Math Phys Anal Geom (2007) 10:1–41 DOI 10.1007/s11040-007-9019-2
Groupoids, von Neumann Algebras and the Integrated Density of States Daniel Lenz · Norbert Peyerimhoff · Ivan Veseli´c
Received: 9 March 2006 / Accepted: 12 March 2007 / Published online: 17 May 2007 © D. Lenz, N. Peyerimhoff and I. Veseli´c 2007
Abstract We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schrödinger operators on manifolds, quantum percolation and quasi– crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure. Keywords Groupoids · Von Neumann algebras · Integrated density of states · Random operators · Schrödinger operators on manifolds · Trace formula Mathematics Subject Classifications (2000) 46L10 · 35J10 · 46L51 · 82B44 1 Introduction The aim of this paper is to review and present a unified treatment of basic features of random (Schrödinger) operators using techniques from Connes’ D. Lenz (B) · I. Veseli´c Fakultät für Mathematik, D-09107 TU Chemnitz, Germany e-mail:
[email protected] URL:www.tu-chemnitz.de/mathematik/mathematische_physik/ URL:www.tu-chemnitz.de/mathematik/schroedinger/ N. Peyerimhoff Department of Mathematical Sciences, Durham University, Durham, UK URL:http://www.maths.du.ac.uk/˜dma0np/ ©2007 by D. Lenz, N. Peyerimhoff and I. Veseli´c. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.
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noncommutative integration theory and von Neumann algebras [22]. Particular emphasis will be laid on an application of the general setting to the example of –
a group action on a manifold proposed by two of the authors [76].
This example merges and extends two situations, viz periodic operators on manifolds as studied first by Adachi/Sunada [1] and random Schrödinger operators on Rd or Zd as studied by various people (s. below) starting with the work of Pastur [71]. In the first situation a key role is played by the geometry of the underlying manifold. In the second situation, the crucial ingredient is the randomness of the corresponding potential. We also apply our discussion to two more examples: – –
Random operators on tilings and Delone sets whose mathematically rigorous study goes back to Hof [38] and Kellendonk [47]. Random operators on site-percolation graphs, see e.g. [14, 17, 25, 90].
As for the above three examples let us already point out the following differences: in the first example the underlying geometric space is continuous and the group acting on it is discrete; in the second example the underlying geometric space is discrete and the group acting on it is continuous; finally, in the third example both the underlying geometric space and the group acting on it are discrete. The use of von Neumann algebras in the treatment of special random operators is not new. It goes back at least to the seminal work of Šubin on almost periodic operators [83]. These points will be discussed in more detail next. Random Schrödinger operators arise in the quantum mechanical treatment of disordered solids. This includes, in particular, periodic operators, almost periodic operators and Anderson type operators on Zd or Rd (cf. the textbooks [20, 24, 51, 75, 85]). In all these cases one is given a family (Hω ) of selfadjoint operators Hω acting on a Hilbert space Hω , indexed by ω in a measure space (, μ) and satisfying an equivariance condition with respect to a certain set of unitary operators (U i )i∈I . While specific examples of these cases exhibit very special spectral features, there are certain characteristics shared by all models. These properties are as follows. (In parentheses we give a reference where the corresponding property is established.) (P1) Almost sure constancy of the spectral properties of Hω given some ergodicity condition. In particular, the spectrum is nonrandom (Theorem 5.1). (P2) Absence of discrete spectrum (Corollary 5.9) and, in fact, a dichotomy (between zero and infinity) for the values of the dimensions of spectral projections. (P3) A naturally arising von Neumann algebra (Section 3) with a canonical trace τ , to which the random operators are affiliated (Section 4).
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(P4) A measure ρ, called the density of states, governing global features of the family (Hω ), in particular, having as its support (Proposition 5.2). This measure is related to the trace of the von Neumann algebra. Let us furthermore single out the following point, which we show for the three abovementioned examples: (P5) A local procedure to calculate ρ via an exhaustion given some amenability condition. This is known as Pastur–Šubin trace formula. It implies the self-averaging property of the density of states (discussed for the examples mentioned above in Sections 6, 7, 8). Let us now discuss these facts for earlier studied models. The interest in property (P5) arouse from the physics of disordered media. First mathematically rigorous results on the (integrated) density of states are due to Pastur [71–73], Fukushima, Nakao and Nagai [32–35, 70], Kotani [56], and Kirsch and Martinelli [52, 54]. In these papers two different methods for constructing the integrated density of states (IDS) can be found (property (P5)). Either one uses the Laplace transform to conclude the convergence of certain normalized eigenvalue counting functions, or one analyzes the counting functions directly via the so called Dirichlet–Neumann bracketing. In our setting the Laplace transform method seems to be of better use, since the pointwise superadditive ergodic theorem [2] used in the Dirichlet–Neumann bracketing approach [52] has no counterpart in the (nonabelian) generality we are aiming at. For the more recent development in the study of the IDS of alloy type and related models, as well as the results on its regularity and asymptotic behaviour, see [20, 75, 85, 88] and the references cited there. For almost periodic differential operators on Rd and the associated von Neumann algebras, a thorough study of the above features (and many more) has been carried out in the seminal papers by Coburn, Moyer and Singer [21] and Šubin [83]. Almost periodic Schrödinger operators on Zd and Rd were then studied by Avron and Simon [4, 5]. An abstract C∗ -algebraic framework for the treatment of almost periodic operators was then proposed and studied by Bellissard [6, 7] and Bellissard, Lima and Testard [9]. While these works focus on K-theory and the so called gap-labeling, they also show (P1)–(P5) for almost periodic Schrödinger type operators on Rd and Zd . Let us emphasize that large parts of this C∗ -algebraic treatment are not confined to almost periodic operators. In fact, (P1)–(P4) are established there for crossed products arising from arbitrary actions of locally compact abelian groups on locally compact spaces X. After the work of Aubry/André [3] and the short announcement of Bellissard/Testard in [10], investigations in this framework, centered around so called spectral duality, were carried out by Kaminker and Xia [40] and Chojnacki [18]. A special one-dimensional version of spectral duality based on [37] can also be found in [59]. An operator algebraic framework of crossed-products (involving von Neumann crossed products) can also be used in the study of general random operators if one considers Rd actions together with operators on L2 (Rd )
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(cf. [59]). However, certain of these models rather use actions of Zd together with operators on L2 (Rd ), like the thoroughly studied alloy or continuous Anderson type models. This presents a difficulty which was overcome in a work by Kirsch [50] introducing a so called suspension construction, see also [9] for related material. This allows to “amplify” these Zd actions to Rd actions and thus reduce the treatment of (P1)–(P4) in the Zd case to the Rd case. In recent years three more classes of examples have been considered. These are random operators on manifolds [60, 61, 76, 86, 87], discrete random operators on tilings [8, 38, 39, 47, 48, 62, 64], and random Hamiltonians generated by percolation processes [14, 55, 89, 90]. In these cases the algebraic framework developed earlier could not be used to establish (P1)–(P5). Note, however, that partial results concerning, e.g., (P1) or restricted versions of (P5) are still available. Note also that continuous operators associated with tilings as discussed in [8, 13] fall within the C∗ -algebraic framework of [6, 7]. A more detailed analysis of the point spectrum of discrete operators associated to tilings and percolation graphs will be carried out in [65]. The model considered in [76] includes periodic operators on manifolds. In fact, it was motivated by work of Adachi and Sunada [1], who establish an exhaustion construction for the IDS as well as a representation as a -trace in the periodic case. For further investigations related to the IDS of periodic operators in both discrete and continuous geometric settings, see e.g. [29–31, 65, 67, 68]. More precisely, our first example concerned with Random Schrödinger operators on Manifolds (RSM) can be described as follows,see [60, 76]: Example (RSM) Let (X, g0 ) be the Riemannian covering of a compact Riemannian manifold M = X/ . We assume that there exists a family (gω )ω∈ of Riemannian metrics on X which are parameterized by the elements of a probability space (, B , P) and which are uniformly bounded by g0 , i.e., there exists a constant A 1 such that 1 g0 (v, v) gω (v, v) Ag0 (v, v) for all v ∈ T X and ω ∈ . A Let λω denote the Riemannian volume form corresponding to the metric gω . We assume that acts ergodically on by measure preserving transformations. The metrics are compatible in the sense that for all γ ∈ the corresponding deck transformations γ : (X, gω ) → (X, gγ ω ) −1
are isometries. Then the induced maps U (ω,γ ) : L2 (X, λγ ω ) → L2 (X, λω ), (U (ω,γ ) f )(x) = f (γ −1 x) are unitary operators. Based on this geometric setting, we consider a family (Hω : ω ∈ ), Hω = ω + Vω , of Schrödinger operators satisfying the following equivariance condition ∗ Hω = U (ω,γ ) Hγ −1 ω U (ω,γ ),
(1)
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for all γ ∈ and ω ∈ . We also assume some kind of weak measurability in ω, namely, we will assume that (2) ω → f (ω, ·), F(Hω ) f (ω, ·)ω := f¯(ω, x) [F(Hω ) f ](ω, x) dλω (x) X
is measurable for every measurable f on × X with f (ω, ·) ∈ L2 (X, λω ), ω ∈ , and every function F on R which is uniformly bounded on the spectra of the Hω . Note that L2 (X, λω ) considered as a set of functions (disregarding the scalar product) is independent of ω. The expectation with respect to the measure P will be denoted by E. This example covers the following two particular cases: (a) A family of Schrödinger operators ( + Vω )ω∈ on a fixed Riemannian manifold (X, g0 ) with random potentials, see [76]. In this case the equivariance condition (1) transforms into the following property of the potentials Vγ ω (x) = Vω (γ −1 x). (b) A family of Laplacians ω on a manifold X with random metrics (gω )ω∈ satisfying some additional assumptions [60]. By the properties of X and M in (RSM), the group is discrete, finitely generated and acts cocompactly, freely and properly discontinuously on X. In the physical literature the equivariance condition (1) is denoted either as equivariance condition, see e.g. [6], or as ergodicity of operators [51, 74], where it is assumed that the measure preserving transformations are ergodic. From the probabilistic point of view this property is simply the stationarity of an operator valued stochastic process. It is our aim here to present a groupoid based approach to (P1)–(P4) covering all examples studied so far. This includes, in particular, the case of random operators on manifolds, the tiling case and the percolation case. Our framework applies also to Schrödinger operators on hyperbolic space (e.g. the Poissonian model considered in [86]). However, our proof of (P5) does not apply to this setting because of the lack of amenability of the isometry group. For the example (RSM) with amenable group action , we will prove (P5) in Section 6. Note that case (a) of (RSM) includes the models treated earlier by the suspension construction. Thus, as a by product of our approach, we get an algebraic treatment of (P5) for these models. As mentioned already, our results can also be applied to further examples. Application to tilings is discussed in [62, 64]. There, a uniform ergodic type theorem for tilings along with a strong version of Pastur–Šubin-formula (P5) is given. The results also apply to random operators on percolation graphs. In particular they provide complementary information to the results in [55, 89, 90], where the integrated density of states was defined rigorously for site and edge percolation Hamiltonians. A basic discussion of these examples and the connection to our
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study here is given in Sections 7 and 8, respectively. This will in particular show that (P5) remains valid for these examples. For further details we refer to the cited literature. The results also apply to random operators on foliations (see [57] for related results). Our approach is based on groupoids and Connes theory of noncommutative integration [22]. Thus, let us conclude this section by sketching the main aspects of the groupoid framework used in this article. The work [22] on noncommutative integration theory consists of three parts. In the first part an abstract version of integration on quotients is presented. This is then used to introduce certain von Neumann algebras (viz. von Neumann algebras of random operators) and to classify their semifinite normal weights. Finally, Connes studies an index type formula for foliations. We will be only concerned with the first two parts of [22]. The starting point of the noncommutative integration is the fact that certain quotients spaces (e.g., those coming from ergodic actions) do not admit a nontrivial measure theory, i.e., there do not exist many invariant measurable functions. To overcome this difficulty the inaccessible quotient is replaced by a nicer object, a groupoid. Groupoids admit many transverse functions, replacing the invariant functions on the quotient. In fact, the notion of invariant function can be further generalized yielding the notion of random variable in the sense of [22]. Such a random variable consists of a suitable bundle together with a family of measures admitting an equivariant action of the groupoid. This situation gives rise to the so called von Neumann algebra of random operators and it turns out that the random operators of the form (Hω ) introduced above are naturally affiliated to this von Neumann algebra. Moreover, each family (Hω ) of random operators gives naturally rise to many random variables in the sense of Connes. Integration of these random variables in the sense of Connes yields quite general proofs for main features of random operators. In particular, an abstract version of the integrated density of states is induced by the trace on the von Neumann algebra. 2 Abstract Setting for Basic Geometric Objects: Groupoids and Random Variables In this section we discuss an abstract generalization for the geometric situation given in example (RSM). The motivation for this generalization is that it covers many different settings at once, such as tilings, percolation, foliations, equivalence relations and our concrete situation, a group acting on a metric space. Let us first introduce some basic general notations which are frequently used in this paper. For a given measurable space (S, B ) we denote the set of measures by M(S) and the corresponding set of measurable functions by F (S). The symbol F + (S) stands for the subset of nonnegative measurable functions. M f denotes the operator of multiplication with a function f . We begin our abstract setting with a generalization of the action of the group on the measurable space (, B ). This generalization, given by G = × in
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the case at hand, is called a groupoid. The main reason to consider it is the fact that it serves as a useful substitute of the quotient space / , which often is a very unpleasant space (e.g., in the case when acts ergodically). The general definition of a groupoid is as follows [79]. Definition 2.1 A triple (G , ·,−1 ) consisting of a set G , a partially defined associative multiplication ·, and an inverse operation −1 : G → G is called a groupoid if the following conditions are satisfied: – – – –
(g−1 )−1 = g for all g ∈ G , If g1 · g2 and g2 · g3 exist, then g1 · g2 · g3 exists as well, g−1 · g exists always and g−1 · g · h = h, whenever g · h exists, h · h−1 exists always and g · h · h−1 = g, whenever g · h exists.
A given groupoid G comes along with the following standard objects. The subset G 0 = {g · g−1 | g ∈ G } is called the set of units. For g ∈ G we define its range r(g) by r(g) = g · g−1 and its source by s(g) = g−1 · g. Moreover, we set G ω = r−1 ({ω}) for any unit ω ∈ G 0 . One easily checks that g · h exists if and only if r(h) = s(g). The groupoids under consideration will always be measurable, i.e., they posses a σ -algebra B such that all relevant maps are measurable. More precisely, we require that · : G (2) → G , −1 : G → G , s, r : G → G 0 are measurable, where G (2) := {(g1 , g2 ) | s(g1 ) = r(g2 )} ⊂ G 2
and G 0 ⊂ G are equipped with the induced σ -algebras. Analogously, G ω ⊂ G are measurable spaces with the induced σ -algebras. As mentioned above, the groupoid associated with (RSM) is simply G = × and the corresponding operations are defined as (ω, γ )−1 = (γ −1 ω, γ −1 ), (ω1 , γ1 ) · (ω2 , γ2 ) = (ω1 , γ1 γ2 ),
(3) (4)
where the left hand side of (4) is only defined if ω1 = γ1 ω2 . It is very useful to γ consider the elements (ω, γ ) of this groupoid as the set of arrows γ −1 ω −→ ω. This yields a nice visualization of the operation · as concatenation of arrows and of the operation −1 as reversing the arrow. The units G 0 = {(ω, ) | ω ∈ } can canonically be identified with the elements of the probability space . Via this identification, the maps s and r assign to each arrow its origin and its destination. Our groupoid can be seen as a bundle over the base space of units with the fibers G ω = {(ω, γ ) | γ ∈ } ∼ = . For simplicity, we henceforth refer to the set of units as also in the setting of an abstract groupoid. The notions associated with the groupoid are illustrated in Fig. 1 for both the abstract case and the concrete case of (RSM). Next, we introduce an appropriate abstract object which corresponds to the Riemannian manifold X in (RSM).
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Fig. 1 Notations of the groupoid G = × in (RSM)
Definition 2.2 Let G be a measurable groupoid with the previously introduced notations. A triple (X , π, J) is called a (measurable) G -space if the following properties are satisfied: X is a measurable space with associated σ -algebra BX . The map π : X → is measurable. Moreover, with X ω = π −1 ({ω}), the map J assigns, to every g ∈ G , an isomorphism J(g) : X s(g) → X r(g) of measurable spaces with the properties J(g−1 ) = J(g)−1 and J(g1 · g2 ) = J(g1 ) ◦ J(g2 ) if s(g1 ) = r(g2 ). Note that a picture similar to Fig. 1 exists for a G -space X . An easy observation is that every groupoid G itself is a G -space with π = r and J(g)h = g · h. The G -space in (RSM) is given by X = × X together with the maps π(ω, x) = ω and J(ω, γ ) : X γ
−1
ω
→ X ω,
J(ω, γ )(γ −1 ω, x) = (ω, γ x).
Similarly to the groupoid G , an arbitrary G -space can be viewed as a bundle over the base with fibers X ω . Our next aim is to exhibit natural measures on these objects. We first introduce families of measures on the fibers G ω . In the case of (RSM), this can be viewed as an appropriate generalization of the Haar measure on . Definition 2.3 Let G be a measurable groupoid and the notation be given as above. (a) A kernel of G is a map ν : → M(G ) with the following properties: – –
the map ω → ν ω ( f ) is measurable for every f ∈ F + (G ), ν ω is supported on G ω , i.e., ν ω (G − G ω ) = 0.
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(b) A transverse function ν of G is a kernel satisfying the following invariance condition f (g · h)dν s(g) (h) = f (k)dν r(g) (k) G s(g)
G r(g)
for all g ∈ G and f ∈ F + (G r(g) ). In (RSM) the discreteness of impliesthat any kernel ν can be identified with a function L ∈ F + ( × ) via ν ω = γ ∈ L(ω, γ )δ(ω,γ ) . For an arbitrary unimodular group , the Haar measure m induces a transverse function ν by ν ω = m for all ω ∈ on the groupoid × via the identification G ω ∼ = . In the next definition we introduce appropriate measures on the base space of an abstract groupoid G . Definition 2.4 Let G be a measurable groupoid with a transverse function ν. A measure μ on the base space (, B ) of units is called ν-invariant (or simply invariant, if there is no ambiguity in the choice of ν) if where (μ ◦ ν)( f ) = f (g−1 ).
μ ◦ ν = (μ ◦ ν)∼ ,
ν ω ( f )dμ(ω) and (μ ◦ ν)∼ ( f ) = (μ ◦ ν)( f˜) with f˜(g) =
In (RSM) it can easily be checked that, with the above choice ν ≡ m , a measure μ on (, B ) is ν-invariant if and only if μ is -invariant in the classical sense, see [22, Cor. II.7] as well. Thus a canonical choice for a m -invariant measure on is P. Analogously to transverse functions on the groupoid, we introduce a corresponding fiberwise consistent family α of measures on the G -space, see the next definition. We refer to the resulting object (X , α) as a random variable in the sense of Connes. These random variables are useful substitutes for measurable functions on the quotient space / with values in X. Measurable functions on / can be identified with -invariant measurable functions on . Note that, in the case of an ergodic action of on , there are no nontrivial -invariant measurable functions, whereas there are usually lots of random variables in the sense of Connes (see below for examples). Definition 2.5 Let G be a measurable groupoid and X be a G -space. A choice of measures α : → M(X ) is called a random variable (in the sense of Connes) with values in X if it has the following properties – – –
the map ω → α ω ( f ) is measurable for every f ∈ F + (X ), α ω is supported on X ω , i.e., α ω (X − X ω ) = 0, α satisfies the following invariance condition f (J(g) p)dα s(g) ( p) = f (q)dαr(g) (q) X s(g)
X r(g)
+
for all g ∈ G and f ∈ F (X
r(g)
).
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To simplify notation, we write gh, respectively, gp for g · h, respectively, J(g) p. The general setting for the sequel consists of a groupoid G equipped with a fixed transverse function ν and an ν-invariant measure μ on , and a fixed random variable (X , α). We use the following notation for the ‘averaging’ of a u ∈ F + (X ) with respect to ν ν ∗ u0 ( p) :=
G π( p)
u0 (g−1 p)dν π( p) (g)
for p ∈ X .
We will need the following further assumptions in order to apply the integration theory developed in [22]. Definition 2.6 Let (G , ν, μ) be a measurable groupoid and (X , α) be a random variable on the associated G -space X satisfying the following two conditions (6) (7)
The σ -algebras BX and B are generated by a countable family of sets, all of which have finite measure, w.r.t. μ ◦ α (respectively w.r.t. μ). There exists a strictly positive function u0 ∈ F + (X ) satisfying ν ∗ u0 ( p) = 1 for all p ∈ X .
Then we call the tupel (G , ν, μ, X , α, u0 ) an admissible setting. Before continuing our investigation let us shortly discuss the relevance of the above conditions: Condition (6) is a strong type of separability condition for the Hilbert space L2 (X , μ ◦ α). It enables us to use the techniques from direct integral theory discussed in Appendix A which are crucial to the considerations in Section 3. Condition (7) is important to apply Connes’ non-commutative integration theory. Namely, it says that (X , J) is proper in the sense of Lemma III.2 and Definition III.3 of [22]. Therefore (X , J) is square integrable by Proposition IV.12 of [22]. This square integrability in turn is a key condition for the applications of [22] we give in Sections 3, 4 and 5. On an intuitive level, (7) can be understood as providing an ‘embedding’ of G into X . Namely, every u ∈ F (X ) with ν ∗ u ≡ 1 gives rise to the fibrewise defined map q = qu : F (G ) → F (X ) by q( f )( p) :=
G π( p)
u(g−1 p) f (g)dν π( p) (g)
(8)
for all p ∈ X . Note that the convolution property of u implies that the map (8) satisfies q(1G ) = 1X . Moreover, q can be used to obtain new functions w ∈ F (X ) satisfying ν ∗ w ≡ 1. This is the statement of the next proposition. Proposition 2.7 Let u ∈ F + (X ) with ν ∗ u ≡ 1 be given, and q be as above. For any function f ∈ F (G ) with ν( f˜) ≡ 1 on we have ν ∗ q( f ) ≡ 1.
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Proof The proof is given by the following direct calculation with ω = π( p) ∈ : q( f )(g−1 p)dν ω (g) (ν ∗ q( f ))( p) = Gω
=
Gω
(ν transverse function) =
Gω
(Fubini) =
Gω
= (ν( f˜) ≡ 1) =
Gω
Gω
G s(g)
Gω
u(h−1 g−1 p) f (h)dν s(g) (h)dν ω (g)
u(k−1 p) f (g−1 k)dν ω (k)dν ω (g)
u(k−1 p) u(k−1 p)
Gω
Gω
f (g−1 k)dν ω (g)dν ω (k) f˜(k−1 g)dν ω (g)dν ω (k)
u(k−1 p)dν ω (k)
= 1. Note that in the above calculation the integration variable g has the property r(g) = π( p) = ω. This finishes the proof.
Remark 2.8 We consider examples of admissible settings in the case (RSM). Recall that we will identify G ω with and X ω with X for all ω ∈ . A transverse function of G is given by copies of the Haar measure: ν ω = m for all ω ∈ . X together with the Riemannian volume forms λω (corresponding to the metrics gω ) on the fibers X ω ≡ X is an example of a random variable. This will be shown next by discussing validity of conditions (7) and (6). Condition (7): Let D be a fundamental domain for the action on X such that γ ∈ γD = X is a disjoint union, c.f. [77, §6.5]. Then every function v ∈ F () with γ ∈ v(γ ) = 1 gives rise to a u0 satisfying ν ∗ u0 ≡ 1 by u0 (ω, x) = v(γ ) where γ ∈ is the unique element with x ∈ γ D. Note that this construction assigns to a strictly positive v, again a strictly positive u0 on X . Hence the setting ( × , m , P, × X, λ, u0 ) satisfies condition (7). Condition (6): This condition is clearly satisfied if the σ -algebra of is countably generated. In the case (RSM) (a) this countability condition can always be achieved by passing to an equivalent version of the defining stochastic process. Namely, given a random potential V : × X → R, we ˜ × X → R with the same finite dimensional construct a stochastic process V˜ : distributions such that (6) is satisfied, c.f. e.g. [36, 75]. Assume at first that the random potential V : × X → R can be written as Vω (x) = fγ (ω, γ −1 x). (9) γ ∈
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Here ( fγ )γ ∈ is a sequence of measurable functions on the probability space with values in a separable Banach space B of functions on X. Such models have been studied by Kirsch in [49]. Note that the Borel-σ -algebra B B is generated by a countable set E ⊂ B B . In this case can be chosen to be the canonical probability space B . Its σ -algebra is generated by a countable family of cylinder sets of the form {ω ∈ B | ∀γ ∈ H n : ω(γ ) ∈ Mγ } where H denotes a finite set of generators of , n ∈ N and Mγ ∈ E for each γ . An appropriate choice for the (separable) Banach space B is:
1/ p 1 p p (L )(X) := f : X → C | f (x)| dλ(x) dm (γ ) < ∞
orL ( )(X) := p
1
γD
p
1/ p −1 f : X → C | f (γ x)|dm (γ ) dλ(x) <∞ . D
Here D denotes an arbitrary -fundamental domain in X. We have the inclusion L p (X) ⊂ 1 (L p (X)) ⊂ L p (1 (X)) and L p (1 (X)) is separable, cf. [49, 51]. More correctly, one could use the notation 1 (, L p (D)) for 1 (L p )(X) and L p (D, 1 ()) for L p (1 )(X). Now we show that actually all models which are of the type (RSM) (a) can be writtenin the way (9). Let u : X → R+ be a bounded measurable function such that γ ∈ u(γ −1 x) = 1 for all x ∈ X. Let φ = Vu. Then V(ω, x) = V(ω, x)u(γ −1 x) = V(γ −1 ω, γ −1 x)u(γ −1 x) γ ∈
=
γ ∈
φ(γ −1 ω, γ −1 x).
γ ∈
Setting fγ (ω, x) = φ(γ −1 ω, x) we have a representation as in (9). p The regularity assumptions on V in (RSM) imply that V(ω, ·) ∈ Lloc,uni f (X) p for all ω ∈ and all p ∈ [1, ∞]. Here Lloc,uni f (X) = ∞ (L p )(X) denotes the set of locally L p -integrable functions f such that the L p -norm of f χγ D is uniformly bounded in γ ∈ . The choice u = χD yields φ(ω, ·) ∈ L p (X). Thus the functions fγ (ω, ·) are in the Banach space B = L p (X). Summarizing the above considerations, we conclude that (RSM) (a) satisfies both condition (6) and (7) (after a suitable modification of the underlying probability space). The case (RSM) (b) can be treated similarly. A crucial fact about the integration of random variables is given in the following lemma, essentially contained in (the proof of) Lemma III.1 in [22]. We include a proof for the convenience of the reader.
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Lemma 2.9 Let G be a groupoid with transverse function ν and ν-invariant measure μ. Let furthermore X be a G -space. (a) For a given transverse function φ on G , the integral φ ω ( f ) dμ(ω) does not depend on f ∈ F + (G ), provided f satisfies ν( f˜) ≡ 1. (b) For a given random variable α with values in X the integral α ω (u) dμ(ω) does not depend on u, provided u ∈ F + (X ) satisfies ν ∗ u ≡ 1. Proof We prove first part (b). So let (X , α) and u, u ∈ F (X ) with ν ∗ u ≡ ν ∗ u ≡ 1 be given. Inserting 1 ≡ ν ∗ u , we calculate ω α (u) dμ(ω) = u( p) · 1 · dα ω ( p) dμ(ω)
Xω
Gω
Gω
Gω
(Fubini) =
(μ inv.) =
(α inv.) =
Xω
u (g−1 p) u( p) dα ω ( p) dν ω (g) dμ(ω)
X s(g)
Xω
u (gp ) u( p ) dα s(g) ( p ) dν ω (g) dμ(ω)
u ( p) u(g−1 p) dα ω ( p) dν ω (g) dμ(ω).
ω Another ω application of Fubini and use of ν ∗ u ≡ 1 then gives α (u)dμ(ω) = α (u )dμ(ω) and the proof of (b) is finished. Note that every groupoid G is a G -space X . Furthermore, in this case the convolution condition ν ∗ u ≡ 1 translates into ν( f˜) ≡ 1. This proves (a).
Finally, we introduce, for a given G -space X , a new G -space X × X . It consists of the set X × X := {( p, q) | π( p) = π(q)} ⊂ X × X .
equipped with the induced σ -algebra. The corresponding maps π and J of X × X are defined in an obvious way. If X is actually a random variable with measures (α ω )ω , then X × X inherits the structure of a random variable by setting (α × α)ω := α ω ⊗ α ω . 3 The von Neumann Algebra N (G, X ) In this section we discuss how a von Neumann Algebra arises naturally for an admissible setting (G , ν, μ, X , α, u0 ), cf. [22]. The random operators we are interested in are affiliated to this von Neumann algebra. Recall that a selfadjoint operator is called affiliated to a von Neumann algebra if its spectral family is contained in the von Neumann algebra. A G -space X is, by definition, a bundle over = G (0) via π : X −→ . This bundle structure of a random variable (X , α) induces a bundle structure of
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L2 (X , μ ◦ α): Using the Fubini Theorem, we can associate with f ∈ L2 (X , μ ◦ α) a family ( fω )ω∈ of functions fω ∈ L2 (X ω , α ω ) such that f (x) = fπ(x) (x)
(10)
for μ-almost all ω ∈ . This way of decomposing will be a key tool in the sequel. On the technical level, this is very conveniently expressed using direct integral theory (see e.g. [28]) and the fact that there is a canonical isomorphism ⊕ L2 (X ω , α ω ) dμ(ω). (11) L2 (X , μ ◦ α)
As direct integral theory tends to be rather technical, we try to avoid it in the sequel and rather give direct arguments which, however, are inspired by the general theory (cf. Appendix A). A special role will be played by those operators which respect the bundle structure of L2 (X , μ ◦ α) given by (10). Namely, we say that the (not necessarily bounded) operator A : L2 (X , μ ◦ α) −→ L2 (X , μ ◦ α) is decomposable if there exist operators Aω : L2 (X ω , α ω ) −→ L2 (X ω , α ω) such that (A f )(x) = ⊕ (Aω fω )(x), for almost every ω ∈ . We then write A = Aω dμ(ω). Let the groupoid (G , μ, ν) and the random variable (X , α) be as in the last section. For g ∈ G , let the unitary operator U g be given by U g : L2 (X s(g) , α s(g) ) −→ L2 (X r(g) ,
αr(g) ), U g f ( p) := f (g−1 p).
A family (Aω )ω∈ of bounded operators Aω : L2 (X ω , α ω ) → L2 (X ω , α ω ) is called a bounded random operator if it satisfies : – – –
ω → gω , Aω fω is measurable for arbitrary f, g ∈ L2 (X , μ ◦ α). There exists a C 0 with Aω C for almost all ω ∈ . The equivariance condition Ar(g) = U g As(g) U g∗ for all g ∈ G is satisfied.
Two bounded random operators (Aω ), (Bω ) are called equivalent, (Aω ) ∼ (Bω ) if Aω = Bω for μ-almost every ω ∈ . Each equivalence class of bounded random operators (Aω ) gives rise to a bounded operator A on L2 (X , μ ◦ α) by A f ( p) := Aπ( p) fπ( p) (cf. Appendix A). This allows us to identify the class of (Aω ) with the bounded operator A. The following is the main theorem on the structure of the space of random operators. Theorem 3.1 The set N (G , X ) of classes of bounded random operators is a von Neumann algebra. Proof This follows immediately from [22, Thm. V.2].
To get some insight we give a proof of Theorem 3.1 for the particular case of (RSM). In this example we can canonically identify L2 (X ω , α ω ) with L2 (X, λω ) and L2 (X , μ ◦ α) with L2 ( × X, P ◦ λ). Under this identification a bounded random operator is just a family of uniformly bounded operators Aω : L2 (X, λω ) → L2 (X, λω ),
ω∈
Groupoids, von Neumann algebras and the IDS
15
with (ω, x) → (Aω fω )(x) measurable and ω → Aω fω ∈ L2 (X, λω ) for every f ∈ L2 ( × X, P ◦ λ), satisfying the equivariance condition ∗ Aω = U (ω,γ ) Aγ −1 ω U (ω,γ ),
(12)
where we write U g as U (ω,γ ) due to the product structure of G = × . In this case we can also associate with (Aω ) an operator A on the Hilbert space L2 ( × X, P ◦ λ) by setting (A f )(ω, x) ≡ (Aω fω )(x). Now, (12) implies γ = U γ A, AU
(13)
γ is the unitary operator on L2 ( × X, P ◦ λ) defined by where U γ f )ω (x) = fγ −1 ω (γ −1 x). (U Conversely, one can show for a decomposable operator A that (13) implies (12) P-almost everywhere. Now, a suitable averaging procedure and a change of the fibers Aω on a set of P-measure zero yields (12) for all ω (cf. [59] or [22, p. 88]). It is well known that a bounded operator A on L2 ( × X) is decomposable, i.e. A = Aω dP(ω) for a suitable family (Aω ), if and only if A commutes with the multiplication operators Mh◦π for every h ∈ L∞ (, P) (see, e.g., [28, Thm. 1 in II.2.5]). Summarizing these considerations, we conclude γ | γ ∈ } ∩ {Mh◦π | h ∈ L∞ (, P)} , N ( × , × X) = {U γ | γ ∈ where S denotes the commutant of a set S of operators. Obviously, {U } and {Mh◦π | h ∈ L∞ (, P)} are closed under taking adjoints. Thus, their commutants are von Neumann algebras. This reasoning shows directly, in the case of (RSM), that the space of random operators is a von Neumann algebra.
4 The Canonical Trace on N (G, X ) In this section we start with an admissible setting (G , ν, μ, X , α, u0 ) and its associated von Neumann algebra N (G , X ) of bounded operators on L2 (X , μ ◦ α). Let N + (G , X ) denote the set of non negative selfadjoint operators in N (G , X ). We will show that every operator A ∈ N + (G , X ) gives rise to a new random variable (X , β A ). Integrating this random variable, we obtain a weight on N (G , X ). Under certain (mild) assumptions this weight can be shown to be a trace. We start by associating a transversal function as well as a random variable with each element in N + (G , X ). In the following, qω ( f ) denotes the restriction of q( f ) as defined in (8) to the fiber X ω .
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Lemma 4.1 Let A ∈ N + (G , X ). (a) Then φ A , given by φ ωA ( f ) := tr(Aω Mqω ( f ) ), f ∈ F (G ω ), defines a transverse function. ω (b) Then β A , given by β A ( f ) := tr(Aω Mf ), f ∈ F (X ω ), is a random variable. Proof This follows by direct calculation using the equivariance properties of
the family Aω and the qω . Let us recall the following definitions. A weight on a von Neumann algebra N is a map τ : N + → [0, ∞] satisfying τ (A + B) = τ (A) + τ (B) and τ (λA) = λτ (A) for arbitrary A, B ∈ N + and λ 0. The weight is called normal if τ (An ) converges to τ (A) whenever An is an increasing sequence of operators (i.e. An An+1 , n ∈ N) converging strongly to A. It is called faithful if τ (A) = 0 implies A = 0. It is called semifinite if τ (A) = sup{τ (B) : B A, τ (B) < ∞}. If a weight τ satisfies τ (CC∗ ) = τ (C∗ C) for arbitrary C ∈ N (or equivalently τ (U AU ∗ ) = τ (A) for arbitrary A ∈ N + and unitary U ∈ N , cf. [28, Cor. 1 in I.6.1]), it is called a trace. Theorem 4.2 For A ∈ N + (G , X ), the expression 1 1 1 1 tr(Aω2 Muω Aω2 )dμ(ω) = tr(Mu2ω Aω Mu2ω )dμ(ω) τ (A) :=
does not depend on u ∈ F + (X ) provided ν ∗ u ≡ 1. (a) The map τ : N + (G , X ) −→ [0, ∞] is a faithful, semifinite normal weight on N (G , X ). (b) If the groupoid G satisfies the freeness condition r−1 (ω) ∩ s−1 (ω) = {ω} for μ-almost all ω ∈ ,
(14)
then τ is a trace.
Proof That τ is independent of u follows easily from Lemma 2.9 and Lemma 4.1. (a) The proof that τ is a normal weight is straightforward. The proof of the semifiniteness of τ is not simple and we refer the reader to Theorem VI.2 in [22]. To show that τ is faithful, assume τ (A) = 0 for A ∈ N + (G , X ). This implies tr(Muω Aω ) = 0 for almost every ω. As the trace tr is faithful, this implies Muω Aω = 0 for almost every ω. Choosing a strictly positive function u (e.g. u = u0 ), we infer Aω = 0 for almost every ω and we see that τ is faithful. (b) The freeness condition (14) easily shows that Corollary VI.7 of [22] is applicable and the statement follows.
Groupoids, von Neumann algebras and the IDS
17
Remark 4.3 (a) Property (14) can be slightly relaxed (cf. [22]). However, some condition of this form is necessary to prove the trace property in the generality addressed in the theorem. It will turn out to be dispensable in many concrete cases. Indeed, we will see below that τ is a trace in the case of (RSM) without any condition of this form. However, when considering factorial type properties this condition again plays an important role. (b) The motivation to call property (14) a freeness condition is easily understood in the context of (RSM). Namely, in this case (14) is equivalent to P(ω | γ −1 ω = ω) = 0 for all γ ∈ \{ }.
(15)
(c) Given condition (14) and some ergodicity assumptions (see the next section), it follows from [22] that the von Neumann algebra is actually a factor. In this case the trace τ is unique (up to a scalar factor). We are now heading towards an alternative direct proof of part (b) of Theorem 4.2 in the particular case (RSM). This proof will be based on a study of τ for certain operators with a kernel. It will in fact be more general, in that it does not use the freeness condition (14) on the action of on . Along our way, we will also find an instructive formula for τ on these operators. Let us emphasize that the approach to prove that τ is a trace given below is in no way restricted to (RSM) (cf. [62, 64] for related material). An operator K on L2 (X , μ ◦ α) is called a Carleman operator (cf. [91] for further details) if there exists a k ∈ F (X × X ) with k( p, ·) ∈ L2 (X π( p) , α π( p) ) for all p ∈ X such that
K f ( p) =
X π( p)
k( p, q) f (q)dα π( p) (q) =: Kπ( p) fπ( p) ( p).
This k is called the kernel of K: Obviously, K = all Carleman operators satisfying for all g ∈ G k(gp, gq) = k( p, q)
⊕
Kω dμ. Let K be the set of
for α π(g) × α π(g) almost all p, q .
(16)
Proposition 4.4 K is a right ideal in N (G , X ). Proof Obviously, K ∈ K is decomposable. Thus, to show that K is a subset of N (G , X ), it suffices to show the equivariance condition. But this is immediate from (16). Thus, it remains to show that K is a right ideal. This follows from a standard calculation in the theory of Carleman operators: K A f ( p) = Kπ( p) Aπ( p) fπ( p) ( p) = k( p, ·), Aπ( p) fπ( p) π( p) ,
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where f1 , f2 ω = given by
f1 (q) f2 (q)dα ω (q). This shows that K A has a kernel k K A
Xω
k K A ( p, q) = A∗π( p) k( p, ·) (q),
which finishes the proof.
For a Carleman operator K the expressions τ (KK∗ ) and τ (K∗ K) can directly be calculated. Proposition 4.5 Let K ∈ K be given. Then we have ∗ τ (K K) = u(q)|k( p, q)|2 dα ω ( p)dα ω (q)dμ(ω) = τ (KK∗ ), Xω
Xω
+
for any u ∈ F (X ) satisfying ν ∗ u ≡ 1. Proof We use the well known formula tr(A∗ A) = S×S |a(x, y)|2 dλ(x)dλ(y) valid for any bounded operator A with kernel a. Note that this formula holds for both a ∈ L2 (S × S, λ ⊗ λ) and a ∈ / L2 (S × S, λ ⊗ λ). In the latter case both sides of the formula are just infinity. Now, we can calculate as follows: τ (K∗ K) = tr(Mu1/2 Kω∗ Kω Mu1/2 )dμ(ω) ω ω
=
Xω
Xω
u(q)|k( p, q)|2 dα ω ( p)dα ω (q)dμ(ω).
This proves the first equation. To show the second equation, we insert ν ∗ u ≡ 1 in the first equation, finding
τ (K∗ K) = u(g−1 p)dν ω (g) u(q)|k( p, q)|2 dα ω (q)dα ω ( p)dμ(ω).
Xω
Xω
Gω
By Fubini and the fact that K is invariant, we infer ··· = u(g−1 p)u(q)|k(g−1 p, g−1 q)|2 dαr(g) ( p)dαr(g) (q)d(μ ◦ ν)(g). G
X r(g)
X r(g)
Invoking invariance of μ, i.e. (μ ◦ ν)∼ = μ ◦ ν, we finally obtain ··· = u(gp)u(q)|k(gp, gq)|2 dα s(g) ( p)dα s(g) (q)d(μ ◦ ν)(g) G
X s(g)
G
X r(g)
=
X s(g)
X r(g)
u( p )u(g−1 q )|k( p , q )|2 dαr(g) ( p )dαr(g) (q )d(μ ◦ ν)(g).
Now, the desired equality follows by reversing the steps from the preceding calculation. This finishes the proof.
The proposition shows that τ has the trace property τ (A A∗ ) = τ (A∗ A) for all Carleman operators A. To show that τ is actually a trace, we need two more steps. In the first step, we show that τ must be a trace if it satisfies this trace
Groupoids, von Neumann algebras and the IDS
19
property for sufficiently many operators. In the second step, we show that the set of Carleman operators is large. Lemma 4.6 Let I be a right ideal in a von Neumann algebra N acting on a separable Hilbert space H with I ∗ I (H) = H. Let τ be a normal weight on N satisfying τ (A∗ A) = τ (A A∗ ) for all A ∈ I . Then τ is a trace. Proof It suffices to show τ (U AU ∗ ) = τ (A) for arbitrary A ∈ N+ and unitary U ∈ N . Let C(I ) be the norm closure of I ∗ I . Obviously, I ∗ I is a dense ideal in C(I ). By [27, Prop. 1.7.2], there exists then an increasing approximate identity (Iλ ) for C(I ) in I . As H is separable, we can choose (Iλ ) to be a sequence (In ). By I ∗ I (H) = H, we infer that In converges monotonously to the identity Id on H. By polarization, every In ∈ I ∗ I can be written as ∗ In = Di,n Di,n − C∗j,n C j,n , (17) for suitable, finitely many Di,n , C j,n ∈ I . Now, let an arbitrary D ∈ I be given. Using that I is a right ideal, we can calculate τ (U A 2 D∗ DA 2 U ∗ ) = τ (DA 2 U ∗ U A 2 D∗ ) = τ (DA 2 A 2 D∗ ) = τ (A 2 D∗ DA 2 ). (18) 1
1
1
1
1
1
1
1
Combining (18), (17) and the fact that τ is normal, one can conclude 1 1 the proof of the Lemma as follows: τ (U AU ∗ ) = limn→∞ τ (U A 2 In A 2 U ∗ ) = 1 1 limn→∞ τ (A 2 In A 2 ) = τ (A).
Proposition 4.7 Consider (RSM) and define, for t 0, the operator S(t) : L2 ( × X) → L2 ( × X), by S(t) fω (x) := (e−t ω fω )(x). Then, S(t) is a selfadjoint bounded random operator for each t 0. For t > 0, the operator S(t) belongs to K. The family t → S(t) is a strongly continuous semigroup. Proof Property (2) gives immediately the necessary measurability of S(t). Moreover, e−t ω is a selfadjoint contraction for every ω ∈ . Therefore, Fubini easily shows S(t) f f for every f ∈ L2 ( × X). Thus, S(t) is a decomposable operator. As its fibres are selfadjoint, it is selfadjoint as well (cf. [78, Thm. XIII.85] and [27, Appendix A.78]). The transformation formula ∗
ω = U (ω,γ ) γ −1 ω U (ω,γ ) shows that S(t) satisfies the equivariance property. Thus, S(t) is indeed a random operator. Using that t → e−t ω is a strongly continuous semigroup of operators with norm not exceeding 1 for every ω ∈ , we can directly calculate that S(t) is a strongly continuous semigroup, as well. For t > 0, every S(t) has a kernel ktω (see, e.g., [16] or [80]). Using selfadjointness and the semigroup property, we can calculate |ktω (x, y)|2 dλω (y) = ktω (x, y)ktω (y, x)dλω (y) = k2t ω (x, x) < ∞. X
X
This shows that S(t) belongs to K.
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Now, the following theorem is not hard to prove. Theorem 4.8 In the example (RSM), the map τ is a normal trace. Proof By Proposition 4.5 and Lemma 4.6, it suffices to show that K∗ K L2 (X × ) is dense in L2 ( × X). But, this follows from the foregoing proposition, which shows that S(t) f converges to f for every f ∈ L2 ( × X), where S(t) ∈ K∗ K by the semigroup property.
5 Fundamental Results for Random Operators In this section we present a comprehensive treatment of the basic features (P1)–(P4) of random operators mentioned in the introduction. This unifies and extends the corresponding known results about random Schrödinger operators in Euclidean space and random operators induced by tilings. As in the previous section, we always assume that (G , ν, μ, X , α, u0 ) is an admissible setting. A function f on is called invariant if f ◦ r = f ◦ s. The groupoid G is said to be ergodic (with respect to μ) if every invariant measurable function f is μ-almost everywhere constant. This translates, in the particular case (RSM), to an ergodic action of on . We will be mostly concerned with decomposable selfadjoint operators on L2 (X , μ ◦ α). For a selfadjoint operator H, we denote by σdisc (H), σess (H), σac (H), σsc (H) and σ pp (H) the discrete, essential, absolutely continuous, singular continuous, and pure point part of its spectrum, respectively. ⊕ Theorem 5.1 Let G be an ergodic groupoid. Let H = Hω dμ(ω) be a selfadjoint operator affiliated to N (G , X ). There exist ⊂ of full measure and , • ⊂ R, • = disc, ess, ac, sc, pp, such that σ (Hω ) = ,
σ• (Hω ) = •
for all ω ∈
for • = disc, ess, ac, sc, pp. Moreover, σ (H) = . Note that σ pp denotes the closure of the set of eigenvalues. Proof The proof is essentially a variant of well known arguments (cf. e.g. [19, 20, 24, 53, 58]). However, as our stetting is different and, technically speaking, involves direct integrals with non constant fibres, we sketch a proof. Let J be the family of finite unions of open intervals in R, all of whose endpoints are rational. Let J k consist of those elements of J which are unions of exactly k intervals. Denote the spectral family of a selfadjoint operator H by E H . It is not hard to see that ω → trE Hω (B) is an invariant measurable function for every B ∈ J (and, in fact, for every Borel measurable B ⊂ R).
Groupoids, von Neumann algebras and the IDS
21
Thus, by ergodicity, this map is almost surely constant. Denote this almost sure value by f B . As J is countable, we find ⊂ of full measure, such that for every ω ∈ , we have trE Hω (B) = f B for every B ∈ J . By σ (H) = {λ ∈ R : E H (B) = 0, for all B ∈ J with λ ∈ B },
(19)
and as tr is faithful, we infer constancy of σ (Hω ) on . By a completely analogous argument, using σess (H) = {λ ∈ R : trE H (B) = ∞ for all B ∈ I with λ ∈ B}, we infer almost sure constancy of σess (Hω ) and thus also of σdisc (Hω ). To show constancy of the remaining spectral parts, it suffices to show measurability of pp
sing
ω → gω , E Hω (B)gω ω , and ω → gω , E Hω (B)gω ω
(20) pp
sing
for every g ∈ L2 (X, μ ◦ α) and every B ∈ J . Here, of course, E H and E H denote the restrictions of the spectral family to the pure point and singular part of the underlying Hilbert space, respectively. To show these measurabilities, recall that for an arbitrary measure μ on R with pure point part μ pp and singular part μsing we have μsing (B) = lim
sup
n→∞ J∈J ,|J|n−1
μ(B ∩ J),
μ pp (B) = lim lim
sup
k→∞ n→∞ J∈J k ,|J|n−1
μ(B ∩ J).
Here, the first equation was proven by Carmona (see [19, 20, 24]), and the second follows by a similar argument. As this latter reasoning does not seem to be in the literature, we include a discussion in Appendix B. Given these equalities, (20) is an immediate consequence of measurability of ω → gω , E Hω (B)gω ω , (which holds by assumption on H). (Note that instead of considering μ pp as above, one could have considered the continuous part μc of μ by a method given in [20].) It remains to show the last statement. Obviously, E H (B) = 0 if and only if E Hω (B) = 0 for almost every ω ∈ . Using this, (19), and almost sure constancy of trE Hω (B), infer that λ ∈ σ (H) if and only if E Hω (B) = 0 for every B ∈ J with λ ∈ B and almost every ω ∈ . This proves the last statement. Alternatively, one could follow the proof of [78, Thm.XIII.85] which is valid in the case of non-constant fibres, too.
Recall that a measure φ on R is a spectral measure for a selfadjoint operator H with spectral family E H if, for Borel measurable B ⊂ R, φ(B) = 0 ⇔ E H (B) = 0. Proposition 5.2 For a Borel measurable B in R and a selfadjoint H affiliated to N (G , X ), let ρ H (B) be defined by ρ H (B) := τ (E H (B)). Then ρ H is a spectral measure for H. Moreover, for a bounded measurable function F : R −→ [0, ∞), the equality τ (F(H)) = ρ H (F) := F(x)dρ H (x) holds.
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Proof As τ is a normal weight, ρ H is a measure. As τ is faithful, ρ H is a spectral measure. The last statement is then immediate for linear combinations of characteristic functions with non negative coefficients. For arbitrary functions it then follows after taking suitable monotone limits and using normality of τ .
Definition 5.3 The measure ρ H is called (abstract) density of states. Corollary 5.4 The topological support supp(ρ H ) = {λ ∈ R | ρ H (]λ − , λ + [) > 0 for all > 0} coincides with the spectrum σ (H) of H for every selfadjoint H affiliated to N (G , X ). If G is, furthermore, ergodic this gives supp(ρ H ) = σ (Hω ) for almost every ω ∈ . Proof The first statement is immediate as ρ H is a spectral measure. The second follows from Theorem 5.1.
Remark 5.5 In general, ρ H is not the spectral measure of the fibre Hω . Actually, there are several examples where the set := {ω ∈ | ρ H is a spectral measure for the operator Hω } has measure zero. We shortly discuss two classes of them (see [5] as well for related material). Firstly we consider the case where (Hω )ω exhibits localization in an energy interval I. This means that for a set loc ⊂ of full measure σ (Hω ) ∩ I = ∅ and σc (Hω ) ∩ I = ∅ for all ω ∈ loc . Examples of such ergodic, random operators on L2 (Rd ) and 2 (Zd ) can be found e.g. in the textbooks [20, 75, 85]. They particularly include random Schrödinger operators in one- and higher dimensional configuration space. If Hω is an ergodic family of Schrödinger operators on 2 (Zd ) or L2 (R) one knows moreover for all energy values E ∈ R P{(E)} = 0, where (E) := {ω ∈ | E is an eigenvalue of Hω }.
Assume that ρ H is a spectral measure of Hω for some ω ∈ loc . Then there exists an eigenvalue E ∈ I of Hω and consequently ρ H ({E}) > 0. Thus ρ H can only be a spectral measure for Hω if ω ∈ (E), but this set has measure zero. Similarly, there are one-dimensional discrete random Schrödinger operators which have purely singular continuous spectrum almost surely. An example is the almost Mathieu operator, cf. [5], for a certain range of parameters. Moreover, in [37] it is proven that the singular continuous components of the spectral measures of these models are almost surely pairwise orthogonal. Thus again, ρ H can be a spectral measure only for a set of ω of measure zero.
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Lemma 5.6 Let G be ergodic with respect to μ with μ() < ∞ such that the following exhaustion property holds: There exists a sequence ( fn ) in F + (G ) with G = {g : fn (g) > 0}, n
fn ∞
→ 0 as n → ∞ and ν( fn ) ≡ 1 for all n ∈ N.
(21)
Then, for every transverse function φ, either φ ω (1) ≡ 0 almost surely or φ ω (1) ≡ ∞ almost surely. ω Proof By ergodicity, φ ω (1) is constant almost surely. ω Assume φ (1) = c < ∞ for almost every ω. Lemma 2.9 (a) implies that φ ( f )dμ(ω) is independent f ) ≡ 1. For such f , we infer of f ∈ F + with ν( Constant = φ ω ( f )dμ f ∞ c μ().
As this is in particular valid for every fn , n ∈ N, we infer Constant = 0. This shows φ ω ( fn ) = 0 for almost every ω and every n ∈ N. By G = n {g : fn (g) > 0}, this gives φ ω (1) = 0 for almost every ω ∈ .
Remark 5.7 Let (G , ν, μ) be a measurable groupoid, (X , α) be an associated random variable and u ∈ F + (X ) such that property (6) holds and ν ∗ u ≡ 1. satisfying condition (21). Then (G , ν, μ, X , α, u0 ) with Let fn be a sequence 1 u0 = q u ( ∞ f ) is an admissible setting. n n n=1 2 Remark 5.8 The exhaustion property (21) can easily be seen to hold in the case (RSM), if is infinite. Namely, we can choose fn (g) = fn (ω, γ ) =
1 χ I (γ ). |In | n
Here, In is an exhaustion of the group . The lemma has an interesting spectral consequence. Corollary ⊕ 5.9 Let the assumptions of Lemma 5.6 be satisfied and the selfadjoint H = Hω dμ(ω) be affiliated to N (G , X ). Then Hω has almost surely no discrete spectrum. Proof We already know that the discrete spectrum is constant almost surely. Let B be an arbitrary Borel measurable subset of R. Then, the map ω → tr(E Hω (B)Mqω (·) ) is a transverse function, by Lemma 4.1 (a). Thus, tr(E Hω (B)Mqω (1) ) = tr(E Hω (B)) equals almost surely zero or infinity. As B is arbitrary, this easily yields the statement.
Remark 5.10 Corollary 5.9 is well known for certain classes of random operators. However, our proof of the key ingredient, Lemma 5.6, is new. It seems to be more general and more conceptual. Moreover, as discussed in the proof of
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Theorem 5.11 below, Lemma 5.6 essentially implies that N (G , X ) is not type I. Thus, the above considerations establish a connection between the absence of discrete spectrum and the type of the von Neumann algebra N (G , X ). Let us finish this section by discussing factorial and type properties of N (G , X ). By [22, Cor. V.8] (cf. Cor. V.7 of [22], as well), the von Neumann algebra N is a factor (i.e. satisfies N ∩ N = C Id) if G is ergodic with respect to μ and the freeness condition (14) holds. There are three different types of factors. These types can be introduced in various ways. We will focus on an approach centered around traces (cf. [23] for further discussion and references). A factor is said to be of type I I I, if it does not admit a semifinite normal trace. If a factor admits such a trace, then this trace must be unique (up to a multiplicative constant) and there are two cases. Namely, either, this trace assumes only a discrete set of values on the projections, or the range of the trace on the projections is an interval of the form [0, a] with 0 < a ∞. In the first case, the factor is said to be of type I. It must then be isomorphic to the von Neumann algebra of bounded operators on a Hilbert space. In the second case, the factor is said to be of type I I. Theorem 5.11 Let the assumptions of Lemma 5.6 and condition (14) be satisfied, then N (G , X ) is a factor of type I I. Proof By Lemma 5.6, there does not exist a bounded transversal function φ whose support {ω : φ ω (1) = 0} has positive μ measure. By [22, Cor. V.9], we infer that N (G , X ) is not type I. On the other hand, as τ is a semifinite normal trace on N by Theorem 4.2, it is not type I I I.
Remark 5.12 (a) In the case (RSM), under the countability and freeness assumptions (6) and (14), we know that N ( × , × X) is actually a factor of type I I∞ : Since the identity on an infinite dimensional Hilbert space has trace equal to infinity, we conclude tr L2 (X) (MχD ) = tr L2 (D) (Id) = ∞, where D is a fundamental domain as in Remark 2.8. Now, choosing uω (x) ≡ χD (x), we have ν ∗ u ≡ 1 and, consequently, τ (Id) = E{tr(MχD )} = ∞. (b) In the case of tiling groupoids and percolation models one finds factors of type I I with a finite value of the canonical trace on the identity. This finite value is determined by geometric features of the underlying tiling, respectively, the percolation process, cf. Sections 7 and 8.
Groupoids, von Neumann algebras and the IDS
25
6 The Pastur–Šubin-Trace Formula for (RSM) The aim of this section is to give an explicit exhaustion construction for the abstract density of states (cf. Section 5) of the model (RSM). The construction shows in particular that the IDS, the distribution function of the density of states, is self-averaging. This means that it can be expressed by a macroscopic limit which is ω-independent, although one did note take the expectation over the randomness. We recall in this section the relevant definitions and results of [60] concerning the example (RSM) and outline the main steps of the proofs. In the following we assume that the group is amenable to be able to apply an appropriate ergodic theorem. The exhaustion procedure yields a limiting distribution function which coincides, at all continuity points, with the distribution function of the abstract density of states. For the calculation of this distribution function the Laplace transformation has proved useful [71, 84]. We refer to the second reference for a detailed description of the general strategy. In Section 3 of [60] the operators Hω of (RSM) are defined via quadratic forms, the measurability of the latter is established, and the validity of the measurability condition (2) is deduced under the following additional hypotheses (22) (23)
The map × T X → R, (ω, v) → gω (v, v) is jointly measurable. There is a Cg ∈ ]0, ∞[ such that Cg−1 g0 (v, v) gω (v, v) Cg g0 (v, v) for all v ∈ T X.
(24)
There is a Cρ ∈ ]0, ∞[ such that |∇0 ρω (x)|0 Cρ for all x ∈ X,
(25)
(26)
where ∇0 denotes the gradient with respect to g0 , ρω is the unique smooth density of λ0 with respect to λω , and |v|20 = g0 (v, v). There is a uniform lower bound K ∈ R for the Ricci curvatures of all Riemannian manifolds (X, gω ). Explicitly, Ric(gω ) Kgω for all ω ∈ and on the whole of X. Let V : × X → R be a jointly measurable mapping such that for all ω ∈ the potential Vω := V(ω, ·) 0 is in L1 (A) for any compact A ⊂ X.
A key technique is an ergodic theorem by Lindenstrauss [66], valid for amenable groups acting ergodically by measure preserving transformations on . This theorem relies on suitable sequences (In )n , In ⊂ , so called tempered Følner sequences introduced by Shulman [82]. For an appropriate fundamental domain D (cf. §3 in [1]), the sequence (In )n induces a sequence (An )n , An ⊂ X by An = int( γ ∈In γ D). The ergodic theorem in [66] together
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with the equivariance property (1) imply that the following limits hold pointwise almost surely and in L1 (, P) sense 1 tr(χ An e−tHω ) = E{tr(χD e−tH• )} n→∞ |In |
(27)
lim
lim
n→∞
1 ω λ (An ) = E(λ• (D)). |In |
Moreover the L∞ (, P) norms of the sequences ω
λ (An ) tr(χ An e−tHω ) , , |In | |In | n n
|In | λω (An )
(28)
n
are uniformly bounded in the variable n ∈ N. Another important ingredient in the proof of (30) below is the following heat kernel lemma (see [60, Lemma 7.2]): Let (An )n be as above. Then we have 1 tr(χ A e−tHω ) − tr(e−tHωn ) = 0. (29) lim sup n n→∞ ω∈ λω (An ) The sets An together with the random family (Hω ) of Schrödinger operators on X are used to introduce the normalized eigenvalue counting functions Nωn (λ) =
|{i | λi (Hωn ) < λ}| , λω (An )
where Hωn denotes the restriction of Hω = + Vω to the domain An with Dirichlet boundary conditions and λi (Hωn ) denotes the ith eigenvalue of Hωn counted with multiplicities. The cardinality of a set is denoted by | · |. Note n that tr(e−tHω ) = e−tλ dNωn (λ). Using (27), (28) and (29) it is shown in [60] that there exists a distribution function N H : R → [0, ∞[, i.e., N H is left continuous and monotone increasing, such that the following P-almost sure pointwise and L1 -convergence of Laplace-transforms holds true: for all t > 0 ∞ ∞ lim N˜ ωn (t) = lim e−tλ dNωn (λ) = e−tλ dN H (λ) = N˜ H (t), (30) n→∞
n→∞ −∞
−∞
and that N˜ H can be identified with the explicit expression τ (e−tH ) E(tr(χD e−tH• )) = . N˜ H (t) = E(λ• (D)) E(λ• (D))
(31)
This implies by the Pastur–Šubin-Lemma [71, 83] the following convergence (for P-almost all ω ∈ ) lim Nωn (λ) = N H (λ)
n→∞
(32)
at all continuity points of N H . Note that N H does not depend on ω. Moreover, N H does not depend on the sequence (An )n as long as (An )n is chosen in the above way. The function N H is called the integrated density of states.
Groupoids, von Neumann algebras and the IDS
27
Now, our trace formula reads as follows. Theorem 6.1 Let the measure ρ H be the abstract density of states introduced in Section 5. Then we have N H (λ) =
ρ H (] − ∞, λ[) E(λ• (D))
at all continuity points λ of N H . Remark 6.2 The theorem implies in particular N H (λ) =
1 E(λ• (D))
E tr χD E Hω (] − ∞, λ[) ,
where D is a fundamental domain of as in Remark 2.8. This alternative localized formula for the IDS is well-known in the Euclidean case. Note that it doesn’t rely on a choice of boundary condition. Proof By the uniqueness lemma for the Laplace transform (see Lemma C.1 in the Appendix) it suffices to show that for all t > 0 • −tλ E(λ (D)) e dN H (λ) = e−tλ dρ H (λ). (33) To this end we observe that by (31) τ (e−tH ) = E(λ• (D)) e−tλ dN H (λ) which leaves to prove that τ (e−tH ) = ρ H (e−tλ ).
(34)
To do so, we will use that the operator H is bounded below, say H C, with a suitable C ∈ R. Define F : R −→ [0, ∞[ by F(λ) = e−tλ if λ C and by F(λ) = 0 otherwise. By spectral calculus, we infer e−tH = F(H) and, in particular, τ (e−tH ) = τ (F(H)). By Proposition 5.2, this implies ρ H (F) = τ (e−tH ). As, again by Proposition 5.2, ρ H is a spectral measure for H, its support is contained in [C, ∞[ and we easily find ρ H (e−tλ ) = ρ H (F). Combining these equalities, we end up with the desired equality (34).
7 Quasicrystal Models In this section we shortly discuss how to use the above framework to study random operators associated to quasicrystals. Quasicrystals are usually modelled by tilings or Delone sets and these two approaches are essentially equivalent. Here, we work with Delone sets and follow [62–64] to which we refer for further details. The investigation of quasicrystals via groupoids goes back to Kellendonk [47, 48] and his study of K-theory and gap labelling in this context (see [8, 62–64] for further discussion of quasicrystal groupoids and [11–13, 41] for recent work proving the so-called gap-labelling conjecture).
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A subset ω of Rd is called Delone if there exist 0 < r R such that r x − y whenever x, y ∈ ω with x = y and ω ∩ {y : y − x R} = ∅ for all x ∈ Rd . Here, · denotes the Euclidean norm on Rd . There is a natural action T of Rd on the set of all Delone sets by translation (i.e. Tt ω = t + ω). Moreover, there is a topology (called the natural topology by some authors) such that T is continuous. Then, (, T) is called a Delone dynamical system if is a compact T-invariant set of Delone sets. In this case G (, T) := × Rd is clearly a groupoid with transversal function ν with ν ω = Lebesgue measure for all ω ∈ . If μ is a T-invariant measure on , it is an invariant measure on G (, T) in the sense discussed above. By the compactness of there exists at least one such non trivial μ by the Krylov– Bogolyubov theorem. In fact, in the prominent examples for quasicrystals, there is a unique such probability measure; these systems are called uniquely ergodic. This notation comes from the fact that this unique T-invarant measure is necessarily ergodic. We now assume that (, T) with invariant measure μ is given. Then, there is a natural space X given by X = {(ω, x) ∈ G (, T) : x ∈ ω} ⊂ G (, T).
Then, X inherits a topology from G (, T) and is in fact a closed subset. The space X is fibred over with fibre map π : X −→ , (ω, x) → ω. Thus, the fibre X ω can naturally be identified with ω. In particular, every g = (ω, x) ∈ G (, T) gives rise to a isomorphism J(g) : X s(g) → X r(g) , J(g)(ω − x, p) = (ω, p + x) and J(g1 g2 ) = J(g1 )J(g2 ) and J(g−1 ) = J(g)−1 . Each fibre ω carries the discrete measure α ω giving the weight one to each point of ω. Then, (X , α) is a random variable. Let furthermore u 0 be a continuous function on Rd with u(t)dt = 1. Then, u gives rise to a function u0 on X via u0 (ω, x) = u(x) and
G (,T)π( p)
u0 (γ −1 p)dν π( p) (γ ) =
Rd
u(t)dt = 1.
Therefore, we are in an admissible setting. The freeness condition that γ −1 ω = ω whenever γ = (ε, ω) is known as aperiodicity. The associated operators are given by families Aω : 2 (ω) −→ 2 (ω), ω ∈ satisfying a measurability and boundedness assumption as well as the equivariance condition ∗ Aω = U (ω,t) Aω−t U (ω,t)
for ω ∈ and t ∈ Rd , where U (ω,t) : 2 (ω − t) −→ 2 (ω) is the unitary operator induced by translation.
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There is a canonical trace on these operators given by tr(Mu0 Aω )dμ(ω). τ (A) :=
It is not hard to see that τ (Id) < ∞ as there exists r > 0 with x − y r whenever x, y ∈ ω for ω ∈ and x = y. If μ is ergodic, then τ (Id) is just the density of points of almost all ω ∈ . In this case, we can conclude from the discussion in Section 5 that the discrete spectrum is absent. The necessary sequence fn is defined by fn (ω, x) =
1 χ B (x), vol(Bn ) n
where Bn is the ball in Rd around the origin with radius n, χ denotes the characteristic function and vol stands for Lebesgue measure. In fact, assuming ergodicity of μ together with aperiodicity, i.e. freeness, we can even conclude from Section 5 that the von Neumann algebra of random operators is a factor of type I I1 . Of course, in the ergodic case the results of Section 5 can be applied. They give almost sure constancy of the spectral components and the possibility to express the spectrum of a random operator A with the help of the measure ρ A defined there by ρ A (ϕ) = τ (ϕ(A)) for continuous ϕ on R with compact support. Theorem 7.1 Let (Aω ) be a selfadjoint random operator in the setting discussed in this section and assume that μ is ergodic. Then there exists ⊂ of full measure and subsets of the real numbers and • , where • ∈ {disc, ess, ac, sc, pp}, such that for all ω ∈ σ (Aω ) =
σ• (Aω ) = •
and
for any • = disc, ess, ac, sc, pp. Moreover, disc = ∅ and coincides with the topological support of ρ A . In the situation of the theorem it is also possible to calculate the distribution function of ρ A by a limiting procedure. Details are discussed in the literature cited above, see [63, 64]. Here, we mention the following results for so called finite range operators (Aω ): Denote by | · | the number of elements of a set and by Aω | Bn the restriction of Aω to ω ∩ Bn . Define the measures μnω on R by C0 (R) ϕ → μnω (ϕ) :=
1 tr ϕ(Aω | Bn ). |Bn ∩ ω|
These are just the measures associated to the eigenvalue counting functions studied so far, i.e. Nωn (λ) := μnω (] − ∞, λ[) =
|{i : λi (Aω | Bn ) < λ}| |Bn ∩ ω|
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with the (ordered) eigenvalues λi (Aω | Bn ) of Aω | Bn . Set D := τ (Id). Then, the measures μnω converge for n → ∞ vaguely to the measure ϕ →
1 1 τ (ϕ(A)) = ρ A (ϕ) D D
for almost every ω ∈ . If (, T) is uniquely ergodic, the convergence holds for all ω ∈ . Assuming further regularity, one can even show convergence of the corresponding distribution functions with respect to the supremum norm. In any case there exists a distribution function N A such that limn→∞ Nωn (λ) = N A (λ) = D1 ρ A (] − ∞, λ[) exists almost surely at all continuity points of N A . Note that amenability is not an issue here since the group Rd is abelian. In fact, instead of the balls Bn we could also consider rather general van Hove sequences. Let us finish this section by giving an explicit example of what may be called a nearest neighbour Laplacian in the context of a Delone set ω. For x ∈ ω, define the Voronoi cell V (x) of x by V (x) := { p ∈ Rd : p − x p − y for all y ∈ ω }.
Then, it is not hard to see that V (x) is a convex polytope for every x ∈ ω. The operator Aω is then defined via its matrix elements by Aω (x, y) = 1 if V (x) and V (y) share a (d − 1)-dimensional face and Aω (x, y) = 0 otherwise.
8 Percolation Models In this section we shortly discuss how to fit percolation operators, more precisely site percolation operators, in our framework. In fact, edge percolation or mixed percolation could be treated along the same lines. Theoretical physicists have been interested in Laplacians on percolation graphs as quantum mechanical Hamiltonians for quite a while [17, 25, 26, 46]. Somewhat later several computational physics papers where devoted to the numerical analysis of spectral properties of percolation Hamiltonians, see e.g [42–45, 81]. More recently there was a series of rigorous mathematical results on percolation models [14, 55, 69, 89, 90]. We have to identify the abstract quantities introduced in the abstract setting in the context of percolation. A graph G may be equivalently defined by its vertex set X and its edge set E, or by its vertex set X and the distance function d : X × X → {0} ∪ N. We choose the second option and tacitly identify the graph with its vertex set. In particular, each graph X gives naturally rise to the so called adjacency matrix A(X) : X × X −→ {0, 1} defined by A(X)(x, y) = 1 if d(x, y) = 1 and A(X)(x, y) = 0 otherwise. This adjacency matrix can be considered to be an operator on 2 (X).
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We assume that the vertex set of the graph is countable. Let be a group acting freely on X such that the quotient X/ is a finite graph, i.e. the -action is quasi-transitive. The associated probability space is given by = {0, 1} X , with the σ -field defined by the finite-dimensional cylinder sets. For simplicity let us consider only independent, identically distributed percolation on the vertices of X. The statements hold analogously for correlated site or bond percolation processes under appropriate ergodicity assumptions. Thus we are given a sequence of i.i.d. random variables ωx : → {0, 1}, for x ∈ X, with distribution measure pδ1 + (1 − p)δ0 . The measure P on the probability space is given by the product P = X ( pδ1 + (1 − p)δ0 ). The groupoid is given by G := × g = (ω, γ ) and the G -space by X := × X.
The operation of the groupoid is the same as defined in (3) and (4), the projection π is again given by π(ω, x) := ω for all (ω, x) ∈ × X, and the map J is the same as described on page 7 for the model (RSM), where x now stands for a vertex of the graph X. Define for ω ∈ the random subset X(ω) := {x ∈ X | ωx = 1} of X. Defining α by α ω := δ X(ω) for all ω ∈ we obtain a random variable (X , α). Similarly as in the case of random Schrödinger operators on manifolds we define the transverse function ν on the groupoid by setting ν ω equal to the counting measure m on the group for every ω ∈ and the ν-invariant measure μ equal to P. We next show that we are in an admissible setting according to Definition 2.6. As for the countability condition in this definition, we proceed as follows: By the countability of the vertex set of the graph X the σ -field BX is countably generated. Since B is generated by the finite dimensional cylinder sets, it is countably generated as well. Denote by D a fundamental domain of the covering graph X and set u0 (ω, x) := χD (x). Then clearly u0 (g−1 p)dν π( p) (g) = u0 (ω, γ x) = 1 for all ω ∈ , x ∈ X. G π( p)
γ ∈
This shows the second condition of Definition 2.6. The corresponding random operators are given by families Hω : 2 (X, α ω ) → 2 (X, α ω ), ω ∈ , satisfying a measurability and a boundedness assumption as well as the equivariance condition ∗ Hω = U (ω,γ ) Hγ −1 ω U (ω,γ ) −1
for ω ∈ and γ ∈ , where U (ω,γ ) : 2 (X, α γ ω ) → 2 (X, α ω ) is the unitary operator mapping φ to φ(γ −1 ·). A special example of such a random operator
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is given by the family (Aω )ω , where each Aω is the adjacency matrix of the induced subgraph of X generated by the vertex set X(ω). We next show that the functional τ (H) := E{tr(Mu0 Hω )} defined in Theorem 4.2 is a trace. As (a) of this Theorem holds by general arguments, we just have to show that the freeness condition (14) is satisfied. This can be shown as follows: Since acts freely, γ x = x for all x ∈ X, γ ∈ \ { }. Thus the set {ω | ωγ x = ωx } has measure smaller than one for all x ∈ X. If the graph is infinite, this poses infinitely many independent conditions on the elements in γ := {ω | γ ω = ω}. Therefore, γ has measure zero for each γ ∈ \ { }. As is countable, γ ∈\{ } γ has P-measure zero as well. Thus the desired freeness statement follows. To show that an associated operator family ω → Hω has almost surely no discrete spectrum for infinite , we define the sequence of functions fn by fn (ω, γ ) :=
1 χ I (γ ). |In | n
Here, In is an exhaustion of the infinite group . The sequence ( fn ) satisfies property (21). Thus, Lemma 5.6 and Corollary 5.9 hold. In fact, the above exhaustion and freeness properties allow us to apply Theorem 5.11 and conclude that the von Neumann algebra is of type I I. More precisely τ (Id) = E{tr(χD )} = p|D| < ∞ shows that the type is I I1 . Besides the discrete, essential, absolutely continuous, singular continuous, and pure point spectrum, σdisc , σess , σac , σsc , σ pp , the set σ f in consisting of eigenvalues which posses an eigenfunction with finite support is a quantity which may be associated with the whole family Hω , ω ∈ . The following two theorems hold for a class of percolation Hamiltonians (Hω )ω which are obtained from a deterministic finite hopping range operator by a percolation process, cf. [89]. In particular the class contains the adjacency operator (Aω )ω introduced above. Recall from Section 5 that the measure ρ H on R is given by ρ H (ϕ) = τ (ϕ(H)) for continuous ϕ on R with compact support. Theorem 8.1 There exists an ⊂ of full measure and subsets of the real numbers and • , where • ∈ {disc, ess, ac, sc, pp, f in}, such that for all ω ∈ σ (Hω ) =
and
σ• (Hω ) = •
Groupoids, von Neumann algebras and the IDS
33
for any • = disc, ess, ac, sc, pp, f in. Moreover, the almost-sure spectrum coincides with the topological support of ρ H . If is infinite, disc = ∅. If the group acting on X is amenable it was shown in [89] that property (P5), too, holds for the family (Hω )ω . More precisely, it is possible to construct the measure ρ H by an exhaustion procedure using the finite volume eigenvalue counting functions Nωn (λ). These are defined by the formula |{i ∈ N | λi (Hωn ) < λ}| , |In | · |D| where In is a tempered Følner sequence, An = γ ∈In γ D, and Hωn is the restriction of the operator Hω to the space 2 (X(ω) ∩ An ). Nωn (λ) :=
Theorem 8.2 Let be amenable and {In } be a tempered Følner sequence of . Then there exists a subset ⊂ of full measure and a distribution function N H , called integrated density of states, such that for all ω ∈ lim Nωn (λ) = N H (λ),
n→∞
(35)
at all continuity points of N H . N H is related to the measure ρ H via the following trace formula N H (λ) =
ρ H (] − ∞, λ[) . |D |
(36)
Appendix A: Some Direct Integral Theory The aim of this appendix is to prove the following lemma and to discuss some of its consequences. By standard direct integral theory [28], the lemma 2 is essentially equivalent ⊕ 2 ω toω the statement that L (X , μ ◦ α) is canonically isomorphic to L (X , α ) dμ(ω). Throughout this section let a measurable groupoid (G , ν, μ) and a random variable (X , α) satisfying condition (6) on the associated G -space be given. Lemma A.1 There exists a measurable function N : −→ N0 ∪ {∞}, a sequence (g(n) )n of measurable functions on X and a subset of of full measure satisfying the following: – – –
(gω(n) : 1 n N(ω)) is an orthonormal basis of L2 (X ω , α ω ) for every ω ∈ . gω(n) = 0 for n > N(ω), ω ∈ . gω(n) = 0 for ω ∈ / .
Proof Let D be a countable generator of the σ -algebra of X such that μ ◦ α(D) < ∞ for every D ∈ D. Such a D exists by condition (6). Let ( f (n) )n∈N be
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the family of characteristic functions of sets in D. By assumption on D, we infer that ( f (n) ) are total in L2 (X , μ ◦ α). By the Fubini Theorem, there exist a set of full measure, such that fω(n) belongs to L2 (X ω , α ω ) for every n ∈ N and every ω ∈ . As D generates the σ -algebra of X and X ω is equipped with the induced σ -algebra, we infer that ( fω(n) )n∈N is total in L2 (X , α ω ) for every ω ∈ . Now, define for n ∈ N the function h(n) ∈ L2 (X , μ ◦ α) by setting h(n) ( p) = (n) f ( p) if π( p) ∈ and h(n) ( p) = 0, otherwise. Applying the Gram–Schmidt-orthogonalization procedure to (h(n) ω )n∈N simultaneously for all ω ∈ , we find N : −→ N0 ∪ {∞} and gω(n) as desired. (This simultaneous Gram-Schmidt procedure is a standard tool in direct integral theory, see [28] for details.) The proof of the Lemma is finished.
Proposition A.2 Let (Aω ) be a family of bounded operators L2 (X ω , α ω ) −→ L2 (X ω , α ω ) such that ω → fω , Aω gω ω is measurable for arbitrary f, g ∈ L2 (X , μ ◦ α).
Aω : (37)
Then, Ah : X −→ C, (Ah)( p) ≡ (Aπ( p) hπ( p) )( p) is measurable for every h : X −→ C measurable with hω ∈ L2 (X ω , α ω ) for every ω ∈ . Proof Let h be as in the assumption. Invoking suitable cutoff procedures, we can assume without loss of generality h ∈ L2 (X , μ ◦ α) as well as Aω C for all ω ∈ ,
(38)
for a suitable C independent of ω. Obviously, (37) implies that ω → fω , Aω gω ω is measurable for every f, g : X −→ C measurable with fω , gω ∈ L2 (X ω , α ω ) for every ω ∈ . Thus, with g(n) , n ∈ N, as in the previous lemma, we see that ω → gω(n) , Aω hω is measurable for every n ∈ N. In particular, (n) (n) X −→ C, p → gπ( p) , Aπ( p) hπ( p) π( p) g ( p) is measurable for every n ∈ N.
As (gω(n) : 1 n N(ω)) is an orthonormal basis in L2 (X ω , α ω ) for every ω ∈ and gω(n) = 0 for n > N(ω) and ω ∈ , we have (Aω hω )( p) =
∞ gω(n) , (Aω hω )ω g(n) ( p) n=1
for almost every ω ∈ . Note that the last equality holds in the L2 (X ω , ω α )-sense. By (38) and the Fubini Theorem, this implies (Aω hω )( p) = ∞ (n) (n) 2 n=1 gω , (Aω hω )ω g ( p) in the sense of L (X , μ ◦ α) and the desired measurability follows.
Groupoids, von Neumann algebras and the IDS
35
Proposition A.3 Let C > 0 and (Aω ) be a family of operators Aω : L2 (X ω , α ω ) −→ L2 (X ω , α ω ) with Aω C for every ω ∈ and p → (Aπ( p) fπ( p) )( p) measurable for every f ∈ L2 (X , μ ◦ α). Let A : L2 (X , μ ◦ α) −→ L2 (X , μ ◦ α), (A f )( p) = (Aπ( p) fπ( p) )( p) be the associated operator. Then, Aω = 0 for μ-almost every ω ∈ if A = 0.
Proof Choose g(n) , n ∈ N, as in Lemma A.1 and let E be a countable dense subset of L2 (, μ). For f ∈ L2 (X , μ ◦ α) with f ( p) = g(n) ( p)ψ(π( p)) for n ∈ N and ψ ∈ E , we can then calculate (n) 0 = A f = ( p → ψ(π( p))(Aπ( p) gπ( p) )( p)).
As E is dense and countable, we infer, for μ-almost all ω ∈ , Aω gω(n) = 0 for all n ∈ N. This proves the statement, as (gω(n) : n ∈ N) is total in L2 (X ω , α ω ) for almost every ω ∈ .
Corollary A.4 Let (Aω ) and (Bω ) be random operators with associated operators A and B, respectively. Then A = B implies (Aω ) ∼ (Bω ). Proof This is immediate from the foregoing proposition.
Appendix B: A Proposition from Measure Theory In this appendix we give a way to calculate the point part of a finite measure on R. Recall that a measure is called continuous if it does not have a point part. We start with the following Lemma. Lemma B.1 Let μ be a continuous finite measure on R. Then limn→∞ μ(In ) = 0 for every sequence (In ) of open intervals whose lengths tend to zero. Proof Assume the contrary. Then there exists a sequence of open intervals (In ) with |In | → 0 and a δ > 0 with μ(In ) δ, n ∈ N. For each n ∈ N choose an arbitrary xn ∈ In . If the sequence (xn ) were unbounded, one could find a subsequence (Ink )k∈N of (In ) consisting disjoint intervals. This would imply the contradic of pairwise ∞ tion μ(R) ∞ μ(I ) n k k=1 k=1 δ = ∞. Thus, the sequence (xn ) is bounded and therefore contains a converging subsequence. Without loss of generality we assume that xn → x for n → ∞. For every open interval I containing x we then have μ(I) μ(In ) for n large enough. This gives μ(I) δ for every such interval. From Lebesgue Theorem, we then infer μ({x}) δ, contradicting the continuity of μ.
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Proposition B.2 Let μ be a finite measure on R with point part μ pp and continuous part μc . Then, for every B ⊂ R, μ pp (B) is given by μ pp (B) = lim lim
sup
k→∞ n→∞ |J|n−1 ,J∈J k
μ(B ∩ J).
Proof Obviously, the limits on the right hand side of the formula make sense. We show two inequalities: “”: Let {xi } be a countable subset of R with μ pp = i μ({xi })δxi , where δx denotes the point measure with mass one at x. Then, we have μ pp (B) = xi ∈B μ({xi }). For every > 0, we can then find a finite subset B of {xi : xi ∈ B} with number of elements #B and μ pp (B) + x∈B μ({x}). This easily gives μ({x}) + μ(B ∩ I) μ pp (B) + x∈B
for suitable J ∈ J #B of arbitrary small Lebesgue measure. As is arbitrary, the desired inequality follows. “”: By the foregoing lemma, we easily conclude for every k ∈ N that limn→∞ sup J∈J k ,|J|n−1 μc (B ∩ J) = 0. Combining this with the obvious inequality μ pp (B) μ pp (B ∩ J) valid for arbitrary measurable B, J ⊂ R, we infer μ pp (B) lim lim
sup
μ pp (B ∩ J)
= lim lim
sup
μ pp (B ∩ J) + lim lim
= lim lim
sup
μ(B ∩ J).
k→∞ n→∞ |J|n−1 ,J∈J k k→∞ n→∞ |J|n−1 ,J∈J k k→∞ n→∞ |J|n−1 ,J∈J k
sup
k→∞ n→∞ |J|n−1 ,J∈J k
μc (B ∩ J)
This finishes the proof.
Appendix C: Uniqueness Lemma for the Laplace Transform The results in [76] use heavily the Laplace transform techniques developed in papers by Pastur and Šubin [71, 84]. In the present paper only the uniqueness lemma is used. In the literature the uniqueness lemma for the Laplace transform is stated mostly for finite measures (e.g., Theorem 22.2 in [15]). For the convenience of the reader we show how to adapt the uniqueness result to our case, where the distribution function N H is unbounded. Lemma C.1 Let f1 , f2 : ]0, ∞[→ R be monotonously increasing function with limλ 0 f1 (λ) = limλ 0 f2 (λ) = 0. Let the integrals ∞ e−tλ df j(λ), j = 1, 2, t > 0 (39) 0
Groupoids, von Neumann algebras and the IDS
be finite and moreover
∞
37
e−tλ df1 (λ) =
0
∞
e−tλ df2 (λ)
(40)
0
for all positive t. Then the sets of continuity points of f1 and f2 coincide and for λ0 in this set we have f1 (λ0 ) = f2 (λ0 ). Proof Choose s > 0 arbitrary. The measures ∞ μ j(g) := g(λ) e−sλ df j(λ) ∞
e−sλ df j(λ) < ∞ by assumption. Since ∞ μ j(e−t· ) = e−(t+s)λ df j(λ) = f˜j(t + s),
are finite, since μ j(1) =
(41)
0
0
(42)
0
we have, by assumption, that the Laplace transforms of the measures μ j coincide for all t > 0: μ1 (e−t· ) = μ2 (e−t· ).
(43)
As we are dealing with finite measures, the Theorem 22.2 in [15] implies μ1 = μ2 . We consider E e−sλ df j(λ) (44) μ j([0, E]) = 0
as a sequence of integrals depending on the parameter s → 0. Since e−s· : [0, λ0 ] → [0, 1]
(45)
converges uniformly and monotonously to the constant function 1, we conclude by Beppo Levi’s theorem λ0 df j(λ). (46) lim μ j([0, λ0 ]) = s 0
0
For a continuity point λ0 of f1 we have λ0 df1 (λ) = f1 (λ0 ),
(47)
0
which implies f1 (λ0 ) = f2 (λ0 ).
Corollary C.2 Under the assumptions of the Lemma C.1 we have ∞ ∞ g(λ) d f1 (λ) = g(λ) d f2 (λ) 0
0
for all continuous functions g with compact support.
(48)
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Acknowledgements It is a pleasure to thank W. Kirsch and P. Stollmann for various stimulating discussions on random operators and hospitality at Ruhr-Universität Bochum, respectively, TU Chemnitz. This work was supported in part by the DFG through the SFB 237, the Schwerpunktprogramm 1033, and grants Ve 253/1 & Ve 253/2 within the Emmy-Noether Programme.
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Math Phys Anal Geom (2007) 10:43–64 DOI 10.1007/s11040-007-9020-9
Well-posedness for Semi-relativistic Hartree Equations of Critical Type Enno Lenzmann
Received: 1 April 2006 / Accepted: 1 December 2006 / Published online: 26 May 2007 © Springer Science + Business Media B.V. 2007
Abstract We prove local and global well-posedness for semi-relativistic, non√ linear Schrödinger equations i∂t u = − + m2 u + F(u) with initial data in H s (R3 ), s 1/2. Here F(u) is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing F(u), which arise in the quantum theory of boson stars, we derive global-in-time existence for small initial data, where the smallness condition is expressed in terms of the L2 -norm of solitary wave ground states. Our proof of well-posedness does not rely on Strichartz type estimates. As a major benefit from this, our method enables us to consider external potentials of a quite general class. Keywords Well-posedness · Cauchy problem · Semi-relativistic Hartree equation · Boson stars Mathematics Subject Classifications (2000) 35Q40 · 35Q55 · 47J35
1 Introduction In this paper we study the Cauchy problem for nonlinear Schrödinger equations with kinetic energy part originating from special relativity. That is, we consider the initial value problem for (t, x) ∈ R1+3 , (1) i∂t u = − + m2 u + F(u),
E. Lenzmann (B) Department of Mathematics, Massachusetts Institute of Technology Building 2, Room 230, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA e-mail:
[email protected]
44
E. Lenzmann
where u(t, x) is complex-valued, m 0 denotes √ a given mass parameter, and F(u) is some nonlinearity. Here the operator − + m2 is defined via its symbol ξ 2 + m2 in Fourier space. Such “semi-relativistic” equations have (though not Lorentz covariant in general) interesting applications in the quantum theory for large systems of self-interacting, relativistic bosons. Equation (1) arises, for instance, as an effective description of boson stars, see, e. g., [4, 5, 12], where F(u) is a focusing Hartree nonlinearity given by λ F(u) = ∗ |u|2 u, (2) |x| with some constant λ < 0 and ∗ as convolution. Motivated by this physical example with focusing self-interaction of Coulomb type, we address the Cauchy problem for (1) and a class of Hartree nonlinearities including (2). In fact, we shall prove well-posedness for initial data u(0, x) = u0 (x) in H s = H s (R3 ), s 1/2; see Theorems 1–3 below. Let us briefly point out a decisive feature of the example cited in (2) above. Apart from its physical relevance, the nonlinearity given by (2) leads to an L2 -critical equation as indicated by the fact that the coupling constant λ has to be dimensionless. In consequence of this, L2 -smallness of the initial datum enters as a sufficient condition for global-in-time solutions. More precisely, we derive for u0 ∈ H s , s 1/2, the following criterion implying global well-posedness 2 |u0 (x)| dx < |Q(x)|2 dx. (3) R3
R3
This condition holds irrespectively of the parameter m 0 in (1); see Theorem 2 below. Here Q ∈ H 1/2 is a positive solution (ground state) for the nonlinear equation √ λ (4) − Q + ∗ |Q|2 Q = −Q, |x| which gives rise to solitary wave solutions, u(t, x) = eit Q(x), for (1) with m = 0. In fact, it can be shown that criterion (3) guaranteeing global-in-time solutions in the focusing case is optimal in the sense that there exist solutions, u(t), with u0 22 > Q22 , which blow up within finite time; see [7]. Physically, this blowup phenomenon indicates “gravitational collapse” of a boson star whose mass exceeds a critical value. Furthermore, criterion (3) can be linked with established results as follows. First, it is reminiscent to a well-known condition derived in [16] for global well-posedness of nonrelativistic Schrödinger equations with focusing, local nonlinearity (see also [13] for Hartree nonlinearities). Second, criterion (3) is in accordance with a sufficient stability condition proved in [12] for the related time-independent problem (i. e., a static boson star); see [6] for more details concerning known results on Hartree equations. We now give an outline of our methods. The proof of well-posedness presented below does not rely on Strichartz (i. e., space-time) estimates for the
Well-posedness for semi-relativistic Hartree equations of critical type √
45
propagator, e−it −+m , but it employs sharp estimates (e. g., Kato’s inequality (17) below) to derive local Lipschitz continuity of L2 -critical nonlinearities of Hartree type. Local well-posedness then follows by standard methods for abstract evolution equations. Furthermore, global well-posedness is derived by means of a priori estimates and conservation of charge and energy whose proof requires a regularization method. This paper is organized as follows. –
– –
2
In Section 2 we introduce a class of critical Hartree nonlinearities including (2). First, we state Theorems 1 and 2 that establish local and global well-posedness in energy space H 1/2 for this class of nonlinearities. In Theorem 3 we extend these results to H s , for every s 1/2. Finally, external potentials are included, i. e., we consider i∂t u = − + m2 + V u + F(u), (5) where V : R3 → R is given. In Theorem 4 we state local and global wellposedness for (5) with initial datum u(0, x) = u0 (x) in the appropriate energy space. Assumption 1 imposed below on V is considerably weak √ and implies that − + m2 + V defines a self-adjoint operator via its form sum. The main results (i. e., Theorems 1–4) are proved in Section 3. Appendices A and B contain some useful facts about fractional derivatives, a discussion of ground states, and some details of the proofs.
Notation Throughout this text, the symbol ∗ stands for convolution on R3 , i. e., ( f ∗ g)(x) := f (x − y)g(y) dy, R3
and L p (R3 ), with norm · p and 1 p ∞, denotes the usual Lebesgue L p -space of complex-valued functions on R3 . Moreover, L2 (R3 ) is associated with the scalar product defined by u(x)v(x) dx. u, v := R3
For s ∈ R and 1 p ∞, we introduce fractional Sobolev spaces (see, e. g., [1]) with their corresponding norms according to H s, p (R3 ) := u ∈ S (R3 ) : u Hs, p := F −1 [(1 + ξ 2 )s/2 F u] p < ∞ , where F denotes the Fourier transform in S (R3 ) (space of tempered distributions). In our analysis, the Sobolev spaces H s (R3 ) := H s,2 (R3 ), with norms · Hs := · Hs,2 , will play an important role.
46
E. Lenzmann
In addition to the common L p -spaces, we also make use of local L p -space, p with 1 p ∞, and weak (or Lorentz) spaces, Lw (R3 ), with 1 < p < ∞ and corresponding norms given by
p Lloc (R3 ),
u p,w := sup ||−1/ p
|u(x)| dx,
where 1/ p + 1/ p = 1 and denotes an arbitrary measurable set with p Lebesgue measure || < ∞; see, e. g., [11] for this definition of Lw -norms. p p 3 3 Note that L (R ) L
w (R ), for 1 < p < ∞. 3 ∂x2i stands for the usual Laplacian on R3 , and The symbol = i=1 √ 2 is defined via its symbol ξ 2 + m2 in Fourier space. Besides the − + m√ 2 operator − + m , we also employ Riesz and Bessel potentials of order s ∈ R, which we denote by (−)s/2 and (1 − )s/2 , respectively; see also Appendix A. Except for theorems and lemmas, we often use the abbreviations L p = p p p L (R3 ), Lw = Lw (R3 ), and H s = H s (R3 ). In what follows, a b always denotes an inequality a cb , where c is an appropriate positive constant that can depend on fixed parameters.
