszul formula then yields: ∇hη =
dη 2
and
∇ h d xi =
ι X i dη ⊗ η + η ⊗ ι X i dη . 2
(4)
In other words, the only non trivial Christoffel coefficients are h(∇ Xh i T, X j ) = h(∇Th X i , X j ) = −h(∇ Xh i X j , T ) =
dη(X i , X j ) . 2
It is important to keep in mind that ∇ h X i and ∇ h T are O(r −τ −1 ). Moreover, a short computation yields h u = −X i · X i · u − T · T · u. Given a local section of the circle bundle, one may consider a local vertical coordinate t and work in the coordinates (x1 , . . . , xm , t). Observe ∂t = T and η = dt + Ai d xi for some functions Ai independent of t. Then X k = ∂k − Ak ∂t and h = −∂kk − ∂tt + 2 Ak ∂kt − A2k ∂tt + ∂k Ak ∂t .
(5)
2. Analysis on Asymptotic Circle Fibrations In this section, we work on a complete Riemannian manifold (M m+1 , g), m ≥ 3, such that for some compact subset K , M\K is diffeomorphic to the total space X of a principal circle fibration π over B := Rm \Bm , with a model metric h as in Sect. 1, such that: g = h + O(r −τ ) and ∇ h g = O(r −τ −1 ).
930
V. Minerbe
Recall τ is assumed to be positive. We intend to study the equation g u = f in weighted L 2 spaces. The presentation is deeply influenced by [HHM and Pac]. Given a real number δ and a subset of M, we first define the weighted Lebesgue space ⎧ ⎫ ⎪ ⎪ ⎨ ⎬
2 L 2δ ( ) := u ∈ L loc u 2 r −2δ dvolh < ∞ ⎪ ⎪ ⎩ ⎭
\K
and endow it with the norm u L 2 ( ) :=
δ
∩K
u 2 dvol g +
u 2 r −2δ dvolh
\K write L 2δ
1 2
.
L 2δ (M).
for The Changing K only produces equivalent norms. We will often m 2 a following should be kept in mind: r ∈ L δ (M\K ) ⇔ δ > 2 + a. Note the Riemannian measures dvol g and dvolh on M\K can always be interchanged, thanks to our asymptotic assumption. For the same reason, the Riemannian connections ∇ g and ∇ h will be completely equivalent outside K . Any function u ∈ L 2loc (M\K ) can be written u = 0 u + ⊥ u, where 0 u is obtained by computing the mean value of u along the fibers: ( 0 u)(x) = L1 π −1 (x) u η. It corresponds to a Fourier series decomposition along the fibers: 0 u is the part in the kernel of −T 2 = −∂tt while ⊥ u lies in the positive eigenspaces of −∂tt . The point is h commutes with the projectors 0 and ⊥ : for any function u, one can use (5) to write locally ( 0 h u)(x) as 1 −∂kk − ∂tt + 2 Ak (x)∂kt − Ak (x)2 ∂tt + ∂k Ak (x)∂t u(x, t)dt. L This simplifies into ( 0 h u)(x) =
− L1
∂x x u(x, t)dt =
−∂x x L1
u(x, t)dt = h
1 L
u(x, t)dt ,
which ensures 0 h = h 0 and also ⊥ h = h ⊥ . Since this decomposition will prove useful, we introduce the following Hilbert spaces, depending on two real parameters δ and :
2
0 u L 2 ( \K ) < ∞ and ⊥ u L 2 ( \K ) < ∞ . L 2δ, ( ) := u ∈ L loc ( ) δ
They are endowed with the following Hilbert norm: 1 2
u L 2 ( ) := u 2L 2 (K ∩ ) + 0 u 2L 2 ( \K ) + ⊥ u 2L 2 ( \K ) . δ,
δ
We also introduce the Sobolev space
2 ∇ g d 0 u 2 Hδ2 := u ∈ Hloc + d 0 u L 2 (K c ) + 0 u L 2 (K c ) < ∞ L δ−2 (K c ) δ δ−1 g and ∇ d ⊥ u L 2 (K c ) + d ⊥ u L 2 (K c ) + ⊥ u L 2 (K c ) < ∞ δ−2
δ−2
δ−2
endowed with the obvious Hilbert norm. In what follows, we will always write A R for the “annulus” defined by R ≤ r ≤ 2R and AκR for 2−κ R ≤ r ≤ 2κ+1 R (κ ≥ 0). Similarly, the “balls” K ∪ {r ≤ R} will be denoted by B R . The letter c will always denote a positive constant whose value changes from line to line. Its precise dependence on parameters will often be clear in the context; otherwise, we will write c(. . . ) to clarify it.
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Fig. 1. The critical exponents
2.1. A priori estimates. 2.1.1. A priori estimates on the kernel of ⊥ . We aim at some a priori estimates for h on X . Let us begin with the kernel of ⊥ : basically, we work with functions defined on exterior domains in Rm and the Laplace operator is the standard one! What follows is therefore standard (it is in [Pac] and can be compared with [LP,Bart,HHM]). We will nonetheless provide details of the proofs, in order to use them later. Lemma 1. Given δ in R, there is a positive number c = c(m, δ) such that for any R0 ≥ 1 and any u in L 2δ ∩ Ker ⊥ , h
du
u
+ ≤ c u
+ c c c 2 2 2 ∇ du 2 h L (B ) L (B ) L (B ) . c L δ−2 (B2R )
δ−1
0
δ−2
2R0
δ
R0
R0
Proof. Scaling the usual Garding inequality for the Laplacian on Rm yields 2 2 2 2
Ddu L 2 (A ) + du L 2 (A ) ≤ c h u L 2 (A1 ) + u L 2 (A1 ) . δ−2
δ−1
R
δ−2
R
δ
R
R
The formulas for the connection of h imply ∇ h du ≤ c |Ddu| + cr −τ −1 |du| so we can replace D by ∇ h in the estimate above. Using it for R = 2i+1 R0 , i ∈ N, and summing over i, we get the desired inequality. To carry on, we need to use a L 2 spectral decomposition for the Laplace operator S on the unit sphere Sm−1 in Rm . Its eigenvalues are λ j := j (m − 2 + j), with j ∈ N, and we denote by E j the corresponding eigenspaces. We also set δ j :=
m 2−m + j and ν ± ± (δ j − 1) j := 2 2
± for j ∈ N. This simply means ν +j = j and ν − j = 2 − m − j. These numbers ν j are usually called “indicial basic property is the following: given an element roots”. Their φ j of E j , we have r ν j φ j (ω) = 0 outside 0. It can easily be seen from the formula
= −∂rr −
m−1 r ∂r
+
1 r2 S
− ∂tt .
(6)
Definition 1. We will say δ is critical if δ = δ j or δ = 2 − δ j for some j in N. Lemma 2. If δ is not critical, there is a positive number c = c(m, δ) such that for any R0 ≥ 1 and any u in L 2δ ∩ Ker ⊥ ,
u L 2 (B c ) ≤ c h u L 2 (B c ) + u L 2 (A R ) . δ
2R0
δ−2
R0
δ
0
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V. Minerbe
Proof. If we first perform a spectral decomposition with respect to S , our problem reduces to find an estimate on solutions u j ∈ L 2 (R+ , r m−1−2δ dr ) to the radial equation −∂rr u j −
m−1 r ∂r u j
+
λj u r2 j
= f j.
Setting r := es , v j (s) := u(r ) and g j (s) = r 2 f j (r ), we turn the equation into d v j (s)e(m−2)s = (λ j v j (s) − g j (s))e(m−2)s , ds
(7)
and we work in L 2 (R+ , dµδ ), with dµδ := e(m−2δ)s ds. Given a radial truncature function χ vanishing on B R0 and equal to 1 outside B2R0 , integration by parts provides ((χ v j ) )2 dµδ = χ 2 v 2j dµδ + (χ 2 v j ) v j e(m−2)s e(2−2δ)s ds 2 2 2 = χ v j dµδ − (χ v j )(λ j v j − g j )dµδ − (1 − δ) χ 2 (v 2j ) e(m−2δ)s ds = χ 2 + (1 − δ)(χ 2 ) v 2j dµδ (χ v j )2 dµδ + χ 2 v j g j dµδ . + (1 − δ)(m − 2δ) − λ j 2 (χ v j )2 dµδ ≤ ((χ v j ) )2 dµδ to One might then use the Hardy inequality (m−2δ) 4 deduce the estimate (δ j − δ)(δ j + δ − 2) (χ v j )2 dµδ ≤ χ 2 + (1 − δ)(χ 2 ) v 2j dµδ + χ 2 v j g j dµδ . For all indices j such that (δ j − δ)(δ j + δ − 2) > 0, Young inequality yields ∞
2R0 2 (χ v j ) dµδ ≤ c v j dµδ + c χ 2 g 2j dµδ , 2
2R0
R0
which is what we need. For the finitely many indices j such as 2 − δ < δ j < δ, we 1 w j,+ − w j,− , with consider the function w j := 2(1−δ j) s w j,± (s) :=
±
eν j (s−σ ) g j (σ )dσ.
(8)
R0
Since
wj,±
=
ν± j w j,±
(m − 2δ)
+ g j , integration by parts leads to
R∞ w 2j,± (s)e(m−2δ)s R0
=
−2ν ± j
R∞ R∞ 2 w j,± dµδ − 2 w j,± g j dµδ + w 2j,± (R∞ )e(m−2δ)R∞ .
R0
R0
≥0
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933
Since m − 2δ + 2ν ± j > 0, it implies (Cauchy-Schwarz inequality): ∞
∞ w 2j,± dµδ
≤c
R0
g 2j dµδ . R0
Now, w j is a solution of (7) as well as v j , so from ODE theory, w j − v j is a linear ±
combination of eν j s . Its L 2 (dµδ )-norm over [R0 , +∞[ can therefore be estimated by its L 2 (dµδ )-norm over [R0 , 2R0 [. It follows that v j can be estimated by ∞
∞ v 2j dµδ
≤c
R0
g 2j dµδ
2R0 +c v 2j dµδ .
R0
R0
The remaining case, δ < δ j < 2 − δ can be dealt with in a similar way. One defines an explicit solution w j as above, replacing only R0 by +∞ in formula (8). The integration m by parts argument still works (because w j,± = o(e(δ− 2 )s )) and the fact that v j and w j belong to L 2 (R+ , dµδ ) forces them to coincide. 2.1.2. A priori estimates on the kernel of 0 . The following lemma shows that estimates are basically better on the kernel of 0 . Lemma 3. Given δ ∈ R and a large number R0 , there is a constant c such that for any function u in L 2δ ∩ Ker 0 , h ∇ du 2 c + du L 2 (B c ) + u L 2 (B c ) ≤ c h u L 2 (B c ) + u L 2 (A R ) . L δ (B2R )
δ
0
δ
2R0
δ
2R0
δ
R0
0
Proof. Given parameters R >> 1 and 0 ≤ κ < κ ≤ 1, we can always choose a smooth 1 nonnegative function χ that is equal to 1 on AκR , vanishes outside Aκ R , is S -invariant and has gradient bounded by c(κ, κ )R −1 . The integration by part formula |d(χ u)|2 dvolh = |dχ |2 u 2 dvolh + χ 2 uh u dvolh can be used together with the Young inequality to obtain
du 2L 2 (Aκ ) ≤ c() h u 2L 2 (Aκ ) + u 2L 2 (Aκ ) , R
R
(9)
R
where can be chosen arbitrarily small provided R is sufficiently large. Since 0 u = 0, we have 2 2 (−T u)u η = du(T ) η ≥ c u 2 η. fiber
fiber
fiber
This implies u L 2 (Aκ ) ≤ c du L 2 (Aκ ) and therefore, with (9): R
R
u 2L 2 (Aκ ) + du 2L 2 (Aκ ) ≤ c() h u 2L 2 (Aκ ) + u 2L 2 (Aκ ) . R
R
R
R
(10)
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V. Minerbe
To get an order two estimate, we first write 2 2 h h 2 h χ ∇ du = (∇ (χ du), ∇ du) − 2 χ (dχ ⊗ du, ∇ h du). The Bochner Laplacian ∇ h,∗ ∇ h and the Hodge Laplacian h = ddh∗ + dh∗ d only differ by the Ricci endomorphism, so an integration by parts provides: 2 2 h 2 2 χ ∇ du = χ (h du, du) − χ Rich (du, du) − 2 χ (dχ ⊗ du, ∇ h du). Since the Hodge Laplacian commutes with d, another integration by part yields χ 2 (h du, du) = (dh u, χ 2 du) = χ 2 |h u|2 − 2 χ (dχ , du)h u. Putting these formulas altogether and using the Young inequality, we obtain: 2 χ 2 ∇ h du ≤ 4 χ 2 |h u|2 + 6 |dχ |2 |du|2 − 2 χ 2 Rich (du, du). The upshot of this formula is the following estimate (observe Rich is bounded): h 2 ∇ du 2 κ ≤ c h u L 2 (Aκ ) + c du 2L 2 (Aκ ) . L (A R )
R
R
With (10), we deduce:
2
u 2L 2 (Aκ ) + du 2L 2 (Aκ ) + ∇ h du 2 R
L (AκR )
R
≤ c() h u 2L 2 (Aκ ) + u 2L 2 (Aκ ) . R
R
We then set = 0.5, κ = 0, κ 1, we multiply these inequalities by R −2δ and sum them for R = 2k R0 , k ∈ N∗ (R0 is chosen large) to find: 2
u 2L 2 (B c ) + du 2L 2 (B c ) + ∇ h du 2 c ≤ c f 2L 2 (B c ) + c u 2L 2 (A ) . δ
δ
2R0
L δ (B2R )
2R0
δ
0
δ
R0
R0
Remark 1. When the fibration is trivial, one can also study the integral kernel of the resolvent, as in [Dav]. Using a Fourier decomposition, we are left to prove that for positive k, the integral kernel 1 + d(o, y) δ P(x, y) := (Rm + k 2 )−1 (x, y) 1 + d(o, x) defines a bounded operator on L 2 (Rm ). Indeed, it is true for instance on a complete manifold with Ric ≥ 0 and ∀t ≥ 1, At ν ≤ vol B(x, t) ≤ Bt ν for some ν > 2. Noticing 2 −1
(Rm + k )
1 =√ π
∞
e−t e−k t 1 + d(o, y) ≤ 1 + d(x, y), dt and √ 1 + d(o, x) t 2
0
we can use the Li-Yau Gaussian estimate on the heat kernel [LY] to get ∞ P(x, y) ≤ c(1 + d(x, y)) 0
d(x,y)2
e− ct e−k t √ √ dt. vol B(x, t) t 2
It implies P(x, y) ≤ ce and therefore P(x, y)d x ≤ c and P(x, y)dy ≤ c, which is enough to ensure the boundedness of P on L p , 1 < p < ∞. − d(x,y) c
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2.1.3. The main estimate Proposition 1. If δ is not critical, a compact set B such that thereis a constant c and 2 for any u in L δ,δ−2 , u H 2 ≤ c g u L 2 + u L 2 (B) . δ
δ−2
Proof. Since h commutes with 0 and ⊥ , we can write the equation g u = f outside a large ball B R0 as follows: h u 0 = f 0 + 0 (h − g )u h u ⊥ = f ⊥ + ⊥ (h − g )u. We have denoted 0 u by u 0 , ⊥ u by u ⊥ , etc, to make the equations easier to read. We apply Lemmata 1 and 2 to the first equation and Lemma 3 to the second equation, which results in h + du 0 L 2 (B c ) + u 0 L 2 (B c ) ∇ du 0 2 c L δ−2 (B2R )
+ ∇ h du ⊥
δ−1
0
c ) L 2δ−2 (B2R
+ du ⊥ L 2
c δ−2 (B2R0 )
c δ−2 (B R0 )
Since
2R0
+ u ⊥ L 2
c δ−2 (B2R0 )
+ c u L 2 (B2R ) + c (h − g )u L 2 0
≤ c f L 2
δ
2R0
δ
c δ−2 (B R0 )
0
.
(h − g )u ≤ c |g − h| ∇ h du + c ∇ g − ∇ h |du| ,
there is a function going to zero at infinity such that −1 h (h − g )u 2 (r )∇ r ≤ du + (r )du c L (B ) c 2 δ−2
L δ−2 (B R )
R0
0
Therefore, (h − g )u L 2 (R0 ) ∇ h du 0
c δ−2 (B R0 )
L 2δ−2 (B Rc )
.
0
can be bounded by
c ) L 2δ−2 (B2R 0
+ (R0 ) du ⊥ L 2
(11)
c δ−2 (B2R0
+ ∇ h du ⊥ h + du ∇ )
c ) L 2δ−2 (B2R 0
L 2δ−2 (A R0 )
+ du 0 L 2
c δ−1 (B2R )
+ du L 2
0
δ−2 (A R0
) .
Using this and choosing R0 large enough, we find h + du 0 L 2 (B c ) + u 0 L 2 (B c ) ∇ du 0 2 c ) δ 2R0 δ−1 2R0 L δ−2 (B2R 0 + ∇ h du ⊥ 2 + du ⊥ L 2 (B c ) + u ⊥ L 2 (B c ) c ) δ−2 2R0 δ−2 2R0 L δ−2 (B2R 0 ≤ c f L 2 (B c ) + c u L 2 (B2R ) + c ∇ h du 2 + c du L 2 (B2R ) . δ−2
0
R0
L (B2R0 )
0
Owing to the asymptotic of the metric, ∇ h can be changed into ∇ g in this estimate. Since the standard Garding inequality provides g ∇ du 2 + du L 2 (B2R ) ≤ c f L 2 (B4R ) + c u L 2 (B4R ) , L (B ) 2R0
0
we can conclude: u H 2 ≤ c f L 2 δ
δ−2,δ−2
0
+ c u L 2 (B4R ) . 0
0
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V. Minerbe
2.2. Mapping properties. We are interested in the unbounded operator Pδ :
D(Pδ ) −→ L 2δ−2,δ−2 u → g u
whose domain D(Pδ ) is the dense subset of L 2δ,δ−2 whose elements u have their Laplacian in L 2δ−2,δ−2 . 2.2.1. Fredholmness. Proposition 1 has a direct consequence. In view of Rellich’s theorem and Peetre’s lemma (cf. [LiM], p. 171), it implies Ker Pδ is finite dimensional (for any δ) and Ran Pδ is closed (for any noncritical δ). The usual L 2 pairing identifies the topological dual space of L 2δ,δ−2 (resp. L 2δ−2,δ−2 ) with L 2−δ,2−δ (resp. L 22−δ,2−δ ). For this identification, the adjoint Pδ∗ of Pδ is Pδ∗ :
D(Pδ∗ ) −→ L 2−δ,2−δ u → g u
where the domain D(Pδ∗ ) is the dense subset of L 22−δ,2−δ whose elements u have their Laplacian in L 2−δ,2−δ . Observing Ker Pδ∗ ⊂ Ker Pη for some large non-critical η, we see that Ker Pδ∗ is always finite dimensional. We have proved Proposition 2. If δ is not critical, then Pδ is Fredholm. 2.2.2. Solving an exterior problem. We first solve the equation h u = f on an exterior domain. Lemma 4. Given a noncritical δ and a large number R0 , there is a bounded operator G h : L 2δ−2 (B Rc 0 ) −→ Hδ2 (B Rc 0 ) such that h ◦ G h = id. Proof. We will define G h in three steps, relying on an orthogonal decomposition of L 2δ−2 (B Rc 0 ). For the first step, we pick a function f in L 2δ−2 ∩ Ker ⊥ and we assume it has no component along the eigenspaces E j of S such that (δ j − δ)(δ j + δ − 2) > 0. Given R >> R0 , we can use standard elliptic theory to solve the equation h u R = f in L 2 (B R \B R0 ), with Dirichlet boundary condition. Adapting the proof of Lemma 2 (on B R \B R0 , with χ = 1), we obtain
u R L 2 (B R \B R δ
0)
≤ c f L 2
δ−2 (B R \B R0 )
,
(12)
with c independent of R. Elliptic regularity then bounds the H 2 norm of u R over compact subsets in terms of f L 2 , so that we can use Rellich theorem and a diagonal δ−2
1 , with u = f argument to extract a sequence u R converging to a function u in Hloc h and u = 0 on ∂ B R0 . Taking a limit in (12), one gets u L 2 (B c ) ≤ c f L 2 (B c ) . δ−2
R0
δ−2
R0
From (1) (plus standard elliptic arguments near ∂ B R0 ), we deduce an estimate on the derivatives:
u H 2 (B c δ
R0 )
≤ c f L 2
c δ−2 (B R0 )
.
(13)
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937
We need to show that such a u is uniquely defined, i.e. independent of the choice of extracted sequence. The difference v between two such functions u obeys Rm v = 0 − + ± so, from ODE theory, its modes read v j (r, ω) = r ν j φ +j (ω) + r ν j φ − j (ω) with φ j in E j . ± 2 Since v j is in L δ and vanishes on the boundary, we have φ j = 0, so v = 0. We can therefore set G h f := u. As a second step, we observe the same approach can be used for a function f in L 2δ−2 ∩ Ker 0 . We can still obtain the functions u R by solving the same problem and the proof of Lemma 3 can be adapted to provide the estimate (12), provided R0 is large enough. This makes it possible to extract a converging subsequence as above, yielding a function u such that h u = f , u = 0 on ∂ B R0 and satisfying (13). As for unicity, we consider the difference v between two such functions u: v is in L 2δ,δ−2 ∩ Ker 0 , vanishes on ∂ B R0 and obeys h v = 0. Given a large number R, we can choose a cutoff function χ that is equal to 1 on A R , vanishes on A1R and has gradient bounded by 10/R. Then 2 2 2 |d(χ v)| = |dχ | v + χ 2 vh v implies dv L 2η (A R ) ≤ c v L 2
1 η−1 (A R )
for any exponent η. Since 0 v = 0, we get
v L 2η (A R ) ≤ c dv L 2η (A R ) ≤ c v L 2
1 η−1 (A R )
for any exponent η.
This improves L 2η estimates into L 2η−1 estimates. So v ∈ L 2δ implies v ∈ L 2η for any η. In particular, v is in L 2 and thus vanishes. So u is well defined and we can set G h f := u. As a third step, we consider those f in L 2δ−2 ∩ Ker ⊥ whose only component in the spectral decomposition of S is in E j , with (δ j − δ)(δ j + δ − 2) < 0. In case 2 − δ < δ j < δ, we set G h f := u, with ⎛ ⎞ r r − − + 1 ⎜ ν +j ⎟ t 1−ν j f (t, ω)dt − r ν j t 1−ν j f (t, ω)dt ⎠ . (14) u(r, ω) := ⎝r 2(1 − δ j ) R0
R0
The proof of Lemma 2 yields
u L 2 (B c
≤ c f L 2
.
(15)
u H 2 (B c
≤ c f L 2
.
(16)
δ
R0 )
c δ−2 (B R0 )
With Lemma 1, we obtain: δ
2R0 )
c δ−2 (B R0 )
Since u vanishes on ∂ B R0 , standard elliptic estimates improve (16) into
u H 2 (B c δ
R0 )
≤ c f L 2
c δ−2 (B R0 )
.
The case δ < δ j < 2 −δ is dealt with similarly, replacing R0 by +∞ in (14). Note u does not vanish on ∂ B R0 in this setting. But it can be bounded by c f L 2 (B c ) on ∂ B R0 ; since S u j = λ j u j , u H 2 (∂ B R
above still works.
0)
is then bounded by c f L 2
δ−2
c δ−2 (B R0 )
R0
, so the argument
A perturbation argument extends this result to a more general setting.
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V. Minerbe
Proposition 3. Given a noncritical δ and a large number R0 , we can define a bounded operator G g : L 2δ−2 (B Rc 0 ) −→ Hδ2 (B Rc 0 ) such that g ◦ G g = id. Proof. Thanks to Lemma 4,we can write g = h id +G h g − h . With (11), we can estimate (h − g )u L 2 (B c ) by (R0 ) u H 2 (B c ) . Since G h is bounded from δ δ,δ−2 R0 R0 c c 2 2 L δ,δ−2 (B R0 ) to Hδ (B R0 ), we deduce that G h g − h defines a bounded operator on Hδ2 (B Rc 0 ), with norm strictly inferior to 1 for R0 large enough. So id + g − h −1 h −1 is an automorphism of Hδ2 (B Rc 0 ) and G g := id +G h g − h G h is a bounded operator from L 2δ,δ−2 (B Rc 0 ) to Hδ2 (B Rc 0 ), with g G g = h G h = id. This lemma can be used to build functions which are harmonic outside a compact set and have some prescribed asymptotics. Corollary 1. Given j ∈ N and φ ∈ E j , there are functions ℵ±j,φ that are harmonic ±
outside a compact set and can be written ℵ±j,φ = r ν j φ + v± with v+ in Hη2 for any η > δ j − τ and v− in Hη2 for any η > 2 − δ j − τ . Proof. For noncritical δ > δ j , since r ν j φ is in L 2δ and h (r ν j φ) = 0, Lemma 1 implies +
ν +j
+
ν +j
r φ is in Hδ2 . With (11), we deduce g (r φ) ∈ r −τ L 2δ−2 = L 2δ−τ −2 . Now we can use Lemma 3 to solve g u = −g (r ν j φ) outside a compact set and put ℵ+j,φ := χ (r ν j φ +u) for some smooth nonnegative function χ which vanishes on a large compact set and is equal to 1 outside a larger compact set. The construction of ℵ−j,φ follows the same lines. +
+
2.2.3. Decay jumps. The following lemma is the key to understand the growth of solutions to our equations. Lemma 5. Suppose h u = f with u in L 2δ (B Rc 0 ) and f in L 2δ −2 (B Rc 0 ) for non-critical exponents δ > δ and a large number R0 . Then there is an element v of L 2δ ,δ −2 (B Rc 0 ) such that u − v is a linear combination of the following functions: – r ν j φ j with φ j in E j and δ < δ j < δ; +
– r
ν− j
φ j with φ j in E j and δ < 2 − δ j < δ.
Proof. We will build v step by step, starting from the solution v˜ of h v˜ = f provided by Lemma 4: v˜ is in L 2δ −2,δ (B Rc 0 ). Consider w := 0 (u − v) ˜ and look at its modes −
± w j . The equation h w j = 0 implies w j = r ν j φ +j + r ν j φ − j with φ j in E j . Observing +
−
r ν j ∈ L 2η ⇔ η > δ j and r ν j ∈ L 2η ⇔ η > 2 − δ j , one can see that each term is either in L 2δ −2,δ , so that we can add it to v˜ and forget it, or satisfies the conditions in the statement. What about z := ⊥ (u − v) ˜ ? This z satisfies h z = 0, 0 z = 0 and z ∈ L 2δ . As in the proof of lemma 4, for any η, L 2η estimates on z improve into L 2η−1 estimates, so z ∈ L 2δ implies z ∈ L 2η for any η and the proof is complete. +
Let us generalize this to g .
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Proposition 4. Suppose g u = f with u in L 2δ (K c ) and f in L 2δ −2 (K c ) for noncritical exponents δ > δ . Then, up to enlarging K , there is an element v of L 2δ ,δ −2 (K c ) such that u − v is a linear combination of the following functions: – ℵ+j,φ j with φ j in E j , if δ < δ j < δ;
– ℵ−j,φ j with φ j in E j , if δ < 2 − δ j < δ. Proof. Thanks to Lemma 1, the equation g u = f , with u in L 2δ,δ−2 and f in L 2δ −2 , ensures u ∈ Hδ2 . With (11), we obtain h u ∈ L 2δ −2 + L 2δ−τ −2 . So if we pick any noncritical η ≥ max(δ , δ − τ ), we have h u ∈ L 2η−2 . Lemma 5 then says u admits a decomposition u = u1 +
%
r ν j+ φ +j+ + +
j+
%
r
ν− j
−
φ− j− ,
j−
where u 1 belongs to L 2η,η−2 , η < δ j+ < δ and 2 − δ < 2 − δ j− < 2 − η. With Lemma 1, we can therefore write % % ℵ+j+ ,φ + + ℵ−j ,φ − u = u2 + j+
j+
j−
−
j−
with u 2 in L 2η,η−2 , η < δ j+ < δ and 2 − δ < 2 − δ j− < 2 − η. If δ − τ ≤ δ , we are done. If not, observe g u 2 = f outside a compact set and u 2 belongs to L 2η,η−2 . So we can repeat the argument with u 2 in the role of u and η in the role of δ. In a finite number of steps, we are in the first case. Corollary 2. Pδ is surjective for any noncritical value δ > 2 −
m 2.
Proof. For any noncritical δ ≥ 1, Ker Pδ∗ consists of harmonic functions u ∈ L 21 . For every large number R, there is a smooth function 0 ≤ χ R ≤ 1 such that χ R = 1 on B R , χ R = 0 outside B2R and |dχ R | ≤ 10 R . Then
|d(χ R u)| = M
|dχ R | u +
2
2
M
χ R2 ug u
2
M
|du| ≤ c 2
implies BR
u 2 r −2
AR
and, letting R go to infinity, one finds du = 0, hence u = 0 : Ker Pδ∗ = {0}. So Pδ is surjective as soon as δ ≥ 1 is noncritical (recall it is Fredholm). Now, choose some δ in ]2 − m2 , 1[ and pick some function f in L 2δ −2 . In particular, f is in L 21−2 , so there is a solution u ∈ L 21,1−2 of g u = f . Proposition 4, with δ = 1 and δ = δ , implies u ∈ L 2δ ,δ −2 , for there is no critical exponent in ]δ , 1[ ! Pδ is therefore surjective.
940
V. Minerbe
2.3. An extension to the Dirac operator. If we endow Rm × S1 with its trivial spin structure, we can define a Dirac operator Dh 0 whose square D2h 0 acts diagonally as the Laplace operator h 0 in a constant trivialization. As a consequence, the a priori estimates that we have proved for h 0 are also available for D2h 0 . We wish to work on a complete spin Riemannian manifold (M m+1 , g), m ≥ 3, such that for some compact subset K , M\K is spin-diffeomorphic to Rm \Bm ×S1 , with: g = h 0 + O(r −τ ), ∇ h 0 g = O(r −τ −1 ) and ∇ h 0 ,2 g = O(r −τ −2 ), τ > 0. A perturbation argument easily yields. Proposition 5. If δ is not critical, there is a constant c and a compact set B such that for any ψ in L 2δ,δ−2 ,
ψ H 2 δ
≤ c D2g ψ
L 2δ−2
+ ψ L 2 (B) .
The functional spaces are defined in the obvious way, using a constant trivialization. The proof is basically the same as that of Proposition 1. Note however the estimate on the second derivative of g: we need this to control the difference between the model Dirac Laplacian D2h 0 and the operator D2g , and more precisely the 0th order term (which of course vanishes for the Laplace operator on functions). Indeed, we bound D2g ψ − D2h 0 ψ by a constant times 2 |g − h 0 | ∇ h 0 ,2 ψ + ∇ h 0 g ∇ h 0 ψ + ∇ g − ∇ h 0 + ∇ h 0 ,2 g |ψ| . The Fredholmness of the corresponding operator Pδ , δ noncritical, follows immediately. The decay jump phenomenons carry over to this setting in exactly the same way. The proof of Corollary 2 can be adapted to get Corollary 3. If Scalg ≥ 0, Pδ is surjective for any noncritical value δ > 2 −
m 2.
Proof. Lichnerowicz formula and the assumption Scalg ≥ 0 ensure the L 21 kernel of D is trivial: just use the integration by part formula
|∇(χ R ψ)|2 = M
|dχ R |2 |ψ|2 +
M
χ R2 (ψ, D2 ψ) −
M
The end of the proof is the same as for the Laplace operator.
χ R2 Scal |ψ|2 .
1 4 M
3. Towards a Mass In this section, we introduce a notion of mass for ALF metrics. The first paragraph contains a few (classic) algebraic computations which are useful in the sequel. The second and third paragraph develop two points of view corresponding to the two standard examples of Dirac type operators.
A Mass for ALF Manifolds
941
3.1. Algebraic preliminaries. Let M n be a Riemannian manifold and let E be a bundle of left modules over the Clifford algebra bundle Cl(T M). For definitions and basic facts about spin geometry, we refer to [LM]. We assume E is endowed with a compatible Euclidean metric (., .) and metric connection ∇, whose curvature tensor is R. This data determines a Dirac type operator D on E and a section R of End E. In a local orthonormal basis (ea )a , these are D=
n %
ea · ∇ea and R =
a=1
n 1 % [ea ·, eb ·]Rea ,eb . 4 a,b=1
Remark 2. Commutators are easier to handle than Clifford products, for the latter are not antisymmetric. The identity [ea ·, eb ·] = 2(δab + ea · eb ·)
(17)
makes the translation. Moreover, brackets are skew-symmetric with respect to the Euclidean metric: ([ea ·, eb ·]ψ, ψ) = 0.
(18)
Given sections α and β of E, we define a one-form ζα,β on M by the following formula (as in [AD,He2,Dai]): ζα,β (X ) := (∇ X α + X · Dα, β). The point is: d ∗ ζα,β = (Dα, Dβ) − (∇α, ∇β) − (Rα, β). We can integrate this Lichnerowicz-type formula over a domain to get
d ζα,β dvol =
[(∇α, ∇β) + (Rα, β) − (Dα, Dβ)] dvol = −
∗
d ∗ ζα,β .
Stokes formula then provides
⎡ ⎣(∇α, ∇β) + (Rα, β) −
⎤ (Dα, Dβ)⎦ dvol =
∗ζα,β .
(19)
∂
If ωα,β is the two-form defined by ωα,β (X, Y ) = ([X ·, Y ·]α, β), one can compute d ∗ ωα,β = 4ζα,β − 4ζβ,α . Stokes theorem thus implies:
∗ζα,β = ∂
∗ζβ,α . ∂
(20)
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3.2. The Gauss-Bonnet case. Let us consider a complete oriented Riemannian manifold (M m+1 , g), m ≥ 3, such that for some compact subset K , M\K is diffeomorphic to the total space of a circle fibration π over Rm \Bm , which we endow with a model metric h = d x 2 + η2 as in Sect. 1. Recall dη = π ∗ ω, with ω = O(r −τ −1 )
Dω = O(r −τ −2 ).
and
(21)
We assume g is asymptotic to h in the following sense: g = h + O(r −τ ), ∇ h g = O(r −τ −1 ) and ∇ h,2 g = O(r −τ −2 ).
(22)
For what follows, it is important to require : τ > m−2 2 . To simplify the statement of Lemma 6, we will also suppose τ ≤ m − 2 (if not, take τ := m − 2). In this paragraph, we work on the exterior bundle M(=: E), endowed with the Levi-Civita connection ∇ g and we use the Gauss-Bonnet operator d + d ∗ as Dirac-type operator D. The Clifford product v· := v − ιv is obtained from the exterior product v := (v, .)∧ and the interior product ιv . In this setting, R preserves the form degree. In degree 1, it reduces to the natural action of the Ricci tensor. Let us introduce the vector space Z spanned by X 1 , . . . , X m . As a first step toward the definition of a mass, we wish to build g-harmonic one-forms that are asymptotic to any element of the dual vector space Z ∗ . We fix a small positive number (say 0 < < 1/2 and < τ − m−2 2 ). Lemma 6. For any element * α in Z ∗ , there is a one-form α on M such that (d + d ∗ )α = 0 and α = * α + β, with β = O(r −τ ) and ∇ g β ∈ L 2−τ −1+ m . 2
Remark 3. It is important to stick to asymptotically horizontal forms: this lemma cannot be generalized by choosing * α = η. A counter-example is provided by the Taub-NUT metric : g = V d x 2 + V1 η2 with V = 1 + 2m r (cf. the end of the paper for more details). In this Riemannian manifold, if there were a one-form α with (d + d ∗ )α = 0 and α asymptotic to η (in the sense above), it would have to be η/V (for it is harmonic and asymptotic to this one-form ; a similar argument can be found in the proof of Corollary 3). But this one-form is not closed, hence is not suitable. This fact explains the difference B and µ D . In the Gauss-Bonnet case, we will define the between the formulas for µG g g mass by taking a trace over the horizontal directions (because of this lemma), whereas we would have needed a full trace to recover the same mass as in the Dirac setting. We will need a Sobolev inequality. Indeed, the assumptions on h and g imply Ricg ≥ cr −2 , so we can use [SC] (or [Mi1]) to ensure the following family of scaled Sobolev inequalities: with n = m +1, any smooth function u with compact support in A R satisfies
u
2n
L n−2
≤c
R 1
(vol B R ) n
du L 2 .
This allows Moser iterations, turning integral estimates into pointwise estimates ([GT]; more precisely, Lemma 3.9 in [TV] and Lemma A.3 in [Mi2]): 2 c
u L ∞ ≤ g u L ∞ . (23) 1 u L 2 + c R (vol B R ) 2
Since Ricg ≥ cr −2 , this estimate also holds for one-forms (g is then replaced by the Hodge Laplacian (d + d ∗ )2 ) and for spinors (with the Dirac Laplacian D2 ).
A Mass for ALF Manifolds
943
Proof. It is enough to prove the claim for * α := d xk . Let us choose a truncature function χ that vanishes on a large compact set and is equal to 1 outside a larger compact set. Since g xk is in L 2δ −2 for any δ > 1+ m2 −τ , we can apply weighted analysis (Corollary 2 ; observe δ > 2 − m2 because of our assumption τ ≤ m − 2) to construct a function u k in Hδ2 such that g u k = −g (χ xk ), for any noncritical δ > 1 + m2 − τ . Setting βk := du k , we therefore obtain (d + d ∗ )βk = −(d + d ∗ )d (χ xk ), with βk ∈ L 2δ and ∇βk ∈ L 2δ−1 , for any δ > m2 − τ such that δ = δ + 1 is not critical. We may choose δ = m2 − τ + . Then the one-form αk := d (χ xk ) + βk obeys (d + d ∗ )αk = 0. The pointwise estimate on βk follows from (23), using βk ∈ L 2δ and g βk = O(r −τ −2 ) (this coincides with dg xk outside a compact set). Now, the basic formula we need is given by (19) and (20): for any one-form α as in Lemma 6 and any large number R, we have g 2 ∇ α + Ricg (α, α) dvol = ∗ζ* ∗ζ* ∗ζβ,β . (24) α ,* α +2 α ,β + ∂ BR
BR
∂ BR
∂ BR
Our next aim consists in understanding the asymptotic behaviour of the right-hand-side as much as possible. The mass is to be the limit of this quantity when the domains B R are larger and larger. Let us tackle the second and third terms on the right-hand side. The assumption τ > m−2 2 kills them at infinity. Lemma 7. There is a sequence (Ri )i going to infinity such that ∗ζ* lim ∗ζβ,β = 0. lim α ,β = 0 and i−→∞ ∂ B Ri
i−→∞ ∂ B Ri
Proof. Since β is in L 2m +−τ and ∇β is in L 2m +−τ −1 , one can find a sequence Ri going 2 2 2 2 g 2 |β| + r ∇ β to infinity such that dvol = o(Rim−2τ +2−1 ). The lemma then ∂ Ri
α = O(r −τ −1 ), with τ > follows from Cauchy-Schwarz inequality and ∇ g*
m−2 2 .
We need to compute the first term in the right-hand side of (24). Observe h identifies Z and Z ∗ , so any Z in Z corresponds to a well defined * α =* α Z in Z ∗ . Lemma 8. ζ* α− α ,* α = −(divh g)(Z )*
1 2
α + d (g(Z , Z ))] + O(r −2τ −1 ). [d(Tr h g)(Z ) *
Proof. Let us g-orthonormalize the frame field (X 1 , . . . , X m , T ) into (ea )a and set ωcd := g(∇ g ec , ed ) − g(∇ h ec , ed ). Since the connection on the tangent bundle T M reads ∇ g = ∇ h + ωcd ec ⊗ ed , the connection on the cotangent bundle satisfies: ∇ g* α = ∇ h* α − ωdc * α (ec )ed . Therefore, we can write ζ* α ,* α = ξ1 + ξ2 , with ξ1 =
1 1 g([ea ·, eb ·]∇ehb * α, * α )ea and ξ2 = − ωdc (eb ) * α (ec )g([ea ·, eb ·]ed , * α )ea . 2 2
We are lead to estimate terms like α ) = g([ea − ιea , eb − ιeb ]ed , * α) g([ea ·, eb ·]ed , * α) = −g([ιea , eb ]ed + [ea , ιeb ]ed , *
944
V. Minerbe
(recall * α is a one-form). A little algebra provides [ea , ιeb ]ed = 2δbd ea − δab ed and therefore: 1 g([ea ·, eb ·]ed , * α ) = δad * α (eb ) − δbd * α (ea ). (25) 2 α , ed [δad * α (eb ) − δbd * α (ea )] ea , namely ξ1 = ∇ Zh * α Formula (25) implies ξ1 reads g ∇ehb * h h ∗ b α, e * α +(dh * α )* α +O(r −2τ −1 ) = O(r −2τ −1 ) is α . With (4), we deduce ξ1 = ∇ Z * −g ∇eb * a negligible term. As for ξ2 , we use (25) again to obtain ξ2 = ωdc (eb ) * α (ec )[δbd * α (ea ) − δad * α (eb )]ea . Since the Koszul formula provides 2ωx y (ew ) = −g(ew , [ex , e y ]) − g(ex , [ew , e y ]) + g(e y , [ew , ex ]) − 2g(∇ehw ex , e y ) = −g(ew , ∇ehx e y ) − g(ex , ∇ehw e y ) + g(e y , ∇ehw ex ) + g(ew , ∇ehy ex ) +g(ex , ∇ehy ew ) − g(e y , ∇ehx ew ) − 2g(∇ehw ex , e y ) = (∇ehx g)(e y , ew ) − (∇ehy g)(ex , ew ) + (∇ehw g)(ex , e y ), we eventually find: 2ξ2 = (∇ehb g)(Z , eb ) * α − (∇ Zh g)(eb , eb ) * α + (∇ehb g)(eb , Z ) * α −(∇eha g)(Z , Z ) ea + (∇ Zh g)(ea , Z ) ea − (∇ Zh g)(ea , Z ) ea . We then use the asymptotic of g and (4) to simplify this expression into the promised −2(divh g)(Z )* α − d(Tr h g)(Z ) * α − d (g(Z , Z )), up to O(r −2τ −1 ). The computations are done, it is time to draw a theorem, which requires a definition. Definition 2. On M\K , we define a one-form qg,h with values in the quadratic forms on Z by the formula qg,h (Z ) = −(divh g)(Z )* αZ −
1 2
α Z + d (g(Z , Z ))] . [d(Tr h g)(Z ) *
The “mass” quadratic form Qg,h is the quadratic form defined on Z by: 1 Qg,h (Z ) := ∗h qg,h (Z ), lim sup ωm L R−→∞ ∂ BR
where ωm is the volume of Sm−1 and L is the asymptotic length of fibers. Why this normalization constant? The factor 1/L is there to make the mass independent of the length of the asymptotic circles. The normalization by the volume of a sphere is more anecdotic. This corresponding positive mass theorem is the following. Theorem 3. Let (M m+1 , g), m ≥ 3, be a complete oriented manifold with nonnegative Ricci curvature. We assume that, for some compact subset K , M\K is the total space of a circle fibration over Rm \Bm , which can be endowed with a model metric h such that (21) and (22) hold with τ > m−2 2 . Then Qg,h is a nonnegative quadratic form. It vanishes exactly when (M, g) is the standard Rm × S1 .
A Mass for ALF Manifolds
945
Proof. Formula (24), together with Ric ≥ 0, Lemma 7 and Lemma 8, provides g 2 ∇ α Z + Ricg (α Z , α Z ) dvol ≤ ωm L Qg,h (Z ).
(26)
M
Since Ric ≥ 0, we deduce Qg,h (Z ) ≥ 0. Now we assume Qg,h = 0. In view of (26), the g-harmonic one-forms α Z are then g-parallel. We have therefore built m g-parallel one-forms α1 , . . . , αm that are asymptotic to d x1 , . . . , d xm . We also put αm+1 := ∗g (α1 ∧ · · · ∧ αm ), so as to obtain m + 1 g-parallel one-forms that are linearly independent outside a compact set, hence linearly independent on the whole M. This yields a global parallel coframe field on M. In particular, (M, g) is flat, so it is a flat vector bundle over a compact flat manifold and g is naturally induced by the flat connection ([CG] p. 281). Since the volume growth of (M, g) is comparable to that of Rm , the fibers of this bundle are m-dimensional. This means (M, g) is a flat Rm -bundle over S1 , namely it is obtained as a quotient of Rm+1 by the group generated by a rigid motion ρ: ρ(x) = Ax + v, with A ∈ O(m + 1), v = 0 and Av = v. For such a manifold, the holonomy group is generated by A. But we have built a global parallel coframe on (M, g), so the holonomy group is trivial: A is the identity. It follows that (M, g) is obtained as a quotient of Rm+1 by a translation: it is isometric to the standard Rm × S1 . B Note Qg,h vanishes if and only if its trace vanishes. So µG g,h := Tr Qg,h plays the role of a numerical mass. We have: 1 B lim sup µG = − ∗h divh g + d Tr h g − 21 d(g(T, T )) . (27) g,h ωm L R−→∞ ∂ BR
In this formula, the integrand can be replaced by the Hodge star of (−∇ Xh j g)(X i , X j ) + X i · g(X j , X j ) − (∇Th g)(X i , T ) + d(g(T,T2 ))(X i ) d xi . This can be simplified for we can use (4) and the closeness of g to h to get (∇Th g)(X i , T )∗h d xi = T · g(X i , T ) ∗h d xi + O(r −2τ −1 ), which can be rephrased as d(g(X i , T ) ∗h (d xi ∧ η)) + X i · g(X i , T ) ∗h η + O(r −2τ −1 ). So this part integrates to zero at infinity! In the same spirit, (∇ Xh j g)(X i , X j ) equals X j · g(X i , X j ) up to O(r −2τ −1 ), so the mass B µG g,h reduces to B µG g,h
=
1 ωm L
lim sup R−→∞
∗h
X j · g(X i , X j ) − X i · g(X j , X j ) d xi −
d(g(T,T )) 2
.
∂ BR
The term between brackets is similar to the usual expression of the mass in the asymptotically Euclidean setting (see (1) in the Introduction). It is the contribution from the base. More precisely, one can average the metric along the fibers and compute the mass of the asymptotically Euclidean metric induced on the base: it is this term. The other term is related to the variation of the length of fibers. We now turn to the geometric invariance of the mass: does the mass really depend on h or is it a Riemannian invariant of g ? To ensure the answer is yes, we replace the assumptions (21) and (22) by D i ω = O(r −τ −1−i ), 0 ≤ i ≤ 2,
(28)
946
V. Minerbe
and ∇ h,i (g − h) = O(r −τ −i ), 0 ≤ i ≤ 3.
(29)
B If there is a model metric h such that these estimates hold, the corresponding mass µG g,h does not depend on h but only on g: it is a Riemannian invariant. Before embarking into the proof, let us explain what the assumptions (28) and (29) are for. If we go back to the crucial Lemma 6, we observe the derivative of β is only controlled in an L 2 norm, which explains why we have to choose a convenient sequence of radii Ri in Lemma 7 and why we define the mass with a limsup instead of a simple limit. Our new assumptions provide a C 1 estimate on β, which allows us to get rid of the limsup.
Lemma 9. If (28) and (29) hold, then the mass can be written as 1 GB lim ∗h qg,h (X k ) µg,h = ωm L R−→∞ ∂ BR
=
1 ωm L
m % |∇αk |2 + (Ric(αk ), αk ) dvol. k=1 M
Proof. As explained above, it amounts to obtain a C 1 bound on the one-form βk obtained in Lemma 6 if α˜ = d xk . Let denote the rough Laplacian ∇ ∗ ∇. A slight computation yields ∇βk = ∇βk + Rm ∇βk + ∇ Rm βk , where denotes any bilinear pairing obtained by contracting with respect to the metric. The Bochner formula βk = (dd ∗ + d ∗ d)βk − Ric(βk ) implies that, outside a compact set, we have βk = −dg xk − Ric(βk ), hence ( − Rm )∇βk = −∇dg xk + ∇ Rm βk = O(r −τ −3 )
(30)
thanks to (28) and (29). Since Rm = O(r −τ −2 ), the term Rm can be treated as a perturbation and we can use a Moser iteration (cf. Lemma A.3 in [Mi2]) in order to find 2 c
∇βk L ∞ (A R ) ≤ ( − Rm )∇βk L ∞ (A ) , 1 ∇βk L 2 (A ) + c R (vol B R ) 2
R
R
with A R = B2R \B R and AR = B2.5R \B0.5R . With (30) and Lemma 6 (to get an L 2 bound on ∇βk ), we end up with: ∇βk = O(r −τ −1 ). With this and Lemma 6, we can deduce from formula (24) and Lemma 8 that 2 |∇αk | + (Ric(αk ), αk ) dvol = ∗h ζd xk ,d xk + o(1) ∂ BR
BR
=
∗h qg,h (X k ) + o(1), ∂ BR
with αk = d xk + βk . Since Ric ≥ 0, the left-hand side admits a limit in [0, +∞], so ∗ ∂ B R h qg,h (X k ) goes to an element of [0, +∞] as R goes to infinity. Summing over k, B we see that the limsup in the definition of µG g,h can be turned into a genuine limit, in [0, +∞], hence the lemma.
A Mass for ALF Manifolds
947
B Proposition 6. If h and h are two model metrics satisfying (28) and (29), then µG g,h = B µG g,h .
Proof. The metrics h = d x 2 + η2 and h = d x 2 + η2 come with two circle fibrations π and π , with connections as above. We can assume (28) and (29) hold for h and h with the same τ . The proof of the unicity of the mass is in three steps (compare [Bart,CH,Chr]). We first prove that, given an adapted (i.e. as above) h-orthonormal frame field (X 1 , . . . , X m , T ), we can find an adapted h -orthonormal frame field , T ) that is r −τ -close to (X , . . . , X , T ). Secondly, we show that the (X 1 , . . . , X m 1 m computation of the mass term does not depend on the choice of “spheres at infinity” (∂ B R = r −1 (R) or ∂ B R = r −1 (R)). And finally, we prove that the difference between the integrands corresponding to h and h is the sum of an exact form and of a negligible term. For the first step, we need to prove that the infinitesimal generators T and T of both circle actions are r −τ -close. Note that, since h and h are asymptotic to g (and because the fibers of π and π have bounded length), the quotient r/r is bounded from above and below outside a compact set and, indeed, both r and r are comparable to the g-distance to a fixed point. Observe also that by assumption: T = 1 + O(r −τ ) = |T |h + O(r −τ ). (31) h For every point x with r (x) sufficiently large, we introduce the smooth loop β defined ˙ = T ; we can push it into a smooth loop γ := π ◦ β on Rm . by β(0) = x and β(t) β(t) Observe L = L is well defined since it is the injectivity radius at infinity. The idea of the proof is the following. Suppose Tx were not close to Tx (nor to −Tx ). Then its h-horizontal component would be substantial, so that the curve γ would have a substantial initial speed vector (π∗ T )x . We will prove that γ has a very small acceleration, so that it remains close to a straight line, which is not compatible with the fact that γ has to come back to its origin in a time L. So Tx has to be close to Tx . To make this argument effective, we decompose T as the sum of its h-horizontal part H and h-vertical part W ; observe (31) bounds the norm of H and W by a constant. Since π is a Riemannian submersion between h and the flat metric on Rm , we have h h h Dπ∗ T π∗ T = π∗ (∇ H H ) = π∗ (∇ H T ) − π∗ (∇ H W ). (32) h T ), is bounded by a constant times ∇ h T , which is O(r −τ −1 ) The first term, π∗ (∇ H h in view of (28), (29) and (4). To bound the second term, we use (4) to compute h h W, X i ) = −h(W, ∇ H Xi ) = h(∇ H
1 η(W )dη(H, X i ). 2
This is O(r −τ −1 ), by (28). With (32), this ensures Dπ∗ T π∗ T = O(r −τ −1 ). Given a point x , the acceleration γ¨ of the corresponding loop γ is Dγ˙ γ˙ = Dπ∗ T π∗ T (by definition of γ ), so it obeys : ∀ t, |γ¨ | ≤ cr (x)−τ −1 . Since γ (L) = γ (0), Taylor formula L provides L γ˙ (0) = 0 t γ¨ (t)dt. With the bound on γ¨ , we deduce |γ˙ (0)| ≤ cr (x)−τ −1 . Since, by definition of γ , γ˙ (0) = (π∗ T )x , we can conclude that the h-horizontal part Hx of Tx is bounded by a constant times r (x)−τ −1 . Therefore, up to an error term of order r −τ −1 , the vector fields T and T are colinear. Since they have the same h-norm up to O(r −τ ) by (31), we may assume T = T + O(r −τ ) (changing T into −T if necessary). Now, given an adapted h-orthonormal frame field (X 1 , . . . , X m , T ), we average the
948
V. Minerbe
vectors π∗ X i along the fibers of π , so as to get a frame field ( Xˇ 1 , . . . , Xˇ m ) on the base Rm of π . Now we can proceed on the base as in the asymptotically Euclidean setting (cf. [Chr]). Since each Xˇ i has derivative bounded by r −1−τ , Xˇ i = vi + O(r −τ ) for some (constant) vector vi on Rm (Cauchy criterion); moreover, (v1 , . . . , vm ) is an orthonormal , T ) is an frame in Rm . We define X i as the h -horizontal lift of vi . Then (X 1 , . . . , X m −τ h -adapted frame field that is r -close to (X 1 , . . . , X m , T ). To complete the second step of the proof, we rely on Lemma 9: 1 GB µg,h = lim ∗h qg,h (X k ) ωm L R−→∞ ∂ B R
1
=
ωm L
m % 2 ∇α + (Ric(α ), α ) dvol. k
k
k
k=1 M
Since the ratio of the distance functions r/r is bounded from above and below (as detailed in [Chr]), the end of the proof of Lemma 9 can be adapted to obtain 2 ∇α + (Ric(α ), α ) dvol = ∗h ζd xk ,d xk + o(1) k k k ∂ BR
BR
=
∗h qg,h (X k ) + o(1).
∂ BR
Letting R go to infinity, we find 2 ∇α + (Ric(α ), α ) dvol = lim k k k M
hence
=
B µG g,h
1
lim
ωm L
R−→∞ ∂ BR
1 ωm L
lim
R−→∞ ∂ BR
∗h
R−→∞ ∂ BR
∗h qg,h (X k ),
∗h qg,h (X k ), which can be computed by
d g(T ,T ) X j · g(X i , X j ) − X i · g(X j , X j ) d xi − ( 2 ) ,
where ∂ B R = r −1 (R) is defined with respect to h. Let us turn to the third step, which consists in proving the equality d g(T ,T ) lim X j · g(X i , X j ) − X i · g(X j , X j ) d xi − ( 2 ) ∗h R−→∞ ∂ BR
= lim
R−→∞ ∂ BR
∗h
X j · g(X i , X j ) − X i · g(X j , X j ) d xi −
d(g(T,T )) 2
First observe the difference d(g(T , T )) − d(g(T, T )) reads
(∇ h g)(T , T ) − (∇ h g)(T, T ) + 2g(∇ h T , T ) − O(r −2τ −1 )
O(r −2τ −1 )
2g(∇ h T, T )
2η (∇ h T )+O (r −2τ −1 )
,
.
A Mass for ALF Manifolds
949
which implies ∗h d(g(T , T )) − ∗h d(g(T, T )) = 2 ∗h η (∇ h T ) + O(r −2τ −1 ). Now, a slight computation (using [X i , T ] = 0) leads to
∗h η (∇ h T ) = η (∇ Xhi T ) ∗h d xi + η (∇Th T ) ∗h η = η (∇Th X i ) ∗h d xi + η (∇Th T ) ∗h η = T · η (X i ) ∗h d xi + η (∇Th T ) ∗h η + O(r −2τ −1 ) = d ∗h (η ∧ η) + X i · η (X i ) ∗h η +η (∇Th T ) ∗h η + O(r −2τ −1 ). Since the terms involving ∗η integrate to zero on ∂ B R , Stokes formula yields ∗h d(g(T , T )) − ∗h d(g(T, T )) = 0. lim R−→∞ ∂ BR
To tackle the remaining term, we expand it as follows: X j · g(X i , X j ) − X i · g(X j , X j ) − X j · g(X i , X j ) − X i · g(X j , X j ) + = (∇ Xh g)(X i , X j ) + g(∇ Xh X i , X j ) + g(X i , ∇ Xh X j ) − (∇ Xh g)(X j , X j ) j j j i , −2g(∇ Xh X j , X j ) − (∇ Xhj g)(X i , X j ) + g(∇ Xhj X i , X j ) i +g(X i , ∇ Xhj X j ) − (∇ Xhi g)(X j , X j ) − 2g(∇ Xhi X j , X j ) . Using the closeness of the frames and ∇ h g = O(r −τ −1 ), one can see that the contribution of the terms involving ∇ h g is of order r −2τ −1 , hence negligible. Besides, using the closeness of g to h and (4), we have g(∇ Xh X j , X k ) = O(r −2τ −1 ), which ensures i many terms above are lower order terms. Another simplification comes from the fact that the commutator [X i , X j ] is π -vertical and of order r −τ −1 : it implies ∇ Xhi X j and ∇ Xhj X i only differ by a lower order term. All in all, we find X j · g(X i , X j ) − X i · g(X j , X j ) − X j · g(X i , X j ) − X i · g(X j , X j ) = g(∇ Xhi X j , X j ) − g(X i , ∇ Xhj X j ) + O(r −2τ −1 ). We now introduce the (m − 1)-form h (X j , X i ) ∗h (d xi ∧ d x j ). Its exterior derivative d h (X j , X i ) ∗h (d xi ∧ d x j ) is g(∇ Xhj X j , X i ) ∗h d xi − g(∇ Xhi X j , X i ) ∗h d x j plus lower order terms. After changing ∇ Xhi X j into ∇ Xhj X i (which costs only a lower order term) and switching summation indices i and j in the second term, we turn this into g(∇ Xhj X j , X i ) ∗h d xi − g(∇ Xhi X j , X j ) ∗h d xi + O(r −2τ −1 ). This computation can be combined with Stokes formula to ensure ∗h X j · g(X i , X j ) − X i · g(X j , X j ) d xi lim R−→∞ ∂ BR
= lim
R−→∞ ∂ BR
∗h X j · g(X i , X j ) − X i · g(X j , X j ) d xi .
950
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B GB We have proved: µG g,h = µg,h .
3.3. The spin case, with a trivial fibration. In this paragraph, we would like to explain a spin analogue to the construction above. The setting is a complete spin manifold (M m+1 , g), m ≥ 3 with nonnegative scalar curvature, such that for some compact subset K , M\K is diffeomorphic to Rm \Bm ×S1 , which we endow with the standard flat metric h 0 = d x 2 + dt 2 . We assume the metric g is asymptotic to h 0 in the following sense: for some τ > m−2 2 , g = h 0 + O(r −τ ),
Dg = O(r −τ −1 ) and D 2 g = O(r −τ −2 ).
As previously, we can choose τ ≤ m − 2. We furthermore assume the spin structure of M coincides outside K with the trivial spin structure on Rm \Bm × S1 . We work on the spinor bundle M := g M(=: E) corresponding to the metric g and we endow it with the pullback of the Levi-Civita connection. Then D is the standard Dirac operator D and R is the multiplication by 14 Scal ([LM]). Outside K , M is not the bundle corresponding to h 0 , but we can identify them in a natural way. To make it precise, we denote by P the unique g-symmetric section of End T M such that h 0 = g(P., P.). One can also see P as a natural bijection between the principal bundles of orthonormal frames so that it lifts as an identification of the spin bundles and therefore identify h 0 (M\K ) with g (M\K ) = (M\K ). The Levi-Civita connection D of h 0 induces a flat metric connection ∇ euc on (T M, g), given by the formula ∇ Xeuc Y = P D X (P −1 Y ). Since it is a metric connection, it induces a metric connection ∇ euc on the spinor bundle g M = M. To sum up, we have three connections on T M: ∇ g is metric for g and torsionless; ∇ euc is metric for g, is flat but has torsion; D is metric for h 0 , flat and torsionless. Only two of them lift to M: ∇ g and ∇ euc . Now, the spinor bundle (M\K ) is trivial and ∇ euc -flat, so we can find a unit ∇ euc parallel spinor field α0 . If is a small positive number, we can adapt Lemma 6. Lemma 10. There is a spinor field α := * α + β such that D α = 0, with * α = α0 outside a compact set, β = O(r −τ ) and ∇β ∈ L 2m +−τ −1 . 2
Proof. If χ is a convenient truncature function χ , we can see * α0 := χ α0 as a section of M. Set γ := − D(χ α0 ). Since γ = O(r −τ −1 ), it belongs to L 2−τ −1+ m . From analysis 2
2 in weighted spaces (Corollary 3), we obtain a solution σ ∈ H−τ of the equation +1+ m
D2 σ = γ . Put β = D σ . The estimates on β follow as in Lemma 6. As in the Gauss-Bonnet case, we can use formula 19 to find g 2 ∇ α + Scalg |α|2 dvol ≤ lim sup ∗ζα0 ,α0 . g M
Lemma 11. ζα0 ,α0 = −
R−→∞
2
(33)
∂ BR
1 d Tr h 0 g + divh 0 g + O(r −2τ −1 ). 4
Proof. The proof is by now standard. We consider the frame field (∂a )a := (∂1 , . . . , ∂m , ∂t ). It is orthonormal for the model metric h 0 . Putting ea := P∂a , we obtain an orthog normal frame field for g. We need to understand ζα0 ,α0 = 21 g([ea ·, eb ·]∇eb α0 , α0 )ea .
A Mass for ALF Manifolds
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Since α0 is ∇ euc -parallel, with ωcd := g(∇ g ec , ed ) and ∇ g − ∇ euc = ([LM]), we obtain ζα0 ,α0 =
1 8 ωcd [ec ·, ed ·]
1 ωcd (eb )g([ea ·, eb ·][ec ·, ed ·]α0 , α0 )ea . 16
(34)
In order to compute the connection one-form, we resort to the Koszul formula: 2ωcd (eb ) = −g(eb , [ec , ed ]) − g(ec , [eb , ed ]) + g(ed , [eb , ec ]). Expanding the brackets thanks to the torsionless connection D, one finds: 2ωcd (eb ) = −(Ded g)(eb , ec ) + (Dec g)(eb , ed ) + g(Deb ec , ed ) − g(Deb ed , ec ). (35) Let us denote by H the g-symmetric endormorphism such that g − h 0 = g(H., .). From De H P 2 = I − H and ec = P∂c , we get Deb ec = − 2b ∂c + O(r −2τ −1 ). And since H is gsymmetric, we have g((Deb H )∂c , ∂d ) = g((Deb H )∂d , ∂c ) + O(r −2τ −1 ). So we deduce g(Deb ec , ed ) − g(Deb ed , ec ) = O(r −2τ −1 ). Plugging this into (35) and then (34), we see that ζα0 ,α0 can be approximated by 1 (∂c gbd − ∂d gbc ) g([ea ·, eb ·][ec ·, ed ·]α0 , α0 )ea + O(r −2τ −1 ) 32 1 = ∂c gbd g([ea ·, eb ·][ec ·, ed ·]α0 , α0 )ea + O(r −2τ −1 ) 16 1 = ∂c gbd g((δab δcd + δab ec ed + δcd ea eb + ea eb ec ed ) · α0 , α0 )ea + O(r −2τ −1 ). 4 In view of (17) and (18), every term like g(ea · eb · α0 , α0 ) can be replaced by −δab |α0 |2 . Moreover, using (17), we can write ∂c gbd g(ea eb ec ed · α0 , α0 ) = − ∂c g2bd g(ea [eb , ed ]ec · α0 , α0 ) + ∂c gbd g(δbd ea ec · α0 , α0 ). Since ∂c gbd is symmetric with respect to b and d whereas [eb , ed ] is antisymmetric, the first term vanishes. These observations lead to: ζα0 ,α0 = 41 ∂c gbd (δab δcd − δab δcd − δcd δab − δbd δac ) |α0 |2 ea + O(r −2τ −1 ), which reduces to ζα0 ,α0 =
1 4
(−∂c gac − ∂a gbb ) ea + O(r −2τ −1 ).
The corresponding positive mass theorem involves 1 D µg,h lim sup =− ∗h divh 0 g + d Tr h 0 g . ωm L R−→∞
(36)
∂ BR
Theorem 4. Let (M m+1 , g), m ≥ 3, be a complete spin manifold with nonnegative scalar curvature. We assume there is a compact set K and a spin preserving diffeomorphism between M\K and Rm \Bm × S1 such that g = gRm ×S1 + O(r −τ ), Dg = O(r −τ −1 ) and D 2 g = O(r −τ −2 ) D with τ > m−2 2 . Then µg,h is nonnegative and vanishes exactly when (M, g) is isometric m to the standard R × S1 .
952
V. Minerbe
D , hence the Proof. Since Scalg ≥ 0, formula 33 leads to M |∇ g α|2 dvol ≤ ωm4 L µg,h D . When µ D vanishes, every constant spinor α in the model gives nonnegativity of µg,h 0 g,h rise to a harmonic and parallel spinor field α that is asymptotic to α0 . This makes it possible to produce a parallel trivialization of the spinor bundle. It follows that (M, g) has trivial holonomy ([MS]) and we can conclude as in Theorem 3. D does not depend Finally, the proof of Proposition 6 can be adapted to prove that µg,h on h but only on g, under the assumptions of Theorem 2.
4. Examples 4.1. Schwarzschild metrics. These are complete Ricci flat metrics on R2 × Sn−2 , n ≥ 4, given by the formula: gγ = dρ 2 + Fγ (ρ)2 dθ 2 + G γ (ρ)2 dω2 . ρ, θ are polar coordinates on the R2 factor, dω2 is the standard metric on Sn−2 , Fγ and G γ are smooth functions defined by n−3 γ 2γ γ n−3 G γ (ρ) = 1 − , G γ (0) = γ and Fγ = 1− , Gγ n−3 Gγ for some positive parameter γ . G γ increases from γ to ∞ and G γ ∼ ρ at infinity ; Fγ 2γ 2γ increases from 0 to n−3 and Fγ ∼ ρ near 0. Setting r := G γ (ρ) and t = n−3 θ , we can write γ n−3 2 dr 2 2 2 dt . gγ = γ n−3 + r dω + 1 − r 1− r In this way, it is apparent that gγ is asymptotic to the flat metric on Rn−1 ×S1 , with circle γ 2 n−2 is spin (with a unique spin length equal to L := 4π n−3 at infinity. Observe M = R ×S structure, since it is simply connected). To compute the masses, we introduce “isotropic” 2 n−3 n−3 coordinates: putting r = u 1 + 41 γu , we get gγ = 1 +
1 4
4 γ n−3 n−3
u
. (du + u dω ) + 2
2
2
1− 1+
/ 1 γ n−3 2 4 u dt 2 1 γ n−3 4 u
for u > 0. We can then choose Cartesian coordinates x1 , . . . , xn−1 corresponding to the polar coordinates (u, ω) and keep in mind the first order terms: n−3 gγ ≈ 1 + γn−3 u 3−n d x 2 + 1 − γ n−3 u 3−n dt 2 . n−1 n−3 B With h 0 = d x 2 + dt 2 , this readily provides µgDγ = γ n−3 and µG . For gγ = 2 γ instance, in dimension n = 4, the masses reduce to the parameter γ (up to a universal constant), which is basically what we hoped (cf. Introduction). The mass is therefore positive, but we cannot deduce it from the “spin” version of the positive mass theorem: the spin structure at infinity is not the trivial spin structure!
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953
Indeed, the spin structure on the asymptotic S1 comes from the spin structure of the unit circle in the R2 factor of R2 × Sn−2 . This spin structure is therefore induced by the (unique and trivial) spin structure on the unit disk: it is the non-trivial spin structure on the circle (see [Bar]). In the next paragraph, we will give examples showing that this non-trivial spin structure at infinity really allows negative mass in nonnegative scalar curvature. But here, since the Ricci curvature of Schwarzschild metrics is nonnegative, we can use the Gauss-Bonnet point of view and explain the positivity of the mass: it is a consequence of Ric ≥ 0 and not of Scal ≥ 0. 4.2. Reissner-Nordström metrics. In four dimensions, the Schwarzschild metric gγ belongs to a broader family of complete scalar flat metrics on R2 ×S2 that are asymptotic to R3 ×S1 , with the non-trivial spin structure at infinity. These Reissner-Nordström metrics [Dai,BrH,CJ,PK] are given by the same ansatz g = dρ 2 + F(ρ)2 dθ 2 + G(ρ)2 dω2 , with 0 0 G =
1−
2m G
−
q2 G2
and F =
G 20 G 0 −m
1−
2m G
−
q2 G2
1 and G(0) := G 0 = m + m 2 + q 2 . The behaviour of G and F is similar to the Schwarzschild analogue. The new feature is m can be chosen negative: the metric is then still complete, provided q is nonzero. Setting r := G(ρ) and t = more familiar formula: g=
dr 2 1−
2m r
−
q2
+ 1−
2m r
−
q2 r2
G 20 G 0 −m θ ,
we can obtain a
dt 2 + r 2 dω2 .
r2
The formulas for doubly-warped products [Pet] (or geometric arguments as in [Bes], 3.F) make it possible to compute the curvature. The eigenvalues of the Ricci tensor are 2 q2 , along ∂ρ and ∂t , and − Gq 4 , along the S2 factor. It therefore has no sign but the scalar G4 curvature vanishes. To compute the mass µgD , we can again use isotropic coordinates. 2 2 The new radial coordinate u is given by r = u 1 + mu + m4u+q2 and we find
g = 1+
m u
+
m 2 +q 2 4u 2
2
⎡ (du 2 + u 2 dω2 ) + ⎣
m 2 +q 2 4u 2 m 2 +q 2 m u + 4u 2
1− 1+
⎤2 ⎦ dt 2 .
Comparing to the Schwarzshild formula, one can see that the asymptotic is the same up to O(u −2 ), so the mass µgD is 2m. Since we can choose m negative, this yields a whole family of metrics with negative mass!
4.3. Multi-Taub-NUT metrics. The Taub-NUT metric is the basic non-trivial example of ALF gravitational instanton. In particular, it is hyperkähler hence Ricci flat. For details, the reader is referred to [Le2,HHM]. Basically, it is a complete metric on R4 which is adapted at infinity to the Hopf fibration R4 \ {0} −→ R3 \ {0}. It can be written outside one point as g = V d x 2 + V1 η2 , where V = 1 + 2m r for some positive parameter m and η is a connection form with curvature ∗R3 d V . Up to a constant, η is the standard contact
954
V. Minerbe
B form on the three-spheres (cf. Sect. 1). With h = d x 2 + η2 , we can compute µG g = 3m: as expected, the mass is essentially the parameter m. The same computations apply to the multi-Taub-NUT metrics (cf. [Le2,HHM]). These metrics are again hyperkähler and their asymptotic is adapted to a principal circle bundle over S2 with Chern number −k. They can be written oustide k points as 2k 2m 1 g = V d x 2 + V1 η2 with V = 1 + i=1 |x−xi | and η is a connection form on a S bundle 3 G B of these metrics is over R \ {x1 , . . . , xk } whose curvature is ∗R3 d V . The mass µ 3km.
Acknowledgements. I would like to thank Gilles Carron, Gary Gibbons and Marc Herzlich for their interest in this work and for many interesting comments. This work benefited from the French ANR grant GeomEinstein.
References [AD]
Andersson, L., Dahl, M.: Scalar curvature rigidity for asymptotically locally hyperbolic manifolds. Ann. Global Anal. Geom. 16(1), 1–27 (1998) [ADM] Arnowitt, R., Deser, S., Misner, C.W.: Coordinate invariance and energy expressions in general relativity. Phys. Rev. 122(2), 997–1006 (1961) [AH] Atiyah, M., Hitchin, N.: The Geometry and Dynamics of Magnetic Monopoles. M. B. Porter Lectures. Princeton, NJ: Princeton University Press, 1988 [BKN] Bando, S., Kasue, A., Nakajima, H.: On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97(2), 313–349 (1989) [Bar] Bär, C.: Dependence of the Dirac spectrum on the Spin structure. In: Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Smin. ´ Congr., 4, Paris: Soc. Math. France, (2000), pp. 17–33 [Bart] Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39, 661–693 (1986) [Bes] Besse, A.: Einstein Manifolds. Berlin: Springer-Verlag, 1987 [BH] Boualem, H., Herzlich, M.: Rigidity at infinity for even-dimensional asymptotically complex hyperbolic spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(2), 461–469 (2002) [BrH] Brill, D., Horowitz, G.T.: Negative energy in string theory. Phys. Lett. B 262(4), 437–443 (1991) [CG] Cheeger, J., Gromov, M.: Collapsing Riemannian manifolds while keeping their curvature bounded. II. J. Diff. Geom. 32(1), 269–298 (1990) [Chr] Chru´sciel, P.T.: Boundary conditions at spatial infinity from a Hamiltonian point of view. In: Topological properties and global structure of space-time (Erice, 1985), NATO Adv. Sci. Inst. Ser. B Phys. 138, New York: Plenum, (1986), pp. 49–59 [CH] Chru´sciel, P.T., Herzlich, M.: The mass of asymptotically hyperbolic riemannian manifolds. Pacific J. Math. 212(2), 231–264 (2003) [CJ] Corley, S., Jacobson, T.: Collapse of kaluza-klein bubbles. Phys. Rev. D 49, R6261–R6263 (1994) [Dah] Dahl, M.: The positive mass theorem for ALE manifolds. In: Mathematics of Gravitation, Part I (Warsaw, 1996), Banach Center Publ. 41, Part I, Warsaw: Polish Acad. Sci., (1997), pp. 133–142 [Dai] Dai, X.: A positive mass theorem for spaces with asymptotic susy compactification. Commun. Math. Phys. 244(2), 335–345 (2004) [Dav] Davies, E.B.: L p spectral independence and L 1 analyticity. J. London Math. Soc. (2) 52(1), 177–184 (1995) [DS] Denisov, V.I., Solovev, V.O.: Energy defined in general relativity on the basis of the traditional Hamiltonian approach has no physical meaning. Theor. Math. Phys. 56(2), 832–841 (1983) [GT] Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Berlin: Springer-Verlag, 2001 [HHM] Hausel, T., Hunsicker, E., Mazzeo, R.: Hodge cohomology of gravitational instantons. Duke Math. J. 122(3), 485–548 (2004) [He1] Herzlich, M.: Théorèmes de masse positive, Sémin. Théor. Spectr. Géom. 16, Univ. Grenoble I, Saint-Martin-d’Hères, année 1997–1998, pp. 107–126 [He2] Herzlich, M.: Scalar curvature and rigidity of odd-dimensional complex hyperbolic spaces. Math. Ann. 312, 641–657 (1998) [Hit] Hitchin, N.: Monopoles, minimal surfaces and algebraic curves, Séminaire de Mathmatiques Supérieures [Seminar on Higher Mathematics] 105. Montreal, QC: Presses de l’Université de Montréal, 1987
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Communicated by G. W. Gibbons
Commun. Math. Phys. 289, 957–993 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0809-1
Communications in
Mathematical Physics
The Davey-Stewartson Equation on the Half-Plane A. S. Fokas Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK. E-mail: [email protected] Received: 2 June 2008 / Accepted: 10 December 2008 Published online: 1 May 2009 – © Springer-Verlag 2009
Abstract: The Davey-Stewartson (DS) equation is a nonlinear integrable evolution equation in two spatial dimensions. It provides a multidimensional generalisation of the celebrated nonlinear Schrödinger (NLS) equation and it appears in several physical situations. The implementation of the Inverse Scattering Transform (IST) to the solution of the initial-value problem of the NLS was presented in 1972, whereas the analogous problem for the DS equation was solved in 1983. These results are based on the formulation and solution of certain classical problems in complex analysis, namely of a Riemann Hilbert problem (RH) and of either a d-bar or a non-local RH problem respectively. A method for solving the mathematically more complicated but physically more relevant case of boundary-value problems for evolution equations in one spatial dimension, like the NLS, was finally presented in 1997, after interjecting several novel ideas to the panoply of the IST methodology. Here, this method is further extended so that it can be applied to evolution equations in two spatial dimensions, like the DS equation. This novel extension involves several new steps, including the formulation of a d-bar problem for a sectionally non-analytic function, i.e. for a function which has different non-analytic representations in different domains of the complex plane. This, in addition to the computation of a d-bar derivative, also requires the computation of the relevant jumps across the different domains. This latter step has certain similarities (but is more complicated) with the corresponding step for those initial-value problems in two dimensions which can be solved via a non-local RH problem, like KPI. 1. Introduction The most well known integrable nonlinear equations in 2 + 1, i.e. in two spatial and one temporal dimensions, are the Davey-Stewartson (DS) and the Kadomtsev-Petviashvili (KP) equations. These equations are physically significant integrable generalizations of the celebrated nonlinear Schrödinger and Korteweg-deVries equations respectively. The Cauchy problem of the DS and KP equations can be analyzed using either a non-local
958
A. S. Fokas
Riemann-Hilbert formalism [1,2], or a d-bar formalism [3,4] (rigorous results can be found in [5–7]). This should be contrasted with the analysis of the Cauchy problem of integrable nonlinear evolution equations in 1 + 1, i.e. in one spatial and one temporal dimensions, which can be analyzed using a local Riemann-Hilbert formalism. A method for analyzing initial-boundary value problems for linear and for integrable nonlinear PDEs in two dimensions was introduced in [8] and further developed by several researchers, see for example [9–17] and the monograph [18]. For equations in 1 + 1 formulated either on the half-line or on the internal, this method yields a local Riemann-Hilbert problem which is formulated on a more complicated contour than the analogous Riemann-Hilbert (RH) problem associated with initial-value problems. In this paper, we extend the method of [8] to three-dimensions and in particular we implement it to the linear equation iqt + qzz + qz¯ z¯ = 0, z = x + i y, x, y ∈ R, t > 0,
(1.1)
as well as the DS system, iqt + qzz + qz¯ z¯ + 4( f + f¯)q = 0, 2 f z¯ − (|q|2 )z = 0.
(1.2)
For simplicity we consider Eq. (1.1) and (1.2) in the half-plane, i.e. in the domain defined by = {0 < t < T, −∞ < x < ∞, 0 < y < ∞}, T > 0.
(1.3)
For linear equations in 1 + 1, the novel integral representation obtained by the new method can be constructed by any one of the following methods: (a) Performing the simultaneous spectral analysis of the two eigenvalue equations forming the associated Lax pair. (b) Applying the usual Fourier transform and then using appropriate contour deformations [19]. (c) Employing the classical Green’s representation and then performing an explicit integration [20]. However, among these approaches, it is only the approach (a) that can be extended to integrable nonlinear PDEs. For linear equations in 2 + 1 (like Eq. (1.1)) the situation is similar. Indeed, the approach (b) above has already been implemented in [19]. Taking into consideration that it is only the approach (a) that can be nonlinearized, it is useful, at least from a didactic point of view, to analyze Eq. (1.1) by performing the simultaneous spectral analysis of the associated Lax pair. This is implemented in Sect. 2 using the fact that Eq. (1.1) is the compatibility condition of the following pair of eigenvalue equations for the scalar function µ(x, y, t, k1 , k2 ): µz¯ + ikµ = −q,
(1.4a)
µt − iµzz + ik µ = −(iqz¯ + kq), k = k1 + ik2 , k1 , k2 ∈ R.
(1.4b)
2
It turns out that an integral representation for q can be obtained by formulating a d-bar problem for a sectionally defined non-analytic function (the jumps along the contours defining the different sectors give rise to non-local contributions similar to those arising in non-local RH problems). In Sect. 3 we analyze the DS system (1.2) in the domain (see Eq. (1.3)) by performing the simultaneous spectral analysis of the following pair of eigenvalue equations
Davey-Stewartson Equation on the Half-Plane
959
for the vector function (µ1 (x, y, t, k1 , k2 ), µ2 (x, y, t, k1 , k2 )): µ1z¯ = −qµ2 , µ2z + ikµ2 = qµ ¯ 1, iµ1t + µ1zz + 2ikµ1z + qµ2z¯ − qz¯ µ2 + 4 f µ1 = 0, ¯ 1z + q¯ z µ1 + 4 f¯µ2 = 0. −iµ2t + µ2z¯ z¯ − ikµ2z − qµ
(1.5a) (1.5b) (1.5c) (1.5d)
The spectral analysis yields the solution q(x, y, t) in terms of a vector-valued sectionally defined non-analytic function satisfying a d-bar problem. Notations, Assumptions and the Pompieu Formula. • The complex variables k, z, ζ will denote the following: k = k1 + ik2 , z = x + i y, ζ = ξ + iη,
(1.6)
where k1 , k2 , x, y, ξ, η are real variables. • A bar on top of a complex variable will denote the complex conjugate of this variable, in particular, k¯ = k1 − ik2 . • C+ and C− will denote the upper and lower half of the complex extension of the ¯ + and C ¯ − will denote the closure of C+ and C− . k2 -plane; C • Let f (x, y) be a smooth function defined for (x, y) ∈ D ⊂ R2 , where D is a piecewise smooth domain. Then Pompieu’s (or d-bar, or Cauchy-Green) formula implies that f admits the following representations: dξ dη ∂ f dζ 1 1 f (ξ, η) (1.7a) (ξ, η) + f (x, y) = − π 2iπ ∂ D ζ − z D ζ − z ∂ζ and 1 f (x, y) = − π
D
1 dξ dη ∂ f (ξ, η) − ¯ζ − z¯ ∂ζ 2iπ
d ζ¯ ∂D
ζ − z¯
f (ξ, η).
(1.7b)
• Subscript “o” will denote evaluation at t = 0 (with the exception of g0 (x, t), see Eq. (1.10) below) and superscript “∧” will denote evaluation at k2 = 0. • e will denote the exponential e(k1 , k2 , x, y, t) = e−2ik1 x+2ik2 y+2i(k1 −k2 )t . 2
2
(1.8)
• q0 (x, y) and {g0 (x, t), g1 (x, t)} will denote the relevant initial and boundary values, namely q0 (x, y) = q(x, y, 0), x ∈ R, 0 < y < ∞.
(1.9)
g0 (x, t) = q(x, 0, t), g1 (x, t) = q y (x, 0, t), x ∈ R, 0 < t < T.
(1.10)
We will assume that q0 ∈ S(R × R+ ), where S denotes the space of Schwartz functions. • Throughout this paper we will assume that there exists a solution q(x, y, t) with ¯ which denotes the closure of the domain . sufficient smoothness and decay in , • The superscript + and − will denote functions defined in the upper and lower half complex k2 -plane. We emphasize that these functions are not necessarily analytic.
960
A. S. Fokas
Imk 2
Rek 2
Fig. 1. The contour ∂ D in the complex k2 -plane
2. The Linearized DS Under the a-priori assumption of existence we will show that the solution q of Eq. (1.1) admits the representation q(x, y, t) =
1 2 2 dk1 dk2 e−2i(k1 x+k2 y)−2i(k1 −k2 )t qˆ0 (k1 , k2 ) 2 π R2 ∞ 1 2 2 dk1 dk2 e−2i(k1 x+k2 y)−2i(k1 −k2 )t g(k ˜ 1 , k2 ), (x, y, t) ∈ , − 2 2iπ −∞ ∂D (2.1)
where the functions qˆ0 , g˜ and the contour ∂ D are defined as follows: qˆ0 (k1 , k2 ) = g(k ˜ 1 , k2 ) =
∞
−∞ ∞ −∞
∞
dx
¯ +, dye2i(k1 x+k2 y) q0 (x, y), k1 ∈ R, k2 ∈ C
(2.2a)
0
dx
T
dτ e2ik1 x+2i(k1 −k2 )τ 2
2
0
¯ × [g1 (x, τ ) − 2ik2 g0 (x, τ )] , k1 ∈ R, k2 ∈ C,
(2.2b)
∂ D is the boundary of the fourth quadrant D of the complex k2 -plane with the orientation that the direction is positive to the left of D, see Fig. 1.
2.1. Preliminary considerations. 1. Equation (1.1) admits the Lax pair formulation (1.4). Indeed, Eq. (1.4) are compatible if and only if ∂t + ik 2 − i∂z2 q − (∂z¯ + ik) (iqz¯ + kq) = 0. Simplifying this equation we find Eq. (1.1).
Davey-Stewartson Equation on the Half-Plane
961
2. The Lax pair (1.4) is equivalent to Eq. (1.4a) supplemented with the equation µt − iµx x + 2kµx + 2ik 2 µ = iqx + q y − 2kq.
(2.3)
Indeed, using the identities 1 1 (∂x − i∂ y ), ∂z¯ = (∂x + i∂ y ), 2 2 Eq. (1.4a) can be rewritten in the form ∂z =
(2.4)
iµ y = −µx − 2ikµ − 2q. Hence, iµ yy = −µx y − 2ikµ y − 2q y = i∂x [−µx − 2ikµ − 2q] − 2k[−µx − 2ikµ − 2q] − 2q y . Substituting the above expressions for iµ y and iµ yy in Eq. (1.4b), i.e. in equation i −µx x + µ yy + 2iµx y = − (iqz¯ + kq), 4 this equation becomes Eq. (2.3). 3. Suppose that Eq. (1.1) is valid for (x, y) ∈ 0 ⊂ R2 , where 0 is a bounded, piecewise smooth domain. Then the following identity holds: 2 2 −2i d xd yei(λz+k z¯ )+i(k +λ )t q(x, y, t) = 0 t i(λz+k z¯ )+i(k 2 +λ2 )t (iqz + λq)d z¯ − (iqz¯ + kq)dz , λ, k ∈ C. (2.5) e µt + ik 2 µ +
∂0
Indeed, it is straightforward to verify that Eq. (1.1) admits the following divergence form formulation: ˘ t = [ E(iq ˘ z + λq)]z + [ E(iq ˘ z¯ + kq)]z¯ , ( Eq)
(2.6a)
2 2 E˘ = ei(λz+k z¯ )+i(k +λ )t .
(2.6b)
where
The Poincaré lemma
0
dW =
∂0
for the 1-form W = Adz + Bd z¯ , yields (Bz − A z¯ )dz ∧ d z¯ = 0
W,
∂0
(Adz + Bd z¯ ),
where dz ∧ d z¯ = −2idxdy. Choosing ˘ z¯ + kq), A = − E(iq
˘ z + λq) B = E(iq
˘ t , we find Eq. (2.5). and replacing Bz − A z¯ by ( Eq)
962
A. S. Fokas
2.2. The direct problem. The solution of the so-called direct problem [18] involves constructing a function µ (in terms of q and its derivatives), which is bounded for all {k ∈ C, (x, y, t) ∈ }. By performing the simultaneous spectral analysis of Eq. (1.4a) and (2.3), we will show that such a solution is given by ⎡ µ+ (x, y, t, k1 , k2 ), k2 ≥ 0 µ(x, y, t, k1 , k2 ) = ⎣ (2.7) µ− (x, y, t, k1 , k2 ), k2 ≤ 0, where k1 ∈ R, (x, y, t) ∈ and µ± are defined as follows: ∞ 1 ∞ q(ξ, η, t) ik(ζ¯ −¯z )+i k(ζ ¯ −z) e dξ dη µ± = π −∞ ζ − z 0 ∞ e−2ky ∞ + dl dξ 2π −2k1 −∞ ⎡ T − t dτ ⎢ 2 2 ⎢ ×⎣ {g1 (ξ, τ ) − (l + 2k)g0 (ξ, τ )}eil(z−ξ )+i(l +2kl+2k )(τ −t) .
t 0 dτ ((2.8)± ) Before deriving the above expressions we first note that the functions µ± are well defined. Indeed, for the function µ+ the parameter k appears in the following exponentials: ¯
¯
eik(ζ −¯z )+i k(ζ −z) , e2i(kl+k
2 )(τ −t)
, e−2ky .
The real part of the first exponential vanishes; the real part of the second exponential equals exp[−2k2 (l + 2k1 )(τ − t)], which is bounded since l + 2k1 ≥ 0 and τ − t ≥ 0; the real part of the third exponential equals exp[−2k1 y], which decays as k1 → ∞ (as k1 → −∞ the integral with respect to dl vanishes). In order to derive Eq. (2.7) and (2.8)± we start with Eq. (1.4a), which we rewrite in the form (eik z¯ µ)z¯ = −eik z¯ q. ¯ (which commutes with ∂ z¯ ) and then we use We multiply this equation with exp[i kz] Pompieu’s formula (1.7a) with D = {−∞ < x < ∞, 0 < y < ∞}. This yields ∞ 1 ϕ(ξ, t, k1 , k2 ) 2ik1 ξ −i(k z¯ +kz) ¯ µ = µ0 + e dξ , k1 , k2 ∈ R, (x, y, t) ∈ , 2πi −∞ ξ −z (2.9) where µ0 denotes the first term in the rhs of equations (2.8)± and ϕ(x, t, k1 , k2 ) = µ(x, 0, t, k1 , k2 ), x ∈ R, 0 < t < T, k1 , k2 ∈ R.