2 Main Results We consider the following initial value problem ⎧ ⎨
−μ|x| λe − + m2 u + ∗ |u|2 u, |x| ⎩ u(0, x) = u0 (x), u : [0, T) × R3 → C, i∂t u =
(6)
where m 0, λ ∈ R, and μ 0 are given parameters. Note that |λ| could be absorbed in the normalization of u(t, x), but we shall keep λ explicit in the following; see also [4] for this convention. Our particular choice of the Hartree type nonlinearities in (6) is motivated by the fact that (6) can be rewritten as the following system of equations
√ i∂t u = − + m2 u + u, (μ2 − ) = 4π λ|u|2 , u(0, x) = u0 (x),
(7)
where = (t, x) is real-valued and (t, x) → 0 as |x| → ∞. This reformulation stems from the observation that e−μ|x| /4π |x| is the Green’s function of (μ2 − ) in R3 ; see Appendix A. System (7) now reveals the physical intuition behind (6), i. e., the function u(t, x) corresponds to a “positive energy wave” with instantaneous self-interaction that is either of Coulomb or Yukawa type depending on whether μ = 0 or μ > 0, respectively. To prove well-posedness we shall, however, use formulation (6) instead, and we refer to facts from potential theory only when estimating the nonlinearity.
Well-posedness for semi-relativistic Hartree equations of critical type
47
2.1 Local Well-posedness Let us begin with well-posedness in energy space, i. e., we assume that u0 ∈ H 1/2 holds in (6). The following Theorem 1 establishes local well-posedness in the strong sense, i. e., we have existence and uniqueness of solutions, their continuous dependence on initial data, and the blow-up alternative. The precise statements is as follows. Theorem 1 Let m 0, λ ∈ R, and μ 0. Then initial value problem (6) is locally well-posed in H 1/2 (R3 ). This means that, for every u0 ∈ H 1/2 (R3 ), there exist a unique solution u ∈ C0 [0, T); H 1/2 (R3 ) ∩ C1 [0, T); H −1/2 (R3 ) , and it depends continuously on u0 . Here T ∈ (0, ∞] is the maximal time of existence, where we have that either T = ∞ or T < ∞ and limt↑T u(t) H1/2 = ∞ holds. Remark Continuous dependence means that the map u0 → u ∈ C0 (I; H 1/2 ) is continuous for every compact interval I ⊂ [0, T). 2.2 Global Well-posedness The local-in-time solutions derived in Theorem 1 extend to all times, by virtue of Theorem 2 below, provided that either λ 0 holds (corresponding to a repulsive nonlinearity) or λ < 0 and the initial datum is sufficiently small in L2 . Theorem 2 The solution of (6) derived in Theorem 1 is global in time, i. e., we have that T = ∞ holds, provided that one of the following conditions is met. (1) λ 0. (2) λ < 0 and u0 22 < Q22 , where Q ∈ H 1/2 (R3 ) is a strictly positive solution (ground state) of √ − Q +
λ 2 ∗ |Q| Q = −Q. |x|
Moreover, we have the estimate Q22 >
(8)
4 . π |λ|
Remarks (1) Notice that condition (2) implies global well-posedness for (6) irrespectively of m 0.
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E. Lenzmann
(2) For a > 0, the function Qa (x) = a3/2 Q(ax) yields another ground state with Qa 2 = Q2 that satisfies √ λ − Qa + ∗ |Qa |2 Qa = −aQa . (9) |x| We refer to Appendix B for a discussion of Q ∈ H 1/2 . (3) Condition (2) resembles a well-known criterion derived in [16] for globalin-time existence for L2 -critical nonlinear (nonrelativistic) Schrödinger equations. (4) It is shown in [7] that criterion (3) for having global-in-time solutions in the focusing case is optimal in the sense that there exist solutions, u(t), with u0 22 > Q22 , which blow up within finite time. 2.3 Higher Regularity We now turn to well-posedness of (6) in H s , for s 1/2, which is settled by the following result. Theorem 3 For every s 1/2, the conclusions of Theorems 1 and 2 hold, where H 1/2 (R3 ) and H −1/2 (R3 ) in Theorem 1 are replaced by H s (R3 ) and H s−1 (R3 ), respectively. Remark For s = 1, this result is needed in [4] for a rigorous derivation of (6) with Coulomb type self-interaction (i. e., μ = 0) from many-body quantum mechanics. 2.4 External Potentials Now we consider the following extension of (6) that arises by adding an external potential: ⎧ −μ|x| λe ⎨ 2 2 i∂t u = ∗ |u| 0 u, − + m + V u + (10) |x| ⎩ u : [0, T) × R3 → C, u(0, x) = u0 (x), where m 0, λ ∈ R, μ 0 are given parameters, and V : R3 → R denotes a preassigned function that meets the following condition. Assumption 1 Suppose that V = V+ + V− holds, where V+ and V− are realvalued, measurable functions with the following properties. (1) V+ ∈ L1loc (R3 ) and V+ 0. √ (2) V− is −-form bounded with relative bound less than 1, i. e., there exist constants 0 a < 1 and 0 b < ∞, such that √ |u, V− u | au, − u + b u, u holds for all u ∈ H 1/2 (R3 ).
Well-posedness for semi-relativistic Hartree equations of critical type
49
√ We mention that Assumption 1 implies that − + m2 + V leads to a self-adjoint operator on L2 via its form sum. Furthermore, the energy space given by X := u ∈ H 1/2 (R3 ) :
R3
V(x) |u(x)| dx < ∞ 2
(11)
∗ is complete with norm · X , and √ its dual space is denoted by X . We refer to 2 Section 3.4 for more details on − + m + V and X. After this preparing discussion, the extension of Theorems 1 and 2 for the initial value problem (10) can be now stated as follows.
Theorem 4 Let m 0, λ ∈ R, μ 0, and suppose that V satisfies Assumption 1. Then (10) is locally well-posed in the following sense. For every u0 ∈ X, there exists a unique solution u ∈ C0 ([0, T); X) ∩ C1 ([0, T); X ∗ ), and it depends continuously on u0 . Here T ∈ (0, ∞] is the maximal time of existence such that either T = ∞ or T < ∞ and limt↑T u(t) X = ∞ holds. Moreover, we have that T = ∞ holds, if one of the following conditions is satisfied. (1) λ 0. (2) λ < 0 and u0 22 < (1 − a)Q22 , where Q is the ground state mentioned in Theorem 2 and 0 a < 1 denotes the relative bound introduced in Assumption 1. Remarks (1) To meet Assumption 1 for V+ , we can choose, for example, V+ (x) = |x|β , with β 0; or even super-polynomial growth such as V+ (x) = e|x| . Note that Assumption 1 for V− is satisfied (by virtue of Sobolev inequalities), if |V− (x)|
c +d |x|1−ε
holds for some 0 < ε 1 and constants 0 c, d < ∞. In fact, we can even admit ε = 0 provided that c < 2/π holds, as can be seen from inequality (17) below. (2) Since we avoid using Strichartz estimates in our well-posedness proof below, we only need that V+ belongs to L1loc . In contrast to this, compare, for instance, the conditions on V in [17] for deriving Strichartz type estimates for e−it(−+V) in order to prove local well-posedness for (nonrelativistic) nonlinear Schrödinger equations with external potentials.
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3 Proof of the Main Results In this section we prove Theorems 1–4. Although Theorem 4 generalizes Theorems 1 and 2, we postpone the proof of Theorem 4 to the final part of this section. 3.1 Proof of Theorem 1 (Local Well-posedness) Let u0 ∈ H 1/2 be fixed. In view of (6) we put A :=
− + m2
F(u) :=
and
λe−μ|x| ∗ |u|2 u, |x|
and we consider the integral equation t −it A u(t) = e u0 − i e−i(t−τ )A F(u(τ )) dτ.
(12)
(13)
0
Here u(t) is supposed to belong to the Banach space YT := C0 [0, T); H 1/2 (R3 ) ,
(14)
with some T > 0 and norm uYT := supt∈[0,T) u(t) H1/2 . The proof of Theorem 1 is now organized in two steps as follows. Step 1: Estimating the Nonlinearity We show that the nonlinearity F(u) is locally Lipschitz continuous from H 1/2 into itself. This is main point of our argument for local well-posedness and it reads as follows. Lemma 1 For μ 0, the map J(u) := uous from H 1/2 (R3 ) into itself with
e−μ|x| |x|
∗ |u|2 u is locally Lipschitz contin-
J(u) − J(v) H1/2 (u2H1/2 + v2H1/2 )u − v H1/2 , for all u, v ∈ H 1/2 (R3 ). Proof (Of Lemma 1) We prove the claim for μ = 0 and μ > 0 in a common way, so let μ 0 be fixed. For s ∈ R, it is convenient to introduce Ds := (μ2 − )s/2 .
Note that due to the equivalence u2 + D1/2 u2 u H1/2 u2 + D1/2 u2 , it is sufficient to estimate the quantities I := J(u) − J(v)2
and
I I := D1/2 [J(u) − J(v)]2 ,
Well-posedness for semi-relativistic Hartree equations of critical type
51
where I is needed only if μ = 0. Using now the identity J(u) − J(v) = 1/2
e−μ|x| ∗ (|u|2 − |v|2 ) (u + v) + |x| −μ|x| e + ∗ (|u|2 + |v|2 ) (u − v) |x|
together with Hölder’s inequality (which we tacitly apply from now on), we find that I
−μ|x| e 2 2 + − |v| ) (u + v) ∗ (|u| |x| 2 −μ|x| e 2 2 ∗ (|u| + + |v| ) (u − v) |x| 2 −μ|x| e 2 2 |x| ∗ (|u| − |v| ) u + v3 + 6 −μ|x| e 2 2 ∗ (|u| + |v| ) u − v2 . + |x| ∞
(15)
Observing that e−μ|x| |x|−1 ∈ L3w holds, the first term of right-hand side of (15) can be bounded by means of the weak Young inequality (see, e. g., [11]) as follows −μ|x| −μ|x| e e 2 2 |x| ∗ (|u| − |v| ) |x| 6
|u|2 − |v|2 6/5 3,w
u + v3 u − v2 .
(16)
The second term in (15) can be estimated by noting that −μ|x| e 2 ∗ |u| |x|
sup ∞
y∈R3
R3
|u(x)|2 dx (−)1/4 u22 , |x − y|
(17)
which follows from the operator inequality |x − y|−1 π2 (−x−y )1/2 (see, e. g., [10, Section V.5.4]) and translational invariance, i. e., we use that x−y = x holds for all y ∈ R3 . Combining now (16) and (17) we find that I u + v23 u − v2 + u2H1/2 + v2H1/2 u − v2 u2H1/2 + v2H1/2 u − v H1/2 , where we make use of the Sobolev inequality u3 u H1/2 in R3 .
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It remains to estimate I I. To do so, we appeal to the generalized (or fractional) Leibniz rule (see Appendix A) leading to −μ|x| 1/2 e 2 2 ∗ (|u| − |v| ) (u + v) I I D + |x| 2 −μ|x| 1/2 e 2 2 ∗ (|u| + |v| ) (u − v) + D |x| 2 1/2 e−μ|x| ∗ (|u|2 − |v|2 ) D u + v3 + |x| 6 −μ|x| e 2 2 1/2 + |x| ∗ (|u| − |v| ) D (u + v)2 + ∞ −μ|x| 1/2 e 2 2 u − v3 + ∗ (|u| + |v| + D |x| 6 −μ|x| e 2 2 1/2 + |x| ∗ (|u| + |v| ) D (u − v)2 . ∞
(18)
−μ|x|
By referring to Appendix A, we notice that e4π |x| ∗ f can be expressed as D−2 f = (μ2 − )−1 f in R3 (here f ∈ S (R3 ) is initially assumed, but our arguments follow by density). Thus, the first term of the right-hand side of (18) is found to be 1/2 e−μ|x| 2 2 D−3/2 (|u|2 − |v|2 )6 D − |v| ) ∗ (|u| |x| 6 μ G3/2 ∗ (|u|2 − |v|2 )6 μ G3/2 2,w |u|2 − |v|2 3/2 u + v3 u − v3 ,
(19)
where we use weak Young’s inequality together with the fact that D−3/2 f μ μ corresponds to G3/2 ∗ f with some G3/2 ∈ L2w (R3 ); see (42). The · ∞ -part of the second term occurring in (18) can be estimated by using the CauchySchwarz inequality and (17) once again: −μ|x| e 2 2 |x| ∗ (|u| − |v| )
∞
1 2 2 ∗ (|u| − |v| ) |x| sup y∈R3
∞
|u(x)| − |v(x)|2 dx |x − y| 2
R3
1 sup (u(x) + v(x)), (u(x) − v(x)) |x − y| 3 y∈R
Well-posedness for semi-relativistic Hartree equations of critical type
53
(−)1/4 (u + v)2 (−)1/4 (u − v)2 (u H1/2 + v H1/2 )u − v H1/2
(20)
The remaining terms in (18) deserve no further comment, since they can be estimated in a similar fashion to all estimates derived so far. Thus, we conclude that J(u) − J(v) H1/2 I + I I (u2H1/2 + v2H1/2 )u − v H1/2 and the proof of Lemma 1 is now complete.
Remarks (1) The proof of Lemma 1 relies on (17) in a crucial way. Employing just the Sobolev embedding H 1/2 ⊂ L2 ∩ L3 (in R3 ) together with the (non weak) −μ|x| Young inequality is not sufficient to conclude that e |x| ∗ |u|2 ∞ < ∞ whenever u ∈ H 1/2 . (2) The proof of Lemma 1 fails for “super-critical” Hartree nonlinearities J(u) = (|x|−α ∗ |u|2 )u, where 1 < α < 3. Thus, the choice α = 1 represents a borderline case when deriving local Lipschitz continuity in energy space H 1/2 . Step 2: Conclusion Returning to the proof of Theorem 1, we note that A defined in (12) gives rise to a self-adjoint operator L2 with domain H 1 . Moreover, its extension to H 1/2 , which we denote by A : H 1/2 → H −1/2 , generates a C0 -group of isometries, {e−it A }t∈R , acting on H 1/2 . Local well-posedness in the sense of Theorem 1 now follows by standard methods for evolution equations with locally Lipschitz nonlinearities. That is, existence and uniqueness of a solution u ∈ YT for the integral equation (13) is deduced by a fixed point argument, for T > 0 sufficiently small. The equivalence of the integral formulation (13) and the initial value problem (6), with u0 ∈ H 1/2 , as well as the blow-up alternative can also be deduced by standard arguments; see, e. g., [3, 14] for general theory on semilinear evolution equations. Finally, note that u ∈ C1 ([0, T); H −1 ) follows by (6) itself. The proof of Theorem 1 is now accomplished. 3.2 Proof of Theorem 2 (Global Well-posedness) The first step taken in the proof of Theorem 2 settles conservation of energy and charge that are given by E[u] := 1/2 u(x) − + m2 u(x) dx + R3
−μ|x| λe + 1/4 ∗ |u|2 (x) |u(x)|2 dx, |x| R3 N[u] := |u(x)|2 dx, R3
(21) (22)
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E. Lenzmann
respectively. After deriving the corresponding conservation laws (where proving energy conservation requires a regularization), we discuss how to obtain a-priori bounds on the energy norm of the solution. Step 1: Conservation Laws Lemma 2 The local-in-time solutions of Theorem 1 obey conservation of energy and charge, i. e., E[u(t)] = E[u0 ] and
N[u(t)] = N[u0 ],
for all t ∈ [0, T).
Proof (Of Lemma 2) Let u be a local-in-time solution derived in Theorem 1, and let T be its maximal time of existence. Since u(t) ∈ H 1/2 holds, we can ¯ and integrate over R3 . Taking then real parts yields multiply (6) by iu(t) d N[u(t)] = 0 for t ∈ [0, T), (23) dt which shows conservation of charge. At a formal level, conservation of energy follows by multiplying (6) with ˙¯ ∈ H −1/2 and integrating over space, but the paring of two elements u(t) of H −1/2 is not well-defined. Thus, we have to introduce a regularization procedure as follows; see also, e. g., [2, 8] for other regularization methods for nonlinear (nonrelativistic) Schrödinger equations. Let us define the family of operators Mε := (ε A + 1)−1 , for ε > 0, (24) √ where the operator A = − + m2 0 is taken from (12). Consider the sequences of embedded spaces
. . . H 3/2 → H 1/2 → H −1/2 → H −3/2 . . . It is easy to see (by using functional calculus) that the following properties hold. (a) For ε > 0 and s ∈ R, we have that Mε is a bounded map from H s into H s+1 . (b) Mε u Hs u Hs whenever u ∈ H s and s ∈ R. (c) For u ∈ H s and s ∈ R, we have that Mε u → u strongly in H s as ε ↓ 0. We shall use tacitly properties (a)–(c) in the following analysis. By means of Mε and noting that E ∈ C1 (H 1/2 ; R), we can compute in a welldefined way for t1 , t2 ∈ [0, T) as follows t2 ˙ dt E[Mε u(t2 )]− E[Mε u(t1 )] = E (Mε u), Mε u =
t1 t2 t1
Re AMε u+ F(Mε u),−iMε (Au+ F(u)) dt
Well-posedness for semi-relativistic Hartree equations of critical type
=
t2
t1
55
Im AMε u, Mε Au +F(Mε u), Mε Au + +AMε u, Mε F(u) +F(Mε u), Mε F(u) dt
=: t1
t2
fε (t) dt,
(25)
where we write u = u(t) for brevity and recall the definition of F from (12). We observe that the first term in fε (t) is the “most singular” part, i. e., if ε = 0 we would have pairing of two H −1/2 -elements. But for ε > 0 we can use the obvious fact that Mε A = AMε holds and conclude that Im AMε u, Mε Au = Im AMε u, AMε u = 0. Notice that this manipulation is well-defined, since AMε u and Mε Au are in H 1/2 whenever u ∈ H 1/2 . After some simple calculations, we find fε (t) to be of the form fε (t) = Im F(Mε u), Mε Au + AMε u, Mε F(u) + + F(Mε u), Mε F(u) = Im A1/2 F(Mε u), A1/2 Mε u + A1/2 Mε u, A1/2 Mε F(u) + + F(Mε u), Mε F(u) , Since Mε u → u strongly in H 1/2 as ε ↓ 0, we can infer, by Lemma 1, that lim fε (t) = Im A1/2 F(u), A1/2 u + A1/2 u, A1/2 F(u) + F(u), F(u) ε↓0
= Im (Real Number) = 0. To interchange the ε-limit with the t-integration in (25), we appeal to the dominated convergence theorem. That is, we seek for a uniform bound on fε (t). In fact, by using the Cauchy–Schwarz inequality and Lemma 1 again we find the following estimate | fε (t)| |A1/2 F(Mε u), A1/2 Mε u | + |A1/2 Mε u, A1/2 Mε F(u) | + + |F(Mε u), Mε F(u) | A1/2 F(Mε u)2 A1/2 Mε u2 +
+ A1/2 Mε u2 A1/2 Mε F(u)2 + F(Mε u)2 Mε F(u)2 u4H1/2 + u6H1/2 ,
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E. Lenzmann
for all ε > 0. Putting now all together leads to conservation of energy, i. e., we find for all t1 , t2 ∈ [0, T) that E[u(t2 )] − E[u(t1 )] = lim E[Mε u(t2 )] − E[Mε u(t1 )] ε↓0
= lim ε↓0
t2
t2
fε (t) dt =
t1
t1
lim fε (t) dt = 0. ε↓0
This completes the proof of Lemma 2.
Step 2: A Priori Bounds To fill the last gap towards the global well-posedness result of Theorem 2, we now discuss how to obtain a priori bounds on the energy norm. By the blowup alternative of Theorem 1, global-in-time existence follows from an a priori bound of the form u(t) H1/2 C(u0 ).
(26)
First, let us assume that λ 0 holds. Then, for all t ∈ [0, T), we find from Lemma 2 and (22) that (−)1/4 u(t)2 E[u(t)] = E[u0 ]. This implies together with charge conservation derived in Lemma 2, i. e., u(t)22 = N[u(t)] = N[u0 ]
(27)
an a priori estimate (26). Therefore condition (1) in Theorem 2 is sufficient for global existence. Suppose now a focusing nonlinearity, i. e., λ < 0 holds, and without loss of generality we assume that λ = −1 is true (the general case follows by rescaling). Now we can estimate as follows. −μ|x| e ∗ |u|2 (x) |u(x)|2 dx E[u] = 1/2(− + m2 )1/4 u22 − 1/4 |x| R3 1 2 1/4 2 2 1/2(− + m ) u2 − 1/4 ∗ |u| (x) |u(x)|2 dx |x| R3 1 (−)1/4 u22 u22 1/2(−)1/4 u22 − 4K 1 = 1/2 − u22 (−)1/4 u22 , 4K
(28)
where K > 0 is the best constant taken from Appendix B. Thus, energy conservation leads to an a priori bound on the H 1/2 -norm of the solution, if u0 22 < 2K holds. In fact, the constant K satisfies K=
Q22 2 > , 2 π
(29)
Well-posedness for semi-relativistic Hartree equations of critical type
57
where Q(x) is a strictly positive (ground state) solution of √ 1 2 ∗ |Q| Q = −Q; − Q − |x|
(30)
see Appendix B. Going back to (29), we find that u0 22 < Q22
(31)
is sufficient for global existence for λ = −1. The assertion of Theorem 2 for all λ < 0 now follows by simple rescaling. The proof of Theorem 2 is now complete. 3.3 Proof of Theorem 3 (Higher Regularity) To prove Theorem 3, we need the following generalization of Lemma 1, whose proof is a careful but straightforward generalization of the proof of Lemma 1. We defer the details to Appendix A.1. −μ|x|
Lemma 3 For μ 0 and s 1/2, the map J(u) := ( e |x| ∗ |u|2 )u is locally Lipschitz continuous from H s (R3 ) into itself with J(u) − J(v) Hs (u2Hs + v2Hs )u − v Hs for all u, v ∈ H s (R3 ). Moreover, we have that J(u) Hs u2Hr u Hs holds for all u ∈ H s (R3 ), where r = max{s − 1, 1/2}. Local well-posedness of (10) in H s√ , for s > 1/2, can be shown now as follows. We note that {e−it A }t∈R , with A = − + m2 , is a C0 -group of isometries on H s . Moreover, since the nonlinearity defined in (12), is locally Lipschitz continuous from H s into itself, local well-posedness in H s follows similarly as explained in the proof of Theorem 1 for H 1/2 . To show global well-posedness in H s , we prove by induction and Lemma 3 that an a priori bound on the H 1/2 norm of solution implies uniform bounds on the H s -norm on any compact interval [0, T∗ ] ⊂ [0, T). This claim follows from (13) and the second inequality stated in Lemma 3 by noting that t −it A s s u(t) H e u0 H + e−i(t−τ )A F(u(τ )) Hs dτ 0
t
u0 Hs +
F(u(τ )) Hs dτ
0
t
C1 + C2
u(τ ) Hs dτ,
0
holds, provided that u(t) Hr 1 for r = max{s − 1, 1/2} < s. Invoking Gronwall’s inequality we conclude that u(t) Hs eC2 T∗ ,
for t ∈ [0, T∗ ] ⊂ [0, T).
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E. Lenzmann
Induction now implies that an a-priori bound on u(t) H1/2 guarantees uniform bounds u(t) Hs on any compact interval I ⊂ [0, T). Thus, the maximal time of existence of an H s -valued solution coincides with the maximal time of existence when viewed as an H 1/2 -valued solution. Therefore sufficient conditions for global existence for H 1/2 -valued solutions imply global-in-time H s -valued solutions. This completes the proof of Theorem 3. 3.4 Proof of Theorem 4 (External Potentials) Let V = V+ + V− satisfy Assumption 1 in Section 2. We introduce the quadratic form Q(u, v) := u, − + m2 v + u, V− v + u, V+ v , (32) which is well-defined on the set (energy space) X := u ∈ L2 (R3 ) : Q(u, u) < ∞ .
(33)
Note that Assumption 1 also guarantees that C0∞ (R3 ) ⊂ X. It easy to show that our assumption on V implies that the quadratic form (32) is bounded from below, i. e., we have Q(u, u) −Mu, u holds for all u ∈ X and some constant M 0. By the semi-boundedness of Q, we can assume from now on (and without loss of generality) that Q(u, u) 0
(34)
holds for all u ∈ X. Since Q(·, ·) is closed (it is a sum of closed forms), the energy space X equipped with its norm (35) u X := u, u + Q(u, u) is complete, and we have the equivalence 1/2
1/2
u H1/2 + V+ u2 u X u H1/2 + V+ u2 .
(36)
Furthermore, there exists a nonnegative, self-adjoint operator A : D(A) ⊂ L2 → L2
(37)
with X = D(A1/2 ), such that u, Av = Q(u, v)
(38)
holds for all u ∈ X and v ∈ D(A); see, e. g., [10]. This operator can be extended to a bounded operator, still denoted by A : X → X ∗ , where X ∗ is the dual space of X. To prove now the assertion about local well-posedness in Theorem 4, we have to generalize Lemma 1 to the following statement.
Well-posedness for semi-relativistic Hartree equations of critical type
59
Lemma 4 Suppose μ 0 and let V satisfy Assumption 1. Then the map J(u) := −μ|x| ( e |x| ∗ |u|2 )u is locally Lipschitz continuous from X into itself with J(u) − J(v) X (u2X + v2X )u − v X for all u, v ∈ X. Proof (Of Lemma 4) By (36), it suffices to estimate J(u) − J(v) H1/2 and 1/2 V+ [J(u) − J(v)]2 separately. By Lemma 1, we know that J(u) − J(v) H1/2 (u2H1/2 + v2H1/2 )u − v H1/2 (u2X + v2X )u − v X . 1/2
It remains to estimate V+ [J(u) − J(v)]2 , which can be achieved by recalling (20) and proceeding as follows. 1/2 e−μ|x| 1/2 2 2 ∗ (|u| − |v| ) (u + v) V+ [J(u) − J(v)]2 V+ + x 2 −μ|x| 1/2 e + ∗ (|u|2 + |v|2 ) (u − v) V+ x 2 −μ|x| e 1/2 2 2 |x| ∗ (|u| − |v| ) V+ (u + v)2 + ∞ −μ|x| e 1/2 2 2 ∗ (|u| + |v| ) V+ (u − v)2 + |x| ∞ 1/2 u + v H1/2 u − v H1/2 V+ (u + v)2 + 1/2
+ (u2H1/2 + v2H1/2 )V+ (u − v)2 u2X + v2X )u − v X .
This completes the proof of Lemma 4.
−it A }t∈R is a Returning to the proof of Theorem 4, we √ simply note that {e 2 C -group of isometries on X, where A = − + m + V is defined in the form sense (see above). By Lemma 4, the nonlinearity is locally Lipschitz on X. Thus, local well-posedness now follows in the same way as for Theorem 1. To establish global well-posedness we have to prove conservation of charge, N[u], and energy, E[u], which is for (10) given by E[u] := 1/2 u(x) − + m2 u(x) dx + 1/2 V(x)|u(x)|2 dx + 0
R3
+ 1/4
R3
−μ|x|
λe |x|
R3
∗ |u|2 (x) |u(x)|2 dx.
(39)
As done in Section 3.2, we have to employ a regularization method using the class of operators Mε := (ε A + 1)−1 ,
for ε > 0,
(40)
60
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where we assume without loss of generality that A 0 holds. The mapping Mε acts on the sequence of embedded spaces . . . X +2 → X +1 → X −1 → X −2 . . . ,
(41)
with corresponding norms given by u X s := (1 + A) u2 . Note that X = X +1 (with equivalent norms) and that its dual space obeys X ∗ = X −1 . By using functional calculus, it is easy to show that Mε exhibits properties that are analog to (a)–(c) in Section 3.2. The rest of the argument for proving conservation of energy carries over from Section 3.2 without major modifications. Finally, we mention that deriving a priori bounds on u(t) X leads to a similar discussion as presented in Section 3.2, while noting that we have to take care that V− has a relative (−)1/2 -form bound, 0 a < 1, introduced in Assumption 1. This completes the proof of Theorem 4. s/2
Acknowledgements The author is grateful to Demetrios Christodoulou, Jürg Fröhlich, Lars Jonsson, and Simon Schwarz for many valuable and inspiring discussions.
Appendix A: Fractional Calculus The following result (generalized Leibniz rule) is proved in [9] for Riesz and Bessel potentials of order s ∈ R, which are denoted by (−)s/2 and (1 − )s/2 , respectively. But as a direct consequence of the Milhin multiplier theorem [1], the cited result holds for Ds := (μ2 − )s/2 , where μ 0 is a fixed constant. Lemma 5 (Generalized Leibniz Rule) Suppose that 1 < p < ∞, s 0, α 0, β 0, and 1/ pi + 1/qi = 1/ p with i = 1, 2, 1 < qi ∞, 1 < pi ∞. Then Ds ( fg) p c(Ds+α f p1 D−α gq1 + D−β f p2 Ds+β gq2 ), where the constant c depends on all of the parameters but not on f and g. A second fact we use in the proof of our main result is as follows. For 0 < α < 3 and μ 0, the potential operator D−α = (μ2 − )−α/2 corresponds to f → Gμα ∗ f , with f ∈ S (R3 ), and we have that Gμα ∈ L3/(3−α) (R3 ). w To see this, we refer to the inequality and the exact formula cα for μ 0 and 0 < α < 3, 0 Gμα (x) G0α (x) = 3−α , |x|
(42)
(43)
with some constant cα ; these facts can be derived from [15, Section V.3.1]. Now 3/σ (42) follows from |x|−σ ∈ Lw (R3 ) whenever 0 < σ < 3. Another observation used in Section 2 is the well-known explicit formula μ
G2 (x) =
e−μ|x| . 4π |x|
(44)
Well-posedness for semi-relativistic Hartree equations of critical type
That is, (μ2 − ) in R3 has the Green’s function conditions.
e−μ|x| 4π |x|
61
with vanishing boundary
A.1 Proof of Lemma 3 Proof (Of Lemma 3) We only show the second inequality derived in Lemma 3, since the first one can be proved in a similar way. Let μ 0 and s 1/2. We put Dα := (μ2 − )α/2 for α ∈ R. By the generalized Leibniz rule and (17), we have that Ds J(u)2 Ds [(D−2 |u|2 )u]2 Ds−2 |u|2 p1 uq1 + D−2 |u|2 ∞ Ds u2 Ds−2 |u|2 p1 uq1 + u2H1/2 u Hs ,
(45)
where 1/ p1 + 1/q1 = 1/2 with 1 < p1 , q1 ∞. The first term of the righthand side of (45) can be controlled as follows, where we introduce r = max{s − 1, 1/2}. (1) For 1/2 s < 3/2, we choose p1 = 3/s and q1 = 6/(3 − 2s) which leads to μ
Ds−2 |u|2 3/s u6/(3−2s) G2−s 3/(1+s),w |u|2 3/2 u Hs u2H1/2 u Hs u2Hr u Hs ,
where we use the weak Young inequality, as well as Sobolev’s inequality u6/(3−2s) u Hs in R3 , and (42) once again. (2) For s 3/2, we choose p1 = 6 and q1 = 3. This yields Ds−2 |u|2 6 u3 Ds−1 |u|2 2 u3 Ds−1 u6 u23 Ds u2 u23 u Hs u2Hr ,
while using twice Sobolev’s inequality f 6 D f 2 in R3 . Putting now all together, we conclude that J(u) Hs J(u)2 + Ds J(u)2 u2H1/2 u2 + u2Hr u Hs u2Hr u Hs .
Appendix B: Ground States We consider the functional (see also [12]) K[u] :=
(−)1/4 u22 u22 , −1 ∗ |u|2 )(x) |u(x)|2 dx R3 (|x|
(46)
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which is well-defined for all u ∈ H 1/2 with u ≡ 0. Note that by using (17) we can estimate the denominator in K[u] as follows. 1 1 π 2 2 2 ∗ |u| (x) |u(x)| dx ∗ |u| u22 (−)1/4 u22 u22 , |x| |x| 2 R3 ∞ (47) which leads to the bound 2 K[u] < ∞. (48) π Indeed, we will see that the estimate from below is a strict inequality. With respect to the related variational problem (49) K := inf K[u] : u ∈ H 1/2 (R3 ), u ≡ 0 we can state the following result. Lemma 6 (Ground States) There exists a minimizer, Q ∈ H 1/2 (R3 ), for (49), and we have the following properties. (1)
Q(x) is a smooth function that can be chosen to be real-valued, strictly positive, and spherically symmetric with respect to the origin. It satisfies √ 1 2 ∗ |Q| Q = −Q, (50) − Q − |x|
and it is nonincreasing, i. e., we have that Q(x) Q(y) whenever |x| |y|. (2) The infimum satisfies K = Q22 /2 and K > 2/π . Proof (Sketch of Proof) We present the main ideas for the proof of the preceding lemma. That (49) is attained at some real-valued, radial, nonnegative and nonincreasing function Q(x) 0 can be proved by direct methods of variational calculus and rearrangement inequalities; see also [16] for a similar variational problem for nonrelativistic Schrödinger equations with local nonlinearities. Furthermore, any minimizer, Q ∈ H 1/2 , has to satisfy the corresponding Euler–Lagrange equation that reads √ λ 2 − Q − ∗ |Q| Q = −Q, (51) |x| after a suitable rescaling Q(x) → aQ(b x) with some a, b > 0. Let us make some comments about the properties of Q. Using an bootstrap argument and Lemma 3 for the nonlinearity, it follows that Q belongs to H s , for all s 1/2. Hence it is a smooth function. To see that Q(x) 0 is strictly positive, i. e., Q(x) > 0, we rewrite (51) such that −1 √ − + 1 W, (52) Q= where W := (|x|−1 ∗ |Q|2 )Q. By functional calculus, we have that ∞ √ √ −1 − + 1 = e−t e−t − dt. 0
(53)
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Next, we notice by the explicit formula for the kernel (in R3 ) √ t e−t − (x, y) = F −1 e−t|ξ | (x − y) = C · 2 , [t + |x − y|2 ]2 with some constant C > 0; see, e. g., [11]. This explicit formula shows that √ e−t√− is positivity improving. This means that if f 0 with f ≡ 0 then √ e−t − f > 0 almost everywhere. Hence ( − + 1)−1 is also positivity improving, by (53), and we conclude that Q(x) > 0 holds almost everywhere, thanks to (52) and W 0. Moreover, we know that Q(x) is a nonincreasing, continuous function. Therefore Q(x) > 0 holds in the strong sense, i. e., for every x ∈ R3 . Finally, to see that (2) holds, we consider the variational problem I N := inf E[u] : u ∈ H 1/2 (R3 ), u22 = N , (54) where N > 0 is a given parameter and E[u] = 1/2(−)
1/4
u22
− 1/4
R3
1 2 ∗ |u| |u(x)|2 dx. |x|
Due to the scaling behavior E[α u(α·)] = α E[u], we have that either I N = 0 or I N = −∞ holds. By noting that N 1 E[u] − (−)1/4 u22 , 2 4K 3/2
and the fact that equality holds if and only if u minimizes K[u], we find that I N = 0 holds if and only if N Nc := 2K. Moreover, I N = 0 is attained if ˜ be such a minimizer with Q ˜ 2 = Nc . Thanks to and only if N = Nc . Let Q 2 ˜ is realthe proof of part (1), we can assume without loss of generality that Q valued, radial, and strictly positive. Calculating the Euler–Lagrange equation for (54), with N = Nc , yields √ ˜ = −θ Q, ˜ ˜ − 1 ∗ | Q| ˜ 2 Q − Q |x| for some multiplier θ, where it is easy to show that θ > 0 holds. Setting now ˜ −1 x), which conserves the L2 -norm, leads to a ground state Q(x) = θ −3/2 Q(θ Q(x) satisfying (51). Thus, we have that ˜ 2 /2 = Q2 /2. K = Q 2 2 To prove that K > 2/π holds, let us assume K = 2/π . This implies that the first inequality in (47) is an equality for u(x) = Q(x) > 0. But this leads to (|x|−1 ∗ |Q|2 )(x) = const., which is impossible.
References 1. Bergh, J., Löfström, J.: Interpolation Spaces. Springer, New York (1976) 2. Cazenave, T.: Semilinear Schrödinger equations. In: Courant Lecture Notes, vol. 10. American Mathematical Society, Providence, RI (2003)
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3. Cazenave, T., Haraux, A.: An introduction to semilinear evolution equations. In: Oxford Lecture Series in Mathematics and Its Applications, vol. 13. Oxford University Press, New York (1998) 4. Elgart, A., Schlein, B.: Mean field dynamics of Boson stars. Comm. Pure Appl. Math. 60, 500– 545 (2007) 5. Fröhlich, J., Jonsson, B.L.G., Lenzmann, E.: Boson stars as solitary waves. Comm. Math. Phys. (Preprint arXiv:math-ph/0512040) (2006) (accepted) 6. Fröhlich, J., Lenzmann, E.: Mean-field limit of quantum Bose gases and nonlinear Hartree equation. Sémin. Équ. Dériv. Partielles (Ecole Polytechnique) XIX, 1–26 (2004) 7. Fröhlich, J., Lenzmann, E.: Blow-Up for nonlinear wave equations describing Boson stars. Comm. Pure Appl. Math. (Preprint avXiv:math-ph/0511003) (2006) (in press) 8. Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of Hartree equations. In: Nonlinear Wave Equations. Contemporary Mathematics, vol. 263, pp. 29–60. American Mathematical Society, Providence, RI (2000) 9. Gulisashvili, A., Kon, M.K.: Exact smoothing properties of Schrödinger semigroups. Amer. J. Math. 118, 1215–1248 (1996) 10. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1980) 11. Lieb, E.H., Loss, M.: Analysis. In: Graduate Studies in Mathematics, 2nd edn., vol. 14. American Mathematical Society, Providence, RI (2001) 12. Lieb, E.H., Yau, H.-T.: The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys. 112, 147–174 (1987) 13. Nawa, H., Ozawa, T.: Nonlinear scattering with nonlocal interaction. Comm. Math. Phys. 146, 269–275 (1992) 14. Pazy, A.: Semi-groups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) 15. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ (1970) 16. Weinstein, M.I.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87, 567–576 (1983) 17. Yajima, K., Zhang, G.: Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity. J. Differential Equations 202, 81–110 (2004)
Math Phys Anal Geom (2007) 10:65–80 DOI 10.1007/s11040-007-9021-8
Feynman’s Operational Calculi: Spectral Theory for Noncommuting Self-adjoint Operators Brian Jefferies · Gerald W. Johnson · Lance Nielsen
Received: 3 November 2006 / Accepted: 16 April 2007 / Published online: 29 June 2007 © Springer Science + Business Media B.V. 2007
Abstract The spectral theorem for commuting self-adjoint operators along with the associated functional (or operational) calculus is among the most useful and beautiful results of analysis. It is well known that forming a functional calculus for noncommuting self-adjoint operators is far more problematic. The central result of this paper establishes a rich functional calculus for any finite number of noncommuting (i.e. not necessarily commuting) bounded, self-adjoint operators A1 , . . . , An and associated continuous Borel probability measures μ1 , · · · , μn on [0, 1]. Fix A1 , . . . , An . Then each choice of an n-tuple (μ1 , . . . , μn ) of measures determines one of Feynman’s operational calculi acting on a certain Banach algebra of analytic functions even when A1 , . . . , An are just bounded linear operators on a Banach space. The Hilbert space setting along with self-adjointness allows us to extend the operational calculi well beyond the analytic functions. Using results and ideas drawn largely from the proof of our main theorem, we also establish a family of Trotter product type formulas suitable for Feynman’s operational calculi.
B. Jefferies School of Mathematics, The University of New South Wales, Sydney 2052, Australia e-mail:
[email protected] G. W. Johnson Department of Mathematics, 333 Avery Hall, The University of Nebraska, Lincoln, Lincoln, NE 68588-0130, USA e-mail:
[email protected] L. Nielsen (B) Department of Mathematics, Creighton University, Omaha, NE 68178, USA e-mail:
[email protected]
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Keywords Noncommuting self-adjoint operators · Spectral theories · Feynman’s operational calculi · Disentangling Mathematics Subject Classifications (2000) Primary 47A13 · 47A60 · Secondary 46J15
1 Introduction Let X be a Banach space and suppose that A1 , . . . , An are noncommuting elements in L(X), the space of bounded linear operators on X. Further, for each i ∈ {1, . . . , n}, let μi be a continuous probability measure defined on B ([0, 1]), the Borel class of [0, 1]. (Recall that a measure μ is continuous provided that μ({s}) = 0 for every single point set {s}.) Such measures determine an operational calculus or ‘disentangling map’ Tμ1 ,...,μn from a commutative Banach algebra D(A1 , . . . , An ), called the ‘disentangling algebra’ of analytic functions into the noncommutative Banach algebra L(X). (See [4] or Definition 1.1 below.) It is natural to seek conditions under which such an operational calculus can be extended beyond the analytic functions in D(A1 , . . . , An ). Theorem 2.2, the main result of this paper will show, in conjunction with results from [6], that when X = H is a Hilbert space and A1 , . . . , An are self-adjoint, the domain of each of the operational calculi is much richer than D(A1 , . . . , An ). Feynman developed ‘rules’ for his operational calculus for noncommuting operators while discovering the famous perturbation series and Feynman graphs of quantum electrodynamics. By the time he wrote [2] he realized that this operational calculus could be developed into a widely applicable mathematical technique. Feynman was aware that his work was far from being mathematically rigorous (see page 108 of [2]), especially with regard to the ‘disentangling’ process, the central operation of his functional calculus. He regarded his operational calculus as a kind of generalized path integral. (See Section 14.3 of [10].) We now give a brief description of Feynman’s heuristic rules: (a) Attach time indices to the operators to keep track of the order of the operators in products. Operators with smaller (or earlier) time indices are to act before operators with larger (or later) time indices no matter how they are ordered on the page. (b) With time indices attached, functions of the operators are formed just as if they were commuting. (c) Finally, the operator expressions are to be restored to their natural order; this is the so-called disentangling process. This final step is often difficult; it consists roughly of manipulating the operator expressions until their order on the page is consistent with the time ordering. How does one accomplish (a)–(c)? There have been several quite varied approaches to this subject. Many of the references can be found in one of
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the books [10, 13]. We also call attention to the recent monograph by B. Jefferies [3] and the 1968 paper by Taylor [18]. Both of these are essentially concerned with the Weyl calculus for a finite number of noncommuting bounded operators. The paper [18] focused on operators which are also selfadjoint. The work begun by Maslov [12] and pursued by him and by several others is the furthest developed. See especially the book [13] by Nazaikinskii, Shatalov, and Sternin. We will follow the approach initiated recently by Jefferies and Johnson ([4–7]) and further developed by them, Nielsen and others ([8, 9, 11, 14]). A large family of operational calculi is defined at one time in this approach. This allows us to study a variety of operational calculi within one framework. It also permits us to solve a wide variety of evolution equations using various exponential functions of sums of noncommuting operators. (This was carried out in [8], Section 4, and we hope to pursue this further in later work.) Finally, one can sometimes get information about one (or one type of) operational calculus by showing that it is the limit of simpler operational calculi. Indeed, the main theorem of this paper will rest in large part on such an argument. Johnson and Nielsen established a stability theorem for Feynman’s operational calculi [11] which will supply one of the central facts that we will need for our main result. We state that theorem now even though the precise definitions of the disentangling algebra and the disentangling map will be postponed until further on in this section. Theorem 1.1 For each i = 1, . . . , n, let μi and μik , k = 1, 2, . . . be continuous probability measures on B ([0, 1]) and suppose that the sequence (μik ) converges weakly to μi (denoted μik μi ) as k → ∞. Then for every f ∈ D(A1 , . . . , An ), Tμ1k ,...,μnk f ( A˜ 1 , . . . , A˜ n ) → Tμ1 ,...,μn f ( A˜ 1 , . . . , A˜ n ) in the operator norm on L(X) as k → ∞. Note: (a) The weak convergence above is meant in the probabilist’s sense (see [1], p. 229). (b) We can alternatively describe the conclusion of Theorem 1.1 as follows: The sequence of operational calculi specified by the sequence of n-tuples {(μ1k , . . . , μnk ) : k = 1, . . . , ∞} converges as k → ∞ to the operational calculus specified by the n-tuple (μ1 , . . . , μn ). We finish this introduction by briefly outlining the essential definitions and some basic facts of the approach to Feynman’s operational calculi initiated in [4, 5]. A discussion of the heuristic ideas behind these operational calculi can be found in Chapter 14 of [10]. Let X be a Banach space and let A1 , . . . , An be nonzero bounded linear operators on X. Except for the numbers A1 , . . . , An , which will serve as weights, we ignore for the present the nature of A1 , . . . , An as operators and introduce a commutative Banach algebra consisting of ‘analytic
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functions’ f ( A˜ 1 , . . . , A˜ n ), where A˜ 1 , . . . , A˜ n are treated as purely formal commuting objects. Consider the collection D = D(A1 , . . . , An ) of all expressions of the form ∞
f ( A˜ 1 , . . . , A˜ n ) =
1 ˜ mn cm1 ,...,mn A˜ m 1 · · · An
(1.1)
m1 ,...,mn =0
where cm1 ,...,mn ∈ C for all m1 , . . . , mn = 0, 1, . . . , and f ( A˜ 1 , . . . , A˜ n ) = f ( A˜ 1 , . . . , A˜ n )D(A1 ,...,An ), :=
∞
|cm1 ,...,mn |A1 m1 · · · An mn < ∞.
(1.2)
m1 ,...,mn =0
As pointed out in [4] the function on D(A1 , . . . , An ) defined by (1.2) makes D(A1 , . . . , An ) into a commutative Banach algebra under pointwise operations ([4], Proposition 1.1). We refer to D(A1 , . . . , An ) as the disentangling algebra associated with the n-tuple (A1 , . . . , An ) of bounded linear operators acting on X. This commutative Banach algebra will provide us with a framework where we can apply Feynman’s ‘rule’ (b) above rigorously rather than just heuristically. Let μ1 , . . . , μn be continuous probability measures defined at least on B ([0, 1]), the Borel class of [0, 1]. The idea is to replace the operators A1 , . . . , An with the elements A˜ 1 , . . . , A˜ n from D and then form the desired function of A˜ 1 , . . . , A˜ n . Still working in D, we time order the expression for the function and then pass to L(X) simply by removing the tildes. Given nonnegative integers m1 , . . . , mn , we let m = m1 + · · · + mn and mn 1 Pm1 ,...,mn (z1 , . . . , zn ) = zm 1 · · · zn .
(1.3)
We are now ready to define the disentangling map Tμ1 ,...,μn which will carry us from our commutative framework to the noncommutative setting of L(X). For j = 1, . . . , n and all s ∈ [0, 1], we take A j(s) = A j (recall that each A j is independent of s) and, for i = 1, . . . , m, we define ⎧ A1 (s) if i ∈ {1, . . . , m1 }, ⎪ ⎪ ⎪ ⎨ A2 (s) if i ∈ {m1 + 1, . . . , m1 + m2 }, Ci (s) := (1.4) . .. ⎪ . ⎪ .. ⎪ ⎩ An (s) if i ∈ {m1 + · · · + mn−1 + 1, . . . , m}. For each m = 0, 1, . . . , let Sm denote the set of all permutations of the integers {1, . . . , m}, and given π ∈ Sm , we let m (π ) = {(s1 , . . . , sm ) ∈ [0, 1]m : 0 < sπ(1) < · · · < sπ(m) < 1}. Finally, we remark that we will use the notation μk to denote μ × · · · × μ.
k times
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Definition 1.2 Tμ1 ,...,μn Pm1 ,...,mn ( A˜ 1 , . . . , A˜ n ) mn 1 Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )(μm := 1 × · · · × μn )(ds1 , . . . , dsm ). π ∈Sm
m (π )
(1.5) Then, for f ( A˜ 1 , . . . , A˜ n ) ∈ D(A1 , . . . , An ) given by f ( A˜ 1 , . . . , A˜ n ) =
∞
1 ˜ mn cm1 ,...,mn A˜ m 1 · · · An ,
(1.6)
m1 ,...,mn =0
we set Tμ1 ,...,μn f ( A˜ 1 , . . . , A˜ n ) equal to ∞
cm1 ,...,mn Tμ1 ,...,μn Pm1 ,...,mn ( A˜ 1 , . . . , A˜ n ) .
(1.7)
m1 ,...,mn =0
Remark 1.3 Even though the A j’s are independent of s, the order of the operator products in each term of (1.5) depends on the s’s and on the measures μ1 , . . . , μn . (If the A j’s do depend on s, we obtain exactly the same expression as seen in (1.5) and then we have a nontrivial integrand. But this situation will concern us only marginally in this paper. For details of the time dependent setting see, for example, the papers [8, 15, 16].) It is worth noting that the disentangling map as defined above is a linear operator of norm one from D(A1 , . . . , An ) to L(X). (See [4].) In the commu1 mn tative setting, the right-hand side of (1.5) gives us Am 1 · · · An , the expected result of the commutative functional calculus [4, Proposition 2.2]. (Of course, commutativity allows us to write the m operators in any desired order.) As is usual, we shall write the operator Tμ1 ,...,μn f in place of Tμ1 ,...,μn ( f ) for an element f of D(A1 , . . . , An ). We shall sometimes write the bounded linear operator Tμ1 ,...,μn f ( A˜ 1 , . . . , A˜ n ) as fμ1 ,...,μn (A1 , . . . , An ), fμ1 ,...,μn ( A) with A denoting the n-tuple (A1 , . . . , An ) of operators, or fμ ( A) with μ denoting the n–tuple (μ1 , . . . , μn ) of measures. In particular, ...mn Pμm11,...,μ ( A) = Tμ1 ,...,μn Pm1 ,...,mn ( A˜ 1 , . . . , A˜ n ) . (1.8) n √ We find it convenient to use i as an index, so i denotes −1. The real part of a complex number z is written as z and the imaginary part as z. For a complex vector ζ = (ζ1 , . . . , ζn ) ∈ Cn , we set ζ = (ζ1 , . . . , ζn ), ζ = (ζ1 , . . . , ζn ), |ζ | = |ζ1 |2 + · · · + |ζn |2 . Remark 1.4 A family of Trotter product type formulas suitable for Feynman’s operational calculi (and mentioned in the abstract) will be established in
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Theorem 2.4. Here the bounded linear operators A1 , . . . , An need not be selfadjoint and X can be a Banach space. However, we will require the probability measures μ1 , . . . , μn to be absolutely continuous with respect to Lebesgue measure λ.
2 The Main Theorem We give a detailed proof that for any n-tuple A = (A1 , . . . , An ) of self-adjoint operators, there exists r > 0 such that A is of ‘Paley–Wiener type’ (0, r, μ) for any μ = (μ1 , . . . , μn ). We will follow the statement of what this means with a brief discussion of its consequences for the enlargement of the domain of the associated operational calculi. Our main interest in this paper is in the Hilbert space setting. However, we state the definition of ‘Paley–Wiener type’ in the more general Banach space setting. Definition 2.1 Let A1 , . . . , An be bounded linear operators acting on a Banach space X. Let μ = (μ1 , . . . , μn ) be an n-tuple of continuous probability measures on B ([0, 1]) and let Tμ1 ,...,μn : D(A1 , . . . , An ) → L(X)
(2.1)
be the disentangling map defined in Definition 1.2. If there exists C, r, s ≥ 0 such that ˜ for all ζ ∈ Cn , (2.2) Tμ1 ,...,μn ei(ζ, A) L(X) ≤ C(1 + |ζ |)s er|ζ | , then the n-tuple A = (A1 , . . . , An ) of operators is said to be of Paley–Wiener ˜ = ζ1 A˜ 1 + · · · + ζn A˜ n .) type (s, r, μ). (Note: Given A and ζ ∈ Cn , (ζ, A) If the estimate (2.2) holds, then there exists a unique L(X)-valued distribution Fμ, A ∈ L(C∞ (Rn ), L(X)) such that
˜ Fμ, A ( f ) = (2π )−n Tμ1 ,...,μn ei(ξ, A) fˆ(ξ ) dξ, (2.3) Rn
for every rapidly decreasing function f ∈ S (Rn ). Here
fˆ(ξ ) = e−i(x,ξ ) f (x) dx Rn
denotes the Fourier transform of f . Moreover, ,...,mn Fμ, A (Pm1 ,...,mn ) = Pμm11,...,μ (A1 , . . . , An ), n
(2.4)
for all nonnegative integers m1 , . . . , mn . Hence we have a rich extension of the functional calculus f −→ fμ ( A) from analytic functions with a uniformly convergent power series in a polydisk, to functions C∞ in a neighbourhood of
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the support γμ ( A) of Fμ, A . In fact, all of the distributions just mentioned are compactly supported and so of finite order k. In the setting of Theorem 2.2, k will be the smallest integer strictly greater than n/2. (For example, k = 2, if n = 2 or 3.) Now let K = nj=1 [−A j, A j]. The functional calculus then extends to all functions f that are k times continuously differentiable on some open set containing K. The support of γμ ( A) of the distribution Fμ, A is defined as the μ–joint spectrum of the n–tuple A = (A1 , . . . , An ). The distribution Fμ, A is called Feynman’s μ–functional calculus for A. The number rμ ( A) = sup{|x| : x ∈ γμ ( A)} is called the μ–joint spectral radius of A. It is shown in [7] via Clifford analysis that the nonempty compact subset γμ ( A) of Rn may be interpreted as the set of singularities of a multidimensional analogue of the resolvent family of a single operator. For more detail on or related to the last two paragraphs, see pages 186–192 and especially Theorem 3.1 and Proposition 3.2 of [6]. If we have more information about the particular operators and measures that are involved, we can sometimes further enlarge the functional calculus. Example 2.1, p.176–178 in [6] is an extreme case. Here A1 and A2 are the 2 by 2, self-adjoint Pauli matrices σ1 and σ3 . The measures μ1 and μ2 are any continuous probability measures with the support of μ1 entirely to the left of the support of μ2 . In this case, fμ1 ,μ2 (A1 , A2 ) makes sense for any f which is defined on the 4 point set {−1, 1} × {−1, 1}. This last set is the product of the ordinary spectrums of σ1 and σ3 . Theorem 2.2 An n-tuple A = (A1 , . . . , An ) of bounded self-adjoint operators acting on a Hilbert space H is of Paley–Wiener type (0, r, μ) with r = (A1 2 + · · · + An 2 )1/2 , for any n-tuple μ = (μ1 , . . . , μn ) of continuous probability measures on B ([0, 1]). Proof One of the keys to the proof is a use of the Martingale Convergence Theorem. We can apply it to the Radon–Nikodyn derivative of probability measures on [0, 1] which are absolutely continuous with respect to Lebesgue measure λ. Such Radon–Nikodyn derivatives are nonnegative functions with L1 (λ)-norm 1. However, we are only assuming that μ1 , . . . , μn are continuous and so, given μi , we will begin by finding a sequence of absolutely continuous probability measures which converge weakly to μi . (We will, but do not need to, do this even for the μi ’s that are absolutely continuous with respect to λ.) Let μ be a continuous Borel probability measure on [0, 1]. Let ρ : [0, 1] → R be a nonnegative continuous function with compact support in [0, 1) and ρ1 = 1. It will be convenient to let ρ(x) = 0 for x < 0. For every ∈ (0, 1], we set ρ (x) = −1 ρ(x/ ), 0 ≤ x ≤ 1. It is easy to check that ρ 1 = 1 for 0 < ≤ 1. Now we let
(ρ ∗ μ)(x) := 0
x
ρ (x − y)μ(dy),
0 ≤ x ≤ 1.
(2.5)
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We assert that (ρ ∗ μ)λ μ as → 0+ . Indeed, using the definition of weak convergence, our assertion follows once we know that, for every continuous function φ : [0, 1] → R,
1
1 (ρ ∗ μ)(x)φ(x)dx → φ(x)μ(dx) as → 0+ . (2.6) 0
0
We omit the proof of the limit (2.6) as it can be carried out using standard techniques. The basic idea of the proof is much like arguments using approximate identities although some of the particular details follow Exercise 10, p. 194 of [17] more closely. Now consider the partition of the interval [0, 1) into nk disjoint intervals Ik, = [( − 1)n−k , n−k ), = 1, . . . , nk each of length n−k . (Below, n will be the number of operator-measure pairs in our problem.) The collection Ak of finite disjoint unions of these intervals is an algebra – in fact a σ -algebra. Note that Ak ⊂ Ak+1 for k = 1, 2, . . . . Let Pk : L1 (λ) → L1 (λ) be the conditional expectation operator [1, p.265] with respect to Ak ; that is, n
k
Pk f = nk
χ Ik,
f dλ.
(2.7)
Ik,
=1
Note that the sequence {Pk f } is adapted with respect to the sequence {Ak } of σ -algebras [1, p. 280]. Then for each f ∈ L1 (λ), by the Martingale Convergence Theorem [1, p. 285–286], Pk f → f in L1 (λ) (and λ-a.e.) as k → ∞. Further, by Theorem 10.1.3 from [1], we have
1
1 Pk f dλ = f dλ, k = 1, 2, . . . . (2.8) 0
0
(Remark The usual notation for Pk f in the probability literature is E( f |Ak ).) Now set fi, j,k := Pk (ρ1/j ∗ μi ) for each i = 1, . . . , n and j, k = 1, 2, . . . . Each such step function fi, j,k is constant on each interval Ik, , = 1, . . . , nk and n
k
fi, j,k := n
k
χ Ik,
ρ1/j ∗ μi dλ.
(2.9)
Ik,
=1
The function f˜i, j,k that we are about to define is a key to the proof: f˜i, j,k := nk
nk−1 −1
χ Ik,mn+i
ρ1/j ∗ μi dλ.
(2.10)
Ik−1,m+1
m=0
Note that this function has support in the finite union Ji,k :=
nk−1 −1
Ik,mn+i
m=0
of disjoint intervals, for each i = 1, . . . , n and Ji,k ∩ J ,k = ∅ for i = . The integral Ik−1,m+1 ρ1/j ∗ μi dλ in the sum (2.10) is a weight factor compensating
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for the omission of terms from the sum (2.9), so that the following equalities hold true:
1
1
1 fi, j,k dλ = ρ1/j ∗ μi dλ = f˜i, j,k dλ. (2.11) 0
0
0
The first equality follows from (2.7), (2.8) and the definition of fi, j,k above. We turn now to the second equality in (2.11):
1
f˜i, j,k dλ = n
nk−1 −1
k
0
m=0
=
1 nk
ρ1/j ∗ μi dλ Ik−1,m+1
ρ1/j ∗ μi dλ + Ik−1,1
ρ1/j ∗ μk dλ
+
=
ρ1/j ∗ μi dλ + . . . Ik−1,2
Ik−1,nk−1 1
ρ1/j ∗ μi dλ.
(2.12)
0
Thus (2.11) is established. Now let φ be a continuous function on [0, 1]. Given > 0, use the uniform continuity of φ to choose k so large that |φ(x) − φ(y)| < whenever |x − y| < 1 1 n−k+1 . We wish to compare the integrals 0 fi, j,k φ dλ and 0 f˜i, j,k φ dλ. We begin with the first of these. The RHS of the 2nd equality in (2.13) below is just another way of writing the sum of the nk terms that appear on the LHS. ⎧ ⎫
1
1⎨
nk ⎬ fi, j,k φ dλ = nk χ Ik, p φ ρ1/j ∗ μi dλ dλ ⎭ 0 0 ⎩ Ik, p p=1
1
=
nk 0
= nk
⎧ n ⎨nk−1 −1 ⎩
χ Ik,mn+ φ
ρ1/j ∗ μi dλ Ik,mn+
m=0 =1
nk−1 n −1
⎭
φ dλ .
ρ1/j ∗ μi dλ Ik,mn+
m=0 =1
⎫ ⎬ dλ
(2.13)
Ik,mn+
On the other hand, starting with (2.10) we have
1 0
f˜i, j,k φ dλ =
1
n
k
nk−1 −1
0
=n
k
=n
χ Ik,mn+i φ
ρ1/j ∗ μi dλ dλ Ik−1,m+1
m=0
nk−1 −1 m=0
k
ρ1/j ∗ μi dλ Ik−1,m+1
nk−1 n −1 m=0 =1
φdλ Ik,mn+i
ρ1/j ∗ μi dλ Ik,mn+
φ dλ . (2.14) Ik,mn+i
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First using (2.13) and (2.14) and then the Mean Value Theorem for Integrals, there exist ξk, ∈ Ik, , = 1, . . . , nk such that 1 ˜ fi, j,k − fi, j,k φ dλ 0
≤
nk−1 n −1 m=0 =1
=
Ik,mn+
nk−1 n −1 m=0 =1
ρ1/j ∗ μi dλ nk
φ dλ − n Ik,mn+
k Ik,mn+i
φ dλ
ρ1/j ∗ μi dλ φ ξk,mn+ − φ ξk,mn+i < .
(2.15)
Ik,mn+
The final inequality follows from the fact that |ξk,mn+ − ξk,mn+i | < n−k+1 for each i ∈ {1, . . . , n} and each j ∈ {1, 2, 3, . . . }. Thus for each i ∈ {1, . . . , n} and each j ∈ {1, . . . , m}, fi, j,k λ − f˜i, j,k λ 0 as k → ∞.
(2.16)
Let μi, j,k := fi, j,k λ, i = 1, . . . , n and j, k = 1, 2, . . . . Next we summarize what we have proved so far about the weak limits and then draw some conclusions. For each i = 1, . . . , n, the following limits obtain. A ρ1/j ∗ μi λ μi as j → ∞. B fi, j,k λ = Pk ρ1/j ∗ μi λ ρ1/j ∗ μi λ as k → ∞. In fact, the probability densities in B converge in L1 -norm and so the measures converge in total variation norm and so certainly converge weakly. C
fi, j,k λ − f˜i, j,k λ 0 as k → ∞.
From B and C we see that D fi, j,k λ = Pk ρ1/j ∗ μi λ ρ1/j ∗ μi λ as k → ∞. Now from D and A we have the iterated weak limits, E lim j→∞ limk→∞ f˜i, j,k λ = lim j→∞ ρ1/j ∗ μi = μi . Since [0, 1] is a separable metric space, it follows from E that there exists a sequence ( f˜i, j,k( j ) )λ such that for each i = 1, . . . , n, F f˜i, j,k( j ) λ μi as j → ∞. (See [1, p. 309–310] and especially the paragraph preceding Theorem 11.3.3.) Note that the increasing sequence k( j ), j = 1, 2, . . . , may be chosen independently of i = 1, . . . , n by observing that the space M([0, 1], Rn ) of Rn -valued Borel measures on [0, 1] is in duality with the space C([0, 1], Rn ) of Rn -valued continuous functions and that closed balls in M([0, 1], Rn ) are compact and metrizable for the associated weak*-topology.
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Note As we continue we will just write j, k but will understand that k = k( j ) as in F above. The measure μi, j,k is given by μi, j,k = f˜i, j,k λ = nk
nk−1 −1
χ Ik,mn+i λ
ρ1/j ∗ μi dλ.
(2.17)
Ik−1,m+1
m=0
Before beginning the calculation below we make some simple comments and introduce some notation. (a) Since exponential functions are entire, the exponential functions involved below are certainly in the domain of the disentangling map. (b) We will write (ζ, A) = ζ1 A1 + · · · + ζn An where ζ = (ζ1 , . . . , ζn ) is an ntuple of complex numbers and A = (A1 , . . . , An ). Similar notation will ˜ be used in connection with A. (c) Since the disentangling algebra is commutative, we have ˜
˜
˜
˜
˜
e(ζ, A) = eζ1 A1 +···+ζn An = eζ1 A1 . . . eζn An . Because of how the function f˜i, j,k is supported (see (2.10)) and since μi, j,k is defined using f˜i, j,k , an extension of Proposition 2.2 from [6] allows us to do the following calculation (much as was done in Example 2.2 of that paper): ˜
Tμ1, j,k ,...,μn, j,k (ei(ζ, A) ) = exp iζn
ρ1/j ∗ μn dλ An . . . Ik−1,nk−1
Ik−1,nk−1
...
!
ρ1/j ∗ μn dλ An . . .
exp iζn Ik−1,2
ρ1/j ∗ μ1 dλ A1
exp iζ1
ρ1/j ∗ μ1 dλ A1
exp iζ1
Ik−1,2
exp iζn
Ik−1,1
exp iζ1
! .
! ρ1/j ∗ μn dλ An . . . ! . ρ1/j ∗ μ1 dλ A1
(2.18)
Ik−1,1
Breaking ζ1 = ξ1 + iη1 into real and imaginary parts, we have iζ1 = −η1 + iξ1 . Further, |eiζ1 | = |eiξ1 ||e−η1 | ≤ e|η1 | = e|ζ1 | . Similarly, |eiζ2 | ≤ e|ζ2 | , . . . , |eiζn | ≤ e|ζn | .
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Now using (2.18), the self-adjointness of Ai , i = 1, . . . , n, the standard multiplicative Banach algebra inequality and the simple computations just above, we can write ˜
Tμ1, j,k ,...,μn, j,k (ei(ζ, A) ) " " " " " " ρ1/j ∗ μn dλ An " . . . ≤ "exp iζn " " Ik−1,nk−1 " " " " " " ρ1/j ∗ μ1 dλ A1 " . . . "exp iζ1 " " Ik−1,nk−1 " !" " " "exp iζn ρ1/j ∗ μn dλ An " " "... " " "exp iζ1 "
Ik−1,2
!" " ρ1/j ∗ μ1 dλ A1 " ". Ik−1,2 " !" " " "exp iζn ρ1/j ∗ μn dλ An " " "... Ik−1,1 " !" " " "exp iζ1 ρ1/j ∗ μ1 dλ A1 " " "
(2.19)
Ik−1,1
|ζn | I
≤e
k−1,nk−1
ρ1/j ∗μn dλ An
|ζn | I
ρ1/j ∗μn dλ An
|ζn | I
ρ1/j ∗μn dλ An
e
k−1,2
e
k−1,1
|ζ1 | I
...e
k−1,nk−1
ρ1/j ∗μ1 dλ A1
|ζ1 | I
ρ1/j ∗μ1 dλ A1
|ζ1 | I
ρ1/j ∗μ1 dλ A1
...e ...e
k−1,2
k−1,1
...
. .
Now we are dealing with numbers and so the noncommutativity of the operators is not an issue. Since
ρ1/j ∗ μ1 dλ + Ik−1,1
ρ1/j ∗ μ1 dλ + . . . Ik−1,2
+
1
ρ1/j ∗ μ1 dλ = Ik−1,nk−1
ρ1/j ∗ μ1 dλ = 1,
0
the product of the terms in the column furthest to the right is e|ζ1 |A1 . A similar thing happens in the other columns and so we obtain ˜
Tμ1, j,k ,...,μn, j,k (ei(ζ, A) ) ≤ e(|ζ1 |A1 +|ζ2 A2 +···+|ζn |An ) ≤ e[|ζ1 | +···+|ζn | 2
]
2 1/2
[A1 2 +···+An 2 ]1/2
= er|ζ | (2.20)
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where r = [A1 2 + · · · + An 2 ]1/2 and |ζ | is the Euclidean norm in Rn of the vector (ζ1 , . . . , ζn ). Theorem 1.1 and the inequality (2.20) show that the bounded selfadjoint operators A1 , . . . , An are of Paley–Wiener type (0, r, μ) where μ = (μ1 , . . . , μn ). Remark 2.3 Following a suggestion of Jefferies, Johnson and Nielsen used the main theorem from [11], that is, Theorem 1.1 of this paper, to prove the special case of Theorem 2.2 where n = 2 and the operational calculus is the Weyl calculus. In the closing remark of [11], Johnson and Nielsen asserted that an argument similar to the one in that paper would take care of the case of the Weyl calculus for any n. In fact, the argument in [11] fails at a critical point for n ≥ 3. The following result is a simple consequence of Theorem 1.1 and the proof of Theorem 2.2. Corollary 2.1 Let A1 , . . . , An be bounded, self-adjoint operators on the Hilbert space H and let μ1 , . . . , μn be absolutely continuous probability measures on B ([0, 1]). Finally let ξ = (ξ1 , . . . ξn ) be an n-tuple of real numbers. Then ˜
˜
˜
Tμ1 ,...,μn (ei(ξ, A) ) = Tμ1 ,...μn (eiξ1 A1 · · · eiξn An ) = 1.
(2.21)
Proof Let ζ from Theorem 2.2 (see also Definition 2.1) equal ξ as above. Then we see that the operator on the RHS of (2.18) is unitary and so has norm 1. But, by Theorem 1.1 and the fact that μi, j,k μi for each i = 1, . . . , n, we have the operator norm convergence of the sequence of operators in (2.18) ˜ to Tμ1 ,...,μn (ei(ξ, A) ). The equality in (2.21) follows. The reader may have noticed that we did not need the self-adjointness of the operators nor even the Hilbert space setting until late in the proof of Theorem 2.2. In fact, we can use some of the early parts of the proof to establish a class of “Trotter product formulas” suitable for Feynman’s operational calculi in the general Banach space setting. Theorem 2.4 Let X be a Banach space over C and let ζ = (ζ1 , . . . , ζn ) be an n-tuple of complex numbers. Further, let μ1 , . . . , μn be probability measures on B ([0, 1]) each of which is absolutely continuous with respect to λ; hence, μ1 = g1 λ, . . . , μn = gn λ where g1 , . . . , gn are nonnegative functions in L1 ([0, 1], B ([0, 1]), λ). Finally, let A = (A1 , . . . , An ) be an n-tuple of bounded linear operators on X. Then ˜
˜
Tμ1 ,...,μn (ei(ζ, A) ) − Tμ1,k ,...,μn,k (ei(ζ, A) )L(X) → 0
(2.22)
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as k → ∞ where ˜
˜
˜
Tμ1,k ,...,μn,k (ei(ζ, A) ) = Tμ1,k ,...,μn,k (eiζ1 A1 . . . eiζn An ) gn dλ An . . . exp iζ1 = exp iζn
Ik−1,nk−1
! gn dλ An . . . exp iζ1
exp iζn
Ik−1,2
!
Ik−1 ,nk−1
Ik−1,1
g1 dλ A1 . . .
! g1 dλ A1
Ik−1,2
gn dλ An . . . exp iζ1
exp iζn
!
! g1 dλ A1 .
(2.23)
Ik−1,1
Proof We will just comment on which parts of the proof of Theorem 2.2 are needed here and which are not. We will also note the point at which Theorem 1.1 is applied. The first part of the proof of Theorem 2.2 is needed only for measures which are continuous but not absolutely continuous. We have no such measures here. The second part of the earlier proof involves the Martingale Convergence Theorem. We need this and we obtain fi,k = Pk gi , i = 1, . . . , n with (2.9) suitably adjusted. (Note that we do not need the index j as our measures are all absolutely continuous.) The function f˜i,k is then defined with the integrands in (2.10) changed to gi . The proof now moves along with no j’s involved and with ρ1/j ∗ μi replaced by gi , i = 1, . . . , n. When we reach the summary A-F, A is not needed. Replace B with fi,k λ = (Pk gi )λ gi λ as k → ∞ for i = 1, . . . , n. Change C to fi,k λ − f˜i,k λ 0 as k → ∞ for i = 1, . . . , n. Based on the revised form of B and C, change D to f˜i,k λ gi λ as k → ∞; that is μi,k μi as k → ∞ for i = 1, . . . , n. (Note that in (2.17) μi, j,k , f˜i, j,k and ρ1/j ∗ μi are replaced by μi,k , f˜i,k and gi , respectively.) The calculation in (2.18) is done in the same way but the subscript j on the LHS is missing and ρ1/j ∗ μi is replaced by gi , i = 1, . . . , n, on the RHS. We can now apply Theorem 1.1 to finish the proof of this theorem. Remark 2.5 (a) The inequalities which concerned us toward the end of the proof of Theorem 2.2 did not concern us in the proof of Theorem 2.4 since we were not trying to show that (A1 , . . . , An ) is of Paley–Wiener type (0, r, (μ1 , . . . , μn )). (b) The Trotter products on the RHS of (2.23) look more like the usual Trotter products when special choices are made for ζ . Some examples: (1) Each ζ j equals i (or ti). (2) Each ζ j equals −1 (or −t). (c) We hope to investigate variations and consequences of Theorem 2.4 in later work.
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Remark 2.6 In view of Corollary 3.1 of the paper [14], we may write (2.22) as
˜
Tλ,...,λ (ei(ζ,[g·A]) ) − Tμ1,k ,...,μn,k (ei(ζ, A) )L(X) → 0 (2.24) where [g · A] := [g 1 · A1 ], . . . , [gn · An ] , that is the time independent operators A1 , . . . , An are replaced by the time dependent operators g1 · A1 , . . . , gn · 1 n , . . . , gn = dμ . An where g1 = dμ dλ dλ We now present two simple examples illustrating Theorem 2.4. Example 2.7 For the first example we assume that, in Theorem 2.4, g1 = · · · = gn = 1; i.e. Lebesgue measure is associated to each operator. It follows from [5, Lemma 5.4] that ˜
˜
Tμ1 ,...,μn (ei(ζ, A) ) = Tλ,...,λ (ei(ζ, A) ) = ei(ζ, A)
(2.25)
and so Theorem 2.4 tells us that ei(ζ, A) = lim
! !!nk−1 ζn ζ1 A . . . exp A n 1 nk−1 nk−1
exp
k→∞
(2.26)
Example 2.8 In the second example, we will consider two operators, A1 and A2 and we will take μ1 = 2t dλ and μ2 = 3t2 dλ. For any nonnegative integer l, we have
2l − 1 g1 dλ = 2k−2 , (2.27) 2 Ik−1,l and
g2 dλ = Ik−1,l
3l 2 − 3l + 1 . 23k−3
(2.28)
Theorem 2.4 tells us that ˜
˜
#
˜
˜
lim Tμ1,k ,μ2,k ei(ζ1 A1 +ζ2 A2 )
k→∞
= lim
k→∞
$ 3 · 22k−2 − 3 · 2k−1 + 1 exp iζ2 A2 · 23k−3 $ # 2 · 2k−1 − 1 exp iζ1 A1 · · · 22k−2 # $ # $ 7 3 exp iζ2 3k−3 A2 exp iζ1 2k−2 A1 · 2 2 $ # $ ! # 1 1 exp iζ2 3k−3 A2 exp iζ1 2k−2 A1 2 2
= Tμ1 ,μ2 ei(ζ1 A1 +ζ2 A2 ) .
(2.29)
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Using Corollary 3.1 of [14], the last expression immediately above can be written as Tλ,λ ei(ζ1 [g1 ·A1 ]+ζ2 [g2 ·A2 ]) where the time dependence is carried by the functions g1 and g2 . It is clear from Example 2.8 that we can produce an infinite number of distinct Trotter product formulas by varying the probability densities g1 and g2 .
References 1. Dudley, R.M.: Real Analysis and Probability. Chapman and Hall Mathematics Series, New York (1989) 2. Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84, 108–128 (1951) 3. Jefferies, B.: Spectral properties of noncommuting operators. In: Lecture Notes in Mathematics, vol. 1843. Springer, Berlin Heidelberg New York (2004) 4. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: definitions and elementary properties. Russian J. Math. Phys. 8, 153–171 (2001) 5. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting systems of operators: tensors, ordered supports and disentangling an exponential factor. Math. Notes 70, 815–838 (2001) 6. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting systems of operators: spectral theory. Infin. Dimens. Anal. Quantum Prob. 5, 171–199 (2002) 7. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: the monogenic calculus. Adv. Appl. Clifford Algebras 11, 233–265 (2002) 8. Jefferies, B., Johnson, G.W., Nielsen, L.: Feynman’s operational calculi for time-dependent noncommuting operators. J. Korean Math. Soc. 38(2), (March 2001) 9. Johnson, G.W., Kim, B.S.: Extracting linear and bilinear factors in Feynman’s operational calculi. Math. Phys. Anal. Geom. 6, 181–200 (2003) 10. Johnson, G.W., Lapidus, M.L.: The Feynman integral and Feynman’s operational calculus. In: Oxford Mathematical Monographs. Oxford University Press, Oxford (2000) 11. Johnson, G.W., Nielsen, L.: A stability theorem for Feynman’s operational calculus. In: Conference Proc. Canadian Math. Soc., Conference in Honor of Sergio Albeverio’s 60th birthday, vol. 29, pp. 351–365 (2000) 12. Maslov, V.P.: Operational Methods. Mir, Moscow (1976) 13. Nazaikinskii, V.S., Shatalov, V.E., Sternin, B.Yu.: Methods of noncommutative analysis. In: Studies in Mathematics, vol. 22. Walter de Gruyter, Berlin, Germany (1996) 14. Nielsen, L.: Effects of absolute continuity in Feynman’s operational calculus. Proc. Amer. Math. Soc. 131, 781–791 ( 2002) 15. Nielsen, L.: Time dependent stability for Feynman’s operational calculus. Rocky Mountain J. Math. 35, 1347–1368 (2005) 16. Nielsen, L.: Stability properties of Feynman’s operational calculus for exponential functions of noncommuting operators. Acta Appl. Math. 74, 265–292 (2002) 17. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) 18. Taylor, M.E.: Functions of several self-adjoint operators. Proc. Amer. Math. Soc. 19, 91–98 (1968)
Math Phys Anal Geom (2007) 10:81–95 DOI 10.1007/s11040-007-9022-7
On the Spectral Behaviour of a Non-self-adjoint Operator with Complex Potential Carmen Martínez Adame
Received: 27 September 2006 / Accepted: 26 May 2007 / Published online: 4 July 2007 © Springer Science + Business Media B.V. 2007
Abstract We consider the non-self-adjoint Anderson operator with a complex potential as a pseudo-ergodic operator in one spatial dimension and use second order numerical ranges to obtain tight bounds on the spectrum of the operator. We also find estimates for the size of possible holes contained in the spectrum of such an operator. Keywords Complex potential · Non-self-adjoint Anderson model · Numerical range · Spectrum Mathematics Subject Classifications (2000) 47B80 · 47A12 · 60H25 · 65F15
1 Introduction and General Context The Anderson model was developed in the early 1960’s as a theoretical model to describe the effect of the presence of an impurity atom in a metal and the non-self-adjoint version of this model originated in [9] motivated by the study of superconductivity and has been studied in greater detail in [10, 11]. Since then however, its purely mathematical treatment has grown extensively. It has been the subject of a series of papers by Davies, [1, 2], and by Goldsheid and Khoruzhenko, [5–7]; and in [13] we looked at the non-self-adjoint Anderson operator with real potential and were able to obtain very tight bounds on the spectrum and in many cases to determine it completely.
C. Martínez Adame (B) Departamento de Matemáticas, Facultad de Ciencias, UNAM, Ciudad Universitaria, 04510, México D.F., Mexico e-mail:
[email protected]
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Our interest in this paper is to study the non-self-adjoint Anderson operator H with complex potential as a pseudo-ergodic operator defined on l 2 (Z) and as such to obtain inner and outer bounds on the spectrum of said operator. In Theorem 5 we give an estimate of the size of the hole in the spectrum when 0 ∈ Spec(H) and Theorems 6 and 10–12 provide precise bounds for the spectrum of the operator, which is always a bounded set of C. Theorem 13 is the main result of this paper, it delimits the spectrum of H and allows us to appreciate the strength of the methods developed to study this operator as we come very close to producing a complete determination of the spectrum. 1.1 Pseudo-ergodic Operators Pseudo-ergodic operators have been treated in recent papers and they are the right context in which to work in this case as they allow us to eliminate all probabilistic aspects of the problem at hand. An account of their spectral theory can be found in [4], for instance, and thus, we limit ourselves below to including the definition of these operators which is all that the reader needs to understand the present paper. Definition 1 Let M1 , M2 , M3 be compact subsets of C and let H be an operator defined on l 2 (Z) such that H (x, y) = 0 if |x − y| > 1, H (x, x) ∈ M1 , H (x, x + 1) ∈ M2 , H (x, x − 1) ∈ M3 . We say that H is (Z, M1 , M2 , M3 ) pseudo-ergodic if for every ε > 0, every finite subset F ⊂ Z and every Wr : F −→ Mr , where r = 1, 2, 3, there exists γ ∈ Z such that |H (γ + x, γ + x) − W1 (x)| < ε, |H (γ + x, γ + x + 1) − W2 (x)| < ε, |H (γ + x, γ + x − 1) − W3 (x)| < ε, for all x ∈ F. If M2 and M3 consist solely of one point we will say that H is (Z, M1 ) pseudo-ergodic. 1.2 Higher Order Numerical Ranges Many of the bounds we find for the spectrum of the operator that interests us depend on results obtained from studying the second order numerical range. This concept was introduced in detail and in greater generality in [3, 12, 13] and it should be noted that it differs entirely from other generalisations of the numerical range as those that appear in [8]. In this paper we restrict the presentation of this subject to the following brief account.
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Let H be a bounded linear operator defined on a Hilbert space and let p (z) : C → C be a polynomial. We define Num ( p, H) = z : p (z) ∈ Num ( p (H)) , (1) Num ( p, H) , (2) Numn (H) = deg( p)n
and we call the set defined by (2) the n-th numerical range of H. It then follows that Spec (H) ⊆ Num ( p, T) and it is clear that for any n ∈ N Spec (H) ⊆ Numn (H) .
(3)
2 The Non-self-adjoint Anderson Operator with Complex Potential We consider the (Z, M) pseudo-ergodic operator given by H fn = ±β fn−1 + vn fn + α fn+1
(4)
acting on l (Z), where α, β ∈ R and M is a compact subset of C. It is easy to see that the case α, β ∈ C can be reduced to this case. To do this we need the following lemma whose proof we omit: 2
Lemma 2 Given any two points z, w ∈ C there exist θ and ϕ in R such that zeiθ−iϕ and we−iθ −iϕ are real. Now, let α and β be any two complex numbers; we rewrite them as |α| eiθα and 1 iθβ |β| e respectively. Let θ = 2 π − θα + θβ and let us also define the unitary operator U θ on l 2 (Z) given by U θ fn = e−niθ fn . It follows that U θ HU θ ∗ has the same spectrum as H and in fact, U θ HU θ ∗ is given by U θ HU θ ∗ fn = βe− 2 (π −θα +θβ ) fn−1 + vn fn + αe 2 (π −θα +θβ ) fn+1 . i
i
i If we now consider the operator defined by K = e− 2 (θα +θβ +π ) U θ HU θ ∗ , it i follows that K fn = − |β| fn−1 + e− 2 (θα +θβ +π ) vn fn + |α| fn+1 and hence the desired result follows. Thus, let H fn = β fn−1 + vn fn + α fn+1 with α, β ∈ R and let M be a compact subset of C. The particular case with which we are interested here is the case M = {γ + iδ, γ − iδ, −γ + iδ, −γ − iδ} with γ , δ > 0. We will consider the cases when the coefficient of fn−1 is either β or −β separately, however we note that it is only when we combine the results obtained in both cases that we get a more complete description of the spectrum of H. Let us concentrate first on the case when
H fn = β fn−1 + vn fn + α fn+1 . We will assume, without loss of generality, that H has been suitably scaled so that α − β = 1 and thus Spec (H) ⊂ {z ∈ C : |y| δ + 1} .
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If we split vn into its real and imaginary parts we may rewrite H in the following forms: (6) H = A + iB = C + Vγ + i (B + Vδ ) = H0 + V where B, C, Vγ , Vδ are self-adjoint; B = 1, Vγ = γ , Vδ = δ; V, Vγ , Vδ are diagonal and H0 is normal with spectrum x2 2 2 {(α + β) cos θ + i sin θ : θ ∈ [0, 2π ]} = (x, y) ∈ R : +y =1 . (2a)2 These operators are defined by A fn = a fn−1 + vn fn + a fn+1 ,
V fn = vn fn ,
B fn = b fn−1 − b fn+1 ,
Vγ fn = Re (vn ) fn ,
C fn = a fn−1 + a fn+1 ,
Vδ fn = Im (vn ) fn .
i 1 (α + β) and b = . 2 2 With regard to these operators we have the following theorem which is a corollary of a more general result that Davies proved in [2]. We state it here as it will be of great use throughout this paper: where a =
Theorem 3 The spectrum of H satisfies the following Spec (H) ⊇ {(α + β) cos θ + i sin θ : θ ∈ [0, 2π ]} + M, Spec (H) ⊆ conv {(α + β) cos θ + i sin θ : θ ∈ [0, 2π ]} + conv (M) , Spec (H) ⊆ Spec (H0 ) + B (0, |γ + iδ|) ,
B (m, α + β) . Spec (H) ⊆ m∈M
where B (0, |γ + iδ|) = {z ∈ C : |z| < |γ + iδ|}. As a corollary to Theorem 3 we have the following result with respect to holes in Spec (H). Corollary 4 If |γ + iδ| > α + β, then 0 ∈ Spec (H). Proof This follows from the last inclusion in Theorem 3.
We can also prove the following theorem which provides a more explicit result. Theorem 5 If |γ + iδ| < 1, then Spec (H) does not intersect the interior of the curve given parametrically by
|γ + iδ| cos t 2a |γ + iδ| sin t 2a cos t − , sin t − 4a2 sin2 t + cos2 t 4a2 sin2 t + cos2 t
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with t ∈ [0, 2π ], or the exterior of the curve
|γ + iδ| cos t 2a |γ + iδ| sin t 2a cos t + , sin t + . 4a2 sin2 t + cos2 t 4a2 sin2 t + cos2 t Proof Theorem 3 implies that Spec (H) is contained in the set {2a cos θ + i sin θ : 0 θ 2π } + B (0, |γ + iδ|), or in other words, Spec (H) is contained in the union of balls of radius |γ + iδ| with centre at each point of the ellipse that defines Spec (H0 ). Since |γ + iδ| < 1 there is a hole in the spectrum of H and the origin is not contained in these balls. Thus, the envelope of this family of circles is the curve that gives an estimate of the size of this hole. By construction, the envelope of this family of circles is a curve which is parallel to the ellipse at a distance |γ + iδ|. That is, a curve which is displaced from the ellipse by |γ + iδ| in the direction of the curve’s normal and given this characterization it is straightforward to see that if the ellipse is given by ( f (t) , g (t)) = (2a cos t, sin t) then the envelope is given parametrically by (F (t) , G (t)) where |γ + iδ| cos t F (t) = 2a cos t − 4a2 sin2 t + cos2 t 2a|γ + iδ| sin t . G (t) = sin t − 4a2 sin2 t + cos2 t
Let us now prove the following theorem that provides outer bounds for Spec (H). Theorem 6 The spectrum of H is contained in the set 1 2 2 2 2 z∈C:x −y 2 γ − 2δ − 2 . 2a + 1 Proof It is straightforward to see that 2 H 2 = C + Vγ − (B + Vδ )2 + i[(C + Vγ )(B + Vδ ) + (B + Vδ )(C + Vγ )] and hence, 2 Re H 2 = C + Vγ − (B + Vδ )2 , = C2 + Vγ2 + Vγ C + CVγ − B2 − Vδ2 − Vδ B − BVδ . Now, for any ϕ and ψ in R \ {0} we can write this equality as follows. 2 2 1 1 Re H 2 = ϕC + Vγ + ψ B − Vδ + 1 − ϕ 2 C2 + ϕ ψ 1 1 + 1 − 2 Vγ2 − 1 + ψ 2 B2 − 1 + 2 Vδ2 ϕ ψ
(7)
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and since B2 +
1 C2 4a2
= I, it follows that
2 2 1 1 Re H 2 = ϕC + Vγ + ψ B − Vδ + 1 − ϕ 2 C2 + ϕ ψ 1 1 1 + 1 − 2 Vγ2 − 1 + ψ 2 I − 2 C2 − 1 + 2 Vδ2 ϕ 4a ψ 1 ψ2 1 1 − ϕ 2 C2 + + 2 C2 + 1 − 2 Vγ2 − 2 4a 4a ϕ 1 − 1 + 2 Vδ2 − 1 + ψ 2 I ψ 2 2 as ϕC + ϕ1 Vγ and ψ B − ψ1 Vδ are non-negative. To optimize this last expression let ϕ 2 = 1−ϕ + 2
1+ψ 4a2
2
2a2 +1 2a2
and ψ = 1. This implies that
= 0 and hence 1 1 Re H 2 1 − 2 Vγ2 − 1 + 2 Vδ2 − 1 + ψ 2 I, ϕ ψ 2 2 ϕ −1 ψ +1 2 = V − Vδ2 − 1 + ψ 2 I, γ 2 2 ϕ ψ 1 = Vγ2 − 2Vδ2 − 2I. 2 2a + 1 However, as Vγ = γ and Vδ = δ this last inequality becomes 2 1 γ 2 I − 2δ 2 I − 2I. Re H 2a2 + 1
That is,
Re H
2
1 2 2 γ − 2δ − 2 I. 2a2 + 1
(8)
the spectrum of H is contained in Num ( p2 , H) Hence, if p2 (z) = z2 , then 2 1 and given that Re H 2a2 +1 γ 2 − 2δ 2 − 2 I it follows that Num ( p2 , H) ⊆ z ∈ C : Re z2 2a21+1 γ 2 − 2δ 2 − 2 and the required result follows.
The curve x2 − y2 = 2a21+1 γ 2 − 2δ 2 − 2 which determines the boundary of the set defined by (7) is a hyperbola (as expected since we are dealing with the second order numerical range) whose shape depends upon the values of γ and δ. In fact, it depends upon the sign of η (γ , δ), where η (γ , δ) :=
1 γ 2 − 2δ 2 − 2. 2a2 + 1
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For η (γ , δ) to be negative, γ and δ must satisfy the inequality 1 γ 2 − δ 2 < 1. +2
4a2
(9)
The set of points which satisfy (9) is a hyperbolic region in the γ δ-plane with boundary 4a21+2 γ 2 − δ 2 = 1. If (γ , δ) satisfy (9) then 2a21+1 γ 2 − 2δ 2 − 2 is negative and the boundary of the hyperbolic region described by (7), which is the hyperbola given by the equation x2 − y2 = 2a21+1 γ 2 − 2δ 2 − 2, opens about 2 + 2δ 2 + 2. the imaginary axis and has vertices at the points ±i 2a−1 2 +1 γ If γ and δ satisfy 1 γ 2 − δ 2 = 1, 4a2 + 2 then η (γ , δ) = 0 and the hyperbola defined in (7) turns into two straight lines given by y = ±x. Finally, if 4a21+2 γ 2 − δ 2 > 1 then 2a21+1 γ 2 − 2δ 2 − 2 > 0 and the hyperbola which determines (7) opens about the real axis and has vertices at the points ±
1 γ2 2a2 +1
− 2δ 2 − 2.
Remark 1 It is worth noting at this point that the case when the coefficient of fn−1 is negative in the definition of H will produce a second hyperbola, that will not only bound the spectrum of that operator but will also provide bounds for the spectrum of the operator we are dealing with here by means of a simple rotation and rescaling. Now, for any for any complex number u = s + it let us define a potential V in terms of Vγ and Vδ as follows by setting v˜n = vn − u. We rewrite V = Vγ − sI + i (Vδ − tI) . V = H0 + V = C + Vγ − sI + i (B + (Vδ − tI)), then Let H if and only if z + u ∈ Spec (H) . z ∈ Spec H
(10)
We can thus prove that there exists a hyperbolic region that contains and hence, a hyperbolic region in which Spec (H) lies, namely a Spec H region determined by a hyperbola centered at u = s + it. 2 1 2 2 |s|) |t|) Theorem 7 Re H − − 2 + − 2 I. (γ (δ 2 2a +1 Proof The proof follows in a straightforward manner from the proof of Theorem 6.
Corollary 8 For any s, t ∈ R, Spec (H) is contained in the set 1 2 2 2 2 z ∈ C : (x − s) − (y − t) 2 (γ − |s|) − 2 (δ + |t|) − 2 . 2a + 1
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Proof This result follows from the previous theorem and (10).
These last results imply that the spectrum of H is contained inside a nonempty region determined by a two-parameter family of curves. To obtain the envelope of such a family we will differentiate the two-parameter function with respect to each of the parameters. Lemma 9 Let s, t ∈ R \ {0} and let F (x, y, s, t) = (x − s)2 − (y − t)2 −
1 (γ − |s|)2 + 2 (δ + |t|)2 + 2. 2a2 + 1
If δ 1, then the envelope of this two-parameter family of curves lies on the ellipses defined by 2 2 1 x − sign (s) γ + y + sign (t) δ = 1 2 4a where sign (x) :=
x |x|
for any x ∈ R \ {0}.
Proof The envelope of the family of curves defined by F is obtained by solving ∂F = ∂∂tF = F = 0. However, given that both |s| and |t| appear in the expression ∂s of F we will do this by cases. Let us consider first the case when both s and t are positive: ∂F 2 = −2 (x − s) + 2 (γ − s) , ∂s 2a + 1 ∂F = 2 (y − t) + 4 (δ + t) ∂t
(11) (12)
and setting both equations equal to 0 and substituting these values into F = 0 gives 2 2 2 2a + 1 1 x− x − 2γ − y − (−2δ − y) − 2 2a 2a 2 2 2 1 2a + 1 1 − 2 γ− x − γ + 2 δ + (−2δ − y) + 2 2 2 2a + 1 2a 2a =
1 2 2 4a δ + 4a2 y2 + 8a2 δy − 4a2 + x2 − 2xγ + γ 2 = 0 2 2a
or equivalently 1 (x − γ )2 + (y + δ)2 = 1. 4a2
(13)
However, given the fact that F describes a two-parameter family of curves we need to determine the specific relation between these parameters that yields (13) as the envelope of F. This is done by solving (11) and (12) with respect to x and y and substituting these values into (13) to obtain a relation between s and t.
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2
Now, x is given by 2a2a2 +1 s + 2a21+1 γ and y = −2δ − t and substituting these values into (13) we obtain 2 1 2a2 1 s+ 2 γ − γ + (−2δ − t + δ)2 = 1. 4a2 2a2 + 1 2a + 1 Simplifying this expression we have that
a2 2a2 + 1
2 2 2 (s − γ ) + (δ + t) = 1.
However, given that t is positive we observe that for this equation to be defined we need to have δ 1. If s > 0 and t < 0 then 1 2a2 + 1 x − 2 γ, 2a2 2a t = −y + 2δ
s=
and the envelope lies on the ellipse given by 1 (x − γ )2 + (y − δ)2 = 1 4a2 with the same requirement that δ 1 as s and t need to satisfy the equation a2 (s − γ )2 + (t − δ)2 = 1. 2 (2a +1)2 The other two cases are analogous, and again, only valid if δ 1. If s < 0 and t > 0 then 1 2a2 + 1 x + 2 γ, 2 2a 2a t = −y − 2δ
s=
and the envelope lies on the ellipse both negative then
1 4a2
(x + γ )2 + (y + δ)2 = 1. If s and t are
2a2 + 1 1 x + 2 γ, 2 2a 2a t = −y + 2δ
s=
and the envelope is contained in
1 4a2
(x + γ )2 + (y − δ)2 = 1.
The fact that the results stated in Lemma 9 rely heavily on the sign of s and t imply that only a certain portion of the ellipses is actually obtained as the envelope of F. We will assume that δ 1 and study the case when both s and t are positive in detail and obtain the other three cases from the symmetry of the problem. In this case we know that the envelope of F lies on 4a12 (x − γ )2 + (y + δ)2 = 1. We will restrict the values of s to be between 0 and γ as Theorem 3 gives us better results than what we would obtain by this method from the cases given by s > γ . In fact, Theorem 3 establishes that Spec (H) is contained
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in the convex hull of the four ellipses given by 4a12 (x ± γ )2 + (y ± δ)2 = 1 and contains these four ellipses and hence, gives the best possible information in certain regions of the complex plane. That is, it determines the boundary of the spectrum completely in the four quarter-planes given by {z ∈ C : |x| |γ | and |y| |δ|}. Theorem 10 Let δ 1 and let μ = max γ − 2a 1 − δ 2 ,
1 γ 2 2a + 1
then Spec (H) does not intersect the subset of points z of C such that μ < x < γ and 1 max −δ − 1, −δ − 4a2 − (x − γ )2 < y < −δ − 1 − 2 (x − γ )2 . 4a
Furthermore, let
= B (γ − iδ, 2a) ∩ {z ∈ C : γ − 2a < x < μ & − δ − 1 < y < −δ} . √ If μ = γ − 2a 1 − δ 2 then Spec (H) does not intersect {z ∈ C : F− (x, y, s0 , t0 ) 0} ∩ and if μ =
1 γ 2a2 +1
then Spec (H) does not intersect {z ∈ C : F− (x, y, s1 , t1 ) 0} ∩
where F (x, y, s, t) is as given in Lemma 9 and F− denotes the bottom (or left) branch of the hyperbola and s0 = γ −
2a2 + 1 1 − δ2, a
t0 = 0, s1 = 0,
a2
t1 = −δ +
1−
Proof Let s > 0 and t > 0 satisfy
a2 2 +1 2 2a ( )
2a2 + 1
2 γ 2 .
(s − γ )2 + (t + δ)2 = 1 which is the
equation obtained in the proof of the last Lemma. Taking into account the remarks made previous to the statement of this this implies theorem that we 2a2 +1 can restrict the interval in which s will vary to max 0, γ − a , γ and we will take t ∈ (0, 1 − δ]. 2 We have that x = 2a2a2 +1 s + 2a21+1 γ , and given the interval of variation of s, this implies that x lies either in (γ −2a, γ ) or 2a21+1 γ , γ . Similarly, y = −2δ − t
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and given that t ∈ (0, 1 − δ] this implies that y varies between −δ − 1 and −2δ. However, as the equation of the envelope given by 1 (x − γ )2 + (y + δ)2 = 1 4a2 provides us with a specific relation between x and y, the fact that y can at most be −2δ implies that we have a bound on x as well. Indeed, let us take y = −2δ, then 1 (x − γ )2 + (−2δ + δ)2 = 1, 4a2
x − γ = ±2a 1 − δ 2 √ and hence, x cannot be less than γ − 2a 1 − δ 2 . In other words the portion of 1 (x − γ )2 + (y + δ)2 = 1 that we can obtain as the envelope of the family 4a2 described by F in Lemma 13 will be obtained as x varies in the interval given by 1 2 max γ , γ − 2a 1 − δ , γ . 2a2 + 1 Thus, in light of Theorem 3, this renders the first statement of the theorem. To prove the other two statements we note that there exists an element of the family described by F that intersects the ellipse 4a12 (x − γ )2 + (y + δ)2 = 1 at the point μ, −δ − 1 − 4a12 (μ − γ )2 . √ Let us suppose that μ = γ − 2a 1 − δ 2 . In this case, the member of the family described by F is obtained by letting t → 0+ . This implies that s → √ 2a2 +1 2 γ− a 1 − δ and hence the hyperbola we are interested in is given by the √ 2 point (s0 , t0 ) = γ − 2a a+1 1 − δ 2 , 0 , that is, by the equation x−γ +
2a2 + 1 1 − δ2 a
Let us now suppose that μ =
2 − y2 +
4a2 + 1 δ 2 − 1 = 0. a2
1 . We proceed in an analogous manner and 2a2 +1 2 → −δ + 1 − 2aa2 +1 γ 2 and the hyperbola that
let s → 0+ . This implies that t yields thelast statement of the theorem is determined by the point (s1 , t1 ) = 0, −δ +
1−
a2 γ2 2a2 +1
x − y+δ− 2
and thus is given by the equation
1−
a2 2a2 + 1
2 2 γ 2
−
4a2 + 1 2 2 γ + 4 = 0. 2a2 + 1
We note that this theorem was obtained by restricting the possible values of s and t in Corollary 8 to R+ . If we now remove this restriction we can obtain the following theorem from the symmetry of the problem.
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√ Theorem 11 Let δ 1 and μ = max γ − 2a 1 − δ 2 , 2a21+1 γ . Spec (H) does not intersect the subset of points of C such that μ < |x| < γ and 1 2 2 max −δ − 1, −δ − 4a2 − (x − γ ) < |y| < −δ − 1 − 2 (x − γ ) . 4a Furthermore, if
B (m, 2a) z ∈ C : γ − 2a < |x| < μ and − δ − 1 < |y| < −δ
= m∈M
where √ M = {γ + iδ, γ − iδ, −γ + iδ, −γ − iδ} as before, then, if μ = γ − 2a 1 − δ 2 , Spec (H) does not intersect the set {z ∈ C : F (x, y, s0 , t0 ) 0} ∩ and if μ =
1 γ, 2a2 +1
then Spec (H) does not intersect the set {z ∈ C : F (x, y, s1 , t1 ) 0} ∩
where F (x, y, s, t) is as given in Lemma 9 and 2a2 + 1 2 1−δ , s0 = ± γ − a t0 = 0, s1 = 0,
t1 = ± −δ +
1−
a2 2a2 + 1
2 γ 2 .
Let us now consider H fn = −β fn−1 + vn fn + α fn+1 ,
(14)
we will assume that H has been scaled so that α + β = 1 and rewrite H as in (6) with a = 12 (α − β) and b = 2i . Spec (H) is thus contained in {z ∈ C : |y| δ + 1} . and it is straightforward to see that Theorem 6, Corollary 8 and Theorem 11 hold in this case with no modifications to the proofs as stated previously. We refrain from stating these results again and proceed to combining the two cases given by (4) to obtain better bounds on the spectrum of a given operator. When the coefficient of fn−1 in the definition of an operator of the type considered here is positive the spectrum contains four ellipses whose major axes are parallel to the real axis, whereas if the coefficient of fn−1 is negative the spectrum still contains four ellipses but now the major axes are parallel to the imaginary axis. This is analogous to what happens in the case of real potentials as we have seen in [13]. We will use these facts to obtain tighter bounds for the spectrum of any given operator H defined by (4). In fact, we have the following result which we state for the case H fn = β fn−1 + vn fn + α fn+1 with α − β = 1. We observe that an analogous result is true for H fn = −β fn−1 + vn fn + α fn+1 ,
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however we omit the proof as it is clear that this case can be reduced to the case we present below. Theorem 12 Let H be given by H fn = β fn−1 + vn fn + α fn+1 with α − β = 1. (Fig. 1). Then 1 2 2 2 2 γ − 2δ − 2 , Spec (H) ⊆ z ∈ C : x − y 2 2a + 1 and
8a2 2 2 2 Spec (H) ⊆ z ∈ C : −x + y δ − 2γ − 8a . 1 + 8a2 2
2
Proof The first inclusion is that of Theorem 6. To prove the second inclusion i we define an auxiliary operator by multiplying H by α+β , however, to simplify our calculations here we consider the operator H whose spectrum is clearly − i equal to Spec α+β H and is given by H− fn = −
β i α fn−1 + vn fn + fn+1 . α+β α+β α+β
(15)
This operator is of the type defined in (14) and satisfies the condition that β α + α+β = 1, which is the condition we have set throughout the paper for α+β operators of this type. Applying the analogue of Theorem 6 to H− it follows that ⎫ ⎧ ⎪ ⎪ ⎨ ⎬ 1 δ2 γ2 Spec (H− ) ⊆ z ∈ C : x2 − y2 − 2 − 2 , 2 ⎪ ⎪ (α + β)2 (α + β)2 ⎩ ⎭ 2 1 1 2 α+β
and hence,
Spec (H− ) ⊆ z ∈ C : x2 − y2
2 1 δ2 − 2 γ 2 − 2 1 + 8a2 2a
(16)
where a is defined in terms of H as 12 (α + β). This in turn yields a result in terms of Spec (H) when we rescale by (α+β) , that is, we obtain a hyperbola i that bounds Spec (H) when we rotate the hyperbola defined in (16) by − π2 and scale by (α + β). In other words, given the hyperbola x2 − y2 =
2 1 δ 2 − 2 γ 2 − 2, 2 1 + 8a 2a
we rotate it to obtain −x2 + y2 =
2 1 δ2 − 2 γ 2 − 2 2 1 + 8a 2a
and scale it by a factor of α + β = 2a which produces 2 1 2 2 −x2 + y2 = (2a)2 δ − γ − 2 1 + 8a2 2a2
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and simplifying this we obtain Spec (H) ⊆ z ∈ C : −x2 + y2
8a2 2 2 2 . δ − 2γ − 8a 1 + 8a2
β i α Now, given the operator H− fn = − α+β fn−1 + α+β vn fn + α+β fn+1 which we constructed in the proof of Theorem 12 we can produce a family of hyperbolae which bound Spec (H) where H is given by H fn = β fn−1 + vn fn + α fn+1 as before. This is done by applying Theorem 7 to H− . In other words, if s, t ∈ R and we define $ H− = H− − (s + it) I, then 2 2 1 2 2 $ Re H− (δ − |s|) − 2 (γ + |t|) − 2 I, 8a2 + 1 2a
and hence Spec (H− ) is contained in the set %
z ∈ C : (x − t)2 − (y − s)2
2 1 2 2 |t|) |s|) + − − 2 . (17) − (γ (δ 8a2 + 1 2a2
This in turn provides a bound on Spec (H) as we see in the following theorem. Theorem 13 Let H be given by H fn = β fn−1 + vn fn + α fn+1 where α − β = 1 and let s, t ∈ R. The spectrum of H is contained in the set 8a2 2 2 2 2 2 z ∈ C : − (x + t) + (y + s) (δ − |s|) − 2 (γ + |t|) − 8a . 1 + 8a2 Proof Given (17) and the relation that exists between H and H− , it follows that for each pair of real numbers s, t, Spec (H) is contained in the region obtained by rotating by − π2 and scaling by α + β the region defined by
(x − t)2 − (y − s)2 8a22+1 (δ − |s|)2 − 2a12 (γ + |t|)2 − 2. The figure below shows how the spectrum of a particular operator H is bounded by several elements of this family of hyperbolae together with hyperbolae obtained from Corollary 8.
Fig. 1 In this example α = 1.5, β = 0.5, γ = 1.8 and δ = 0.8 and Spec (H) is bounded by two families of hyperbolae which are obtained via Theorem 12 and by varying s and t in Corollary 8 and Theorem 13
4 3 2 1 0 –1 –2 –3 –4 –6
–4
–2
0
2
4
6
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References 1. Davies, E.B.: Spectral properties of random non-self-adjoint matrices and operators. Proc. Roy. Soc. London Ser. A 457, 191–206 (2001) 2. Davies, E.B.: Spectral theory of pseudo-ergodic operators. Comm. Math. Phys. 216, 687–704 (2001) 3. Davies, E.B.: Spectral bounds using higher order numerical ranges. LMS J. Comput. Math. 8, 17–45 (2005) 4. Davies, E.B., Simon, B.: L1 properties of intrinsic schrödinger operators. J. Funct. Anal. 65, 126–146 (1986) 5. Goldsheid, I.Y., Khoruzhenko, B.A.: Distribution of eigenvalues in non-hermitian Anderson model. Phys. Rev. Lett. 80, 2897–2901 (1998) 6. Goldsheid, I.Y., Khoruzhenko, B.A.: Eigenvalue curves of asymmetric tridiagonal random matrices. Electron. J. Probab. 5(16), 1–28 (2000) 7. Goldsheid, I.Y., Khoruzhenko, B.A.: Regular spacings of complex eigenvalues in the onedimensional non-hermitian Anderson model. Comm. Math. Phys. 238, 505–524 (2003) 8. Gustafson, K.E. Rao, D.K.M.: Numerical Range. The field of values of linear operators and matrices. Springer, New York (1997) 9. Hatano, N., Nelson, D.R.: Localization transitions in non-hermitian quantum mechanics. Phys. Rev. Lett. 77, 570–573 (1996) 10. Hatano, N., Nelson, D.R.: Vortex pinning and non-hermitian quantum mechanics. Phys. Rev. B 56, 8651–8673 (1997) 11. Hatano, N., Nelson, D.R.: Non-hermitian localization and Eigenfunctions. Phys. Rev. B 58, 8384–8390 (1998) 12. Martínez, C.: Spectral Properties of Tridiagonal Operators, PhD Dissertation, King’s College London, London (2005) 13. Martínez, C.: Spectral estimates for the one-dimensional non-self-adjoint Anderson model. J. Operator Theory 56(1), 59–88 (2006)
Math Phys Anal Geom (2007) 10:97–122 DOI 10.1007/s11040-007-9023-6
Positivity of Lyapunov Exponents for a Continuous Matrix-Valued Anderson Model Hakim Boumaza
Received: 28 February 2007 / Accepted: 6 June 2007 / Published online: 11 July 2007 © Springer Science + Business Media B.V. 2007
Abstract We study a continuous matrix-valued Anderson-type model. Both leading Lyapunov exponents of this model are proved to be positive and distinct for all energies in (2, +∞) except those in a discrete set, which leads to absence of absolutely continuous spectrum in (2, +∞). This result is an improvement of a previous result with Stolz. The methods, based upon a result by Breuillard and Gelander on dense subgroups in semisimple Lie groups, and a criterion by Goldsheid and Margulis, allow for singular Bernoulli distributions. Keywords Lyapunov exponents · Anderson model Mathematics Subject Classification (2000) 34F05
1 Introduction We will study the question of separability of Lyapunov exponents for a continuous matrix-valued Anderson–Bernoulli model of the form: d2 01 HAB (ω) = − 2 + + 10 dx ω(n) χ[0,1] (x − n) 0 1 + (1) 0 ω2(n) χ[0,1] (x − n) n∈Z
H. Boumaza (B) Institut de Mathématiques de Jussieu, Université Paris, 7 Denis Diderot, 2 place Jussieu, 75251 Paris, France e-mail:
[email protected]
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acting on L2 (R) ⊗ C2 . This question is coming from a more general problem on Anderson–Bernoulli models. Indeed, localization for Anderson models in dimension d 2 is still an open problem if one look for arbitrary disorder, especially for Bernoulli randomness. A possible approach to the localization for d = 2 is to discretize one direction. It leads to consider onedimensional continuous Schrödinger operators, no longer scalar-valued, but now N × N matrix-valued. Before considering N× N matrix-valued continuous Schrödinger operators, we start with the model (1) corresponding to N = 2. What is already well understood is the case of dimension one scalarvalued continuous Schrödinger operators with arbitrary randomness including Bernoulli distributions (see [6]) and discrete matrix-valued Schrödinger operators also including the Bernoulli case (see [7] and [10]). We aim at combining existing techniques for these cases to prove that for our model (1), the Lyapunov exponents are all positive and distinct for all energies outside a discrete set, at least for energies in (2, +∞) (see Theorem 3). It is already proved in [3] that for model (1), the Lyapunov exponents are separable for all energies except those in a countable set, the critical energies. Due to Kotani’s theory (see [11]) this result already implies the absence of absolutely continuous spectrum in the interval (2, +∞). But the techniques used in [3] didn’t allow us to avoid the case of an everywhere dense countable set of critical energies and we have to keep in mind that we want to be able to use our result to prove Anderson localization and not only the absence of absolutely continuous spectrum. The separability of Lyapunov exponents can be seen as a first step in order to follow a multiscale analysis scheme. The next step would be to prove some regularity on the integrated density of states, like local Hölder-continuity and then to prove a Wegner estimate and an initial length scale estimate to start the multiscale analysis (see [13]). To prove the local Hölder-continuity of the integrated density of states, we need to have the separability of the Lyapunov exponents on intervals (see [5] or [6]). But, if like in [3] we can get an everywhere dense countable set of critical energies, we will not be able to prove local Hölder-continuity of the integrated density of states. This is the main reason of doing the present improvement of the result of [3]. Our approach of the separability of Lyapunov exponents is based upon an abstract criterion in terms of the group generated by the random transfer matrices. This criterion has been provided by Gol’dsheid and Margulis in [7] and was used to prove Anderson localization for discrete strips (see [10]). It is also interesting because it allows for singularly distributed random parameters, including Bernoulli distributions. We had the same approach in [3]; what changes here is the way to apply the criterion of Gol’dsheid and Margulis. We have to prove that a certain group is Zariski-dense in the symplectic group Sp2 (R). In [3] we were constructing explicitly a family of ten matrices linearly independent in the Lie algebra sp2 (R) of Sp2 (R). This construction was only possible by considering energies in (2, +∞) and an everywhere dense countable set of critical energies. By using a result of group theory by Breuillard and Gelander (see [4]), we are here able
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to prove that the group involved in Gol’dsheid–Margulis’ criterion is dense in Sp2 (R) for all energies in (2, +∞), except those in a discrete subset. We start in Section 2 with a presentation of the necessary background on products of i.i.d. symplectic matrices and with a statement of the criterion of Gol’dsheid and Margulis. We also present the result of Breuillard and Gelander in this section. Then, in Section 3 we specify the assumptions made on the model (1) and we make explicit the transfer matrices associated to this model. In Section 4 we give the proof of our main result, Theorem 3 by following the steps given by the assumptions of Theorem 2 by Breuillard and Gelander. We finish by mentioning that different methods have been used in [9] to prove localization properties for random operators on strips. They are based upon the use of spectral averaging techniques which did not allow to handle with singular distributions of the random parameters. So even if the methods used in [9] (which only considers discrete strips) have potential to be applicable to continuous models, one difference between these methods and the ones used here is that, like in [3], we handle singular distributions, in particular Bernoulli distributions.
2 Criterion of Separability of Lyapunov Exponents We start with a review of some results about Lyapunov exponents and how to prove their separability. These results hold for general sequences of i.i.d. random symplectic matrices. Even if we will only use them for symplectic matrices in M4 (R), we will write these results for symplectic matrices in M2N (R) for arbitrary N. Let N be a positive integer. Let Sp N (R) denote the group of 2N × 2N real symplectic matrices, i.e.: Sp N (R) = {M ∈ GL2N (R) | t M J M = J} where J=
0 −I I 0
.
Here, I = I N is the N × N identity matrix. Definition 1 (Lyapunov exponents) Let (Aωn )n∈N be a sequence of i.i.d. random matrices in Sp N (R) with E(log+ ||Aω1 ||) < ∞.
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The Lyapunov exponents γ1 , . . . , γ2N associated with (Aωn )n∈N are defined inductively by p
1 E(log || ∧ p (Aωn . . . Aω1 )||). n→∞ n
γi = lim
i=1
Here, ∧ (Aωn . . . Aω1 ) (Aωn . . . Aω1 ), acting on p
denotes the p-th exterior power of the matrix the p-th exterior power of R2N . For more details about these p-th exterior powers, see [2].
One has γ1 . . . γ2N . Moreover, the random matrices (An )n∈N being symplectic, we have the symmetry property γ2N−i+1 = −γi , for i = 1, . . . , N (see [2], Proposition 3.2). We say that the Lyapunov exponents of a sequence (Aωn )n∈N of i.i.d. random matrices are separable when they are all distinct: γ1 > γ2 > . . . > γ2N . We now give a criterion of separability of the Lyapunov exponents. For the definitions of L p -strong irreducibility and p-contractivity we refer to [2], Definitions A.IV.3.3 and A.IV.1.1, respectively. Let μ be a probability measure on Sp N (R). We denote by Gμ the smallest closed subgroup of Sp N (R) which contains the topological support of μ, supp μ. Now we can set forth the main result on separability of Lyapunov exponents, which is a generalization of Furstenberg’s theorem to the case N > 1. Proposition 1 Let (Aωn )n∈N be a sequence of i.i.d. random symplectic matrices of order 2N and p be an integer, 1 p N. Let μ be the common distribution of the Aωn . If (a) Gμ is p-contracting and L p -strongly irreducible, (b) E(log Aω1 ) < ∞, then the following holds: 1. γ p > γ p+1 2. For any non zero x in L p : 1 lim E log ∧ p Aωn . . . Aω1 x = γi . n→∞ n i=1 p
Proof See [2], Proposition 3.4. Corollary 1 If (a) Gμ is p-contracting and L p -strongly irreducible for p = 1, . . . , N, (b) E(log Aω1 ) < ∞, then γ1 > γ2 > . . . > γ N > 0.
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Proof Use Proposition 1 and the symmetry property of Lyapunov exponents. For explicit models like (1), it can be quite difficult to check the pcontractivity and the L p -strong irreducibility for all p. To avoid this difficulty, we will use the Gol’dsheid–Margulis theory presented in [7] which gives an algebraic criterion to check these assumptions. The idea is the following: if the group Gμ is large enough in an algebraic sense then Gμ is p-contractive and L p -strongly irreducible for all p. We first recall the definition of the Zariski topology on M2N (R). We identify 2 M2N (R) to R(2N) by identifying a matrix to the list of its entries. Then for S ⊂ R[X1 , . . . , X(2N)2 ], we set: V(S) = {x ∈ R(2N) | ∀P ∈ S, P(x) = 0} 2
So, V(S) is the set of common zeros of the polynomials in S. These sets V(S) 2 are the closed sets of the Zariski topology on R(2N) . Then, on any subset of M2N (R) we can define the Zariski topology as the topology induced by the Zariski topology on M2N (R). In particular we define in this way the Zariski topology on Sp N (R). The Zariski closure of a subset G of Sp N (R) is the smallest Zariski closed subset that contains G. We denote it by ClZ (G). In other words, if G is a subset of Sp N (R), its Zariski closure ClZ (G) is the set of zeros of polynomials vanishing on G. A subset G ⊂ G is said to be Zariski-dense in G if ClZ (G ) = ClZ (G), i.e., each polynomial vanishing on G vanishes on G. Being Zariski-dense is the meaning of being large enough for a subgroup of Sp N (R) to be p-contractive and L p -strongly irreducible for all p. More precisely, from the results of Gol’dsheid and Margulis one gets: Theorem 1 (Gol’dsheid–Margulis criterion, [7]) If Gμ is Zariski dense in Sp N (R), then for all p, Gμ is p-contractive and L p -strongly irreducible. Proof It is explained in [3] how to get that criterion from the results of Gol’dhseid and Margulis stated in [7]. As we can see in [3], it is not easy to check directly that the group Gμ E introduced there is Zariski-dense. In [3] we were reconstructing explicitly the Zariski closure of Gμ E . But this construction was possible only for energies not in a dense countable subset of R. We will now give a way to prove more systematically the Zariski-density of a subgroup of Sp N (R). It is based on the following result of Breuillard and Gelander: Theorem 2 (Breuillard, Gelander [4]) Let G be a real, connected, semisimple Lie group, whose Lie algebra is g. Then there is a neighborhood O of 1 in G, on which log = exp−1 is a well defined diffeomorphism, such that g1 , . . . , gm ∈ O generate a dense subgroup whenever log g1 , . . . , log gm generate g.
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We will use this theorem in the sequel to prove that the subgroup generated by the transfer matrices associated to our operator is dense, hence Zariskidense, in Sp N (R). In the next section we will specify the assumptions on model (1) and give the statement of our main result.
3 A Matrix-Valued Continuous Anderson Model Let ω(n) χ[0,1] (x − n) d2 0 1 HAB (ω) = − 2 + V0 + dx 0 ω2(n) χ[0,1] (x − n) n∈Z
(2)
be a random Schrödinger operator acting in L2 (R) ⊗ C2 . Here • • •
χ[0,1] denotes the characteristic function of the interval [0, 1], 01 , V0 is the constant-coefficient multiplication operator by 10 (ω1(n) )n∈Z , (ω2(n) )n∈Z are two independent sequences of i.i.d. random variables with common distribution ν such that {0, 1} ⊂ supp ν. 2
2
d d This operator is a bounded perturbation of − dx 2 ⊕ − dx2 . Thus it is self-adjoint 2 2 on the Sobolev space H (R) ⊗ C . For the operator HAB (ω) defined by (2) we have the following result:
Theorem 3 Let γ1 (E) and γ2 (E) be the positive Lyapunov exponents associated to HAB (ω). There exists a discrete subset SB ⊂ R such that γ1 (E) > γ2 (E) > 0 for all E > 2, E ∈ / SB . Corollary 2 HAB (ω) has no absolutely continuous spectrum in the interval (2, +∞). We will first specify some notations. We consider the differential system: HAB u = Eu,
E ∈ R.
(3) (n)
For a solution u = (u1 , u2 ) of this system we define the transfer matrix Aωn (E), n ∈ Z from n to n + 1 by the relation ⎞ ⎛ ⎞ ⎛ u1 (n) u1 (n + 1) ⎜ u2 (n) ⎟ ⎜ u2 (n + 1) ⎟ ω(n) ⎟ ⎜ ⎟ ⎜ ⎝ u1 (n + 1) ⎠ = An (E) ⎝ u 1 (n) ⎠ . u 2 (n + 1) u 2 (n)
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(n)
The sequence {Aωn (E)}n∈Z is a sequence of i.i.d. random matrices in the symplectic group Sp2 (R). This sequence will determine the Lyapunov exponents at energy E. In order to use Proposition 1, it is necessary to define a measure on (n) Sp2 (R) adapted to the sequence {Aωn (E)}n∈Z . The distribution μ E is given by: (0) μ E () = ν ω(0) = ω1(0) , ω2(0) ∈ (supp ν)2 | Aω0 (E) ∈ (0)
for any Borel subset ⊂ Sp2 (R). The distribution μ E is defined by Aω0 (E) (n) alone because the random matrices Aωn (E) are i.i.d. Definition 2 We denote by Gμ E the smallest closed subgroup of Sp2 (R) generated by the support of μ E . Since {0, 1} ⊂ supp ν, A(0,0) (E), A(1,0) (E), A(0,1) (E), A(1,1) (E) ∈ Gμ E . We 0 0 0 0 need to work with explicit forms of these four transfer matrices. First, we set: Mω(0) =
ω1(0) 1 1 ω2(0)
.
(4)
(0)
We start by writing Aω0 (E) as an exponential. We associate to the second order differential system (3) the following first order differential system: Y =
0 Mω(0)
I2 −E 0
Y
(5)
with Y ∈ M4 (R). If Y is the solution with initial condition Y(0) = I4 , then (0) Aω0 (E) = Y(1). Solving (5), we get: (0)
Aω0 (E) = exp
0 Mω(0)
I2 −E 0
.
(6)
To compute this exponential, we have to compute the successive powers of Mω(0) . To do this, we diagonalize the real symmetric matrix Mω(0) by an orthogonal matrix Sω(0) : Mω(0) =
ω1(0) 1 1 ω2(0)
= Sω(0)
(0)
λω1 0 (0) 0 λω2
S−1 , ω(0)
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(0)
the eigenvalues λω2 λω1 of Mω(0) being real. We can compute these eigenvalues and the corresponding matrices Sω(0) for the different values of ω(0) ∈ {0, 1}2 . We get: 1 1 1 S(0,0) = √ , λ(0,0) = 1, λ(0,0) = −1, 1 2 2 1 −1 λ(1,1) = 2, 1
S(1,1) = S(0,0) , ⎛ ⎞ √ 2 √ √ 2 √ 10−2 5 10+2 √ 5 ⎠, S(1,0) = ⎝ −1+√5 −1− 5 √ √ √ √ ⎛ S(0,1) = ⎝
10−2 5
√
2
√ 10−2 √ 5 √1− 5√ 10−2 5
10+2 5
√
2
√ 10+2 √ 5 √1+ 5√ 10+2 5
λ(1,0) 1
√ 1+ 5 = , 2
λ(1,0) 2
√ 1− 5 = , 2
λ(0,1) 1
√ 1+ 5 = , 2
λ(0,1) 2
√ 1− 5 = . 2
⎞ ⎠,
We also define the block matrices:
Rω(0) =
λ(1,1) = 0, 2
Sω(0) 0 0 Sω(0)
.
Let E > 2 be larger than all eigenvalues of all Mω(0) . With the abbreviation (0) rl = rl (E, ω(0) ) := E − λlω , l = 1, 2, the transfer matrices become ⎛
cos r1
⎜ 0 (0) ⎜ Aω0 (E) = Rω(0) ⎜ ⎝ −r1 sin r1 0
0
sin r1 r1
0
⎞
sin r2 ⎟ cos r2 0 ⎟ −1 r2 ⎟ R (0) . 0 cos r1 0 ⎠ ω −r2 sin r2 0 cos r2
(7)
4 Proof of Theorem 3 We will show in the last part of this section that Theorem 3 can be easily deduced from the following proposition: Proposition 2 There exists a discrete subset SB ⊂ R such that Gμ E = Sp2 (R) for all E > 2, E ∈ / SB . Sections 4.1–4.3 are devoted to the proof of Proposition 2. To prove this proposition, we will follow Theorem 2 for G = Sp2 (R). Let O be a neighborhood of the identity in G = Sp2 (R) as in Theorem 2. 4.1 Elements of Gμ E in O To apply Theorem 2 we need to work with elements in the neighborhood O of the identity. We will work with the four matrices A(0,0) (E), A(1,0) (E), A(0,1) (E) 0 0 0
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and A(1,1) (E) which are in Gμ E . We will prove that by taking a suitable power 0 of each of these matrices we find four matrices in Gμ E which lies in an arbitrary small neighborhood of the identity and thus in O. For this purpose we will use a simultaneous diophantine approximation result. Theorem 4 (Dirichlet [12]) Let α1 , . . . , α N be real numbers and let M > 1 be an integer. There are integers y, x1 , . . . , x N in Z such that 1 y M and |αi y − xi | < M− N 1
for i = 1, . . . , N. From this theorem we deduce the proposition: Proposition 3 Let E ∈ (2, +∞). For all ω(0) ∈ {0, 1}2 , there is an integer mω (E) 1 such that: (0)
Aω0 (E)mω (E) ∈ O. r1 Proof We fix ω(0) ∈ {0, 1}2 . Let M > 1 be an integer. Theorem 4 with α1 = 2π r2 and α2 = 2π leads to the existence of y, x1 , x2 ∈ Z, such that 1 y M and r r 1 1 1 2 y − x1 < M − 2 , y − x2 < M − 2 , 2π 2π
which be can be written as: |r1 y − 2x1 π | < 2π M− 2 , 1
|r2 y − 2x2 π | < 2π M− 2 . 1
(8)
Let θi = yri − 2π xi , i = 1, 2. Then we have: ⎛
⎞ sin yr1 cos yr1 0 0 r1 sin yr2 ⎟ ⎜ (0) 0 cos yr2 0 ⎟ R−1(0) r2 Aω0 (E) y = Rω(0) ⎜ ⎝ −r sin yr ⎠ ω 0 cos yr 0 1 1 1 0 −r2 sin yr2 0 cos yr2 ⎛ ⎞ sin θ1 cos θ1 0 0 r1 sin θ2 ⎟ ⎜ 0 cos θ2 0 ⎟ R−1(0) r2 = Rω(0) ⎜ ⎝ −r sin θ 0 cos θ1 0 ⎠ ω 1 1 0 −r2 sin θ2 0 cos θ2 by 2π -periodicity of sinus and cosinus. Let ε > 0. If we choose M large enough, 1 M− 2 will be small enough to get: ⎛ ⎞ sin θ1 cos θ1 0 0 r1 sin θ2 ⎟ ⎜ 0 0 cos θ2 ⎟ − I4 < ε. ⎜ r2 ⎝ −r sin θ ⎠ 0 cos θ1 0 1 1 0 −r2 sin θ2 0 cos θ2
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The matrices Sω(0) being orthogonal, so are also the matrices Rω(0) . Then conjugating by Rω(0) does not change the norm and: (0)
Aω0 (E) y − I4 < ε. As O depends only on the semisimple group Sp2 (R), we can choose ε such that B(I4 , ε) ⊂ O. So if we set y = mω (E), we have 1 mω (E) M and: (0)
Aω0 (E)mω (E) ∈ O.
Remark It is important to note that the neighborhood O does not depend neither on E nor on ω(0) . So the integer M > 1 also does not depend neither on E nor on ω(0) . It will be crucial in some step of the proof to say that even if the integer mω (E) depends on E and ω(0) , it belongs always to an interval of integers {1, . . . , M} independent of E and ω(0) . To apply Theorem 2, we need to show that the logarithms of the matrices (0) Aω0 (E)mω (E) generate the Lie algebra sp2 (R) of Sp2 (R). A first difficulty is to (0) compute the logarithm of Aω0 (E)mω (E) which belongs to log O. 4.2 Computation of the Logarithm of Aω0 (E)mω (E) We fix ω(0) ∈ {0, 1}2 . We assume E > 2. Let ϑi = mω (E)ri , i = 1, 2. To compute the logarithm of Aω0 (E)mω (E) , we start from its expression: ⎛
cos ϑ1
⎜ 0 (0) ⎜ Aω0 (E)mω (E) = Rω(0) ⎜ ⎝ −r1 sin ϑ1 0
0
sin ϑ1 r1
0
⎞
sin ϑ2 ⎟ cos ϑ2 0 ⎟ −1 r2 ⎟ R (0) . 0 cos ϑ1 0 ⎠ ω −r2 sin ϑ2 0 cos ϑ2
We can always permute the vectors of the orthonormal basis defined by the columns of Rω(0) . So there exists a permutation matrix Pω(0) (thus orthogonal) such that: (0)
Aω0 (E)mω (E)
⎛
⎞ sin ϑ1 0 0 cos ϑ1 r1 ⎜ −r sin ϑ cos ϑ 0 0 ⎟ 1 1 ⎜ 1 ⎟ −1 −1 = Rω(0) Pω(0) ⎜ ⎟ Pω(0) Rω(0) . sin ϑ 2 ⎝ ⎠ 0 0 cos ϑ2 r2 0 0 −r2 sin ϑ2 cos ϑ2
Recall that we can choose mω (E) such that Aω0 (E)mω (E) is arbitrarily close to the identity in Sp2 (R). Particularly we can assume that: (0) ω A0 (E)mω (E) − I4 < 1 .
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So we can use the power series of the logarithm: (0)
log Aω0 (E)mω (E) =
(−1)k+1 k
k1
k (0) Aω0 (E)mω (E) − I4 .
(9)
To simplify our computations we will also use the complex forms of sinus and cosinus. We set: ⎞ ⎛ i i − r1 r1 0 0 ⎜ 1 1 0 0⎟ ⎟ ⎜ Qω(0) = ⎜ ⎟. ⎝ 0 0 − ri ri ⎠ 2
0 Hence:
0
⎛
Q−1 ω(0)
ir1 1⎜ −ir 1 = ⎜ 2⎝ 0 0
1
2
1
1 0 1 0 0 ir2 0 −ir2
⎞ 0 0⎟ ⎟ 1⎠ 1
Let κl± = e±imω (E)rl ,
l = 1, 2.
(10)
Then we have: (0)
Aω0 (E)mω (E) − I4
⎞ κ1+ − 1 0 0 0 ⎜ 0 κ− − 1 0 0 ⎟ 1 ⎟ Q−1 P−1 R−1 . = Rω(0) Pω(0) Qω(0) ⎜ ⎝ 0 0 κ2+ − 1 0 ⎠ ω(0) ω(0) ω(0) 0 0 0 κ2− − 1 ⎛
So by using (9) we only have to compute: +∞ (−1)k+1 k=1
k
(κl± − 1)k .
Let Ln be the main determination of the complex logarithm defined on C \ R− . We want to write, for l = 1, 2: +∞ (−1)k+1
k
k=1
(κl± − 1)k = Ln κl± .
To write this, we have to assume that rl = introduce the discrete set (0)
E − λlω
(0)
∈ / π + 2π Z. So we
S1 = {E > 2 | E = −λlω + π + 2 jπ for j ∈ Z, l = 1, 2, ω(0) ∈ {0, 1}2 }.
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If we choose E > 2, E ∈ / S1 we can write: (0)
log Aω0 (E)mω (E)
⎛
⎞ 0 0 Ln κ1+ 0 ⎜ 0 Ln κ − 0 0 ⎟ 1 ⎟ Q−1 P−1 R−1 . = Rω(0) Pω(0) Qω(0) ⎜ ⎝ 0 0 Ln κ2+ 0 ⎠ ω(0) ω(0) ω(0) 0 0 0 Ln κ2−
Therefore, we are left with computing Ln κl± . We do this for l = 1, the computation will be the same for l = 2. We have: Ln κ1+ = i Arg κ1+ = i Arcsin sin ϑ1 mω (E)r1 mω (E)r1 1 + 12 = i mω (E)r1 − π + (−1) π π 2
(11)
where · in (11) denotes the integer part. We recall that by (8), mω (E)r1 can 1 is arbitrarily be chosen arbitrarily close to 2π Z., i.e. we can assume that mω (E)r π − 12 close to an even integer. It suffices to choose M such that 2M < 12 to have mω (E)r1 1 even and more precisely equal to 2x1 . Thus (11) becomes: + π 2 mω (E)r1 1 + . Ln κ1+ = i mω (E)r1 − π π 2
We have the corresponding equation for the conjugate logarithm: m (E)r mω (E)r1 1 − ω π 1 + 12 − Ln κ1 = i −mω (E)r1 − π − + (−1) . π 2 Then:
− ri1 ri1 1 1 1 = 2
×
Ln κ1+ 0 0 Ln κ1−
+
Ln κ1−
and, for all x ∈ R:
×
1 2
ir1 1 −ir1 1
= −iπ
x+
(13)
Ln κ1+ + Ln κ1− − ri1 Ln κ1+ − Ln κ1− ir1 Ln κ1+ − Ln κ1− Ln κ1+ + Ln κ1−
By (12) and (13) we have: Ln κ1+
(12)
(14)
mω (E)r1 1 mω (E)r1 1 + + − + π 2 π 2
1 1 1 if x ∈ 12 + Z, + −x = 0 otherwise. 2 2
l We can assume that mω (E)r is arbitrarily close to an even number, hence we π l can assume that for l = 1, 2, mω (E)r does not belong to 12 + Z. So we have: π
Ln κ1+ + Ln κ1− = 0
(15)
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and: Ln κ1+ − Ln κ1−
1 mω (E)r1 1 mω (E)r1 − − − + = 2imω (E)r1 − iπ π 2 π 2 mω (E)r1 1 = 2imω (E)r1 − 2iπ − . π 2
Let, for l = 1, 2: 1 xl = xl (E, ω) := 2
(16)
mω (E)rl 1 − . π 2
αl = −mω (E)rl2 + 2πrl xl , βl = mω (E) −
2π xl . rl
(17)
Putting (15) and (16) into (14), and doing the same for the block corresponding to r2 , we get: ⎛ ⎞ 0 β1 0 0 ⎜ α1 0 0 0 ⎟ −1 −1 (0) ⎟ log Aω0 (E)mω (E) = Rω(0) Pω(0) ⎜ ⎝ 0 0 0 β2 ⎠ Pω(0) Rω(0) 0 0 α2 0 ⎛ ⎞ 0 0 β1 0 ⎜ 0 0 0 β2 ⎟ −1 ⎟ = Rω(0) ⎜ ⎝ α1 0 0 0 ⎠ Rω(0) 0 α2 0 0 We set: (0)
L Aω(0) := log Aω0 (E)mω (E) . We can summarize the computations we have done in this section. For all E > 2, E ∈ / S1 : ⎛ ⎞ 0 0 β1 0 ⎜ 0 0 0 β2 ⎟ −1 ⎟ L Aω(0) = Rω(0) ⎜ (18) ⎝ α1 0 0 0 ⎠ Rω(0) . 0 α2 0 0 We have now to prove that the four matrices L Aω(0) , for ω(0) ∈ {0, 1}2 , generate the whole Lie algebra sp2 (R). 4.3 The Lie Algebra la2 (E) For E ∈ (2, +∞) \ S1 , we denote by la2 (E) the Lie subalgebra of sp2 (R) generated by the L Aω(0) for ω(0) ∈ {0, 1}2 . We will use the expressions of λiω(0) and Sω computed in Section 3.
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4.3.1 Notations We set:
√ m(0,0) (E) E − 1 1 a1 = x1 (E, (0, 0)) = + π 2 √ m(0,0) (E) E + 1 1 + a2 = x2 (E, (0, 0)) = π 2 ⎢ √ ⎢ ⎢ m(1,0) (E) E − 1+2 5 b1 = x1 (E, (1, 0)) = ⎣ + π ⎢ ⎢ ⎢ m(1,0) (E) E − b2 = x2 (E, (1, 0)) = ⎣ π
and
⎢ ⎢ ⎢ m(0,1) (E) E − c1 = x1 (E, (0, 1)) = ⎣ π ⎢ ⎢ ⎢ m(0,1) (E) E − c2 = x2 (E, (0, 1)) = ⎣ π
√ 1− 5 2
√ 1+ 5 2
√ 1− 5 2
⎥ ⎥ 1⎥ ⎦ 2
⎥ ⎥ 1⎥ + ⎦ 2 ⎥ ⎥ 1⎥ + ⎦ 2 ⎥ ⎥ 1⎥ + ⎦ 2
√ m(1,1) (E) E 1 d1 = x1 (E, (1, 1)) = + π 2 √ m(1,1) (E) E − 2 1 + . d2 = x2 (E, (1, 1)) = π 2 We denote by M[i, j ] the (i, j ) entry of a matrix M. We also set: √ √ r100 = E − 1, r200 = E + 1, √ √ r111 = E − 2, r211 = E, √ √ 1 + 1− 5 5 10 01 10 01 r1 = r1 = E − , r2 = r2 := E − , 2 2 and finally we set:
√ √ √ 1+ 5 1− 5 E− D1 (E) = E − 1 E + 1 E − , 2 2 √ √ √ √ 1+ 5 1− 5 . D2 (E) = E E − 2 E − E− 2 2 √
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To prove that la2 (E) = sp2 (R), we will find a family of 10 matrices linearly independent in la2 (E). First we will consider the subspace generated by the Lie brackets [L Aω(0) , L Aω˜ (0) ]. 4.3.2 The Subspace V1 Generated by the [L Aω(0) , L Aω˜ (0) ] A direct computation shows that each Lie bracket [L Aω(0) , L Aω˜ (0) ] is of the form
A 0 0 −t A
(19)
for some A ∈ M2 (R). Let V1 be the 4-dimensional subspace of sp2 (R) of matrices of the form (19). We will show that outside a discrete set of energies E, the four Lie brackets ϒ1 = [L A(1,0) , L A(0,0) ],
ϒ2 = [L A(1,0) , L A(1,1) ],
ϒ3 = [L A(0,1) , L A(0,0) ],
ϒ4 = [L A(0,1) , L A(0,0) ]
generate V1 . Expression of ϒ1 = [L A(1,0) , L A(0,0) ]. We give the expressions of the entries. By (19) it suffices to give the entries corresponding to the first diagonal 2 × 2 block. 1 ϒ1 [1, 1] = − √ × 4 5D1 (E) √ × − π a1r200 + a2r100 + 2m00r100r200 π b1 (1 + 5)r210 − −π b2 (1 − ϒ1 [1, 2] =
√ 10 √ ! 5)r1 − 2 5m10r110r210
10 ! π2E b1r2 − b2r110 a1r200 − a2r100 √ 2 5D1 (E)
π × ϒ1 [2, 1] = − √ 4 5D1 (E) × π a2r100 − 5a1r200 + 4m00r100r200 b1r210 + b2r110 + √ ! + a1r200 − a2r100 2 5m10r110r210 + 2π E b1r210 − b2r110 ϒ1 [2, 2] =
10 ! π2 b1r2 − b2r110 a1r200 − a2r100 . √ 2 5D1 (E)
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Expression of ϒ2 = [L A(0,1) , L A(0,0) ]. We have: 1 × 20D1 (E) √ √ × 10 5π m00r100r200 − 5π 2 a2r100 + 3a1r200 c1r210 − c2r110 + + 5 π 2 a1r200 − 3a2r100 + 2π m00r100r200 c1r210 + c2r110 − ! −10 π m01 a1r200 − 3a2r100 + 2m00 m01r100r200 r110r210
ϒ2 [1, 1] = −
1 ϒ2 [1, 2] = − √ × 2 5D1 (E) × π 2 a1r200 − 3a2r100 + π 2 E a1r200 − a2r100 + √ +(2 + 2 5)π m00r100r200 c1r210 − c2r110 − √ − 5π 2 a1r200 + a2r100 c1r210 + c2r110 + √ ! +2 5 π m01 a1r200 + a2r100 − 2m00 m01r100r200 r110r210 1 × 20D1 (E) × 5π 2 a1r200 + 3a2r100 − 20π m00r100r200 c1r210 + c2r110 + √ + 5π 2 (2E − 5) a1r200 − a2r100 c1r210 − c2r110 − ! − 10 π m01 a1r200 + 3a2r100 − 4m00 m01r100r200 r110r210 π × ϒ2 [2, 2] = − 10D1 (E) √ × 5π a1r200 − a2r100 c1r210 + c2r110 + 2 5 π a1r200 + a2r100 + + 2m00r100r200 c1r210 − c2r110 + ! + 10m01 a1r200 + a2r100 r110r210 . ϒ2 [2, 1] = −
Expression of ϒ3 = [L A(1,0) , L A(1,1) ]. We have: π × 10D2 (E) √ × 2 5 2m11r111r211 − π d1r211 + d2r111 b1r210 − b2r110 + ! + 5π d2r111 − d1r211 b1r210 + b2r110 + 10m10 d1r211 − d2r111 r110r210
ϒ3 [1, 1] = −
ϒ3 [1, 2] =
1 × 20D2 (E) √ × 5π 2 d1r211 − d2r111 (2E − 3) b1r210 − b2r110 +
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+ 5π 2 d1r211 + 3d2r111 − 20π m11r111r211 b1r210 + b2r110 + ! + 40m11 m10r111r211 − 10π m10 d1r211 + 3d2r111 r110r210 1 × 10D2 (E) √ √ × 2π 2 5 2d2r111 − d1r211 + π 2 5E d1r211 − d2r111 − √ −2 π 5m11r111r211 b1r210 − b2r110 + + 10π m11r111r211 − 5π 2 d1r211 + d2r111 b1r210 + b2r110 + ! + 10π m10 d1r211 + d2r111 − 20m11 m00r111r211 r110r210
ϒ3 [2, 1] = −
ϒ3 [2, 2] = −
1 × 20D2 (E) √ √ × 10π 5m11r111r211 − π 2 5 3d1r211 + 7d2r111 b 1r210 − b 2r110 + + 5π 2 3d2r111 − d1r211 − 10π m11r111r211 b1r210 + b2r110 + ! + 10π m10 d1r211 − 3d2r111 + 20m11 m00r111r211 r110r210 .
Expression of ϒ4 = [L A(0,1) , L A(1,1) ]. We have: ! 11 π2 ϒ4 [1, 1] = √ d1r2 − d2r111 c1r210 − c2r110 2 5D2 (E) π × ϒ4 [1, 2] = √ 4 5D2 (E) × π d1r211 + 3d2r111 + 2π E d1r211 − d2r111 − 4m11r111r211 × √ × c1r210 − c2r110 + 5π d2r111 − d1r211 c1r210 + c2r110 √ ! + 2 5m01 d1r211 − d2r111 r110r210 ! π2 (E − 1) d1r211 − d2r111 c1r210 − c2r110 ϒ4 [2, 1] = − √ 2 5D2 (E) 1 ϒ4 [2, 2] = − √ × 4 5D2 (E) √ × 2m11r111r211 − π d1r211 + d2r111 2 5m01r110r210 + √ ! + π c1r210 − c2r110 − 5π c1r210 + c2r110 . To prove that la2 (E) = sp2 (R), we will build a family of 10 matrices linearly independent in la2 (E). First we will consider the subspace generated by the Lie brackets [L Aω(0) , L Aω˜ (0) ].
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Then we consider the determinant of these entries: ⎛
⎞ ϒ1 [1, 1] ϒ2 [1, 1] ϒ3 [1, 1] ϒ4 [1, 1] ⎜ ϒ1 [1, 2] ϒ2 [1, 2] ϒ3 [1, 2] ϒ4 [1, 2] ⎟ ⎟ det ⎜ ⎝ ϒ1 [2, 1] ϒ2 [2, 1] ϒ3 [2, 1] ϒ4 [2, 1] ⎠ ϒ1 [2, 2] ϒ2 [2, 2] ϒ3 [2, 2] ϒ4 [2, 2] = f1 (E) = f˜1 (a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 , m00 , m01 , m10 , m11 , E) (20) where f˜1 (X1 , . . . , X12 , Y) is a polynomial function in X1 , . . . , X12 , analytic in jk Y. Indeed, the determinant (20) is a rational function in the ri which are analytic functions in E not vanishing on the interval (2, +∞). Note that all coefficients a1 , . . . , d2 , m00 , . . . , m11 depend also on E and are not analytic in E. Hence f1 is a priori not analytic in E. We will now explain how to avoid this difficulty. We recall that for all E and ω, 1 mω (E) M with M independent of E and ω. Thus mω (E) only take a finite number of values in the set {1, . . . , M}. Then we consider the sequence of intervals I2 = (2, 3], I3 = [3, 4], and for all k 3, Ik = [k, k + 1]. These intervals cover (2, +∞). We fix k 2 and we assume that E ∈ Ik . Then the integers xiω (E)
=
" mω (E) E − λiω 1 + π 2
are bounded by a constant depending only on M and Ik . Indeed, the eigenvalues λiω are all in the fixed interval [−2, 2], mω (E) take its values in {1, . . . , M} and E ∈ Ik . So the integers xiω (E) take only a finite number of values in a set {0, . . . , Nk }. To study the zeros of the function f1 on Ik , we have only to study the zeros of a finite number of analytic functions: f˜1, p,l : E → f˜1 ( p1 , . . . , p8 , l1 , . . . , l4 , E) for pi ∈ {0, . . . , Nk } and l j ∈ {1, . . . , M}. We have to show that the functions f˜1, p,l do not vanish identically on Ik . In fact, the only bad case is when all the xiω are zero. Indeed, f˜1 (0, . . . , 0, X9 , . . . , X12 , Y) is identically zero. But if we look at the values of xiω for E > 2 and mω (E) 1, we get that a2 1. We can compute the term of the determinant (20) involving only a2 . We get: m210 m201 m211 π 2 a22 π2 > 0. E+1 E+1
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By observing all entries of (20), this term is the only one involving E only by this power of E + 1 = (r200 )2 and no other power of the rklj . So this term cannot be cancelled uniformly in E by another term of the development of the determinant (20), whatever values taken by the integers a1 , b1 , . . . , d2 and m00 , . . . , m11 . So the only case where f˜1, p,l could identically vanish does not happen. We set: J1 = {(a1 , b1 , . . . , d2 , m00 , . . . , m11 ) | 0 a1 , c1 , . . . , d2 Nk , 1 b1 Nk , 1 mij M}. Then, as (a1 , . . . , m11 ) ∈ J1 the set of zeros of f1 in Ik is included in the following finite union of discrete sets: {E ∈ Ik | f1 (E) = 0} ⊂
#
{E ∈ Ik | f˜1, p,l (E) = 0}
( p,l )∈J1
Thus this set is also discrete in Ik . We finally get that: {E ∈ (2, +∞) | f1 (E) = 0} =
#
{E ∈ Ik | f1 (E) = 0}
k2
is discrete in (2, +∞). We set: S2 = {E > 2 | f1 (E) = 0}.
Let E > 2, E ∈ / S1 ∪ S2 . As the determinant (20) is not zero, it follows that the four matrices ϒ1 , . . . , ϒ4 are linearly independent in the subspace V1 ⊂ sp2 (R) of dimension 4. Thus, they generate V1 . We deduce that: for all E ∈ (2, +∞) \ (S1 ∪ S2 ),
V1 ⊂ la2 (E)
(21)
We now have to find another family of six matrices linearly independent in a complement of V1 in sp2 (R). 4.3.3 The Orthogonal V2 of V1 in sp2 (R) We begin by giving the expressions of the three matrices L A(1,0) − L A(0,0) , L A(1,0) − L A(1,1) , L A(0,1) − L A(0,0) .
(22)
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Looking at the form of L Aω(0) given by (18) we already know that all these differences are of the form: ⎛
00 ⎜0 0 ⎜ ⎝a c cb
⎞ g f⎟ ⎟ 0⎠ 0
e g 0 0
(23)
for (a, b , c, e, f, g) ∈ R6 . Let V2 ⊂ sp2 (R) be the 6-dimensional subspace of matrices of the form (23). We have the direct sum decomposition sp2 (R) = V1 ⊕ V2 . By (23) it suffices to compute the [3, 1], [3, 2], [4, 2], [1, 3],[1, 4] and [2, 4] entries of the three matrices (22). Expression of 1 = L A(1,0) − L A(0,0) . We have: 1 [3, 1] = m10 (1 − E) + m00 E −
π 00 a1r2 + a2r100 + 2
π 10 π b1r2 + b2r110 + √ b1r210 − b2r110 + 2 2 5 1 [3, 2] = m10 − m00 +
π π 00 a2r1 − a1r200 + √ b1r210 − b2r110 2 5
1 [4, 2] = (m00 − m10 )E −
π 00 a1r2 + a2r100 − 2
π 10 π − √ b1r210 − b2r110 + b1r2 + b2r110 2 2 5 π a1 a2 + − 1 [1, 3] = m10 − m00 + 2 r200 r100 π + √ 2 5 π 1 [1, 4] = 2
b2 b1 − 10 10 r1 r2
a1 a2 − 00 00 r2 r1
1 [2, 4] = m10 − m00 + π + √ 2 5
π 2
−
π +√ 5
π 2
b2 b1 − 10 10 r1 r2
a1 a2 + 00 00 r2 r1
b1 b2 − 10 10 r2 r1
b2 b1 + 10 10 r2 r1
π − 2
+ b2 b1 + 10 10 r2 r1
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Expression of 2 = L A(1,0) − L A(1,1) . We have: π 11 d1r2 + d2r111 + 2 π 10 π b1r2 + b2r110 + √ b1r210 − b2r110 + 2 2 5
2 [3, 1] = m10 + (m11 − m10 )E −
π π 11 d2r1 − d1r211 + √ b1r210 − b2r110 2 5 π 11 2 [4, 2] = (m11 − m10 )E − d1r2 + d2r111 − 2 π 10 π 10 − √ b1r2 − b2r110 + b1r2 + b2r110 2 2 5 π d1 d2 2 [1, 3] = m10 − m11 + + − 2 r200 r100 b1 π b1 b2 b2 π − − + − √ 2 r210 r110 2 5 r210 r110 π π d1 d2 b2 b1 2 [1, 4] = − − − √ 2 r200 r100 5 r210 r110 π d1 d2 2 [2, 4] = m10 − m11 + + + 2 r200 r100 π π b1 b1 b2 b2 + √ − − + 2 r210 r110 2 5 r210 r110 2 [3, 2] = m10 + m11 +
Expression of 3 = L A(0,1) − L A(0,0) . We have: π 00 a1r2 + a2r100 + 2 π π 10 + √ c1r210 − c2r110 + c1r2 + c2r110 2 2 5
3 [3, 1] = m01 + (m00 − m01 )E −
3 [3, 2] = − (m00 + m01 ) +
π 00 a2r1 − a1r200 + 2 10
π + √ c2r110 − c1r2 5
π 00 a1r2 + a2r100 + 2 π 10 π 10 c1r2 + c2r110 + √ c1r2 − c2r110 − 2 2 5
3 [4, 2] = (m00 − m01 )E −
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π a1 a2 3 [1, 3] = m01 − m00 + + + 2 r200 r100 c2 π c1 π c1 c2 − + √ − + 2 r210 r110 2 5 r110 r210 π a1 a2 c1 π c2 3 [1, 4] = − − −√ 2 r200 r100 5 r110 r210 π a1 a2 3 [2, 4] = m01 − m00 + + − 2 r200 r100 c2 π c1 c2 π c1 − √ − + − 2 r210 r110 2 5 r110 r210 Now we assume that E ∈ (2, +∞) \ (S1 ∪ S2 ). Then V1 ⊂ la2 (E) and in particular the following matrices are in la2 (E): ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 10 0 0 000 0 01 0 0 ⎜0 0 0 0⎟ ⎜0 1 0 0 ⎟ ⎜0 0 0 0⎟ ⎟ ⎟ ⎟ Z1 = ⎜ Z2 = ⎜ Z3 = ⎜ ⎝ 0 0 −1 0 ⎠ , ⎝0 0 0 0 ⎠, ⎝0 0 0 0⎠. 00 0 0 0 0 0 −1 0 0 −1 0 So we can consider the three matrices of la2 (E): [L A(1,0) − L A(0,0) , Z 1 ],
[L A(1,0) − L A(1,1) , Z 2 ],
[L A(0,1) − L A(0,0) , Z 3 ].
We can check that in general the Lie bracket of an element of V1 and an element of V2 is still in V2 . So, to write this three matrices we will only have to give explicitly six of their entries. Expression of 4 = [L A(1,0) − L A(0,0) , Z 1 ]. We have: 4 [3, 1] = 2m10 + 2(m00 − m10 )E − π a1r200 + a2r100 + π + π b1r210 + b2r110 + √ b1r210 − b2r110 5 00 π 4 [3, 2] = m10 − m00 + π a2r1 − a1r200 + √ b1r210 − b2r110 5 4 [4, 2] = 0 a1 a2 4 [1, 3] = 2(m00 − m10 ) − π 00 + 00 + r2 r1 b1 b2 π b1 b2 + π 10 + 10 + √ − r2 r1 5 r210 r110 b1 π a2 a1 π b2 4 [1, 4] = − +√ − 2 r100 r200 5 r210 r110 4 [2, 4] = 0.
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Expression of 5 = [L A(1,0) − L A(1,1) , Z 2 ]. We have: 5 [3, 1] = 0
π π 11 d2r1 − d1r211 + √ b1r210 − b2r110 2 5 11 5 [4, 2] = 2(m11 − m10 )E − 2m11 − π d1r2 + d2r111 + π + π b1r210 + b2r110 − √ b1r210 − b2r110 5 5 [1, 3] = 0 π π d2 d1 b2 b1 + 5 [1, 4] = − − √ 2 r111 r211 5 r210 r110 d1 d2 5 [2, 4] = 2(m11 − m10 ) − π 11 + 11 + r2 r1 b2 π b2 b1 b1 − + π 10 + 10 − √ r2 r1 5 r210 r110 5 [3, 2] = m10 + m11 +
Expression of 6 = [L A(0,1) − L A(0,0) , Z 3 ]. We have: 6 [3, 1] = 0
π 00 a1r2 + a2r100 + 2 π 10 π 10 10 + c1r2 + c2r1 + √ c1r2 − c2r110 2 2 5 2π 6 [4, 2] = −2(m00 + m01 ) + π a2r100 − a1r200 − √ c1r210 − c2r110 5 a2 a1 c2 2π c1 − 10 6 [1, 3] = π 00 − 00 − √ 10 r1 r2 r1 5 r2 π a1 a2 + 6 [1, 4] = m00 − m01 − + 2 r200 r100 π c1 π c1 c2 c2 + − √ + − 2 r210 r110 2 5 r210 r110 6 [3, 2] = m01 + (m00 − m01 )E −
6 [2, 4] = 0. It remains to check that these six matrices are linearly independent, at least for all E > 2 except those in a discrete set. We denote by f2 (E) the determinant of the 6 × 6 matrix whose columns are representing the 6 matrices we just compute. Each column is made of the 6 entries we compute for each matrix. We also set: f2 (E) = f˜2 (a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 , m00 , m01 , m10 , m11 , E)
(24)
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where f˜2 (X1 , . . . , X12 , Y) is polynomial in the coefficients X1 , . . . , X12 and analytic in Y. We define the functions f˜2, p,l as we defined the functions f˜1, p,l . We can show that the f˜2, p,l do not vanish identically on Ik . More precisely we can look at the term in the development of the determinant (24) involving only a2 : m10 (m11 − m10 )π 2 a22 × 4(E + 1)3 √ × πa2 m11 (10 E + 1E3 − 8(E + 1)3/2 E3 √ − 9(E + 1)7/2 + (E + 1)5/2 + 28 E + 1E2 + 14(E + 1)3/2 E − √ − 2(E + 1)3/2 E2 − 11(E + 1)5/2 E + 8(E + 1)7/2 E + 26 E + 1E + √ + 8(E + 1)3/2 + 8 E + 1) + πa2 m10 (10(E + 1)5/2 + 2(E + 1)7/2 + √ + 8(E + 1)3/2 E3 + 14(E + 1)5/2 E − 8(E + 1)7/2 E − 29 E + 1E2 − √ − (E + 1)3/2 E + 10(E + 1)3/2 E2 − 28 E + 1E − 3(E + 1)3/2 − √ √ ! − 9 E + 1−10 E + 1E3 )+m10 m11 (16E4 +32E3 −16E2 −64E − 32) . This term is different from 0 for a2 1, m10 1, m11 1 and m10 = m11 . But we can always assume that these two integers are distinct. Indeed, in the proof of Proposition 3, we can replace m10 by 2m10 and multiply by 2 the integers x10 1 and x10 1 . And of course m10 and 2m10 cannot be both equal to m11 . The term we just computed is the only one in the development of the determinant (24) involving exactly those powers of E and E + 1 in the numerator and in the denominator. So this term cannot be cancelled uniformly in E by another term of the development of the determinant (24). As before, the functions f˜2, p,l do not vanish identically on Ik whenever ( p, l ) ∈ J2 with: J2 = {( p1 , . . . , p8 , l1 , . . . , l4 ) | 0 p1 , p3 , . . . , p8 Nk , 1 p2 Nk , 1 l j M, l3 = l4 }. As we have justified that (a1 , . . . , m11 ) ∈ J2 , we have: # {E ∈ Ik | f2 (E) = 0} ⊂ {E ∈ Ik | f˜2, p,l (E) = 0}. ( p,l)∈J2
So the set of zeros of f2 is a discrete subset in (2, +∞). If we set: S3 = {E > 2 | f2 (E) = 0}, S3 is discrete, and for E > 2, E ∈ / S1 ∪ S2 ∪ S3 , then f2 (E) = 0. For these energies, the matrices
L A(1,0) − L A(0,0) ,
L A(1,0) − L A(1,1) ,
[L A(1,0) − L A(0,0) , Z 1 ],
L A(0,1) − L A(0,0) ,
[L A(1,0) − L A(1,1) , Z 2 ],
[L A(0,1) − L A(0,0) , Z 3 ]
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are linearly independent in the 6-dimensional subspace V2 . Thus, for all E > 2,
E∈ / S1 ∪ S2 ∪ S3 ,
V2 ⊂ la2 (E).
Finally, we set SB = S1 ∪ S2 ∪ S3 .
We fix E > 2, E ∈ / SB . We have V1 ⊂ la2 (E) and V2 ⊂ la2 (E). As V1 ⊕ V2 = sp2 (R), we get: for all E > 2,
E∈ / SB ,
sp2 (R) ⊂ la2 (E)
E∈ / SB ,
sp2 (R) = la2 (E).
We have proven: for all E > 2,
This ends our study of the Lie algebra la2 (E). We actually have proven that for E > 2, E ∈ / SB , we can apply Theorem 2 to the four matrices A(0,0) (E)m00 (E) , A(1,0) (E)m10 (E) , A(0,1) (E)m01 (E) , A(1,1) (E)m11 (E) . 0 0 0 0 Indeed, they are all in O and their logarithms generate the whole Lie algebra sp2 (R). So this achieves the proof of Proposition 2. 4.4 End of the Proof of Theorem 3 It remains to explain how we deduce Theorem 3 from Proposition 2. Let E > 2, E∈ / SB be fixed. By Proposition 2, Gμ E is dense, therefore Zariski-dense, in Sp2 (R). So, applying Theorem 1, we get that Gμ E is p-contractive and L p strongly irreducible for all p. Then applying Corollary 1 we get the separability of the Lyapunov exponents of the operator HAB (ω) and the positivity of the two leading exponents. Thus we obtain Theorem 3: for all E > 2, E ∈ / SB , we have γ1 (E) > γ2 (E) > 0. 4.5 Proof of Corollary 2 Corollary 2 says that HAB (ω) has no absolutely continuous spectrum in (2, +∞). For this we refer to Kotani’s theory in [11]. Note that [11] considers R-ergodic systems, while our model is Z-ergodic. But we can use the suspension method provided in [8] to extend the Kotani’s theory to Z-ergodic operators. So, non-vanishing of all Lyapunov exponents for all energies except those in a discrete set allows to show the absence of absolutely continuous spectrum via Theorem 7.2 of [11]. Acknowledgements The author would like to thank Anne Boutet de Monvel and Günter Stolz for numerous helpful suggestions and remarks, and also for their constant encouragements during this work.
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References 1. Benoist, Y.: Sous-groupes discrets des groupes de Lie. European Summer School in Group Theory (1997) 2. Bougerol, P., Lacroix, J.: Products of random matrices with applications to Schrödinger operators. Prog. Probab. Stat., vol. 8. Birkhäuser, Boston (1985) 3. Boumaza, H., Stolz, G.: Positivity of Lyapunov exponents for Anderson-type models on two coupled strings. Electron. J. Differential Equations 2007(47), 1–18 (electronic) 4. Breuillard, E., Gelander, T.: On dense free subgroups of Lie groups. J. Algebra 261(2), 448–467 (2003) 5. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators, Probability and Its Applications. Birkhäuser, Boston (1990) 6. Damanik, D., Sims, R., Stolz, G.: Localization for one-dimensional, continuum, BernoulliAnderson models. Duke Math. J. 114, 59–99 (2002) 7. Gol’dsheid, I.Ya., Margulis, G.A.: Lyapunov indices of a product of random matrices. Russian Math. Surveys 44(5), 11–71 (1989) 8. Kirsch, W.: On a class of random Schrödinger operators. Adv. in Appl. Math. 6, 177–187 (1985) 9. Kirsch, W., Molchanov, S., Pastur, L., Vainberg, B.: Quasi 1D localization: deterministic and random potentials. Markov Process. Related Fields 9, 687–708 (2003) 10. Klein, A., Lacroix, J., Speis, A.: Localization for the Anderson model on a strip with singular potentials. J. Funct. Anal. 94, 135–155 (1990) 11. Kotani, S., Simon, B.: Stochastic Schrödinger operators and Jacobi matrices on the strip. Comm. Math. Phys. 119(3), 403–429 (1988) 12. Schmidt, W.: Diophantine approximation. Lecture Notes in Mathematics, vol. 785. Springer, Berlin (1980) 13. Stollmann, P.: Caught by disorder – bound states in random media. Progress in Mathematical Physics, vol. 20. Birkhäuser (2001) 14. Stolz, G.: Strategies in localization proofs for one-dimensional random Schrödinger operators. Proc. Indian Acad. Sci. 112, 229–243 (2002)
Math Phys Anal Geom (2007) 10:123–134 DOI 10.1007/s11040-007-9024-5
Super D-Differentiation for R∞ -Supermanifolds D. Hurley · M. Vandyck
Received: 27 April 2007 / Accepted: 13 July 2007 / Published online: 31 July 2007 © Springer Science + Business Media B.V. 2007
Abstract The concept of ‘D-Differentiation’, which, in the context of smooth manifolds, generalises Lie and covariant differentiation, is extended to R∞ -supermanifolds under the name of ‘Super D-Differentiation’. This is done by defining new (non-linear) mappings, called ‘μ-mappings’ and by relating their non-linearity to the Leibniz rule that a derivation must satisfy when it acts on a tensor product. The resulting axiomatics, which is basis-independent and coordinate-free, is then expressed in a general basis (not necessarily holonomic). Super Lie and Super covariant differentiation are, amongst others, special cases of Super D-Differentiation. In particular, the transformation rules for the connection coefficients and the commutation coefficients of non-holonomic bases are obtained. These special cases are found to be in agreement with the DeWitt Super covariant and Super Lie derivatives. Keywords Graded manifolds · R∞ -supermanifolds · D-differentiation Mathematics Subject Classifications (2000) 58C20 · 58A50 · 58C50
1 Introduction Since the introduction of anticommuting variables in Physics, much effort has been devoted to the construction of a fully comprehensive framework for ‘supergeometry’, be it under the name of a ‘superspace’, a ‘supermanifold’ or a
D. Hurley (B) Mathematics Department, National University of Ireland, Cork, Ireland e-mail:
[email protected] M. Vandyck Physics Department, National University of Ireland, Cork, Ireland
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‘graded manifold’. This is not the place for a detailed description of the various methods successively employed to reach this goal (See, for instance, [1–3] for a comprehensive list of references, and a careful analysis). Let us only recall that the first satisfactory attempt [1], from the mathematical point of view, was that of Berezin, Leites and Kostant [4, 9], and was based on algebraic geometry. Unfortunately, in the context of physical applications, such ‘graded manifolds’ (also called ‘supermanifolds’ in the Russian literature), led to real-valued spinor fields, which is incompatible with the requirements of supersymmetry. A different approach, proposed by DeWitt [5] and Rogers [10, 11], defined a ‘supermanifold’ by analogy with an ordinary (C ∞ ) manifold, using an atlas. It turned out, however, that this created difficulties, because of the delicate nature of the concept of differentiability of the transition functions between coordinate patches [1]. In particular, the choice of the ‘ground algebra’ of ‘supernumbers’ plays a role in the differentiability issue. Rothstein [1, 12] introduced then a set of axioms to single out ‘wellbehaved’ supermanifolds (called here ‘R-supermanifolds’ for brevity). It was thought that R-supermanifolds reduced to the Berezin–Leites–Kostant graded manifolds when is commutative, and to certain extensions of the Rogers G∞ -supermanifolds when is a finite-dimensional exterior algebra. However, the situation was re-investigated later [2], and both statements were found to be false. In the course of the analysis, Bartocci, Bruzzo, Hernández-Ruipérez and Pestov defined a new structure [2], called an ‘R∞ -supermanifold’, which requires to be a ‘Banach algebra of Grassmann origin’ (BGO-algebra). Examples of BGO-algebras are the finite-dimensional Grassmann algebras, the infinite-dimensional algebra B∞ employed by Rogers, and (trivially) the real (or complex) numbers, but DeWitt’s original algebra ∞ is not of that kind. R∞ -supermanifolds based on R or C coincide with Berezin–Leites–Kostant graded manifolds, whereas they generalise Rogers’s G∞ -supermanifolds (in a well-defined sense [2]) when the ground algebra is B∞ . As a consequence, R∞ -supermanifolds resolve the problems encountered with Rothstein’s Rsupermanifolds. Owing to the desirability to include DeWitt’s ∞ in the set of acceptable ground-algebras, R∞ -supermanifolds were finally developed [3] over ‘Arens–Michael algebras of Grassmann origin’ (AMGO-algebras), which are generalisations of BGO-algebras that do contain ∞ . In spite of the fact that AMGO-algebras are exceedingly general (so much so that all algebras ever used for supernumbers belong [3] to this category), R∞ -supermanifolds still constitute ‘workable’ structures, in the sense that they allow for a substantial amount of theory to be developed within their framework. For instance, they shed light [3] on DeWitt supermanifolds. Let us now return to ordinary differential geometry. In this context, one often introduces Lie differentiation L and covariant differentiation ∇ as two fundamentally important derivations of the space of tensors. However, one seldom avails of the possibility [7] of defining them as special cases of the much
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larger family of derivations called ‘D-differentiation’. Not only is this alternative method more economical, but it also provides greater conceptual insight into L and ∇, through generalisations of the notions of torsion, curvature, metricity, geodesics, etc. In addition, certain operators of D-differentiation, which are neither L nor ∇, sometimes prove beneficial in the study of some areas of Physics, such as the dynamics of rigid bodies, the semi-classical motion of electrons in crystals, or optics in General Relativity [6–8]. It would therefore seem reasonable to try to construct an operator of ‘super D-differentiation’ that would, in an R∞ -supermanifold, play the same role as D-differentiation does in an ordinary C ∞ manifold. If this programme is successful, it will automatically provide super Lie differentiation and super covariant differentiation (amongst other operators), and it will illustrate further that R∞ -supermanifolds are rich enough to support a considerable amount of structure. Such will be our aim in the present article. However, to avoid cumbersome technicalities, we shall restrict attention (loosely speaking) to super D-differentiation of vectors. The extension of the formalism to tensors will not be presented here. In the same spirit, we shall confine ourselves to the local construction of D-differentiation. It is well known [1] that covariant differentiation does not always exist globally, so that the same must be true of D-differentiation, which contains it as a special case. To fix the notation, we shall begin by a brief summary of R∞ -supermanifolds in Section 2. Then, we shall define, in Section 3, non-linear mappings (called ‘μ-mappings’), which will be closely related to super D-differentiation. The operator D itself will be defined in Section 4, and some of its properties will be studied. Examples will be provided in Section 5, which will concentrate on Lie and covariant differentiation. Finally, future developments will be contemplated in the conclusion. 2 Summary of R∞ -Supermanifolds We do not wish to present here a detailed introduction to R∞ -supermanifolds based on an AMGO-algebra; the reader is referred to [2, 3], in which all the details are available. We shall rather summarise the fundamental axioms in a manner that will emphasise exclusively the aspects required for our construction. Let be a graded-commutative algebra (and technically an AMGOalgebra) over the real numbers. (It would be possible to have over C.) A graded module over is said to be ‘free of rank m|n’ if it is isomorphic [3] to a module of the form m|n ≡ ⊗ Rm|n , where Rm|n is the standard graded real vector space of dimension m|n. The ‘linear superspace’ m,n of dimension (m, n) is then the even part of m|n . It can be endowed with two fundamentally different topologies: the fine (or Rogers) one and the coarse (or DeWitt) one (see [3]).
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A superspace (not necessarily linear) over is a triple (X, A, ev), where X is a paracompact topological space, A denotes a sheaf of graded-commutative algebras over , and ev is a morphism of sheaves between A and the sheaf C X of -valued continuous functions on X. Let p be a point of X, and let U be an open neighbourhood containing p. A graded derivation V of A|U is an endomorphism of sheaves V : A|U → A|U satisfying the graded Leibniz rule: V( f · g) = V( f ) · g + (−1)| f ||V| f · V(g),
(2.1)
where vertical bars around an object denote the parity (even or odd) of this object, assumed homogeneous. Derivations form a presheaf, which, after appropriate completion, yields the sheaf DerA of graded modules over A. The elements of DerA(U) will informally be called ‘vectors’. The sheaf DerA possesses a dual, denoted by Der∗ A, which is also a sheaf of graded modules over A. The elements of Der∗ A(U) will informally be called ‘forms’. For all homogeneous elements f of A(U), it is possible to define the form df by (d f )[V] ≡ (−1)| f ||V| V( f ),
(2.2)
and d constitutes a morphism of sheaves from A to Der∗ A. An R∞ -supermanifold of dimension (m, n) over is now introduced as a superspace (X, A, ev) over , satisfying the following four axioms: Axiom 1 Der∗ A is a locally-free graded module of rank (m, n). For every point p of X there exist an open neighbourhood U of p and sections xi , 1 i m + n of A, such that xμ , 1 μ m (respectively x A , m + 1 A m + n), belongs to the even part A(U)0 of A(U) (respectively the odd part A(U)1 of A(U)), with the property that the set {dxi , 1 i m + n} ≡ {dxμ (1 μ m), dx A (m + 1 A m + n)} forms a graded basis of Der∗ A(U) over A(U). As one can see, Greek (respectively capital Latin) indices lie in the range {1, · · · , m} (respectively {m + 1, · · · , m + n}), whereas lower-case Latin ones take values in {1, · · · , m + n}. By definition, the parity of a Greek index is zero, that of a capital Latin index is one, and that of a lower-case Latin index is either zero or one, according as it belongs to the set {1, · · · , m} or {m + 1, · · · , m + n}. We shall also systematically adopt Einstein’s summation convention, with the amendment that two repeated indices, of which one is surrounded by vertical bars (indicating parity), are not summed over, unless an additional repeated index is present, which is not surrounded by vertical bars. For instance, there is no sum over a in (−1)|a| αa , whereas the index a does undergo summation in (−1)|a| αa X a . Moreover, given a basis {dxi , 1 i m + n}, an arbitrary set of linear combinations of {dxi , 1 i m + n} that forms a basis will be denoted by {e(i ) , 1 i m + n}. Of course, it will be assumed that the basic elements e(i )
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are numbered in such a manner that the first m vectors are even, and the last n ones are odd. In addition, it follows from the local freeness of Der∗ A that DerA is also locally free of rank (m, n). The basis of DerA(U) to which {e(i ) , 1 i m + n} is dual will be denoted by {(i ) e, 1 i m + n}. In the special case where e(i ) = dxi , the corresponding (i ) e will be denoted by ∂/∂ xi . Axiom 2 Given a so-called ‘coordinate chart’, namely a collection (U, (dx1 , · · · , dxm+n )) as in Axiom 1, the assignment p → ((ev(x1 ))( p), · · · , (ev(xm+n ))( p))
(2.3)
is a homeomorphism of U onto an open subset of m,n . Axiom 3 Let I p denote the graded ideal of the stalk A p of A at p defined as I p ≡ {φ ∈ A p : (ev(φ ))( p) = 0}.
(2.4)
Then, I p is finitely generated for all p in X. It is shown in [2] that this axiom is closely related to the existence of a Taylor-like expansion for the germs φ in A p . Axiom 4 For every open subset U of X, the algebra A(U) is complete and Hausdorff. This axiom requires, obviously, that a topology be specified in A(U). For our purposes, It suffices to recall that the ground-algebra possesses a topology, which is determined by continuous submultiplicative prenorms π . If f is an element of A(U), the action L( f ) of a differential operator L on f also belongs to A(U), because such an operator is an element of the graded A-module D(A) generated multiplicatively by DerA over A. By applying the evaluation morphism ev to L( f ), one obtains a -valued function, so that (ev(L( f )))( p) belongs to , for all p in X. The topology on A(U) is then described in terms of π((ev(L( f )))( p)). Details may be found in [3]. At this stage, we are ready to construct super D-differentiation. Loosely speaking, one would like DV W to be a vector when V and W are vectors. Thus, one is led to consider vector-valued mappings μ taking two vectors, V and W, as arguments. Such mappings, having suitable properties, will be obtained in the following section. Later, in Section 4, we shall re-interpret some of them in terms of an operator of super D-differentiation. Remarks 1. When, after Axiom 1, we described the bases {e(i ) , 1 i m + n} and {(i ) e, 1 i m + n} as being dual, the graded dual was understood, in the sense e(a) [(b ) e] = (−1)|a| b δ a ,
(2.5)
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where b δ a stands for the usual Kronecker symbol (with its indices staggered, in accordance with the conventions of [5]). In the graded situation that occupies us, it is convenient to introduce ‘double’ duality, which consists in allowing a vector X to act on a form α as X[α] ≡ (−1)|X||α| α[X].
(2.6)
From this new point of view, the duality-relation (2.5) of the bases may be reformulated as (a) (b ) e[e ]
= b δa ,
(2.7)
which is simpler than (2.5). Furthermore, with the same notation, the defining-relation (2.2) of the exterior differential df becomes V[d f ] = (−1)| f ||V| (df )[V] = V( f ).
(2.8)
On the left-hand side, the square brackets emphasise that V is acting on a form (by double duality), whereas, on the right-hand side, V acts as a derivation on the element f of the algebra A. 2. There is no difficulty in establishing that, by virtue of the definition (2.2), the expression of the exterior differential df reads, in a pair of dual bases, df = (df )a e(a) = (−1)|a|(| f |+1)
(a) e(
f ) e(a) ,
(2.9)
which will be useful later. 3 Construction of μ-Mappings As in the previous section, let p be a point of X, and let U denote an open set containing p. We now wish to consider mappings μ defined on DerA(U) × DerA(U), with values in DerA(U), satisfying the axioms μ(V + W, Z ) = μ(V, Z ) + μ(W, Z )
(3.1)
μ(Z , V + W) = μ(Z , V) + μ(Z , W)
(3.2)
| f |(|V|+|W|)
μ(V, W · f ) = μ(V, W) · f + (−1)
A(df, V, W)
μ( f · V, W) = f · μ(V, W) − (−1)|V||W| B(df, W, V),
(3.3) (3.4)
where A and B are even vector-valued (graded) multilinear mappings. In the context of vector-valued mappings, linearity means (by definition) that the -valued mappings A and B constructed from A and B as A : (α, V, W, β) → A (α, V, W, β) ≡ (A(α, V, W))[β]
(3.5)
B : (α, V, W, β) → B (α, V, W, β) ≡ (B(α, V, W))[β]
(3.6)
are (graded)-linear in the entries α, V, W and β, for all V and W in DerA(U), and all α and β in Der∗ A(U). Note that, in (3.5) and (3.6), the vectors A(α, V, W) and B(α, V, W) act on β in accordance with (2.6).
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The presence of A and B in (3.3) and (3.4) prevents μ from being linear, in general. However, the ‘linearity-breaking’ objects A and B are, themselves, linear mappings. In this sense, one may consider μ as being next in simplicity after linear mappings. One readily establishes, from (3.1–3.4), the following propositions, which we state without proof: Proposition 1 The parity of μ is given by |μ(V, W)| = |V| + |W|,
(3.7)
for all homogeneous V and W. Proposition 2 For all vectors V and W, and for all elements k of , one has μ(V, W · k · 1) = μ(V, W) · k · 1,
μ(k · 1 · V, W) = k · 1 · μ(V, W),
(3.8)
where 1 denotes the unit of the algebra A(U). The comparison of (3.8) with (3.3) and (3.4) expresses in what sense μ is ‘-linear’, but not ‘A(U)-linear’. Proposition 3 Let the commutator [V, W] be defined by [V, W] ≡ V ◦ W − (−1)|V||W| W ◦ V.
(3.9)
Then, [V, W] is a mapping of the type (3.1–3.4), with A and B given by A(α, V, W) = B(α, V, W) = α[V] · W ≡ L A(α, V, W).
(3.10)
In this proposition, we have introduced the definition of the mapping L A, which will be convenient later. At this stage, we have focussed attention on the family (3.1–3.4) of mappings, which we shall call ‘arbitrary μ-mappings’. This family is, however, too large for our purposes (although it can be studied in full generality). Therefore, in the next section, we shall restrict ourselves to the subfamily of the ‘generalisable μ-mappings’. 4 Generalisable μ-Mappings Let us now specialise μ by requiring that, in (3.3), the mapping A be given by L A, which was defined in (3.10). It is then obvious that (3.3) is replaced by μ(V, W · f ) = μ(V, W) · f + (−1)|V||W| W · V( f ).
(4.1)
This relationship, which is investigated in a broader context in the appendix, enables one to consider μ(V, W) as the ‘rate of change’ DV W of W along V, also called the ‘D-derivative of W along V’. Indeed, in terms of the symbol D, (4.1) reads DV (W · f ) = (DV W) · f + (−1)|V||W| W · V( f ).
(4.2)
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The right-hand side of (4.2) corresponds to that produced by an operator DV (for V fixed) satisfying the (graded) Leibniz rule for the product W · f of W by the element f of A(U). With the same notation, the axiom (3.4) becomes D f ·V W = f · DV W − (−1)|V||W| B(df, W, V),
(4.3)
which specifies the behaviour of the operator D with respect to its ‘differentiating slot’. This behaviour depends on the properties of the mapping B, so that each operator D is characterised by its own B. (Examples will be provided in the next section.) The axioms that we have adopted imply that the operator D is entirely determined. Its local expression in a basis is given by the following theorem. Theorem 1 Let {(i ) e, 1 i m + n} be a basis of DerA(U). For all V and W, the D-derivative DV W reads DV W = V(W i ) + (−1)|V||W| W j ji (V) (i ) e (4.4) i j (V)
≡ (−1)(|V|+|b |)| j| V b jb λi −(−1)(|V|+|b |)(|a|+| j|)+|a| (a) e(V b )
in which the elements
i jb λ
and
a
i jb B
i jb B ,
(4.5)
of A(U) uniquely satisfy
(−1)| j||b |
i jb λ (i ) e
| j||b | a
i jb B (i ) e
−(df )a (−1)
a
= D(b ) e ( j ) e
(4.6)
= D{ f ·(b ) e} ( j ) e − f · D(b ) e ( j ) e.
(4.7)
In other words, the operator D is determined by its action D(i ) e ( j ) e on a basis, which is enciphered in the coefficients ijλk , and by the relationship between a D{ f ·(i ) e} ( j ) e and f · D(i ) e ( j ) e, which is represented by the components ij Bb . a
Under a change of basis, the quantities ij Bb transform in an obvious fashion, because they are the components of the linear mapping B of (4.3). On the other hand, the coefficients ijλk transform as follows: Theorem 2 Let {(i ) e , 1 i m + n} be a basis of DerA(U), related to the basis {( j ) e, 1 j m + n} by (i ) e
= iMj
( j )e ,
(i ) e
= iNj
MN = N M = 1.
( j ) e,
(4.8)
Then, the coefficients λ are given in terms of λ by ij λ
k
= (−1)|a|(|b |+| j|) i N a j N b
c k ab λ c M
−(−1)(|a|+|c|)(| j|+|d|)+|a| i N c
+ (−1)|i|| j| j N a
(a) e( j N
)
d a
cd B
b
bM
k
(a) e(i N
.
b
) b Mk − (4.9)
It goes without saying that Theorem 1 and Theorem 2 reduce to the analogous results for ordinary D-differentiation in C ∞ manifolfds when all the objects involved are purely even. The reader is referred to [7] or [8] for details. After these considerations valid for an arbitrary operator D, we are now going to investigate two special cases: super Lie differentiation L and super
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covariant differentiation ∇. This will enable us to establish a closer contact with the supermanifold literature.
5 Examples Let us begin with the case of the commutator [V, W] ≡ LV W. By virtue of (4.6), we find successively (−1)| j||b |
i jb λ (i ) e
= [(b ) e, ( j ) e]
(5.1)
≡ b jC
(5.2)
i
(i ) e
= −(−1)| j||b |
i jb C (i ) e,
(5.3)
where we have introduced the ‘commutation coefficients’ ijCk of the basis, and we have exploited their property [5] of graded antisymmetry. As a consequence of (5.3), the value of jb λi pertaining to the commutator reads i jb λ
= − jb Ci ,
(5.4)
from which the transformation rule for the commutation coefficients jb Ci is also obtainable, through (4.9). Moreover, the expression (2.9) of the exterior differential in components, a combined with the definition (4.7) of the quantities jb Bi and the properties of the commutator, yields − (−1)|a|(| f |+1)
(a) e(
f)
a
i jb B (i ) e
= −(df )a
a
i jb B (i ) e
(5.5) = (−1)| j||b | [ f · (b ) e, ( j ) e]− f · [(b ) e, ( j ) e] (5.6) = −(df )[( j ) e] · (b ) e = −(−1)| f || j| | f ||a|
= −(−1)
(5.7)
( j ) e(
f ) (b ) e
(a) e(
f) δj bδ a
(5.8) i
(i ) e,
(5.9)
which implies a
i jb B
= (−1)|a| a δ j b δ i .
(5.10)
The result (5.10) is the translation in coordinate-language of the coordinatefree statement (3.10). a When the values (5.4) and (5.10) of jb λi and jb Bi are inserted in the decomposition (4.4), (4.5), the final expression for the commutator becomes [V, W] =
LV W = V(W i ) − (−1)|V||W| W(V i ) + (−1)(|W|+| j|)|k| V k W j kjCi (i ) e. (5.11)
In (5.11), the term involving b jCi arises from the failure of two basic vectors to commute, in general. The special bases for which all the b jCi vanish are called ‘holonomic bases’, ‘natural bases’ or ‘coordinate bases’ [5].
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Apart from trivial differences in the notation, (5.11) coincides with the DeWitt Lie derivative of a vector W. (Note that, in [5], the formula for L is only displayed in a holonomic basis.) On the other hand, the case of covariant differentiation is fundamentally different from that of Lie differentiation, in the sense that ∇ is linear in its differentiating slot: ∇ f ·V = f · ∇V , which compels definition
a
cd B
b
(5.12)
to vanish, by virtue of (4.7). Furthermore, with the ∇(i ) e ( j ) e ≡ ji k
(k) e,
(5.13)
where the symbols are the so-called ‘connection’ coefficients, the coordinateexpressions (4.4) and (4.5) become ∇V W = V(W i ) + (−1)|k|(|W|+| j|) V k W j jk i (i ) e. (5.14) In addition, the transformation rule (4.9) simplifies as ij
k
= (−1)| j|(|a|+|i|) i N a j N b
c k ab c M
+ j Na
(a) e(i N
b
) b Mk .
(5.15)
The results (5.14) and (5.15) reproduce those of the DeWitt super covariant derivative [5], apart from minor differences in the notation.
6 Conclusion In this article, we adopted the framework of R∞ -supermanifolds, and we constructed locally an operator of differentiation, called ‘(super) Ddifferentiation’. This operator, as presented here, acts on vectors. Its extension to tensors will be investigated elsewhere. We showed how (super) Lie and (super) covariant differentiation are contained in (super) D-differentiation as special cases. Lie and covariant differentiation had, separately, been considered before in the literature, for instance by DeWitt [5], within his own theory of supermanifolds, and by Bartocci, Bruzzo and Hernández-Ruipérez, for the so-called ‘G-supermanifolds’. Both kinds of supermanifold are included within the category of R∞ -supermanifolds, so that D-differentiation provides a unified approach of a considerable generality. In physical applications, the DeWitt formalism is frequently employed. The present construction may be considered as an illustration of how some of the outcomes of this formalism (such as the transformation rule for connection coefficients) may be obtained from the point of view of R∞ -supermanifolds. Therefore, not only do R∞ -supermanifolds provide the benefit of unifying various supermanifold theories, but they are also a natural context for some of the DeWitt formalism, which is familiar to, and employed by, Physicists.
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Appendix In order to interpret the two kinds of μ-mappings from a deeper point of view, let us assume that we have at our disposal an operator of D-differentiation that acts on tensors. For a vector V fixed, DV must satisfy the Leibniz rule for the tensor product ⊗, namely DV (T1 ⊗ T2 ) = (DV T1 ) ⊗ T2 + (−1)|V||T1 | T1 ⊗ DV T2 .
(7.1)
Furthermore, DV must be ‘compatible’ with V, in the sense DV f = V( f ),
(7.2)
for all elements f of A. Let us now consider f as a tensor of rank (0, 0). Then, with T2 = f , the Leibniz rule (7.1) reduces to DV (T · f ) = DV (T ⊗ f ) = (DV T) ⊗ f + (−1)|V||T| T ⊗ DV f = (DV T) · f + (−1)|V||T| T · DV f .