(2.10)
We will determine ϕ by demanding that it satisfies Eq. (2.3) evaluated at y = 0, i.e. equation ϕt − iϕx x + 2kϕx + 2ik 2 ϕ = Q,
(2.11a)
Davey-Stewartson Equation on the Half-Plane
963
where Q(x, t, k) = ig0x (x, t) − 2kg0 (x, t) + g1 (x, t).
(2.11b)
In order to perform the spectral analysis of Eq. (2.11a) we let ∞ 1 ϕ(x, t, k1 , k2 ) = dleilx ϕ(l, ˆ t, k1 , k2 ), (k1 , k2 ) ∈ R2 , x ∈ R, 0 < t < T. 2π −∞ (2.12) The function ϕˆ satisfies
ϕˆ t + i(l + 2kl + 2k )ϕˆ = 2
2
∞
−∞
dξ e−ilξ Q(ξ, t, k), l ∈ R.
(2.13)
Using the representation (2.12) in the second term of the rhs of Eq. (2.9) we find that this term can be simplified as follows: ∞ ∞ 1 1 −i(k z¯ +kz) ei(2k1 +l)ξ ¯ e dl ϕ(l, ˆ t, k1 , k2 ) dξ 2iπ 2π ξ −z −∞ −∞ ∞ −2ky e = dleilz ϕ(l, ˆ t, k1 , k2 ), 2π −2k1 where we have used Cauchy’s theorem in the complex extension of the ξ -plane to compute explicitly the integral with respect to dξ ; if l + 2k1 > 0 we apply Cauchy’s theorem in the upper half complex ξ -plane and we find a term proportional to exp[i(2k1 + l)z], whereas if 2k1 + l < 0, by employing Cauchy’s theorem in the lower half complex ξ -plane we find that the integral vanishes. Hence, Eq. (2.9) becomes e−2ky ∞ µ = µ0 + dleilz ϕ(l, ˆ t, k1 , k2 ). (2.14) 2π −2k1 Evaluating this equation at y = 0 and using the identity −2k1 ¯ ¯ ei(k ζ +kζ )−2ik1 x 2kη = −ie dleil(x−ζ ) , ζ = ξ + iη, η > 0, ζ −x −∞ Eq. (2.14) yields ∞ −2k1 ∞ i dl dξ dηq(ξ, η, t)eil(x−ζ )+2kη π −∞ −∞ 0 ∞ 1 + dleilx ϕ(l, ˆ t, k1 , k2 ). 2π −2k1
ϕ(x, t, k1 , k2 ) = −
(2.15)
We note that the real part of the exponential appearing in the double integral of the rhs of Eq. (2.15) equals exp[(2k1 + l)η], thus this exponential is bounded. Comparing Eq. (2.15) with the rhs of Eq. (2.12) it follows that Eq. (2.9) does not impose any restriction on ϕˆ for −2k1 < l < ∞, where the value of ϕˆ for −∞ < l < −2k1 is fixed by q, namely ∞ ∞ ϕ(l, ˆ t, k1 , k2 ) = −2i dξ dηq(ξ, η, t)e−ilζ +2kη , −∞ < l < −2k1 . −∞
0
(2.16)
964
A. S. Fokas
We will now use Eqs. (2.13) and (2.16) to determine ϕ. ˆ The integration of Eq. (2.13) gives rise to the exponential E(k, l, τ − t) = ei(l
2 +2kl+2k 2 )(τ −t)
,
(2.17)
whose real part equals exp[−2k2 (l + 2k1 )(τ − t)]. This implies that we must distinguish two different cases, namely k2 ≥ 0 and k2 ≤ 0. (a) ϕˆ + (l, t, k1 , k2 ). We denote by ϕˆ + a solution of Eq. (2.13) valid for k2 ≥ 0. Integrating Eq. (2.13) from t = 0 we find t ∞ + ϕˆ (l, t, k1 , k2 ) = dτ dξ e−ilξ E(k, l, τ − t)U (ξ, τ, k, l) 0
−∞
+E(k, l, −t)ϕˆ + (l, 0, k1 , k2 ),
(2.18)
where we have used integration by parts with respect to dξ to replace Q by U , which is defined by U (x, t, k, l) = g1 (x, t) − (l + 2k)g0 (x, t).
(2.19)
In the two exponentials E appearing in Eq. (2.18), k2 ≥ 0 and τ − t ≤ 0, −t ≤ 0; thus these exponentials are bounded for l + 2k1 ≤ 0. Hence, computing ϕˆ + (l, 0, k1 , k2 ) by evaluating Eq. (2.16) at t = 0, Eq. (2.18) becomes t ∞ ϕˆ + = dτ dξ e−ilξ E(k, l, τ − t)U (ξ, τ, k, l) 0 −∞ ∞ ∞ −2i E(k, l, −t) dξ dηq0 (ξ, η)e−ilζ +2kη , −∞ < l < −2k1 . 0
0
(2.20a) For −2k1 < l < ∞, the boundness of E requires τ − t ≥ 0, thus we integrate Eq. (2.13) from t = T and we choose ϕˆ + (l, T, k1 , k2 ) = 0. This yields T ∞ ϕˆ + = − dτ dξ e−ilξ E(k, l, τ − t)U (ξ, τ, k, l), −2k1 < l < ∞. (2.20b) t
−∞
Substituting this expression in the rhs of Eq. (2.14) we find Eq. (2.8)± . (b) ϕˆ − (l, t, k1 , k2 ). We denote by ϕˆ − a solution of Eq. (2.13) valid for k2 ≤ 0. Integrating Eq. (2.13) from t = T we find T ∞ − ϕˆ = − dτ dξ e−ilξ E(k, l, τ −t)U (ξ, τ, k, l)+ E(k, l, T −t)ϕˆ − (l, T, k1 , k2 ). t
−∞
Both exponentials E appearing in the rhs of this equation are bounded for l +2k1 ≤ 0. Hence, computing ϕˆ − (l, T, k1 , k2 ) by evaluating Eq. (2.16) at t = T , the above equation yields T ∞ ϕˆ − = − dτ dξ e−ilξ E(k, l, τ − t)U (ξ, τ, k, l) t −∞ ∞ ∞ −2i E(k, l, T − t) dξ dηq(ξ, η, T )e−ilζ +2kη , −∞ < l < −2k1 . −∞
0
(2.21a)
Davey-Stewartson Equation on the Half-Plane
965
For −2k1 < l < ∞, the boundness of E requires τ − t ≤ 0, thus we integrate Eq. (2.13) from t = 0 and we choose ϕˆ − (l, 0, k1 , k2 ) = 0. This yields t ∞ ϕˆ − = dτ dξ e−ilξ E(k, l, τ − t)U (ξ, τ, k, l), −2k1 < l < ∞. (2.21b) −∞
0
Substituting this expression in the rhs of Eq. (2.14) we find Eq. (2.8)± . Remark 2.1. Comparing Eq. (2.20a) with the Fourier transform of Eq. (2.15) we find ∞ ∞ −2i dξ dηq(ξ, η, t)e−ilζ +2kη −∞ 0 ∞ ∞ 2 2 = −2i dξ dηq0 (ξ, η)e−ilζ +2kη−i(l +2kl+2k )t −∞ 0 t ∞ 2 2 + dτ dξ [g1 (ξ, τ ) − (l + 2k)g0 (ξ, τ )]e−ilξ +i(l +2kl+2k )(τ −t) , 0
−∞
−∞ < l < −2k1 .
(2.22)
This identity can be verified directly: Equation (2.5) with λ = −(k + l) and 0 the upper half complex z-plane yields ∞ ∞ −ilζ +2kη+i(l 2 +2kl+2k 2 )t −2i dξ dηq(ξ, η, t)e −∞ 0 t ∞ 2 2 = dξ [g1 (ξ, t) − (l + 2k)g0 (ξ, t)] e−ilξ +i(l +2kl+2k )t . (2.23) −∞
Integrating this equation from t = 0 we find Eq. (2.22). Similarly, integrating Eq. (2.23) from t = T we find the identity obtained by comparing Eq. (2.21a) with the Fourier transform of Eq. (2.15). 2.3. The inverse problem. The solution of the so-called Inverse Problem [18] involves expressing µ in terms of {qˆ0 , g}. ˜ Using the fact that the function µ given by Eqs. (2.7) and (2.8)± is well defined for all k ∈ C it follows that µ admits the representation ∞ dk1 1 (µ+ − µ− )(x, y, t, k1 , 0) µ(x, y, t, k1 , k2 ) = 2iπ −∞ k1 − k ∞ dk2 ∂µ+ 1 ∞ dk1 − (x, y, t, k1 , k2 ) π −∞ k − k 0 ∂k 0 dk2 ∂µ− (2.24) + (x, y, t, k1 , k2 ) , k2 = 0, −∞ k − k ∂k where k1 ∈ R and (x, y, t) ∈ . In what follows we will compute ∂µ± /∂ k¯ and µ+ − µ− . Computing the ∂ k¯ derivative of Eq. (2.8)± we find t ∞ i ∂µ− 1 2 2 ¯ q(k ˆ 1 , k2 , t)+ dξ dτ U˜ (ξ, τ, k2 )e2ik1 ξ +2i(k1 −k2 )(τ −t) , = e−i(k z¯ +kz) π 2π −∞ ∂ k¯ 0 (2.25)
966
A. S. Fokas
where
q(k ˆ 1 , k2 , t) =
∞
−∞
∞
dξ
¯ ¯
dηq(ξ, η, t)ei(k ζ +kζ )
(2.26)
0
and U˜ (x, t, k2 ) = g1 (x, t) − 2ik2 g0 (x, t).
(2.27)
Equation (2.5) with λ = k¯ and 0 the upper half complex z-plane yields ∞ 2 2 2 2 − 2i e2i(k1 −k2 )t q(k ˆ 1 , k2 , t) = U˜ (x, t, k2 )e2ik1 x+2i(k1 −k2 )t d x. t
−∞
(2.28)
Integrating this equation with respect to t it follows that the curly bracket appearing in the rhs of Eq. (2.25) can be expressed in terms of qˆ0 and hence ∂µ− ∂k
=
i e(k ˘ 1 , k2 , x, y, t)qˆ0 (k1 , k2 ), k1 ∈ R, k2 ≤ 0, (x, y, t) ∈ , (2.29) π
where e(k ˘ 1 , k2 , x, y, t) = e−2i(k1 x+k2 y)−2i(k1 −k2 )t . 2
2
(2.30)
The ∂ k¯ derivative of Eqs. (2.8)± implies that ∂µ+ /∂ k¯ satisfies an equation similar with
t
T Eq. (2.25) where 0 is replaced by − t . Hence, using T t T − = − , (2.31) t
it follows that ∂µ+ ∂k
0
0
= e(k ˘ 1 , k2 , x, y, t)
i 1 qˆ0 (k1 , k2 ) − g(k ˜ 1 , k2 ) , k1 ∈ R, k2 ≤ 0, (x, y, t) ∈ . π 2π (2.32)
Let
µ(x, y, t, k1 ) = (µ+ − µ− )(x, y, t, k1 , 0). Letting k2 = 0 in Eqs. (2.8)± and then subtracting the resulting equations it follows that ∞ T ∞ 1 2 2
µ = − dl dξ dτ U (ξ, τ, k, l)e−2k1 y+il(z−ξ )+i(l +2k1 l+2k1 )(τ −t) , 2π −2k1 −∞ 0 where U is defined in (2.19). Letting l = λ − 2k1 , the above expression becomes ∞ ∞ T 1 2 2 ( µ)=− dλ dξ dτ U˜ (ξ, τ, λ)e−2ik1 x+iλz−i(λ−2k1 )ξ +i(λ +2k1 −2k1 λ)(τ −t) , 2π 0 −∞ 0 where U˜ is defined in (2.27). The exponential appearing in the above expression can be rewritten in the form e
λ 2 iλ 2 −2i(k1 − λ2 )(x−ξ )−2i(− iλ 2 )y+2i (k1 − 2 ) −(− 2 ) (τ −t)
.
Davey-Stewartson Equation on the Half-Plane
967
Hence, using the substitutions k1 −
λ iλ = kˆ1 , − = kˆ2 , 2 2
we find ( µ)(x, y, t, kˆ1 + i kˆ2 ) =
i π
0 −i∞
d kˆ2 e( ˘ kˆ1 , kˆ2 , x, y, t)g( ˜ kˆ1 , kˆ2 ).
The first term of the rhs of Eq. (2.24) can be written in the form ∞ d kˆ1 1 ˆ ( µ)(x, y, t, k), kˆ = kˆ1 + i kˆ2 . 2iπ −∞ kˆ − k
(2.33)
(2.34)
¯ ∂µ+ /∂ k, ¯ µ+ − µ− are given by Eqs. (2.29), (2.32), (2.33). In summary: ∂µ− /∂ k, Substituting these equations in Eq. (2.24) we find 0 ∞ 1 g( ˜ kˆ1 , kˆ2 ) ˆ1 µ= d k d kˆ2 e( ˘ kˆ1 , kˆ2 , x, y, t) 2 2π kˆ − k −∞ −i∞ ∞ ∞ g(k ˜ 1 , k2 ) dk1 dk2 e(k ˘ 1 , k2 , x, y, t) + k −k −∞ 0 ∞ ∞ qˆ0 (k , k ) i (2.35) − 2 dk1 dk2 e(k ˘ 1 , k2 , x, y, t) 1 2 . π −∞ k −k −∞ Substituting this expression in Eq. (1.4a) we find Eq. (2.1). 3. The DS System Under the a-priori assumption of existence we will show that the solution of the DS system (1.2) can be expressed through a d-bar formalism involving sectionally defined non-analytic functions. 3.1. Preliminary considerations. 1. The DS system admits the Lax pair formulation (1.5). Indeed, it is straightforward to verify that the DS system (1.2) is the compatibility condition of the following equations for the vector-valued function ( 1 (x, y, t), 2 (x, y, t)):
i 1t + 1zz −i 2t + 2z¯z¯
1z¯ 2z + q 2z¯ − qz¯ 2 + 4 f 1 − q ¯ 1z + q¯ z 1 + 4 f¯ 2
= −q 2 , = q ¯ 1, = 0, = 0.
Making the substitutions 2
j = µ j eikz−ik t ,
j = 1, 2,
it follows that the vector-valued function (µ1 (x, y, t, k1 k2 ), µ2 (x, y, t, k1 , k2 )) satisfies Eqs. (1.5).
968
A. S. Fokas
2. Equations (1.5) are equivalent with Eqs. (1.5a) and (1.5b) supplemented with the following two equations: iµ1t + µ1x x + 2ikµ1x + 2qµ2x + 2ikqµ2 + (qx − iq y )µ2 + 4 f µ1 = 0, (3.1a) 2 −iµ2t + µ2x x + 2ikµ2x − 2k µ2 − 2qµ ¯ 1x − (q¯ x + i q¯ y + 2ik q)µ ¯ 1 + 4 f¯µ2 = 0. (3.1b) Indeed, Eqs. (1.5a) and (1.5b) imply µ1 y = iµ1x + 2iqµ2 , µ2 y = −iµ2x + 2kµ2 + 2i qµ ¯ 1. The derivatives with respect to z and z¯ appearing in Eqs. (1.5c) and (1.5d) can be expressed in terms of derivatives with respect to x and y (see the identities (2.4)); using the above expressions for µ1 y and µ2 y to eliminate the derivatives with respect to y, Eqs. (1.5c) and (1.5d) become Eqs. (3.1). 3. Suppose that the DS system (1.2) is valid for (x, y) ∈ 0 ⊂ R2 and let (µ1 , µ2 ) satisfy Eqs. (1.5). Then, the following identity holds for all k ∈ C: i(kz+k¯ z¯ )−i(k 2 +k¯ 2 )t 2 d xd y(qµ ¯ 1 )e 0 t [(q¯ z − ik q)µ = ¯ 1 − qµ ¯ 1z + 4 f¯µ2 ]d z¯ − [(q¯ z¯ − i k¯ q)µ ¯ 1 + qµ ¯ 1z¯ ]dz ∂0
¯
×ei(kz+k z¯ )−i(k
2 +k¯ 2 )t
.
(3.2a)
Indeed, this equation is the direct consequence of the following identity (see the analogous discussion in Sect. 2): i(qµ ¯ 1 e) ¯ t = [(q¯ z − ik q)µ ¯ 1 − qµ ¯ 1z + 4 f¯µ2 ]e¯ z + [(q¯ z¯ − i k¯ q)µ ¯ 1 + qµ ¯ 1z¯ ]e¯ z¯ , (3.2b) where ¯
e = e−i(kz+k z¯ )+i(k
2 +k¯ 2 )t
= e−2i(k1 x−k2 y)+2i(k1 −k2 )t . 2
2
In order to derive Eq. (3.2b), we note that i(qµ ¯ 1 e) ¯ t = (i q¯t )µ1 e¯ + q(iµ ¯ ¯ 1 (−i)(k 2 + k¯ 2 )e¯ 1t )e¯ + i qµ ¯ = eµ ¯ 1 [q¯z¯ z¯ + q¯ zz + 4 f q¯ + 4 f¯q]
+ e¯q[−µ ¯ ¯ 1 (−i)(k 2 + k¯ 2 )e, ¯ 1zz − 2ikµ1z − qµ2z¯ + qz¯ µ2 − 4 f µ1 ] + i qµ
where we have used Eqs. (1.2a) and (1.5c) to replace q¯t and µ1t . Using the equations eµ ¯ 1 q¯z¯ z¯ = (eµ ¯ 1 q¯ z¯ )z¯ − eµ ¯ 1z¯ q¯ z¯ − i k¯ e¯q¯ z¯ µ1 , eµ ¯ 1 q¯ zz = (eµ ¯ 1 q¯ z )z − e¯q¯ z µ1z − ik e¯q¯ z µ1 , ¯ 4 f eµ ¯ 1 q¯ = 4e¯ f¯(µ2z + ikµ2 ) = (4e¯ f¯µ2 )z − 4e¯ f¯z µ2 , −e¯qµ ¯ 1zz = −(e¯qµ ¯ 1z )z + e¯q¯ z µ1z + ik e¯qµ ¯ 1z ,
Davey-Stewartson Equation on the Half-Plane
969
replacing −eµ ¯ 1z¯ q¯ z¯ by eq ¯ q¯ z¯ µ2 and 4e¯ f¯z µ2 by 2e|q| ¯ 2z¯ µ2 , it follows that the only terms in the equation for i(qµ ¯ 1 e) ¯ t which are not total derivatives with respect to either z or z¯ are the terms ¯ e¯q) −e(q ¯ qµ ¯ 2 )z¯ − i k( ¯ z¯ µ1 . By adding to these two terms the term −i k¯ e¯ q(qµ ¯ 2 + µ1z¯ ) (which vanishes due to Eq. (1.5a)), the above two terms also yield a total derivative with respect to z¯ . Collecting the derivatives with respect to ∂z and to ∂z¯ and replacing −e|q| ¯ 2 µ2 by e¯qµ ¯ 1z¯ , we find Eq. (3.2b). 3.2. The direct problem. Proposition 3.1. Under the a-priori assumption of the existence of solutions of the DS system, Eqs. (1.5) admit a solution which is bounded for all {k ∈ C, (x, y, t) ∈ }. This vector-valued solution is given by ⎡ (µ+ (x, y, t, k1 , k2 ), µ+2 (x, y, t, k1 , k2 )), k2 ≥ 0, k1 ∈ R, (µ1 , µ2 ) = ⎣ 1 (3.3) − (µ− 1 (x, y, t, k1 , k2 ), µ2 (x, y, t, k1 , k2 )), k2 ≤ 0, k1 ∈ R, where the functions µ±j , j = 1, 2, are defined as follows: µ± 1
∞
∞
q(ξ, η, t) ± µ2 (ξ, η, t, k1 , k2 ) dη ζ −z −∞ 0 ⎡ ∞ ∞ T − dτ 1 {U1 (ξ, τ, k + l)ϕ2± (ξ, τ, k1 , k2 ) dl dξ ⎣ t t + 2iπ 0 −∞ dτ 0
1 = 1+ π
dξ
−4 f (ξ, 0, τ )ϕ1± (ξ, τ, k1 , k2 )}E 1 (k, l, τ − t, z − ξ ) and µ± 2 =−
1 π
∞
∞
q(ξ, ¯ η, t)
((3.4)± )
¯ ¯
ik(ζ −z)+i k(ζ −¯z ) µ± 1 (ξ, η, t, k1 , k2 )e ζ − z¯ ⎡ ∞ T − dτ e2ky −2k1 − dl dξ ⎣ t t {U2 (ξ, τ, k + l)ϕ1± (ξ, τ, k1 , k2 ) 2iπ −∞ −∞ dτ 0 −∞
dξ
dη
0
−4 f¯(ξ, 0, τ )ϕ2± (ξ, τ, k1 , k2 )}E 2 (k, l, τ − t, z¯ − ξ ),
((3.5)± )
with E 1 (k, l, τ, z) = eilz+i(l +2kl)τ , E 2 (k, l, τ, z¯ ) = eil z¯ −i(l +2kl+2k U1 (ξ, τ, k) = g0ξ (ξ, τ ) + ig1 (ξ, τ ) − 2ikg0 (ξ, τ ), U2 (ξ, τ, k) = −g¯ 0ξ (ξ, τ ) + i g¯ 1 (ξ, τ ) + 2ik g¯ 0 (ξ, τ ) 2
2
2 )τ
,
(3.6) (3.7)
and ± ϕ± j (x, t, k1 , k2 ) = µ j (x, 0, t, k1 , k2 ),
j = 1, 2.
(3.8)
970
A. S. Fokas
Proof. We begin with Eqs. (1.5a), (1.5b) and we look for a vector-valued solution such that 1 , z → ∞. (µ1 , µ2 ) ∼ 1, O z Thus, 1 µ1 −1= π
∞ −∞
∞
dξ
and
0
dη(qµ2 ) 1 (ξ, η, t, k1 , k2 )+ ζ −z 2iπ
∞ −∞
dξ [ϕ1 (ξ, t, k1 , k2 ) − 1] ξ −z (3.9)
∞ dη(qµ ¯ 1) 1 ∞ ¯ ¯ (ξ, η, t, k1 , k2 )eik(ζ −z)+i k(ζ −¯z ) dξ π −∞ ζ − z¯ 0 ∞ dξ ϕ2 (ξ, t, k1 , k2 ) 2ik1 ξ −i(kz+k¯ z¯ ) 1 − . e 2iπ −∞ ξ − z¯
µ2 = −
(3.10)
The functions ϕ j satisfy Eqs. (3.1) evaluated at y = 0, i.e. equations iϕ1t + ϕ1x x + 2ikϕ1x = Q 1
(3.11a)
− iϕ2t + ϕ2x x + 2ikϕ2x − 2k 2 ϕ2 = Q 2 ,
(3.12a)
Q 1 = − 2g0 ϕ2x + 2ikg0 ϕ2 + (g0x − ig1 )ϕ2 + 4 f (x, 0, t)ϕ1
(3.11b)
Q 2 = 2g¯ 0 ϕ1x + g¯ 0x + i g¯ 1 + 2ik g¯ 0 ϕ1 − 4 f¯(x, 0, t)ϕ2 .
(3.12b)
and
where
and
The Spectral Analysis of the Equation for ϕ1 . Letting ∞ 1 ϕ1 (ξ, t, k1 , k2 ) − 1 = dleilξ ϕˆ1 (l, t, k1 , k2 ), k1 , k2 ∈ R, 2π −∞
(3.13)
Eq. (3.11a) yields
ϕˆ1 ei(l
2 +2kl)t
t
= −i
∞ −∞
dξ Q 1 (ξ, τ, k1 , k2 )e−ilξ +i(l
2 +2kl)t
.
(3.14)
Using the representation (3.13) in the second term of the rhs of Eq. (3.9) we find that this term can be simplified as follows: 1 1 2iπ 2π
∞
−∞
dl ϕˆ1 (l, t, k1 , k2 )
∞ −∞
dξ
1 eilξ = ξ −z 2π
∞
dleilz ϕˆ1 (l, t, k1 , k2 ).
0
(3.15)
Davey-Stewartson Equation on the Half-Plane
971
Replacing the second term of the rhs of Eq. (3.9) by the rhs of Eq. (3.15), evaluating the resulting equation at y = 0, and using the identity 0 1 dleil(x−ζ ) , ζ = ξ + iη, η > 0, = −i ζ −x −∞ it follows that Eq. (3.9) evaluated at y = 0 can be rewritten as follows: ∞ ∞ 0 i ϕ1 = 1 − dl dξ dη(qµ2 )(ξ, η, t, k1 , k2 )eil(x−ζ ) π −∞ −∞ 0 ∞ 1 ilx + dle ϕˆ 1 (l, t, k1 , k2 ). 2π 0
(3.16)
Comparing the Fourier transform of this equation with the definition (3.13) it follows that Eq. (3.9) does not impose any restriction on ϕˆ1 for 0 < l < ∞, whereas ∞ ∞ ϕˆ1 (l, t, k1 , k2 ) = −2i dξ dη(qµ2 )(ξ, η, t, k1 , k2 )e−ilζ , −∞ < l < 0. −∞
0
(3.17) In what follows we will determine ϕˆ1 by employing Eqs. (3.14) and (3.17). The former equation gives rise to the exponential exp[2ilk(τ − t)], whose real part equals exp[−2k2 l(τ − t)]. This implies that we should distinguish the cases of k2 ≥ 0 and k2 ≤ 0. (a) ϕˆ1+ (l, t, k1 , k2 ). We denote by ϕˆ1+ a solution of Eq. (3.14) valid for k2 ≥ 0. For −∞ < l < 0 we integrate Eq. (3.14) from t = 0 and we compute ϕˆ1+ (l, 0, k1 , k2 ) by evaluating Eq. (3.17) at t = 0. For 0 < l < ∞ we integrate Eq. (3.14) from t = T and we choose ϕˆ 1 (l, T, k1 , k2 ) = 0. This implies the following: t ∞ 2 dτ dξ G +1 (ξ, τ, l, k1 , k2 )e−ilξ +i(l +2kl)(τ −t) ϕˆ 1+ = −i 0 −∞ ∞ ∞ 2 −2ie−i(l +2kl)t dξ dη(qµ+2 )(ξ, η, 0, k1 , k2 )e−ilζ , −∞ < l < 0 −∞
0
(3.18a) and ϕˆ1+
=i
T
dτ t
∞ −∞
dξ G +1 (ξ, τ, l, k1 , k2 )e−ilξ +i(l
2 +2kl)(τ −t)
, 0 < l < ∞,
(3.18b)
where G +1 is defined below by Eq. (3.19)± , ± ± G± 1 (ξ, τ, l, k1 , k2 ) = U1 (ξ, τ, k + l)ϕ2 (ξ, τ, k1 , k2 ) − 4 f (ξ, 0, τ )ϕ1 (ξ, τ, k1 , k2 ), ((3.19)± )
and we have used integration by parts with respect to dξ to replace Q +1 by G +1 . We note that the real part of exp[−ilζ ] equals exp[lη] which is bounded for l < 0. Replacing the second term of the rhs of Eq. (3.9) by the rhs of Eq. (3.15) where ϕˆ 1+ is defined by (3.18b) we find (3.4)± . (b) ϕ1− (l, t, k1 , k2 ). We denote by ϕˆ1− a solution of Eq. (3.14) valid for k2 ≤ 0. For −∞ < l < 0 we integrate Eq. (3.14) from t = T and we compute ϕˆ1− (l, T, k1 , k2 ) by
972
A. S. Fokas
evaluating Eq. (3.17) at t = T . For 0 < l < ∞ we integrate Eq. (3.14) from t = 0 and we choose ϕˆ1− (l, 0, k1 , k1 ) = 0. This implies the following: ϕˆ 1−
T
∞
−ilξ +i(l +2kl)(τ −t) dξ G − 1 (ξ, τ, l, k1 , k2 )e t −∞ ∞ ∞ i(l 2 +2kl)(T −t) −ilζ −2ie dξ dη(qµ− , −∞ < l < 0 2 )(ξ, η, T, k1 , k2 )e
=i
2
dτ
−∞
0
(3.20a) and ϕˆ1−
t
= −i
dτ 0
∞ −∞
−ilξ +i(l dξ G − 1 (ξ, τ, l, k1 , k2 )e
2 +2kl)(τ −t)
, 0 < l < ∞. (3.20b)
Replacing the second term of the rhs of Eq. (3.9) by the rhs of Eq. (3.15), where ϕˆ1− is defined by (3.20b) we find (3.4)± . The Spectral Analysis of the Equation for ϕ2 . Letting ∞ 1 ϕ2 (ξ, t, k1 , k2 ) = dleilξ ϕˆ2 (l, t, k1 , k2 ), k1 , k2 ∈ R, (3.21) 2π −∞ Eq. (3.12a) yields {ϕˆ 2 e−i(l
2 +2kl+2k 2 )t
}t = i
∞
−∞
dξ Q 2 (ξ, τ, k1 , k2 )e−ilξ −i(l
2 +2kl+2k 2 )t
.
(3.22)
Using the representation (3.21) in the second term of the rhs of Eq. (3.10) we find that this term can be simplified as follows: ∞ ¯ 1 e−i(kz+k z¯ ) ∞ ei(l+2k1 )ξ − dl ϕˆ2 (l, t, k1 , k2 ) dξ 2iπ 2π ξ − z¯ −∞ −∞ −2k1 1 ¯ = dl ϕˆ2 (l, t, k1 , k2 )e−i(kz+k z¯ )+(l+2k1 )¯z . 2π −∞
(3.23)
Replacing the second term of the rhs of Eq. (3.10) by the rhs of Eq. (3.23), evaluating the resulting equation at y = 0, and using the identity ¯¯
ei(kζ +k ζ )−2ik1 x ζ −x
= ie−2kη
∞
−2k1
¯
dleil(x−ζ ) , ζ = ξ + iη, η > 0,
it follows that Eq. (3.10) evaluated at y = 0 can be written in the form ∞ ∞ ∞ i ¯ ϕ2 = − dl dξ dη(qµ ¯ 1 )(ξ, η, t, k1 , k2 )e−2kη+il(x−ζ ) π −2k1 −∞ 0 −2k1 + dleilx ϕˆ2 (l, t, k1 , k2 ). −∞
(3.24)
Davey-Stewartson Equation on the Half-Plane
973
Comparing the Fourier transform of this equation with the definition (3.21) it follows that Eq. (3.10) does not impose any restriction on ϕˆ2 for −∞ < l < −2k1 , whereas ∞ ∞ ¯ dξ dη(qµ ¯ 1 )(ξ, η, t, k1 , k2 )e−2kη−il ζ , −2k1
0
(3.25) Equation (3.22) gives rise to the exponential exp[−2i(kl + k 2 )(τ − t)], whose real part equals exp[2k2 (l + 2k1 )τ − t)]. This implies that we should distinguish the cases of k2 ≥ 0 and k2 ≤ 0. (a) ϕˆ2+ (l, t, k1 , k2 ). Proceeding as in the case for ϕˆ1+ we find the following: t ∞ 2 2 + ϕˆ2 = i dτ dξ G +2 (ξ, τ, l, k1 , k2 )e−ilξ +i(l +2kl+2k )(t−τ ) 0 −∞ ∞ ∞ 2 ¯ i(l +2kl+2k 2 )t −2ie dξ dη(qµ ¯ +1 )(ξ, η, 0, k1 , k2 )e−2kη−il ζ , −2k1
0
(3.26a) and ϕˆ 2+
T
= −i
dτ
∞ −∞
t
dξ G +2 (ξ, τ, l, k1 , k2 )e−ilξ +i(l
2 +2kl+2k 2 )(t−τ )
, −∞ < l < −2k1 , (3.27b)
where G +2 is defined below by Eqs. 3.28, ± ± ¯ G± 2 (ξ, τ, l, k1 , k2 ) = U2 (ξ, τ, k + l)ϕ1 (ξ, τ, k1 , k2 ) − 4 f (ξ, 0, τ )ϕ2 (ξ, τ, k). ((3.28)± )
Replacing the second term of the rhs of Eq. (3.10) by the rhs of Eq. (3.23), where ϕˆ 2+ is defined by (3.27b) we find (3.5)± . (b) ϕˆ2− (l, t, k1 , k2 ). Proceeding as in the case of ϕˆ1− we find the following: T ∞ −ilξ +i(l 2 +2kl+2k 2 )(t−τ ) ϕˆ2− = −i dτ dξ G − 2 (ξ, τ, l, k1 , k2 )e t −∞ ∞ ∞ i(l 2 +2kl+2k 2 )(t−T ) −2ie dξ dη(qµ ¯ − 1 )(ξ, η, T, k1 , k2 ) e and ϕˆ2− = i
t
dτ 0
−2kη−il ζ¯
∞ −∞
−∞
0
, −2k1 < l < ∞
−ilξ +i(l dξ G − 2 (ξ, τ, l, k1 , k2 )e
(3.28a)
2 +2kl+2k 2 )(t−τ )
, −∞
Replacing the second term of the rhs of Eq. (3.10) by the rhs of Eq. (3.23), where ϕˆ2− is defined by (3.28b) we find q (3.5)± . We note that the rhs of Eq. (3.26a) and (3.28a) involves the exponential exp[−2kη − il ζ¯ ] whose real part equals exp[−(l + 2k1 )η], which is bounded for l + 2k1 ≥ 0.
974
A. S. Fokas
Remark 3.1. Comparing Eqs. (3.17) and (3.18a) we find the identity ∞ ∞ 2 dξ dη(qµ+2 )(ξ, η, t, k1 , k2 )e−ilζ −∞ 0 ∞ ∞ −i(l 2 +2kl)t = 2e dξ dη(qµ+2 )(ξ, η, 0, k1 , k2 )e−ilζ −∞ 0 t ∞ 2 + dτ dξ G +1 (ξ, τ, l, k1 , k2 )e−ilξ +i(l +2kl)(τ −t) , −∞ < l < 0. (3.29) 0
−∞
Similarly, comparing Eqs. (3.25) and (3.26a) we find the identity ∞ ∞ ¯ 2 dξ dη(qµ ¯ +1 )(ξ, η, t, k1 , k2 )e−2kη−il ζ −∞ 0 ∞ ∞ ¯ i(l 2 +2kl+2k 2 )t = 2e dξ dη(qµ ¯ +1 )(ξ, η, 0, k1 , k2 )e−2kη−il ζ −∞ 0 t ∞ 2 2 + − dτ dξ G 2 (ξ, τ, l, k1 , k2 )e−ilξ +i(l +2kl+2k )(t−τ ) , −2k1 < l < ∞. 0
−∞
(3.30) Equations (3.29) and (3.30), as well as the analogous equations involving (qµ− 2) − (ξ, η, T, k1 , k2 ), (qµ ¯ 1 )(ξ, η, T, k1 , k2 ), can be verified directly using identities similar to Eq. (3.2a). This implies that the Eqs. (3.18), (3.20), (3.26), (3.28) defining ϕˆ ± j , j = 1, 2, are consistent with Eqs. (3.4)± , (3.5)± evaluated at y = 0. 3.3. The inverse problem. In order to express µ in terms of the nonlinear analogue of {qˆ0 , g} ˜ we first derive the analogue of Eqs. (2.29), (2.32) and (2.33). Proposition 3.2. Define the vector-valued function (µ1 , µ2 ) by Eq. (3.3) where {µ±j }2j=1 are defined by Eqs. (3.4)± and (3.5)± . Then, ∂µ± 2 ∂k
∂µ± 1 ∂k
(x, y, t, k1 , k2 ) = e−2i(k1 x−k2 y)+2i(k1 −k2 )t γ ± (k1 , k2 )µ¯ ± 1 (x, y, t, k1 , k2 ), 2
2
((3.31)± )
(x, y, t, k1 , k2 ) = −e−2i(k1 x−k2 y)+2i(k1 −k2 )t γ ± (k1 , k2 )µ¯ ± 2 (x, y, t, k1 , k2 ), 2
2
((3.32)± )
where (x, y, t) ∈ , k1 ∈ R, {k2 ≥ 0 for (3.31)± , (3.32)± }, {k2 ≤ 0 for (3.31)± , (3.32)± } and γ ± are defined as follows: γ + = β + + α + , k1 ∈ R, k2 ≥ 0,
((3.33)+ )
γ − = β − , k1 ∈ R, k2 ≤ 0,
((3.33)− )
Davey-Stewartson Equation on the Half-Plane
β ± (k1 , k2 ) = −
i π
∞
−∞
∞
dx 0
975
2i(k1 x−k2 y) dy q¯0 (x, y)µ± , 1 (x, y, 0, k1 , k2 )e
((3.34)± )
∞ T 1 α (k1 , k2 ) = − dτ d x [i g¯ 1 (x, τ ) − g¯ 0x (x, τ ) − 2i k¯ g¯ 0 (x, τ )] 2iπ 0 −∞ × ϕ1+ (x, τ, k1 , k2 ) 2 2 (3.35) −4 f¯(x, 0, τ )ϕ2+ (x, τ, k1 , k2 ) e2ik1 x−2i(k1 −k2 )τ . +
Proof. We denote Eqs. (3.4)± as (3.5)± as follows: ∞ qµ± 1 ± 1 ∞ 2 µ± + J {U1 ϕ2± − 4 f ϕ1± }E 1 ((3.36)± ) = 1 + dξ dη 1 π −∞ ζ − z 2iπ 1 0 and µ± 2
1 =− π
∞ −∞
∞
dξ
dη 0
qµ ¯ ± 1 ζ − z¯
¯ ¯
eik(ζ −z)+i k(ζ −¯z ) −
e2ky ± J {U2 ϕ1± − 4 f¯ϕ2± }E 2 . 2iπ 2 ((3.37)± )
For the derivation of Eqs. (3.31)± and (3.32)± we will also use the following equivalent forms of Eqs. (3.36)± and (3.37)± : ∞ qµ± 1 ∞ 2 µ± = 1 + dξ dη 1 π −∞ ζ −z 0 e 2k¯ y ± ¯ ± 2ik1 ξ −2i(k 2 −k 2 )τ 2 2 1 2 e J2 −U2 ϕ2 e − 4 f ϕ1± e2ik1 ξ −2i(k1 −k2 )τ E¯ 2 + 2iπ ((3.38)± ) and µ± 2
∞ qµ ¯ ± 1 ∞ ¯ ζ¯ −¯z ) 1 ik(ζ −z)+i k( e =− dξ dη π −∞ ζ − z¯ 0 e ± ¯ ± 2ik1 ξ −2i(k 2 −k 2 )τ 2 2 1 2 J1 −U1 ϕ1 e − 4 f¯ ϕ2± e2ik1 ξ −2i(k1 −k2 )τ E¯ 1 , − 2iπ ((3.39)± )
where e is defined in Eq. (1.8). Equations (3.38)± and (3.39)± follow from Eqs. (3.36)± and (3.37)± by replacing l with −(l + 2k1 ). For example, using this transformation in Eq. (3.37)± we find U2 → −g¯ 0ξ + i g¯ 1 + 2i(k − l − 2k1 )g¯ 0 = −U¯ 1 and J2± E 2 → e−2ik1 z¯ +2i(k1 −k2 )t J1± E¯ 1 e2ik1 ξ −2i(k1 −k2 )τ . 2
2
2
2
976
A. S. Fokas
We next compute the ∂k¯ derivative of Eqs. (3.4)± and (3.5)± . Equations (3.4)± do ¯ thus Eqs. (3.4)± yield equations with no forcing. not involve explicit dependence on k, ± On the other hand Eqs. (3.5) yield the following forcing: ∞ i −i(kz+k¯ z¯ ) ∞ i(kζ +k¯ ζ¯ ) − e dξ dηqµ ¯ ± 1e π −∞ 0 ⎡ T − t dτ e2ky ∞ U2 ϕ1± − 4 f¯ϕ2± E 2 + dξ ⎣ t . 2iπ −∞ 0 dτ l=−2k1
Writing the integral from t to T in the form of Eq. (2.31) the forcing becomes 2 2 ¯ e−i(kz+k z¯ ) B ± + e2i(k1 −k2 )t α + H (k2 ) ,
(3.40)
where H denotes the Heaviside function, α + is defined by Eq. (3.35) and the functions B ± are defined by ∞ ∞ i ± i(kζ +k¯ ζ¯ ) B =− dξ dηqµ ¯ ± 1e π −∞ 0 ∞ 2 −k 2 )t t 2i(k e 1 2 2 2 + dτ dξ U˜ 2 ϕ1± − 4 f¯ϕ2± e2ik1 ξ −2i(k1 −k2 )τ , 2iπ 0 −∞ ((3.41)± ) where U˜ 2 denotes U2 evaluated at l = −2k1 . Equation (3.2a) with 0 the upper half complex z-plane implies that B ± = β ± exp[2i(k12 − k22 )t], where β ± are defined by ¯ j = 1, 2 satisfy the following equations: Eqs. (3.34)± . Hence, ∂µ±j /∂ k, ∂µ± 2
1 =γ e− π ∂k ±
−
e2ky 2iπ
J2±
and ∂µ± 1
1 = π ∂k
∞ −∞
∞
dξ 0
∞
dξ
∞
dη
q¯
∂µ± 1
ζ − z¯ ∂k ∂ϕ2± U2 − 4 f¯ E2 ∂k ∂k
−∞
¯
eik(ζ −z)+i k(ζ −¯z )
0
∂ϕ1±
((3.42)± )
± ± ∂ϕ ∂ϕ q ∂µ± 1 2 J ± U1 2 − 4 f 1 E 1 . dη + ζ − z ∂k 2iπ 1 ∂k ∂k ((3.43)± )
Taking the complex conjugate of Eqs. (3.38)± and (3.39)± and multiplying the resulting equations by e we find the following equations: ∞ q( ¯ µ¯ ± 1 ∞ ¯ ζ¯ −¯z ) ± 2 e) ik(ζ −z)+i k( eµ¯ 1 = e + e dξ dη π −∞ ζ − z¯ 0 1 2ky ± 2 2 ± −2ik1 ξ +2i(k12 −k22 )τ e J2 −U2 ϕ¯ 2 e − 4 f¯ ϕ¯1± e−2ik1 ξ +2i(k1 −k2 )τ E 2 − 2iπ ((3.44)± )
Davey-Stewartson Equation on the Half-Plane
and − eµ¯ ± 2 =
1 π
∞
−∞
∞
dξ
dη 0
977
q (µ¯ ± e) ζ −z 1
1 ± ± −2ik1 ξ +2i(k 2 −k 2 )τ 2 2 1 2 J1 U1 ϕ¯1 e − 4 f −ϕ¯2± e−2ik1 ξ +2i(k1 −k2 )τ E 1 . + 2iπ ((3.45)± ) Multiplying Eqs. (3.44)± and (3.45)± by γ ± and comparing the resulting equations with Eqs. (3.42)± and (3.43)± we find Eqs. (3.31)± and (3.32)± . Proposition 3.3. Define the vector-valued function (µ1 , µ2 ) by Eq. (3.3) where {µ±j }2j=1 are defined by Eqs. (3.4)± and (3.5)± . Then, (µ+1 − µ− 1 )(x, y, t, k1 , 0) = ( µ1 )(x, y, t, k1 ) + (δµ1 )(x, y, t, k1 ),
(µ+2 − µ− 2 )(x, y, t, k1 , 0) = ( µ2 )(x, y, t, k1 ) + (δµ2 )(x, y, t, k1 ), k1 ∈ R, (x, y, t) ∈ ,
where
λ iλ x, y, t, k1 + , ,
µ1 = − 2 2 0 ∞ λ iλ ,
µ2 = dλeλ χ1 (k1 , λ)µ¯ − 1 x, y, t, k1 + , 2 2 0 ∞ iλ λ , δµ1 = eˆ dλE λ χ2 (k1 , λ)µ− 1 x, y, t, k1 + , − 2 2 0 ∞ iλ λ , δµ2 = eˆ dλE λ χ2 (k1 , λ)µ− 2 x, y, t, k1 + , − 2 2 0 k1 ∈ R, (x, y, t) ∈ ,
∞
dλeλ χ1 (k1 , λ)µ¯ − 2
(3.46)
(3.47a) (3.47b) (3.47c)
(3.47d)
and e, ˆ E λ , eλ , χ1 , χ2 are defined as follows: eˆ = e−2ik1 x+2ik1 t , eλ = e−i(2k1 +λ)x−λy+i(2k1 +2k1 λ+λ )t , 2
2
2
E λ = ei(2k1 +λ)x−λy−i(2k1 +2k1 λ+λ )t , 2
χ1 (k1 , λ) +
∞ 0
χ2 (k1 , λ) +
f 1 (k1 , λ, l)= −
1 iπ
dlχ2 (k1 , l + λ) f 2 (k1 , λ + l, l) = A+2 (k1 , λ),
∞ −∞
dlχ1 (k1 , l + λ) f 1 (k1 , λ + l, l) = −A+1 (k1 , λ),
∞
0
2
∞
dξ 0
(3.48)
(3.49a)
(3.49b)
iλ −ilξ −lη λ ξ, η, 0, k , − e dηq¯0 (ξ, η)µ¯ − + , l > 0, 1 2 2 2 (3.50a)
978
A. S. Fokas
f 2 (k1 , λ, l) =
A+1 (k1 , λ) =
1 iπ
1 2iπ ×
A+2 (k1 , λ) = − ×
∞ −∞
∞
dξ 0
T
dτ 0
iλ ilξ −lη λ ξ, η, 0, k , − e dηq0 (ξ, η)µ− + , l > 0, 1 2 2 2 (3.51a)
∞
−∞
dx
i g¯ 1 (x, τ ) − g¯ 0x (x, τ ) − 2i(k1 + λ)g¯ 0 (x, τ )
ϕ1+ (x, τ, k1 , 0) − 4 f¯(x, 0, τ )ϕ2+ (x, τ, k1 , 0) 1 2iπ
T
∞
dτ
ei(2k1 +λ)x−i(2k1 +2k1 λ+λ )τ , (3.52a) d x ig1 (x, τ ) + g0x (x, τ ) − 2i(k1 + λ)g0 (x, τ ) 2
0 −∞ 2 + ϕ2 (x, τ, k1 , 0) − 4 f (x, 0, τ )ϕ1+ (x, τ, k1 , 0) e−iλx+i(2k1 λ+λ )τ .
2
(3.52b)
Proof. Evaluating Eqs. (3.36)± and (3.37)± at k2 = 0 and then subtracting the resulting − + equations we find two equations for µ+1 − µ− 1 and µ2 − µ2 both of which involve a forcing. Let ( µ1 , µ2 ) denote the solution corresponding to the forcing from Eqs. (3.37)± and let (δµ1 , δµ2 ) denote the solution corresponding to the forcing from Eqs. (3.36)± ; then linearity implies Eqs. (3.46). (a) ( µ1 , µ2 ). Subtracting Eqs. (3.37)± evaluated at k2 = 0 we find that the relevant forcing equals e2k1 y 2iπ
−2k1 −∞
dl
∞
−∞
T
dξ 0
dτ Uˆ 2 ϕˆ1+ − 4 f¯ϕˆ2+ Eˆ 2 ,
where throughout the paper superscript ∧ denotes evaluation at k2 = 0. Letting l = −2k1 − λ, the forcing becomes ∞ dλeλ A+1 (k1 , λ), 0
A+1 (k1 , λ)
where eλ and are defined in Eqs. (3.48) and (3.52a). Indeed, under the transformation l = −2k1 − λ the term k1 + l in U2 becomes −(λ + k1 ), whereas the term exp[2k1 y] Eˆ 2 becomes eλ (k1 , λ, x, y, t)eλ−1 (k1 , λ, ξ, 0, τ ). It is important to note that eλ can be obtained by evaluating e at (k1 + λ/2, iλ/2): eλ = e(k1 + λ/2, iλ/2, x, y, t). We note that A+1 (k1 , λ) can be obtained from the evaluation of −α + (k1 , k2 ) (see Eq. (3.35)) at (k1 + λ/2, iλ/2) provided that ϕ1+ and ϕ2+ are evaluated at (k1 , 0). By subtracting Eqs. (3.36)± , (3.37)± evaluated at k2 = 0 and by ignoring the forcing obtained from Eqs. (3.36)± we find that the vector ( µ1 , µ2 ) satisfies the following equations: ∞ 1 ∞ q¯
µ2 =
µ1 e2ik1 (ξ −x) dλeλ A+1 (k1 , λ) − dξ dη π ζ − z ¯ 0 −∞ e2k1 y ˆ− ˆ − (3.53a) J {U2 ϕ1 − 4 f¯ ϕ2 } Eˆ 2 2iπ 2
Davey-Stewartson Equation on the Half-Plane
979
and
µ1 =
1 π
∞
−∞
dξ
∞
dη 0
1 ˆ− ˆ q µ2 + J {U1 ϕ2 − 4 f ϕ1 } Eˆ 1 . ζ − z 2iπ 1
(3.53b)
In order to relate µ2 with µ¯ − 1 it is imperative to replace the second term of the rhs of Eqs. (3.38)± by the sum of two terms by employing the following identity: 1 e−iλ(z−ζ ) = −i ζ −z ζ −z
−2k1 +λ −2k1
dle−i(l+2k1 )(z−ζ ) .
(3.54)
We then take the complex conjugate of the resulting equation and we also multiply by e (which is defined by Eq. (1.8)) to obtain the following equation: eµ¯ − 1 =e+
1 π
∞ −∞
∞
dξ
dη 0
q(e ¯ µ¯ − 2 ) iλ(¯z −ζ )−2ik1 (x−ξ )+2ik2 (y−η) e ζ¯ − z¯
e2ky
J − {−U2 (e˜ϕ¯2− ) − 4 f¯(e˜ϕ¯ 1− )}E 2 2iπ 2 ∞ −2k1 +λ ∞ i i(l+2k1 )(¯z −ζ¯ )−2ik1 (x−ξ )+2ik2 (y−η) + dl dξ dηq(e ¯ µ¯ − , 2 )e π −2k1 −∞ 0 (3.55) −
where throughout this paper the superscript ∼ denotes evaluation at y = 0 (or η = 0). We next evaluate Eq. (3.55) at (k1 + λ/2, iλ/2). In this respect we note the following: λ iλ = k1 , thus (U j , E j ) → (Uˆ j , Eˆ J ), j = 1, 2. • k = k1 + ik2 → k1 + + i 2 2 • J2− gives rise to Jˆ2− as well as to an additional integral with respect to dl from −2k1 to −2k1 − λ. • The exponent of the exponential appearing in the second term of the rhs of Eq. (3.55) simplifies to λ ¯ iλ(¯z − ζ ) − 2i k1 + (x − ξ ) − λ(y − η) = −2ik1 (x − ξ ). 2 • The exponent of the exponential appearing in the fourth term of the rhs of Eq. (3.55) simplifies to λ ¯ i(l + 2k1 + λ)(¯z − ζ ) − 2i k1 + (x − ξ ) − λ(y − η) 2 = il(x − ξ ) + (l + 2k1 )(y − η). Hence, using the notation
λ iλ M j (x, y, t, k1 , λ) = x, y, t, k1 + , , 2 2 λ iλ x, y, t, k j (x, y, t, k1 , λ) = ϕ − , , + 1 j 2 2 µ¯ −j
(3.56) j = 1, 2,
980
A. S. Fokas
Eq. (3.55) becomes eλ M1 = eλ + F1 + −
1 π
∞
∞
dξ
−∞
dη
q(e ¯ λ M2 )
0
ζ − z¯
e−2ik1 (x−ξ )
e2k1 y ˆ− J2 −Uˆ 2 (e˜λ 2 ) − 4 f¯(e˜λ 1 ) Eˆ 2 , 2iπ
(3.57a)
where −2k1 1 2 2 F1 = − dleilx+(l+2k1 )y−i(l +2k1 l+2k1 )t 2iπ −2k1 −λ ∞ ∞ 2 2 2 dξ dηq¯ M2 eλ e−ilξ −(l+2k1 )η−i(l +2k1 l+2k1 )t −∞ 0 ∞ t 2 2 − dξ dτ −Uˆ 2 2 − 4 f¯1 e˜λ e−ilξ −i(l +2k1 l+2k1 )τ . −∞
(3.58)
0
We take the complex conjugate of Eqs. (3.39)± and then we multiply the resulting equation by e; this implies that the first term of the rhs of Eqs. (3.39)± becomes q(eµ¯ − 1 )/(ζ − z). Evaluating the resulting equation at (k1 + λ/2, iλ/2) we find ∞ 1 ˆ− ˆ 1 ∞ q(eλ M1 ) + − eλ M2 = dξ dη J {U1 (e˜λ 1 ) − 4 f (−e˜λ 2 )} Eˆ 1 . π −∞ ζ −z 2iπ 1 0 (3.57b) Equations (3.57a) and (3.57b) are similar with Eqs. (3.53a) and (3.53b) except for the forcing eλ + F1 which will be analyzed in (c) below. (b) (δµ1 , δµ2 ). Subtracting Eqs. (3.36)± evaluated at k2 = 0 we find that the relevant forcing equals −
1 2iπ
∞
dl 0
∞ −∞
T
dξ 0
dτ {Uˆ 1 ϕˆ2+ − 4 f ϕˆ1+ } Eˆ 1 .
This expression can be rewritten in the form ∞ −2ik1 x+2ik12 t e dλE λ A+2 (k1 , λ), 0
where E λ and A+2 are defined in Eqs. (3.48) and (3.52b). It is important to note that E λ can be obtained by evaluating e−1 at (k1 +λ/2, −iλ/2). For the determination of (δµ1 , δµ2 ), it is more convenient to use Eqs. (3.38)± and (3.39)± instead of Eqs. (3.36)± and (3.37)± . By subtracting Eqs. (3.38)± , (3.39)± evaluated at k2 = 0 and by ignoring the forcing obtained from Eqs. (3.39)± we find that the vector (δµ1 , δµ2 ) satisfies the following equations: eˆ−1 δµ1 =
∞ 0
+
e2k1 y 2iπ
∞
∞
q(eˆ−1 δµ2 ) 2ik1 (x−ξ ) e ζ −z −∞ 0 (3.59a) Jˆ2− −U¯ˆ 2 (eˆ−1 δ2 ) − 4 f (eˆ−1 δ1 ) Eˆ 2
dλE λ A+ (k1 , λ) +
1 π
dξ
dη
Davey-Stewartson Equation on the Half-Plane
and
981
∞ 1 ˆ− 1 ∞ q( ¯ eˆ−1 δµ1 ) − dξ dη J π −∞ 2iπ 1 ζ¯ − z¯ 0 × −U¯ˆ 1 (eˆ−1 δ1 ) − 4 f¯(eˆ−1 δ2 ) E¯ˆ 1 .
eˆ−1 δµ2 = −
(3.59b)
As in the case (a) earlier it is imperative to replace the second term of the rhs of Eqs. (3.38)± by the two terms obtained via the identity (3.59). We then multiply by e−1 and evaluate the resulting equation at (k1 + λ/2, −iλ/2). In this respect we note that ¯k = k1 − ik2 → k1 + λ − i − iλ = k1 , thus (U¯ j , E¯ j ) → (Uˆ j , Eˆ j ), j = 1, 2. 2 2 Hence, using the notation
λ iλ , N j (x, y, t, k1 , λ) = µ−j x, y, t, k1 + , − 2 2 λ iλ j (x, y, t, k1 , λ) = ϕ − x, y, t, k , + , − 1 j 2 2
(3.60) j = 1, 2,
in analogy with Eq. (3.57a) we now have the following equation: ∞ 1 ∞ q(E λ N2 ) 2ik1 (x−ξ ) E λ N1 = E λ + F2 + e dξ dη π −∞ ζ −z 0 e2k1 y ˆ− ˆ ˜ + J2 −U2 ( E λ 2 ) − 4 f ( E˜ λ 1 ) Eˆ 2 , 2iπ where F2 =
−2ki 1 2 2 dle−ilx+(l+2k1 )y+i(l +2k1 l+2k1 )t 2iπ −2k1 −λ ∞ ∞ 2 2 × 2 dξ dηq N2 E λ eilξ −(l+2k1 )η−i(l +2k1 l+2k1 )t −∞ 0 ∞ t 2 2 − dξ dτ −U¯ˆ 2 2 − 4 f 1 E˜ λ eilξ +i(l +2k1 l+2k1 )τ . −∞
(3.61a)
(3.62)
0
Similarly, multiplying Eqs. (3.39)± by e−1 and evaluating the resulting equation at (k1 + λ/2, −iλ/2) we find ∞ 1 ∞ q(E ¯ λ N1 ) E λ N2 = − dξ dη π −∞ ζ − z¯ 0 1 ˆ− ˆ ˜ − (3.61b) J1 −U 1 ( E λ 1 ) − 4 f¯( E˜ λ 2 ) Eˆ 1 . 2iπ Equations (3.61a) and (3.61b) are similar with Eqs. (3.59a) and (3.59b) except for the forcing E λ + F2 which will be analyzed in (c) below. (c) The Analysis of F1 and F2 . The first and the second integrals in the curly bracket in the definition of F1 (Eq. (3.58)) involve the following exponentials: e˜λ e−ilξ −i(l
2 +2k
2 1 l+2k1 )τ
= e−i(l+λ+2k1 )ξ +i(λ−l)(l+λ+2k1 )τ
982
A. S. Fokas
and eλ e−ilξ −(l+2k1 )η−i(l
2 +2k
2 1 l+2k1 )t
= e−i(l+λ+2k1 )ξ −(l+λ+2k1 )η+i(λ−l)(l+λ+2k1 )t .
Replacing l + λ + 2k1 by l the above exponentials become e−ilξ +il(2λ+2k1 −l)τ and e−il z¯ +il(2λ+2k1 −l)t ; furthermore the term 2k1 + 2l in Uˆ 2 becomes 2(l − λ − k1 ). Thus the curly bracket in (3.58) is precisely the expression that results from integrating Eq. (A.8) from t = 0. Hence, λ 2 2 2 F1 = dlei(l−λ−2k1 )x+(l−λ)y+i(l +λ +2k1 +2k1 λ−2k1 l−2lλ)t f 1 (k1 , λ, l), (3.63) 0
where f 1 (k1 , λ, l) is defined in Eq. (3.50a). We will now determine a function χ1 (k1 , λ) such that ∞ ∞ + dλeλ (k1 , λ)A1 (k1 , λ) = dλeλ (k1 , λ)χ1 (k1 , λ) 0 0 λ ∞ dλ dl e(k ˘ 1 , λ, l)χ1 (k1 , λ) f 1 (k1 , λ, l) , 0
(3.64)
0
where e˘ denotes the exponential appearing in Eq. (3.63) and for convenience of notation we have suppressed the (x, y, t) of eλ and e. ˘ After changing the order of the integration in the second term of the rhs of Eq. (3.64) we find that this term equals ∞ ∞ dl dλe(k ˘ 1 , λ, l)χ1 (k1 , λ) f 1 (k1 , λ, l) 0 l ∞ ∞ dl dλe(k ˘ 1 , λ + l, l)χ1 (k1 , λ + l) f 1 (k1 , λ + l,l). = 0
0
It is remarkable that e(k ˘ 1 , λ + l, l) = eλ (k1 , λ), hence Eq. (3.64) yields Eq. (3.49a). Multiplying Eqs. (3.57) by χ1 , integrating with respect to dλ and comparing the resulting equations with Eqs. (3.53) we find Eqs. (3.47a) and (3.47b). The derivation of Eqs. (3.47c) and (3.47d) is similar. 3.4. The d-bar problem. In what follows we will formulate the basic d-bar problem associated with the solution of the DS system in . In addition, we will briefly discuss how the a-priori assumption of existence can be eliminated. Let us consider the initial-boundary value problem defined by q0 (x, y) = q(x, y, 0), x ∈ R, 0 < y < ∞, g0 (x, t) = q(x, 0, t), f 0 (x, t) = f (x, 0, t), x ∈ R, 0 < t < T, where the given functions {q0 , g0 , f 0 } have sufficient smoothness and decay.
(3.66)
Davey-Stewartson Equation on the Half-Plane
983
Let g1 (x, t) denote the unknown Neumann boundary value q y (x, 0, t). This function can be characterized in terms of the given data via the analysis of the associated global relation, see Sect. 3.5. We will now define the map {q0 (x, y), g0 (x, t), g1 (x, t), f 0 (x, t)} → {α + (k1 , k2 ), β ± (k1 , k2 ), χ1 (k1 , λ), χ2 (k1 , λ)},
(3.67)
where (x, y, t) ∈ , k1 ∈ R, λ > 0 and k2 ≥ 0 for {α + , β + }, whereas k2 ≤ 0 for β − . 1. q0 → {ψ1− (x, y, k1 , k2 ), ψ2− (x, y, k1 , k2 )}, x ∈ R, 0 < y < ∞, k1 ∈ R, k2 ≤ 0. The vector (ψ1− , ψ2− ) is defined in terms of q0 via the linear integral equations ∞ q0 ψ2− 1 ∞ , ψ1− = 1 + dξ dη π −∞ ζ −z 0 (3.68) ∞ q¯0 ψ1− 1 ∞ − ψ2 = − dξ dη . π −∞ ζ − z¯ 0 2. {q0 , g0 , g1 , f 0 } → {ψ1+ (x, y, k1 , k2 ), ψ2+ (x, y, k1 , k2 )}, {ϕ1+ (x, t, k1 , k2 ), ϕ2+ (x, t, k1 , k2 )}, where (x, y, t) ∈ , k1 ∈ R, k2 ≥ 0. The vectors (ψ1+ , ψ2+ ) and (ϕ1+ , ϕ2+ ) are defined in terms of {q0 , g0 , g1 , f 0 } through the following linear integral equations: ∞ ∞ T q0 ψ2+ 1 2 dl dτ dξ G +1 eil(z−ξ )+i(l +2kl)τ , − ζ −z 2iπ 0 −∞ 0 0 −∞ (3.69) ∞ T ∞ q¯0 ψ1+ e2ky −2k1 1 ∞ 2 2 + ψ2+ = − dξ dη dl dτ dξ G +2 eil(¯z −ξ )−i(l +2kl+2k )τ π −∞ 2iπ ζ − z¯ 0 0 −∞ −∞ ψ1+ = 1 +
and
1 π
∞
dξ
∞
dη
∞ ∞ T i 2 = 1+ dl dτ dξ G +1 eil(x−ξ )+i(l +2kl)(τ −t) 2π 0 t −∞ t ∞ 0 i 2 − dl dτ dξ G +1 eil(x−ξ )+i(l +2kl)(τ −t) 2π −∞ 0 −∞ ∞ ∞ 0 i 2 − dl dξ dηq0 ψ2+ e−i(l +2kl)t+lη , π −∞ −∞ 0 ∞ −2k1 T i 2 2 + ϕ2 = − dl dτ dξ G +2 eil(x−ξ )+i(l +2kl+2k )(t−τ ) 2π −∞ t −∞ t ∞ ∞ i 2 2 + dl dτ dξ G +2 eil(x−ξ )+i(l +2kl+2k )(t−τ ) 2π −2k1 0 −∞ ∞ ∞ ∞ i 2 2 − dl dξ dηq¯0 ψ1+ ei(l +2kl+2k )t−(2k+l)η , π −2k1 −∞ 0
ϕ1+
(3.70)
where G +1 (ξ, τ, k1 , k2 , l) = [g0ξ (ξ, τ ) + ig1 (ξ, τ ) − 2i(k + l)g0 (ξ, τ )]ϕ2+ (ξ, τ, k1 , k2 ) G +2 (ξ, τ, k1 , k2 , l)
−4 f 0 (ξ, τ )ϕ1+ (ξ, τ, k1 , k2 ), = [−g¯ 0ξ (ξ, τ ) + i g¯ 1 (ξ, τ ) + 2i(k + l)g¯ 0 (ξ, τ )]ϕ1+ (ξ, τ, k1 , k2 ) (3.71) −4 f¯0 (ξ, τ )ϕ2+ (ξ, τ, k1 , k2 ).
984
A. S. Fokas
The above definitions can be motivated from the results presented in Sect. 3.2. Indeed, letting ± ψ± j (x, y, k1 , k2 ) = µ j (x, y, 0, k1 , k2 ),
j = 1, 2
and evaluating Eqs. (3.4)± and (3.5)± at t = 0, we find Eqs. (3.68). Similarly, evaluating Eqs. (3.4)± and (3.5)± at t = 0, we find Eqs. (3.69). Furthermore, replacing in Eq. (3.13) ϕˆ1 by the expressions in Eqs. (3.18), we find Eq. (3.70a). Similarly, replacing in Eq. (3.21) ϕˆ 2 by the expressions in Eq. (3.21), we find Eq. (3.70b). The definitions (3.71) follows from the definitions of G +1 and G +2 in Eqs. (3.19)± and (3.28)± , as well as from the definitions of U1 and U2 in Eqs. (3.7). 3. {q0 , g0 , g1 , f 0 } → {β ± (k1 , k2 ), α + (k1 , k2 )}, k1 ∈ R, where k2 ≥ 0 for α + and β + , whereas k2 ≤ 0 for β − . The functions β ± and α ± are defined by Eqs. (3.34)± and (3.35), i.e. by the following equations: i π
β ± (k1 , k2 ) = −
∞
dx
1 α (k1 , k2 ) = − 2iπ +
∞
−∞ T
0
dτ
0
dy q¯0 ψ1± e2i(k1 x−k2 y) ,
∞
−∞
(3.72)
d x{[−g¯ 0x + i g¯ 1 − 2i k¯ g¯ 0 ]ϕ1+ − 4 f¯0 ϕ2+ }
×e2ik1 x−2i(k1 −k2 )τ . 2
2
(3.73)
4. {q0 , g0 , g1 , f 0 } → { f j (k1 , λ, l), A+j (k1 , λ)}2j=1 , k1 ∈ R, λ > 0, l > 0. The functions f j and A+j are defined by Eqs. (3.50) and (3.51), i.e. by the following equations: ∞ ∞ 1 λ iλ −ilx−ly − e f 1 (k1 , λ, l) = − dx dy q¯0 (x, y)ψ2 x, y, k1 + , , iπ −∞ 2 2 0 ∞ ∞ iλ ilx−ly 1 λ f 2 (k1 , λ, l) = dx dyq0 (x, y)ψ2− x, y, k1 + , − e iπ −∞ 2 2 0 (3.74) and A+1 (k1 , λ) =
1 2iπ
T
dτ 0
∞
−∞
d x [−g¯ 0x + i g¯ 1 − 2i(k1 + λ)g¯ 0 ]ϕ1+ (x, τ, k1 , 0)
−4 f¯0 ϕ2+ (x, τ, k1 , 0) ∞ T 1
A+2 (k1 , λ) = −
2iπ
dτ
2 +2k
ei(2k1 +λ)x−i(λ
2 1 λ+2k1 )τ
,
d x [g0x + ig1 − 2i(k1 + λ)g0 ]ϕ2+ (x, τ, k1 , 0)
0 −∞ 2 + −4 f 0 ϕ1 (x, τ, k1 , 0) e−iλx+i(2k1 λ+λ )τ .
(3.75)
5. { f j , A j }2j=1 → {χ j (k1 , λ)}2j=1 , k1 ∈ R, λ > 0. The functions χ j are defined in terms of f j and A j , j = 1, 2, via the linear Volterra integral Eqs. (3.49).
Davey-Stewartson Equation on the Half-Plane
985
The function µ2 satisfies Eq. (2.24); the function µ1 satisfies a similar equation with the addition of the term 1 in the rhs of Eq. (2.24). Using in these equations the results of Propositions 3.2 and 3.3, we find that {µ j }2j=1 satisfy a system of two linear integral equations involving the spectral functions {β ± , α + , χ1 , χ2 }. The terms involving {µ+j − µ−j }2j=1 can be further simplified using a simple change of variables. In particular, the terms µ j , j = 1, 2, yield ∞ ∞ dk1 λ iλ − eλ , j = 1, 2, (3.76a) σ dλχ1 (k1 , λ)µ j+1 x, y, t, k1 + , 2 2 −∞ k1 − k 0 where σ = −1 for j = 1 and σ = 1 for j = 2 and µ3 = µ1 . The substitutions λ iλ λ ˆ = kˆ1 , = kˆ2 , i.e. k1 = kˆ1 − = kˆ1 + i kˆ2 k, 2 2 2 transform the expression (3.76a) to ∞ i∞ ˆ d k2 ˆ −2i kˆ2 )µ− (x, y, t, kˆ1 , kˆ2 ) d kˆ1 −2iσ χ1 (k, j+1 kˆ − k −∞ 0 k1 +
ˆ
ˆ
ˆ2 ˆ2
×e−2i(k1 x−k2 y)−2i(k2 −k2 )t ,
j = 1, 2.
Similarly, the terms δµ j , j = 1, 2, yield ∞ ∞ dk1 iλ λ − x, y, t, k e ˆ , − Eλ. dλχ (k , λ)µ + 2 1 1 j 2 2 −∞ k1 − k 0
(3.76b)
(3.77a)
The substitutions λ −iλ λ ˆ = kˆ2 , i.e. k1 = kˆ1 − = kˆ1 − i kˆ2 k, k1 + = kˆ1 , 2 2 2 yield −i∞ ˆ ∞ 2 d k2 − −2i kˆ 1 x+2i kˆ 1 t ˆ ˆ ˆ ˆ ˆ d k1 2i χ2 (k, 2i k2 ) µ j (x, y, t, k1 , k2 )e −∞ 0 kˆ − k ˆ
ˆ
ˆ2 ˆ2
×e2i k1 x−2i k2 y+2i(k1 −k2 )t .
(3.77b)
Using Eqs. (3.76b) and (3.77b) and replacing ∧ by prime we find the following equations for {µ j }21 : ∞ ∞ dk2 1 e[β + (k1 , k2 ) + α + (k1 , k2 )]µ¯ +2 dk1 µ1 = 1 + π k − k −∞ 0 ∞ ∞ dk2 eβ − (k1 , k2 )µ− + dk1 2 −k k −∞ 0 ∞ i∞ dk2 eχ (k , −2ik2 )µ− + dk1 2 k − k 1 −∞ 0 −i∞ ∞ dk2 + dk1 ˆ − ] (3.78a) eχ ¯ 2 (k¯ , 2ik2 )[eµ 1 k − k −∞ 0
986
A. S. Fokas
and
1 µ2 = − π +
∞ −∞
dk1
∞
0
dk2 e[β + (k1 , k2 ) + α + (k1 , k2 )]µ¯ +2 k − k
dk2 eβ − (k1 , k2 )µ− 1 k − k −∞ 0 ∞ i∞ dk2 + dk1 eχ1 (k , −2ik2 )µ− 1 k − k −∞ 0 ∞ −i∞ dk2 − ¯ − dk1 ˆ 2] , eχ ¯ 2 (k , 2ik2 )[eµ k − k −∞ 0 ∞
dk1
∞
(3.78b)
where e is defined by Eq. (1.8) (with k1 and k2 replaced with k1 and k2 ) and
2
eˆ = e2ik x−2ik t .
(3.78c)
Substituting the representations µ(1) 1 1 2 (x, y, t) , µ2 = µ1 = 1 + O +O k k k2 (1)
in Eq. (1.5b), it follows that q¯ = iµ2 . Hence, Eq. (3.78b) implies the following integral representation for q, ¯ i q(x, ¯ y, t) = π
−∞
∞ +
∞
∞
−∞
dk1
0
dk2 e(β + + α + )µ+1 +
dk1 eχ1 (k , −2ik2 )µ− 1 −
∞ −∞
∞ −∞
∞ dk1
−i∞
dk1
0
0
dk2 eβ − µ− 1
dk2 eχ ¯ 2 (k¯ , 2ik2 )(eµ ˆ − 2)
.
(3.79) Remark 3.2. In the linear limit of small q, χ2 → 0 and µ1 → 1, thus Eq. (3.78b) becomes 1 ∞ ∞ dk2 e[β L+ (k1 , k2 ) + α +L (k1 , k2 )] dk1 µ2 ∼ − π −∞ k − k 0 dk2 1 ∞ eβ L− (k1 , k2 ), − dk1 (3.80a) π −∞ ∂I k − k where β L±
α +L = −
1 2iπ
i =− π
T
dτ 0
∞ −∞
∞
−∞
dx
∞
((3.80b)± )
dy q¯0 e2i(k1 x−k2 y) ,
0
d x[−g¯ 0x + i g¯ 1 − 2ik g¯ 0 ]e2ik1 x−2i(k1 −k2 )τ , 2
2
(3.80c)
and we have used the fact that χ1 L (k , −2ik2 ) = −A+1L (k , −2ik2 ) = −β L− (k1 , k2 ). It can be verified that q¯ = i limk→∞ kµ2 solves the complex conjugate of Eq. (1.1).
Davey-Stewartson Equation on the Half-Plane
987
Remark 3.3. The rigorous analysis of the above formalism involves the following steps. 1. Show that the linear integral Eqs. (3.68) (defining ψ − j , j = 1, 2), as well as (3.69) and + + (3.70) (defining ψ j , ϕ j , j = 1, 2), have a unique solution. This is straightforward if the given data satisfy appropriate “small norm” conditions. 2. Show that the linear integral Eqs. (3.78a) and (3.78b) (defining µ j , j = 1, 2) have a unique solution. This is straightforward if the spectral functions {β ± , α + , χ1 , χ2 } satisfy appropriate small norm assumptions. 3. Prove that the function q defined by Eq. (3.79) solves the DS equation and furthermore show that q(x, y, 0) = q0 (x, y), q(x, 0, t) = g0 (x, t), q y (x, 0, t) = g1 (x, t). This can be achieved using the so-called “dressing method” [21] as well as techniques introduced in [22]. 4. Show that the solution q(x, y, t) for 0 < t < T∗ , where 0 < T∗ < T , depends only on the boundary values for t between 0 and T∗ (for the analogous results in 1 + 1 see [22]). 3.5. The global relation. For the solution of initial-boundary value problems the most difficult step is the characterization of the unknown boundary values in terms of the given boundary conditions. For linear evolution PDEs and for certain particular nonlinear boundary value problems this step can be simplified as follows: Instead of characterizing the unknown boundary values, it is possible to obtain directly the unknown spectral functions. This is indeed the case for the Dirichlet problem of the linear PDE (1.1), i.e. for the case that q is prescribed at t = 0 and at y = 0, q0 (x, y) = q(x, y, 0), x ∈ R, 0 < y < ∞; g0 (x, y) = q(x, 0, t), x ∈ R, 0 < t < T.
(3.81)
In this case the analysis of the associated global relation implies that q(x, y, t) is given by Eq. (2.1), where the function qˆ0 (k1 , k2 ) is defined in terms of q0 (x, y) by Eq. (2.2a) and the function g(k ˜ 1 , k2 ) is defined in terms of qˆ0 and g0 by the expression ∞ T 2 2 g(k ˜ 1 , k2 ) = 2i qˆ0 (k1 , −k2 ) − 4ik2 dτ d xg0 (x, τ )e2ik1 x+2i(k1 −k2 )τ , 0
−∞
3π ≤ arg k2 ≤ 2π. k1 ∈ R, 2
(3.82)
Indeed, let g˜ 0 (k1 , k22 ) denote the double integral appearing in Eq. (3.82) and let g˜ 1 (k1 , k22 ) denote the analogous integral with g0 replaced by g1 . Then, the function g(k ˜ 1 , k2 ) defined by Eq. (2.2b) takes the form g(k ˜ 1 , k2 ) = g˜ 1 (k1 , k22 ) − 2ik2 g˜ 0 (k1 , k22 ).
(3.83)
The associated global relation is given by − 2i qˆ0 (k1 , k2 ) + g(k ˜ 1 , k2 ) = −2i qˆ T (k1 , k2 )e2i(k1 −k2 )T , k1 ∈ R, k2 ∈ C+ , 2
2
(3.84)
where qˆ T is defined by an expression similar to qˆ0 with q0 (x, y) replaced by q(x, y, T ). The global relation can be obtained from the integration of Eq. (2.5) from t = 0 to
988
A. S. Fokas
¯ The restriction on k2 is t = T , where 0 is the upper half complex z-plane and λ = k. needed in order for qˆ0 and qˆ T to make sense. Equation (2.1) involves an integral along the boundary of the fourth quadrant of the complex k2 -plane, thus in order to be able to employ Eq. (3.84) in the fourth quadrant we replace in Eq. (3.84) k2 by −k2 . This yields (after using Eq. (3.83)) the equation −2i qˆ0 (k1 , −k2 ) + g˜ 1 (k1 , k22 ) + 2ik2 g˜ 0 (k1 , k22 ) −2i qˆ T (k1 , −k2 )e2i(k1 −k2 )T , k1 ∈ R, k2 ∈ C− . 2
2
Solving this equation for g˜ 1 and then substituting the resulting expression in Eq. (3.83) we find g(k ˜ 1 , k2 ) = 2i qˆ0 (k1 , −k2 ) − 4ik2 g˜ 0 (k1 , k22 ) −2i qˆ T (k1 , −k2 )e2i(k1 −k2 )T , k1 ∈ R, k2 ∈ C− . 2
2
This expression implies Eq. (3.82), after using the crucial fact that qˆ T does not contribute to the solution q. Indeed, the contribution of qˆ T equals 1 π2
∞
−∞
dk1
dk2 e−2i(k1 x+k2 y)+2i(k1 −k2 )(T −t) 2
∂D
(x, y, t) ∈ .
2
∞
−∞
∞
dξ
dηq(ξ, η, T )e2i(k1 ξ −k2 η) ,
0
The exponentials exp[−2ik2 y] and exp[−2ik2 η] are bounded in the lower half of the complex k2 -plane, whereas the exponential exp[−2ik22 (T − t)] is bounded in the union of the second and the fourth quadrant of the complex k2 -plane. Hence, these three exponentials are bounded in the fourth quadrant of the complex k2 -plane and then Cauchy’s theorem implies that the integral along ∂ D vanishes. The global relation for the DS.. The equation B + = β + exp[2i(k12 − k22 )t], where B + is defined by Eqs. (3.41)± , yields the following global relation:
∞
−∞
∞
dx
0
dy q¯0 ψ1+ e2i(k1 x+k2 y) ∞
2 2 d x (−g¯ 0x + i g¯ 1 − 2i k¯ g¯ 0 )ϕ1+ − 4 f¯0 ϕ2+ e2ik1 x−2i(k1 −k2 )τ 0 −∞ ∞ ∞ 2 2 −2i(k1 −k2 )T =e dx dye2i(k1 x+k2 y) (qµ ¯ +1 ) |t=T , k1 ∈ R, k2 ∈ C− . (3.85) 1 − 2
T
dτ
−∞
0
In the linear limit of small q, µ+1 → 1, ϕ1+ → 1, ϕ2+ → 0,
f 0 → 0,
and Eq. (3.85) reduces to an equation similar to Eq. (3.84). The characterization of g1 through the analysis of the global relation (3.85) will be presented elsewhere.
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989
4. Conclusions We have presented an extension of the method of [8] from evolution equations in one spatial dimension to evolution equations in two spatial dimensions. The basic ingredients of the relevant methodology remain the simultaneous spectral analysis of both parts of the Lax pair, as well as the characterization of the unknown boundary values via the analysis of the global relation. However, the implementation of these ingredients to PDEs in three dimensions presents several highly non-trivial novel features: 1. The eigenfunction defining the contribution of the boundary values is now coupled with the eigenfunction defining the contribution of the initial conditions, see for example Eqs. (3.69) and (3.70). 2. The basic eigenfunctions are now sectionally non-analytic, as opposed to sectionally analytic functions that occur in 1 + 1. Thus, the relevant inverse problem is not a RH problem but a d-bar problem involving two different contributions, namely the derivative of µ with respect to k¯ and the jumps of µ across the different sectors. 3. The computation of the relevant jumps (for the nonlinear case) is complicated and involves rewriting µ in a form which is difficult to motivate. The relevant computations have certain similarities (but are substantially more complicated) with the corresponding computations for those initial-value problems in 2 + 1 that can be solved via a non-local RH problem like KPI and DSI. 4. The computation of the spectral functions requires the employment of several versions of the global relation (see Eqs. (3.2a) and (A.8)). The case of q0 (x, y) = 0. In order to emphasize the contribution of the boundary values we consider the case of q0 = 0. Then β ± = f 1 = f 2 = 0. Hence, Eqs. (3.78a) and (3.78b) characterizing the inverse problem are uniquely defined in terms of the spectral functions α + (k1 , k2 ), χ1 (k1 , λ) = −A+1 (k1 , λ), χ2 (k1 , λ) = A+2 (k1 , λ), which are defined by Eqs. (3.73) and (3.75) in terms of {g0 , g1 , f 0 } and the vector-valued eigenfunction (ϕ1+ , ϕ2+ ); furthermore, this eigenfunction is defined in terms of {g0 , g1 , f 0 } by equations (3.70) with q0 = 0 (which eliminates the dependence on {ψ +j }21 ). In more details, given g0 , characterize g1 via the global relation. Then define φ1+ and + φ2 via the solution of the following linear integral equations:
ϕ1+ = 1 + −
i 2π
i 2π
∞
dl 0 0
dτ t t
dl
−∞ −2k1
T
dτ
0
∞
−∞ ∞
−∞
dξ G +1 eil(x−ξ )+i(l
dξ G +1 eil(x−ξ )+i(l
2 +2kl)(τ −t)
2 +2kl)(τ −t)
,
T ∞ i 2 2 dl dτ dξ G +2 eil(x−ξ )+i(l +2kl+2k )(t−τ ) 2π −∞ t −∞ t ∞ ∞ i 2 2 + dl dτ dξ G +2 eil(x−ξ )+i(l +2kl+2k )(t−τ ) , 2π −2k1 0 −∞
ϕ2+ = −
990
A. S. Fokas
where G +1 and G +2 are defined in (3.71). Given φ1+ and φ2+ define α + by Eq. (3.73) and A+1 , A+2 by Eq. (3.75). Given {α + , A+1 , A+2 } define {µ1 , µ2 } by
∞ i∞ dk2 dk2 + + (k , k ) µ ¯ − dk eα e A+ (k , −2ik2 )µ− 1 2 2 1 2 k − k k − k 1 −∞ 0 −∞ 0 ∞ −i∞ dk2 e¯ A+2 (k¯ , 2ik2 )[eµ + dk1 ˆ − 1 ] , k ∈ C, k −k −∞ 0
µ1 = 1 +
1 π
and 1 π −
µ2 = −
∞
∞
−∞ ∞
−∞
dk1
dk1
dk1
∞
∞
0 −i∞
0
dk2 eα + (k1 , k2 )µ¯ +2 − k −k dk2
k − k
∞ −∞
dk1
i∞
0
dk2 e A+ (k , −2ik2 )µ− 1 −k 1
k
e¯ A+2 (k¯ , 2ik2 )[eµ ˆ − 2 ] , k ∈ C,
where e is defined by Eq. (1.8). Then q¯ is given by ∞ ∞ ∞ i q(x, ¯ y, t) = dk1 dk2 eα + µ+1 − dk1 e A+1 (k , −2ik2 )µ− 1 π −∞ 0 −∞ −i∞ ∞ − dk1 dk2 e¯ A+2 (k¯ , 2ik2 )(eµ ˆ − 2) . −∞
0
The main advantage of the methodology of [8] is that it yields integral representations in the complex plane. This has both computational [23–25] and analytical advantages; in particular it provides the basis for obtaining asymptotic results. The efficiency of the formalism of [8] for computing both the large time and the small dispersion limit of evolution PDEs in 1 + 1 using the Deift-Zhou [26] and the Deift-Zhou-Venakides [27] techniques has been demonstrated in several publications. The employment of the integral representation presented here (Eq. (3.79)) for the computation of the large t asymptotics using the techniques of [28 and 29] is work in progress. Several important problems remain open, including: 1. The rigorous investigation of the formalism presented here and in particular the investigation of the existence of “vanishing lemmas” (which are needed in order to eliminate the “small norm” assumptions). 2. The formal and rigorous analysis of the global relation. 3. The investigation of the existence of coherent structures (solitons). In particular Eqs. (3.11) and (3.12) are similar to the equations describing the evolution of the scattering data for the case that DS1 supports dromions [30]. Thus, we expect that the formalism presented here supports coherent structures (see also [31]). These coherent structures should arise from homogeneous solutions of the equations for (ϕ1+ , ϕ2+ ). Acknowledgement. The author acknowledges support from EPSRC.
Appendix Proposition A.1. Let q, f satisfy Eqs. (1.2) and let µ satisfy Eqs. (1.5). Then
2 2 i q¯ µ¯ 2 e−ilz+i(2k1 l+2λl−l )t = (q¯ z µ¯ 2 − 4 f¯µ¯ 1 − q¯ µ¯ 2z )e−il z¯ +i(2k1 l+2λl−l )t t z −il z¯ +i(2k l+2λl−l 2 )t 1 + (q¯ z¯ + i(l − 2k1 − 2λ)q) ¯ µ¯ 2 + q¯ µ¯ 2z¯ e . z¯
(A.1)
Davey-Stewartson Equation on the Half-Plane
991
Furthermore, suppose that the DS system is valid for (x, y) ∈ 0 ⊂ R2 . Then −ilz+i(2k1 l+2λl−l 2 )t 2 q¯ z µ¯ 2 − 4 f¯µ¯ 1 − q¯ µ¯ 2z d z¯ d xd y q¯ µ¯ 2 e = 0
t
∂0
2 − (q¯ z¯ + i(l − 2k1 − 2λ)q) ¯ µ¯ 2 + q¯ µ¯ 2z¯ dz e−il z¯ +i(2k1 l+2λl−l )t .
(A.2)
Proof. Let E = e−il z¯ +i(2k1 l+2λl−l
2 )t
.
(A.3)
Then ¯ µ¯ 2 )t i(q¯ µ¯ 2 e)t = E(i q¯t )µ¯ 2 + E q¯ µ¯ 2 (l 2 − 2k1l − 2λl) + E q(i 2 ¯ = E µ¯ 2 [q¯z¯ z¯ + q¯ zz + 4( f + f )q] ¯ + E q¯ µ¯ 2 (l − 2k1l − 2λl) −E q¯ µ¯ 2zz + i(k1 + λ)µ¯ 2z¯ − q µ¯ 1z¯ + qz¯ µ¯ 1 + 4 f µ¯ 2 ,
(A.4)
where we have used Eqs. (1.2a) and (1.5d) to replace q¯t and µ¯ 2t . Using the equations E µ¯ 2 q¯z¯ z¯ = (E µ¯ 2 q¯ z¯ )z¯ + il E µ¯ 2 q¯ z¯ − E µ¯ 2z¯ q¯ z¯ , E µ¯ 2 q¯ zz = (E µ¯ 2 q¯ z )z − E µ¯ 2z q¯ z , 4 f¯ E µ¯ 2 q¯ = 4E f¯[−µ¯ 1z ] = −(4E f¯µ¯ 1 )z + 4E f¯z µ¯ 1 , −E q¯ µ¯ 2zz = −(E q¯ µ¯ 2z )z + E q¯ z µ¯ 2z ,
(A.5a) (A.5b) (A.5c) (A.5d)
as well as the equations 4E f¯z µ¯ 1 = 2E µ¯ 1 |q|2z¯ , −E µ¯ 2z¯ q¯ z¯ = −E q¯ z¯ [q µ¯ 1 + i(k1 + λ)µ¯ 2 ], it follows that the terms in (A.4) which are not total derivatives with respect to either z or z¯ are the following: E il[−il q¯ µ¯ 2 + q¯ z¯ µ¯ 2 ] − i(k1 + λ)[q¯ µ¯ 2z¯ + q¯ z¯ µ¯ 2 − il q¯ µ¯ 2 ] − i(k1 + λ)(−il)q¯ µ¯ 2 + 2[µ¯ 1 |q|2z¯ + µ¯ 1z¯ |q|2 ] − [q q¯ µ¯ 1z¯ − qq ¯ z¯ µ¯ 1 − q q¯ z¯ µ¯ 1 ] , (A.6) where we have written |q|2 µ¯ 1z¯ as 2|q|2 µ¯ 1z¯ − q q¯ µ¯ 1z¯ and also have written the term −2l(k1 + λ)q¯ µ¯ 2 as two different terms each with coefficient 1. The terms in (A.6) yield il(E q) ¯ z¯ µ¯ 2 − i(k1 + λ)(q¯ µ¯ 2 E)z¯ − i(k1 + λ)(−il)q¯ µ¯ 2 E (A.7) +2E(|q|2 µ¯ 1 )z¯ − E(q q¯ µ¯ 1 )z¯ . Replacing i(k1 + λ)µ¯ 2 by q µ¯ 1 − µ¯ 2z¯ the expression in (A.7) becomes −i(k1 + λ)(q¯ µ¯ 2 E)z¯ + (E|q|2 µ¯ 1 )z¯ + il(E q¯ µ¯ 2 )z¯ . The sum of these terms as well as of the terms in (A.5) involving total derivatives yields the rhs of Eq. (A.1), where we have used ¯ µ¯ 1 ) = q¯ µ¯ 2z¯ − i(k1 + λ)µ¯ 2 . |q|2 µ¯ 1 = q(q
992
A. S. Fokas
In the particular case that 0 is the upper half complex z-plane after using integration by parts to eliminate µ¯ 2z Eq. (A.2) becomes ∞ ∞ −ilz+i(2k1 l+2λl−l 2 )t 2 dx dy q¯ µ¯ 2 e = −∞ 0 t ∞ 2 d x − −g¯ 0x + i g¯ 2 + 2i(l − k1 − λ)g¯ 0 ϕ¯2 − 4 f¯ϕ¯ 1 e−ilx+i(2k1 l+2λl−l )t , −∞
(A.8) where we have used integration by parts to eliminate µ¯ 2x .