(7.3) (7.4)
Even if T is restricted to being a vector W, the right-hand side of (7.4) has no meaning in terms of μ-mappings, because such mappings cannot act on f . On the other hand, if (7.2) is inserted in (7.4), one finds DV (T · f ) = (DV T) · f + (−1)|V||T| T · V( f ).
(7.5)
When T is a vector W, the requirement (7.5) is expressible in the language of μ-mappings, by putting μ(V, W · f ) = μ(V, W) · f + (−1)|V||W| W · V( f ) μ(V, W) ≡ DV W.
(7.6) (7.7)
The observation that (7.6) coincides with (4.1) implies that, for a μ-mapping to be the restriction to vectors of an operator of D-differentiation defined on tensors, it is necessary that this mapping be of the type called ‘generalisable’ in Section 4. If, in addition, the general operator D is assumed to preserve tensor rank and to commute with tensor contractions, then D can uniquely be constructed from a generalisable μ-mapping, so that the above condition is also sufficient. In practice, this statement means that the components of the D-derivative of i any tensor are (locally) determined by the quantities ijλk and jk Bl appearing in (4.5). However, we shall not present these developments here.
References 1. Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D.: The Geometry of Supermanifolds. Kluwer Academic Publishers, Dordrecht (1991) 2. Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D., Pestov, V.G.: Foundations of supermanifold theory: the axiomatic approach. Differential Geom. Appl. 3, 135–155 (1993)
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3. Bruzzo, U., Pestov, V.G.: On the structure of DeWitt supermanifolds. J. Geom. Phys. 30, 147–168 (1999) 4. Berezin, F.A., Leites, D.A.: Supermanifolds. Soviet Math. Dokl. 16, 1218–1222 (1975) 5. DeWitt, B.S.: Supermanifolds. Cambridge University Press, London (1984) 6. Hurley, D., Vandyck, M.: An application of D-differentiation to solid-state Physics. J. Phys. A 33, 6981–6991 (2000) 7. Hurley, D., Vandyck, M.: A unified framework for Lie and covariant differentiation. J. Math. Phys. 42, 1869–1886 (2001) 8. Hurley, D., Vandyck, M.: Topics in Differential Geometry; A New Approach using Ddifferentiation. Springer-Praxis, Chichester (2002) 9. Kostant B.: Graded manifolds, graded Lie theory, and prequantization. In: Differential Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol. 570, pp. 177–306. Springer, Berlin (1977) 10. Rogers, A.: A global theory of supermanifolds. J. Math. Phys. 21, 1352–1365 (1980) 11. Rogers, A.: Graded manifolds, supermanifolds and infinite-dimensional Grassmann algebras. Comm. Math. Phys. 105, 375–384 (1986) 12. Rothstein, M.J.: The axioms of supermanifolds and a new structure arising from them. Trans. Amer. Math. Soc. 297, 159–180 (1986)
Math Phys Anal Geom (2007) 10:135–154 DOI 10.1007/s11040-007-9026-3
L p -Spectral Theory of Locally Symmetric Spaces with Q-Rank One Andreas Weber
Received: 15 March 2007 / Accepted: 26 July 2007 / Published online: 12 September 2007 © Springer Science + Business Media B.V. 2007
Abstract We study the L p -spectrum of the Laplace–Beltrami operator on certain complete locally symmetric spaces M = \X with finite volume and arithmetic fundamental group whose universal covering X is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces can be generalized to manifolds with cusps of rank one. Keywords Arithmetic lattices · Heat semigroup on L p -spaces · Laplace–Beltrami operator · Locally symmetric space · L p -spectrum · Manifolds with cusps of rank one Mathematics Subject Classifications (2000) Primary 58J50 · 11F72 · Secondary 53C35 · 35P05
1 Introduction Our main concern in this paper is to study the L p -spectrum σ ( M, p ), p ∈ (1, ∞), of the Laplace–Beltrami operator on a complete non-compact locally symmetric space M = \X with finite volume, such that (1) X is a symmetric space of non-compact type, (2) ⊂ Isom0 (X) is a torsion-free arithmetic subgroup with Q-rank() = 1. We also treat the case of manifolds with cusps of rank one which are more general than the locally symmetric spaces defined above.
A. Weber (B) Institut für Algebra und Geometrie, Universität Karlsruhe (TH), Englerstr. 2, 76128 Karlsruhe , Germany e-mail:
[email protected]
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Whether the L p -spectrum of a complete Riemannian manifold M depends on p or not is related to the geometry of M. More precisely, Sturm proved in [23] that the L p -spectrum is p-independent if the Ricci curvature of M is bounded from below and the volume of balls in M grows uniformly subexponentially (with respect to their radius). This is for example true if M is compact or if M is the n-dimensional euclidean space Rn . On the other hand, if the Ricci curvature of M is bounded from below and the volume density of M grows exponentially in every direction (with respect to geodesic normal coordinates around some point p ∈ M with empty cut locus) then the L p -spectrum actually depends on p. More precisely, Sturm showed that in this case inf Re σ ( M,1 ) = 0 whereas inf σ ( M,2 ) > 0. An example where this happens is M = Hn , the n-dimensional hyperbolic space. In the latter case and for more general hyperbolic manifolds of the form M = \Hn where denotes a geometrically finite discrete subgroup of the isometry group of Hn such that either M has finite volume or M is cusp free, the L p -spectrum was completely determined by Davies, Simon, and Taylor in [8]. They proved that σ ( M, p ) coincides with the union of a parabolic region P p and a (possibly empty) finite subset {λ0 , . . . , λm } of R0 that consists of 2 eigenvalues for M, p . Note, that we have P2 = (n−1) ,∞ . 4 Taylor generalized this result in [24] to symmetric spaces X of non-compact type, i.e. he proved that the L p -spectrum of X coincides with a certain parabolic region Pp (now defined in terms of X) that degenerates in the case p = 2 to the interval [||ρ||2 , ∞), where a definition of ρ can be found in Section 2.2. He also showed that the methods from [8] can be used in order to prove the following: Proposition 1.1 (cf. Proposition 3.3 in [24]) Let X denote a symmetric space of non-compact type and M = \X a locally symmetric space with finite volume. If σ ( M,2 ) ⊂ {λ0 , . . . , λm } ∪ [||ρ||2 , ∞),
(1.1)
where λ j ∈ [0, ||ρ||2 ) are eigenvalues of finite multiplicity, then we have for p ∈ [1, ∞): σ ( M, p ) ⊂ {λ0 , . . . , λm } ∪ Pp . However, for non-compact locally symmetric spaces \X with finite volume the assumption (1.1) is in general not fulfilled: If X is a symmetric space of non-compact type and ⊂ Isom0 (X) an arithmetic subgroup such that the quotient M = \X is a complete, non-compact locally symmetric space, the continuous L2 -spectrum of M contains the interval [||ρ||2 , ∞) but is in general strictly larger. Another upper bound for the L p -spectrum σ ( M, p ) is the sector | p − 2| z ∈ C \ {0} : | arg(z)| arctan √ ∪ {0} 2 p−1
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Fig. 1 The parabolic region P p if p = 3
which is indicated in Fig. 1. This actually holds in a much more general setting, i.e. for generators of so-called sub-Markovian semigroups (cf. Section 2.1). We are going to prove in Section 3 that a certain parabolic region (in general different from the one in Proposition 1.1) is contained in the L p -spectrum σ ( M, p ) of a locally symmetric space M = \X with the properties mentioned in the beginning. In the case where X is a rank one symmetric space it happens that our parabolic region and the one in Taylor’s result coincide. Therefore, we are able to determine explicitly the L p -spectrum in the latter case. In Section 4 we briefly explain, how the results from Section 3 can be generalized to manifolds with cusps of rank one. For these manifolds, every cusp defines a parabolic region that is contained in the L p -spectrum. In contrast to the class of locally symmetric spaces however, these parabolic regions need not coincide. This is due to the fact that the volume growth in different cusps may be different in manifolds with cusps of rank one whereas this can not happen for locally symmetric spaces as above. Consequently, the number of (different) parabolic regions in the L p -spectrum σ ( M, p ), p = 2, of a manifold with cusps of rank one seems to be a lower bound for the number of cusps of M. As in the case p = 2 the Laplace–Beltrami operator is self-adjoint, we obtain in this case only the trivial lower bound one. Therefore, it seems that more geometric information is encoded in the L p -spectrum for some p = 2 than in the L2 -spectrum. Note however, that nothing new can be expected for compact manifolds as the L p -spectrum does not depend on p in this case. For results concerning the L p -spectrum of locally symmetric spaces with infinite volume see [26, 27].
2 Preliminaries 2.1 Heat Semigroup on L p -spaces In this section M denotes an arbitrary complete Riemannian manifold. The Laplace–Beltrami operator M := −div(grad) with domain Cc∞ (M) (the set
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of differentiable functions with compact support) is essentially self-adjoint and hence, its closure (also denoted by M ) is a self-adjoint operator on the Hilbert space L2 (M). Since M is positive, − M generates a bounded analytic semigroup e−t M on L2 (M) which can be defined by the spectral theorem for unbounded self-adjoint operators. The semigroup e−t M is a submarkovian semigroup (i.e., e−t M is positive and a contraction on L∞ (M) for any t 0) and we therefore have the following: (1) The semigroup e−t M leaves the set L1 (M) ∩ L∞ (M) ⊂ L2 (M) invariant and hence, e−t M | L1 ∩L∞ may be extended to a positive contraction semigroup T p (t) on L p (M) for any p ∈ [1, ∞]. These semigroups are strongly continuous if p ∈ [1, ∞) and consistent in the sense that T p (t)| L p ∩Lq = Tq (t)| L p ∩Lq . (2) Furthermore, if p ∈ (1, ∞), the semigroup T p (t) is a bounded analytic . semigroup with angle of analyticity θ p π2 − arctan 2|√p−2| p−1 For a proof of (1) we refer to [7, Theorem 1.4.1]. For (2) see [19]. In general, the semigroup T1 (t) needs not be analytic. However, if M has bounded geometry T1 (t) is analytic in some sector (cf. [6, 25]). In the following, we denote by − M, p the generator of T p (t) (note, that M = M,2 ) and by σ ( M, p ) the spectrum of M, p . Furthermore, we will write e−t M, p for the semigroup T p (t). Because of (2) from above, the L p -spectrum σ ( M, p ) has to be contained in the sector
π − θ p ∪ {0} ⊂ 2 | p − 2| ⊂ z ∈ C \ {0} : | arg(z)| arctan √ ∪ {0}. 2 p−1
z ∈ C \ {0} : | arg(z)|
If we identify as usual the dual space of L p (M), 1 p < ∞, with L p (M), 1 + p1 = 1, the dual operator of M, p equals M, p and therefore we always p have σ ( M, p ) = σ ( M, p ). 2.2 Symmetric Spaces Let X denote always a symmetric space of non-compact type. Then G := Isom0 (X) is a non-compact, semi-simple Lie group with trivial center that acts transitively on X and X = G/K, where K ⊂ G is a maximal compact subgroup of G. We denote the respective Lie algebras by g and k. Given a corresponding Cartan involution θ : g → g we obtain the Cartan decomposition g = k ⊕ p of g into the eigenspaces of θ. The subspace p of g can be identified with the tangent space TeK X. We assume, that the Riemannian metric ·, · of X in p ∼ = TeK X coincides with the restriction of the Killing form B(Y, Z ) := tr(adY ◦ adZ ), Y, Z ∈ g, to p.
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For any maximal abelian subspace a ⊂ p we refer to = (g, a) as the set of restricted roots for the pair (g, a), i.e. contains all α ∈ a∗ \ {0} such that hα := {Y ∈ g : ad(H)(Y) = α(H)Y for all H ∈ a} = {0}.
These subspaces hα = {0} are called root spaces. a+ in a is chosen, we denote by + the subset Once a positive Weyl chamber of positive roots and by ρ := 12 α∈ + (dim hα )α half the sum of the positive roots (counted according to their multiplicity). 2.2.1 Arithmetic Groups and Q-rank Since G = Isom0 (X) is a non-compact, semi-simple Lie group with trivial center, we can find a connected, semi-simple algebraic group G ⊂ GL(n, C) defined over Q such that the groups G and G(R)0 are isomorphic as Lie groups (cf. [9, Proposition 1.14.6]). Let us denote by TK ⊂ G (K = R or K = Q) a maximal K-split algebraic torus in G. Remember that we call a closed subgroup T of G a torus if T is diagonalizable over C, or equivalently if T is abelian and every element of T is semi-simple. Such a torus T is called R-split if T is diagonalizable over R and Q-split if T is defined over Q and diagonalizable over Q. All maximal K-split tori in G are conjugate under G(K), and we call their common dimension K-rank of G. It turns out that the R-rank of G coincides with the rank of the symmetric space X = G/K, i.e. the dimension of a maximal flat subspace in X. Since we are only interested in non-uniform lattices ⊂ G, we may define arithmetic lattices in the following way (cf. [29, Corollary 6.1.10] and its proof): Definition 2.1 A non-uniform lattice ⊂ G in a connected semi-simple Lie group G with trivial center and no compact factors is called arithmetic if there are (1) a semi-simple algebraic group G ⊂ GL(n, C) defined over Q and (2) an isomorphism ϕ : G(R)0 → G such that ϕ(G(Z) ∩ G(R)0 ) and are commensurable, i.e. ϕ(G(Z) ∩ G(R)0 ) ∩ has finite index in both ϕ(G(Z) ∩ G(R)0 ) and . For the general definition of arithmetic lattices see [29, Definition 6.1.1]. A well-known and fundamental result due to Margulis ensures that this is usually the only way to obtain a lattice. More precisely, every irreducible lattice ⊂ G in a connected, semi-simple Lie group G with trivial center, no compact factors and R-rank(G) 2 is arithmetic ([20, 29]). Further results due to Corlette (cf. [5]) and Gromov and Schoen (cf. [12]) extended this result to all connected semi-simple Lie groups with trivial center except SO(1, n) and SU(1, n). In SO(1, n) (for all n ∈ N) and in SU(1, n) (for n = 2, 3) actually non-arithmetic lattices are known to exist (see e.g. [11, 20]).
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Definition 2.2 (Q-rank of an arithmetic lattice) Suppose ⊂ G is an arithmetic lattice in a connected semi-simple Lie group G with trivial center and no compact factors. Then Q-rank() is by definition the Q-rank of G, where G is an algebraic group as in Definition 2.1. The theory of algebraic groups shows that the definition of the Q-rank of an arithmetic lattice does not depend on the choice of the algebraic group G in Definition 2.1. A proof of this fact can be found in [28, Corollary 9.12]. We already mentioned a geometric interpretation of the R-rank: The R-rank of G as above coincides with the rank of the corresponding symmetric space X = G/K. For the Q-rank of an arithmetic lattice that acts freely on X there is also a geometric interpretation in terms of the large scale geometry of the corresponding locally symmetric space \X: Let us fix an arbitrary point p ∈ M = \X. The tangent cone at infinity of M is the (pointed) Gromov–Hausdorff limit of the sequence M, p, n1 d M of pointed metric spaces. Heuristically speaking, this means that we are looking at the locally symmetric space M from farther and farther away. The precise definition can be found in [22, Chapter 10]. We have the following geometric interpretation of Q-rank(). For a proof see [13, 18] or [28]. Theorem 2.3 Let X = G/K denote a symmetric space of non-compact type and ⊂ G an arithmetic lattice that acts freely on X. Then, the tangent cone at infinity of \X is isometric to a Euclidean cone over a finite simplicial complex whose dimension is Q-rank(). An immediate consequence of this theorem is that Q-rank() = 0 if and only if the locally symmetric space \X is compact. 2.2.2 Siegel Sets and Reduction Theory Let us denote in this subsection by G again a connected, semi-simple algebraic group defined over Q with trivial center and by X = G/K the corresponding symmetric space of non-compact type with G = G0 (R). Our main references in this subsection are [1, 4, 16]. Langlands decomposition of rational parabolic subgroups. Definition 2.4 A closed subgroup P ⊂ G defined over Q is called rational parabolic subgroup if P contains a maximal, connected solvable subgroup of G. (These subgroups are called Borel subgroups of G.) For any rational parabolic subgroup P of G we denote by NP the unipotent radical of P, i.e. the largest unipotent normal subgroup of P and by NP := NP (R) the real points of NP . The Levi quotient LP := P/NP is reductive and both NP and LP are defined over Q. If we denote by SP the maximal Q-split torus in the center of LP and by AP := SP (R)0 the connected component of
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SP (R) containing the identity, we obtain the decomposition of LP (R) into AP and the real points MP of a reductive algebraic group MP defined over Q: LP (R) = AP MP ∼ = AP × MP . After fixing a certain basepoint x0 ∈ X, we can lift the groups LP , SP and MP into P such that their images LP,x0 , SP,x0 and MP,x0 are algebraic groups defined over Q (this is in general not true for every choice of a basepoint x0 ) and give rise to the rational Langlands decomposition of P := P(R): P∼ = NP × AP,x0 × MP,x0 . More precisely, this means that the map P → NP × AP,x0 × MP,x0 ,
g → (n(g), a(g), m(g))
is a real analytic diffeomorphism. Denoting by XP,x0 the boundary symmetric space XP,x0 := MP,x0 /K ∩ MP,x0 we obtain, since the subgroup P acts transitively on the symmetric space X = G/K (we actually have G = PK), the following rational horocyclic decomposition of X: X∼ = NP × AP,x0 × XP,x0 . More precisely, if we denote by τ : MP,x0 → XP,x0 the canonical projection, we have an analytic diffeomorphism μ : NP × AP,x0 × XP,x0 → X, (n, a, τ (m)) → nam · x0 .
(2.1)
Note, that the boundary symmetric space XP,x0 is a Riemannian product of a symmetric space of non-compact type by a Euclidean space. For minimal rational parabolic subgroups, i.e. Borel subgroups P, we have dim AP,x0 = Q-rank(G). In the following we omit the reference to the chosen basepoint x0 in the subscripts. Q-Roots. Let us fix some minimal rational parabolic subgroup P of G. We denote in the following by g, aP , and nP the Lie algebras of the (real) Lie groups G, AP , and NP defined above. Associated with the pair (g, aP ) there is –similar to Section 2.2 – a system (g, aP ) of so-called Q-roots. If we define for α ∈
(g, aP ) the root spaces gα := {Z ∈ g : ad(H)(Y) = α(H)(Y) for all H ∈ aP },
we have the root space decomposition g = g0 ⊕
α∈ (g,aP )
gα ,
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where g0 is the Lie algebra of Z (SP (R)), the center of SP (R). Furthermore, the minimal rational parabolic subgroup P defines an ordering of (g, aP ) such that
nP = gα . α∈ + (g,aP )
The root spaces gα , gβ to distinct positive roots α, β ∈ + (g, aP ) are orthogonal with respect to the Killing form: B(gα , gβ ) = {0}. In analogy to Section 2.2 we define ρP :=
(dim gα )α.
α∈ + (g,aP )
Furthermore, we denote by ++ (g, aP ) the set of simple positive roots. Recall, that we call a positive root α ∈ + (g, aP ) simple if 12 α is not a root. Remark 2.5 The elements of (g, aP ) are differentials of characters of the maximal Q-split torus SP . For convenience, we identify the Q-roots with characters. If restricted to AP we denote therefore the values of these characters by α(a), (a ∈ AP , α ∈ (g, aP )) which is defined by α(a) := exp α(log a). Siegel sets. Since we will consider in the succeeding section only (nonuniform) arithmetic lattices with Q-rank() = 1, we restrict ourselves from now on to the case Q-rank(G) = 1.
For these groups we summarize several facts in the next lemma. Lemma 2.6 Assume Q-rank(G) = 1. Then the Following holds: (1) For any proper rational parabolic subgroup P of G, we have dim AP = 1. (2) All proper rational parabolic subgroups are minimal. (3) The set ++ (g, aP ) of simple positive Q-roots contains only a single element:
++ (g, aP ) = {α}. For any rational parabolic subgroup P of G and any t > 1, we define AP,t := {a ∈ AP : α(a) > t}, where α denotes the unique root in ++ (g, aP ). If we choose a0 ∈ AP with the property α(a0 ) = t, the set AP,t is just a shift of the positive Weyl chamber AP,1 by a0 : AP,t = AP,1 a0 .
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Before we define Siegel sets, we recall the rational horocyclic decomposition of the symmetric space X = G/K: X∼ = NP × AP × XP . Definition 2.7 Let P denote a rational parabolic subgroup of the algebraic group G with Q-rank(G) = 1. For any bounded set ω ⊂ NP × XP and any t > 1, the set SP,ω,t := ω × AP,t ⊂ X
is called Siegel set. Precise reduction theory. We fix an arithmetic lattice ⊂ G = G(R) in the algebraic group G with Q-rank(G) = 1. Recall, that by a well known result due to A. Borel and Harish–Chandra there are only finitely many -conjugacy classes of minimal parabolic subgroups (see e.g. [1]). Using the Siegel sets defined above, we can state the precise reduction theory in the Q-rank one case as follows: Theorem 2.8 Let G denote a semi-simple algebraic group defined over Q with Q-rank(G) = 1 and an arithmetic lattice in G. We further denote by P1 , . . . , Pk representatives of the -conjugacy classes of all rational proper (i.e. minimal) parabolic subgroups of G. Then there exist a bounded set 0 ⊂ X and Siegel sets ω j × AP j ,t j ( j = 1, . . . , k) such that the following holds: (1) Under the canonical projection π : X → \X each Siegel set ω j × AP j ,t j is mapped injectively into \X, i = 1, . . . , k. (2) The image of ω j in ( ∩ Pj)\NP j × XP j is compact ( j = 1, . . . , k). (3) The subset 0 ∪
k
ω j × AP j ,t j
j=1
is an open fundamental domain for . In particular, \X equals the closure
of π(0 ) ∪ kj=1 π(ω j × AP j ,t j ). Geometrically this means that the closure of each set π(ω j × AP j ,t j ) corresponds to one cusp of the locally symmetric space \X and the numbers t j are chosen large enough such that these sets do not overlap. Then the interior of the bounded set π(0 ) is just the complement of the closure of
k j=1 π(ω j × AP j ,t j ) cf. Fig. 2. Since in the case Q-rank(G) = 1 all rational proper parabolic subgroups are minimal, these subgroups are conjugate under G(Q) (cf. [1, Theorem 11.4]). Therefore, the root systems (g, aP j ) with respect to the rational proper
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Fig. 2 Disjoint decomposition of a Q-rank-1 space
parabolic subgroups P j, j = 1 . . . k, are canonically isomorphic (cf. [1, 11.9]) and moreover, we can conclude ||ρP1 || = . . . = ||ρPk ||. 2.2.3 Rational Horocyclic Coordinates For all α ∈ + (g, aP ) we define on nP = form hα by hα :=
α∈ + (g,aP )
gα a left invariant bilinear
·, · , on gα 0, else,
where Y, Z := −B(Y, θ Z ) denotes the usual Ad(K)-invariant bilinear form on g induced from the Killing form B. We then have (cf. [2, Proposition 1.6] or [3, Proposition 4.3]): Proposition 2.9 ∼ NP × XP × AP the tangent spaces at x to (a) For any x = (n, τ (m), a) ∈ X = the submanifolds {n} × XP × {a}, {n} × {τ (m)} × AP , and NP × {τ (m)} × {a} are mutually orthogonal. (b) The pullback μ∗ g of the metric g on X to NP × XP × AP is given by ds2(n,τ (m),a) =
1 2
e−2α(log a) hα ⊕ d(τ (m))2 ⊕ da2 .
α∈ + (g,aP )
If we choose orthonormal bases {N1 , . . . , Nr } of nP , {Y1 , . . . , Yl } of some tangent space Tτ (m) XP and H ∈ a+ P with ||H|| = 1, we obtain rational horocyclic coordinates ϕ : NP × XP × AP → Rr × Rl × R, ⎞ ⎛ ⎞ ⎞ ⎛ ⎛ r l ⎝exp ⎝ x j N j⎠ , exp ⎝ x j+r Yj⎠ , exp(yH)⎠ → (x1 , . . . , xr+l , y). j=1
j=1
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In the following, we will abbreviate (x1 , . . . , xr+l , y) as (x, y). The representation of the metric ds2 with respect to these coordinates is given by the matrix ⎛ ⎞ 1 −2α1 (log a) e 2 ⎜ ⎟ .. ⎜ ⎟ 0 . ⎜ ⎟ ⎜ ⎟ 1 −2αr (log a) e ⎜ ⎟ 2 ⎜ ⎟ (gij)i, j(n, τ (m), a) = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ hkm 0 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 1 where the positive roots αi ∈ + (g, aP ) appear according to their multiplicity and the (l × l)-submatrix (hkm )lk,m=1 represents the metric d(τ (m))2 on the boundary symmetric space XP . Corollary 2.10 The volume form of NP × XP × AP with respect to rational horocyclic coordinates is given by r/2 1 det(gij)(n, τ (m), a) dxdy = det(hkm (τ (m)) e−2ρP (log a) dxdy 2 r/2 1 = det(hkm (τ (m)) e−2||ρP ||y dxdy, 2 where log a = yH. A straightforward calculation yields. Corollary 2.11 The Laplacian on NP × XP × AP in rational horocyclic coordinates is = −2
r j=1
e2α j
∂2 ∂ ∂2 + XP − 2 + 2||ρP || , 2 ∂y ∂y ∂xj
(2.2)
where XP denotes the Laplacian on the boundary symmetric space XP and e2α j is short hand for the function (x, y) → e2yα j (H) . 3 L p -Spectrum In this section X = G/K denotes again a symmetric space of non-compact type whose metric coincides on TeK (G/K) ∼ = p with the Killing form of the Lie algebra g of G. Furthermore, ⊂ G is an arithmetic (non-uniform) lattice with Q-rank() = 1. We also assume that is torsion-free. The corresponding locally symmetric space M = \X has finitely many cusps and each cusp corresponds to a -conjugacy class of a minimal rational parabolic subgroup P ⊂ G. Let P1 , . . . , Pk denote representatives of the
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-conjugacy classes. Since these subgroups are conjugate under G(Q) and the respective root systems are isomorphic (cf. Section 2.2.2), we consider in the following only the rational parabolic subgroup P := P1 . We denote by ρP as in the preceding section half the sum of the positive roots (counted according to their multiplicity) with respect to the pair (g, aP ). We define for any p ∈ [1, ∞) the parabolic region ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ 4||ρP ||2 1 y2 P p := z = x + iy ∈ C : x 1− + 2 ⎪ ⎪ p p ⎩ 4||ρP ||2 1 − 2p ⎭ if p = 2 and P2 := [||ρP ||2 , ∞). Note, that the boundary ∂ P p of P p is parametrized by the curve 1 4||ρP ||2 2 2 1− + s + 2i||ρP ||s 1 − R → C, s → p p p 2||ρP || 2||ρP || = + is 2||ρP || − − is p p and that this parabolic region coincides with the one in Proposition 1.1 if and only if ||ρP || = ||ρ||. Our main result in this chapter reads as follows: Theorem 3.1 Let X = G/K denote a symmetric space of non-compact type and ⊂ G an arithmetic lattice with Q-rank() = 1 that acts freely on X. If we denote by M := \X the corresponding locally symmetric space, the parabolic region P p is contained in the spectrum of M, p , p ∈ (1, ∞): P p ⊂ σ ( M, p ). Lemma 3.2 Let M denote a Riemannian manifold with finite volume. For 1 p q < ∞, we have e−t M,q M,q ⊂ M, p e−t M,q . Proof Since the volume of M is finite, it follows by Hölder’s inequality Lq (M) → L p (M), i.e. Lq (M) is continuously embedded in L p (M). Therefore, we obtain the boundedness of the operators e−t M,q : Lq (M) → L p (M).
(3.1)
To prove the lemma, we choose an f ∈ dom( M,q ) = dom(e−t M,q M,q ). Because of e−t M,q f ∈ L p (M) ∩ dom( M,q ) and the consistency of the
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semigroups e−t M, p , p ∈ [1, ∞), we have e−s M, p e−t M,q f = e−(t+s) M,q f and obtain by using (3.1): " " " 1 −s " " (e M, p e−t M,q f − e−t M,q f ) − e−t M,q M,q f " "s " p L " " " 1 −s " + M,q C" f − f ) − M,q f " " s (e " q → 0 (s → 0 ). L Thus, the function e−t M,q f is contained in the domain of M, p and we also have the equality e−t M,q M,q f = M, p e−t M,q f. The following proposition follows from the preceding lemma as in [15, Proposition 3.1] or [14, Proposition 2.1]. For the sake of completeness we work out the details. Proposition 3.3 Let M denote a Riemannian manifold with finite volume. For 2 p q < ∞, we have the inclusion σ ( M, p ) ⊂ σ ( M,q ). Proof The statement of the proposition is obviously equivalent to the reverse inclusion for the respective resolvent sets: ρ( M,q ) ⊂ ρ( M, p ). We are going to show that for λ ∈ ρ( M,q ) ∩ ρ( M, p ) the resolvents coincide on Lq (M) ∩ L p (M). From Lemma 3.2 above, we conclude for these λ (λ − M, p )−1 e−t M,q = (λ − M, p )−1 e−t M,q (λ − M,q )(λ − M,q )−1 = (λ − M, p )−1 (λ − M, p ) e−t M,q (λ − M,q )−1 = e−t M,q (λ − M,q )−1 ,
(3.2)
where the equality is meant between bounded operators from L (M) to L p (M). If t → 0, we obtain # # (λ − M, p )−1 # Lq ∩L p = (λ − M,q )−1 # Lq ∩L p . q
For q1 + q1 = 1 (in particular, this implies q p q) and λ ∈ ρ( M,q ) = ρ( M,q ) we have by the preceding calculation # # (λ − M,q )−1 # Lq ∩Lq = (λ − M,q )−1 # Lq ∩Lq . The Riesz–Thorin interpolation theorem implies that (λ − M,q )−1 is bounded if considered as an operator Rλ on L p (M). In the remainder of the proof we show that Rλ coincides with (λ − M, p )−1 and hence ρ( M,q ) ⊂ ρ( M, p ). Notice, that (3.2) implies (λ − M, p )e−t M,q (λ − M,q )−1 f = e−t M,q f,
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for all f ∈ L p (M) ∩ Lq (M). Since M, p is a closed operator, we obtain for t → 0 the limit (λ − M, p )Rλ f = f. As Lq (M) ∩ L p (M) is dense in L p (M) and M, p is closed, it follows (λ − M, p )Rλ f = f for all f ∈ L p (M). Therefore, (λ − M, p ) is onto. If we assume that (λ − M, p ) is not one-to-one, λ would be an eigenvalue of M, p . Assume f = 0 is an eigenfunction of M, p for the eigenvalue λ. Then it follows from Lemma 3.2: λe−t M, p f = M,q e−t M, p f. Since e−t M, p is strongly continuous there is a t0 > 0 such that e−t0 M, p f = 0 and e−t0 M, p f is therefore an eigenfunction of M,q for the eigenvalue λ. But this contradicts λ ∈ ρ( M,q ) = ρ( M,q ). We finally obtain Rλ = (λ − M, p )−1 . Proposition 3.4 For 1 p < ∞ the boundary ∂ P p of the parabolic region P p is contained in the approximate point spectrum of M, p : ∂ P p ⊂ σapp ( M, p ). Proof In this proof we construct for any z ∈ ∂ P p a sequence fn of differentiable functions in L p (X) with support in a fundamental domain for such that || X, p fn − zfn || L p →0 (n → ∞). || fn || L p Since such a sequence ( fn ) descends to a sequence of differentiable functions in L p (M) this is enough to prove the proposition. Recall that a fundamental domain for is given by a subset of the form 0 ∪
k
ωi × APi ,ti ⊂ X
i=1
(cf. Theorem 2.8), and each Siegel set ωi × APi ,ti is mapped injectively into \X. Furthermore, the closure of π(ωi × APi ,ti ) fully covers an end of \X (for any i ∈ {1, . . . , k}). Now, we choose some 2||ρP || 2||ρP || z = z(s) = + is 2||ρP || − − is ∈ ∂ P p . p p Furthermore, we take the Siegel set ω × AP,t := ω1 × AP1 ,t1 where AP,t = {a ∈ AP : α(a) > t}, and define a sequence fn of smooth functions with support in ω × AP,t with respect to rational horocyclic coordinates by
2
||ρ ||+is y
fn (x, y) := cn (y)e p P , log t where cn ∈ Cc∞ ||α|| ,∞ is a so-far arbitrary sequence of differentiable log t , ∞ . Since ω is bounded, each fn is clearly functions with support in ||α||
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log t contained in L p (X). Furthermore, the condition supp(cn ) ⊂ ||α|| , ∞ ensures that the supports of the sequence fn are contained in the Siegel set ω × AP,t . Using formula (2.2) for the Laplacian in rational horocyclic coordinates, we obtain after a straightforward calculation X, p fn (x, y) − zfn (x, y) 2||ρ || P +is y 2 = −cn (y) + 2||ρP || − 2 cn (y) e p , ||ρP || + is p and therefore p
|| X, p fn − zfn || L p $ = | X, p fn − zfn | p dvol X ω×AP,t
=
r/2 $ 1 | X, p fn (x, y) − zfn (x, y)| p det(hkm (τ (m)) e−2||ρP ||y dxdy 2 ω×AP,t
# #p # # 2||ρP || # =C + is cn (y)## dy, #−cn (y) + 2||ρP || − 2 p 0 r/2 % √ det(hkm (τ (m))dx < ∞ because ω ⊂ NP × XP is bounded. where C := 12 ω This yields after an application of the triangle inequality $ ∞ 1/ p $ ∞ 1/ p |c n (y)| p dy + C2 |c n (y)| p dy . || X, p fn − zfn || L p C1 $
∞
0
0
By an analogous calculation we obtain $ ∞ 1/ p p || fn || L p = C3 |cn (y)| dy . 0
Cc∞ (R),
not identically zero, with supp(ψ) ⊂ (1, 2), We choose a function ψ ∈ a sequence rn > 0 with rn → ∞ (if n → ∞), and we eventually define y . cn (y) := ψ rn log t For large enough n, we have supp(cn ) ⊂ ||α|| , ∞ . An easy calculation gives $ 0
$ $
∞
∞ 0 ∞ 0
$ |cn (y)| p dy = rn 1
|c n (y)| p dy1 = rn1− p
2
|ψ(u)| p du,
$
|c n (y)| p dy1 = rn1−2 p
2
|ψ (u)| p du,
1
$
2 1
|ψ (u)| p du.
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In the end, this leads to the inequality || X, p fn − zfn || p C4 C5 + 2 −→ 0 || fn || p rn rn
(n → ∞),
where C4 , C5 > 0 denote positive constants, and the proof is complete.
Proof of Theorem 3.1 The inclusion P p ⊂ σ ( M, p ) for p ∈ [2, ∞) follows immediately from Proposition 3.3 and Proposition 3.4 by observing & Pp = ∂ Pq . q∈[2, p]
The inclusion for all p ∈ (1, ∞) follows by duality as P p = P p if
1 p
+
1 p
= 1.
Up to now, we considered non-uniform arithmetic lattices ⊂ G with Qrank one. We made no assumption concerning the rank of the respective symmetric space X = G/K of non-compact type. However, if rank(X) = 1, we are able to sharpen the result of Theorem 3.1 considerably. In the case Q-rank() = rank(X) = 1, the one dimensional abelian subgroup AP of G (with respect to some rational minimal parabolic subgroup) defines a maximal flat subspace, i.e. a geodesic, AP · x0 of X. Hence, the Q-roots coincide with the roots defined in Section 2.2 and for any rational minimal parabolic subgroup P we have in particular ||ρP || = ||ρ||. Corollary 3.5 Let X = G/K denote a symmetric space of non-compact type with rank(X) = 1. Furthermore, ⊂ G denotes a non-uniform arithmetic lattice that acts freely on X and M = \X the corresponding locally symmetric space. Then, we have for all p ∈ (1, ∞) the equality σ ( M, p ) = {λ0 , . . . , λm } ∪ P p , where 0 = λ0 , . . . , λm ∈ 0, ||ρ||2 are eigenvalues of M,2 with finite multiplicity. Proof Langlands’ theory of Eisenstein series implies (see e.g. [17] or the surveys in [16] or [4]) ( ' σ ( M,2 ) = λ0 , . . . , λm ∪ ||ρ||2 , ∞ , where 0 = λ0 , . . . , λm ∈ 0, ||ρ||2 are eigenvalues of M,2 with finite multiplicity. Thus, we can apply Proposition 1.1 and obtain σ ( M, p ) ⊂ {λ0 , . . . , λm } ∪ P p .
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As in the proof of [8, Lemma 6] one sees that the discrete part of the L2 -spectrum {λ0 , . . . , λm } is also contained in σ ( M, p ) for any p ∈ (1, ∞). Together with Theorem 3.1 and the remark above this concludes the proof. As remarked in [24] one can prove as in [8] that every L2 -eigenfunction of the Laplace–Beltrami operator M,2 with respect to the eigenvalue λ j, j = 0, . . . , m, lies in L p (M) if λ j is not contained in P p .
Remark 3.6 Because of the description of fundamental domains for general lattices in semi-simple Lie groups with R-rank one (see [10]) it seems that the arithmeticity of in Corollary 3.5 is not needed.
4 Manifolds with Cusps of Rank One In this chapter we consider a class of Riemannian manifolds that is larger than the class of Q-rank one locally symmetric spaces. This larger class consists of those manifolds which are isometric – after the removal of a compact set – to a disjoint union of rank one cusps. Manifolds with cusps of rank one were probably first introduced and studied by W. Müller (see e.g. [21]). 4.1 Definition Recall, that we denoted by ω × AP,t ⊂ X Siegel sets of a symmetric space X = G/K of non-compact type. The projection π(ω × AP,t ) of certain Siegel sets to a corresponding Q-rank one locally symmetric space \X is a cusp and every cusp of \X is of this form (cf. Section 2.2.2). Definition 4.1 A Riemannian manifold is called cusp of rank one if it is isometric to a cusp π(ω × AP,t ) of a Q-rank one locally symmetric space. Definition 4.2 A complete Riemannian manifold M is called manifold with cusps of rank one if it has a decomposition M = M0 ∪
k &
Mj
j=1
such that the following holds: (1) M0 is a compact manifold with boundary. (2) The subsets M j, j ∈ {0, . . . , k}, are pairwise disjoint. (3) For each j ∈ {1, . . . , k} there exists a cusp of rank one isometric to M j. Such manifolds certainly have finite volume as there is only a finite number of cusps possible and every cusp of rank one has finite volume.
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From Theorem 2.8 it follows that any Q-rank one locally symmetric space is a manifold with cusps of rank one. But since we can perturb the metric on the compact manifold M0 without leaving the class of manifolds with cusps of rank one, not every such manifold is locally symmetric. Of course, they are locally symmetric on each cusp and we can say that they are locally symmetric near infinity. 4.2 L p -Spectrum and Geometry Precisely as in Proposition 3.4 one sees that we can find for every cusp M j, j ∈ ) {1, . . . , k} of a manifold M = M0 ∪ kj=1 M j with cusps of rank one a parabolic ( j)
( j)
region P p such that the boundary ∂ P p is contained in the approximate point spectrum of M, p . Here, the parabolic regions are defined as the parabolic region in the preceding section, where the constant ||ρP || is replaced by an analogous quantity, say ||ρP j ||, coming from the respective cusp M j. That is to say, we have the following lemma: Lemma 4.3 Let M denote a manifold with cusps of rank one. Then we have for p ∈ [1, ∞) and j = 1, . . . , k: ∂ P(pj) ⊂ σapp ( M, p ). Since the volume of a manifold with cusps of rank one is finite, we can apply Proposition 3.3 in order to prove (cf. the proof of Theorem 3.1) the following: Theorem 4.4 Let M = M0 ∪
)k j=1
M j denote a manifold with cusps of rank ( j)
one. Then, for p ∈ (1, ∞), every cusp M j defines a parabolic region P p that is contained in the L p -spectrum (cf. Fig. 3): k & j=1
Fig. 3 The union of two (1) parabolic regions P p and (2) P p if p = 2
P(pj ) ⊂ σ ( M, p ).
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Of course, the compact submanifold M0 contributes some discrete set to the L p -spectrum, and 0 is always an eigenvalue as the volume of M is finite. It seems to be very likely that besides some discrete spectrum the union of the parabolic regions in Theorem 4.4 is already the complete spectrum. But at present, I do not know how to prove this result. The methods used in [8] or [24] to prove a similar result need either that the manifold is homogeneous or that the injectivity radius is bounded from below, and it is not clear how one could adapt the methods therein to our case. Nevertheless, given the L p -spectrum for some p = 2, we have the following geometric consequences: Corollary 4.5 Let M = M0 ∪ such that
)k j=1
M j denote a manifold with cusps of rank one
σ ( M, p ) = {λ0 , . . . , λr } ∪ P p , for some p = 2 and some parabolic region P p . Then every cusp M j is of the form π(ω j × AP j ,t j ) with volume form r j /2 1 e−2yc dxdy, 2 where c is a positive constant. ( j)
Proof Since all parabolic regions P p induced by the cusps M j coincide, the quantities ||ρP j || coincide. Therefore, we can take c := ||ρP1 ||. This result generalizes to the case where the continuous spectrum consists of a finite number of parabolic regions in an obvious manner.
References 1. Borel, A.: Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341. Hermann, Paris (1969) MR MR0244260 (39 #5577) 2. Borel, A.: Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. J. Differential Geom. 6, 543–560 (1972) MR MR0338456 (49 #3220) 3. Borel, A.: Stable real cohomology of arithmetic groups. Ann. Sci. École. Norm. Sup. 7(4), 235–272 (1974) MR MR0387496 (52 #8338) 4. Borel, A., Ji, L.: Compactifications of symmetric and locally symmetric spaces. Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA (2006) MR MR2189882 5. Corlette, K.: Archimedean superrigidity and hyperbolic geometry. Ann. of Math. (2) 135(1), 165–182 (1992) MR MR1147961 (92m:57048) 6. Brian Davies, E.: Pointwise bounds on the space and time derivatives of heat kernels. J. Operator Theory 21(2), 367–378 (1989) MR MR1023321 (90k:58214) 7. Brian Davies, E.: Heat kernels and spectral theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press (1990) MR MR1103113 (92a:35035) 8. Brian Davies, E., Simon, B., Taylor, M.E.: L p spectral theory of Kleinian groups. J. Funct. Anal. 78(1), 116–136 (1988) MR MR937635 (89m:58205) 9. Eberlein, P.B.: Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1996) MR MR1441541 (98h:53002)
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10. Garland, H., Raghunathan, M.S.: Fundamental domains for lattices in (R-)rank 1 semisimple Lie groups. Ann. of Math. 92(2), 279–326 (1970) MR MR0267041 (42 #1943) 11. Gromov, M., Piatetski-Shapiro, I.I.: Nonarithmetic groups in Lobachevsky spaces. Inst. Hautes Études Sci. Publ. Math. (66), 93–103 (1988) MR MR932135 (89j:22019) 12. Gromov, M., Schoen, R.: Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Inst. Hautes Études Sci. Publ. Math. (76), 165–246 (1992) MR MR1215595 (94e:58032) 13. Hattori, T.: Asymptotic geometry of arithmetic quotients of symmetric spaces. Math. Z. 222(2), 247–277 (1996) MR MR1429337 (98d:53061) 14. Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in L p (Rν ) is p-independent. Comm. Math. Phys. 104(2), 243–250 (1986) MR MR836002 (87h:35247) 15. Hempel, R., Voigt, J.: On the L p -spectrum of Schrödinger operators. J. Math. Anal. Appl. 121(1), 138–159 (1987) MR MR869525 (88i:35114) 16. Ji, L., MacPherson, R.: Geometry of compactifications of locally symmetric spaces. Ann. Inst. Fourier (Grenoble) 52(2), 457–559 (2002) MR MR1906482 (2004h:22006) 17. Langlands, R.P.: On the functional equations satisfied by Eisenstein series. Lecture Notes in Mathematics, vol. 544. Springer-Verlag, Berlin (1976) MR MR0579181 (58 #28319) 18. Leuzinger, E.: Tits geometry, arithmetic groups, and the proof of a conjecture of Siegel. J. Lie Theory 14(2), 317–338 (2004) MR MR2066859 (2006a:53040) 19. Liskevich, V.A., Perel’muter, M.A.: Analyticity of sub-Markovian semigroups. Proc. Amer. Math. Soc. 123(4), 1097–1104 (1995) MR MR1224619 (95e:47057) 20. Margulis, G.A.: Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17. SpringerVerlag, Berlin (1991) MR MR1090825 (92h:22021) 21. Müller, W.: Manifolds with cusps of rank one. Lecture Notes in Mathematics, vol. 1244, Spectral theory and L2 -index theorem. Springer-Verlag, Berlin (1987) MR MR891654 (89g:58196) 22. Petersen, P.: Riemannian geometry. Graduate Texts in Mathematics, vol. 171. SpringerVerlag, New York (1998) MR MR1480173 (98m:53001) 23. Sturm, K.-T.: On the L p -spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118(2), 442–453 (1993) MR MR1250269 (94m:58227) 24. Taylor, M.E.: L p -estimates on functions of the Laplace operator. Duke Math. J. 58(3), 773–793 (1989) MR MR1016445 (91d:58253) 25. Varopoulos, N.Th.: Analysis on Lie groups. J. Funct. Anal. 76(2), 346–410 (1988) MR MR924464 (89i:22018) 26. Weber, A.: Heat kernel estimates and L p -spectral theory of locally symmetric spaces. Dissertation, Universitätsverlag Karlsruhe (2006) 27. Weber, A.: L p -spectral theory of locally symmetric spaces with small fundamental group (2007) (Submitted) 28. Witte Morris, D.: Introduction to arithmetic groups. URL-Address: http://www.math.okstate. edu/∼dwitte, February 2003 29. Zimmer, R.J.: Ergodic theory and semisimple groups. Monographs in Mathematics, vol. 81. Birkhäuser Verlag, Basel (1984) MR MR776417 (86j:22014)
Math Phys Anal Geom (2007) 10:155–196 DOI 10.1007/s11040-007-9028-1
Asymptotic Behaviour of the Spectrum of a Waveguide with Distant Perturbations Denis Borisov
Received: 23 May 2007 / Accepted: 11 August 2007 / Published online: 1 September 2007 © Springer Science + Business Media B.V. 2007
Abstract We consider a quantum waveguide modelled by an infinite straight tube with arbitrary cross-section in n-dimensional space. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators “localized” in a certain sense. We study the asymptotic behaviour of the discrete spectrum of such system as the distance between the “supports” of localized perturbations tends to infinity. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We provide a list of the operators, which can be chosen as distant perturbations. In particular, the distant perturbations may be a potential, a second order differential operator, a magnetic Schrödinger operator, an arbitrary geometric deformation of the straight waveguide, a delta interaction, and an integral operator. Keywords Distant perturbation · Waveguide · Asymptotics · Eigenvalue · Eigenfunction Mathematics Subject Classifications (2000) 35P05 · 35B20 · 35B40
The research was supported by Marie Curie International Fellowship within 6th European Community Framework Programm (MIF1-CT-2005-006254). The author is also supported by the Russian Foundation for Basic Researches (No. 07-01-00037) and by the Czech Academy of Sciences and Ministry of Education, Youth and Sports (LC06002). The author gratefully acknowledges the support from Deligne 2004 Balzan prize in mathematics and the grant of Republic Bashkortostan for young scientists and young scientific collectives. D. Borisov (B) ˇ near Prague 25068, Czechia, Nuclear Physics Institute, Academy of Sciences, Rež Bashkir State Pedagogical University, October rev. st. 3a, 450000 Ufa, Russia e-mail:
[email protected]
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1 Introduction The multiple well problem for the Schrödinger operator attracted much attention of many researches. The example of such operator with two wells is as follows, − + V1 (x − a1 ) + V2 (x − a2 ),
(1.1)
where ai are some points, and Vi are real-valued functions satisfying certain smoothness conditions and being either compactly supported or decaying sufficiently fast at infinity. A lot of works were devoted to the study of the discrete spectrum in the semi-classical case, i.e., as → 0 (see, for instance, [8, 11, 22], and references therein). The results obtained in this case are close in a certain sense to that obtained in the regime when the distances between the wells tend to infinity, for instance, as = 1 and |a1 − a2 | → +∞ in (1.1). We mention the papers [14, 16, 18] as well as the book [12, Sec. 8.6] devoted to such problems (see also bibliography of these works). The main result of these works was a description of the asymptotic behaviour of the eigenvalues and the eigenfunctions as the distance between wells tended to infinity. In [15] a double-well problem for the Dirac operator with large distance between wells was studied. The convergence and certain asymptotic results were established. We also mention the paper [19], where the usual potential was replaced by a delta-potential supported by a curve. The results of this paper imply the asymptotic estimate for the lowest spectral gap in the case the curve consists of several disjoint components and the distances between components tend to infinity. One of the possible ways to generalize the mentioned problems is to consider them not in Rn , but in some other unbounded domains. An example of physical relevance is an infinite tube, since such domain arises in the waveguide theory. An additional motivation is that the tube is infinite in one dimension only, and it simplifies the considerations. One of such problems has already been treated in [4]. Here we considered the Dirichlet Laplacian in a two-dimensional straight infinite strip. The perturbation consisted of two segments of the same length on the boundary, on which the Dirichlet condition was switched to the Neumann one. As the distance between the segments tended to infinity, a convergence result and the asymptotic expansions for the eigenvalues and the eigenfunctions were obtained. The technique used in [4] employed essentially the symmetry of the problem. In the present paper we consider an infinite straight tube in n-dimensional space and consider the Dirichlet Laplacian −(D) in this tube. The perturbation consists of two arbitrary operators L± “localized” in a certain sense and satisfying certain sufficiently weak conditions. The distance between their “supports” is assumed to be a large parameter. In what follows we will call these operators distant perturbations. The main distinction to the articles cited is that we don not specify the nature of these operators. Their precise description will be given in the next section; we only say here that a lot of
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interesting examples are particular cases of these operators. In particular, the classical example, a compactly supported potential can be chosen as one of the distant perturbation; the same is true for a second order differential operators with compactly supported coefficients. In addition, one can also consider integral operators as a distant perturbation, a delta interaction or a geometric deformation of the tube can be treated as well. We refer to Section 7 for detailed presentation of these examples. Our main results are as follows. First we prove the convergence for the eigenvalues of the perturbed operator, and show that the limiting values for these eigenvalues are the discrete eigenvalues of the limiting operators −(D) + L± and the threshold of the essential spectrum. The most nontrivial result of the article is the asymptotic expansions for the perturbed eigenvalues. Namely, we obtain a scalar equation for these eigenvalues. Based on this equation, we construct the leading terms of their asymptotic expansions. We also characterize the asymptotic behaviour of the perturbed eigenfunctions. If the distant perturbations consisted of two potential wells as in (1.1), and one of the wells can be translated into the other, it is well-known that each limiting eigenvalue splits into pair of two perturbed eigenvalues one of which being larger than a limiting eigenvalue while the other being less. It is also known that the next to leading terms of their asymptotics are exponentially small w.r.t. the distance between the wells, and have the same modulus but different signs. In the present article we show that the similar phenomenon occur in the general situation, too (see Theorem 2.8). Let us discuss the technique used in the paper. The core of the approach is a scheme which allows us to reduce the eigenvalue equation for the perturbed operator to an equivalent operator equation in an auxiliary Hilbert space. The main motivation of such reduction is that the original distant perturbations are replaced by an operator which is meromorphic w.r.t. the spectral parameter and is multiplied by a small parameter. We solve this equation explicitly by the modification of the Birman–Schwinger technique suggested in [13], and in this way we obtain the described results. We stress that our approach requires no symmetry restriction in contrast to [4]. One of the advantages of our approach is that it is independent of the type of boundary condition, for instance, similar problem for the Neumann Laplacian can be solved effectively, too. Moreover, our technique can applied to other problems with distant perturbations. We can refer to [5] where we studied the Dirichlet Laplacian in two adjacent straight infinite strips; the perturbation consisted of two finite segments cut out in the common boundary and separated by a large distance. This problem was a natural generalization of that treated in [4] and it can not be described by the operators L± considered in the present paper. At the same time, we showed in [5] that the main ideas of the present paper could be adapted to the aforementioned problem with minor changes. We also cite [6] where we considered the Laplacian in a multi-dimensional space with a finite number of distant perturbations. We followed the ideas of the present paper to treat this problem. On the other hand, the waveguide we consider here is infinite in one dimension only, while in [6] the domain was infinite in many dimensions.
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Because of this we had to employ an additional technique; we should also say that the results of [6] are less explicit than those in the present paper. The article is organized as follows. In the next section we give precise statement of the problem and formulate the main results. In the second section we prove that the essential spectrum of the perturbed operator is invariant w.r.t. the perturbations and occupies a real semi-axis, while the discrete spectrum contains finitely many eigenvalues. The third section is devoted to the study of the limiting operators; we collect there some preliminary facts required for the proof of the main results. In the fourth section we provide the aforementioned scheme transforming the original perturbed eigenvalue equation to an equivalent operator equation. Then we solve this equation explicitly. The fifth section is devoted to the proof of the convergence result. The asymptotic formulas for the perturbed eigenelements are established in the sixth section. The final seventh section contains some examples of L± .
2 Statement of the Problem and Formulation of the Results Let x = (x1 , x ) and x = (x2 , . . . , xn ), be Cartesian coordinates in Rn and Rn−1 , respectively, n 2, and let ω be a bounded domain in Rn−1 having infinitely differentiable boundary. By we denote an infinite tube R × ω. Given any bounded domain Q ⊂ , by L2 (, Q) we denote the subset of the functions from L2 () whose support lies inside Q. For any domain ⊆ and any (n − j 1)-dimensional manifold S ⊂ the symbol W2,0 (, S) will indicate the subset j of the functions from W2 () vanishing on S. If S = ∂, we will write shortly j W2,0 (). Let ± be a pair of bounded subdomains of defined as ± := (−a± , a± ) × ω, where a± ∈ R are fixed positive numbers. We let γ± := (−a± , a± ) × ∂ω. j By L± we denote a pair of bounded linear operators from W2,0 (± , γ± ) into j L2 (, ± ). We assume that for all u1 , u2 ∈ W2,0 (± , γ± ) the identity (L± u1 , u2 ) L2 (± ) = (u1 , L± u2 ) L2 (± )
(2.1)
holds true. We also suppose that the operators L± satisfy the estimate (L± u, u) L2 (± ) −c0 ∇u2L2 (± ) − c1 u2L2 (± )
(2.2)
2 for all u ∈ W2,0 (± , γ± ), where the constants c0 , c1 are independent of u, and
c0 < 1.
(2.3)
2 () on ± belongs to Since the restriction of each function u ∈ W2,0 2 W2,0 (± , γ± ), we may also regard the operators L± as unbounded ones in 2 (). L2 () with the domain W2,0 By S (a) we denote a shift operator in L2 () acting as (S (a)u)(x) := u(x1 + a, x ), and for any l > 0 we introduce the operator
Ll := S (l)L− S (−l) + S (−l)L+ S (l).
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Clearly, this operator depends on values its argument takes on the set {x : (x1 + l, x ) ∈ − } ∪ {x : (x1 − l, x ) ∈ + }. As l → +∞, this set consists of two components separated by the distance 2l. This is why we can regard the operator Ll as the distant perturbations formed by L− and L+ . We introduce the operator Hl := −(D) + Ll in L2 () with the domain 2 W2,0 (). Here −(D) indicates the Laplacian in L2 () with the domain 2 W2,0 (). We suppose that the operators L± are such that the operator Hl is self-adjoint. The main aim of this paper is to study the behaviour of the spectrum of Hl as l → +∞. In order to formulate the main results we need to introduce additional notations. We will employ the symbols σ (·), σess (·), σdisc (·) to indicate the spectrum, the essential spectrum and the discrete one of an operator. We denote H± := −(D) + L± , and suppose that these operators with the domain 2 W2,0 () are self-adjoint in L2 (). Remark 2.1 We note that the assumptions (2.1), (2.2), (2.3) do not imply the self-adjointness of Hl and H± , and we can employ here neither the KLMN theorem no the Kato–Rellich theorem. On the other hand, if the operators L± satisfy stricter assumption and are −(D) -bounded with the bound less than one, it implies the self-adjointness of Hl and H± . Let ν1 > 0 be the lowest eigenvalue of the Dirichlet Laplacian in ω. Our first result is Theorem 2.1 The essential spectra of Hl , H+ , H− coincide with the semi-axis [ν1 , +∞). The discrete spectra of Hl , H+ , H− consist of finitely many real eigenvalues. We denote σ∗ := σdisc (H− ) ∪ σdisc (H+ ). Let λ∗ ∈ σ∗ be a p− -multiple eigenvalue of H− and p+ -multiple eigenvalue of H+ , where we set p± equal to zero, if λ∗ ∈ σdisc (H± ). In this case we will say that λ∗ is ( p− + p+ )-multiple. Theorem 2.2 Each discrete eigenvalue of Hl converges to one of the numbers in σ∗ or to ν1 as l → +∞. Theorem 2.3 If λ∗ ∈ σ∗ is ( p− + p+ )-multiple, the total multiplicity of the eigenvalues of Hl converging to λ∗ equals p− + p+ . In what follows we will employ the symbols (·, ·) X and · X to indicate the scalar product and the norm in a Hilbert space X. Suppose that λ∗ ∈ σ∗ is ( p− + p+ )-multiple, and ψi± , i = 1, . . . , p± , are the eigenfunctions of H± associated with λ∗ and orthonormalized in L2 (). If p− 1, we denote φ i (·, l) := (0; L+ S (2l)ψi− ) ∈ L2 (− ) ⊕ L2 (+ ), T1(i) f := ( f− , ψi− ) L2 (− ) ,
i = 1, . . . , p− ,
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where f := ( f− ; f+ ) ∈ L2 (− ) ⊕ L2 (+ ). If p+ 1, we denote φ i+ p− (·, l) := (L− S (−2l)ψi+ ; 0) ∈ L2 (− ) ⊕ L2 (+ ), (i+ p− )
T1
f := ( f+ , ψi+ ) L2 (+ ) ,
i = 1, . . . , p+ .
In the fourth section we will show that the operator T2 (λ, l) f := L− S (−2l)(H+ − λ)−1 f+ ; L+ S (2l)(H− − λ)−1 f−
(2.4)
in L2 (− ) ⊕ L2 (+ ) satisfies the relation T2 (λ, l) = −
p 1 φ (·, l)T1(i) + T3 (λ, l), λ − λ∗ i=1 i
(2.5)
for λ close to λ∗ , where p := p− + p+ , and the norm of T3 tends to zero as l → +∞ uniformly in λ. We introduce the matrix ⎛ ⎞ A11 (λ, l) . . . A1 p (λ, l) ⎜ ⎟ .. .. A(λ, l) := ⎝ ⎠, . . A p1 (λ, l) . . . A pp (λ, l)
where Aij(λ, l) := T1(i) (I + T3 (λ, l))−1 φ j(·, l). Theorem 2.4 Let λ∗ ∈ σ∗ be ( p− + p+ )-multiple, and let λi = λi (l) −−−−→ λ∗ , l→+∞
i = 1, . . . , p, p := p− + p+ , be the eigenvalues of Hl taken counting multiplicity and ordered as follows 0 |λ1 (l) − λ∗ | |λ2 (l) − λ∗ | . . . |λ p (l) − λ∗ |.
(2.6)
These eigenvalues solve the equation det (λ − λ∗ )E − A(λ, l) = 0, and satisfy the asymptotic formulas 2 4l √ λi (l) = λ∗ + τi (l) 1 + O l p e− p ν1 −λ∗ ,
(2.7)
l → +∞.
(2.8)
Here
√ τi = τi (l) = O e−2l ν1 −λ∗ , l → +∞, (2.9) are the zeroes of the polynomial det τ E − A(λ∗ , l) taken counting multiplicity and ordered as follows 0 |τ1 (l)| |τ2 (l)| . . . |τ p (l)|.
(2.10)
The eigenfunctions associated with λi satisfy the asymptotic representation ψi (x, l) =
p− i=1
ki, jψ − j (x1 + l, x ) +
p+
√ −2l ν1 −λ∗ ki, j+ p− ψ + (x − l, x ) + O e , 1 j
i=1
(2.11)
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as l → +∞ in W22 ()-norm. The numbers ki, j are the components of the vectors t ki = ki (l) = ki,1 (l) . . . ki, p (l) solving the system (λ − λ∗ )E − A(λ, l) k = 0 (2.12) for λ = λi (l), and satisfying the condition
1, if i = j, √ (ki , k j)C p = −2l ν1 −λ∗ O le , if i = j.
(2.13)
According to this theorem, the leading terms of the asymptotics expansions for the eigenvalues λi are determined by the matrix A(λ∗ , l). On the other hand, in applications it could be quite complicated to calculate this matrix explicitly. This is why in the following theorems we provide one more way of calculating the asymptotic expansions. We will say that a square matrix A(l) satisfies the condition (A), if it is diagonalizable and the determinant of the matrix formed by the normalized eigenvectors of A(l) is separated from zero uniformly in l large enough. Theorem 2.5 Let the hypothesis of Theorem 2.4 hold true. Suppose that the matrix A(λ∗ , l) can be represented as A(λ∗ , l) = A0 (l) + A1 (l),
(2.14)
where the matrix A0 satisfies the condition (A), and A1 (l) → 0 as l → +∞. In this case the eigenvalues λi of Hl satisfy the asymptotic formulas √ λi = λ∗ + τi(0) 1 + O l 2 e−4l ν1 −λ∗ + O(A1 (l)), l → +∞. (2.15) Here τi(0) = τi(0) (l) are the roots of the polynomial det τ E−A0 (l) taken counting multiplicity and ordered as follows 0 τ1(0) (l) τ2(0) (l) . . . τ p(0) (l). Each of these roots satisfies the estimate τi(0) (l) = O(A0 (l)),
l → +∞.
(2.16)
This theorem states that the leading terms of the asymptotics for the eigenvalues can be determined by that of the asymptotics for A(λ∗ , l). We observe that the estimate for the error term in (2.15) can be worse than that in (2.8). In the following theorem we apply Theorem 2.5 to several important particular cases. Let ν2 > ν1 be the second eigenvalue of the negative Dirichlet Laplacian in ω, and φ1 = φ1 (x ) be the eigenvalue associated with ν1 and normalized in L2 (ω). In the fifth section we will prove
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Lemma 2.1 Let the hypothesis of Theorem 2.4 hold true, and p± 1. Then the functions ψi± can be chosen so that √ √ ψ1± (x) = β± e± ν1 −λ∗ x1 φ1 (x ) + O e± ν2 −λ∗ x1 , √ ψi± (x) = O e± ν2 −λ∗ x1 , (2.17) as x1 → ∓∞, i = 2, . . . , p± , β± are some number, and the functions ψi± are orthonormalized in L2 (). In what follows we assume that the functions ψi± are chosen in accordance with this lemma. Theorem 2.6 Let the hypothesis of Theorem 2.5 hold true, and p+ = 0. Then the eigenvalues λi satisfy the asymptotic formulas √ √ λi (l) = λ∗ + O e−2l( ν1 −λ∗ + ν2 −λ∗ ) , i = 1, . . . , p − 1, √ √ − e−4l ν1 −λ∗ + λ p (l) = λ∗ − 2 ν1 − λ∗ |β− |2 β √ √ √ + O e−2l( ν1 −λ∗ + ν2 −λ∗ ) + l 2 e−6l ν1 −λ∗ ,
(2.18)
− is determined uniquely by the identity where the constant β √ √ − e− ν1 −λ∗ x1 φ1 (x ) + O e ν2 −λ∗ x1 , U + (x) = β x1 → −∞, √ U + : = (H+ − λ∗ )−1 L+ e− ν1 −λ∗ x1 φ1 (x ) .
(2.19)
Theorem 2.7 Let the hypothesis of Theorem 2.5 hold true, and p− = 0. Then the eigenvalues λi satisfy the asymptotic formulas √ √ λi (l) = λ∗ + O e−2l( ν1 −λ∗ + ν2 −λ∗ ) , i = 1, . . . , p − 1, √ √ + e−4l ν1 −λ∗ + λ p (l) = λ∗ − 2 ν1 − λ∗ |β+ |2 β √ √ √ + O e−2l( ν1 −λ∗ + ν2 −λ∗ ) + l 2 e−6l ν1 −λ∗ ,
(2.20)
+ is uniquely determined by the identity where the constant β √ √ + e ν1 −λ∗ x1 φ1 (x ) + O e ν2 −λ∗ x1 , U + (x) = β x1 → +∞, √ U + := (H− − λ∗ )−1 L+ e− ν1 −λ∗ x1 φ1 (x ) These two theorems treat the first possible case when the number λ∗ ∈ σ is the eigenvalue of one of the operators H± only. The formulas (2.18), (2.20) give the asymptotic expansion for the eigenvalue λ p , and the asymptotic estimates for the other eigenvalues. At the same time, given generic L± and an eigenvalue λ∗ of H± , this eigenvalue is simple. In this case p = 1, and by
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Theorem 2.3 there exists the unique perturbed eigenvalue converging to λ∗ , and Theorems 2.6, 2.7 provide its asymptotics. Theorem 2.8 Let the hypothesis of Theorem 2.4 hold true, and p± 1. Then the eigenvalues λi satisfy the asymptotic formulas √ λi (l) = λ∗ + O le−4l ν1 −λ∗ , i = 1, . . . , p − 2, √ √ √ λ p−1 (l) = λ∗ − 2|β− β+ | ν1 − λ∗ e−2l ν1 −λ∗ + O le−4l ν1 −λ∗ , √ √ √ (2.21) λ p (l) = λ∗ + 2|β− β+ | ν1 − λ∗ e−2l ν1 −λ∗ + O le−4l ν1 −λ∗ , as l → +∞. This theorem deals with the second possible case when the number λ∗ ∈ σ is an eigenvalue of both operators H± . Similarly to Theorems 2.6, 2.7, the formulas (2.21) give the asymptotic expansions for λ p−1 and λ p , and the asymptotic estimates for the other eigenvalues. The most probable case is that λ∗ is a simple eigenvalue of H+ and H− . In this case there exist only two perturbed eigenvalues converging to λ∗ , and Theorem 2.8 give their asymptotic expansions. Suppose now that under the hypothesis of Theorem 2.8 the inequality β− β+ = 0 holds true. In this case the next to leading terms of the asymptotic expansions for λ p−1 and λ p have the same modulus but different signs. Moreover, these eigenvalues are simple. This situation is similar to what one has when dealing with a double-well problem with symmetric wells. We stress that in our case we assume no additional restrictions except the belonging λ∗ ∈ σdisc (H− ) ∩ σdisc (H+ ). Hence, the latter condition is sufficient for the mentioned phenomenon to occur regardless of whether L+ and L− are equal or not. We also observe that the formulas (2.21) allow us to estimate the spectral gap between λ p−1 and λ p , √ √ λ2 (l) − λ1 (l) = 4|β− β+ | ν1 − λ∗ e−2l ν1 −λ∗ + O(le−4l ν1 −λ∗ ), l → +∞. Finally we note that it is possible to calculate the asymptotic expansions for the eigenvalues λi , i p − 1, in Theorem 2.6, 2.7, and for λi , i p − 2, in Theorem 2.8. In order to do it, one should employ the technique of the proofs of the mentioned theorems and extract the next-to-leading term of the asymptotic for A(λ∗ , l). Then these terms should be treated as the matrix A0 (l) in (2.14). We refrain from presenting such calculations here in order not to overburden the text by quite bulky and technical details.
3 Proof of Theorem 2.1 Let be a bounded non-empty subdomain of defined as := (−a, a) × ω, where a ∈ R, a > 0, γ := (−a, a) × ∂ω, and let L be an arbitrary bounded
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2 operator from W2,0 (, γ ) into L2 (, ). Assume that for all u, u1 , u2 ∈ 2 W2,0 (, γ ) the relations
(Lu1 , u2 ) L2 () = (u1 , Lu2 ) L2 () , (Lu, u) L2 () −c0 ∇u2L2 () − c1 u2L2 ()
(3.1)
hold true, where c0 , c1 are constants, and c0 obeys (2.3). As in the case of the operators L± , we can also regard L as an unbounded operator in L2 () with 2 (). Suppose also that the operator HL := −(D) + L with the domain W2,0 2 the domain W2,0 () is self-adjoint in L2 (). Lemma 3.1 σess (HL ) = [ν1 , +∞). Proof Let λ ∈ [ν1 , +∞). Since λ ∈ σess (−(D) ), it is well-known that there exists a Weyl sequence of −(D) at λ and this sequence of compactly supported functions; without loss of generality we can also assume that the supports of these functions do not intersect with . Now it is easy to check that the mentioned sequence is also a Weyl sequence for HL at λ. Therefore, [ν1 , +∞) ⊆ σess (HL ). To complete the proof, it is sufficient to show that inf σess (HL ) = ν1 . The threshold of the essential spectrum of HL is given by Agmon–Persson formula (HL u, u) L2 () inf σess (HL ) = lim inf , (3.2) R→+∞ u∈C0∞ (−R ∪+R ) u2L2 () where ±R := ∩ {x : ±x1 > ±R}. The proof of this formula for the Schrödinger operator was given in [21]; the case of more general elliptic operator was treated in [1, Ch. 3, Th. 3.2] and also in [23, Ch. 14, Sec. 14.3, Prop. 14.8]. In our case the proof of this formula reproduces word by word the proof given in [23]; this is why we do not give it here. Since (HL u, u) L2 () = ∇u2L2 () , if u ∈ C0∞ (−R ∪ +R ) and R > a, it follows from (3.2) that inf σess (HL ) = inf σess (−(D) ) = ν1 . Lemma 3.2 The discrete spectrum of HL consists of finitely many eigenvalues. Proof Let χ = χ(t) ∈ C∞ (R) be a cut-off function which takes values in [0, 1], is identically equal to one for t < 0, and vanishes for t > 1. Due to (3.1) we have (u, HL u) L2 () ∇u2L2 () − c0 ∇u2L2 () − c1 u2L2 () ∇u, (1 − c0 χ(|x1 | − a))∇u L2 () − u, c1 χ(|x1 | − a)u L2 () .
(3.3)
± ± By HL we denote the Laplacian in L2 (a+1 ) whose domain consists of the ± ± 2 functions in W2,0 (a+1 , ∂a+1 \ (±a ± 1) × ω) satisfying Neumann boundary 0 condition on (±a ± 1) × ω. The symbol HL denotes the operator − div 1 − c0 χ(|x1 | − a) ∇ − c1 χ(|x1 | − a)
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in L2 (0 ), 0 := (−a − 1, a + 1) × ω, with the domain formed by the functions 2 in W2,0 (0 , (−a − 1, a + 1) × ∂ω) satisfying Neumann condition on {−a − 1} × ω and {a + 1} × ω. The inequality (3.3) implies that + 0 L := H− ⊕ HL HL H ⊕ HL . L
(3.4)
(±) ± ± It is easy to see that HL are self-adjoint operators and σ (HL ) = σess (HL )= 0 [ν1 , +∞). The self-adjoint operator HL is lower semibounded due to (2.3), its spectrum is purely discrete and this is why it has finitely many eigenvalL contains finitely many ues in (−∞, ν1 ]. Hence, the discrete spectrum of H eigenvalues. Due to (3.4) and the minimax principle we can claim that the k-th L is estimated from above by the k-th eigenvalue of HL . The eigenvalue of H former having finitely many discrete eigenvalues, the same is true for HL .
The statement of Theorem 2.1 follows from Lemmas 3.1, 3.2, if one chooses L = L+ , = + ; L = L− , = − ; L = Ll , = ω × (−l − a− , l + a+ ).
4 Analysis of H± In this section we establish certain properties of the operators H± which will be employed in the proof of Theorems 2.2–2.7. By Sδ we indicate the set of all complex numbers separated from the halfline [ν1 , +∞) by a distance greater than δ. We choose δ so that σdisc (H± ) ⊂ Sδ . 2 Lemma 4.1 The operator (H± −λ)−1 : L2 () → W2,0 () is bounded and meromorphic w.r.t. λ ∈ Sδ . The poles of this operator are the eigenvalues of H± . For any λ close to p-multiple eigenvalue λ∗ of H± the representation
(H± − λ)−1 = −
p ± ψ± j (·, ψ j ) L2 () j=1
λ − λ∗
+ T4± (λ)
(4.1)
± holds true. Here ψ ± j are the eigenfunctions associated with λ and orthonormal2 () is a bounded operator being ized in L2 (), while T4± (λ) : L2 () → W2,0 holomorphic w.r.t. λ in a small neighbourhood of λ± . The relations T4± (λ) f, ψ ± = 0, j = 1, . . . , p, (4.2) j L2 ()
are valid. Proof According to [17, Ch. V, Sec. 3.5], the resolvent (H± − λ)−1 considered as an operator L2 () is bounded and meromorphic w.r.t. λ ∈ Sδ and its poles are the eigenvalues of H± . The same is true, if we consider the resolvent as the 2 (); this fact follows from the obvious identity operator from L2 () into W2,0 (H± − λ − η)−1 − (H± − λ)−1 = η(H± − λ)−1 (I − η(H± − λ)−1 )−1 (H± − λ)−1 .
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Consider λ ranging in a small neighbourhood of λ∗ . The formula (3.21) in [17, Ch. V, Sec. 3.5] gives rise to (4.1), where the operator T4± (λ) : L2 () → L2 () is bounded and holomorphic w.r.t. λ. The self-adjointness of H± and (4.1) yield that for any f ∈ L2 () the identities (4.2) and T4± (λ) f = (H± − λ)−1 f,
f = f−
p
± ψ± j ( f, ψ j ) L2 ()
(4.3)
j=1
hold true. Let ⊥ be a subspace of the functions in L2 () which are orthogonal to ψ ± j , j = m, . . . , m + p − 1. The identities (4.3) mean that T4± (λ) ⊥ = (H± − λ)−1 ⊥ .
(4.4)
The operator (H± − λ)−1 ⊥ is holomorphic w.r.t. λ as an operator from ⊥ 2 into W2,0 () ∩ ⊥ . This fact is due to holomorphy in λ of the operator (H± − 2 λ) : W2,0 () ∩ ⊥ → ⊥ and the invertibility of this operator (see [17, Ch. VII, Sec. 1.1]). Therefore, the restriction of T4± (λ) on ⊥ is holomorphic w.r.t. λ as 2 an operator into W2,0 (). Taking into account (4.3), we conclude that for any f ∈ L2 () the function T4± (λ) f is holomorphic w.r.t. λ. The holomorphy in a weak sense implies the holomorphy in the norm sense [17, Ch. VII, Sec. 1.1], and we arrive at the statement of the lemma. Let 0 < ν1 < ν2 . . . ν j . . . be the eigenvalues of the negative Dirichlet Laplacian in ω taken in a non-decreasing order counting multiplicity, and φi = φi (x ) bethe associated eigenfunctions orthonormalized in L2 (ω). We denote s (λ) := ν j − λ, where the branch of the roof is specified by the requirement √j 1 = 1. We remind that a± := ∩ {x : ±x1 > ±a}. Lemma 4.2 Suppose that u ∈ W21 (a± ) is a solution to the boundary value problem ( + λ)u = 0,
x ∈ a± ,
±
u = 0,
x ∈ ∂ ∩ a ,
where λ ∈ Sδ . Then the function u can be represented as u(x) =
∞
α je−s j (λ)(±x1 −a) φ j(x ),
j=1
αj =
u(a, x )φ j(x ) dx .
(4.5)
ω
± ± The series (4.5) converges in W2 (± b ) for any b ⊂ a , p 0. The coefficients α j satisfy the identity p
∞ j=1
|α j|2 = u(·, a)2L2 (ω) .
(4.6)
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Proof In view of the obvious change of variables it is sufficient to prove the + ∞ lemma for + 0 . It is clear that v ∈ C (0 \ ω0 ), where ω0 := {0} × ω. Since + + 1 v ∈ W2 (0 ) and v(x, λ) = 0 as x ∈ ∂0 \ ω0 , the representation u(x, λ) =
∞
φ j(x )u j(x1 , λ),
u j(x1 , λ) :=
j=1
u(x1 , t, λ)φ j(t) dt, ω
holds true for any x1 0 in L2 (ω). We have the identity ∞
|u j(x1 , λ)|2 = u(x1 , ·, λ)2L2 (ω)
(4.7)
j=1
for any x1 0. Employing the equation for u we obtain d2 u j =− dx21
φ j(x ) (x + λ) u(x, λ) dx
ω
=−
u(x, λ) (x + λ) φ j(x ) dx = s2j u j
ω
as x1 > 0. Since v → W21 (+ 0 ), the identity obtained implies that u j(x1 , λ) = α je−
√
ν j −λx1
.
(4.8)
We have employed here that the functions u j are continuous at 0 since t ∂u ∂u u j(t, λ) − u j(0, λ) φ j dx |ω||t| . ∂ x1 ∂ x1 L2 (+0 ) ω
0
The identities (4.7), (4.8) yield (4.6). For x1 b > 0 the coefficients of the series in (4.5) decays exponentially as j → +∞, that implies the convergence p of this series in W2 (+ b ), p 0. For l a− + a+ we introduce the operators T6± (λ, l) : L2 (, ∓ ) → L2 (, ± ), T6± (λ, l) := L± S (±2l)(H∓ − λ)−1 .
(4.9)
Lemma 4.3 The operator T6± is bounded and meromorphic w.r.t. λ ∈ Sδ . For any compact set K ⊂ Sδ separated from σdisc (H∓ ) by a positive distance the estimates i ± ∂ T6 i −2l Re s1 (λ) , i = 0, 1, λ ∈ K, (4.10) ∂λi Cl e
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D. Borisov
hold true, where the constant C is independent of λ ∈ K and l a− + a+ . For any λ close to a p-multiple eigenvalue λ∗ of H∓ the representation T6± (λ, l)
=−
p ϕ ∓j (·, ψ ∓ j ) L2 (∓ )
λ − λ∗
j=1
+ T7± (λ, l),
ϕ ∓j := L± S (±2l)ψ ∓ j ,
(4.11)
is valid. Here ψ ∓ j are the eigenfunctions associated with λ∗ and orthonormalized in L2 (), while the operator T7± (λ, l) : L2 (, ∓ ) → L2 (, ± ) is bounded and holomorphic w.r.t. λ close to λ∗ and satisfies the estimates i ± ∂ T7 i+1 −2l Re s1 (λ) , i = 0, 1, (4.12) ∂λi Cl e where the constant C is independent of λ close to λ∗ and l a− + a+ . The identities T7± (λ∗ , l) = L± S (±2l)T4∓ (λ∗ )
(4.13)
hold true. Proof We prove the lemma for T6+ only; the proof for T6− is similar. Let f ∈ L2 (, − ), and denote u := (H− − λ)−1 f . The function f having a compact support, by Lemma 4.2 the function u can be represented as the series (4.5) for x1 a− . Hence,
∞ S (2l)u (x) = α je−2s j (λ)l e−s j (λ)(±x1 −a) φ j(x ). j=1
Employing this representation and (4.6), we obtain ∞ −2s j (λ)l −s j (λ)(x1 −a− ) α je e φ j(x ) W22 (+ )
j=1
C
∞
|α j|e−2l Re s j (λ) e−s j (λ)(·−a− ) W22 (−a+ ,a+ ) ×
j=1
× x φ j L2 (ω) + φ j L2 (ω) C
∞
(4.14)
|α j||s j(λ)|3/2 ν je−(2l−a− −a+ ) Re s j (λ)
j=1
⎛ C⎝
∞
⎞1/2 ⎛ |α j|2 ⎠
j=1
C 1 + |λ|
⎞1/2 ∞ 7/2 ⎝ ν j + |λ|7/2 e−2(2l−a− −a+ ) Re s j (λ) ⎠ j=1
7/4
e−(2l−a− −a+ ) Re s1 (λ) u(a, ·) L2 (ω) ,
(4.15)
where the constant C is independent of λ ∈ Sδ and l a− + a+ . Here we have also applied the well-known estimate [20, Ch. IV, Sec. 1.5, Theorem 5] 2
2
cj n−1 ν j Cj n−1 .
(4.16)
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169
By direct calculations we check that ∞ αj ∂ S (2l)u (x, λ, l) = (x1 − a− + 2l)e−2s j (λ)l e−s j (λ)(x1 −a− ) φ j(x ). ∂λ 2s (λ) j j=1 Proceeding in the same way as in (4.15) we obtain ∂ S (2l)u Cl 1 + |λ|5/4 e−(2l−a− −a+ ) Re s1 (λ) u(a, ·) L2 (ω) , ∂λ 2 W2 (+ )
(4.17)
where the constant C is independent of λ ∈ Sδ and l a− + a+ . In the same way we check that 2 ∂ S (2l)u Cl 1 + |λ|3/4 e−(2l−a− −a+ ) Re s1 (λ) u(a, ·) L2 (ω) , (4.18) ∂λ2 2 W2 (+ ) where the constant C is independent of λ ∈ Sδ and l a− + a+ . Lemma 4.1 implies u(a− , ·) = −
p − ψ− j (a− , ·)( f, ψ j ) L2 () j=1
λ − λ∗
+ T4− (λ) f (a− , ·).
This representation and (4.6), (4.15), (4.17), (4.18) lead us to the statement of the lemma.
5 Reduction of the Eigenvalue Equation for Hl In this section we reduce the eigenvalue equation Hl ψ = λψ
(5.1)
to an operator equation in L2 (− ) ⊕ L2 (+ ). The reduction will be one of the key ingredients in the proofs of Theorems 2.2–2.7. Hereafter we assume that l a− + a+ . Let f± ∈ L2 (, ± ) be a pair of arbitrary functions, and let functions u± satisfy the equations (H± − λ)u± = f± ,
(5.2)
where λ ∈ Sδ . We choose δ so that σdisc (H± ) ⊂ Sδ . We construct a solution to (5.1) as ψ = S (l)u− + S (−l)u+ .
(5.3)
Suppose that the function ψ defined in this way satisfies (5.1). We substitute (5.3) into (5.1) to obtain 0 = (Hl − λ)ψ = (Hl − λ)S (l)u− + (Hl − λ)S (−l)u+ .
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By direct calculations we check that (Hl − λ)S (l)u− = S (l)(−(D) + L− − λ)S (−l)S (l)u− + + S (−l)L+ S (2l)u− = S (l) f− + S (−l)L+ S (2l)u− , (Hl − λ)S (−l)u+ = S (−l) f+ + S (l)L− S (−2l)u+ . Hence, S (l) f− + L− S (−2l)u+ + S (−l) f+ + L+ S (2l)u− = 0. Since the functions f± + L± S (±2l)u∓ are compactly supported, it follows that the functions S (∓l) f± + L± S (±2l)u∓ are compactly supported, too, and their supports are disjoint. Thus, the last equation obtained is equivalent to the pair of the equations f− + L− S (−2l)u+ = 0,
f+ + L+ S (2l)u− = 0.
(5.4)
These equations are equivalent to (5.1). The proof is the subject of Lemma 5.1 To any solution f := ( f− , f+ ) ∈ L2 (− ) ⊕ L2 (+ ) of (5.4) and functions u± solving (5.2) there exists the unique solution of (5.1) given by (5.2), (5.3). For any solution ψ of (5.1) there exists the unique f ∈ L2 (− ) ⊕ L2 (+ ) solving (5.4) and the unique functions u± satisfying (5.2) such that ψ is given by (5.2), (5.3). The equivalence holds for any λ ∈ Sδ . Proof It was shown above that if f ∈ L2 (− ) ⊕ L2 (+ ) solves (5.4), and the functions u± are the solutions to (5.2), the function ψ defined by (5.3) solves (5.1). Suppose that ψ is a solution of (5.1). This functions satisfies the equation (− − λ)ψ = 0,
−l + a− < x1 < l − a+ ,
x ∈ ω,
(5.5)
and vanishes as −l + a− < x1 < l − a+ , x ∈ ∂ω. Due to standard smoothness improving theorems (see, for instance, [20, Ch. IV, Sec. 2.2]) it implies that ψ ∈ C∞ ({x : −l + a− < x1 < l − a+ , x ∈ ω}). Hence, the numbers 1 ∂ψ ± ± , ρ j = ρ j (l, λ) := (1/2) ϒ± (x , l, λ)φ j(x ) dx , ϒ± := ψ ± s j ∂ x1 x1 =0 ω
are well-defined. Employing (5.5), the identity ψ = 0 as x ∈ ∂, and the smoothness of ψ we integrate by parts, ρ ±j = −
1 2ν j
1 = 2ν j
ϒ± x φ j dx = −
ω
φj ω
1 2ν j
φ jx ϒ± dx
ω
∂ 1 + λ ϒ± dx = p 2 ∂ x1 2ν j 2
φj ω
∂2 +λ ∂ x21
p
ϒ± dx
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171
for any p ∈ N. In view of (4.16) it yields that
∞ j=1
j p |ρ ±j | < ∞ for any p ∈ N.
Now we introduce the functions u± , u± (x1 ∓ l, x , l) :=
∞
ρ ±j (l)e±s j (λ)x1 φ j(x ),
x ∈ ∓ 0,
j=1
x ∈ ± 0,
u± (x1 ∓ l, x , l) := ψ(x, l) − u∓ (x1 ± l, x , l), and conclude that they satisfy (5.3), and
± ± 2 u− (x1 + l, x , l), u+ (x1 − l, x , l) ∈ C∞ ± 0 ∩ W2,0 0 , ∂ ∩ ∂0 .
The smoothness of ψ gives rise to the representations ∞ ∞ + ∂ψ − + − ψ x1 =0 = ρ s jφ j. ρ j + ρ j φ j, = − ρ j j ∂ x1 x1 =0 j=1 j=1 These relations and the aforementioned smoothness of u± imply that u± ∈ 2 (). We define a vector f = ( f− ; f+ ) ∈ L2 (− ) ⊕ L2 (+ ) by f− := −L− u+ W2,0 (x1 − 2l, x , l), f+ := −L+ u− (x1 + 2l, x , l). Let us check that the functions u± satisfy (5.2); it will imply that f solves (5.6). The definition of u± yields ( + λ)u± (x1 ∓ l, x , l) = 0,
∓
x ∈ 0 .
Thus, ( + λ)u− = 0,
+
x ∈ l ,
( + λ)u+ = 0,
−
x ∈ −l .
These equations, the equation for ψ, and the definitions of u− (x1 + l, x , l) for − x ∈ − 0 , follow that for x ∈ l (− − λ + L− )u− (x, l) = (− − λ + L− ) ψ(x1 − l, x , l) − u+ (x1 − 2l, x , l) = = −(− − λ + L− )u+ (x1 − 2l, x , l) = −L− u+ (x1 − 2l, x , l) = f− (x). Therefore, (H− − λ)u− = f− . The relation (H+ − λ)u+ = f+ can be established in the same way. Suppose that λ ∈ Sδ \ σ∗ . In this case u± = (H± − λ)−1 f± . This fact together with the definition (4.9) of T6± implies that L± S (±2l)u∓ = T6± (λ, l) f± . Substituting the identity obtained into (5.4), we arrive at the equation f + T2 (λ, l) f = 0,
(5.6)
where f := ( f− ; f+ ) ∈ L2 (− ) ⊕ L2 (+ ), where the operator T2 : L2 (− ) ⊕ L2 (+ ) → L2 (− ) ⊕ L2 (+ ) is defined in (2.4). Proof of Theorem 2.2 The inequality (3.3) yields that the operator Hl is lower semibounded with lower bound −c1 . Together with Theorem 2.1 it implies that
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D. Borisov
the discrete eigenvalues of the operator are located in [−c1 , ν1 ). The set Kδ := [−c1 , ν1 − δ) \ (λ − δ, λ + δ) satisfies the hypothesis of Lemma 4.3, and the λ∈σ∗
estimate (4.10) implies T2 (λ, l) C(δ)e−2l Re s1 (λ) < 1, if l is large enough and λ ∈ Kδ . Therefore, (5.6) has no nontrivial solutions, if l is large enough and λ ∈ Kδ . Since Kδ ∩ σ∗ = ∅, the identity f = 0 implies that u± = (H± − λ)−1 f± = 0, i.e., ψ = 0. Thus, (5.1) has no nontrivial solution for λ ∈ Kδ , i.e., Kδ ∩ σdisc (Hl ) = ∅, if l is large enough. The number δ being arbitrary, the last identity completes the proof. Assume that λ∗ ∈ σ∗ is ( p− + p+ )-multiple. Lemma 4.3 implies that for λ close to λ∗ the representation (2.5) holds true, where T3 (λ, l) f := (T7− (λ, l) f+ ; T6+ (λ, l) f− ),
if
p− = 0,
(T6− (λ, l) f+ ; T7+ (λ, l) f− ),
if
p+ = 0,
T3 (λ, l) f := (T7− (λ, l) f+ ; T7+ (λ, l) f− ),
if
p± = 0.
T3 (λ, l) f :=
Lemma 4.3 yields also that the operator T3 (λ, l) is bounded and holomorphic w.r.t. λ in a small neighbourhood of λ∗ , and satisfies the estimate i ∂ T3 i+1 −2l Re s1 (λ) , i = 0, 1, (5.7) ∂λi Cl e where the constant C is independent of l a− + a+ and λ close to λ∗ . Suppose that λ = λ∗ is an eigenvalue of Hl converging to λ∗ . In this case the identity f = 0 leads us to the relations u± = (H± − λ)−1 f± = 0, ψ = 0. Thus, the corresponding equation (5.6) has a nontrivial solution. Let us solve this equation. We substitute (2.5) into (5.6), and obtain p 1 f− φ T (i) f + T3 f = 0. λ − λ∗ i=1 i 1
(5.8)
In view of the estimate (5.7) the operator (I + T3 (λ, l)) is invertible, and the operator (I + T3 )−1 is bounded and holomorphic w.r.t. λ in a small neighbourhood of λ∗ . Applying this operator to the last equation gives rise to one more equation, p 1 f = i T1(i) f , λ − λ∗ i=1
(5.9)
where i = i (·, λ, l) := (I + T3 (λ, l))−1 φ i (·, l). We denote ki = ki (λ, l) := ( j) (λ − λ∗ )−1 T1(i) f , and apply the functionals T1 , j = 1, . . . , p, to the equation (5.9) that leads us to (2.12), where k := (k1 . . . k p )t . Given a non-trivial solution of (5.6), the associated vector k is nonzero, since otherwise the definition of ki , and (5.9) would imply that f = 0.
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173
Therefore, if λ = λ∗ is an eigenvalue of Hl , the system (2.12) has a non-trivial solution. It is true, if and only if (2.7) holds true. Thus, each eigenvalue of Hl converging to λ∗ and not coinciding with λ∗ should satisfy this equation. Let us show that if λ∗ is an eigenvalue of Hl , it satisfies (2.7) as well. In this case λ∗ the associated eigenfunction is given by (5.3) that is due to Lemma 5.1. The self-adjointness of H± implies f± , ψi± L2 (± ) = u± , (H± − λ∗ ) ψi± L2 (± ) = 0, (5.10) i = 1, . . . , p± . Therefore, the functions u± can be represented as u− (·, l) = T4− (λ∗ ) f− −
p−
ki ψi− ,
u+ (·, l) = T4+ (λ∗ ) f+ −
i=1
p+
ki+ p− ψi+ ,
i=1
(5.11) where ki are numbers to be found. Employing the relation (4.13) and substituting (5.11) into (5.4), we obtain f + T3 (λ∗ , l) f =
p
ki φ i (·, l),
f =
i=1
p
ki i (·, λ∗ , l).
(5.12)
i=1
The relations (5.10) can be rewritten as T1(i) f = 0, i = 1, . . . , p, that together with the second identity in (5.12) implies the system (2.12) for λ = λ∗ . The vector k is non-zero since otherwise the second relation in (5.12) and (5.11) would imply f = 0, u± = 0, ψ = 0. Thus, det A(λ∗ , l) = 0, which coincides with (2.7) for λ = λ∗ . Let λ be a root of (2.7), converging to λ∗ as l → +∞. We are going to prove that in this case (5.1) has a non-trivial solution, i.e., λ is an eigenvalue of Hl . The equation (5.1) being satisfied, it follows that the system (2.12) has a nontrivial solution k. We specify this solution by the requirement kC p = 1, and define f :=
p
(5.13)
ki i (·, λ, l) ∈ L2 (− ) ⊕ L2 (+ ). The system (2.12) and the
i=1
definition of T1(i) give rise to the identities ( f− , ψi− ) L2 (− ) = T1(i) f = (λ − λ∗ )ki , ( f+ , ψi+ ) L2 (− ) =
(i+ p ) T1 −
i = 1, . . . , p− ,
f = (λ − λ∗ )ki+ p− , i = 1, . . . , p+ .
(5.14)
Taking these identities into account and employing (4.1), in the case λ = λ∗ we arrive at the formulas u− = −
p− i=1
ki ψi−
+
T4− (λ) f− ,
u+ = −
p+
ki+ p− ψi+ + T4+ (λ) f+ .
(5.15)
i=1
In the case λ = λ∗ we adopt these formulas as the definition of the functions u± that is possible due to (4.4) and (5.14) with λ = λ∗ .