References 1. Fokas, A.S.: On the inverse scattering of first order systems in the plane related to nonlinear multidimensional equations. Phys. Rev. Lett. 51, 3–6 (1983) 2. Fokas, A.S., Ablowitz, M.J.: On the inverse scattering of the time dependent Schrödinger equation and the associated KPI equation. Stud. Appl. Math. 69, 211–228 (1983) 3. Ablowitz, M.J., BarYaacov, D., Fokas, A.S.: On the inverse scattering transform for the Kadomtsev-Petvisvhili equation. Stud. Appl. Math. 69, 135–143 (1983) 4. Fokas, A.S., Ablowitz, M.J.: On the inverse scattering transform of multi-dimensional nonlinear equations related to first order systems in the plane. J. Math. Phys. 25, 2494–2505 (1984) 5. Zhou, X.: Inverse scattering transform for the time dependent Schrödinger equation with application to the KPI equation. Commun. Math. Phys. 128, 551–564 (1990) ¯ 6. Beals, R., Coifman, R.R.: Linear spectral problems, nonlinear equations and the ∂-method. Inverse Problems 5, 87 (1989) 7. Fokas, A.S., Sung, L.Y.: On the solvability of the N-wave, the Davey-Stewartson and the KadomtsevPetviashvili equation. Inverse Problems 8, 673–708 (1992) 8. Fokas, A.S.: A unified transform method for solving linear and certain nonlinear PDE’s. Proc. R. Soc. Lond. A 453, 1411–1443 (1997) 9. Pelloni, B.: The spectral representation of two-point boundary value problems for linear evolution equations. Proc. Roy. Soc. Lond. A 461, 2965–2984 (2005) 10. Fokas, A.S., Schultz, P.F.: The long-time asymptotics of moving boundary problems using an ehrenpreis-type representation and its Riemann-Hilbert nonlinearisation. Comm. Pure Appl. Maths LVI, 1–40 (2002) 11. Fokas, A.S., Sung, L.Y.: Generalised fourier transforms, their nonlinearisation and the imaging of the brain, Notices of the AMS. Feature Article 52, 1176–1190 (2005) 12. Fokas, A.S.: Two dimensional linear PDEs in a convex polygon. Proc. R. Soc. Lond. A 457, 371–393 (2001) 13. Fokas, A.S.: A generalised dirichlet to neumann map for certain nonlinear evolution PDEs. Comm. Pure Appl. Math. LVIII, 639–670 (2005) 14. Fokas, A.S.: Integrable nonlinear evolution equations on the half-line. Commun. Math. Phys. 230, 1–39 (2002) 15. Boutetde Monvel, A., Fokas, A.S., Shepelsky, D.: The analysis of the global relation for the nonlinear Schrödinger equation on the half-line. Lett. Math. Phys. 65, 199–212 (2003) 16. Boutetde Monvel, A., Fokas, A.S., Shepelsky, D.: Integrable nonlinear evolution equations on the interval. Commun. Math. Phys. 263, 133–172 (2006) 17. Biondini, G., Hwang, G.: Initial-boundary value problems for differential-difference evolution equations: Discrete linear and integrable nonlinear Schrödinger equations. http://arxiv.org/abs/0810.1300v1[nlin. SI], 2008, to appear in Inverse Problems 18. Fokas, A.S.: A Unified Approach to Boundary Value Problems. Philadelphia, PA: SIAM, 2008 19. Fokas, A.S.: A new transform method for evolution PDEs. IMA J. Appl. Math. 67, 1–32 (2002) 20. Fokas, A.S., Zyskin, M.: The fundamental differential form and boundary-value problems. Q. J. Mech. App. Maths. 55, 457–479 (2002) 21. Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of inverse scattering problem, Part I. Funct. Annal. Appl. 8, 43 (1974); A Scheme for Integrating the Nonlinear Equations of Mathematical Physics by the Method of Inverse Scattering Problem, Part II, Funct. Annal. Appl. 13, 13 (1979)
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22. Fokas, A.S., Its, A.R., Sung, L.Y.: The nonlinear schroedinger equation on the half-line. Nonlinearity 18, 1771–1822 (2005) 23. Flyer, N., Fokas, A.S.: A hybrid analytical numerical method for solving evolution partial dierential equations. I: The Half-Line. Proc. R. Soc. 464, 1823–1849 (2008) 24. Zheng, C.: Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equation. J. Comput. Phys. 215, 552–565 (2006); C. Zheng, Numerical Simulation of a Modified KdV Equation On the Whole Real Axis. Numer. Math. 105, 315–335 (2006) 25. Smitheman, S., Spence, E.A., Fokas, A.S.: A spectral collocation method for the laplace and the Modified Helmholtz equation in a convex polygon. IMA J. Num. Anal., in press 26. Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann-Hilbert problems. Bull. Am. Math. Soc. 20, 119 (1992) 27. Deift, P., Venakides, S., Zhou, X.: New results in the small dispersion KdV by an extension of the method of steepest descent for Riemann-Hilbert problems. IMRN 6, 285–299 (1997) 28. Sung, L.Y.: Long-time decay of the solutions of the Davey-Stewartson II equations. J. Nonlinear Sc. 5, 433–452 (1995) 29. Kiselev, O.M.: Asymptotics of a solution of the Kadomtsev-Petviashvili-2 equation. Proc. Steklov Institute of Mathematics 1, 107–139 (2001) 30. Fokas, A.S., Santini, P.M.: Dromions and a boundary value problem for the Davey-Stewartson I Equation. Physica D 44, 99–130 (1990) 31. Fokas, A.S.: The Davey Stewartson I equation on the quarter plane with homogeneous Dirichlet boundary conditions. J. Math. Phys. 44, 3226–3244 (2003) Communicated by P. Constantin
Commun. Math. Phys. 289, 995–1021 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0815-3
Communications in
Mathematical Physics
A Tropical Analogue of Fay’s Trisecant Identity and the Ultra-Discrete Periodic Toda Lattice Rei Inoue1 , Tomoyuki Takenawa2 1 Faculty of Pharmaceutical Sciences, Suzuka University of Medical Science,
3500-3 Minami-Tamagaki, Suzuka, Mie, 513-8670, Japan. E-mail: [email protected] 2 Faculty of Marine Technology, Tokyo University of Marine Science and Technology, 2-1-6 Etchu-jima, Koto-ku, Tokyo, 135-8533, Japan. E-mail: [email protected] Received: 20 June 2008 / Accepted: 10 February 2009 Published online: 30 April 2009 – © Springer-Verlag 2009
Abstract: We introduce a tropical analogue of Fay’s trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann’s theta function. 1. Introduction 1.1. Background and main results. Fay’s trisecant identity is an important and special identity satisfied by Riemann’s theta functions for the Jacobian of a curve [1,12], and plays a crucial role in studying classical integrable systems. For instance, Fay’s trisecant identity gives a solution to the Hirota-Miwa equation [2,11] from which many soliton equations derive. Recently a tropical analogue of Riemann’s theta function was introduced in some contexts ([8,10], in [14] the tropical Riemann’s addition formula is introduced). The first aim of this paper is to introduce a tropical version of Fay’s trisecant identity (Theorem 2.4). Due to a technical difficulty, our result is restricted to the case of some special hyperelliptic tropical curves, but we expect that it also holds for general tropical curves. On the other hand, in [4] we studied the ultra-discrete periodic Toda lattice (UD-pToda) and proposed a method to deal with integrable cellular automata via the tropical algebraic geometry. Since the UD-pToda is closely related to an important soliton cellular automata called the box and ball system [15], it is regarded as a fundamental object in the studies of integrable automata. The UD-pToda is defined by the following piecewise-linear map t t t Q t+1 Wnt+1 = Q tn+1 + Wnt − Q t+1 n = min[Wn , Q n − X n ], n , k t (Wn−l − Q tn−l ) , with X nt = min k=0,...,g
l=1
(1.1)
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on the phase space: T = {(Q, W ) = (Q 1 , . . . , Q g+1 , W1 , . . . , Wg+1 ) ∈ R2(g+1) |
g+1 n=1
Qn <
g+1
Wn }.
n=1
(1.2) t Here we fix g ∈ Z>0 and assume periodicity Q tn+g+1 = Q tn and Wn+g+1 = Wnt . This system has a tropical spectral curve , and we conjectured that its general isolevel set is isomorphic to the tropical Jacobian J () of (the cases of g = 1, 2, 3 were proved). It is expected that the solution for the UD-pToda is written in terms of the tropical Riemann’s theta function associated with as the classical cases, which is another aim of this paper. For this purpose we transform (1.1) into an equation for the quasi-periodic function Tnt ,
Tnt+2 + Tnt = min
k=0,...,g
t+1 t t t k L + 2Tn−k + Tn+1 + Tnt − (Tn−k+1 + Tn−k ) ,
(1.3)
via t+1 t + Tn+1 − Tnt − Tnt+1 + Cg , Wnt = L + Tn−1 t t+1 Q tn = Tn−1 + Tnt+1 − Tn−1 − Tnt + Cg .
(1.4)
Here L and Cg are determined by {Q tn , Wnt }n=1,...,g+1 and preserved by the map (1.1). We show that Eq. (1.3) is essentially equivalent to a tropical bilinear equation (Proposition 3.4): t−1 t+1 + Tn+1 + L], Tnt−1 + Tnt+1 = min[2 Tnt , Tn−1
(1.5)
which turns out to be a particular case of Fay’s trisecant identity for the tropical Riemann’s theta function (Corollary 2.13). Finally we obtain the general solution for the UD-pToda in terms of the tropical Riemann’s theta function (Theorem 3.5). In the following three subsections, we introduce some fundamental notions related to tropical geometry used in this article and the background results on the relation between the discrete periodic Toda lattice (D-pToda) and the UD-pToda.
1.2. Tropical Jacobian. Definition 1.1 [9]. A finite connected graph → R2 is called a (plane) tropical curve if the weight we ∈ Z>0 is defined for each edge e and the following is satisfied: (i) The tangent vector of each edge is rational. (ii) For each vertex v, let e1 , . . . , en be the oriented edges outgoing from v. Then the primitive tangent vectors ξek of ek satisfy nk=1 wek ξek = 0. Further, a tropical curve is called smooth if the following is satisfied: (iii) All the weights are 1. (iv) Each vertex v is 3-valent and the primitive tangent vectors satisfy |ξei ∧ ξe j | = 1 for i = j ∈ {1, 2, 3}.
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
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Let ˜ be a smooth tropical curve, := ˜ \ {infinite edges} be the maximal compact ˜ g be the genus of , i.e. g = dim H1 (, Z), and B1 , . . . , Bg be a basis of subset of , H1 (, Z). Following [10, Sect. 3.3], we equip with the structure of a metric graph. For points x, y on some edge e in , we define the weighted distance d(x, y) by d(x, y) =
x−y , ξe
where . denotes the Euclidean norm in R2 . With this distance the tropical curve becomes a metric graph. The metric on defines a symmetric bilinear form ·, · on the space of paths on as follows: for a non-self-intersecting path ρ, set ρ, ρ := lengthd (ρ), and extending it to any pairs of paths bilinearly. Definition 1.2 [10, Sect. 6.1]. The tropical Jacobian of is a g dimensional real torus defined as J () = H1 (, R)/H1 (, Z) Rg /K Zg . Here K ∈ Mg (R) are given by K i j = Bi , B j .
(1.6)
Since ·, · is nondegenerate, K is symmetric and positive definite. In particular, J () is called principally polarized. Fix P0 ∈ . Let Div() be the set of divisors on . Following [10] we define the tropical Abel-Jacobi map η : Div() → J () by i
ki Pi →
Pi
ki
i
:=
P0
ki ( PP0i , B1 , . . . , PP0i , Bg ),
(1.7)
i
where ki is a nonzero integer and PP0i is a path from P0 to Pi on . For example, the divisor P2 − P1 is mapped to η(P2 − P1 ) =
P2 P1
= ( PP12 , B1 , . . . , PP12 , Bg ).
1.3. Ultra-discrete limit. The ultra-discrete limit (UD-limit) links discrete dynamical systems to cellular automata, and algebraic curves to tropical curves. This limit is also called tropicalization. We define a map Logε : R>0 → R with an infinitesimal parameter ε > 0 by Logε : x → −ε log x. X
(1.8)
For x > 0, we define X ∈ R by x = e− ε . Then the limit ε → 0 of Logε (x) converges X to X . The procedure limε→+0 Logε with the scale transformation as x = e− ε is called the UD-limit. We summarize this procedure in more general setting:
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Proposition 1.3. For A, B, C ∈ R and ka , kb , kc > 0, set A
B
C
a = ka e− ε , b = kb e− ε , c = kc e− ε . Then the UD-limit of the equations (i) a + b = c,
(ii) ab = c,
(iii) a − b = c
yields the following: (i) min[A, B] = C, (ii) A + B = C, A=C (if A < B, or A = B and ka > kb ) . (iii) contradiction (otherwise) 1.4. From D-pToda lattice to UD-pToda. We briefly review the D-pToda and the way to obtain the UD-pToda. Fix g ∈ Z>0 . The (g + 1)-periodic Toda lattice of discrete time t ∈ Z [3] is given by the rational map on the phase space U = {(I1 , . . . , Ig+1 , V1 , . . . , Vg+1 )} C2(g+1) : t+1 Int+1 = Int + Vnt − Vn−1 ,
Vnt+1 =
t Vt In+1 n , Int+1
(1.9)
t t where we assume the periodicity In+g+1 = Int and Vn+g+1 = Vnt . This system has the (g + 1) by (g + 1) Lax matrix: ⎞ ⎛ bt a1t 1 (−1)g y1 ⎟ ⎜ ⎟ ⎜ b2t a2t 1 ⎟ ⎜ t ⎟. ⎜ . . . (1.10) L (y) = ⎜ .. .. .. ⎟ ⎟ ⎜ t t ⎝ bg ag 1 ⎠ t t bg+1 ag+1 (−1)g y t + V t , bt = I t V t . The evolution Here y ∈ C is a spectral parameter, and we set ant = In+1 n n n n (1.9) preserves the algebraic curve γ given by f (x, y) = y det(x E + L t (y)) = 0. When we fix a polynomial f (x, y):
f (x, y) = y 2 + y(x g+1 + cg x g + · · · + c1 x + c0 ) + c−1 ,
(1.11)
the isolevel set Uc for (1.9) is Uc = {(Int , Vnt )n=1,...,g+1 ∈ U | y det(x E + L t (y)) = f (x, y)}. It is known that for generic f (x, y), Uc is isomorphic to an affine part of the Jacobian Jac(γ ) of γ [13]. Equations (1.9) are rewritten as [7] 1−
Int+1 = Vnt + Int 1+ Vnt+1 =
t Vt In+1 n Int+1
.
t Vn−1 t In−1
g+1 n=1 g+1
Vnt
t n=1 In t ···V t Vn−1 n−g t ···I t In−1 n−g
+ ··· +
, (1.12)
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
999
Fig. 1. Tropical curve ˜
g+1 t t n=1 In > n=1 Vn , the UD-limit of (1.12) with the scale transQt Wt − εn − εn t t formation In = e , Vn = e gives the UD-pToda (1.1). The UD-pToda preserves the tropical curve ˜ ⊂ R2 given by [4]:
Under the condition
g+1
˜ = {(X, Y ) ∈ R2 | F(X, Y ) is not smooth}, (1.13) where F(X, Y ) = min[2Y, Y + min[(g + 1)X, g X + Cg , . . . , X + C1 , C0 ], C−1 ]. Here Ci ’s are regarded as tropical polynomials on T (1.2). For generic (Q, W ) ∈ T, C = (C−1 , C0 , . . . , Cg ) ∈ Rg+2 satisfies C−1 > 2C0 , Ci + Ci+2 > 2Ci+1 (i = 0, . . . , g − 2), Cg−1 > 2Cg ,
(1.14)
and we refer to it as the generic condition. With this condition, ˜ becomes a smooth tropical curve. We show the shape of ˜ with the basis Bi ’s of H1 (, Z) at Fig. 1.
1.5. Content. In Sect. 2, we introduce a tropical analogue of Fay’s trisecant identity (Theorem 2.4) and obtain the bilinear form of the UD-pToda (1.5) as its particular case. In Sect. 3, we study the relation between the UD-pToda (1.1) and the bilinear form (1.5) (Lemma 3.2), and obtain the general solutions in terms of the tropical Riemann’s theta function (Theorem 3.5). Appendix A is devoted to prove Theorem 2.6 which is a key to Theorem 2.4.
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1.6. Notations. We use the following notations of vectors in Rg : g ei I Ik
= (g, g − 1, . . . , 1), : the i th vector of standard basis of Rg , = (1, 1, . . . , 1) = e1 + · · · + eg , = (1, . . . , 1, 0, . . . , 0) = e1 + · · · + ek .
2. Fay’s Trisecant Identity and its Tropicalization 2.1. Fay’s trisecant identity for hyperelliptic curves. Let γ be the hyperelliptic curve 2g+2 given by v 2 = i=1 (u − u i ), which defines the two-sheeted covering u ± of u with branches [u 2k+1 , u 2k+2 ] (k = 0, 1, 2, . . . , g). Choose the basis a1 , . . . , ag , b1 , . . . , bg of H1 (γ , Z) as usual as (a) ak goes the circuit around the branch [u 2k+1 , u 2k+2 ] on u + , (b) bk goes on the upper half of u + from [u 1 , u 2 ] to [u 2k+1 , u 2k+2 ] and goes on the lower half of u − from [u 2k+1 , u 2k+2 ] to [u 1 , u 2 ]. Let ω1 , . . . , ωg be a basis of the holomorphic differentials H 0 (γ , 1 ) normalized so that the period matrix with respect to a1 , . . . , ag , b1 , . . . , bg has the form (I, ), where I is the g × g identity matrix and ∈ Mg (C) is the symmetric matrix whose imaginary part is positive definite. We write Jac(γ ) = Cg /(Zg I + Zg ) for the Jacobian of γ . We define a generalization of Riemann’s theta function as √ θ [α, β](z) := exp{π –1(β β ⊥ + 2β(z + α)⊥ )}θ (z + β + α) for α, β ∈ Rg and z ∈ Cg , where Riemann’s theta function θ (z) is √ θ (z) = θ [0, 0](z) = exp{π –1(m m⊥ + 2mz⊥ )}. m∈Z
(2.1)
g
We set Q i = (u i , 0) ∈ γ , and take Q 1 as a base point of the Abel-Jacobi map, Pi η : Div(γ ) → Jac(γ ); ki Pi → ki ( ω j ) j=1,...,g , (2.2) i
i
Q1
where ki ∈ Z, Pi ∈ γ . Let K γ be the canonical divisor on γ and φ be the hyperelliptic involution of γ (interchanging the two sheets u ± ). One sees that := −Q 1 + Q 3 + Q 5 + · · · + Q 2g+1 ∈ Picg−1 (γ ) is a theta characteristic (i.e. 2 = K γ in Pic2g−2 (γ )) and that D := P +φ(P) ∈ Pic2 (γ ) for P ∈ γ satisfies η(D) = 0. Via the Abel-Jacobi map (2.2), the theta characteristics correspond to the half-periods of Jac(γ ). For instance we have 1 g g − 1 ··· 1 , η() = 2 1 1 ··· 1
α where = α I + β ∈ Cg for α, β ∈ Rg . β
{1, 2, . . . , 2g + 2}. The For m = 0, 1, . . . , [ g+1 2 ], let {i 1 , . . . , i g+1−2m } be a subset of g+1 following is known [1, pp 12-15]: for m = 0, η( + D − k=1 Q ik ) are the nong−1 singular even half-periods, while for m = 1, η( − k=1 Q ik ) are the non-singular g+1−2m odd half-periods, and for m > 1, η( − (m − 1)D − k=1 Q ik ) are the even (odd) singular half-periods of multiplicity m when m is even (odd). By using the formulae:
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
η(Q 2 ) =
1 2
1 1 ··· 1 0 0 ··· 0
η(Q 2k+2 ) =
⎛
⎛
1001
, η(Q 2k+1 ) = ⎞
k
k
⎞
1⎜ ∨ ⎟ ⎝0 ··· 0 1 1 ··· 1⎠ , 2 0 ··· 0 1 0 ··· 0
1⎜ ∨ ⎟ ⎝ ⎠ , 2 0 ··· 0 0 1 ··· 1 0 ··· 0 1 0 ··· 0
we obtain the following: Proposition 2.1. For a hyperelliptic curve γ and a half integer vector β(= 0) ∈ 1 g 1 g g 2 Z / Z , set α ∈ 2 Z as i ∨
α = (0, . . . , 0,
, 0, . . . , 0), α where βi = 21 . Then is a non-singular odd half-period of Jac(γ ). β
1 2
Proof. It is elementarily shown by using the fact that singular half-periods have at least two nonzero entries in α. α , and denote the corresponding (non-singular Fix a non-singular odd half-period β
odd) theta characteristic by δ. Define the half order differential h δ (x) on γ by h 2δ (x) =
g ∂θ [α, β] i=1
∂z i
(0)ωi (x).
Here h δ (x) is the holomorphic section of the line bundle corresponding to δ. Then the prime form is defined by E(x, y) =
θ [α, β](η(y − x)) h δ (x)h δ (y)
(2.3)
for x, y ∈ γ . We do not use h δ (x) in this paper except in the following theorem. See [1] or [12] for general settings other than the hyperelliptic case. Theorem 2.2 [1, Eq. (45)]. Let γ be a hyperelliptic curve, θ (z) be the Riemann’s theta function (2.1), and E(x, y) be the prime form (2.3). Then for P1 , P2 , P3 , P4 in the universal covering space of γ and z ∈ Cg , the formula P3 P4 θ (z + )θ (z + )E(P3 , P2 )E(P1 , P4 ) P1
+θ (z + = θ (z + holds, where
Pj Pi
P2 P3
)θ (z +
P2 P3 +P4
P4
)E(P3 , P1 )E(P4 , P2 )
P1
)θ (z)E(P3 , P4 )E(P1 , P2 )
P1 +P2
denotes η(P j − Pi ).
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R. Inoue, T. Takenawa
By eliminating the common denominator, we have P3 P4 P2 θ (z + )θ (z + )θ [α, β]( )θ [α, β]( P1
+ θ (z + = θ (z +
P2 P3
)θ (z +
P2 P3 +P4
P3 P4
)θ [α, β](
P1
)θ (z)θ [α, β](
P1 +P2
)
P1 P1
)θ [α, β](
P3 P4
P4
)
(2.4)
P4
P2
)θ [α, β](
P3
P2
).
P1
2.2. Tropical analogue of Fay’s identity. For a positive definite symmetric matrix K ∈ Mg (R) and β ∈ Rg we define qβ (m, Z) =
1 mK m⊥ + m(Z + β K )⊥ 2
(Z ∈ Rg , m ∈ Zg ),
and write the tropical Riemann’s theta function as (Z) = min q0 (m, Z) m∈Zg
(Z ∈ Rg ).
Let us introduce a generalization of the tropical Riemann’s theta function: [β](Z) :=
1 β Kβ ⊥ + βZ⊥ + ming qβ (m, Z). 2 m∈Z
We write argm∈Zg qβ (m, Z) for m ∈ Zg , where qβ (m, Z) takes the minimum value. Proposition 2.3. The function [β](Z) satisfies the following properties: (i) the periodicity in β: [β + l](Z) = [β](Z) (l ∈ Zg ), (ii) the quasi-periodicity in Z: 1 [β](Z + lK ) = − lK l⊥ − lZ⊥ + [β](Z) (l ∈ Zg ), 2 (iii) the symmetry in β and Z: [β](−Z) = [−β](Z). (iv) If β ∈
1 2
Zg / Zg , then [β](Z) is an even function with respect to Z and β.
Proof. (i) and (ii). Replace m by m − l in 1 1 [β + l](Z)= (β + l)K (β + l)⊥ + (β + l)Z⊥ + ming [ mK m⊥ + m(Z + (β + l)K )⊥ ] 2 m∈Z 2 and [β](Z + lK ) =
1 1 β Kβ ⊥ + β(Z + lK )⊥ + ming [ mK m⊥ + m(Z + lK + β K )⊥ ]. 2 m∈Z 2
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
1003
(iii) Replace m by −m in [β](−Z) =
1 1 β Kβ ⊥ − βZ⊥ + ming [ mK m⊥ + m(−Z + β K )⊥ ]. 2 m∈Z 2
(iv) By (i) and (ii), if β ∈
Zg / Zg , we have
1 2
[β](−Z) = [−β](Z) = [−β + 2β](Z) = [β](Z). In the rest of this section, let us restrict ourselves to the case of a family of hyperelliptic tropical curves given by (1.13) with the generic condition (1.14). We define L, λ0 , λ1 , . . . , λg and p1 , . . . , pg by L = C−1 − 2(g + 1)Cg , λi = Cg−i − Cg−i+1 ,
pi = L − 2
g
λ0 = C g ,
min[λi − λ0 , λ j − λ0 ] (for 1 ≤ i ≤ g, ).
j=1
(2.5) = (λ1 − λ0 , . . . , λg − λg−1 ). Due to the condition (1.14) one sees We set λ λ0 < λ1 < λ2 < · · · < λg ,
0 < pg < pg−1 < · · · < p1 ,
2
g
(λi − λ0 ) < L ,
i=1
= g · λ
g (λi − λ0 ) = C0 − (g + 1)Cg .
(2.6)
i=1
We show the maximal compact subset of ˜ in Fig. 2, where a scalar on each edge denotes its lengthd . The period matrix K = (K i j ) (1.6) for becomes ⎧ ⎪ pi−1 + pi + 2(λi − λi−1 ) > 0, where p0 = L (i = j) ⎪ ⎪ ⎨− p < 0 ( j = i + 1) i Ki j = ⎪ − p j < 0 (i = j + 1) ⎪ ⎪ ⎩0 (otherwise).
(2.7)
Theorem 2.4 (Tropical analogue of Fay’s trisecant identity). Let ˜ be a smooth tropical curve given by (1.13) with the generic condition (1.14). For β ∈ 21 Zg (β = 0 mod Zg ), set α ∈ 21 Zg as i ∨
α = (0, . . . , 0, − 21 ,
1 2
, 0, . . . , 0),
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R. Inoue, T. Takenawa
Fig. 2. and its metric
where β j = 0 for 1 ≤ j ≤ i − 1 and βi = 0. For P1 , P2 , P3 , P4 on the universal covering space of , we define the sign si ∈ {±1} (i = 1, 2, 3) as si = (−1)ki , where P2 P4 k1 = 2α · argm∈Zg qβ (m, ) + argm∈Zg qβ (m, ) , g k2 = 2α · argm∈Z qβ (m,
P3 P1
) + argm∈Z qβ (m,
P1 P2
g
P3
k3 = 1 + 2α · argm∈Zg qβ (m,
P4 P3
P4
) ,
) + argm∈Zg qβ (m,
P2
) .
P1
g
For Z ∈ R , set F1 , F2 , F3 ∈ R as P3 F1 = (Z + ) + (Z + F2 = (Z + F3 = (Z +
P1 P3
) + (Z +
P2 P3 +P4
P4
P2 P4
) + [β]( ) + [β](
P1
) + (Z) + [β](
P1 +P2
P3 P1
P3
Then, the formula Fi = min[Fi+1 , Fi+2 ]
P4
) + [β]( ) + [β](
P3
P4
holds if si = ±1, si+1 = si+2 = ∓1 for i ∈ Z /3 Z.
P2
) + [β](
P1 P2 P4
P2 P1
).
), ),
(2.8)
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
Remark 2.5.
1005
(i) The case of s1 = s2 = s3 does not occur.
P (ii) The sign si is not determined when the corresponding argm∈Zg qβ (m, P jk ) is not unique. Sometimes it is possible to move the points P j ’s slightly so that si is determined. But it can not be done always. (See Example 2.12.) The following theorem is the key to the proof of Theorem 2.4, which links the Abel-Jacobi map on γ to that on : Theorem 2.6. By the UD-limit with the scale transformation |x| = e−X/ε , |y| = e−Y /ε , z = −
K˜ i j Z , i j = − √ , √ 2π –1 ε 2π –1 ε
(2.9)
the Abel-Jacobi map (2.2) on γ becomes the tropical Abel-Jacobi map (1.7) on as η : Div() → J ();
i
m i Pi →
m i ( PP0i , B˜ j ) j=1,...,g ,
i
where P0 ∈ is a base point and B˜ j = Bg− j+1 + Bg− j+2 + · · · + Bg . In this limit, K˜ becomes K˜ i j = B˜ i , B˜ j . We will prove it analytically as a variation of the result in [6]. Since it is straightforward but tedious, we give it in the Appendix. Remark 2.7. (i) The cycles B˜ j ’s are obtained from B j ’s by the base change as ( B˜ 1 , . . . , B˜ g ) = (B1 , . . . , Bg )T with ⎛ ⎞ 0 1 ⎜ ⎟ T = ⎝ . . . ... ⎠ ∈ Mg (C). 1 ... 1 Thus the tropical Abel-Jacobi map (1.7) is obtained from the complex Abel-Jacobi map (2.2) through the UD-limit. (ii) We have K˜ = T K T which corresponds to the base change of H1 (γ , Z) as (ai )i → (ai )i T and (bi )i → (bi )i T −1 = (bg−i+1 − bg−i )i . Thus α I + β of Theorem 2.4 corresponds to αT + βT −1 which is a nonsingular odd half period of Jac(γ ) from Proposition 2.1. Proof of Theorem 2.4. By changing the basis of H1 (γ , Z) from (ai )i , (bi )i to (ai )i T, (bi )i T −1 , the limit of (bi )i T −1 becomes (Bi )i . By the scale transformation of z and
as (2.9) the theta function θ [α, β](z) becomes √
1 1 exp{− ( β Kβ ⊥ + βZ⊥ )} ε 2 √ 1 1 ⊥ e2π –1 mα exp{− ( mK m⊥ + m(Z + Kβ)⊥ )}. × ε 2 g
θ [α, β](z) = e2π
–1 βα ⊥
m∈Z
(2.10)
1006
R. Inoue, T. Takenawa √
√
–1 βα ⊥
⊥
= ±1 is a common factor and e2π –1 mα = ±1, we can set P3 P4 P2 P4 √ ⊥ f 1 := e−4π –1 βα θ (z + )θ (z + )θ [α, β]( )θ [α, β]( ) = f 1+ − f 1− ,
Step 1. Since e4π
P1
P2
P3
P1
(2.11) √
√
⊥
⊥
where we take f 1+ > 0 and f 1− > 0 as the part of e2π –1 mα = 1 and of e2π –1 mα = −1 respectively. Similarly we set P3 P4 P1 P2 √ −4π –1 βα ⊥ θ (z + )θ (z + )θ [α, β]( )θ [α, β]( ) = f 2+ − f 2− , f 2 := e f 3 := e−4π
√
–1 βα ⊥
θ (z +
P2 P3 +P4
P1
)θ (z)θ [α, β](
P1 +P2
P3
P4 P3
P4
P2
)θ [α, β](
P1
) = f 3+ − f 3− .
Then f 1+ + f 2+ + f 3− = f 1− + f 2− + f 3+ holds from (2.4). We write Fi± for the UD-limit of f i± . Then we obtain Fi = min[Fi+ , Fi− ] and min[F1+ , F2+ , F3− ] = min[F1− , F2− , F3+ ]. Now we have the following cases: If F1+ < F1− , F2+ < F2− and F3+ < F3− , then min[F1 , F2 ] = F3 . If F1+ > F1− , F2+ > F2− and F3+ > F3− , then min[F1 , F2 ] = F3 . If F1+ < F1− , F2+ > F2− and F3+ < F3− , then min[F2 , F3 ] = F1 . If F1+ > F1− , F2+ < F2− and F3+ > F3− , then min[F2 , F3 ] = F1 . If F1+ < F1− , F2+ > F2− and F3+ > F3− , then min[F3 , F1 ] = F2 . If F1+ > F1− , F2+ < F2− and F3+ < F3− , then min[F3 , F1 ] = F2 . If F1+ < F1− , F2+ < F2− and F3+ > F3− , then f 1+ + f 2+ + f 3− > f 1− + f 2− + f 3+ for sufficiently small ε, which is a contradiction. (viii) If F1+ > F1− , F2+ > F2− and F3+ < F3− , then f 1+ + f 2+ + f 3− < f 1− + f 2− + f 3+ for sufficiently small ε, which is a contradiction. (i) (ii) (iii) (iv) (v) (vi) (vii)
Step 2. We check the definition of si . By (2.10) and (2.11), s1 = 1 means e
π
√
P –1 α·argm∈Zg q0 (m, P 2 ) 3
·e
π
√ P –1 α·argm∈Zg q0 (m, P 4 ) 1
= 1,
and thus F1 = F1+ . Similarly we have F1 = F1− if s1 = −1, F2 = F2+ if s2 = 1, F2 = F2− if s2 = −1, F3 = F3+ if s3 = −1 and F3 = F3− if s3 = 1. From Steps 1 and 2 the claim follows. 2.3. Bilinear equation of Toda type. We return to the definition of the tropical Riemann’s theta function and investigate the fundamental regions. We define the fundamental region Dm of (Z) as Dm = {Z ∈ Rg | (Z) = 21 mK m⊥ + mZ⊥ }, which is explicitly written as 1 Dm = {Z ∈ Rg | − lZ⊥ ≤ lK (m + l)⊥ for any l ∈ Zg }. 2 Note that argm∈Zg q0 (m, Z) = m if and only if Z is in the interior of Dm . See Fig. 3 for the g = 2 case. We easily see the following:
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
1007
Fig. 3. Fundamental regions of g = 2 (Z = (Z 1 , Z 2 ))
Lemma 2.8. The period matrix K (2.7) satisfies the following properties: (i) IK = pg eg + Le1 + 2λ,
(ii)
g
K i j > 0,
(iii) g K = (g + 1)Le1 .
j=1
Lemma 2.9. Dm is written as 1 Dm = {Z ∈ Rg | − lZ⊥ ≤ lK (m + l)⊥ 2 for any l = ±(e j + e j+1 + · · · + ek ) such that 1 ≤ j ≤ k ≤ g}.
(2.12)
Proof. Since Dm = D0 − mK from the definition, it is enough to show for D0 . We show that if Z ∈ Rg satisfies −lZ⊥ >
1 lK l⊥ 2
for some l ∈ Zg , then there exists l = ±(e j + · · · + ek ) for 1 ≤ j ≤ k ≤ g, which satisfies 1 − l Z⊥ > l K (l )⊥ . (2.13) 2 For a vector v ∈ Rg , let v ≥ 0 denote that all elements of v are greater than or equal to zero. This lemma is shown by checking the following three claims (a)-(c): (a) There exists l ≥ 0 or l ≤ 0 which satisfies (2.13). Decompose l as l = l+ + l− such that l+ ≥ 0, l− ≤ 0 and l+ · l− = 0. Then we have −(l+ + l− )Z⊥ >
1 1 ⊥ ⊥ (l+ + l− )K (l+ + l− )⊥ = (l+ K l+⊥ + l− K l− ) + l+ K l− . 2 2
1008
R. Inoue, T. Takenawa
Since K i j ≤ 0 for i = j, 1 ⊥ ≥0 l+ K l− 2 holds. We have −(l+ + l− )Z⊥ >
1 ⊥ (l+ K l+⊥ + l− K l− ) 2
⊥. and hence −l+ Z⊥ > 21 l+ K l+⊥ or −l− Z⊥ > 21 l− K l− (b) If l ≥ 0 or l ≤ 0, then there exists l ∈ ±{0, 1}g which satisfies (2.13). Without loss of generality we can assume that l ≥ 0. For simplicity we also assume that l = (l1 , l2 , . . . , lg ) satisfies l1 ≥ l2 ≥ · · · ≥ lg . One can prove similarly other cases. We use the induction on k := l1 . Assume that if l1∗ ≤ k − 1 for l∗ satisfying the above assumptions for l, then there exists l ∈ {0, 1}g satisfying (2.13). We set a natural number r as l1 = l2 = · · · = lr > lr +1 ≥ · · · ≥ lg and a vector l∗ as l∗ = l − (l1 − lr +1 )Ir . Then l is decomposed as l = (l1 − lr +1 )Ir + l∗ and we have
−lZ⊥ > = where
1 lK l⊥ 2
1 1 (l1 − lr +1 )2 Ir K Ir⊥ + l∗ K (l∗ )⊥ + (l1 − lr +1 )Ir K (l∗ )⊥ , 2 2 ⎛
⎞ lr +1 K 11 + · · · + lr +1 K 1r + lr +1 K 1 r +1 + · · · + lg K 1g .. ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ ⎜ lr +1 K r 1 + · · · + lr +1 K rr + lr +1 K r r +1 + · · · + lg K r g ⎟ ∗ ⊥ Ir K (l ) = Ir · ⎜ ⎟ ∗ ⎜ ⎟ ⎜ ⎟ .. ⎝ ⎠ . ∗ ≥ lr +1 ((K 11 + · · · + K 1g ) + · · · + (K r 1 + · · · + K r g )) ≥ 0.
Thus we obtain −((l1 − lr +1 )Ir + l∗ )Z⊥ >
1 1 (l1 − lr +1 )2 Ir K Ir⊥ + l∗ K (l∗ )⊥ , 2 2
and therefore −Ir Z⊥ >
1 Ir K Ir⊥ or 2
− l∗ Z⊥ >
1 ∗ l K (l∗ )⊥ , 2
since l1 −lr +1 ≥ 1. If the former holds, we obtain the claim by setting l = Ir . Otherwise, since l1∗ ≤ k − 1, there exists l ∈ {0, 1}g satisfying (2.13) by the induction hypothesis. (c) Suppose l ≥ 0 or l ≤ 0 and ls = 0 for some 1 < s < g, and decompose l as l = l L + l R in such a way that (l L )i = 0 for s ≤ i ≤ g and (l R )i = 0 for 1 ≤ i ≤ s. Then l = l L or l = l R satisfies (2.13). We can assume l ≥ 0. We have 1 (l L + l R )K (l L + L R )⊥ 2 1 1 ⊥ ⊥ = l L K l⊥ L + l R K l R + lL K l R , 2 2
−(l L + l R )Z⊥ >
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
1009
where l L K l⊥ R
=
g s−1
li l j K i j = 0.
i=1 j=s+1
Thus the claim follows. Proposition 2.10. The tropical Riemann’s theta function satisfies + L] = (Z + 2λ) + (Z). (Z − Le1 ) + (Z + Le1 + 2λ) min[2(Z + λ), Proof. Take β = − 21 I and 1 1 P1 = (λg , L + pg + (g + 1)Cg ), 2 2 P3 = (λ0 , L + (g + 1)Cg ), Then we have
P3
P1 P4 P1
=
P4
, =λ
P2
= Le1 + 2λ,
P2 = (λ0 , (g + 1)Cg ),
1 1 P4 = (λg , L − pg + (g + 1)Cg ). 2 2
P3
P2 P4 P3
= −Le1 ,
P2
=
= Le1 + λ.
P1
We use the following lemma which will be proved after this proof. Lemma 2.11. Set β = − 21 I. Then the following are satisfied: (i) the point Z = β K is on the boundary ∂ D0 ; (n = 0, 1, t = 0, 1, 2) except β K are in the interior of (ii) all of Z = β K + n Le1 + t λ D0 ; (iii) if β K + v is in the interior of D0 , then β K − v is in the interior of DI . = 0 and argm∈Zg qβ (m, −n L From (ii) and (iii), we have argm∈Zg qβ (m, n Le1 + t λ) = I for n = 0, 1, t = 0, 1, 2 except for n = t = 0. Further, from the definition e1 − t λ) of [β](Z), we have the following: P2 P2 1 1 ⊥ g [β]( P3 ) = − 2 L + 2 β Kβ , argm∈Z qβ (m, ) = 0, [β]( [β]( [β](
P1 P3
P2 P4
P4 P1
) = − 21 L − (λg − λ0 ) + 21 β Kβ ⊥ ,
) = [β](
P3
) = [β](
[β]( [β](
P4 P3
P2 P1
P1
P4 P2
argm∈Zg qβ (m,
) = − 21 (λg − λ0 ) + 21 β Kβ ⊥ ,
argm∈Zg qβ (m,
) = − 21 (λg − λ0 ) + 21 β Kβ ⊥ ,
argm∈Zg qβ (m,
) = − 21 L − 21 (λg − λ0 ) + 21 β Kβ ⊥ ,
argm∈Zg qβ (m,
) = − 21 L − 21 (λg − λ0 ) + 21 β Kβ ⊥ ,
argm∈Zg qβ (m,
P3 P4
P1 P1 P3 P2 P4 P4 P3 P2 P1
) = 0, ) = I, ) = I, ) = 0, ) = 0,
1010
R. Inoue, T. Takenawa
thus s1 = 1, s2 = 1 and s3 = −1 hold. Substituting them into the tropical Fay’s identity (2.8), we obtain the claim. Proof of Lemma 2.11. In this proof we use 1 f m (Z, l) := lZ⊥ + lK (m + l)⊥ , 2 which is “r.h.s - l.h.s” of the conditional equation for Dm (2.12). (i) For l = I and m = 0, we have f 0 (β K , I) = (−I)( 21 K I)⊥ + I( 21 K I)⊥ = 0. For l ∈ {0, 1}g \ {0, I}, we have f 0 (β K , l) =
1 1 1 lK l⊥ + lKβ ⊥ = − lK (I − l)⊥ = − εi j K i j , 2 2 2 i= j
where εi j = 1 if “li = 1 and l j = 0” and otherwise εi j = 0. Since εi,i±1 is not zero by the assumption, f 0 (β K , l) is positive. For l ∈ {0, −1}g \ {0}, we have f 0 (β K , l) =
1 1 1 lK l⊥ − lK I⊥ ≥ lK l⊥ > 0. 2 2 2
Thus we see f 0 (β K , l) > 0 for l ∈ {0, 1}g ∪ {0, −1}g \ {0, I} and f 0 (β K , I) = 0. This implies β K ∈ ∂ D0 from Lemma 2.9. (n = 0, 1, t = 0, 1, 2). For l = I, we have (ii) Set Z = β K + n Le1 + t λ 1 ⊥ = I(n Le1 + t λ) ≥ 0, (by (i)), IK I⊥ + I(β K + n Le1 + t λ) 2
f 0 (Z, I) =
where the equality holds iff n = t = 0. For l ∈ {0, 1}g \ {0, I}, we have f 0 (Z, l) =
1 ⊥ > l(n Le1 + t λ) ≥ 0, (by (i)). lK l⊥ + l(β K + n Le1 + t λ) 2
From Lemma 2.8, we have 1 1 = − pg eg + (n − )Le1 + (t − 1)λ β K + n Le1 + t λ 2 2 . = −β K − pg eg − (1 − n)Le1 − (2 − t)λ Thus, for l ∈ {0, −1}g \ {0}, f 0 (Z, l) becomes 1 )⊥ lK l⊥ + l(−β K − pg eg − (1 − n)Le1 − (2 − t)λ 2 > −l( pg eg + (1 − n)Le1 + (2 − t)λ) ≥ 0 (n = 0, 1, t = 0, 1, 2). Then the claim follows from Lemma 2.9. (iii) When β K + v is in the interior of D0 , f 0 (β K + v, l) = 21 lK l⊥ + l(β K + v)⊥ > 0 holds for all l ∈ Zg . Thus we see that f I (β K − v, −l) > 0 for all l ∈ Zg , since 1 f I (β K − v, −l) = −lK (I − l)⊥ − l(β K − v)⊥ = f 0 (β K + v, l), 2 for all l ∈ Zg .
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
1011
Example 2.12 (Counter example). Set as g = 2, C2 = 0, L = 11, λ1 = 2, λ2 = 3, β = (− 21 , 0) and take P1 , P2 , P3 , P4 as Proposition 2.10, then 18 −3 K = , P1 = (3, 6), P2 = (0, 0), P3 = (0, 11), P4 = (3, 5), −3 6 P3 P4 P3 P4 P4 P2 = = (2, 1), = (−11, 0), = (15, 2), = = (13, 1), P1
P2
P2
P1
P3
P1
P hold, and thus β K + P14 = (6, 3.5) is on the boundary D(0,−1) ∩ D(−1,−1) . Therefore we cannot determine the sign s1 , while at Z = (0, 0) we obtain F1 = (2, 1)+(2, 1)+β Kβ ⊥ +β(11, 0)⊥ +(2, 1.5)+β(15, 2)⊥ +(6, 3.5) = −9, F2 = (−11, 0) + (15, 2) + β Kβ ⊥ + 2β(−2, −1)⊥ + 2(−11, 0.5) = −7.5, F3 = (4, 2)+(0, 0)+β Kβ ⊥ +β(13, 1)⊥ +(4, 2.5)+β(13, 1)⊥ +(4, 2.5) = −8.5, and thus Fay’s type identities do not hold in this case. From Proposition 2.10 we have the following. Corollary 2.13. For Z0 ∈ Rg , the function Tnt given by Tnt = (Z0 − n Le1 + λt)
(2.14)
satisfies the tropical bilinear equation (1.5). 3. Solution of UD-pToda 3.1. τ -function for UD-pToda. Fix a positive integer g. The UD-pToda is defined by the piecewise-linear map t+1 ϕT : (Q tn , Wnt )n=1,...,g+1 → (Q t+1 n , Wn )n=1,...,g+1
given by (1.1) on the phase space T (1.2). The map ϕT preserves the tropical polynomials Ci (Q, W ) (i = −1, 0, . . . , g) on T. Fix C = (C−1 , C0 , . . . , Cg ) ∈ Rg+2 as (1.14), and define the tropical curve ˜ (1.13) and the isolevel set TC as TC = {(Q, W ) ∈ T | Ci (Q, W ) = Ci (i = −1, 0, . . . , g)}.
(3.1)
See [4] for a detail of Ci (Q, W ). Let St (t ∈ Z) be a subset of infinite dimensional space: St = {Tnt ∈ R | n ∈ Z}, t = Tnt + cnt , where cnt satisfies where Tnt has a quasi-periodicity; i.e. Tnt satisfies Tn+g+1
cnt = an + bt + c
(3.2)
for some a, b, c ∈ R. Fix L ∈ R such that 2b − a < (g + 1)L
(3.3)
1012
R. Inoue, T. Takenawa
and define a map ϕS from St × St+1 to St+1 × St+2 as ϕS : (Tnt , Tnt+1 )n∈Z → (Tnt+1 , Tnt+2 )n∈Z with (g)
Tnt+2 = 2Tnt+1 − Tnt + X n+1,t ,
(3.4)
where we define a function on St × St+1 : (k) t+1 t t t+1 t t + T + T − (2T + T + T ) X n,t = min j L + 2Tn− n n−1 n−1 j−1 n− j n− j−1 , j=0,...,k
(3.5)
t < (g + 1)L for k ∈ Z≥0 . Note that it follows from (3.2) and (3.3) that 2cnt+1 − cnt − cn+1 (k) for all n ∈ Z. The function X n,t has the following properties:
Lemma 3.1.
(k)
(i) X n,t satisfies a recursion relation:
(k) (k−1) t+1 t+1 t+1 t + X n,t = min 2Tn−1 , L + 2Tn−2 + Tnt − Tn−2 + X n−1,t , 2Tn−1
for k ≥ 1. (3.6)
(g+1)
(ii) X n,t
(g)
= X n,t . (k)
Proof. (i) It is shown just by the definition of X n,t . (k) (ii) For simplicity we rewrite (3.5) as X n,t = min j=0,··· ,k [a tj ], where t+1 t t t+1 t t a tj = j L + 2Tn− j−1 + Tn + Tn−1 − (2Tn−1 + Tn− j + Tn− j−1 ).
By making use of (3.2) we obtain t t+1 t t+1 t t ag+1 = (g + 1)L + 2Tn−g−2 + Tnt + Tn−1 − (2Tn−1 + Tn−g−1 + Tn−g−2 ) t t t+1 = (g + 1)L + cn−g−2 + cn−g−1 − 2cn−g−2 > 0.
At the same time we have a0t = 0. Thus we obtain (g+1)
X n,t
(g)
(g)
t = min[X n,t , ag+1 ] = X n,t .
Let σt be a map σt : St × St+1 → T given by (1.4). It is easy to check that σt is well-defined where (3.2) assures the periodicity of (Q tn , Wnt )n , and (3.3) assures the g+1 t t condition g+1 n Qn < n Wn of the phase space T. (g)
Lemma 3.2. The relation σt (X n,t ) = X nt holds, and the following diagram is commutative: σt
St × St+1 → ↓ϕ S σt+1 St+1 × St+2 →
T ↓ϕ T . T
(3.7)
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
1013
(g)
Proof. By direct calculation we can check σt (X n,t ) = X nt . To check ϕT ◦σt = σt+1 ◦ϕS, it is enough to calculate Q t+1 n in the image of each map. We have t t t Q t+1 n = min[Wn , Q n − X n ] (g)
t+1 t t t+1 = min[L +Tn−1 +Tn+1 −Tnt −Tnt+1 , Tn−1 +Tnt+1 −Tn−1 −Tnt − X n,t ] (by (1.4)),
for ϕT ◦ σt , and for σt+1 ◦ ϕS, t+2 t+1 t+2 Q t+1 + Tn−1 − Tn−1 − Tnt+1 n = Tn (g)
t+1 t+2 = 2Tnt+1 + X n+1,t + (Tn−1 − Tnt − Tn−1 − Tnt+1 ) (by (3.4)) (g+1)
= 2Tnt+1 + X n+1,t (g)
t+1 t +(−X n,t − Tn−1 − Tnt − Tnt+1 + Tn−1 )
(by Lemma 3.1(ii) and (3.4)) (g)
t+1 t t = min[2Tnt+1 , L + 2Tn−1 + Tn+1 − Tn−1 + X n,t ] + (
)
(by (3.6)).
It is easy to see these two expressions of Q t+1 n coincide. From Lemma 3.1, the next proposition immediately follows. Proposition 3.3. If {Tnt }n,t ∈ t St satisfies (3.4), then it satisfies (1.5). Proof. (g)
t+2 t t+1 t t + Tn+1 + L] = min[2 Tnt+1 , (2Tn−1 − Tn−1 + X n,t ) + Tn+1 + L] min[2 Tnt+1 , Tn−1 (g+1)
(g)
= 2Tnt+1 + X n+1,t = 2Tnt+1 + X n+1,t = Tnt + Tnt+2 . Conversely the following proposition holds, which is proved after Theorem 3.5 is proved. Proposition 3.4. Let {Tnt }n,t ∈ t St satisfy (1.5) and At denote a set: t+2 t {n ∈ Z | Tnt + Tnt+2 = 2Tnt+1 , i.e. 2Tnt+1 ≤ Tn−1 + Tn+1 + L}.
Then we have the following: (i) n ∈ At ⇔ n + g + 1 ∈ At , (ii) At = ∅, (iii) (3.4) holds for all n, t ∈ Z2 . The following is the main result of this section. Theorem 3.5. Let ιt : Rg → St × St+1 be a map: Tnt+1 = (Z0 − n Le1 + (t + 1)λ)) n∈Z . Z0 → (Tnt = (Z0 − n Le1 + t λ), Then the following diagram is commutative: ιt
σt
Rg → St × St+1 → TC ↓id. ↓ϕ S ↓ϕ T . σt+1 ιt+1 Rg → St+1 × St+2 → TC
(3.8)
1014
R. Inoue, T. Takenawa
satisfies (3.4) and gives a solution In short, for any Z0 ∈ Rg , Tnt = (Z0 −n Le1 +t λ) of (1.1) through (1.4). satisfies the quasiProof of Theorem 3.5. We first check that Tnt = (Z0 − n Le1 + t λ) periodicity (3.2) and (3.3). From Prop. 2.3 and Lemma 2.8 (iv), we have 1 − g K )t . cnt = an + bt + c = g · (Z0 − n L e 1 + t λ 2
t = Tnt + cnt , Tn+g+1
(3.9)
Then from (2.6) we obtain g 1 (λi − λ0 ), c = g · Z0 − g(g + 1)L , a = −gL , b = 2 2 i=1
(g + 1)L − 2b + a = (g + 1)L − 2
g (λi − λ0 ) + gL = pg > 0. i=1
Due to Prop. 3.4 (iii), the left part of the diagram (3.8) is commutative. Since (Z) is associated to , the image of σt is in TC , while the commutativity of the right part of (3.8) follows from Lemma 3.2. Remark 3.6. If At = ∅, the map “σt+1 ◦ ϕS ◦ σt−1 ” induces a map given by (Q tn , t t+1 t t+1 Wnt )i=1,...,g+1 → (Q n = Wn , Wn = Q n+1 )n=1,...,g+1 , which does not preserve the inequality n Q tn < n Wnt . Proof of Proposition 3.4. t+2 t + Tn+1 +L (i) n ∈ At ⇔ Tnt + Tnt+2 = Tn−1 t t+2 t+2 t+2 t t ⇔ (Tn+g+1 − cnt ) + (Tn+g+1 − cnt+2 ) = (Tn+g − cn−1 ) + (Tn+g+2 − cn+1 )+L t t+2 t+2 t ⇔ Tn+g+1 + Tn+g+1 = Tn+g + Tn+g+2 +L
⇔ n + g + 1 ∈ At . (ii) Assume that Tnt satisfies the quasi-periodicity (3.2) and At = ∅. Then we have t+2 t Tnt + Tnt+2 = Tn−1 + Tn+1 + L for all n, t+2 t+2 t ⇒ Tn+g − Tn−1 = Tn+g+1 − Tnt + (g + 1)L ⇒ −a + 2b + c = c + (g + 1)L ⇔ 2b − a = (g + 1)L .
This contradicts the claim of the quasi-periodicity. t+2 = 2T t+1 − T t + (iii) From (i) and (ii), it is enough to show that if n ∈ At then Tn+s n+s n+s (g) X n+s+1,t holds for all s ≥ 0. We show it by induction on s. In case s = 0, due to (1.5), t+2 t+1 t Tn− j ≤ 2Tn− j − Tn− j , t+2 Tn− j
≤L+
t+2 Tn− j−1
+
t Tn− j+1
(3.10) −
t Tn− j
(3.11)
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
1015
hold for all j. Taking the sum of (3.11) for j = 0, 1, . . . , k − 1 and using (3.10), we have t+2 t t Tnt+2 ≤ k L + Tn−k + Tn+1 − Tn−k+1 t+1 t t t ≤ k L + (2Tn−k − Tn−k ) + Tn+1 − Tn−k+1
= 2Tnt+1 − Tnt + akt , (g)
(g)
for all k ≥ 0, where X n+1,t = mink=0,1,...,g [akt ], and thus, Tnt+2 ≤ 2Tnt+1 − Tnt + X n+1,t (g)
holds. Since Tnt+2 = 2Tnt+1 − Tnt from the assumption n ∈ At and X n+1,t ≤ a0t = 0, we (g)
have X n+1,t = 0, which yields the claim for s = 0. (g)
t+2 = 2T t+1 − T t + X Assume Tn+s n+s n+s n+s+1,t holds for some s ≥ 0: t+2 t+1 t t+2 t t+1 = 2Tn+s+1 − Tn+s+1 + min[0, Tn+s + Tn+s+2 + L − 2Tn+s+1 ] (by (1.5)) Tn+s+1 (g)
t+1 t t+1 t t t+1 −Tn+s+1 + min[0, (2Tn+s − Tn+s + X n+s+1,t ) + Tn+s+2 + L−2Tn+s+1 ] = 2Tn+s+1 (g+1)
t+1 t − Tn+s+1 + X n+s+2,t (by Lemma 3.1) = 2Tn+s+1 (g)
t+1 t − Tn+s+1 + X n+s+2,t . = 2Tn+s+1
Thus, we have that the claim holds for s + 1.
Corollary 3.7. We write ϕλ for the translation on the Jacobian J () as ϕλ : [Z0 ] → Let ισ : J () → TC be the map induced by σt ◦ ιt : Rg → T. Then the [Z0 + λ]. following diagram is commutative: ισ
J () → TC ↓ϕλ ↓ϕ T . ισ J () → TC Proof. The commutativity follows from (3.7) and (3.8). The well-definedness, ισ (Z0 ) = ισ (Z0 + lK ) for any l ∈ Zg , is guaranteed by the quasi-periodicity (3.2) t) satisfies. Tnt = (Z0 − n Le1 + λ Further, by concrete computation, we conjecture that the map ισ is isomorphic 1 . Appendix A. Proof of Theorem 2.6 (UD-Limit of Abel-Jacobi Map) We assume (c−1 , c0 , · · · , cg ) ∈ Rg+2 . By setting u := −x, (u) := (−1)g+1 u g+1 + (−1)g cg u g + · · · + (−1)c1 u + c0 , v := 2y + (u),
(A.1)
the spectral curve γ (1.11) becomes the standard form of a hyperelliptic curve: γ1 :
v 2 = (u)2 − 4c−1 .
1 We proved this conjecture after submitting the present paper [5]. The proof needs rather complicated combinatorial discussion.
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R. Inoue, T. Takenawa
It is known that (u) has properties [16,17]: special g (i) the zeros of (u) = i=0 (u i − u) are simple and positive, ordered as 0 < u0 < u1 < · · · < ug .
(A.2)
g (ii) The zeros of (u)2 − 4c−1 = i=0 (u − u i+1 )(u − u i−1 ) are positive and they can −1 +1 +1 +1 −1 +1 be ordered as 0 < u −1 0 < u 0 ≤ u 1 < u 1 ≤ · · · u g−1 ≤ u g < u g ,
(iii) We have u i−1 < u i < u i+1 . Further, since |c−1 | is very small when we consider the UD-limit of γ , we can assume that the zero locus of (u)2 − 4c−1 are simple and positive, i.e. −1 +1 +1 −1 +1 0 < u −1 0 < u0 < u1 < u1 < · · · < ug < ug
(A.3)
holds. +1 Take the two-sheeted covering u + , u − of u with branches [u −1 j , u j ] ( j=0, 1, 2, . . . , g) and choose the homology basis a1 , . . . , ag , b1 , . . . , bg and the basis of the holomorphic differentials ω1 , . . . , ωg ∈ H 0 (γ , 1 ) as usual as in Sect. 2.1. i.e. ω j ’s are written in the form w j,g−1 u g−1 + w j,g−2 u g−2 + · · · + w j,0 du (w j,k ∈ C) v and satisfy ( ai ω j )i, j = I , ( bi ω j )i, j = . ωj =
Lemma A.1. When Ci satisfy the generic condition (1.14), for 0 ≤ i < i + 2 ≤ j ≤ g, Ci + C j > Ci+1 + C j−1 holds. Proof. Consider the sum of the equations Ci+k + Ci+k+2 > 2Ci+k+1 for k = 0, 1, . . . , g − 2. Proposition A.2. By the UD-limit with the scale transformation |u| = e
− Xε
Y ε
, |y| = e , u j = e
−
Xj ε
,
u ±1 j
=e
−
X ±1 j ε
Ci
, ci = e− ε ,
(A.4)
the following hold: +1 (i) X j = X −1 j = X j = C j − C j+1 . (ii) The limit of cycle bi is B˜ i = Bg−i+1 + Bg−i+2 + · · · + Bg .
Proof. Notice that X g < X g−1 < · · · < X 0 holds by the definition of (A.4). (i) Since the even g case can be shown similarly, we only show the odd g case. By the scale transformation, the equations (u) = 0 and (u)2 − 4c−1 = 0 are respectively written as e−
C0 ε
+ e−
2X +C2 ε
+ · · · + e−
(g+1)X ε
= e−
X +C1 ε
+ e−
3X +C3 ε
+ · · · + e−
g X +Cg ε
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
Fig. 4. Spectral curve with (u) =
5
k=1 (u − (2k − 1))
and C−1 2C0 2X +C0 +C2 2X +2C1 2(g+1)X e− ε − 4e− ε + 2e− ε + · · · + e− ε + e− ε (2g+1)X +Cg X +C0 +C1 3X +C0 +C3 3X +C1 +C2 ε + 2e− ε + · · · + 2e− = 2e− ε + 2e− ε . By Lemma A.1, taking the UD-limit of both sides, we have min[C0 , 2X + C2 , . . . , (g + 1)X ] = min[X + C1 , 3X + C3 , . . . , g X + Cg ]
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R. Inoue, T. Takenawa
and min[2C0 , 2X + 2C1 , . . . , (2g + 2)X ] = min[X + C0 + C1 , 3X + C1 + C2 , . . . , (2g + 1)X + Cg ]. By solving these, we obtain the claim. (ii) It is easily shown. Proposition A.3. Set ω0j as ω0j
1 = √ 2π –1
1 1 du − u −uj u − u0
for j = 1, . . . , g and define u j,k ∈ C by ω0j =
u j,g−1 u g−1 + u j,g−2 u g−2 + · · · + u j,0 du. (u)
ω˜ j =
u j,g−1 u g−1 + u j,g−2 u g−2 + · · · + u j,0 du (u)2 − 4c−1
Set ω˜ j as
and take the scale transformation (A.4), then lim ω˜ j = lim ω0j = δi, j ε→+0 ai
ε→+0 ai
(A.5)
hold. Proof. Since 2C0 < C−1 from (1.14), we have limε→+0 ω˜ j /ω0j = 1. Further, by the residue theorem, ai ω0j = δi, j holds for any ε > 0. ω0j
By this proposition, denoting the normalized 1-form on γ 1 as ω j, we have limε→+0 ω j / = 1.
Remark A.4. By the scale transformation (A.4), the integral paths ai and bi are also transformed. Although the integral paths converge to zero in the variables u(= exp(−X/ε)), u i (= exp(−X i /ε)), they converge to nonzero paths of finite length in the variables X , X i . Since we can exchange the order of the UD-limit and the integration, we will consider by the variables X ’s in the following. Proposition A.5. Suppose that there exist the limits limε→+0 Logε (u a , ya ) = (X a , Ya ) and limε→+0 Logε (u b , yb ) = (X b , Yb ). (i) If (X a , Ya ) and (X a , Yb ) are on the same edge of , and X i+1 < X a , X b < X i for some i, then ⎧ ( j ≤ i) ⎪ ub ⎨0 √ ( j > i and Ya , Yb ≥ 21 C−1 ) − 2π –1 ε lim ω j = (X a − X b ) ⎪ ε→+0 u a ⎩ −(X a − X b ) ( j > i and Ya , Yb ≤ 21 C−1 ) (A.6) holds.
Fay’s Trisecant Identity and Ultra-Discrete Periodic Toda Lattice
(ii) If X a = X b = X i for some i, then − 2π
√
–1 ε lim
yb
ε→+0 ya
1019
⎧ ⎨ −(Ya − Yb ) ω j = (Ya − Yb ) ⎩0
(i = 0) (i = j) (i = 0, j)
(A.7)
holds. +1 Remark A.6. The sign of ω j is changed by passing through the branches [u −1 k , u k ]. At √ 1 the branch point u = u ±1 k , we have v = 0 and y = − 2 (u) = ± c−1 , where the sign −1 +1 is + if and only if u = u k (k is odd) or u = u k (k is even).
Proof. For simplicity, we omit the sign of ω j if there is no possibility of misunderstanding. Since limε→0 ω j /ω0j = 1, the integrals are not changed by substituting ω j by ω0j . (i) Without loss of generality we can assume Ya , Yb ≥ 21 C−1 , which corresponds to the sign of ω j being +: ub ub √ √ 1 1 1 0 du lim −2π –1 ε ω j = lim −2π –1 ε − √ ε→+0 ε→+0 u − u0 ua u a 2π –1 u − u j (u b − u j ) − log (u b − u 0 ) = − lim ε log (u − u ) ε→+0 (u a − u j ) a 0 (X b − X a ) − (X b − X a ) = 0 ( j ≤ i) = Xa − Xb ( j > i) (limε→+0 Logε (u − u ) = X if u > u > 0). (ii) We divide into two cases (ii-1) Ya , Yb ≤ 21 C−1 or Ya , Yb ≥ 21 C−1 and (ii-2) Ya < 21 C−1 < Yb or Ya > 21 C−1 > Yb . (ii-1) Without loss of generality we can assume Ya , Yb ≥ 21 C−1 . Substituting 2y + (u) dy, du = − y (u) g !
(u) =
(u − u k )
= (u)
k=0
g k=0
1 , u − uk
and (u) = −
y 2 + c−1 , y
into ω0j , we have ω0j
=
2π
1 √
–1
1 1 − u −uj u − u0
g k=0
1 u − uk
!−1
Here, if limε→+0 Logε u = X i ,
2π
1 √
–1
1 1 − u −uj u − u0
g k=0
1 u − uk
!−1 =
1 2y − + 2 y y + c−1
⎧ 1 ⎪ ⎨ − 2π √–1 ⎪ ⎩ 2π 0
1√
–1
dy.
(i = 0) (i = j) (i = 0, j).
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Thus we have −2π
√
–1 ε lim
yb
ε→+0 ya
ωj
⎧ c−1 yb ⎪ )] ya lim ε[log(y + (i = 0) ⎪ ⎨ ε→+0 y c −1 y = lim −ε[log(y + )] yab (i = j) ⎪ ⎪ y ⎩ ε→+0 0 (i = 0, j) ⎧ ⎨ − min[Yb , C−1 − Yb ] + min[Ya , C−1 − Ya ] = −(Ya − Yb ) = min[Yb , C−1 − Yb ] − min[Ya , C−1 − Ya ] = (Ya − Yb ) ⎩0
(i = 0) (i = j) (i = 0, j).
(ii-2) Without loss of generality we can assume Ya < 21 C−1 < Yb . The ultra-discrete √ limit of |y| = c−1 is Y = 21 C−1 . From (ii-1) and Remark A.6, 1 ! yb yb − 2 (u) √ √ ω j = −2π –1 ε lim ωj + ωj −2π –1 ε lim ε→+0
ya
ε→+0
⎧ 1 ⎨ −( 2 C−1 − Ya ) + ( 21 C−1 − Yb ) = Ya − Yb = ( 21 C−1 − Ya ) − ( 21 C−1 − Yb ) = −Ya + Yb ⎩ 0
ya
− 21 (u)
(i = 0) (i = j) (i = 0, j).
Theorem 2.6 immediately follows from Proposition A.5 and Proposition A.2 (ii). Actually, by Proposition A.5, Theorem 2.6 holds if the points P and Q are on a common edge of . The general case is shown by the additivity of the Abel-Jacobi map η for paths. By this fact together with Proposition A.2 (ii), the UD-limit of i j becomes K˜ i j = B˜ i , B˜ j . Acknowledgement. T. T. appreciates the assistance from the Japan Society for the Promotion of Science. R. I. is supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B) (19740231).
References 1. Fay, J.D.: Theta Functions on Riemann Surfaces. Lecture notes in Math. 352, Berlin-New York: SpringerVerlag, 1973 2. Hirota, R.: Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan 50(11), 3785–3791 (1981) 3. Hirota, R., Tsujimoto, S., Imai, T.: Difference scheme of soliton equations. In: Future directions of nonlinear dynamics in physical and biological systems, Christiansen, P.L., Eilbeck, J.C., Parmentier, R.D. (eds.) New York: Plenum Press, 1993; R. Hirota, S. Tsujimoto: Conserved quantities of a class of nonlinear difference-difference equations. J. Phys. Soc. Japan 64(9), 3125–3127 (1995) 4. Inoue, R., Takenawa, T.: Tropical spectral curves and integrable cellular automata. Int. Math. Res. Not., Vol. 2008, rnn019, 27 pp (2008) 5. Inoue, R., Takenawa, T.: Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic. http://arXiv.org/abs/0902.0448v1[nlin.SI], 2009 6. Iwao, S., Tokihiro, T.: Ultradiscretization of the theta function solution of pd Toda. J. Phys. A: Math. Theor. 40, 12987–13021 (2007)
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7. Kimijima, T., Tokihiro, T.: Initial-value problem of the discrete periodic Toda equations and its ultradiscretization. Inverse Problems 18, 1705–1732 (2002) 8. Kuniba, A., Sakamoto, R.: The bethe ansatz in a periodic box-ball system and the ultradiscrete Riemann theta function. J. Stat. Mech. Theory Exp., no. 9, P09005, 12 pp (2006) 9. Mikhalkin, G.: Enumerative tropical algebraic geometry in R2. J. Amer. Math. Soc. 18, 313–377 (2005) 10. Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and theta functions. http://arXiv.org/abs/ math.AG/0612267, 2006 11. Miwa, T.: On Hirota’s difference equations. Proc. Japan Acad. 58, Ser. A, 9–12 (1982) 12. Mumford, D.: Tata Lectures on Theta I, II . Boston: Birkhäuser, 1984 13. van Moerbeke, P., Mumford, D.: The spectrum of difference operators and algebraic curves. Acta Math. 143, 93–154 (1979) 14. Nobe, A.: Periodic multiwave solutions to the Toda-type cellular automaton. J. Phys. A 38, L715– L723 (2005) 15. Takahashi, D., Satsuma, J.: A soliton cellular automaton. J. Phys. Soc. Japan 59, 3514–3519 (1990) 16. Date, E., Tanaka, S.: Analogue of inverse scattering theory for the discrete Hill’s equation and exact solutions for the periodic Toda lattice. Progr. Theoret. Phys. 55, 457–465 (1976) 17. Toda, M.: Theory of Nonlinear Lattices. Berlin-New York: Springer-Verlag, 1981 Communicated by L. Takhtajan
Commun. Math. Phys. 289, 1023–1055 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0822-4
Communications in
Mathematical Physics
Spectral Measure of Heavy Tailed Band and Covariance Random Matrices Serban Belinschi1, , Amir Dembo2, , Alice Guionnet3,4 1 Department of Mathematics and Statistics, University of Saskatchewan,
and Institute of Mathematics of the Romanian Academy, 106 Wiggins Road, Saskatoon, Saskatchewan S7N 5E6, Canada. E-mail: [email protected] 2 Stanford University, Stanford, CA 94305, USA. E-mail: [email protected] 3 Ecole Normale Supérieure de Lyon, Unité de Mathématiques pures et appliquées, UMR 5669, 46 Allée d’Italie, 69364 Lyon Cedex 07, France. E-mail: [email protected]_lyon.fr; [email protected] 4 Miller Institute for Basic Research in Science, University of California Berkeley, Berkeley, USA Received: 11 July 2008 / Accepted: 3 February 2009 Published online: 19 May 2009 – © Springer-Verlag 2009
Abstract: We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure µˆ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix YσN whose (i, j) entry is σ Ni , Nj xi j , where (xi j , 1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, α ∈ (0, 2), and σ is a deterministic function. For random diagonal D N independent of YσN and with appropriate rescaling a N , we prove that µˆ a −1 Yσ + D N converges in mean towards a N N limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries. 1. Introduction We study the asymptotic behavior of the spectral measure of large band random real symmetric matrices with independent (apart from symmetry) heavy tailed entries. Specifically, with (xi j , 1 ≤ i ≤ j < ∞) an infinite array of i.i.d real variables, let X N denote the N × N symmetric matrix given by X N (i, j) = xi j
if i ≤ j, x ji otherwise.
Fixing σ : [0, 1] × [0, 1] → R, a (uniformly over 1/N -lattice grids) square integrable measurable function such that σ (x, y) = σ (y, x), we denote by YσN the N × N symmetric matrix with entries Y Nσ (i, j) = σ ( Ni , Nj )xi j . These matrices are sometime called “band matrices” after the choice of σ (x, y) = 1|x−y|≤b for some 0 < b < 1 (cf. Supported in part by a Discovery grant from the Natural Sciences and Engineering Research Council of Canada and a University of Saskatchewan start-up grant. Research partially supported by NSF grant #DMS-0806211.
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S. Belinschi, A. Dembo, A. Guionnet
Remark 1.9). Another important special case, σ (x, y) = 1(x−1/2)(1/2−y)>0 yields the spectral measure of empirical covariance matrices X N XtN (as shown in Sect. 5.1). For i.i.d. entries (xi j , 1 ≤ i ≤ j ≤ N ) of finite second moment, it was proved by Berezin that the spectral measure of AσN := N −1/2 YσN converges almost surely weakly (see a rigorous proof in [7]). More precisely, for any z ∈ C\R the matrices G N (z) := (zI N − AσN )−1 are such that for any bounded continuous function φ, 1 N 1 i lim G N (z)ii = φ φ(u)K uσ (z)du a.s. N →∞ N N 0 i=1
1 with K xσ (z) the unique solution of K xσ (z) = (z − 0 |σ (x, v)|2 K vσ (z)dv)−1 such that 1 z → 0 φ(u)K uσ (z)du is analytic in C\R. In particular, taking constant φ(·) we have the almost sure convergence of the spectral measure of AσN to the probability measure µσ2 whose Cauchy-Stieltjes transform is G σ2 (z)
=
1 dµσ (λ) = z−λ 2
0
1
K vσ (z)dv.
(1.1)
We consider here the case of heavy tailed entries, where the common distribution of the absolute values of the xi j ’s is in the domain of attraction of an α-stable law, for α ∈]0, 2[. That is, there exists a slowly varying function L(·) such that for any u > 0, P(|xi j | ≥ u) = L(u)u −α .
(1.2)
1 , a N := inf u : P[|xi j | ≥ u] ≤ N
(1.3)
The normalizing constants
are then such that a N = L 0 (N )N 1/α for some (other) slowly varying function L 0 (·). σ having eigenvalues Hereafter, let AσN denote the normalized matrix AσN := a −1 N Y N
N (λ1 , . . ., λ N ) and the corresponding spectral measure µˆ AσN := N1 i=1 δλi (and when σ the choice of σ (·) is clear we also use the notations Y N and A N for Y N and AσN , respectively). Predictions about the limiting spectral measure in case σ (·, ·) ≡ 1 (the heavy tail analog of Wigner’s theorem) have been made in [2] and rigorously verified in [1] (cf. [1, Sect. 8]). We follow here the approach of [1], which consists of proving the convergence of the resolvent, i.e. of the mean of the Cauchy-Stieltjes transform of the spectral measure, outside of the real line, by proving tightness and characterizing uniquely the possible limit points. In the latter task, for each α ∈ (0, 2) the limiting spectral measure of AσN is characterized in terms of the entire functions ∞ α α gα (y) := t 2 −1 e−t exp{−t 2 y}dt, (1.4) 0 ∞ α α e−t exp{−t 2 y}dt = 1 − ygα (y). (1.5) h α (y) := 2 0 We define for any α ∈ (0, 2) the usual branch of the power function x → xα , which πα is the analytic function on C\R− such that (i)α = ei 2 . This amounts to choosing
Heavy Tailed Band Matrices
1025
xα = r α eiαθ when x = r eiθ with θ ∈] − π, π [. We also adopt throughout the notation x−α for (x−1 )α . With these notations in place, recall [1, Theorem 1.4] that in case σ (·, ·) ≡ 1, the limiting spectral measure µα for Wigner matrices with entries in the domain of attraction of an α-stable law has for z ∈ C+ = {z ∈ C : (z) > 0}, the Cauchy-Stieltjes transform 1 1 dµα (x) = h α (Y (z)), G α (z) := (1.6) z−x z where Y (z) is the unique analytic on C+ solution of z α Y (z) = Cα gα (Y (z))
(1.7)
tending to zero at infinity, and Cα := i α (1 − α2 )/ ( α2 ). In [1, Theorem 1.6] it is further shown that µα has a smooth symmetric density ρα outside a compact set of capacity zero, and that t α+1 ρα (t) → α/2 as t → ∞. In addition to considering the more general case of band matrices, we devote some effort to the analysis of the limiting Cauchy-Stieltjes transform as (z) → 0 and its consequences on existence and regularity of the limiting density. For example, as a by product of our analysis we prove the following about µα of [1], showing in particular that it has a uniformly bounded density. Proposition 1.1. The unique analytic on C+ solution Y (z) of (1.7) tending to zero at infinity takes values in the set Kα := {Reiθ : |θ | ≤ απ 2 , R ≥ 0} on which gα (·) is uniformly bounded. Its continuous extension to R\{0} is analytic except possibly at the finite set Dα = {0, ±t : t α = Cα gα (y) > 0, y ∈ Kα , gα (y) = ygα (y)}. Further, the symmetric uniformly bounded density of µα is ρα (t) = −
1 α|t|α−1 −α (h α (Y (t))) = i Y (|t|)2 , πt 2|Cα |π
(1.8)
continuous at t = 0, real-analytic outside Dα and non-vanishing on any open interval. Remark 1.2. It is noted in [1, Remark 1.5] that α → µα is continuous on (0, 2) with respect to weak convergence of probability measures. We further show in Lemma 5.2 that as α → 2 the measures µα converge to the semi-circle law µ2 . Let C denote the set of piecewise constant functions σ (x, y) such that for some finite q, some 0 = b0 < b1 < · · · < bq = 1 and a q × q symmetric matrix of entries {σr s , 1 ≤ r, s ≤ q}, σ (x, y) = σr s
for all (x, y) ∈ (br −1 , br ] × (bs−1 , bs ].
(1.9)
Our next result provides the weak convergence of the spectral measures for AσN and characterizes the Cauchy-Stieltjes transform of their limit, in case σ ∈ C . Even for σ (·, ·) ≡ 1 it goes beyond the results of [1] by strengthening the weak convergence of the expected spectral measures E[µˆ A N ] to the weak convergence of µˆ A N holding with probability one. A special interesting case of σ is when q = 2 and σr s = 1|r −s|=1 , out t of which we get the spectral measure of the empirical covariance matrices a −2 N XN XN (cf. Theorem 1.10 and its proof in Sect. 5).
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S. Belinschi, A. Dembo, A. Guionnet
Theorem 1.3. Fixing σ ∈ C , let r = br − br −1 for r = 1, . . . , q. With probability one, the sequence µˆ AσN converges weakly towards the non-random, symmetric probability measure µσ . The limiting measure has a continuous density ρ σ on R\{0} which is bounded off zero, and its Cauchy-Stieltjes transform is, for any z ∈ C+ , q 1 1 σ dµ (x) = G α,σ (z) := s h α (Ys (z)), (1.10) z−x z s=1
where Y (z) ≡ (Yr (z), 1 ≤ r ≤ q) is the unique solution of α
z Yr (z) = Cα
q
|σr s |α s gα (Ys (z)),
(1.11)
s=1
composed of functions that are analytic on z ∈ C+ and tend to zero as |z| → ∞. Moreover, z α Y (z) is uniformly bounded on C+ , both G α,σ (z) and Y (z) ∈ (Kα )q have continuous, algebraic extensions to R \ {0}, and for some R = R(σ ) finite the
mapping q Y (z) extends analytically through the subset (R, ∞), where ρ σ (t) = − π1t s=1 s (h α (Ys (t))) is real-analytic. Finally, the map z → Y (z) is injective whenever σ ≡ 0. Remark 1.4. The measure µσ may have an atom at zero when q > 1. Indeed, Theorem 1.10 provides one such example in case q = 2. Remark 1.5. While we do not pursue it here, similarly to [1, Section 9], one can apply the moment method developed by Zakharevich [9], to characterize µσ as the weak limit B → ∞ of the limiting spectral measures for appropriately truncated matrices Aσ,B N . As done in Lemma 5.2 for σ ≡ 1, we expect this to yield the continuity of µσ with respect to α → 2, for each fixed σ ∈ C , i.e. to connect the limiting measures of Theorem 1.3 to µσ2 of (1.1). Let L 2 ([0, 1]2 ) denote the space of equivalence classes with respect to the semi-norm f := lim sup f (n −1 nx , n −1 ny ) 2 , n→∞
on the space of functions on [0, 1]2 for which · is finite. For each measurable 1 f : [0, 1]2 → R let f := 0 | f (x, v)|dv ∞ denote the associated operator norm, where · ∞ denotes hereafter the usual (essential-sup) norm of L ∞ ((0, 1]). We consider the subset Fα of those symmetric measurable functions σ ∈ L 2 ([0, 1]2 ) with |σ |α finite which are each the L 2 -limit of some sequence σ p ∈ C such that lim |σ p |α − |σ |α = 0.
p→∞
(1.12)
In fact, to verify that σ ∈ Fα it suffices to check that |σ |α is finite and find L 2 -approximation of σ (·, ·) by bounded continuous symmetric functions σ p (·, ·) for which (1.12) holds. Obviously Fα contains √ all bounded continuous symmetric functions on [0, 1]2 (but for example σ (x, y) = 1/ x + y ∈ L 2 ([0, 1]2 ) is not in Fα ). Remark 1.6. Things are a bit simpler if in the definition of the matrix YσN one replaces the sample σ ( Ni , Nj ) by the average of σ (·, ·) with respect to Lebesgue measure on j−1 j i ( i−1 N , N ] × ( N , N ], for then we can replace throughout this paper the semi-norm · and the space L 2 ([0, 1]2 ) by the usual L 2 -norm and space.
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We further say that σ ∈ Fα is equivalent to σ ∈ C if for the relevant finite partition 0 = b0 < b1 < · · · < bq = 1 we have for any 1 ≤ r, s ≤ q that
bs
|σ (x, v)|α dv = | σr s |α
for all x ∈ (br −1 , br ].
bs−1
Extending Theorem 1.3 we next characterize the Cauchy-Stieltjes transform of µσ for any σ ∈ Fα . Theorem 1.7. Given σ ∈ Fα , the sequence E[µˆ AσN ] converges weakly towards the symmetric probability measure µσ such that for some R = R(σ ) finite,
1 1 dµσ (x) = z−x z
0
1
h α (Yvσ (z))dv
(1.13)
and Y σ is the unique analytic mapping Y σ : C+ → L ∞ ((0, 1]; Kα ) such that if |z| ≥ R then for almost every x ∈ (0, 1], z α Yxσ (z) = Cα
0
1
|σ (x, v)|α gα (Yvσ (z))dv.
(1.14)
The measure µσ has a density ρ σ on R\{0} which is bounded off zero and such that t α+1 ρ σ (t) → α2 |σ (x, v)|α dxdv as t → ∞. σ. Further, if σ ∈ Fα is equivalent to σ ∈ C then µσ = µ Remark 1.8. A similar invariance applies in the case of entries with bounded variance, where the kernel K xσ (z) that characterizes the limit law in (1.1) is the same across each equivalence class of F2 . Also note that for α = 2 we have C2 = −1 and g2 (y) = h 2 (y) = 1/(y+1) is well defined when (y) > −1. Plugging the latter expressions into (1.13) and (1.14) indeed coincide with (1.1) upon setting z K xσ (z) = g2 (Yxσ (z)) = 1/(1 + Yxσ (z)), whereas (1.6) and (1.7) result for α = 2 with Y (z) = − 1z G 2 (z) and the Cauchy-Stieltjes √ transform G 2 (z) = (z − z 2 − 4)/2 of the semi-circle law µ2 (upon properly choosing the branch of the square root). σ ∈ C is often quite useful. For Remark 1.9. The equivalence between σ ∈ Fα and example, if ϕ : [−1, 1] → R is any even, periodic function of period one and finitely many jump discontinuities, then σ (x, y) = ϕ(x − y) ∈ Fα and is equivalent to the con1 σ ·) of [1] and stant σ = [ 0 |ϕ(v)|α dv]1/α . Consequently, in this case µσ equals µα ( hence has the symmetric, uniformly bounded, continuous off zero, density σ −1 ρα (t/ σ) with respect to Lebesgue measure on R. t Consider next the empirical covariance matrices W N ,M = a −2 N +M X N ,M X N ,M , where X N ,M is an N × M matrix with heavy tailed entries xi j , 1 ≤ i ≤ N , 1 ≤ j ≤ M, the law of which satisfies (1.2) (and B t denotes throughout the transpose of the matrix B). Taking N → ∞ and M/N → γ ∈ (0, 1] the scaling constant a N is chosen per (1.3) (so 2 from (1.2) we have that a 2N +M ∼ N α (1 + γ )2/α L 1 (N ) for some slowly varying function L 1 (·)). In this setting we show the following about the limiting spectral measure of W N ,M .
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S. Belinschi, A. Dembo, A. Guionnet
Theorem 1.10. If N → ∞ and M N → γ ∈ (0, 1] then with probability one, the spectral γ measures µˆ W N ,M converge to a non-random probability measure µα . The probability measure µ1α is absolutely continuous with the density √ ρα1 (t) = 21/α t −1/2 ρα (21/α t) on (0, ∞). Fixing γ ∈ (0, 1) let (Y1 (z), Y2 (z)) denote the unique analytic functions of z ∈ C+ tending to zero at infinity, such that z α Y1 (z) =
γ Cα gα (Y2 (z)), 1+γ
z α Y2 (z) =
1 Cα gα (Y1 (z)). 1+γ
(1.15)
The functions Y1 (z) and Y2 (z) extend continuously to functions on (0, ∞) that are anaγ γ lytic through (R, ∞) for some finite R = Rα . The probability measure µα then has an atom at zero of mass 1 − γ and the continuous density √ 1 ραγ (t) = − h α (Y1 ( t ) , (1.16) πt on (0, ∞) which is real-analytic on (R, ∞), bounded off zero, does not vanish in any γ αγ neighborhood of zero and such that t 1+α/2 ρα (t) → 2(1+γ ) as t → ∞. γ
Remark 1.11. Note the contrast between the non-vanishing near zero density ρα and the γ Pastur-Marchenko law µ2 which vanishes throughout [0, 1 − γ ] (cf. [8]). We also consider diagonal perturbations of heavy tailed matrices. Namely, the limit of the spectral measures µˆ AσN + D N , where D N is a diagonal N × N matrix, whose entries {D N (k, k), 1 ≤ k ≤ N } are real valued, independent of the random variables (xi j , 1 ≤ i ≤ j < ∞) and identically distributed, of law µ D which has a finite second moment. In this setting we have the following extension of Theorem 1.3 and Theorem 1.7. α := {R0 eiϕ : − απ ≤ ϕ ≤ 0, R0 ≥ 0}. Given σ ∈ Fα , the Theorem 1.12. Let K 2 sequence E[µˆ AσN + D N ] converges weakly towards the probability measure µσ, D whose Cauchy-Stieltjes transform at z ∈ C+ is 1 α 1 D dµ D (λ) (z) = h α ((λ − z)− 2 (1.17) G α,σ X vσ (z))dv, z−λ 0 α ) for some R = R(σ ) finite and the unique analytic mapping X σ : C+ → L ∞ ((0, 1]; K such that if (z) ≥ R(σ ), then for almost every x ∈ (0, 1], 1 α α σ α X x (z) = C α |σ (x, v)| (λ − z)− 2 gα (λ − z)− 2 X vσ (z) dµ D (λ) dv. (1.18) 0
X xσ (z) takes the same value X r (z) for all x ∈ (br −1 , br ], where If σ ∈ C , then α such ( X r (z), 1 ≤ r ≤ q) is the unique collection of analytic functions from C+ to K that q α α α X r (z) = C α |σr s | s (λ − z)− 2 gα (λ − z)− 2 X s (z) dµ D (λ) (1.19) s=1 α
and | X r (z)| ≤ c((z))− 2 for some finite c and all r ∈ {1, . . . , q}.
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Remark 1.13. The substitution of g2 (y) = h 2 (y) = 1/(1+y) in (1.18) and (1.19) leads to D (z) = (λ−z− the prediction G 2,σ X vσ (z))−1 dvdµ D (λ) with X xσ (z) = |σ (x, v)|2 (λ− X xσ (z) = G 2D (z) z− X vσ (z))−1 dvdµ D (λ) which in particular for σ (·, ·) ≡ 1 results with independent of x that to the celebrated free-convolution of µ D and µ2 . corresponds D D Namely, G 2 (z) = (λ − z − G 2 (z))−1 dµ D (λ). While beyond the scope of this paper, it is of interest to study the behavior of the eigenvectors of large random matrices of heavy tailed entries (such as AσN or W N ,M ), and in particular, to find out if they concentrate on indices associated with the entries of extreme values or are rather “spread-out”. After devoting the next section to the truncation and approximation tools used in our work, we proceed to prove our main results, starting with the proof of Theorem 1.3 in Sect. 3. This is followed by the proof of Theorem 1.7 in Sect. 4, the specialization to covariance matrices (i.e. proof of Theorem 1.10) in Sect. 5 and the generalization to diagonal perturbations (i.e. proof of Theorem 1.12) in Sect. 6. 2. Truncation, Tightness and Approximations As the second moment of entries of our random matrices is infinite, we start by providing appropriate truncated matrices, whose spectral measures approximate well (in the limit N → ∞) the spectral measures µˆ A N . Specifically, let Y NB denote the N × N symmetric matrix with entries σ ( Ni , Nj )xi j 1|xi j | 0. We further consider the N × N symmetric matrix YκN with entries σ ( Ni , Nj )xi j 1|xi j | 0, and the corresponding normalized matrices, −1 κ B B κ AN := a −1 N Y N , A N := a N Y N .
It is easy to adapt the proof of [1, Lemma 2.4] to our setting and deduce that for every > 0, there exists B() finite and δ(, B) > 0 when B > B(), such that P(rank(Y N − Y NB ) ≥ N ) ≤ e−δ(,B)N . Likewise, for κ > 0, and a ∈]1 − ακ, 1[ there exists a finite constant C = C(α, κ, a) such that P(rank(Y N − YκN ) ≥ N a ) ≤ e−C N
a
log N
(and both bounds are independent of σ (·, ·)). By Lidskii’s theorem it then readily follows that P d1 (µˆ A N , µˆ A B ) ≥ 2 ≤ e−δ(,B)N , (2.1) N a P d1 (µˆ A N , µˆ AκN ) ≥ 2N a−1 ≤ e−C N log N , (2.2) where the metric d1 (µ, ν) :=
sup
f BL ≤1, f ↑
f dν − f dµ
on the set P(R) of Borel probability measures on R is compatible with the topology of weak convergence (for example, see [1, Lemma 2.1]), and throughout f BL denotes the standard Bounded Lipschitz norm on R. Just as in [1, Lemmas 3.1], we have the following tightness result.
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Lemma 2.1. The sequence (E[µˆ A N ]; N ∈ N) is tight for the topology of weak convergence on P(R). Further, for every B > 0 and κ > 0, the sequences (E[µˆ A B ]; N ∈ N) N and (E[µˆ AκN ]; N ∈ N) are also tight in this topology. Proof. Recall that
N 1 1 i j 2 B 2 2 E σ ( E |x | 1 tr((A N ) ) = , ) i j |x |
(2.3)
As the latter expectation does not depend on i, j and using the key estimate E[|xi j |ζ 1|xi j |
α ζ B ζ −α a N N −1 , ζ −α
(2.4)
for any ζ > α, we deduce that since σ is in L 2 ([0, 1]2 ), 1 α B 2 tr((A N ) ) ≤ B 2−α σ 2 < ∞. lim E N →∞ N 2−α
(2.5)
This implies the tightness of (E[µˆ A B ], N ∈ N) which upon using (2.1) and (2.2) proN vides also the tightness of (E[µˆ A N ], N ∈ N) and (E[µˆ AκN ], N ∈ N), respectively (for more details, see the proof of [1, Lemma 3.1]). We next show that it suffices to prove the convergence of the spectral measures E[µˆ AσN ] for σ (·, ·) in any given dense subset of L 2 ([0, 1]2 ). Proposition 2.2. Suppose that a sequence (σ p , p ∈ N) converges in L 2 ([0, 1]2 ) towards σ and that for all p ∈ N, lim E[µˆ Aσ p ] = µσ p .
N →∞
(2.6)
N
Then, µσ p converges weakly as p → ∞ towards some Borel probability measure µσ and E[µˆ AσN ] converges weakly towards µσ as N → ∞. Proof. Note that for some finite constant c = c(α, B), independent of N and σ , 1 B,σ B,σ tr (A N − A N p )2 ≤ c2 σ − σ p 2 . (2.7) E[d1 (µˆ A B,σ , µˆ B,σ p )]2 ≤ E AN N N Indeed, the leftmost inequality is based on Lidskii’s theorem (see [3, (2.16)]), whereas the rightmost one is obtained by an application of (2.3)–(2.5) with σ replaced by σ −σ p . Next, from the triangle inequality for the d1 -metric, we have that d1 (E[µˆ AσN ], µσ p ) ≤ d1 (E[µˆ AσN ], E[µˆ A B,σ ]) + d1 (E[µˆ A B,σ ], E[µˆ N
+ d1 (E[µˆ
B,σ p
AN
B,σ p
AN σp
N
])
], E[µˆ Aσ p ]) + d1 (E[µˆ Aσ p ], µ ). N
N
By our hypothesis (2.6), the last term converges to zero as N → ∞. Further, by (2.1) and the boundedness and convexity of d1 , we find that for some (B) → 0 as B → ∞, independently of σ and σ p , lim sup d1 (E[µˆ AσN ], E[µˆ A B,σ ]) + lim sup d1 (E[µˆ N →∞
N
N →∞
B,σ p
AN
], E[µˆ Aσ p ]) ≤ 8(B). N
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Moreover, by the convexity of d1 and (2.7), we have that lim sup d1 (E[µˆ A B,σ ], E[µˆ N →∞
N
B,σ p
AN
]) ≤ c(α, B) σ − σ p .
Upon combining these estimates we deduce that for any p ∈ N and B > 0, lim sup d1 (E[µˆ AσN ], µσ p ) ≤ 8(B) + c(α, B) σ − σ p . N →∞
(2.8)
In particular, we get the bound sup d1 (µσ p , µσq ) ≤ 16(B) + 2c(α, B)δ(r )2 ,
p,q≥r
where by hypothesis δ(r )2 := sup p≥r σ − σ p converges to zero as r → ∞. Taking r and B going to infinity such that c(α, B) ≤ δ(r )−1 we conclude that (µσ p , p ∈ N) is d1 -Cauchy and hence converges to some µσ ∈ P(R) (recall that (B) and c(α, B) are independent of σ ). By this convergence, combining (2.8) and the triangle inequality for the d1 -metric, we deduce upon taking p → ∞ and then B → ∞, that E[µˆ AσN ] also converges towards µσ as N → ∞. Remark 2.3. By our assumptions, when dealing with σ ∈ Fα we may and shall take in κ,σ Proposition 2.2 some σ p ∈ C . Since the rank of the matrix E[A N p ] is then uniformly κ,σ bounded in N , as in [1, Remark 2.5] we may and shall recenter A N p without changing its limiting spectral distribution. We conclude by showing an interpolation property of µˆ AσN in case σ ∈ C . That is, the weak convergence of µˆ AσN follows once we have it along a suitable sub-sequence φ(n). Lemma 2.4. Suppose σ ∈ C and the increasing function φ : N → N is such that φ(n − 1)/φ(n) → 1. If µˆ Aσφ(n) converges weakly to some probability measure µσ then so does µˆ AσN . Proof. For any N ∈ (φ(n − 1), φ(n)] set M = φ(n) and let AσN denote the M × M dimensional matrix whose upper left N × N corner equals (a N /a M )AσN and having zero entries everywhere else. Letting 0 = b0 < b1 < · · · < bq = 1 denote the σ (i, j) = Aσ (i, j) unless either partition that corresponds to σ ∈ C , observe that A N M i ∈ (br N , br M] or j ∈ (br N , br M] for some r = 0, 1, . . . , q. As the latter applies for at most (q + 1)(M − N + 1) values of 1 ≤ i ≤ M and at most (q + 1)(M − N + 1) values of 1 ≤ j ≤ M, it follows that rank( AσN − AσM ) ≤ 2(q + 1)(M − N + 1), so by Lidskii’s theorem N −1 φ(n − 1) , d1 (µˆ ˆ AσM ) ≤ 4(q + 1)(1 − ) ≤ 4(q + 1) 1 − AσN , µ M φ(n) which converges to zero as N → ∞ (hence n → ∞). Therefore, by the triangle inequality for the d1 -metric, our assumption that d1 (µˆ Aσφ(n) , µσ ) → 0 implies that
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S. Belinschi, A. Dembo, A. Guionnet
σ σ d1 (µˆ AσN , µ ) → 0 as N → ∞. Next note that the eigenvalues of A N are those of (a N /a M )AσN augmented by M − N zero eigenvalues. Fixing a monotone non-decreasing bounded Lipschitz function f (·), we have thus seen that N N σ ) f (0) + f d µˆ = (1 − x)d µ ˆ (x) → f dµσ , (2.9) f (β σ N AN AN M M
when N → ∞, where 1 ≥ β N := a N /a M ≥ aφ(n−1) /aφ(n) (as both φ(·) and ak are non-decreasing, see (1.3)). Since φ(n − 1)/φ(n) → 1 the same applies for N /M ∈ (φ(n − 1)/φ(n), 1]. Further, ak = L 0 (k)k 1/α with L 0 (·) a slowly varying function, hence also aφ(n−1) /aφ(n) → 1 when n → ∞ and consequently β N → 1 as N → ∞. Fixing > 0, since f (·) is monotone and bounded, there exists K = K () finite such that | f (x) − f (y)| ≤ whenever min(x, y) ≥ K or max(x, y) ≤ −K . Thus, for any β ∈ (0, 1], K sup | f (x) − f (βx)|dν(x) ≤ + (1 − β) f L . β ν∈P (R) In particular, since β N → 1, for any > 0, f d µˆ AσN − f (β N x)d µˆ AσN (x)| ≤ , lim | N →∞
which in view of (2.9) results with f d µˆ AσN → f dµσ . This holds for each monotone non-decreasing bounded Lipschitz function f (·), which is equivalent to our thesis that µˆ AσN converges weakly to µσ . 3. Induction and the Limiting Equations We consider throughout this section σ ∈ C . That is, there exist 0 = b0 < b1 < · · · < bq = 1 and a q × q symmetric matrix of entries σr s for 1 ≤ r, s ≤ q such that σ (x, y) = σr s for all (x, y) ∈ (br −1 , br ] × (bs−1 , bs ].
(3.1)
Associated with such σ are the random matrix AσN and the N × N piecewise constant matrix σ N of entries σ N (i, j) = σr s for [N br −1 ] < i ≤ [N br ] and [N bs−1 ] < j ≤ [N bs ]. 3.1. Characterization of limit points. For each z ∈ C+ = {z ∈ C : (z) > 0} we define, as in [1, Sect. 4], the matrices G N (z) := (zI N − A N )−1 and the probability measure L zN on C such that for f ∈ Cb (C), N 1 z f (G N (z)kk ) . LN( f ) = E (3.2) N k=1
q as a weighted sum L zN = r =1 N ,r L zN ,r , It is useful for our purpose to represent where L zN ,r are the probability measures on C given by ⎡ ⎤ [N br ] 1 f (G N (z)kk )⎦ , (3.3) L zN ,r ( f ) := E ⎣ [N br ] − [N br −1 ] L zN
k=[N br −1 ]+1
Heavy Tailed Band Matrices
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and N ,r := N −1 ([N br ] − [N br −1 ]) → r as N → ∞. Since each term G N (z)kk belongs to the compact set K(z) := {x ∈ C− : |x| ≤ |(z)|−1 }, the probability measures L zN ,r are supported on K(z) for all N ∈ N and 1 ≤ r ≤ q. We denote by G κN (z) and L z,κ N ,r the corresponding objects when A N is replaced by the truncated matrix AκN . Similarly to [1, Lemma 4.4] we next show that 1 Lemma 3.1. For 0 < κ < 2(2−α) , any 1 ≤ r ≤ q and Lipschitz function f on K(z),
N
z,κ κN ([N br ], k)2 G κN (z)kk )−1
= 0, lim E L N ,r ( f ) − E f (z − A
N →∞ k=1
where
AκN
is an independent copy of
AκN .
Proof. Without loss of generality, it suffices to prove the lemma for r = 1 (the general ¯ κ denote an (N + 1) × (N + 1) case follows by permuting indices). To this end, let A N +1 κ κ (0, k) = A κ (k, 0) symmetric matrix obtained by adding to A N a first row and column A N N κ (0, 0) = κ (0, k), k ≥ 1) is an independent copy of (Aκ (1, k), k ≥ 1) and A such that ( A N N N −1 ¯κ ¯κ κ σ N (1, 1)a −1 N x00 1|x00 |
(3.4)
As in [1], the key to our proof is Schur’s complement formula ⎛ ⎞−1 N κN (0, 0) − κN (l, 0)G κN (z)kl ⎠ , κN (0, k) A G¯ κN +1 (z)00 = ⎝z − A A k,l=1
from which we thus get that ⎡ ⎛⎛ ⎞−1 ⎞⎤ N ⎥ ⎢ ⎜⎝ κN (l, 0)G κN (z)kl ⎠ ⎟ κN (0, k) A κ E[ L¯ z,κ A ⎠⎦ . N +1,1 ( f )] = E ⎣ f ⎝ z − A N (0, 0) − k,l=1
(3.5) Recall that the entries of AκN are centered (see Remark 2.3), and independent of the κ matrix G N (z). Further, as the entries of the matrix σ N are uniformly bounded, the statement and proof of [1, Lemma 4.3] extends readily to our setting, showing that the off diagonal terms in the right hand side of (3.5) are small with overwhelming probability (this is simply based on a computation of the variance of this term, which is possible thanks to the cut-off κ). As shown proof of [1, Lemma 4.4], this allows us to
in the κ (0, 0) and k =l A κ (l, 0)G κ (z)kl in (3.5), resulting with κ (0, k) A neglect the terms A N N N N
N
¯ z,κ κ 2 κ −1 N (0, k) G N (z)kk ) lim E[ L N +1,1 ( f )] − E f (z − A
= 0. (3.6)
N →∞ k=1
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S. Belinschi, A. Dembo, A. Guionnet
Further, with σ N uniformly bounded, adapting the proof of [1, Lemma 4.1] to our setting, we deduce that −η ¯ z,κ lim P(d1 (L z,κ ) = 0, N ,1 , L N +1,1 ) > N
N →∞
z,κ for any 0 < η < 21 (1 − κ(2 − α)). Consequently, |E[ L¯ z,κ N +1,1 ( f )] − E[L N ,1 ( f )]| → 0 as N → ∞ and (3.6) finishes the proof of the lemma.
Identifying C with R2 , recall [1, Defn. 5.1]. Namely, Definition 3.2. Given α ∈ (0, 2) and a compactly supported probability measure µ on C, let P µ denote the probability measure on C whose characteristic function at t ∈ R2 is π α α , eit,x d P µ (x) = exp −vµ, α2 (t) 2 1 − iβµ, α2 (t) tan 4 R2 where
vµ,α (t) = [vα−1 |t, z|α dµ(z)]1/α , ∞ (2 − α) cos( π2α ) sin x −1 , vα = dx = xα 1−α 0 |t, z|α sign(t, z)dµ(z) βµ,α (t) = , |t, z|α dµ(z)
and βµ,α (t) = 0 whenever vµ,α (t) = 0. In particular, if µ is supported in the closure of C− , then so does P µ . Equipped with this definition, our next proposition characterizes the set of possible limit points of {E[L z,κ N ,r ], 1 ≤ r ≤ q}. 1 and z ∈ C+ , any limit point (µrz , 1 ≤ r ≤ q) of Proposition 3.3. For 0 < κ < 2(2−α) z,κ the sequence {(E[L N ,r ], 1 ≤ r ≤ q), N ∈ N} consists of probability measures on K(z) that satisfy the system of equations q q 2 " z f dµrz = f (z − σr2s sα xs )−1 d P µs (xs ) (3.7) s=1
s=1
for r ∈ {1, . . . , q} and every bounded continuous function f on K(z). The following concentration result is key to the proof of Proposition 3.3. 1 Lemma 3.4. For κ ∈ 0, 2−α let = 1 − κ(2 − α) > 0. There exists c < ∞ so that for z ∈ C+ , s ∈ {1, . . . , q}, δ > 0, N ∈ N and any Lipschitz function f on K(z),
c f 2BL −
z,κ ( f ) − E[L ( f )] N , P L z,κ
≥δ ≤ N ,s N ,s |(z)|4 δ 2 with f BL denoting here the Bounded Lipschitz norm of f restricted to K(z).
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Proof. Fixing s ∈ {1, . . . , q} and z ∈ C+ , note that the value of f outside the compact set K(z) on which all probability measures L z,κ N ,s are supported, is irrelevant. We thus assume without loss of generality that f is bounded and continuously differentiable and as in the proof of [1, Lemma 5.4], let FN (A) := L z,κ N ,s ( f ) =
1 N
[b s N]
f (G κN (z)kk ),
k=[bs−1 N ]+1
a smooth function of the n = N (N − 1)/2 independent, centered, random variables AκN (k, l) for 1 ≤ k ≤ l ≤ N . By a classical martingale decomposition we see that E[(FN − E[FN ])2 ] ≤
∂ A(i, j) FN 2∞ E[(AκN (i, j) − E[AκN (i, j)])2 ].
(3.8)
1≤i≤ j≤N
Moreover, similarly to the proof of [1, Lemma 5.4] we have here that 1 ∂ A(m,l) FN (A) = N
[N bs ]
f (G κN (z)kk )(G κN (z)kl G κN (z)mk + G κN (z)km G κN (z)lk )
k=[N bs−1 ]+1
$ 1 # κ [G N (z) Ds ( f )G κN (z)]ml + [G κN (z) Ds ( f )G κN (z)]lm = N with Ds ( f ) the N -dimensional diagonal matrix of entries Ds ( f )kk := f (G κN (z)kk )1[N bs−1 ]
Further, with σ N uniformly bounded, from (2.4) (for ζ = 2), we get that for some c0 finite and all N , sup 1≤i≤ j≤N
E[|AκN (i, j)|2 ] ≤ c0 N κ(2−α)−1 .
As = 1 − κ(2 − α) > 0, substituting these bounds into (3.8) we find that E[(FN − E[FN ])2 ] ≤ 4c0 f 2BL |(z)|−4 N − , and conclude the proof by Chebychev’s inequality. Proof of Proposition 3.3. The sequence of q-tuples of probability measures (E[L z,κ N ,r ], 1 ≤ r ≤ q) N ∈N , each supported in the compact set K(z), is clearly tight. Considerz,κ ing a subsequence (E[L φ(N ),r ], 1 ≤ r ≤ q) N ∈N that converges weakly to a limit point z (µr , 1 ≤ r ≤ q), passing to a further subsequence still denoted φ(N ), we have by Lemma z,κ z 3.4 that (L φ(N ),r , 1 ≤ r ≤ q) N ∈N also converges almost surely to (µr , 1 ≤ r ≤ q), a q-tuple of probability measures on K(z).
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S. Belinschi, A. Dembo, A. Guionnet
By Lemma 3.1, fixing r ∈ {1, . . . , q}, it suffices to show that U N (z, r ) :=
N
κN ([br N ], k)2 G κN (z)kk A
k=1
q 2/α is such that Uφ(N ) (z, r ) converges in law towards s=1 σr2s s xs , where (xs , 1 ≤ s ≤ q) z are independent, with xs ∈ distributed according to P µs for s = 1, . . . , q.
C q Note that U N (z, r ) = s=1 σr2s W N (z, s), where [N bs ]
W N (z, s) :=
κN ([br N ], k)2 G κN (z)kk , A
k=[N bs−1 ]+1
κ ([br N ], k) = A κ ([br N ], k)/σr s are independent of and the i.i.d. random variables A N N G κN (z) and correspond to taking σ ≡ 1. Next let a N (s) = inf u : P(|xi j | ≥ u) ≤
1 N N ,s
,
noting that by (1.2), a N (s) 1/α = s . N →∞ a N lim
(3.9)
Further, applying [1, Theorem 10.4] for X k = x2[N br ]k , a N = a N (s)2 and (N ) = z,κ 2 2κ (a N /a N (s)) N → ∞, on the subsequence φ(N ) and subject to the event that L φ(N ),s z converges to µsz , we deduce that (a N /a N (s))2 W N (z, s) converges in law to P µs . By the conditional independence of W N (z, s) for 1 ≤ s ≤ q (per fixed G κN (z)), and (3.9) we arrive at the stated convergence in law of Uφ(N ) (z, r ). We next derive the analog of [1, Theorem 5.5]. 1 Proposition 3.5. For 0 < κ < 2(2−α) any subsequence of the functions (X N ,r (z) := z,κ α/2 + E[L N ,r (x )], 1 ≤ r ≤ q) from C to Cq has at least one limit point (X r (z), 1 ≤ r ≤ q) such that z → X r (z) are analytic in C+ , |X r (z)| ≤ ((z))−α/2 and for all z ∈ C+ ,
∞
X r (z) = C(α)
α
α
t −1 (it) 2 eit z exp{−(it) 2 X r (z)} dt,
(3.10)
0 πα
with C(α) =
e−i 2 and ( α2 ) q α X r (z) := 1 − |σr s |α s X s (z). 2 s=1
(3.11)
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Proof. The proof is an easy adaptation of [1, Theorem 5.5]. In fact, for each 1 ≤ r ≤ q, the analytic functions X N ,r (z) on C+ are uniformly bounded by ((z))−α/2 (hence uniformly bounded on compacts). Consequently, by Montel’s theorem, any subsequence (X φ(N ),r (z), 1 ≤ r ≤ q) has a limit point (X r (z), 1 ≤ r ≤ q) (with respect to uniform convergence on compacts), consisting of analytic functions on C+ (cf. [4, Theorem 17.21]), that obviously are also bounded by ((z))−α/2 . Fixing z ∈ C+ and passing to a further sub-subsequence along which the compactly supported probability measures E[L z,κ ] converge weakly to µrz for all 1 ≤ r ≤ q, it follows by definition that N ,rα z X r (z) = x 2 dµr (x) (as x → xα/2 is in Cb (K(z))). Next, we prove (3.10) by applying [1, Lemma 5.6] which states that for all z ∈ C+ , ∞ α α t −1 (it) 2 eit z dt. (3.12) z − 2 = C(α) 0
Indeed, combining (3.7) and (3.12) we see that − α2 q q 2 " z X r (z) = z− σr2s sα xs d P µs (xs ) s=1
s=1
∞
= C(α)
t
−1
α 2
(it) exp{it (z −
0
q s=1
2
σr2s sα xs )}dt
q "
d P µs (xs ). z
s=1
Recall [1, Theorem 10.5] that for α ∈ (0, 2) and any probability measure ν compactly supported in the closure of C− , α α α −itx ν (it) 2 e d P (x) = exp − 1 − (3.13) x 2 dν(x) . 2 Since z ∈ C+ and (xs ) ≤ 0, by Fubini’s Theorem and (3.13) we deduce that ∞ q 2 " α z X r (z) = C(α) t −1 (it) 2 eit z exp −itσr2s sα xs d P µs (xs ) dt 0
∞
= C(α)
α
t −1 (it) 2 eit z
0
s=1 q " s=1
% & α α (it) 2 |σr s |α s X s (z) dt, exp − 1 − 2
as claimed. 3.2. Properties of the functions (X r , 1 ≤ r ≤ q). We provide now key information about X r (z) of Proposition 3.5. z ∈ C+ , if X s (z) is as in Proposition 3.5 and as are α q non-negative for s ∈ {1, . . . , q}, then (−z)− 2 s=1 as X s (z) is in the set Kα := {Reiθ : |θ | ≤ απ 2 , R ≥ 0} on which for each β > 0, the entire function ∞ β α gα,β (y) := t 2 −1 e−t exp{−t 2 y}dt, (3.14)
Lemma 3.6. For 0 < κ <
1 2(2−α) ,
0
is uniformly bounded. In particular, this applies to gα = gα,α , to h α = gα,2 and their derivatives of all order.
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S. Belinschi, A. Dembo, A. Guionnet
− Proof. Recall that for z ∈ C+ the measures L z,κ N ,s are each supported on C . Hence, by definition each of the functions X N ,s (z) is in the closed cone % & απ θ α := R0 ei K ≤ θ ≤ 0, R0 ≥ 0 , :− (3.15) 2 α q and thus so is any limit point X s (z) of X N ,s (z). Setting w := (−z)− 2 s=1 as X s (z), it thus follows that for any z ∈ C+ and non-negative as ,
0 ≤ arg(w) +
απ α arg(z) ≤ . 2 2
(3.16)
In particular, w ∈ Kα , as claimed. Key to the boundedness of gα,β (·) on this set is the identity of [1, Eq. (40)], where it is shown that ∞ β α α (−z)−β/2 gα,β (y) = t −1 (it) 2 eit z exp −(−z) 2 (it) 2 y dt, (3.17) 0
for any z ∈ enough so
C+
and y ∈ C. Indeed, for each α ∈ (0, 2) set η = η(α) ∈ (0, π/2] small ϕ :=
πα α π + η< 4 2 2
and let z = eiη ∈ C+ when (y) ≥ 0 while z = ei(π −η) ∈ C+ otherwise. Either way, (z) = sin(η) > 0 and if y = Reiθ ∈ Kα , that is |θ | ≤ απ/2, then α α πα α + η ≥ R cos(ϕ) > 0. (−z) 2 (i) 2 y = R cos |θ | − 4 2 Setting ξ := ξ(α) = cos(ϕ)/(sin(η))α/2 > 0 we thus deduce from (3.17) that for any β > 0, ∞ α β t 2 −1 e−t sin(η) exp −t 2 |y| cos(ϕ) dt |gα,β (y)| ≤ 0
= (sin(η))−β/2 gα,β (ξ |y|) ≤ (sin(η))−β/2 gα,β (0),
(3.18)
is uniformly bounded on Kα . Recall that a mapping f : U → Cq defined on some open U ⊆ Cn is holomorphic on U if each of its coordinates admits a convergent power series expansion around each point of U. Proposition 3.5 suggests viewing (X r (z), 1 ≤ r ≤ q) as an implicit mapping from C+ into Cq that is defined in terms of the zero set of the holomorphic f = ( fr (z, w1 , . . . , wq ), 1 ≤ r ≤ q), where
∞
fr (z, w1 , . . . , wq ) = wr − C(α) 0
α
α
t −1 (it) 2 eit z exp{−(it) 2
q
cr s ws } dt,
s=1
$ # and cr s = 1 − α2 |σr s |α s . Key properties of (X r (z), 1 ≤ r ≤ q) are then consequences of the rich theory of zero sets of holomorphic mappings. We shall employ this α X r (z) is X r (z) and strategy, but for Y (z) ≡ (Y1 (z), . . . , Yq (z)), where Yr (z) := (−z)− 2
Heavy Tailed Band Matrices
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given by (3.11). Indeed, our next result, extending [1, Theorem 6.1], characterizes Y (z) as implicitly defined for u = z −α via u → V (u) such that F(u, V (u)) = 0.
(3.19)
With ar s = Cα |σr s |α s , the holomorphic mapping F : C × Cq → Cq is given for u ∈ C and y = (y1 , . . . , yq ) ∈ Cq by Fr (u, y) = yr − u
q
ar s gα (ys )
1≤r ≤q
(3.20)
s=1
Proposition 3.7. Setting Eα := {u ∈ C : −π α < arg(u) < 0}, there exist ε = ε(σ ) > 0 and a unique analytic solution y = V (u) of F(u, y) = 0 on the open set Eα,ε := Eα ∪ B(0, ε). Further, there exists a unique collection of analytic functions (X r (z), 1 ≤ α r ≤ q) on C+ such that |X r (z)| ≤ ((z))− 2 and for which (3.10) holds. The functions α Yr (z) = (−z)− 2 X r (z) are then the unique solution of (1.11) analytic on z ∈ C+ and each tending to zero as |z| → ∞. Moreover, Yr (z) = Vr (z −α ) ∈ Kα are for r = 1, . . . , q such that Yr (−z) = Yr (z) and have an analytic continuation through (R, ∞) for some α finite R = R(σ ), whereas z 2 X r (z) (hence z α Yr (z)), are uniformly bounded on C+ . α
α
Proof. First, with (−z) 2 (−z)− 2 = 1, we deduce from (3.17) that (3.10) is equivalent to α
X r (z) = C(α)(−z)− 2 gα,α (Yr (z)),
(3.21)
which in combination with (3.11) shows that (Yr (z), 1 ≤ r ≤ q) satisfies (1.11). The existence of analytic solutions (X r (z), 1 ≤ r ≤ q) and (Yr (z), 1 ≤ r ≤ q) such that α |X r (z)| ≤ ((z))− 2 is thus obvious from Proposition 3.5. This solution of (1.11) consists by Lemma 3.6 of analytic functions from C+ to Kα . Further, by the boundedness of gα (·) on Kα we know that |X r (z)| ≤ κ|z|−α/2 and |Yr (z)| ≤ κ|z|−α for some finite constant κ, all z ∈ C+ and r ∈ {1, . . . , q}. We turn to prove the uniqueness of the analytic solution of (1.11) tending to zero as (z) → ∞ (hence the uniqueness of such solutions tending to zero as |z| → ∞). To this end, considering F of (3.20) note that F(0, 0) = 0 and the complex Jacobian matrix of y → F(0, y) at y = 0 has a non-zero determinant (since ∂ys Fr (0, 0) = δr s , with determinant one). Consequently, by the local implicit function theorem there are positive constants ε, δ and an analytic solution y = V (u) of F(u, y) = 0 on B(0, ε) which for any |u| < ε is also the unique solution with y < δ. Identifying C+ with Eα via the analytic function u = z −α , note that Y (z) solves (1.11) for z ∈ C+ if and only if V (u) = Y (z) satisfies (3.19) for u ∈ Eα . Consequently, setting R = ε−1/α finite, any two solutions Y i (z), i = 1, 2 of (1.11) that tend to zero as (z) → ∞ coincide once (z) > R is large enough to assure that maxi=1,2 Y i (z) < δ. The uniqueness of the analytic solution z → Y (z) of (1.11) on C+ tending to zero as (z) → ∞ then follows by the identity theorem. By (3.21) this implies also the uniqueness of the solution of α (3.10) which is analytic and bounded by ((z))− 2 throughout C+ . Moreover, by the identity theorem, u → V (u) extends uniquely to an analytic solution of (3.19) on Eα,ε and Y (z) = V (z −α ) has an analytic extension through (R, ∞). Next, recall that Aκ,−σ = −Aκ,σ N N are real-valued matrices, hence by definition κ,σ κ,−σ ( f (x)) = G N (z) = −G N (−z) for any z ∈ C+ , implying by (3.3) that L z,κ,−σ N ,s
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S. Belinschi, A. Dembo, A. Guionnet
L −z,κ,σ ( f (−x)). If x ∈ K(z) then so is −x and xα/2 = i α (−x)α/2 . It thus follows N ,s −σ
from Proposition 3.5 that X s (z) = i α X sσ (−z) for any z ∈ C+ and 1 ≤ s ≤ q. Since (X rσ (z), 1 ≤ r ≤ q) are uniquely determined by Eqs. (3.10) which are invariant under σ → −σ and (−z)α/2 = i α (z)α/2 for all z ∈ C+ , we thus deduce from (3.11) that Yr (z) = Yr (−z) for all 1 ≤ r ≤ q and z ∈ C+ . To recap, for some ε > 0 we got the existence of a unique analytic solution y = V (u) of F(u, y) = 0 on Eα,ε for the holomorphic mapping F : C × Cq → Cq of (3.20). We proceed to show that V (u) has a continuous algebraic extension to E α,ε , and in particular to (0, ∞) (by algebraic extension we mean that (3.19) holds throughout E α,ε ). As we show in the sequel, this yields the claimed continuity of the density ρ σ in Theorem 1.3. To this end, recall that M ⊆ Cn is an embedded complex manifold (in short, a manifold), of dimension p if for each a ∈ M there exist a neighborhood U of a in Cn and a holomorphic mapping f : U → Cn− p such that M ∩ U = {z ∈ U : f (z) = 0} and the complex Jacobian matrix of f (·) is of rank n − p at a (in short, ranka ( f ) = n − p, cf [5, Def. 2, Sect. A.2.2]). Indeed, our claim is merely an application of the following general extension result for the mapping F of (3.20), taking u 0 = 0 in the nonempty open simply connected set O = Eα,ε of piecewise smooth boundary. Proposition 3.8. Suppose F : C × Cq → Cq is a holomorphic mapping and F(u, V (u)) = 0 for analytic V : O → Cq and a nonempty open connected O ⊆ C. Suppose further that the graph V := {(u, V (u)) : u ∈ O}
(3.22)
of V is a one-dimensional complex manifold and the Jacobian determinant det[∂y F] is non-zero at some v 0 = (u 0 , V (u 0 )) with u 0 ∈ O. Then, V (·) has a continuous extension at boundary points x ∈ O, where O is locally connected and V is locally uniformly bounded (i.e. O ∪ {x} admits a local basis of connected relative neighborhoods and V is uniformly bounded on U ∩ O for some neighborhood U of x in C). Moreover, F(x, V (x)) = 0 at any such point. Deferring the proof of Proposition 3.8 to the end of this section, we next collect all properties needed for applying it in our setting. Lemma 3.9. Assuming σ ≡ 0, the mapping u → V (u) of Proposition 3.7 is injective on Eα,ε (and consequently, so is the map z → Y (z) = V (z −α )). Further, in this case V := {(u, V (u)) : u ∈ Eα,ε } is a one-dimensional complex manifold containing the point (0, 0), where [∂y F] is the identity matrix, and V (u) 2 ≤ K |u| for some finite constant K = K (σ ) and all u ∈ Eα,ε . Proof. First note that if F(u, y) = F( u , y) = 0 for some y = 0 then by (3.20) necessarily u = u . Further,
by excluding σ ≡ 0 we made sure that if F(u, 0) = 0 then u = 0 (since gα (0) > 0 and s ar s = 0 for some r ). In particular, u → V (u) is injective. By the same reasoning, V (u) = 0. Indeed, (3.19) amounts to Vr (u) − u
q s=1
ar s gα (Vs (u)) = 0,
1 ≤ r ≤ q,
(3.23)
Heavy Tailed Band Matrices
1041
and differentiating this identity in u, we see that if V (u) = 0 then necessarily q
ar s gα (Vs (u)) = 0,
1 ≤ r ≤ q.
(3.24)
s=1
Clearly, if (3.24) holds then it follows from (3.23) that V (u) = 0 and as we have already seen, for σ ≡ 0 it is then impossible for (3.24) to hold. Next we show that V ⊆ C × Cq is a complex one-dimensional manifold, by finding for any point u ∈ Eα,ε , a suitable holomorphic mapping from a neighborhood U of v = (u, V (u)) in Cq+1 to Cq having a Jacobian of rank q at v. Indeed, as it is not possible to have V1 (u) = · · · = Vq (u) = 0, we may assume without loss of generality that, for a given u, Vq (u) = 0. Then, by the inverse function theorem there exists a neighborhood U ⊆ Eα,ε of u with Vq (·) having an analytic inverse on the neighborhood Vq (U ) of Vq (u). Thus, on the neighborhood U = U × Cq−1 × Vq (U ) of v in Cq+1 we have the holomorphic mapping f : U → Cq , where fr (w, y) = yr − Vr (Vq−1 (yq )) for 1 ≤ r ≤ q − 1 and f q (w, y) = Vq (w) − yq . Clearly, f (w, y) = 0 for (w, y) ∈ U if and only if y = V (w) and w ∈ U , hence {(w, y) ∈ U : f (w, y) = 0} is precisely V ∩ U. Further, since ∂yr f s = δr s for 1 ≤ r ≤ q − 1 and ∂w f s = Vq (w)δqs , the Jacobian determinant at v of f (·, yq ) with yq fixed is Vq (u) = 0. We conclude that rankv ( f ) = q and V is a one dimensional complex manifold, as claimed. Finally, while proving Proposition 3.7 we found that det[∂y F](0, 0) = 1, that V (u) ∈ (Kα )q for all u ∈ Eα , and that V (·) is uniformly bounded on B(0, ε). With gα (·) uniformly bounded on Kα (and on compacts), it follows from (3.23) that V (u) 2 ≤ K |u| for some finite constant K = K (σ ) and all u ∈ Eα,ε . Remark 3.10. The assumptions of Proposition 3.8 do not yield a unique extension of V around boundary points of O. That is, the extension provided there may well be non-analytic. For example, the Cauchy-Stieltjes transform y = G 2 (z) of the semi-circle law µ2 at z = u −1 is specified in terms of zeros of the holomorphic function F(u, y) = y−u(y 2 +1) on C2 . It is not hard to check that for any positive ε < 1/2 √ the unique analytic solution y = V (u) of F(u, y) = 0 on E1,ε is then V (u) = (1 − 1 − 4u 2 )/(2u) for u = 0 and V (0) = 0. Following the arguments of Lemma 3.9, one finds that this injective function is uniformly bounded in the neighborhood of any boundary point of E1,ε and its graph V is a one-dimensional manifold containing the origin (where ∂ F/∂y = 1). However, V (x) does not have an analytic extension at x = 1/2 as the corresponding density ρ2 (t) is not real-analytic at t = ±2. For the convenience of the reader, we summarize, following reference [5], the terminology and results about analytic functions of several complex variables which we use in proving Proposition 3.8. A (local) analytic set is a subset A of a complex manifold M such that for any a ∈ A there exists a neighborhood U of a in M and a holomorphic mapping f : U → Cn such that A ∩ U = {z ∈ U : f (z) = 0} (in contrast with a manifold, there is no condition on the rank of the Jacobian of the mapping f ). We call A ⊆ M an analytic subset of the complex manifold M if this further applies at all a ∈ M (and not only at the points a in A), and say that A is a proper analytic subset of M if A = M. In particular, any embedded complex manifold is an analytic set (of Cq ), but, unless it is closed in Cq , it cannot be an analytic subset of Cq . For example, H = {z ∈ Cq : z 2 < 1, z 1 = 0} is a manifold (of dimension q − 1), a (local) analytic set in Cq , but not an analytic subset
1042
S. Belinschi, A. Dembo, A. Guionnet
of Cq . However, as observed in [5, Sect. 1.2.1], every (local) analytic set on a complex manifold M is an analytic subset of a certain neighborhood of M (for example, H is an analytic subset of the open unit ball in Cq ). A point of an analytic set A (on Cq ) is called regular if it has a neighborhood U (in Cq ) so that A ∩ U is a manifold in Cq . Clearly, the set regA of regular points of an analytic set A is a union of manifolds (alternatively, an analytic set is a manifold around each of its regular points). Topologically, most points of an analytic set are regular. That is, for an arbitrary analytic set A the set regA of regular points is everywhere dense in A (cf. [5, Sect. 1.2.3]). Thus, the dimension dima A of A at a point a ∈ A is defined as the dimension of the manifold around a if a ∈reg A and in general by dima A =
lim sup dim z A.
z→a, z∈regA
The dimension of the analytic set A, denoted dimA is then the largest such number when a runs through A and an analytic set A is called p-dimensional if dimA = p (see [5, Sect. 1.2.4]). An essential ingredient of our proof is the notion of irreducibility and of irreducible components for analytic sets [5, Sect. 1.5.3]. An analytic subset A of a complex manifold M is reducible in M if there exist two analytic subsets A1 , A2 of M so that A = A1 ∪ A2 and A1 = A = A2 . Otherwise A is called irreducible. For example, A = {z ∈ C3 : z 1 z 2 = z 1 z 3 = 0} is a reducible set, being the union of A1 = {z ∈ C3 : z 2 = z 3 = 0} (a one dimensional manifold), and A2 = {z ∈ C3 : z 1 = 0} (a two dimensional manifold). An irreducible analytic subset A of an analytic set A is called an irreducible component of A if every analytic subset A of A such that A A is reducible. It is known [5, Theorem, Sect. 1.5.4] that any analytic subset A of a complex manifold M has a unique decomposition into countably (or finitely) many irreducible components S j , which are the closures (in M) of the partition {S j } of regA into disjoint connected components. Further, dimS j = dimS j [5, Theorem, Sect. 1.5.1] and by definition of regular points each connected component S j is a manifold (in case M = Cq ). The importance of irreducibility for us resides in the following ‘uniqueness’ result: if A, A are analytic subsets of a complex manifold, A is irreducible and A ⊆ A , then dimA ∩ A < dimA [5, Sect. 1.5.3, Cor. 1]. Topological properties simplify considerably when a set A is contained in a proper analytic subset of a connected complex manifold M. Indeed, in this context M\A is arc-wise connected and in case of a one dimensional manifold M (that is, a Riemann surface), we further have that A is locally finite, i.e. A ∩ K is a finite set for any compact K ⊆ M (see [5, Sect. 1.2.2]). Proof of Proposition 3.8. Clearly, V is a connected set (being the graph of a continuous function on the connected set O). Further, by our assumptions, the connected one-dimensional complex manifold V is contained in the analytic subset A = {(u, y) ∈ C × Cq : F(u, y) = 0}, of C × Cq given by the zeros of the holomorphic mapping F. We proceed to show the crux of our argument, that the closure V of V (in Cq+1 ) is part of a one-dimensional irreducible component of A. To this end, consider the analytic subset D = {(u, y) ∈ C × Cq : det[∂y F](u, y) = 0}
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of C × Cq . By definition, V ∩ D is an analytic subset of V, and it is a proper subset, for we know that v 0 = (u 0 , V (u 0 )) ∈ V\D. Further, by the implicit function theorem, the analytic subset A is regular at any a ∈ A\D, so V\ regA is contained in the proper analytic subset V ∩ D of the Riemann surface V. Consequently, by [5, Prop. 2, Sect. 1.2.2], we deduce that V is an ‘almost regular’ part of A. That is, V\ regA is locally finite and consequently the closure V of V in Cq+1 is the same as the closure of V ∩ reg A. Further, by [5, Prop. 3, Sect. 1.2.2], V∩reg A is arc-wise connected, hence included in one connected component S of regA, with V = V ∩ regA thus contained in the closure S of S (in Cq+1 ). Recall that by definition the connected component S of regA is a manifold and since v 0 is in V∩ reg A, we have that S contains the manifold V ∩ U for some neighborhood U of v 0 (in Cq+1 ). The connected manifold V ∩ U has an accumulation point in both V and S, so all three have the same dimension, that is, dimS = dimV = 1 (see [5, Sect. 1.2.2]). As shown in [5, Theorem, Sect. 1.5.4], the irreducible component of A containing S is its closure S, which by the definition of an irreducible component is an analytic subset of Cq+1 and further by [5, Theorem, Sect. 1.5.1] dimS = dimS = 1. We claim that if V (u) is uniformly bounded on O ∩ U for some neighborhood U (in C) of a boundary point x ∈ O where O is locally connected, then the existence of the one-dimensional irreducible analytic subset S of Cq+1 insures that V (·) extends continuously at x such that F(x, V (x)) = 0. Indeed, since V is uniformly bounded on O ∩ U , by the continuity of V (·) on O the cluster set Cl(x) of all limit points of {V (u) : u ∈ O} as u → x, is a non-empty, compact, connected subset of Cq (see the proof given in [6, Theorem 1.1] for q = 1). Clearly, {x} × Cl(x) is contained in the analytic subset A(x) = {(u, y) ∈ A : u = x} of C × Cq as well as in the closure V of V (in Cq+1 ). With V ⊆ S, we thus deduce that {x} × Cl(x) ⊆ A(x) ∩ S. Recall that S is a one-dimensional, irreducible analytic subset of Cq+1 . Since S ⊆ A(x) (as v 0 ∈ V and u 0 is not a boundary point of O), by [5, Cor. 1, Sect. 1.5.3] we have that dimA(x) ∩ S = 0. Thus, A(x) ∩ S is a discrete (analytic) set, so its connected subset {x} × Cl(x) must be a single point, i.e. V extends continuously at x. Moreover, A is a closed subset of C × Cq (by continuity of F), hence the extension V (x) of V satisfies F(x, V (x)) = 0, as claimed. 3.3. Limiting spectral measures: proof of Theorem 1.3. Fixing z ∈ C+ let (µsz , 1 ≤ s ≤ q) denote some limit point of the compactly supported (E[L z,κ N ,s ], 1 ≤ s ≤ q). Then, on α the corresponding subsequence, X N ,r (z) converges for r = 1, . . . , q to x 2 dµrz which by Propositions 3.5 and 3.7 thus coincides with the unique analytic solution X r (z) of α (3.10) bounded by ((z))− 2. By (3.7) (for f (x) = x bounded and continuous on ∞ K(z)), the identity z −1 = −i 0 eit z dt, and Fubini’s theorem, we deduce that for each r ∈ {1, . . . , q}, ∞ q " 2 z xdµrz = −i eit z exp{−itσr2s sα xs }d P µs (xs ) dt
0
∞
= −i 0
s=1 α
eit z exp{−(it) 2 X r (z)}dt = z −1 gα,2 (Yr (z)),
(3.25)
where we get the latter equality from (3.13) and the definition (3.11) of X r (z), followed by the application of (3.17) with β = 2. In particular, by Proposition 3.7 xdµrz are
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uniquely determined (for all r and z ∈ C+ ), hence E[L z,κ N ,s (x)] converges as N → ∞ to the right side of (3.25). Next, by Lemma 2.1 the sequence E[µˆ Aκ,σ ] is tight for the topology of weak converN gence. Further, recall that for any z ∈ C+ and all N ,
1 N ,s L z,κ d µˆ Aκ,σ (x) = N ,s (x). N z−x q
s=1
Hence, any limit point
µσ
of E[µˆ Aκ,σ ] is such that for each z ∈ C+ ,
1 s dµσ (x) = z−x
N
q
xdµsz (x).
(3.26)
s=1
Recall that gα,2 = h α , so combining (3.25) and (3.26) we thus arrive at the stated formula (1.10) for the values of the Cauchy-Stieltjes transform G α,σ (z) of the probability measure µσ on the real line, at all z ∈ C+ . Since h α is uniformly bounded on the closed set Kα (see Lemma 3.6), and Ys (z) ∈ Kα for all z ∈ C+ and 1 ≤ s ≤ q, we deduce from (1.10) that G α,σ (z) is uniformly bounded on C+ ∩ B(0, δ)c for each δ > 0. By the Stieltjes-Perron inversion formula, it follows that the density ρ σ of the probability measure µσ with respect to Lebesgue measure on R\{0} is bounded on (−δ, δ)c for any δ > 0. With G α,σ (z) uniquely determined, we conclude that so is the weak limit µσ of E[µˆ Aκ,σ ]. Further, applying Lemma 3.4 for f (x) = x and considering the union bound N over 1 ≤ s ≤ q, we find that, with = 1 − κ(2 − α) > 0, for some c(z) finite on C+ , any z ∈ C+ , δ > 0 and N ∈ N,
qc(z) − 1 1 d µˆ Aκ,σ (x) − E[ d µˆ Aκ,σ (x)]
≥ δ ≤ N . P
N N z−x z−x δ2 Consequently, setting φ(n) = [n γ ] for γ = 2/, by the Borel-Cantelli lemma, with probability one, as n → ∞, 1 G n (z) := d µˆ κ,σ (x) → G α,σ (z). z − x Aφ(n) Since G n (z) ≤ ((z))−1 for all n and z ∈ C+ , applying this for a countable collection z k with a cluster point in C+ we deduce by Vitali’s convergence theorem that with probability one, G n (z) → G α,σ (z) for all z ∈ C+ . Such convergence of the CauchyStieltjes transforms implies of course that µˆ Aκ,σ converges weakly to µσ and by (2.2) φ(n) we deduce after yet another application of the Borel-Cantelli lemma, that with probability one µˆ Aσφ(n) converges weakly to µσ . Finally, since φ(n − 1)/φ(n) → 1 we have from Lemma 2.4 that the same weak convergence to µσ holds for µˆ AσN . With h α (y) = h α (y), combining the identities Yr (−z) = Yr (z) of Proposition 3.7 with the formula (1.10) for the Cauchy-Stieltjes transform G α,σ of the probability measure µσ on R we find that G α,σ (−z) = −G α,σ (z) = −G α,σ (z) for all z ∈ C+ , hence necessarily µσ (·) = µσ (−·) is symmetric about zero. Further, as shown in Proposition 3.7, z α Y (z) is uniformly bounded and extends analytically through the subset (R, ∞), where Y (z) = V (z −α ) ∈ (Kα )q is the unique analytic solution of (1.11) on z ∈ C+ that
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tend to zero as |z| → ∞ (and as shown in Lemma, 3.9, z → Y (z) is injective when σ ≡ 0). If σ ≡ 0 then V (u) = 0 is analytic on C. Turning to σ ≡ 0, in view of Lemma 3.9 the function V is uniformly bounded on Eα,ε ∩ K for any compact K ⊆ C. Thus, combining Lemma 3.9 with Proposition 3.8 we find that V (u) has a continuous, algebraic extension to (0, ∞). As Yr (−z) = Yr (z), this yields the continuous, algebraic extension of Y (z) to R \ {0}, analytic on (R, ∞), from which we get by (1.10) and the analyticity of h α (·) the corresponding continuous/algebraic/analytic extension of G α,σ (z). Recall the Plemelj formula, that for x = 0, the limit as ↓ 0 of −π −1 (G α,σ (x + i)) is then precisely the continuous density ρ σ (x) of µσ with respect to Lebesgue measure on R\{0}, and ρ σ (x) is real-analytic on (R, ∞). 4. Proof of Theorem 1.7 We start with the following consequence of Proposition 2.2 and Theorem 1.3. Corollary 4.1. For any σ ∈ Fα , the probability measures E[µˆ AσN ] converge weakly towards some symmetric probability measure µσ . Proof. We approximate σ in L 2 ([0, 1]2 ) by a sequence of piecewise constant functions σ p . Applying Theorem 1.3 for σ = σ p we deduce that hypothesis (2.6) holds. Hence, by Proposition 2.2 E[µˆ AσN ] converges weakly towards the limit µσ of the corresponding measures µσ p . We have seen already that µσ p are symmetric measures, hence so is their limit µσ . Fixing σ ∈ Fα we proceed to characterize the limiting measure µσ . To this end, recall that kσ := |σ |α is finite and fix a sequence σ p ∈ C that converges to σ in L 2 , satisfying (1.12) and such that sup p∈N |σ p |α ≤ 2kσ . For each p ∈ N let p p p 0 = b0 < b1 < · · · < bq(σ p ) = 1 denote the finite partition of [0, 1] induced by σ p and per z ∈ C+ consider the piecewise constant function Y.σ p (z) : (0, 1] → Kα such that σ
Yx p (z) = Ys (z)
p
p
for x ∈ (bs−1 , bs ]
and
s = 1, . . . , q(σ p ),
where Ys (z) ∈ Kα is the unique collection of (analytic) functions of z ∈ C+ that satisfy p p (1.11) for the q × q matrix of entries σr s := σ p (br , bs ), as in Theorem 1.3. This way (1.14) holds for σ = σ p and each p ∈ N (being precisely (1.11)). We next show the existence of R = R(σ ) finite such that if |z| ≥ R then (Y.σ p (z), p ∈ N) is a Cauchy sequence for the L ∞ -norm. To this end, it is convenient to view (1.14) (at each z ∈ C+ ) as the fixed point equation in L ∞ ((0, 1]; Kα ), 1 σ σ −α Y. = Fz (σ, Y. ), Fz (σ, Y ) := Cα z |σ (·, v)|α gα (Yv )dv. (4.1) 0
Then, with gα Kα := sup{|gα (y)| : y ∈ Kα } finite by Lemma 3.6, bounding the L ∞ -norm of Fz (σ, Y ) for Y ∈ Kα we deduce from (4.1) that for any > 0, sup sup Y.σ p ∞ ≤ 2kσ −α |Cα | gα Kα =: rσ
|z|≥ p∈N
(4.2)
∞ ≤ r and measurable ∞ ≤ r , Y σ (·, ·), σ (·, ·), is finite. Note that for Y ' ( ) − Fz ( ) ∞ ≤ |z|−α gα r | − Y ∞ , Fz ( σ |α − | σ,Y σ,Y σ |α + | σ |α Y (4.3)
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S. Belinschi, A. Dembo, A. Guionnet
where gα r is the sum of the supremum and Lipschitz norms of y → Cα gα (y) on the σ ball {y ∈ C : |y| ≤ r }. Suppressing hereafter the dependence of Yx p (z) on z, since σ p (σ p , Y. ), p ∈ N, satisfy (4.1), from (4.2) and (4.3) we have that for any p, q ∈ N and |z| ≥ , ' ( Y.σq − Y.σ p ∞ ≤ |z|−α gα rσ |σq |α − |σ p |α + 2kσ Y.σq − Y.σ p ∞ . Taking R = R(σ ) ≥ finite such that R −α gα rσ kσ ≤ 1/3, this implies that for |z| ≥ R, Y.σq − Y.σ p ∞ ≤ 3|z|−α gα rσ |σq |α − |σ p |α . In view of (1.12), we conclude that (Y.σ p , p ∈ N) is for each |z| ≥ R a Cauchy sequence in L ∞ (0, 1]), which thus converges in this space to a bounded measurable function Y.σ from (0, 1] to the closed set Kα . Further, then Y.σ ∞ ≤ rσ (see (4.2)), so from (4.3) and (1.12) we deduce that ' ( Fz (σ, Y.σ ) − Fz (σ p , Y.σ p ) ∞ ≤ −α gα rσ |σ |α − |σ p |α + kσ Y.σ − Y.σ p ∞ → 0, as p → ∞. With (4.1) holding for the pairs (σ p , Y.σ p ), p ∈ N, it follows that the same applies for (σ, Y.σ ), thus establishing (1.14). Turning to show the uniqueness of the solution to (1.14), suppose Y j = Fz (σ, Y j ) for σ (·, ·) such that kσ = |σ |α is finite, some |z| ≥ R(σ ) and measurable Y j : (0, 1] → Kα , j = 1, 2. Then, as in the derivation of (4.2) we have that Y j ∞ ≤ rσ for j = 1, 2. So, applying (4.3) once more, Y1 −Y2 ∞ = Fz (σ, Y1 )− Fz (σ, Y2 ) ∞ ≤ |z|−α gα rσ kσ Y1 −Y2 ∞ ≤
1 Y1 −Y2 ∞ , 3
and necessarily Y1 = Y2 almost everywhere on (0, 1]. To recap, the sequence of holomorphic mappings Y σ p from C+ to the closed subset F := L ∞ ((0, 1]; Kα ) of the complex Banach space L ∞ ((0, 1]; C) is such that Y σ p (z) → Y σ (z) in F at each point z of the non-empty open subset B(0, R)c ∩ C+ . Further, in view of (4.2) we have that (Y σ p , p ∈ N) is locally uniformly bounded on C+ , hence by Vitali’s convergence theorem for vector-valued holomorphic mappings, it converges at every z ∈ C+ to an analytic mapping Y σ : C+ → F (see [4, Theorem 14.16]). We also characterized Y σ (z) for each |z| ≥ R as the unique solution in F of (1.14), so by the identity theorem for vector-valued holomorphic mappings (see [4, Exercise 9C]), we have thus uniquely determined Y σ : C+ → F. Next, note that the identity (1.13) holds for σ = σ p ∈ C , p ∈ N, in which case it is merely the formula (1.10). Recall Proposition 2.2 that due to the L 2 -convergence of σ p to σ , for each z ∈ C+ the left-hand side of these identities converge as p → ∞ to G α,σ (z) := (z − x)−1 dµσ (x). If in addition |z| ≥ R(σ ), then Y.σ p − Y.σ ∞ → 0 and by dominated convergence the right hand sides of the same identities converge to the corresponding expression for Y.σ (z). Thus, (1.13) holds also for σ ∈ Fα and |z| ≥ R(σ ). With µσ a probability measure on R, the left side of (1.13) is obviously an analytic function of z ∈ C+ . Further, the entire function h α (·) and its first two derivatives are uniformly bounded on the set Kα (see Lemma 3.6), in which the analytic mapping 1 Y σ : C+ → F takes values. Hence, it is not hard to see that z → 0 h α (Yvσ (z))dv is also analytic on C+ . We thus deduce by the identity theorem that (1.13) holds for all z ∈ C+ . 1 Consequently, with 0 h α (Yvσ (z))dv uniformly bounded on C+ , the Cauchy-Stieltjes
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transform of µσ is uniformly bounded on C+ ∩ B(0, δ)c . This in turn implies (by the Stieltjes-Perron inversion formula), that the density ρ σ of µσ with respect to Lebesgue measure on R\{0} is bounded outside any neighborhood of zero. We have seen already that Y σ ∞ ≤ c(σ )|z|−α for some c(σ ) finite and all |z| ≥ R. Hence, for z ∈ C+ such that |z| ≥ R, we have from (1.13) and (1.14) that 1 1 h α (0) + h α (0) G α,σ (z) = Yxσ (z)dx + O(|z|−2α ) z 0 1 1 1 −α α −2α h α (0) + z Cα h α (0)gα (0) |σ (x, v)| dxdv + O(|z| ) . = z 0 0 # $ Recall the Plemelj formula, that ρ σ (t) is the limit of −π −1 G α,σ (t + i) as ↓ 0. Thus, as h α (0) ∈ R, it follows that t α+1 ρ σ (t) → L α |σ (x, v)|α dxdv as t → ∞ and it is not hard to check that L α = −π −1 h α (0)gα (0)(Cα ) equals α2 (by Euler’s reflection formula for the Gamma function). Turning to verify the last statement of the theorem, note that the equivalence between σ (z) : (0, 1] → K we have σ ∈ Fα and σ ∈ C implies that the piecewise constant Y. α constructed before out of (Ys (z), 1 ≤ s ≤ q) satisfies (1.14) for any x ∈ (0, 1] and all σ (z) = Y σ (z) z ∈ C+ . It then follows by the uniqueness of such solution of (1.14) that Yx x + for all z ∈ C such that |z| ≥ R(σ ) and almost every x ∈ (0, 1]. In view of (1.13), the Cauchy-Stieltjes transform of µσ coincides for such z with the Cauchy-Stieltjes trans σ form G α, σ (z) of µ . As such information uniquely determines the probability measure σ. in question, it follows that µσ = µ 5. Proof of Proposition 1.1 and Theorem 1.10 γ
5.1. Convergence to µα and its characterization. Consider the (N + M)-dimensional square matrix X N ,M 0 a −1 N +M A N ,M = , t a −1 0 N +M X N ,M noting that B N ,M = A2N ,M is then of the form
B N ,M =
t a −2 0 N +M X N ,M X N ,M −2 t 0 a N +M X N ,M X N ,M
=:
0 W N ,M N ,M 0 W
N ,M augmented and that the eigenvalues of W N ,M consist of the M eigenvalues of W by N − M zero eigenvalues. Therefore, µˆ B N ,M =
2N M−N µˆ W N ,M + δ0 . N+M N+M
(5.1)
We next show that with probability one µˆ B N ,M converges weakly. Since B N ,M = A2N ,M , for any f (·) bounded and continuous, f (x2 )d µˆ A N ,M , (5.2) f (x)d µˆ B N ,M =
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so that it is enough to prove the convergence of µˆ A N ,M . To this end, consider AσN +M for σ (x, y) =
1 1 , 1] × (0, 1+γ ] 1 if x, y ∈ ( 1+γ 0 otherwise.
)
1 1 (0, 1+γ ] × ( 1+γ , 1]
.
(5.3)
Note that with M/N → γ and
N+M − N
, rank(A N ,M − AσN +M ) ≤ 2
1+γ it follows by Lidskii’s Theorem that d1 (µˆ A N ,M , µˆ AσN +M ) → 0 as N → ∞. Therefore, applying Theorem 1.3 we deduce that with probability one µˆ A N ,M converges weakly to the non-random probability measure µσ . By (5.2) and (5.1) this implies that µˆ B N ,M and µˆ W N ,M also converge weakly to non-random probability measures, µ B :=
2 γ −1 µγ + δ0 1+γ α γ +1
(5.4)
γ
and µα , respectively. γ We proceed to show that for z ∈ C+ the Cauchy-Stieltjes transform of µα is G γα (z) =
√ √ 1 1−γ γ h α (Y1 ( z )) = + h α (Y2 ( z )). z z z
(5.5)
Indeed, note that (Y1 (z), Y2 (z)) of (1.15) are precisely the solution of (1.11) considered in Proposition 3.7 for z ∈ C+ and our special choice of σ (·, ·). Theorem 1.3 thus asserts that the Cauchy-Stieltjes transform G α,σ of µσ is then such that, for any z ∈ C+ , zG α,σ (z) =
1 γ h α (Y1 (z)) + h α (Y2 (z)). 1+γ 1+γ
(5.6)
Moreover, by (5.2) and the symmetry of the law µσ (see Corollary 4.1), we have that
1 dµ B (x) = z−x
1 1 dµσ (x) = √ 2 z−x z
√
1 dµσ (x). z−x
γ
From this and formula (5.4) relating µ B to µα , we deduce that G γα (z) =
√ 1+γ 1−γ . √ G α,σ ( z ) + 2 z 2z
(5.7)
Multiplying the left identity of (1.15) by Y2 (z) and the right identity of (1.15) by Y1 (z) we find that Y1 (z)gα (Y1 (z)) = γ Y2 (z)gα (Y2 (z)) and hence h α (Y1 (z)) = 1 − γ + γ h α (Y2 (z)). Upon combining (5.6), (5.7) and (5.8) we get the formula (5.5).
(5.8)
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5.2. Analysis of the limiting measures. In case γ = 1, the function σ (x, y) of (5.3) is σ has the equivalent to the constant σ = 2−1/α , which as in Remark 1.9 implies that µ√ √ σ 1/α 1/α 1 density ρ (t) = 2 ρα (2 t). Further, we see from (5.7) that√ G α (z) = G α,σ ( z )/ z, √ so the probability measure µ1α on (0, ∞) has the density ρ σ ( t)/ t, as stated. Considering hereafter γ ∈ (0, 1), observe that by Theorem 1.3, Y1 (z) and Y2 (z) extend continuously to functions on (0, ∞) that are analytic outside of some bounded set. By the analyticity of h α (·) and (5.5) we have the corresponding continuous extension γ γ γ of G α (z), whereby the Plemelj formula provides the density ρα (t) = −π −1 (G α (t)) √ γ γ 1+γ σ of µα with respect to Lebesgue measure, as in (1.16). In particular, ρα (t) = 2√t ρ ( t ) γ
by (5.7), with σ (·, ·) of (5.3), so we read the tail behavior of ρα out of that of ρ σ (per Theorem 1.7). γ γ Turning next to the behavior near zero of the probability measure µα , recall that G α (z) γ γ is analytic outside the support [0, ∞) of µα and the non-tangential limit of zG α (z) at the boundary point z = 0 (i.e., its limit as |z| → 0 while θ0 ≤ arg(z) ≤ 2π − θ0 for some fixed θ0 > 0), exists and equals to the mass at zero of√this measure. Further, the identity (5.5) extends by continuity to z = −x2 , x > 0 and z = ix ∈ C+ , hence µγα ({0}) = lim zG γα (z) = lim h α (Y1 (ix )) = 1 − γ + γ lim h α (Y2 (ix )). z→0
x↓0
x↓0
(5.9)
Since Ys (−z) = Ys (z) for s = 1, 2 and all z ∈ C+ (see Proposition 3.7), we have in particular that Y1 (ix) and Y2 (ix) are real-valued for all x > 0. As gα (y) > 0 for y ∈ R, it further follows from (1.15) that Ys (iR+ ) ⊆ R+ for s = 1, 2. With h α : R+ → R+ monotone decreasing and h α (y) → 0 as (y) → ∞, it thus follows from (5.8) that h α (Y1 (ix)) ≥ 1 − γ for all x > 0 and consequently, that (Y1 (ix), x > 0) is uniformly bounded. This of course implies that (ix)α Y1 (ix) → 0 as x ↓ 0 which in view of (1.15) requires that gα (Y2 (ix)) → 0 as well. As gα : R+ → R+ is bounded away from zero on compacts, we deduce that Y2 (ix) → ∞ as x ↓ 0, hence h α (Y2 (ix)) → 0 and µγα ({0}) = lim h α (Y1 (ix)) = 1 − γ , x↓0
+ as claimed. Moreover, from the preceding Y1 (ix) → h −1 α (1 − γ ) := b ∈ R as x ↓ 0. + Since Y1 (z) is a Kα -valued continuous function of z ∈ C , its cluster set Cl(0) at the boundary point z = 0 of C+ is a closed, connected subset of Kα (see [6, Theorem 1.1]). Further, Cl(0) contains b ∈ R+ , so its boundary ∂Cl(0) must intersect [0, ∞). We have seen that Y1 (z) extends continuously on (0, ∞) which due to the relation Y1 (−z) = Y1 (z) implies that it also extends continuously on (−∞, 0) with Y1 (−t) = Y1 (t) for all t > 0. In particular, since the cluster set of Y1 (t) for non-zero, real-valued √ t → 0 contains ∂Cl(0) (see [6, Theorem 5.2.1]), necessarily the cluster set of Y1 ( t ) at the boundary point t = 0 of R+ also intersects [0, ∞). Using the bound sin(ζ )/ζ ≥ 1−ζ 2 /6, we deduce from (1.5) that if (h α (x+iy)) = 0 for y = 0, then y 2 ≥ 6h α (x)/ h that this function α (x), and direct calculation shows √ of x is positive and monotone√non-decreasing. Thus, with√Y1 ( t ) ∈ Kα there exists δ > 0√such that if (h α (Y1 ( t ))) = 0, then either Y1 ( t ) ≥ 0 is real-valued, or |(Y1 ( t ))| ≥ δ. By (1.16), the latter property applies whenever t > 0 is such that γ γ ρα (t) = 0. Moreover, by the continuity α vanishes √ of Y1 (·) on (0, ∞), if the density ρ√ on an open interval I, then either Y1 ( t ) ≥ 0 for all t ∈ I or inf t∈ √I |(Y1 ( t ))| ≥ δ. For I = (0, ) we have already√seen that the cluster set of Y1 ( t ) as t ↓ 0 intersects [0, ∞), so necessarily Y1 ( t ) ∈ [0, ∞) for all t ∈ I. Since (1.15) extends to
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√ z ∈ R+ and gα (R) ⊆ R+ this in turn implies that Y2 ( t ) = i α r (t) for some continuous function r : I → R+ such that r (t) → ∞ as t ↓ 0. The entire function f α,θ (z) := 2i1 [h α (eiθ z) − h α (e−iθ z)] is then by (5.5) such that √ f α,θ (r (t)) = h α (eiθ r (t)) = h α (Y2 ( t )) = 0 (0) = sin(θ )h (0) = 0 contradicts the for θ = π α/2 and all t ∈ I, which with f α,θ α γ identity theorem. We thus conclude that ρα does not vanish on any non-empty interval (0, ).
5.3. Properties of µα . Proof of Proposition 1.1. Taking σ ≡ 1 we deduce from Theorem 1.3 that Y (z) of (1.7) is in Kα hence uniformly bounded on C+ \{z : |z| < δ}. Similarly to the argument of Sect. 5.2, if y ∈ Cl(t) at t > 0 then y ∈ Kα and F(t, y) := t α y − Cα gα (y) = 0, so from the analyticity of y → F(t, y) and uniform boundedness of gα (·) on Kα we deduce by the identity theorem that Y (z) extends continuously to a function Y (t) on (0, ∞). Moreover, t → Y (t) is real-analytic on (0, ∞) outside the set of those t > 0, where both ∂y F(t, y) = 0 and F(t, y) = 0 at y = Y (t). The latter set is clearly contained in the set Dα+ of t > 0 such that t α = Cα gα (y) > 0 for some y ∈ Kα at which ygα (y) − gα (y) = 0. Note that the set Dα+ is discrete since ygα (y) − gα (y) is an entire function of y. Further, Dα+ is a bounded set (by the uniform boundedness of gα (·) on Kα , see Lemma 3.6). Consequently, Dα+ is a finite set. We already saw that Y (−z) = Y (z) for all z ∈ C+ , so Y (−z) extends continuously to Y (−t) = Y (t) for any t > 0 at which Y (·) extends continuously. We thus deduce that the exceptional set where t → Y (t) may be non-analytic is contained in the finite set {0, ±t : t ∈ Dα+ }, as claimed. With h α an entire function, it then follows that G α (·) extends continuously to R\{0} with the formula (1.8) for the symmetric density ρα (t) on R\{0} that is real-analytic outside Dα (to verify the right-most expression in (1.8) note that h α (Y (z)) = 1 − 2Cα α z α Y (z)2 by (1.5) and (1.7)). If the symmetric density ρα vanishes on an open interval, then it also vanishes on some open interval I ⊆ R+ , where the continuous function t → Y (t) is the limit of Y (z) as arg(z) ↓ 0, hence arg(Y (t)) ∈ [0, απ 2 ] (see (3.16)). Further, the right-most expression in (1.8) tells us that sin(2arg(Y (t)) − απ 2 ) = 0 for all t ∈ I, so necessarily απ iθ Y (t) = e r (t) for θ = 4 and the continuous r : I → [0, ∞). Since (1.7) extends to t ∈ I and gα (0) = 0, we see that Y (t) = 0 is injective on I, so r (I) contains an accumulation point. Finally, as argued at the end of Sect. 5.2, from (1.8) we also have that f α,θ (r (t)) = (h α (Y (t))) = 0 for the entire function f α,θ (·) and all t ∈ I, yielding a contradiction. Consequently, the density ρα does not vanish on any open interval, as claimed. It remains to show that µα has a uniformly bounded density. We get this by proving the stronger statement that G α (z) is uniformly bounded on the connected set C+∗ := C+ ∪R+ . To this end, let Cl∗ (0) denote the cluster set of the continuous function Y (z) at the boundary point z = 0 of C+∗ . If y ∈ C is in Cl∗ (0) then there exists z n ∈ C+∗ such that z n → 0 and Y (z n ) → y, hence gα (y) = 0 by (1.7). Whereas Cl∗ (0) is a closed connected subset of C ∪ {∞} (by [6, Theorem 1.1]), the set of zeros of the entire function gα (·) is discrete, so necessarily Cl∗ (0) is a single point. Taking z = ix, x > 0 we have that Y (ix) ∈ R+ , hence Y (ix) → ∞ by (1.7) and the boundedness of gα (R+ ), from which we deduce
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that Cl∗ (0) = {∞}. Considering (3.18) for β = 2, we note that |h α (y)| ≤ c0 h α (ξ |y|) for some ξ = ξ(α) > 0, c0 = c0 (α) finite and all y ∈ Kα . In particular, for z ∈ C+∗ such that |z| → 0 we already know that Y (z) ∈ Kα and |Y (z)| → ∞, hence by the preceding bound and the decay to zero of h α (r ) as r ∈ R+ goes to infinity, we have that h α (Y (z)) → 0. That is, Y (z)gα (Y (z)) → 2/α (see (1.5)). Next, observing that h α (r ) ≤ c1r −2/α for some positive, finite c1 and all r ∈ R+ , we deduce from (1.6) and (1.7) that for some finite constants ci = ci (α) and all z ∈ C+∗ , |G α (z)| = |z|−1 |h α (Y (z))| ≤ c0 |z|−1 h α (ξ |Y (z)|) ≤ c2 (|z α Y (z)2 |)−1/α = c3 |Y (z)gα (Y (z))|−1/α .
(5.10)
For any δ > 0 we have the uniform boundedness of G α (z) on C+∗ ∩ B(0, δ)c (from the uniform boundedness of h α on Kα ). Further, for z ∈ C+∗ converging to zero the right side of (5.10) remains bounded (by c3 (2/α)−1/α ), hence G α (z) is uniformly bounded on C+∗ , as stated. Remark 5.1. We saw that√ Y (ix) ∈ R+ and xα Y (ix)2 = |Cα |Y (ix)gα (Y (ix)) → 2|Cα |/α as x ↓ 0. With ζ = 2|C α |/α, it then follows by dominated convergence that ∞ π −1 x−1 h α (Y (ix)) → π −1 0 exp(−ζ u α/2 )du finite and positive. This is of course the value of ρα (0), provided ρα is continuous at t = 0. Lemma 5.2. The measures µα converge weakly to µ2 when α ↑ 2. Proof. Applying the method of moments, as developed by Zakharevich [9], it is shown in [1, Theorem 1.8] that for any B < ∞ fixed, E[µˆ A B ] converges to some non-random N
µαB as N → ∞ (for instance, when xi j are stable variables of index α). Examining the dependence of C(B) of [1, Eq. (13)] on α, we see that (2.1) applies for some δ(, B) > 0, all B > B(, α0 ) and any α ∈ (α0 , 2). For such B and α we thus have, in view of the almost sure convergence of µˆ A N to µα , that P(d1 (µα , µˆ A B ) ≥ 3) → 0 as N → ∞, N from which we deduce by the boundedness and convexity of d1 that d1 (µα , µαB ) = lim d1 (µα , E[µˆ A B ]) ≤ lim sup E d1 (µα , µˆ A B ) ≤ 3. N →∞
N
N →∞
N
Fixing B < ∞ it further follows from [1, Lemmas 9.1 and 9.2] that µαB converges weakly to the semi-circle µ2 when α → 2. Hence, fixing α0 > 0, > 0 and B > B(, α0 ), by the triangle inequality, d1 (µα , µ2 ) ≤ d1 (µα , µαB ) + d1 (µαB , µ2 ) ≤ 3 + d1 (µαB , µ2 ) → 3 as α ↑ 2. Taking ↓ 0 we thus conclude that µα → µ2 when α ↑ 2. 6. Diagonal Perturbation: Proof of Theorem 1.12 6.1. The extension of Theorem 1.3. We shall prove the convergence of the expected spectral measures E[µˆ A N + D N ] and characterize their limit in case σ ∈ C is given as in Sect. 3 by (3.1) for some q ∈ N, 0 = b0 < b1 < · · · < bq = 1 and σr s = σsr with the corresponding random matrix A N = AσN and the N × N piecewise constant matrix σ N . To this end, recall that D N is a diagonal N × N matrix, whose entries {D N (k, k), 1 ≤ k ≤ N } are real valued, independent of the random variables (xi j , 1 ≤
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i ≤ j < ∞) and identically distributed, of law µ D having a finite second moment. In view of the assumed finite second moment of µ D , the proof of (2.2) and Lemma 2.1 also show that the sequences (E[µˆ A N + D N ]; N ∈ N), (E[µˆ A B + D N ]; N ∈ N) and N (E[µˆ AκN + D N ]); N ∈ N) are tight for the topology of weak convergence on P(R), and that (E[µˆ AκN + D N ]); N ∈ N) has the same set of limit points as (E[µˆ A N + D N ]; N ∈ N). Setting now G N (z) = (zI N − D N − A N )−1 , we define for z ∈ C+ the probability measures L zN and L zN ,r on C as in (3.2) and (3.3), with G κN (z) and L z,κ N ,r denoting again the corresponding objects when A N is replaced by AκN . 1 For 0 < κ < 2(2−α) any 1 ≤ r ≤ q and bounded Lipschitz function f , we then have similarly to Lemma 3.1 that as N → ∞,
⎡ ⎛⎛ ⎞−1 ⎞⎤
N
⎥ ⎢ ⎜ z,κ κ ([N br ], k)2 G κ (z)kk ⎠ ⎟ A
E L N ,r ( f ) − E ⎣ f ⎝⎝z − D N (0, 0) − ⎠⎦ → 0, N N
k=1
(6.1) where AκN denotes an independent copy of AκN which is also independent of D N while D N (0, 0) of law µ D is independent of all other variables. Indeed, focusing w.l.o.g. on ¯ κN +1 (z) = (zI N +1 − D ¯ κ )−1 (with D ¯ N +1 − A ¯ N +1 denoting the r = 1 and taking G N +1 diagonal matrix of entries D N (k, k), k = 0, . . . , N ), we get (3.4) by the invariance ¯ κ to symmetric permutations of its first [N b1 ] + 1 rows and ¯ N +1 + A of the law of D N +1 columns. Schur’s complement formula then leads to the identity (3.5) with D N (0, 0) κ (0, 0) on its right side. All eigenvalues (and diagonal terms) of G κ (z) are added to A N N in the compact set K(z), regardless of the value of D N , and the centered entries of AκN κ are independent of both G N (z) and D N (0, 0). Thus, as in the proof of Lemma 3.1 we
κ (0, 0) and k =l A κ (l, 0)G κ (z)kl in (3.5) and get (3.6) κ (0, k) A can neglect both A N N N N except for changing here z to z − D N (0, 0) in its right side. Equipped with the latter 1 version of (3.6), fixing 0 < κ < 2(2−α) , we arrive at (6.1) upon adapting [1, Lemma κ ¯ N +1 and G κ (while taking there the corresponding 4.1] and its proof to our matrices G N κ ˆ N = (zI N +1 − D ˆ κ )−1 ). ¯ N +1 − A matrices G N The concentration result of Lemma 3.4 holds in the presence of the diagonal matrix D N of i.i.d. entries. Indeed, its proof is easily adapted to the current setting by considering κ for f continuously differentiable L z,κ N ,s ( f ) := FN (D N (l, l), A N (k, l), 1 ≤ k ≤ l ≤ N ), and noting that for 1 ≤ l ≤ N , 1 ∂ D(l,l) FN = [G κN (z) Ds ( f )G κN (z)]ll . N The spectral radius of G κN (z) Ds ( f )G κN (z) is again bounded by f ∞ /|(z)|2 , so supl ∂ D(l,l) FN ∞ ≤ f BL (N |(z)|2 )−1 . There are only N such variables {D N (l, l)} to consider, each having the same finite second moment, so using the same martingale bound as in (3.8), their total effect on E[(FN − E[FN ])2 ] is taken care of by enlarging the finite constant c0 . Equipped with this concentration result and replacing Lemma 3.1 with (6.1), we follow the proof of Proposition 3.3 to deduce that in our current setting, for r ∈ {1, · · · , q} and every bounded continuous function f on K(z), q q 2 " z z 2 −1 α f dµr = f (z − λ − σr s s xs ) d P µs (xs )dµ D (λ). (6.2) s=1
s=1
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Following the proof of Proposition 3.5 we find that this in turn implies that any α/2 )] has at least one limit point subsequence of the functions X N ,r (z) = E[L z,κ N ,r (x (X r (z), 1 ≤ r ≤ q) composed of analytic functions on C+ that are bounded by ((z))−α/2 and satisfy the following generalization of (3.10): ∞ α α X r (z) = C(α) t −1 (it) 2 eit (z−λ) exp{−(it) 2 X r (z)} dt dµ D (λ), 0
α of (3.11). for the analytic functions X r : C+ → K We proceed to extend Proposition 3.7 to the setting of AσN + D N . Indeed, fixing z ∈ C+ , upon applying per λ ∈ R the identity (3.17) for β = α, y = (λ − z)−α/2 X r (z) and with z − λ ∈ C+ replacing z, we see that the preceding generalization of (3.10) is equivalent to α α X r (z) = C(α) (λ − z)− 2 gα,α ((λ − z)− 2 X r (z)) dµ D (λ). By (3.11) we thus deduce that ( X r (z), 1 ≤ r ≤ q) satisfy (1.19). Namely, it is a solu α , where F z (·) = tion of x = F z ( x) composed of analytic functions from C+ to K (Fz,r (·), 1 ≤ r ≤ q) and x) := C α Fz,r (
q
ar s
α α xs dµ D (λ), (λ − z)− 2 gα (λ − z)− 2
s=1
α , then (λ − z)− α2 for ar s = |σr s |α s . Note that if xs ∈ K xs is in Kα so such soluα q ar s | finite and all tions must have | xr | ≤ c((z))− 2 for c := |Cα | gα Kα maxr s=1 | z ∈ C+ . Consequently, if (z) ≥ 1 then maxr | xr | ≤ c. Thus, for such z, any 1 ≤ r ≤ q α )q , and any two fixed points x and y of F z (·) in (K x) − Fz,r ( y )| ≤ max{| ar s |} gα c ((z))−α x − y 1 |Fz,r ( r,s
q (where gα c and gα Kα are as in the proof of Theorem 1.7 and x 1 := s=1 | xs |). x) − F z ( y ) 1 ≤ 21 x− y 1 resulting Thus, for some k0 finite, if (z) ≥ k0 then F z ( α )q . This in turn implies the stated with uniqueness of the fixed point of F z (·) in (K α . uniqueness of such fixed point composed of analytic functions z → xs from C+ to K To complete the proof of Theorem 1.12 in case σ ∈ C , we adapt our proof of Theorem 1.3, where instead of (3.25), combining (6.2) for f (x) = x with (3.13), here the limit points µsz of (E[L z,κ N ,s ], 1 ≤ s ≤ q) are such that for each r ∈ {1, . . . , q},
xdµrz (x)
= −i
D
dµ (λ)
∞
α
eit (z−λ) exp{−(it) 2 X r (z)}dt.
(6.3)
0
X r (z) we deduce that the sequence In particular, since xdµrz is uniquely determined by (x)] converges as N → ∞ to the right side of (6.3). So, with E[L z,κ N ,r E
q 1 d µˆ AκN + D N (x) = N ,s E[L z,κ N ,s (x)] z−x s=1
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for any z ∈ C+ , it follows that
1 dµσ, D (x) = −i s z−x q
dµ D (λ)
∞
α
eit (z−λ) exp{−(it) 2 X s (z)}dt,
0
s=1
D of for any limit point µσ, D of E[µˆ AκN + D N ]. With the Cauchy-Stieltjes transform G α,σ µσ, D ∈ P(R) uniquely determined, we deduce that E[µˆ AκN + D N ] converges to µσ, D , hence so does E[µˆ A N + D N ]. Finally, for z ∈ C+ we arrive at the formula
D G α,σ (z) =
α 1 s h α ((λ − z)− 2 X s (z)) dµ D (λ), z−λ
q
(6.4)
s=1
α
by applying (3.17) with β = 2, y = (λ − z)− 2 X s (z) and z − λ instead of z. 6.2. The extension of Theorem 1.7. Setting σ ∈ Fα we adapt the proof of Theorem 1.7 to the current setting. Indeed, using the same approximating sequence σ p ∈ C of σ ∈ Fα as in the proof of Theorem 1.7, we have shown already that (1.18) holds for α , p ∈ N, where each of the piecewise constant functions X .σ p (z) : (0, 1] → K σ X s (z) X x p (z) =
for
p
p
x ∈ (bs−1 , bs ]
and s = 1, . . . , q(σ p ),
α are the unique collections of (analytic) functions of z ∈ C+ we have and X s (z) ∈ K constructed in Sect. 6.1. Similarly to the proof of Theorem 1.7, we get the existence of a bounded measurable α of (1.18) whenever (z) ≥ R = R(σ ) by showing that solution X .σ (z) : (0, 1] → K for such z the fixed points ( X .σ p (z), p ∈ N) of the mappings 1 α α α Fz (σ, X ) := C α |σ (·, v)| (λ − z)− 2 gα (λ − z)− 2 X v dµ D (λ) dv, 0
at σ = σ p form a Cauchy sequence in L ∞ ((0, 1]). To this end, recall that gα Kα is X ) for finite (by Lemma 3.6), so fixing ∈ (0, 1) and bounding the L ∞ -norm of Fz (σ, α we deduce that X ∈K X .σ p ∞ ≤ rσ of (4.2) for all p ∈ N, whenever (z) ≥ . It is easy to verify that for such z our mapping Fz (·, ·) satisfies the inequality (4.3) except for replacing there |z|−α by ((z))−α/2 . Consequently, with X .σ p fixed points of this σ p mapping, our uniform bound on X . ∞ implies that ' ( X .σ p ∞ ≤ ((z))−α/2 gα rσ |σq |α − |σ p |α + 2kσ X .σq − X .σ p ∞ , X .σq − for any p, q ∈ N and (z) ≥ . Thus, setting R ≥ such that R −α/2 gα rσ kσ ≤ 1/3, we conclude in view of (1.12) that ( X .σ p , p ∈ N) is a Cauchy sequence in L ∞ (0, 1]; C) + whenever z is in C R := {z : (z) > R}. As in the proof of Theorem 1.7, the L ∞ -norm of its limit X .σ is at most rσ so by (1.12) and the modified inequality (4.3) X .σ (z) must be a fixed point of Fz (σ, ·). Further, equipped with the latter inequality, the uniqueness (almost everywhere) of such a solution to (1.18) is obtained by a re-run of the relevant argument from the proof of Theorem 1.7. We have seen that the holomorphic mappings α ) of L ∞ ((0, 1]; C) are locally uni X σ p from C+ to the closed subset F := L ∞ ((0, 1]; K formly bounded. Hence, their L ∞ -convergence to X σ extends by Vitali’s convergence
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theorem from the non-empty open subset C+R to all of C+ , with X σ : C+ → F an analytic mapping which is uniquely determined by the uniqueness of the solution in F of (1.18) for each z ∈ C+R (and the identity theorem). Next, with the same proof as in Proposition 2.2 we have from the L 2 -convergence D (z) → G D (z) as p → ∞, for each z ∈ C+ . If z ∈ C+ then also of σ p to σ that G α,σ α,σ R p X .σ p − X .σ ∞ → 0. As the identity (1.17) holds for σ = σ p ∈ C , p ∈ N (being then merely the formula (6.4)), taking p → ∞ we deduce by dominated convergence that (1.17) holds for σ ∈ Fα and z ∈ C+R . For all x ∈ (0, 1], λ ∈ R and z ∈ C+ the argument (λ − z)−α/2 X xσ (z) of the entire function h α is in the set Kα where h α and its derivatives are uniformly bounded. Further, for such λ the mapping z → (λ − z)−α/2 X σ (z) from C+ to L ∞ ((0, 1]; C) is analytic, out of which one can verify that the right side of (1.17) D also analytic on C+ the validity of (1.17) extends from C+ is analytic on C+ . With G α,σ R + to C (by the identity theorem). Acknowledgement. We thank Ofer Zeitouni for telling us about the interpolation approach to convergence almost surely (as done here in Lemma 2.4), and Gerard Ben Arous for proposing that we investigate the gap γ at zero for the support of the measure µα and suggesting the argument for the continuity of α → µα at α = 2. We are also grateful to Alexey Glutsyuk and Jun Li for their help in forming Proposition 3.8 and to the anonymous referee for pointing out the problem in using an L 2 -norm instead of the L 2 semi-norm.
References 1. Ben Arous, G., Guionnet, A.: The spectrum of heavy tailed random matrices. Commun. Math. Phys. 278(3), 715–751 (2008) 2. Bouchaud, J., Cizeau, P.: Theory of Lévy matrices. Phys. Rev. E 50, 1810–1822 (1994) 3. Bryc, W., Dembo, A., Jiang, T.: Spectral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab. 34(1), 1–38 (2006) 4. Chae, S.B.: Holomorphy and Calculus in Normed Spaces. Vol. 92 of Monographs and Textbooks in Pure and Applied Mathematics. New York: Marcel Dekker Inc., 1985 (With an appendix by Angus E. Taylor) 5. Chirka, E.M.: Complex Analytic Sets, Vol. 46 of Mathematics and its Applications (Soviet Series), Dordrecht: Kluwer Academic Publishers Group, 1989, translated from the Russian by R. A. M. Hoksbergen 6. Collingwood, E.F., Lohwater, A.J.: The Theory of Cluster Sets. Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge: Cambridge University Press, 1966 7. Khorunzhy, A., Khoruzhenko, B., Pastur, L., Shcherbina, M.: The large-n limit in statistical mechanics and the spectral theory of disordered systems. In: Phase transition and critical phenomena, Vol. 15, London-New York: Academic Press, 1992, p. 73 8. Khorunzhy, A.M., Khoruzhenko, B.A., Pastur, L.A.: Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37(10), 5033–5060 (1996) 9. Zakharevich, I.: A generalization of Wigner’s law. Commun. Math. Phys. 268(2), 403–414 (2006) Communicated by B. Simon
Commun. Math. Phys. 289, 1057–1086 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0824-2
Communications in
Mathematical Physics
Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem Christian B. Mendl1 , Michael M. Wolf1,2 1 Max-Planck-Institute for Quantum Optics, Garching, Germany.
E-mail: [email protected]; [email protected]
2 Niels Bohr Institute, Copenhagen, Denmark
Received: 13 July 2008 / Accepted: 5 March 2009 Published online: 26 May 2009 – © Springer-Verlag 2009
Abstract: The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can become elements of this set by either taking two copies of them or supplementing with a completely depolarizing channel. These scenarios imply that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable. Contents I. II.