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D. Borisov
If λ = λ∗ , we employ (2.12) to check by direct calculations that f solves (5.9), and thus (5.4). If λ = λ∗ , the identity (4.13) implies
L− S (−2l)u+ ; L+ S (2l)u− = T3 (λ∗ , l) f − ki φ i (·, l) = T3 (λ∗ , l) f − p
i=1
−
p
ki (I + T3 (λ∗ , l))i (·, λ∗ , l)
i=1
= T3 (λ∗ , l) f − (I + T3 (λ∗ , l)) f = − f . Hence, (5.4) holds true. With Lemma 5.1 in mind we therefore conclude that the function ψ defined by (5.3) solves (5.1). Let us prove that ψ ≡ 0; it will imply that λ is an eigenvalue of Hl . Lemma 2.1 implies that the functions ϕi± obey the estimate ϕi± L2 (∓ ) Ce−2ls1 (λ∗ ) ,
(5.16)
where the constant C is independent of l. This estimate together with (5.7) gives rise to the similar estimates for i : i L2 (− )⊕L2 (+ ) Ce−2ls1 (λ∗ ) , ∂ i Cl 2 e−2l(s1 (λ∗ )+s1 (λ)) , ∂λ
(5.17)
L2 (− )⊕L2 (+ )
where the constant C is independent of λ and l. The latter inequality is based on the formula ∂ T3 ∂ (I + T3 )−1 = −(I + T3 )−1 (I + T3 )−1 . ∂λ ∂λ Hence, the asymptotics (2.11) is valid. The vector k being non-zero, it implies ψ ≡ 0. We summarize the results of the section in Lemma 5.2 The eigenvalues of Hl converging to a ( p− + p+ )-multiple λ∗ ∈ σ∗ coincide with the roots of (2.7) converging to λ∗ . The associated eigenfunctions are given by (5.3), (5.11), (5.15), where the coefficients ki form non-trivial solutions to (2.12). If λ(l) is an eigenvalue of Hl converging to λ∗ as l → +∞, its multiplicity coincides with the number of linear independent solutions of (2.12) taken for λ = λ(l). The associated eigenfunctions satisfy (2.11). 6 Proof of Theorem 2.3 Throughout this and next sections the parameter λ is assumed to belong to a small neighbourhood of λ∗ , while l is supposed to be large enough. We begin with the proof of Lemma 2.1. Proof of Lemma 2.1 We will prove the lemma for ψi− only, the case of ψi+ is completely similar. According to Lemma 4.2 the functions ψi− can be represented as the series (4.5) in a+− . Let be the space of the L2 ()-functions
Asymptotic behaviour of the spectrum...
175
spanned over ψi− . Each function from this space satisfies the representation (4.5) in a+− . We introduce two quadratic forms in this finite-dimensional space, the first being generated by the scalar product in L2 (), while the other is defined as q(u, v) := α1 [u]α1 [v], where the α1 [u], α1 [v] are the first coefficients in the representations (4.5) for u and v in a+− . By the theorem on simultaneous diagonalization of two quadratic forms, we conclude that we can choose the basis in so that both these forms are diagonalized. Denoting this basis as ψi− , we conclude that these functions are orthonormalized in L2 (), and = 0, if i = j. (6.1) q ψi− , ψ − j Suppose that for all the functions ψi− the coefficient α1 [ψi− ] is zero. In this case we arrive at (2.17), where β− = 0. If at least one of the functions ψi− has a nonzero coefficient α1 , say ψ1− , the identity (6.1) implies that α1 [ψi− ] = 0, i 2, and we arrive again at (2.17). The definition of Aij and (5.17) imply the estimates ∂ Aij −2ls1 (λ∗ ) (λ, l) Cl 2 e−2l(s1 (λ∗ )+s1 (λ)) , |Aij(λ, l)| Ce , ∂λ
(6.2)
where the constant C is independent of λ and l. The holomorphy of T3 w.r.t. λ yields that the functions Aij are holomorphic w.r.t. λ. This fact and the estimate (6.2) allow us to claim that the right hand side of (2.7) reads as follows, F(λ, l) := det (λ − λ∗ )E + A(λ, l) = (λ − λ∗ ) p + Pi (λ, l)(λ − λ∗ )i , p−1
i=0
where the functions Pi are holomorphic w.r.t. λ, and obey the estimate |Pi (λ, l)| Ce−2( p−i)ls1 (λ∗ )
(6.3)
with the constant C independent of λ and l. Given δ > 0, this estimate implies p−1 i p P (λ, l)(λ − λ ) as |λ − λ∗ | = δ, i ∗ < |λ − λ∗ | i=0
if l is large enough. Now we employ Rouché theorem to infer that the function F(λ, l) has the same amount of the zeroes inside the disk {λ : |λ − λ∗ | < δ} as the function λ → (λ − λ∗ ) p does. Thus, the function F(λ, l) has exactly p zeroes in this disk (counting their orders), if l is large enough. The number δ being arbitrary, we conclude that (2.7) has exactly p roots (counting their orders) converging to λ∗ as l → +∞. Lemma 6.1 Suppose that λ1 (λ) and λ2 (λ) are different roots of (2.7), and k1 (l) and k2 (l) are the associated non-trivial solutions to (2.12) normalized by (5.13). Then k1 (l), k2 (l) C p = O le−2ls1 (λ∗ ) , l → +∞.
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D. Borisov
Proof According to Lemma 5.2, the numbers λ1 (l) = λ2 (l) are the eigenvalues of Hl , and the associated eigenfunctions ψi (x, l), i = 1, 2, are generated by (5.3), (5.11), (5.15), where ki are components of the vectors k1 , k2 , respectively. − ± Using the representations (4.5) for ψi+ in ± ±a+ and for ψi in ±a− , by direct calculations one can check that + −2ls1 (λ∗ ) ψi , S (2l)ψ − , l → +∞. j L2 () = O le The operator being self-adjoint, the eigenfunctions ψi (x, l) are orthogonal in L2 (). Now by Lemma 5.2 and the last identity we obtain 0 = (ψ1 , ψ2 ) L2 () =
p
ki(1) ki(2) + O e−2ls1 (λ∗ ) ,
l → +∞,
i=1
that completes the proof.
For each root λ(l) −−−−→ λ∗ of (2.7) the system (2.12) has a finite number of l→+∞
linear independent solutions. Without loss of generality we assume that these solutions are orthonormalized in C p . We consider the set of all such solutions associated with all roots of (2.7) converging to λ∗ as l → +∞, and indicate these vectors as ki = ki (l), i = 1, . . . , q. In view of the assumption for ki just made and Lemma 6.1 the vectors ki satisfy (2.13). For the sake of brevity we denote B(λ, l) := (λ − λ∗ )E − A(λ, l). Lemma 6.2 Let λ(l) −−−−→ λ∗ be a root of (2.7) and ki , i = N, . . . , N + m, m l→+∞
0, be the associated solutions to (2.12). Then for any h ∈ C p the representation −1
B (λ, l)h =
N+m i=N
T8(i) (l)h ki (l) + T9 (λ, l)h λ − λ(l)
holds true for all λ close to λ(l). Here T8(i) : C p → C are functionals, while the matrix T9 (λ, l) is holomorphic w.r.t. λ in a neighbourhood of λ(l). Proof The matrix B being holomorphic w.r.t. λ, the inverse B−1 is meromorphic w.r.t. λ and has a pole at λ(l). The residue at this pole is a linear combination of ki , and for any h ∈ C p we have B−1 (λ, l)h =
N+m 1 ki T8(i) (l)h + O (λ − λ(l))−s+1 , s (λ − λ(l))
λ → λ(l),
i=N
(6.4) where s 1 is the order of the pole, and T8(i) : C p → C are some functionals. We are going to prove that s = 1. Let g± = g± (x) ∈ L2 (, ± ) be arbitrary functions. We consider the equation (Hl − λ)u = g := S (−l)g+ + S (l)g− ,
(6.5)
Asymptotic behaviour of the spectrum...
177
where λ is close to λ(l) and λ = λ(l), λ = λ∗ . The results of [17, Ch. VII, Sec. 3.5] imply that for such λ the function u can be represented as u=−
N+m 1 (g, ψi ) L2 () ψi + T10 (λ, l)g, λ − λ(l) i=N
(6.6)
where ψi are the eigenfunctions of Hl associated with λ(l) and orthonormalized in L2 (), while the operator T10 (λ, l) is bounded and holomorphic w.r.t. λ close to λ(l) as an operator in L2 (). The eigenvalue λ(l) of the operator Hl is (m + 1)-multiple by the assumption and Lemma 5.2. Completely by analogy with the proof of Lemma 5.1 one can make sure that (6.5) is equivalent to f + T2 (λ, l) f = g,
(6.7)
where g := (g− , g+ ) ∈ L2 (− ) ⊕ L2 (+ ), and the solution of (6.5) is given by u± = (H± − λ)−1 f± .
u = S (l)u− + S (−l)u+ ,
(6.8)
Proceeding as in (5.6), (5.8), (5.9), one can solve (6.7), f =
p
U i i + f,
U = B−1 (λ, l)h,
f := (I + T3 )−1 g,
(6.9)
i=1
where U i := T1(i) f , and the vector U := (U 1 . . . U p )t is a solution to B(λ, l)U = h, h = (h1 . . . h p )t ,
(6.10)
hi := T1(i) (λ)(I + T3 (λ, l))−1 g
(6.11)
Knowing the vectors U and f , we can restore the functions u± by (4.1), p−
u− (·, λ, l) = −
U i (λ, l)ψi−
−
i=1
u+ (·, λ, l) = −
U i (λ, l)T4− (λ, l)i− + T4− (λ) f− ,
i=1
p+
p
U i+ p− (λ, l)ψi+ −
p
i=1
U i (λ, l)T4+ (λ, l)i+ + T4+ (λ) f+ ,
i=1
f± are introduced as i = (i− ; i+ ), f = ( f− ; f+ ). In these where i± and formulas we have also employed (6.10) in the following way: (
f− , ψ − j ) L2 ()
=
( j) T1
f = hj +
p
A ji U i = (λ − λ∗ )U j,
j = 1, . . . , p− ,
i=1 ( j+ p− )
( f+ , ψ + j ) L2 () = T1
f = h j+ p− +
p
A ji U i = (λ−λ∗ )U j+ p− ,
i=1
The estimates (5.17) allow us to infer that
T4± (λ, l)i± L2 () = O e−2ls1 (λ∗ ) ,
j= 1, . . . , p+ .
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while in view of holomorphy of T4± and (5.7) we have T4± (λ) f± L2 () Cg L2 (− )⊕L2 (+ ) , where the constant C is independent of λ. Now we use the first of the relations (6.8) and obtain u(·, λ, l) = −
p−
U i (λ, l) S (l)ψi− + O e−2ls1 (λ∗ )
i=1
−
p+
U i+ p− (λ, l) S (−l)ψi+ + O e−2ls1 (λ∗ ) + O g L2 (− )⊕L2 (+ ) .
i=1
(6.12) Now we compare (6.6) and (6.12) and conclude that the coefficients U i (λ, l), i = 1, . . . , p have a simple pole at λ(l). By (6.9) it implies that the vector B−1 (λ, l)h has a simple pole at λ(l), where h is defined by (6.11). It follows from (5.7) and (6.11) that for each h ∈ C p there exists g ∈ L2 (− ) ⊕ L2 (+ ) so that h is given by (6.11). Together with (6.4) it completes the proof. Lemma 6.3 The number λ(l) −−−−→ λ∗ is a m-th order zero of F(λ, l) if and l→+∞
only if it is a m-multiple eigenvalue of Hl . Proof Let λ(i) (l) −−−−→ λ∗ , i = 1, . . . , M, be the different zeroes of F(λ, l), and l→+∞
ri , i = 1, . . . , M, be the orders of these zeroes. By Lemma 5.2, each zero λ(i) (l) is an eigenvalue of Hl ; its multiplicity will be indicated as mi 1. To prove the lemma it is sufficient to show that mi = ri , i = 1, . . . , M. Let us prove first that mi ri . In accordance with Lemma 5.2 the multiplicity mi coincides with a number of linear independent solutions of (2.12) with λ = λ(i) (l). Hence, rank B(λ(l), l) = p − mi .
(6.13)
By the assumption ∂j det B(λ, l) = 0, ∂λ j
j = 1, . . . , ri − 1,
∂ ri det B(λ, l) = 0, ∂λri
as λ = λ(i) (l). Let B j = B j(λ, l) be the columns of the matrix B, i.e., B = (B1 , . . . , B p ). Employing the well-known formula ∂ ∂ B1 ∂ B2 det B = det B2 . . . B p + det B1 . . . Bp + ∂λ ∂λ ∂λ ∂ Bp , + det B1 B2 . . . ∂λ
Asymptotic behaviour of the spectrum...
179
one can check easily that for each 0 j mi − 1 ∂j det B(λ, l) = cς det Bς , λ=λ(i) (l) ∂λ j ς where cς are constants, and at least ( p − mi + 1) columns of each matrix Bς are those of B. In view of (6.13) these columns are linear dependent, and therefore det Bς = 0 for each ς. Thus, mi − 1 ri − 1 that implies the desired inequality. M M mi = ri = p to prove that mi = ri . It is sufficient to check that q = i=1
i=1
Lemma 6.2 yields that for a given fixed δ small enough and l large enough q T8(i) (l)h B (λ, l)h = ki (l) + T9 (λ, l)h, λ − λ(i) (l) i=1 −1
(6.14)
for any h ∈ C p and λ so that |λ − λ∗ | = δ. It is also assumed that |λ(i) (l) − λ∗ | < δ for the considered values of l. We integrate this identity to obtain 1 2π i
−1
B (λ, ls )h dλ = |λ−λ∗ |=δ
Due to (6.2) we conclude that 1 1 B−1 (λ, l) dλ −−−−→ l→+∞ 2π i 2π i |λ−λ∗ |=δ
q
ki T8(i) (l)h.
(6.15)
i=1
|λ−λ∗ |=δ
E dλ = E. λ − λ∗
(6.16)
Hence, the right hand side of (6.15) converges to h. By (2.13) it implies that q = p for l large enough. The statement of Theorem 2.3 follows from the proven lemma.
7 Asymptotics for the Eigenelements of Hl In this section we prove Theorems 2.4–2.7. Throughout the section the hypothesis of Theorem 2.4 is assumed to hold true. Theorem 2.3 implies that the number of the vectors ki introduced in the previous section equals p. Let S = S(l) be the matrix with columns ki (l), i = 1, . . . , p, i.e., S(l) = (k1 (l) . . . k p (l)). Without loss of generality we can assume that det S(l) 0. Lemma 7.1 det S(l) = 1 + O(le−2ls1 (λ∗ ) ), as l → +∞. Proof The relations (2.13) yield S2 (l) = E + O(le−2ls1 (λ∗ ) ), l → +∞, that implies det2 S(l) = 1 + O(le−2ls1 (λ∗ ) ). The last identity proves the lemma.
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Lemma 7.1 implies that there exists the inverse matrix S−1 (l) for l large enough. Lemma 7.2 The matrix R(λ, l) := S−1 (l)A(λ, l)S(l) reads as follows ⎛ λ1 − λ∗ + (λ − λ1 )r11 (λ − λ2 )r12 ... (λ − λ p )r1 p ⎜ (λ − λ )r λ − λ + (λ − λ )r . . . (λ − λ p )r2 p 1 21 2 ∗ 2 22 ⎜ R=⎜ .. .. .. ⎝ . . . (λ − λ1 )r p1
(λ − λ2 )r p2
⎞ ⎟ ⎟ ⎟. ⎠
. . . λ p − λ∗ + (λ − λ p )r pp
where λi = λi (l), while the functions rij = rij(λ, l) are holomorphic w.r.t. λ close to λ∗ and obey the uniform in λ and l estimates |rij(λ, l)| Cl 2 e−2l(s1 (λ∗ )+s1 (λ)) .
(7.1)
Proof The system (2.12) implies A(λ, l)ki = A(λi (l), l)ki + A(λ, l) − A(λi (l), l) ki = (λi (l) − λ∗ )ki + A(λ, l) − A(λi (l), l) ki . The matrix A(λ, l) − A(λi (l), l) is holomorphic w.r.t. λ, and λi (l), l) := A(λ, l) − A(λi (l), l) = A(λ,
λ λi (l)
∂A (z, l) dz. ∂λ
(7.2)
Due to (6.2) and (2.12) the last identity implies that A(λ, l)ki = (λi (l) − λ∗ )ki + (λ − λi )K i (λ, l),
(7.3)
where the vectors K i (λ, l) are holomorphic w.r.t. λ close to λ∗ and satisfy the uniform in λ and l estimate K i C p Cl 2 e−2l(s1 (λ∗ )+s1 (λ)) . By Lemma 6.3 the vectors ki , i = 1, . . . , p, form a basis in C p . Hence, K i (λ, l) =
p
rij(λ, l)k j(l),
t ri1 (λ, l) . . . rip (λ, l) = S−1 (l)K i (λ, l).
(7.4)
j=1
Due to Lemma 7.1, the relations (2.13), and the established properties of K i we infer that the functions rij are holomorphic w.r.t. λ and satisfy (7.1). Taking into account (7.3) and (7.4), we arrive at the statement of the lemma. Lemma 7.3 The polynomial det τ E − A(λ∗ , l) has exactly p roots τi = τi (l), i = 1, . . . , p counting multiplicity which satisfy (2.9).
Asymptotic behaviour of the spectrum...
181
Proof Since det τ E − A(λ∗ , l) is a polynomial of p-th order, it has p roots τi (l), i = 1, . . . , p, counting multiplicity. It is easy to check that det τ E − A(λ∗ , l) = τ p + P j(λ∗ , l)τ j, p−1
j=0
where the functions P j(λ∗ , l) satisfy (6.3). We make a change of variable τi = zi e−2ls1 (λ∗ ) , and together with the representation just obtained it leads us to the equation for zi , zp +
p−1
e−2l( j− p)s1 (λ∗ ) P j(λ∗ , l)z j = 0.
(7.5)
j=0
Due to (6.3) the coefficients of this equation are bounded uniformly in l. By Rouché theorem it implies that all the roots of (7.5) are bounded uniformly in l. This fact yields (2.9). In what follows the roots τi are supposed to be ordered in accordance with (2.10). We denote μi (l) := λi (l) − λ∗ , i = 1, . . . , p. Proof of Theorem 2.4 The formulas (2.9), (2.11) were established in Lemma 7.3. Let us prove (2.8). Namely, let us prove that for each l large enough the roots of (2.7) can be ordered so that 2 4l τi (l) = μi (l) 1 + O l p e− p s1 (λ∗ ) , l → +∞. (7.6) Assume that this not true on a sequence λs → +∞. We introduce an equivalence relation ∼ on {μi (ls )}i=1,..., p saying that μi ∼ μ j, if 2 4l p − s s1 (λ∗ ) p μi (ls ) = μ j(ls ) 1 + O ls e , ls → +∞. This relation divides all μi (ls ) into disjoint groups, q {λmi (l), . . . , λmi+1 −1 (l)}, {λ1 (l), . . . , λ p (l)} = i=1
where 1 = m1 < m2 < . . . < mq+1 = p + 1, λk ∼ λt , k, t = mi , . . . , mi+1 − 1, i = 1, . . . , q, and λk ∼ λt , if mi k mi+1 − 1, m j t m j+1 − 1, i = j. Extracting if needed a subsequence from {ls }, we assume that mi and q are independent of {ls }. Given k ∈ {1, . . . , p}, there exists i such that mi k := mi+1 , m := m − m. mi+1 − 1. For the sake of brevity we denote m := mi , m To prove (2.8) it issufficient to show that m roots τ , i = m, . . . , m − 1, counting i multiplicity of det τ E − A(λ∗ , l) satisfy (7.6). Since det τ E − A(λ∗ , l) = det S−1 (l)(τ E − A(λ∗ , l))S(l) = det τ E − R(λ∗ , l) ,
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D. Borisov
due to Lemma 7.2 the equation for τi can be rewritten as τ − μ1 (1 − r11 ) −μ2r12 ... −μ pr1 p −μ1r21 τ − μ2 (1 − r22 ) . . . −μ pr2 p = 0, .. .. .. . . . −μ2r p2 . . . τ − μ p (1 − r pp ) −μ1r p1
(7.7)
where μ j = μ j(l), r jk = r jk (λ∗ , l). Assume first that μk (ls ) = 0. In view of (2.6) = m2 , and μ j(ls ) = 0, j = m, . . . , m − 1. In this case it implies that m = m1 , m (7.7) becomes τ . . . 0 −μm −μ pr1 p r1 m ... .. .. .. .. . . . . 0 . . . τ −μm = 0, −μ prm rm −1 −1 p m ... .. .. .. .. . . . . 0 . . . 0 −μm r . . . τ − μ (1 − r ) p m p pp and it implies that zero is the root of det τ E − A(λ∗ , l) of multiplicity at least . In this case the identities (7.6) are obviously valid. m Assume now that μk (ls ) = 0. We seek the needed roots as τ = μk (ls )(1 + z). We substitute this identity into (7.7) and divide then first ( m − 1) columns by −μk (ls ), while the other columns are divided by the functions −μ j(ls ) corresponding to them. This procedure leads us to the equation z1 − z .. . μ1 r −11 μk m μ1 μk r m 1 .. . μ1 μ r p1 k
.. . . . . zm rm . . . rm −1 − z −1 −1 p m = 0, μm μk −1 . . . μk r m z . . . rm m −1 zm − μm p .. .. .. . . . . . . μμmk−1 r p r p . . . z p − μμkp z m−1 m ...
zi = zi (l) :=
μm −1 r m−1 μk 1
r1 m .. .
...
r1 p .. .
μi (l) − 1, (1 − rii (λ∗ , l)) − 1, i = 1, . . . , m μk (l)
zi = zi (l) := 1 −
μk (l) − rii (λ∗ , l), μi (l)
, . . . , p, i=m
where the arguments of all the functions are l = ls , λ = λ∗ . Due to (2.6) all the fractions in this determinant are bounded uniformly in ls . Using this fact and
Asymptotic behaviour of the spectrum...
183
(7.1), we calculate this determinant and write the multiplication of the diagonal separately, F1 (z, ls )F2 (z, ls )F3 (z, ls ) − F4 (z, ls ) = 0, F1 (z, ls ) :=
m−1
(zi (ls ) − z) ,
F2 (z, ls ) :=
(7.8) m −1
i=1
F3 (z, ls ) :=
p i= m
(zi (ls ) − z) ,
i=m
μi (ls ) zi (ls ) − z , μk (ls )
where the coefficients ci obey the estimate ci (ls ) = O ls2 e−4ls s1 (λ∗ ) ,
F4 (z, ls ) :=
p−2
zi ci (ls )
i=0
ls → +∞.
(7.9)
As it follows from the definition of the equivalence relation, |μi − μk | C0 ζ, |μk | |μi − μk | θζ, |μk |
− 1, i = m, . . . , m
ls → +∞,
− 1}, i ∈ {m, . . . , m
ζ = ζ (ls ) := ls2 e−4ls s1 (λ∗ ) ,
θ = θ(ls ) → +∞,
ls → +∞,
where the constant C0 is independent of ls . These estimates and (7.1), (2.6) imply |F1 (z, l)| C(θζ )m−1 ,
m+1 |F3 (z, l)| C(θζ ) p− ,
|z| 2C0 ζ.
(7.10)
m ζ The zeroes zi (ls ) of F2 (z, ls ) satisfy |zi (ls )| C0 ζ , and hence |F2 (z, ls )| C0m as |z| = C0 ζ . Employing this estimate, (7.10) and rewriting (7.8) as
F2 (z, ls ) −
F4 (z, ls ) = 0, F1 (z, ls )F3 (z, ls )
roots by Rouché theorem we conclude that the last equation has exactly m counting multiplicity in the disk {|z| < 2C0 ζ }. We denote these roots as z( j) (ls ), − 1. It follows from (7.8), (7.9), (7.10) that j = m, . . . , m m F4 (z( j) , ls ) ( j) ( j) min |z − zi | |F2 (z , ls )| = ( j) ( j) i=m,..., m−1 F1 (z , ls )F3 (z , ls ) C
ζm (ls ) Cζ m (ls ), p− m θ (ls )
where z( j) = z( j) (ls ), and the constant C is independent of ls . Hence, for each j there exists index i, depending on ls , such that z( j) = zi + O(ζ ),
ls → +∞.
This identity, the definition of the equivalence relation and (7.1) imply τ ( j) = μk (1 + z( j) ) = μ j(1 − r jj) + O(μk ζ ) = μ j + O(μ jζ ) that yields (7.6).
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D. Borisov
In the proof of Theorem 2.5 we will employ the following lemma. Lemma 7.4 For λ close to λ∗ and h ∈ C p the representation B−1 (λ, l)h =
p T (i) (λ, l)h 10
i=1
λ − λi (l)
ki
holds true. Here T10(i) (λ, l) : C p → C are functionals bounded uniformly in λ and l.
Proof Lemma 6.2 implies that for λ close to λ∗ the identity (6.14) holds true for any h ∈ C p , where ls should be replaced by l, q = p and λ(i) = λi . We introduce the vectors ki⊥ (l)
ki (l) , := ki (l)
ki := ki −
p
ki , k j
Cp
k j.
j=1 j =i
The relations (2.13) implies that the vectors ki⊥ satisfy these relation as well. Moreover, the vectors ki⊥ form the orthogonal basis for {ki }. Bearing this fact in mind, we multiply the relation (6.15) by k⊥j with ls replaced by l, q = p and obtain ⎞ ⎛ 1 ⎜ ⎟ ( j) B−1 (λ, l)h dλ, k⊥j ⎠ = T8 (l)h. ⎝ 2π i |λ−λ∗ |=δ
Cp
( j)
Due to (6.16) we conclude that the functionals T8 are bounded uniformly in l. Let us prove that the matrix T9 (λ, l) is bounded uniformly in λ and l. Due to (6.16) and the convergences λi (λ) → λ∗ we have p T8(i) (l)h −1 ki (l) ChC p , T9 (λ, l)hC p = B (λ, l)h − λ − λi (l) Cp
i=1
as |λ − λ∗ | = δ, if l is large enough. The constant C here is independent of h and λ such that |λ − λ∗ | = δ. The matrix T9 being holomorphic w.r.t. λ, by the maximum principle for holomorphic functions the estimate holds true for |λ − λ∗ | < δ, too. Thus, the matrix T9 is bounded uniformly in λ and l. We can expand T9 h in terms of the basis {ki }, T9 (λ, l)h =
p
ki T11(i) (λ, l)h,
( p)
T11(1) h . . . T11 h
t
= S−1 (l)T9 (λ, l)h,
(7.11)
i=1
where the functionals T11(i) : C p → C are bounded uniformly in λ and l. Substituting (7.11) into (6.14) with ls replaced by l, q = p, λ(i) = λi , we arrive at the desired representation, where T10(i) (λ, l) = T8(i) (l) + (λ − λi (l))T11(i) (λ, l).
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185
Proof of Theorem 2.5 Since the matrix A0 (l) satisfies the condition (A), it has p eigenvalues τi(0) , i = 1, . . . , p, counting multiplicity. By hi = hi (l), i = 1, . . . , p, we denote the associated eigenvectors normalized in C p . Completely by analogy with the proof of (2.9) in Lemma 7.3 one can establish (2.16). It is easy to check that the vectors hi satisfy the identities λ∗ , λ∗ + τ (0) , l hi (l) := B(λ∗ + τi(0) , l)hi (l) = −A1 (l)hi (l) − τi(0) A hi (l), i was introduced in (7.2). Now we employ (6.2) where, we remind, the matrix A to obtain hi (l)C p A1 (l) + C|τi(0) |l 2 e−2l(s1 (λ∗ )+s1 (λi (l)))
A1 (l) + C|τi(0) |l 2 e−4ls1 (λ∗ ) ,
(7.12)
where the constant C is independent of λ and l. Here we have also used the identities λi (l) = λ∗ + O e−2ls1 (λ∗ ) , e−2ls1 (λi (l)) = O e−2ls1 (λ∗ ) , l → +∞, which are due to (2.8), (2.9). Since hi (l) = B−1 (λ∗ + τi(0) (l), l) hi (l), Lemma 7.4 implies h j(l) =
p T10(i) (λ∗ + τ j(0) (l), l) hi (l) i=1
τi(0) (l) + λ∗ − λ j(l)
ki (l),
j = 1, . . . , p.
The fractions in these identities are bounded uniformly in λ and l since h j are normalized and Q ji (l) :=
T10(i) (λ∗ + τ j(0) (l), l) h j(l)
τi(0) (l)
+ λ∗ − λi (l)
= h j(l), ki⊥ (l) C p ,
where, we remind, the vectors ki⊥ were introduced in the proof of Lemma 7.4. We are going to prove that the roots τ j(0) can be ordered so that the formulas (2.15) hold true. The matrices ⎛ ⎞ Q11 . . . Q1 p ⎜ .. ⎟ , K = k⊥ . . . k⊥ , H = h . . . h t , Q := ⎝ ... ⎠ 1 p 1 p . Q p1 . . . Q pp satisfy the identity HK = Q. The basis {ki⊥ } being orthogonal to {ki }, we conclude that Kt = S−1 . Now Lemma 7.1 and the condition (A) for A0 (l) imply the uniform in l estimate | det Q| C0 > 0. Hence, there exists a permutation 1 ... p η = η(l), η = such that η1 . . . η p p i=1
|Qiηi |
C0 . p!
(7.13)
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This fact is proved easily by the contradiction employing the definition of the determinant. The normalization of hi and the identities (2.13) for ki⊥ imply that |Qiηi | hi C p k⊥ ηi C p = 1. Hence, by (7.13) we obtain C0 |Qiηi | 1, |Q jη j | p! i=1 p
C0 |Q jη j |. p!
i = j
Substituting the definition of Qiηi in this estimate, we have (0) 2 −4ls1 (λ∗ ) . |λi − λ∗ − τi(0) | C T10(i) (λ∗ + τη(0) , l) h i C A1 + |τηi |l e i Here we have also used the boundedness of T10(i) (see Lemma 7.4) and (7.12). , we complete the proof. Rearranging the roots τi(0) as τi(0) := τη(0) i Proof of Theorem 2.8 Let us prove first that the matrix A(λ∗ , l) obeys the representation (2.14), where all the elements of A0 are zero except ones standing on the intersection of the first row and of ( p− + 1)-th column and ( p− + 1)-th column and the first row, and these elements are given by A0,1 p− +1 (l) = A0, p− +11 (l) = 2s1 (λ∗ )β − β+ e−2ls1 (λ∗ ) . Also we are going to prove that the corresponding matrix A1 satisfies the estimate A1 (l) = O(le−4l
√ ν1 −λ∗
),
l → +∞.
(7.14)
The definition of i , the formulas (4.11) for ϕ ±j , (5.16), (5.7) imply i (·, λ∗ , l) = φ i (·, l) + O le−4ls1 (λ∗ ) ,
l → +∞,
(7.15)
in L2 (− ) ⊕ L2 (+ )-norm. The identity (7.15) and the definition of Aij and T1(i) yield that for i, j = 1, . . . , p− Aij(λ∗ , l) = T1(i) j(·, λ∗ , l) = T1(i) φ j(·, l)+ O le−4ls1 (λ∗ ) = O le−4ls1 (λ∗ ) ,
(7.16)
since T1(i) φ j(·, l) = 0, i, j = 1, . . . , p− . In the same way one can check easily the same identity for i, j = p− + 1, . . . , p. Taking into account the definition (4.11) of ϕ1− , and (2.17) by direct calculations we obtain A1 p− +1 (λ∗ , l) = ψ1+ , ϕ1− (·, l) L2 (+ ) = ψ1+ , L+ S (2l)ψ1− L2 (+ ) = = β − e−2ls1 (λ∗ ) ψ1+ , L+ e−s1 (λ∗ )x1 φ1 L2 (+ ) + O e−2(s1 l(λ∗ )+s2 (λ∗ )) ,
Asymptotic behaviour of the spectrum...
187
as l → +∞. Now we use the definition of L+ , (2.17), and integrate by parts, + ψ1 , L+ e−s1 (λ∗ )x1 φ1 L2 (+ ) = ψ1+ , (− − λ∗ + L+ )e−s1 (λ∗ )x1 φ1 L2 () ∂ψ1+ −s1 (λ∗ )x1 e = lim φ1, − x1 →−∞ ∂ x1 L2 (ω) ∂ −s1 (λ∗ )x1 + − ψ1 , e φ1 ∂ x1 L2 (ω) = 2s1 (λ∗ )β+ ,
(7.17)
which together with previous formula implies A1 p− +1 (λ∗ , l) = 2β − β+ s1 (λ∗ )e−2ls1 (λ∗ ) + O e−2l(s1 (λ∗ )+s2 (λ∗ )) ,
(7.18)
as l → +∞. In the same way one can show that A1 j(λ∗ , l) = O e−2l(s1 (λ∗ )+s2 (λ∗ )) , j = p− + 2, . . . , p, Aij(λ∗ , l) = O e−4ls2 (λ∗ ) , i = 2, . . . , p− , j = p− + 1, . . . , p, A p− +11 (λ∗ , l) = 2β− β + s1 (λ∗ )e−2ls1 (λ∗ ) + O e−2l(s1 (λ∗ )+s2 (λ∗ )) , A p− +1 j(λ∗ , l) = O e−2l(s1 (λ∗ )+s2 (λ∗ )) , j = 2, . . . , p− , Aij(λ∗ , l) = O e−4ls2 (λ∗ ) , i = p− + 2, . . . , p, j = 1, . . . , p− , as l → +∞. The formulas obtained and (7.16), (7.18) lead us to the representation (2.14), where the matrix A0 is as described, while the matrix A1 satisfies (7.14). The matrix A0 being hermitian, it satisfies the condition (A). This fact can be proved completely by analogy with Lemma 7.1. Let us calculate the roots of det(τ E − A0 ). For the sake of brevity within the proof we denote c := −2β − β+ s1 (λ∗ )e−2ls1 (λ∗ ) . Expanding the determinant det(τ E − A0 ) w.r.t. the first column, one can make sure that det(τ E − A0 ) = τ p−2 τ 2 − |c|2 . ( p−2)
( p−1)
Thus, τ0(1) (l) = . . . = τ0 (l) = 0 is a root of multiplicity ( p − 2), and τ0 ( p) −2|c|, τ0 (l) = 2|c|. Applying Theorem 2.5, we arrive at (2.21).
(l) =
Proof of Theorem 2.6 As in the proof of Theorem 2.8, we begin with the proving (2.14). Namely, we are going to show that A0 (l) = diag{A11 (λ∗ , l), 0, . . . , 0}, A1 (l) = O e−2l(s1 (λ∗ )+s2 (λ∗ )) . (7.19) The definition of A implies Aij(λ∗ , l) = − T6− (λ∗ , l)(I − T7+ (λ∗ , l)T6− (λ∗ , l))−1 ϕi− (·, l), ψ − j L2 () .
(7.20)
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Employing the definition (4.9) of T6− , for any f ∈ L2 (, + ) we check that − −1 T6 (λ∗ , l) f, ψ − f, ψ − j L2 () = L− S (−2l)(H+ − λ∗ ) j L2 () = S (−2l)(H+ − λ∗ )−1 f, L− ψ − j L2 () = (H+ − λ∗ )−1 f, S (2l)( + λ∗ )ψ − j L2 () = − (H+ − λ∗ )−1 f, (H+ − λ∗ − L+ )S (2l)ψ − j L2 () − = f, (H+ − λ∗ )−1 L+ S (2l)ψ − j − S (2l)ψ j L2 () . Using this identity, (7.20), (4.10), (4.12), and Lemma 2.1, we obtain that for (i, j) = (1, 1) the functions Aij satisfy the relation |Aij| (I − T7+ T6− )−1 ϕi− L2 (+ )
−2l(s1 (λ∗ )+s2 (λ∗ )) − , (H+ − λ∗ )−1 L+ S (2l)ψ − j − S (2l)ψ j L2 (+ ) = O e
as l → +∞, where in the arguments λ = λ∗ , and the operator (H+ − λ∗ )−1 is bounded since p+ = 0. The identities (7.19) therefore hold true. We apply now Theorem 2.5 and infer that the formulas (2.18) are valid for the eigenvalues λi , i = 1, . . . , p − 1, while the eigenvalue λ p satisfies λ p (l) = λ∗ + A11 (λ∗ , l) 1 + O(l 2 e−2ls1 (λ∗ ) ) + O e−2l(s1 (λ∗ )+s2 (λ∗ )) , (7.21) as λ → +∞. Now it is sufficient to find out the asymptotic behaviour of A11 (λ∗ , l). The formula (7.20) for A11 , Lemma 2.1, and (4.10), (4.12) yield A11 (λ∗ , l) = − T6− (λ∗ , l)ϕ1− (·, l), ψ1− L2 () + O le−8ls1 (λ∗ ) = β− e−2ls1 (λ∗ ) T6− (λ∗ , l)L+ e−s1 (λ∗ )x1 , ψ1− L2 () + + O e−2l(s1 (λ∗ )+s2 (λ∗ )) + le−8ls1 (λ∗ ) , − T6 (λ∗ , l)L+ e−s1 (λ∗ )x1 , ψ1− L2 () = L− S (−2l)(H+ − λ∗ )−1 L+ e−s1 (λ∗ )x1 , ψ1− L2 (− ) = S (−2l)(H+ − λ∗ )−1 L+ e−s1 (λ∗ )x1 , L− ψ1− L2 (− ) = S (−2l)U + , L− ψ1− L2 (− ) . Lemma 4.2 implies that the function U satisfies (2.19) that determines the − uniquely. Employing (2.19), we continue our calculations, constant β − e−2ls1 (λ∗ ) es1 (λ∗ )x1 φ1 (x ), L− ψ1− S (−2l)U + , L− ψ1− L2 (− ) =β + L2 () −2ls2 (λ∗ ) . +O e Integrating by parts in the same way as in (7.17), we obtain s1 (λ∗ )x1 e φ1 (x ), L− ψ1− L2 () = L− es1 (λ∗ )x1 φ1 (x ), ψ1− L2 () = −2s1 (λ∗ )β − .
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Therefore,
− e−4ls1 (λ∗ ) + O e−2l(s1 (λ∗ )+s2 (λ∗ )) + le−8ls1 (λ∗ ) . A11 (λ∗ , l) = 2s1 (λ∗ )|β− |2 β
Substituting this identity into (7.21), we arrive at the required formula for λ p . The proof of Theorem 2.7 is completely analogous to that of Theorem 2.6.
8 Examples In this section we provide some examples of the operators L± . In what follows we will often omit the index “±” in the notation L± , H± , ± , a± , writing L, H, , a instead. 1. Potential. The simplest example is the multiplication operator L = V, where V = V(x) ∈ C() is a real-valued compactly supported function. Although this example is classical one for the problems in the whole space, to our knowledge, the double-well problem in waveguide has not been considered yet. 2. Second order differential operator. This is a generalization of the previous example. We introduce the operator L as L=
n i, j=1
∂2 ∂ + bi + b 0, ∂ xi ∂ x j ∂ xi i=1 n
b ij
(8.1)
where the complex-valued functions b ij = b ij(x) are piecewise continuously differentiable in , b i = b i (x) are complex-valued functions piecewise continuous in . These functions are assumed to be compactly supported. The only restriction to the functions are the conditions (2.1), (2.2); the self-adjointness of H and Hl is implied by these conditions. One of the possible way to choose the functions in (8.1) is as follows n ∂ ∂ L = div G∇ + i − b i + b 0, (8.2) bi ∂ xi ∂ xi i=1 where G = G(x) is n × n hermitian matrix with piecewise continuously differentiable coefficients, b i = b i (x) are real-valued piecewise continuously differentiable functions, b 0 = b 0 (x) is a real-valued piecewise continuous function. The matrix G and the functions b i are assumed to be compactly supported and (G(x)y, y)Cn −c0 y2Cn ,
x ∈ ,
y ∈ Cn .
The constant c0 is independent of x, y and satisfies (2.3). The matrix G is not necessarily non-zero. In the case G = 0 one has an example of a first order differential operator. 3. Magnetic Schrödinger operator. This is the example with a compactly supported magnetic field. The operator L is given by (8.2), where G = 0. The coefficients b j form a magnetic real-valued vector-potential b =
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(b 1 , . . . , b n ) ∈ C1 (), and b 0 = b2Cn + V, where V = V(x) ∈ C() is a compactly supported real-valued electric potential. The main assumption is the identities ∂b j ∂b i = , ∂ xi ∂xj
x ∈ \ ,
i, j = 1, . . . , n.
(8.3)
To satisfy the conditions required for L, the magnetic vector potential should have a compact support. We are going to show that one can always achieve it by employing the gauge invariance. The operator −(D) + L can be represented as −(D) + L = (i∇ + b)2 + V, and for any β = β(x) ∈ C2 () the identity e−iβ (i∇ + b)2 eiβ = (i∇ + b − ∇β)2 holds true. In view of (8.3) we conclude that there exist two functions β± = β± (x) belonging to C2 ( ∩ {x : ±x1 > a}) such that ∇β± = b, x ∈ ∩ {x : ± x1 > a}. We introduce now the function β as ⎧ ⎪ ⎪ χ(a + x1 + 1)β− (x), x1 ∈ (−∞, −a), x ∈ ω, ⎨ x1 ∈ [−a, a], x ∈ ω, β(x) = 0, ⎪ ⎪ ⎩ x ∈ ω, χ(a − x1 + 1)β+ (x), x1 ∈ (a, +∞), where, we remind, the cut-off function χ was introduced in the proof of Lemma 3.2. Clearly, β ∈ C2 (), and ∇β = b, x ∈ \ {x : |x1 | < a + 1}. Therefore, the vector b − ∇β has compact support. If one of the distant perturbations in the operator Hl , say, the right one, is a compactly supported magnetic field, it is sufficient to employ the gauge (x) to satisfy the conditions for L+ . transformation ψ(x) → eiβ(x1 −l,x ) ψ 4. Curved and deformed waveguide. One more interesting example is a geometric perturbation. Quite popular cases are local deformation of the boundary and curving the waveguide (see, for instance, [3, 9, 10], and references therein). Here we consider the case of general geometric perturbation, which includes in particular deformation and curving. Namely, let x = G (x) ∈ C2 () xn ), G (x) = (G1 (x), . . . , Gn (x)). We be a diffeomorphism, where x = ( x1 , . . . , := G (), P := G −1 . By P = P(x) we indicate the matrix denote ⎛ ∂ P1 P1 ⎞ . . . ∂∂ ∂ x1 xn ⎜ .. .. ⎟ , P := ⎝ . . ⎠ ∂ Pn ∂ x1
...
∂ Pn ∂ xn
while the symbol p = p(x) denotes the corresponding Jacobian, p(x) := det P(x). The function p(x) is supposed to have no zeroes in . The main assumption we make is P(x) = const,
Pt P = E,
x ∈ \ .
(8.4)
It implies that outside the mapping P acts as a combination of a shift and a given by G ((a, +∞) × ω) is also a tubular domain rotating. Hence, the part of
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being a direct product of a half-line and ω. The same is true for G ((−∞, −a) × is given on Fig. 1. ω). The typical example of the domain (D) be the negative Dirichlet Laplacian in L2 ( ) with the domain Let − 2 W2,0 (). It is easy to check that the operator U : L2 () → L2 () defined as (8.5) (U v)(x) := p−1/2 (x)v P −1 (x) 2 (D) U −1 has W2,0 is unitary. The operator H := −U () as the domain and is self-adjoint in L2 (). It can be represented as H = −(D) + L, where L is a second order differential operator
L = −p1/2 divx p−1 Pt P∇x p1/2 + x .
(8.6)
Indeed, for any u1 , u2 ∈ C0∞ () (D) −1 U u1 , U −1 u2 (Hu1 , u2 ) L2 () = − ) L2 ( 1/2 = ∇x p1/2 u1 , ∇x p1/2 u2 L2 ( u1 , p−1 P∇x p1/2 u2 L2 () ) = P∇x p = − p1/2 divx p−1 Pt P∇x p1/2 u1 , u2 L2 () . (8.7) The assumption (8.4) yields that p = 1 holds for x ∈ \ , and therefore the coefficients of the operator L have the support inside . We are going to check the conditions (2.1), (2.2), (2.3) for the operator L introduced by (8.6). The symmetricity is obvious, while the estimates follow from (8.6), (8.7), (Lu, u) L2 () = p−1/2 P∇x p1/2 u2L2 () − ∇u2L2 () C∇x p1/2 u2L2 () − ∇u2L2 () = C p1/2 ∇x u2L2 () +u∇x p1/2 2L2 () + 2(p1/2 ∇x u, u∇p1/2 ) L2 () − 1 1/2 − ∇u2L2 () C p ∇x u2L2 () − u∇x p1/2 2L2 () − 2
− ∇x u2L2 () C ∇x u2L2 () − Cu2L2 () , − 1− 2 where C > 0 is a constant.
Fig. 1 Geometric perturbation
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If both the operators L± are the geometric perturbations described by the diffeomorphisms G ± (x), without loss of generality we can assume that G ± (x) ≡ x as ±x1 < −a± . We introduce now one more diffeomorphism
G + (x1 − l, x ) + (l, 0, . . . , 0), x1 0, Gl (x) := G − (x1 + l, x ) − (l, 0, . . . , 0), x1 0, l := Gl () can This mapping is well-defined for l > max{a− , a+ }. The domain be naturally regarded as a waveguide with two distant geometric perturbations. ), we obtain easily an Considering the negative Dirichlet Laplacian in L2 ( unitary equivalent operator in L2 (). The corresponding unitary operator is defined by (8.5) with P replaced by Gl−1 and the similar replacement for p is required. Clearly, the obtained operator in L2 () is the operator Hl generated by the operators L± associated with G ± . 5. Delta interaction. Our next example is the delta interaction supported by a manifold. Let be a closed bounded C3 -manifold in of the codimension one and oriented by a normal vector field ν. It is supposed that ∩ ∂ = ∅. The manifold can consist of several components. By ξ = (ξ1 , . . . , ξn−1 ) we denote coordinates on , while will indicate the distance from a point to measured in the direction of ν = ν(ξ ). We assume that is so that the coordinates ( , ξ ) are well-defined in a neighbourhood of . Namely, we suppose that the mapping ( , ξ ) = P ( x) is C3 -diffeomorphism, where x = ( x1 , . . . , xn ) are the coordinates in . Let b = b (ξ ) ∈ C3 ( ) be a real-valued function. The operator in question is the negative Laplacian defined on the functions v ∈ 1 W22 ( \ , ∂) ∩ W2,0 () satisfying the condition ! ∂v [v]0 = 0, = 0, [u]0 := u =+0 − u =−0 . (8.8) ∂ 0 We indicate this operator as H . An alternative way to introduce H is via associated quadratic form q (v1 , v2 ) := (∇x v1 , ∇x v2 ) L2 () + (b v1 , v2 ) L2 ( ) ,
(8.9)
1 where u ∈ W2,0 () (see, for instance, [2, Appendix K, Sec. K.4.1], [7, Remark 4.1]). It is known that the operator H is self-adjoint. x) such We are going to show that there exists a diffeomorpism x = P ( that a unitary operator U defined by (8.5) maps L2 () onto L2 (), and 2 the operator UH U −1 has W2,0 () as the domain. We will also show that −1 (D) UH U = − + L, where L is given by (8.1). First we introduce an auxiliary mapping as
( P x) := P −1 ( , ξ ) ,
:= + 1/2 | |b (ξ ),
( , ξ ) = P (x).
The coordinates ( , ξ ) are well-defined in a neighbourhood of , which can be described as {x : | | < δ}, where δ is small enough. Indeed, i ( i ( P x) = xi + | |P x),
(8.10)
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i (x) are continuously differentiable in a neighbourhood where the functions P i tends to one as of . Therefore, the Jacobian of P → 0 uniformly in ξ ∈ . Now we construct the required mapping as follows ( x = P ( x) := 1 − χ ( /δ) x+χ ( /δ)P x). (8.11) The symbol χ (t) indicates an even infinitely differentiable cut-off function being one as |t| < 1 and vanishing as |t| > 2. We assume also that δ is chosen so that supp χ ( /δ) ∩ ∂ = ∅. Let us prove that P is a C1 -diffeomorhism and P () = . It is obvious that P ∈ C1 (). If χ ( /δ) = 0, the mapping P acts as an identity mapping, and therefore for such x the Jacobian p = p( x) of P equals one. The function χ is non-zero only in a small neighbourhood { x : | | < 2δ} of , where the identities (8.10) are applicable. These identities together with the definition of P imply that p( x) = 1 + p1 ( x), where the function p1 ( x) is bounded uniformly in δ and x as | | 2δ. Hence, we can choose δ small enough so that p 1/2 as | | 2δ. Therefore, P is C1 -diffeomorphism. As x close to ∂, the diffeomorphism P acts as the identity mapping. It yields that P (∂) = ∂, P () = . Since = 0 as = 0, it follows that P ( ) = . The function → is continuously differentiable and its second derivative is piecewise continuous. Thus, the second derivatives of Pi ( x) are piecewise continuous as well. The same is true for the inverse mapping P −1 . We introduce the unitary operator U by (8.5), where P is the diffeomorphism defined by (8.11). It is obvious that U maps L2 () onto itself. 2 Let us prove that it maps the domain of H onto W2,0 (). In order to do it, we have to study the behaviour of p in a vicinity of . It is clear that p( x) ∈ C2 ( \ ) ∩ C(), and this function has discontinuities at only. By p = p ( , ξ ) we denote the Jacobian corresponding to P . It is obvious that p ∈ C2 ({( , ξ ) : | | δ}). Employing the well-known properties of the Jacobians, we can express p in terms of p , p( x) =
(1 + | |b (ξ ))p ( , ξ ) p ( , ξ )(1 + | |b (ξ )) = , p ( , ξ ) p ( + 1/2 | |b (ξ ), ξ )
| | δ.
This relation allows us to conclude that p ∈ C2 ({( , ξ ) : 0 δ}), p ∈ C2 ({( , ξ ) : −δ 0}), and ! ∂p1/2 1/2 [p ]0 = 0, = b. (8.12) ∂ 0 2 Given u = u(x) ∈ W2,0 (), we introduce the function v = v( x) := (U −1 u)( x) = 2 1 1/2 x)u(P ( x)). Due to the smoothness of p, v( x) ∈ W2 ( \ ) ∩ W2,0 (). The p ( identities (8.12) and the belonging u(x) ∈ W22 () imply the condition (8.8) for v. 1 Suppose now that a function v = v( x) ∈ W22 ( \ ) ∩ W2,0 () satisfies (8.8). Due to the smoothness of P it is sufficient to check that the function U v 2 regarded as depending on x belongs to W2,0 (), i.e., u( x) := p−1/2 ( x)v( x) ∈ 2 2 1 W2,0 (). It is clear that u ∈ W2 ( \ ) ∩ W2,0 (). Hence, it remains to check the belonging u ∈ W22 ({ x : | | δ}). The functions Pi being twice piecewise
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continuously differentiable, it is sufficient to make sure that the function u treated as depending on and ξ is an element of W22 ({( , ξ ) : | | < δ, ξ ∈ }). 1 , ξ ) : | | < δ, ξ ∈ }, We have u ∈ W2 (R), u ∈ W22 (R+ ), u ∈ W22 (R− ), R := {( R± := {( , ξ ) : 0 < ± < ±δ, ξ ∈ }. Hence, we have to prove the existence of the generalized second derivatives for u belonging to L2 (R). The condition (8.8) and the formulas (8.12) yield that ∂u ∂u u =+0 = u =−0 , = . ∂ =+0 ∂ =−0 Since u =±0 ∈ W21 ( ), the first of these relations implies that ∂u ∂u = , i = 1, . . . , n − 1. ∂ξi =+0 ∂ξi =−0 Having the obtained relations in mind, for any ζ ∈ C02 (R) we integrate by parts, ∂ 2ζ ∂ 2ζ ∂ 2ζ u, 2 = u, 2 + u, 2 ∂ L2 (R) ∂ L2 (R− ) ∂ L2 (R+ ) 2 2 ∂ u ∂ u = , ζ + , ζ , ∂ 2 ∂ 2 L2 (R− ) L2 (R+ ) and in the same way we obtain 2 2 ∂ 2ζ ∂ u ∂ u u, = ,ζ + ,ζ , ∂ ∂ξi L2 (R) ∂ ∂ξi ∂ ∂ξi L2 (R− ) L2 (R+ ) 2 2 ∂ u ∂ u ∂ 2ζ = ,ζ + ,ζ . u, ∂ξi ∂ξ j L2 (R) ∂ξi ∂ξ j ∂ξi ∂ξ j L2 (R− ) L2 (R+ ) Thus, the generalized second derivatives of the function u exist and coincide with the corresponding derivatives of u regarded as an element of W22 (R− ) and W22 (R+ ). Let us show that UH U −1 = −(D) + L, where L is given by (8.1). Proceeding in the same way as in (8.7) and using (8.9), for any u1 , u2 ∈ C0∞ () we obtain (UH U −1 u1 , u2 ) L2 () = ∇x p1/2 u1 , p−1 Pt P∇x p1/2 u2 L2 () + (b u1 , u2 ) L2 ( ) . We have used here that p ≡ 1 as x ∈ and P ( ) = . Employing (8.12) and having in mind that P = E due to (8.10), we integrate by parts, ! ∂ 1/2 −1 (p u1 ) − b u1 , u2 − (UH U u1 , u2 ) L2 () = ∂ ρ 0 L2 ( ) − p1/2 divx p−1 Pt P∇x p1/2 u1 , u2 L2 (\ ) = − p1/2 divx p−1 Pt P∇x p1/2 u1 , u2 L2 (\ ) .
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Here \ means that in a neighbourhood of we partition the domain of integration into two pieces one being located in the set {x : | | < 0}, while the other corresponds to | | > 0. Such partition is needed since the first derivatives of p have jump at and therefore the second derivatives of p are not defined at . At the same time, the function p has continuous second derivatives as <0 and these derivatives have finite limit as → −0. The same is true for > 0. The matrix P is piecewise continuously differentiable, and therefore UH U −1 = −p1/2 divx p−1 Pt P∇x p1/2 , where the second derivatives of p are treated in the aforementioned sense. This operator is obviously self-adjoint. Outside the set {x : | | < 2δ} the diffeomorphism P acts as the identity mapping. It yields that at such points P = E, p = 1. Therefore, L is a differential operator having compactly supported coefficients, and is a particular case of (8.1). It is also clear that the operator L satisfies (2.1). The inequality (2.2) can be checked in the same way how it was proved in the previous subsection. 6. Integral operator. The operator L is not necessary to be either a differential operator or reducible to a differential one. An example is an integral operator Lu = L(·, y)u(y) dy.
The kernel L ∈ L2 ( × ) is assumed to be symmetric, i.e., L(x, y) = L(y, x). It is clear that the operator L satisfies (2.1), (2.2). It is also (D) -compact and therefore the operators H and Hl are self-adjoint. In conclusion we stress that not all possible examples of L are exhausted by the list given above. For instance, combinations of these examples are possible like compactly supported magnetic field with delta interaction, delta interaction in a deformed waveguide, integro-differential operator, etc. Moreover, the operators L− and L+ need not necessarily to be of the same nature. For example, L− can be a potential, while L+ may describes compactly supported magnetic field with delta interaction. Acknowledgements I am grateful to Pavel Exner who attracted my attention to the problem studied in this article. I thank him for stimulating discussion and the attention he paid to this work. I also thank the referee for useful remarks.
References 1. Agmon, S.: Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of N-body Schrödinger operator. Mathematical Notes. Princeton University Press (1982) 2. Albeverio, S., Gesztesy, S., Høegh-Krohn, H. Holden, R.: Solvable Models in Quantum Mechanics, 2nd edn. AMS Chelsea Publishing. Providence, Rhode Island (2005) 3. Borisov, D., Exner, P., Gadyl’shin, R., Krejˇciˇrík, D.: Bound states in weakly deformed strips and layers. Ann. H. Poincaré. 2, 553–572 (2001) 4. Borisov, D., Exner, P.: Exponential splitting of bound states in a waveguide with a pair of distant windows. J. Phys. A. 37, 3411–3428 (2004) 5. Borisov, D., Exner, P.: Distant perturbation asymptotics in window-coupled waveguides. I. The non-threshold case. J. Math. Phys. 47, 2113502-1–113502-24 (2006)
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6. Borisov, D.: Distant perturbations of the Laplacian in a multi-dimensional space. Ann. H. Poincare (2007) (in press) 7. Brasche, J.F., Exner, P., Kurepin, Yu.A., Šeba, P.: Schrödinger operator with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994) 8. Briet, Ph., Combes, J.M., Duclos, P.: Spectral stability under tunneling. Comm. Math. Phys. 126, 133–156 (1989) 9. Bulla, W., Gesztesy, F., Renger, W., Simon, B.: Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 125, 1487–1495 (1997) 10. Chenand, B., Duclos, P., Freitas, P., Krejˇciˇrík, D.: Geometrically induced spectrum in curved tubes. Differential Geom. Appl. 23, 95–105 (2005) 11. Combes, J.M., Duclos, P., Seiler,R.: Krein’s formula and one-dimensional multiple well. J. Funct. Anal. 52, 257–301 (1983) 12. Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1995) 13. Gadyl’shin, R.: On local perturbations of Schrödinger operator in axis. Theor. Math. Phys. 132, 976–982 (2002) 14. Harrel, E.M.: Double wells. Comm. Math. Phys. 75, 239–261 (1980) 15. Harrel, E.M., Klaus, M.: On the double-well problem for Dirac operators. Ann. Inst. H. Poincaré, Sect. A. 38, 153–166 (1983) 16. Høegh-Krohn, R., Mebkhout, M.: The 1r expansion for the critical multiple well problem. Comm. Math. Phys. 91, 65–73 (1983) 17. Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966) 18. Klaus, M., Simon, B.: Binding of Schrödinger particles through conspiracy of potential wells. Ann. Inst. H. Poincaré, Sect. A. 30, 83–87 (1979) 19. Kondej, S., Veseli´c I.: Lower bounds on the lowest spectral gap of singular potential Hamiltonians. Ann. H. Poincaré. 8, 109–134 (2007) 20. Mikhajlov, V.P.: Partial Differential Equations. Mir Publishers, Moscow (1978) 21. Persson, A.: Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8, 143–153 (1960) 22. Simon, B.: Semiclassical analysis of low-lying eigenvalues. I. Non-generate minima: asymptotic expansions. Ann. Inst. H. Poincaré, Sect. A. 38, 295–308 (1983) 23. Weidmann, J.: Mathematische Grundlagen der Quantummechanik II. Frankfurt: Fachbereich Mathematik der Universität Frankfurt (1995)
Math Phys Anal Geom (2007) 10:197–203 DOI 10.1007/s11040-007-9025-4
Asymptotic Inverse Problem for Almost-Periodically Perturbed Quantum Harmonic Oscillator Alexis Pokrovski
Received: 12 July 2007 / Accepted: 13 July 2007 / Published online: 13 September 2007 © Springer Science + Business Media B.V. 2007
d 2 2 Abstract Let {μn }∞ n=0 be the spectrum of − dx2 + x + q(x) in L (R), where q is an even almost-periodic complex-valued function with bounded primitive and 1 derivative. It is known that μn = μ0n + O(n− 4 ), where {μ0n }∞ n=0 is the spectrum of the unperturbed operator. Suppose that the asymptotic approximation to 1 the first asymptotic correction μn = μn − μ0n + o(n− 4 ) is given. We prove the formula that recovers the frequencies and the Fourier coefficients of q in terms of μn . 2
Keywords Almost-periodic perturbation · Inverse problem · Quantum harmonic oscillator · Spectral asymptotics Mathematics Subject Classifications (2000) 34L20 · 81Q15
1 Introduction and Main Result Consider the operator describing perturbed quantum harmonic oscillator d2 + x2 + q(x) in L2 (R) (1.1) dx2 with theperturbation q(x) from the class B = {q : q ∞ + Q∞ < ∞}, where x Q(x) = 0 q dt and · ∞ denotes the norm in L∞ (R). It was proved in [1] that − 13 0 1 ), where the spectrum {μn }∞ n=0 of A has the asymptotics μn = μn + μn + O(n 1 μ0n = 2n + 1 and μ1n = O(n− 4 ). A=−
A. Pokrovski (B) Laboratory of Quantum Networks, Institute for Physics, St-Petersburg State University, St. Petersburg 198504, Ulyanovskaya 1, Russia e-mail:
[email protected]
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For the perturbations that are sum of almost-periodic and decaying terms we study the problem of recovering of the almost-periodic part from the first asymptotic correction μ1n . Specifically, we consider the perturbations q = p+r ∈ B ,
p∈ B1 , 1 T→∞ 2T
r B1 ≡ lim
p(−x) = p(x) T −T
|r(x)|dx = 0,
and (1.2)
functions [2] (closure where B1 is the Besikovitch space N of almost-periodic of trigonometric polynomials k=0 ak eiνk x , νk real, in the norm · B1 ). It is sufficient to recover p in terms of its Fourier transform [2]. Here is the main result. Theorem 1.1 Let {μn }∞ n=N approximates the first asymptotic correction to the spectrum of the operator (1.1), (1.2): 1 μn = μn − μ0n + o n− 4 . (1.3) Then the spectrum and the Fourier coefficients of the almost-periodic part p can be recovered from the relation lim
L→∞
L−1 1 μn Gν (xn , x L )(xn+1 −xn ) x L n=N
1 = lim T→∞ T
T
p(t) cos νt dt,
ν 0,
(1.4)
0
T ϕ (t)dt √ where xn = μ0n = 2n + 1, Gν (x, T) = −x x √ν,Tt2 −x2 , ϕν,T (t) = η(t −T) cos νt and η ∈ C2 (R) is a smoothed step function such that η(t) = 1 for x ∈ (−∞, −1], η(t) = 0 for x ∈ [0, ∞) and η (0) = 0. Asymptotic inverse spectral problem for quantum harmonic oscillator with slowly decaying perturbation was considered by Gurarie [3]. He studied the operator (1.1) with real q(x) ∼ |x|−α am cos ωm x for |x| → ∞, where α > 0 and the sum in the numerator is finite. The approach in [3] is based on the spectral asymptotics √
am q( 2n) 1 μn = 1/4+α/2 + O √ , q(x) = const cos(ωm x − π/4) √ n ωm n which exhibits linear relation between the leading asymptotic terms of q and μn . However, the technique of [3] does not cover the case α = 0. We consider just this case in a slightly more general setting (almost-periodic functions vs. finite trigonometric sums). Our method also allows complexvalued q. Technically, the result is based on the recent proof [1] of the spectral asymptotics π 1 1 0 0 q (1.5) μn = μn + μn sin ϑ dϑ + O n− 3 , for q ∈ B . 2π −π
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Thus the proof of Theorem 1.1 follows from the asymptotic behavior of the integral in (1.5), which is analyzed in Lemmas 2.1 and 2.3.
2 Properties of the Schlömilch Integral The integral in the spectral asymptotics (1.5) is the Schlömilch integral [4] gq (x) =
2 π
π/2
q+ (x sin ϑ)dϑ =
0
2 π
x
0
q+ (t)dt , √ x2 − t 2
q+ (x) = (q(x) + q(−x))/2 (2.6)
√ evaluated at the points xn = 2n + 1. In the next Lemma we estimate the integral and its derivatives. Then in Lemma 2.3 we prove similar estimates for the inverse Schlömilch integral. (We could not find in the literature the results of these Lemmas for the specific class B .) Using the two Lemmas, we prove Theorem 1.1. Everywhere below C denotes an absolute constant. Lemma 2.1 Let f ∈ B and g(x) = |g(x)| C
F∞ + f ∞ , √ 1+x
where F(x) =
x 0
π/2 0
f (x sin ϑ)dϑ. Then
|g (x)| C
f ∞ + f ∞ , √ 1+x
x > 0, (2.7)
f dt.
Proof For x 1 the result is evident, so we consider only the case x > 1. Using 1 f (xt)dt and split it as the change of variables t = sin ϑ, we write g(x) = 0 √ 1−t2 g = I1 + I2 ,
1−ε
I1 = 0
f (xt)dt , √ 1 − t2
where ε = 1/x. Using the notation (. . .)t for
I2 =
1 1−ε
f (xt)dt , √ 1 − t2
(2.8)
∂ (. . .) ∂t
and choosing the prim1−ε dt of f , satisfying F(x(1 √ = itive F − ε)) = 0, we have I1 = 1x F(xt) t 1−t2 0 t=1−ε 1−ε F(xt)t F(xt) dt 1 √ + . Therefore, x (1−t2 )3/2 1−t2 t=0
0
⎛ ⎞
1 dt ⎠ 1 F∞ ⎝ F∞ 1+ 1 + |I1 | 2 C . √ x t3/2 x ε ε
(2.9)
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ε √ Now we substitute the estimate |I2 | f ∞ 0 √dtt = 2 f ∞ / x and (2.9) into (2.8). This gives the first inequality in (2.7). We prove the second one in a similar way, writing 1−ε 1 t f (xt)dt t f (xt)dt g = I1 + I2 , I1 = , I2 = , ε = 1/x. √ √ 1 − t2 1 − t2 0 1−ε (2.10) We integrate by parts in I1 , choosing the primitive 1−ε f (xt) dt This gives I1 = − 1x 0 , hence (1−t2 )3/2
f (xt) = f (xt) − f (x(1 − ε)).
1 (2.11) 1+ √ . ε √ We substitute the estimate |I2 | C f ∞ / x and (2.11) in (2.10). This gives the second inequality in (2.7).
|I1 | C
f ∞ x
Remark 2.2 The rate of decay x−1/2 as x → ∞ in (2.7) cannot be improved. The example f (x) = cos x gives the Bessel function J0 . Lemma 2.3 Let T > 2, ϕ, ϕ ∈ L∞ ([0, T]) and ϕ(T) = 0. Then the equation T T ϕ(t) = π2 t √g(x)dx has the unique solution g(x) = −x x √ϕt(t)dt for x ∈ [0, T], 2 −x2 x2 −t2 such that √ for x > 1. (2.12) |g(x)| C(ϕ∞ + ϕ ∞ ) x, If, in addition, ϕ (T) = 0, then
√ |g (x)| C(ϕ ∞ + ϕ ∞ ) x,
for
x > 1.
(2.13)
√ √ g( x) √ and ϕ (t) = ϕ( t) the equation on g becomes 2 x T2 the Abel equation ϕ (t) = π2 t g√(s)s−tds . Its solution for absolutely continuous ϕ T2 2 ϕ (T ) (u)du ϕ is g(s) = √T 2 −s − s √u−s (see Chap. 1, paragraph 2 of [5]). Using ϕ(T) = 0,
Proof In terms of g(x) =
we obtain the required formula for g. Consider (2.12). For x ∈ [T − 1, T] the inequality follows from the direct ∞ T √ dt | ϕ√2x estimate | g(x) T−1 t−(T−1) . For x ∈ [0, T − 1] write x x+1 T g(x) ϕ (t)dt ϕ (t)dt − = I1 + I2 , I1 = , I2 = (2.14) √ √ x t 2 − x2 t 2 − x2 x x+1 t=T T and integrate I2 by parts. We have I2 = √ϕ(t) − x+1 ϕ(t) ∂t∂ √t21−x2 dt. 2 2 t −x Therefore,
t=x+1
ϕ∞ (−1) t=∞ 2ϕ∞ + ϕ∞ √ √ . (2.15) |I2 | √ 2 2 1 + 2x 1 + 2x t − x t=x+1 x+1 √ Now we substitute the estimate |I1 | ϕ ∞ x √t2dt−x2 ϕ ∞ 2/x and (2.15) into (2.14). This gives (2.12), as required.
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Next consider (2.13). By g (x) = x(g(x)/x) + g(x)/x and (2.12), it is sufficient to estimate x(g(x)/x) . Using ϕ (T) = 0, we obtain T xsϕ (xs) ds tϕ (t) dt = . (2.16) √ √ 2 s −1 t 2 − x2 1 x √ T √ (t)| dt For x ∈ [T − 1, T] we have |x(g(x)/x) | √2x2x T−1 √|ϕt−(T−1) 2 2ϕ ∞ x. For x ∈ [0, T − 1] write x+1 T tϕ (t)dt tϕ (t)dt x(g(x)/x) = I1 + I2 , I1 = , I2 = √ √ t 2 − x2 t 2 − x2 x x+1 Tϕ (T) x(g(x)/x) = − √ + T 2 − x2
T/x
(2.17)
(x+1) √ and take the integral for I2 by parts. We have I2 = − (x+1)ϕ + (x+1)2 −x2 T ∂ t ∂ √ t √ x+1 ϕ (t) ∂t t2 −x2 dt. Hence, using ∂t t2 −x2 0 we obtain
|I2 |
(1 + x)ϕ ∞ √ + ϕ ∞ 1 + 2x
∞ x+1
√ ∂ (−t) dt 2ϕ ∞ 1 + x. √ 2 2 ∂t t − x
Now we substitute the estimate |I1 | ϕ ∞ (2.18) into (2.17). This gives (2.13).
x+1 x
√ tdt t2 −x2
(2.18)
√ ϕ ∞ 1 + 2x and
Remark 2.4 Note that the condition ϕ(T) = 0 is necessary for (2.12). If ϕ(T) = 0, then g(x) is unbounded due to the non-integral term in the inversion formula for the Abel equation. The term is O( √ϕ(T) ) for x ↑ T. Similarly, the condition T−x ϕ (T) = 0 is necessary for (2.13). Proof of Theorem 1.1 Compare (1.3) with the asymptotis (1.5). It is clear that 1 √ (2.19) μn = gq (xn ) + o n− 4 , xn = μ0n = 2n + 1, where the Schlömilch integral gq is given by (2.6). The proof is based on the fact that the set {xn }∞ n=0 becomes arbitrarily dense as n → ∞, 1 so that Riemann sums T xn+1 T Gν (xn , T)gq (xn )(xn+1 − xn ) approximate 1 T Gν (x, T)gq (x)dx, provided the integrand is smooth enough. Using the T 0 inversion formula for the Schlömilch integral, we choose Gν such that the last T expression tends to T1 0 q+ (t) cos νt dt as T → ∞. By Lemma 2.3, we have
T T T 2 Gν (x, T)dx Gν (x, T)gq (x)dx = q+ (t) dt √ π t x2 − t 2 0 0 T q+ (t)ϕν,T (t) dt. (2.20) = 0
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Let us show that the integrand in the left-hand side of (2.20) is sufficiently smooth. By Lemma 2.1, |gq (x)| C
Q∞ + q∞ , √ 1+x
|gq (x)| C
q∞ + q ∞ , √ 1+x
x 0, (2.21)
where C is an absolute constant. Similarly, since ϕν,T satisfies the hypothesis of Lemma 2.3, uniformly in T x √ |Gν (x, T)| C(1 + ν) x, where we used ϕν,T ∞ C,
√ ∂ ∂ x Gν (x, T) C(1 + ν)2 x, ϕν,T ∞
x 1, (2.22)
C(1 + ν) . Therefore, for any fixed 2
def
ν the function hT (x) = Gν (x, T)gq (x) and its x-derivative are uniformly bounded for x 1 and T x. Hence, for T → ∞ we have
1 T 1 T 1 1 ln T hT (xn )(xn+1 − xn )− hT (x) dx = O dx = O → 0, T T 0 T 0 x T xn+1 T
(2.23) where we used xn+1 − xn = O(x−1 n ). Now, by (2.23), (2.19) and the first estimate in (2.22), 1 1 T lim μn Gν (xn , T)(xn+1 − xn ) = lim Gν (x, T)gq (x) dx. T→∞ T T→∞ T 0
(2.24)
xn+1 T
Next we divide (2.20) by T and take the limit T → ∞. By (2.24) and T T
lim T1 0 q+ ϕν,T dt = lim T1 0 p(t) cos νt dt, this gives (1.4). T→∞
T→∞
Remark 2.5 We have (νx)−1 Gν (x, ∞) = Bessel function (see e.g. [6]).
2 π
∞ x
sin νt dt √ t2 −x2
= J0 (νx), where J0 is the
Acknowledgements The author is thankful to E. Korotyaev and S. Naboko for fruitful discussions and valuable advice.
References 1. Klein, M., Korotyaev, E., Pokrovski, A.: Spectral asymptotics of the harmonic oscillator perturbed by bounded potentials. Ann. Henri Poincare 6, 747–789 (2005) (arxiv.org/math-ph/0312066) 2. Besikovitch, A.S.: Almost Periodic Functions. Dover Publ Inc. (1954) 3. Gurarie, D.: Asymptotic inverse spectral problem for anharmonic oscillators. Comm. Math. Phys. 112, 491–502 (1987)
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4. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, vol. 1. Cambridge University Press (1927) 5. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Integrals and derivatives of fractional order and their applications (in Russian), Nauka i Tehnika, Minsk (1987) 6. Gradstein, I.S., Ryzhik, I.M.: Tables of integrals, sums, series and products (in Russian), GIFML, Moscow (1963)
Math Phys Anal Geom (2007) 10:205–225 DOI 10.1007/s11040-007-9027-2
Cuspons and Smooth Solitons of the Degasperis–Procesi Equation Under Inhomogeneous Boundary Condition Guoping Zhang · Zhijun Qiao
Received: 16 January 2007 / Accepted: 3 August 2007 / Published online: 26 September 2007 © Springer Science + Business Media B.V. 2007
Abstract This paper is contributed to explore all possible single peakon solutions for the Degasperis–Procesi (DP) equation mt + mx u + 3mux = 0, m = u − uxx . Our procedure shows that the DP equation either has cusp soliton and smooth soliton solutions only under the inhomogeneous boundary condition lim|x|→∞ u = A = 0, or possesses the regular peakon solutions ce−|x−ct| ∈ H 1 (c is the wave speed) only when lim|x|→∞ u = 0 (see Theorem √ 4.1). In particular, 1,1 we first time obtain the stationary cuspon solution u = 1 − e−2|x| ∈ Wloc of the DP equation. Moreover we present new cusp solitons (in the space of 1,1 Wloc ) and smooth soliton solutions in an explicit form. Asymptotic analysis and numerical simulations are provided for smooth solitons and cusp solitons of the DP equation. Keywords Soliton · Integrable system · Analysis · Traveling wave Mathematics Subject Classifications (2000) 35D05 · 35G30 · 35Q53 · 37K10 · 37K40 · 76B15 · 76B25
G. Zhang · Z. Qiao (B) Department of Mathematics, The University of Texas – Pan American, 1201 W University Drive, Edinburg, TX 78541, USA e-mail:
[email protected] G. Zhang Department of Mathematics, Morgan State University, 1700 E Cold Spring Ln, Baltimore, MD 21239, USA e-mail:
[email protected]
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1 Introduction The b -weight-balanced wave equation mt + mx u + bmux = 0,
m = u − uxx
(1.1)
was proposed in [8] in 2003, and recently has arisen a lot of attractive attention. This family is proved to be integrable when b = 2, 3 by using symmetry approach [11]. More deeper mathematical attributes on the CH equation was shown by the constraint method and r-matrix structure [14, 15], inverse spectral theory [3], Riemann–Hilbert problem [12], and the global conservative solution [1]. In an earlier paper [18], Qiao and Zhang discussed the traveling wave solutions for the b = 2 equation – the Camassa–Holm (CH) equation [2] under the inhomogeneous boundary condition lim|x|→∞ u = A (A is a nonzero constant), and found new soliton solutions both smooth and cusped. In the paper [19], Vakhnenko and Parkes presented the periodic and loop soliton solutions for the b = 3 equation – the Degasperis–Procesi (DP) equation [6] from the mathematical point of view. Their solutions were expressed in a implicit form. Later in [10], Lenells also studied the traveling wave solitary solutions of the DP equation, which decay to zero at both infinities, but did not give explicit soliton solutions, either. An important issue to study both the CH equation and the DP equation is to find their new solutions through investigating their intrinsic mathematical structures. The DP equation has Lax pair [7] (therefore is integrable), and is able to be extended to a whole integrable hierarchy of equations with parametric solutions under some constraints [13, 17]. Also, the DP equation admits the global weak solution and blow-up structure [9]. In this paper, we give all possible single peak soliton solutions of the DP equation mt + mx u + 3mux = 0,
m = u − uxx ,
(1.2)
through setting the traveling wave mode under the boundary condition u → A (A is a constant) as x → ±∞. Our procedure shows that the DP equation either has cusp soliton and smooth soliton solutions only under the boundary condition lim|x|→∞ u = A = 0, or possesses the regular peakon solutions only when lim|x|→∞ u = 0 (see Theorem 4.1). In particular, we first time obtain a stationary cuspon solution of the DP equation. Moreover we present new cusp solitons and smooth soliton solutions in an explicit form [see formulas (3.7), (5.11) and (5.16)]. Asymptotic analysis and numerical simulations are provided for smooth solitons and cusp solitons of the DP equation. In the literature [4], cusp soliton was also called breaking wave. Due to some complex notations and definitions, we shift the statement of our main result to Section 4 (see Theorem 4.1).
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2 Traveling Wave Setting Let us consider the traveling wave solution of the DP equation (1.2) through a generic setting u(x, t) = U(x − ct), where c is the wave speed. Let ξ = x − ct, then u(x, t) = U(ξ ). Substituting it into the DP equation (1.2) yields (U − c)(U − U ) + 3U (U − U ) = 0,
(2.1)
where U = U ξ , U = U ξ ξ , U = U ξ ξ ξ . If U − U = 0, then (2.1) has general solutions of U(ξ ) = c1 eξ + c2 e−ξ with any real constants c1 , c2 . Of course, they are the solutions of the DP equation (1.2). This result is not so interesting for us. On the other hand, the DP equation has the well-known peakon solution [8] u(x, t) = U(ξ ) = ce−|x−ct−ξ0 | (ξ0 = x0 − ct0 ) with the following properties U(ξ0 ) = c,
U (ξ0 −) = c,
U(±∞) = 0,
U (ξ0 +) = −c,
(2.2)
where U (ξ0 −) and U (ξ0 +) represent the left-derivative and the rightderivative at ξ0 , respectively. Let us now assume that U is neither a constant function nor satisfies U − U = 0. Then (2.1) can be changed to 3U (U − U ) = . U − U c−U Taking the integration twice on both sides leads to (U )2 = U 2 −
(2.3)
C2 + C1 , (c − U)2
(2.4)
where C2 = 0, C1 ∈ R are two integration constants. Let us solve (2.4) with the following boundary condition lim U = A,
(2.5)
ξ →±∞
where A is a constant. Equation (2.4) can be cast into the following ODE: (U )2 =
(U − A)2 [(U − c + A)2 − cA] . (U − c)2
(2.6)
The fact that both sides of (2.6) are nonnegative implies (U − c + A)2 − cA 0. If cA 0, we denote B1 = c − A +
√ cA,
B2 = c − A −
(2.7) √ cA.
(2.8)
Apparently, B1 B2 .
3 Smooth Solution and Weak Solution Let Ck () denote the set of all k times continuously differentiable functions p on the open set . Lloc (R) refers to be the set of all functions whose restriction
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1,1 1 on any compact subset is L p integrable. Hloc (R) and Wloc (R) stand for 1,1 1 2 2 1 1 Hloc (R) = {u ∈ Lloc (R)| u ∈ Lloc (R)} and Wloc (R) = {u ∈ Lloc (R)| u ∈ Lloc (R)}, respectively.
Definition 3.1 A function u(x, t) = U(x − ct) is said to be a single peak soliton solution for the DP equation (1.2) if U satisfies the following conditions (C1) U(ξ ) is continuous on R and has a unique peak point, denoted by ξ0 , where U(ξ ) attains its global maximum or minimum value; (C2) U(ξ ) ∈ C3 (R − {ξ0 }) satisfies (2.6) on R − {ξ0 }; (C3) U(ξ ) satisfies the boundary condition (2.5). Without losing the generality, from now on we assume ξ0 = 0. Lemma 3.2 Assume that u(x, t) = U(x − ct) is a single peak soliton solution of the DP equation (1.2) at the peak point ξ0 = 0. Then we have a) if cA < 0, then U(0) = c; b) if cA 0, then U(0) = c or U(0) = B1 or U(0) = B2 . Moreover, we have the following solutions classification: (1) if U(0) = c, then U(ξ ) ∈ C∞ (R), and u is a smooth soliton solution. (2) if U(0) = c and A = 0, then u is a cusp soliton and U has the following asymptotic behavior U(ξ ) − c = λ|ξ |1/2 + O(|ξ |), U (ξ ) = 1/2λ|ξ |−1/2 sign(ξ ) + O(1),
ξ → 0; ξ → 0;
√ 1 where λ = ± 2|c − A| A2 − cA. Thus U(ξ ) ∈ / Hloc (R). (3) if U(0) = c and A = 0, then u gives the regular peaked soliton ce−|x−ct| . Proof (3) is obvious. Let us prove (1), a), b) and (2) in order. (1) If U(0) = c, then U(ξ ) = c for any ξ ∈ R since U(ξ ) ∈ C3 (R − {0}). Differentiating both sides of (2.4) yields U ∈ C∞ (R). a) When cA < 0, if U(0) = c, by (1) we know that U is smooth and U (0) = 0. However, by (2.6) we must have U(0) = A, which contradicts the fact that 0 is the unique peak point. b) When cA 0, if U(0) = c, by (2.4) we know U (0) exists. So, U (0) = 0 since 0 is a peak point. But, by (2.6) we obtain U(0) = B1 or U(0) = B2 , since U(0) = A contradicts the fact that 0 is the unique peak point. (2) If U(0) = c and A = 0, then by the definition of the single peak soliton we have A = c, thus (U − c + A)2 − cA doesn’t contain the factor U − c. From (2.6) we obtain U − A U = sign(c − A) (3.1) (U − c + A)2 − cAsign(ξ ). U −c
Cuspons and smooth solitons of the Degasperis–Procesi equation
Let h(U) =
sign(c−A) √
(U−A)
(U−c+A)2 −cA
, then h(c) =
209
√1 , |c−A| A2 −cA
and
h(U)(U − c)dU =
sign(ξ )dξ.
(3.2)
Inserting h(U) = h(c) + O(U − c) into (3.2) and using the initial condition U(0) = c, we obtain h(c) (U − c)2 (1 + O(U − c)) = |ξ |, 2 thus
(3.3)
2 2 1/2 −1/2 U −c=± |ξ | (1 + O(U − c)) =± |ξ |1/2 (1 + O(U − c)) h(c) h(c) (3.4) which implies U − c = O(|ξ |1/2 ). Therefore we have 2 U(ξ ) = c ± ξ → 0, |ξ |1/2 + O(|ξ |) = c + λ|ξ |1/2 + O(|ξ |), h(c) 2 = ± 2|c − A| A2 − cA, λ=± h(c)
and U (ξ ) = (1/2)λ|ξ |−1/2 sign(ξ ) + O(1),
ξ → 0.
1 (R). So, U ∈ / Hloc
Let us rewrite (2.6) in the following form [U (U − c)]2 = (U − A)2 [(U − c + A)2 − cA].
(3.5)
Then, we have the following proposition. Proposition 3.3 If u(x, t) = U(x − ct) is a single peak soliton for the DP equation (1.2), then U must be a weak solution of (3.5) in the distribution sense. In this sense we say u is a weak solution for the DP equation (1.2). Proof By the asymptotic estimates in Lemma 3.2, we have U (U − c) is 1 , thus the left hand side of (3.5) bounded, which implies [U (U − c)]2 ∈ Lloc does make sense. Since U is bounded, the right hand side of (3.5) is also 1 in Lloc . Thus we may define the distribution function L(U) = [U (U − c)]2 − (U − A)2 [(U − c + A)2 − cA]. By the definition condition (C2), we know that suppL(U) ⊂ {0}. Thus L(U) must be a linear combination of Dirac function 1 δ(ξ ) and its derivatives. However the previous analysis shows L(U) ∈ Lloc (R). Therefore L(U) = 0.
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If u ∈ H 1 , the DP equation (1.2) can be equivalently cast into the following nonlocal conservation law form [1, 4] L(u) ≡ ut + uux + ∂x (1 − ∂x2 )−1
3 2 u 2
= 0.
(3.6)
However, if u ∈ / H 1 , (3.6) is no longer equivalent to the DP equation (1.2). Actually, through a direct calculation we can verify that the following function (a stationary cusp soliton) u(x, t) =
1,1 1 − e−2|x| ∈ Wloc (but ∈ / H1)
(3.7)
satisfies the DP equation (1.2) for any x = 0. However, the solution (3.7) does not solve the nonlocal conservation system (3.6) because L(u) = e−|x| sign(x).
Therefore it is impossible to find the cusp soliton of the DP equation (1.2) through investigating (3.6). Therefore, we conclude this section with the following remark. Remark 3.4 The nonlocal DP equation (3.6) is equivalent to the standard DP 1,1 equation (1.2) under the H 1 -norm instead of Wloc norm. In fact, the regular −|x−ct| 1 peakon u(x, t) = ce ∈ H (c is the wave speed), which is a stable solution of the CH equation [5], also satisfies equation (3.6) and √ both the nonlocal 1,1 the DP equation (1.2). But u(x, t) = 1 − e−2|x| ∈ Wloc is a solution of the DP equation (1.2) in the sense of our definition, but does not satisfy the nonlocal DP equation (3.6).
4 New Single Peak Solitons Lemma 3.2 (3) gives a classification for all single peak soliton solutions for the DP equation (1.2). In this section we will present all possible single peak soliton solutions and find some explicit solution in the case of specific c and A. We will discuss three cases: cA = 0, cA > 0 and cA < 0. 4.1 Case I: cA = 0 (1) If A = 0, then the only possible single peak soliton is the regular peakon soliton. (2) If A = 0 and c = 0, there is the following stationary cusp soliton solution 1,1 . u(x, t) = A 1 − e−2|x| ∈ Wloc
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4.2 Case II: cA > 0 By virtue of Lemma 3.2 any single peak soliton for the DP equation (1.2) must satisfy the following initial and boundary values problem (IBVP) ⎧ (U − A)2 (U − B1 )(U − B2 ) ⎪ 2 ⎪ ⎪ ; ⎨ (U ) = g(U) = (U − c)2 (4.1) U(0) ∈ {c, B1 , B2 }; ⎪ ⎪ ⎪ lim U(ξ ) = A. ⎩ |ξ |→∞
g(U) 0 and the boundary condition (2.5) imply U B1 ,
or U B2 ,
and (A − B1 )(A − B2 ) 0.
(4.2)
By (2.8), (4.2) is equivalent to (c − A)(c − 4A) 0.
(4.3)
Since A = 0, introducing the constant α = c/A yields (α − 1)(α − 4) 0,
(4.4)
which implies: 0 < α < 1;
α > 4;
α = 1;
α = 4.
From the standard phase analysis and Lemma 3.2 we know that if U is a single peak soliton of the DP equation, then U − B2 U−A sign(ξ ), (4.5) U =− (U − B1 ) U −c U − B1 and
U(0) =
max(c, B1 ), min (c, B2 ),
if U(0) is a minimum, if U(0) is a maximum.
Let U −c h(U) = (U − A)(U − B1 )
U − B1 , U − B2
(4.6)
(4.7)
then taking the integration of both sides of (4.5) leads to h(U)dU = −|ξ |. After a lengthy calculation of integral, we obtain (α = 4, i.e. c = 4A) c− A INT2 (U) − K ≡ H(U) − K, h(U)dU = sign(A − c)INT1 (U) − c − 4A (4.8)
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where
B + B
2
1
− x − (x − B1 )(x − B2 ) , INT1 (x) = ln 2
(4.9)
(A − B )(x − B )+(A − B )(x − B )+2(A − B )(A − B )(x − B )(x − B )
1 2 2 1 1 2 1 2 INT2 (x) = ln
, x− A
(4.10) and K is an arbitrary integration constant. Thus we obtain the implicit solution U defined by H(U) = −|ξ | + K,
(4.11)
where H(U) is very complicated. But its derivative h(U) is simple so that we may get all single peak soliton through our monotonicity analysis [18]. Apparently,
B −B 2
1 INT1 (B1 ) = INT1 (B2 ) = ln INT2 (B1 ) = INT2 (B2 ) = ln|B1 − B2 |.
, 2 So, for U(0) = B1 or B2 , the constant K0 = H(U(0)) is defined by ⎛ ⎞ √ c − A ⎠ ln cA − c − A ln2 ∈ R, (4.12) K0 = − ⎝sign(A) + c − 4A c − 4A and for U(0) = c,
K0 = sign(A − c)INT1 (c) −
c− A INT2 (c) ∈ R. c − 4A
(4.13)
4.2.1 Case II.1: 0 < α < 1 1.
If A > 0, then B2 < B1 < c < A and U B1 . By standard phase analysis, we have U(0) = c,
c < U < A,
and
H(U) = INT1 (U) −
c− A INT2 (U). c − 4A
(4.14)
H(U) is strictly decreasing on the interval [c, A), thus H1 (U) = H|[c,A) (U)
(4.15)
will give an single peak soliton. Apparently, H1 (c) = K0 ,
lim H1 (U) = −∞.
U→A
Therefore U 1 (ξ ) = H1−1 (−|ξ | + K0 ) is the solution satisfying U 1 (0) = c,
lim U 1 (ξ ) = A,
ξ →±∞
(4.16)
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and U 1 (±0) = ±∞. So, U 1 (ξ ) is a kind of cusp soliton solution. 2. If A < 0, then B1 > B2 > c > A and U B2 . A similar analysis gives c− A H(U) = −INT1 (U) − INT2 (U) (4.17) c − 4A is strictly decreasing on the interval (A, c]. Thus H1 (U) = H|(A,c] (U)
(4.18)
has the inverse denoted by U 1 (ξ ) = H1−1 (−|ξ | + K0 ). U 1 (ξ ) gives a kind of cusp soliton solution satisfying U 1 (0) = c,
U 1 (±0) = ∓∞.
lim U 1 (ξ ) = A,
ξ →±∞
4.2.2 Case II.2: α > 4 1. If A > 0, then c > 4A and A < B2 < c < B1 ,
U(0) = B2 ,
A < U B2 .
Since H(U) is strictly increasing on the interval (A, B2 ], H2 (U) = H|(A,B2 ] (U)
(4.19)
gives a smooth soliton solution. Moreover, U 2 (ξ ) = H2−1 (−|ξ | + K0 ) is the unique soliton satisfying the IBVP (4.1) with U 2 (0) = B2 and U 2 (0) = 0. 2. If A < 0, then c < 4A and B2 < c < B1 < A,
U(0) = B1 ,
B1 U < A.
Through a similar analysis, we know that the strictly increasing on the interval [B1 , A) H2 (U) = H|[B1 ,A) (U)
(4.20)
gives a smooth soliton solution U 2 (ξ ) = H2−1 (−|ξ | + K0 ) satisfying the IBVP (4.1) with U 2 (0) = B2 and U 2 (0) = 0. 4.2.3 Case II.3: α = 1 In this case A = c, (2.6) becomes U = − U 2 − A2 sign(A)sign(ξ ),
U(±∞) = A.
A direct calculation shows that there is no solution for the above boundary condition.
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4.2.4 Case II.4: α = 4 If A > 0, then B1 = 5A, B2 = A, B2 < c < B1 and if A < 0, then B1 = A, B2 = 5A, B2 < c < B1 . For both subcases, there is no single peak soliton. 4.3 Case III: cA < 0 In this case, U(0) = c (see Lemma 3.2). Let us separate two subcases to discuss: (1) c < 0 < A and (2) c < 0 < A. (1) If A < 0 < c, we have c U < A and U − A U = − (U − c + A)2 − cAsign(ξ ). (4.21) U −c Let X = U − c + A,
p = A,
q = 2A − c,
r=
√ −cA,
then (4.21) becomes f (X)dX ≡
dX X−p = −sign(ξ )dξ. √ X − q X 2 + r2
(4.22)
Integration of both sides of (4.22) gives F(X) = −|ξ | + K
(4.23)
where
F(X) = ln(X + X 2 − cA) − √ √ (c− A)
(2A−c)X −cA+ (c− A)(c−4A) X 2 −cA
− ln
+ln2 (c−4A) X +c−2A (4.24)
F(X) is strictly decreasing on the interval [A, 2A − c) and lim X→2A−c F(X) = −∞. Define F1 (X) = F|[A,2A−c) (X).
(4.25)
F1 (X) = K0 − |ξ |,
(4.26)
Then
where
K0 = F(X(0)) = F(A) = ln(A + A2 − cA ) + c− A +√ ln(2A + 4A2 − cA ) + ln2 ∈ R. (4.27) (c − A)(c − 4A)
Since F1 is a strictly decreasing function from [A, 2A − c) onto (−∞, K0 ] we can solve for X uniquely from (4.26) and obtain U(ξ ) = F1−1 (K0 − |ξ |) + c − A.
(4.28)
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215
It is easy to check that U satisfies U(0) = c,
lim U(ξ ) = A,
|ξ |→∞
U (0+) = ∞,
U (0−) = −∞.