Introduction . . . . . . . . . . . . . . . . . Unital Quantum Channels . . . . . . . . . A Preliminaries . . . . . . . . . . . . . . B Representations . . . . . . . . . . . . C Extreme points . . . . . . . . . . . . . III. Mixtures of Unitary Channels . . . . . . . A Separation witnesses . . . . . . . . . . B A negativity measure . . . . . . . . . . IV. Covariant Channels . . . . . . . . . . . . . A O(d) covariance . . . . . . . . . . . . B A complete picture . . . . . . . . . . . V. Restoring Birkhoff’s Theorem . . . . . . . A Two copies of a channel . . . . . . . . B Help from a noisy friend . . . . . . . . C Environment-assisted error correction . VI. Discussion . . . . . . . . . . . . . . . . . . VII. Appendix A–Matrix Optimization Problems
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A B C
Minimizing tr[A A] subject to fixed singular values Maximizing |tr[U to fixed tr[UU ] . . . . ]| subject T2 subject to fixed tr[UU ] . Minimizing tr Us Us T2 . . . . . . . . . . . . . . . D Minimizing tr U U VIII. Appendix B–A Special Extremal Channel . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction Quantum channels are the most general input-output relations which the framework of quantum mechanics allows for arbitrary inputs. Physically, they describe any transmission in space, e.g., through optical fibres, and/or evolution in time, as in quantum memories, from a general open-systems point of view. Mathematically, they are characterized by linear, completely positive maps acting (in the Schrödinger picture) on density operators in a trace-preserving manner. The present work investigates the particular class of quantum channels which leaves the maximally mixed (chaotic or infinite-temperature) state invariant. These channels are called unital or doubly-stochastic (referring to unital and trace-preserving) and they appear naturally in contexts with an irreducible symmetry. Apart from their practical relevance the interest in these channels has various origins: (i) they exhibit many special properties, e.g., regarding contractivity [1] or fixed points [2] — often allowing for a more geometric intuition, (ii) for small dimensions their additional constraint is strong enough to considerably simplify problems [3], and (iii) for sufficiently large dimensions problems on general channels can often be reduced to their unital counterparts [4–6]. The line of interest taken up by this article concerns the convex structure of the set of unital channels and, in particular, its relation to the subset of mixtures of unitary channels. This question was addressed and touched upon in [7–9] where a crucial difference between the classical and the quantum case was realized: whereas, by Garrett Birkhoff’s theorem [10], every doubly stochastic matrix (describing a classical channel) is a convex combination of reversible ones (i.e., permutations), not every doubly-stochastic quantum channel has to be a mixture of unitaries. The latter phenomenon became more significant when it was realized in Ref.[11] that a quantum channel allows for perfect environment-assisted error correction if and only if it is a mixture of unitaries. Another remarkable step was made in Ref.[12] where evidence has been provided that asymptotically many copies of a unital channel might always be well approximated by a mixture of unitaries—a conjectured restoration of Birkhoff’s theorem in the asymptotic limit. We will (in Sec.V) provide a proof of this conjecture for special instances. This will result as a simple corollary from a thorough investigation of the convex structure of the sets of unital channels and of mixtures of unitary conjugations—in particular under symmetries. Along the way we obtain various other results on the mentioned convex structures which may be interesting in their own right. An outline of the paper and a summary of its results follows: • In Sec.II we provide two characterizations of unital channels: (i) as channels which are convex combinations of unitaries acting on Hilbert-Schmidt space, and (ii) as channels which are affine combinations of unitary channels. Moreover, we show that extreme points of the set of unital channels need not be extremal within the set of all channels.
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• In Sec.III computable criteria for the separation of unital channels from the set of mixtures of unitaries are provided and a respective negativity measure is introduced. • In Sec.IV we focus on covariant channels (in particular w.r.t. O(d)) and show how symmetry enables us to explicitly determine the above sets and to compute the negativity measure. • In Sec.V we apply the acquired tools in order show that families of covariant channels outside the convex hull of unitary channels can come into this set by either taking several copies of them or supplementing with a completely depolarizing channel. II. Unital Quantum Channels A. Preliminaries. We begin with introducing some notation and basic concepts. Throughout we will work in the Schrödinger picture and consider quantum channels T with finite and equal input and output dimensions, i.e., T : Md → Md is a linear map on d × d (density)matrices. Complete positivity enables a Kraus decomposition † T (ρ) = Ai ρ Ai† , Ai Ai = 1, (1) i
i
where the second relation expresses the trace preserving property. A channel is called unital if T (1) = 1 and as we include the trace preserving property in the definition of a channel, a unital channel is a doubly-stochastic completely positive map. It is often convenient to regard Md as a vector space which, when equipped with the inner product A, B := tr[A† B], forms the Hilbert-Schmidt Hilbert space Hd . Every channel is thus a linear map on this space and has as such a respective matrix representation Tˆ ∈ Md 2 B(H).1 We will occasionally use a (non-orthogonal) basis for H which is obtained from embedded Pauli-matrices in the form jk
σx := | j k| + |k j| for all j < k, jk
σ y := −i (| j k| − |k j|) for all j < k, j
σz := | j j| − | j + 1 j + 1| ∀ j = 1, . . . , d − 1,
(2)
together with the identity matrix. Another useful concept is the state-channel duality introduced by Jamiolkowski [13] which assigns a density operator ρT ∈ Md 2 to every channel T via d 1 | j, j, ρT = (id ⊗ T )(| |), | = √ d j=1
where is a maximally entangled state. The states ρT corresponding to unital channels are exactly those with reduced density matrices tr 1 [ρT ] = tr 2 [ρT ] = 1/d.
(3)
Note that due to the linearity of the correspondence the convex structure of channels is entirely reflected by the convex structure of the set of their dual states. Depending on what is more convenient we will switch back and forth between T and ρT . 1 B(X ) denotes the space of bounded linear operators on X .
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B. Representations. In the remainder of this subsection we will prove the following characterization of unital channels: Theorem 1 (Characterization of unital channels). Let T : Md → Md be a quantum channel. Then the following are equivalent: 1. T is unital (i.e., doubly-stochastic), 2. T is a convex combination of unitaries on Hd , i.e., Tˆ = α pα Wα with the p’s being probabilities and each Wα ∈ Md 2 unitary, 3. T (·) = i λi Ui · Ui† is an affine combination of unitary channels, i.e., the λ’s are real and sum up to one and each Ui ∈ Md is unitary. In order to see 1⇔2 we use a result from [1]: for any p > 1 a positive trace-preserving map T is a contraction in the sense2 of T p→ p ≤ 1 iff3 T is unital. In addition we have
T 2→2 = Tˆ ∞ so that T is unital iff Tˆ is a contraction with respect to the operator norm. The set of these contractions in turn is the convex hull of unitaries (as can be seen from the singular value decomposition) which completes 1⇔2. As 3⇒1 is obvious it remains to show 1⇒3. To this end we introduce X := A ∈ Md : A = A† , tr A = 0 , V := A → U AU † : U ∈ Md unitary ⊂ B(X ). That is, X is a real linear subspace of H containing all Hermitian operators orthogonal to 1 and V are the unitary conjugations on X . Note that the real linear span of V is invariant under composition and that the set in Eq. (2) (without the identity) forms a basis of X . The idea is now to show first how B(X ) can be obtained from V and then to extend this to the claimed implication 1⇒3 in Thm.1. Denote the subspace of real linear combinations of vectors {x1 , . . . , xn } such that the coefficients sum to zero by zerospanR {x1 , . . . , xn } := λi xi : λi ∈ R, λi = 0 . i
i
Lemma 2. For each basis vector B ∈ X in (2) there exists a T ∈ zerospanR V which maps B to itself and all other basis vectors to zero. Proof. We explicitly construct such a T w.l.o.g. for σx12 . Set
12 1
† ρ + U1 ρU1 , U1 := T1 (ρ) := , 2 −1
σy 1
† ρ − U2 ρU2 ∈ zerospanR V, U2 := T2 (ρ) := . 2 1 Then for all α ∈ R4 and σ ≡ σx , σ y , σz , 12 , T1 T2 α · σ B∗ α·σ 0 α1 σx + α3 σz 0 A := . → → B C 0 C 0 0 2 The norms are defined as T
† p/2 ] 1/ p . p→q = sup A T (A) q / A p with A p = tr[(A A) 3 As usual ‘iff’ should be read ‘if and only if’.
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In a similar vein we can finally map α3 to zero by a T3 , defined as T2 only with σ y in U2 replaced by σz . Then T := T3 ◦ T2 ◦ T1 is the desired operator which satisfies T (A) = α1 σx12 , and T ∈ zerospanR V as T2 ∈ zerospanR V. Clearly, the same type of construction works for all basis vectors in (2). Lemma 3. For every pair of basis vectors B1 , B2 ∈ X in (2), there is a T ∈ V such that T (B1 ) = B2 . Proof. As B1 and B2 are Hermitian, there are unitaries U1 and U2 such that U †j B j U j ( j = 1, 2) are both diagonal. These can in turn be mapped onto each other by a permutation in V since they both have eigenvalues (1, −1, 0, . . . , 0). Exploiting that V forms a group we can compose these steps to obtain T (B1 ) = B2 . Proposition 4. V zero-spans all linear operators on X , that is, zerospanR V = B(X ). Proof. For any two basis vectors B1 , B2 in (2), by the above lemmas there is a T ∈ zerospanR V which maps B1 to B2 and all other basis vectors to zero, so that a linear combination of these T ’s generates any linear map on X . This immediately implies 1⇒3 in Thm.1 as for every unital quantum channel T we have that T − id ∈ B(X ) so that we can write T (ρ) = ρ + i λi Ui ρUi† with the λ’s summing up to zero. Note that Thm.1 implies that mixtures of unitaries form a set of non-zero measure within the set of doubly stochastic channels. Conversely, assuming that they have nonzero measure implies the equivalence of 1 and 3.4
C. Extreme points. The set of all unital quantum channels on Md is convex and compact. That is, every unital channel T can be decomposed as T =
pi Ti ,
(4)
i
where the p’s are probabilities and the Ti ’s are extremal unital channels, i.e., those which cannot be further decomposed in a non-trivial way. Despite considerable effort [7,9,15–17] not much is known about the explicit structure of these extreme points beyond d = 2 (in which case they are all unitary conjugations [7]). The small contribution of this subsection is to review the existing results and to apply them in order to show that channels which are extremal within the set of unital channels are not necessarily extremal within the convex set of all channels. To the best of our knowledge all known examples so far were extremal within both sets—although the numerical results stated in [17] already indicate that this might not be generally true. The main ingredient is the following theorem which is stated in [9] and based on [18]. 4 The fact that they have non-zero measure was proven independently (after the present paper has been made public) in [14].
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Theorem 5. (Extremal channels). Consider a quantum channel with Kraus operators {Ai }i=1,...,N . It is an extreme point within the convex set of quantum channels iff the set of matrices (5) A†k Al k,l=1...N
is linearly independent. Assume further that the channel is unital. Then it is extremal within the convex set of unital channels iff A†k Al ⊕ Al A†k (6) k,l=1...N
is linearly independent. We will exploit the fact that (5) allows less linearly independent √ operators than (6): while (5) gives the simple bound N ≤ d, the set (6) yields N ≤ 2d.5 For our example we choose dimension d = 3 and N = 4 linearly independent Kraus operators. The former ensures that there are non-trivial extreme points, and the latter already implies that (5) can never be linearly independent as N ≤ d. We start with an Ansatz for the Jamiolkowski state of the sought channel of the form 6
ρT = (id ⊗ T )(| |) =
xi j |ψi ψ j ,
i, j=1
where the (|ψi )i span the orthogonal complement of (|kk)k , namely 1 1 1 |ψ1 = √ (|12 + |21), |ψ2 = √ (|13 + |31), |ψ3 = √ (|23 + |32), 2 2 2 1 1 1 |ψ4 = √ (|12 − |21), |ψ5 = √ (|13 − |31), |ψ6 = √ (|23 − |32), 2i 2i 2i and the Hermitian matrix X ≡ xi j is given by ⎛
1 2
0
0
i µ1 i µ4 (i − 2) µ3 0
⎜ 0 ⎜ 1 ⎜ i µ1 X := ⎜ 3⎜ ⎜−i µ3 ⎝−i µ4
1 2
−i µ1 −i µ1 1 2
i µ3 −i µ4 0
0 0 2 µ2 − i µ3
1 2
i µ4 −(2 + i) µ3 0 0
0 i µ1
−i µ1
1 2
⎞ 0 ⎟ 0 ⎟ 2 µ2 + i µ3 ⎟ ⎟. −i µ1 ⎟ ⎟ i µ1 ⎠ 1 2
(7) The latter is chosen such that ρT satisfies the conditions (3) corresponding to a unital and trace-preserving map. It remains to choose algebraic numbers µ1 , . . . , µ4 ∈ R such that X is positive semidefinite with rank N = 4, and that at the same time (6) is linearly independent when plugging in the corresponding Kraus operators. A possible choice for such a set of parameters is provided in Appendix VIII. 5 In fact, in [16] it was shown that N ≤
when applied to integer N .
√ 2d 2 − 1 which is, however, practically the same as N ≤ 2d
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III. Mixtures of Unitary Channels This section deals with the class of unital channels which can be represented as T (ρ) =
N
pi Ui ρUi† , Ui Ui† = 1,
pi > 0 ∀ i.
(8)
i=1
The Jamiolkowski states corresponding to these mixtures of unitary conjugations are exactly the states which are convex combinations of maximally entangled states. The rank of the Jamiolkowski state ρT gives a simple bound [19] for the minimal N as there exists always a decomposition with N ≤ (rank ρT )2 . For d = 2 we can achieve equality in the general lower bound N ≥ rankρT and, as mentioned before, every unital channel on M2 is a mixture of unitaries [7]. For d ≥ 3 the question whether a given unital channel allows for such a representation was investigated and reformulated in [17] but a general operational way of deciding it remains to be found. The approach in the following subsection provides a class of easily computable necessary conditions which, when applied to covariant channels, will later be extended to necessary and sufficient criteria. A. Separation witnesses. Since the set (8) of mixtures of unitary channels is convex and compact, every unital channel which lies outside this set can be separated from it by a hyperplane — a witness. As this can most easily be expressed on the level of Jamiolkowski states we introduce the corresponding sets S := {ρT : ρT = (id ⊗ T ) (| |) , T : Md → Md cp, tp, unital} = ρ ∈ Md 2 : ρ ≥ 0, tr 1 ρ = tr 2 ρ = 1/d ,
U := conv (1 ⊗ U ) | | 1 ⊗ U † : UU † = 1 , which we will, with some abuse of notation, occasionally also use for channels, i.e., we will write ‘T ∈ S’ meaning ρT ∈ S. The following shows that we may impose some structure on the witnesses — they can be taken from the affine span of U. Proposition 6. (Separation witnesses). Let ρ ∈ S characterize a unital quantum channel. Then ρ ∈ U, i.e., it is a mixture of maximally entangled states, iff tr [Wρ] ≥ 0 for all Hermitian operators W ∈ Md 2 which satisfy tr 1 W = tr 2 W = 1/d, tr [W σ ] ≥ 0 ∀ σ ∈ U.
(9)
Proof. We have to show that if ρ ∈ / U, then there exists such a W with tr [Wρ] < 0. First note that X := A ∈ Md 2 : A = A† , tr 1 A = tr 2 A = 0 is a real linear space and S −1/d 2 ⊂ X . Set ρ˜ := ρ −1/d 2 ∈ X . Using the Hahn-Banach separation theorem [20, Theorem 1.C in Chap. 1] we find a W˜ ∈ X with
tr W˜ ρ˜ < −1/d 2 , but tr W˜ σ˜ ≥ −1/d 2 ∀ σ˜ ∈ U − 1/d 2 . Setting W := W˜ + 1/d 2 yields the sought witness.
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To simplify matters we will in the following also consider Hermitian witnesses which do not fulfill the l.h.s. of (9) as long as the r.h.s. is satisfied. A class of this kind which turns out to be particularly useful are operators constructed from the flip operator F : |k, l → |l, k in the form W = (1 ⊗ B) F 1 ⊗ B † + w(B) 1, B ∈ Md , (10) where w(B) ∈ R is a constant depending on B such that W fulfills the r.h.s. in (9). Before we determine this dependence let us note that replacing (1 ⊗ B) by (A ⊗ B) in Eq. (10) won’t lead to a more general class of witnesses since (A ⊗ B) F A† ⊗ B † = 1 ⊗ B A† F 1 ⊗ AB † . The sharpest constant w(B) for which (10) fulfills the witness condition tr[Wρ] ≥ 0 for all ρ ∈ U is obtained from w(B) = − min tr (1 ⊗ B)F 1 ⊗ B † (1 ⊗ U ) | | 1 ⊗ U † U 1 = − min tr B † U B T U d U 1 = − min tr A A : σ (A) = σ (B) , d A where U is unitary, A ∈ Md and σ (A) denotes the singular values of A. We solve this matrix optimization problem in Appendix VII A arriving at the following result. Theorem 7. (Tight witnesses). For any B ∈ Md with singular values σ1 ≥ · · · ≥ σd the operator in Eq. (10) is a separation witness iff d/2 d even 1 2 i=1 σ2i−1 σ2i , (d−1)/2 (11) w(B) ≥ 2 , d odd. − σ σ σ d 2 2i−1 2i d i=1 Note in particular that for B = 1 and d odd we get w ≥ 1 − 2/d while for d even w ≥ 1. Hence, for even d no channel is separated from U by such a witness (since F + 1 ≥ 0). However, we will see in Sec. IV B that for d odd it becomes a powerful tool.
B. A negativity measure. There are several possible ways of quantifying the deviation of a channel T ∈ S\U from being a mixture of unitary channels: one may for instance follow [17], use the entanglement of assistance [12,21] or the minimal distance to the set U w.r.t. some distance measure. The representation Thm.1 enables a very natural alternative approach—a base norm (inspired by [22]). That is, the deviation is quantified by the smallest negative contribution when representing T as an affine combination of terms in U. More formally: Definition 8. (Negativity). For all ρ ∈ S the base norm associated with U is
ρ U := inf α p + αn : ρ = α p σ p − αn σn , α p,n ≥ 0, σ p,n ∈ U , and the corresponding negativity is given by NU (ρ) := inf αn : ρ = α p σ p − αn σn , α p,n ≥ 0, σ p,n ∈ U .
(12)
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For tr[ρ] = 1 the two are related via ρ U = 1 + 2NU (ρ) and obviously NU (ρ) = 0 iff ρ ∈ U. The base norm behaves nicely under concatenation and convex combination. Writing T U := ρT U we get Proposition 9. Let Ti ∈ S be a set of quantum channels and pi ≥ 0 probabilities, then
Ti U , and pi Ti ≤ pi Ti U . Ti ≤ U
U
Both can easily be proven from the definition. The latter can be interpreted as coming from triangle inequality and homogeneity of the norm. Note also that the above norm is unitarily invariant in the sense of T V U = V T U = T U for every unitary conjugation V . As always measures are easy to define but hard to compute. For covariant channels we will show the calculation in Sec. IV B. IV. Covariant Channels In order to arrive at more explicit results we need some help—coming in the form of symmetries imposed on the channels. Consider any subgroup G ⊂ U (d) with elements g ∈ G and two unitary representations Vg , V˜g on Cd . We say that a channel T : Md → Md is G-covariant w.r.t. these representations if for all g ∈ G:
T Vg · Vg† = V˜g T (·)V˜g† . (13) In this sense the action of the channel ‘commutes with the symmetry’. If V˜ is an irreducible representation then T is unital as T (1) = dg T (Vg Vg† ) = dg V˜g T (1)V˜g† = 1 by invoking Schur’s Lemma (where dg is the Haar measure). In order to express Eq. (13) in terms of the Jamiolkowski state ρT we introduce G = {V g ⊗ V˜g }g∈G and its commutant G = {X ∈ Md 2 : ∀ Ug ∈ G : [X, Ug ] = 0}. Covariance of the channel translates then simply to ρT ∈ G . As we will see below most of the analysis can w.l.o.g. be restricted to this commutant which considerably simplifies matters as dim G is for a sufficiently large symmetry group much smaller than d 4 , the dimensionality we would have to deal with otherwise. The map ! P(A) := dg Ug AUg† defines a projection in B(Md 2 ), often called twirl, which maps every matrix A into G and acts as the identity on G . Moreover, since G is an algebra it is spanned by a set of minimal projections {Pi }. These are orthogonal if G is abelian (which happens for large enough symmetry groups) so that every X ∈ G can be written as X = i xi Pi with xi = tr[X Pi ]/tr[Pi ]. In this case we can easily determine6 tr[A Pi ] Pi . P(A) = (14) tr[Pi ] i
6 Here we have used that tr[ P(A)P ] = tr[A P(P )] = tr[A P ]. i i i
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If G fails to be abelian a similar reasoning still applies—for a detailed exposition of these matters we refer to [22]. In order to see how covariance helps for our purposes let us denote the set of witnesses by W := {W = W † : ∀σ ∈ U : tr[W σ ] ≥ 0}. Proposition 10 (Reduction to the commutant). Let ρ ∈ S ∩ G be the Jamiolkowski state corresponding to a covariant unital channel. Then ρ ∈ U iff tr[Wρ] ≥ 0 for all W ∈ W ∩ G . Moreover,
ρ U = inf α p + αn : ρ = α p σ p − αn σn , α p,n ≥ 0, σ p,n ∈ U ∩ G , which equivalently holds for the negativity NU . Proof. The crucial point for both parts is that σ ∈ U implies P(σ ) ∈ U which in turn means that P(W ) ∈ W for every W ∈ W. Therefore due to tr[ρ P(W )] = tr[ P(ρ)W ] = tr[ρW ] the set W can w.l.o.g. be restricted to G . Regarding the base norm we arrive at the stated result when starting with any optimal decomposition ρ = α p σ p − αn σn and applying the twirl to both sides of the equation. This suggests the program for the next subsections: fix a symmetry group, identify the commutant G and determine U, · U and NU by exploiting the reduction to G . A. O(d) covariance. The symmetry we will consider is the one of the real orthogonal group, i.e., G = {O ⊗ O : O ∈ Md real orthogonal}. The most prominent non-trivial example of a channel having this symmetry is T (ρ) = tr[ρ] 1 − ρ T /(d − 1), (15) which (for d = 3) gained some popularity as a steady source of counterexamples: for the multiplicativity of the output p-norm [23], the additivity of the relative entropy of entanglement [24] and, most relevant in our context, the quantum analogue of Birkhoff’s theorem [9]. On the level of Jamiolkowski states we can make use of the analysis in [24] where the commutant G was shown to be abelian and spanned by F , G = span 1, F, " where " F := d | |. From there the minimal projections can be identified as 1" F = | |, d 1 P1 = (1 − F), 2 1 1 F, P2 = (1 + F) − " 2 d
P0 =
where (1 ± F)/2 are the projections onto the symmetric and anti-symmetric subspace, respectively. Consequently, every density operator in G is in the convex hull of the corresponding normalized density matrices ρi = Pi /tr[Pi ] of which ρ1 corresponds to the Werner-Holevo channel in (15), ρ0 is the ideal channel and i Pi /d 2 corresponds to the completely depolarizing channel T (ρ) = tr[ρ] 1/d. Clearly, all of them are unital, i.e., elements of S. Every state ρ ∈ G is completely characterized by its “coordinates” # F , " F . F ρ ≡ tr [ρF] , tr ρ"
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Fig. 1. The set of orthogonal covariant channels PS in the Jamiolkowski representation (outer/green triangle) and the convex hull of unitary channels P U (blue/shaded area), which is described in analytic terms by Proposition 12. Note that the Werner-Holevo channel ρ− is “furthest away” from the unitaries. The orange line (from ρ− to ρ+ ) depicts the U (d) covariant channels, and the dotted unit-square corresponds to entanglement-breaking channels. Compare with Ref. [24, Fig. 2]
Especially for the extreme points ρi we obtain (see Fig. 1) state coords
ρ0 (1, d)
ρ1 (−1, 0)
ρ2 (1, 0)
(16)
B. A complete picture. We will now determine the subset U of mixtures of unitary channels within the set of O(d)-covariant channels. Following the above considerations this amounts to identifying the corresponding region in the two-dimensional parameter space # U ∩ G ∼ F ρ : ρ∈U = F , " # (17) = conv F , " F U : U unitary , where the index U stands for the expectation value w.r.t. (1⊗U ) | which parameterizes an extreme point within U. A short calculation reveals that
1 FU ≡ tr (1 ⊗ U ) | | 1 ⊗ U † F = tr UU , d
1 # " F = |tr U |2 . (18) F U ≡ tr (1 ⊗ U ) | | 1 ⊗ U † " d The picture depends crucially on whether d is even or odd.
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Theorem 11 (Even dimension). If d is even then U ∩ G = S ∩ G , i.e., every O(d)covariant channel is a mixture of unitary channels. Proof. It suffices to note that the expectation values (16,18) with respect to ρi and Ui coincide for U0 = 1, U1 = diag σ y , . . . , σ y and U2 = diag (σz , . . . , σz ), just by plugging in (18). So the interesting structure only emerges for d odd (see Fig. 1 for d = 3), for which we need the following result proven in Appendix VII B: Proposition 12. Let d ≥ 1 be odd. Then for all x ∈ −1 + d2 , 1 there exists a unitary U ∈ U (d) such that tr UU /d = x, and max |tr U | /d : U ∈ U (d), tr UU /d = x %1/2 $ 1 2 1 1 1− 1− +x =: m(x). (19) = + 2 d d d Theorem 13 (Odd dimension). Let d ≥ 3 be odd. Then the extreme points of the set (17) corresponding to mixtures of unitary channels are
x, d (m(x))2 : x ∈ [−1 + 2/d, 1] . (20) (−1 + 2/d, 0) , (1, 0) and Proof. “(17) ⊂ conv(20)”: For all unitary U ∈ U (d), 1 tr UU ∈ [−1 + 2/d, 1], d which follows from the fact that for any matrix A the spectrum of A A is symmetric with respect to the real axis, the eigenvalues λ, λ have the same algebraic multiplicity, and the algebraic multiplicity of all negative eigenvalues of A A (if any) is even, see [25]. Together with Prop. 12 we obtain the stated bounds on (17). “(20) ⊂ (17)”: Set ⎛ ⎞ 0 1 − i −1 − i 1 1 ⎠ Q 0 := ⎝−1 + i −i 2 1+i 1 i and ϕ := exp (2πi/3), then the coordinates(20) are obtained by U0 = diag σ y , . . . , σ y , Q 0 , U1 = diag σz , . . . , σz , ϕ, ϕ 2 , 1 and the unitary matrices which solve the maximization problem (19) (explicitly given in Appendix VII B). The fact that according to Prop. 10, we can restrict to decompositions within the two-dimensional parameter space, together with the explicit characterization of the set U ∩ G enables us now to compute the negativity NU , as follows. We show first that in Eq. (12), σn = ρ2 always obtains the infimum, as illustrated in Fig. 2. Since U ∩ G is convex and closed, the optimal σ p,n in (12) are on the boundary of U ∩ G , and σn lies either on the segment joining ρ2 and ρ0 or the one joining ρ2 and the covariant state with coordinates (−1 + 2/d, 0). We may w.l.o.g. assume the former, i.e., σn = λ ρ0 + (1 − λ)ρ2 for a λ ∈ [0, 1]. Considering decompositions ρ = α p σ p − αn σn
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Fig. 2. Negativity as distance measure, exemplified by orthogonal covariant channels. NU is constant along red lines, which are obtained – in geometric terms – by a uniform scaling of the unitary channel boundary about ρ2 as origin. σ p,n ∈ P U are the optimal states in (12) given ρ ∈ PS\ P U
with optimal σ p (depending on λ) given ρ and σn via λ, both α p,n are already determined by the x-coordinates of ρ and σ p,n due to α p + αn = 1. Note that the x-coordinates of ρ and σn remain fixed for all values of λ whereas the x-coordinate of σ p is non-increasing as λ decreases, and so is αn . That is, λ = 0 or equivalently σn = ρ2 minimizes αn . It follows that a uniform scaling of the boundary of U ∩ G by a factor (1 + αn ) starting from ρ2 as origin yields precisely the set of points with negativity αn . We may write each ρ ∈ S\U ∩ G in terms of a convex combination of the ρi listed in Table 16, that is, ρ = i qi ρi with qi ≥ 0 and i qi = 1. From Fig. 2 it is evident that q1 > 0 for all ρ ∈ / U. Set q := q0 /q1 and distinguish the following two cases due to the particular shape of U ∩ G . 1 • q > d(d−1) . This corresponds exactly to the area above the dashed line in Fig. 2. Applying the scaling (1 + αn ) to the curve in Theorem 13, an explicit calculation shows that
& d −1 1 d −2 2 2 q −2 q q1 − 1, q + NU (ρ) = d −1+ d + d −2 d −2 d −2 d −1 ρ=
3 i=1
qi ρi ∈ S\U with qi ≥ 0,
i
qi = 1, q := q0 /q1 .
1070
• q≤
C. B. Mendl, M. M. Wolf 1 d(d−1) .
In this case the negativity does not depend on q, and we get NU (ρ) =
d q1 − 1. d −1
In particular, NU (ρ) is maximal exactly for the Werner-Holevo channel ρ− , namely NU (ρ− ) = 1/(d − 1). V. Restoring Birkhoff’s Theorem Measures quantifying the deviation of a unital channel from being a mixture of unitary channels are known to be not additive (or multiplicative). That is, a naive extrapolation from the ‘distance’ between a given T ∈ S\U and U typically leads to an overestimation of the respective quantity for T ⊗n , i.e., several copies of the channel. This effect was studied in detail in the context of the entanglement of assistance [12,21] E a (ρ) := sup
pi S (tr 1 i ) : ρ =
i
pi | i i | ,
i
where S(ρ) = −tr[ρ log ρ] is the von Neumann entropy, and the supremum has to be taken over all convex decompositions of the given state ρ ∈ B(Cd ⊗ Cd ) into pure ones. As S (tr 1 i ) ≤ log d with equality iff i is a maximally entangled state we have that E a (ρ) ≤ log d with equality iff ρ ∈ U. It was shown in [12] that ∀ρ ∈ S :
1 ⊗n Ea ρ = log d, n→∞ n lim
which suggests that the approximation of ρ ⊗n by an element of U improves as n increases. This would mean a restoration of Birkhoff’s theorem in the asymptotic limit. Whether this statement is valid in general when formulated in terms of norm distances (either for channels or, supposedly weaker, for states) remains an open problem [26]. In the following subsections we will prove it in the strongest possible sense for a class of O(d)-covariant channels. We will see that at least for these cases neither the asymptotic limit nor an approximation is required—a remarkable effect from the perspective of environment-assisted error correction (Sec. V C). More specifically we will show that for a T ∈ U we find T ⊗ T˜ ∈ U when choosing T : Md → Md , T (ρ) =
1+δ tr[ρ] 1 − δ ρ T , d odd, d
(21)
with appropriate δ and either T˜ = T (Sec. V A) or T˜ : ρ ∈ M D → tr[ρ] 1 ∈ M D completely depolarizing (Sec. V B). The symmetry of the channels will help us in two stages: (i) we can use Thm.13 which tells us that T ∈ U for δ > 1/(d + 1), and (ii) it circumvents having to find an explicit decomposition in terms of unitary channels for T ⊗ T˜ : if the convex hull of the relevant G expectation values of any set of unitary channels contains the ones of T ⊗ T˜ , then the twirling projection P does the rest of the job.
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A. Two copies of a channel. As usual we switch to the Jamiolkowski representation, where the family of channels in Eq. (21) becomes 1 1 ρ− + − ρ+ , d odd, ρT = 1 − + d 2 d 2 where ρ± are the normalized projections onto the symmetric and anti-symmetric subspace, respectively. The parametrization is chosen such that ρT ∈ S\U for ∈ (0, 2/d] since 2 tr [ρT F] = −1 + − , d and F + (1 − 2/d) 1 is a tight separation witness according to Thms.7,13. To exploit the full symmetry coming from T˜ = T we follow Sect. V.B of [24] and increment the tensor product symmetry group7 {(U ⊗ U ) ⊗ (V ⊗ V )} by a flip operator which interchanges the tensor factors in the product T ⊗ T˜ . This results in a larger symmetry group G, thus yielding a smaller commutant G ⊂ Md 4 which is spanned by 1 and F :=
1 (1 ⊗ F + F ⊗ 1) , 2
F12 := F ⊗ F.
That is, every state ρ ∈ G is now completely characterized by the expectation values/coordinates (F , F12 )ρ ≡ (tr [ρ F] , tr [ρ F12 ]) .
Especially for any unitary channel described by U ∈ Md 2 , setting Us := 21 U + U T and denoting partial transposes by Ti gives
1 F12 U = tr F12 (1 ⊗ U ) | | 1 ⊗ U † = 2 tr U U , d
† FU = tr F (1 ⊗ U ) | | 1 ⊗ U T 1 T2 1 = tr U U + U 2 d2 1 1 T T = 2 tr U Us 2 = 2 tr Us Us 2 . (22) d d T
The last equation uses the fact that Us 2 is again symmetric. The ranges of the expectation values in Eq. (22) are studied in Appendix VII C. In particular for d = 3 we provide an explicit construction for the coordinates 8 1 − (cos ϑ + 1)2 + 3, 16 cos2 ϑ − 7 , ϑ ∈ [0, π/2] (23) (F, F12 )ϑ = 9 3 corresponding to convex combinations of unitary channels. Now matching the coordinates of T ⊗ T ,
2 (F, F12 )T = tr [ρT F] , tr [ρT F]2 = x, x 2 , x := −1 + − d 7 The aim is to use the full symmetry group. That is, we use V ⊗ V , V ∈ U (d) for ρ which already T allows us to discard Fˆ from the commutant. We use further that the commutant of a tensor product is the tensor product of the commutants and that Prop. 10 remains true when adding the additional flip operator.
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C. B. Mendl, M. M. Wolf
Fig. 3. Twofold tensor products of covariant channels for d = 3 (outer/green triangle). The mixtures of unitary channels correspond to the blue/shaded region, the orange parabola to tensor products of a channel with itself and the red part of that curve to the elements of U on the single-channel level. Compare with Fig. 9 in [24]
with (23) yields =
√ √ √ 2
1 4−3 2− 3+ 6 ≈ √ 3 10
as shown in Fig. 3. The blue area corresponds to convex combinations of unitary channels,8 i.e., elements of U, the orange curve to coordinates of single-channel tensor products and the red part of this curve to the elements of U on the single-channel level. The state at the lower corner is ρm =
1 (ρ− ⊗ ρ+ + ρ+ ⊗ ρ− ) . 2
As can be seen from direct inspection, each point on the curve (23) is an extreme point of the blue area. The remaining extreme points (1, 1) and 19 , − 79 are realized by the ⊗4 unitaries 1 and 1 −1 ⊕ 1, respectively. B. Help from a noisy friend. Instead of adding a second copy of the channel, we will now supplement it by a completely depolarizing T˜ . The Jamiolkowski representation of 8 A proof that (23) really solves the minimization problem (32) as acclaimed is still outstanding, but strongly supported by numerical tests. In any case, the set of convex combinations of unitary channels is at least as big as the blue area.
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Table 1. Numerical minimization of (25). Note that the table is asymmetric w.r.t d ↔ D, and that the lower bound −1 can apparently be obtained for d = 5 and D = 4, that is, each channel-tensor product T ⊗ T˜ (with T˜ the completely depolarizing channel) becomes then a convex combination of unitaries d
D=1
2
3
4
5
1
1
1
1
1
1
3
− 13
− 97
23 − 27
− 45
37 − 45
5
− 35 − 57
23 − 25 47 − 49
−0.929 . . .
−1
−0.976 . . .
7
the completely depolarizing channel T˜ : B (K1 ) → B (K2 ) , Ki = C D is
ρT˜ ≡ id ⊗ T˜ (| |) = 1/D 2 . Let H be the corresponding symmetry group of all local unitaries V1 ⊗ V2 with Vi ∈ B (Ki ), then H = span{1}. Using again that (G ⊗ H ) = G ⊗ H (see Example 7 in Sect. II.D of [24]) we get that every element of the commutant is completely characterized by the expectation value of Y = FH1 ⊗H2 ⊗ 1K1 ⊗K2 , yielding 2 tr ρT ⊗ ρT˜ Y = tr [ρT F] = −1 + − . d
(24)
Since −1 ≤ Y ≤ 1 every normalized state ρ satisfies tr [ρY ] ∈ [−1, 1]. In order to obtain the subinterval of [−1, 1] covered by convex combinations of unitary channels, we have to calculate Y U for unitary U : H1 ⊗ K1 → H2 ⊗ K2 ,
1 T tr U U 2 . tr (1 ⊗ U ) | | 1 ⊗ U † Y = dD
(25)
As U = 1 reaches the upper bound 1, the hard part is the lower bound which is treated in Appendix VII D. The results suggest that for D = 2, Eq. (25) gives $
% 2 −1 + 2 , 1 d
(26)
for the range in which T ⊗ T˜ ∈ U while, recall, T ∈ U only within [−1 + 2/d, 1]. The interval (26) can be related to the conjectured existence of a certain quaternion matrix, which we construct explicitly for d = 3 and d = 5. This means that in this case (25) covers this range at least. In particular, for ≤ 2(d − 1)/d 2 , the expectation value (24) lies within this interval such that ρT ⊗ ρT˜ becomes then indeed a convex combination of maximally entangled states. For higher values of D, we reproduce Table 2 from Appendix VII D.
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C. B. Mendl, M. M. Wolf
Fig. 4. The correction scheme as in [11], applied to the simultaneous usage of two noisy channels T : B(H1 ) → B(H2 ) and T˜ : B(K1 ) → B(K2 ). The channels are represented by unitary couplings U and U˜ to an environment which is initially in a pure product state. The classical result α of the measurement on the global environment is used by the receiver who chooses the recovery operation Rα (again a quantum channel) accordingly. As discussed in the text, T ⊗ T˜ can become perfectly correctable (i.e., Tcorr = id) although neither T nor T˜ is so
C. Environment-assisted error correction. The above results become especially remarkable from the point of view of environment-assisted error correction—a concept introduced in [11]. There it was studied which channels allow complete correction, given a suitable feedback of classical information from the environment (see Fig. 4). The class of perfectly correctable channels was identified with the set of convex combinations of unitary channels. In this way the above observations yield examples of channels which are not perfectly correctable on their own but become so when either taking several copies, or supplementing with a completely depolarizing channel.
VI. Discussion The presented investigation of the set of unital quantum channels is to a large extent based on and inspired by methods and ideas from entanglement theory. The tools acquired in this context could be directly applied to the Jamiolkowski representation of the channel. This approach as such leads to questions about further analogies between the two fields. It would in particular be interesting whether a useful counterpart to positive maps, i.e., powerful non-linear criteria can be found for the separation of the set of mixtures of unitary channels.9 Clearly, the asymptotic Birkhoff conjecture[26] remains an important open problem for which the present work might be regarded as supporting evidence as it provides the first class of examples for which there is a rigorous proof. In this context it might be interesting to investigate T ⊗ T˜D with T˜D a D-dimensional maximally depolarizing channel, as studied in Sec. V B. Is there a dense subset of unital channels such that a finite D makes T ⊗ T˜D a mixture of unitary channels? Acknowledgements. We thank R.F. Werner for valuable discussions and the F, and J.I. Cirac for many useful comments and the steady support along the way. M.M.W. acknowledges financial support by QUANTOP and the Danish Natural Science Research Council (FNU). 9 Here it might be helpful to replace positivity by some form of contractivity.
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VII. Appendix A–Matrix Optimization Problems A. Minimizing tr[A A] subject to fixed singular values. We solve the minimization problem posed in Sect. III A which deals with a special class of separation witnesses. Considering any complex d × d matrix A, we denote its singular values by σi (A) (counting multiplicity) such that σ1 (A) ≥ · · · ≥ σd (A). Our main result is the following proposition, which is similar to Theorem 7.4.10 in [27]. Proposition 14. Let A ∈ Md and σ1 ≥ · · · ≥ σd denote the singular values of A. If A A is Hermitian, then there exists a permutation τ of {1, . . . , d}, an even r ≤ d and a function ρ : {1, . . . , r/2} → {0, 1} with r/2 d (−1)ρ(i) στ (2i−1) στ (2i) + στ2(i) . tr A A = 2
i=1
(27)
i=r +1
Conversely, given any such τ, r, ρ and nonnegative numbers σ1 ≥ · · · ≥ σd , there exists an A ∈ Md such that σi (A) = σi for all i and (27) holds. Proof. We split the “⇒” part into the following steps: 1. tr A A and the singular values of A are invariant under A → U AU T for any unitary U . Note that this map sends A A → U A AU † , so by the spectral theorem, w.l.o.g. A A real diagonal. 2. It follows that A A = A A = A A, i.e. A commutes with A, and each of the eigenspaces of A A is invariant under A and A. Stated differently, A is block diagonal, each block corresponding to an eigenspace of A A, so w.l.o.g. A A = λ 1 for a λ ∈ R. 3. The case λ = 0: Applying the singular value decomposition yields unitary matrices U, V such that A1 := U † AV = diag (σ1 , . . . , σd ). Using A = λA−1 , we have
−1
−1 −1 . = λA−1 = λ diag σ , . . . , σ A2 := V † AU = λ U † AV 1 1 d the same singular values translates to {σ1 , . . . , σd } = |λ| A1 and A2 sharing −1 −1 σ1 , . . . , σd , so there is a permutation τ of {1, . . . , d} with '
στ (2i−1) στ (2i) = |λ| στ (2i−1) στ (2i) = |λ|
for i = 1, . . . , d2 , d even 2 |λ| for i = 1, . . . , d−1 , σ = , d odd. τ (d) 2
Note that d cannot be odd if λ < 0 as the negative eigenvalues of A A are of even algebraic multiplicity (see e.g. [27], pp. 252, 253). Concluding, tr A A can always be written in the form (27). 4. The case λ = 0: Let r denote the number of nonzero singular values of A. A A = 0 means that range A ⊆ kern(A), so r = rank(A) = dim range A ≤ dim kern(A) = d − r, i.e. 2r ≤ d, and there is a permutation τ such that each summand in the right-hand side of (27) is zero.
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C. B. Mendl, M. M. Wolf
To prove the “⇒” part, set A := diag A1 , . . . , Ar/2 , στ (r +1) , . . . , στ (d) with
(−1)ρ(i) στ (2i) . Ai := σ 0 0 τ (2i−1)
Corollary 15. Given any nonnegative numbers σ1 ≥ · · · ≥ σd , min tr A A : A ∈ B (H) , σi (A) = σi ∀ i d/2 −2 i=1 σ2i−1 σ2i , d even = d−1/2 −2 i=1 σ2i−1 σ2i + σd2 , d odd.
(28)
Proof. What remains to be shown is A A being Hermitian for optimal A; then Proposition 14 guarantees optimality. Exploiting invariance under A → V AV T for unitary V we can w.l.o.g. assume that A = U D with D = diag (σ1 , . . . , σd ) and U unitary. Now vary U to minimize tr A A ; the unitary constraint translates via
U dU † + dU U † = d U U † = 0 to X := i dU U † being Hermitian. We have d tr DU DU dU = 2 Re tr D dU DU = 2 Im tr U DU D X . !
0=
This has to hold true for any Hermitian matrix X . Decomposing U DU D = B1 + i B2 , B1 and B2 Hermitian, it follows that B2 = 0. B. Maximizing |tr[U ]| subject to fixed tr[UU ]. In this subsection we prove Proposition 12 from Sect. IV B, i.e. we calculate the analytic solution of max |tr U | for fixed tr UU over all unitaries U ∈ U (d). The motivation for this optimization problem comes from Eq. (18), which characterizes the convex hull of unitary channels within the set of orthogonal-covariant channels. We need the following lemma first. Lemma 16. Let U ∈ U (2) be a unitary 2 × 2 matrix and Us := 21 U + U T be the symmetric part of U . Then the trace-norm of Us equals (
Us 1 ≡ σ j (Us) = tr UU + 2. j=1,2
Proof. There are α, β ∈ C such that up to an unimportant phase factor α β U= with |α|2 + |β|2 = 1. −β α
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1077
Direct calculation shows that
tr UU = 2 |α|2 − Re β 2 = 4 |α|2 + (Im β)2 − 2,
α i Im β , UsUs† = |α|2 + (Im β)2 1, Us = i Im β α ( ( (30) (29) σ j (Us) = 2 |α|2 + (Im β)2 = tr UU + 2.
(29) (30)
j=1,2
Proposition 17. Let d ≥ 1 be odd. Then for all x ∈ −1 + d2 , 1 , there exists a unitary U ∈ U (d) such that d −1 tr UU = x, and max d −1 |tr U | : U ∈ U (d), d −1 tr UU = x %1/2 $ 1 2 1 1 1− 1− +x + . = 2 d d d
(31)
√ 1/2 1 d 1 Proof. Set α := d−1 ∈ [0, 1], β := 1 − α 2 and 1 − d2 + x 2 1− d
α β −1 σ := −β α , then a short calculation shows that U := diag (σ, . . . , σ, 1) satisfies d tr UU = x with d −1 |tr U | equal to the right hand side of (31). So what remains to be shown is an upper bound on d −1 |tr U |. Let U ∈ U (d) be a unitary matrix with d −1 tr UU = x. By the Youla theorem [25], given any conjugate-normal matrix A (that is, A A† = A† A), there exists a unitary V such that V AV T is a block diagonal matrix with diagonal blocks of order 1 × 1 and 2 × 2, the 1 × 1 blocks corresponding to the real nonnegative eigenvalues of A A and the 2 × 2 blocks corresponding either to pairs of equal negative eigenvalues of A A or to conjugate pairs of non-real eigenvalues of A A. Applying this to U , there is a unitary V with U = V DV T , the block diagonal matrix D as described. Let r2 be the number of 2 × 2 blocks (with even r ≤ d) and denote these blocks by Di . As U and V are unitary, so must be D, i.e. Di ∈ U (2) for all i. UU unitary guarantees |λ| = 1 for each eigenvalue λ of UU , so each real nonnegative be 1. Alto eigenvalue of UU must gether we have D = diag D1 , . . . , D r2 , 1, . . . , 1 . Set ci := 21 tr Di Di ∈ [−1, 1] and Ds := 21 D + D T . Using Lemma 16 and the fact that V T V is unitary and symmetric, d |tr U | = tr DV T V = tr DsV T V ≤ σ j (Ds) j=1
=
r/2
i=1 j=1,2
σ j Di,s + (d − r ) = 2 ·
r/2 i=1
&
ci + 1 + d − r. 2
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C. B. Mendl, M. M. Wolf
Let c := d · x ∈ [−d + 2, d]. Some elementary analysis shows that the problem ⎧ ⎨
max 2 · ⎩ subject to
2·
r/2
r/2 i=1
&
⎫ ⎬ ci + 1 +d −r ⎭ 2
ci + d − r = c,
i=1
ci ∈ [−1, 1] for all i, r even, r ≤ d c−1 has optimal solution r = d − 1, ci = d−1 ∀ i and the obtained maximum is d 21 1 − d1 1/2 1 − d2 + dc + 1. This upper bound on |tr U | corresponds exactly to the right-hand side of (31). T C. Minimizing tr Us Us 2 subject to fixed tr[UU ]. Motivated by Eq. (22) in Sect. V A, we investigate '
min
1
1 T U + UT , tr Us Us 2 : Us = d1 d2 2 / 1 U ∈ B (H ⊗ K) unitary with tr UU = y d1 d2
(32)
for Hilbert spaces H and K with dimensions d1 and d2 , respectively, and provided y ∈ [−1 + 2/d1 d2 , 1], where d1 ≥ 3 is odd. The partial transposes T1 and T2 are defined w.r.t. a fixed product basis by the linear extension of (A ⊗ B)T1 = A T ⊗ B and (A ⊗ B)T2 = A ⊗ B T , A ⊗ B ∈ B (H ⊗ K), respectively. Note that for any A and B, T tr A B 2 = tr A T1 B † , and for any real or complex A with A T = A, the partial transposes are on equal footing, i.e. A T1 = A T2 , so (32) is inherently symmetric with respect to d1 ↔ d2 . We identify B (H ⊗ K) ∼ = Cd1 d2 ×d1 d2 by means of the ordered computational basis (|11 , |12 , . . . |1d2 , . . . |d1 d2 ). All quantities in (32), especially the minimizers, will stay invariant if we send U → (W1 ⊗ W2 ) U (W1 ⊗ W2 )T with arbitrary unitaries W1 ∈ B(H) and W2 ∈ B(K).
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Since every unitary matrix is also conjugate-normal (that is, U U † = U † U ), the Youla-theorem10 states that there exists a unitary matrix V ∈ B (H ⊗ K) such that U = V D VT with D real block-diagonal and blocks of size 1 × 1 and 2 × 2, the former non-negative and the latter of the form σz −z with σ ≥ 0. Since D must also be unitary, this equals σ cos ϑ − sin ϑ for a ϑ ∈ R, and all 1 × 1 blocks are 1. Note that sin ϑ cos ϑ 1
D + D T ≥ 0 diagonal, Us = V Ds V T , Ds := 2 in particular, Ds contains the singular values of Us. Moreover, tr UU ≡ tr D D is independent of V , so D fixes y in (32) and we may freely vary V . Conversely, Takagi’s theorem [27] asserts that every complex-symmetric matrix A ∈ Cn×n can be decomposed into A = V diag (σ1 , . . . , σn ) V T
(33)
with unitary V and σi ≥ 0 for all i, so identifying A ≡ Us and diag (σ1 , . . . , σn ) ≡ Ds, the minimization problem (32) can be reduced to the following problem and a subsequent optimization over Ds: ' / 1 T2 T min : A ∈ B(H ⊗ K), A = A, σi (A) = σi ∀ i . tr A A (34) d1 d2 This closely resembles (28), and we have effectively decoupled the target function from the peculiar unitary constraint in (32). Proposition 18. Every (local) minimizer A of (34) satisfies A A
T2
Hermitian.
Proof. Denote the derivative w.r.t. V in (33) by dV ; since V is unitary,
V dV † + dV V † = d V V † = 0 so X :=
1 i
dV V † must be Hermitian. Plugging
d A = dV diag (σi ) V T + V diag (σi ) dV T = i X A + AX T dV into the target function (34) yields
T d T tr A A 2 = i tr X A + AX T A 2 dV $ % T2 −i tr A X A + AX T $
† % T = 2 i tr X A A 2 − i tr A T2 X A + AX T
T = 2 i tr X A A 2 − A T2 A
T ! = 2 i tr X A A 2 − h.c. = 0.
10 Refer in particular to Thm. 4 in [25]. The Youla-form corresponds to the Schur-form w.r.t. unitary congruence transformations A → V AV T and is a generalization of Takagi’s factorization [27].
1080
C. B. Mendl, M. M. Wolf T
As this must hold for any Hermitian X , the last equation can only be fulfilled if A A 2 is Hermitian. It is instructive to rewrite the target function as follows, setting σ := σ1 , . . . , σd1 d2 . Denote the columns of V by vi , i.e. V = v1 | . . . |vd1 d2 , then A = i σi vi viT and 1 T tr A A 2 = σ | G σ , d1 d2 G = gi j Hermitian with (35) % $ T2 1 . gi j = tr v j v Tj vi viT d1 d2 Writing vi =: k |k ⊗ xik , xik ∈ K the last expression becomes $
# † %
# 1 T T l l| ⊗ x jl x jl k k| ⊗ xik xik gi j = tr d1 d2 k,k ,l,l 1 = tr K x jk x Tjl xik xil† d1 d2 k,l
# # 1 xik | x jl xil | x jk = d1 d2 k,l # 1 = tr si2j , si j := xik | x jl k,l=1,...,d . 1 d1 d2 # # ! V being unitary translates to tr si j = k xik | x jk = vi | v j = δi j . In what follows, we provide an explicit upper bound11 of (32) for d1 = d2 =: d and d = 3. Start with the Ansatz that all 2 × 2 blocks in D belong to the same phase, i.e. ⊗4 cos ϑ − sin ϑ ⊕ 1, ϑ ∈ [0, π/2] , so D= sin ϑ cos ϑ (36) 1
1 1 ! 2 16 cos ϑ − 7 = y, tr UU = 2 tr D D = d2 d 9 and set ⎛ ⎞ √i √1 − √i − √i 2 2 6 2 2 3 ( ⎜ ⎟ 3 1 ⎜ √i ⎟ √ − ⎜ 2 ⎟ 8 2 2 ⎜ ⎟ √1 √i ⎜ ⎟ 2 2 ⎜ ⎟ ( ⎜ i ⎟ 3 1 ⎜√ ⎟ √ − 8 ⎜ 2 ⎟ 2 2 ⎜ ⎟ i i 1 V =⎜ √ √ √ ⎟ 2 6 3⎟ ⎜ ⎜ ⎟ √1 √i ⎜ ⎟ 2 2 ⎜ ⎟ 1 i √ −√ ⎜ ⎟ 2 2 ⎜ ⎟ 1 i ⎜ ⎟ √ −√ ⎝ ⎠ 2 2 √1 − √i − √i − √i 2
2 6
2 2
3
11 Numeric tests strongly suggest that this is the actual minimum. Most interestingly, the acclaimed
minimizer V does not depend on ϑ!
Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem
1081
independent of ϑ! Then Ds = diag (σ1 , . . . , σ9 ) with σ1 = · · · = σ8 = cos ϑ, σ9 = 1, and G in (35) becomes ⎛ 3 ⎞ 9 0 − 11 0 0 −1 2 8 0 −8 0 ⎜ ⎟ 1 0 38 − 23 0 − 23 −1 ⎟ ⎜ 0 23 8 ⎜ ⎟ ⎜ 11 1 ⎟ 3 1 1 1 1 − −1 − −1 ⎜− 8 8 ⎟ 2 4 8 4 8 ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 − 41 23 − 43 0 − 23 0 −1 ⎟ ⎜ ⎟ 1 ⎜ 9 3 9×9 G= − 8 8 −1 − 43 23 83 − 43 83 −1 ⎟ ⎜ ⎟∈Q . 27 ⎜ ⎟ ⎜ 0 − 23 18 0 38 23 0 − 23 −1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 − 41 − 23 − 43 0 23 0 −1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 − 23 18 0 38 − 23 0 23 −1 ⎠ −1 −1 −1 −1 −1 −1 −1 −1 1 Finally evaluating the target function provides the supposed minimum 1 8 1 T2 2 σ − = | G σ = tr U U + 3 ϑ + 1) (cos s s d2 9 3
(37)
with ϑ defined by (36). Interestingly, the smallest eigenvalue − d12 of G is of algebraic multiplicity 1 with corresponding eigenvector σ (ϑ) evaluated at ϑ = π/3 and coordinates − 13 (1, 1). Furthermore, (37) is minimal w.r.t. ϑ exactly for ϑ = 0, which corresponds to maximal y = d12 tr UU = 1 and σ1 = · · · = σd 2 = 1. In this case, D is the identity and U = V V T complex symmetric. Comparing with Appendix VII D, notice that we obtain the same minimum value − 23 27 . T D. Minimizing tr U U 2 . We explore the following minimization problem posed by Eq. (25) in Sect. V B. ' / 1 T min (38) tr U U 2 : U ∈ B (H ⊗ K) unitary , d1 d2 where H ⊗ K is the tensor product of two Hilbert spaces with dimensions d1 = dim H and d2 = dim K, respectively, d1 being odd. The partial transposition is introduced in VII C. Note that any transformation
U → V T ⊗ W1† U V ⊗ W2 for unitary V ∈ B (H) and unitary W1 , W2 ∈ B (K) leaves the target function invariant. If we allowed tensor products only, i.e. U = U1 ⊗U2 , the target function would collapse to d11 tr U1 U1 ≥ −1 + d21 , which is in general strictly greater than (38), see below. It is worth mentioning that (38) is inherently asymmetric w.r.t. d1 ↔ d2 , as opposed to the previous Sect. VII C. Proposition 19. U U
T2
is Hermitian for every minimizer U of (38).
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Proof. As in previous sections, we differentiate the target function with respect to U . As U is unitary, X := 1i dU U † must be Hermitian, and we get d T T tr U U 2 = i tr XUU 2 − i tr U T1 (XU )† dU
T ! = i tr X UU 2 − h.c. = 0. This holds for any Hermitian X , so UU
T2
must be Hermitian, too.
Disassembly and reformulation. Let X = B (K) be the Hilbert space equipped with the Hilbert-Schmidt inner product and induced Frobenius norm. By partitioning U as U = i,d1j=1 |i j| ⊗ u i j with u i j ∈ X , we can now reformulate the target function (38) as # T tr U U 2 = u i j | u ji . i, j
The condition U unitary translates to † UU † = 1 ⇔ u i j u ik = δ jk 1 ∀ j, k = 1, . . . , d1 ⇔
i
# u i j | u ik x = δ jk 1 | x ≡ δ jk tr x ∀ j, k; x ∈ X
i
and the condition in Proposition 19 to UU
T2
Hermitian ⇔
⇔
u ki u i†j =
i
#
u i j | x u ki =
u ik u †ji
i
i
u ji | x u ik
#
∀ j, k; x ∈ X .
i
Note that these equations can be rewritten in terms of the Hilbert-Schmidt inner product as shown. Quaternion structure. In this paragraph we assume d2 = 2 and set d = d1 . Numeric tests suggest that in this case, the minimum value (38) is −1+
2 . d2
(39)
Interestingly, there emerges a substructure which is best described by quaternions. Recall that quaternions H = {x0 + x1 i + x2 j + x3 k : x0 , . . . , x3 ∈ R} are a non-abelian division ring and form a 4-dimensional normed division algebra over the real numbers. We regard R and C as subalgebras of H and denote the quaternionconjugate of q = x0 + x1 i + x2 j + x3 k ∈ H by q ∗ . Furthermore, define Re q := x0 and q := q − Re q = x1 i + x2 j + x3 k.
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To bridge the gap between quaternions and operators on Hilbert spaces, employ the identification12 i 0 0 1 0i , j ↔ i σy = , k ↔ i σx = . i ↔ i σz = 0 −i −1 0 i 0 Note that in this representation x0 + i x1 q ↔ qˆ = −x2 + i x3
x2 + i x3 x0 − i x1
∈ C2×2 ,
(40)
the quaternion conjugate is the Hermitian conjugate of the corresponding matrix, and the quaternion norm is the square root of the determinant, q ∗ ↔ qˆ † , q = det(q). ˆ Let Hd be the d-dimensional “vector space” over H with multiplication from the right, then each linear transformation on Hd can be represented by a d × d matrix A ∈ Hd×d . The identification (40) provides an algebra isomorphism between Hd×d ∼ = Rd×d ⊗ H and the complex 2d × 2d matrices consisting of 2 × 2 blocks (40); to obtain Ax, define u, v ∈ Cd by x =: u − j v and set xˆ = (u 1 , v1 , . . . , u d , vd )T ∈ C2d ; then Ax corresponds exactly to Aˆ x. ˆ In the following it will be clear from context which representation is employed. For all A ∈ Hd×d , the component-wise quaternion conjugate A∗ and the quaternion conjugate-transpose A† are intuitively translated to T2
A∗ ↔ Aˆ ,
A† ↔ Aˆ † .
Consequently, we say that A is Hermitian if A† = A. As in [28], call λ ∈ C, Im λ ≥ 0 an eigenvalue of A with the corresponding eigenvector x ∈ Hd if Ax = xλ. These are exactly the eigenvalues of Aˆ which have nonnegative imaginary part. Note that most of the well-known linear algebra theorems can be generalized straightforward to quaternions. Proposition 20. Let A ∈ Hd×d be Hermitian with eigenvalues −1 ± d, the algebraic d−1 multiplicity of (−1 + d) being d+1 2 and of −(1 + d) being 2 , respectively. Suppose A can be chosen such that Re A = 0, then there exists a unitary U ∈ B (H ⊗ K) with 1 2 T tr U U 2 = −1 + 2 . 2d d Proof. Since Re A = 0, we have A∗ = −A, and consequently A T = −A. Set U = d1 (1 + A), embedding Hd×d into Cd×d ⊗ C2×2 as described above, then U will be Hermitian and unitary since the eigenvalues satisfy λ(U ) = d1 (1 + (−1 ± d)) = ±1. T2
≡ d1 (1 + A∗ ) = d1 (1 − A), we get
1 1 1
1 2 T 2 2 = 1 − d tr 1 − A − 1 = −1 + 2 . tr U U 2 = 2 2 2d 2d d d d
Using U
The isomorphism (40) introduces an additional factor 2 into the trace, which cancels in the last equation. 12 We adhere here to a different convention than e.g. [28].
1 2
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Table 2. Numeric solutions of (38); the columns correspond to different values of d2 . The case d2 = 1 is treated analytically and is a special case of Sec. VII A. Note that d2 = 2 is in agreement with (39), and for d1 = d2 = 3 we obtain the same value as the minimum of (37) with respect to ϑ d1
d2 = 1
2
3
4
5
1
1
1
1
1
1
3
− 31
− 79
23 − 27
− 45
37 − 45
5
− 53 − 75
− 23 25 − 47 49
−0.92915 . . .
−1
−0.97632 . . .
7
Note that the conditions on A can be rephrased as follows. A ∈ Hd×d is Hermitian with Re A = 0 such that A2 + 2 A − (d 2 − 1)1 = 0. The requirement Re A = 0 implies the respective eigenvalue multiplicities via tr A = 0. To ensoul Proposition 20, we provide explicit examples of A meeting all requirements for d = 3 and d = 5, namely ⎞ ⎛ 0 −i j ⎟ ⎜ ⎜ i 0 −k ⎟ ⎟ ⎜ A = 2⎜ ⎟ ⎠ ⎝ −j k 0 and ⎛
0
√ −2i − 12 j
2k
−2 j
⎞
⎜ ⎟ ⎜ 2i 0 0 −2 j 4k ⎟ ⎜ ⎟ ⎜ ⎟ √ ⎜√ ⎟ ⎜ 12 j 0 ⎟ 0 − 12i 0 A=⎜ ⎟ ⎜ ⎟ ⎜ −2k 2 j √12i 0 −2i ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 2 j −4k 0 2i 0 T
respectively. Note that for these quaternion models, U is Hermitian, as well as U U 2 , in agreement with Proposition 19. Higher dimensions. Table 2 contains numerical results for different values of d1 and d2 . We have simply employed U = exp [i X ] with Hermitian X to represent unitary matrices. The local convergence error is about 10−6 , but it is still difficult to find the global minimizers. Quite remarkably, it seems that the lower bound −1 can be obtained for d1 = 5 and d2 = 4, even if we restrict to real orthogonal matrices. VIII. Appendix B–A Special Extremal Channel The following algebraic values for the coefficients µ1 , . . . , µ4 of X in (7) are appropriate; we have obtained them basically by guessing and suppose that at least polynomial
Unital Quantum Channels – Convex Structure and Revivals of Birkhoff’s Theorem
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degree 3 is required. µ1 = =
1/2 1 Root 1 −356 + 312x − 66x 2 + 3x 3 6 ⎛
√ 43 · 22/3 1 + i 3 √ 1/3 3 977 + 213 i 7 ⎞1/2
√ 1/3
977 + 213 i 7 √ ⎟ 1−i 3 ⎠ − 2/3 3·2
1 ⎜ 22 ⎝ − 6 3
. = 0.21821, µ2 = Root 1 −1 + 432x 2 + 2592x 3 √ √
√ 1/3 1 1+i 3 1
=− − 1 − i 3 1 + 3i 7
1/3 − √ 18 72 18 1 + 3 i 7 . = −0.14937, 1 µ3 = , 6 µ4 = Root 2 1 − 6x + 18x 3
√
√ 1/3 1 + i 3 −3 + i 7 =− − 6 · 22/3 . = 0.18595.
√ 3
√ 1/3 3 2 −3 + i 7 1−i
The eigenvalues of X as calculated by a computer algebra program are then √ / ' 1 1 1± α λ(X ) = 0, 0, , , 3 3 6 / ' 1 1 . = 0, 0, , , 0.23604, 0.097285 3 3 with
α = Root 1 −25957 + 163107x − 78003x 2 + 6561x 3
√
√ √ 1/3 2/3 1 + i 3 104 · 2 13 1 − i 3 67 + 23i 7 107 − =
1/3 − 2/3 √ 27 27 · 2 27 67 + 23i 7 . = 0.17329.
In particular, X is positive semidefinite with rank equal to 4. Similarly, the linear independence of (6) can be verified explicitly.
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References 1. Pérez-García, D., Wolf, M.M., Petz, D., Ruskai, M.B.: Contractivity of positive and trace-preserving maps under L p norms. J. Math. Phys. 47(8), 083506 (2006) 2. Arias, A., Gheondea, A., Gudder, S.: Fixed points of quantum operations. J. Math. Phys. 43(12), 5872–5881 (2002) 3. King, C.: Additivity for unital qubit channels. J. Math. Phys. 43, 4641–4653 (2002) 4. Fukuda, M.: Simplification of additivity conjecture in quantum information theory. Quant. Inf. Comp. 6, 179–186 (2007) 5. Fukuda, M., Wolf, M.M.: Simplifying additivity problems using direct sum constructions. J. Math. Phys. 48, 2101 (2007) 6. Rosgen, B.: Additivity and distinguishability of random unitary channels. J. Math. Phys. 49, 102107 (2008) 7. Tregub, S.L.: Bistochastic operators on finite-dimensional von-Neumann algebras. Soviet Math. 30, 105 (1986) 8. Kümmerer, B., Maassen, H.: The essentially commutative dilations of dynamical semigroups on Mn . Commun. Math. Phys. 109, 1–22 (1987) 9. Landau, L.J., Streater, R.F.: On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras. Lin. Alg. Appl. 193, 107–127 (1993) 10. Birkhoff, G.: Three observations on linear algebra. Univ. Nac. Tucuan. Revista A 5, 147–151 (1946) 11. Gregoratti, M., Werner, R.F.: Quantum lost and found. J. Mod. Opt. 50, 915–933 (2003) 12. Smolin, J.A., Verstraete, F., Winter, A.: Entanglement of assistance and multipartite state distillation. Phys. Rev. A 72, 052317 (2005) 13. Jamiolkowski, A.: Linear transformations which preserve trace and positive semidefiniteness of operators. Rep. Math. Phys. 3, 275–278 (1972) 14. Watrous, J.: Mixing doubly stochastic quantum channels with the completely depolarizing channel. http://arXiv.org/abs/0807.2668v1[quant-ph], 2008 15. Rudolph, O.: On extremal quantum states of composite systems with fixed marginals. J. Math. Phys. 45(11), 4035–4041 (2004) 16. Parthasarath, K.R.: Extremal quantum states in coupled systems. Ann. l’Inst. H. Poincaré (B) Prob. Stat. 41, 257–268 (2005) 17. Audenaert, K.M.R., Scheel, S.: On random unitary channels. New J. Phys. 10, 023011 (2008) 18. Choi, M.-D.: Completely positive linear maps on complex matrices. Lin. Alg. Appl. 10, 285–290 (1975) 19. Buscemi, F.: On the minimal number of unitaries needed to describe a random-unitary channel. Phys. Lett. A 360, 256–258 (2006) 20. Zeidler, E.: Applied Functional Analysis. Main Principles and Their Applications. New York: Springer, 1995 21. DiVincenzo, D.P., Fuchs, C.A., Mabuchi, H., Smolin, J.A., Thapliyal, A.V., Uhlmann, A.: Entanglement of assistance. In: Proc. Quantum Computing and Quantum Communications, First NASA Intl. Conf., Palm Springs, Berlin-Heidelberg-New York: Springer, 1999, pp. 247–257 22. Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002) 23. Werner, R.F., Holevo, A.S.: Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43, 4353–4357 (2002) 24. Vollbrecht, K.G.H., Werner, R.F.: Entanglement measures under symmetry. Phys. Rev. A 64, 062307 (2001) 25. Faßbender, H., Ikramov, Kh.D.: Some observations on the Youla form and conjugate-normal matrices. Lin. Alg. Appl. 422, 29–38 (2006) 26. Open problems in QIT. http://www.imaph.tu-bs.de/qi/problems/ 27. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge, Cambridge University Press, 1990 28. Bunse-Gerstner, A., Byers, R., Mehrmann, V.: A quaternion qr algorithm. Numer. Math. 55, 83–95 (1989) Communicated by M.B. Ruskai
Commun. Math. Phys. 289, 1087–1098 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0825-1
Communications in
Mathematical Physics
Universal Coding for Classical-Quantum Channel Masahito Hayashi Graduate School of Information Sciences, Tohoku University, Sendai, 980-8579, Japan. E-mail: [email protected] Received: 18 July 2008 / Accepted: 12 February 2009 Published online: 13 May 2009 – © The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract: We construct a universal code for a stationary and memoryless classicalquantum channel as a quantum version of the universal coding by Csiszár and Körner. Our code is constructed utilizing a combination of irreducible representations, a decoder introduced through the quantum information spectrum, and the packing lemma. 1. Introduction How to transmit information via a noisy communication channel is one of the most important problems for current information network systems. The first big step in this direction was Shannon’s channel coding theorem [1], in which he proved that there exists a code enabling reliable communication whose transmission rate is the capacity of the channel, i.e., the maximum of mutual information between the input and output systems. In his formulation, Shannon treated the channel as a stochastic matrix. In the present paper, we consider the ultimate transmission rate for sending classical messages, when the communication channel is given as a pair of a fixed optical fiber and a fixed modulator. In this case, the input system is described by a set X of classical alphabets, and the output system is described by a quantum system. Therefore, the channel is given as a map from a classical alphabet to a quantum state (i.e., a density matrix), which is called a classical-quantum channel. In contrast, a stochastic matrix is called a classical channel. When all output density matrices commute with each other, the original coding theorem of Shannon can be trivially extended to the quantum case. However, in the general case, there is a serious non-commutative difficulty for its quantum extension. Although it is not so difficult to extend the mutual information to this non-commutative quantum framework, there have been several obstacles to the establishment of the channel coding theorem, even for the classical-quantum channel. The crucial obstacle was first resolved by Holevo [2] and Schumacher-Westmoreland [3]. They showed that there exists a reliable code realizing transmission of the maximum value of quantum mutual information. In contrast, in 1970s studies by Holevo [4,5], it
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was shown that there does not exist a reliable code overcoming the maximum value of quantum mutual information. The combination of the additivity1 of the maximum of the mutual information and these results yields the capacity theorem for the classicalquantum channel. That is, it implies that this maximum value is equal to the maximum reliable transmission rate, which is called the capacity. After their achievement, Ogawa and Nagaoka [9] and Hayashi and Nagaoka [10] systematically constructed other codes which realized capacity transmission using the information spectrum method. However, since these existing codes depended on the form of the channel, they were not robust with respect to disagreements between the sender’s and receiver’s coordinate systems. In the classical system, Csiszár and Körner [11] constructed a universal channel coding, whose construction does not depend on the channel and depends only on the mutual information and the ‘type’ of the input system, i.e., the empirical distribution of code words. (The notion of type will be explained in Sect. 3.) Here, we should remark that a universal channel code can universally realize not the capacity but the mutual information because the constructed code is based on an (empirical) distribution on the input classical system whereas universal data compression can universally realize the minimum compression rate for both variable-length settings [12,13] and fixed-length settings [11]. In order to extend Csiszár-Körner’s universal coding to the quantum case, we have to overcome the non-commutative obstacle. Concerning the quantum system, Jozsa et al. [14] constructed a universal fixed-length source coding, which depended only on the compression rate and realized the minimum compression rate. Hayashi [15] discussed the exponentially decreasing rate of the decoding error. Further, Hayashi and Matsumoto [16] constructed a universal variable-length source coding for the quantum system. Hence, we can expect to establish a quantum version of universal channel coding. For example, even if the receiver cannot synchronize his coordinate system with the sender’s coordinate system, universal coding guarantees reliable communication. In the present paper, we construct a universal coding for a classical-quantum channel, which enables transmission of the quantum mutual information and which depends only on the coding rate and the ‘type’ of the input system. Unfortunately, the capacity cannot be attained universally because its construction depends on the distribution of the input system. In the proposed construction, the following three factors play essential roles for resolving the non-commutative obstacle. One is the decoder given by the proof of the information spectrum method. In the information spectrum method, the decoder is constructed by the square root measurement of the projectors given by the quantum analogue of the likelihood ratio between the signal state and the mixture state [10,17]. The second factor is the irreducible decomposition of the dual representation of the special unitary group and the permutation group, which is known as Schur-duality. The method of irreducible decomposition provides the universal protocols in the quantum setting [14,16,18–22]. However, even in the classical case, the universal channel coding requires the conditional type as well as the type [11]. In the present paper, we introduce a quantum analogue of the conditional type, which is the most essential part of the present paper. 1 Holevo [5] mentioned this type of additivity, whose proof is written in Fujiwara and Nagaoka [6] (Lemma 3) and Holevo [7]. While Fujiwara and Nagaoka [6] treats the case when the input set X is given as the set of density matrices of the input quantum system, their proof is valid even when the input set is given as an arbitrary finite set. This is because the key point is essentially shown by the chain rule of classical mutual information. Holevo [7] shows this kind of additivity in a more general setting, in which, he regards this kind of channel as a special case of a channel with a quantum input system.
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The third factor is the packing lemma, which yields a suitable combination of the signal states independent of the form of the channel in the classical case [11]. This method plays the same role in the present paper. An independent work, Bjelakovic and Boche [28], treats a code for a classicalquantum channel that universally realizes transmission of quantum mutual information. However, the result [28] is different from the present paper with respect to the following points. Firstly, the present paper explicitly gives the pair of the encoder and the decoder that universally attains transmission of maximal mutual information. Secondly, the present paper provides an upper bound (26) for the average error probability whose decreasing speed is exponential, whereas the paper [28] does not give such an upper bound. Thirdly, the present paper makes use of Schur-duality, which can be regarded as a kind of quantum extension of the method of type by Csiszár and Körner. This fact suggests that the proposed method can be applied to another topic in Csiszár and Körner. Further, our construction of encoder does not depend on the dimension of the output system. Only the decoder depends on the dimension of the output system. Note that Csiszár and Körner’s construction and Bjelakovic et al’s construction depend on the output system. The present paper employs Packing lemma in the construction of encoder as well as Csiszár and Körner. However, the present paper uses this lemma in a way different from Csiszár and Körner. Hence, even if the obtained result is restricted to the classical case, it contains a new result in this point. The remainder of the present paper is organized as follows. In Sect. 2, we explain the notation used herein and the main result including the existence of a universal coding for a classical-quantum channel. In this section, we present the exponential decreasing rate of the error probability of our universal code. In Sect. 3, the notation for group representation theory is presented and a quantum analogue of conditional type is introduced. In Sect. 4, we provide a code that works well universally. In Sect. 5, the exponentially decreasing rate mentioned in Sect. 2 is proven by using the property given in Sect. 3. 2. Main Result For the classical-quantum channel (see Fig. 1), we focus on the set of input alphabets X := {1, . . . , k} and the representation space H of the output system, whose dimension is d. Then, a classical-quantum channel is given as a map from X to the set of density matrices on H of the form i → W (i). The n-fold discrete memoryless extension is given as the map from X n to the set of density matrices on the n th tensor product system H⊗n . That is, this extension maps the input sequence i = (i 1 , . . . , i n ) to the state Wn (i n ) := W (i 1 ) ⊗ · · · ⊗ W (i n ). Sending the message {1, . . . , Mn } requires an encoder and a decoder. The encoder is given as a map ϕn from the set of messages {1, . . . , Mn } Mn to the set of alphabets X n , and the decoder is given by a POVM Y n = {Yin }i=1 . Thus,
Encoder
ϕn
Classical input
Modulator
Optical fiber
Quantum output
Classical-Quantum Channel
W Fig. 1. Figure of classical-quantum channel
Measurement (Decoder)
Yn
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the triplet n := (Mn , ϕn , Y n ) is called a code. Its performance is evaluated by the size |n | := Mn and the average error probability, given by ε[n , W ] :=
Mn 1 Tr Wn (ϕn (i))(I − Yin ). Mn i=1
The following theorem is known as the classical-quantum channel coding theorem. The optimal reliable transmission rate is equal to the capacity max I ( p, W ), p
k where the mutual information I ( p, W ) is defined for p = { pi }i=1 on the set of input alphabets X := {1, . . . , k} as
I ( p, W ) :=
k
pi Tr W (i)(log W (i) − log W p ),
i=1
W p :=
k
pi W (i).
i=1
As stated in the following main theorem, there exists a reliable code that depends only on the coding rate R and the distribution p on the input system when the coding rate R is smaller than the mutual information I ( p, W ). Note that this theorem does not imply the universal achievement of the capacity max p I ( p, W ) because our construction depends on the input distribution p. k on the set of input alphabets Theorem 1. For any distribution p = { pi }i=1 X := {1, . . . , k} and any real number R, there is a sequence of codes {n }∞ n=1 such that
lim
n→∞
φW, p (t) − t R −1 log ε[n , W ] ≥ max , 0≤t≤1 n 1+t 1 lim log |n | = R n→∞ n
for any classical-quantum channel W , where φW, p (t) is given by φW, p (t) := −(1 − t) log Tr
k
1 1−t
pi W (i)1−t
.
i=1
Note that the code {n }∞ n=1 does not depend on the channel W , and depends only on the distribution p and the coding rate R. The derivative of φW, p (t) is given as φW, p (0) = I ( p, W ).
When the transmission rate R is smaller than the mutual information I ( p, W ), φW, p (t) − t R >0 0≤t≤1 1+t max
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because there exists a parameter t ∈ (0, 1) such that φW, p (t) − t R > 0. That is, the average error probability ε[n , W ] goes to zero. This fact implies that without knowledge of the channel W we can mathematically construct a reliable code based only on the input distribution p when the coding rate R is smaller than the mutual information I ( p, W ). Therefore, in order to construct a code attaining the capacity max p I ( p, W ), we need to know only the value of argmax p I ( p, W ). We do not require complete knowledge of the classical-quantum channel i → W (i). For example, we may not be able to identify the coordinates of the output system, i.e., we cannot identify only the unitary U in the classical-quantum channel i → U W (i)U † ; however we are able to identify the maximizing input distribution argmax p I ( p, W ). In this case, we can construct a code that realizes the capacity max p I ( p, W ). Furthermore, the proposed code is robust with respect to small disturbances, in other words, our evaluation (26) of the average error probability guarantees reliable communication under the proposed code even if the true channel is a little different from the estimated channel. 3. Group Representation Theory In this section, we focus on the dual representation of the n-fold tensor product space by the special unitary group SU(d) and the n th symmetric group Sn .2 For this purpose, we focus on the Young diagram and the ‘type’. The former is a key concept in group representation theory and the latter is the corresponding notion in information theory [11]. When the vector of integers n = (n 1 , n 2 , · · · , n d ) satisfies the condition d n 1 ≥ n 2 ≥ · · · ≥ n d ≥ 0 and i=1 n i = n, the vector n is called a Young diagram (frame) of size n and depth d; the set of such vectors is denoted as Ynd . When the vector d of integers n satisfies the condition n i ≥ 0 and i=1 n i = n, the vector p = nn is called a ‘type’ of size n; the set of these vectors is denoted as Tnd . Further, for p ∈ Tnd , a subset of X n is defined by: T p := {x ∈ X n |The empirical distribution of x is equal to p}. The cardinalities of these sets are constrained as follows: |Ynd | ≤ |Tnd | ≤ (n + 1)d−1 , (n + 1)
−d n H ( p)
e
≤ |T p |,
(1) (2)
d where H ( p) := − i=1 pi log pi [11]. Using the Young diagram, the irreducible decomposition of the above representation can be characterized as follows: H⊗n = Un ⊗ Vn , (3) n∈Ynd
where Un is the irreducible representation space of SU(d) characterized by n, and Vn is the irreducible representation space of n th symmetric group Sn characterized by n. Here, the representation of the n th symmetric group Sn is denoted as V : s ∈ Sn → Vs . Hence, Eq. (3) gives the irreducible decomposition of the representation of the group 2 Christandl [23] contains a good survey of representation theory for quantum information.
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SU(d) × Sn , which is called Schur-duality. For n ∈ Ynd , the dimension of Un is evaluated by dim Un ≤ n
d(d−1) 2
.
(4)
Then, denoting the projection to the subspace Un ⊗ Vn as In , we define the following: ρn := ρU,n
1
In , dim Un ⊗ Vn 1 ρn . := |Ynd | d
(5) (6)
n∈Yn
Any state ρ and any Young diagram n ∈ Ynd satisfy the following: dim Un ρn ≥ In ρ ⊗n In . Thus, (1), (4), and (6) yield the inequality n
d(d−1) 2
|Ynd |ρU,n ≥ ρ ⊗n .
(7)
Next, we focus on two systems X and Y = {1, . . . , l}. When the distribution of X is given by a probability distribution p = ( p1 , . . . , pd ) on {1, . . . , d}, and the conditional distribution on Y with the condition on X is given by V , we denote the joint distribution on X × Y by pV and the distribution on Y by p · V . When the empirical distribution of x ∈ X n is ( nn1 , . . . , nnd ), the sequence of types V = (v 1 , . . . , v d ) ∈ Tnl1 × · · · × Tnld is called a conditional type for x [11]. We denote the set of conditional types for x by V (x, Y). For any conditional type V for x, we define the subset of Y n : The empirical distribution of , TV (x) := y ∈ Y n ((x1 , y1 ), . . . , (xn , yn )) is equal to pV . where p is the empirical distribution of x. We define the state ρ x for x ∈ X n . For this purpose, we consider a special element x = (1, . . . , 1, 2, . . . , 2, . . . , k, . . . , k ). The state ρ x is defined as ρ x := ρU,m 1 ⊗ m1
m2
mk
ρU,m 2 ⊗ · · · ⊗ ρU,m k . For a general element x ∈ X n , we choose a permutation s ∈ Sn such that x = s x . Then, we define a state ρ x by ρ x := Us ρ x Us† , where Us is the unitary representation of Sn . This state plays a similar role to the conditional type in the classical case. Using the inequality (7), we have n
kd(d−1) 2
|Ynd |k ρ x ≥ Wn (x).
(8)
As is shown here, the density matrix ρ x := ρU,m 1 ⊗ ρU,m 2 ⊗ · · · ⊗ ρU,m k commutes with ρU,n . For simplicity, we show commutativity between ρU,m 1 ⊗ρU,m 2 and ρU,m 1 +m 2 , first. In order to prove this fact, it is sufficient to show the existence of a resolution of the identity by the projections {E i }i such that ∃{ai }, ρU,m 1 ⊗ ρU,m 2 = ai E i , (9) ∃{bi }, ρU,m 1 +m 2 =
i
i
bi E i .
(10)
Universal Coding for Classical-Quantum Channel
1093
When the resolution {E i1 } of the identity is given as projections to the irreducible spaces of the representation of the group SU(d) × Sm 1 +m 2 , the resolution {E i1 } satisfies the condition (10) because of the construction of ρU,m 1 +m 2 . Similarly, the resolution {E i2 } of the identity is given as projections to the irreducible spaces of the representation of the group SU(d) × SU(d) × Sm 1 × Sm 2 = (SU(d) × Sm 1 ) × (SU(d) × Sm 2 ), and the resolution {E i2 } satisfies the condition (9). The group SU(d) × Sm 1 × Sm 2 is a subgroup of SU(d) × Sm 1 +m 2 , and it is also a subgroup of SU(d) × SU(d) × Sm 1 × Sm 2 via the correspondence (g, s1 , s2 ) → (g, g, s1 , s2 ). Now, the resolution {E 3j } j∈J of the identity is given as projections to the irreducible spaces of the representation of the group SU(d) × Sm 1 × Sm 2 . For any E i1 ∈ {E i1 }, there is a subset Ji of J such that E i1 = j∈Ji E 3j . The same fact holds for {E i2 }. Therefore, the resolution {E 3j } j∈J satisfies (10) and (9). Thus, the density matrix ρU,m 1 ⊗ ρU,m 2 commutes with ρU,m 1 +m 2 . Applying the same discussion to the group SU(d) × Sm 1 × Sm 2 × · · · × Sm k , we can show that ρ x := ρU,m 1 ⊗ ρU,m 2 ⊗ · · · ⊗ ρU,m k commutes with ρU,n . This property is essential for the construction of the proposed decoder. 4. Construction of the Code According to Csiszár and Körner [11], the proposed code is constructed as follows. The main point of this section is to establish that Csiszár-Körner’s Packing lemma provides a code whose performance is essentially equivalent to the average performance of random coding in the sense of (12). In the following discussion, we treat the conditional type in the case when the system Y coincides with the other system X . Lemma 1. For a positive number δ > 0, a type p ∈ Tnd , and a real positive number R < H ( p), there exist Mn := en(R−δ) distinct elements Mn := {x 1 , . . . , x Mn } ⊂ T p such that their empirical distributions are p and |TV (x) ∩ (Mn \ {x})| ≤ |TV (x)|e−n(H ( p)−R) for x ∈ Mn ⊂ T p and V ∈ V (x, X ). This lemma can be shown by substituting the identical map into Vˆ in Lemma 5.1 in Csiszár and Körner [11], which is known as the Packing lemma. Since Csiszár and Körner proved Lemma 5.1 using the random coding method, we can replace δ by √1n . √
That is, there exist Mn := en R− n distinct elements Mn := {x 1 , . . . , x Mn } ⊂ T p such that their empirical distributions are p and |TV (x) ∩ (Mn \ {x})| ≤ |TV (x)|e−n(H ( p)−R)
(11)
for x ∈ Mn ⊂ T p and V ∈ V (x, X ). Note that this encoder Mn does not depend on the output system because the employed Packing lemma treats the conditional types from the input system to the input system. Now, we transform the property (11) to a more useful form. Using the encoder Mn , we can define the distribution PMn as 1 x ∈ Mn pMn (x) = |Mn | 0 x∈ / Mn .
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M. Hayashi
For any x ∈ X n , we define an invariant subgroup Sx ⊂ Sn : Sx := {s ∈ Sn |s(x) = x}. Since x ∈ T p implies that pn (x ) = e−n H ( p) , any element x ∈ TV (x) ∩ Mn ⊂ T p satisfies 1 1 |TV (x) ∩ (Mn \ {x})| |TV (x) ∩ Mn | pMn ◦ s(x ) = · = |Sx | |TV (x)| |Mn | |TV (x)||Mn | s∈Sx ≤ e−n H ( p) e
√
n
= pn (x )e
√
n
(12)
when the conditional type V is not identical. Relation (12) holds for any x (= x) ∈ Mn because there exists a conditional type V such that x ∈ TV (x) and V is not identical. Next, for any x ∈ X n and any real number Cn , we define the projection P(x) := {ρ x − Cn ρU,n ≥ 0}, where {X ≥ 0} presents the projection i:xi ≥0 E i for a Hermitian matrix X with the diagonalization X = i xi E i . Remember that the density matrix ρ x commutes with the other density matrix ρU,n . Using the projection P(x), we define the decoder:
−1 −1 Y x := P(x) P(x ) P(x) . x ∈Mn
x ∈Mn √
In the following, the above-constructed code (en R− U,n ( p, R).
n , M , {Y } n x x ∈Mn )
is denoted by
5. Exponential Evaluation Hayashi and Nagaoka [10] showed that I − Y x ≤ 2(I − P(x )) + 4
P(x).
x (= x )∈Mn
Then, the average error probability of U,n ( p, R) is evaluated by 1 Tr Wn (x )(I − Y x ) |Mn | x ∈Mn
2 4 Tr Wn (x )(I − P(x )) + Tr Wn (x ) |Mn | |Mn |
≤
x ∈Mn
2 Tr Wn (x)(I − P(x)) |Mn | x ∈Mn ⎛ ⎡ 1 P(x) ⎝ + 4 Tr ⎣ |Mn |
x ∈Mn
P(x)
x (= x )∈Mn
=
x ∈Mn
x (= x )∈Mn
⎞⎤ Wn (x )⎠⎦ .
(13)
Universal Coding for Classical-Quantum Channel
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Since the density matrix ρ x commutes with the density matrix ρU,n , we have t (I − P(x)) = {ρ x − Cn ρU,n < 0} ≤ ρ x−t Cnt ρU,n
(14)
for 0 ≤ t ≤ 1. Since the density matrix ρ x commutes with the density matrix Wn (x), Wn (x)ρ x−t is a Hermite matrix and (8) implies that Wn (x)ρ x−t ≤ n
ktd(d−1) 2
|Ynd |kt Wn (x)1−t .
(15)
Using (14) and (15), we have t Tr Wn (x)(I − P(x)) ≤ Tr Wn (x)ρ x−t ρU,n Cnt
≤n
ktd(d−1) 2
t |Ynd |kt Cnt Tr Wn (x)1−t ρU,n .
(16)
Since the quantity Tr Wn (x)(I − P(x)) is invariant with respect to the action of the permutation and the relation (2) implies that (n + 1)−d |T p |
pn (x) = e−n H ( p) ≥
(17)
for x ∈ T p , we obtain Tr Wn (x)(I − P(x)) =
≤ (n + 1)d
1 Tr Wn (x )(I − P(x )) |T p | x ∈T p
pn (x ) Tr Wn (x )(I − P(x ))
x ∈X n
≤ (n + 1)d n
ktd(d−1) 2
|Ynd |kt Cnt Tr(
(18)
t pn (x )Wn (x )1−t )ρU,n
x ∈X n
d+ ktd(d−1) 2
≤ (n + 1)
|Ynd |kt Cnt max Tr σ
⎛ ≤ (n + 1)d+
ktd(d−1) 2
⎜ |Ynd |kt Cnt ⎝Tr
(19)
⊗n p(x)Wn (x)1−t
σt
x∈X
1 ⊗n 1−t
p(x)Wn (x)1−t
⎞1−t ⎟ ⎠
(20)
x∈X
⎛ d+ ktd(d−1) 2
= (n + 1)
|Ynd |kt Cnt ⎝Tr
1 1−t
p(x)Wn (x)1−t
⎞n(1−t) ⎠
x∈X
= (n + 1)
d+ ktd(d−1) 2
|Ynd |kt Cnt e−nφW, p (t) ,
(21)
where (18) and (19) follow from (17) and (16), respectively. The inequality (20) can be checked in the following way: When X is a positive semi-definite matrix, σ is a density matrix, 0 ≤ t ≤ 1, p = 1/(1 − t), and q = 1/t, the Hölder inequality implies that 1
1
1
Tr X σ t ≤ Tr |X σ t | ≤ (Tr X p ) p (Tr σ tq ) q = (Tr X 1−t )1−t because
1 p
+
1 q
= 1. This inequality yields (20).
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M. Hayashi
Next, we evaluate the second term of (13) using the invariant property of Sx : ⎞⎤ ⎡ ⎛ 1 Wn (x )⎠⎦ Tr ⎣ P(x) ⎝ |Mn | x (= x )∈Mn ⎡ ⎤ = Tr ⎣ P(x) pMn (x )Wn (x )⎦ ⎡
x (= x )∈Mn
⎤ 1 = Tr ⎣ P(x) pMn (x )Vs Wn (x )Vs∗ ⎦ |Sx | s∈Sx x (= x )∈Mn ⎡ ⎤ 1 = Tr ⎣ P(x) pMn ◦ s −1 (x )Wn (x )⎦ |S | x x (= x )∈Mn s∈Sx ⎡ ⎤ √ ≤ Tr ⎣ P(x) pn (x )e n Wn (x )⎦
(22)
x (= x )∈Mn
Tr P(x)W ⊗n p √ d(d−1) ≤ e n Tr P(x)n 2 |Ynd |ρU,n √ d(d−1) ≤ e n Tr P(x)n 2 |Ynd |Cn−1 ρ x d(d−1) √ √ d(d−1) ≤ e n Tr n 2 |Ynd |Cn−1 ρ x = e n n 2 |Ynd |Cn−1 ,
=e
√
n
(23) (24) (25)
where (22), (23), and (24) follow from (12), (7), and the inequality P(x)(ρU,n − Cn−1 ρ x ) ≤ 0. √ For any t ∈ (0, 1) and R > 0, we choose |Mn | := en R− n , Cn := en(R+r (t)) , and φ (t)−t R r (t) := W, p1+t . Since r (t) = φW, p (t) − t (R + r (t)), from (13), (21) and (25), the average error probability can be evaluated as ε(U,n ( p, R), W ) ≤ 2(n + 1)d+
ktd(d−1) 2
|Ynd |kt e−n(φW, p (t)−t (R+r (t))) + 4n
d(d−1) 2
|Ynd |e−nr (t) .
(26)
Then, its exponentially decreasing rate is characterized by φW, p (t) − t R −1 log ε(U,n ( p, R), W ) ≥ min{φW, p (t) − t (R + r (t)), r (t)} = . n→∞ n 1+t lim
That is, when we choose t0 := argmaxt∈(0,1) Cn := en(R+r (t0 )) , we obtain lim
n→∞
φW, p (t)−t R , 1+t
φW, p (t) − t R −1 log ε(U,n ( p, R), W ) ≥ max t∈(0,1) n 1+t
for any channel W . Therefore, we obtain Theorem 1.
√
|Mn | := en R−
n,
and
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6. Discussion We have constructed a universal code realizing the transmission of quantum mutual information by combining the information spectrum method with group representation theory and using the Packing lemma. The code that we have developed works well because any tensor product state ρ ⊗n is close to the state ρU,n . Indeed, Krattenthaler and Slater [24] demonstrated the existence of a state σn such that n1 D(ρ ⊗n σn ) → 0 for any state ρ in the qubit system as a quantum analogue of Clarke and Barron’s result [25]. Its d-dimensional extension is discussed in another paper [26]. Further, Hayashi [27] derived another exponentially decreasing rate of error probability for a classical-quantum channel, which is maxt:0≤t≤1 −(log i pi Tr[W (i)1−t W tp ])− t R. Since φW, p (t)−t (R+r (t)) 1+t e− = e−(φW, p (t)−t (R+r (t))) = et (R+r (t)) max Tr( pi W (i)1−t )σ t σ
≥e
tR
Tr(
pi W (i)
i
1−t
)(
pi W (i)) = e t
−(−(log
i
i pi Tr[W (i)1−t W tp ])−t R)
,
i
we obtain max −(log
t:0≤t≤1
i
pi Tr[W (i)1−t W tp ]) − t R ≥ max
t:0≤t≤1
φW, p (t) − t R . 1+t
That is, the obtained exponentially decreasing rate is smaller than that of Hayashi [27]. However, according to Csiszár and Körner [11], the exponentially decreasing rate of the universal coding is the same as the optimal exponentially decreasing rate in the classical case when the rate is close to the capacity. Hence, if a more sophisticated analysis were to be applied, a better exponentially decreasing rate could be expected. Such an analysis is left as a future problem. The proposed encoder does not depend on the output system. Such a construction is realized by employing the Packing lemma in a way different from that of Csiszár and Körner. In the present paper, the Packing lemma treats the conditional types from the input system to the input system. We hope that such a style of application of the Packing lemma yields another new result on information theory in the future. Acknowledgement. This research was partially supported by a Grant-in-Aid for Scientific Research on Priority Area ‘Deepening and Expansion of Statistical Mechanical Informatics (DEX-SMI)’, No. 18079014 and a MEXT Grant-in-Aid for Young Scientists (A) No. 20686026. The author thanks the referees and the editor for helpful comments concerning this manuscript. He also acknowledges Professor Hiroshi Nagaoka for an interesting discussion. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
References 1. Shannon, C.E.: A mathematical theory of communication. Bell System Technical Journal 27, 623–656 (1948) 2. Holevo, A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inform. Theory 44, 269–273 (1998)
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3. Schumacher, B., Westmoreland, M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997) 4. Holevo, A.S.: Bounds for the quantity of information transmitted by a quantum communication channel. Probl. Inform. Transm. 9, 177–183 (1973) 5. Holevo, A.S.: On the capacity of quantum communication channel. Probl. Inform. Transm. 15(4), 247–253 (1979) 6. Fujiwara, A., Nagaoka, H.: Capacity of a memoryless quantum communication channel. Mathematical Engineering Technical Reports, METR 94-22, the University of Tokyo, http://www.keisu.t.u-tokyo.ac.jp/ research/techrep/1994.html, 1994 7. Holevo, A.S.: Quantum Coding Theorems. Russ. Math. Surv. 53:6, 1295–1331 (1999); Coding Theorems for quantum channels, http://arXiv.org/abs/quant-ph/9809023, 1998 8. Hayashi, M.: Quantum Information: An Introduction. Berlin: Springer, 2006 9. Ogawa, T., Nagaoka, H.: Making good codes for classical-quantum channel coding via quantum hypothesis testing. IEEE Trans. Inform. Theory 53, 2261–2266 (2007) 10. Hayashi, M., Nagaoka, H.: General formulas for capacity of classical-quantum channels. IEEE Trans. Infor. Theory 49, 1753–1768 (2003) 11. Csiszár, I., Körner, J.: Information Theory: Coding Theorems for Discrete Memoryless Systems. London-New York: Academic Press, 1981 12. Lynch, T.J.: Sequence time coding for data compression. Proc. IEEE 54, 1490–1491 (1966) 13. Davisson, L.D.: Comments on Sequence time coding for data compression. Proc. IEEE 54:2010, 1966 14. Jozsa, R., Horodecki, M., Horodecki, P., Horodecki, R.: Universal quantum information compression. Phys. Rev. Lett. 81, 1714 (1998) 15. Hayashi, M.: Exponents of quantum fixed-length pure state source coding. Phys. Rev. A 66, 032321 (2002) 16. Hayashi, M., Matsumoto, K.: Quantum universal variable-length source coding. Phys. Rev. A 66, 022311 (2002) 17. Verdú, S., Han, T.S.: A general formula for channel capacity. IEEE Trans. Inform. Theory 40, 1147–1157 (1994) 18. Hayashi, M.: Asymptotics of quantum relative entropy from a representation theoretical viewpoint. J. Phys. A: Math. and Gen. 34, 3413–3419 (2001) 19. Keyl, M., Werner, R.F.: Estimating the spectrum of a density operator. Phys. Rev. A 64, 052311 (2001) 20. Hayashi, M.: Optimal sequence of POVMs in the sense of Stein’s lemma in quantum hypothesis. J. Phys. A: Math. and Gen. 35, 10759–10773 (2002) 21. Bjelakovi´c, I., Deuschel, J.-D., Kruger, T., Seiler, R., Siegmund-Schultze, R., Szkoła, A.: A quantum version of sanov’s theorem. Commun. Math. Phys. 260, 659–671 (2005) 22. Matsumoto, K., Hayashi, M.: Universal distortion-free entanglement concentration. Phys. Rev. A 75, 062338 (2007) 23. Christandl, M.: The structure of bipartite quantum states - insights from group theory and cryptography. PhD thesis, February, University of Cambridge, http://arxiv.org/abs/quant-ph/0604183, 2006 24. Krattenthaler, C., Slater, P.: Asymptotic redundancies for universal quantum coding. IEEE Trans. Inform. Theory 46, 801–819 (2000) 25. Clarke, B.S., Barron, A.R.: Information-theoretic asymptotics of Bayes methods. IEEE Trans. Inform. Theory 36, 453–471 (1990) 26. Hayashi, M.: Universal approximation of multi-copy states and universal quantum lossless data compression. http://arxiv.org/abs/:0806.1091v2[quant-ph], 2008 27. Hayashi, M.: Error exponent in asymmetric quantum hypothesis testing and its application to classicalquantum channel coding. Phys. Rev. A 76, 062301 (2007) 28. Bjelakovic, I., Boche, H.: Classical capacities of averaged and compound quantum channels. http://arxiv. org/abs/0710.3027v2[quant-ph], 2009 Communicated by M.B. Ruskai
Commun. Math. Phys. 289, 1099–1129 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0788-2
Communications in
Mathematical Physics
Boundary Conditions for Coupled Quasilinear Wave Equations with Application to Isolated Systems H.-O. Kreiss1,2 , O. Reula3,4 , O. Sarbach5 , J. Winicour2,6 1 NADA, Royal Institute of Technology, 10044 Stockholm, Sweden 2 Albert Einstein Institute, Max Planck Gesellschaft, Am Mühlenberg 1,
D-14476 Golm, Germany
3 FaMAF Universidad Nacional de Córdoba, Córdoba, Argentina 5000 4 Instituto de Física Enrique Gaviola, CONICET, Córdoba, Argentina 5 Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,
Edificio C-3, C.P. 58040 Morelia, Michoacán, Mexico. E-mail: [email protected]
6 Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh,
Pennsylvania 15260, USA Received: 21 July 2008 / Accepted: 8 December 2008 Published online: 18 April 2009 – © Springer-Verlag 2009
Abstract: We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form [0, T ] × , where is a compact manifold with smooth boundaries ∂. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on ∂. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell’s equations in the Lorentz gauge and Einstein’s gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems. I. Introduction and Main Results Motivated in part by the numerical computation of spacetimes on a finite domain with artificial boundaries, the initial-boundary value problem (IBVP) in general relativity has started to receive a lot of attention during the last few years (see [1] for a review). A well posed IBVP for Einstein’s vacuum field equations was formulated for the first time by Friedrich and Nagy [2] based on tetrad fields and the theory of quasilinear, symmetric hyperbolic systems with maximal dissipative boundary conditions [3–5]. More recently, Kreiss and Winicour [6] formulated a well posed IBVP for the harmonic gauge formulation of the Einstein vacuum equations which casts the field equations into a set of ten coupled quasilinear wave equations subject to four constraints. There are two key ideas behind the result of [6]. The first one is the realization that the wave equations, when viewed as first order pseudodifferential equations, have a non-characteristic boundary matrix. This allows application of the boundary value theory for such systems developed by Kreiss in the 1970’s [7]. The second idea is the formulation of boundary conditions for the frozen coefficient form of the harmonic Einstein equations which ensures constraint propagation and satisfies the estimates required by the Kreiss theory. The well posedness
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H.-O. Kreiss, O. Reula, O. Sarbach, J. Winicour
of the system and the generalization to the quasilinear case can then be established using the theory of pseudodifferential operators (see, for instance, [8]). In a subsequent paper [9], similar results were obtained via more mundane energy estimates which follow by integration by parts, without resort to the pseudodifferential calculus. For this, a non-standard energy norm is constructed which is based upon the choice of a particular time-like direction adapted to the boundary conditions being imposed. With respect to this energy the Kreiss-Winicour boundary conditions are maximal dissipative and so standard well posedness theorems apply even in the quasilinear case [5,10]. Besides being a simpler proof, or at least a proof that can be followed completely by a reader not familiar with the pseudodifferential techniques, it implies similar results for the stability of finite difference approximations to Einstein’s equations in the harmonic gauge. This follows from considering the semidiscrete system of ordinary differential equations in time obtained by substituting finite differences for spatial derivatives. If the semidiscrete system is stable, then for appropriate time discretizations the fully discrete system is guaranteed to be stable [11]. The stability of the semidiscrete system can be established by the use of finite difference operators satisfying summation by parts [12], the counterpart of integration by parts, by mimicking the steps leading to the continuum energy estimate. A summation by parts algorithm based upon the standard energy norm for the harmonic Einstein problem was developed in [13] and verified to be stable in numerical tests [14]. The non-standard energy norm employed here and in [9] provides the basis to formulate a summation by parts algorithm whose numerical stability follows from established theory. In this paper we present a more general and geometric version of the foregoing results which applies to coupled systems of quasilinear wave equations with a certain class of boundary conditions. The strong well posedness of the resulting IBVP is established by reducing the wave system to first order symmetric hyperbolic equations subject to maximal dissipative boundary conditions. This allows us to identify the structure in first order systems which can be used to establish boundary stability. This structure arises from the non-absolute nature of time in Lorentzian physics, whereby a Lorentz boost gives rise to a new conserved energy and so to a different symmetrizer.1 Realizing this, we are able to restate and prove our earlier results in terms of standard maximal dissipative boundary conditions for symmetric hyperbolic systems. As we show, our class of boundary conditions is sufficiently flexible for obtaining well posed IBVP formulations for different models of isolated systems in physics, including the wave equation, Maxwell’s equations and the Einstein field equations. In what follows, we present the main mathematical result in Sect. I A with two applications in Sects. I B and I C. The corresponding proof of strong well posedness is given in Sects. II and III. We then show in Sect. IV that these results can be applied to electromagnetic and gravitational theory to formulate boundary conditions of practical value for the numerical treatment of isolated systems. A. Main theorem. Let T > 0, and denote by a d-dimensional compact manifold with smooth boundaries ∂. The type of system our results apply to is a set of quasilinear wave equations on M = [0, T ] × coupled both by lower order terms and in the principal part, by a change in the characteristic directions via a metric which can depend on the local value of the fields involved. More precisely, let π : E → M be a vector bundle 1 In theories such as hydrodynamics, the four-velocity determines a preferred time direction and thereby a unique symmetrizer.
Boundary Conditions for Coupled Quasilinear Wave Equations
1101
over M with fibre R N , let ∇a be a fixed, given connection on E and let gab = gab () be a Lorentz metric on M with inverse g ab () which depends pointwise and smoothly on a set of fields = { A } A=1,2,...N parameterizing a local section of E. Our signature convention for gab is (−, +, . . . , +). We shall also assume that each time-slice t = {t} × is space-like and that the boundary T = [0, T ] × ∂ is time-like with respect to gab (). In the following, we will refer to local sections in E as vector-valued functions over M. We will also assume the existence of a positive-definite fibre metric h AB on E. We consider a system of quasilinear wave equations of the form g ab ()∇a ∇b A = S A (, ∇),
(1)
where S A (, ∇) is a vector-valued function which depends pointwise and smoothly on its arguments. The wave system (1) is subject to the initial conditions n b ∇b A = 0A , (2) A = 0A , 0
0
where 0A and 0A are given vector-valued functions on 0 , and where n b = n b () denotes the future-directed unit normal to 0 with respect to gab . In order to describe the boundary conditions, let T a = T a ( p, ) be a future-directed vector field which is tangent to T and which is normalized with respect to gab and let N a = N a ( p, ) be the unit outward normal to T with respect to the metric gab . We consider boundary conditions on T of the following form2 : T b + α N b ∇b A = ca A B ∇a B + d A B B + G A , (3) T
T
T
where α = α( p, ) > 0 is a strictly positive, smooth function, G A = G A ( p) is a given, vector-valued function on T and the matrix coefficients ca A B = ca A B ( p, ) and d A B = d A B ( p, ) are smooth functions of their arguments. Furthermore, we assume that ca A B can be made arbitrarily small in the following sense: Given a local trivialization ϕ : U × R N → π −1 (U ) of E such that U¯ ⊂ M is compact and contains a portion U of the boundary T , and given ε > 0, there exists a smooth map J : U → G L(N , R), p → (J A B ( p)) such that the transformed matrix coefficients D c˜a A B := J A C ca C D J −1 B satisfy the condition h AB c˜a A C ()c˜b B D ()Va C Vb D ≤ εh AB eab ()Va A Vb B ,
(4)
for all vector-valued one-forms VaA on U, where here and in the following, eab refers to the Euclidean metric eab = gab + 2Ta Tb which is defined for points on T . The main result of this paper is: Theorem 1. The IBVP (1,2,3) is well posed. Given T > 0 and sufficiently small and smooth initial and boundary data 0A , 0A and G A satisfying the usual compatibility conditions at ∂0 , there exists a unique smooth solution on M satisfying the evolution equation (1), the initial condition (2) and the boundary condition (3). Furthermore, the solution depends continuously on the initial and boundary data. 2 We adopt the Einstein summation convention for the lower case Latin abstract spacetime indices a, b, c, ... as well as for the Capital indices A, B, C, ... on the fibre of E.
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A common situation in which condition (4) is automatically satisfied is given in the following: Lemma 1. Let U ⊂ T be an open and bounded subset of T . Assume there exists a smooth map J : U → G L(N , R), p → (J A B ( p)) over U such that the transformed D matrix coefficients c˜a A B := J A C ca C D J −1 B are in upper triangular form with zeroes on the diagonal, that is c˜a A B = 0,
B ≤ A.
Then, the condition (4) is satisfied on U. Proof (cf. The proof of the Liapunov stability theorem). In order to simplify the notation we use a matrix notation and write c˜a = J ca J −1 . Let δ > 0, and define Dδ := diag(1, δ, δ 2 , ..., δ N −1 ) and Jδ := Dδ−1 J . Then, cδa := Jδ ca Jδ−1 = Dδ−1 c˜a Dδ has the components (cδa ) A B = δ B−A c˜a A B , where here, δ B−A refers to the (B − A)th power of δ. Since c˜a A B = 0 for B ≤ A we have cδa = O(δ), and cδa satisfies condition (4) provided δ > 0 is chosen small enough. The proof of Theorem 1 is given in Sects. II and III. In order to illustrate the ideas on a simpler example, we start in Sect. II with the wave equation on a fixed background metric gab , and analyze the general case in Sect. III. Since many physical systems can be described by systems of wave equations, Theorem 1 should have many applications. In the following, we mention two such applications for the initial-boundary value formulation of isolated systems with constraints. The physical motivation for the choice of nonreflecting boundary conditions in these examples is described in detail in Sect. IV. B. Maxwell’s equations in the Lorentz gauge. The first application describes an electromagnetic field on the manifold M = [0, T ] × with a fixed background metric gab and corresponding Levi-Civita connection ∇a . As before, we assume that each time-slice t = {t} × is space-like and that the boundary T = [0, T ] × ∂ is time-like. In the Lorentz gauge C := ∇b Ab = 0, where Ab denotes the 4-vector potential, Maxwell’s equations assume the form of a system of wave equations, g ab ∇a ∇b Ac = R c d Ad − J c ,
(5)
where Rab denotes the Ricci tensor belonging to the metric gab and J c is the four-current. Equation (5) implies that the constraint variable C obeys the following equation: g ab ∇a ∇b C = −∇ c Jc .
(6)
Therefore, the imposition of the boundary condition C|T = 0 and the satisfaction of the continuity equation ∇ c Jc = 0 imply that any smooth enough solution of (5) with initial data satisfying n a ∇a C = 0, C|0 = 0, 0
satisfies the constraint C = 0 on M since in this case the constraint propagation system (6) is homogeneous.
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1103
Asymptotically nonreflecting boundary conditions at T = [0, T ] × , in the sense of Sect. IV, can be formulated by first introducing a null tetrad {K a , L a , Q a , Q¯ a } which is adapted to the boundary. Let T a be a future-directed time-like vector field tangent to T normalized such that gab T a T b = −1. For example, one can define T a by orthogonal projection of the future-directed normal to the time-slices t onto T . Next, let N a denote the unit outward normal to T with respect to gab and complete T a and N a to an orthonormal basis {T a , N a , V a , W a } of T p M at each point p ∈ T . Then, we define the null vectors K a := T a + N a ,
L a := T a − N a ,
Q¯ a := V a − i W a ,
Q a := V a + i W a ,
√ where i = −1. Finally, let r denote the areal radius of the cross sections ∂t . The following boundary conditions are motivated from the considerations in Sect. IV B: 2 b K K b ∇a A + K b A = q K , r T a a b K Q b − Q K b ∇a A = q Q , T a a a ¯ a ¯ K L b + L K b − Q Q b − Q Q b ∇a Ab a
b
(7) (8) T
= 0,
(9)
where q K and q Q are given real and complex scalars on T . The first condition is a gauge condition, the second condition controls the electromagnetic radiation through T and the third condition enforces the constraint C = g ab ∇a Ab = 0 on T . For the special case of a flat background with a spherical boundary, these boundary conditions reduce to the ones proposed in Sect. IV B which are shown to yield small spurious reflections. Therefore, we expect them to yield small spurious reflections also in the case of asymptotically flat curved spacetimes as long as the boundary is nearly spherical and located far into the wave zone. The evolution equation (5) has the form (1) where E is the tangent bundle over M, and the boundary conditions (7,8,9) have the form (3) with α = 1,
1 (a ¯ c) 2Q Q K d + L a K c K d − K c Q a Q¯ d + Q¯ a Q d , 2 1 c c G = −L q K + Q¯ c q Q + Q c q¯ Q . 2
ca c d =
dcd =
1 c L Kd , r
Since ca c d K d = 0, ca c d Q d = −Q a K c , ca c d Q¯ d = − Q¯ a K c , ca c d L d = −L a K c − Q¯ a Q c − Q a Q¯ c , the matrix elements ca c d are in upper triangular form with zeroes in the diagonal when expressed in terms of the basis {K a , Q a , Q¯ a , L d }. Therefore, the assumptions of Lemma 1 are satisfied and we obtain a well posed IBVP.
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H.-O. Kreiss, O. Reula, O. Sarbach, J. Winicour
C. Einstein’s equations in harmonic coordinates. As a second application of our theorem we consider Einstein’s field equations in (generalized) harmonic coordinates. For this, we follow [15,16] and choose a fixed background metric g˚ ab on M = [0, T ] × with the property that each time-slice t = {t} × is space-like and the boundary T = [0, T ] × ∂ is time-like with respect to g˚ ab . We impose the following gauge condition on the dynamical metric gab : c ˚c C c := g ab c ab −
ab − H = 0.
(10)
˚ c are the Christoffel symbols Here, H c is a given vector field on M and c ab and
ab corresponding to the dynamical and background metrics, respectively. In the particular case where H c = 0 and where the background metric is the Minkowski metric in ˚ c vanishes, and the condition C c = 0 reduces to the standard Cartesian coordinates,
ab usual condition for harmonic coordinates x µ = 0 for µ = t, x, y, z. However, the advantage of the condition (10) is that it maintains the covariance of the theory since C c is the difference between the two Christoffel symbols, ˚ c = 1 g cd ∇˚ h + ∇˚ h − ∇˚ h C c ab ≡ c ab −
ab a bd b ad d ab , 2
(11)
where h ab = gab − g˚ ab denotes the difference between the dynamical and the background metric. With the condition (10), Einstein’s field equations are equivalent to the wave system e g cd ∇˚ c ∇˚ d h ab = 2 ge f g cd C e ac C f bd + 4 C c d(a gb)e C e c f g d f − 2 g cd R˚ cd(a gb)e
1 +16π G Tab − gab g cd Tcd + 2 ∇(a Hb) , (12) 2 a where R˚ bcd denotes the curvature tensor with respect to g˚ ab , Tab the stress-energy tensor and G denotes Newton’s constant. Solutions of this equation which are smooth enough imply that the constraint variable Ca satisfies
g cd ∇c ∇d Ca = −Ra b Cb − 16π G∇ b Tab .
(13)
Therefore, the imposition of the boundary condition Ca |T = 0 implies that any smooth enough solution of (12) with initial data satisfying Ca |0 = 0, n a ∇a Cb = 0, 0
satisfies the constraint Ca = 0 on M provided the stress-energy tensor is divergence free, ∇ b Tab = 0. In order to formulate asymptotically nonreflecting boundary conditions we first construct an adapted local null tetrad {K a , L a , Q a , Q¯ a } as in the electromagnetic case. Notice that here these quantities are defined with respect to the dynamical metric gab and not the background metric g˚ ab and as a consequence, they depend on gab . However, it is important to note that these vectors do not depend on derivatives of gab . A radial function r on T is defined as the areal radius of the cross sections ∂t with
Boundary Conditions for Coupled Quasilinear Wave Equations
1105
respect to the background metric. The boundary conditions which are motivated from the considerations in Sect. IV C are the following: 2 K a K b K c ∇˚ a h bc + K b K c h bc = −q K K , (14) r T 1 1 K a K b L c ∇˚ a h bc + K b L c h bc + Q b Q¯ c h bc = −q Q Q¯ , (15) r r T 2 K a K b Q c ∇˚ a h bc + K b Q c h bc = −q K Q , (16) r T (17) K a Q b Q c ∇˚ a h bc − Q a Q b K c ∇˚ a h bc = −q Q Q , T K a Q b Q¯ c + L a K b K c − Q a K b Q¯ c − Q¯ a K b Q c ∇˚ a h bc = −2K a Ha T , (18) T a b c a b c a b c a b c ˚ ¯ K L Q + L K Q − Q K L + Q Q Q ∇a h bc = −2Q a Ha T , (19) T K a L b L c + L a Q b Q¯ c − Q a Q¯ b L c − Q¯ a Q b L c ∇˚ a h bc = −2L a Ha T , (20) T
where q K K and q Q Q¯ are real-valued given functions on T and q K Q and q Q Q are complex-valued given functions on T . The evolution equation (12) has the form (1) where E is the vector bundle of symmetric, covariant tensor fields on M and the boundary conditions (14–20) have the form (3), where α=1 and ca bc de is in upper triangular form when expressed in terms of the basis {K b K c ,K (b L c) ,K (b Q c) ,Q b Q c , Q (b Q¯ c) , L (b Q c) , L b L c }. For the case where g˚ ab is the Minkowski metric and h ab is treated as a linear perturbation thereof, the boundary conditions (14–17) reduce to the ones proposed in Sect. IV C for a spherical boundary. As in the preceding application to electrodynamics, we expect these boundary conditions to yield small spurious reflections in the case of a nearly spherical boundary in the wave zone of an asymptotically flat curved spacetime. Their content can be clarified by considering the case of a wave incident on a plane boundary. The discussion in Sect. IV A shows that the first three conditions (14),(15) and (16) are related to the gauge freedom; and the condition (17) controls the gravitational radiation. The remaining conditions (18),(19) and (20) enforce the constraint Ca = 0 on the boundary. II. The Wave Equation on a Curved Background In this section we prove Theorem 1 for the case of a single wave equation g ab ∇a ∇b φ = S
(21)
on M = [0, T ] × . For simplicity, we also assume that gab and S are independent of φ and that ∇a is the Levi-Civita connection with respect to gab . The IBVP consists in finding solutions of (21) subject to the initial conditions n b ∇b φ = π0 , (22) φ|0 = φ0 , 0
where φ0 and π0 are given functions on 0 , and the boundary conditions T b ∇b φ + α N b ∇b φ = G, T
(23)
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H.-O. Kreiss, O. Reula, O. Sarbach, J. Winicour
where G is a given function on T . Here, n b and N b denote the future-directed unit vector field to the time-slices t and the outward unit normal vector field to T , respectively, T b is an arbitrary future-directed time-like vector field which is tangent to the boundary surface T and α is a strictly positive function on T . Without loss of generality, we assume that T a is normalized such that gab T a T b = −1. Furthermore, by redefining φ and S if necessary, we may also assume that the boundary data G vanishes identically. In order to show well posedness for this problem, we use a geometric reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions [3,4,17]. First, introducing the variables Va = ∇a φ, the wave equation can be rewritten as the first order system ∇a φ = Va , g ∇a Vb = S, ∇a Vb − ∇b Va = 0. ab
(24) (25) (26)
Next, we specify any future-directed time-like vector field u a and contract the first and the last equation with it. This yields the evolution system £u φ = u a Va ≡ , g ab ∇a Vb = S, £u Vb = ∇b ,
(27) (28) (29)
where £u denotes the Lie derivative with respect to u a . This system is subject to the initial and boundary conditions φ|0 = φ0 , n b Vb = π0 , ι∗0 Vb = ι∗0 ∇b φ0 , (30) 0 T b Vb + α N b Vb = 0, (31) T
where ι0 : 0 → M is the inclusion map, and subject to the constraint Ca = 0, where the constraint variable Ca is defined as Ca = Va − ∇a φ. The evolution equations (27) and (29) imply that Ca is Lie-dragged by the time evolution vector field u a , £u Ca = 0. In the following, we assume that u a is pointing away from the domain at the boundary. This implies that a solution of (27,28,29) with constraint-satisfying initial data automatically satisfies the constraints everywhere on M, and no extra boundary conditions are needed in order to ensure that the constraint Ca = 0 propagates. Still, there is a huge freedom in choosing the evolution vector field u a ; different choices lead to first order evolution systems (27,28,29) which are inequivalent to each other if the solution is off the constraint surface Ca = 0. In this work we exploit this freedom in order to obtain energy estimates which allow for an appropriate control of the fields not only in the bulk but also on the boundary of the domain (see the estimate (37) below). In order to analyze this, following [17] we rewrite the evolution system (28,29) in the form Aa bc ∇a V c ≡ −u a (∇a Vb − ∇b Va ) + u b ∇a V a = u b S,
Boundary Conditions for Coupled Quasilinear Wave Equations
1107
where the symbol is given by Aa bc = −u a gbc + 2δ a (b u c) . Since Aa bc is symmetric in bc and since u a Aa bc = −u a u a gbc + 2u b u c is positive definite, the evolution system is symmetric hyperbolic. In particular, the evolution equations imply that ∇a (Aa bc V b V c ) = (∇a Aa bc )V b V c + 2(u b V b )S. Integrating both sides of this equation over the manifold M = [0, T ] × and using Gauss’ theorem, one obtains3
n a Aa bc V b V c = T
n a Aa bc V b V c +
0
Na Aa bc V b V c T
(∇a Aa bc )V b V c + 2(u b V b )S . −
(32)
M
The following two conditions from the theory of symmetric linear operators (see [4]) guarantee that the first order IBVP (27,28,29,30,31) is well posed: (i) n a Aa bc is positive definite. (ii) For each p ∈ T , the subspace N− ( p) ⊂ T p M consisting of the vectors V b ( p) satisfying the boundary condition (31) at p is maximal non-positive. This means that Na Aa bc ( p)V b ( p)V c ( p) ≤ 0 for all V b ( p) ∈ N− ( p) and that N− ( p) does not possess a proper extension with this property. For the following, we choose the time evolution vector field u a such that u a is everywhere future-directed and time-like on M and such that u a lies in the plane spanned by T a and N a at each point of the boundary, more specifically, u a T = T a + δ N a , with 0 < δ < 1 a function on T . The following two lemmas imply the satisfaction of the conditions (i) and (ii) for an appropriate choice of δ. Lemma 2. n a Aa bc ( p) is positive definite for all p ∈ M. Proof Let h ab = gab + n a n b be the induced metric on t and expand u a = µ(n a + u¯ a ), where µ = −n a u a . Since u a is future-directed and time-like, µ > 0 and u¯ a u¯ a < 1. Therefore, n a Aa bc = µ h bc + n b n c + 2n (b u¯ c) is positive definite.
Lemma 3. Let 0 < δ ≤ α(1 + α 2 )−1 . Then, the boundary spaces N− ( p) are maximal non-positive for all p ∈ T . 3 Notice that since n a is future directed, its flow increases t; hence in coordinates (t, x i ), where t parametrizes [0, T ] and x i are local coordinates on , we have n t > 0 and n t < 0.
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Proof (cf. Appendix B in Ref. [9]). Fix a point p ∈ T , and let V b ∈ T p M. We have Na Aa bc V b V c = δ Tb Tc + δ Nb Nc + 2T(b Nc) − δ Hbc V b V c = −δ (T b Vb )2 + (N b Vb )2 + Hbc V b V c +2 δ(T b Vb )2 + δ(N b Vb )2 + (T b Vb )(N c Vc ) , where Hbc = gbc + Tb Tc − Nb Nc is the induced metric on the orthogonal complement of the plane spanned by T b and N b . Eliminating the terms (T b Vb ) in the second square bracket on the right-hand side using the boundary condition (31) we obtain Na Aa bc V b V c = −δ (T b Vb )2 + (N b Vb )2 + Hbc V b V c + 2 δ(α 2 + 1) − α (N b Vb )2 . (33) The last term on the right-hand side is non-positive by the assumption of the lemma. Therefore, Na Aa bc is negative-definite on the subspace of vectors V a satisfying the boundary condition. Finally, we observe that N− ( p) is maximal since its dimension is d = dim T p M − 1, while the symmetric bilinear form Na Aa bc has signature (1, d). If we relax the assumption of homogeneous boundary data and replace the condition (31) by the condition T b Vb + α N b Vb = G, (34) T
we obtain, instead of (33), Na Aa bc V b V c = −δ (T b Vb )2 + (N b Vb )2 + Hbc V b V c + 2 δ(α 2 + 1) − α (N b Vb )2 +2(1 − 2δα)(N b Vb )G + 2δ G 2 . Let 0 < ρ < 1 and set δ = (1 − ρ)α(1 + α 2 )−1 . Then, we have (cf. Appendix B in Ref. [9]) (1 − 2δα)2 G 2 . (35) Na Aa bc V b V c ≤ −δ (T b Vb )2 + (N b Vb )2 + Hbc V b V c + 2δ + 2αρ This and the positivity of n a Aa bc implies the existence of strictly positive constants C1 and C2 (depending on δ and ρ) such that Na Aa bc V b V c ≤ −C1 n a Aa bc V b V c + C2 G 2 .
(36)
Using this in the identity (32) we obtain the estimate n a Aa bc V b V c ≤ n a Aa bc V b V c − C1 n a Aa bc V b V c + C2 G 2 t
0
t +C3 0
Tt
⎡
⎢ ⎣
s
na A
a
bc V
b
c
V + s
⎤ ⎥ S 2 ⎦ ds
Tt
Boundary Conditions for Coupled Quasilinear Wave Equations
1109
for all 0 ≤ t ≤ T , where C1 , C2 and C3 are strictly positive constants which are independent of V b , and Tt := [0, t] × ∂. Applying Gronwall’s lemma4 to the function t n a Aa bc V b V c ds we obtain from this y(t) := 0 s
Lemma 4. Let T > 0. There is a constant C = C(T ) ≥ 1 such that all smooth enough solutions to the IBVP (28,29,30,34) satisfy the inequality a b c n a A bc V V + n a Aa bc V b V c t
Tt
⎡
⎢ ≤C⎣
n a Aa bc V b V c +
0
t G2 +
Tt
0
⎛ ⎞ ⎤ ⎥ ⎜ 2⎟ ⎝ S ⎠ ds ⎦ ,
(37)
s
for all 0 ≤ t ≤ T , where Tt := [0, t] × ∂. Since any solution of this problem also satisfies u a Ca = u a Va − £u φ = 0, £u Ca = 0 and ι∗0 Ca = ι∗0 (Va − ∇a φ) = 0, and since u a points outward from the domain at T , the constraint Ca = 0 is satisfied everywhere on M. From this and the previous lemma, we have established: Theorem 2. The second order problem (21,22,23) is strongly well posed: given smooth initial and boundary data φ0 , π0 and G satisfying the usual compatibility conditions at ∂0 , there exists a unique smooth solution satisfying the estimate (37) with V a replaced by ∇ a φ. Remark 1. The important feature of the estimate (37) is the second term on the lefthand side which yields a L 2 boundary estimate for the gradient of φ. This estimate is obtained by choosing the time evolution vector field u a in such a way that the boundary matrix Na Aa bc is negative definite on the subspace of vectors satisfying the boundary conditions. As we will see (Lemma 6 in the next section), this property is important for systems of wave equations since it allows the coupling of the boundary conditions through small enough terms involving first derivatives of the fields. If, on the other hand, u a is chosen to be tangent to the boundary, the boundary matrix has a nontrivial kernel and one does not obtain an estimate for the full gradient of φ on the boundary from the first order system. However, this does not affect the strong well posedness of the second order system which is independent of u a . As an example, consider the wave equation on the half-plane = R+ × R2 with the flat metric g = −dt 2 + d x 2 + dy 2 + dz 2 . In this case, we have n a ∂a = ∂t ,
N a ∂a = −∂x
T a ∂a =
1 ∂t − β y ∂ y − β z ∂z , p
with (β y )2 + (β z )2 < 1 and p := 1 − (β y )2 − (β z )2 , and the boundary condition (23) reduces to (38) φt + pαφx − β y φ y − β z φz x=0 = pG, 4 See, for instance, Lemma 3.1.1 in Ref. [18].
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H.-O. Kreiss, O. Reula, O. Sarbach, J. Winicour
where φt := ∂t φ etc. Choosing u a = p(T a + δ N a ) with 0 < δ < 1, the energy norm for this problem reads
∞ ∞ ∞ φt2 + φx2 + φ y2 + φz2 + 2φt δpφx + β y φ y + β z φz d y dz d x. n a A bc V V = a
t
b
c
0 −∞ −∞
This is similar to the norm we used in Ref. [9] for obtaining an a priori energy estimate for the second order wave equation with boundary condition (38). III. Systems of Wave Equations and Proof of Main Theorem In order to show that the system (1,2,3) yields a well posed IBVP, we follow the arguments given in Sect. II and reduce it to a first order symmetric hyperbolic system with maximal dissipative boundary conditions. Let Va A := ∇a A , and let u a ( p, ) denote a future-directed time-like vector field on M such that u a T = T a + δ N a , with 0 < δ < 1 a function on T to be determined. Then (1) can be rewritten as the first order evolution system u a ∇a A = u a Va A , g ()∇a Vb A = S A (, V ), u a ∇a Vb A − ∇b Va A = u a R A Bab B , ab
(39) (40) (41)
where R A Bab denotes the curvature belonging to the connection ∇a . At this point, we stress that the connection ∇a is a fixed background connection on the vector bundle E, and not the Levi-Civita connection belonging to the metric gab (), so that R A Bab does not depend on nor its derivatives. The system (39,40,41) is subject to the constraint Cb A = 0, where Cb A := ∇b A − Vb A . Equations (39,41) imply that the constraint variable Cb A is Lie-dragged by u a : £u Cb A ≡ u a ∇a Cb A + (∇b u a )Ca A = 0. Therefore, any smooth enough solution of the first order problem (39,40,41) belonging to initial data with Cb A = 0 satisfies the constraint Cb A = 0 everywhere it is defined. The initial condition is n b VbA = 0A , ι∗0 VbA = ι∗0 ∇b 0A , (42) A = 0A , 0
0
and the boundary condition (3) reads T b Vb + α N b Vb = ca A B Va B + d A B B + G A . T
T
T
(43)
In order to analyze the well posedness of the first order IBVP (39,40,41,42,43) we first linearize the system by replacing the coefficients gab (), S A (, ∇), T b (), N b (), α(), ca A B (), d A B () by smooth functions gab , S A , T b , N b , α, ca A B , d A B , respectively. Local in time well posedness for the original quasilinear system follows
Boundary Conditions for Coupled Quasilinear Wave Equations
1111
by iteration from the well posedness result for the linear system with enough differentiability5 . Next, we use a partition of unity in order to localize the problem. With this, it is sufficient to consider a local trivialization ϕ : U × R N → π −1 (U ) of E such that U¯ ⊂ M is compact and contains a portion U of the boundary T . Let ε > 0. According to the assumption there exists a smooth map Jε : U → G L(N , R), p → (Jε ( p)) such that the transformed matrix coefficients c˜a := Jε ca Jε−1 satisfy the condition (4) for all vector-valued one-forms Va on U. Setting h AB (ε) := (JεT h Jε ) AB = h C D (Jε )C A (Jε ) D B , we can reformulate this condition by stating that h AB (ε)ca A C ()cb B D ()Va C Vb D ≤ εh AB (ε)eab ()Va A Vb B ,
(44)
for all vector-valued one-forms VaA on U. The system (39,40,41) can be written in the form B
0 −h AB (ε)u a ∇a = S(, V ), (45) 0 h AB (ε)Aa bc ∇a Vc B where > 0 is to be determined, Aa bc = −u a gbc + 2δ a (b u c) and
−h AB (ε)u a Va B . S(, V ) = −h AB (ε)R B Cab C u a + h AB (ε)u b S B (, V ) Let B(n a ; (, W ), (, V )) denote the bilinear form belonging to the principal symbol of (45), that is, for an arbitrary one-form wa on M define B(wa ; (, W ), (, V )) := −u a wa h AB (ε) A B + h AB (ε)wa Aa bc W b A V c B . We have Lemma 5. Let > 0. Then, B(n a ; (, W ), (, V )) is symmetric in (, W ), (, V ) and positive definite for wa = u a and wa = n a . Therefore, the system (45) is symmetric hyperbolic. Proof The symmetry property follows immediately from the symmetry of h AB (ε) and the symmetry of Aa bc in bc. In order to check the positivity statements, let wa = u a , √ γ := −u a u a and uˆ a := γ −1 u a . Since Aa bc u a = γ 2 gbc + 2uˆ b uˆ c , we find B(u a ; (, V ), (, V )) = γ 2 h AB (ε) A B + (gab + 2uˆ a uˆ b )h AB (ε)Va A V b B which is manifestly positive definite. The proof that B(n a ; (, V ), (, V )) is positive definite is similar to the proof of Lemma 2. As in the previous section we obtain well posedness of the linearized system provided we can show that each boundary space N− ( p) := {(, V ) ∈ R N × R(d+1)N : T b ( p) + α( p)N b ( p) Vb A = ca A B ( p)Va B + d A B ( p) B },
p ∈ U,
is maximal non-positive with respect to B(Na ; (, V ), (, V )). This is the statement of the next lemma. 5 See, for instance, [10,18].
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Lemma 6. Set δ := α(1+α 2 )−1 /2 and κ := 2[2δ +(1−2δα)2 /α]2 . Choose ε > 0 small enough such that κε < δ and > 0 large enough such that 2κh AB (ε)d A C d B D C D ≤ δh AB (ε) A B for all ∈ R N . Then, the boundary space N− ( p) is maximal nonpositive for all p ∈ U. Proof Let p ∈ U. We have, as in the proof of Lemma 3, B(Na ; (, V ), (, V )) = −u a Na h AB (ε) A B + h AB (ε)Na Aa bc V b A V c B = −δh AB (ε) (T a T b + N a N b + H ab )Va A Vb B + A B +2 δ T a T b + δ N a N b + T a N b h AB (ε)Va A Vb B . (46) Let ( A , Va A ) ∈ N ( p). Then, T a Va A = −α N a Va A + G˜ A with G˜ A := ca A B Va B + d A B B , and we may use this equation in order to eliminate the terms (T a Va A ) in the second bracket on the right-hand side of (46). This yields B(Na ; (, V ), (, V )) ≤ −δh AB (ε) (T a T b + N a N b + H ab )Va A Vb B + A B
(1 − 2δα)2 h AB (ε)G˜ A G˜ B , + 2δ + α where we have set δ := α(1+α 2 )−1 /2 and used the boundary estimate (35) with ρ = 1/2. Now, h AB (ε)G˜ A G˜ B ≤ 2h AB (ε)ca A C Va C cb B D Vb D + 2h AB (ε)d A C C d B D D ≤ 2εh AB (ε)eab Va A Vb B + 2h AB (ε)d A C d B D C D , (47) where we have used the estimate (44) in the last step. Recalling that eab = g ab +2T a T b = T a T b + N a N b + H ab and the definition of κ in the assumption of the lemma we find B(Na ; (, V ), (, V )) ≤ −δh AB (ε) eab Va A Vb B + A B + κ εh AB (ε)eab Va A Vb B + h AB (ε)d A C d B D C D . The non-positivity of N− ( p) now follows from the assumptions on ε and . Finally, we observe that an element in N− ( p) is characterized by N linear conditions in a (d + 2)N dimensional vector space which implies that dim N− ( p) ≥ (d + 1)N . On the other hand, from Eq. (46) we see that the signature of B(Na ; ., .) is given by (N , (d + 1)N ). Therefore, dim N− ( p) = (d + 1)N and the maximality of N− ( p) follows. IV. Boundary Conditions for Isolated Systems We consider here boundary conditions for an isolated system emitting radiation. If, for computational purposes, the evolution domain of such a system has a finite (artificial) boundary, some artificial boundary condition must be imposed. If one knew the correct boundary data for the analytic problem, then in principle one could use any boundary condition corresponding to a well posed IBVP. However, the determination of the correct boundary data is in general a global problem, in which the boundary data must be determined by extending the solution to infinity either by matching to an exterior (linearized or nonlinear) solution obtained by some other means. The matching approach
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has been reviewed elsewhere [19]. Here we consider an alternative approach in which homogeneous boundary data can be assigned in such a way that the accuracy of the boundary condition becomes exact in the limit that the boundary is extended to infinity. (Such boundary conditions would also be beneficial to the matching approach because the corresponding boundary data would be small so that numerical or other error would also have a small effect.) Artificial boundary conditions for an isolated radiating system for which homogeneous data is approximately valid are commonly called absorbing boundary conditions (see e.g. [20–25]), or nonreflecting boundary conditions (see e.g. [26–28]) or radiation boundary conditions (see e.g. [29]). Such boundary conditions are advantageous for computational use. However, local artificial boundary conditions are not perfectly nonreflecting in general. Here, to be more precise, we consider nonreflecting boundary conditions in the sense of boundary conditions for a well posed problem for which homogeneous data produces no spurious reflection in the limit that the boundary approaches an infinite sphere. The extensive literature on improved versions of nonreflecting boundary conditions involves higher order and nonlocal methods. Our interest here is to investigate the optimal choice of local first order homogeneous boundary conditions on a spherical boundary for the constrained Maxwell and linearized Einstein problems expressed in terms of the gauge dependent variables Aµ and γ µν . See [30–32] for the construction of higher-order and higher-accurate boundary conditions for Einstein’s equations. We base our discussion on waves from an isolated system satisfying a system of flat space wave equations. We use Greek indices to denote standard inertial coordinates x µ = (t, x, y, z) in which the components of the Minkowski metric ηµν are diag(−1, 1, 1, 1). In the case of a scalar field , we thus consider the wave equation ηαβ ∂α ∂β = −∂t2 + ∂x2 + ∂ y2 + ∂z2 = S, where the source S has compact support. Outside the source, we assume that the solution has the form =
f (t − r, θ, φ) g(t − r, θ, φ) h(t, r, θ, φ) + + , r r2 r3
(48)
where (r, θ, φ) are standard spherical coordinates and f , g and h and their derivatives are smooth bounded functions. These assumptions determine the exterior retarded field of a system emitting outgoing radiation. The simplest case is the monopole radiation =
f (t − r ) r
which satisfies (∂t + ∂r )(r ) = 0. This motivates the use of a Sommerfeld condition 1 (∂t + ∂r )(r )| R = q(t, R, θ, φ) r on a finite boundary r = R. The resulting Sommerfeld boundary data q in the general case (48) falls off as 1/R 3 , so that a homogeneous Sommerfeld condition introduces an error which is vanishingly small for increasing R. As an example, for the dipole solution
f (t − r ) f (t − r ) f (t − r ) cos θ =− + Di pole = ∂z r r r2
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we have q=
f (t − r ) cos θ . R3
(49)
˜ Di pole A homogeneous Sommerfeld condition at r = R would lead to a solution containing a reflected ingoing wave. For large R, ˜ Di pole ∼ Di pole + κ
F(t + r − 2R) cos θ , r
where ∂t f (t) = F(t) and the reflection coefficient has asymptotic behavior κ = O(1/R 2 ). More precisely, the Fourier mode eiω(t−r ) eiω(t+r −2R) ˜ Di pole (ω) = ∂z + κω , r r ˜ Di pole )(ω)| R = 0 with reflecsatisfies the homogeneous boundary condition (∂t +∂r )(r tion coefficient κω =
1 1 . ∼ 2ω2 R 2 + 2iω R − 1 2ω2 R 2
(50)
Note that (50) and (49) satisfy κ ∼ q R.
(51)
In the case of a system of equations κ will have N components corresponding to the number of modes generated in the reflected wave. The boundary conditions lead to a system of simultaneous equations relating κ to the components of the Sommerfeld data q. If these equations are nondegenerate then (51) continues to hold. However, degeneracies could conceivably lead to weaker asymptotic falloff of κ. (It would be interesting to determine whether such cases exist.) In any case, (51) gives the optimum allowable behavior of the reflection coefficients so that the asymptotic behavior of the Sommerfeld data q is a good indicator of the quality of the boundary condition. This forms the basis of our investigation of the Maxwell and linearized Einstein equations with a spherical boundary in Sects. IV B and IV C. A. A plane boundary. The key ideas in the above example are that (i) the Sommerfeld condition is only satisfied exactly by waves traveling in the radial direction and (ii) in the asymptotic limit r → ∞ all waves from an isolated system propagate in the radial direction. This allows us to reformulate our discussion of the Sommerfeld condition by considering a wave propagating in the domain x < 0, which is incident on a plane boundary at x = 0 with the boundary condition K α ∂α |x=0 = 0, where K α ∂α = ∂t + ∂x is the characteristic direction determined by the outward normal to the boundary ∂x and the time direction ∂t . This homogeneous condition is satisfied for plane waves = G(t + k x x + k y y + k z z) incident on the boundary only for the single case (k x , k y , k z ) = (1, 0, 0), i.e. a plane wave propagating in the outgoing normal direction. Plane waves in the normal direction pass through the boundary, whereas plane
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waves incident in other directions on the boundary give rise to a reflected wave. We will take advantage of this simplification of the plane wave case in discussing boundary conditions for electromagnetic and gravitational waves. The results then suggest how to formulate boundary conditions for an isolated electromagnetic or gravitational system with a spherical boundary of radius R, where in the limit R → ∞ all radiation is incident normally. For the electromagnetic case, we describe the field by means of a vector potential Aµ satisfying the Lorentz gauge condition. Maxwell’s equations in a flat spacetime with Minkowski metric ηµν then reduce to the wave equations ηαβ ∂α ∂β Aµ = 0 subject to the constraint C := ∂µ Aµ = 0 introduced by the Lorentz gauge condition. This constraint keeps us from requiring that each component of Aµ satisfy a homogeneous Sommerfeld condition, in contrast to the scalar example. The electromagnetic case also differs from the scalar case because of the remaining gauge freedom allowed by the Lorentz condition. An electromagnetic plane wave incident in the outgoing normal direction can be described by the real part of the vector potential Aµ = F(t − x)Q µ + G(t − x)K µ , where F(t − x) is complex, Q µ = Y µ + i Z µ is a complex null polarization vector, G(t − x) represents gauge freedom and K µ = T µ + X µ , in terms of the orthonormal tetrad (T µ , X µ , Y µ , Z µ ) aligned with the coordinate axes satisfying ηµν = −Tµ Tν + X µ X ν + Yµ Yν + Z µ Z ν . In order to formulate a gauge invariant boundary condition we consider the corresponding electromagnetic field tensor Fµν = ∂µ Aν − ∂ν Aµ = −F (t − x)(K µ Q ν − Q µ K ν ). Here we adopt the notation ∂u F(u) = F (u). For this plane wave, all components of Fµν satisfy K µ Fµν = 0. However, this condition rules out the possibility of a static electric field oriented normal to the boundary. For the purpose of formulating a boundary condition which only restricts propagating waves it suffices to consider the weaker condition K µ Q ν Fµν = 0.
(52)
In terms of the electric and magnetic field components tangential to the boundary, (52) corresponds to the plane wave relations Etan · Btan = 0 and |Etan | = |Btan |, with the corresponding Poynting vector in the outward normal direction. We can incorporate (52) into the following homogeneous Sommerfeld boundary conditions for the vector potential: K ν K µ ∂µ Aν = 0, ν µ µ A ν = K Q ∂µ A ν .
Qν K µ∂
(53) (54)
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The remaining boundary condition can be expressed in Sommerfeld form by rewriting the constraint as C=
1 ν µ −L K − K ν L µ + Q ν Q¯ µ + Q¯ ν Q µ ∂µ Aν = 0, 2
(55)
where L µ = T µ − X µ . Here (K µ , L µ , Q µ ) form a null tetrad according to the conventions ηµν = −K (µ L ν) + Q (µ Q¯ ν) .
(56)
We assume throughout the following that the spin transformation freedom Q µ → eiα Q µ has been restricted according to K µ ∂µ α = 0. The Sommerfeld boundary conditions (53), (54) and (55) have the required hierarchical, upper triangular form for a well posed IBVP, see Lemma 1. For the purpose of extending this approach to the gravitational case, we write the linearized Einstein vacuum equations in the form ηαβ ∂α ∂β γ µν = 0
(57)
subject to the harmonic constraints C ν := −∂µ γ µν = 0.
(58)
√ Here, to linearized accuracy, we set −gg µν = ηµν + γ µν so that γµν = −h µν + 21 ηµν h represents the densitized version of the metric perturbation gµν = ηµν + h µν . (Indices of linearized objects are raised and lowered with the Minkowski metric.) The corresponding linearized curvature tensor is 2Rµνρσ = ∂ρ ∂ν h µσ − ∂σ ∂ν h µρ − ∂ρ ∂µ h νσ + ∂σ ∂µ h νρ .
(59)
In the linear approximation, the diffeomorphism freedom reduces to the gauge freedom h µν → h µν + 2∂(µ ξν) , which leaves Rµνρσ invariant. A plane wave incident on the boundary in the outgoing normal direction is given by the real part of h µν = F(t − x)Q µ Q ν + 2∂(µ ξν) (t − x) = F(t − x)Q µ Q ν − 2K (µ ξν) (t − x), (60) where the ξν (t − x) term describes a pure gauge wave. Similarly, a plane wave incident on the boundary in the ingoing normal direction is given by h µν = F(t + x)Q µ Q ν − 2L (µ ξν) (t + x).
(61)
In these plane waves, F describes the gravitational radiation. The curvature tensors corresponding to (60) and (61) are, respectively, Rµνρσ = 2F (t − x)K [µ Q ν] Q [ρ K σ ]
(62)
Rµνρσ = 2F (t + x)L [µ Q ν] Q [ρ L σ ] .
(63)
and
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The analogue of the boundary conditions (14),(15),(16) and (17) for a plane boundary are K µ K ρ K σ ∂µ h ρσ = −q K K , K µ K ρ L σ ∂µ h ρσ = −q Q Q¯ ,
K µ K ρ Q σ ∂µ h ρσ = −q K Q , (K µ Q ρ Q σ − Q µ Q ρ K σ )∂µ h ρσ = −q Q Q .
(64) (65) (66) (67)
The outgoing plane wave (60) satisfies the homogeneous boundary conditions q K K = q Q Q¯ = q K Q = q Q Q = 0. For the ingoing plane wave (61), q K K = −8K σ ξσ (t + x), q Q Q¯ = −4L σ ξσ (t + x), q K Q = −4Q σ ξσ (t + x), q Q Q = −4 F¯ (t + x),
all evaluated on the boundary at x = 0. Thus the boundary conditions (64),(65) and (66) control the gauge waves entering through the boundary; and the condition (67) controls the gravitational waves entering. In order to formulate a boundary condition with gauge invariant meaning analogous to (52) in the Maxwell case, we consider the linearized curvature tensor. Outgoing wave boundary conditions on the curvature tensor could be imposed by requiring that the Newman-Penrose component 0 = K µ Q ν Q ρ K σ Rµνρσ vanish on the boundary. (See [2] for a discussion of the appropriateness of this boundary condition.) However, this requirement involves second derivatives in the normal direction when expressed in terms of γµν . Instead, we require := K µ Q ν Q ρ T σ Rµνρσ = 0 on the boundary. The condition = 0 is equivalent to 0 = 0 if the Ricci component Rµν Q µ Q ν = 0, e.g. if the vacuum Einstein equations are satisfied. A straightforward calculation leads to 1 − 2 = K µ Q ν Q ρ T σ (∂ρ ∂ν γµσ − ∂σ ∂ν γµρ − ∂ρ ∂µ γνσ + ∂σ ∂µ γνρ ) + Q ν Q ρ ∂ν ∂ρ γ 2 = K µ Q ν Q ρ T σ −∂σ ∂ν γµρ − ∂ρ ∂µ γνσ + ∂σ ∂µ γνρ 1 + K µ K σ + Q µ Q¯ σ Q ν Q ρ ∂ν ∂ρ γµσ 2
1 µ σ K K + Q µ Q¯ σ Q ρ ∂ρ γµσ − K µ Q ρ T σ ∂µ γσρ = Q ν ∂ν 2 (68) +T σ ∂σ −K µ Q ν Q ρ ∂ν γµρ + K µ Q ν Q ρ ∂µ γνρ . Thus, besides containing no second derivatives normal to the boundary, the condition = 0 can be reduced to two first order conditions by factoring out the Q ν ∂ν and T σ ∂σ derivatives in (68) which are tangential to the boundary. There are many ways this can be done. In order to obtain first order conditions which fit into a hierarchy of Sommerfeld
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conditions, we modify (68) according to the steps
1 µ σ 1 ν − 2 = Q ∂ν (K K + Q µ Q¯ σ )Q ρ ∂ρ γµσ − K ρ Q µ L σ ∂ρ γµσ 2 2 1 µ ρ σ − K Q K ∂µ γσρ 2 (69) +T σ ∂σ −K µ Q ν Q ρ ∂ν γµρ + K µ Q ν Q ρ ∂µ γνρ 1 = Q ν ∂ν (K µ K σ Q ρ + Q µ K σ L ρ − Q µ Q σ Q¯ ρ )∂ρ γµσ 2 −2Q µ Cµ − K µ Q ρ K σ ∂µ γσρ (70) +T σ ∂σ −K µ Q ν Q ρ ∂ν γµρ + K µ Q ν Q ρ ∂µ γνρ 1 = Q ν ∂ν (K µ K σ Q ρ − Q µ Q σ Q¯ ρ )∂ρ γµσ − 2K µ Q ρ K σ ∂µ γσρ − 2Q µ Cµ 2 +T σ ∂σ K µ Q ν Q ρ ∂µ γνρ . (71) Thus since the derivatives Q ν ∂ν and T ν ∂ν are tangential to the boundary, we can enforce = 0 on the boundary through the first order boundary conditions Q α Q β K µ ∂µ γαβ = 0, K α Qβ K µ∂
µ γαβ
−
1 α β µ 2 K K Q ∂µ γαβ
+
1 α β ¯µ 2 Q Q Q ∂µ γαβ
(72) = 0.
(73)
These two boundary conditions can then be included in a hierarchical set of Sommerfeld boundary conditions, according to the example K α K β K µ ∂µ γαβ = 0,
(74)
Qα Qβ K µ∂
(75)
µ γαβ
= 0,
Q α Q¯ β K µ ∂
K α Qβ K µ∂
µ γαβ
−
µ γαβ = 0, 1 α β µ 1 α β ¯µ 2 K K Q ∂µ γαβ + 2 Q Q Q ∂µ γαβ
(76) = 0.
(77)
The constraints Cρ = 0, which determine the remaining boundary conditions, can be cast in the Sommerfeld form Cρ =
1 ν µ L K + K ν L µ − Q¯ ν Q µ − Q ν Q¯ µ ∂µ γνρ = 0, 2
which can also be incorporated into the hierarchy. However, there are many alternative possibilities to (74)–(77) which preserve the hierarchical Sommerfeld structure and lead to a well posed IBVP. In the absence of a clear geometric approach, we next examine the boundary conditions appropriate to an isolated system by considering the resulting reflection off a spherical boundary.
B. Application to Maxwell fields with a spherical boundary. In the case of a general retarded solution for a massless scalar wave equation, we found that a Sommerfeld boundary condition on a spherical boundary of radius R required data q = O(1/R 3 ). Homogeneous Sommerfeld data gave rise to an ingoing wave with reflection coefficient κ = O(1/R 2 ), as in (50). This is the best that can be achieved with a local first order
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homogeneous boundary condition on a spherical boundary. We now investigate the corresponding result for the constrained Maxwell equations expressed in terms of a vector potential Aµ . In doing so, we associate spherical coordinates (r, x A ), x A = (θ, φ), in a standard way with the Cartesian coordinates x i = (x, y, z), e.g. z = r cos θ . As in (56) we introduce a null tetrad (K µ , L µ , Q µ ) adapted to the boundary, where now K µ ∂µ = ∂t + ∂r , L µ ∂µ = ∂t − ∂r , and we fix the spin-rotation freedom in the complex null vector Q µ = (0, Q i ) by setting Qi =
∂xi A Q , ∂x A
where
θ
Q = Q ,Q A
φ
(78)
i 1 1, . = r sin θ
We describe outgoing waves in terms of the retarded time u = t − r . In order to investigate the vector potential describing the exterior radiation field emitted by an isolated system we introduce a Hertz potential with the symmetry 1 H µν = H [µν] + ηµν H. 4 Then the vector potential Aµ = ∂ν H µν satisfies the Lorentz gauge condition and generates a solution of Maxwell’s equations provided the Hertz potential satisfies the wave equation. The trace H represents pure gauge freedom. We consider outgoing dipole waves oriented with the z-axis. Other dipole waves can be generated by a rotation. Higher multipole waves can be generated by taking spatial derivatives. [µν] = 0 gives rise to the dipole gauge wave The choice H = Z α ∂α F(u) r , H
F (u) F (u) 2F (u) 3F(u) cos θ K µ + + Aµ = + r r2 r2 r3
F (u) F(u) Zµ + 3 × cos θ ∂µr − r2 r
with components F (u) 2F(u) cosθ, K Aµ = + r2 r3
F (u) F(u) Q µ Aµ = + 3 sin θ. r2 r µ
(79)
In Appendix V we give some useful formulae underlying the calculation leading to (79) and the following results.
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The choice H µν = (T µ Z ν − Z µ T ν ) f (u) r gives rise to a dipole electromagnetic wave
f (u) f (u) f (u) Tµ cos θ − Aµ = − + 2 Zµ r r r with components f (u) cosθ, r2 f (u) Aµ Q µ = sin θ. r Aµ K µ =
(80)
The choice H µν = (X µ Y ν − Y µ X ν ) f (u) r gives rise to a dipole electromagnetic wave with the dual polarization
y Xµ xYµ f (u) f (u) + 2 − Aµ = − r r r r with components Aµ K µ = 0,
f (u) f (u) µ sin θ. A Qµ = i + 2 r r
(81)
We wish to formulate boundary conditions which generalize the Sommerfeld hierarchy (53) and (54) to a spherical boundary of radius R in a way which minimizes reflection. By inspection of (79), (80) and (81), we consider the choice 1 µ K ∂µ (r 2 K ν Aν ) = q K , r2
(82)
1 µ K ∂µ (r Q ν Aν ) − Q µ ∂µ (K ν Aν ) = q Q , r
(83)
chosen to minimize the asymptotic behavior of the Sommerfeld data. As before, the constraint determines the remaining boundary condition as part of the Sommerfeld hierarchy. For the dipole gauge wave (79), qK = −
2F(u) cos θ , R4
q Q = 0;
for the dipole electromagnetic wave (80), q K = 0,
qQ =
f (u) sin θ ; R3
and for the dual dipole electromagnetic wave (81) q K = 0,
qQ =
−i f (u) sin θ. R3
Overall this implies q K = O(1/R 4 ) and q Q = O(1/R 3 ). We have checked that homogeneous Sommerfeld data leads to reflection coefficients with overall behavior κ = O(1/R 2 ) in accordance with (51).
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Note that the relations (A1) and (A8) allow us to express (82) and (83) in the form 1 ν µ K K ∂µ (r 2 Aν ) = q K , r2 Q ν K µ ∂µ A ν − K ν Q µ ∂µ A ν = q Q ,
(84) (85)
which correspond to (7) and (8) when ∂µ is generalized to the connection ∇a in a curved space background with K a ∇a r = 1. Here (85) is equivalent to the gauge invariant condition Q ν K µ Fµν = q Q .
(86)
C. Application to linearized gravitational fields with a spherical boundary. The gravitational case is more complicated than the electromagnetic case because the geometry of the boundary is coupled with the boundary condition. Additionally, there are no gauge invariant quantities, analogous to (86) in the electromagnetic case, on which to base first order boundary conditions. We begin with a discussion of how to adapt to a curved boundary the first order version of the boundary condition given in Sect. IV A for a plane boundary. In the nonlinear treatment of a curved boundary with unit outer normal N a we can decompose the metric according to gab = τab + Na Nb , where τab is the metric intrinsic to the time-like boundary. Let Da denote the covariant derivative associated with τab . The extrinsic curvature of the boundary is Nab = τa c ∇c Nb . We complete an orthonormal basis by setting τab = −Ta Tb + Q (a Q¯ b) in terms of a time-like vector T a and complex null vector Q a tangent to the boundary. We decompose := K a Q b Q c T d Rabcd = T + N and the Weyl component 0 = K a Q b Q c K d Rabcd = T + N + 2T N , where K a = T a + N a and T = T a Q b Q c T d Rabcd , N = N a Q b Q c T d Rabcd , T N = T a Q b Q c N d Rabcd .
(87) (88) (89)
When the vacuum Einstein equations are satisfied the Riemann curvature tensor may be replaced by the Weyl tensor whose symmetry implies T N = 0. Therefore, in this case, = 0 implies the vanishing of the Newman-Penrose Weyl component 0 = 0. A short calculation gives the embedding formulae N = Q b Q c T d (Dd Nbc − Db Ncd ) and T = T a Q b Q c T d
(3)
Rabcd − Nac Nbd + Nbc Nad ,
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where (3) Rabcd is the intrinsic curvature to the boundary, i.e. T a Q b Q c T d (3) Rabcd = Q b Q c T d (Dd Dc − Dc Dd )Tb . (These are the embedding equations for the Cauchy problem corrected for the space-like character of the normal to the boundary.) We now apply these results to a spherical boundary r = R in linearized theory off a Minkowski background, i.e. gµν = ηµν +h µν in standard inertial coordinates x µ , where is the linearization parameter. We choose Tµ = ∂µ t + O() and Nµ = ∂µr + O(). Then Dµ Tν = O() and Nµν = R −1 Q µν + O(), where Q µν = Q (µ Q¯ ν) is the metric of a 2-sphere of radius R. We choose the basis to satisfy T µ Dµ Tν = 0 and T µ Dµ Q ν = 0, so that T = T µ Q ν Q ρ T σ (3) Rµνρσ + O( 2 ) = T σ Dσ (Q ν Q ρ Dρ Tν ) + O( 2 ) and N = T σ Dσ (Q ν Q ρ Nρν ) − Q ρ Dρ (Q ν T σ Nσ ν ) + +
1 ρ Q (Dρ Q µ ) Q¯ µ Q ν T σ Nσ ν 2
1 ν ρ Q Q Dρ Tν + O( 2 ). R
Thus the boundary conditions Q ν Q ρ (Nρν + Dρ Tν ) = 0, Q ν T ρ Nρν = 0,
(90)
imply to linearized accuracy that =
1 ν ρ Q Q Dρ Tν . R
(91)
This gives a geometric formulation of the first differential order version of the requirement that → 0 in the asymptotic limit R → ∞. However, 0 = O(1/R 5 ) in an asymptotically flat space-time, whereas (91) leads to = O(1/R 2 ). This is an indication that the boundary conditions (90) might lead to more reflection than desirable. Can this be remedied by the introduction of, say, lower order terms in the boundary conditions? We investigate this question in the context of a well posed IBVP based upon the harmonic version of the linearized Einstein equations (57) and (58), where γ µν = −h µν + 21 ηµν h. For this purpose, we now consider linearized outgoing waves in the harmonic gauge which are incident on a spherical boundary. We model our discussion on the Maxwell case by using the gravitational analogue of a Hertz potential H µανβ [33,34], which has the symmetries H µανβ = H [µα]νβ = H µα[νβ] = H νβµα and satisfies the flat space wave equation ∂ σ ∂σ H µανβ = 0. Then the densitized metric perturbation γ µν = ∂α ∂β H µανβ
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satisfies the linearized Einstein equations in the harmonic gauge. Outgoing waves can be generated from the potential H µανβ =
f µανβ (u) , r
and its spatial derivatives. The incidence of such an outgoing wave on a boundary r = R leads to reflection, with the asymptotic falloff of the reflection coefficients depending upon the choice of boundary conditions. We limit our calculation of reflection coefficients to the case of outgoing quadrupole waves, which can be obtained from the Hertz potential H µανβ = K µανβ
f (u) , r
(92)
where K µανβ is a constant tensor. (All higher multipoles can be constructed by taking spatial derivatives.) K µανβ has 21 independent components. However, the choice K µανβ = µανβ leads to γ µν = 0 so there are only 20 independent waves. These can be further reduced to pure gauge waves, corresponding to the trace terms in K µανβ , e.g. K µανβ = ηαν ηβµ − ηµν ηαβ leads to a monopole gauge wave. Linearized gravitational waves arise from the trace-free part of K µανβ . There are ten independent quadrupole gravitational waves, corresponding to spherical harmonics with ( = 2, −2 ≤ m ≤ 2) in the two independent polarization states. The other ten independent potentials comprise two monopole gauge waves, three dipole gauge waves and five quadrupole gauge waves, for which the linearized Riemann tensor vanishes. It suffices to consider the following examples of waves with quadrupole dependence aligned with the z-axis. Other quadrupole waves can be obtained by rotation and have similar asymptotic behavior. Reflection coefficients from the other monopole and dipole gauge waves are smaller and provide no further useful information. The Hertz potential (92) gives rise to the perturbation f (u) . r Appendix V lists useful formula for the calculations underlying the following results. γ µν = K µανβ ∂α ∂β
1. Quadrupole-monopole gauge wave. The Hertz potential f (u) H µανβ = Z µ ηαν Z β + Z ν ηβµ Z α − Z µ ηαβ Z ν − Z β ηνµ Z α r gives rise to a combination monopole-quadrupole gauge wave with components
f (u) f (u) α β sin2 θ, Q Q γαβ = −2 + 3 r2 r
f (u) 2 f (u) 2 f (u) α ¯β Q Q γαβ = −2 cos2 θ, + + r r2 r3 f (u) K α Q β γαβ = − 3 sin θ cos θ,
r f (u) 2 f (u) cos2 θ, K α K β γαβ = 2 + r2 r3
2 f (u) f (u) f (u) (1 − 3 cos2 θ ). γ =− cos2 θ + 2 + r r2 r3
(93)
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H.-O. Kreiss, O. Reula, O. Sarbach, J. Winicour
Here the sin2 θ dependence of the spin-weight 2 component Q α Q β γαβ is a pure 2 Y20 spin-weighted spherical harmonic; the sin θ cos θ dependence of the spin-weight 1 component K α Q β γαβ is a pure 1 Y20 harmonic; and the remaining spin-weight 0 components are mixtures of Y00 and Y20 . 2. Quadrupole gravitational wave. The trace-free Hertz potential H µανβ = (T µ Z α − Z µ T α )(X ν Y β − Y ν X β ) + (X µ Y α − Y µ X α )(T ν Z β − Z ν T β )
f (u) r
(94)
gives rise to a perturbation with γ = 0 and components Q α Q β γαβ = 2i sin2 θ
Q α Q¯ β γαβ = 0, K α Q β γαβ = i cos θ sin θ
f (u) f (u) , + 2 r r
2 f (u) 3 f (u) , + r2 r3
(95)
K α K β γαβ = 0, which have spin-weighted = 2, m = 0 dependence. 3. Dual quadrupole gravitational wave The trace-free Hertz potential H µανβ = (T µ Z α − Z µ T α )(T ν Z β − Z ν T β ) − (X µ Y α − Y µ X α )(X ν Y β − Y ν X β ) 1 f (u) , + (ηµν ηαβ − ηµβ ηνα ) 3 r obtained from the dual of (94), gives gives rise to a perturbation with γ = 0 and components α
β
Q Q γαβ Q α Q¯ β γαβ K α Q β γαβ K α K β γαβ
f (u) f (u) f (u) , + 2 + 3 = 2 sin θ r r r
f (u) f (u) 1 , = 4(cos2 θ − ) + 3 3 r2 r
2 f (u) f (u) , = cos θ sin θ + 3 r2 r 1 f (u) = 2(cos2 θ − ) 3 , 3 r
2
which have spin-weighted = 2, m = 0 dependence.
(96)
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4. Sommerfeld-type boundary conditions. Sommerfeld boundary conditions consistent with a well posed harmonic IBVP have wide freedom regarding (i) partial derivative terms consistent with the hierarchical upper triangular structure of the boundary condition and (ii) lower differential order terms. Here we consider three choices of boundary conditions and compare their reflection coefficients. One basic idea common to these choices has already been used in the scalar and Maxwell cases, i.e by inspecting the asymptotic behavior of the waves (93), (95) and (96) we use the property K α ∂α f (u) = 0 to introduce the appropriate powers of r that lead to the smallest asymptotic behavior in the resulting Sommerfeld data. Our first choice of boundary conditions is the mathematically simplest choice 1 α β µ K K K ∂µ (r 2 γαβ ) r2 1 α β µ Q Q K ∂µ (r γαβ ) r 1 α ¯β µ Q Q K ∂µ (r γαβ ) r 1 α β µ K Q K ∂µ (r 2 γαβ ) r2
= qK K ,
(97)
= qQ Q ,
(98)
= q Q Q¯ ,
(99)
= qK Q .
(100)
This was the choice adopted in numerical tests verifying the stability of the harmonic IBVP with a plane boundary [14]. The powers of r in (97)-(100) are based upon the leading asymptotic behavior of the components for the gauge wave (93) and the gravitational waves (95) and (96). These choices lead to boundary data with the asymptotic behavior f (u) , R4 f (u) ∼ , R3 f (u) ∼ , R3 f (u) ∼ . R4
qK K ∼ qQ Q q Q Q¯ qK Q
Thus the behavior of q Q Q and q Q Q¯ imply that the resulting reflection coefficients have overall asymptotic dependence no weaker than κ = O(1/R 2 ). Our second choice, which is partially suggested by the electromagnetic case (83) and leads to weaker reflection, consists of the modifications 1 α β µ K K K ∂µ (r 2 γαβ ) r2 1 α β µ K Q K ∂µ (r 2 γαβ ) r2 1 α ¯β µ γ Q Q K ∂µ (r 2 γαβ ) − r2 r Q α Q β K µ ∂µ γαβ − Q α K β Q µ ∂µ γαβ
= qK K ,
(101)
= qK Q ,
(102)
= q Q Q¯ ,
(103)
= qQ Q .
(104)
Now q.. ∼ f (u)/R 4 for both gravitational quadrupole waves. For the gauge waves, q Q Q¯ ∼ f (u)/R 3 . Using the Regge-Wheeler-Zerilli perturbative formulation and the
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H.-O. Kreiss, O. Reula, O. Sarbach, J. Winicour
metric reconstruction method described in [35] we have independently checked that this leads to reflection coefficients κ = O(1/R 3 ) for the gravitational waves and κ = O(1/R 2 ) for the gauge waves in accord with (51). After replacing γµν = −h µν + h ˚ µ 2 ηµν , observing that K ∂µ r = 1 and identifying ∂µ with the connection ∇a of the background metric g˚ ab , (101)-(104) correspond to the boundary conditions (14)-(17) discussed in Sect. I C. Our third choice of boundary conditions, motivated by the first order version of the 0 boundary condition (77), is K µ ∂µ (r 2 K α K β γαβ ) = q K K , µ
α
(105)
β
K ∂µ (r Q Q γαβ ) = q Q Q , (106) µ α ¯β K ∂µ (r Q Q γαβ ) = q Q Q¯ , (107) 1 µ 1 1 K ∂µ (r 2 K α Q β γαβ ) − Q µ ∂µ K α K β γαβ + Q¯ µ Q α Q β ∂µ γαβ = q K Q . (108) r2 2 2 However, for the gravitational quadrupole wave (95), this leads to q K Q ∼ f (u)/R 2 and so it results in much stronger reflection than the first two choices. Thus, as might have been anticipated by the discussion following (91), the first order version of the boundary condition is not as effective as (104)-(101) in the case of a spherical boundary. V. Conclusion We have considered the IBVP for a coupled system of quasilinear wave equations and established (local in time) well posedness for a large class of boundary conditions. In particular, this allows for the formulation of a well posed IBVP for quasilinear wave systems in the presence of constraints on finite domains with artificial, nonreflecting boundaries. Therefore, we anticipate that our results will have application to a wide range of problems in computational physics. Furthermore, since our proof is based on a reduction to a symmetric hyperbolic system with maximal dissipative boundary conditions, it also lays the path for constructing stable finite difference discretizations for such systems. Our work has been motivated by the importance of the computation of gravitational waves from the inspiral and merger of binary black holes, which has enjoyed some recent success [36–40]. At present, however, none of the simulations of the binary black hole problem have been based upon a well posed IBVP. The closest example is the harmonic approach of the Caltech-Cornell group [41–43] which incorporates the freezing 0 boundary condition in second order form and has been shown to be well posed in the generalized sense in the high frequency limit [16]. Our results have potential application to improving the binary black hole simulations. However, many of these simulations are carried out using the BSSN formulation [44,45] of Einstein’s equations, which differs appreciably from the harmonic formulation considered here. Although our results constitute a complete analytic treatment of the IBVP for the harmonic formulation of Einstein’s equations, the extension to the BSSN formulation is not immediately evident. For this purpose, it would be useful to reformulate the boundary data for the harmonic problem in terms of the intrinsic geometry and extrinsic curvature of the boundary, as has been done for the initial data for the Cauchy problem. Such a geometric reformulation remains an outstanding problem.
Boundary Conditions for Coupled Quasilinear Wave Equations
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Acknowelegements. The work of O. R. was supported in part by CONICET, SECYT-UNC and NSF Grant INT0204937 to Louisiana State University. The work of O. S. was supported in part by grant CIC 4.19 to Universidad Michoacana, PROMEP UMICH-PTC-195 from SEP Mexico and CONACyT grant No. 61173. The work of J. W. was supported by NSF grant PH-0553597 to the University of Pittsburgh. During the course of this research we have profited from many discussions with H. Friedrich.
Appendix A: Some Useful Formulae Here we give a short summary of the formulae and conventions underlying the calculational results of Sects. IV B and IV C. We have ∂α f (u) = − f (u)K α , u = t − r,
K α ∂α K β = 0,
(A1)
so that f (u) f (u) 2 f (u) f (u) rα rβ = K α K β + 2 (K α rβ + rα K β ) + r r r r3
∂α ∂β
−(
f (u) f (u) + 2 )rαβ r r
(A2)
and K µ ∂µ ∂α ∂β
f (u) f (u) 2 f (u) = − 2 Kα Kβ − (K α rβ + rα K β ) r r r3
2 f (u) 3 f (u) 6 f (u) rαβ , + − 4 rα rβ + r r2 r3
(A3)
where rα := ∂α r and rαβ := ∂α ∂β r . The spatial components are ri =
xi x j δi j xi = (sin θ cos φ, sin θ sin φ, cosθ ), ri j = − 3 . r r r
(A4)
Our conventions for the polarization dyad give rise to the Cartesian components (Q x , Q y , Q z ) = (cos θ cos φ − i sin φ, cos θ sin φ + i cos φ, − sin θ ),
(A5)
which satisfy (Q x )2 + (Q y )2 = − sin2 θ, Qx
Qx
y x − Q y = −i sin θ, r r
y x + Q y = sin θ 2 cos θ cos φ sin φ + i(cos2 φ − sin2 φ) r r
(A6)
and Q j ri j =
Qi , r
Q j ∂ j Qi =
cot θ i Q, r
Q j ∂ j Q¯ i = −
cot θ ¯ i 2r j . Q − r r
(A7)
From these follow the necessary commutation relations such as [r Q µ ∂µ , K ν ∂ν ] = 0.
(A8)
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References 1. Sarbach, O.: Absorbing boundary conditions for Einstein’s field equations. J. Phys. Conf. Ser. 91, 012005 (2007) 2. Friedrich, H., Nagy, G.: The initial boundary value problem for Einstein’s vacuum field equations. Commun. Math. Phys. 201, 619–655 (1999) 3. Friedrichs, K.O.: Symmetric positive linear differential equations. Commun. Pure Appl. Math. 11, 333–418 (1958) 4. Lax, P.D., Phillips, R.S.: Local boundary conditions for dissipative symmetric linear differential operators. Commun. Pure Appl. Math. 13, 427–455 (1960) 5. Secchi, P.: Well-posedness of characteristic symmetric hyperbolic systems. Arch. Rat. Mech. Anal. 134, 155–197 (1996) 6. Kreiss, H.O., Winicour, J.: Problems which are well-posed in a generalized sense with applications to the Einstein equations. Class. Quant. Grav. 23, S405–S420 (2006) 7. Kreiss, H.O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23, 277–298 (1970) 8. Taylor, M.E.: Partial Differential Equations II, Qualitative Studies of Linear Equations. BerlinHeidelberg-New York: Springer, 1996 9. Kreiss, H.O., Reula, O., Sarbach, O., Winicour, J.: Well-posed initial-boundary value problem for the harmonic Einstein equations using energy estimates. Class. Quant. Grav. 24, 5973–5984 (2007) 10. Rauch, J.B., Massey, F.J. III.: Differentiability of solutions to hyperbolic initial-boundary value problems. Trans. Am. Math. Soc. 189, 303–318 (1974) 11. Kreiss, H-O., Wu, L.: On the stability definition of difference approximations for the initial boundary value problem. Appl. Num. Math. 12, 213–227 (1993) 12. Kreiss, H-O., Scherer, G.: Method of lines for hyperbolic differential equations. SIAM J. Numer. Anal. 29, 640–646 (1992) 13. Babiuc, M.C., Szilagyi, B., Winicour, J.: Harmonic initial-boundary evolution in general relativity. Phys. Rev. D 73, 064017(1)–064017(23) (2006) 14. Babiuc, M.C., Kreiss, H.-O., Winicour, J.: Constraint-preserving Sommerfeld conditions for the harmonic Einstein equations. Phys. Rev. D 75, 044002(1)–044002(13) (2007) 15. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space Time. Cambridge: Cambridge University Press, 1973 16. Ruiz, M., Rinne, O., Sarbach, O.: Outer boundary conditions for Einstein’s field equations in harmonic coordinates. Class. Quant. Grav. 24, 6349–6378 (2007) 17. Geroch, R.: Partial differential equations of physics. In: General Relativity: Proceedings. Edited by G.S. Hall, J.R. Pulham. Edinburgh: IOP Publishing, 1996, p. 19 18. Kreiss, H.O., Lorenz, J.: Initial-Boundary Value Problems and the Navier-Stokes Equations. London-New York: Academic Press, 1989 19. Winicour, J.: Characteristic evolution and matching. Liv. Rev. Rela. 28, 10 (2005) 20. Engquist, B., Majda, A.: Absorbing boundary conditions for the numerical simulation of waves. Math. Comp. 31, 629–651 (1977) 21. Higdon, R.L.: Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation. Math. Comput. 47(176), 437–459 (1986) 22. Trefethen, L.N., Halpern, L.: Well-posedness of one-way wave equations and absorbing boundary conditions. Math. Comput. 47, 421–435 (1986) 23. Blaschak, J., Kriegsmann, G.: A comparative study of absorbing boundary conditions. J. Comput. Phys. 77, 109–139 (1988) 24. Jiang, H., Wong, Y.S.: Absorbing boundary conditions for second order hyperbolic equations. J. Comput. Phys. 88(1), 205–231 (1990) 25. Renaut, R.A.: Absorbing boundary conditions, difference operators, and stability. J. Comput. Phys. 102(2), 236–251 (1992) 26. Hedstrom, G.W.: Nonreflecting boundary conditions for nonlinear hyperbolic systems. J. Comput. Phys. 30(2), E222–E237 (1979) 27. Givoli, D.: Non-reflecting boundary conditions. J. Comput. Phys. 94(1), 1–29 (1991) 28. Grote, M.J., Keller, J.B.: Nonreflecting boundary conditions for Maxwell’s equations. J. Comput. Phys. 139(2), 327–342 (1998) 29. Bayliss, A., Turkel, E.: Radiation boundary conditions for wavelike equations. Commun. Pure Appl. Math. 33, 707–725 (1980) 30. Lau, S.R.: Analytic structure of radiation boundary kernels for blackhole perturbations. J. Math. Phys. 46, 102503(1)–102503(21) (2005) 31. Buchman, L.T., Sarbach, O.C.A.: Towards absorbing outer boundaries in general relativity. Class. Quant. Grav. 23, 6709–6744 (2006)
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32. Buchman, L.T., Sarbach, O.C.A.: Improved outer boundary conditions for Einstein’s field equations. Class. Quant. Grav. 24, S307–S326 (2007) 33. Sachs, R., Bergmann, P.G.: Structure of particles in linearized gravitational theory. Phys. Rev. 112, 674–680 (1958) 34. Boardman, J., Bergmann, P.G.: Spherical gravitational waves. Phys. Rev. 115, 1318–1324 (1959) 35. Sarbach, O., Tiglio, M.: Gauge invariant perturbations of Schwarzschild black holes in horizon penetrating coordinates. Phys. Rev. D 64, 084016(1)–084016(15) (2001) 36. Pretorius, F.: Evolution of binary black-hole spacetimes. Phys. Rev. Lett. 95, 121101(1)–121101(4) (2005) 37. Campanelli, M., Lousto, C.O., Marronetti, P., Zlochower, Y.: Accurate evolutions of orbiting black-hole binaries without excision. Phys. Rev. Lett. 96, 111101(1)–111101(4) (2006) 38. Baker, J.G., Centrella, J., Choi, D.-I., Koppitz, M., van Meter, J.: Gravitational-wave extraction from an inspiraling configuration of merging black holes. Phys. Rev. Lett. 96, 111102(1)–111102(4) (2006) 39. Gonzalez, J.A., Sperhake, U., Bruegmann, B., Hannam, M., Husa, S.: Total recoil: the maximum kick from nonspinning black-hole binary inspiral. Phys. Rev. Lett. 98, 091101 (2007) 40. Szilagyi, B., Pollney, D., Rezzolla, L., Thornburg, J., Winicour, J.: An explicit harmonic code for blackhole evolution using excision. Class. Quant. Grav. 24, S275–S293 (2007) 41. Lindblom, L., Scheel, M.A., Kidder, L.E., Owen, R., Rinne, O.: A new generalized harmonic evolution system. Class. Quant. Grav. 23, S447–S462 (2006) 42. Rinne, O., Lindblom, L., Scheel, M.A.: Testing outer boundary treatments for the Einstein equations. Class. Quant. Grav. 24, 4053–4078 (2007) 43. Pfeiffer, H.P., Brown, D.A., Kidder, L.E., Lindblom, L., Lovelace, G., Scheel, M.A.: Reducing orbital eccentricity in binary black hole simulations. Class. Quant. Grav. 24, S59–S82 (2007) 44. Shibata, M., Nakamura, T.: Evolution of three-dimensional gravitational waves: Harmonic slicing case. Phys. Rev. D 52, 5428–5444 (1995) 45. Baumgarte, T.W., Shapiro, S.L.: On the numerical integration of Einstein’s field equations. Phys. Rev. D 59, 024007(1)–024007(7) (1999) Communicated by G. W. Gibbons
Commun. Math. Phys. 289, 1131–1150 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0764-x
Communications in
Mathematical Physics
Equidistribution of Eisenstein Series in the Level Aspect Shin-ya Koyama The Institute of Mathematical Sciences, Ewha Womans University, Daehyun-dong 11-1, Sedaemoon-ku, 120-750 Seoul, South Korea. E-mail: [email protected] Received: 29 July 2008 / Accepted: 4 December 2008 Published online: 28 February 2009 – © Springer-Verlag 2009
Abstract: We prove an equidistribution property of the Eisenstein series for congruence subgroups as the level goes to infinity. This is an analogy of the phenomenon called quantum ergodicity.