Therefore, the solution U defined by (4.28) is a cusp soliton solution for the DP equation. (2) If A < 0 < c, we have A < U c and U − A (U − c + A)2 − cAsign(ξ ). U −c
U =
(4.29)
Similarly, we get a strictly decreasing function F(X) on the interval (2A − c, A] satisfying: F(X) = |ξ | + K
(4.30)
where F(X) is defined by equation (4.24). Let F1 (X) = F|(2A−c,A] (X),
(4.31)
then F1 is a strictly decreasing function from (2A − c, A] onto [K0 , ∞) so that we can solve for X and obtain U(ξ ) = F1−1 (|ξ | + K0 ) + c − A.
(4.32)
It is easy to check that U satisfies U(0) = c,
lim U(ξ ) = A,
|ξ |→∞
U (0+) = −∞,
U (0−) = +∞.
Therefore, the solution U defined by (4.32) is also a cusp soliton solution for the DP equation. Let us summarize our results in the following theorem. Theorem 4.1 Assume that the single peak soliton u(x, t) = U(x − ct) (without losing the generality, we assume 0 is the unique peak point of U) of the DP equation (1.2) satisfies the boundary condition (2.5). Then we have (1) if A = 0, the single peak soliton u(x, t) is only the following peakon u(x, t) = U(x − ct) = ce−|x−ct| , with the properties: U(0) = c,
U(±∞) = 0,
U (0+) = −c,
U (0−) = c;
(2) if A = 0, let α = c/A, then (a) if 1 α 4, there is no soliton for the DP equation (1.2); (b) if α < 0 (cA < 0), the single peak soliton can be uniquely expressed as (see Figs. 1 and 2) u(x, t) = U(x − ct) = F1−1 (K0 − sign(A)|x − ct|),
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Fig. 1 2D graphic for a cusp soliton with A = 1, c = −1
with the property: U(0) = c,
U (0+) = sign(A)∞,
U(±∞) = A,
U (0−) = −sign(A)∞,
where F1 and K0 are defined by (4.25) (if A > 0), (4.31) (if A < 0) and (4.27) respectively. In this case, the single peak soliton is a cusp soliton.
Fig. 2 2D graphic for a cusp soliton with A = −1, c = 1
Cuspons and smooth solitons of the Degasperis–Procesi equation
217
Fig. 3 2D graphic for a cusp soliton with A = 8, c = 5
(c) if α = 0 (c = 0), there is the following stationary cusp soliton 1,1 ; u(x, t) = A 1 − e−2|x| ∈ Wloc (d) if 0 < α < 1, the single peak soliton (see Figs. 3 and 4) of the DP equation (1.2) can be uniquely expressed as u(x, t) = U(x − ct) = H1−1 (−|x − ct| + K0 ), with the property: U(0) = c,
U (0+) = sign(A)∞,
U(±∞) = A,
U (0−) = −sign(A)∞,
where H1 and K0 are defined by (4.15) (if A > 0), (4.18) (if A < 0) and (4.13) respectively. In this case, the single peak soliton is a cusp soliton. (e) if α > 4, the DP equation (1.2) has the following traveling soliary wave solutions (see Figs. 5 and 6) u(x, t) = U(x − ct) = H2−1 (−|x − ct| + K0 ) with the properties: √ U(0) = c − A + sign(A) cA,
U(±∞) = A,
U (0) = 0,
where H2 and K0 are defined by (4.19)(if A > 0), (4.20)(if A > 0) and (4.12) respectively. In this case, the soliton solution is smooth.
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Fig. 4 2D graphic for a cusp soliton with A = −8, c = −5
5 Explicit Solutions In the previous section we constructed all possible single peak soliton solutions in our main theorem (Theorem 4.1). But, usually it is very hard to find an explicit formula of the solution based on the implicit functions H(U) we
Fig. 5 2D graphic for a smooth soliton with A = 1, c = 5
Cuspons and smooth solitons of the Degasperis–Procesi equation
219
Fig. 6 2D graphic for a smooth soliton with A = −1, c = −5
obtained. However, in some specific cases, we do have explicit solutions. For this purpose, let us rewrite (4.5) as follows U −c U − B1 dU = −sign(ξ )dξ. (5.1) (U − A)(U − B1 ) U − B2 Let
X=
U − B1 , U − B2
a=
A − B1 , A − B2
then U = B2 −
B1 − B2 , X2 − 1
dU =
2(B1 − B2 )XdX , (X 2 − 1)2
and (5.1) is converted to dX dX dX dX r − + − = −sign(ξ )dξ, X −a X +a X +1 X −1
(5.2)
where r=
c− A . a(B2 − A)
Taking integration on both sides, we arrive at
X − a
r
X + 1 −|ξ |
X + a X − 1 = C0 e ,
(5.3)
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where C0 is a positive constant. In the following, we try to find explicit formulas of cusp soliton and smooth soliton for some specific number r. Case 1 0 < α = c/A < 1
In this case, the solution U is a cusp soliton.
1. If A > 0, then we have 0 < c < A,
B2 < B1 < c < U < A,
U(0) = c,
therefore 0 < X < a < 1. Equation (5.3) becomes a− X r 1+ X = C0 e−|ξ | , a+ X 1− X where C0 =
a − X(0) a + X(0)
r
(5.4)
1 + X(0) , 1 − X(0)
X(0) =
√ cA √ . A + cA A−
For general r, (5.4) is not algebraically solvable. But, a specific value of r may make (5.4) solvable for X. To this end let us write r as follows A−c 1−α = r=√ , 0 < α < 1. 4−α (c − A)(c − 4A) A simple analysis shows that the range of r is 0 < r < 1/2. 2. If A < 0, a similar analysis yields X > a > 1, and
X −a X +a
r
X +1 X −1
= C0 e−|ξ | ,
where C0 =
X(0) − a X(0) + a
r
X(0) + 1 , X(0) − 1
(5.5)
X(0) =
√ cA √ A + cA A−
The range of r is 0 < r < 1/2. Notice that r = 1/2 corresponds α = 0 (c = 0). In this case, there is an explicit stationary cusp soliton u(x, t) = A 1 − e−2|x| . If taking r = 1/3, then we obtain √ 11 − 2 10sign(A) c = 5/8A, a= , 9
√ √ 2 6 + 15sign(A) X(0) = , 3
Cuspons and smooth solitons of the Degasperis–Procesi equation
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and (X − a)(X + 1)3 = b (ξ )(X + a)(X − 1)3 ,
(5.6)
where b (ξ ) = C03 e−3|ξ | . Equation (5.6) is able to be algebraically solvable. But that is a very complicated procedure. We omit it here. Case 2 α = c/A > 4 In this case, the solution U is a smooth soliton. 1. If A > 0, then we have A < U < B2 < c < B1 ,
c > 4A > 0, X > a > 1,
U(0) = B2 ,
X(0) = ∞,
and
X −a X +a
r
X +1 X −1
−|ξ |
=e
,
r=
α−1 , α−4
1 < r < ∞.
(5.7)
Let us choose r = 2, then c = 5A,
√ 3+ 5 a= , 2
and (5.7) becomes (X − a)2 (X + 1) = e−|ξ | (X + a)2 (X − 1). Substituting a =
√ 3+ 5 2
into (5.8), we obtain
√ √ √ 7+3 5 1+ 5 2 X+ b = 0, X − (2 + 5)bX + 2 2 3
(5.8)
b=
1 + e−|ξ | . 1 − e−|ξ | (5.9)
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With the help of Maple it is easy to check (5.9) has the following real root ⎡ √ 7+3 5 ⎣ X(ξ ) = − b+ 3 ⎤1/3 √ √ √ √ 2 + 5 517 + 231 5 2 521 + 233 5 4 ⎦ 38 +17 5 3 b+ + b − b + + 27 27 54 54 ⎡
√ √ 7+3 5 38 + 17 5 3 ⎣ + − b+ b − 3 27
⎤1/3 √ √ √ √ 2 + 5 517 + 231 5 2 521 + 233 5 4 ⎦ 2+ 5 − + b − b b. + 27 54 54 3 (5.10) Hence, we obtain an explicit formula of the smooth soliton solution (see Fig. 7) √ √ 2 5 U(ξ ) = A (4 − 5) − , A > 0, (5.11) X(ξ )2 − 1 where X(ξ ) is defined by (5.10).
Fig. 7 2D graphic for a smooth soliton with A = 1, c = 5
Cuspons and smooth solitons of the Degasperis–Procesi equation
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2. If A < 0, then we have A > U > B1 > c > B2 ,
c < 4A < 0, 0 < X < a < 1,
U(0) = B1 ,
X(0) = 0,
and
a− X a+ X
r
1+ X 1− X
−|ξ |
=e
,
r=
α−1 , α−4
1 < r < ∞.
(5.12)
Let us take r = 2, then c = 5A,
a=
√ 3− 5 . 2
In a similar way, we obtain √ √ √ 1− 5 7−3 5 2 X + ( 5 − 2)bX + X+ b = 0, 2 2 3
(5.13)
where b=
1 − e−|ξ | . 1 + e−|ξ |
Fig. 8 2D graphic for a smooth soliton with A = −1, c = −5
(5.14)
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Solving (5.13) leads to ⎡ √ √ 7−3 5 38 − 17 5 3 2⎣ X(ξ ) = ω − b+ b + 3 27 +
⎤1/3 √ √ √ 2 − 5 517 − 231 5 2 521 − 233 5 4 ⎦ + b − b + 27 54 54
⎡
√ √ 7 − 3 38 − 17 5 5 3 + ω ⎣− b+ b − 3 27
⎤1/3 √ √ √ 2 − 5 517 − 231 5 2 521 − 233 5 4 ⎦ + b − b − + 27 54 54 √ 2− 5 b, 3
+
(5.15) √ −1+ 3i . 2
where b is defined by (5.14) and ω = Thus we obtain another explicit form of smooth soliton solution (see Fig. 8) √ √ 2 5 , A < 0, (5.16) U(ξ ) = A (4 + 5) + X(ξ )2 − 1 where X(ξ ) is defined by equation (5.15). If we take r = 3, then 35 A, c= 8
√ 19 + 2 70sign(A) a= . 9
We can repeat the above procedure to get explicit soliton solutions corresponding to r = 3. This is left for reader’s practice.
6 Conclusions In this paper, we investigate the DP equation under the inhomogeneous boundary condition. Through the traveling wave setting, the DP equation is converted to the ODE (2.6), which we solve for all possible single soliton solutions of the DP equation. Actually, the ODE (2.6) has a physical meaning and can be cast into the Newton equation U 2 = V(U) − V(A) of a particle with a new potential V(U) V(U) = U 2 +
A(c + A)(c − A)2 4cA(c − A) + . U −c (U − c)2
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In the paper, we successfully solve the Newton equation U 2 = V(U) − V(A) and give a single peak cusp [including a stationary cusp soliton, see (3.7)] and smooth soliton solutions in an explicit formula [see (5.11) and (5.16)]. Our smooth solutions (5.11) and (5.16) are orbitally stable, but we do not know if our new cuspon (4.32) and the cuspon, defined by equation (4.18), are stable. Very recently, we found a new integrable equation with no classical (smooth) soliton, only possessing weak solutions, such as cuspons and W/M-shape peak solitons, see the details in paper [16].
Acknowledgement UTPA-URI.
Qiao’s work was partially supported by the UTPA-FRC and the
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Math Phys Anal Geom (2007) 10:227–236 DOI 10.1007/s11040-007-9029-0
Topological Classification of Morse Functions and Generalisations of Hilbert’s 16-th Problem Vladimir I. Arnold
Received: 30 August 2007 / Accepted: 18 September 2007 / Published online: 3 January 2008 © Springer Science + Business Media B.V. 2007
Abstract The topological structures of the generic smooth functions on a smooth manifold belong to the small quantity of the most fundamental objects of study both in pure and applied mathematics. The problem of their study has been formulated by A. Cayley in 1868, who required the classification of the possible configurations of the horizontal lines on the topographical maps of mountain regions, and created the first elements of what is called today ‘Morse Theory’ and ‘Catastrophes Theory’. In the paper we describe this problem, and in particular describe the classification of Morse functions on the 2 sphere and on the torus. Keywords Classification of maps · Morse functions Mathematics Subject Classifications (2000) 57R99 · 58D15 · 58E05
The topological structures of the generic smooth functions on a smooth manifold belong to the small quantity of the most fundamental objects of study both in pure and applied mathematics. The problem of their study has been formulated by A. Cayley in 1868, who required the classification of the possible configurations of the horizontal lines on the topographical maps of mountain regions, and created the first elements of what is called today ‘Morse Theory’ and ‘Catastrophes Theory’.
V. I. Arnold (B) Steklov Mathematical Institut, Moscow, Russia e-mail:
[email protected] V. I. Arnold Université Paris Dauphine, Paris, France
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M. Morse has told me, in 1965, that the problem of the description of the possible combinations of several critical points of a smooth function on a manifold looks hopeless to him. L.S. Pontrjagin and H. Whitney were of the same opinion. I formulate below some recent results in this domain. Thus the classification of the Morse functions on a circle S1 leads to the Taylor coefficients of the tangent function. On the two-dimensional sphere the number of topologically different Morse functions with T saddle points (that is, having 2T + 2 critical points) grows with T as T 2T . The tangent function is replaced in this study by some elliptic integral (discovered by L. Nicolaescu, while he was continuing Arnold’s calculation of the number of topologically different Morse functions having 4 saddle points on the sphere S2 —that number is equal to 17 746). Replacing the 2-sphere by the two-dimensional torus, one obtains an infinite number of topologically different Morse functions with a fixed number of critical points, provided that the topological equivalence is defined by the action of the identity connected component of the group of diffeomorphisms (or homeomorphisms), that is, if the corresponding mappings of the torus are supposed to be homotopic to the identity map. However, if one accepts mappings permuting the parallels and the meridians of the torus, the classification becomes finite and similar to that of the Morse functions on the 2-sphere. The topological classification of the functions on the 2-sphere is related to one of the questions of the 16-th Hilbert’s problems (on the arrangements of the planar algebraic curves of fixed degree, defined on the real plane R2 ). A real polynomial of fixed degree, defined on the real plane R2 , generates a smooth function on S2 (with one more critical point at infinity), and the topological structure of these functions on the 2-sphere influences the topological properties of the arrangements of the real algebraic curves where the initial polynomials vanish. It seems that only a small part of the topological equivalence classes of the Morse functions having T saddle points on S2 , contains polynomial representatives of corresponding degree. For instance, in the case of 4 saddles (T = 4) the polynomials are of degree 4 in two real variables, and provide less that 1000 topological types, from the total number of different topologic types, which is 17 746: the majority of the classes of smooth Morse functions with 4 saddle points have no polynomial representatives. The classification of the Morse functions on the torus is related similarly to the topological investigation of the trigonometric polynomials. The degree of a polynomial is replaced in the case of the functions on T 2 by the Newton polygon of a trigonometric polynomial (which is the convex hull of the set of the wave-vectors of the harmonics forming the trigonometric polynomial). When such a convex polygon is fixed, its trigonometric polynomials (formed by the harmonics whose wave-vectors belong to the Newton polygon) realize only a finite subset of the infinite set of the topologically different classes of functions, classified up to those transformations of the torus which are homotopic to the identity map. This finite subset depends on the Newton
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polygon, and is small when the polygon is small. It is finite for the following reason: In the torus transformations needed to reduce the initial trigonometric polynomial to the finite set of classes, discussed above, parallels and meridians are sent only to linear combinations of the parallel and meridian classes whose coefficients are not large. The number of the homotopy classes of such transformations of the torus is finite, making finite the resulting set of topological equivalence classes of trigonometric polynomials with fixed Newton polygon (classified up to the torus transformations homotopic to the identity map). Definition 1 The graph of a Morse function f : M → R on a manifold M is the space whose points are the connected components of the level hypersurfaces f −1 (c) ⊂ M. This space is a finite complex (at least when M is compact). Example 1 The twin peak mountain Elbrouz (whose height function f is considered continued to the sphere S2 with a minimum D at the antipodal point) generates, as the graph of the height function f , the character ‘Y’ (Fig. 1). The topographical ‘bergshtrechs’ at the horizontals map show the antigradient vector directions (followed by the grass of the mountain after the rain). Example 2 The volcanic Vesuvius mountain has a maximum height point A, a height local minimum point B (crater), and a saddle point C, defining the graph of Fig. 2. The vertices of the graphs (the end-points A, B, D and the triple point C) represent the critical points of the function f . They form a finite set for a Morse function on a compact manifold M. We see that the graphs of the two preceding examples are topologically equivalent, while the two corresponding functions are not. Taking this into account, we will distinguish the graphs, ordering the vertices by the critical values of the height.
D Fig. 1 The twin peak mountain Elbrouz
D
D
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D D
D
Fig. 2 The volcanic Vesuvius mountain
For instance, for a Morse function with n critical points, we may fix the n critical values to be 1 < 2 < · · · < n. Then the first (Elbrouz) graph would obtain the numbering of its vertices { f (D) = 1, f (C) = 2, f (B) = 3, f (A) = 4} and the second (Vesuvius) graph’s vertex numbering is { f (D) = 1, f (B) = 2, f (C) = 3, f (A) = 4} Therefore we will consider the graphs of the Morse functions to be labeled by the above ‘height’ numbering (1, 2, . . . , n) of the vertices (ordering then maxima, minima, and saddle points). Examples 1 and 2 show that the heights of the neighbours of a triple vertex can’t be all three higher or all three lower than the triple vertex itself (Fig. 3). The ordering of the vertices verifying this restriction, will be called regular. The regularly ordered graph is a topological invariant of a Morse function. For Morse functions f on the spheres (M = Sm , m > 1, f : M → R) the graphs are trees. For Morse functions on a surface of genus g, the graph has g independent cycles (1 for the torus surface T2 , two for the sphere with two handles, three for the bretzel surface and so on). In the case of the sphere of dimension 2 this regularly ordered graph is the only topological invariant of a Morse function f : S2 → R (up to diffeomorphisms or up to homeomorphisms of S2 , if we fix the critical values to be {1, 2, . . . , n}). All regular orderings (of any tree) with n vertices are realized as the graphs of such Morse functions. Counting (in 2005) the number ϕ(T) of such regularly ordered trees with T triple points (and of diffeomorphism classes of Morse functions having
Fig. 3 Possible, impossible cases
Possible cases
Impossible cases
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values {1, 2, . . . , n} at their 2T + 2 critical points, including the T saddle points) Arnold proved the following: Theorem 1 There exists positive constants a and b such that for any T the inequalities aT T < ϕ(T) < b T 2T hold. The article [1] claims that the upper bound of Theorem 1 is closer to the genuine asymptotics of the function ϕ, mentioning some arguments for this conjecture, based on (unproved) ergodic properties of random graph ordering. This conjecture was then proved by L. Nicolaescu [6] who replaced the recurrent inequality used by Arnold to prove his upper bound theorem by an exact nonlinear recurrent relation (similar to the Givental’s mirror symmetry proof in quantum field theory). Nicolaescu’s recurrent relation involves 43 terms of which Arnold had used only the first leading term to prove his inequality. Using the computer, Nicolaescu solved this nonlinear recurrence explicitly, representing ϕ in terms of the coefficients of the power series expansion of some elliptic integral. Studying the behaviour in the complex domain of these integrals, Nicolaescu succeeded to prove the Arnold conjecture on the growth of ϕ (leaving unproved however, the conjecture on the ergodic theory of random graphs which had led Arnold to his conjecture on the asymptotics of ϕ). The first values of the function ϕ (calculated in Arnold [1]) are T
1
2
3
4
ϕ(T)
2
19
428
17 746
Already the easy computation of ϕ(2) = 19 provides some of the ideas of the general structure theory. According to Nicolaescu’s computer ϕ(5) = 1178792. Arnold was unable to draw all the shapes of functions with T = 5 saddles, drawing only those with T = 4. 10 In the case T = 5 the upper bound T 2T = 10 ≈ 1010−3 is only 10 times 2 higher than the exact value provided by Nicolaescu. A Morse polynomial f of degree m in two variables has at most (m − 1)2 critical points. Let m = 2k be even. Suppose that the leading form at infinity (of degree m) is positive. In this case the polynomial provides a Morse function f : S2 → R having exactly 4k2 − 4k + 2 critical points (taking that at infinity into account). The relation 2T + 2 = 4k2 − 4k + 2 provides the value T = 2k(k − 1) for the number T of the saddles. For the polynomials of degree m = 4 we find k = 2 and T = 4. Therefore, the topological types of these 4-th degree polynomials (with 9 critical points in R2 ) are included in the set of 17 746 classes of topologically equivalent smooth Morse functions on the sphere S2 , with T = 4 saddles. The topological equivalence is defined here by the actions of the homeomorphisms (or diffeomorphisms) both of the sphere, where the functions are
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defined, and the orientation preserving homeomorphisms of the axis of values. One also needs the mappings of the axis of values, if the critical values are not fixed e.g. as {1, 2, . . . , 9}, but may move. According to my calculations, among the 17 746 topologically different classes of the Morse functions on S2 with 4 saddles at most, one thousand of classes have a polynomial representative of degree 4. I have no conjecture on the growth rate of the number of classes realizable by polynomials of the corresponding degree, for a growing value of the number T of saddles. A trigonometric Morse polynomial of degree n has at most 2n critical points on the circle S1 . For the case where all the 2n critical values are different, the table (from ‘serpent’ theory [4]) provides the following numbers ϕ of the topologically different functions f : S1 → R with 2n critical points (considered up to the orientation preserving homeomorphisms or diffeomorphisms of S1 and of R) 2n
2
4
6
8
10
ϕ
1
2
16
272
7 936
Every such class of smooth functions includes some trigonometric polynomial of degree n. Theorem 2 The numbers ϕ(n) of classes of the oriented topological equivalence of Morse functions with 2n critical points and 2n critical values on the circle, provide the Taylor series of the tangent function (as of the exponential generating function): tan t =
1 2 16 272 7 7936 9 t + t3 + t5 + t + t + ... 1! 3! 5! 7! 9!
We consider next the trigonometric polynomials of two variables as some smooth functions on the 2-torus, f : T2 → R. 2 affine Coxeter group of trigonometric polynomials of Example 3 The A degree 1 form the following 6-parameter vector space of trigonometric polynomials in two variables x and y: (∗) f = a cos x + b sin x + c cos y + d sin y + p cos(x + y) + q sin(x + y). This family of trigonometric polynomials has been studied in the articles by Arnold [2, 3]. The number of critical points of such Morse polynomials (∗) on T2 does not exceed 6. In the general case of the arbitrary trigonometric polynomials the upper bound of the number of critical points on T2 is provided by the doubled area of the Newton polygon (n! the volume of the Newton polyhedron for the n-torus case Tn ) (Fig. 4).
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Fig. 4 Newton polygon
T∗T2
area = 3 saddles number, T = 3
Newton polygon
Definition 2 The Diff-equivalence of smooth functions on T2 is the belonging to the same orbit of the natural action on functions on the torus of the group Diff (T2 ) of the diffeomorphisms of the torus, accompanied by the orientation preserving diffeomorphisms of the axis of the values. The Diff 0 -equivalence is defined similarly, replacing, however, the group Diff(T2 ) by its connected subgroup Diff 0 (T2 ), formed by the diffeomorphisms homotopic to the identity, which is the connected component of the identity in Diff(T2 ) (whose elements may interchange the meridian and parallel classes). The articles by Arnold [2, 3] contain the proof of Theorem 3 The number of equivalence classes of the C∞ Morse functions with 6 critical points on the two-dimensional torus T2 and of the trigonometric 2 have the following values: polynomials (∗) of the affine Coxeter group A Diff
Diff 0
C∞
16
∞
2 A
2
6
The 16 topologically different types of the smooth Morse functions with 6 critical points and values on the torus are described in Arnold [3] by the 16 regularly ordered graphs having 3 triple points (), three end points () and one cycle (for each graph). These 16 graphs, ordered by the natural height function, are shown in Fig. 5. Among all these topological equivalence classes only two classes (marked by the sign (∗)) contain some of the trigonometric polynomials. All the other 14 cases are eliminated by the algebraic geometry of the elliptic level curves of functions (∗), as is explained in articles by Arnold [2, 3]. This finite set of the 16 Diff -equivalence classes provides an infinity of different Diff 0 -classes for the following reason. A generic point P on the cycle of the graph does not separate the graph. Hence the level line component, represented by P, does not separate the torus T2 into two parts, and therefore this closed curve is not homologous to zero on the torus. Its homology class does not depend on the choice of the point P on the cycle, since any pair of such points {P, P} separates the graph into two parts, and hence the cycle P − P is the boundary of a domain on the torus. This 1-dimensional homology class is an invariant of the Diff 0 -equivalence (at least up to the choice of its sign).
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Fig. 5 The 16 graphes
For suitable choices of Morse functions with 6 critical points and values on the torus, it can take an infinity of different values. All these 1homology classes are Diff -equivalent, each of them being represented by a non-bounding, non-self-intersecting closed curve on the torus. Thus we obtain an infinite set of the Diff 0 -equivalence classes of the Morse functions having 6 critical points and 6 critical values on the 2-torus T2 . However the trigonometric polynomials (∗) provide only a finite part of this infinite set of the Diff 0 -equivalence classes. Namely, in Arnold [2] it is proved that the algebraic geometry of real elliptic curves restricts the homology class of the level line component P on the torus: it may attain only 3 values (or 6 if we take the orientation into account). These 6 realizable classes form in the one-dimensional homology group of 2 the torus the Dynkin diagram of the Coxeter group A2 (whence the name A of the class (∗) of trigonometric polynomials) (Fig. 6). For the other classes (say defined by arbitrary Newton polygons) the number of Diff -equivalence and Diff 0 -equivalence classes are unknown. I do not know how fast the number of Diff -equivalence classes grows when the number T of saddle points is large (neither for the smooth Morse functions, nor for the trigonometric polynomials having a fixed Newton polygon). Even
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Fig. 6 H1 (T2 , Z)
the finiteness proof for the number of Diff -equivalence classes (for each value of T) is not published. This abundance of unsolved problems was the reason to choose these questions of real algebraic geometry and of Morse functions statistics for this paper: I hope that the readers will go further than me in this rapidly evolving domain at the intersection of all the branches of mathematics. It is rather strange that the computer contribution to real algebraic geometry is still almost negligible while theoretical mathematicians have made a lot. The only real contribution that I know, is the recent result of a former student of the Université Paris-Jussieu, Adriana Ortiz-Rodriguez, working at Mexico (and having started these works in Paris). This result describes the topology of the parabolic curves on algebraic surfaces of the projective space, and belongs to the intersection of real algebraic geometry and symplectic geometry. The graph {z = f (x, y)} of a polynomial f in two real variables x and y is a smooth surface in R3 . The parabolic lines of this surface are projected to the {(x, y)}-plane as algebraic curves of degree 2n − 4. The question studied by A. Ortiz-Rodriguez is to evaluate the possible numbers of closed parabolic curves for polynomials of a given degree: how large may this number of parabolic curves be? And how large may be the number of connected components of the corresponding algebraic curve of degree 2n − 4 in the real projective plane? The classical Harnack theorem of real algebraic geometry implies that for the polynomial f of degree n = 4 the number M of closed parabolic curves cannot exceed 4. It is not too difficult to construct examples of polynomials f of degree 4, for which the number of closed parabolic curves attains the value M = 3. But the problem of wether the case of 4 closed parabolic curves is realizable by some polynomial of degree 4 resisted to all the attempts of mathematicians, and only the computer helped to solve it. Namely, in a year of uninterrupted calculations, the Mexico computer of A. Ortiz-Rodriguez has studied 50 millions of polynomials of degree 4. Among all these polynomials, just 3 polynomials have, each of them, 4 closed parabolic curves on their graphs.
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For the polynomials f of degree n, A. Ortiz-Rodriguez has proved (in her thesis, preceding the experiment described above) the inequalities an2 M bn2 for the maximal number M(n) of closed parabolic curves on the graph of a polynomial of degree n: for any such polynomial, the number of parabolic curves is at most bn2 , and there exist polynomials of degree n for which the parabolic curve number is at least an2 . Unfortunately, b is higher than a, and thus the question of the genuine asymptotics of M(n) is waiting for the courageous researchers (and computer experiments). The situation is even more difficult with another similar problem (also studied by A. Ortiz-Rodriguez). Consider in the 3-dimensional projective space R P3 a smooth algebraic surface of degree n. How large may be the number of its closed parabolic curves? Here the Ortiz-Rodriguez boundaries are An3 M Bn3 , but the coefficient B is a dozen of times higher than A and the genuine asymptotic behaviour of the maximal number M(n) of the connected components of the parabolic curves on smooth real projective hypersurfaces of degree n remains unknown. These real algebraic geometry versions of the higher dimensional Plücker formulae remain a challenge for modern algebraic geometry, which seems, however, to be unable to study the real things. Of course the celebrated Tarski-Seidenberg theorem implies the existence of an algorithm providing the needed answers, but the required time for the real application is usually much larger than the whole life span of the universe.
References 1. Arnold, V.I.: Smooth functions statistics. Funct. Anal. Other Math. 1(2), 125–133 (2006) (ICTP Preprint IC2006/012, 1–9 (2006)) 2. Arnold, V.I.: Topological classifications of trigonometric polynomials, related to affine Coxeter group A˜ 2 . ICTP Preprint IC2006/039, 1–15 (2006) 3. Arnold, V.I.: Dynamical systems: modeling, optimization, and control. Proc. Steklov Inst. Math. Suppl. 1, 13–23 (2006) 4. Arnold, V.I.: Snake calculus and the combinatorics of the Bernoulli, Euler and Springer numbers. Russian Math. Surveys 47(1), 1–51 (1992) 5. Arnold, V.I.: Topological classification of real trigonometric polynomials and cyclic serpents polyhedron. The Arnold-Gelfand Seminars, pp. 101–106. Boston, Birkhäuser (1996) 6. Nicolaescu, L.I.: Morse functions statistics. Funct. Anal. Other Math. 1(1), 97–103 (2006) (Counting Morse functions on the 2-sphere, Preprint math.GT/0512496)
Math Phys Anal Geom (2007) 10:237–249 DOI 10.1007/s11040-007-9030-7
Limit Cycles Bifurcating from a k-dimensional Isochronous Center Contained in Rn with k n Jaume Llibre · Marco Antonio Teixeira · Joan Torregrosa
Received: 12 January 2007 / Accepted: 16 October 2007 / Published online: 7 December 2007 © Springer Science + Business Media B.V. 2007
Abstract The goal of this paper is double. First, we illustrate a method for studying the bifurcation of limit cycles from the continuum periodic orbits of a k-dimensional isochronous center contained in Rn with n k, when we perturb it in a class of C 2 differential systems. The method is based in the averaging theory. Second, we consider a particular polynomial differential system in the plane having a center and a non-rational first integral. Then we study the bifurcation of limit cycles from the periodic orbits of this center when we perturb it in the class of all polynomial differential systems of a given degree. As far as we know this is one of the first examples that this study can be made for a polynomial differential system having a center and a non-rational first integral. Keywords Limit cycle · Periodic orbit · Center · Isochronous center · Averaging method · Generalized Abelian integral Mathematics Subject Classifications (2000) 34C29 · 34C25 · 47H11
The first and third authors are partially supported by a MCYT/FEDER grant MTM2005-06098-C01, and by a CIRIT grant number 2005SGR-00550. The second author is partially supported by a FAPESP–BRAZIL grant 10246-2. The first two authors are also supported by the joint project CAPES–MECD grant HBP2003-0017. J. Llibre · J. Torregrosa (B) Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain e-mail:
[email protected] J. Llibre e-mail:
[email protected] M. A. Teixeira Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, São Paulo, Brazil e-mail:
[email protected]
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1 Introduction and Statement of the Results We say that a singular point p ∈ Rn of a differential system is a k-dimensional center if there exists a k-dimensional submanifold M of Rn with k n such that p ∈ M, M is invariant under the flow of the differential system, and all the orbits M \ { p} are periodic. Moreover, we say that the k-dimensional center p is isochronous if all its periodic orbit have the same period. In the first part of this paper we illustrate a method for studying the limit cycles bifurcating from the periodic orbits of a k-dimensional isochronous center contained in Rn with k n, by studying with all the details an example with k = 2 and n = 4. In recent years equations of the form x IV + bx I I + ax = ψ t, x, x I , x I I , x I I I arise in many contexts. For example, the simplest cases when ψ = x2 and ψ = x3 describe the travelling waves solutions of some Korteweg–de Vries equations (KdV) and nonlinear Schrödinger equations, see [3, 8, 9]. On the other hand, the existence of periodic solutions is discussed in [4] for equations modeling undamped oscillators and having the form x I I + ω02 x = φ(t) where ω0 > 0, and φ is a continuous periodic function whose period is normalized to 2π . In this paper we deal with a particular differential equation of order four of the form d4 x + α x + ψ(x, t) = 0. dt4 This class of equations have been studied in Peletier and Troy [10] and Sanchez [12]. Here we will analyze the particular differential equation d4 x − x − ε sin(x + t) = 0, dt4 or equivalently the differential system x˙ = y, y˙ = z, z˙ = w, w˙ = x + ε sin(x + t),
(1)
where the dot denotes the derivative with respect to the time variable t. Our main result is the content of the following theorem. Theorem 1 For |ε| = 0 sufficiently small the differential system (1) has an arbitrary number of limit cycles bifurcating from the continuum of the periodic orbits of the 2-dimensional isochronous center that the system has for ε = 0. The proof of Theorem 1 is given in Section 3, and uses the averaging theory, more precisely the proof uses Theorem 3.
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In the second part we deal with the homogeneous polynomial differential system x˙ = −y(3x2 + y2 ), y˙ = x(x2 − y2 ),
(2)
of degree 3 that has the non-rational first integral 2 2x2 2 H(x, y) = x + y exp − 2 . x + y2 Theorem 2 The homogeneous polynomial differential system (2) has a global center at the origin (i.e. all the orbits contained in R2 \ {(0, 0)} are periodic). Let P(x, y) and Q(x, y) be two polynomials of degree at most m. Then, for convenient polynomials P and Q, the polynomial differential system x˙ = −y 3x2 + y2 + ε P(x, y), y˙ = x x2 − y2 + ε Q(x, y), (3) has [(m − 1)/2] limit cycles bifurcating from the periodic orbits of the global center (2), where [·] denotes the integer part function. As far as we know this is one of the first examples for which the limit cycles bifurcating from the periodic orbits of a 2-dimensional center of a polynomial differential system having a non-rational first integral have been studied. The unique other example that we know was given recently in [6]. The proof of Theorem 2 is given in Section 4. Again we use averaging. More precisely, we will apply Theorem 4, which gives a method to determine bifurcation of periodic solutions from isochronous centers. We note that the center of system (2) is not isochronous; but, after a change of variables, it can be transformed to an isochronous center. We also show in Section 4 that Theorem 2 can be proved using the theory based on the generalized Abelian integrals, see a definition of these integrals at the end of Section 2.
2 Basic Results In this section first we present the basic results from the averaging theory and Abelian integrals that we shall need for proving the main results of this paper. We consider the problem of the bifurcation of T-periodic solutions from the differential system x (t) = F0 (t, x) + εF1 (t, x) + ε2 F2 (t, x, ε),
(4)
with ε = 0 to ε = 0 sufficiently small. Here, the functions F0 , F1 : R × → Rn and F2 : R × × (−ε0 , ε0 ) → Rn are C 2 functions, T-periodic in the first
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variable, and is an open subset of Rn . One of the main assumptions is that the unperturbed system x (t) = F0 (t, x),
(5)
has a submanifold of periodic solutions. A solution of this problem is given using the averaging theory. For a general introduction to the averaging theory see the books of Sanders and Verhulst [13], and of Verhulst [14]. Let x(t, z) be the solution of the unperturbed system (5) such that x(0, z) = z. We write the linearization of the unperturbed system along the periodic solution x(t, z) as y = Dx F0 (t, x(t, z))y.
(6)
In what follows we denote by Mz (t) some fundamental matrix of the linear differential system (6), and by ξ : Rk × Rn−k → Rk the projection of Rn onto its first k coordinates; i.e. ξ(x1 , . . . , xn ) = (x1 , . . . , xk ). Theorem 3 Let V ⊂ Rk be open and bounded, and let β0 : Cl(V) → Rn−k be a C 2 function. We assume that (i) Z = {zα = (α, β0 (α)) , α ∈ Cl(V)} ⊂ and that for each zα ∈ Z the solution x(t, zα ) of (5) is T-periodic; (ii) for each zα ∈ Z there is a fundamental matrix Mzα (t) of (6) such that the matrix Mz−1 (0) − Mz−1 (T) has in the upper right corner the k × (n − k) α α zero matrix, and in the lower right corner a (n − k) × (n − k) matrix α with det( α ) = 0. We consider the function F : Cl(V) → Rk
T
F (α) = ξ 0
Mz−1 (t)F1 (t, x(t, zα ))dt α
.
(7)
If there exists a ∈ V with F (a) = 0 and det ((dF /dα) (a)) = 0, then there is a T-periodic solution ϕ(t, ε) of system (4) such that ϕ(0, ε) → za as ε → 0. Theorem 3 goes back to Malkin [7] and Roseau [11], for a shorter proof see Buica˘ et al. [2]. We assume that there exists an open set V with Cl(V) ⊂ such that for each z ∈ Cl(V), x(t, z, 0) is T-periodic, where x(t, z, 0) denotes the solution of the unperturbed system (5) with x(0, z, 0) = z. The set Cl(V) is isochronous for the system (4); i.e. it is a set formed only by periodic orbits, all of them having the same period. Then, an answer to the problem of the bifurcation of T–periodic solutions from the periodic solutions x(t, z, 0) contained in Cl(V) is given in the following result.
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Theorem 4 (Perturbations of an isochronous set) We assume that there exists an open and bounded set V with Cl(V) ⊂ such that for each z ∈ Cl(V), the solution x(t, z) is T-periodic, then we consider the function F : Cl(V) → Rn
T
F (z) = 0
Mz−1 (t, z)F1 (t, x(t, z))dt.
(8)
If there exists a ∈ V with F (a) = 0 and det ((dF /dz) (a)) = 0, then there exists a T-periodic solution ϕ(t, ε) of system (4) such that ϕ(0, ε) → a as ε → 0. For a proof of Theorem 4 see Corollary 1 of Buica˘ et al. [2]. Now we summarize the results on generalized Abelian integrals that we shall use. Suppose that the unperturbed system x˙ = f (x, y), y˙ = g(x, y),
(9)
has a first integral H(x, y) with an integrating factor 1/R(x, y). Assume that the origin of this system is a center and that the periodic orbits of this center are given by the family of ovals γh contained in the level curves {H(x, y) = h}. Now we consider the perturbed system x˙ = f (x, y) + ε P(x, y), y˙ = g(x, y) + ε Q(x, y), which can be written into the form x˙ = −
∂H (x, y)R(x, y) + ε P(x, y), ∂y
y˙ =
∂H (x, y)R(x, y) + ε Q(x, y). ∂x
Then the generalized Abelian integral associated to this system is P(x, y)dy − Q(x, y)dx I(h) = . R(x, y) γh
(10)
(11)
Since I(h) gives the first order approximation in ε of the displacement function, we get the following result. Theorem 5 The simple zeros of the function I(h) provide limit cycles for the perturbed system (10) which bifurcate from the periodic orbits of the unperturbed system (9). For more details about (generalized) Abelian integrals and the proof of Theorem 5 see Li [5].
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3 Perturbation of a 2-Dimensional Isochronous Center in R4 In this section we prove Theorem 1. The linear part at the origin of the matrix ⎛ 0 1 ⎜0 0 ⎜ ⎝0 0 1 0
differential system (1) is given by the 0 1 0 0
⎞ 0 0⎟ ⎟, 1⎠ 0
(12)
and its eigenvalues are ±1 and ±i. Doing the change of variables (x, y, z, w) → (X, Y, Z , W) given by ⎛ ⎞ ⎛ ⎞⎛ ⎞ X 1 −1 −1 1 x ⎜ Y ⎟ ⎜ −1 −1 ⎟⎜ y ⎟ 1 1 ⎜ ⎟=⎜ ⎟⎜ ⎟, ⎝Z⎠ ⎝ 1 1 1 1⎠⎝ z ⎠ W −1 1 −1 1 w system (1) becomes ˙ = −Y + ε sin((4t + X − Y + Z − W)/4), X Y˙ =
X + ε sin((4t + X − Y + Z − W)/4),
Z˙ =
Z + ε sin((4t + X − Y + Z − W)/4),
˙ = −W + ε sin((4t + X − Y + Z − W)/4). W
(13)
Note that the differential of this system at the origin is the real normal Jordan form of the matrix (12). Now we shall apply Theorem 3 to the differential system (13) taking x = (X, Y, Z , W), F0 (t, x) = (−Y, X, Z , −W), F1 (t, x) = (A, A, A, A), F2 (t, x, ε) = 0, = R4 ,
(14)
where A = sin((4t + X − Y + Z − W)/4). Clearly system (13) with ε = 0 has a linear center at the origin in the (X, Y)plane. We remark that all linear centers are isochronous. Using the notation of Section 2 (mainly the notation related with the statement of Theorem 3), the periodic solution x(t, z) of this center with z = (X0 , Y0 , 0, 0) is X(t) = X0 cos t − Y0 sin t, Y(t) = Y0 cos t + X0 sin t, Z (t) = 0, W(t) = 0,
(15)
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with period T = 2π . The V and α of Theorem 3 are V = {(X, Y, 0, 0) : 0 < X 2 + Y 2 < ρ}, for some real number ρ > 0, and α = (X0 , Y0 ) ∈ V. For the function F0 given in (14) and the periodic solution x(t, z, 0) given in (15) the fundamental matrix M(t) of the differential system (6) such that M(0) is the identity matrix of R4 is ⎞ ⎛ cos t − sin t 0 0 ⎜ sin t cos t 0 0 ⎟ ⎟. M(t) = ⎜ ⎝ 0 0 et 0 ⎠ 0 0 0 e−t We remark that for system (13) with ε = 0 the fundamental matrix does not depend on the particular periodic orbit x(t, z); i.e. it is independent of the initial conditions z. Therefore, an easy computation shows that ⎛ ⎞ 0 0 0 0 ⎜0 0 0 0 ⎟ ⎟. M−1 (0) − M−1 (2π ) = ⎜ ⎝ 0 0 1 − e−2π 0 ⎠ 0 0 0 1 − e2π Consequently all the assumptions of Theorem 3 are satisfied. Therefore, we must study the zeros in V of the system F (α) = 0 of two equations and two unknowns, where F is given by (7). More precisely, we have F (α) = (F1 (X0 , Y0 ), F2 (X0 , Y0 )) where
2π (X0 − Y0 ) cos t − (X0 + Y0 ) sin t F1 = (cos t + sin t) sin t + dt, 4 0
2π (X0 − Y0 ) cos t − (X0 + Y0 ) sin t F2 = (cos t − sin t) sin t + dt. 4 0 After a tedious calculation (which can be checked using an algebraic processor) and the change of variables (X0 , Y0 ) → (r, s) given by X0 − Y0 = 4r cos s, X0 + Y0 = −4r sin s, we obtain for gj(r, s) = Fj(2r(cos s − sin s), −2r(cos s + sin s)), with j = 1, 2, that g1 (r, s) = π [J0 (r) + J2 (r)(cos 2s − sin 2s)], g2 (r, s) = −π [J0 (r) + J2 (r)(cos 2s + sin 2s)], where Jμ(r) is the μ-th Bessel function of first kind (see Abramowitz and Stegun [1]).
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Adding and subtracting the two equations g j(r, s) = 0, for j = 1, 2, we obtain the system h1 (r, s) = J2 (r) sin 2s = 0, h2 (r, s) = J0 (r) + J2 (r) cos 2s = 0.
(16)
It is known that the zeros of the functions Jμ (r) are distinct for different μ’s, then either s = 0, or s = π/2. We are not interested in all the solutions of this system, we are only interested to show that it has as many solutions as we want satisfying the assumptions of Theorem 3. So, in what follows we only study the solutions with s = 0. Consequently, from the second equation of (16) we obtain J0 (r) + J2 (r) = 0. Since J0 (r) + J2 (r) = 2J1 (r)/r, and the function J1 (r) has infinitely many positive zeros tending to be uniformly distributed when r → ∞, because the √ asymptotic behavior of J1 (r) is 2/(πr) cos(r − 3π/4), it follows that system (16) has infinitely many solutions of the form (r0 , 0) being r0 a positive zero of J1 (r). Then, (X0 , Y0 ) = (2r0 , −2r0 ) is a solution of the system Fj(X0 , Y0 ) = 0 for j = 1, 2. Moreover, the determinant of ∂(F1 , F2 )/∂(X0 , Y0 ) at the point (2r0 , −2r0 ) is
π2 2 r0 H 3, −r02 /2 H 3, −r02 /2 − H 2, −r02 /2 . det(r0 ) = 8 where H is the regularized hypergeometric function, see Abramowitz and Stegun [1]. Using the formula Jμ (z) =
z2 2 H μ + 1, −z /4 , 2μ (μ + 1)!
we get det(r0 ) =
72 π 2 J2 (r0 )2 . r02
Since the zeros of J1 (r) and J2 (r) are different, we get that det(r0 ) = 0. Hence, by Theorem 3 for each (2r0 , −2r0 ) contained in V we have a periodic orbit of system (13) with |ε| = 0 sufficiently small. Finally, for a given positive integer N we can fix ρ in the definition of V in such a way that the interval (0, ρ) contains exactly N zeros of the function J1 (r). Then taking |ε| = 0 small enough, Theorem 3 guarantees the existence of N periodic orbits for system (13). Moreover, choosing |ε| = 0 smaller if necessary, since system (13) with ε = 0 has its periodic orbits strongly stable and unstable in the directions Z and W respectively, it follows that the N periodic orbits for system (13) obtained using Theorem 3 are limit cycles; i.e. they are isolated in the set of all periodic orbits. This completes the proof of Theorem 1.
Limit cycles bifurcating from an isochronous center
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4 Perturbation of a 2-Dimensional Center having a Non-rational First Integral First we show that the homogeneous polynomial differential system (2) has a global center at the origin. In polar coordinates (r, θ ) defined by x = r cos θ, y = r sin θ, system (2) becomes r˙ = −r3 sin 2θ, θ˙ = r2 . Of course, to study this system is equivalent to study the differential equation dr = −r sin 2θ, dθ whose solution r(θ, z) satisfying r(0, z) = z is r(θ, z) = z exp − sin2 θ .
(17)
(18)
Therefore all the solutions of the differential equation (17) and consequently all the solutions of the homogeneous polynomial differential system (2) are periodic with the exception of the origin which is a singular point. Hence it is proved that the origin of system is a global center. Now we want to study the limit cycles of the perturbed system (3) for |ε| = 0 sufficiently small, which bifurcate from the periodic orbits of the center of system (2). We write the polynomial P(x, y) of degree m of system (2) as P(x, y) =
m
Pl (x, y),
l=0
where Pl (x, y) is the homogeneous part of degree l of P(x, y). We do the same for the polynomial Q(x, y). Doing for system (13) the same changes of variables that we have done to system (2), we obtain that system (2) can be written as dr = −r sin 2θ + εF1 (θ, r) + O ε2 , dθ
(19)
where F1 (θ, r) =
m l=0
rl − 2 [(cos θ + cos 3θ)Pl (cos θ, sin θ)+ + (3 sin θ + sin 3θ)Ql (cos θ, sin θ)].
Now we shall apply Theorem 4 to the differential equation (19) taking k = n = 1 and x = r, t = θ, F0 (θ, x) = −r sin 2θ, = (0, ∞).
(20)
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Clearly the differential equation (19) is T = 2π periodic in the variable θ. Moreover this equation for ε = 0 has all its solutions 2π -periodic and given by (18). The V and α of Theorem 4 are V = {r : 0 < r < ρ}, for some real number ρ > 0, and α = z ∈ V. For the function F0 given in (20) and the periodic solution r(θ, z) given in (18) the 1 × 1 fundamental matrix M(θ) of the differential equation (19) with ε = 0 such that M(0) = (1) is 2 M(θ) = e− sin θ . We remark that for system (20) the fundamental matrix does not depend on the particular periodic orbit r(θ, z); i.e. it is independent of the initial condition z. Therefore 2 M−1 (θ) = esin θ . Since all the assumptions of Theorem 4 are satisfied, we must study the zeros in V of the function F (z), where F is given by (8). More precisely, we have F (z) =
m
zl−2 Il ,
l=0
where
2π
Il =
e(3−l) sin
2
θ
0
[(cos θ + cos 3θ)Pl (cos θ, sin θ)+ + (3 sin θ + sin 3θ)Ql (cos θ, sin θ)]dθ.
By symmetry the integral Il = 0 if l is even. So, if m = 2ν + 1 then ν
F (z) =
1 2l z I2l+1 . z
(21)
l=0
Hence the polynomial F (z) at most can have ν = [(m − 1)/2] positive real roots. If m = 2ν then ν−1
F (z) =
1 2l z I2l+1 . z
(22)
l=0
Therefore, again the polynomial f (z) at most can have ν − 1 = [(m − 1)/2] positive real roots. In short, using Theorem 4 we at most can get [(m − 1)/2] limit cycles of system (3) bifurcating from the periodic orbits of system (2). We shall prove that when m = 2ν + 1 the function (21) for the perturbed system (3) with P(x, y) =
ν l=0
a2l+1 x2l+1 ,
Q(x, y) = 0,
(23)
Limit cycles bifurcating from an isochronous center
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can be chosen in order that it has exactly ν = [(m − 1)/2] positive arbitrary zeros. In a similar way it can be proved that when m = 2ν the function (22) for the perturbed system (3) with P(x, y) =
ν−1
a2l+1 x2l+1 ,
Q(x, y) = 0,
l=0
can be chosen in order that it has exactly ν − 1 = [(n − 1)/2] positive arbitrary zeros. Hence Theorem 4 will be proved. Now for m = 2ν + 1 we consider system (3) with P and Q given by (23). Then for its corresponding function (21) we have
2π
I2l+1 = a2l+1
e2(1−l) sin θ (cos θ + cos 3θ) cos2l+1 θ dθ. 2
0
Again after a tedious computation (that we can help with an algebraic manipulator as mathematica or maple) we obtain that I2l+1 is equal to L(l) = (12l 2 − 28l + 19) H(l + 1/2, l + 3, 2l − 2) + +(l + 2)(6l − 5) H(l − 1/2, l + 2, 2l − 2),
(24)
multiplied by the constant 2L − 1 √ π (l − 1/2) e2−2l , (l + 2)! where H(a, b, z) is the Kummer confluent hypergeometric function and (z) is the Gamma function, see Abramowitz and Stegun [1]. The value of L(l) is nonzero for all non-negative integer l, see the Appendix. Hence in the polynomial (21) we always can choose the coefficients al+1 conveniently and alternating the sign (by the Descartes rule) in order that the polynomial has the maximum possible number of positive roots ν = [(m − 1)/2]. This completes the proof of Theorem 2. Finally we remark that if we compute the generalized Abelian integral (11) for the system (3) taking x(θ) = ze− sin
2
θ
cos θ,
y(θ) = ze− sin
2
θ
sin θ,
as a parametrization of the center when ε = 0 and integrating with respect to the variable θ between 0 and 2π , we obtain that I(z) =
z F (z). e2
Hence both functions have the same positive zeros, and in this case the method based in Theorem 3 and the method based in the generalized Abelian integral coincide.
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Appendix The Kummer confluent hypergeometric function is given by the series H(a, b, z) =
∞ (a + μ)(b ) μ z . (a)(b + μ)μ! μ=0
(25)
We shall prove that L(l) defined in (24) is always positive for any integer l 0. For showing that we will compute the coefficients of the series expansion of L(l) and we will see that all of them are positive. Using (25) we obtain that L(l) =
∞
Lμ,l (2l − 2)μ ,
μ=0
where the coefficient Lμ,l is equal to (36l 2 − 72l + 43)μ + 3(2l − 1)(6l 2 − 7l + 3)(l + 2)! (l + μ − 1/2) . (l + 1/2)(l + μ + 2)! μ! For l > 0 since 36l 2 − 72l + 43 > 0, 6l 2 − 7l + 3 > 0, (l + μ − 1/2) > 0 and (l + 1/2) > 0 it follows that Lμ,l > 0 for μ = 0, 1, 2 . . . and l > 0. Therefore L(l) > 0 if l > 0. Finally, from (25) we have L(0) = 2eπ(J0 (1) − 2J1 (1)) ≈ 2.31849804 > 0.
References 1. Abramowitz, M., Stegun, I.A.: Bessel Functions J and Y, 9.1. In: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, pp. 358–364. Dover, New York (1972) ˘ A., Françoise, J.P., Llibre, J.: Periodic solutions of nonlinear periodic differential 2. Buica, systems with a small parameter. Commun. Pure Appl. Anal. 6, 103–111 (2007) 3. Champneys, A.R.: Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics. Phys. D 112, 158–186 (1998) 4. Fabry, C., Mawhin, J.: Properties of solutions of some forced nonlinear oscillations at resonance. Progress in Nonlinear Analysis. In: Proc. of the Second Conference on Nonlinear Analysis, pp 103–118. Tianjin, China (1999) 5. Li, C.: Abelian integrals and applications to weak Hilbert’s 16th problem. In: Christopher, C., Li, C. (eds.) Limit Cycles of Differential Equations. Advanced Courses in Mathematics, pp. 91–162. CRM Barcelona, Birkhaüser, Basel (2007) 6. Li, J.: Limit cycles bifurcated from a reversible quadratic center. Qual. Theory Dyn. Syst. 6, 205–216 (2005) 7. Malkin, I.G.: Some problems of the theory of nonlinear oscillations. Gosudarstv. Izdat. Tehn.Teor. Lit., Moscow (1956) (Russian) 8. Ostrovski, L., et al.: On the existence of stationary solitons. Phys. Lett. A 74, 177–170 (1979) 9. Peletier, L.A., Troy, W.C.: Spatial patterns described by the extended Fisher–Komolgorov equation: Kinks. Differential Integral Equations 8, 1279–1304 (1995) 10. Peletier, L.A., Troy, W.C.: Spatial Patterns. Higher Order Models in Physics and Mechanics. Progress in Nonlinear Differential Equations and their Applications, vol. 5. Birkhaüser, Boston (2001)
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11. Roseau, M.: Vibrations non linéaires et théorie de la stabilité, (French) Springer Tracts in Natural Philosophy, vol.8. Springer, Berlin Heidelberg New York (1966) 12. Sanchez, L.: Boundary value problems for some fourth order ordinary differential equations. Appl. Anal. 38, 161–177 (1990) 13. Sanders, J.A., Verhulst, F.: Averaging methods in nonlinear dynamical systems. Appl. Math. Sci. 59, 1–247 (1985) 14. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, Universitext. Springer, Berlin Heidelberg New York (1991)
Math Phys Anal Geom (2007) 10:251–259 DOI 10.1007/s11040-007-9031-6
On the Uniqueness of Gravitational Centre Irmina Herburt
Received: 25 October 2006 / Accepted: 10 December 2007 / Published online: 3 January 2008 © Springer Science + Business Media B.V. 2007
Abstract The dual volume of order α of a convex body A in Rn is a function which assigns to every a ∈ A the mean value of α-power of distances of a from the boundary of A with respect to all directions. We prove that this function is strictly convex for α > n or α < 0 and strictly concave for 0 < α < n (for α = 0 and for α = n the function is constant). It implies that the dual volume of a convex body has the unique minimizer for α > n or α < 0 and has the unique maximizer for 0 < α < n. The gravitational centre of a convex body in R3 coincides with the maximizer of dual volume of order 2, thus it is unique. Keywords Gravitational centre · Gravitational potential · Convex body · Dual volume · Radial centre Mathematics Subject Classifications (2000) 52A20 · 52A40 · 51P05 · 85A25 · 86A20
1 Introduction A point mass M in R3 placed at a distance r from the reference point x is the source of the gravitational potential at x, which is given by (x) = −
GM , r
I. Herburt (B) Department of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661, Warsaw, Poland e-mail:
[email protected]
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where G is the gravitational constant. Hence, a uniformly distributed mass in a region A ⊂ R3 will create the gravitational potential at the point x according to the following formula: γ dλ3 A (x) = −G , r A where r is the distance from the point x and γ is the mass density (here constant). The minimum of the gravitational potential defines the place where the test particle can stay at rest, i.e., a stable equilibrium. For A being a convex body we can apply techniques for convex sets (comp. [6]). The class Kn0 of convex bodies consists of compact, convex subsets of Rn with nonempty interiors. Let us recall that for every convex body A with 0 ∈ A the radial function A : Sn−1 → R is defined by A (u) := sup{λ 0 | λu ∈ A}. Evidently, if x ∈ int A, then for every u ∈ Sn−1 A−x (u) = x − a, where a is the point of bd A ∩ {x + {λu | λ > 0} (comp. [3] (A.55) or [10]). Thus for A ∈ K03 we obtain Gγ 2 (u)dσ u, A (x) = − 2 S2 A−x where σ is the spherical Lebesgue measure (see Section 2 or [6]). Hence the minimum of the function A coincides with the maximum of the function αA : A −→ R given by α A (a) := αA−a (u)dσ (u) (1) Sn−1
for n = 3 and α = 2. Functions defined by (1) are called dual volumes of order α of a convex body A (compare [3, 10]). Therefore, the dual volume of order α of a convex body A is a function αA : A −→ R which assigns (up to a constant factor) to every a ∈ A the mean value of α-power of distances of a from the boundary of A with respect to all directions. These important functionals have been studied by many authors (see [1, 2, 4, 7–9]). In [6] it is proved that for every A ∈ K0n , if α ∈ (0, 1], then the function A defined by (1) is strictly concave. Extremizers of dual volumes of order α are called radial centers of order α. Therefore, for any convex body a radial centre of order α for α ∈ (0, 1] is unique. In this paper we extend this result to each α ∈ R. We shall prove that the dual volume of order α is strictly convex for α > n or α < 0 and strictly concave for 0 < α < n. Obviously, the dual volume
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of order 0 is a constant function. The dual volume of order n is equal to n times the volume of A, for every A ∈ K0n and every a ∈ A (see [11], Example 3.2). Since the gravitational centre of a convex body A is the radial centre of order 2 of A it is unique. Radial centres have other interesting physical interpretations. In [6] it is also proved that – The light intensity radiated by a transparent medium (like stars uniformly filling some region in the universe) attains its maximum at the radial centre of order 1. – Results analogous to gravitational centre hold if instead of gravitational interactions one considers electric forces. In [5] the notion of dual volume of order α is used to describe the propagation of electromagnetic waves from a transmitter, chosen randomly and according to the uniform distribution in a convex body A, to a receiver fixed at the origin. The authors investigate the case when the origin coincides with the centre of an inscribed ball, which in many cases is not the optimal choice. To maximize the mean received power, the receiver should be located at the radial centre of suitable order. We follow, in principle, terminology and notation used in [13]. In particular: – The k-dimensional Lebesgue measure in Euclidean k-dimensional space is λk (in integrals with respect to λ1 we shall abbreviate dλ1 (x) to dx). – Bn is the unit ball in Rn and Sn−1 is its boundary in Rn . – [x, y] is the closed segment with the end points x and y in Rn .
2 Convexity of Dual Volumes For every nonnegative measurable function f in Rn ∞ f (x)dλn (x) = f (tu)tn−1 dtdσ (u). Rn
Sn−1
0
(comp. [14], formula (5.2.3) in Theorem 5.2.2). Thus for every α > 0, A ∈ K0n and a ∈ A we get Sn−1
αA−a (u)dσ (u) = α
Sn−1
αA−a (u) 0
tα−1 dtdσ (u) = α
A
1 dλn (x). x − an−α (2)
α Hence,1 for α > 0, the convexity of A (·) is equivalent to the convexity of A x−(·)n−α dλn (x). For every α ∈ R and a ∈ int A the integral αA (a) is finite. However the 1 integral A x−a n−α dx is divergent for α < 0. Thus, the case α < 0 needs more careful treatment.
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Lemma 2.1 Let A ∈ K0n . If for every x0 ∈ int A and u ∈ Sn−1 there exists ε > 0 such that the function αA |{x0 + tεu : −1 < t < 1} is convex, then the function αA is convex on int A. Proof The proof is based on two facts concerning convex functions: – A function is convex on int A if and only if for any a, b ∈ int A its restriction to relint[a, b ] is convex – A function of one variable is convex on an open interval I if and only if it is locally convex on I. Lemma 2.2 Let A ∈ K0n and let f be a similarity of Rn . If αA is convex (concave) on A, then αf (A) is convex (concave) on f (A). Take an arbitrary x0 ∈ int A and ε such that ε Bn + x0 ⊂ int A. Then 1 dλn (x) + α(ε Bn +x0 ) (a). αA (a) = α n−α A\(ε Bn +x0 ) x − a
(3)
By Lemma 2.1,to investigate the convexity of αA it is enough to investigate the 1 α convexity of α A\(ε Bn +x0 ) x−(·) n−α dλn (x) and (ε Bn +x ) restricted to {x0 + tεu : 0 −1 < t < 1}, for every u ∈ Sn−1 . s Let F K be defined by s FK (a) = x − as dλn (x). K
In case s + n > 0 we take K = A and a ∈ A. In case s + n < 0 we take K = A \ (ε Bn + x0 ) and a ∈ ε Bn + x0 . s We are going to prove that the function F K is strictly convex for s > 0 and strictly concave for s < 0. We shall need the following lemmas. Easy technical parts of their proofs will be omitted. s Lemma 2.3 Let A ∈ K0n . For every isometry f of Rn and a in the domain of F K s FK (a) = F sf (K) ( f (a)).
Lemma 2.4 Let A be a convex set in Rn . Let μ be a measure on a σ -field F and B ∈ F . Let f (a, ·) : B −→ R be integrable with respect to μ for every a in A. If f (·, b ) is convex (concave) on A for almost all b ∈ B then B f (·, b )dμ(b ) is convex (concave) on A. s Lemma 2.5 Let A ∈ K0n and a, b in the domain of F K . Then for every λ ∈ [0, 1] s FK (λa + (1 − λ)b )
> (<)
s s λF K (a) + (1 − λ)F K (b )
On the uniqueness of gravitational centre
255
if and only if there exists an isometry f : Rn −→ Rn such that f (a) =
(a1 , a2 , . . . , an ), f (b ) = (b 1 , a2 , . . . , an ) and
F sf (K) (λa1 + (1 − λ)b 1 , a2 , . . . , an ) > (<)
λF sf (K) (a1 , a2 , . . . , an ) + (1 − λ)F sf (K) (b 1 , a2 , . . . , an ). s Proof =⇒ By Lemma 2.3 the function F K is equivariant under the isometries
n of R , thus it is enough to take an isometry f which maps [a, b ] to [a , b ]
parallel to the first axis, and so ai = b i for i = 2, 3, ..., n.
⇐= Isometry f −1 maps (λa1 + (1 − λ)b 1 , a2 , . . . , an ) onto λa + (1 − λ)b .
Lemma 2.6 Let A ∈ K0n . Then there exist functions fi , hi , for i = 1, . . . , n − 1, and numbers α1 , α2 such that for every x ∈ A fi (xi+1 , xi+2 , . . . , xn ) xi hi (xi+1 , xi+2 , . . . , xn ) and α1 xn α2 . Proof The proof is by induction with respect to the number of coordinates and based on the observation that if we take the natural ordering of a line in any direction, then there exists the first and the last element in the intersection of the line with the convex body A. s Theorem 2.7 F K is strictly convex for s > 0 and strictly concave for s < 0.
Proof By the definition, s (a) = x − as dλn (x) FK K
=
(x1 − a1 )2 + . . . + (xn − an )2
2s
dλn (x).
K
By Lemma 2.5, it is enough to prove the convexity with respect to the first coordinate a1 with a2 , a3 , . . . , an being fixed. For K = A and s + n > 0, by Lemma 2.6, we obtain α2 hn−1 h1 p s FK (a) = ··· (x1 − a1 )2 + . . . + (xn − an )2 dx1 . . . dxn , (4) α1
fn−1
f1
s where p = FK it is enough to prove the convexity of the function F given by h1 p F(a1 ) = (x1 − a1 )2 + c dx1 , s . By Lemma 2.4, to prove the convexity of 2
f1
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where c = (x2 − a2 )2 + . . . + (xn − an )2 is fixed. After a substitution t = h1√−a1 we obtain (up to a constant factor) the integral f1√−ac1 (t2 + 1) p dt.
x1√ −a1 c
c
By standard calculations we can check that the sign of its second derivative s with respect to a1 is equal to the sign of p. Therefore, for s > 0 the function F K is strictly convex and for −n < s < 0 it is strictly concave. For K = A \ (ε Bn + x0 ) and s + n < 0 we have to replace the integral in (4) by the finite sum of integrals of such form, and then the proof follows. Lemma 2.8 The function f : [−1, 1] −→ R defined, for k > 0, by the formula π α 2 f (t) = −t cos ϕ + 1 − t2 (sin ϕ)2 (cos ϕ)k dϕ − π2
is strictly convex for α < 0. Proof By standard calculations we show that
d2 f dt2
is positive.
Lemma 2.9 For every v ∈ Sn−1 and every α < 0 the function αBn |(Bn ∩ lin(v)) is strictly convex. Proof Let (e1 , e2 , . . . , en ) be the canonical basis in Rn . We may assume v = en . Then for u ∈ Sn−1 we have u = (cos ϕn−1 ) · u¯ + (sin ϕn−1 ) · en , where u¯ ∈ Sn−1 ∩ lin(e1 , . . . , en−1 ) and ϕn−1 in [− π2 , π2 ] is the measure of the angle between u and its projection onto lin(e1 , e2 , . . . , en−1 ). Since Sn−1 ∩ lin(e1 , . . . , en ) = Sn−2 {0} we get, by induction a parametrization p of Sn−1 p : (ϕ1 , ϕ2 , . . . , ϕn−1 ) −→ u ∈ Sn−1 . Let pi =
δp δϕi
for i = 1, . . . , n − 1 and let G := ( pi ◦ p j)i, j=1,...,n−1 . Moreover, δ p¯ ¯ := ( p¯ i ◦ ¯ p¯ i = δϕ let p¯ : (ϕ1 , ϕ2 , . . . , ϕn−2 ) −→ u, for i = 1, . . . , n − 2 and let G i p¯ j)i, j=1,...,n−2 . Then √ ¯ detG = (cos ϕn−1 )n−2 · detG. In general √ detG = 1 for n = 2, √ detG = cos ϕ2 for n = 3 and √ detG = (cos ϕn−1 )n−2 · . . . · (cosϕ3 )2 · cos ϕ2 for n > 3.
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For every t ∈ [−1, 1], a = tv and ϕ = ϕn−1 we get αBn (a) =
=
Sn−1
αBn −a (u)dσ (u)
Sn−2 {0}
π 2
− π2
−t cos ϕ +
α ¯ ¯ 1 − t2 (sin ϕ)2 (cos ϕ)n−2 detGdϕdσ (u).
Thus, by Lemma 2.8 and Lemma 2.4 we get the claim.
Theorem 2.10 Let A ∈ K0n . The dual volume αA is strictly convex for α > n or α < 0 and strictly concave for 0 < α < n. Proof Let s = α − n. By (2) and Theorem 2.7 the dual volume of order α is strictly convex for α > n and strictly concave for 0 < α < n. By (3), Theorem 2.7, Lemma 2.9 and Lemma 2.2, the dual volume of order α is strictly convex for α < 0 with values ∞ for points in the bd A. By Theorem 2.10 we immediately obtain Corollary 2.11 Let A ∈ K0n . The dual volume αA has the unique minimizer for α > n or α < 0 and has the unique maximizer for 0 < α < n. Example 2.12 Let As be a triangle with vertices (1, 0), (0, −s), (0, s) in R2 for some s ∈ R+ . We shall calculate approximate coordinates of radial center rα (As ) of order α of As for some α and s. Since L = {(t, 0) : t ∈ R} is the symmetry line of the triangle As and rα (As ) is equivariant under the isometries of R2 it follows that rα (As ) ∈ L. Thus we calculate αAs (a) only for a = (t, 0) and t ∈ [0, 1]. Let γ1 (t, s) = π2 − arctan st , γ2 (t, s) = π2 − arctan 1s − arctan st , γ3 (t, s) = π2 − arctan s. Then, as is easy to check, αAs (a) = 2 fsα (t), where fsα (t) = tα
γ1 (t,s) 0
1 dφ (cos φ)α
+ ((1 − t) sin arctan(s))α + ((1 − t) sin arctan(s))α
γ2 (t,s) 0 γ3 (t,s) 0
1 dφ (cos φ)α 1 dφ. (cos φ)α
In Fig. 1 there are graphs of t → fsα (t) for chosen values of parameters α and s and for t close to the argument of extremum. The values of the function t → fsα (t) are found by means of numerical methods.
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Fig. 1 Graphs of t → 12 αAs ((t, 0)) for various α and s
Finally let us mention that the notion of dual volumes can be generalized on star bodies (see [11] or [12]). In the first version of the present paper we gave an example of star body whose radial center of order 1 is not unique. However there was a mistake in calculations and the example is wrong. This example was cited in [12], p. 195. For non-convex bodies the problem of uniqueness of extremizers of dual volumes remains open.
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References 1. Gardner, R.J., Jensen, E.B., Volcic, A.: Geometric tomography and local stereology. Adv. in Appl. Math. 30, 397–423 (2003) 2. Gardner, R.J.: Geometric tomography. Notices Amer. Math. Soc. 42(4), 422–429 (1995) 3. Gardner, R.J.: Geometric Tomography. Cambridge University Press, Cambridge (1995) 4. Gardner, R.J., Soranzo, A., Volcic, A.: On the determination of star and convex bodies by section functions. Discrete Comput. Geom. 21(1), 69–85 (1999) 5. Hansen, J., Reitzner, M.: Electromagnetic wave propagation and inequalities for moments of chord lengths. Adv. in Appl. Probab. 36(4), 987–995 (2004) ´ ´ 6. Herburt, I., Moszynska, M., Peradzynski, Z.: Remarks on radial centres of convex bodies. MPAG 8(2), 157–172 (2005) 7. Klain, D.A.: Star valuations and dual mixed volumes. Adv. Math. 121, 80–101 (1996) 8. Klain, D.A.: Invariant valuations on star-shaped sets. Adv. Math. 125, 95–113 (1997) 9. Ludwig, M.: Dual mixed volumes. Pacific J. Math. 58, 531–538 (1975) 10. Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. Math. 71(2), 232–261 (1988) ´ 11. Moszynska, M.: Looking for selectors of star bodies. Geom. Dedicata 81, 131–147 (2000) ´ 12. Moszynska, M.: Selected Topics in Convex Geometry, Birkhäuser (2005) 13. Schneider, R.: Convex Bodies: the Brunn–Minkowski theory. Cambridge University Press, Cambridge (1993) 14. Stroock, D.W.: A Concise Introduction to the Theory of Integration, Birkhäuser (1990)
Math Phys Anal Geom (2007) 10:261–270 DOI 10.1007/s11040-007-9032-5
Estimating Eigenvalue Moments via Schatten Norm Bounds on Semigroup Differences M. Hansmann
Received: 10 August 2007 / Accepted: 11 December 2007 / Published online: 4 January 2008 © Springer Science + Business Media B.V. 2007
Abstract We derive new bounds on the moments of the negative eigenvalues of a selfadjoint operator B. The moments of order 0 < γ 1 are estimated in terms of Schatten-norm bounds on the difference of the semigroups generated by B and a reference operator A which is assumed to be nonnegative and selfadjoint. The estimate in the case γ = 1 is sharp. Keywords Eigenvalues · Moments · Schatten ideals · Selfadjoint operators · Semigroup difference · Spectral shift function Mathematics Subject Classifications (2000) 47A10 · 47A75 · 81Q10
1 Introduction The spectral analysis of selfadjoint operators plays an important role in many areas of mathematical physics. Topics such as the stability of the absolutely continuous spectrum, absence of singular continuous spectrum or perturbations of eigenvalues have been investigated in great detail and a lot of results are known, not only in the context of Schrödinger operators. Recently, Demuth and Katriel [5] proposed a new method to derive bounds on the moments of the negative eigenvalues of a selfadjoint operator. Given two selfadjoint operators A 0 and B −cB , cB > 0 in a Hilbert space H and assuming that D = e−B − e−A ∈ S1 , where S1 denotes the ideal of
M. Hansmann (B) Institute of Mathematics, Technical University of Clausthal, 38678 Clausthal-Zellerfeld, Germany e-mail:
[email protected]
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trace-class operators, they showed that the following inequality is valid for every γ > 1 |λ|γ Ctr (γ )DS1 . (1) λ∈σ (B)∩(−∞,0)
Here σ (B) means the spectrum of B, Ctr (γ ) is some suitable constant with Ctr (γ ) → ∞ as γ → 1 and each eigenvalue of B is repeated according to its multiplicity. A similar result has been shown in case that D is a Hilbert– Schmidt operator, see [5]. The method used in [5] involves complex function theory (e.g. Jensen’s identity) and is rather general, in particular it is possible to extend this result to other Schatten-classes Sp , p 1 (Demuth and Katriel, private communication). Recall that D ∈ Sp if 1/ p ∞ p DSp := |μk (D)| < ∞, 0 < p < ∞, k=1
where μk (D) is the k−th singular value of the compact operator D. Note that we write DSp even in the case p < 1 where .Sp does not fulfill the triangle inequality. There exists an alternative way to proof estimates similar to (1) suggested by M. Solomyak. Theorem 1 [Solomyak, private communication] Let A and B be as above and suppose that D = e−B − e−A ∈ Sp where 1 < p < ∞. Then the following estimates are valid p |λ| p 2Cp DSp (2) λ∈σ (B)∩(−∞,0)
and in particular
|λ|2 D2S2 .
(3)
λ∈σ (B)∩(−∞,0)
Here Cp → ∞ as p → 1 or p → ∞. Since the idea of proof will be used in the sequel, we provide a short sketch. Proof Define the Lipschitz continuous function 0 , x1 h(x) = . ln(x) , x>1 Note that this function admits the representation x η(s) ds h(x) = h(0) + 0
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with a suitable left-continuous function η of bounded total variation. Using the theory of double operator integrals developed by Birman and Solomyak, one can show that −B
h e − h e−A Sp 2Cp e−B − e−A Sp , where Cp is a suitable constant, see [2, Theorem 8.6]. By the spectral theorem
h e−B − h e−A = B− , where B− = 12 (|B| − B) and this implies the validity of (2). In the same manner (3) follows from the inequality B− S2 DS2 which is a consequence of [2, Theorem 8.1]. Note that in the case of Hilbert-Schmidt perturbations inequality (3) is sharp, i.e. there can not be a constant c < 1 such that |λ|2 cD2S2 . (4) λ∈σ (B)∩(−∞,0)
This can be seen by considering a compact operator B such that B = −diag(bn ) in some suitable basis and A = 0. Then the inequality
2 b2n c · ebn − 1 n
n
holds in general only if c 1. Regarding the estimates (1) and (2) which give estimates on |λ|γ λ∈σ (B)∩(−∞,0)
in the case γ > 1, it is quite natural to ask what can be said in the case 0 < γ 1. The aim of this article is to show that estimates like (1) and (2) are valid in this case as well. To be more explicit we will proof the following two theorems. Theorem 2 Let A and B be as above and suppose that D = e−B − e−A ∈ S1 . Then for every γ 1 the following estimate holds |λ|γ C1 (γ )DS1 , (5) λ∈σ (B)∩(−∞,0)
where C1 (γ ) = γ (γ − 1)(γ −1) e1−γ . In particular |λ| DS1 . λ∈σ (B)∩(−∞,0)
Note that as in the case of (3) the estimate in (6) is sharp.
(6)
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Theorem 3 Let A and B be as above and suppose that D = e−B − e−A ∈ Sp where 0 < p < 1. Then for every γ > p the following estimate holds γ γ −p p |λ|γ 2cB (7) DSp , γ−p λ∈σ (B)∩(−∞,0)
where B −c B . The main ingredient in the proof of Theorem 3 is the following proposition which is of interest in its own. Proposition 1 Let A and B be as above and suppose that D = e−B − e−A ∈ Sp where 0 < p < 1. Then N B (−s) 2 s− p DSp , p
s>0
where N B (−s) gives the number of eigenvalues of B in (−∞, −s] and each eigenvalue is counted according to its multiplicity. Our proofs will neither depend on complex function theory nor on estimates of the type used by Solomyak. Instead we will apply the theory of Krein’s spectral shift function (SSF) which has been developed by M. G. Krein with further contributions by Birman, Solomyak, Peller and other authors. See [3, Birman et al.] for an extensive review including references to the original literature. In the next section we will provide some basic facts about the SSF necessary to understand the proofs of Theorem 2 and Theorem 3 given in Sections 3 and 4 respectively. Finally, in Section 5 we gather some remarks concerning possible applications and extensions of the given results.
2 Some Information on Krein’s SSF We refer to [8, Yafaev, Chapter 8] and [3, Birman et al.] as general references for the results in this section. The SSF for the pair e−B , e−A is defined via the corresponding perturbation determinant from scattering theory, that is
−1 , ξ(x) := ξ(x, e−B , e−A ) := π −1 lim+ arg det 1 + D e−A − x − iε ε→0
−B
where D = e
−A
−e
∈ S1 . Given this definition the following holds ξ(x) dx = TrD R
and
(8)
R
|ξ(x)| dx DS1 .
(9)
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Furthermore, for suitable f , D ∈ S1 implies that f (e−B ) − f (e−A ) ∈ S1 and −B
−A Tr f e − f e = f (x)ξ(x) dx. (10) R
This equality is known as Krein’s trace formula. In particular, it is valid for differentiable f with Hölder continuous derivative, see [1, Birman et al.]. We will also need some information from the L p -theory of the SSF, see [4, Combes et al.]. Namely |ξ(x)|r dx DS 1 , r 1. (11) R
r
3 Proof of Theorem 2 Modifying the idea of Solomyak’s proof we define a function h in the following way 0 ,x 1 , γ > 1. h(x) = ln(x)γ , x > 1 It is not difficult to see that h ∈ C1 (R ) and that h is Hölder continuous. Hence, using (10) we get h e−B − h e−A ∈ S1 and
Tr h e−B − h e−A = h (x)ξ(x) dx. (12) R
By the spectral theorem h e−B = (B− )γ and h e−A = 0. Thus (12) can be rewritten in the following way |λ|γ = h (x)ξ(x) dx R
λ∈σ (B)∩(−∞,0)
=γ 1
Since maxx∈[1,∞) x−1 ln(x)(γ −1) =
∞
ln(x)(γ −1) ξ(x) dx. x
(γ −1)(γ −1) , e(γ −1)
we get for γ > 1
|λ|γ γ (γ − 1)(γ −1) e(1−γ )
∞
|ξ(x)| dx.
In particular we can take the limit γ → 1+ on both sides such that ∞ |λ| |ξ(x)| dx. λ∈σ (B)∩(−∞,0)
(13)
1
λ∈σ (B)∩(−∞,0)
1
The desired result now follows from (13) and (14) using estimate (9).
(14)
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4 Proof of Theorem 3 We take some arbitrary function g ∈ C1 (R) with Hölder continuous derivative such that 0 g 1, g(x) = 0 if x 0 and g(x) = 1 if x 1. Set t−a , t ∈ R, a < b. ha,b (t) := g b −a Then ha,b inherits the smoothness properties of g and ⎧ 0 , t ∈ (−∞, a] ∪ [b , ∞) ⎪ ⎨ d ha,b (t) = 1 t−a ⎪ dt g , t ∈ (a, b ). ⎩ b −a b −a
(15)
Next we select some specific values for the constants a and b . Let s > 0 and 0 < ε < 1 and set ⎧ 0 , t e(1−ε)s ⎪ ⎨ h(t) := he(1−ε)s ,es (t) = h(t) , t ∈ (e(1−ε)s , es ) (16) ⎪ ⎩ 1 , t es . By the spectral theorem h(e−A ) = 0 and furthermore −(1−ε)s −s
−B h e d(EB (x) f, f ) + h(e−x ) d(EB (x) f, f ) f, f =
−∞
−s −∞
−s
d(EB (x) f, f )
= (EB ((−∞, −s]) f, f ) ,
f ∈ H.
(17)
Let N B (−s) be the number of eigenvalues of B smaller than −s, s > 0. By the min-max-principle (17) implies
N B (−s) = Tr E(−∞,−s] (B)
Tr h e−B
= Tr h e−B − h e−A . Hence, using (10) we get
N B (−s) Tr h e−B − h e−A = h (x)ξ(x) dx R
=
es
e(1−ε)s
h (x)ξ(x) dx.
Let q, r > 1 and q−1 + r−1 = 1. By Hölder’s inequality es 1/q 1/r N B (−s) |h (x)|q dx |ξ(x)|r dx . e(1−ε)s
R
(18)
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To simplify the following computations we substitute a = e(1−ε)s and b = es . We have already seen in (15), that x−a 1 g . h (x) = b −a b −a Hence we can compute
b
a
1 |h (x)| dx = (b − a)q
b
q
=
a
1 (b − a)q−1 g Lq (0,1)
x − a q g dx b −a 1
|g (u)|q du
0
q
=
(b − a)q−1
.
In conclusion we get N B (−s)
g Lq (0,1)
1/r |ξ(x)|r dx
1
R (es − e(1−ε)s )1− q 1/r g Lq (0,1) r |ξ(x)| dx , 1 (εs) r R
where we have used that es − e(1−ε)s εs and r−1 = 1 − q−1 . Using (11) we arrive at N B (−s) Choosing r =
1 p
g Lq (0,1) (εs)
1 r
1
DSr 1 . r
and taking the limit ε → 1− it follows that
N B (−s)
g L(1− p)−1 (0,1) sp
p
DS p ,
0 < p < 1.
The number of negative eigenvalues N B (−s) is connected to the corresponding moments via cB 1 sγ −1 N B (−s) ds = |λ|γ , γ > 0. (19) γ λ∈σ (B)∩(−∞,0) 0 Using (19) we can finally conclude that for every γ > p λ∈σ (B)∩(−∞,0)
γ −p
|λ|γ c B
γ p g L(1− p)−1 (0,1) DSp , γ−p
(20)
where B −c B . Note that the choice of g is free and that it is elementary to construct a function g with g ∞ 2.
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5 Remarks 1. In applications it is often more usual to posses bounds on the Schatten norms of differences of powers of the resolvents corresponding to A and B. It is not difficult to see that our method can be adjusted to this case. For example in the case γ = 1 one can derive
λ∈σ (B)∩(−∞,0)
|λ|
(1 + r)r−1 1+r a (B + a)−r − (A + a)−r S1 , r
r 1, (21)
provided a > cB . The moments of order 0 < γ < 1 can be handled as well. 2. The method described above translates the problem of estimating the moments of a semibounded selfadjoint operator B to the problem of finding bounds on the p-th Schatten norm of the semigroup difference D = e−B − e−A where A is some arbitrary nonnegative selfadjoint operator. For concrete operators B this opens the opportunity to find an optimal operator A = A(B) adjusted to B, e.g. to improve constants, etc. 3. For the important class of Schrödinger operators, i.e. B = − + V where V is in the Kato-class, the problem of estimating the p-th Schatten norm of D has already been subject of intensive study and we would like to mention just a few known results. For A = − and B as above trace norm estimates are well known and we refer to [6, Demuth et al.] where a thorough account on this topic can be found. In the case p < 1 it has recently been shown in [7, Hundertmark et al.] that supp(V) < ∞ implies that D ∈ Sp for every p > 0 and this result can be generalized to potentials V with unbounded support given some suitable decay conditions on V at infinity. 4. For Schrödinger operators it is interesting to compare our estimates with estimates of the Lieb–Thirring type. We refer to [5, Demuth et al.] where some results on this topic can be found. 5. In some situations moment estimates in terms of Schatten norm bounds can be inadequate and our estimates can be improved. In particular, this is possible in case that more information on the decay rate of the singular values of D is available. As an example we consider the results of the afore mentioned paper [7], where A = − and B = − + V with V of bounded support (note that in [7] the case of magnetic Schrödinger operators was considered). The authors of [7] showed that the singular values μn (D) decay almost 1/d exponentially, i.e. μn (D) Ce−cn , and they used this result to derive an
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integral bound on the SSF for the pair A, B. In the following we shortly sketch the proof of the corresponding result in [7] with ξ(s, A, B) replaced
by ξ s, e−A , e−B . Define the convex function Ft : [0, ∞) → [0, ∞) x
Ft (x) = exp ty1/d − 1 dy, t>0 0
Ft (0)
= 0. Using the decay rate of μn (D) it can be shown that and note that for t small enough
Ft |ξ s, e−B , e−A | ds Kt , R
where Kt > 0 is a suitable constant. Let Gt be the Legendre transform of Ft . Then log(1 + y) d Gt (y) y , y 0. t Using Young’s inequality
−A −B f (x)ξ(x) dx Ft |ξ x, e , e | dx + Gt (| f (x)|) dx Kt + t−d {log(1 + f ∞ )}d f L1 ,
(22)
where f is any bounded integrable function. Coming back to our problem we note that in the proof of Theorem 3 we showed that es h (x)ξ(x) dx, 0 < ε < 1, s > 0, N B (−s) e(1−ε)s
where h was defined in (16). Since h ∞ cs−1 and h L1 = 1 estimate (22) implies that N B (−s) c log(1 + s−1 ) as s → 0. This is an improvement compared to N B (−s) cs−γ , γ > 0 which can be derived directly from Theorem 3. Acknowledgements I would like to thank M. Demuth, G. Katriel and I. Veseli´c for useful hints and discussions. Further I would like to thank M. Solomyak whose ideas initiated the work on this article.
References 1. Birman, M.Sh., Solomyak, M.Z.: Remarks on the spectral shift function. J. Sov. Math. 3, 408–419 (1975) 2. Birman, M.Sh., Solomyak, M.Z.: Double operator integrals in a Hilbert space. Integral Equations Operator Theory 47, 131–168 (2003) 3. Birman, M.Sh., Yafaev, D.: The spectral shift function. The papers of M.G. Krein and their further development. (Russian) Algebra i Analiz 4(5), 1–44 (1992)
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4. Combes, J.M., Hislop, P.D., Nakamura, S.: The L p -theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators. Comm. Math. Phys. 218(1), 113–130 (2001) 5. Demuth, M., Katriel, G.: Eigenvalue Inequalities in Terms of Schatten Norm Bounds on Differences of Semigroups, and Application to Schrödinger Operators. To appear in Annales Henri Poincare. http://arxiv.org/abs/math/0612279 6. Demuth, M., van Casteren, J.: Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration Approach. Birkhäuser Verlag, Basel (2000) 7. Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veseli´c, I.: Bounds on the spectral shift function and the density of states. Comm. Math. Phys. 262(2), 489–503 (2006) 8. Yafaev, D.: Mathematical Scattering Theory. General Theory. Translations of Mathematical Monographs, vol. 105. American Mathematical Society, Providence, RI (1992)
Math Phys Anal Geom (2007) 10:271–295 DOI 10.1007/s11040-007-9033-4
Weak Convergence and Vector-Valued Functions: Improving the Stability Theory of Feynman’s Operational Calculi Lance Nielsen
Received: 3 August 2007 / Accepted: 12 December 2007 / Published online: 23 January 2008 © Springer Science + Business Media B.V. 2007
Abstract In this paper we present a theorem that establishes a relation between continuous, norm-bounded functions from a metric space into a separable Hilbert space and weak convergence of sequences of probability measures on the metric space. After establishing this result, it’s application to the stability theory of Feynman’s operational calculi will be illustrated. We will see that the existing time-dependent stability theory of the operational calculi will be significantly improved when the operator-valued functions take their values in L(H), H a separable Hilbert space. Keywords Weak convergence · Disentangling · Feynman’s operational calculi · Stability theory Mathematics Subject Classifications (2000) 47A13 · 47A60 · 47A56 · 47N50 · 60F99 1 Introduction The two topics under consideration in this paper are a relation between weak convergence of probability measures and Hilbert space valued functions and the consequences of this relation when applied to Feynman’s operational calculus. As indicated, we will first state and prove a theorem (Theorem 2.3 below) that establishes a relation between weak convergence of sequences of probability measures on a metric space S and functions f : S → H, H a separable Hilbert space, that are continuous and norm bounded. The utility of having such a theorem in hand was clear to the author during his development
L. Nielsen (B) Department of Mathematics, Creighton University, Omaha, NE 68178, USA e-mail:
[email protected]
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of the stability theory for Feynman’s operational calculi in his thesis [15] as well as the papers [8, 13, 14], and [12]. However, it was not until recently that the theorem was discovered. After stating and proving Theorem 2.3, we state a modification of the theorem to accommodate not only a weakly convergent sequence of probability measures, but also a corresponding uniformly convergent sequence of H-valued functions. The second focus of this paper is the application of Theorem 2.3 to improving the stability theory of Feynman’s operational calculi. It is possible, however, that Theorem 2.3 may be of independent interest. Before proceeding any further the reader may find a short discussion of Feynman’s operational calculus useful. Feynman’s operational calculus originated with the 1951 paper [3] and concerns itself with the formation of functions of non-commuting operators. Indeed, even functions as simple as f (x, y) = xy are not well-defined if x and y do not commute. Indeed, some possibilities are f (x, y) = yx, f (x, y) = 1 + yx), and f (x, y) = 13 xy + 23 yx. One then has to decide, usually with a 2 (xy particular problem in mind, how to form a given function of non-commuting operators. One method of dealing with this problem is the approach developed by Jefferies and Johnson in the series of papers [4–7] and expanded on in the papers [8, 11], and others. The Jefferies–Johnson approach to the operational calculus uses measures on intervals [0, T] to determine the order of operators in products. In the original setting used by Jefferies and Johnson, only continuous measures were used. However, Johnson and the current author extended the operational calculus to measures with both continuous and discrete parts in [11]. [The reader can see the difference in the operational calculus that results when moving from continuous time ordering measures to time ordering measures with a non-zero discrete part if they compare (3.19) on page 14 to equations (3.15) and (3.17) on page 12.] The discussion above, then, begs the question of how measures can be used to determine the order of operators in products. Feynman’s heuristic rules for the formation of functions of non-commuting operators give us a starting point. 1) Attach time indices to the operators to specify the order of operators in products. 2) With time indices attached, form functions of these operators by treating them as though they were commuting. 3) Finally, “disentangle” the resulting expressions; i.e. restore the conventional ordering of the operators. As is well known, the central problem of the operational calculus is the disentangling process. Indeed in his 1951 paper, [3], Feynman points out that “The process is not always easy to perform and, in fact, is the central problem of this operator calculus.” We first address rule (1) above. It is in the use of this rule that we will see measures used to track the action of operators in products. First, it may be that the operators involved may come with time indices naturally attached. For example, we might have operators of multiplication by time dependent potentials. However, it is also commonly the case that the operators used are
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independent of time. Given such an operator A, we can (as Feynman most often did) attach time indices according to Lebesgue measure as follows: 1 t A= A(s) ds t 0 where A(s) := A for 0 s t. This device does appear a bit artificial but does turn out to be extremely useful in many situations. We also note that mathematical or physical considerations may dictate that one use a measure different from Lebesgue measure. For example, if μ is a probability measure on the interval [0, T], and if A is a linear operator, we can write A= A(s) μ(ds) where once again A(s) := A for 0 s T. When we write A in this fashion, we are able to use the time variable to keep track of when the operator A acts. Indeed, if we have two operators A and B, consider the product A(s)B(t) (here, time indices have been attached). If t < s, then we have A(s)B(t) = AB since here we want B to act first (on the right). If, on the other hand, s < t, then A(s)B(t) = B A since A has the earlier time index. In other words, the operator with the smaller (or earlier) time index, acts to the right of (or before) an operator with a larger (or later) time index. (It needs to be kept in mind that these equalities are heuristic in nature.) For a much more detailed discussion of using measures to attach time indices, see Chapter 14 of the book [9] and the references contained therein. Concerning the rules (2) and (3) above, we mention that, once we have attached time indices to the operators involved, we calculate functions of the non-commuting operators as if they actually do commute. These calculations are, of course, heuristic in nature but the idea is that with time indices attached, one carries out the necessary calculations giving no thought to the operator ordering problem; the time indices will enable us to restore the desired ordering of the operators once the calculations are finished; this is the disentangling process and is typically the most difficult part of any given problem. We now move on to discuss, in general terms, how the operational calculus can be made mathematically rigorous. Suppose that Ai : [0, T] → L(H), i = 1, . . . , n, are given and that we associate to each Ai (·) a Borel probability measure μi on [0, T]; this is the so-called time ordering measure and, as mentioned above, serves to keep track of when a given operator or operatorvalued function acts in products. We construct a commutative Banach algebra (the disentangling algebra) DT (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ of functions analytic on a certain polydisk. With this commutative Banach algebra in hand, we can carry out the disentangling calculations called for by Feynman’s “rules” in a mathematically rigorous fashion. Once the disentangling is carried out in the algebra DT , we map the result to the non-commutative setting of L(X) using the so-called disentangling map TμT1 ,...,μn ; it is the image under the disentangling map that is the disentangled operator given by the application of Feynman’s “rules”. We note that changing the n-tuple of time ordering measures will,
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in general, change the operational calculus, as it will usually change the action of the disentangling map. Of course, a change in the operators will also generally change the operational calculus. The stability theory for the Jefferies–Johnson formulation of the operational calculus was developed initially in [15] and expanded on in [10, 13, 14], and [12]. In particular, stability with respect to the time ordering measures, one focus of the current paper, can be described as follows. We select sequences {μik }∞ k=1 of Borel probability measures on [0, T] such that μik μi as k → ∞. We then have, for each k ∈ N, a particular operational calculus, given by the action of TμT1k ,...,μnk , indexed by the n-tuple (μ1k , . . . , μnk ) of measures and thus a sequence of operational calculi. The stability question is then the question of whether the sequence of operational calculi has a limiting operational calculus as k → ∞. The stability theory, in its current form, is not entirely satisfactory, at least in the time dependent setting. For example, consider the following stability theorem in the time dependent setting (in the setting of the current paper, the space X will be a separable Hilbert space): Theorem Let Ai : [0, T] → L(X), i = 1, . . . , n, be continuous with respect to the usual topology on [0, T] and the norm topology on L(X). Associate to each Ai (·) a continuous Borel probability measure μi on [0, T]. Let {μik }∞ k=1 , i = 1, . . . , n, be sequences of Borel probability measures on [0, T] such that, for each i = 1, . . . , n, μik μi . Construct the direct sum Banach algebra U D := DT (A1 (·), μ1k )∼ , . . . , (An (·), μnk )∼ k∈N∪{0}
where for k = 0 the summand is DT (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ . Denote by · U D the norm on U D . Then lim TμT1k ,...,μnk (πk (θ f )) − TμT1 ,...,μn (π0 (θ f )) = 0 k→∞
for all ∈ L(X)∗ and all θ f = ( f, f, f, . . .) ∈ U D .