1. Introduction Equidistribution of Eisenstein series was discovered by Luo and Sarnak [LS] in 1995. In the remarkable paper they showed for the Eisenstein series E(z, s) for the modular surface X = PSL(2, Z)\H 2 with H 2 the upper half plane that lim A
t→∞
B
|E(z, 21 + it)|2 d V (z) |E(z,
1 2
+ it)|2 d V (z)
=
vol(A) vol(B)
(1.1)
1 d xd y with A, B compact Jordan measurable subsets of X and d V = vol(X ) y 2 the normalized d xd y π volume measure with vol(X ) = X y 2 = 3 . The phenomenon (1.1) is an analogue of so-called quantum ergodicity, which is described as follows. Let u j be eigenfunctions of the Laplacian and all Hecke oper ators with the normalization X u i (z)u j (z)d V (z) = δi, j . Quantum ergodicity means the property that the probability measures dµ j = |u j (z)|2 d V (z) converge weak-∗ to d V (z). For any X = \P S L 2 (R) with any congruence lattice, Lindenstrass [L] proved quantum ergodicity for compact cases. He also proved for non-compact cases, that is when is a congruence subgroup, that any limit point of the sequence µ j is a multiple of d V .
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S. Koyama
The result (1.1) is regarded as a version of quantum ergodicity for continuous spectra. They actually proved 2 E(z, 1 + it) d V (z) ∼ 48 vol(A) log t 2 π vol(X ) A as t → ∞. In the proof they essentially used a subconvexity estimate for the automorphic L-function for a Maass cusp form u: 1
L( 21 + it, u) = O(t 3 +ε )
(t → ∞),
due to Meurman [M]. Such equidistribution was generalized to three-dimensional hyperbolic spaces by the author [K] by use of the subconvexity estimate of Petridis and Sarnak [PS]. According to various aspects of estimates of L-functions, it is possible to consider analogues of such equidistribution. For example, Rudnick and Sarnak [RS] posed a conjecture of quantum ergodicity in the weight aspect ([KMV] Conjecture 1.4), and Sarnak [S] obtained a subconvexity bound for the Rankin-Selberg convolution L-function along this direction. An analogue of this conjecture in the level aspect is formulated for holomorphic cusp forms by Kowalski, Michel and Vanderkam in [KMV] as follows. Here we put πq : X q → X 1 to be the canonical projection with X q = 0 (q)\H 2 . Conjecture 1.1 ([KMV] Conjecture 1.5). For k ≥ 2 even and fixed, let { f j } j≥1 be any sequence of primitive holomorphic forms of weight k with increasing levels q j . As j → ∞ the sequence of probability measure πq j ,∗ (µ f j ), j ≥ 1 converge weakly to the Poincaré measure d V (z) on X 1 , where µ f j = Petersson inner product.
3 | f (z)|2 k d xd y π ( f, f ) y y 2
and ( f, f ) is the
In this paper we deal with a relevant problem for the Eisenstein series E q,κ (z, s). It is defined for 0 (q) at a cusp κ as (Imσκ−1 γ z)s , E q,κ (z, s) := γ ∈q,κ \0 (q)
where z ∈ H 2 , σκ ∞ = κ and q,κ is the stabilizer of the cusp κ in 0 (q). It is absolutely convergent in Re(s) > 1 and has a meromorphic continuation to s ∈ C. For each prime number q there exist only two inequivalent cusps κ = 0, i∞. A result corresponding to Conjecture 1.1 is summarized as follows, which is a consequence from our main result (Theorem 1.3). Theorem 1.2. Let q be a prime number or q = 1. For a congruence surface X q = 0 (q)\H 2 , we denote by πq : X q → X 1 the natural projection from X q onto X 1 . For a subset A = A1 ⊂ X 1 , we denote Aq := πq−1 (A). For an arbitrary sequence of cusps {κ(q) | κ(q) = 0, i∞ for each prime q} and any compact Jordan measurable subsets A, B ⊂ X 1 , we have 1 2 vol(Aq ) vol(A) Aq |E q,κ(q) (z, 2 + it)| d Vq (z) = , (1.2) = lim lim 1 2 q→∞ q→∞ vol(Bq ) vol(B) B |E q,κ(q) (z, 2 + it)| d Vq (z) q
Equidistribution of Eisenstein Series in the Level Aspect
1133
d xd y 1 where d Vq = vol(X is the normalized volume measure on X q with q ) y2 vol(X q ) = X q d V (z).
This statement is an analog of Conjecture 1.1 in the sense that it concerns the average behavior of the Eisenstein series on the domain Aq . The goal of this paper is to establish the following more precise result. Theorem 1.3. Let q be a prime number or q = 1. For a congruence surface X q = 0 (q)\H 2 , we denote by πq : X q → X 1 the natural projection from X q onto X 1 . For a connected compact Jordan measurable subset A = A1 ⊂ X 1 , we put {Aq0 | q : prime} to be an arbitrary sequence of connected components of πq−1 (A). For an arbitrary sequence of cusps {κ(q) | κ(q) = 0, i∞ for each prime q} and any connected compact Jordan measurable subsets A, B ⊂ X 1 , we have 1 2 vol(Aq0 ) Aq0 |E q,κ(q) (z, 2 + it)| d Vq (z) vol(A) = , (1.3) = lim lim 1 0 2 q→∞ q→∞ vol(Bq ) vol(B) B 0 |E q,κ(q) (z, 2 + it)| d Vq (z) q
where d Vq = X q d V (z).
1 d xd y volX q y 2
is the normalized volume measure on X q with vol(X q ) =
Actually we prove an asymptotic Aq0
0 E q,κ(q) (z, 1 + it)2 d Vq (z) ∼ 24vol(A ) log q ∼ 24vol(A) log q 2 vol(X ) vol(X ) q
(q → ∞).
The proofs basically follow the method of [LS]. We compute the integral f (z)|E q,κ(q) (z, 21 + it)|2 d Vq (z) Xq
for f (z) ∈ L 2 (X q ) for the proof of Theorem 1.3. If we carry out the calculation under the restriction to f ∈ L 2old (X q ), we obtain Theorem 1.2. Here L 2old (X q ) is the space of oldforms. By the spectral decomposition and the standard approximation argument used in [LS], the problem is reduced to the cases when f (z) is a cusp form u(z) or an incomplete Eisenstein series Fq,κ(q) (z|h) defined later. Indeed we will prove Proposition 1.4 below for cusp forms, and Proposition 1.5 for incomplete Eisenstein series: Proposition 1.4. Let u j,q be a normalized Hecke eigen Maass cusp form for 0 (q) with u j,q = λ j,q u j,q = ( 41 + r 2j )u j,q (λ j,q ≥ 0) with the Laplacian with the normali zation X q u i,q (z)u j,q (z)d Vq (z) = δi, j . We also assume that u j,q is diagonalized with respect to the antiholomorphic involution ι : H 2 x + i y → −x + i y ∈ H 2 . Then we have u j,q (z)|E q,κ(q) (z, 21 + it)|2 d Vq (z) = o(q −1 ) (q → ∞). Xq
More precisely, for any sequence in {( j, q) | j ∈ Z, j > 0, q : prime}
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S. Koyama
with q → ∞, it holds that u j,q (z)|E q,κ(q) (z, 21 + it)|2 d Vq (z) Xq
⎧ 3 ⎨O q− 2 (u j,q ∈ L 2old (X q ) for all ( j, q)) = 2 ⎩ O q −1− 23041 +ε (u j,q ∈ L 2 (X q )).
2 The exponent − 23041 comes from the bound of Duke-Friedlander-Iwaniec [DFI].
Proposition 1.5. Let Fq ∈ C ∞ (X q ) be of the form Fq,κ(q) (z|h). Then as q → ∞,
2
log q 1 log q . +O Fq (z) E q,κ(q) (z, + it) d Vq (z) = 24 Fq (z)d Vq 2 q q2 Xq Xq 2. Contribution from Cusp Forms In this section we carry out the calculation of u j (z)|E q,κ(q) (z, 21 + it)|2 d Vq (z), Xq
where u j = u j,q is a Hecke eigen Maass cusp form for 0 (q) with u j = λ j u j = ( 41 + r 2j )u j (λ j ≥ 0) for the Laplacian with the normalization X q u i (z)u j (z)d Vq (z) = δi, j . We also assume that u j is diagonalized with respect to the antiholomorphic involution ι : H 2 x + i y → −x + i y ∈ H 2 . We fix t ∈ R and put u j (z)E q,κ(q) (z, 21 + it)E q,κ(q) (z, 21 − it)d Vq (z) J j (q) := Xq
and
I j (q, s) := Xq
u j (z)E q,κ(q) (z, 21 + it)E q,κ(q) (z, s)d Vq (z).
We note J j (q) = I j (q, 21 − it). By unfolding we have u j (z)E q,κ(q) (z, 21 + it) I j (q, s) = =
0 (q)\H 2
q,κ \H 2
Replacing σκ−1 z by z, I j (q, s) = =
(Imσκ−1 γ z)s d Vq (z)
γ ∈q,κ \0 (q)
u j (z)E q,κ(q) (z, 21 + it)(Imσκ−1 z)s d Vq (z).
q,∞ \H 2 ∞ 1 0
0
u j (σκ z)E q,κ(q) (σκ z, 21 + it)(Imz)s d Vq (z)
u j (σκ z)E q,κ(q) (σκ z, 21 + it)y s
d xd y . vol(X q )y 2
(2.1)
Equidistribution of Eisenstein Series in the Level Aspect
1135
The Fourier expansion of E q,κ(q) (z, s) at cusp κ is known ([I, Theorem 3.4]): 1
1
E q,κ (σκ z, 21 + it) = y 2 +it + ϕq,κ,κ ( 21 + it)y 2 −it +
n =0
ϕq,κ,κ (n, 21 + it)W 1 +it (nz) 2
(2.2) √ with W 1 +it (nz) = 2 |n|y K it (2π |n|y)e(nx), where 2
π 2 (s − 21 ) Sq,κ,κ (0, 0, c) , ϕq,κ,κ (s) := (s) c2s c 1
ϕq,κ,κ (n, s) :=
π s n s−1 Sq,κ,κ (0, n, c) , (s) c c2s
(2.3) (2.4)
with Sq,κ,κ (0, n, c) :=
e
na
a ∗ ∈B\σ −1 (q)σ /B 0 κ κ cd
c
(2.5)
and B=
1b 01
: b∈Z .
Lemma of Sq,κ,κ (0, n, √ 2.1. The value √ c) vanishes unless c ∈ Z for κ = κ , and c ∈ qZ for κ = κ . Putting γ := c/ q, we have for n = 0, ⎧ c ⎨ µ( (c,n) )ϕ(c) (q | c) c ϕ( (c,n) ) Sq,κ,κ (0, n, c) = , ⎩0 (q | c)
and for κ = κ , Sq,κ,κ (0, n, c) =
⎧ ⎨0 ⎩
µ( (γγ,n) )ϕ(γ ) ϕ( (γγ,n) )
(q | γ ) (q | γ )
with µ and ϕ the Möbius and the Euler functions. For n = 0 it holds that ϕ(c) (q | c) Sq,κ,κ (0, 0, c) = , 0 (q | c) and for κ = κ Sq,κ,κ (0, 0, c) =
0 (q | γ ) ϕ(γ ) (q | γ ).
1136
S. Koyama
Proof. a ∗ We first compute the case κ = κ . When q c, there does not exist a matrix c d ∈ 0 (q), and so Sq,κ,∞ (0, n, c) = 0. Hereafter we compute for the case q | c. The set of representatives of B\σκ−1 0 (q)σκ /B is obtained by [I, Theorem 2.7]:
10 01
∗∗ ∪ ∈ 0 (q) c > 0, 1 ≤ d < c, (c, d) = 1 . cd
(2.6)
By the formula in [IK, (3.3)], we compute that Sq,∞,∞ (0, n, c) is the Ramanujan sum: Sq,∞,∞ (0, n, c) =
e
1≤d
na c
=
µ
c δ
δ|(c,n)
δ=
c µ( (c,n) )ϕ(c) c ϕ( (c,n) )
.
Next we treat the case κ = κ . The set of representatives of B\σa−1 0 (q)σκ /B is obtained from [I, Theorem 2.7]: √ √ α √q β√ q −1 (2.7) ∈ σ0 0 (q) αδq ≡ 1 (mod γ ) . γ q δ q We deduce that αδq ≡ 1
(mod γ ) ⇔ q |γ ⇔ q |γ
αδ ≡ q −1 (mod γ ) δ ∈ (Z/γ Z)∗ ,
and and
to get the representatives as
∗ ∗ √ √ γ qδ q
∈ σ0−1 0 (q) γ > 0, q |γ , 1 ≤ δ < γ , (γ , δ) = 1 .
Thus we again have the Ramanujan sum. When n = 0, the results follow from (2.6) and (2.7). Lemma 2.2. If we put an (c) :=
c µ( (c,n) )ϕ(c) c ϕ( (c,n) )
,
we have L n (s) :=
∞ an (c) c=1
cs
=
σ1−s (n) . ζ (s)
Proof. By an (1) = 1 and multiplicativity of µ and ϕ, we find an (c) is multiplicative as a function in c. So we have an Euler product
an ( p) an ( p 2 ) L n (s) = 1+ + + ··· . ps p 2s p: prime
Equidistribution of Eisenstein Series in the Level Aspect
1137
Here a( p j ) ( j = 1, 2, 3, . . .) is calculated as follows. Put α = α p to be the exponent such that p α p n. Then j
an ( p ) = j
=
µ( ( ppj ,n) )ϕ( p j ) j
ϕ( ( ppj ,n) )
⎧ ⎪ µ( p j ) ⎪ ⎨
( p |n, i.e. α = 0)
µ(1)ϕ( p j ) ϕ(1) ⎪ ⎪ )ϕ( p j ) ⎩ µ( p j−αj−α ϕ( p )
⎧ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨0 = p j−1 ( p − 1) ⎪ ⎪ ⎪ − pα ⎪ ⎪ ⎩ 0
( p α n, j ≤ α) ( p α n, 1 ≤ α < j) ( p |n, ( p |n, ( p α n, ( p α n, ( p α n,
j = 1) j ≥ 2) j ≤ α) j = α + 1) j ≥ α + 2).
For prime divisors of n, the Euler factors are taken as follows:
p − 1 p( p − 1) + + ps p2s p|n 1 1 1 + s−1 + 2(s−1) = p p p|n 1 1 1 + s−1 + 2(s−1) = p p p|n
1 1− s . = σ1−s (n) p 1+
p2 ( p − 1) pα−1 ( p − 1) pα + · · · + − pαs p3s p(α+1)s
1 1 1 1 1 + · · · + α(s−1) − + + + · · · + ps p2s−1 p3s−2 p p(α+1)s−α
1 1 1− s + · · · + α(s−1) p p
p|n
Therefore
1 σ1−s (n) 1− s = . L n (s) = σ1−s (n) p ζ (s) p primes
Lemma 2.3. Let α = α p satisfy that q α n. Then it holds that ⎧
σ1−2s ( qnα ) ⎪ ⎪ ⎪ σ1−2s (n) − (κ = κ ) ⎪ −2s ⎪ ∞ 1 − q ⎨ Sq,κ,κ (0, n, c) 1 × = c2s ζ (2s) ⎪ ⎪ c=1 ⎪ σ1−2s ( qnα ) ⎪ ⎪ ⎩ (κ = κ ). q s (1 − q −2s ) Namely, we have ϕq,κ,κ (n, s) =
n s−1 ηq,κ,κ (n, s), ζˆ (2s)
1138
S. Koyama
where
ηq,κ,κ (n, s) =
⎧ σ1−2s ( qnα ) ⎪ ⎪ ⎪ σ (n) − (κ = κ ) ⎪ 1−2s ⎪ ⎨ 1 − q −2s ⎪ ⎪ σ1−2s ( qnα ) ⎪ ⎪ ⎪ ⎩ s q (1 − q −2s )
(κ = κ )
with ζˆ (s) = π −s/2 (s/2)ζ (s). Proof. We first deal with the case κ = κ . By the previous lemma putting c = γ have ∞ Sq,κ,κ (0, n, c) 1 an (γ ) L n (2s) . = s = s 2s 2s c q γ q × (the q-Euler factor)
√ q we
q |γ
c=1
As calculated in the proof of the previous lemma, (the q-Euler factor) = 1 − q −2s σ1−2s (q α ).
(2.8)
Hence we have the result for κ = κ . When κ = κ we compute ∞ Sq,κ,κ (0, n, c) c=1
c2s
=
an (c) q|c ∞
c2s
an (c) an (c) − c2s c2s c=1 q |c
1 . = L n (2s) 1 − (the q-Euler factor) =
We again appeal to (2.8) and reach the conclusion.
The Fourier expansion of u j at the cusp i∞ is given by √ ρ j (n)K ir j (2π ny)e(nx) u j (z) = y
(2.9)
n =0
with ρ j (n) the Fourier coefficient and λ j = 14 + r 2j . When u j is a newform, put ρ˜ j (n) = ρ j (n)/ρ j (1), which is multiplicative. In this case the automorphic L-function is defined by L(s, u j ) =
∞ ρ˜ j (n) n=1
ns
for Re(s) > 1. It is meromorphic in s ∈ C and is expressed as an Euler product −1 1 − ρ˜ j ( p) p −s + χ0 ( p) p −2s , L(s, u j ) = where χ0 ( p) =
p: prime
0 ( p = q) 1 ( p = q).
Equidistribution of Eisenstein Series in the Level Aspect
1139
On the other hand, when u j is an oldform, either u j (z) is a cusp form for SL(2, Z), or there exists a cusp form v j for S L(2, Z) such that u j (z) = v j (qz). In the former case, ρ˜ j (n) is similarly defined and L(s, u j ) has an Euler product as above. In the latter case, putting √ v j (z) = y τ j (n)K ir j (2π ny)e(nx), (2.10) n =0
we have u j (z) = v j (qz) =
√ qy τ j (n)K ir j (2πqny)e(qnx), n =0
and thus
ρ j (n) =
τ j (n/q) (q|n) . 0 (otherwise)
Hence for an oldform u j , ∞ ρ j (n) n=1
ns
=
∞ ρ j (qn) n=1
(qn)s
= q −s
∞ τ j (n) n=1
ns
= q −s τ j (1)
∞ τ˜ j (n) n=1
ns
= q −s τ j (1)L(s, v j )
−1 = q −s τ j (1) 1 − τ˜ j ( p) p −s + χ0 ( p) p −2s . p: prime
Lemma 2.4. (1) When u j is either a newform for 0 (q) or a cusp form for S L(2, Z), we have ∞ ρ˜ j (n)σν (n) n=1 ∞
ns
=
ρ˜ j (n)σν ( qnα ) ns
n=1
L(s, u j )L(s − ν, u j ) , ζ (2s − ν)(1 − q −2s+ν )
=
L(s, u j )L(s − ν, u j ) 1 − ρ˜ j (q)q −(s−ν) × . ζ (2s − ν) 1 − q −2s+ν
(2.11)
(2.12)
(2) When u j is an oldform for 0 (q) and expressed by u j (z) = v j (qz) with v j a cusp form for S L(2, Z), we have ∞ ρ j (n)σν (n) n=1
ns
=
τ j (1)L(s, v j )L(s − ν, v j ) 1 − τ˜ j (q)q −s+ν − q 2ν (1 − τ˜ j (q)q −s ) , q s ζ (2s − ν)(1 − q ν ) 1 − q −2s+ν (2.13)
and ∞ ρ (n)σ ( n ) j ν qα n=1
ns
=
τ j (1)σν (q)(1 − τ˜ j (q)q −(s−ν) ) L(s, v j )L(s − ν, v j ) . q s (1 − q −2s+ν ) ζ (2s − ν) (2.14)
1140
S. Koyama
Proof. The first identity (2.11) is shown by ∞ ρ˜ j (n)σν (n)
ns
n=1
∞ ρ˜ j ( p k )σν ( p k ) p ks
=
p: prime k=0
∞ ρ˜ j ( p k )(1 + p ν + · · · + p kν ) p ks
=
p: prime k=0
∞ ρ˜ j ( p k )(1 − p ν(k+1) ) p ks (1 − p ν )
=
p: prime k=0
=
p: prime
=
p: prime
− =
1 1 − pν
∞ ρ˜ j ( p k )
p ks
k=0
∞ ρ˜ j ( p k ) −p p k(s−ν)
ν
k=0
1 1 − ρ˜ j
( p) p −s
+ χ0 ( p) p −2s
pν −(s−ν) 1 − ρ˜ j ( p) p + χ0 ( p) p −2(s−ν)
p: prime
=
1 1 − pν
(1 − ρ˜ j ( p) p −s
1 − p −2s+ν χ0 ( p) + χ0 ( p) p −2s )(1 − ρ˜ j ( p) p −(s−ν) + χ0 ( p) p −2(s−ν) )
L(s, u j )L(s − ν, u j ) . ζ (2s − ν)(1 − q −2s+ν )
The second identity (2.12) is proved by (2.11) as ∞ ρ˜ (n)σ ( n ) j ν qα n=1
ns
⎞
∞ ∞ k )σ ( p k ) k ρ ˜ ( p j ν ⎟ ρ˜ j (q ) ⎜ =⎝ ⎠ p ks q ks p: prime ⎛
p =q
=
k=0
k=0
L(s, u j )L(s − ν, u j ) 1 − ρ˜ j (q)q −(s−ν) × . ζ (2s − ν) 1 − q −2s+ν
The third identity (2.13) is shown as follows: ∞ ρ j (n)σν (n) n=1
ns
=
∞ ρ j (qn)σν (qn)
(qn)s
n=1
= q −s
∞ τ j (n)σν (qn) n=1
ns
Equidistribution of Eisenstein Series in the Level Aspect
= q −s τ j (1)σν (q)
1141 ∞ τ˜ j (n)σ˜ ν (qn)
ns
n=1
=q
∞ τ˜ j ( p k )σ˜ ν (qp k ) τ j (1)σν (q) , p ks
−s
p: prime k=0
where σ˜ ν (qn) = σν (qn)/σν (q) which is multiplicative in n. Here ⎧ ν 2ν (k+1)ν ⎨1 + q + q + · · · + q ( p = q) k σ˜ ν (qp ) = 1 + qν ⎩ ν 2ν kν 1 + p + p + ··· + p ( p = q) ⎧ (k+2)ν ⎪ ⎪ 1−q ( p = q) ⎨ ν )(1 − q ν ) . = (1 + q(k+1)ν 1− p ⎪ ⎪ ⎩ ( p = q) 1 − pν Thus ∞ ρ j (n)σν (n)
ns
n=1
⎛
∞
⎜ = q −s τ j (1)σν (q) ⎝ p: prime p =q
⎛
=
τ j (1) ⎝ qs
k=0
⎞ p (k+1)ν
( pk )
τ˜ j 1− ks p 1 − pν
∞
k (k+2)ν ⎟ τ˜ j (q ) 1 − q ⎠ q ks (1 + q ν )(1 − q ν ) k=0
⎞ ∞ ∞ k τ˜ j ( p ) τ˜ j (q k ) 1 ⎠⎜ (k+1)ν ⎟ (1− p ) (1−q (k+2)ν ). ⎠ ⎝ ks ks 1− p ν p q p: prime ⎞⎛
p: prime
p =q
k=0
k=0
Since for p = q, ∞ τ˜ j ( p k ) k=0
p ks
(1 − p (k+1)ν ) =
∞ τ˜ j ( p k ) k=0
p ks
−
∞ τ˜ j ( p k ) p k(s−ν)−ν k=0
1 pν = − −s −2s −s+ν 1 − τ˜ j ( p) p + p 1 − τ˜ j ( p) p + p −2s+2ν =
(1 − p −2s+ν )(1 − p ν ) , (1 − τ˜ j ( p) p −s + p −2s )(1 − τ˜ j ( p) p −s+ν + p −2s+2ν )
and for p = q, ∞ τ˜ j (q k ) k=0
q ks
(1 − q (k+2)ν ) =
∞ τ˜ j (q k ) k=0
=
q ks
−
∞ k=0
τ˜ j (q k ) q k(s−ν)−2ν
1 q 2ν − , 1 − τ˜ j (q)q s 1 − τ˜ j (q)q −s+ν
1142
S. Koyama
it holds that ∞ ρ j (n)σν (n) ns
n=1
⎞ −2s+ν ν τ j (1)ζ (−ν) ⎜ (1 − p )(1 − p ) ⎟ = ⎝ −s + p −2s )(1 − τ˜ ( p) p −s+ν + p −2s+2ν ) ⎠ qs (1 − τ ˜ ( p) p j j p: prime ⎛
p =q
1 q 2ν × − 1 − τ˜ j (q)q s 1 − τ˜ j (q)q −s+ν τ j (1)L(s, v j )L(s − ν, v j ) = q s ζ (2s − ν)
(1 − τ˜ j (q)q −s )(1 − τ˜ j (q)q −s+ν ) 1 q 2ν × − (1 − q −2s+ν )(1 − q ν ) 1 − τ˜ j (q)q −s 1 − τ˜ j (q)q −s+ν =
τ j (1)L(s, v j )L(s − ν, v j ) 1 − τ˜ j (q)q −s+ν − q 2ν (1 − τ˜ j (q)q −s ) . q s ζ (2s − ν)(1 − q ν ) 1 − q −2s+ν
Finally (2.14) is shown from (2.12) as follows: ∞ ρ (n)σ ( n ) j ν qα n=1
ns
=
∞ ρ (qn)σ ( qn ) j ν qα n=1
= σν (q)
(qn)s ∞ τ (n)σ ( n ) j ν qα n=1
(qn)s ∞
=
τ j (1)σν (q) τ˜ j (n)σν ( q α ) qs ns n
n=1
τ j (1)σν (q)(1 − τ˜ j (q)q −(s−ν) ) L(s, v j )L(s − ν, v j ) . = q s (1 − q −2s+ν ) ζ (2s − ν) Lemma 2.5. Assume Re(s) = 1/2. Then we estimate as q → ∞ that I j (q, s) ⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ρ j (1)L(s − it, u j )L(s + it, u j ) ⎪ ⎪ ⎪ ⎪ ⎨ vol(X q )q 21 ζˆ (1 + 2it)ζ (2s) ∞ 1 = s dy ⎪ × 1 + O q− 4 K (2π y)K (2π y)y ⎪ it ir j ⎪ ⎪ y 0 ⎪ ⎪ 1 ⎪ 1 ⎪ ⎪ ⎩ O q− 2 vol(X q )
(if u j is an odd form)
(if u j is an even newform) (if u j is an even oldform).
Proof. Applying (2.9) and (2.2) into (2.1), we find that ϕq,κ,κ (n, 1 + it)ρ j (n) ∞ dy 2 2 K it (2π y)K ir j (2π y)y s , I j (q, s) = 1 s− vol(X q ) y 0 |n| 2 n =0
Equidistribution of Eisenstein Series in the Level Aspect
1143
where by the previous lemma 1
n − 2 +it ηq,κ,κ (n, 21 + it) 1 . ϕq,κ,κ (n, + it) = 2 ζˆ (1 + 2it) When u j is odd, then ρ j (n) = −ρ j (−n) gives I j (q, s) = 0. Hereafter we assume u j is even. The sum over n is ∞ ∞ ϕq,κ,κ (n, 21 + it)ρ j (n) ηq,κ,κ (n, 21 + it)ρ j (n) 1 = . (2.15) 1 n s−it ζˆ (1 + 2it) n=1 n s− 2 n=1 Since σ−2it ( qnα ) 1 + it) = σ−2it (n) − 2 1 − q −1−2it by Lemma 2.3, we compute from Lemma 2.4 for u j being either a newform for 0 (q) or a cusp form for SL(2, Z) that
∞ n ∞ ρ (n)σ ρ j (n)σ−2it (n) j −2it ( q α ) 1 1 (2.15) = − n s−it 1 − q −1−2it n s−it ζˆ (1 + 2it) n=1 n=1
1 − ρ˜ j (q)q −(s+it) L(s − it, u j )L(s + it, u j ) 1− = ρ j (1) (1 − q −1−2it )(1 − q −2s ) ζˆ (1 + 2it)ζ (2s) L(s − it, u j )L(s + it, u j ) ρ˜ j (q)q −(s+it) 1 + O q −1 . = ρ j (1) ζˆ (1 + 2it)ζ (2s) In particular, when u j is a cusp form for SL(2, Z), the value L(s − it, u j )L(s + it, u j ) is bounded as q → ∞, and thus 1 (2.15) = O q − 2 , ηq,κ,κ (n,
when Re(s) = 1/2. On the other hand, when u j is an oldform expressed by u j (z) = v j (qz) with v j a cusp form of S L(2, Z), we compute
∞ n ∞ ρ (n)σ ρ j (n)σ−2it (n) j −2it ( q α ) 1 1 (2.15) = − n s−it 1 − q −1−2it n s−it ζˆ (1 + 2it) n=1
n=1
τ j (1)L(s − it, v j )L(s + it, v j ) = q s−it ζˆ (1 + 2it)ζ (2s)(1 − q −2s )
1− τ˜ j (q)q −s−it − q −4it (1− τ˜ j (q)q −s+it ) σ−2it (q)(1 − τ˜ j (q)q −(s+it) ) − × 1 − q −2it 1 − q −1−2it τ j (1)L(s − it, v j )L(s + it, v j ) q s−it ζˆ (1 + 2it)ζ (2s)(1 − q −2s ) × 1 + q −2it − τ˜ j (q)q −s−it − σ−2it (q)(1 − τ˜ j (q)q −(s+it) ) 1 + O(q −1 ) τ j (1)L(s − it, v j )L(s + it, v j ) τ˜ j (q) 1 + O(q −1 ) = q 2(s+it) ζˆ (1 + 2it)ζ (2s)(1 − q −2s ) 3 = O q− 4 , =
1144
S. Koyama
where the last identity holds for Re(s) = 1/2, and where we used the fact that the value 1 L(s −it, v j )L(s +it, v j ) does not grow as q → ∞, and that τ˜ j (q) = O(q 4 ) as q → ∞. Proposition 2.6. Assume Re(s) = 1/2. Then we have lim vol(X q )I j (q, s) = 0.
q→∞
More precisely, as q → ∞,
⎧ 1 ⎨O q− 2 (u j ∈ L 2old (X q )) vol(X q )I j (q, s) = 2 +ε ⎩ O q − 23041 (u j ∈ L 2 (X q )).
Proof. When u j ∈ L 2old (X q ), the result follows from the previous lemma. When u j ∈ L 2 (X q ), the subconvexity estimate due to Duke-Friedlander-Iwaniec [DFI] gives 1
L(s, u j ) = O(q 4 −δ ) 1 . 23041 The Hoffstein-Lockhart estimate [HL]
with δ =
ρ j (1) = O(q ε ) leads to vol(X q )I j (q, s) = O(q −2δ+ε ). Since vol(X q ) =
π 3 (q
+ 1) for q > 1, Proposition 1.4 follows.
3. Contribution from Incomplete Eisenstein Series Let h(y) ∈ C ∞ (R+ ) be a rapidly decreasing function at 0 and ∞, that is h(y) = O N (y N ) as y → 0 or ∞ and N ∈ Z. Let H (s) be its Mellin transform ∞ dy h(y)y −s . H (s) = y 0 We see H (s) is entire in s and is of Schwartz class in t = Im(s) for fixed σ = Re(s). The inversion formula gives 1 h(y) = H (s)y s ds 2πi (σ ) for any σ ∈ R. In this section we compute the integral 2 1 Fq,κ(q) (z|h) E q,κ(q) (z, + it) d Vq (z), 2 Xq
Equidistribution of Eisenstein Series in the Level Aspect
1145
where Fq,κ(q) (z|h) is the incomplete Eisenstein series at cusp κ: h(Im(σκ−1 γ z)) Fq,κ(q) (z|h) = γ ∈q,κ(q) \0 (q)
=
1 2πi
(2)
H (s)E q,κ(q) (z, s)ds.
Proposition 3.1. Let Fq ∈ C ∞ (X q ) be of the form Fq,κ(q) (z|h). Then as q → ∞,
2
log q log q 1 . +O Fq (z) E q,κ(q) (z, + it) d Vq (z) = 24 Fq (z)d Vq 2 q q2 Xq Xq Proof. It holds that Fq,κ(q) (z|h)(z) is rapidly decreasing in the cusps. Hence 2 1 Fq,κ(q) (z|h) E q,κ(q) (z, + it) d Vq (z) 2 Xq 2 1 1 = H (s)E q,κ(q) (z, s)ds E q,κ(q) (z, + it) d Vq (z) 2πi X q (2) 2 2 1 1 −1 s = H (s) (Imσκ γ z) ds E q,κ(q) (z, + it) d Vq (z) 2πi X q (2) 2 γ ∈q,κ \0 (q) 2 1 1 H (s)(Imσκ−1 z)s ds E q,κ(q) (z, + it) d Vq (z) = 2πi q,κ \H 2 (2) 2 2 1 ∞ 1 d xd y 1 s E q,κ(q) (σκ z, + it) = H (s)y ds , 2πi 0 2 vol(X q )y 2 (2) 0 where in the last identity we replaced σκ−1 z with z. It equals by the Fourier expansion
2 1 +it 1 vol(X q )−1 ∞ 1 s −it 2 2 H (s)y ds y + ϕq,κ,κ ( + it)y 2πi 2 (2) 0 2 ∞ dy 1 + . ϕq,κ,κ (n, 2 + it)W 21 +it (nz) y2 n=1
We put this as vol(X q )−1 (I1 + I2 ), where 2 ∞ dy 1 1 1 1 H (s)y s ds y 2 +it + ϕq,κ,κ ( + it)y 2 −it 2 I1 = 2πi 0 2 y (2) ∞ 1 2 1 + |ϕq,κ,κ ( 2 + it)| + 2Re(ϕq,κ,κ ( 21 + it)y 2it ) 1 dy = H (s)y s ds 2πi 0 y (2) and 1 I2 = 2πi
0
2 ∞ dy 1 H (s)y ds ϕq,κ,κ (n, 2 + it)W 21 +it (nz) y 2 . (2)
∞
s
n=1
1146
S. Koyama
We first compute I1 . By the definition (2.3) and Lemma 2.1, we have ϕq,κ,κ (w) = π 1/2
∞ (w − 21 ) ϕ(qc) (w) (qc)2w c=1
(w − 21 ) ζ (2w − 1) q 2w − q = π 1/2 (w) ζ (2w) q 2w − 1 ζˆ (2w − 1) q 2w − q . = ζˆ (2w) q 2w − 1 When w =
1 2
+ it, we estimate that
ζˆ (2it) 2| sin(t log q)| 1 ϕq,κ,κ ( + it) = 1 + O(q −1 ) = O(q −1 ) (κ = 0, ∞). 2 q ζˆ (1 + 2it) Then we have 2 ϕq,κ,κ ( 1 + it) = O(q −2 ) 2 and that
1 2it = O(q −1 ) Re ϕq,κ,κ ( + it)y 2 as q → ∞. Therefore we prove that I1 = =
1 2πi ∞ 0
∞
0
(2)
H (s)y s ds
dy + O(q −1 ) y
h(y) dy + O(q −1 ) y
as q → ∞. Next we deal with I2 . By the definition (2.4) we have 2 2 4y K it (2π ny)ηq,κ,κ (n, 21 + it) ϕq,κ,κ (n, 1 + it)W 1 (nz) = , 2 2 +it 2 ˆ ζ (1 + 2it) where ηq,κ,κ (n, 21 + it) is given by Lemma 2.3. Then I2 =
2
2 πi ζˆ (1 + 2it)
∞ |ηq,κ,κ (n, 21 + it)|2 ∞ dy |K it (2π ny)|2 y s ds . H (s) s n y (2) 0 n=1
Equidistribution of Eisenstein Series in the Level Aspect
1147
We decompose the integer n as n = q α m to get 2 1 σ−2it (q α ) − 1 − q −1−2it α=0 q |m 2 ∞ |σ−2it (m)|2 1 1 − q −2(α+1)it 1 = − ms q αs 1 − q −2it 1 − q −1−2it
∞ ∞ |ηq,κ,∞ (n, 21 + it)|2 |σ−2it (m)|2 = ns (q α m)s n=1
α=0
q |m
=
+ 2it)ζ (s − 2it)(1 − q −s+2it )(1 + q −s−2it ) ζ (2s)(1 + q −s )|1 − q −2it |2 |1 − q −1−2it |2 2 ∞ (1 − q −2(α+1)it )(1 − q −1−2it ) − 1 + q −2it × . q αs ζ (s)2 ζ (s
α=0
(3.1) Here putting θ = tα log q and θ = t log q, 2 (1 − q −2(α+1)it )(1 − q −1−2it ) − 1 + q −2it 2 = q −2it − q −1−2it − q −2(α+1)it (1 − q −1−2it ) 2 = q −2it − q −2(α+1)it − q −1 (q −2it − q −2(α+2)it ) 2 = q (α+1)it − q −(α−1)it − q −1 (q (α+1)it − q −(α+1)it ) 2 = cos(θ + θ ) + i sin(θ + θ ) − (cos(θ − θ ) − i sin(θ − θ )) − 2iq −1 sin(θ + θ ) 2 = (cos(θ + θ ) − cos(θ − θ ))2 + sin(θ + θ ) + sin(θ − θ ) − 2q −1 sin(θ + θ ) 2 = (−2 sin θ sin θ )2 + 2 sin θ cos θ − 2q −1 sin(θ + θ ) = 4 sin2 θ − 8q −1 sin θ cos θ sin(θ + θ ) + 4q −2 sin2 (θ + θ ). The contribution of the first term 4 sin2 θ to the sum over α in (3.1) is 4
∞ ∞ sin2 θ q 2itα + q −2itα − 2 = − αs q q αs
α=0
=−
α=0 ∞
q α(2it−s) + q α(−2it−s) − 2q −αs
α=0
1 2 1 − + 2it−s −2it−s 1−q 1−q 1 − q −s −s −s 2it q (1 + q )(1 − q )2 = − 2it q (1 − q 2it−s )(1 − q −2it−s )(1 − q −s ) q −s (1 + q −s )|1 − q −2it |2 =− . (1 − q 2it−s )(1 − q −2it−s )(1 − q −s ) =−
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S. Koyama
The second and third terms are similarly treated, so that they are estimated as O(q −1 ) as q → ∞. Then from (3.1) we have ∞ |ηq,κ,∞ (n, 21 + it)|2 ns n=1
ζ (s)2 ζ (s + 2it)ζ (s − 2it)(1 + q −s−2it ) = ζ (2s)|1 − q −1−2it |2
−q −s −1 + O(q ) . (1 − q −2it−s )(1 − q −s )
Consequently, I2 =
2 2 πi ζˆ (1 + 2it) ×
(2)
H (s)
ζ (s)2 ζ (s + 2it)ζ (s − 2it)(1 + q −s−2it ) ζ (2s)|1 − q −1−2it |2
(s/2)2 ( 2s − it)( 2s + it) −q −s ds + O(q −1 ). (1 − q −2it−s )(1 − q −s ) (s)
If we put it by I2 =
2
2 πi ζˆ (1 + 2it)
(2)
B(s)ds + O(q −1 ),
then we compute by shifting the contour to Re(s) = 1/2 that
2 I2 = B(s)ds + O(q −1 ). 2 2πiRess=1 B(s) + ˆ (1/2) πi ζ (1 + 2it) This is justified by the Stirling formula and that H (σ + it) is rapidly decreasing in t. The integral (1/2) B(s)ds converges for any q, and the resulting value is O(1) as q → ∞. Therefore we have 4 I2 = 2 Ress=1 B(s) + O(1) ˆ ζ (1 + 2it) as q → ∞. It remains to estimate the residue term. We put B(s) = ζ (s)2 G(s), where G(s) is a holomorphic function at s = 1 defined by G(s) =
−q −s H (s)ζ (s + 2it)ζ (s − 2it)(1 + q −s−2it ) −1−2it 2 −2it−s ζ (2s)|1 − q | (1 − q )(1 − q −s ) (s/2)2 ( 2s − it)( 2s + it) × . (s)
Then since we know as s → 1 that ζ (s) =
1 + γ + O(s − 1) s−1
Equidistribution of Eisenstein Series in the Level Aspect
1149
with γ the Euler constant, we have
G (1) . Ress=1 B(s) = G(1) 2γ + G
We compute G (1) = − log q + O(1) G as q → ∞. We conclude that 4G(1) I2 = 2 (log q + O(1)) . ˆ ζ (1 + 2it) As π 2 |ζˆ (1 + 2it)|2 (1 + q −1−2it ) −q −1 ζ (2)|1 − q −1−2it |2 (1 − q −2it−1 )(1 − q −1 ) 6H (1)|ζˆ (1 + 2it)|2 + O(q −2 ), =− q
G(1) = H (1)
we have log q I2 = 24H (1) +O q
log q q2
.
We find that
∞
H (1) =
h(y)
0
=
∞ 1
dy y2
h(y) 0
=
0
d xd y y2
Fq,κ(q) (z|h)d V Xq
= vol(X q )
Fq,κ(q) (z|h)d Vq , Xq
which shows
I2 = 24
Fq,κ(q) (z|h)d Vq log q + O Xq
This completes the proof of the proposition.
log q q
.
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S. Koyama
References [DFI] [HL] [I] [IK] [K] [KMV] [L] [LS] [R] [RS] [M]
[PS] [S]
Duke, W., Friedlander, J.B., Iwaniec, H.: The subconvexity problem for artin L-functions. Invent. Math. 149, 489–577 (2002) Hoffstein, J., Lockhart, P.: Coefficients of maass forms and the siegel zero. Ann. Math. 140, 161–181 (1994) Iwaniec, H.: Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, Vol. 53, Providence, RI: Amer. Math. Soc., Iwaniec, H., Kowalski, E.: Analytic Number Theory. Colloquium Publications 53, Providence, RI: Amer. Math. Soc., 2004 Koyama, S.: Quantum ergodicity of eisenstein series for arithmetic 3-manifolds. Commun. Math. Phys. 215, 477–486 (2000) Kowalski, E., Michel, P., Vanderkam, J.: Rankin-selberg L-functions in the level aspect. Duke Math. J. 114, 123–191 (2002) Lindenstrass, E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. 163, 165–219 (2006) Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on P S L 2 (Z)\H 2 . Publ. I.H.E.S. 81, 207–237 (1995) Ramanujan, S.: Some formulae in the arithmetic theory of numbers. Messenger of Math. 45, 81–84 (1916) Rudnick, Z., Sarnak, P.: The behavior of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys 161, 195–213 (1994) Meurman, T.: On the order of the Maass L-function on the critical line. In: Number theory, Vol. I (Budapest, 1987), Colloq. Math. Soc. János Bolyai, 51, Amsterdams: North-Holland, 1990, pp. 325–354 Petridis, Y., Sarnak, P.: Quantum ergodicity for S L(2, o)\H 3 and estimates for L-functions. J. Evol. Equ. 1, 277–290 (2001) Sarnak, P.: Estimates for rankin-selberg L-functions and quantum unique ergodicity. J. Funct. Anal. 184, 419–445 (2001)
Communicated by S. Zelditch
Commun. Math. Phys. 289, 1151–1169 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0789-1
Communications in
Mathematical Physics
Local Heterotic Torsional Models Ji-Xiang Fu1 , Li-Sheng Tseng2,3 , Shing-Tung Yau2 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China 2 Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.
E-mail: [email protected]
3 Center for the Fundamental Laws of Nature, Jefferson Physical Laboratory,
Harvard University, Cambridge, MA 02138, USA Received: 29 July 2008 / Accepted: 23 December 2008 Published online: 2 April 2009 – © Springer-Verlag 2009
Abstract: We present a class of smooth supersymmetric heterotic solutions with a non-compact Eguchi-Hanson space. The non-compact geometry is embedded as the base of a six-dimensional non-Kähler manifold with a non-trivial torus fiber. We solve the non-linear anomaly equation in this background exactly. We also define a new charge that detects the non-Kählerity of our solutions. 1. Introduction In this paper, we study six-dimensional supersymmetric non-compact solutions of the ten-dimensional heterotic supergravity. Non-compact solutions can have different physical interpretations in string theory. They may be local models of a compact solution or they may correspond to the supergravity descriptions of solitonic objects of the theory. We demonstrate the existence of six-dimensional smooth solutions on T 2 bundles over an AL E space. For the base being the minimally resolved C2 /Z2 , we work out the solution in detail using the Eguchi-Hanson metric [1]. In solving this solution, we work in complex coordinates and exploit the SU (2) global symmetry of the EguchiHanson metric. Importantly, the symmetry reduces the anomaly equation to a first-order non-linear differential equation which we solve exactly. Our solutions are 1/2 BPS and are asymptotically RP3 × T 2 . These local nonKähler models are closely related to the compact heterotic models of T 2 bundle over K 3 described in [2,3] (see also [4,5]). They give an explicit local description of the sixdimensional compact solution near an A1 orbifold singularity of the base K 3. Moreover, it may be possible that our local solutions can be consistently glued-in to resolve in a nonKähler manner singular compact manifolds such as T 4 /Z2 × T 2 or even K 3/Z2 × T 2 . Alternatively, the local solutions we construct can be interpreted to describe a heterotic five-brane that is wrapped around a torus and transverse to an Eguchi-Hanson space. Heterotic five-brane solutions with a tranverse Eguchi-Hanson space [6,7] or wrapped over an S 1 [8,9] have been discussed previously in the literature. Solutions of this type
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J.-X. Fu, L.-S. Tseng, S.-T. Yau
differ from the original five-brane solution [10–13] in that the five-brane charge can be sourced by a non-trivial U (1) gauge field instead of an SU (2) instanton. Here, we point out that both the Eguchi-Hanson geometry and the non-trivial fibered torus induce non-trivial H fluxes. And of particular importance for the heterotic string is that their presence introduces highly non-linear terms in the anomaly differential equation. A main purpose of this paper is to demonstrate that the induced fluxes can be carefully balanced to give smooth non-compact solutions that solve the heterotic supergravity exactly at one-loop order. The outline of the paper is as follows. In Sect. 2, we review the supersymmetry conditions and the solution ansatz we will use. In Sect. 3, we write down explicitly the solution with an Eguchi-Hanson space and the differential equation that must be solved from the anomaly equation. In Sect. 4, we solve the differential equation exactly. In Sect. 5, we write down our solutions in general form and discuss their physical characteristics. Though our smooth solutions have zero five-brane charge, they are in general non-zero under a new charge which we define that detects the non-Kählerity of the solutions. 2. Supersymmetry Conditions and Solution Ansatz We start from the ten-dimensional heterotic supergravity on the product manifold, M 3,1 × X 6 , a four-dimensional Minkowski spacetime times a six-dimensional manifold. Preserving supersymmetry requires that X 6 is complex and has an SU (3) holonomy with respect to a torsional connection. The heterotic solution on X 6 can be described by a hermitian metric J , a holomorphic (3, 0)-form , and a stable gauge bundle E ⊂ S O(32) or E 8 × E 8 with curvature F. The additional conditions from supersymmetry and the consistency of anomaly cancellation are d( J J ∧ J ) = 0, F (2,0) = F (0,2) = 0, Fmn J mn = 0, α 2i ∂ ∂¯ J = [tr(R ∧ R) − tr(F ∧ F)], 4
(2.1) (2.2) (2.3)
where ¯ = i∧
4 2J J ∧ J ∧ J. 3
(2.4)
Following Strominger [14], we take the curvature R in (2.3) to be defined by the hermitian connection. Though the type of connection is not specified physically at one-loop order,1 the hermitian connection is the unique metric connection that is compatible with the complex structure and whose torsion tensor does not contain a (1,1) component. Furthermore, the resulting tr(R ∧ R) is always a (2,2)-form.2 The above equations define what is called the Strominger system in the mathematical literature. It consists of a conformally balanced condition for the hermitian metric J , a hermitian Yang-Mills condition for the bundle curvature F, and an anomaly condition relating the difference of the two Pontryagin classes, p1 (R) and p1 (F). The relations to the physical fields - the 1 Physical relationships between different connections have been discussed in [15–17]. 2 tr(R∧ R) for non-hermitian connections will generally contain (3,1) and (1,3) components. Since the other
two terms in the anomaly equation in (2.3) are (2,2)-forms, the presence of these additional components will likely over-constrain the system of differential equations as they must be set to zero. We note that nilmanifold solutions with different connections have been discussed recently in [18].
Local Heterotic Torsional Models
1153
metric g, the antisymmetric three-form field H , and the scalar dilaton field φ - are given as follows: gmn = Jmr I r n , H = i(∂¯ − ∂)J, e−2φ = J , (2.5) where I is the complex structure determined by the holomorphic three-form . There is a much-studied solution ansatz on the T 2 bundle over a Calabi-Yau two-fold [2–5,19]. The metric takes the form i ¯ (2.6) J = eu JCY2 + (dz + β) ∧ (d z¯ + β), 2 where u is a function of the base Calabi-Yau and the torus curvature ω = dθ ≡ d(dz +β) satisfies the quantization and primitivity conditions ω ω ∧ JCY2 = 0. (2.7) √ ∈ H 1,1 (M) ∩ H 2 (M, Z), 2π α 2,0 Taking the holomorphic three-form to be 3,0 = CY ∧ θ which is a closed (3, 0)-form 2 by (2.7), it is straightforward to check that the conformally balanced condition is satisfied for any function u. We note that with the metric and three-form ansatz, the conformal factor eu = e2φ which follows from the third equation of (2.5), u 2φ −1 J =e =e .
(2.8)
Further, choosing a hermitian Yang-Mills curvature, F, pull-backed from the base CY2 , the anomaly equation (2.3) reduces to a non-linear second-order differential equation for u (or equivalently the dilaton field) that must be solved. Below, we analyze the case in which the base Calabi-Yau two-fold is taken to be a noncompact AL E space. In particular, we shall work out the case with the Eguchi-Hanson metric in detail. 3. Eguchi-Hanson Base Solution Consider C2 with coordinates (z 1 , z 2 ) and an involution, σ : (z 1 , z 2 ) → (−z 1 , −z 2 ). Let M be the blow up of C2 /σ at the origin by a P1 . Then M is biholomorphic to OP1 (−2) = T ∗ P1 , the cotangent bundle of P1 . The Eguchi-Hanson metric [1,20] is an explicit complete, smooth Ricci-flat metric on M. Outside the origin of C2 /σ , the metric is SU (2) invariant and depends only on the radial coordinate r 2 = |z 1 |2 + |z 2 |2 . Being Kähler, the metric can be expressed as i J E H = ∂ ∂¯ K(r 2 ) 2 i ¯ 2 ¯ 2 , k ∂ ∂r + k ∂r 2 ∧ ∂r (3.1) = 2 where the Kähler potential K, the function k(r 2 ) = dK/dr 2 , and its derivative k (r 2 ) = dk/dr 2 are given by r2 2 4 4 , (3.2) K = r + a + a log √ r 4 + a4 + a2 a4 a2 a2 r4 k = 1 + 4 = 2 1 + 4, k = − . (3.3) 4 r r a r4 1 + r a4
The constant a > 0 is a measure of the diameter of the central
P1 .
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J.-X. Fu, L.-S. Tseng, S.-T. Yau
On M, there is a normalizable anti-self-dual closed (1, 1)-form. It corresponds to the curvature of the line bundle of the P1 and has the form up to a constant c, h h ¯ 2+ ¯ 2 . ∂ ∂r η = i∂ ∂¯ lnh = i ∂r 2 ∧ ∂r (3.4) h h The function (h / h) can be found by imposing the primitivity condition, ω ∧ J E H = 0 . This gives the differential equation
h h 2 = 0, (3.5) k+ kr h h which has the solution, modulo a multiplicative integration constant, 4 2 ar 4 + 1 1 1 h h = 4 = , =− . 4 4 h r k h a 2 r 4 (1 + ar 4 )3/2 a 2 r 2 1 + ar 4
(3.6)
We can now write down explicitly the T 2 bundle over the Eguchi-Hanson space metric ansatz i (3.7) J = eu J E H + θ ∧ θ¯ . 2 For the curvature of the torus bundle, we utilized the anti-self-dual (1, 1)-form, ⎧ ⎫ r4 ⎨ ⎬ + 1 2 ic 1 4 a ¯ 2− ¯ 2 , (3.8) ω = dθ = ic ∂ ∂¯ lnh = 2 ∂ ∂r ∂r 2 ∧ ∂r 4 ⎭ a ⎩ r 2 1 + r4 r 4 (1 + ar 4 )3/2 a4 having inserted (3.6) into (3.4) and allowed for an overall complex constant c . The constant c is quantized since ω√ ∈ H 1,1 (M) ∩ H 2 (M, Z). We can obtain the 2π α
quantization condition by integrating the curvature ω over the P1 at the origin. Working in the coordinate chart (y2 = 0), z1 y2 = z 22 , r 2 = |z 1 |2 + |z 2 |2 = |y2 |(1 + |y1 |2 ), (3.9) y1 = , z2 we integrate ω over P1 parametrized by y1 in the limit y2 → 0. We can rewrite 1 ic 2 ω= 2 + O(|y2 | ) dy1 ∧ d y¯1 + · · · , (3.10) a (1 + |y1 |2 )2 where we have only written out only the dy1 ∧ d y¯1 term. Therefore, 1 1 1 ic ω= dy1 ∧ d y¯1 √ √ a 2 (1 + |y1 |2 )2 2π α P1 2π α ∞ 2π c d x 2 1 c = = √ . √ 2 (1 + x 2 )2 2 a 2π α 0 a α The quantization requirement imposes √ √ c = a 2 α n ≡ a 2 α (n 1 + in 2 ), n 1 , n 2 ∈ Z.
(3.11)
(3.12)
Having written down explicitly the metric which is conformally balanced by construction, we now proceed to discuss the gauge connection and the anomaly equation.
Local Heterotic Torsional Models
1155
3.1. Hermitian Yang-Mills connections and curvature. By convention, our gauge curvature F is imaginary and the Hermitian Yang-Mills condition requires that it is also (1, 1) anti-self dual. F takes value in the Lie algebra of S O(32) or E 8 × E 8 . Hermitian Yang-Mills connections on Eguchi-Hanson space has been studied by Kronheimer and Nakajima for various rank bundles. In this paper, we will limit the discussion explicitly to the U (1) case. For the rank one or U (1) gauge bundle, we note that there is only the line bundle over P1 so F must be proportional to η in (3.4). In general, we can have a direct sum of U (1) bundles. The curvature for each U (1) bundle takes the form (3.4) h h ¯ 2+ ¯ 2 , ∂ ∂r F = c ∂ ∂¯ ln h = c ∂r 2 ∧ ∂r (3.13) h h where c is a real number. We then have
2 h h 2 ¯ 2 2 2 2 2 2 ¯ + ¯ ∧ ∂r ∧ ∂r ¯ F∧F =c ∂ ∂r ∧ ∂ ∂r ∂ ∂r h h ¯ 2 + F ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2, ¯ 2 ∧ ∂ ∂r ≡ F ∂ ∂r
(3.14)
where F =c
2
h h
2 =
c2 a 4 r 4 (1 +
The U (1) gauge bundle also has a quantization: computation in (3.10)-(3.11), this implies c = a 2 m,
iF 2π
r4 ) a4
.
(3.15)
∈ H 1,1 ∩ H 2 (Z). Following the
m ∈ Z.
(3.16)
3.2. Anomaly equation. With the metric ansatz (3.7), the anomaly equation is explicitly (see [2] for derivation) α (tr[R ∧ R] − tr[F ∧ F]) 2 g −1 α ¯ ∧ ∂ ∂u ¯ +2 ∂ ∂[e ¯ −u tr(∂¯ B ∧ ∂ B ∗ E H )]−tr[F ∧ F] , = × tr[R E H ∧ R E H ]+2 ∂ ∂u 2 2
2i∂ ∂¯ J =
(3.17)
B1 ¯ 1 dz 1 +B2 dz 2 ) = ω . locally defined such that ∂(B B2 Note that each term is a closed (2, 2)-form on the base. Since the solution has SU (2) ¯ 2 ∧ ∂ ∂r ¯ 2 global symmetry, we can express each term in terms of a combination of ∂ ∂r 2 2 2 ¯ ¯ and ∂ ∂r ∧ ∂r ∧ ∂r . We now proceed to calculate each term below. A. d H = 2i∂ ∂¯ J term. Using (3.7) for J , we find where B is a column vector B =
¯ u ∧ J E H − ω ∧ ω, 2i∂ ∂¯ J = 2i∂ ∂e ¯
(3.18)
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J.-X. Fu, L.-S. Tseng, S.-T. Yau
and ¯ u ∧ J E H =− (eu ) k ∂ ∂r ¯ 2 ∧ ∂ ∂r ¯ 2 + [(eu ) k] ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2 , (3.19) 2i∂ ∂e
2 h h 2 ¯ 2 2 2 2 2 2 ¯ + ¯ ∧ ∂r ∧ ∂r ¯ −ω ∧ ω¯ = |c| ∂ ∂r ∧ ∂ ∂r ∂ ∂r . (3.20) h h Combining the two terms, we can write ¯ 2 ∧ ∂ ∂r ¯ 2 + J ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2, 2i∂ ∂¯ J ≡ J ∂ ∂r
(3.21)
where J = −(eu ) k + |c|2
h h
2
a2 = −(eu ) 2 r
1+
r4 |c|2 + a 4 a 4 r 4 (1 +
As will be needed shortly, we note here that −ω ∧ ω¯ = ω2
J E2 H 2!
r4 ) a4
.
implies
h 2 h 2 8|c|2 ω = −4|c| 2 + r2 = . 4 h h a 8 (1 + ar 4 )2 2
(3.22)
2
(3.23)
B. tr[R E H ∧ R E H ] term. The curvature tensor is written in terms of metric (g E H )a b¯ = −i(J E H )a b¯ in (3.1). For the hermitian curvature, we find R E H = ∂¯ (∂g E H ) g −1 EH
k 3 k k 3k ¯ 2 ∧ ∂r 2 ¯ 2+ I − 2 M ∂∂r = I− 2 M ∂r k r k k r k k ¯ k 3k 2 ¯ + ∂r ∧ ∂ M − 2 ∂¯ M ∧ ∂r 2 + ∂∂ M k r k k
(3.24)
with the 2 × 2 matrix I = δi j and Mi j = z¯ i z j . A long calculation results in tr[R E H
k 2 2 k 2 ∧ RE H ] = 6 2 + r dz 1 ∧ d z¯ 1 ∧ dz 2 ∧ d z¯ 2 k k
2 k 2 ¯ 2 k ¯ 2+ ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2 =6 ∂ ∂r ∧ ∂ ∂r ∂ ∂r k k ¯ 2 + R ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2 ¯ 2 ∧ ∂ ∂r ≡ R ∂ ∂r
(3.25)
with 2 k R=6 = k
6 r4 1 +
r4 a4
2 .
(3.26)
Local Heterotic Torsional Models
1157
Alternatively, we can express tr[R E H ∧ R E H ] = −
24 dz 1 ∧ d z¯ 1 ∧ dz 2 ∧ d z¯ 2 . 4 3 a 4 1 + ar 4
(3.27)
¯ 2 term can be formally written as C. Other trace R 2 terms. The (∂ ∂u) 2 ¯ 2 2 2 2 2 2 ¯ ¯ ¯ ¯ ¯ 2∂ ∂u ∧ ∂ ∂u = 2 (u ) ∂ ∂r ∧ ∂ ∂r + (u ) ∂ ∂r ∧ ∂r ∧ ∂r ¯ 2 ∧ ∂ ∂r ¯ 2 + U ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2, ≡ U ∂ ∂r
(3.28)
where U = 2(u )2 .
(3.29)
As for the remaining term, we use a formula in [2], e−u tr(∂¯ B ∧ ∂ B ∗
g −1 |c|2 EH )] = i ω2 J E H 2 4 |c|2 ¯ 2 2 ¯ 2 k ∂ ∂r = −e−u + k ∂r ∧ ∂r (1 + r 4 )2 ¯ 2 + H2 ∂r 2 ∧ ∂r ¯ 2, ≡ H1 ∂ ∂r
(3.30)
where H1 = −e−u
|c|2 a 6r 2 (1 +
r 4 3/2 ) a4
,
H2 = e−u
|c|2 a 6r 4 (1 +
r 4 5/2 ) a4
.
(3.31)
This implies g −1 ¯ −u tr(∂¯ B ∧ ∂ B ∗ E H )] 2 ∂ ∂[e 2 2 ¯ ¯ 2 + (H1 − H2 ) ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2 = 2 (H1 − H2 )∂ ∂r ∧ ∂ ∂r ¯ 2 + H ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2, ¯ 2 ∧ ∂ ∂r ≡ H ∂ ∂r
(3.32)
where H=
2(H1
2 −u
− H2 ) = 2|c| e
u a 6r 2 (1 +
r 4 3/2 ) a4
+
4 a 10 (1 +
r 4 5/2 ) a4
.
(3.33)
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J.-X. Fu, L.-S. Tseng, S.-T. Yau
3.3. The resulting anomaly differential equation. We can now write the anomaly equation (3.17) as 2i∂ ∂¯ J −
α ¯ 2 + A ∂ ∂r ¯ 2 ∧ ∂r 2 ∧ ∂r ¯ 2 ¯ 2 ∧ ∂ ∂r (tr[R ∧ R] − tr[F ∧ F]) ≡ A ∂ ∂r 2 1 = 2 A(r 2 ) r 4 dz 1 ∧ d z¯ 1 ∧ dz 2 ∧ d z¯ 2 , r (3.34)
where A=J +
α α F − (R + U + H), 2 2
(3.35)
written in terms of functions defined in (3.22), (3.15), (3.26), (3.29), and (3.33). The anomaly condition is therefore solved setting A = 0 . With the quantization conditions (3.12) and (3.16), A = 0 leads to the first order differential equation a2 −u eu 2 r =α
1+ 3
r 4 (1 +
αm i 2 r4 α |n|2 + + 4 4 a r 4 (1 + r 4 ) 2r 4 (1 + r 44 ) a a 2
r4 2 ) a4
2 −u
+ (u ) + α |n| e
u
a 2 r 2 (1 +
r 4 3/2 ) a4
+
4 4
a 6 (1+ ar 4 )5/2
, (3.36)
where |n|2 = n 21 + n 22
and
n 1 , n 2 , m i ∈ Z.
(3.37)
In m i , we have allowed for the possibility of multiple U (1) gauge bundles denoted by the index i . Heterotic string allows for at most a rank 16 gauge bundle so m i2 should be 2 taken to denote 16 j=1 m j . 2
For |n|2 + m2i = 3, we find that the differential equation has a smooth solution for u for all values of aα2 > 0. Explicitly, it takes the form eu =
∞
ak 4
k
(1 + ar 4 ) 2 ⎤ ⎡ 2 3
2 2 + 9/7) 1 |n| (|n| α α α = a0 ⎣1− 2 + 2 + + ···⎦ r 4 27 a a0 (1 + r 4 ) 23 a a0 (1 + r 44 )2 a 2 a0 (1 + ) a a4 a4 k=0
(3.38)
which converges for a10 aα2 < 1 sufficiently small. In the next section, we will derive the solution showing how the constants ak can be found iteratively and that the series converges to an exact solution of the differential equation (3.36).
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4. Solving the Anomaly Equation To solve the differential equation, we first rewrite (3.36) in a more convenient form in a few steps. To start, multiplying (3.36) by 1/a 2 and re-arranging terms gives 2 2 3 − (|n|2 + m2i ) |n|2 + m2i u u r4 α 2 e 1+ 4 + − u + 4 4 r2 a a2 r 4 (1 + ar 4 )2 a 4 (1 + ar 4 )2 2 1 4 α 2 −u u |n| e + + = 0, (4.1) a2 r 2 (1 + r 44 )3/2 a 4 (1 + r 44 )5/2 a
Setting
α a2
= α, |n|2 +
m i2 2
a
= 3 and replacing u eu with (eu ) , we find
(eu ) r4 3α 1 + − α(eu ) (e−u ) − 4 4 r2 a4 a (1 + ar 4 )2 −α 2 |n|2
1 (e−u ) e−u 2 2 + 4α |n| = 0. 3 4 5 r 2 (1 + r 4 ) 2 a 4 (1 + r ) 2 a4
(4.2)
a4
√
4
d And lastly, defining eu = v(s) , s = ar 4 , with drd 2 eu = 2 a 2s ds v and multiplying through 4 2 by a v , we arrive at the final form of the differential equation D(α, v) which we will solve 1
D(α, v) = 2(1 + s) 2 v 2 v + 4α(1 + s)v 2 − 4αv 2 3α 2α 2 |n|2 4α 2 |n|2 − v2 + v + v = 0. 3 5 2 (1 + s) (1 + s) 2 (1 + s) 2
(4.3)
In writing D(α, v), we have emphasized the dependence of the differential equation on the parameter α. The solution function v = v(s, α) of course depends on the coordinate s but should also vary with α. The presence of the parameter α is actually rather useful. Together with v, we see that D(α, v) is indeed homogenous under the scaling D(λα, λv) = λ3 D(α, v),
for λ ∈ R+ .
(4.4)
This is important as it means that if we find a solution D(α0 , v0 ) = 0 at a given value α = α0 , then for any other value α = α˜ = λα0 , there is also a solution given by v = λv0 . Taking advantage of this fact, we will solve D(α, v) for α < 1 and sufficiently small (which we shall make precise later). The scaling of (4.4) then implies a solution for all α > 0. The form of (4.3) suggests that we look for a solution of the type v=
∞
ak
k=0
(1 + s) 2
k
,
(4.5)
with the coefficients αk ’s possibly depending on the constants α and |n|2 . Since the four-dimensional base metric in (2.6) should be asymptotic to the flat metric as s → ∞, we must have a0 > 0 . This positive constant a0 can be identified as a parameter of the
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solution space of v(s, α) for a given α.3 For notational simplicity, we shall set a0 = 1 and find solutions for this case. At the end of this section, we shall show how solutions with a0 = 1 can be easily obtained from those of a0 = 1 via a scaling argument. With the differential equation (4.3) and the solution ansatz (4.5), we proceed now to give a method to determine all the coefficients ak . We shall show that our prescription for the ak ’s results in v being a convergent series for α sufficiently small. We then prove that v indeed converges to the solution D(α, v) = 0. 4.1. Determining the coefficients ak . For specifying the ak ’s, we consider the finite series vk =
k
al
l=0
(1 + s) 2
l
.
(4.6)
We introduce the error function E(vk (s)) = D(α, vk ), or explicitly 1
E(vk ) = 2(1 + s) 2 vk2 vk + 4α(1 + s)vk2 − 4αvk2 −
3α 2α 2 |n|2 4α 2 |n|2 2 v + v + v . k 3 k 5 k (1 + s)2 (1 + s) 2 (1 + s) 2
(4.7)
Thus for example, E(v0 ) = −
3α 4α|n|2 + . (1 + s)2 (1 + s) 25
(4.8)
And making the choice a1 = a2 = 0 and a3 = −α leads to E(v0 ) = E(v1 ) = E(v2 ),
(4.9)
and E(v3 ) =
4α 2 |n|2 (1 + s)
5 2
−
α 3 |n|2 − 9α 3 9α 3 − . (1 + s)4 (1 + s)5
(4.10)
Thus far, the error functions follow the form E(vk ) =
bk+2 (1 + s)
k+2 2
+ ··· ,
(4.11) k+3
with bk+2 = 0 for k = 0, 1, 2 and we have omitted terms of O((1 + s)− 2 ). In fact, we can iteratively choose ak+1 such that (4.11) also holds for any k > 3. To show this, we first write E(vk+1 ) = E(vk ) + (E(vk+1 ) − E(vk )) .
(4.12)
We observe that E(vk+1 ) − E(vk ) =
−(k + 1)ak+1 (1 + s)
k+2 2
+ ··· ,
(4.13)
3 From the string theory perspective, a = e2φ0 is the string coupling g at the asymptotic infinity of the s 0 Eguchi-Hanson space.
Local Heterotic Torsional Models
1161 1
which comes from the first term 2(1 + s) 2 vk2 vk+1 in (4.7). Comparing (4.11) and (4.13), we can set
ak+1 = which would cancel the
bk+2 (1+s)
k+2 2
bk+2 , k+1
(4.14)
term and gives us for (4.12),
E(vk+1 ) =
bk+3 (1 + s)
k+3 2
+ ··· .
(4.15)
We shall choose each ak ’s similarly and thereby ensure (4.11) is valid for all k. We have thus given an algorithm to determine each ak from those ai ’s with i < k. Explicitly, the coefficients are given by ⎧ k−2 k−3 1 ⎨ 2 2 ak+1 = ai a j −α i j ai a j −α |n| (k − 7)ak−3 − 3 α k + 1⎩ i, j=0 i+ j=k−2 i, j=1 i+ j=k−2 ⎫ k−1 k k ⎬ +α i j ai a j − l ai a j al . (4.16) ⎭ i, j=1 i+ j=k
i, j=0 l=1 i+ j+l=k+1
Using this formula, we find for instance
9 , a4 = α 2 |n|2 , a5 = 0 , a6 = 0 , a7 = α 3 |n|2 + 7
16 a8 = −α 4 |n|4 + 3|n|2 , a9 = α 3 −1 + α 2 |n|4 , 9
(4.17)
and so on.
4.2. Estimates for ak and convergence. Being able to iteratively generate the coefficients of each term of the series (4.5), we can now show that the series converges when α < 1 is sufficiently small. Since |a3 | = α < 1 is small, we can write |a3 | =
α03 , 33 C
(4.18)
for some large constant C and small α0 < 1. For a fixed α0 < 1 and with (4.16) and (4.18), we shall prove by induction that when C is sufficiently large, |ak | ≤
α0k . k3 C
This estimate then immediately implies that the series
(4.19) ∞
k=0
ak k
(1+s) 2
converges for any
s ≥ 0 since α0 < 1 . We proceed now with the induction proof of (4.19). Let us assume that (4.19) is true for 1 ≤ k ≤ N and N ≥ 3. We shall prove that (4.19) is then also true for k = N + 1. We show this by deriving explicit estimates for
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J.-X. Fu, L.-S. Tseng, S.-T. Yau
all five terms in the expression for ak in (4.16) for k = N + 1. As convention, we take as definition 0k = 1 below. Starting with the first term of (4.16), we find the estimate | α 2 |n|2 (N − 7)a N −3 | α 2 |n|2 |N − 7| α0N −3 ≤ N +1 N +1 (N − 3)3 C N +1 2 α0 |n|2 |N − 7|(N + 1)2 α0 ≤ (N + 1)3 C C 2 36 (N − 3)3 N +1 2 α0 α0 ≤ C1 3 (N + 1) C C 2
(4.20)
with the constant C1 = sup i≥3
|n|2 |i − 7|(i + 1)2 . 36 (i − 3)3
(4.21)
For the estimate of the second term in (4.19) for k = N + 1, we find 3α N +1
| ai a j | ≤
i+ j=N −2 i, j≥0
≤
3α α0N −2 N + 1 C2
i+ j=N −2 i, j≥0
α0N +1 6αα0−3 (N + 1)C C
1 i3 j3
j=N −2−i
1 j3
N −2 i≥[ N 2−2 ]
N −2−[ N 2−2 ] + 1)2
≤
α0N +1 1 16(N (N + 1)3 C C 2 9(N − 2)3
≤
α0N +1 (N + 1)3 C
j=0
1 i3 (4.22) 1 j3
C2 C2
with the constant ∞ 28 1 C2 = . 9 j3
(4.23)
j=0
The estimates for the third and fourth term are found similarly. For the third term, we find α N +1
i j | ai a j |≤
i+ j=N −2 i, j≥1
α0N +1 C3 (N + 1)3 C C 2
(4.24)
with the constant C3 =
∞ 26 1 , 33 j2 j=1
(4.25)
Local Heterotic Torsional Models
1163
and for the fourth term α0N +1 C4 α i j | ai a j |≤ N +1 (N + 1)3 C C 2
(4.26)
∞ 26 1 C4 = 5 . j2 3
(4.27)
i+ j=N i, j≥1
with the constant
j=1
Lastly, we estimate the fifth term in (4.16) for k = N + 1. From direct calculation, we obtain 1 l | ai a j al | N +1 i, j≥0,l≥1 i+ j+l=N +1
≤
≤
α0N +1 (N + 1)C 3
i, j≥0,l≥1 i+ j+l=N +1
(N
9α0N +1 (N + 1)3 C 3
≤
27α0N +1 (N + 1)3 C 3
≤
α0N +1 (N + 1)3 C
⎜ ⎝
⎞
⎟ ⎜ + ⎝ ⎠
i+ j=N −1−l
⎛
≤
1 l 2i 3 j 3
⎛
α0N +1 + 1)C 3
l≥[ N3+1 ]
N +1−[ N3+1 ]
i+ j=0 N +1−[ N3+1 ]
i+ j=0
l<[ N3+1 ]
1 +2 i3 j3
1 l 2i 3 j 3 ⎞
j=N +1−i−l i≥[ N +1 ] l≤[ N +1 ] 3
1 ⎟ ⎠ l 2i j 3
3
1 i3 j3
C5 C2
(4.28)
with the constant C5 = 27
∞ i+ j=0
1 i3 j3
.
(4.29)
Now let C0 = max{C1 , C2 , C3 , C4 , C5 }. For α0 < 1 , we choose the constant (4.30) C ≥ 5 C0 . By summing over the five estimates in (4.20), (4.22), (4.24), (4.26), and (4.28), we obtain the estimate |a N +1 | ≤
α0N +1 . (N + 1)3 C
(4.31)
And by induction, we have proven the desired estimate (4.19) and therefore v(s) = ∞ ak k converges for any s. k=0 (1+s) 2
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J.-X. Fu, L.-S. Tseng, S.-T. Yau
Having shown that the series v converges, we still need to make sure that v = eu > 0. This positivity condition will give us a bound on α for solutions with a0 = 1. Clearly for any s ≥ 0, α03 α 1 k =1− v >1− 3 α0 = 1 − 3 . (4.32) 3 C 3 (1 − α0 )C 1 − α0 k≥3
Since, 0 < α0 < 1, (4.32) gives the condition α ≤ 1 − α0 < 1
(4.33)
to ensure v(s) > 0 . Let α˜ > 0 be the solution of the equation (1 − α) ˜ 3 . √ 33 5 C0
α˜ =
(4.34)
Then by (4.18), (4.30), and (4.33), for any 0 < α ≤ α, ˜ v(s) =
∞
k=0
and v(s) > 0 for all s ≥ 0 .
ak k
(1+s) 2
converges
4.3. Proving the series solves the differential equation. Finally, having established that v is a convergent series, we now prove that v is indeed a solution to the differential equation (4.3). This is equivalent to showing that the error vanishes for the entire series, i.e. lim E(vk ) = 0.
(4.35)
k→∞
Since the leading term is (1 + s)−
k+2 2
, we can write 3k+1
2
cp
p=k+2
(1 + s) 2
E(vk ) =
p
,
(4.36)
with ck+2 = bk+2 . By direct computation, we find c p = −α 2 |n|2 ( p − 9)a p−5 − 3α
k
ai a j − α
i, j=0 i+ j= p−4
+α
k
k k
i j ai a j −
i, j=1 i+ j= p−2
k
i j ai a j
i, j=1 i+ j= p−4
l ai a j al ,
(4.37)
i, j=0 l=1 i+ j+l= p−1
and the first term is zero if p > k + 5. Similar to the estimate for |ak | in (4.19), we find the estimate for |c p |, p−1
α0 , (4.38) ( p − 1)2 the summation of absolute values of every term in a p−1 .
|c p | ≤ C( p − 1) a p−1 ≤ where we denote a p−1 Therefore,
3k+1
|E(vk )| ≤
2
p=k+2
|c p | (1 + s)
as k → ∞. This proves E(v) = 0.
p 2
≤
α0k+1 (1 + s)
k+2 2
3k+1
2
p=k+2
1 → 0, ( p − 1)2
(4.39)
Local Heterotic Torsional Models
1165
4.4. Solution and parameter space. We have shown that the differential equation D(α, v) = 0 in (4.3) is solved by the convergent series v(s, α) =
∞ k=0
) * α 2 |n|2 α 3 |n|2 + 97 =1− + + + ··· , 3 3 k 2 (1 + s) 2 (1 + s) (1 + s) 2 (1 + s) 2 ak
α
(4.40)
for α ≤ α˜ and ak given by (4.16). We can now use the scale invariance of D(α, v) = 0 in (4.4) to demonstrate a one parameter family of solution for any given value of α. We first show this for α = α˜ as defined in (4.34) for a0 = 1 solutions. Let α0 < α˜ and write α0 = α/λ ˜ for a real constant λ > 1. At α = α0 , we have the solution v(s, α0 ) given in (4.40). Making use of the scaling of (4.4), we obtain 1 1 α˜ α˜ 1 0 = D(α0 , v(s, α0 )) = D( α, ˜ λ v(s, )). ˜ λ v(s, )) = 3 D(α, λ λ λ λ λ
(4.41)
This implies a family of solutions parametrized by λ at α = α˜ given by α˜ vλ (s, α) ˜ = λ v(s, ) λ 2 3 ) 2 9 * |n| + 7 1 α˜ |n|2 α˜ α˜ = λ 1− + + + ··· , 3 λ (1 + s) 23 λ (1 + s)2 λ (1 + s) 2
(4.42)
with λ = [1, ∞) . To show a family of solutions for any value of α = µ α˜ for any real constant µ, we apply the scaling of (4.4) again to obtain ˜ = µλ v(s, vλ (s, α) = µ vλ (s, α) In terms of the original expansion v =
∞
k=0
ak k
(1+s) 2
α ). µλ
(4.43)
, we find that a0 = µλ and we
have convergence to a solution for a0 = [µ, ∞) . More simply, we write the convergent solution as 2 3 ) 2 9 * |n| + 7 1 |n|2 α α α v(s, α) = a0 1 − + + + ··· , 3 a0 (1 + s) 23 a0 (1 + s)2 a0 (1 + s) 2 (4.44) with the condition aα0 = αλ˜ < 1 sufficiently small (since α˜ < 1 and λ ≥ 1). In summary, we have found a one-parameter family of solutions for the anomaly equation for any value of α = α /a 2 . 5. Discussion We have constructed a class of smooth non-compact solutions that exactly solve the heterotic supergravity supersymmetry constraints to first order in α . We write below
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the solution in the most general form, introducing the complex moduli τ = τ1 + iτ2 (in z = x + τ y) and area A of the torus as parameters: i A ¯ J = eu J E H + (dz + β) ∧ (d z¯ + β), 2τ ⎤ ⎡ 2 2 2 4 1 i a r ¯ 2 a ¯ 2 ⎦, − 4 ∂r 2 ∧ ∂r J E H = ⎣ 2 1 + 4 ∂ ∂r 2 r a r r4 1 + a4 ⎤ ⎡ r4 √ + 1 2 1 4 a ¯ 2− ¯ 2⎦, ∂ ∂r ∂r 2 ∧ ∂r ω = dβ = i α (n 1 + τ n 2 )⎣ 3 4 (1+ r 4 ) 2 r4 2 r r 1 + a4 a4 ⎡ ⎤ 4 2 ar 4 + 1 1 ¯ 2− ¯ 2 ⎦, ∂ ∂r ∂r 2 ∧ ∂r Fi = m i ⎣ 4 3 4 r 4 r r (1 + a 4 ) 2 r2 1 + 4
(5.1) (5.2)
(5.3)
(5.4)
a
e2φ = eu =
∞ k=0
⎡
ak (1 +
r 4 k2 ) a4
1 α = e2φ0 ⎣1 − 2φ 2 + e 0 a (1 + r 4 ) 23 a4
α e2φ0 a 2
2
A τ2 |n 1
+ τ n 2 |2
(1 +
r4 2 ) a4
⎤ + · · · ⎦ (5.5)
for A mi 2 = 3, |n 1 + τ n 2 |2 + τ2 2
and
n 1 , n 2 , m i ∈ Z,
(5.6)
and eφ0 is the string coupling at asymptotic spatial infinity r → ∞. From (5.3) and (5.4), we see that both the torus twist curvature ω and the U (1) gauge fields curvature F are localized around the origin of the Eguchi-Hanson space and vanish in the asymptotic limit of r → ∞. The expression for e2φ in (5.5) is obtained from (4.44) by replacing |n|2 → τA2 |n 1 + τ n 2 |2 and setting a0 = e2φ0 . The condition for the convergence of the series then becomes
α 1 < 1, (5.7) 2 a gs2 and sufficiently small. Clearly, our solution is consistent in the supergravity limit of gs 1 and α/a 2 1 for sufficiently large a 2 . We observe that our solution with non-zero H fluxes have moduli which may be constrained but are not fixed. Certainly the string coupling, gs = eφ0 , and the size of the resolved P1 as measured by a 2 are not fixed. Together, they are constrained by (5.7). As for the torus, Eq. (5.6) gives only one constraint for the torus area A and complex structure moduli τ combined. Thus, we are free to vary τ with a compensating variation of A.4 Nevertheless, if n 1 and n 2 are not both zero, the area of the torus is constrained to be of O(α ) (as A is normalized with respect to α in (5.6)), 4 In the compact case of T 2 bundle over K 3 base as discussed in [21], the torus complex structure moduli can be fixed with appropriately chosen ω = ω1 + τ ω2 ∈ H 2,0 ⊕ H 1,1 . Here, the Eguchi-Hanson base is special in that it has only one normalizable two-form.
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1167
If we treat our solution as a solitonic object, we should determine its five-brane charge. This charge can be obtained by integrating H = d c J at the spatial infinity of the transverse Eguchi-Hanson space, E H . However, because of the non-trivial fibering, the Eguchi-Hanson space is not a four-dimensional submanifold of X 6 and so taking the spatial infinity limit of E H is ill-defined in X 6 . Thus, to be rigorous, we should pull-back RP3 (r ) at the radial coordinate r in E H to a T 2 bundle over RP3 (r ) which is a submanifold over X 6 . Denoting this five-submanifold by S(r ), we define the five-brane charge in X 6 as5
i 1 1 ¯ θ ∧ θ H ∧ J = lim H ∧ r →∞ (4π 2 α )2 S(r ) r →∞ (4π 2 α )2 S(r ) 2 1 1 = lim H = lim i(∂¯ − ∂)eu ∧ J E H r →∞ 4π 2 α RP3 (r ) r →∞ 4π 2 α RP3 (r ) 2 1 a r4 ¯ 2 u ¯ 2 + ∂r 2 ∧ ∂∂r ¯ 2) = − lim (e ) 1 + (∂r ∧ ∂ ∂r r →∞ 8π 2 α RP3 (r ) r2 a4 ⎡ ⎤ 4 1 ⎣ 4 u a = − lim r (e ) 1 + 4 ⎦ (5.8) r →∞ 2α r
Q 5 = lim
having used (5.1) and (5.2). Plugging in the expression for eu in (5.5), we find that the total net charge is zero. This is perhaps as expected since in imposing the condition (5.6), we have effectively cancelled the negative charge contribution from the curvature of the Eguchi-Hanson space with the positive charge contribution from the torus twist and gauge fields. A non-zero five-brane charge would likely require a singular solution. Being zero, the five-brane charge can not distinguish between different torus curvature ω which when non-zero makes X 6 a non-Kähler manifold. We can however define a new charge 1 ˜ dH ∧ J Q= (4π 2 α )2 X 6
* ) i 1 u ¯ ¯ θ ∧θ , (5.9) 2i∂ ∂e ∧ J E H − ω ∧ ω¯ ∧ = (4π 2 α )2 X 6 2 where we have used the primitivity condition ω ∧ J E H = 0 . Now, the first term on the right-hand side, integrates to zero since it is a total derivative with zero boundary contribution as in (5.8). The second term reduces to an integral on E H, 1 ω ∧ ω¯ Q˜ = − 2 4π α E H 2 α |n|2 1 1 =− 2 dz 1 ∧ d z¯ 1 ∧ dz 2 ∧ d z¯ 2 = |n|2 . (5.10) 4π α C2/Z2 a 4 (1 + r 44 )2 2 a
Therefore, when ω = 0 and X 6 is non-Kähler, Q˜ = 0. 5 For simplicity, we have set A = 1 and τ = i for the moduli of the torus in the discussion. The area of the √ torus is conventionally normalized to (2π α )2 .
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A motivation for considering the charge Q˜ is that for the Kähler case where d J = 0, Stokes’s theorem implies Q˜ = Q 5 (compare (5.8) with (5.9)). Note that d H corresponds to the source density of the five-brane. But when J is not Kähler, we have H ∧ d J = −2i ∂ J ∧ ∂¯ J. (5.11) Q˜ − Q 5 = X6
X6
Hence, the difference between Q˜ and Q 5 implies non-Kählerity. We also note that for the compact case, Q˜ is well-defined for J as a class in the ∂ ∂¯ cohomology. That is, Q˜ is ¯ , where γ is (1, 0)-form. This may be relevant as the invariant under J → J + ∂ γ¯ + ∂γ anomaly equation (2.3) is locally a ∂ ∂¯ equation [21]. It is expected that as higher order α corrections to the supergravity constraints are taken into account, the explicit form of our solutions will be corrected. The explicit form as in the series expansion of (3.38) suggests that the corrections can probably be incorporated order by order in α . Alternatively, one would like to have a worldsheet conformal field theory description of the geometrical model. Such has been presented in [22] using the gauged linear sigma model formalism of [23]. We have given a detailed study of the solution of a torus bundle over a non-compact Eguchi-Hanson space with U (1) gauge bundles. This can be considered the simplest case of a more general class of solutions that involve non-Abelian gauge bundles and more general AL E base geometry. Investigations on these more general solutions are interesting and we plan to report on them elsewhere. Acknowledgements. We thank A. Adams, M. Becker, J. Harvey, J.T. Liu, A. Strominger, and A. Tomasiello for discussion. J.-X. Fu is supported in part by NSFC grant 10771037, LMNS, and Fan Fund. L.-S. Tseng is supported in part by NSF grant PHY-0714648. And S.-T. Yau is supported in part by NSF grants DMS-0306600 and PHY-0714648.
References 1. Eguchi, T., Hanson, A.J.: Asymptotically flat selfdual solutions to Euclidean gravity. Phys. Lett. B 74, 249 (1978) 2. Fu, J.-X., Yau, S.-T.: The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation. J. Diff. Geom. 78, 369 (2008) 3. Becker, K., Becker, M., Fu, J.-X., Tseng, L.-S., Yau, S.-T.: Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory. Nucl. Phys. B 751, 108 (2006) 4. Dasgupta, K., Rajesh, G., Sethi, S.: M theory, orientifolds and G-flux. JHEP 9908, 023 (1999) 5. Becker, K., Dasgupta, K.: Heterotic strings with torsion. JHEP 0211, 006 (2002) 6. Cvetic, M., Lu, H., Pope, C.N.: Nucl. Phys. B 600, 103 (2001) 7. Cvetic, M., Gibbons, G.W., Lu, H., Pope, C.N.: Ricci-flat metrics, harmonic forms and brane resolutions. Commun. Math. Phys. 232, 457 (2003) 8. Lu, H., Vazquez-Poritz, J.F.: Resolution of overlapping branes. Phys. Lett. B 534, 155 (2002) 9. Gauntlett, J.P., Martelli, D., Waldram, D.: Phys. Rev. D 69, 086002 (2004) 10. Strominger, A.: Heterotic solitons. Nucl. Phys. B 343, 167 (1990) [Erratum-ibid. B 353, 565 (1991)] 11. Duff, M.J., Lu, J.X.: Elementary five-brane solutions of D = 10 supergravity. Nucl. Phys. B 354, 141 (1991) 12. Callan, C.G., Harvey, J.A., Strominger, A.: World sheet approach to heterotic instantons and solitons. Nucl. Phys. B 359, 611 (1991) 13. Callan, C.G., Harvey, J.A., Strominger, A.: Worldbrane actions for string solitons. Nucl. Phys. B 367, 60 (1991) 14. Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274, 253 (1986) 15. Hull, C.M.: Anomalies, ambiguities and superstrings. Phys. Lett. B 167, 51 (1986) 16. Sen, A.: (2, 0) supersymmetry and space-time supersymmetry in the heterotic string theory. Nucl. Phys. B 278, 289 (1986) 17. Kimura, T., Yi, P.: Comments on heterotic flux compactifications. JHEP 0607, 030 (2006)
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18. Fernandez, M., Ivanov, S., Ugarte, L., Villacampa, R.: Non-Kähler heterotic string compactifications with non-zero fluxes and constant dilaton. http://arXiv.org/abs/0804.1648v4[math.DG], 2008 19. Goldstein, E., Prokushkin, S.: Geometric model for complex non-Kähler manifolds with SU(3) structure. Commun. Math. Phys. 251, 65 (2004) 20. Eguchi, T., Hanson, A.J.: Selfdual solutions to Euclidean gravity. Ann. Phys. 120, 82 (1979) 21. Becker, M., Tseng, L.-S., Yau, S.-T.: Moduli space of torsional manifolds. Nucl. Phys. B 786, 119 (2007) 22. Adams, A.: Talk at String Workshop on String Theory: From LHC Physics to Cosmology, College Station, TX March 9–12, 2008, paper to appear 23. Adams, A., Ernebjerg, M., Lapan, J.M.: Linear models for flux vacua. Adv. Theor. Math. Phys. 12, 817–851 (2008) Communicated by G. W. Gibbons
Commun. Math. Phys. 289, 1171–1210 (2009) Digital Object Identifier (DOI) 10.1007/s00220-009-0828-y
Communications in
Mathematical Physics
Dynamical Formation of Correlations in a Bose-Einstein Condensate László Erd˝os , Alessandro Michelangeli , Benjamin Schlein Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany. E-mail: [email protected] Received: 2 August 2008 / Accepted: 12 February 2009 Published online: 23 May 2009 – © Springer-Verlag 2009
Abstract: We consider the evolution of N bosons interacting with a repulsive short range pair potential in three dimensions. The potential is scaled according to the Gross-Pitaevskii scaling, i.e. it is given by N 2 V (N (xi − x j )). We monitor the behaviour of the solution to the N -particle Schrödinger equation in a spatial window where two particles are close to each other. We prove that within this window a short-scale interparticle structure emerges dynamically. The local correlation between the particles is given by the two-body zero energy scattering mode. This is the characteristic structure that was expected to form within a very short initial time layer and to persist for all later times, on the basis of the validity of the Gross-Pitaevskii equation for the evolution of the Bose-Einstein condensate. The zero energy scattering mode emerges after an initial time layer where all higher energy modes disperse out of the spatial window. We can prove the persistence of this structure up to sufficiently small times before three-particle correlations could develop. 1. Introduction and Main Result We consider a three-dimensional system of N indistinguishable spinless bosons coupled with a pairwise repulsive interaction VN . The Hamiltonian of this system is HN =
N i=1
(−i ) +
VN (xi − x j )
(1.1)
1i< j N
acting on L 2sym (R3N , dx), the symmetric subspace of the tensor product of N copies of the one-particle space L 2 (R3 ). Here x = (x1 , x2 , . . . , x N ) ∈ R3N denotes the Partially supported by the SFB-TR12 of the German Science Foundation. Supported by a Kovalevskaja Award from the Humboldt Foundation. On leave from Cambridge University.