Remark 1.1 This theorem appears as Theorem 3.1 of [14]. We see that the presence of the sequences of measures gives us a countable family of disentangling algebras DT (A1 (·), μ1k )∼ , . . . , (An (·), μnk )∼ . Of ∞ T course, we also obtain a sequence Tμ1k ,...,μnk k=1 of disentangling maps (or, if you like, a sequence of operational calculi). Now, consider the conclusion of the theorem. In order to obtain the conclusion we have to choose a functional from the dual of L(H); this is not terribly satisfactory. In fact, it would be much more preferable to obtain a conclusion such as
T
T
T
μ1k ,...,μnk (πk (θ f ))φ − Tμ1 ,...,μn (π0 (θ f ))φ H → 0
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for every φ ∈ H; i.e. strong operator convergence. Indeed, in an applied setting, say in non-relativistic quantum mechanics, strong operator convergence is desirable. It is at this point that the main theorem, Theorem 2.3 below, comes into play. The reason that a functional from L(H)∗ had to be used in the theorem as stated above is due to the weak convergence hypotheses on the sequence of measures—a bounded and continuous real or complex valued function is needed in order to apply the standard theorems on weak convergence of probability measures. With Theorem 2.3 in hand, we can obtain the preferred strong operator convergence. While the proof of the theorem as stated above has appeared in [14], we will outline the proof below noting where Theorem 2.3 comes into play. As a second example of applying Theorem 2.3, we will state and prove a theorem similar in nature to the theorem above but where the time ordering measures are allowed to have both continuous and discrete parts. This theorem has not previously appeared in print though the proof is similar to the proof of the theorem above; however, the presence of the discrete measures changes the proof in some significant ways.
2 The Weak Convergence Theorem In this section we will state and prove the weak convergence theorem. Throughout this section H will be a separable Hilbert space. (While this assumption on the Hilbert space is, strictly speaking, not necessary, it does simplify the exposition. We could make the assumption that our Hilbert space valued functions take their values in a separable subpace of H and proceed accordingly.) Also, S will be an arbitrary metric space. Before stating the theorem, we will remind the reader of the definition of weak convergence of a sequence of probability measures. This definition can be found in many places, for example [1]. Definition 2.1 Let S be a metric space and let {μk }∞ k=1 be a sequence of Borel probability measures on S. We say that this sequence converges weakly to the Borel probability measure μ on S if
f (s) μk (ds) =
lim
k→∞
S
f (s) μ(ds) S
for every bounded continuous real-valued function f on S. Weak convergence of μk to μ will be denoted by μk μ. Remark 2.2 Of course, weak convergence in this probabilistic sense is, from a functional analytic viewpoint, weak-∗ convergence. We now state the theorem.
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Theorem 2.3 Let H be a separable Hilbert space and let S be a metric space. Let {μk }∞ k=1 and μ be Borel probability measures on S such that μk μ. Then lim f (s) μk (ds) = f (s) μ(ds) (2.1) k→∞
S
S
in norm on H for every continuous f : S → H such that sups∈S f (s)H < ∞. Proof Let f : S → H be continuous and such that sups∈S f (s)H < ∞. Note that, as f is continuous, it is Bochner integrable with respect to each of our probability measures. Since H is separable, there is a countable orthonormal basis {en }∞ n=1 for H. For n ∈ N define the orthogonal projection Pn : H → span {e1 , . . . , en } by ⎛ ⎞ ∞ n α je j⎠ := α je j. (2.2) Pn ⎝ j=1
j=1
Next define the map ψn : span {e1 , . . . , en } → Cn by ⎛ ⎞ n ψn ⎝ α je j⎠ := (α1 , . . . , αn ) .
(2.3)
j=1
Then, clearly, ψn is linear. Moreover, since
⎛
2 ⎞ 2
n n
2 2
ψn ⎝ ⎠ α je j = |α1 | + · · · + |αn | =
α je j
,
j=1
n
j=1 C
(2.4)
H
it follows that ψn , n ∈ N, is an isometry. We consider the map ψn ◦ Pn ◦ f : S → Cn . Because f is continuous and norm bounded, and since Pn and ψn are continuous and of norm 1, the map ψn ◦ Pn ◦ f is continuous and bounded from S into Cn . For convenience write ψn ◦ Pn ◦ f (s) := (g1 (s), . . . , gn (s))
(2.5)
for bounded and continuous complex-valued functions g1 , . . . , gn on the metric space S. We can write, for j = 1, . . . , n, g j(s) = u j(s) + iv j(s) where u j and v j are continuous and bounded real-valued functions on S. It follows that lim g j(s) μk (ds) = lim u j(s) μk (ds) + i v j(s) μk (ds) k→∞
k→∞
S
S
S
u j(s) μ(ds) + i
=
S
=
g j(s) μ(ds) S
v j(s) μ(ds) S
(2.6)
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and so, applying the above result component-wise, we have ψn ◦ Pn ◦ f (s) μk (ds) = ψn ◦ Pn ◦ f (s) μ(ds). lim k→∞
S
(2.7)
S
Because of the linearity of ψn , we may write ψn ◦ Pn ◦ f (s) μk (ds) − ψn ◦ Pn ◦ f (s) μ(ds) S
S
= ψn
Pn ◦ f (s) μ(ds)
Pn ◦ f (s) μk (ds) − S
(2.8)
S
and so, because ψn is an isometry,
ψn Pn ◦ f (s) μk (ds) − Pn ◦ f (s) μ(ds)
S
Cn
S
. =
P ◦ f (s) μ (ds) − P ◦ f (s) μ(ds) n k n
S
(2.9)
H
S
We have shown above that the quantity on the left-hand-side of (2.9) vanishes in the limit as k → ∞ and therefore it follows that
(2.10) lim Pn ◦ f (s) μk (ds) − Pn ◦ f (s) μ(ds)
= 0. k→∞
S
S
We now move on to investigate limn→∞ is clear that, for each s ∈ S, we have
H
S
Pn ◦ f (s) μk (ds) for any k ∈ N. It
lim Pn ◦ f (s) − f (s)H = 0.
(2.11)
Pn ◦ f (s) − f (s)H 2 f (s)H 2 sup f (s)H
(2.12)
n→∞
Since s∈S
and since sups∈S f (s)H < ∞, we can take h(s) := 2 sups∈S f (s)H for the scalar dominating function that is needed in order to apply the dominated convergence theorem for Bochner integrals. This function is integrable with respect to each of the measures μk and μ and the value of the integral is the same for each measure since they are all probability measures. Applying the dominated convergence theorem for Bochner integrals leads to the statement that lim Pn ◦ f (s) μk (ds) = f (s) μk (ds) (2.13) n→∞
S
S
in norm on H. Note that this limit is, in fact, uniform in k ∈ N. We have now established the following limit statements. First, Pn ◦ f (s) μk (ds) = f (s) μk (ds) lim n→∞
S
S
(2.14)
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in norm on H, uniformly in k ∈ N, and second, lim Pn ◦ f (s) μk (ds) = Pn ◦ f (s) μ(ds) k→∞
S
(2.15)
S
in norm on H for each n ∈ N. We now apply a theorem of E. H. Moore on iterated limits (see [2], page 28). Since we have established the limit on n uniformly in k and the limit on k for each fixed n, Moore’s theorem tells us that lim Pn ◦ f (s) μk (ds) = lim lim Pn ◦ f (s) μk (ds) n→∞ k→∞
n→∞ k→∞
S
S
Pn ◦ f (s) μk (ds)
= lim lim
k→∞ n→∞
=
S
f (s) μ(ds)
(2.16)
in norm on H. The proof of the theorem is finished.
S
We can modify the previous theorem somewhat to obtain a result that we will find useful below. Indeed, we will be concerned below not only with a weakly convergent sequence of probability measures, but also with a corresponding uniformly convergent sequence of H-valued functions. (The realvalued version of this theorem appears as Lemma 3.2 of [12].) The following theorem addresses this situation. Theorem 2.4 Let H be a separable Hilbert space. Let μk , μ be, for k ∈ N, Borel probability measures on the metric space S. Let fk , f , k ∈ N, be continuous norm bounded H-valued functions on S. If μk μ and if fk → f uniformly in H-norm on S, then fk dμk = f dμ (2.17) lim k→∞
E
E
in norm for any Borel set E ⊂ S with μ(∂ E) = 0 (that is, E is a μ-continuity set). Proof Fix a Borel set E in S such that μ(∂ E) = 0. For any k ∈ N, we may write
fk dμk − f dμ fk − f H dμk +
f dμk − f dμ
. (2.18) E
E
H
E
E
E
H
Since fk → f uniformly in H-norm, given > 0, there is a k0 ∈ N such that if k k0 , sup fk (s) − f (s)H < s∈E
2
(2.19)
Weak convergence and vector-valued functions
and so, for k k0 ,
279
fk − f H dμk < E
μk (E) . 2 2
(2.20)
The remainder of the proof amounts to showing that the second term in (2.18) vanishes as k → ∞. We will proceed much as in the proof of Theorem 2.3. Indeed, taking any orthonormal basis e j j∈N , we define, for every n ∈ N, the orthogonal projection Pn : H → span {e1 , . . . , en } exactly as in the proof of Theorem 2.3. Next, we define ψn : span {e1 , . . . , en } → Cn exactly as in the proof of Theorem 2.3. Then ψn is a linear isometry for every n ∈ N. We now consider
ψ ◦ P ◦ f (s) dμ (s)− ψ ◦ P ◦ f (s) dμ(s) lim
n n k n n
n
k→∞ E E C
= lim χ E · ψn ◦ Pn ◦ f (s) dμk (s)− χ E · ψn ◦ Pn ◦ f (s) dμ(s)
(2.21)
k→∞
S
Cn
S
Since χ E · ψn ◦ Pn ◦ f has its discontinuities contained in ∂ E and since μ(∂ E) = 0, Theorem 5.2 of [1] (applied component-wise) shows that
lim χ E · ψn ◦ Pn ◦ f (s) dμk (s)− χ E · ψn ◦ Pn ◦ f (s) dμ(s)
= 0. (2.22) k→∞
S
Cn
S
Because ψn is a linear isometry, it follows at once that, for each n ∈ N,
Pn ◦ f dμk − Pn ◦ f dμ
E
H
E
= ψn Pn ◦ f dμk − Pn ◦ f dμ
E
Cn
E
= ψn ◦ Pn ◦ f dμk − ψn ◦ Pn ◦ f dμ
E
Cn
E
→0
(2.23)
as k → ∞. Next, we use the fact that Pn ◦ f (s) − f (s)H → 0 as n → ∞ for each s ∈ S. Since
Pn ◦ f dμk = χ E · Pn ◦ f dμk
E
H
H
S
sup |χ E (s)| f (s)H sup f (s)H , s∈S
(2.24)
s∈S
(uniformly in k ∈ N) we are free to use the dominated convergence theorem for Bochner integrals to obtain lim Pn ◦ f dμk = f dμ (2.25) n→∞
E
E
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in norm on H. Moreover, this limit is uniform in k. The limit results in equations (2.23) and (2.25) enable us to apply the theorem of E. H. Moore ([2], page 28) on iterated limits exactly as in the proof of Theorem 2.3 to obtain
=0 f dμ − f dμ (2.26) lim
k k
k→∞
E
H
E
and therefore we can choose a k1 ∈ N so that
f dμ − f dμ k k
E
E
H
<
2
(2.27)
whenever k > k1 . Let k2 = max (k0 , k1 ). Then, for k > k2 , we have
f dμ − f dμ fk − f H dμk + k k
E
H
E
+
f dμ − f dμ k
E
E
H
E
<
+ = . 2 2
3 The Disentangling Map We now move on to a discussion of the disentangling map. Before defining the map, however, we need some preliminary definitions and notation (see [4, 8, 11]). (In fact, we follow the paper [11] quite closely here even though that paper was concerned only with the time independent setting and we are here concerned with the time dependent setting.) We begin by introducing two commutative Banach algebras AT and DT . These algebras are closely related and play an important role in the rigorous development of the operational calculus. Given n ∈ N and n positive real numbers r1 , . . . , rn , let AT (r1 , . . . , rn ) or, more briefly AT , be the space of complex-valued functions (z1 , . . . , zn ) → f (z1 , . . . , zn ) of n complex variables that are analytic at the origin and are such that their power series expansion f (z1 , . . . , zn ) =
∞
mn 1 am1 ,...,mn zm 1 · · · zn
(3.1)
m1 ,...,mn =0
converges absolutely at least in the closed polydisk |z1 | r1 , . . . , |zn | rn . All of these functions are analytic at least in the open polydisk |z1 | < r1 , . . . , |zn | < rn . We remark that the entire functions of (z1 , . . . , zn ) are in AT (r1 , . . . , rn ) for any n-tuple (r1 , . . . , rn ) of positive real numbers. For f ∈ AT given by (3.1) above, we let f = f AT :=
∞ m1 ,...,mn =0
|am1 ,...,,mn |r1m1 · · · rnmn .
(3.2)
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281
This expression is a norm on AT and turns AT into a commutative Banach algebra (see Proposition 1.1 of [4]; in fact AT is a weighted 1 -space). We now turn to the construction of the Banach algebra DT . To give the most general definition, we will let X be a separable Banach space. Let Ai : [0, T] → L(X), i = 1, . . . , n, be measurable in the sense that Ai−1 (E) is a Borel set in [0, T] for every strongly open subset E of L(X). Associate to each Ai (·) a Borel probability measure μi on [0, T] with μi = λi + ηi
(3.3)
for i = 1, . . . , n where λi is a continuous measure for each i and ηi is a finitely supported discrete measure for each i. Let {τ1 , . . . , τh } be the set obtained by taking the union of the discrete measures η1 , . . . , ηn and write ηi =
h
pijδτ j
(3.4)
j=1
for each i = 1, . . . , n. (We will assume, for convenience, that τ1 < τ2 < · · · < τh .) With this notation it may well be that many of the pij’s are equal to zero. We now define n positive real numbers r1 , . . . , rn by ri := Ai (s)L(X) μi (ds) (3.5) [0,T]
for each i = 1, . . . , n. These real numbers will serve as weights and we ignore for the present the nature of the Ai (·) as operators and introduce a commutative Banach algebra DT (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ (the disen- tangling algebra) of “analytic functions” f (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ or, more briefly written, f (A1 (·)∼ , . . . , An (·)∼ ) where the objects (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ or, more briefly, A1 (·)∼ , . . . , An (·)∼ replace the indeterminants z1 , . . . , zn . For brevity, we will usually refer to the disentangling algebra as DT . (We write (Ai (·), μi )∼ for the objects replacing z1 , . . . , zn to stress that these objects depend not only on the operator-valued functions but also on the measures we associate with them.) It is worth noting here that the operatorvalued functions do not have to be distinct though we will still consider the formal objects obtained from them to be distinct in the Banach algebra DT . All of this having been said, we take DT (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ to be the collection of all expressions of the form ∼
∼
f (A1 (·) , . . . , An (·) ) =
∞
am1 ,...,mn (A1 (·)∼ )m1 · · · (An (·)∼ )mn
(3.6)
am ,...,m rm1 · · · rmn . 1 n n 1
(3.7)
m1 ,...,mn =0
with the norm defined by f DT =
∞ m1 ,...,mn =0
Via coordinate-wise addition and multiplication of such expressions it easily follows that (3.7) is a norm. Similarly, coordinate-wise addition and multiplication of the expressions seen in (3.6) causes us to observe that DT is a
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commutative Banach algebra (see Proposition 1.2 of [4]). Moreover the Banach algebras AT and DT can be identified (see Proposition 1.3 of [4]; the proof of Proposition 1.3 in [4] is, of course, given in the time independent setting although it turns out that the proof in the time dependent setting is the same). We work here in the commutative setting of the disentangling algebra DT . The definition of the disentangling map will depend on the disentangling of the monomial Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) := (A1 (·)∼ )m1 · · · (An (·)∼ )mn .
(3.8)
Also, it is the disentangling of the monomial that shows best the connection between Feynman’s ideas and this theory. We now introduce the notation that is necessary for the disentangling map. For m ∈ N, let Sm be the set of all permutations of the integers {1, . . . , m} and given π ∈ Sm , we let m (π ) = (s1 , . . . , sm ) ∈ [0, T]m : 0 < sπ(1) < · · · < sπ(m) < T . (3.9) When π is the identity permutation it is common to write m (π ) as m . For j = 1, . . . , n and all s ∈ [0, T], we let A j(s)∼ = A j(·)∼ ;
(3.10)
that is, we discard the time dependence of the operator-valued functions though we will use the time index to keep track of when a given operator acts. Next, given nonnegative integers m1 , . . . , mn and letting m = m1 + · · · + mn , we define ⎧ i ∈ {1, . . . , m1 } A1 (s)∼ if ⎪ ⎪ ⎪ ⎨ A2 (s)∼ if i ∈ {m1 + 1, . . . , m1 + m2 } i (s) = (3.11) C .. .. ⎪ . . ⎪ ⎪ ⎩ i ∈ {m1 + · · · + mn−1 + 1, . . . , m} An (s)∼ if for i = 1, . . . , m and s ∈ [0, T]. Even though Ci (s)∼ clearly depends on the nonnegative integers m1 , . . . , mn , we will suppress this dependence in our notation to ease the presentation. Next, in order to accommodate the use of discrete measures below, we will need a refined version of the time ordered sets m (π ) given above in (3.9). Let τ1 , . . . , τh ∈ [0, T] be such that 0 < τ1 < · · · < τh < T. (Of course, in our setting, the τi will be the elements of the union of the supports of the discrete measures ηi defined above.) Given m ∈ N, π ∈ Sm , and nonnegative integers r1 , . . . , rh+1 such that r1 + · · · + rh+1 = m, define m;r1 ,...,rh+1 (π ) = {(s1 , . . . , sm ) ∈ [0, 1]m : 0 < sπ(1) < · · · < sπ(r1 ) < τ1 < sπ(r1 +1) < · · · < sπ(r1 +r2 ) < τ2 < sπ(r1 +r2 +1) < · · · < sπ(r1 +···+rh ) < τh < sπ(r1 +···+rh +1) < · · · < sπ(m) < 1}.
(3.12)
Note: Cases where τ1 = 0 and/or τh = T are sometimes of interest. A comment on these cases can be found after the statement of the proposition below.
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283
We may have different operators evaluated at the same τi when we carry out time-ordering calculations. When this happens, Feynman’s “rules” above do not specify the order of operation and so a choice needs to be made. (This situation arises in certain physical problems where a particular choice may be natural for the problem at hand.) We can make any choice for the ordering and, indeed, can make different choices at each τi . For convenience, we will make a definite choice, letting A1 (·) act first, A2 (·) act second, etc. We are now prepared to time-order the monomial Pm1 ,...,mn according to the directions provided by the measures μ1 , . . . , μn . We note that the calculation leading to the time-ordered expression below are much more complicated than the corresponding calculation found in Proposition 2.2 of [4] for continuous time ordering measures. The details of the calculation can be found in [11] and we simply quote the result here. Proposition Let m1 , . . . , mn ∈ N be given. Then the monomialPm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) is given in time ordered form by Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) × ··· = q11 +q12 =m1 q21 +q22 =m2
×
qn1 +qn2 =mn
·
m1 ! · · · mn ! q11 !q12 !q21 !q22 ! · · · qn1 !qn2 !
×
π ∈Sq11 +q21 +···+qn1 r1 +···+rh+1 =q11 +q21 +···+qn1 j11 +···+ j1h =q12 j21 +···+ j2h =q22
···
jn1 +···+ jnh =qn2
·
q12 !q22 ! · · · qn2 ! j11 ! · · · j1h ! j21 ! · · · j2h ! · · · jn1 ! · · · jnh !
×
×
π(q11 +q21 +···+qn1 ) (sπ(q11 +q21 +···+qn1 ) ) ·C
q11 +q21 +···+qn1 ;r1 ,··· ,rh+1 (π )
π(r1 +···+rh +1) (sπ(r +···+r +1) ) · pnh An (τh )∼ jnh · · · p2h A2 (τh )∼ j2h × ···C 1 h j π(r1 +···+rh ) (sπ(r1 +···+rh ) ) · · · C π(r1 +1) (sπ(r1 +1) ) × × p1h A1 (τh )∼ 1h · C
j j j π(r1 ) (sπ(r1 ) ) × pn1 An (τ1 )∼ n1 · · · p21 A2 (τ1 )∼ 21 p11 A1 (τ1 )∼ 11 C q11 π(1) (sπ(1) ) · λ × · · · × λqn1 ds1 , . . . , dsq11 +q21 +···+qn1 ···C n 1
(3.13) where λi is the continuous part of μi [see (3.3)]. Note: If τ1 = 0, the first element to act on the right-hand side of (3.13) j above would be p11 A1 (τ1 )∼ 11 . If τh = 1, the last element to act would be j pnh An (τh )∼ nh . Now that we have the time ordered monomial in hand, we can define the disentangling map TμT1 ,...,μn which will take us from the commutative setting of
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L. Nielsen
the disentangling algebra DT to the non-commutative setting of L(X). All that i (s) by the corresponding operator-valued we need to do is replace the objects C functions. This amounts to erasing the tildes; to be precise we define ⎧ i ∈ {1, . . . , m1 } A1 (s) if ⎪ ⎪ ⎪ ⎨ A2 (s) if i ∈ {m1 + 1, . . . , m1 + m2 } (3.14) Ci (s) = .. .. ⎪ . . ⎪ ⎪ ⎩ i ∈ {m1 + · · · + mn−1 + 1, . . . , m} An (s) if Similarly, in the evaluations at τ1 ,. . ., τh we replace Aj(·)∼ with Aj(·) for each j. Definition 3.1 We define the action of the disentangling map TμT1 ,...,μn on the monomial Pm1 ,...,mn by TμT1 ,...,μn Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) × m1 ! · · · mn ! = ··· × q11 !q12 !q21 !q22 ! · · · qn1 !qn2 ! q +q =m q +q =m q +q =m 11
12
×
1
21
22
n1
2
·
n2
n
π ∈Sq11 +q21 +···+qn1 r1 +···+rh+1 =q11 +q21 +···+qn1 j11 +···+ j1h =q12 j21 +···+ j2h =q22
···
·
jn1 +···+ jnh =qn2
q12 !q22 ! · · · qn2 ! j11 ! · · · j1h ! j21 ! · · · j2h ! · · · jn1 ! · · · jnh !
×
×
· Cπ(q11 +q21 +···+qn1 ) (sπ(q11 +q21 +···+qn1 ) )
q11 +q21 +···+qn1 ;r1 ,··· ,rh+1 (π )
j j · · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) · pnh An (τh ) nh · · · p2h A2 (τh ) 2h × j × p1h A1 (τh ) 1h · Cπ(r1 +···+rh ) (sπ(r1 +···+rh ) ) · · · Cπ(r1 +1) (sπ(r1 +1) ) × j j j × pn1 An (τ1 ) n1 p21 A2 (τ1 ) 21 p11 A1 (τ1 ) 11 Cπ(r1 ) (sπ(r1 ) ) q · · · Cπ(1) (sπ(1) ) · λ1 11 × · · · × λqnn1 ds1 , . . . , dsq11 +q21 +···+qn1 . (3.15) Then, for f (A1 (·)∼ , . . . , An (·)∼ ) ∈ DT (A1 (·)∼ , . . . , An (·)∼ ), given by f (A1 (·)∼ , . . . , An (·)∼ ) =
∞
am1 ,...,mn (A1 (·)∼ )m1 · · · (An (·)∼ )mn
(3.16)
m1 ,...,mn =0
we set TμT1 ,...,μn f (A1 (·)∼ , . . . , An (·)∼ )
=
∞ m1 ,...,mn =0
am1 ,...,mn TμT1 ,...,μn Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) . (3.17)
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The disentangling map as defined here does indeed give a bounded linear operator (in fact, a contraction) from DT to L(X). For a proof of this, see Theorem 2 of [11]. (The proof given there is for the time independent setting although the proof for the time dependent setting is essentially identical except for the weights used.) Remark 3.2 We mention here that we will sometimes use fμ1 ,...,μn (A1 (·), . . . , An (·)) in place of TμT1 ,...,μn f (A1 (·)∼ , . . . , An (·)∼ ) . The definition of the disentangling map applied to the monomial Pm1 ,...,mn , quite complicated when the time ordering measures have discrete parts, simplifies significantly when the time ordering measures are all continuous. (We recall that the measure μ on S is continuous when μ({x}) = 0 for all s ∈ S.) Indeed, Proposition 1 of [11] shows that if μ1 , . . . , μn are continuous, then TμT1 ,...,μn Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) = Cπ(m1 +···+mn ) (sπ(m1 +···+mn ) ) · · · Cπ(1) (sπ(1) ) × π ∈Sm1 +···+mn
m1 +···+mn (π )
mn 1 × (μm 1 × · · · × μn )(ds1 , . . . , dsm1 +···+mn ),
(3.18)
and this is the disentangling of the monomial that is found in [4] and [8]. The disentangling of an arbitrary element of DT is found by applying the disentangling of the monomial term-by-term in the power series. We state the end result below: fμ1 ,...,μn (A1 (·)∼ , . . . , An (·)∼ ) =
∞
am1 ,...,mn
m1 ,...,mn =0
Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) ) ·
π ∈Sm T m (π )
1 mn (ds1 , . . . , dsm ). · μm 1 × · · · × μn
(3.19)
where m = m1 + · · · + mn .
4 Improving the Operational Calculus Now that we have proved Theorem 2.3 giving a relation between weak convergence of sequences of probability measures and Hilbert space valued functions in Section 2 and have outlined the operational calculus in the time dependent setting in Section 3, we are ready to discuss how Theorem 2.3 improves the operational calculus, at least when we are working in a separable Hilbert space. Remark 4.1 As mentioned above, the restriction to a separable Hilbert space, while not as general as might be hoped, is often not at all a handicap. Indeed,
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when working in the standard nonrelativistic quantum mechanical setting, we work in the separable Hilbert space L2 (Rd ). We will first address the theorem stated in the introduction above (from [14]). For the reader’s convenience we will state the theorem again: Theorem 4.2 Let Ai : [0, T] → L(X), i = 1, . . . , n, be continuous with respect to the usual topology on [0, T] and the norm topology on L(X). Associate to each Ai (·) a continuous Borel probability measure μi on [0, T]. Let {μik }∞ k=1 , i = 1, . . . , n, be sequences of Borel probability measures on [0, T] such that, for each i = 1, . . . , n, μik μi . Construct the direct sum Banach algebra U D := DT (A1 (·), μ1k )∼ , . . . , (An (·), μnk )∼ k∈N∪{0}
where for k = 0 the summand is DT (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ . Then lim TμT1k ,...,μnk (πk (θ f )) − TμT1 ,...,μn (π0 (θ f )) = 0 k→∞
for all ∈ L(X)∗ and all θ f = ( f, f, f, . . .) ∈ U D . With Theorem 2.3 in hand, the statement of the theorem above can be changed to the following: Theorem 4.3 For i = 1, . . . , n let Ai : [0, T] → L(H), H a separable Hilbert space, be continuous with respect to the usual topology on [0, T] and the norm topology on L(H). Associate to each Ai (·) a continuous Borel probability measure μi on [0, T]. For each i = 1, . . . , n, let {μik }∞ k=1 be a sequence of continuous Borel probability measures on [0, T] such that μik μ as k → ∞. Construct the direct sum Banach algebra U D := DT (A1 (·), μ1k )∼ , . . . , (An (·), μnk )∼ k∈N∪{0}
where for k = 0 the summand is DT (A1 (·), μ1 )∼ , . . . , (An (·), μn )∼ . Then, for any θ f := ( f, f, f, . . .) ∈ U D and any φ ∈ H, we have
lim TμT1k ,...,μnk (πk (θ f ))φ − TμT1 ,...,μn (π0 (θ ( f ))φ H = 0 (4.1) k→∞
where πk is the canonical projection of U D onto the disentangling algebra indexed by the measures μ1k , . . . , μnk . Of course, we can think of the conclusion in (4.1) as
lim TμT1k ,...,μnk ( f )φ − TμT1 ,...,μn ( f )φ H = 0. k→∞
Remark 4.4 Comparing the statement of Theorem 4.3 to the statement of Theorem 4.2, we see the difference quite clearly. We obtain strong operator
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convergence in Theorem 4.3, a much stronger conclusion than in Theorem 4.2. The proofs are quite similar and below we will sketch the proof, going into some detail concerning the use of Theorem (2.3). Proof Let m1 , . . . , mn ∈ N and let φ ∈ H. We show first that lim TμT1k ,...,μnk Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) φ −
k→∞
− TμT1 ,...,μn Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) φH = 0.
(4.2)
NOTE: It is in proving this assertion that the difference in the proof of this theorem as compared to Theorem 4.2 arises. We need only choose a vector from H instead of a linear functional on L(H) (or a linear functional λ ∈ H∗ and a vector φ ∈ H). Using the definition of the disentangling map, remembering that we are in the continuous measure setting of the operational calculus, we can write the norm difference above as
1 mn Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )φ μm 1k × · · · × μnk (ds1 , . . . , dsm ) −
π ∈Sm (π ) m
−
Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )φ
π ∈Sm (π ) m
mn 1 μm (ds1 , . . . , dsm )
1 ×· · ·×μn
. H (4.3)
We now note that, since the operator-valued functions Ai (·) are all continuous, the function fm : [0, T]m → H given by fm (s1 , . . . , sm ) = Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )φ
(4.4)
is then a continuous function. Moreover, it is a norm-bounded function into the Hilbert space, since each of the Ai (·) is a continuous function on a compact 1 subset of R. Also, since [0, T]m is a separable metric space, we have μm 1k × mn m1 mn · · · × μnk μ1 × · · · × μn as k → ∞ (see [1], Theorem 3.2). It follows from Theorem 2.3 that
1 mn Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )φ μm lim
1k × · · · × μnk (ds1 , . . . , dsm ) −
k→∞ π ∈Sm (π ) m
−
Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )φ
π ∈Sm (π ) m
This establishes our assertion.
mn 1 μm (ds1 ,. . ., dsm )
1 ×· · ·×μn
= 0. H (4.5)
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We now sketch the remainder of the proof, reminding the reader that it follows the proof of Theorem 3.1 of [14] very closely. Let θ f = ( f, f, f, . . .) ∈ U D and write f as in (3.6) above. For φ ∈ H, we can write TμT1k ,...,μnk (πk (θ f ))φ − TμT1 ,...,μn (π0 (θ f ))φH
∞
=
a Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )φ × m ,...,m 1 n
m1 ,...,mn =0
π ∈Sm (π ) m
1 mn × μm 1k × · · · × μnk (ds1 , . . . , dsm ) −
∞
am1 ,...,mn
m1 ,...,mn =0
m1 mn × Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) )φ μ1 × · · · × μn (ds1 , . . . , dsm )
∞
φH
|am1 ,...,mn |
m1 ,...,mn =0
×
π ∈Sm (π ) m
H
Cπ(m) (sπ(m) )
π ∈Sm (π ) m
1 mn · · · Cπ(1) (sπ(1) ) μm 1k × · · · × μnk (ds1 , . . . , dsm ) + ∞ |am1 ,...,mn | Cπ(m) (sπ(m) ) · · · Cπ(1) (sπ(1) ) × + m1 ,...,mn =0
π ∈Sm (π ) m
1 mn × μm × · · · × μ (ds , . . . , ds ) = φH f Dk + f D0 1 m n 1
(4.6)
where the subscript Dk refers to the kth - disentangling algebra in the direct sum algebra U D . (The last line is arrived at via the standard Banach algebra inequality xy xy which results in a product of real-valued and consequently commutative functions. The disentangling is then “unraveled” or reversed to obtain the last line.) Recall that the norm on U D is {g }∞ =1 U D = sup g D . ∈N∪{0}
Let > 0 be given. There is a k0 ∈ N such that θ f U D < f k0 + . Using (4.6) we therefore have TμT1k ,...,μnk (π0 (θ f ))φ − TμT1 ,...,μn (π0 (θ f ))φH φH f k0 + f 0 + (4.7) We see, then, that a summable scalar-valued dominating function for ∞
n |am1 ,...,mn | Pμm1k1 ,...,m ,...,μnk (A1 (·), . . . , An (·)) φ −
m1 ,...,mn =0
,...,mn − Pμm11,...,μ (A1 (·), . . . , An (·))φ H n
(4.8)
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is
289
φ H mn m1 m1 mn (m1 , . . . , mn ) → φH |am1 ,...,mn | r1,k · · · r + r · · · r + n 1 n,k 0 0 2m
where the weights ri,k0 are
(4.9)
ri,k0 = and, similarly,
[0,T]
Ai (s) μik0 (ds)
(4.10)
Ai (s) μi (ds).
(4.11)
ri :=
[0,T]
We can therefore apply the dominated convergence theorem for Bochner integrals and pass the limit on the index k through the sum over m1 , . . . , mn . Using (4.2) we finish the proof.
We now move onto the theorem concerning the stability of the disentangling map with respect to time-ordering measures that have discrete parts. In the time dependent setting, this theorem has not appeared in print. A time independent version of this theorem has, however, appeared in [12]. We remark here that the proof in the time independent setting is more straight forward than the proof in the time dependent setting. The main difficulty is, besides the obvious combinatoric complexity of the combined continuous/discrete setting, that we once again have a countably infinite family of disentangling algebras. The strategy of the proof, as one would expect, is the same as the proof just above. We will show that the limit of the norm difference of disentangled monomials vanishes and then verify that we can use the dominated convergence theorem for Bochner integrals to pass the limit on the sequence of measures through the sum over m1 , . . . , mn . The presence of discrete parts to the time ordering measures will cause some complications, as one would expect. Before stating the theorem, however, we need a lemma concerning weak convergence of the types of discrete measures that we will be using. (This lemma is Lemma 3.1 of [12]; the proof will not be presented here.) h Lemma 4.5 η := i=1 pi δτi be a purely discrete probability measure on [0, T] with finite support. Assume that 0 < τ1 < · · · < τh < T. Let αi := min (τi − τi−1 , τi+1 − τi ) for i = 1, . . . , h where we take τ0 = 0 and τh+1 = T. In each interval (τi − αi , τi + αi ), i = 1, . . . , h, choose sequences {τik }∞ k=1 . For each i = 1, . . . , h choose a sequence { pik }∞ such that k=1 ηk =
h i=1
pik δτik
(4.12)
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is a probability measure for each k. Then ηk η if and only if
pik → pi and τik → τi if pi = 0 pik → pi and {τik }∞ bounded if p i =0 k=1
(4.13)
for i=1,. . . ,h. We now state the stability theorem. Theorem 4.6 Let Ai : [0, T] → L(H), i = 1, . . . , n, be continuous with respect to the norm topology on L(H). Associate to Ai (·), i = 1, . . . , the continuous Borel probability measure μi on [0, T] and associate to Ai (·), i = + 1, . . . , n, the purely discrete probability measure ηi with finite support. Let {τ1 , . . . , τh } be the union of the supports of the ηi , assume that 0 < τ1 < · · · < τh < T and write ηi =
h
pijδτ j
(4.14)
j=1
for each i = + 1, . . . , n. (As observed above, some of the numbers pij may ∞ ∞ be zero.) For each i = + 1, . . . , n choose sequences τ jk k=1 and pijk k=1 ∞ as in Lemma 4.5. (In particular, we will assume that each sequence τ jk k=1 converges.) Then ηik =
h
pijk δτ jk
(4.15)
j=1
converges weakly to ηi for each i = +1, . . . , n. Also, choose sequences {μik }∞ k=1 , i = 1, . . . , , of continuous Borel probability measures such that μik μi . 4.3 where the Define the direct sum Banach algebra U D as in Theorem disentangling algebras forming the summands are DT (A1 (·), μ1k )∼ , . . . , ∼ (A (·), μ k )∼ , A +1 (·), η +1,k , . . . , (An (·), ηnk )∼ for k = 0 and DT ((A1 (·), μ1 )∼ , . . . , (A (·), μ )∼ , (A +1 (·), η +1 )∼ , . . . , (An (·), ηn )∼ for k = 0. We conclude that, for any θ f = ( f, f, , f, . . .) ∈ U D and φ ∈ H, lim TμT1k ,...,μ k ,η +1,k ,...,ηnk (πk (θ f ))φ − TμT1 ,...,μ ,η +1 ,...,ηn (π0 (θ f ))φH = 0. (4.16)
k→∞
Proof Let φ ∈ H. For m1 , . . . , mn ∈ N, we first show that lim TμT1k ,...,μ k ,η +1,k ,...,ηnk Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) φ −
k→∞
− TμT1 ,...,μ ,η +1 ,...,ηn Pm1 ,...,mn (A1 (·)∼ , . . . , An (·)∼ ) φH = 0.
(4.17)
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As before, Theorem 2.3 will be crucial. We write (4.17) using the definition of the disentangling map for the combined continuous/discrete setting. Indeed, (3.15) lets us write m1 ,...,mn n Pμm1k1 ,...,m ,...,μ k ,η +1,k ,...,ηnk (A1 (·),. . ., An (·))− Pμ1 ,...,μ ,η +1 ,...,ηn (A1 (·),. . ., An (·)) H
j +1,1 +···+ j +1,h =m +1
×
···
jn1 +···+ jnh =mn
π ∈Sm1 +···+m r1 +···+rh+1 =m1 +···+m
m +1 ! · · · mn ! × j +1,1 ! · · · j +1,h ! · · · jn1 ! · · · jnh !
Cπ(m1 +···+m ) (sπ(m1 +···+m ) )
T m (π ) 1 +···+m ;r1 ,...,rh+1
jnh j +1,h · · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) pknh An (τhk ) · · · pk +1,h A +1 (τhk ) × k jn1 ×Cπ(r1 +···+rh ) (sπ(r1 +···+rh ) ) · · · Cπ(r1 +1) (sπ(r1 +1) ) pn1 An (τ1k ) j +1,1 1 m · · · pk +1,1 A +1 (τ1k ) Cπ(r1 ) (sπ(r1 ) ) · · · Cπ(1) (sπ(1) )φ μm 1k ×· · ·×μ k × × ds1 , . . . , dsm1 +···+m − Cπ(m1 +···+m ) (sπ(m1 +···+m ) )× T m
1 +···+m ;r1 ,...,rh+1
(π )
j j · · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) pnh An (τh ) nh · · · p +1,h A +1 (τh ) +1,h × j ×Cπ(r1 +···+rh ) (sπ(r1 +···+rh ) ) · · · Cπ(r1 +1) (sπ(r1 +1) ) pn1 An (τ1 ) n1 j · · · p +1,1 A +1 (τ1 ) +1,1 Cπ(r1 ) (sπ(r1 ) ) · · · Cπ(1) (sπ(1) )φ ×
1 m
ds × · · · × μ , . . . , ds (4.18) × μm 1 m1 +···+m
1 H
Now, for a nonnegative integer k and any nonnegative integers m1 , . . . , mn , j +1,1 ,. . ., j +1,h , jn1 ,. . ., jnh , r1 ,. . ., rh+1 such that j +1,1 +· · ·+ j +1,h = m +1 , . . . , jn1 + · · · + jnh = mn , r1 + · · · + rh+1 = m1 + · · · + m and any π ∈ Sm1 +···+m , define fk : [0, T]m1 +···+m → H by fk s1 , . . . , sm1 +···+m := Cπ(m1 +···+m ) (sπ(m1 +···+m ) ) · · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) · n−1 ! k jn−α,h · pn−α,h An−α (τhk ) Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) α=0
n−1
· · · Cπ(r1 +1) (sπ(r1 +1) ) ·
jn−α,1 pkn−α,1 An−α (τ1k )
! ×
α=0
× Cπ(r1 ) (sπ(r1 ) ) · · · Cπ(1) (sπ(1) )φ.
(4.19)
(The functions fk are nothing other than the k-dependent integrands in the norm difference above.) Note that the dependence on the index k is only present in the evaluations at the support points of the discrete measures.
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Since each Ai (·) is continuous, Ai (τ jk ) → Ai (τ j) for all i = 1, . . . , n and j = 1, . . . , h. It is clear, then, that fk → f uniformly on [0, T]m1 +···+m where f s1 , . . . , sm1 +···+m := Cπ(m1 +···+m ) (sπ(m1 +···+m ) ) · · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) × n−1 ! jn−α,h × pn−α,h An−α (τh ) Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) α=0
n−1
· · · Cπ(r1 +1) (sπ(r1 +1) ) ·
j pn−α,1 An−α (τ1 ) n−α,1
! ×
α=0
× Cπ(r1 ) (sπ(r1 ) ) · · · Cπ(1) (sπ(1) )φ.
(4.20)
(Of course, the function f is the integrand from the norm difference above that does not depend on the index k.) Applying Theorem 2.4, we have
Cπ(m1 +···+m ) (sπ(m1 +···+m ) ) · · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) ×
T m
1 +···+m ;r1 ,...,rh+1
(π )
jnh j +1,h × pknh An (τhk ) · · · pk +1,h A +1 (τhk ) Cπ(r1 +···+rh ) (sπ(r1 +···+rh ) ) k jn1 k j +1,1 · · · Cπ(r1 +1) (sπ(r1 +1) ) pn1 An (τ1k ) · · · p +1,1 A +1 (τ1k ) × m1 × Cπ(r1 ) (sπ(r1 ) ) · · · Cπ(1) (sπ(1) )φ μ1k × · · · × μm ds1 , . . . , dsm1 +···+m − k Cπ(m1 +···+m ) (sπ(m1 +···+m ) ) − T m
1 +···+m ;r1 ,...,rh+1
(π )
· · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) × j j × pnh An (τh ) nh · · · p +1,h A +1 (τh ) +1,h Cπ(r1 +···+rh ) (sπ(r1 +···+rh ) ) j j · · · Cπ(r1 +1) (sπ(r1 +1) ) pn1 An (τ1 ) n1 · · · p +1,1 A +1 (τ1 ) +1,1 × × Cπ(r1 ) (sπ(r1 ) )· · ·Cπ(1) (sπ(1) )φ
m 1 μm ds1 ,. . ., dsm1 +···+m
1 ×· · ·×μ
→0 H (4.21)
as k → ∞. This in turn shows at once that
n lim Pμm1k1 ,...,m ,...,μ k ,η +1,k ,...,ηnk (A1 (·), . . . , An (·)) φ − k→∞
,...,mn − Pμm11,...,μ (A1 (·), . . . , An (·)) φ H = 0 ,η +1 ,...,ηn
(4.22)
We now let θ f = ( f, f, f, . . .) ∈ UD and write f as in (3.16). Even though we have discrete measures here, the rest of the proof goes through in an
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almost identical manner as the proof of the corresponding part of the proof of Theorem 4.3. We first write
fμ
(A1 (·), . . . , An (·)) φ −
− fμ1 ,...,μ ,η +1 ,...,ηn (An (·), . . . , An (·)) φ H 1k ,...,μ k ,η +1,k ,...,ηn,k
∞
m1 ,...,mn =0
m ,...,m 1 n am ,...,m
(·), . . . , An (·)) φ − 1 n Pμ1k ,...,μ k ,η +1,k ,...,ηnk (A1
,...,mn − Pμm11,...,μ (A1 (·), . . . , An (·)) φ H ,η +1 ,...,ηn ∞
m1 ,...,mn =0
m ,...,m
1 n am ,...,m
(·), . . . , A (·)) φ (A
P
+ 1 n 1 n μ1k ,...,μ k ,η +1,k ,...,ηnk H
,...,mn + Pμm11,...,μ (A1 (·), . . . , An (·)) φ H . ,η +1 ,...,ηn
(4.23)
Now note that
m1 ,...,mn
Pμ1k ,...,μ k ,η +1,k ,...,ηnk (A1 (·), . . . , An (·)) φ
···
j +1,1 +···+ j +1,h =m +1
·
H
jn1 +···+ jnh =mn
π ∈Sm1 +···+m r1 +···+rh+1 =m1 +···+m T m
m +1 ! · · · mn ! · j +1,1 ! · · · j +1,h ! · · · jn1 ! · · · jnh !
· Cπ(m1 +···+m ) (sπ(m1 +···+m ) )
1 ,...,m ;r1 ,...,rh+1
· · · Cπ(r1 +···+rh +1) (sπ(r1 +···+rh +1) ) ·
(π )
n−1
pkn−α,h
An−α (τhk )
jn−α,h
! ×
α=0
× Cπ(r1 +···+rh ) (sπ(r1 +···+rh ) ) · · · Cπ(r1 +1) (sπ(r1 +1) ) × n−1 !
k jn−α,1
Cπ(r ) (sπ(1) ) · · · Cπ(1) (sπ(1) ) φH × × pn−α,1 An−α (τ1k ) 1
α=0
m 1 × μm (ds1 , . . . , dsm1 +···+m ). 1 × · · · × μ
(4.24)
Of course, we obtain essentially the same expression for the other monomial disentangling. The only difference is that there is no dependence on the index k. Once we have arrived at the expression above (and the corresponding expression for the other monomial), we see that with the norms around all of
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our operators, we have products of commutative real-valued functions. Hence, “unraveling” the disentangling we obtain
m1 ,...,mn
Pμ1k ,...,μ k ,η +1,k ,...,ηnk (A1 (·), . . . , An (·)) φ
H m1 m A1 (s) μ1k (ds) ··· A (s) μ k (ds) × [0,T]
×
[0,T]
[0,T]
m +1 A +1 (s) η +1,k (ds) ···
[0,T]
mn An (s) ηnk (ds) φH (4.25)
and
m ,...,m
n
P 1
μ1 ,...,μ ,η +1 ,...,ηn (A1 (·), . . . , An (·)) φ H m1 m A1 (s) μ1 (ds) ··· A (s) μ (ds) × [0,T]
×
[0,T]
[0,T]
m +1 A +1 (s) η +1 (ds) ···
[0,T]
mn An (s) ηn (ds) φH . (4.26)
The integrals just above are the weights for the disentangling algebras under consideration in this theorem and therefore we have
fμ ,...,μ ,η ,...,η (A1 (·),. . ., An (·)) φ− fμ ,...,μ ,η ,...,η (An (·),. . ., An (·)) φ
1k k +1,k n,k 1 +1 n H ⎛ ⎞ ∞ am ,...,m rm1 · · · rmn + rm1 · · · rmn ⎠ φH ⎝ 1 n n 1 1,k n,k m1 ,...,mn =0
= φH ( f k + f 0 ) φH θ f U D + f 0
(4.27)
where the notation is as seen in the proof of Theorem 4.3 above. We can use the equation just above to obtain a scalar-valued summable bound for the expression seen in the second line of (4.23). Indeed, since the norm on U D is {gk } U D = supk∈N gk k , we can, given > 0, choose a k0 ∈ N such that θ f U D < f k0 + . Then φH θ f U D + f 0 < φH f k0 + f 0 +
(4.28) and so our summable dominating function is m1 mn m1 mn + · · · r + r · · · r (m1 , . . . , mn ) −→ φH am1 ,...,mn r1,k n 1 n,k0 0 +
2m1 +···+mn
φH .
(4.29)
We can therefore pass the limit on k through the sum over m1 , . . . , mn in the second line of (4.23) and the theorem is proved.
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Remark 4.7 The reader has probably noticed that the theorem above concerned the stability of the disentangling map in the situation where some of the time ordering measures are continuous and some are purely discrete. While it would be desirable to have a stability theorem in this case, such a theorem has not been found. The difficulty in obtaining such a theorem seems to be with the fact that, given a sequence {μk }∞ k=1 of probability measures (with finitely supported discrete parts) converging weakly to a probability measure μ, the limit measure μ may be continuous, purely discrete, or have both continuous and discrete parts. Moreover, with a general sequence of measures, it may be the case that the supports of the continuous and discrete parts of the individual measures may change. In the current formulation the Feynman’s operational calculus, we are unable to accommodate these features. It may be possible, however, to “smooth out" the sequence of probability measures and so obtain a convergence result for general time ordering measures.
References 1. Billingsley, P.: Convergence of probability measures, 2nd edn., Wiley Series in Probability and Statistics, John Wiley and Sons, Inc., New York, (1968) 2. Dunford, N., Schwartz, J.: Linear Operators, Part I, General Theory, Interscience Publishers, Inc, New York (1958) 3. Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys. Rev. 84, 108–128 (1951) 4. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: definitions and elementary properties. Russ. J. Math. Phys. 8, 153–178 (2001) 5. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: tensors, ordered support and disentangling an exponential factor. Math. Notes 70, 744–764 (2001) 6. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: spectral theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, 171–199 (2002) 7. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: the monogenic calculus. Adv. Appl. Clifford Algebras 11, 233–265 (2002) 8. Jefferies, B., Johnson, G.W., Nielsen, L.: Feynman’s operational calcului for time-dependent noncommuting operators. J. Korean Math. Soc. 38, 193–226 (2001) 9. Johnson, G.W., Lapidus, M.L.: The Feynman Integral and Feynman’s Operational Calculus. Oxford Science Publications, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford and New York (2000) 10. Johnson, G.W., Nielsen, L.: A stability theorem for Feynman’s operational calculus. Stochastic processes, physics and geometry: new interplays, II, pp. 351–365 (Leipzig, 1999) 11. Johnson, G.W., Nielsen, L.: Feynman’s operational calculi: blending instantaneous and continuous phenomena in Feynman’s operational calculi. Stochastic analysis and mathematical physics (SAMP/ANESTOC 2002), pp. 229–254. World Sci. Publ., River Edge, NJ (2004) 12. Nielsen, L.: Stability properties for Feynman’s operational calculus in the combined continuous/discrete setting. Acta Appl. Math. 88, 47–49 (2005) 13. Nielsen, L.: Stability properties of Feynman’s operational calculus for exponential functions of noncommuting operators. Acta Appl. Math. 74, 265–292 (2002) 14. Nielsen, L.: Time dependent stability for Feynman’s operational calculus. Rocky Mountain J. Math. 35, 1347–1368 (2005) 15. Nielsen, L.: Stability properties of Feynman’s operational calculus, Ph.D. Dissertation, Mathematics, University of Nebraska Lincoln (1999)
Math Phys Anal Geom (2007) 10:297–312 DOI 10.1007/s11040-008-9034-y
Reproducing Kernels and Coherent States on Julia Sets K. Thirulogasanthar · A. Krzy˙zak · G. Honnouvo
Received: 21 June 2007 / Accepted: 23 January 2008 / Published online: 13 March 2008 © Springer Science + Business Media B.V. 2008
Abstract We construct classes of coherent states on domains arising from dynamical systems. An orthonormal family of vectors associated to the generating transformation of a Julia set is found as a family of square integrable vectors, and, thereby, reproducing kernels and reproducing kernel Hilbert spaces are associated to Julia sets. We also present analogous results on domains arising from iterated function systems. Keywords Coherent states · Reproducing kernel · Julia sets · IFS · Attractor Mathematics Subject Classifications (2000) Primary 81R30 · 46E22
1 Introduction Hilbert spaces are the underlying mathematical structure of many areas of physics and engineering such as quantum physics and signal analysis. An
The research of the first two authors was supported by Natural Sciences and Engineering Research Council of Canada. K. Thirulogasanthar (B) · A. Krzyz˙ ak Department of Computer Science and Software Engineering, Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8 e-mail:
[email protected] A. Krzyz˙ ak e-mail:
[email protected] G. Honnouvo Department of Mathematics and Statistics, Concordia University, 1455 De Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8 e-mail:
[email protected]
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important family of vectors of the Hilbert space of a physical problem is known as coherent states, CS for short. This family of vectors is very useful in describing quantum phenomena [1]. In this paper we let {φm }∞ m=0 to be an orthonormal basis of an abstract separable Hilbert space H. The well-known canonical coherent states are defined by: 2
− r2
| z = e
∞ zm √ φm ∈ H, m! m=0
where z ∈ C, the complex plane, z = reiθ . The definition of canonical coherent states has been generalized as follows: Let D be an open subset of C. For z ∈ D set | z = N (|z|)− 2 1
∞
zm φm ∈ H, √ ρ(m) m=0
(1.1)
where {ρ(m)}∞ m=0 is a positive sequence of real numbers and N (|z|) is the normalization factor ensuring that z | z = 1. If in addition {| z, z ∈ D} satisfy | zz | dμ = IH , (1.2) D
where dμ is an appropriately chosen measure on D and IH is the identity operator on H, then {| z, z ∈ D} is said to be a set of coherent states on D. This generalization has produced families of vectors which have been successfully applied to physical problems [1, 6, 11, 12, 14, 16]. In general, CS can be constructed as follows: Let (, μ) be a measure space H) and H be a closed subspace of L2 (, μ). Let {m }dim( m=0 , dim(H) denotes the dimension of H, be an orthonormal basis of H satisfying: dim( H)
|m (x)|2 < ∞
m=0
for all x ∈ . Let H be another Hilbert space such that dim(H) = dim(H). Let dim(H) {φm }m=0 be an orthonormal basis of H. Define K(x, y) =
dim( H)
m (x)m (y).
(1.3)
m=0
Then K(x, y) is a reproducing kernel and H is the corresponding reproducing kernel Hilbert space. For x ∈ , define | x = K(x, x)− 2 1
dim( H) m=0
m (x)φm .
(1.4)
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Therefore, x | x = K(x, x)−1
dim( H)
m (x)m (x) = 1,
m=0
and 1
W : H −→ H with W φ(x) = K(x, x) 2 x | φ
is an isometry. Then, for φ, ψ ∈ H we have W φ(x)W ψ(x)dμ(x) φ | ψH = W φ | W ψH = = and
φ | xx | ψK(x, x)dμ(x),
| xx | K(x, x)dμ(x) = IH ,
(1.5)
where K(x, x) is a positive weight function. Thus, the set of states {| x : x ∈ } forms a set of CS. H) In the case where {m }dim( is an orthogonal basis of H, one can define m=0 2 ρ(m) = m ; m = 0, ..., dim(H), and obtain an orthonormal basis dim(H) m √ ρ(m) m=0 of H. Then, setting − 12
| x = K(x, x)
dim( H) m=0
m (x) φm ∈ H, √ ρ(m)
(1.6)
one obtains the desired result which is analogous to (1.1). The above discussion motivates the following definition. Definition 1.1 Let D be an open subset of C. Let m : D −→ C,
m = 0, 1, 2, . . . ,
be a sequence of complex functions. Define | z = N (|z|)− 2 1
∞ m (z) φm ∈ H; z ∈ D, √ ρ(m) m=0
(1.7)
where N (|z|) is a normalization factor and {ρ(m)}∞ m=0 is a sequence of nonzero positive real numbers. The set of vectors in (1.7) is said to form a set of CS if (a) z | z = 1 for all z ∈ D; (b) The states {| z : z ∈ D} satisfy a resolution of the identity: | zz | dμ = I, D
(1.8)
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where dμ is an appropriately chosen measure and I is the identity operator on H. For m (z) = zm , z ∈ C, the states (1.7), for different ρ(m)’s, were studied extensively and applied successfully in quantum theories. Interesting links have been established with group representation, classical polynomials and Lie algebras [1, 5, 7, 12, 16]. Another class of CS was introduced by Gazeau and Klauder for √ Hamiltonians with discrete and continuous spectrum by setting m (J, α) = ( J)m eiem α and ρ(m) = e1 e2 ...em , where em ’s are the spectrum of the Hamiltonian arranged in a suitable way [6]. Recently, in [2, 14], using definition (1.7), CS were presented by setting m (Z ) = Z m , where Z is a n × n matrix valued function. In this note we set m (z) = gm (z)χ Am (z) (see Sections 3 and 4 for precise definitions), where g is a complex function. Using iteration, we build classes of CS on domains arising from dynamical systems with definition (1.7), namely, on Julia sets and fractal sets. In particular, an orthonormal family of vectors associated to the generating transformation of a Julia set is found as a family of square integrable vectors, and, thereby, reproducing kernels and reproducing kernel Hilbert spaces are associated to Julia sets. We also present analogous results on domains arising from iterated function systems. To the best of our knowledge, such a problem has not been addressed in the literature. In Section 2, we motivate definition (1.7) by building CS on a general iterated function system. Section 3 is the main section of the note. In this section, we build CS on Julia sets using iterations. Moreover, it contains the construction of reproducing kernels and reproducing kernel Hilbert spaces. In Section 4, we build CS on fractal sets using iterated function systems. Section 5 discusses the case when zero is contained in the Julia set or in the attractor of an iterated function system.
2 CS on Iterated Function System: General Case In order to motivate definition (1.7) and to set a stage for the construction of CS on Julia sets, we present a class of CS on a set which is invariant under the iterates of a map. Let Q : C → C be a mapping such that for some D ⊆ C we have Qn (D) ⊆ D,
∀n ∈ N.
Let μ be a probability measure on D and {Am }∞ m=0 be a partition of D such that ∞ D = m=0 Am a.e μ, An ∩ Am = ∅ for m = n, and μ(Am ) > 0
∀m 0
(2.1)
Assume that D ⊆ {z : A |z|2 B} for some positive constants A and B. Then by the invariance of D under the iterates of Q, we have A |Qm (z)|2 B.
(2.2)
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Let
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ρ(m) =
|Qm (z)|2 dμ(z).
(2.3)
Am
Then by (2.1) and (2.2) we have 0 < ρ(m) < ∞ and {ρ(m)}∞ m=0 is a positive sequence of real numbers. In the following we present a class of CS on D in the form of (1.7) with m (z) = Qm (z)χ Am (z), where χ Am is the characteristic function of Am , i.e., 1 if z ∈ Am χ Am (z) = 0 if z ∈ / Am Theorem 2.1 For z ∈ D, the collection of vectors | z = N (z)− 2 1
∞ Qm (z)χ Am (z) φm ∈ H √ ρ(m) m=0
(2.4)
forms a set of CS. Proof The normalization condition z | z = 1 holds trivially by (2.1) and (2.2) with ∞ |Qm (z)χ Am (z)|2 < ∞. 0 < N (z) = ρ(m) m=0 We now find a resolution of the identity. Let dν(z) = N (z)dμ(z) be a measure on D. Then, ∞ ∞ Qm (z)Ql (z)χ Am (z)χ Al (z) | zz | dν(z) = dν(z) | φl φm | N (z)ρ(m) D D m=0 l=0
= =
∞ m=0 D ∞ m=0
=
|Qm (z)|2 χ Am (z) dμ(z) | φm φm | ρ(m)
∞
Am
|Qm (z)|2 dμ(z) | φm φm | ρ(m)
| φm φm |= IH ,
m=0
where we have used (2.3) together with the fact that χ Am (z)χ Al (z) = χ Am δml , δml is the Kronecker delta function. Since Julia sets and fractals have rich dynamics and greater interest in mathematics and physics, as an application of the above theory, in the following section, we build CS on Julia sets and fractal sets. Even though a reproducing kernel can be associated with the CS in (2.4), since our primary aim is focused on Julia sets, we shall associate them with the CS of the Julia sets.
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3 CS on Julia Sets In this section, we define the m ’s of definition (1.7) in terms of the iterates of a rational function of the Riemann sphere. Thus, naturally, the domain of interest falls under Julia sets. Julia sets have useful dynamical and topological properties. We exploit these properties in our proofs. 3.1 Holomorphic Dynamics and the Julia Set In this section, we state some definitions and well know results which will be used in the sequel. All the results can be found in the references [4, 9, 10]. Definition 3.1 A family of analytic functions having a common domain of definition is called normal if every sequence in this family contains a locally uniformly convergent subsequence. ˆ the Riemann sphere. Let Q : C ˆ →C ˆ be a rational Definition 3.2 Denote by C map of degree greater than one. We denote the nth iterate of Q by Qn ; i.e., Qn = Q ◦ Q · · · ◦ Q .
n times
The Fatou set is defined by: F (Q) = {z : ∃ a neighborhood U z s.t. {Qn }∞ n=1 is a normal family in U z }.
ˆ \F (Q). The Julia set is defined by J = J(Q) = C The Julia set has the following properties: 1. The Julia set is a compact set. 2. The Julia set is a perfect set. 3. The Julia set is completely invariant under Q; i.e., for any z ∈ J(Q) we have Qn (z) ∈ J(Q), n = 0, ±1, ±2, . . . . Since the Julia set is Q-invariant, we can define Q : J(Q) → J(Q). Let μ be a probability measure defined on the Julia set: dμ(z) = 1. (3.1) J
A famous probability measure which is associated with Julia sets is called conformal measure [13]. We assume: A |Qm (z)|2 B for all m = 0, 1, 2.. and for all z ∈ J.
(3.2)
where A and B are positive constants. We partition the Julia set as follows: J=
∞
Am
a.e μ,
Am ∩ An = ∅ for m = n, μ(Am ) > 0 ∀m 0.
m=0
(3.3)
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Let
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ρ(m) =
|Qm (z)|2 dμ(z).
(3.4)
Am
Thus, by (3.2) and (3.3), 0 < ρ(m) < ∞ and {ρ(m)} is a positive sequence of real numbers. Now in view of Theorem (2.1) we present a class of CS on Julia sets in the form of (1.7) with m (z) = Qm (z)χ Am (z) and z ∈ J as follows: | z = N (z)− 2 1
∞ Qm (z)χ Am (z) φm ∈ H. √ ρ(m) m=0
(3.5)
Remark 3.3 Our condition (3.2) is satisfied by polynomial Julia sets, for instance, by the quadratic family Q(z) = z2 + c (see the examples below and compare with Section 5 to see how we deal with the case when 0 is an element of the Julia set). We now associate a reproducing kernel and a reproducing kernel Hilbert space with Julia sets. Consider the sequence of functions {m }∞ m=0 with m (z) = Qm (z)χ Am (z) as L2 (J, dμ) functions. Since m | n = m (z)n (z)dμ(z)
J
Qm (z)Qn (z)χ Am (z)χ An (z)dμ(z)
= J
|Qm (z)|2 dμ(z)
= δmn Am
= δmn ρ(m), ∞ ∞ √m the family {m }∞ m=0 is orthogonal, and consequently { ρ(m) }m=0 = { m }m=0 is def
H = L2 (J, dν), then by the orthonormal in L2 (J, dμ). Let dν = N (z)dμ(z) and resolution of the identity, for φ ∈ H, the functions
: J −→ C defined by (z) = z | φ are elements of H. Define : H −→ W H by φ → . Using the resolution of the identity we have 2 2 W φ = = (z)(z)dν(z)
J
=
φ | zz | φdν(z) = φ2 . J
is a partial isometry. Thus, W
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, then H K is a closed subspace of Let H K = WH H. Define K(z, z ) = z | z = N (z)− 2 N (z )− 2 1
1
∞
m (z) m (z ).
(3.6)
m=0
Now, for any ∈ H K , using the resolution of the identity, we have:
z | z z | φdν(z )
(z) =
J
K(z, z )(z )dν(z ).
= J
Thus, the reproducing property is satisfied by any ∈ H K . It follows that K is a reproducing kernel and H K is the corresponding reproducing kernel Hilbert space. One can easily see that the kernel K satisfies: (a) Hermiticity, K(z, z ) = K(z , z); (b) Positivity, K(z, z) > 0; (c) Idempotence,
K(z, z )K(z , z )dν(z ) = K(z, z ). J
From this point to Example (3.4) assume that the Julia set is symmetric with respect to the real axis of the complex plane (note: Julia sets of rational maps with real coefficients are symmetric with respect to the real axis [9, 10]). The following argument is standard in the theory of CS [1], however for the sake of completeness we present it for Julia sets in brief. Now we realize a resolution of the identity on a closed subset of L2 (J, dμ) which is isomorphic to the reproducing kernel Hilbert space Hk . For φ ∈ H, set 1
f (z) = N (z) 2 z | φ.
(3.7)
Using the resolution of the identity, we have: f | f =
f (z) f (z)dμ(z)
J
=
φ | zz | φdν(z) = φ2 , J
i.e., f ∈ L2 (J, dμ). One can also characterize a reproducing kernel and a reproducing kernel Hilbert space in terms of the orthonormal family { m }∞ m=0 .
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For this we define 1
W : H −→ L2 (J, dμ) with W φ(z) = N (z) 2 z | φ.
Let WH = H. Then W is an isometry from H to H given by: 1
W φm (z) = N (z) 2 φm | z
= N (z) 2 N (z)− 2 1
=
1
∞ Qm (z)χ Am (z) φm | φm √ ρ(m) m=0
Qm (z)χ Am (z) = m (z); √ ρ(m)
∀z ∈ J.
In fact, H is the closed linear span of the orthonormal family {m }∞ m=0 . Further, H is the reproducing kernel Hilbert space corresponding to the reproducing kernel ∞ z ) = K(z, m (z) m (z ). m=0
Using (3.6) we also obtain z ) = N (z) 12 N (z ) 12 K(z, z ). K(z, A reproducing kernel Hilbert space is uniquely determined by its reproducing kernel. Thus, the Hilbert spaces H K and H are isomorphic. Let 1
ηz = N (z) 2 W | z.
(3.8)
Using the isometry property of W and using (3.7), for any f ∈ H and z ∈ J, we can write f (z) = ηz | f H .
(3.9)
Therefore, the reproducing kernel is given by: ηz | ηz H = N (z) 2 N (z ) 2 z | z = 1
1
∞
z ). m (z) (z ) = K(z,
(3.10)
m=0
is the reproducing kernel for the reproducing kernel Hilbert space H, Since K for any f ∈ H and z ∈ J, we have z ) f (z )dμ(z ) = f (z). K(z, (3.11) J
Using (3.9), (3.10) and (3.11) we obtain: | ηz ηz | dμ(z) = IH ;
(3.12)
J
i.e., a resolution of the identity for the Hilbert space H. Now, we present examples of families of rational maps which satisfy condition (3.2). These examples are well studied in the complex dynamics literature [4, 9, 10].
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Example 3.4 Let Q(z) = eiα kj=1 1−a j zj , where |a j| < 1 and a j ∈ C, j = 1, . . . , k. Q(z) is known as Blaschke product. The following is well know fact about the Julia sets of Blaschke products [10]: The Julia set of a Blaschke product is S1 if and only if there exists z0 ∈ D such that Q(z0 ) = z0 , where D is the open unit disc and S1 is the unit circle. Thus, if Q(z0 ) = z0 for some z0 ∈ D, Q(z) clearly satisfies condition (3.2). For the probability measure μ, we take normalized Lebesgue arc measure. We partition the unit circle [0, 2π ) = ∪∞ m=0 Am as follows:
2π 2π Am = m+1 , m , m = 0, 1, . . . 2 2 Then, 1 μ(Am ) = 2π
2π 2m 2π 2m+1
dθ =
1 2m+1
∀m 0.
Consequently, one can easily derive the explicit formulas of ρ(m) and N (z). Example 3.5 The quadratic family, Q(z) = z2 + c, is the most famous family in holomorphic dynamics. Although it is the simplest non-linear example, it has very rich and complicated dynamics. Its Julia set is always symmetric with respect to the origin and contained within the circle |z| = 2. Replacing c by its conjugate has the effect of reflecting Jc through the horizontal axis [4, 9, 10]. Now, we give examples of Julia sets of the quadratic family that satisfy condition (3.2). The simplest case is when c = 0, where the Julia set is the unit circle. The Julia set of Q(z) = z2 + c, for a very small c, is a quasi-circle. For c = i/4 the Julia set is shown in Fig. 1. It is symmetric with respect to the origin which does not belong to the Julia set. Hence, we can draw a neighborhood N0 around the origin and inside the Julia set which is bounded inside the circle |z| = 2. Therefore, Q(z) satisfies (3.2).
Fig. 1 Q(z) = z2 + i/4
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Fig. 2 Q(z) = z2 + i
We now present an example of the quadratic family which does not satisfy condition (3.2). Example 3.6 Let Q(z) = z2 + i. The Julia set of this function, which is called Dendrite, is shown in Fig. 2. Here we cannot have a neighborhood N0 of the origin such that N0 ∩ J = ∅ since 0 is an element of the Julia set. Thus, Q(z) = z2 + i does not satisfy condition (3.2).
4 CS on Fractal Sets 4.1 Iterated Function System Let (X, d) be a complete metric space and τ1 , τ2 , ..., τ K be a collection of transformations from X into itself. We assume that τ1 , τ2 , ..., τ K are contractions; i.e., for k = 1, . . . , K max d(τk (x), τk (y)) ≤ α · d(x, y), k
where 0 < α < 1. An iterated function system, IFS for short, with state dependent probabilities , T = {τ1 , . . . , τ K ; p1 (x), K. . . , p K (x)} is defined by choosing pk (x) = 1. The iterates of T are τk (x) with probability pk (x), pk (x) ≥ 0, k=1 given by: T m (x) = τkm ◦ τkm−1 ◦ · · · ◦ τk1 (x)
(4.1)
with probability pkm (τkm−1 ◦ · · · ◦ τk1 (x)) · pkm−1 (τkm−2 ◦ · · · ◦ τk1 (x)) · · · pk1 (x). Let (H(X), h(d)) denote the space of nonempty compact subsets corresponding to (X, d) with the Hausdorff metric h(d) [3]. T : H(X) → H(X) is given by: K T(B) = ∪k=1 τk (B)
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for all B ∈ H(X). The Banach contraction theorem furnishes an attractor to the dynamical system (X, T) [3]. In the special case when X = R2 , the attractor of an IFS is called a fractal [3]. The presence of the probabilities allows more weight on some transformations over the others. Thus, the fractal may have parts which are more “dense" than the remaining parts. We consider X = R2 . Then, the transformations are two component objects, i.e., T m (x) = ( f m (x), gm (x)). We interpret T m (x) as the complex number f m (x) + igm (x) and denote it by T m (x). We consider a Hilbert space H over complex numbers, thus the object T m (x)φ ∈ H for all φ ∈ H. Let A denote the attractor of the IFS and B the Borel subsets of (A, d). Let μ be a probability measure on A: dμ(x) = 1. (4.2) A
In particular, one can use the probabilities to construct a measure on the attractor. For example, if the probabilities are constants, the measure can be constructed in the following way: μ(A) = 1, μ(τk (A)) = pk , μ(τl ◦ τk (A)) = pl · pk , and so on. 4.2 Construction of CS In this section, we assume that X = R2 and d = | · | is the Euclidean metric on R2 . We also assume that there exists a neighborhood, N0 , of the origin such that N0 ∩ A = ∅.
(4.3)
Remark 4.1 Observe that if T satisfies (4.3), we have: A |T m (x)| B for all m = 0, 1, 2.. and for all x ∈ A,
(4.4)
where A and B are positive constants. The contraction property of the IFS, which guarantees the existence of the attractor, guarantees the existence of the upper bound B. This is essential in the proof of our theorems. We partition the attractor as follows: A=
∞
Am a.e
Am ∩ An = ∅ for m = n, μ(Am ) > 0
∀m 0. (4.5)
m=0
Let
ρ(m) =
|T m (x)|2 dμ(x).
(4.6)
Am
Thus, by (4.4) and (4.5), 0 < ρ(m) < ∞ and {ρ(m)} is a positive sequence of real numbers.
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Now in view of Theorem (2.1) we present a class of CS on fractal sets in the form of (1.7) with m (x) = T m (x)χ Am (x) and x ∈ A as follows: | x = N (x)− 2 1
∞ T m (x)χ Am (x) φm ∈ H. √ ρ(m) m=0
(4.7)
forms a set of CS. Remark 4.2 Following the procedure discussed in Section 3, we can associate a reproducing kernel and a reproducing kernel Hilbert space for the attractor. (a) The set of vectors m (x) = T m (x)χ Am (x) is an orthogonal family in the m Hilbert space L2 (A, dμ) and the family { √ }∞ = { m }∞ m=0 is orthoρ(m) m=0 normal in the same Hilbert space. (b) There is a partial isometry, W : H −→ L2 (A, dμ),
1
defined by W φ(x) = N (x) 2 φ | x.
(b) WH = H ⊂ L2 (A, dμ) is a reproducing kernel Hilbert space and the associated reproducing kernel is y) = K(x,
∞
m (x) m (y).
m=0
The construction depends on the invariance of the attractor under iteration and on condition (4.3). We present an example which satisfies condition (4.3). Example 4.3 Let T = {τ1 , τ2 , τ3 }, x = (t1 , t2 ) ∈ R2 , where τ1 = 12 t1 , 12 t2 + √ 1 , 0 , τ2 = 12 t1 , 12 t2 + (1, 0) and τ3 = 12 t1 , 12 t2 + 34 , 43 . Notice that T is a 2 contraction with a contractivity factor 12 . The attractor of T is shown in Fig. 3 Fig. 3 Sierpinski triangle, the attractor of the IFS in Example 4.3 and Example 4.4
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with the origin outside the attractor. For this example, the vertices of the √ 3 3 triangle are (1, 0), (2, 0) and 2 , 2 . Thus, T satisfies condition (4.3). Now, we present an example which does not satisfy condition (4.3). Example 4.4 Let T = {τ1 , τ2 , τ3 }, x = (t1 , t2 ) ∈ R2 , where τ1 = 12 t1 , 12 t2 , τ2 = √ 1 1 1 t , t + 2 , 0 and τ3 = 12 t1 , 12 t2 + 14 , 43 . Notice that T is a contraction 2 1 2 2 with a contractivity factor 12 . The attractor of T is shown in Fig. 3. For this √ example, the vertices of the triangle are (0, 0), (1, 0) and 12 , 23 . Thus, T does not satisfy condition (4.3).
5 CS on Julia Sets When the Origin is in the Julia Set This section is motivated by Examples 3.6 and 4.4. We build CS on Julia sets when the origin is an element of the Julia set. For this purpose we use the distance between a reference point and the orbit of an element of the Julia set. Let J be the Julia set of the transformation Q and F be its Fatou set. We assume that F = ∅. Let A be a positive constant and d be a metric on C. Fix z0 ∈ F such that inf
z∈J m=0,1,2..
d(Qm (z), z0 ) A > 0,
(5.1)
and assume that there exists another positive constant B such that A d(Qm (z), z0 ) B for all z ∈ J and m = 0, 1, 2, ... Let {Am }∞ m=0 and μ be as in Section 3. Define 2 ρ(m) = d(Qm (z), z0 ) dμ(z).
(5.2)
(5.3)
Am
Thus, by (5.2) and (3.3), 0 < ρ(m) < ∞ and {ρ(m)} is a positive sequence of real numbers. Once again, in view of Theorem (2.1) we present a class of CS on Julia sets in the form of (1.7) with m (z) = d(Qm (z), z0 )χ Am (z) and z ∈ J as follows:
| z = N (z)− 2 1
∞ d(Qm (z), z0 )χ Am (z) φm ∈ H. √ ρ(m) m=0
(5.4)
Example 5.1 Consider the Julia set of Example 3.6. Since the Julia set is inside the circle {z ∈ C : |z| = 2}, one can fix z0 in the Fatou set, for example, with |z0 | = 3. Thus, condition (5.2) is satisfied.
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Remark 5.2 The above theory can be directly extended to the attractors of Section 4 when the origin is inside the attractor. For instance, in Example 4.4, one can pick up an x0 in the big hole of the attractor.
6 A Hamiltonian Formalism We briefly adapt a Hamiltonian formalism described in [1], with the orthonormal basis of the Hilbert space of CS (3.5). (the same method can also be directly adapted to other sets of CS discussed in this article). For this, let xm =
ρ(m) ρ(m − 1)
∀m 1,
where ρ(m) is given by (3.4), and assume x0 := 0. Then ρ(m) = x1 x2 · · · xm = xm !. Define a Hamiltonian by H=
∞
xm |φm φm |.
(6.1)
m=0
Then we have H|φm = xm |φm for m 0. If we calculate the average energy of the CS (3.5) with the Hamiltonian H, we get E(z) = z|H|z =
∞ 1 |Qm+1 (z)|2 χ Am+1 (z) < ∞, N (z) m=0 xm !
i.e., the CS (3.5) are CS of the Hamiltonian H with finite energy. Remark 6.1 Since, in general, the component Qm (z)χ Am (z) appearing in CS (3.5) cannot be written as Qm (z)χ Am (z) = f (z)Qm−1 (z)χ Am−1 (z) for some function f (z) (independent of m), we cannot define an annihilation operator, a such that a | z = f (z) | z. Thereby, an oscillator algebra cannot be associated with CS (3.5). For detailed explanation along this line of argument, see [15]. Acknowledgement K. Thirulogasanthar would like to thank Wael Bahsoun for helpful discussions about dynamical systems.
References 1. Ali, S.T., Antoine, J-P., Gazeau, J-P.: Coherent States, Wavelets and Their Generalizations. Springer, New York (2000) 2. Ali, S.T., Englis, M., Gaseau, J-P.: Vector coherent states from Plancherel’s theorem, Clifford algebras and matrix domains. J. Phys. A 37, 6067–6089 (2004) 3. Barnsley, M.: Fractals Everywhere. Academic Press, London (1988) 4. Beardon, A.: Iteration of Rational Functions. Springer-Verlag (1991)
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5. Borzov, V.V.: Orthogonal polynomials and generalized oscillator algebras. Integral Transform. Spec. Funct. 12, 115–138 (2001) 6. Gazeau, J-P., Klauder, J.R.: Coherent states for systems with discrete and continuous spectrum. J. Phys. A 32, 123–132 (1999) 7. Klauder, J.R., Skagerstam, B.S.: Coherent States, Applications in Physics and Mathematical Physics. World Scientific, Singapore (1985) 8. Klauder, J.R., Penson, K.A., Sixdeniers, J-M.: Constructing coherent states through solutions of Steieljes and Hausdorff moment problems. Phys. Rev. A 64, 013817 (2001) 9. McMullen, C.: Complex Dynamics and Renormalization. Princeton University Press (1994) 10. Milnor, J.: Dynamics in One Complex Variable: Introductory Lectures. SUNY Stony Brook (1990) 11. Novaes, M., Gazeau, J-P.: Multidimensional generalized coherent states. J. Phys. A 36, 199–212 (2003) 12. Pérélomov, A.M.: Generalized Coherent States and Their Applications. Springer-Verlag, Berlin (1986) 13. Przytycki, F., Urbanski, M.: Fractals in the Plane—The Ergodic Theory Methods. Cambridge Univestity Press (preprint) 14. Thirulogasanthar, K., Twareque Ali, S.: A class of vector coherent states defined over matrix domains. J. Math. Phys. 44, 5070–5083 (2003) 15. Thirulogasanthar, K., Honnouvo, G.: Coherent states with complex functions. Internat. J. Theoret. Phys. 43, 1053–1071 (2004) 16. Rowe, D.J., Repca, J.: Vector coherent state theory as a theory of induced representations. J. Math. Phys. 32, 2614–2634 (1991)
Math Phys Anal Geom (2007) 10:313–358 DOI 10.1007/s11040-008-9035-x
Scattering Theory for Open Quantum Systems with Finite Rank Coupling Jussi Behrndt · Mark M. Malamud · Hagen Neidhardt
Received: 25 June 2007 / Accepted: 23 January 2008 / Published online: 6 March 2008 © Springer Science + Business Media B.V. 2008
Abstract Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator AD in a Hilbert space H is used to describe an open quantum system. In this case the minimal self-adjoint of AD can be regarded as the Hamiltonian of a closed system which dilation K is necessarily not semibounded contains the open system {AD , H}, but since K from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family {A(μ)} of maximal dissipative operators depending on energy μ, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of
Dedicated to Valentin A. Zagrebnov on the occasion of his 60th birthday. Jussi Behrndt gratefully acknowledges support by DFG, Grant 3765/1. Hagen Neidhardt gratefully acknowledges support by DFG, Grant 1480/2. J. Behrndt Technische Universität Berlin, Institut für Mathematik, MA 6-4, Straße des 17. Juni 136, 10623 Berlin, Germany e-mail:
[email protected] M. M. Malamud Department of Mathematics, Donetsk National University, Universitetskaya 24, 83055 Donetsk, Ukraine e-mail:
[email protected] H. Neidhardt (B) WIAS Berlin, Mohrenstr. 39, 10117 Berlin, Germany e-mail:
[email protected]
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single pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm–Liouville operators arising in dissipative and quantum transmitting Schrödinger–Poisson systems. Keywords Scattering theory · Open quantum system · Maximal dissipative operator · Pseudo-Hamiltonian · Quasi-Hamiltonian · Lax–Phillips scattering · Scattering matrix · Characteristic function · Boundary triplet · Weyl function · Sturm–Liouville operator Mathematics Subject Classifications (2000) 47A40 · 47A55 · 47B25 · 47B44 · 47E05 1 Introduction Quantum systems which interact with their environment appear naturally in various physical problems and have been intensively studied in the last decades, see e.g. the monographes [22, 25, 41]. Such an open quantum system is often modeled with the help of a maximal dissipative operator, i.e., a closed linear operator AD in some Hilbert space H which satisfies Im (AD f, f ) 0,
f ∈ dom(AD ),
and does not admit a proper extension in H with this property. The dynamics in the open quantum system are described by the contraction semigroup e−it AD , t 0. In the physical literature the maximal dissipative operator AD is usually called a pseudo-Hamiltonian. It is well known that AD admits a self-adjoint in a Hilbert space K which contains H as a closed subspace, that is, dilation K is a self-adjoint operator in K and K − λ −1 H = (AD − λ)−1 PH K holds for all λ ∈ C+ := {z ∈ C : Im (z) > 0}, cf. [42]. Since the operator K is self-adjoint it can be regarded as the Hamiltonian or so-called quasiHamiltonian of a closed quantum system which contains the open quantum system {AD , H} as a subsystem. In this paper we first assume that an open quantum system is described by a single pseudo-Hamiltonian AD in H and that AD is an extension of a closed densely defined symmetric operator A in H with finite equal deficiency indices. can be realized as a self-adjoint extension Then the self-adjoint dilation K of the symmetric operator A ⊕ G in K = H ⊕ L2 (R, H D ), where H D is finitedimensional and G is the symmetric operator in L2 (R, H D ) given by d g, dom(G) = g ∈ W21 (R, H D ) : g(0) = 0 , dx see Section 3.1. If A0 is a self-adjoint extension of A in H and G0 denotes the usual self-adjoint momentum operator in L2 (R, H D ), Gg := −i
G0 g := −i
d g, dx
dom(G) = W21 (R, H D ),
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can be regarded as a singular perturbation (or, more then the dilation K precisely, a finite rank perturbation in resolvent sense) of the ‘unperturbed operator’ K0 := A0 ⊕ G0 , cf. [8, 49]. From a physical point of view K0 describes a situation where both subsystems {A0 , H} and {G0 , L2 (R, H D )} do takes into account an interaction of the subsystems. Since not interact while K the spectrum σ (G0 ) of the momentum operator is the whole real axis, standard are = σ (K0 ) = R and, in particular, K0 and K perturbation results yield σ K are often necessarily not semibounded from below. For this reason K0 and K called quasi-Hamiltonians rather than Hamiltonians. K0 is a complete scattering system in K = H ⊕ L2 (R, H D ), The pair K, that is, the wave operators K0 := s- lim eit Ke−itK0 Pac (K0 ) W± K, t→±∞
exist and are complete, cf. [9, 17, 72, 73]. Here Pac (K0 ) denotes the orthogonal projection in K onto the absolutely continuous subspace Kac (K0 ) of K0 . The scattering operator K0 := W+ K, K0 ∗ W− K, K0 S K, K0 regarded as an operator in Kac (K0 ) is unitary, of the scattering system K, commutes with the absolutely continuous part K0ac of K0 and is unitarily equivalent to a multiplication operator induced by a (matrix-valued) function S(λ) λ∈R in a spectral representation L2 (R, dλ, Kλ ) of K0ac = Aac 0 ⊕ G0 , cf. [17]. The family S(λ) is called the scattering matrix of the scattering K0 and is one of the most important quantities in the analysis system K, of scattering processes. In our setting the scattering matrix S(λ) decomposes into a 2 × 2 block matrix function in L2 (R, dλ, Kλ ) and it is one of our main goals in Section 3 to show that the left upper corner in this decomposition coincides with the scattering matrix {S D (λ)} of the dissipative scattering system {AD , A0 }, cf. [63, 65, 66]. The right lower corner of S(λ) can be interpreted as the LP Lax–Phillips scattering matrix {S (λ)} corresponding to the Lax–Phillips scat D− , D+ . Here D± := L2 (R± , H D ) are so-called incoming tering system K, we refer to [17, 57] for details on and outgoing subspaces for the dilation K; Lax–Phillips scattering theory. The scattering matrices S(λ) , {S D (λ)} and {S LP (λ)} are all explicitely expressed in terms of an ‘abstract’ Titchmarsh– Weyl function M(·) and a dissipative matrix D which corresponds to the maximal dissipative operator AD in H and plays the role of an ‘abstract’ boundary condition. With the help of this representation of {S LP (λ)} we easily recover the famous relation S LP (λ) = W AD (λ − i0)∗ found by Adamyan and Arov in [2–5] between the Lax–Phillips scattering matrix and the characteristic function WAD (·) of the maximal dissipative operator AD , cf. Corollary 3.11. We point out that M(·) and D are completely determined by the operators A ⊂ A0 and AD from the inner system. This is
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interesting also from the viewpoint of inverse problems, namely, the scattering K0 , in particular, the Lax–Phillips scattering matrix matrix S(λ) of K, {S LP (λ)} can be recovered having to disposal only the dissipative scattering system {AD , A0 }, see Theorem 3.6 and Remark 3.7. We emphasize that this simple and somehow straightforward embedding method of an open quantum system into a closed quantum system by choosing of the pseudo-Hamiltonian AD is very convenient a self-adjoint dilation K for mathematical scattering theory, but difficult to legitimate from a physical and K0 are necessarily not point of view, since the quasi-Hamiltonians K semibounded from below. In the second part of the paper we investigate open quantum systems which are described by an appropriate chosen family of maximal dissipative operators {A(μ)}, μ ∈ C+ , instead of a single pseudo-Hamiltonian AD . Similarly to the first part of the paper we assume that the maximal dissipative operators A(μ) are extensions of a fixed symmetric operator A in H with equal finite deficiency indices. Under suitable assumptions on the family {A(μ)} there exists a symmetric operator T in a Hilbert space G and a self-adjoint extension L of L = A ⊕ T in L = H ⊕ G such that −1 −1 H = A(μ) − μ , μ ∈ C+ , (1.1) L−μ PH holds, see Section 4.2. For example, in one-dimensional models for carrier transport in semiconductors the operators A(μ) are regular Sturm–Liouville differential operators in L2 ((a, b )) with μ-dependent dissipative boundary conditions and the ‘linearization’ L is a singular Sturm–Liouville operator in L2 (R), cf. [14, 38, 43, 53] and Section 4.4. We remark that one can regard and interpret relation (1.1) also from an opposite point of view. Namely, if a selfadjoint operator L in a Hilbert space L is given, then the compression of the resolvent of L onto any closed subspace H of L defines a family of maximal dissipative operators {A(μ)} via (1.1), so that each closed quantum system L, L naturally contains open quantum subsystems {{A(μ)}, H} of the type we investigate here. Nevertheless, since from a purely mathematical point of view both approaches are equivalent we will not explicitely discuss this second interpretation. If A0 and T0 are self-adjoint extensions of A and T in H and G, respectively, then again L can be regarded as a singular perturbation of the self-adjoint operator L0 := A0 ⊕ T0 in L. As above L0 describes a situation where the L takes into account a subsystems {A0 , H} and {T0 , G} do not interact while certain interaction. We note that if A and T have finite deficiency indices, then the operator L is semibounded from below if and only if A and T are semibounded from below. Well-known results imply that the pair L, L0 is a complete scattering system in the closed quantum system and again the scattering matrix S(λ) decomposes into a 2 × 2 block matrix function which can be calculated in terms of abstract Titchmarsh–Weyl functions. In this framework the problem is quite similar to the problem of zero-range potentials with internal structure investigated by Pavlov and his group in the eighties, see for example [55, 69, 70]. However, using boundary triplets and abstract
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Titchmarsh–Weyl functions we present here a general framework in which singular perturbation problems can be embedded and solved. On the other hand it can be shown that the family {A(μ)}, μ ∈ C+ , admits a continuation to R, that is, the limit A(μ + i0) exists for a.e. μ ∈ R in the strong resolvent sense and defines a maximal dissipative operator. The family A(μ + i0), μ ∈ R, can be regarded as a family of energy dependent pseudoHamiltonians in H and, in particular, each pseudo-Hamiltonian A(μ + i0) μ in H ⊕ L2 (R, Hμ ), a complete scattering gives rise K to a quasi-Hamiltonian system Kμ , A0 ⊕ −i(d/dx) and a corresponding scattering matrix Sμ (λ) as illustrated in the first part of the introduction. One of our main observations in Section 4 is that the scattering matrix S(λ) of the scattering system L, L0 in H ⊕ G is related to the scattering ma μ , A0 ⊕−i(d/dx) , μ ∈ R, in H ⊕ L2 (R, Hμ ) via trices Sμ (λ) of the systems K S(μ) = Sμ (μ)
for a.e. μ ∈ R. (1.2) In other words, the scattering matrix S(λ) of the scattering system L, L0 can be completely recovered from scattering matrices of scattering systems for single quasi-Hamiltonians. Furthermore, under certain continuity properties of the abstract Titchmarsh–Weyl functions this implies S(λ) ≈ Sμ (λ) for all λ in a sufficiently small neighborhood of the fixed energy μ ∈ R, which justifies the concept of single quasi-Hamiltonians for small energy ranges. Similarly to the case of a single pseudo-Hamiltonian the diagonal entries of S(μ) or Sμ (μ) can be interpreted as scattering matrices corresponding to energy dependent dissipative scattering systems and energy-dependent Lax– Phillips scattering systems. Moreover, if SμLP (λ) is the scattering matrix of μ , L2 (R± , Hμ ) and W A(μ) (·) denote the the Lax–Phillips scattering system K characteristic functions of the maximal dissipative operators A(μ), then an energy-dependent modification SμLP (μ) = W A(μ) (μ − i0)∗ of the classical Adamyan-Arov result holds for a.e. μ ∈ R, cf. Section 4.3. The paper is organized as follows. In Section 2 we give a brief introduction into extension and spectral theory of symmetric and self-adjoint operators with the help of boundary triplets and associated Weyl functions. These concepts will play an important role throughout the paper. Furthermore, we recall a recent result on the representation of the scattering matrix of a scattering system consisting of two self-adjoint extensions of a symmetric operator from [18], see also [6]. Section 3 is devoted to open quantum systems described by a single pseudo-Hamiltonian AD in H. In Theorem 3.2 a minimal self in H ⊕ L2 (R, H D ) of the maximal dissipative operator AD adjoint dilation K is explicitely constructed. Section 3.2 and Section 3.3 deal with the scattering K0 and the interpretation of the diagonal entries as scattering matrix of K, matrices of the dissipative scattering system {AD , A0 } and the Lax–Phillips L2 (R± , H D ) . In Section 3.4 we give an example of a scattering system K, pseudo-Hamiltonian which arises in the theory of dissipative Schrödinger–
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Poisson systems, cf. [15, 16, 50]. In Section 4 the family {A(μ)} of maximal dissipative operators in H is introduced and, following ideas of [29], we construct a self-adjoint operator L in a Hilbert space L, H ⊂ L, such that (1.1) holds. After some preparatory work the relation (1.2) between the scattering matrices of L, L0 and the scattering systems consisting of quasiHamiltonians is verified in Section 4.3. Finally, in Section 4.4 we consider a socalled quantum transmitting Schrödinger–Poisson system as an example for an open quantum system which consists of a family of energy-dependent pseudoHamiltonians, cf. [14, 20, 23, 38, 43, 53]. We note that the scattering theory for open quantum systems developed in this paper is applicable to various quantenmechanic problems, e.g. quantum pumps, see [10–12, 46]. Notations Throughout this paper (H, (·, ·)) and (G, (·, ·)) denote separable Hilbert spaces. The linear space of bounded linear operators defined on H with values in G will be denoted by [H, G]. If H = G we simply write [H]. The set of closed operators in H is denoted by C (H). The resolvent set ρ(S) of a closed linear operator S ∈ C (H) is the set of all λ ∈ C such that (S − λ)−1 ∈ [H], the spectrum σ (S) of S is the complement of ρ(S) in C. The notations σ p (S), σc (S), σac (S) and σr (S) stand for the point, continuous, absolutely continuous and residual spectrum of S, respectively. The domain, kernel and range of a linear operator are denoted by dom(·), ker(·) and ran (·), respectively. 2 Self-adjoint Extensions and Scattering Systems In this section we briefly review the notion of abstract boundary triplets and associated Weyl functions in the extension theory of symmetric operators, see e.g. [31, 32, 34, 45]. For scattering systems consisting of a pair of self-adjoint extensions of a symmetric operator with finite deficiency indices we recall a result on the representation of the scattering matrix in terms of a Weyl function proved in [18]. 2.1 Boundary Triplets and Closed Extensions Let A be a densely defined closed symmetric operator in the separable Hilbert space H with equal deficiency indices n± (A) = dim ker(A∗ ∓ i) ∞. We use the concept of boundary triplets for the description of the closed extensions A ⊆ A∗ of A in H. Definition 2.1 A triplet = {H, 0 , 1 } is called a boundary triplet for the adjoint operator A∗ if H is a Hilbert space and 0 , 1 : dom(A∗ ) → H are linear mappings such that the ‘abstract Green identity’ (A∗ f, g) − ( f, A∗ g) = (1 f, 0 g) − (0 f, 1 g) holds for all f, g ∈ dom(A∗ ) and the map := (0 , 1 ) : dom(A∗ ) → H × H is surjective.