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position of the N particles. We assume that the interaction potential VN scales with N according to the scaling introduced by Lieb, Seiringer and Yngvason in [6]; that is, we fix a repulsive potential V and we rescale it by defining VN (x) := N 2 V (N x).
(1.2)
We assume that the unscaled potential V : R3 → R is non-negative, smooth, spherically symmetric, and compactly supported. The wave function of the system at time t is denoted by N ,t ∈ L 2sym (R3N ) with N ,t 2 = 1. It evolves according to the Schrödinger equation i∂t N ,t = H N N ,t
(1.3)
with a given initial condition N ,t=0 = N . We will be interested in the evolution of initial data exhibiting complete Bose-Einstein condensation. For a given wave function N ∈ L 2sym (R3N ) we define the one-particle marginal (1)
(1)
γ N = γ N associated with ψ N to be the positive trace-class operator on L 2 (R3k ) with kernel given by (1) γ N (x; x ) := N (x, z 2 , . . . , z N ) N (x , z 2 , . . . , z N ) dz 2 · · · dz N . (1.4) R3(N −1)
We say that a sequence { N } N ∈N of N -body wave functions exhibits complete Bose-Einstein condensation in the one-particle state ϕ ∈ L 2 (R3 ), ϕ2 = 1, if (1)
γ N −→ |ϕϕ|
(1.5)
in the trace-norm topology, as N → ∞. Here |ϕϕ| denotes the orthogonal projection operator onto ϕ (Dirac notation). On the level of the one-particle density matrix, the condition (1.5) is a signature that almost all particles (up to a fraction vanishing in the limit N → ∞) occupy the same one-particle state, described by the orbital ϕ (condensate wave function). Suppose now that a trapping external potential is added to the Hamiltonian, i.e., instead of (1.1) consider the Hamiltonian trap
HN
=
N
(−i + U (xi )) +
VN (xi − x j )
(1.6)
1i< j N
i=1
with U (x) → ∞ as |x| → ∞. In [6], Lieb, Seiringer and Yngvason proved that the trap ground state energy of H N divided by the number of particle N (the ground state energy per particle) converges, as N → ∞, to the minimum of the Gross-Pitaevskii energy functional |∇ϕ|2 + U |ϕ|2 + 4πa|ϕ|4 (1.7) E(ϕ) = R3
over all ϕ ∈ with ϕ2 = 1. The coupling constant a is the scattering length of the unscaled potential V . Later, in [5], Lieb and Seiringer also showed that the ground trap state of H N exhibits complete Bose-Einstein condensation into the minimizer of (1.7). We recall that the scattering length of a potential V is defined as 1 a := lim |x| ω(x) = dxV (x) (1 − ω(x)), (1.8) |x|→∞ 8π R3 L 2 (R3 )
where 1 − ω is the unique non-negative solution of the zero-energy scattering equation
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1173
− + 21V (1 − ω) = 0
(1.9)
with boundary conditions ω(x) → 0 as |x| → ∞. Several properties of ω are collected in Appendix C. It follows from the definition that the scattering length of the scaled potential VN is a N = a/N and that the zero-energy scattering solution associated with − + 21 VN is the function 1 − ω N with ω N (x) = ω(N x). In particular, ω N has a built-in structure at the scale N −1 . The time evolution of a condensate after removing the trap, i.e. setting U ≡ 0, can be described by the solution of the Gross-Pitaevskii equation i∂t ϕt = −ϕt + 8πa|ϕt |2 ϕt (1.10) with initial condition ϕt t=0 = ϕ. The first rigorous derivation of the Gross-Pitaevskii equation (1.10) from many-body Schrödinger dynamics (1.3) has been obtained in [2] under the condition that the unscaled interaction potential V is sufficiently small. This result was later extended to large interaction potentials in [3]. More precisely, it has been shown that if the family of wave functions { N } N ∈N has finite energy per particle (in the sense that N , H N N C N ), and if it exhibits complete Bose-Einstein condensation in the one-particle state ϕ in the sense (1.5), then, at any time t = 0, N ,t = e−i HN t N still exhibits complete Bose-Einstein condensation. Moreover, the condensate wave function ϕt at later times is determined by the solution of the Gross-Pitaevskii equation (1.10) with initial data ϕt=0 = ϕ. At first sight, the condition (1.5) may indicate that wave functions describing condensates are very close to be factorized (for N = ϕ ⊗N , (1.5) holds as an equality). However, the structure of the evolved wave function N ,t is much more complicated. In fact, it turns out that N ,t is characterized by a short-scale correlation structure which plays an important role in the derivation of the Gross-Pitaevskii equation. If this short-scale structure were lacking along the evolution and the wave function N ,t were essentially constant in the relative coordinates xi −x j whenever |xi −x j | N1 , then the condensate wave function ϕt would solve Eq. (1.10) with coupling constant b (this is the first Born approximation of 8πa); we discuss this problem in more details in Appendix E. If the initial state N has a short-scale structure characterized by 1 − ω N (xi − x j ) for |xi − x j | N1 , for example, it is the ground state of the system before removing the traps, then this structure seems to persist along the time evolution. The condensate wave function ϕt lives on order one scales and it changes with time, while the correlation structure on the scale N −1 is conserved. However, Theorem 2.2 of [2] states that even if the initial state does not have a built-in short-scale structure, the time evolution of the orbital wave function is still given by (1.10). This indicates that the characteristic short-scale structure not only persists but dynamically emerges within a very short initial time layer. The main result of this paper is a description of the dynamical formation of this short-scale structure. Consistently with the underlying Gross-Pitaevskii equation, which is derived at any fixed t in the limit N → ∞, correlations in a large but finite system with N particles are expected to form in a short transient time (i.e., o(1) in N ) and then to be preserved at any macroscopic time (i.e., times of order 1). In the previous works the presence of the local structure was captured by the H N2 -energy estimate 2 2 H (x) N N dx C N , ∇i ∇ j N , (1.11) 1 − ω N (xi − x j ) N2 R3N
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valid for every N ∈ L 2 (R3N ) and every fixed indices i = j in {1, . . . N }. This inequality has been proven in [2] for small interaction potential (for large interaction potential, another a-priori bound on the solution N ,t of the N -particle Schrödinger equation has been obtained in [3]). Under very general conditions on the trapping potential U , the r.h.s of (1.11) is of trap order one for the ground state N of H N . If N did not carry a short-scale structure (1 − ω N (xi − x j )) when |xi − x j | ∼ N −1 , say, it was essentially constant on this scale, then the main term on the l.h.s of (1.11) would be of order 2 1 | N (x)|2 dx ∇i ∇ j 1 − ω N (xi − x j ) |xi −x j |1/N ∼ |∇ 2 ω N (x)|2 dx ∼ O(N ). |x|1/N
More generally, this shows that the short-distance behaviour of any N for which the inequality N , H N2 N C N 2 (1.12) holds, is asymptotically given by (1−ω N (xi −x j )) in the regime where |xi −x j | ∼ N −1 . Since H N2 is conserved along the dynamics, N ,t , H N2 N ,t = N , H N2 N , the same conclusion holds for N ,t at any later time t > 0, thus for such initial data the local structure is preserved in time. The same argument cannot be used to detect the presence and the formation of the short-scale structure for states that do not satisfy the bound (1.12). This is the case for the completely uncorrelated initial state ϕ ⊗N , with some ϕ ∈ H 2 (R3 ), because ϕ ⊗N , H N2 ϕ ⊗N ∼ N 3 (see Lemma B.1). However, the validity of the Gross-Pitaevskii equation for the product initial state (Theorem 2.2 of [2]) indicates that the short-scale structure, although initially not present, still forms within a very short time interval. The length of this transient time interval must vanish as N → ∞ since after the N → ∞ limit, the Gross-Pitaevskii Eq. (1.10) is valid for any positive time t > 0. The proof of Theorem 2.2. of [2] still relied on (1.12) after an energy cutoff H N κ N in the manybody Hilbert space. The energy cutoff artificially introduced the short-scale structure in the low energy regime. Although it was showed that, in the case κ 1, the high energy regime H N κ N essentially does not influence the evolution of the condensate, the proof did not reveal whether the short-scale structure is indeed formed in the full wave function without the cutoff. The goal of the present analysis is to prove the dynamical formation of the expected pattern of correlations in the time evolution of an initially uncorrelated many-body state N = ϕ ⊗N . The above discussion suggests to compare N ,t with N
ϕt (xr ) 1 − ω N (xi − x j ) 1i< j N
(1.13)
r =1
after the transient time. Although (1.13) has the expected built-in short-scale structure in each relative variable, the true N -body wave function at later time t > 0 is much more complicated and it is beyond our reach to describe it precisely. The main reason is
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1175
that it quickly develops higher order correlations as well; the typical distance between neighboring particles is of order N −1/3 , so within time t of order one, each particle collides with many of its neighbors. Moreover, the energy of the product initial state, b ⊗N ⊗N 2 4 ϕ , H N ϕ ≈ N |∇ϕ| + |ϕ| 2 R3 R3 is much bigger than the energy predicted by the Gross-Pitaevskii functional, by recalling that b > 8πa. Thus there is an excess energy of order N in the system that remains unaccounted for. This excess energy must live on intermediate length scales N −γ for 0 < γ < 1 not to be detected either on the local structure of order N −1 or on the order one scale of the condensate. Moreover, these excess modes must be sufficiently incoherent not to influence the evolution of the condensate. We cannot describe the evolution of these intermediate modes yet, we can only focus on the formation of the local structure (1 − ω N (xi − x j )). This is selected in a scattering process by having the smallest local energy. Our study is restricted to a spatial window where |xi − x j | < for some intermediate between the length scale N −1 of the expected local structure and the macroscopic order one scale of the system. If the initial datum is essentially constant in the variable xi − x j as far as |xi − x j | < , then a shortscale structure emerges corresponding to the zero-energy mode 1 − ω N together with all higher energy modes; these quickly leave the -window while only the short-scale structure remains. We can thus monitor the formation of the local structure (1 − ω N (xi − x j )) in this window and prove its persistence for a while after its emergence. The time scale must be sufficiently large compared to the window size so that all higher energy modes disperse, but it has to be sufficiently small so that no three-particle correlations could develop yet. We provide a rigorous version of this picture in Theorem 1.1. We focus on the local structure in the relative variable x1 − x2 ; by symmetry the same result holds for any pairs of relative variables xi − x j . We define a cutoff function |x| (1.14) θ (x) := χ with
∞
χ ∈ C (R+ ) ,
χ (r ) :=
1 if 0 r 1 0 if r 2.
We consider the following time-dependent quantity: 2 N ,t (x) − N (x) dx, F N (t) := θ (x1 − x2 ) 1 − ω N (x1 − x2 ) R3N
(1.15)
(1.16)
where 1 − ω N is the zero energy scattering solution of − + 21 VN , and N (x) = N j=1 ϕ(x j ) is the initial N -body wave function. The cutoff scale is always assumed to satisfy 1 1 √ . N N
(1.17)
Here and throughout in the sequel we will make the convention that by A N B N one means that 0 < (log N )k A N C B N with a sufficiently large k and C.
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The decrease of F N (t) has the natural interpretation of formation of a local structure at the scale N −1 within the -window. By controlling this quantity we demonstrate that the conjectured short-scale structure indeed forms within a short time of order N −2 and then it is preserved for longer times in a time window that is essentially 1 N −2 t N −(2− 10 ) . Our main result is the following: Theorem 1.1. Let V : R3 → R be a non-negative, smooth, spherically symmetric, and compactly supported potential. Let VN (x) := N 2 V (N x) for N ∈ N. Let ϕ ∈ L 2 (R3 ) with ϕ2 = 1 and ϕ4,∞,α
4 α m x ∇ ϕ < ∞ := ∞
(1.18)
m=0
for some α > 3. Consider the Hamiltonian HN =
N
(−i ) +
VN (xi − x j )
(1.19)
1i< j N
i=1
acting on L 2 (R3N ), the initial datum N = ϕ ⊗N , and its time evolution N ,t := e−i HN t ϕ ⊗N . Consider the function F N (t) defined in (1.16). Then 4 6 (log N ) 5 (N 2 t)2 (N )4 2 F N (t) C F N (0) + log N t (1.20) 1 N N 2t N5 for all times t such that 0 < t N −1 . As a consequence, F N (t) F N (0)
for
1
1
(N )4 N 2 t N 10 (N ) 2 .
(1.21)
Remark 1. Equation (1.20) does not look dimensionally correct (in our coordinates, time has the dimension of a length squared). The reason is that, to simplify the notation, we consider the length scale λ characterizing the initial wave function ϕ, the radius R of the support of the (unscaled) potential V , and the scattering length a of V as dimensionless constants of order one. More generally, if, for λ > 0 we set ϕ (λ) (x) = λ−3/2 ϕ(x/λ) the bound Eq. (1.20) assumes the (dimensionally correct) form 4 6 1 (log N ) 5 (N 2 t)2 1 (N )4 2 2 + 2 log(N t/λ ) F N (t) C F N (0) , λ3 N 15 N λ N 2t where the dimensionless constant C depends on the ratios R/λ and a/λ (a more refined analysis would also allow to compute the precise dependence on R and a). Remark 2. Since the interaction potential has a length scale 1/N , it will be convenient to introduce the length L = N , expressing the size of the window relative to the interaction range. The two particle scattering process takes place on a time scale 1/N 2 (see Sect. 2), thus it is also natural to introduce the rescaled time T = N 2 t. The appearance
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1177
of the combinations of N and N 2 t in the theorem is motivated by the fact that we actually describe a long time and large distance scattering process in terms of the rescaled variables T 1, L 1, in the regime where L 4 T N 1/10 L 1/2 (in the spirit of Remark 1 above, we consider the regime (L/λ)4 (T /λ2 ) N 1/10 (L/λ)1/2 ). Remark 3. To be concrete, choosing = is established for times
1 N,
the short-scale structure given by 1 − ω N
1 1 t . 1 2− 10 N2 N
(1.22)
The first inequality corresponds to the formation of the short-scale structure beyond the scattering time scale within a window comparable with the interaction range. The second inequality expresses the persistence of this short-scale structure within this spatial window up to times much longer than the scattering time. Remark 4. Note that in the definition of F N (t) we compared N ,t /(1 − ω N ) with the initial product state and not with the evolved product state ϕt⊗N , which would have been more natural. For the relatively short time scales that we can consider, this difference is essentially does not move. One can directly check that irrelevant; the⊗Ncondensate θ (x1 − x2 )|ϕ (x) − ϕt⊗N (x)|2 dx F N (0), hence the modification does not influence (1.21). To investigate the persistence of the local structure up to times of order 1 in N , the definition of F N (t) should, of course, contain ϕt⊗N instead of the initial state ϕ ⊗N . However, at such large times, the i th and j th particles also interact with other particles. Our analysis does not control consecutive multiple collisions, although the validity of the Gross Pitaevskii equation still gives an indirect evidence that the correlation structure is preserved even for times of order one. Remark 5. In [2] the local structure was identified by the L 2 norm of the mixed derivative ∇1 ∇2 [ N ,t /(1 − ω N )]. For initially factorized states, the integral R3N
N ,t (x) θ (x1 − x2 ) ∇1 ∇2 1 − ω (x − x N
1
2
2 dx )
(1.23)
is of the order N at time t = 0, and it is expected to be of order one for times t N −2 . Proving this decay would establish the formation of the local structure in a much stronger norm than the local L 2 norm used for Theorem 1.1. Unfortunately, due to the very singular interaction potential VN , we cannot bound (1.23) effectively (estimating it by the expectation of H N2 produces a bound proportional to N for all times, see Lemma B.1). Remark 6. Condition (1.18) encodes all regularity and decay that we assume on ϕ, although it is not optimal and we do not aim at finding most general conditions on ϕ. In particular, (1.18) implies that ϕ ∈ H 4 (R3 ), hence ϕ ∈ C 2 (R3 ). Notation. By C we will mean a constant depending only on the unscaled potential V and the initial one-body wave function ϕ. Constants denoted by c p,q,... are meant to depend also on the indices p, q, etc.
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2. Proof of Main Theorem In this section we present the main steps of the proof of Theorem 1.1. Proof of Theorem 1.1. Let F N (t) be the quantity defined in (1.16). To evaluate F N , we introduce a dynamics where particles 1 and 2 are decoupled from the others. We define h(1,2) := −1 − 2 + VN (x1 − x2 ), N (3)
H N :=
N
(−i ) +
VN (xi − x j ),
(2.2)
VN (x1 − x j ) + VN (x2 − x j ) ,
(2.3)
3i< j N
i=3 (1,2)
UN
:=
N
(2.1)
j=3
N := h(1,2) + H (3) = H N − U (1,2) . H N N N Then we have N ,t := e−i HN t ϕ ⊗N = (1,2)
with ψt = e−i hN
(1,2)
e−i hN
t ⊗2
ϕ
(2.4)
(3) ⊗ e−i HN t ϕ ⊗(N −2) = ψt ⊗ t ,
(2.5)
(3)
and t = e−i HN t ϕ ⊗(N −2) . Thus, N ,t (x) 2 N ,t (x) − dx F N (t) θ (x1 − x2 ) 1 − ω N (x1 − x2 ) R3N 2 N
N ,t (x) − θ (x1 − x2 ) ϕ(xi ) dx + 3N 1 − ω (x − x ) N 1 2 R t ⊗2 ϕ
(2.6)
i=1
C ( G N (t) + K N (t)) , where
G N (t) := K N (t) :=
R3N
R3N
2 N ,t (x)2 dx, θ2 (x1 − x2 ) N ,t (x) − 2 N
N ,t (x) − θ (x1 − x2 ) ϕ(xi ) dx 1 − ω N (x1 − x2 )
(2.7) (2.8)
i=1
2 (x) has been used). (notice that θ (x) < θ2 G N measures the L 2 -distance, in the spatial region where |x1 − x2 | , between the N ,t of the initial state ϕ ⊗N with the dynamics given by H N and evolutions N ,t and N , respectively. By construction, G N (0) = 0 and at later times G N (t) deteriorates as H N ,t separate. We control such behaviour in Sect. 3: the result the two vectors N ,t and (Proposition 3.1) is 4
G N (t) C (N log N ) 5 t 2 ,
(2.9)
provided that N −2/5 and for all times t 0. Notice that, if we had followed the dependence on the length scale λ characterizing the initial wave function ϕ (as explained
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1179
in Remark 1 after Theorem 1.1), the bound (2.9) would have taken the dimensionally correct form G N (t) C(N log N )4/5 (t/λ2 )2 ; similar remarks would also apply to all estimates in the sequel which are apparently dimensionally incorrect. N ,t = ψt ⊗ t is preserved in time, see On the other hand, since factorisation (2.5), then K N turns out to be essentially a two-body quantity, that is, an integral only in variables x1 and x2 , which involves the Schrödinger evolution of the initial datum ϕ ⊗2 with interaction VN (x1 − x2 ). Reduction of K N to a two-body integral and a remainder is done in Sect. 4. The result (Proposition 4.1) is e−i hN t ψη (x) K N (t) C 2 dη dx θ2 (x) ∇x 1 − ω N (x) R 3 ×R 3 (log N 2 t)6 3 2 3 5 + t + Nt + +C N 2t
2 (2.10)
for all times t > 0. Here we defined h N = −2x + VN (x),
and ψη (x) = ϕ(η + x/2)ϕ(η − x/2)
(2.11)
(η and x denote, in other words, the center of mass and, respectively, the relative coordinate of particle one and two). Last, we consider the two-body integral contained in our bound (2.10) to K N . It has the natural interpretation of a quantity which tracks the dynamical formation of a short-scale structure in the evolution of the two-body initial factor state ϕ ⊗2 . We study this problem in Sect. 5. The result (Corollary 5.2) is
e−i hN t ψη (x) dη dx θ (x) ∇x 1 − ω N (x) R 3 ×R 3
2 2 6 C (log N t) N 2 3 + N t 2 + 3 N 2t (2.12)
for all times t > 0. Bounds (2.9), (2.10), and (2.12) complete estimate (2.6) for F N . It takes the form 4 (log N 2 t)6 3 2 + t + 3 N t + 5 F N (t) C (N log N ) 5 t 2 + N 2t (log N 2 t)6 2 5 2 2 (2.13) N + Nt + N 2t 2
for t > 0 and N −1 < N − 5 . Using that, by Lemma 2.1, F N (0) ∼ /N 2 and restricting to N −1/2 , t N −1 , one has 4 (log N ) 5 (N 2 t)2 (N )4 2 6 + (log N t) . F N (t) C F N (0) (2.14) 1 N N 2t N5 We compute now the asymptotics of F N at time t = 0.
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Lemma 2.1. Let F N be the quantity defined in (1.16). Assume that scales with N in such a way that N → c0 when N → ∞, where c0 ∈ (0, +∞]. Then there exists a constant Cχ , depending only on the cut-off function χ defined in (1.15) and on the potential V , such that N2 F N (0) = Cχ ϕ44 .
(2.15)
N2 F N (0) = 4πa 2 χ L 1 (R) ϕ4L 4 (R3 ) .
(2.16)
lim
N →∞ N →c0
Moreover, if N → ∞, lim
N →∞ N →∞
Proof. One has 2 ϕ(x1 )ϕ(x2 ) N2 N2 F N (0) = − ϕ(x1 )ϕ(x2 ) dx1 dx2 θ (x1 − x2 ) R6 1 − ω N (x1 − x2 ) ω N (x) 2 N2 |ϕ(x2 + x)|2 = dx2 |ϕ(x2 )|2 dx θ (x) R3 1 − ω N (x) R3 1 |x| ω(x) 2 x 2 = dx2 |ϕ(x2 )|2 dx χ ϕ x2 + N R3 N 1 − ω(x) N R3
(2.17)
N →∞
−−−−→ Cχ ϕ4L 4 (R3 )
by continuity of ϕ and by dominated convergence, where 1 |x| ω(x) 2 Cχ := lim dx χ . N →c0 N R3 N 1 − ω(x) The above limit clearly exists if c0 is finite. If, instead, N → ∞, one has 1 |x| ω(x) 2 N →∞ dx χ −−−−→ 4πa 2 χ L 1 (R) . N R3 N 1 − ω(x)
(2.18)
(2.19)
To prove (2.19), one sees that integration for |x| > 1 gives the leading contribution, since 1 |x| ω(x) 2 1 4π ω2∞ N →∞ dx χ −−−−→ 0 (2.20) N |x|<1 N 1 − ω(x) N 3 (1 − cρ)2 while, when |x| > 1, since ω(x) = a/|x|, χ ( N|x| ) a2 |x| ω(x) 2 1 dx χ = dx N |x|>1 N 1 − ω(x) N |x|>1 (|x| − a)2 2 r2 = 4πa 2 dr χ (r ) (r − Na )2 (N )−1 N →∞
−−−−→ 4πa 2 χ L 1 (R) , and (2.16) is proved.
(2.21)
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1181
3. Many-Body Problem for Short Times In this section we control the growth of the nonnegative quantity G N defined in (2.7). Proposition 3.1. Assume that N −2/5 . Then, 4
G N (t) C (N log N ) 5 t 2
(3.1)
for all times t 0. Proof. For arbitrary ˜ 2 we have 2 N ,t (x)2 dx G N (t) = θ2 (x1 − x2 ) N ,t (x) − 3N R N (t). N ,t (x)2 dx =: G θ 2˜ (x1 − x2 ) N ,t (x) − R3N
(3.2)
The parameter ˜ will be fixed later on (we will choose ˜ = (N log N )−2/5 ). Let us denote by ·, · the scalar product in L 2 (R3N ) and by θ12 , ω12 , and Vi j the operators of multiplication by θ ˜(x1 − x2 ), ω N (x1 − x2 ), and VN (xi − x j ) respectively. Then N (t) d dG 2 N ,t , θ12 N ,t ) = N ,t − ( N ,t − dt dt 2 N ,t , [i H N , θ12 N ,t ) ] ( N ,t − = N ,t − 2 N ) N ,t ) , (H N − H N ,t + 2 Im θ12 ( N ,t − (1)
(2)
≡ J N (t) + J N (t). Since 2 2 ] = [−1 − 2 , θ12 ] = −2 [H N , θ12
(3.3)
(∇r · (∇r θ12 ) θ12 + θ12 (∇r θ12 ) · ∇r ),
r =1,2
(3.4) J N(1)
the summand on the r.h.s. of (3.3) takes the form 2 N ,t , [i H N , θ12 N ,t ) N ,t − ] ( N ,t − ∇r δ N ,t , (∇r θ12 ) θ12 δ N ,t − δ N ,t , θ12 (∇r θ12 )∇r δ N ,t = 2i r =1,2
(3.5) N ,t . Then having set δ N ,t := N ,t − (1) (∇r θ12 ) · (∇r δ N ,t ), θ12 δ N ,t J N (t) 4 r =1,2
N (t) 8 G
R3N
2 dx |∇θ12 |2 (∇1 δ N ,t )(x)
N (t) ∇θ12 ∞ =8 G N (t) ˜−1 . C G
R3N
1 2
2 dx (∇1 δ N ,t )(x)
1 2
(3.6)
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On the last line above we used R3N
2 dx (∇1 δ N ,t )(x) C.
Indeed, by the symmetry of N ,t , and because of (B.1), 2 H N ⊗N C, ϕ dx ∇1 N ,t = N ,t , (−1 ) N ,t ϕ ⊗N , N R3N while, due to the factorization (2.5), 2 (1,2) N ,t 2 = dx ∇1 dx1 dx2 (∇1 e−it hN ϕ ⊗2 )(x1 , x2 ) R3N R6 ⊗2 ≤ C. ϕ ⊗2 , h(1,2) ϕ N
(3.7)
(3.8)
(3.9)
(2)
Let us now examine the summand J N in the r.h.s. of (3.3). One has 2 (2) N ) N ,t ) , (H N − H N ,t ( N ,t − J N (t) 2 θ12 (1,2) 2 = 2 δ N ,t , θ12 UN N ,t ⎛ ⎞ N 2 ⎝ ⎠ 4 δ N ,t , θ12 V1 j N ,t j=3
(3.10)
(last line following by the permutational symmetry 1 ↔ 2), whence, by Schwarz inequality, ⎞1 ⎛ N 1 2 2 (2) 2 2 N ,t ⎠ N ,t , θ12 δ N ,t ⎝ V1k V1 j J N (t) 4 δ N ,t , θ12 j,k=3
2 2 N (t) N N ,t , θ12 C G V13 N ,t 1 2 N ,t 2 . N ,t , θ12 V13 V14 +N 2
(3.11)
The first term on the r.h.s. is estimated by 2 2 N ,t , θ12 V13 N ,t = θ 2˜ (x1 − x2 ) VN2 (x1 − x3 ) |ψt (x1 , x2 )|2 |t (x3 , . . . , x N )|2 dx 3N R ψt 2∞ θ 2˜ (x1 − x2 ) N 4 V 2 (N (x1 − x3 )) |t (x3 , . . . , x N )|2 dx R3N 2 4 ˜3 = Cψt ∞ N V 2 (N x) |t (x3 , . . . , x N )|2 dxdx3 · · · dx N = C = C
ψt 2∞ ψt 2∞
˜3
R3(N −1) t 22
N N ˜ 3 .
(3.12)
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1183
It follows by Proposition A.1 that, under the assumption (1.18), ψt 2L ∞ (R6 ,dx
1 dx2 )
C(log N ),
(3.13)
and thus that
2 2 N ,t , θ12 V13 N ,t
C N (log N )2 ˜ 3 .
(3.14)
On the other hand, the second term on the r.h.s. of (3.11) is estimated as 2 N ,t N ,t , θ12 V13 V14 = θ 2˜ (x1 −x2 ) VN (x1 − x3 )VN (x1 −x4 ) |ψt (x1 , x2 )|2 |t (x3 , . . . , x N )|2 dx R3N C ψt 2∞ ˜ 3 VN (x1 −x3 )VN (x1 −x4 ) |t (x3 , . . . , x N )|2 dx1 dx3 · · · dx N 3(N −1) R 2 2 ˜3 C ψt ∞ VN 3 dx3 · · · dx N 1 − 3 t (x3 , . . . , x N ) 2 R3(N −2) × dx1 VN (x1 − x4 ) R3 (3) HN ˜ 3 2 C ψt ∞ V 3 V 1 t , 1 + t 2 N N −2 2 3(N −2) L (R
C
N )2
(log N
)
˜ 3 .
(3.15)
In the last inequality we used again bound (3.13) and the asymptotics (B.1). Thus, (3.11) reads (2) N (t) N ˜ 23 log N . J N (t) C G
(3.16)
Altogether, (3.3), (3.6), and (3.16) give G (t) C G N (t) ˜ −1 + N ˜ 23 log N . N
(3.17)
Letting ˜ = (N log N )−2/5 , we get G (t) C G N (t) (N log N ) 25 , N N (0) = 0. which implies (3.1) by the Gronwall Lemma, because G
(3.18)
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4. Reduction to the Two-Body Problem The goal of this section is to reduce the study of the quantity K N , defined in (2.8), to the analysis of a two-body term (which will then be controlled in Sect. 5). We will use, in this section, the coordinates (η, x) defined by η = (x1 + x2 )/2 x = x2 − x1
(center of mass coordinate), (relative coordinates).
(4.1)
introduced in (2.1) takes the form In these coordinates, the two-body Hamiltonian h(1,2) N (1,2)
hN
= −η /2 + h N ,
with
h N = −2x + VN (x).
Note, also, that the two-body initial data ϕ ⊗2 is given by x x ϕ η− . ψ(η, x) = ψη (x) = ϕ η + 2 2
(4.2)
(4.3)
Proposition 4.1. Suppose that the assumptions of Theorem 1.1 are satisfied. Let h N = −2 + VN (x) and ψη (x) be defined as in (4.3). Then, if K N (t) is defined as in (2.6), we have 2 e−i hN t ψη (x) K N (t) C 2 dη dx θ2 (x) ∇x 1 − ω N (x) R 3 ×R 3 (log N 2 t)6 3 2 (4.4) + t + 3 N t + 5 +C N 2t for all times t > 0. Proof. We divide the proof in four steps. Step 1. We have 2 ψt (x1 , x2 ) K N (t) 2 − ϕ(x1 )ϕ(x2 ) dx1 dx2 θ (x1 − x2 ) 3 3 1 − ω (x − x ) N 1 2 R ×R
+ C 3 N t,
(4.5)
⊗2 is defined in (2.5). where ψt (x1 , x2 ) = exp(−ith(1,2) N )ϕ
N ,t = ψt ⊗ t and we split To prove (4.5), we use that 2 N
N ,t (x) − θ (x1 − x2 ) ϕ(xi ) dx K N (t) = 1 − ω N (x1 − x2 ) R3N i=1 2 ψt (x1 , x2 ) − ϕ(x1 )ϕ(x2 ) dx1 dx2 θ (x1 − x2 ) 2 1 − ω N (x1 − x2 ) R 3 ×R 3 2 +2 θ (x1 − x2 ) ϕ ⊗2 ⊗ t (x) − ϕ ⊗N (x) dx, (4.6)
R3N
Dynamical Formation of Correlations in a Bose-Einstein Condensate
where R3N
1185
2 θ (x1 − x2 ) ϕ ⊗2 ⊗ t (x) − ϕ ⊗N (x) dx 2 θ (x1 − x2 )|ϕ(x1 )|2 |ϕ(x2 )|2 t − ϕ ⊗(N −2) 2
=
R 3 ×R 3 C 3 ϕ4L 4 (R3 )
L (R3(N −2) )
dx1 dx2
t − t=0 2L 2 (R3(N −2) ) .
(4.7)
Equation (4.5) follows now from t − 0 22 C N t.
(4.8)
2 d (3) (3) t − 0 2 0 , H N t 2 0 , H N 0 C N . 2 dt
(4.9)
To show (4.8), observe that
The last inequality follows from (B.1) with N replaced by N − 2 (recall the definition (3) of H N from (2.2)). Step 2. We have 2 ψt (x1 , x2 ) − ϕ(x1 )ϕ(x2 ) dx1 dx2 3 3 1 − ω N (x1 − x2 ) R ×R 2 e−i hN t ψη (x) + C R(1) (t) + R(2) (t) C 2 dη dx θ2 (x) ∇x N N 1 − ω N (x) R 3 ×R 3
θ (x1 − x2 )
(4.10)
with e−i hN t ω N ψη (y) (1) R N (t) := dη dx θ (x) dy θ (y) 1 − ω N (y) R3 R3 R3
2
(4.11)
and (2) R N (t)
η := dη dx θ (x) e−i 2 t ψ(η, x) R3 R3 e−i hN t (1 − ω N )ψη (y) − dy θ (y) 1 − ω N (y) R3
2 .
(4.12)
Here we use the notation θ (y) :=
θ (y) . θ 1
(4.13)
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To prove (4.10), we observe that
dx1 dx2 θ (x1 − x2 )
2 ψt (x1 , x2 ) − ϕ(x1 )ϕ(x2 ) 1 − ω N (x1 − x2 ) R 3 ×R 3 2 i η t −i hN t e 2 (e ψ )(x) η − ψ(η, x) = dη dx θ (x) 1 − ω N (x) R 3 ×R 3 i η t −i hN t e 2 (e ψη )(x) 2 dη dx θ (x) 1 − ω N (x) R 3 ×R 3 2 η ei 2 t (e−i hN t ψη )(y) dy θ (y) (4.14) − 1 − ω N (y) R3 2 i 2η t −i hN t (e ψ )(y) e η +2 dη dx θ (x) ψ(η, x) − dy θ (y) 1 − ω N (y) R 3 ×R 3 R3 2 (e−i hN t ψ )(x) (e−i hN t ψη )(y) η 2 dη dx θ (x) dy θ (y) − 1 − ω N (x) 1 − ω N (y) R 3 ×R 3 R3 2 (e−i hN t ψη )(y) −i 2η t +2 dη dx θ (x) e ψ(η, x) − dy θ (y) 1 − ω N (y) R 3 ×R 3 R3 (1)
by unitarity of eiη t/2 . The second term can be clearly bounded by the sum of R N (t) (2) and R N (t). For every fixed η ∈ R3 , the first term on the r.h.s. of the last equation can be estimated using the Poincaré inequality as
2 (e−i hN t ψ )(x) (e−i hN t ψη )(y) η − dx θ (x) dy θ (y) 1 − ω N (x) 1 − ω N (y) R3 R3 2 (e−i hN t ψ )(x) (e−i hN t ψη )(y) η − dx dy θ (y) 1 − ω N (x) 1 − ω N (y) supp θ supp θ (e−i hN t ψ )(x) 2 η 2 C dx ∇x 1 − ω N (x) supp θ (e−i hN t ψ )(x) 2 η 2 C dx θ2 (x) ∇x (4.15) . 1 − ω N (x) R3 (1)
Step 3. Suppose that R N (t) is defined as in (4.11). Then (1)
R N (t) C 3
(log N 2 t)6 . N 2t
(4.16)
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1187
To show (4.16), we note that, from (4.11), 2 (1) R N (t) C 3 dη e−i hN t ω ψη ∞
L (R3 ,dx)
R3
.
(4.17)
Let N be the wave operator associated with the Hamiltonian h N , defined as the strong limit N = s − lim ei hN t e2it . t→∞
Then, by the intertwining property (D.1) and Yajima’s bound (D.2), 2 2 −i hN t ω N ψη C e 2it ∗N ω N ψη . e ∞
∞
(4.18)
In Proposition 5.3 we prove that 2 2it ∗ N ω N ψη e
∞
cs
|||ψη |||2 ∀s ∈ (3, +∞], (N 2 t)3/s
(4.19)
where cs ∼ (s − 3)−6 as s → 3+ and where we defined the norm |||ψη ||| = ψη W 3,1 + ψη W 3,∞ .
(4.20)
Optimizing in s > 3, we have (1)
R N (t) C 3
(log N 2 t)6 dη |||ψη |||2 , N 2t R3
(4.21)
and thus, since |||ψη ||| Cη−α for some α > 3 (by the definition (4.3), and the condition (1.18)), we obtain (4.16). Step 4. Assume that R(2) N (t) is defined as in (4.12). Then (2) R N (t) C t 2 + 3 t + 5 .
(4.22)
First we rewrite e−it hN (1 − ω N )ψη (y) = dy θ (y) dy θ (y) e−2it L N ψη (y) (4.23) 1 − ω N (y) R3 R3 by means of the operator L N := − + 2
∇ω N · ∇. 1 − ωN
(4.24)
In fact, since h N (1 − ω N ) = (−2 + VN )(1 − ω N ) = 0, one has 1 h N (1 − ω N )φ = 2L N φ ∀φ ∈ H 2 (R3 ), 1 − ωN
(4.25)
whence e−i hN t (1 − ω N )φ = (1 − ω N )e−2i L N t φ
∀φ ∈ L 2 (R3 ).
(4.26)
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It is worth noticing that φ, L N ψ• = L N φ, ψ• = ∇φ, ∇ψ• ,
(4.27)
where f, g• =
dx (1 − ω N (x))2 f (x) g(x).
It follows that the operator L N is self-adjoint on the weighted Hilbert space L 2 (R3 , (1 − ω N (x))2 dx) (L N is the Laplacian on the weighted space). It is also important to note that, because of the properties of ω N , the norm · • defined by the weighted product ·, ·• is comparable with the standard L 2 -norm, in the sense that cφ2 φ• φ2 , with an appropriate constant c > 0. From (4.12), we find 2 η dx θ (x) dη e−i 2 t ψ(η, x) − ψ(η, x) 3 R3 R 2 +C dη dx θ (x) ψη (x) − ψη (0)
(2) R N (t) C
R3
R3
2 dx θ (x) ψη (0) − dy θ (y)ψη (y) R3 R3 R3 2 −2it L N +C dη dx θ (x) dy θ (y) e − 1 ψη (y) . (4.28)
+C
dη
R3
R3
R3
To estimate the first summand on the r.h.s. of (4.28) we use that 2 d −i 2η t ψ(η, x) − ψ(η, x) 2 ∇η ψ(η, x)2L 2 (R3 ,dη) C e dt 2 3 L (R ,dη) (4.29) uniformly in x ∈ R3 (the last inequality follows from (A.18)), whence
R3
dx θ (x)
R
2 η dη e−i 2 t ψ(η, x) − ψ(η, x) C 3 t. 3
(4.30)
To control the second summand on the r.h.s. of (4.28), we observe that ψη (x) − ψη (0) =
1
ds 0
d |x| ψη (sx) |x| ∇x ψη ∞ C ds ηα
(the last inequality follows from (A.18)), and thus 2 dη dx θ (x) ψη (x) − ψη (0) C R3
R3
R3
(4.31)
dη dx θ (x) |x|2 C 5 . η2α R3 (4.32)
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1189
The third summand on the r.h.s. of (4.28) can be bounded by 2 dη dx θ (x) ψη (0) − dy θ (y) ψη (y) 3 3 3 R R R 2 = dη dx θ (x) dy θ (y) ψη (y) − ψη (0)
R3
R3
R3
dη C dx θ (x) 2α 3 η R R3
R3
dy θ (y)|y|
2
C 5 ,
(4.33)
where (4.31) has been used. Finally, to estimate the fourth summand on the r.h.s. of (4.28), we observe that, for fixed η ∈ R3 ,
R3
2 2 −2it L N θ , e−2it L N − 1 ψη dy θ (y) (e − 1) ψη (y) =
2 C −2it L N e − 1 ψη . 3
(4.34)
Expanding −2it L N − 1 ψ = −2i e η
t 0
ds e−2is L N L N ψη ,
we obtain
2 Ct t dy θ (y) (e−2it L N − 1) ψη (y) 3 dse−2is L N L N ψη 2 0 R3 Ct t 3 dse−2is L N L N ψη 2• 0 Ct 2 3 L N ψη 2• . (4.35)
To bound the r.h.s. of the last equation we observe that L N ψη ψη + 2 ∇ω N · ∇ψη • −α Cη + dx |∇ω N (x)|2 |∇x ψη (x)|2 Cη−α
(4.36)
for α > 3. Here we used the definition (4.24) of L N and Eq. (A.18). Plugging (4.34) and (4.36) into the fourth summand on the r.h.s. of (4.28), we find
R3
dη
R
dx θ (x) 3
R3
2 −2it L N dy θ (y) (e − 1) ψη (y) C t 2 .
The results of Steps 1–4 complete the proof of (4.4).
(4.37)
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5. Dynamical Formation of Correlations Among Two Particles In this section we estimate the quantity
−i hN t ψ e (x) η dη dx θ2 (x) ∇x 1 − ω N (x) R 3 ×R 3
2
(5.1)
which arises in the bound (4.4). To control the integral (5.1), we are going to make use of the following proposition, which is stated in terms of macroscopic coordinates. Proposition 5.1. Suppose that V is a non-negative, smooth, spherically symmetric, and compactly supported potential. Let h = −2 + V and denote by 1 − ω the solution to the zero energy scattering equation h(1 − ω) = 0 with boundary condition ω(X ) → 0 as |X | → ∞. Moreover, let θ L be defined as in (1.14) for some L > 1. Consider ψ ∈ W 3,1 (R3 ) ∩ W 3,∞ (R3 ), with ψ = 1. Then, for 1, define ψ as ψ (X ) := ψ(X/)
(5.2)
with L. Define 2 e−i hT ψ (X ) dX. θ L (X ) ∇ F,L (T ) := 1 − ω(X ) R3
Then
F,L (T ) C |||ψ|||
2
(log T )6 3 T 2 + L3 L + T 2
(5.3)
(5.4)
for all T > 0 and L sufficiently large. Here we used the notation |||ψ||| := ψW 3,1 + ψW 3,∞ .
(5.5)
Remark 1. It is simple to check F,L (0) ∼ 1 + L 3 /2 . Thus, Proposition 5.1 states that F,L (T ) F,L (0), if L 3 2 , and times T such that 1 T .
(5.6)
This fact can be interpreted as a sign for the formation of the local structure 1 − ω in e−i hT ψ . Remark 2. Taking formally = ∞, (5.4) describes the relaxation of a constant initial data towards the solution 1 − ω of the zero-energy scattering equation. Remark 3. For < ∞ the function ψ can be thought of as cutting off ψ∞ ≡ 1 at distances . Since the energy hence the velocity of ψ is of order 1 in , a time of order is necessary for the effects of the cut-off to reach the window of size L: this explains why we can only prove that F,L (T ) F,L (0) for times T .
Dynamical Formation of Correlations in a Bose-Einstein Condensate
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Applying Proposition 5.1, we immediately obtain the following bound for the integral (5.1): Corollary 5.2. Under the assumptions, and using the notation introduced in Proposition 4.1, 2 2 6 e−i hN t ψη (x) C (log N t) N 2 3 + N t 2 + 3 dη dx θ2 (x) ∇x 1 − ω N (x) N 2t R 3 ×R 3 (5.7) for all times t > 0. Proof. Changing coordinates to X L = 2N and = N ), e−i hN t ψη (x) dx θ2 (x) ∇x 1 − ω N (x) R3
= N x, we obtain, from Proposition 5.1 (with 2 2 6 C |||ψη |||2 (log N t) N 2 3 + N t 2 + 3 . N 2t (5.8)
Since, by (A.18), |||ψη ||| C η−α for some α > 3, (5.7) follows from (5.8), integrating over η. Remark. By Corollary 5.2, the integral (5.1) can be shown to be smaller than its value at time t = 0, for all times t in the interval N −2 t N −1 . Note that in the many body setting of Theorem 1.1, on the other hand, we can only prove that F N (t)/F N (0) 1 1 up to times t N −(2− 10 ) (if N −1 ); this is due to the lack of control of the many body effects for larger times. Next, we prove Proposition 5.1. As we will see, the two main tools in the proof are Yajima’s bounds on the wave operator associated with the one-particle Hamiltonian − + 21 V and a new generalized dispersive estimate for initial data which are slowly decreasing at infinity but have some regularity. The dispersive estimate is presented in Sect. 6. The definition and the most important properties of the wave operator are collected in Appendix D. Proof of Proposition 5.1. We start by splitting (1) (2) F,L (T ) 2 F,L (T ) + 2 F,L (T ),
(5.9)
where (1) F,L (T ) := θ L (X ) ∇ 3 R (2) F,L (T ) := θ L (X ) ∇ R3
2 e−i hT ωψ (X ) dX, 1 − ω(X ) 2 e−i hT (1 − ω)ψ (X ) dX. 1 − ω(X )
(5.10)
(5.11)
1192
L. Erd˝os, A. Michelangeli, B. Schlein (1)
Let us first estimate F,L . Let be the wave operator associated with − X + 21 V (X ) (that is, our 21 h) as defined in Proposition D.1. Then, by the intertwining relation (D.1), 2 e 2i T ∗ ωψ (X ) dX θ L (X ) ∇ 1 − ω(X ) R3 2 2 . C ∇ω22 e 2i T ∗ ωψ + L 3 ∇ e 2i T ∗ ωψ (1) F,L (T ) =
∞
∞
(5.12)
Here we have also used that 1 − ω(X ) const > 0 (see Lemma C.1). By Yajima’s bound (D.2), we have 2 2 (5.13) e 2i T ∗ ωψ C e 2i T ∗ ωψ ∞
∞
and 2 2 2 ∇ e 2i T ∗ ωψ C e 2i T ∗ ωψ ∞ + ∇ e 2i T ∗ ωψ ∞ . ∞
(5.14) Therefore, using (C.3) to bound ∇ω2 , 2 2 2i T ∗ (1) 3 2i T ∗ . ωψ + L ∇ e ωψ F,L (T ) C e ∞
∞
(5.15)
From Proposition 5.3 below, we get (1) F,L (T ) cs |||ψ|||2
L3 ∀s ∈ (3, ∞], T 3/s
(5.16)
where cs ∼ (s − 3)−6 as s → 3+ . (2) Let us now estimate the term F,L defined in (5.11). We rewrite it conveniently by means of the operator L := − + 2
∇ω ·∇ 1−ω
already introduced in (4.24) (in microscopic variables). Analogously to (4.26), we have e−i hT (1 − ω)φ = (1 − ω)e−2i LT φ ,
∀φ ∈ L 2 (R3 )
and φ, Lψ• = Lφ, ψ• = ∇φ, ∇ψ• ,
where bulleted scalar products here are in the Hilbert space L 2 R3 , (1 − ω(X ))2 dX . On this space L acts then as a selfadjoint operator. So 2 (2) F,L (T ) = dX θ L (X ) ∇e−2i LT ψ (X ) 3 R 2 dX θ L (X ) | ∇ψ (X ) |2 + dX ∇(e−2i LT − 1)ψ (X ) . (5.17) R3
R3
Dynamical Formation of Correlations in a Bose-Einstein Condensate
The first summand in the r.h.s. of (5.17) is bounded as L3 L3 dX θ L (X ) | ∇ψ (X ) |2 C 2 ∇ψ2∞ C |||ψ|||2 2 . R3
1193
(5.18)
The second summand in the r.h.s. of (5.17) is bounded by 2 dX ∇(e−2i LT − 1)ψ (X ) R3
=
∇
R3
C
R3
2 dS e L ψ (X ) 0 T 2 dX (1 − ω(X ))2 ∇ dS e−2i L S L ψ (X ) T
−2i L S
0
2 dS ∇e−2i L S L ψ 0 • 2 −2i L S 2 C T sup ∇e L ψ C
T
•
S∈[0,T ]
CT
2
∇L ψ 2•
(5.19)
because, by (4.27), 2 −2i L S L ψ = ∇e−2i L S L ψ , ∇e−2i L S L ψ • ∇e •
= = = =
e−2i L S L ψ , L e−2i L S L ψ • L ψ , L2 ψ • ∇L ψ , ∇L ψ • ∇L ψ 2• .
(5.20)
In turn, ∇L ψ 2• is estimated as ∇L ψ 2•
2 ∇ω ∇ · ∇ψ • 1−ω • 2 2 ∇ω 2 ∇ 3 ψ + 2 ∇ 1 − ω · ∇ψ , 2 2 2 2 ∇ 3 ψ + 2
(5.21)
where 2 ∇ 3 ψ 22 |||ψ|||2 3 ∇ ψ = 2 3 3 and
(5.22)
2 2 2 ∇ω 2 ∇ · ∇ψ C |∇ ω| |∇ψ | + |∇ω|2 |∇ψ | 2 2 1−ω 2 2 + |∇ω| |∇ 2 ψ | 2
C
|||ψ|||2 2
.
(5.23)
1194
Thus,
L. Erd˝os, A. Michelangeli, B. Schlein
2 T2 dX ∇(e−2i LT − 1)ψ (X ) C 2 |||ψ|||2 . R3
(5.24)
Altogether, we find (2)
F,L (T ) C |||ψ|||2 (1)
T 2 + L3 . 2
(5.25) (2)
Finally, plugging estimates (5.16) for F,L and (5.25) for F,L into (5.9), one gets L3 T 2 + L3 2 ∀s ∈ (3, ∞]. (5.26) F,L (T ) C |||ψ||| cs 3/s + T 2 Optimizing in s (since cs (s − 3)−6 as s → 3+ ), we find (log T )6 3 T 2 + L3 2 L + F,L (T ) C |||ψ||| T 2
(5.27)
and (5.4) is proved. The following proposition, which played an important role in the proof of Proposition 5.1 and was also used in Step 3 of the proof of Proposition 4.1, provides an estimate for the decay (in L ∞ ) of the evolution generated by the Hamiltonian h = −2 + V (x) on initial data decaying only as |x|−1 . It is based on new dispersive bounds for the free evolution which are presented in Sect. 6. Proposition 5.3. Under the same assumptions, and using the same notation as in Proposition 5.1, we have, for every T ∈ R, |||ψ||| −i T ∗ ωψ 1,∞ cs ∀s ∈ (3, +∞] , (5.28) e 3 W T 2s uniformly in . Here cs ∼ (s − 3)−3 as s → 3+ . Remark. For our purposes, the bound (5.28) is better than the standard L 1 → L ∞ dispersive estimate, because it is uniform in and we eventually want to let → ∞ (on the other hand, since ω only decays as |x|−1 , the standard L 1 → L ∞ dispersive bound would diverge like 2 ). Proof. Let be the wave operator associated with − X + 21 V (X ). We will use the dispersive estimate (6.4) for s ∈ (3, ∞], which gives 2 −i T ∗ ωψ e ∞ 2 ∗ C 2 ∗ ωψ 2 + ∇∗ ωψ 23s + ∇ (5.29) ωψ 3s s s+3 T 3/s 2s+3 and
2 −i T ∗ ωψ ∇e ∞ 2 2 C 3 ∗ 2 ∗ ∇∗ ωψ 2 + ∇ ∇ , (5.30) ωψ + ωψ 3s L 3s s T 3/s s+3 2s+3
Dynamical Formation of Correlations in a Bose-Einstein Condensate
respectively. For convenience we set 3s 3 v := ∈ ,3 s+3 2
and z :=
1195
3s 3 ∈ 1, 2s + 3 2
.
(5.31)
We shall make use of the following bounds, which are a consequence of the properties of ω (see Lemma C.1), and are proven separately in Lemma 5.4: ∇ m (ωψ ) p c p,m |||ψ||| ∀ p ∈ [1, ∞], ∀m ∈ {0, 1, 2, 3} s.t. p(m + 1) > 3, (5.32) ∇ m (V ωψ ) p C |||ψ||| ∀ p ∈ [1, ∞], ∀m ∈ {0, 1}. (5.33) One sees that ∗ ωψ ∈ L s (R3 )∀s > 3 because, by Yajima’s bound (D.2) and by estimate (5.32), ∗ ωψ cs ωψ s cs |||ψ|||. (5.34) s Here cs ∼ (s − 3)−1/3 for s → 3+ . To treat ∇∗ ωψ v we use the property √ ∇ f p c p − f
p
∀ p ∈ (1, ∞),
(5.35)
√ valid for every f ∈ L 2 such that − f ∈ L 2 (see Lemma 5.5). This is the case for f = ∗ ωψ , due to Yajima’s bound, although the corresponding norms are not bounded uniformly in . In our application we need p ∈ ( 23 , 3], so we are never at the borderline case and c p remains uniformly bounded. Thus, we get √ ∇∗ ωψ cv − ∗ ωψ = cv ∗ h ωψ v v v ∞ 1 dk h ωψ cv h ωψ cv √ v k+h k 0 v ∞ 1 dk |h ωψ | cv (5.36) √ , k − + k 0 v where we have used intertwining relation (D.1), Yajima’s bound (D.2), the fact that h generates a positivity-preserving semigroup, and 1 ∞ dk 1 h. (5.37) h = √ π 0 k +h k We notice that estimates (5.32) and (5.33) give h ωψ a − (ωψ )a + V ωψ a ca |||ψ||| a ∈ (1, ∞].
(5.38)
Thus, for any b ∈ [1, 3) ∩ [1, v), Young’s inequality gives √ 1 − k |X −Y | 1 e |h ωψ h ωψ | = dY (Y ) − + k 3 4π |X − Y | R v v √ e− k | · | cv,b h ωψ (1+ 1 − 1 )−1 v b |·| b
= cv,b k
1 3 2 − 2b
h ωψ (1+ 1 − 1 )−1 . v
b
(5.39)
1196
L. Erd˝os, A. Michelangeli, B. Schlein
We use this bound to control the r.h.s. of (5.36), applying it with two different values of b in the region |k| 1 and |k| 1. For v ∈ ( 23 , 3), we find ∇∗ ωψ cv v Here cv ∼ (v − 23 )2 when v → ∇∗ ωψ 3
3+ 2 .
h ωψ
3v 2 12v−9−v 2
+ h ωψ v .
(5.40)
For v = 3 we find, on the other hand, C
µ(1 − µ)
1 3
h ωψ
3 2+µ
+ h ωψ 3
(5.41)
for any µ ∈ (0, 1). Hence, by (5.38), ∇∗ ωψ 3s cs |||ψ|||
(5.42)
3+s
uniformly in and ∀s ∈ (3, ∞], and cs ∼ (s − 3)−3 as s → 3+ . To treat ∇ 2 ∗ ωψ z we use the Calderon-Zygmund inequality ∇ 2 f p c p f p ∀ p ∈ (1, ∞)
(5.43)
valid for any compactly supported f in W 2, p (R3 ) (see [4], Theorem 9.9). The behaviour of the constant c p as p → 1+ can be computed from the constant in the Marcinkiewicz interpolation theorem (see [4], Theorem 9.8) and one has c p ∼ ( p − 1)−1 . Although supp(∗ ωψ ) is not compact, we can apply (5.43) to the compactly supported function χA ∗ ωψ , where we have introduced the cut-off function χA(Y ) := χ
|Y | A
(5.44)
at the scale A, with χ defined as in (1.15). Then one has 2 ∗ ∇ ωψ ∇ 2 χA ∗ ωψ + ∇ 2 (1 − χA )∗ ωψ z z z 2 ∗ ∗ cz χA ωψ + ∇ (1 − χA ) ωψ z z ∗ ∗ cz ωψ z + cz (1 − χA ) ωψ z 2 ∗ + ∇ (1 − χA ) ωψ z ∗ cz ωψ z + cz ∇ 2 (1 − χA )∗ ωψ
(5.45)
lim ∇ 2 (1 − χA )∗ ωψ = 0
(5.46)
z
and
A→+∞
z
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1197
because 2 ∇ (1 − χA )∗ ωψ (∇ 2 χA )∗ ωψ + 2 (∇χA ) · ∇∗ ωψ z z z 2 ∗ + (1 − χA )∇ ωψ z
∗ ωψ z ∇∗ ωψ z C + C A2 A 2 ∗ + (1 − χA )∇ ωψ z
A→+∞
−−−−−→ 0.
(5.47)
The last summand above, in particular, vanishes as A → +∞ by dominated convergence: in fact we have (1 − χA )∇ 2 ∗ ωψ → 0 pointwise and is in L z (R3 ) uniformly in A because (1 − χA )∇ 2 ∗ ωψ ∇ 2 ∗ ωψ ∗ ωψ W 2,z z
z
cz ωψ W 2,z < ∞. Thus,
2 ∗ ∇ ωψ cz ∗ ωψ z z
∀z ∈ (1, +∞).
(5.48)
(5.49)
Using (5.49), the intertwining relation (D.1), and Yajima’s bound (D.2), one has (with 3s z = 2s+3 , s > 3) 2 ∗ ∇ ωψ cz ∗ ωψ z cz ∗ h ωψ z z cz (ωψ )z + V ωψ z cs |||ψ||| (5.50) uniformly in and ∀s ∈ (3, ∞], where (5.32) and (5.33) have been used in the last line. Following the blow-up of the various constants, we see that in (5.50) cs ∼ (s − 3)−2 as s → 3+ . From (5.34), (5.42), and (5.50), the dispersive estimate (5.29) takes the form 2 −i T ∗ ωψ e
∞
cs
|||ψ|||2 ∀s ∈ (3, ∞] T 3/s
(5.51)
with cs ∼ (s − 3)−6 as s → 3+ . Next we treat the r.h.s. of (5.30) in analogy to what we have done so far for the r.h.s. of (5.29). The first term on the r.h.s. of (5.30) is bounded as ∇∗ ωψ C ∗ ωψ 1,s cs ωψ 1,s cs |||ψ||| (5.52) s W W for any s > 3 and uniformly in : the second inequality above follows from Yajima’s bound (D.2) and the last one follows from estimates (5.32) and (5.33). The second summand on the r.h.s. of (5.30) is estimated analogously to (5.50) and gives 2 ∗ (5.53) ∇ ωψ cs |||ψ||| v
1198
L. Erd˝os, A. Michelangeli, B. Schlein
uniformly in . Finally, the third term on the r.h.s. of (5.30) can be bounded, similarly to (5.49), by 3 3 3 ∗ 2 ∗ ∂i ∗ ωψ . ∇ ∇ ωψ ∂ ωψ c i z z z
z
i=1
(5.54)
i=1
Hence, by means of the intertwining relation (D.1), of Yajima’s bound (D.2), and of the estimates (5.32) and (5.33), we find ∂i ∗ ωψ z = cz ∂i ∗ h ωψ z cz h ωψ W 1,z cz (ωψ )z + V ωψ z + ∇ 3 (ωψ )z + ∇(V ωψ )z cs |||ψ|||
(5.55)
uniformly in . Thus, from (5.52), (5.53), and (5.55), the dispersive estimate (5.30) takes the form 2 |||ψ|||2 −i T ∗ ωψ cs 3/s ∀s ∈ (3, ∞]. (5.56) ∇e ∞ T Following the blow-up of the various constants, we see that in (5.56) cs ∼ (s − 3)−4 as s → 3+ . In the following two lemmas, we prove estimates which were used in the proof of Proposition 5.3. Lemma 5.4. Let V , ω, ψ, and ψ be as in the hypothesis of Proposition 5.1. Then, for any m ∈ N and any p ∈ [1, ∞] such that p(m + 1) > 3, there exists a constant c p,m such that m ∇ (ωψ ) c p,m |||ψ|||. (5.57) p −1
3 + ) . Moreover, for any m = 0, 1 Here c p,m blows up as ( p (m + 1) − 3) p as p → ( m+1 and any p ∈ [1, ∞], one has m ∇ (V ωψ ) C |||ψ||| (5.58) p
for some constant C depending on V . Proof. We recall from Appendix C that ω satisfies the bounds ∇ n ωq < ∞
∀q ∈ [1, ∞] , ∀n ∈ N s.t. q(n + 1) > 3
(5.59)
with ∇ n ωq ∼ (q(n + 1) − 3)
− q1
3 + as q →( n+1 ) .
(5.60)
Moreover, ψ satisfies the scaling 3
∇ ν ψ q = q
−ν
∇ ν ψq .
(5.61)
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1199
Pick m ∈ N and p ∈ [1, ∞] such that p(m + 1) > 3: then, keeping into account (5.59) and (5.61), one obtains by the Hölder inequality, m ∇ (ωψ )
p
c p,m
m ν ∇ ψ p(m+1)
(5.62)
ν
ν=0 −1
3 + for some constants c p,m blowing up as ( p (m + 1) − 3) p when p → ( m+1 ) . (When p(m+1) ν = 0 it is understood that ν = ∞.) By interpolation,
m ∇ (ωψ )
p
c p,m
3 ν ∇ ψ p(m+1) c p,m ψW 3,1 + ψW 3,∞ , (5.63) ν=0
ν
that is, we obtain (5.57). Equation (5.58) can be proven similarly. √ Lemma 5.5. Let f ∈ L 2 (R3 ) such that − f ∈ L 2 (R3 ). Then √ ∇ f p c p − f
p
(5.64)
for any p ∈ (1, +∞). Proof. The boundedness of the singular integral operator (see, e.g., [8], Chap. III, Theorem 4, and [9], Chap. 13, Theorem 5.1) implies that ∇ √ 1 g Cg p . (5.65) − p √ √ Let g := − f . Then g ∈ L 2 (Rn ). On L 2 (Rn ) the operator − has a trivial kernel, therefore 1 √ − f = f. √ − Then
(5.66)
√ ∇ f p c p − f . p
(5.67)
6. Dispersive Estimate for Regular, Slowly Decaying Initial Data The standard dispersive estimate C it f s e f q 3 1−1 t s 2
q=
s ∈ [1, 2] ∈ [2, +∞]
s s−1
(6.1)
for the free Schrödinger evolution is not suited for functions that decay slowly at infinity. In this section we prove a dispersive estimate which holds for f ∈ L s for any s ∈ [ 23 , ∞], if additionally some L p bound is known on the derivatives of f .
1200
L. Erd˝os, A. Michelangeli, B. Schlein
3q Proposition 6.1. Let s ∈ [ 23 , ∞], q ∈ [max{s, 3}, ∞], and r ∈ [1, 3+2q ]. Let s 3 f ∈ L (R ) such that 3s
∇ f ∈ L s+3 (R3 ),
(6.2)
∇ 2 f ∈ L r (R3 ). Then eit f ∈ L q (R3 ) and C it e f 3 1 1 f s + ∇ f 3s + s+3 q − t2 s q t
3 2
C 1 1 r −q
−1
∇ 2 f r
(6.3)
for some constant C which is independent of s, q, r . Remark. We use this estimate in the proof of Proposition 5.3 and of Proposition A.1, 3s with q = ∞, s ∈ [3, ∞] and r = 3+2s . In this case, (6.3) reads C it (6.4) e f 3 f s + ∇ f 3s + ∇ 2 f 3s . s+3 2s+3 ∞ t 2s Proof of Proposition 6.1. It is enough to prove (6.3) for f ∈ C0∞ (R3 ); then the estimate can be extended by a density argument. For q 1, we have q 1/q i|x−y|2 1 it 4t dx dy e f (y) . (6.5) e f = 3/2 3 3 q (4π t) R R We split the above integral for small and large values of |x − y|: in the latter regime integration by parts will provide the necessary decay at infinity. We introduce R > 0 and we define the smooth cutoff function |x| θ R (x) := χ (6.6) R with χ defined in (1.15). The following scaling properties of θ R will be needed: θR 3 p −m if mp < 3, (6.7) | · |m = Am, p R p 1 − θR 3 p −m if mp > 3, (6.8) | · |m = Bm, p R p ∇θ R 3 p −m−1 ∀ p 1 , ∀ m ∈ R. (6.9) | · |m = Cm, p R p Inserting the cut-off in (6.5), we find q 1 q i|x−y|2 C it 4t dx dy θ R (x − y) e f (y) e f 3/2 q t R3 R3 q 1 q i|x−y|2 C 4t + 3/2 dx dy (1 − θ R (x − y)) e f (y) t R3 R3 ≡ (I ) + (I ), (6.10)
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1201
where the first summand in the r.h.s. is immediately estimated by Young’s inequality as 1
1
3(1+ q − s ) cq,s R (I ) 3/2 f s θ R (1+ 1 − 1 )−1 cq,s f s q s t t 3/2
1 s q.
(6.11)
To estimate the summand (I ), we use e
i|x−y|2 4t
= 2it
i|x−y|2 x−y 4t · ∇ e y |x − y|2
(6.12)
to write
(I ) =
C
t 1/2
R
dx 3
q 1 q i|x−y|2 x−y 4t dy (1 − θ R (x − y)) f (y) · ∇ e . y |x − y|2 R3 (6.13)
Then, with integration by parts and since ∇y ·
x−y 1 =− , 2 |x − y| |x − y|2
(6.14)
we bound (I ) as 1 x − y q q (I ) 1/2 dy f (y) e ∇θ R (x − y) · t |x − y|2 R R3 q 1 q i|x−y|2 C f (y) 4t + 1/2 dx dy (1 − θ R (x − y)) e t |x − y|2 R3 R3 q 1 q i|x−y|2 x−y C 4t + 1/2 dx dy (1 − θ R (x − y)) e · ∇ f (y) t |x − y|2 R3 R3 ≡ (II ) + (III ) + (I V ). (6.15)
C
dx 3
i|x−y|2 4t
The term (II ) is estimated by Young’s inequality and (6.9): ∇θ R cq,s (II ) 1/2 f s |·| t
(1+ q1 − 1s )−1
= cq,s
R
1+ q3 − 3s
t 1/2
f s
1 s q. (6.16)
To control the summands (III), (IV), we integrate by parts once more. From (6.12), (6.14), and ∇y we find
1 x−y =2 , 2 |x − y| |x − y|4
(6.17)
1202
L. Erd˝os, A. Michelangeli, B. Schlein
i|x−y|2 f (y) dy (1 − θ R (x − y)) e 4t 2 |x − y| R3 ! i|x−y|2 f (y) 4t = −2it dy (1 − θ R (x − y)) e |x − y|4 R3 i|x−y|2 x−y + dy (1 − θ R (x − y)) e 4t · ∇ f (y) 3 |x − y|4 " R x − y i|x−y|2 4t dy ∇y θ R (x − y) · e f (y) . − |x − y|4 R3
(6.18)
Therefore q 1 q i|x−y|2 f (y) 4t (III ) C t dy (1 − θ R (x − y)) e 4 3 |x − y| R R q 1 q i|x−y|2 1 x−y +Ct2 dx dy (1 − θ R (x − y)) e 4t · ∇ f (y) 4 |x − y| R3 R3 1 i|x−y|2 1 x − y q q +Ct2 dx dy e 4t f (y) ∇θ R (x − y) · |x − y|4 R3 R3 ⎛ 1 − θR 1 − θR 1 ⎝ 2 cq,s t + ∇ f 3s f s 3+s | · |4 1 1 −1 | · |3 2 1 1 −1 dx 3
1 2
(1+ q − s )
(3+q −s )
⎞
∇θ R ⎠ + f s | · |3 1 1 −1 (1+ − ) q
t
= cq,s
1 2
1+ 3 − 3 R s q
f s + ∇ f
s
3s 3+s
3 2 s q,
(6.19)
where Young’s inequality and (6.8) and (6.9) have been used. Analogously, to handle the term (IV), we perform a second integration by parts. We observe that R3
dy
i|x−y|2 x − y · ∇ f (y) (1 − θ R (x − y)) e 4t |x − y|2 ! i|x−y|2 x−y dy (1 − θ R (x − y)) e 4t · ∇ f (y) = 2it 3 |x − y|4 R i|x−y|2 x−y x−y dy (1 − θ R (x − y)) e 4t · ∇ · ∇ f (y) − y |x − y|2 |x − y|2 R3 " i|x−y|2 x−y x−y dy e 4t · ∇ f (y) · ∇θ (x − y) . (6.20) + R |x − y|2 |x − y|2 R3
Therefore 1
(I V ) C t 2
+C t
dx R3
1 2
R3
dx R3
dy (1 − θ R (x − y))
|∇ f (y)| |x − y|3
q q1
|∇ 2 f (y)| dy (1 − θ R (x − y)) 3 |x − y|2 R
q q1
Dynamical Formation of Correlations in a Bose-Einstein Condensate
1 |∇ f (y)| q q +C t dx dy |∇θ R (x − y)| |x − y|2 R3 R3 1 − θR 1 1 2 ∇ 2 f cq,s t 2 ∇ f 3s + c t q,r −1 3+s 2 1 1 r | · |3 3+q −s ∇θ R 1 + cq,s t 2 ∇ f 3s 3+s | · |2 2 1 1 −1 1 2
1203
t
= cq,s R
1
1+ 3s − q3
∇ f
3s 3+s
1+ q1 − r1
−1
−s
3+q
1 2
1 − θR | · |2
t2
+ cq,r R
3 3 r − q −1
2 ∇ f
(6.21)
r
for 3/2 s q ∞ , q = 3/2 and 1 r < 3q/(3 + q). Here Young’s inequality and the scaling properties (6.8) and (6.9) have been used. Summarizing, it e f (I ) + (II ) + (III ) + (I V ) q 3+ 3 − 3 1 1+ 3 − 3 R q s R q s t2 + + cq,s f s 3 1 1+ 3 − 3 t2 t2 R s q 1
t2
+ cq,s R + cq,r
1+ 3s − q3
t 3
Rr
1 2
− q3 −1
∇ f 3s
3+s
2 ∇ f
(6.22)
r
for any R > 0, 3/2 s √ q ∞, q = 3/2, 1 r < 3q/(3 + q). Optimizing the choice of R leads to R = t , so that (6.22) reads cq,r cq,s it f s + ∇ f 3s + 3 1 1 ∇ 2 f r . (6.23) e f 3 1 1 s+3 ( − ) ( − q )−1 q 2 s q 2 r t t For the r.h.s. of (6.23) to stay bounded in time, we need 1 r q 3. This completes the proof of the proposition.
3q 2q+3 ,
which requires
A. Pointwise Bounds on the Two-Body Wave Function In this section we investigate the boundedness properties in time and in space of the evolution of the two-body wave function ϕ ⊗2 when the two particles are coupled by the interaction VN (x1 − x2 ), that is, where the dynamics is generated by the Hamiltonian (1,2)
hN
= −1 − 2 + VN (x1 − x2 )
acting on L 2 (R3 × R3 , dx1 dx2 ), as defined in (2.1). In spirit, these bounds are similar to the ones proved in Proposition 5.3 (in particular in (5.28) for s = ∞; the initial data however, is different). Proposition A.1. Let V be a non-negative, smooth, spherically symmetric, and com(1,2) pactly supported potential. Let VN (x) = N 2 V (N x) and h N = −1 − 2 + VN (x1 −
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x2 ). Let ψt = e−it hN ϕ ⊗2 , for some ϕ ∈ L 2 (R3 ). Then, for every α > 3, there exists C > 0 such that ψt ∞ C ϕ24,∞,α log N ,
(A.1)
where 4 α m 2 x ∇ ϕ . ∞
ϕ4,∞,α =
(A.2)
m=0
(η, x) = ψ(η + x/2, η − x/2), and Proof. Let ψ t (η, x) = ψt (η + x/2, η − x/2) = e−iη /2t e−i hN t ψ (η, x), ψ where the operator h N = −2 + VN (x1 − x2 ) only acts on the relative variable x. Then ψt 2L ∞ (R6 ,dx
1 dx2 )
t 2 ∞ 6 = ψ L (R ,dη dx) t (·, x)2 2 3 C sup ψ H (R ,dη) x∈R3
C C C
2
sup
3 m=0 x∈R 2
R3
sup
3 m=0 x∈R 2
m=0
R3
R3
2 dη ∇ηm ψ t (η, x) 2 (η, x) dη e−i hN t ∇ηm ψ
2 (η, ·) dη e−i hN t ∇ηm ψ ∞
L (R3 ,dx)
. (A.3)
It is useful to switch to the macroscopic coordinate X = N x. With the short hand notation (m) (m) X ψη,N (X ) := ψη , with (A.4) N x x (η, x) = ∇ηm ϕ η + ψη(m) (x) := ∇ηm ψ ϕ η− , (A.5) 2 2 Eq. (A.3) reads ψt 2L ∞ (R6 ,dx
1 dx2 )
C
2 m=0
R3
2 (m) 2 dη e−i hN t ψη,N ∞
L (R3 ,dX )
,
(A.6)
because h N = N 2 (−2 X + V (X )). To bound the r.h.s. of the last equation, we use the modified dispersive estimate (6.4) with s = ∞. We obtain −i hN 2 t (m) 2 2 ∗ (m) 2 2 ∗ (m) 2 ψ N ,η ∞ 3 C ∗ ψ N(m) e ,η ∞ + ∇ X ψ N ,η 3 + ∇ X ψ N ,η 3 , L (R ,dX )
2
(A.7)
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for every m = 0, 1, 2, where is the wave operator associated with −+ 21 V (x) = h/2, as defined in Proposition D.1. To estimate the terms on the r.h.s. of (A.7) we use the same strategy as followed in the proof of Proposition 5.3. However, since the initial data considered here (the wave (m) function ψ N ,η ) is different from the one considered in Proposition 5.3, the bounds for the r.h.s. of (A.7) cannot be directly inferred from the analogous estimates for the terms of the r.h.s. of (5.29). By Yajima’s bound (D.2), we have (m) (m) ∗ ψη,N ∞ C ψ N(m) ,η ∞ = C ψη ∞ .
(A.8)
Analogously to the treatment of ∇∗ ωψ v after (5.36), we obtain (m)
∇X ∗ ψη,N 3
C (m) (m) h ψ + h ψ 3 3 η,N 2+ε η,N ε(1 − ε)1/3
ε ∈ (0, 1). (A.9)
Since (m)
X ψη,N
3 2+ε
= N ε x ψη(m)
(m) 3 = X ψη,N
one has ∗
∇X
(m) ψη,N 3
C 1
ε(1 − ε) 3
N
ε
3 2+ε
,
1 x ψη(m) 3 , N
x ψη(m) 3 2+ε
(A.10) (A.11)
1 (m) (m) + x ψη 3 + ψη ∞ . N (A.12)
Similarly to the treatment of ∇ 2 ∗ ωψ z in (5.50), we find (m) (m) (m) ∇X2 ∗ ψη,N 3 C X ψη,N 3 + V ψη,N 3 2 2 2 (m) (m) C x ψη 3 + ψη ∞ . 2
(A.13)
Using (A.8), (A.12), and (A.13) to bound the r.h.s. of (A.7), we find −i hN 2 t (m) 2 ψ N ,η ∞ 3 e L (R ,dX ) 2 Nε 1 (m) 2 (m) 2 (m) 2 (m) 2 x ψη 3 + x ψη 3 + x ψη 3 + ψη ∞ . C 1 N 2+ε 2 ε(1 − ε) 3 (A.14) Now it remains to integrate the r.h.s. of the last equation over η ∈ R3 . By the assumptions on ϕ, it follows that |∇ ν ϕ(x)| < ϕ4,∞,α
1 xα
(A.15)
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for any ν = 0, . . . , 4 and some α > 3. This means that for any p ∈ [1, +∞) and n, m ∈ {0, 1, 2} one has n (m) ∇x ψη
1 x p p x ϕ η− dx ∇xn ∇ηm ϕ η + 2 2 R3 1 m+n x p m+n−µ x p p (∇ C dx (∇ µ ϕ) η + ϕ) η − 2 2 R3
=
p
µ=0
C
ϕ24,∞,(α)
1 1 dx x αp 3 η + 2 η − x2 αp R
1
p
,
(A.16)
and the above integral can be estimated as R3
dx
1 1 C x αp x αp η + 2 η − 2 ηαp
(A.17)
as long as αp > 3, which is always the case due to the assumption α > 3. Then ϕ44,∞,α n (m) 2 ∇x ψη C p η2α
n, m ∈ {0, 1, 2}
(A.18)
independently of p ∈ [1, +∞]. Due to (A.18), when plugging (A.8), (A.12), and (A.13) into the r.h.s. of estimate (A.7), and the latter into the r.h.s. of (A.3), integrability in η is guaranteed by the assumption α > 3 and one gets ψt ∞ ϕ24,∞,α
Nε 1
ε(1 − ε) 3
After choosing ε = (log N )−1 , estimate (A.1) is proved.
ε ∈ (0, 1).
(A.19)
B. Estimates for the Energy of a Factorized Data In the following lemma we prove that the expectation of the Hamiltonian H N in a factorized state ϕ ⊗N , for some ϕ ∈ H 2 (R3 ), is of the order N . This estimate is used in Sect. 3. Moreover, we also show that the expectation of H N2 (in the state ϕ ⊗N ) is of the order N 3 . Lemma B.1. Let H N be defined as in (1.19) and let ϕ ∈ H 2 (R3 ). Then 1 N →∞ N 1 lim N →∞ N 3 lim
ϕ ⊗N, H N ϕ ⊗N = ∇ϕ22 + 21V 1 ϕ44 ∀ ϕ ∈ H 1 (R3 ), ϕ ⊗N, H N2 ϕ ⊗N = 21ϕ44 V 22 ∀ ϕ ∈ H 2 (R3 ).
(B.1) (B.2)
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Proof. We will prove Eq. (B.2), the proof of (B.1) is similar but simpler, so we will omit it. Using the permutation symmetry of ϕ ⊗N , we find that 1 ⊗N N (N − 1) ⊗N 2 2 ⊗N ⊗N ϕ , VN (x1 − x2 )ϕ ϕ , H N ϕ − 3 N 2 N −1 ϕ ⊗N , 1 2 ϕ ⊗N + N −2 ϕ ⊗N , 21 ϕ ⊗N +ϕ ⊗N , (−1 )VN (x2 − x3 )ϕ ⊗N + ϕ ⊗N , 1 VN (x1 − x2 )ϕ ⊗N +N ϕ ⊗N , VN (x1 − x2 )VN (x3 − x4 )ϕ ⊗N +ϕ ⊗N , VN (x1 − x2 )VN (x2 − x3 )ϕ ⊗N .
(B.3)
Since ϕ ∈ H 2 (R3 ) and ϕ = 1, it follows that ϕ ⊗N, 1 2 ϕ ⊗N ϕ4H 1 < ∞
and
ϕ ⊗N , 21 ϕ ⊗N ϕ2H 2 < ∞.
(B.4)
Moreover, using Lemma B.2, we have ϕ ⊗N, (−1 )VN (x2 − x3 )ϕ ⊗N C VN 1 ϕ6H 1 C N −1 V 1 ϕ6H 1 and, with a Schwarz inequality and Sobolev embedding, ⊗N ϕ , 1 VN (x1 − x2 )ϕ ⊗N ϕ ⊗N , 1 VN (x1 − x2 )1 ϕ ⊗N 1/2 ×ϕ ⊗N , VN (x1 − x2 )ϕ ⊗N 1/2 C N −1 V 1 ϕ4H 2 . Finally, using again Lemma B.2, we observe that ϕ ⊗N, VN (x1 − x2 )VN (x3 − x4 )ϕ ⊗N C ϕ8H 1 VN 21 C N −2 V 21 ϕ8H 1 and that ϕ ⊗N , VN (x1 − x2 )VN (x2 − x3 )ϕ ⊗N C ϕ4H 1 V 23/2 . Inserting all these bounds in the r.h.s. of (B.3), it follows that 1 ⊗N 2 ⊗N N (N − 1) ⊗N 2 ⊗N ϕ ϕ lim , H ϕ − , V (x − x )ϕ 1 2 N N = 0. 3 N →∞ N 2
(B.5)
(B.6)
Equation (B.2) now follows because 1 1 ⊗N 2 ⊗N ϕ , VN (x1 − x2 )ϕ = dx1 dx2 N 3 V 2 (N (x1 − x2 ))|ϕ(x1 )|2 |ϕ(x2 )|2 2N 2 V 2 1 → ϕ44 2 as N → ∞ (the convergence follows by a Poincaré inequality, since ϕ ∈ H 2 (R3 )).
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Lemma B.2 (Sobolev-type inequalities). Let ψ ∈ L 2 (R6 , dx1 dx2 ). If V ∈ L 3/2 (R3 ), we have |ψ, V (x1 − x2 )ψ| CV 3/2 ψ, (1 − 1 )ψ.
(B.7)
If V ∈ L 1 (R3 ), then |ψ, V (x1 − x2 )ψ| CV 1 ψ, (1 − 1 )(1 − 2 )ψ.
(B.8)
The first bound follows from a Hölder inequality followed by a standard Sobolev inequality (in the variable x1 , with fixed x2 ). A proof of the second bound can be found, for example, in [1, Lemma 5.3]. C. Properties of the One-Body Scattering Solution 1 − ω(x) Lemma C.1. Let V : R3 → R be non negative, smooth, spherically symmetric, compactly supported and with scattering length a. Let 1 − ω(x) be the solution of − + 21 V (1 − ω) = 0 with ω(x) → 0 as |x| → ∞. (C.1) depending on the potential V such that Then there exist constants Cm , C a (C < 1), , |ω(x)| C |x| a , |∇ m ω(x)| Cm , |∇ m ω(x)| Cm |x|m+1 |ω(x)| C0
(C.2) (C.3)
for every nonnegative integer m. As a consequence, for every p ∈ [1, ∞] and every nonnegative integer m such that p(m + 1) > 3, one has ∇ m ω p < Cm, p < ∞.
(C.4)
Proof. Inequalities (C.2) and (C.3) follow immediately from the fact that, out of the support of the potential V one has ω(x) =
a |x|
|x| > R ,
(C.5)
while, inside the support, ∇ m ω is bounded by elliptic regularity and compactness for in (C.2) is strictly smaller than 1 is any nonnegative m. The fact that the constant C proved in Lemma B.1 of [2]. By scaling, one immediately has the following. Corollary C.2. Let V be as in Lemma C.1. Let VN (x) = N 2 V (N x) and let 1 − ω N be the corresponding solution of the zero-energy scattering equation. Then ω N (x) = ω(N x)
(C.6)
whence ∇ m ω N p = N
m− 3p
∇ m ω p
under the same condition for the validity of (C.4).
(C.7)
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D. Properties of the Wave Operator We denote by the wave operator associated with the one-particle Hamiltonian h := − + 21 V , that is, h = 21 h in our previous notation. Its existence and most important properties are stated in the following proposition. Proposition D.1. Suppose V 0, with V ∈ L 1 (R3 ). Then: i) (Existence of the wave operator). The limit = s − lim ei ht eit t→∞
exists. ii) (Completeness of the wave operator). is a unitary operator on L 2 (R3 ) with ∗ = −1 = s − lim e−it e−i ht . t→∞
iii) (Intertwining relations). On D(h) = D(−), we have ∗ h = −.
(D.1)
iv) (Yajima’s bounds). Suppose moreover that V (x) Cx−σ , for some σ > 5. Then, for every 1 p ∞, and ∗ map L p (R3 ) into L p (R3 ), that is, L p →L p < ∞ for all 1 p ∞.
(D.2)
If moreover V ∈ C k (R3 ), we have W m, p →W m, p < ∞ for all m k. Proof. The proof of i), ii), and iii) can be found in [7]. Part iv) is proved in [10,11].
E. Correlation Structure and Gross-Pitaevskii Equation As remarked in the Introduction, the correlation structure developed by the solution to the N -particle Schrödinger equation N ,t = e−i HN t N (with H N defined as in (1.1)) for initial states exhibiting complete Bose-Einstein condensation plays a very important role in the derivation of the Gross-Pitaevskii equation (1.10); more precisely, the emergence of the scattering length in the coupling constant in front of the nonlinearity is a consequence of the presence of the short-scale correlation structure. The goal of this Appendix is to explain this connection in some more details (see also Sect. 3 in [2]). From the Schrödinger equation (1.3), it is simple to obtain an evolution equation for (1) the one-particle density γ N ,t : (1) (1) i∂t γ N ,t (x1 ; x1 ) = −x1 + x1 γ N ,t (x1 ; x1 ) + (N − 1) dx2 VN (x1 − x2 ) − VN (x1 − x2 ) (2)
× γ N ,t (x1 , x2 ; x1 , x2 ).
(E.1)
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This is not a closed equation for γ N ,t because it also depends on the two-particle density γ N(2),t associated with N ,t (the two-particle density is defined similarly to (1.4), integrating however only over the last (N − 2) particles); actually, (E.1) is the first equation of a hierarchy of N coupled equations, known as the BBGKY hierarchy, for the marginal (1) (2) densities of N ,t . From γ N ,t → |ϕt ϕt | as N → ∞, it follows that γ N ,t → |ϕt ϕt |⊗2 . (1)
(2)
If we replace, in (E.1), the densities γ N ,t and γ N ,t by these limit points, and if we replace (N − 1)VN (x) N 3 V (N x) by its (formal) limit bδ(x) with b = V , we obtain a closed equation for the condensate wave function ϕt , which has the same form as the Gross-Pitaevskii equation (1.10), but with a coupling constant in front of the nonlinearity given by b instead of 8πa. The reason why this naive argument leads to the wrong (2) coupling constant is that the two-particle density γ N ,t contains a short-scale correlation structure (inherited by the N -particle wave function N ,t ) which varies on exactly the same length scale N −1 characterizing the interaction potential. Describing correlations by the solution to the zero energy scattering equation, we can approximate, for large but (2) finite N , the two-particle density γ N ,t by (2)
γ N ,t (x1 , x2 ; x1 , x2 ) (1 − ω N (x1 − x2 ))
(1 − ω N (x1 − x2 )) ϕt (x1 )ϕt (x2 )ϕ t (x1 )ϕ t (x2 ).
Inserting this ansatz in the second term on the r.h.s. of (E.1), and using (1.8), we obtain the correct Gross-Pitaevskii equation for ϕt . The emergence of the scattering length in the Gross-Pitaevskii equation is therefore a consequence of the singular correlation structure developed by N ,t (and then inherited by the two particle marginal density). References 1. Erd˝os, L., Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001) 2. Erd˝os, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. Ann. Math. (to appear) http://arXiv.org/abs/math-ph/0606017v3, 2006 3. Erd˝os, L., Schlein, B., Yau, H.-T.: Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential. J. Amer. Math. Soc. (to appear) http://arxiv.org/abs/0802.3877v3[math-ph], 2008 4. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics, Berlin: Springer-Verlag, 2001 5. Lieb, E.H., Seiringer, R.: Proof of Bose-Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409-1-4 (2002) 6. Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2000) 7. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. New York: Academic Press, 1979 8. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: Monographs in Harmonic Analysis, III, Vol. 43 of Princeton Mathematical Series, Princeton, NJ: Princeton University Press, 1993 9. Taylor, M.E.: Partial Differential Equations. III, Vol. 117 of Applied Mathematical Sciences, New York: Springer-Verlag, 1997 10. Yajima, K.: The W k, p -continuity of wave operators for Schrödinger operators. Proc. Japan Acad. Ser. A Math. Sci. 69, 94–98 (1993) 11. Yajima, K.: The W k, p -continuity of wave operators for Schrödinger operators. J. Math. Soc. Japan 47, 551–581 (1995) Communicated by H.-T. Yau