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We refer to [32] and [34] for a detailed study of boundary triplets and recall only some important facts. First of all a boundary triplet = {H, 0 , 1 } for A∗ exists since the deficiency indices n± (A) of A are assumed to be equal. Then n± (A) = dim H and A = A∗ ker(0 ) ∩ ker(1 ) holds. We note that a boundary triplet for A∗ is not unique. In order to describe the closed extensions A ⊆ A∗ of A with the help of a boundary triplet = {H, 0 , 1 } for A∗ we have to consider the set C(H) of closed linear relations in H, that is, the set of closed linear subspaces of H × H. We usually use a column vector notation for the elements in a linear relation . A closed linear operator in H is identified with its graph, so that the set C (H) of closed linear operators in H is viewed as a subset of C(H), in particular, a linear is an operator if and only if the multivalued relation part mul() = f : f0 ∈ is trivial. For the usual definitions of the linear operations with linear relations, the inverse, the resolvent set and the spectrum we refer to [36]. Recall that the adjoint relation ∗ ∈ C(H) of a linear relation in H is defined as
k h ∗
= : (h , k) = (h, k ) for all ∈ , k h and is said to be symmetric (self-adjoint) if ⊂ ∗ ( = ∗ , respectively). Note that this definition extends the definition of the adjoint operator. For a self-adjoint relation = ∗ in H the multivalued part mul() is the orthogonal complement of dom() in H. Setting Hop := dom() and H∞ = mul() one verifies that can be written as the direct orthogonal sum of a self-adjoint operator op in the Hilbert space Hop and the ‘pure’ relation ∞ = f0 : f ∈ mul() in the Hilbert space H∞ . A linear relation in H is called dissipative if Im (h , h) 0 holds for all (h, h ) ∈ and is called maximal dissipative if it is dissipative and does not admit proper dissipative extensions in H; then is necessarily closed, ∈ C(H). We remark that a linear relation is maximal dissipative if and only if is dissipative and some λ ∈ C+ (and hence every λ ∈ C+ ) belongs to ρ(). A description of all closed (symmetric, self-adjoint, (maximal) dissipative) extensions of A is given in the next proposition. Proposition 2.2 Let A be a densely defined closed symmetric operator in H with equal deficiency indices and let = {H, 0 , 1 } be a boundary triplet for A∗ . Then the mapping (2.1) → A := A∗ f ∈ dom(A∗ ) : (0 f, 1 f ) ∈ establishes a bijective correspondence between the set C(H) and the set of closed extensions A ⊆ A∗ of A where (·, ·) is the transposed vector. Furthermore, (A )∗ = A∗ holds for any ∈ C(H). The extension A in (2.1) is symmetric (self-adjoint, dissipative, maximal dissipative) if and only if is symmetric (self-adjoint, dissipative, maximal dissipative, respectively).
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It follows immediately from this proposition that if = {H, 0 , 1 } is a boundary triplet for A∗ , then the extensions A0 := A∗ ker(0 )
and
A1 := A∗ ker(1 )
are self-adjoint. In the sequel usually the extension A0 corresponding to the boundary mapping 0 is regarded as a ‘fixed’ self-adjoint extension. We note that the closed extension A in (2.1) is disjoint with A0 , that is dom(A ) ∩ dom(A0 ) = dom(A), if and only if ∈ C (H). In this case (2.1) takes the form A = A∗ ker 1 − 0 . (2.2) Without loss of generality we will often restrict ourselves to simple symmetric operators. Recall that a symmetric operator is said to be simple if there is no nontrivial subspace which reduces it to a self-adjoint operator. By [54] each ⊕ As symmetric operator A in H can be written as the direct orthogonal sum A in the Hilbert space of a simple symmetric operator A H = clospan{ker(A∗ − λ) : λ ∈ C\R} and a self-adjoint operator As in H H. Here clospan{·} denotes the closed linear span. Obviously A is simple if and only if H coincides with H. Notice ∗ the adjoint that if = {H, 0 , 1 } is a boundary triplet for A of a non-simple symmetric operator A = A ⊕ As , then = H, 0 , 1 , where ∗ ∗ and 1 := 1 dom A 0 := 0 dom A ∗ ∈C H such that the extension is a boundary triplet for the simple part A ∗ ∗ (−1) := A (−1) ∈ , ∈ C (H), in H is given by A ⊕ As , A A = A = C H , and the Weyl functions and γ -fields of = {H, 0 , 1 } and H, 0 , 1 coincide. We say that a maximal dissipative operator is completely non-self-adjoint if there is no nontrivial reducing subspace in which it is self-adjoint. Note that each maximal dissipative operator can be decomposed orthogonally into a selfadjoint part and a completely non-self-adjoint part, see e.g. [42]. 2.2 Weyl Functions, γ -fields and Resolvents of Extensions Let, as in Section 2.1, A be a densely defined closed symmetric operator in H with equal deficiency indices. If λ ∈ C is a point of regular type of A, i.e. (A−λ)−1 is bounded, we denote the defect subspace of A by Nλ = ker(A∗ −λ). The following definition can be found in [31, 32, 34]. Definition 2.3 Let = {H, 0 , 1 } be a boundary triplet for A∗ . The operator valued functions γ (·) : ρ(A0 ) → [H, H] and M(·) : ρ(A0 ) → [H] defined by −1 γ (λ) := 0 Nλ and M(λ) := 1 γ (λ), λ ∈ ρ(A0 ), (2.3) are called the γ -field and the Weyl function, respectively, corresponding to the boundary triplet .
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˙ Nλ , λ ∈ ρ(A0 ), where as It follows from the identity dom(A∗ ) = ker(0 )+ above A0 = A∗ ker(0 ), that the γ -field γ (·) and the Weyl function M(·) in (2.3) are well defined. Moreover, both γ (·) and M(·) are holomorphic on ρ(A0 ) and the relations γ (λ) = I + (λ − μ)(A0 − λ)−1 γ (μ), λ, μ ∈ ρ(A0 ), and M(λ) − M(μ)∗ = (λ − μ)γ (μ)∗ γ (λ),
λ, μ ∈ ρ(A0 ),
(2.4)
are valid (see [32]). The identity (2.4) yields that M(·) is a Nevanlinna function, that is, M(·) is a ([H]-valued) holomorphic function on C\R and M(λ) = M(λ)∗
and
Im (M(λ)) 0 Im (λ)
(2.5)
hold for all λ ∈ C\R. The union of C\R and the set of all points λ ∈ R such that M can be analytically continued to λ and the continuations from C+ and C− coincide is denoted by h(M). Besides (2.5) it follows also from (2.4) that the Weyl function M(·) satisfies 0 ∈ ρ(Im (M(λ))) for all λ ∈ C\R; Nevanlinna functions with this additional property are sometimes called uniformly strict, cf. [30]. Conversely, each [H]-valued Nevanlinna function τ with the additional property 0 ∈ ρ(Im (τ (λ))) for some (and hence for all) λ ∈ C\R can be realized as a Weyl function corresponding to some boundary triplet, we refer to [32, 56, 58] for further details. Let again = {H, 0 , 1 } be a boundary triplet for A∗ with corresponding γ -field γ (·) and Weyl function M(·). The spectrum and the resolvent set of the closed (not necessarily self-adjoint) extensions of A can be described with the help of the function M(·). More precisely, if A ⊆ A∗ is the extension corresponding to ∈ C(H) via (2.1), then a point λ ∈ ρ(A0 ) belongs to ρ(A ) (σi (A ), i = p, c, r) if and only if 0 ∈ ρ( − M(λ)) (resp. 0 ∈ σi ( − M(λ)), i = p, c, r). Moreover, for λ ∈ ρ(A0 ) ∩ ρ(A ) the well-known resolvent formula −1 ∗ (2.6) (A − λ)−1 = (A0 − λ)−1 + γ (λ) − M(λ) γ λ holds, cf. [31, 32, 34]. Formula (2.6) is a generalization of the known Krein formula for canonical resolvents. We emphasize that it is valid for any closed extension A ⊆ A∗ of A with a nonempty resolvent set. 2.3 Self-adjoint Extensions and Scattering Let A be a densely defined closed symmetric operator with equal finite deficiency indices, i.e., n+ (A) = n− (A) < ∞. Let = {H, 0 , 1 }, A0 := A∗ ker(0 ), be a boundary triplet for A∗ and let A be a self-adjoint extension of A which corresponds to a self-adjoint ∈ C(H). Since here dim H is finite by (2.6) (A − λ)−1 − (A0 − λ)−1 ,
λ ∈ ρ(A ) ∩ ρ(A0 )
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is a finite rank operator and therefore the pair {A , A0 } performs a so-called complete scattering system, that is, the wave operators W± (A , A0 ) := s- lim eit A e−it A0 Pac (A0 ) t→±∞
exist and their ranges coincide with the absolutely continuous subspace Hac (A ) of A , cf. [17, 52, 72, 73]. Pac (A0 ) denotes the orthogonal projection onto the absolutely continuous subspace Hac (A0 ) of A0 . The scattering operator S(A , A0 ) of the scattering system {A , A0 } is then defined by S(A , A0 ) := W+ (A , A0 )∗ W− (A , A0 ). If we regard the scattering operator as an operator in Hac (A0 ), then S(A , A0 ) is unitary, commutes with the absolutely continuous part ac Aac 0 := A0 dom(A0 ) ∩ H (A0 )
of A0 and it follows that S(A , A0 ) is unitarily equivalent to a multiplication operator induced by a family {S (λ)} of unitary operators in a spectral representation of Aac 0 , see e.g. [17, Proposition 9.57]. This family is called the scattering matrix of the scattering system {A , A0 }. We note that if the symmetric operator A is not simple, then the Hilbert ⊥ H⊕ H (cf. the end of Section 2.1) such space H can be decomposed as H = , A 0 ⊕ I, that the scattering operator is given by the orthogonal sum S A ⊕ As and A0 = A 0 ⊕ As , and hence it is sufficient to consider where A = A simple symmetric operators A in the following. Since the deficiency indices of A are finite the Weyl function M(·) corresponding to the boundary triplet = {H, 0 , 1 } is a matrix-valued Nevanlinna function. By Fatous theorem (see [37, 44]) then the limit M(λ + i0) := lim M(λ + i )
→+0
(2.7)
from the upper half-plane exists for a.e. λ ∈ R. We denote the set of real points where the limit in (2.7) exits by M and we agree to use a similar notation for arbitrary scalar and matrix-valued Nevanlinna functions. Furthermore we will make use of the notation H M(λ) := ran Im (M(λ)) , λ ∈ M, (2.8) and we will usually regard H M(λ) as a subspace of H. The orthogonal projection and restriction onto H M(λ) will be denoted by P M(λ) and H M(λ) , respectively. Note that for λ ∈ ρ(A0 ) ∩ R the Hilbert space H M(λ) is trivial by (2.4). Again we agree to use a notation analogous to (2.8) for arbitrary Nevanlinna functions. The family {P M(λ) }λ∈ M of orthogonal projections in H onto H M(λ) , λ ∈ M , is measurable and defines an orthogonal projection in the Hilbert space L2 (R, dλ, H); sometimes we write L2 (R, H) instead of L2 (R, dλ, H). The range of this projection is denoted by L2 (R, dλ, H M(λ) ). Besides the Weyl function M(·) we will also make use of the function −1 λ → N (λ) := − M(λ) , λ ∈ C\R, (2.9)
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where ∈ C(H) is the self-adjoint relation corresponding to the extension A via (2.1). Since λ ∈ ρ(A0 ) ∩ ρ(A ) if and only if 0 ∈ ρ( − M(λ)) the function N (·) is well defined. It is not difficult to see that N (·) is an [H]-valued Nevanlinna function and hence N (λ + i0) = lim →0 N (λ + i ) exists for almost every λ ∈ R, we denote this set by N . We claim that −1 N (λ + i0) = − M(λ + i0) , λ ∈ M ∩ N , (2.10) holds. In fact, if is a self-adjoint matrix then (2.10) follows immediately from N (λ)( − M(λ)) = ( − M(λ))N (λ) = IH , λ ∈ C+ . If ∈ C(H) has a nontrivial multivalued part we decompose as = op ⊕ ∞ , where op is a self-adjoint matrix in Hop = dom(op ) and ∞ is a pure relation in H∞ = H Hop , cf. Section 2.1, and denote the orthogonal projection and restriction in H onto Hop by Pop and Hop , respectively. Then we have −1 Pop , λ ∈ C\R, λ → N (λ) = op − Pop M(λ) Hop (see e.g. [56, page 137]) and from −1 N (λ + i0) = op − Pop M(λ + i0) Hop Pop for all λ ∈ M ∩ N we conclude (2.10). Observe that R\( M ∩ N ) has Lebesgue measure zero. The next representation theorem of the scattering matrix is essential in the following, cf. [18, Theorem 3.8]. Since the scattering matrix is only determined up to a set of Lebesgue measure zero we choose the representative of the equivalence class defined on M ∩ N . Theorem 2.4 Let A be a densely defined closed simple symmetric operator with finite deficiency indices in the separable Hilbert space H, let = {H, 0 , 1 } be a boundary triplet for A∗ with corresponding Weyl function M(·) and define the spaces H M(λ) , λ ∈ M , as in (2.8). Furthermore, let A0 = A∗ ker(0 ) and let A = A∗ (−1) , ∈ C(H), be a self-adjoint extension of A. Then the following holds. Aac 0 is unitarily equivalent to the multiplication operator with the free variable in L2 (R, dλ, H M(λ) ). 2. In L2 (R, dλ, H M(λ) ) the scattering matrix {S (λ)} of the complete scattering system {A , A0 } is given by −1 Im (M(λ)) H M(λ) S (λ) = IH M(λ) + 2iP M(λ) Im (M(λ)) − M(λ) 1.
for all λ ∈ M ∩ N , where M(λ) := M(λ + i0). A similar representation of the S-matrix for point interactions was obtained in [6], see also [8]. In order to show the usefulness of Theorem 2.4 and to make the reader more familiar with the notion of boundary triplets and associated Weyl functions we calculate the scattering matrix of the scattering system {−d2 /dx2 + δ, −d2 /dx2 } in the following simple example.
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Example 2.5 Let us consider the densely defined closed simple symmetric operator (A f )(x) := − f
(x), dom(A) = f ∈ W22 (R) : f (0) = 0 , in L2 (R), see e.g. [7]. Clearly A has deficiency indices n+ (A) = n− (A) = 1 and it is well-known that the adjoint operator A∗ is given by (A∗ f )(x) = − f
(x), dom(A∗ ) = f ∈ W22 (R\{0}) : f (0+) = f (0−), f
∈ L2 (R) . It is not difficult to verify that = {C, 0 , 1 }, where 0 f := f (0+) − f (0−) and 1 f := − f (0+),
f ∈ dom(A∗ ),
is a boundary triplet for A∗ and A0 = A∗ ker(0 ) coincides with the usual self-adjoint second order differential operator defined on W22 (R). Moreover the defect space ker(A∗ − λ), λ ∈ [0, ∞), is spanned by the function x → ei
√
λx
χR+ (x) + e−i
√
λx
χR− (x),
λ ∈ [0, ∞),
where √ C with a cut along [0, ∞) and fixed by √ the square root is defined on Im λ > 0 for λ ∈ [0, ∞) and by λ 0 for λ ∈ [0, ∞). Therefore we find that the Weyl function M(·) corresponding to = {C, 0 , 1 } is given by M(λ) =
i 1 fλ = √ , 0 fλ 2 λ
fλ ∈ ker(A∗ − λ),
λ ∈ [0, ∞).
Let α ∈ R\{0} and consider the self-adjoint extension A−α−1 corresponding to the parameter −α −1 , A−α−1 = A∗ ker(1 + α −1 0 ), i.e. (A−α−1 f )(x) = − f
(x) dom(A−α−1 ) = f ∈ dom(A∗ ) : α f (0±) = f (0+) − f (0−) . This self-adjoint operator is often denoted by −d2 /dx2 + αδ, see [7]. It follows immediately from Theorem 2.4 that the scattering matrix {S(λ)} of the scattering system {A−α−1 , A0 } is given by √ 2 λ − iα S(λ) = √ , λ > 0. 2 λ + iα We note that scattering systems of the form −d2 /dx2 +αδ , −d2 /dx2 , α ∈ R, can be investigated in a similar way as above. Other examples can be found in [18].
3 Dissipative and Lax–Phillips Scattering Systems In this section we regard scattering systems {AD , A0 } consisting of a maximal dissipative and a self-adjoint extension of a symmetric operator A with finite deficiency indices. In the theory of open quantum system the maximal dissipative operator AD is often called a pseudo-Hamiltonian. We shall explicitely
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of AD and calculate construct a dilation (or so-called quasi-Hamiltonian) K the scattering matrix of the scattering system K, A0 ⊕ G0 , where G0 is a selfadjoint first order differential operator. The diagonal entries of the scattering matrix then turn out to be the scattering matrix of the dissipative scattering system {AD , A0 } and of a so-called Lax–Phillips scattering system, respectively. We emphasize that this efficient and somehow straightforward method for the analysis of scattering processes for open quantum systems has the essential and A0 ⊕ G0 are necessarily not disadvantage that the quasi-Hamiltonians K semibounded from below. 3.1 Self-adjoint Dilations of Maximal Dissipative Operators Let in the following A be a densely defined closed simple symmetric operator in the separable Hilbert space H with equal finite deficiency indices n± (A) = n < ∞, let = {H, 0 , 1 }, A0 = A∗ ker(0 ), be a boundary triplet for A∗ and let D ∈ [H] be a dissipative n × n-matrix. Then the closed extension AD = A∗ ker(1 − D0 ) of A corresponding to = D via (2.1)–(2.2) is maximal dissipative and C+ belongs to ρ(AD ). Note that here we restrict ourselves to maximal dissipative extensions AD corresponding to dissipative matrices D instead of maximal dissipative relations in the finite dimensional space H. This is no essential restriction, see Remark 3.3 at the end of this subsection. For λ ∈ ρ(AD ) ∩ ρ(A0 ) the resolvent of the extension AD is given by ∗ (AD − λ)−1 = (A0 − λ)−1 + γ (λ)(D − M(λ))−1 γ λ ,
(3.1)
cf. (2.6). Write the dissipative matrix D ∈ [H] as D = Re (D) + iIm (D), decompose H as the direct orthogonal sum of the finite dimensional subspaces ker(Im (D)) and H D := ran (Im (D)), H = ker(Im (D)) ⊕ H D ,
(3.2)
and denote by P D and H D the orthogonal projection and restriction in H onto H D . Since Im (D) 0 the self-adjoint matrix −P D Im (D) H D ∈ [H D ] is strictly positive and the next lemma shows how −iP D Im (D) H D (and iP D Im (D) H D ) can be realized as a Weyl function of a differential operator. Lemma 3.1 Let G be the symmetric first order differential operator in the Hilbert space L2 (R, H D ) defined by (Gg)(x) = −ig (x),
dom(G) = g ∈ W21 (R, H D ) : g(0) = 0 .
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Then G is simple, n± (G) = dim H D and the adjoint operator G∗ g = −ig is defined on dom(G∗ ) = W21 (R− , H D ) ⊕ W21 (R+ , H D ). Moreover, the triplet G = {H D , ϒ0 , ϒ1 }, where − 1 1 ϒ0 g := √ −P D Im (D) H D 2 g(0+) − g(0−) , 2 1 i ϒ1 g := √ −P D Im (D) H D 2 g(0+) + g(0−) , 2 g ∈ dom(G∗ ), is a boundary triplet for G∗ and G0 := G∗ ker(ϒ0 ) is the usual self-adjoint first order differential operator in L2 (R, H D ) with domain dom(G0 ) = W21 (R, H D ) and σ (G0 ) = R. The Weyl function τ (·) corresponding to the boundary triplet G = {H D , ϒ0 , ϒ1 } is given by
−iP D Im (D) H D , λ ∈ C+ , (3.3) τ (λ) = iP D Im (D) H D , λ ∈ C− . Proof Besides the assertion that G = {H D , ϒ0 , ϒ1 } is a boundary triplet for G∗ with Weyl function τ (·) given by (3.3) the statements of the lemma are well-known. We note only that the simplicity of G follows from [1, VIII.104] and the fact that G can be written as a finite direct orthogonal sum of first order differential operators on R− and R+ . A straightforward calculation shows that the identity (G∗ g, k) − (g, G∗ k) = i(g(0+), k(0+)) − i(g(0−), k(0−)) = (ϒ1 g, ϒ0 k) − (ϒ0 g, ϒ1 k) holds for all g, k ∈ dom(G∗ ). Moreover, the mapping (ϒ0 , ϒ1 ) is surjective. Indeed, for an element (h, h ) ∈ H D × H D we choose g ∈ dom(G∗ ) such that 1 − 1 1 g(0+) = √ −P D Im (D) H D 2 h − i −P D Im (D) H D 2 h 2 and
1 − 1 1 g(0−) = √ − −P D Im (D) H D 2 h − i −P D Im (D) H D 2 h 2
holds. Then a simple calculation shows ϒ0 g = h, ϒ1 g = h and therefore G = {H D , ϒ0 , ϒ1 } is a boundary triplet for G∗ . It is not difficult to check that the defect subspace Nλ = ker(G∗ − λ) is
λ ∈ C+ , span x → eiλx χR+ (x)ξ : ξ ∈ H D , Nλ = iλx λ ∈ C− , span x → e χR− (x)ξ : ξ ∈ H D , and hence we conclude that the Weyl function of G = {H D , ϒ0 , ϒ1 } is given by (3.3). Let AD be the maximal dissipative extension of A in H from above and let G be the first order differential operator from Lemma 3.1. Clearly K := A ⊕ G
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is a densely defined closed simple symmetric operator in the separable Hilbert space K := H ⊕ L2 (R, H D ),
with equal finite deficiency indices n± (K) = n± (A) + n± (G) < ∞ and the adjoint is K∗ = A∗ ⊕ G∗ . The elements in dom(K∗ ) = dom(A∗ ) ⊕ dom(G∗ ) will be written in the form f ⊕ g, f ∈ dom(A∗ ), g ∈ dom(G∗ ). In the next of K in K which is a minimal theorem we construct a self-adjoint extension K self-adjoint dilation of the dissipative operator AD in H. The construction is based on the idea of the coupling method from [29]. It is worth to mention that in the case of a (scalar) Sturm–Liouville operator with real potential and dissipative boundary condition our construction coincides with the one proposed by B.S. Pavlov in [68, 71], cf. Example 3.5 below. Theorem 3.2 Let A, = {H, 0 , 1 } and AD be as in the beginning of this section, let G and G = {H D , ϒ0 , ϒ1 } be as in Lemma 3.1 and K = A ⊕ G. Then ⎧ ⎫ P D 0 f − ϒ0 g = 0,⎬ ⎨ = K∗ f ⊕ g ∈ dom(K∗ ) : (1 − P D )(1 − Re (D)0 ) f = 0, , (3.4) K ⎩ ⎭ P D (1 − Re (D)0 ) f + ϒ1 g = 0 is a minimal self-adjoint dilation of the maximal dissipative operator AD , that is, for all λ ∈ C+ − λ −1 H = (AD − λ)−1 PH K − λ −1 H : λ ∈ C\R is holds and the minimality condition K = clospan K = R. satisfied. Moreover, σ K Proof Let γ (·), ν(·) and M(·), τ (·) be the γ -fields and Weyl functions of the boundary triplets = {H, 0 , 1 } and G = {H D, ϒ0 , ϒ1 }, respectively. Then , = H it is straightforward to check that 0 , 1 , where 0 1 − Re (D)0 H := H ⊕ H D , 0 := and 1 := , (3.5) ϒ0 ϒ1 is a boundary triplet for K∗ = A∗ ⊕ G∗ and the corresponding Weyl function and γ -field M(·) γ (·) are given by M(λ) − Re (D) 0 (3.6) M(λ) = , λ ∈ C\R, 0 τ (λ) and
γ (λ) =
γ (λ) 0 , 0 ν(λ)
λ ∈ C\R,
respectively. Note also that K0 := K∗ ker 0 = A0 ⊕ G0 holds.
(3.7)
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= ker(Im (D)) ⊕ H D ⊕ H D of H With respect to the decomposition H (cf. (3.2)) we define the linear relation by
(u, v, v) . := (3.8) : u ∈ ker(Im (D), v, w ∈ ∈ C H H D (0, −w, w) is self-adjoint. Hence it follows from We leave it to the reader to check that ∗ is a self-adjoint extension Proposition 2.2 that the operator K (−1) = K of the symmetric operator K = A ⊕ G in K = H ⊕ L2 (R, H D ) and one verifies from (3.4), K = K without difficulty that this extension coincides with K . −1 − λ , λ ∈ C\R, we use the block matrix In order to calculate K decomposition D D M11 (λ) M12 (λ) (3.9) M(λ) − Re (D) = ∈ ker(Im (D)) ⊕ H D D D M21 (λ) M22 (λ) in (3.8) and (3.6) imply of M(λ) − Re (D) ∈ [H]. Then the definition of ⎧⎛⎛ ⎫ ⎞⎞ D D ⎪ ⎪ (λ)u − M12 (λ)v −M11 ⎪ ⎪ ⎪ ⎪ ⎜ ⎟⎟ ⎬ D D −1 ⎨⎜ u ∈ ker(Im (D)) −w − M (λ)u − M (λ)v ⎜ ⎟ ⎝ ⎠ 21 22 − M(λ) = ⎜ ⎟: v, w ∈ H D ⎪ ⎪⎝ ⎪ ⎠ w − τ (λ)v ⎪ ⎪ ⎪ ⎩ ⎭ (u, v, v) ∩ ρ(K0 ), K0 = A0 ⊕ G0 , it follows and since every λ ∈ C\R belongs to ρ K −1 − M(λ) that , λ ∈ C\R, is the graph of a bounded everywhere defined −1 − M(λ) in a more explicit form we set operator. In order to calculate D D x := −M11 (λ)u − M12 (λ)v, D D y := −w − M21 (λ)u − M22 (λ)v, z := w − τ (λ)v.
This yields
(3.10)
D D (λ) M12 (λ) M11 x u =− D D y+z v M21 (λ) M22 (λ) + τ (λ)
and by (3.3) and (3.9) we have D D M11 (λ) M12 (λ) D − M(λ), = − D D M21 (λ) M22 (λ) + τ (λ) D∗ − M(λ),
λ ∈ C+ , . λ ∈ C−
(3.11)
−1 x −1 u = D − M(λ) + D − M(λ) H D z v y
(3.12)
Hence for λ ∈ C+ we find −1 u x = D − M(λ) v y+z which implies
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−1 x −1 H D z. v = P D D − M(λ) + P D D − M(λ) y
(3.13)
Therefore, by inserting (3.10), (3.12) and (3.13) into the above expression for −1 − M(λ) we obtain −1 (D − M(λ))−1 H D (D − M(λ))−1 − M(λ) (3.14) = P D (D − M(λ))−1 P D (D − M(λ))−1 H D admits for all λ ∈ C+ and by (2.6) the resolvent of the self-adjoint extension K the representation −1 ∗ − λ −1 = (K0 − λ)−1 + − M(λ) K γ (λ) γ λ , (3.15) λ ∈ C\R. It follows from K0 = A0 ⊕ G0 , (3.7) and (3.14) that for λ ∈ C+ the onto H is given by compressed resolvent of K − λ −1 H = (A0 − λ)−1 + γ (λ) D − M(λ) −1 γ λ ∗ , PH K where PH denotes the orthogonal projection in K onto H. Taking into account (3.1) we get − λ −1 H = (AD − λ)−1 , λ ∈ C+ , PH K is a self-adjoint dilation of AD . Since σ (G0 ) = R it follows from and hence K = R holds. well-known perturbation results and (3.15) that σ K satisfies the minimality condition It remains to show that K − λ −1 H : λ ∈ C\R . K = H ⊕ L2 (R, H D ) = clospan K (3.16) −1 − it First of all s-limt→+∞ (−it) K = IK implies that H is a subset of the right hand side of (3.16). The orthogonal projection in K onto L2 (R, H D ) is denoted by P L2 . Then we conclude from (3.7), (3.14) and (3.15) that for λ ∈ C+ − λ −1 H = ν(λ)P D D − M(λ) −1 γ λ ∗ P L2 K (3.17) holds and this gives ! " − λ −1 H = ker(G∗ − λ), ran P L2 K
λ ∈ C+ .
From (3.11) it follows that similar to the matrix representation (3.14) the −1 − M(λ) left lower corner of is given by P D (D∗ − M(λ))−1 for λ ∈ C− . Hence, the analogon of (3.17) for λ ∈ C− implies that − λ −1 H = ker(G∗ − λ) ran P L2 K is true for λ ∈ C− . Since by Lemma 3.1 the symmetric operator G is simple it follows that L2 (R, H D ) = clospan ker(G∗ − λ) : λ ∈ C\R holds, cf. Section 2.1, and therefore the minimality condition (3.16) holds.
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Remark 3.3 We note that also in the case when the parameter D is not a dissipative matrix but a maximal dissipative relation in H a minimal self-adjoint dilation of AD can be constructed in a similar way as in Theorem 3.2. Indeed, let A and = {H, 0 , 1 } be as in the beginning of this section and can be written as ∈ C(H) be a maximal dissipative relation in H. Then D let D the direct orthogonal sum of a dissipative matrix Dop in H op := H mul( D) 0 ∞ := y : y ∈ mul( D) . It and an undetermined part or ‘pure relation’ D follows that ∞ = A∗ (−1) D B := A∗ (−1) 0y : y ∈ mul( D) is a closed symmetric extension of A and Hop , 0 dom(B∗ ) , Pop 1 dom(B∗ ) is a boundary triplet for
B∗ = A∗ f ∈ dom(A∗ ) : (1 − Pop )0 f = 0
with A∗ ker(0 ) = B∗ ker(0 dom(B∗ ) ). In terms of this boundary triplet the (−1) maximal dissipative extension A D D coincides with the extension = ∗ op 0 dom(B∗ ) , BD op = B ker Pop 1 dom(B∗ ) − D op ∈ [Hop ] of D. corresponding to the operator part D Remark 3.4 In the special case ker(Im D) = {0} the relations (3.4) take the form 0 f − ϒ0 g = 0 and
(1 − Re (D)0 ) f + ϒ1 g = 0,
is a coupling of the self-adjoint operators A0 and G0 corresponding so that K to the coupling of the boundary triplets A = {H, 0 , 1 − Re (D)0 } and G = {H, ϒ0 , ϒ1 } in the sense of [29]. In the case ker(Im D) = {0} another is based on the concept of boundary relations (see [30]). construction of K for a scalar Sturm–Liouville operator with A minimal self-adjoint dilation K a complex (dissipative) boundary condition has originally been constructed by B.S. Pavlov in [68]. For the scalar case (n = 1) the operator in (3.20) in the following example coincides with the one in [68]. Example 3.5 Let Q+ ∈ L1loc (R+ , [Cn ]) be a matrix valued function such that Q+ (·) = Q+ (·)∗ , and let A be the usual minimal operator in H = L2 (R+ , Cn ) associated with the Sturm–Liouville differential expression −d2 /dx2 + Q+ , d2 + Q+ , dom(A) = f ∈ Dmax,+ : f (0) = f (0) = 0 , dx2 where Dmax,+ is the maximal domain defined by A=−
Dmax,+ = f ∈ L2 (R+ , Cn ) : f, f ∈ AC(R+ , Cn ), − f
+ Q+ f ∈ L2 (R+ , Cn ) .
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It is well known that the adjoint operator A∗ is given by d2 + Q+ , dom(A∗ ) = Dmax,+ . dx2 In the following we assume that the limit point case prevails at +∞, so that the deficiency indices n± (A) of A are both equal to n. In this case a boundary triplet = {Cn , 0 , 1 } for A∗ is A∗ = −
0 f := f (0),
1 f := f (0),
f ∈ dom(A∗ ) = Dmax,+ .
(3.18)
For any dissipative matrix D ∈ [Cn ] we consider the (maximal) dissipative extension AD of A determined by AD = A∗ ker(1 − D0 ),
Im D 0.
(3.19)
(a) First suppose 0 ∈ ρ(Im D). Then H D = Cn and by Theorem 3.2 and of the operator AD is Remark 3.4 the (minimal) self-adjoint dilation K a self-adjoint operator in K = L2 (R+ , Cn ) ⊕ L2 (R, Cn ) defined by f ⊕ g) = − f
+ Q+ f ⊕ −ig , K( ⎫ ⎧ f ∈ Dmax,+ , g ∈ W21 (R− , Cn ) ⊕ W21 (R+ , Cn )⎪ ⎪ ⎬ ⎨ (3.20) = f (0) − Df (0) = −i(−2Im D)1/2 g(0−), . dom K ⎪ ⎪ ⎩ ⎭ f (0) − D∗ f (0) = −i(−2Im D)1/2 g(0+) (b) Let now ker(Im D) = {0}, so that H D = ran (Im D) = Ck = Cn . According of the operator AD to Theorem 3.2 the (minimal) self-adjoint dilation K in K = L2 (R+ , Cn ) ⊕ L2 (R, Ck ) is defined by f ⊕ g) = − f
+ Q+ f ⊕ −ig , K( ⎧ ⎫ f ∈ Dmax,+ , g ∈ W21 (R− , Ck ) ⊕ W21 (R+ , Ck ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ P D [ f (0) − Df (0)] = −i(−2P D Im (D) H D )1/2 g(0−), ⎬ = dom K .
∗ ⎪ P [ f (0) − D f (0)] = −i(−2P Im (D) H D )1/2 g(0+),⎪ D D ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ f (0) − Re(D) f (0) ∈ H D 3.2 Dilations and Dissipative Scattering Systems Let, as in the previous section, A be a densely defined closed simple symmetric operator in H with equal finite deficiency indices and let = {H, 0 , 1 } be a boundary triplet for A∗ , A0 = A∗ ker 0 , with corresponding Weyl function M(·). Let D ∈ [H] be a dissipative matrix and let AD = A∗ ker(1 − D0 ) be the corresponding maximal dissipative extension in H. Since the function C+ λ → M(λ) − D is a Nevanlinna function the limits M(λ + i0) − D = lim M(λ + i ) − D
→+0
and
−1 N D (λ + i0) = lim N D (λ + i ) = lim D − M(λ + i )
→+0
→+0
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exist for a.e. λ ∈ R. We denote these sets of real points λ by M and N D . Then we have −1 N D (λ + i0) = D − M(λ + i0) , λ ∈ M ∩ ND , (3.21) cf. Section 2.3. Let G be the symmetric first order differential operator in L2 (R, H D ) and let G = {H D , ϒ0 , ϒ1 } be the boundary triplet from Lemma 3.1. Then G0 = G∗ ker(ϒ0 ) is the usual self-adjoint differentiation operator in L2 (R, H D ) and K0 = A0 ⊕ G0 is self-adjoint in K = H ⊕ L2(R, H D). K0 , In the next theorem we consider the complete scattering system K, is the minimal self-adjoint dilation of AD in K from Theorem 3.2. where K Theorem 3.6 Let A, = {H, 0 , 1 }, M(·) and AD be as above and define be the minimal selfH M(λ) , λ ∈ M , as in (2.8). Let K0 = A0 ⊕ G0 and let K adjoint dilation of AD from Theorem 3.2. Then the following holds. K0ac = Aac 0 ⊕ G0 is unitarily equivalent to the multiplication operator with the free variable in L2 (R, dλ, H M(λ) ⊕ H D ). 2. In L2 (R, dλ, H M(λ) ⊕ H D ) the scattering matrix S(λ) of the complete K0 is given by scattering system K, 1.
S(λ) =
11 (λ) T IH M(λ) 0 + 2i 21 (λ) 0 IH D T
12 (λ) T ∈ [H M(λ) ⊕ H D ] 22 (λ) T
for all λ ∈ M ∩ N D , where 11 (λ) = P M(λ) Im (M(λ)) D − M(λ) −1 Im (M(λ)) H M(λ) , T 12 (λ) = P M(λ) Im (M(λ)) D − M(λ) −1 −Im (D) H D , T 21 (λ) = P D −Im (D) D − M(λ) −1 Im (M(λ)) H M(λ) , T 22 (λ) = P D −Im (D) D − M(λ) −1 −Im (D) H D T and M(λ) = M(λ + i0). = H ⊕ HD, Proof Let K = A ⊕ G and let 0 , 1 be the boundary triplet for K∗ from (3.5). Note that since A and G are densely defined closed simple symmetric operators also K is a densely defined closed simple symmetric = H ⊕ HD, operator. Recall that for λ ∈ C+ the Weyl function of 0 , 1 is given by M(λ) − Re (D) 0 M(λ) = . (3.22) 0 −iP D Im (D) H D Then Theorem 2.4 implies that , H M(λ) L2 R, dλ, H M(λ) = H M(λ) ⊕ H D ,
λ ∈ M,
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performs a spectral representation of the absolutely continuous part K0ac = K0 dom(K0 ) ∩ Kac (K0 ) = A0 ⊕ G0 dom(A0 ) ∩ Hac (A0 ) ⊕ L2 (R, H D ) = Aac 0 ⊕ G0 K0 is of K0 such that the scattering matrix S(λ) of the scattering system K, given by # # −1 S(λ) = IH M(λ) + 2iP M(λ) Im ( M(λ)) − M(λ) Im ( M(λ)) H M(λ) (3.23)
for all λ ∈ M ∩ N , where P M(λ) and H M(λ) are the projection and restriction in H = H ⊕ H D onto H M(λ) . Here is the self-adjoint relation from (3.8), the function N is defined analogously to (2.9) and + i0) −1 − M(λ N (λ + i0) =
holds for all λ ∈ M ∩ N , cf. (2.10). By (3.22) we have √ # Im (M(λ + i0)) + i0) = √ 0 Im M(λ 0 P D −Im (D) H D
for all λ ∈ M = M and (3.14) yields −1 (D − M(λ + i0))−1 H D (D − M(λ + i0))−1 + i0) − M(λ = P D (D − M(λ + i0))−1 P D (D − M(λ + i0))−1 H D for λ ∈ M ∩ N . It follows that the sets M ∩ N and M ∩ N D , see (3.21), coincide and by inserting the above expressions into (3.23) we conclude S(λ) is a two-by-two block that for each λ ∈ M ∩ N D the scattering matrix operator matrix with respect to the decomposition H M(λ) = H M(λ) ⊕ H D ,
λ ∈ M ∩ ND ,
with the entries from assertion (2).
Remark 3.7 It is worth to note that the scattering matrix S(λ) of the scatter K0 in Theorem 3.6 depends only on the dissipative matrix D ing system K, and the Weyl function M(·) of the boundary triplet = {H, 0 , 1 } for A∗ . In other words, the scattering matrix S(λ) is completely determined by objects corresponding to the operators A, A0 and AD in H. Let AD and A0 be as in the beginning of this section. In the following we will focus on the so-called dissipative scattering system {AD , A0 } and we refer the reader to [26, 27, 59–65] for a detailed investigation of such scattering systems. We recall only that the wave operators W± (AD , A0 ) of the dissipative scattering system {AD , A0 } are defined by ∗
W+ (AD , A0 ) = s- lim eit AD e−it A0 Pac (A0 ) t→+∞
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and W− (AD , A0 ) = s- lim e−it AD eit A0 Pac (A0 ), t→+∞
where e−it AD := s-limn→∞ (1 + (it/n)AD )−n , see e.g. [52, Section IX]. The scattering operator S D := W+ (AD , A0 )∗ W− (AD , A0 ) of the dissipative scattering system {AD , A0 } will be regarded as an operator in Hac (A0 ). Then S D is a contraction which in general is not unitary. Since S D and Aac 0 commute it follows that S D is unitarily equivalent to a multiplication operator induced by a family {S D (λ)} of contractive operators in a spectral representation of Aac 0 . With the help of Theorem 3.6 we obtain a representation of the scattering matrix of the dissipative scattering system {AD , A0 } in terms of the Weyl function M(·) of = {H, 0 , 1 } in the following corollary, cf. Theorem 2.4. Corollary 3.8 Let A, = {H, 0 , 1 }, A0 = A∗ ker(0 ), M(·) and AD be as above and define H M(λ) , λ ∈ M , as in (2.8). Then the following holds. Aac 0 is unitarily equivalent to the multiplication operator with the free variable in L2 (R, dλ, H M(λ) ). 2. In L2 (R, dλ, H M(λ) ) the scattering matrix {S D (λ)} of the dissipative scattering system {AD , A0 } is given by 1.
−1 S D (λ) = IH M(λ) + 2iP M(λ) Im (M(λ)) D − M(λ) Im (M(λ)) H M(λ)
for all λ ∈ M ∩ N D , where M(λ) = M(λ + i0). be the minimal self-adjoint dilation of AD from Theorem 3.2. Proof Let K Since for t 0 we have −n it −n it H = s- lim 1 + AD PH e−it K H = s- lim PH 1 + K n→∞ n→∞ n n = e−it AD it follows that the wave operators W+ (AD , A0 ) and W− (AD , A0 ) coincide with K0 H = s- lim PH eit K e−itK0 Pac (K0 ) H PH W+ K, t→+∞
= s- lim PH eit K H e−it A0 Pac (A0 ) t→+∞
and
K0 H = s- lim PH eit Ke−itK0 Pac (K0 ) H PH W− K, t→−∞
= s- lim PH e−it K H eit A0 Pac (A0 ), t→+∞
the respectively. This implies that the scattering operator S D coincides with K0 Hac (A0 ) of the scattering operator S K, K0 compression PHac (A0 ) S K,
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onto Hac (A0 ). Therefore the scattering matrix S D (λ) of the dissipative scattering system is given by the upper left corner 11 (λ) , λ ∈ M ∩ N D , IH M(λ) + 2i T K0 , see of the scattering matrix S(λ) of the scattering system K, Theorem 3.6. 3.3 Lax–Phillips Scattering Systems Let again A, = {H, 0 , 1 }, {AD , A0 } and G, G0 , G = {H D , ϒ0 , ϒ1 } be as in the previous subsections. In Corollary 3.8 we have shown that the scattering in matrix of the dissipative scattering system {AD , A0 } is the left upper corner the block operator matrix representation of the scattering matrix S(λ) of the K0 , where K is a minimal self-adjoint dilation of AD in scattering system K, 2 K = H ⊕ L (R, H D ) and K0 = A0 ⊕ G0 , cf. Theorem 3.6. In the following we are going to interpret the right lower corner of S(λ) as the scattering matrix corresponding to a Lax–Phillips scattering system, see e.g. [17, 57] for further details. To this end we decompose the space L2 (R, H D ) into the orthogonal sum of the subspaces D− := L2 (R− , H D )
and D+ := L2 (R+ , H D ).
(3.24)
Then clearly K = H ⊕ D− ⊕ D+ and we agree to denote the elements in K in the form f ⊕ g− ⊕ g+ , f ∈ H, g± ∈ D± and g = g− ⊕ g+ ∈ L2 (R, H D ). By J+ and J− we denote the operators J+ : L2 (R, H D ) → K,
g → 0 ⊕ 0 ⊕ g+ ,
J− : L2 (R, H D ) → K,
g → 0 ⊕ g− ⊕ 0,
and
respectively. Note that J+ + J− is the embedding of L2 (R, H D ) into K. In the next lemma we show that D+ and D− are so-called outgoing and incoming in K. subspaces for the self-adjoint dilation K be the self-adjoint operator from Theorem 3.2, let D± be as Lemma 3.9 Let K in (3.24) and let A0 = A∗ ker(0 ) be as above. Then $ e−it K D± ⊆ D± , t ∈ R± , and e−it K D± = {0} t∈R
and, if in addition σ (A0 ) is singular, then % % . e−it K D+ = e−it KD− = Kac K t∈R
(3.25)
t∈R
Proof Let us first show that
e−it K D± = J± e−itG0 D± ,
t ∈ R± ,
(3.26)
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holds. In fact, since e−itG0 is the right shift group we have e−itG0 (dom(G) ∩ D± ) ⊆ dom(G) ∩ D± ,
t ∈ R± ,
where dom(G) ∩ D± = {W 1,2 (R, H D ) : f (x) = 0, x ∈ R± }. Let us fix some t ∈ R± and denote the symmetric operator A ⊕ G by K. Since J± dom(G) ∩ D± ⊂ dom(K) ⊂ dom K the function
ft,± (s) := ei(s−t) K J± e−isG0 D± f± ,
s ∈ R± ,
f± ∈ dom(G) ∩ D± ,
is differentiable and d − 0H ⊕ G0 J± e−isG0 D± f± = 0, ft,± (s) = iei(s−t) K K ds
t ∈ R± ,
holds. Hence we have ft,± (0) = ft,± (t) and together with the observation that the set dom(G) ∩ D± is dense in D± this immediately implies (3.26). Then we obtain e−it K D± ⊆ D± , t ∈ R± , and $ $ $ e−it K D± ⊆ e−it K D± = J± e−itG0 D± = {0}. t∈R
t∈R±
t∈R±
Let us show (3.25). Since A has finite deficiency indices the wave operators A0 ⊕ G0 exist and are complete, i.e., W± K, A0 ⊕ G0 = Kac K ran W± K, holds. Since A0 is singular we have A0 ⊕ G0 = s- lim eit K(J+ + J− )e−itG0 L2 W± K, t→±∞
A0 ⊕ G0 f± = f± for f± ∈ D± , so and it follows from (3.26) that W± K, for t ∈ R± . Assume now that that in particular D± and e−itG0 D± ∈ Kac K 2 g ∈ L (R, H D ) vanishes identically on some open interval (−∞, α). Then for r > 0 sufficiently large e−irG0 g ∈ D+ and by (3.26) for t > r
eit K (J+ + J− )e−i(t−r)G0 e−irG0 g = eir K J+ e−irG0 g. Since the elements g ∈ L2 (R, H D ) which vanish on intervals (−∞, α) form a A0 ⊕ G0 is complete dense set in L2 (R, H D ) and the wave operator W+ K, we conclude that % eir K D+ (3.27) r∈R+
. A similar argument shows that the set (3.27) with is a dense set in Kac K . This R+ and D+ replaced by R− and D− , respectively, is also dense in Kac K implies (3.25).
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D− , D+ is a Lax–Phillips scattering According to Lemma 3.9 the system K, system and in particular the Lax–Phillips wave operators
± := s- lim eit K J± e−itG0 : L2 (R, H D ) → K t→±∞
exist, cf. [17]. We note that s-limt→±∞ J∓e−itG0 = 0 and therefore the restric K0 , tions of the wave operators W± K, K0 of the scattering system K, K0 = A0 ⊕ G0 , onto L2 (R, H D ), K0 L2 = s- lim eit K (J+ + J− )e−itG0 , W± K, t→±∞
coincide with the Lax–Phillips wave operators ± . Hence the Lax–Phillips scattering operator S LP := ∗+ − admits the representation K0 L2 , S LP = P L2 S K, K0 = W+ K, K0 ∗ W− K, K0 is the scattering operator of the where S K, K0 . The Lax–Phillips scattering operator S LP is a conscattering system K, traction in L2 (R, H D ) and commutes with the self-adjoint differential operator G0 . Hence S LP is unitarily equivalent to a multiplication operator induced by a family {S LP (λ)} of contractive operators in L2 (R, H D ), this family is called the Lax–Phillips scattering matrix. The above considerations together with Theorem 3.6 immediately imply the following corollary on the representation of the Lax–Phillips scattering matrix. D− , D+ be the Lax–Phillips scattering system considCorollary 3.10 Let K, ered in Lemma 3.9 and let A, = {H, 0 , 1 }, AD , M(·) and G0 be as in the previous subsections. Then G0 = Gac 0 is unitarily equivalent to the multiplication operator with the free variable in L2 (R, H D ) = L2 (R, dλ, H D ) and the Lax– Phillips scattering matrix {S LP (λ)} admits the representation −1 S LP (λ) = IH D + 2iP D Im (−D) D − M(λ) Im (−D) H D (3.28) for λ ∈ M ∩ N D , where M(λ) = M(λ + i0). Let again AD be the maximal dissipative extension of A corresponding to the maximal dissipative matrix D ∈ [H] and let H D = ran (Im (D)). By [33] the characteristic function W AD of the completely non-self-adjoint part of AD is given by W AD : C− → [H D ] −1 μ → IH D − 2iP D −Im (D) D∗ − M(μ) −Im (D) H D .
(3.29)
Comparing (3.28) and (3.29) we obtain the famous relation between the Lax–Phillips scattering matrix and the characteristic function found by Adamyan and Arov in [2–5].
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Corollary 3.11 Let the assumption be as in Corollary 3.10. Then the Lax– Phillips scattering matrix {S LP (λ)} and the characteristic function W AD of the maximal dissipative operator AD are related by S LP (λ) = W AD (λ − i0)∗ ,
λ ∈ M ∩ ND .
Next we consider the special case that the spectrum σ (A0 ) of the selfadjoint extension A0 = A∗ ker(0 ) is purely singular, Hac (A0 ) = {0}. As usual let M(·) be the Weyl function corresponding to = {H, 0 , 1 }. Then we have H M(λ) = ran (Im (M(λ + i0))) = {0} for a.e. λ ∈ M , cf. [21], and if even σ (A0 ) = σ p (A0 ) then H M(λ) = {0} for all λ ∈ M . Therefore Theorem 3.6 and Corollaries 3.10 and 3.11 imply the following statement. Corollary 3.12 Let the assumption be as in Corollary 3.10, let K0 = A0 ⊕ G0 and assume in addition that σ (A0 ) is purely singular. Then the scattering K0 coincides with the matrix S(λ) of the complete scattering system K, LP Lax–Phillips scattering matrix {S (λ)} of the Lax–Phillips scattering system D− , D+ , that is, K, S(λ) = S LP (λ) = W AD (λ − i0)∗
(3.30)
for a.e. λ ∈ R. If even σ (A0 ) = σ p (A0 ), then (3.30) holds for all λ ∈ M ∩ N D . 3.4 1D-Schrödinger Operators with Dissipative Boundary Conditions In this subsection we consider an open quantum system consisting of a self-adjoint and a maximal dissipative extension of a symmetric regular Sturm–Liouville differential operator. Such maximal dissipative operators or pseudo-Hamiltonians are used in the description of carrier transport in semiconductors, see e.g. [13, 15, 38, 43, 50, 51, 53]. Assume that −∞ < xl < xr < ∞ and let V ∈ L∞ ((xl , xr )) be a real valued function. Moreover, let m ∈ L∞ ((xl , xr )) be a real function such that m > 0 and m−1 ∈ L∞ ((xl , xr )). It is well-known that (A f )(x) := − ⎧ ⎨
1 d 1 d f (x) + V(x) f (x), 2 dx m(x) dx
⎫ f, m1 f ∈ W21 ((xl , xr )) ⎬ dom(A) := f ∈ L2 ((xl , xr )) : f (xl )= f (xr ) = 0 , ⎩ ⎭ 1 1 f ) = f ) = 0 (x (x l r m m is a densely defined closed simple symmetric operator in the Hilbert space H := L2 ((xl , xr )). The deficiency indices of A are n+ (A) = n− (A) = 2 and the adjoint operator A∗ is given by 1 d 1 d f (x) + V(x) f (x), 2 dx m(x) dx
1 ∗ 1 dom(A ) = f ∈ H : f, f ∈ W2 ((xl , xr )) . m
(A∗ f )(x) = −
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It is straightforward to verify that = {C2 , 0 , 1 }, where 1
(x f ) f (xl ) l 2m1 , 0 f := and 1 f := f (xr ) f (xr ) − 2m
339
(3.31)
f ∈ dom(A∗ ), is a boundary triplet for A∗ . Note that the self-adjoint extension A0 = A∗ ker(0 ) corresponds to Dirichlet boundary conditions, that is,
1 dom(A0 ) = f ∈ H : f, f ∈ W21 ((xl , xr )), f (xl ) = f (xr ) = 0 . m It is well known that A0 is semibounded from below and that σ (A0 ) consists of eigenvalues accumulating to +∞. As usual we denote the Weyl function corresponding to = {C2 , 0 , 1 } by M(·). Here M(·) is a two-by-two matrixvalued function which has poles at the eigenvalues of A0 and in particular we have H M(λ) = ran Im (M(λ)) = {0} for all λ ∈ M . (3.32) If ϕλ , ψλ ∈ L2 ((xl , xr )) are fundamental solutions of 1 1 f + V f = λf − 2 m satisfying the boundary conditions 1 ϕ (xl ) = 0, ϕλ (xl ) = 1, m λ
ψλ (xl ) = 0,
then M can be written as 1 −ϕλ (xr ) 1 , M(λ) = 1 − m1 ψλ (xr ) 2ψλ (xr )
1 ψ (xl ) = 1, m λ
(3.33)
λ ∈ ρ(A0 ).
(3.34)
Im (κr ) 0.
(3.35)
We are interested in maximal dissipative extensions AD = A∗ ker(1 − D0 ) of A where D ∈ [C2 ] has the special form −κl 0 D= , Im (κl ) 0, 0 −κr
Of course, if both κl and κr are real constants then H D = ran (Im (D)) = {0} and AD is self-adjoint. In this case AD can be identified with the self-adjoint acting in H ⊕ L2 (R, {0}) =H, cf. Theorem 3.2. dilation K Let us first consider the situation where both κl and κr have positive imagi from Theorem 3.2 is nary parts. Then H D = C2 and the self-adjoint dilation K given by 1 1
+ V f ⊕ −ig− ⊕ −ig+ , K( f ⊕ g− ⊕ g+ ) = − f 2 m 1
0 f − ϒ0 g = 0, f, m f ∈ W21 ((xl , xr )), dom K = . : (1 − Re (D)0 ) f + ϒ1 g = 0 g± ∈ W21 (R± , C2 )
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Here G = {C2 , ϒ0 , ϒ1 } is the boundary triplet for first order differential operator G ⊂ G∗ in L2 (R, C2 ) from Lemma 3.1 and we have decomposed the elements f ⊕ g in H ⊕ L2 (R, C2 ) as agreed in the beginning of Section 3.3. Let us set
gl (0−) g− (0−) = gr (0−)
and
gl (0+) g+ (0+) = . gr (0+)
Then a straightforward calculation using the definitions of = {C2 , 0 , 1 } 3.1, respectively, shows that an and G = {C2 , ϒ0 , ϒ1 } in (3.31) and Lemma if and only if element f ⊕ g− ⊕ g+ belongs to dom K
1 f (xl ) + κl f (xl ) = −i 2Im (κl )gl (0−), 2m 1 f (xl ) + κ l f (xl ) = −i 2Im (κl )gl (0+), 2m 1 f (xr ) − κr f (xr ) = i 2Im (κr )gr (0−), 2m 1 f (xr ) − κ r f (xr ) = i 2Im (κr )gr (0+) 2m is isomorphic in the sense of [42, Section I.4] hold. We note that this dilation K to those used in [15, 16, 50, 51]. Theorem 3.6 and the fact imply that that σ (A0 ) is singular (cf. (3.32)) K0 , K0 = A0 ⊕ G0 , the scattering matrix S(λ) of the scattering system K, coincides with −1 S LP (λ) = IC2 + 2i −Im (D) D − M(λ) −Im (D) ∈ [C2 ] for all√λ ∈ σ p (A0 ) ∩ R, where M(λ) = M(λ + i0) (cf. √Corollary 3.12). √ By (3.35) here −Im (D) is a diagonal matrix with entries Im (κl ) and Im (κr ). We leave it to the reader to compute S LP (λ) explicitely in terms of the fundamental solutions ϕλ and ψλ in (3.33). According to Corollary 3.11 the continuation of the characteristic function W AD of the completely non-self-adjoint pseudoHamiltonian AD from C− to R\{σ p (A0 )} coincides with S LP (λ)∗ , −1 −Im (D) W AD (λ − i0) = IC2 − 2i −Im (D) D∗ − M λ = S LP (λ)∗ . Next we consider briefly the case where one of the entries of D in (3.35) is real. Assume e.g. κl ∈ R. In this case H D = C= {0} ⊕ C, P D is the orthogonal
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projection onto the second component in C2 and G is a first order differential is operator in L2 (R, C). The self-adjoint dilation K 1 1
K( f ⊕ g− ⊕ g+ ) = − f + V f ⊕ −ig− ⊕ −ig+ , 2 m ⎫ ⎧ P D 0 f − ϒ0 g = 0, ⎬ ⎨ 1 1 2 ((xl , xr )), : (1 − P )( − Re (D) ) f = 0, , = f, m f ∈ W dom( K) D 1 0 ⎭ ⎩ g± ∈ W21 (R± , C2 ) P D (1 − Re (D)0 ) f + ϒ1 g = 0 if and explicitely this means that an element f ⊕ g− ⊕ g+ belongs to dom K and only if √ 1 f (xr ) − κ r f (xr ) = i 2Im (κr )g+ (0+), 2m √ 1 f (xr ) − κr f (xr ) = i 2Im (κr )g− (0−), 2m 1 f (xl ) + κl f (xl ) = 0 2m K0 is given by hold. The scattering matrix of K, −1 S LP (λ) = IH D + 2iIm (κr )P D D − M(λ) H D , λ ∈ M , which is now a scalar function, and is related to the characteristic function of the maximal dissipative operator AD by S LP (λ) = W AD (λ − i0)∗ .
4 Energy Dependent Scattering Systems In this section we consider families {A−τ (λ) , A0 } of scattering systems, where τ (·) is a matrix Nevanlinna function and {A−τ (λ) } is a family of maximal dissipative extensions of a symmetric operator A with finite deficiency indices. Such scattering systems arise naturally in the description of open quantum systems, see e.g. Section 4.4 where a problem arising in model the theory of a so-called quantum transmitting Schrödinger–Poisson system is described. Following ideas in [29] (see also [19, 24, 35, 47, 48]) the family {A−τ (λ) } is ‘linearized’ in an abstract way, that is, we construct a self-adjoint extension L of A which acts in a larger Hilbert space H ⊕ G and satisfies −1 −1 PH L − λ H = A−τ (λ) − λ , so that, roughly speaking, the open quantum system is embedded into a closed system. The corresponding Hamiltonian L is semibounded if and only if A0 is semibounded and τ (·) is holomorphic on some interval (−∞, η). The essential observation here is that the scattering matrix of L, L0 , where L0 is the direct orthogonal sum of A0 and a self-adjoint operator connected with τ(·), K0 as pointwise coincides with the scattering matrix of a scattering system K,
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investigated in the previous section. From a physical point of view this under for suitable justifies continuity assumptions the use of quasi-Hamiltonians K the analysis of scattering processes in suitable small energy ranges. 4.1 The Štraus Family and its Characteristic Functions Let A be a densely defined closed simple symmetric operator in the separable Hilbert space H with equal finite deficiency indices n± (A) = n < ∞ and let = {H, 0 , 1 } be a boundary triplet for A∗ . Assume that τ (·) is an [H]-valued Nevanlinna function and consider the family {A−τ (λ) }, λ ∈ C+ , A−τ (λ) := A∗ ker 1 + τ (λ)0 , of closed extension of A. Sometimes it is convenient to consider A−τ (λ) for all λ ∈ h(τ ), that is, for all λ ∈ C\R and all real points λ where τ is holomorphic, cf. Section 2.2. Since Im τ (λ) 0 for λ ∈ C+ it follows that each A−τ (λ) , λ ∈ C+ , is a maximal dissipative extension of A in H. The family {A−τ (λ) }λ∈C+ is called the Štraus family of A associated with τ (cf. [67] and e.g. [28, Section 3.3]) and for brevity we shall often call {A−τ (λ) } simply Štraus family. Since H is finite dimensional Fatous theorem (see [37, 44]) implies that the limit τ (λ + i0) = lim →+0 τ (λ + i ) from the upper half-plane exists for a.e. λ ∈ R. As in Section 2.3 we denote set of real points λ where this limit exists by τ . If there is no danger of confusion we will usually write τ (λ) instead of τ (λ + i0) for λ ∈ τ . Obviously, the Lebesgue measure of R \ τ is zero. Hence the Štraus family {A−τ (λ) }λ∈C+ admits a continuation to C+ ∪ τ which is also denoted by {A−τ (λ) }, λ ∈ C+ ∪ τ . We remark that in the case Im (τ (λ)) = 0 for some λ ∈ C+ ∪ τ the maximal dissipative operator A−τ (λ) is self-adjoint. Let M(·) be the Weyl function corresponding to the boundary triplet = {H, 0 , 1 }. Then M(·) is an [H]-valued Nevanlinna function and Im (M(λ)) is strictly positive for λ ∈ C+ . Therefore −1 N−τ (λ) (λ) := − τ (λ) + M(λ) , λ ∈ C+ , is a well-defined Nevanlinna function, see also (2.9). The set of all real λ where the limit −1 N−τ (λ+i0) (λ + i0) = lim − τ (λ + i ) + M(λ + i )
→+0
exists will for brevity be denoted by N . Furthermore, for fixed λ ∈ τ we define an [H]-valued Nevanlinna function Q−τ (λ) (·) by −1 Q−τ (λ) (μ) := − τ (λ) + M(μ) , μ ∈ C+ , (4.1) and denote by Qλ the set of all real points μ where the limit Q−τ (λ) (μ + i0) = lim Q−τ (λ) (μ + i )
→+0
(4.2)
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exists. Note that the complements R \ N and R \ Qλ are of Lebesgue measure zero. The next lemma will be used in Section 4.3. Lemma 4.1 Let A, = {H, 0 , 1 }, M(·) and τ (·) be as above. Then the following assertions (1)–(3) are true. 1. If λ ∈ τ and μ ∈ M ∩ Qλ , then the operator τ (λ) + M(μ) is invertible and −1 −1 = lim τ (λ) + M(μ + i ) . (4.3) τ (λ) + M(μ)
→+0
2. If λ ∈ τ ∩ M ∩ N , then the operator τ (λ) + M(λ) is invertible and −1 −1 τ (λ) + M(λ) = lim τ (λ + i ) + M(λ + i ) . (4.4)
→+0
3. If λ ∈ τ ∩ M ∩ N , then λ ∈ Qλ and −1 −1 τ (λ) + M(λ) = lim τ (λ) + M(λ + i ) .
→+0
(4.5)
Proof 1. If λ ∈ τ , μ ∈ M , then lim τ (λ) + M(μ + i ) = τ (λ) + M(μ).
→+0
Since
τ (λ) + M(μ + i ) Q−τ (λ) (μ + i ) = Q−τ (λ) (μ + i ) τ (λ) + M(μ + i ) = −IH
for all > 0, we get −IH = τ (λ) + M(μ) Q−τ (λ) (μ) = Q−τ (λ) (μ) τ (λ) + M(μ) for λ ∈ τ and μ ∈ M ∩ Qλ which proves (4.3). 2. For λ ∈ τ ∩ M clearly lim τ (λ + i ) + M(λ + i ) = τ (λ) + M(λ)
→+0
exists. Since (τ (λ) + M(λ))N−τ (λ) (λ) = N−τ (λ) (λ)(τ (λ) + M(λ)) = −IH for all λ ∈ C+ we have −IH = τ (λ) + M(λ) N−τ (λ) (λ) = N−τ (λ) (λ) τ (λ) + M(λ) for λ ∈ τ ∩ M ∩ N which verifies (4.4). 3. Let λ ∈ τ ∩ M ∩ N . Let us show that λ ∈ Qλ , i.e., we have to show that lim →+0 (τ (λ) + M(λ + i ))−1 exists. Since τ (λ) + M(λ) is boundedly invertible and τ (λ) + M(λ + i ), > 0, converges in the operator norm to
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τ (λ) + M(λ) the family {(τ (λ) + M(λ + i ))−1 } >0 is uniformly bounded. Using −1 −1 τ (λ) + M(λ + i ) − τ (λ) + M(λ) −1 −1 = − τ (λ) + M(λ + i ) M(λ + i ) − M(λ) τ (λ) + M(λ) ,
> 0, one obtains the existence of lim →+0 (τ (λ) + M(λ + i ))−1 and (4.5). Let A, = {H, 0 , 1 } and M(·) be as in the beginning of this section and let as above τ (·) be a matrix Nevanlinna function with values in [H]. For each maximal dissipative operator from the Štraus family {A−τ (λ) }λ∈C+ the characteristic function W A−τ (λ) is given by (4.6) W A−τ (λ) : C− → [Hτ (λ) ] −1 μ → IHτ (λ) + 2iPτ (λ) Im (τ (λ)) τ (λ)∗ + M(μ) Im (τ (λ)) Hτ (λ) , (see [33] and (3.29)), where we have used Hτ (λ) = ran (Im (τ (λ))), λ ∈ τ , and denoted the projection and restriction onto Hτ (λ) by Pτ (λ) and Hτ (λ) , respectively. If we regard the Štraus family {A−τ (λ) } on the larger set C+ ∪ τ , then for λ ∈ τ the characteristic function W A−τ (λ) (·) is defined as in (4.6). Note that in the case Im (τ (λ)) = 0 for λ ∈ τ the characteristic function of the self-adjoint extension A−τ (λ) of A is the identity operator on the trivial space Hτ (λ) = {0}. Since the characteristic functions W A−τ (λ) (·), λ ∈ C+ ∪ τ , are contractive [Hτ (λ) ]-valued functions in the lower half-plane, the limits W A−τ (λ) (μ − i0) = lim W A−τ (λ) (μ − i )
→+0
exist for a.e. μ ∈ R, cf. [42]. The next proposition is a simple consequence of Lemma 4.1. Proposition 4.2 Let A, = {H, 0 , 1 } and M(·) be as above and let τ (·) be an [H]-valued Nevanlinna function. Let {A−τ (λ) }λ∈C+ ∪ τ be the Štraus family of maximal dissipative extensions of A and let W A−τ (λ) (·) be the corresponding characteristic functions. Then the following holds. 1. If λ ∈ τ and μ ∈ M ∩ Qλ , then the limit W A−τ (λ) (μ − i0) exists and W A−τ (λ) (μ − i0)
= IHτ (λ) + 2iPτ (λ) Im (τ (λ))(τ (λ)∗ + M(μ)∗ )−1 Im (τ (λ)) Hτ (λ) .
2. If λ ∈ τ ∩ M ∩ N , then the limit W A−τ (λ) (λ − i0) exists and W A−τ (λ) (λ − i0)
= IHτ (λ) + 2iPτ (λ) Im (τ (λ))(τ (λ)∗ + M(λ)∗ )−1 Im (τ (λ)) Hτ (λ) .
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4.2 Coupling of Symmetric Operators and Coupled Scattering Systems Let, as in the previous subsection, A be a densely defined closed simple symmetric operator in H with equal finite deficiency indices n ± (A) = n and let = {H, 0 , 1 } be a boundary triplet for A∗ with corresponding Weyl function M(·). Let τ (·) be an [H]-valued Nevanlinna function and assume in addition that τ can be realized as the Weyl function corresponding to a densely defined closed simple symmetric operator T in some separable Hilbert space G and a suitable boundary triplet T = {H, ϒ0 , ϒ1 } for T ∗ . It is worth to note that the Nevanlinna function τ (·) has this property if and only if Im (τ (λ)) is invertible for some (and hence for all) λ ∈ C+ and 1 (4.7) lim τ (iy)h, h = 0 and lim y Im τ (iy)h, h = ∞ y→∞ y y→∞ hold for all h ∈ H, h = 0, (see e.g. [56, Corollary 2.5 and Corollary 2.6] and [32, 58]). In the following the function −τ (·) and the Štraus family (4.8) A−τ (λ) = A∗ ker 1 + τ (λ)0 are in a certain sense the counterparts of the dissipative matrix D ∈ [H] and the corresponding maximal dissipative extension AD from Section 3.1. Similarly to Theorem 3.2 we construct an ‘energy dependent dilation’ in Theorem 4.3 below, that is, we find a self-adjoint operator L such that −1 −1 PH L−λ H = A−τ (λ) − λ holds. First we fix a separable Hilbert space G, a densely defined closed simple symmetric operator T ∈ C (G) and a boundary triplet T = {H, ϒ0 , ϒ1 } for T ∗ such that τ (·) is the corresponding Weyl function. We note that T and G are unique up to unitary equivalence and the resolvent set ρ(T0 ) of the self-adjoint operator T0 := T ∗ ker(ϒ0 ) coincides with the set h(τ ) of points of holomorphy of τ , cf. Section 2.2. Since the deficiency indices of T are n+ (T) = n− (T) = n it follows that L := A ⊕ T,
dom(L) = dom(A) ⊕ dom(T),
is a densely defined closed simple symmetric operator in the separable Hilbert space L := H ⊕ G with deficiency indices n± (L) = n± (A) + n± (T) = 2n. The following theorem has originally been proved in [29, Section 5]. For the sake of completeness we present a direct proof here, cf. [19]. Theorem 4.3 Let A, = {H, 0 , 1 }, M(·) and τ (·) be as above, let T be a densely defined closed simple symmetric operator in G and T = {H, ϒ0 , ϒ1 } be a boundary triplet for T ∗ with Weyl function τ (·). Then
0 f − ϒ0 g = 0 ∗ ∗ L = L f ⊕ g ∈ dom(L ) : (4.9) 1 f + ϒ1 g = 0
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is a self-adjoint operator in L such that −1 −1 PH H = A−τ (λ) − λ L−λ holds for all λ ∈ ρ(A0 ) ∩ h(τ ) ∩ h(−(M + τ )−1 ) and the minimality condition −1 L = clospan L − λ H : λ ∈ C\R is satisfied. Moreover, L is semibounded from below if and only if A0 is semibounded from below and (−∞, η) ⊂ h(τ ) for some η ∈ R. = H ⊕ H, 0 := (0 , ϒ0 ) and Proof It is easy to see that 0 , 1 , where ∗ ∗ ∗ 1 := (1 , ϒ1 ) , is a boundary triplet for L = A ⊕ T . If γ (·) and ν(·) denote the γ -fields of = {H, 0 , 1 } and T = {H, ϒ0 , ϒ1}, respectively, then the γ of = H ⊕ H, field γ and Weyl function M 0 , 1 are given by γ (λ) 0 M(λ) 0 λ → γ (λ) = and λ → M(λ) = , 0 ν(λ) 0 τ (λ) λ ∈ ρ(A0 ) ∩ ρ(T0 ), A0 = A∗ ker(0 ), T0 = T ∗ ker(ϒ0 ). A simple calculation shows that the relation
(v, v) H (4.10) := : v, w ∈ ∈ C(H ⊕ H) (w, −w) (−1) is a self-adjoint is self-adjoint in H ⊕ H, hence the operator L = L∗ extension of L in L = H ⊕ G and L coincides with L in (4.9). Hence, with 0 = A0 ⊕ T0 we have L0 = L∗ ker −1 −1 ∗ L−λ = (L0 − λ)−1 + γ (λ) − M(λ) γ λ (4.11) for all λ ∈ ρ L ∩ ρ(L0 ) by (2.6). Note that the difference of the resolvents of L and L0 is a finite rank operator and therefore by well-known perturbation results L is semibounded if and only if L0 is semibounded, that is, A0 and T0 are both semibounded. From ρ(T0 ) = h(τ ) we conclude that the last assertion of the theorem holds. Similar considerations as in the proof of Theorem 3.2 show that −1 (M(λ) + τ (λ))−1 (M(λ) + τ (λ))−1 (4.12) − M(λ) =− (M(λ) + τ (λ))−1 (M(λ) + τ (λ))−1 holds for all λ ∈ ρ L ∩ ρ(L0 ). Therefore the compressed resolvent of L has the form −1 ∗ −1 H = (A0 − λ)−1 − γ (λ) M(λ) + τ (λ) γ λ PH L − λ and coincides with (A−τ (λ) − λ)−1 for all λ belonging to L = ρ(A0 ) ∩ h(τ ) ∩ h −(M + τ )−1 , ρ(L0 ) ∩ ρ see Section 2.2. The minimality condition follows from the fact that T is simple, clospan{ker(T ∗ − λ) : λ ∈ C\R} and (4.11) in a similar way as in the proof of Theorem 3.2
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Example 4.4 Let A be the symmetric Sturm–Liouville differential operator from Example 3.5 and let = {Cn , 0 , 1 } be the boundary triplet for A∗ defined by (3.18). Besides the operator A we consider the minimal operator T in G = L2 (R− , Cn ) associated with the Sturm–Liouville differential expression −d2 /dx2 + Q− , T=−
d2 + Q− , dx2
dom(T) = g ∈ Dmax,− : g(0) = g (0) = 0 .
Analogously to Example 3.5 it is assumed that Q− ∈ L1loc (R− , [Cn ]) satisfies Q− (·) = Q− (·)∗ , that the limit point case prevails at −∞ and the maximal domain Dmax,− is defined in the same way as Dmax,+ in Example 3.5 with R+ and Q+ replaced by R− and Q− , respectively. It is easy to see that T = {Cn , ϒ0 , ϒ1 }, where ϒ0 g := g(0),
ϒ1 g := −g (0),
g ∈ dom(T ∗ ) = Dmax,− ,
(4.13)
is a boundary triplet for T ∗ . For f ∈ dom(A∗ ) and g ∈ dom(T ∗ ) the conditions 0 f − ϒ0 g = 0 and 1 f + ϒ1 g = 0 in (4.9) stand for f (0+) = g(0−)
and
f (0+) = g (0−),
so that the operator L in Theorem 4.3 is the self-adjoint Sturm–Liouville operator
d2 Q+ (x), x ∈ R+ , L = − 2 + Q, Q(x) = dx Q− (x), x ∈ R− , in L2 (R, Cn ). Let A, = {H, 0 , 1 }, M(·) and T, T = {H, ϒ0 , ϒ1 }, τ (·) be as in the beginning of this subsection. We define the families {H M(λ) }λ∈ M and {Hτ (λ) }λ∈ τ of Hilbert spaces H M(λ) and Hτ (λ) by H M(λ) = ran Im (M(λ + i0)) and Hτ (λ) = ran Im (τ (λ + i0)) (4.14) for all real points λ belonging to M and τ , respectively, cf. Section 2.3. As usual the projections and restrictions in H onto H M(λ) and Hτ (λ) are denoted by P M(λ) , H M(λ) and Pτ (λ) , Hτ (λ) , respectively. The next theorem is the counterpart of Theorem 3.6 in the present framework. We consider the complete scattering system L, L0 consisting of the self-adjoint operators L from Theorem 4.3 and A0 = A∗ ker(0 ), T0 = T ∗ ker(ϒ0 ), and express the scattering matrix S(λ) in terms of the function M(·) and τ (·). L0 := A0 ⊕ T0 ,
Theorem 4.5 Let A, = {H, 0 , 1 }, M(·) and T, T = {H, ϒ0 , ϒ1 }, τ (·) be as above. Define H M(λ) , Hτ (λ) as in (4.14) and let L0 = A0 ⊕ T0 and L be as in Theorem 4.3. Then the following holds. 1.
ac ac Lac 0 = A0 ⊕ T0 is unitarily equivalent to the multiplication operator with the free variable in L2 (R, dλ, H M(λ) ⊕ Hτ (λ) ).
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2. In L2 (R, dλ, H M(λ) ⊕ Hτ (λ) ) the scattering matrix S(λ) of the complete scattering system L, L0 is given by 11 (λ) T 12 (λ) T (4.15) S(λ) = IH M(λ) ⊕Hτ (λ) − 2i 22 (λ) ∈ [H M(λ) ⊕ Hτ (λ) ] T21 (λ) T for all λ ∈ M ∩ τ ∩ N , where 11 (λ) = P M(λ) Im (M(λ)) M(λ) + τ (λ) −1 Im (M(λ)) H M(λ) , T 12 (λ) = P M(λ) Im (M(λ)) M(λ) + τ (λ) −1 Im (τ (λ)) Hτ (λ) , T 21 (λ) = Pτ (λ) Im (τ (λ)) M(λ) + τ (λ) −1 Im (M(λ)) H M(λ) , T 22 (λ) = Pτ (λ) Im (τ (λ)) M(λ) + τ (λ) −1 Im (τ (λ)) Hτ (λ) T and M(λ) = M(λ + i0), τ (λ) = τ (λ + i0). = H ⊕ H, Proof Let L = A ⊕ T and let 0 , 1 be the boundary triplet for is L∗ from the proof of Theorem 4.3. The corresponding Weyl function M M(λ) 0 λ → M(λ) = (4.16) , λ ∈ ρ(A0 ) ∩ ρ(T0 ), 0 τ (λ) and since L is a densely defined closed simple symmetric operator in the separable Hilbert space L = H ⊕ G we can apply Theorem 2.4. First of all we immediately conclude from
λ ∈ M = M ∩ τ ,
H M(λ) = H M(λ) ⊕ Hτ (λ) ,
ac ac that the absolutely continuous part Lac 0 = A0 ⊕ T0 of L0 is unitarily equivalent to the multiplication operator with the free variable in the direct integral L2 (R, dλ, H M(λ) ⊕ Hτ (λ) ). Moreover # # −1 S(λ) = IH Im ( M(λ)) − M(λ) Im ( M(λ)) H M(λ) (4.17) λ + 2iP M(λ)
holds for λ ∈ M ∩ N , where is the self-adjoint relation from (4.10), the set N is defined as in Section 2.3 and P M(λ) and H M(λ) denote the projection and restriction in H ⊕ H onto H M(λ) , respectively. For λ ∈ M ∩ N we have + i0) −1 , + i ) −1 = − M(λ lim − M(λ
→+0
and
−1 (M(λ) + τ (λ))−1 (M(λ) + τ (λ))−1 − M(λ) =− (M(λ) + τ (λ))−1 (M(λ) + τ (λ))−1
cf. (4.12). This implies that the sets M ∩ N and M ∩ τ ∩ N coincide. −1 , λ ∈ M ∩ τ ∩ Moreover, by inserting the above expression for − M(λ) N , into (4.17) and taking into account (4.16) we find that the scattering matrix S(λ) of the scattering system L, L0 has the form asserted in (2).
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The following corollary, which is of similar type as Corollary 3.12, is a simple consequence of Theorem 4.5 and Proposition 4.2. Corollary 4.6 Let the assumptions be as in Theorem 4.5, let W A−τ (λ) (·) be the characteristic function of the extension A−τ (λ) in (4.8) and assume in addition that σ (A0 ) is purely singular. Then Lac 0 is unitarily equivalent to the multiplication operator with the free variable in L2 (R, dλ,Hτ (λ) ) and the scattering matrix S(λ) of the complete scattering system L, L0 is given by S(λ) = W A−τ (λ) (λ − i0)∗ −1 √ √ = IHτ (λ) − 2iPτ (λ) Im (τ (λ)) M(λ) + τ (λ) Im (τ (λ)) Hτ (λ) for a.e. λ ∈ R. In the special case σ (A0 ) = σ p (A0 ) this relation holds for all points λ ∈ M ∩ τ ∩ N . Corollary 4.7 Let the assumptions be as in Corollary 4.6 and suppose that the defect of A is one, n± (A) = 1. Then M(λ) + τ (λ) S(λ) = W A−τ (λ) (λ − i0)∗ = M(λ) + τ (λ) holds for a.e. λ ∈ R with Im τ (λ + i0) = 0. 4.3 Scattering Matrices of Energy Dependent and Fixed Dissipative Scattering Systems Let A, = {H, 0 , 1 }, A0 = A∗ ker(0 ) and τ (·) be as in the previous subsections and let {A−τ (λ) } be the Štraus family associated with τ from (4.8). In the following we first fix some μ ∈ C+ ∪ τ and consider the fixed dissipative scattering system {A−τ (μ) , A0 }. Notice that if μ ∈ τ it may happen that A−τ (μ) μ the minimal self-adjoint dilation of the is self-adjoint. Let us denote by K maximal dissipative extension A−τ (μ) in H ⊕ L2 (R, dλ, Hτ (μ) ) constructed in Theorem 3.2. Here the fixed Hilbert space Hτ (μ) = ran (Im (τ (μ))) coincides with H if μ ∈ C+ or Hτ (μ) is a (possibly trivial) subspace of H if μ ∈ τ . Furthermore, if K0 = A0 ⊕ G0 , where G0 is the first order differential operator in L2 (R, dλ, Hτ (μ) ) from Lemma 3.1, then according to Theorem 3.6 the absolutely continuous part K0ac = Aac 0 ⊕ G0 of K0 is unitarily equivalent to the multiplication operator with the free variable in the direct integral Sμ (λ) of the scattering L2 (R, dλ, H M(λ) ⊕ Hτ (μ) ) and the scattering matrix μ , K0 is given by system K 11,μ (λ) T 12,μ (λ) T Sμ (λ) = IH M(λ) ⊕Hτ (μ) − 2i 22,μ (λ) ∈ H M(λ) ⊕ Hτ (μ) T21,μ (λ) T
(4.18)
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for all λ ∈ M ∩ Qμ , where 11,μ (λ) = P M(λ) Im (M(λ)) τ (μ) + M(λ) −1 Im (M(λ)) H M(λ) , T 12,μ (λ) = P M(λ) Im (M(λ)) τ (μ) + M(λ) −1 Im (τ (μ)) Hτ (μ) , T 21,μ (λ) = Pτ (μ) Im (τ (μ)) τ (μ) + M(λ) −1 Im (M(λ)) H M(λ) , T 22,μ (λ) = Pτ (λ) Im (τ (μ)) τ (μ) + M(λ) −1 Im (τ (μ)) Hτ (μ) T and M(λ) = M(λ + i0). Here the set Qμ and the corresponding function λ → Q−τ (μ) (λ) defined in (4.1)–(4.2) replace N D and λ → (D − M(λ))−1 in Theorem 3.6, respectively. The following theorem is one of the main results of this paper. Roughly speaking it says that the scattering matrix of the scattering system L, L0 from Theorem 4.5 pointwise coincides with scattering matrices of scattering systems μ , K0 of the above form. K Theorem 4.8 Let A, = {H, 0 , 1 }, M(·) and T, T = {H, ϒ0 , ϒ1 }, τ (·) be as in the beginning of Section 4.2 and let L0 = A0 ⊕ T0 and L be as in Theorem 4.3. For μ ∈ τ denote the minimal self-adjoint dilation of A−τ (μ) in μ and let K0 = A0 ⊕ G0 , where G0 is the self-adjoint first H ⊕ L2 (R, Hτ (μ) ) by K order differential operator in L2 (R, Hτ (μ) ). Then for each μ ∈ M ∩ τ ∩ N the value of the scattering matrix Sμ (λ) , K at energy λ = μ coincides with the value of the of the scattering system K μ 0 scattering matrix S(λ) of the scattering system L, L0 at energy λ = μ, that is, S(μ) = Sμ (μ)
for all μ ∈ M ∩ τ ∩ N .
(4.19)
Proof According to Lemma 4.1 (3) each real μ ∈ M ∩ τ ∩ N belongs also to the set Qμ . Therefore, by comparing Theorem 4.5 with the scattering μ , K0 at energy λ = μ in (4.18) we conclude (4.19). matrix Sμ (λ) of K Remark 4.9 The statements of Theorem 4.5 and Theorem 4.8 are also interesting from the viewpoint of inverse problems. Namely, if τ (·) is a matrix Nevanlinna function, satisfying ker(Im (τ (λ))) = 0, λ ∈ C+ , and the conditions (4.7), and if {A−τ (λ) , A0 } is a family of energy dependent dissipative scattering systems as considered above, then in general the Hilbert space G and the operators T ⊂ T0 are not explicitely known, and hence also the scattering system L, L0 is not explicitely known. However, according to Theorem 4.5 the scattering matrix S(λ) can be expressed in terms of τ (·) and the Weyl function M(·), and by Theorem 4.8 S(λ) can be obtained with the help of the μ , K0 . scattering matrices Sμ (λ) of the scattering systems K In the following corollary the scattering matrices {S−τ (μ) (λ)} of the energy dependent dissipative scattering systems {A−τ (μ) , A0 }, μ ∈ τ , are evaluated at energy λ = μ.
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Corollary 4.10 Let the assumptions be as in Theorem 4.8 and let μ ∈ M ∩ τ ∩ N . Then the scattering matrix {S−τ (μ) (λ)} of the dissipative scattering system {A−τ (μ) , A0 } at energy λ = μ coincides with theupper left corner of the scattering matrix S(λ) of the scattering system L, L0 at energy λ = μ. μ be the minimal self-adjoint dilation of the maximal dissipative Let again K operator A−τ (μ) in H ⊕ L2 (R, dλ, Hτ(μ) ). In the next corollary we focus on the Lax–Phillips scattering matrices SμLP (λ) of the Lax–Phillips scattering μ , D−,μ , D+,μ , where systems K D−,μ := L2 R− , Hτ (μ) and D+,μ := L2 R+ , Hτ (μ) μ , cf. Lemma 3.9. If W A−τ (μ) (·) is the are incoming and outgoing subspaces for K characteristic function of A−τ (μ) , cf. (4.6), then according to Corollaries 3.10 and 3.11 we have SμLP (λ) = W A−τ (μ) (λ − i0)∗ −1 √ √ = IHτ (λ) − 2iPτ (λ) Im (τ (λ)) τ (μ) + M(λ) Im (τ (λ)) Hτ (λ) for all λ ∈ M ∩ Qμ , cf. Proposition 4.2 and Corollary 4.6. Statements (2) and (3) of the following corollary can be regarded as generalizations of the classical Adamyan-Arov result, cf. [2–5], and Corollary 3.11. Corollary 4.11 Let the assumptions be as in Theorem 4.8 and let μ ∈ M ∩ τ ∩ N . 1. The scattering matrix SμLP (λ) of the Lax–Phillips scattering system μ , D−,μ , D+,μ at energy λ = μ coincides with the lower right corner of K the scattering matrix S(λ) of the scattering system L, L0 at λ = μ. 2. The characteristic function W A−τ (μ) (·) of A−τ (μ) satisfies SμLP (μ) = W A−τ (μ) (μ − i0)∗ −1 = IHτ (μ) − 2iPτ (μ) Im (τ (μ)) τ (μ) + M(μ) Im (τ (μ)) Hτ (μ) . 3. If σ (A0 ) is purely singular, then S(μ) = SμLP (μ) = W A−τ (μ) (μ − i0)∗ holds for a.e. μ ∈ R. In the special case σ (A0 ) = σ p (A0 ) this is true for all μ ∈ M ∩ τ ∩ N . 4.4 1D-Schrödinger Operators with Transparent Boundary Conditions As an example we consider an open quantum system of similar type as in Section 3.4. Instead of a single pseudo-Hamiltonian AD here the open quantum system is described by a family of energy dependent pseudo-Hamiltonians {A−τ (λ) } which is sometimes called a quantum transmitting family.
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Let, as in Section 3.4, (xl , xr ) ⊂ R be a bounded interval and let A be the symmetric Sturm–Liouville operator in H = L2 ((xl , xr )) given by (A f )(x) = −
dom(A) =
⎧ ⎪ ⎨ ⎪ ⎩
1 d 1 d f (x) + V(x) f (x), 2 dx m(x) dx f, m1 f ∈ W21 ((xl , xr ))
⎫ ⎪ ⎬
, f ∈ H : f (xl ) = f (xr ) = 0 ⎪ 1 1 ⎭ f (xl ) = m f (xr ) = 0 m
where V, m, m−1 ∈ L∞ ((xl , xr )) are real functions and m > 0. Let vl , vr be real m ∈ L∞ (R) by constants, let ml , mr > 0 and define V, ⎧ ⎪ x ∈ (−∞, xl ], ⎨vl V(x) := V(x) x ∈ (xl , xr ), (4.20) ⎪ ⎩ vr x ∈ [xr , ∞), and
⎧ ⎪ ⎨ml (x) := m(x) m ⎪ ⎩ mr
x ∈ (−∞, xl ], x ∈ (xl , xr ), x ∈ [xr , ∞),
(4.21)
respectively. We choose the boundary triplet = {C2 , 0 , 1 }, 1 f (xl ) f (xl ) 2m , 1 f = , f ∈ dom(A∗ ), 0 f = 1 f (xr ) − 2m f (xr ) from (3.31) for A∗ . In the following we consider the Štraus family A−τ (λ) = A∗ ker 1 + τ (λ)0 ,
λ ∈ C+ ∪ τ ,
associated with the 2 × 2-matrix Nevanlinna function ⎛ # ⎞ l i λ−v 0 2ml ⎠; # λ → τ (λ) = ⎝ r 0 i λ−v 2mr
(4.22)
here √the square root is defined on√C with a cut along [0, ∞) and fixed by Im λ > 0 for λ ∈ [0, ∞) and by λ 0 for λ ∈ [0, ∞), cf. Example 2.5, so that indeed Im (τ (λ)) > 0 for λ ∈ C+ and τ (λ) = τ (λ), λ ∈ C\R. Moreover it is not difficult to see that τ (·) is holomorphic on C\[min{vl , vr }, ∞) and τ = R. The Štraus family {A−τ (λ) }, λ ∈ C+ ∪ τ , has the explicit form 1 d 1 d f (x) + V(x) f (x), A−τ (λ) f (x) := − 2 dx m dx ⎫ ⎧ f, m1 f ∈ W21 ((xl , xr )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ # ⎬ 1 ⎨ λ−vl f (x ) = −i f (x ), l l dom A−τ (λ) = f ∈ H : 2m . 2ml ⎪ ⎪ # ⎪ ⎪ ⎪ ⎪ 1 ⎩ r f (xr ) = i λ−v f (xr ) ⎭ 2m 2mr
(4.23)
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The operator A−τ (λ) is self-adjoint if λ ∈ (−∞, min{vl , vr }] and maximal dissipative if λ ∈ (min{vl , vr }, ∞). We note that the Štraus family in (4.23) plays an important role for the quantum transmitting Schrödinger–Poisson system in [14] where it was called the quantum transmitting family. For this open quantum system the boundary conditions in (4.23) are often called transparent or absorbing boundary conditions, see e.g. [39, 40]. We leave it to the reader to verify that the Nevanlinna function τ (·) in (4.22) satisfies the conditions (4.7). Hence by [32, 56, 58] there exists a separable Hilbert space G, a densely defined closed simple symmetric operator T in G and a boundary triplet T = {C2 , ϒ0 , ϒ1 } for T ∗ such that τ (·) is the corresponding Weyl function. Here G, T and T = {C2 , ϒ0 , ϒ1 } can be explicitly described. Indeed, as Hilbert space G we choose L2 ((−∞, xl ) ∪ (xr , ∞)) and frequently we identify this space with L2 ((−∞, xl )) ⊕ L2 ((xr , ∞)). An element g ∈ G will be written in the form g = gl ⊕ gr , where gl ∈ L2 ((−∞, xl )) and gr ∈ L2 ((xr , ∞)). The operator T in G is defined by 1 d 1 d 0 − 2 dx ml dx gl (x) + vl gl (x) (Tg)(x) := , d 1 d 0 − 21 dx g (x) + vr gr (x) mr dx r
g ∈ W22 ((−∞, xl )) ⊕ W22 ((xr , ∞)) dom(T) := g = gl ⊕ gr ∈ G : , gl (xl ) = gr (xr ) = gl (xl ) = gr (xl ) = 0 and it is well-known that T is a densely defined closed simple symmetric operator in G with deficiency indices n+ (T) = n− (T) = 2. The adjoint operator T ∗ is given by 1 d 1 d − 2 dx ml dx gl (x) + vl gl (x) 0 ∗ (T g)(x) = , d 1 d 0 − 12 dx g (x) + v g (x) r r r mr dx ∗ 2 dom(T ) = g = gl ⊕ gr ∈ G : W2 ((−∞, xl )) ⊕ W22 ((xr , ∞)) . We leave it to the reader to check that T = {C2 , ϒ0 , ϒ1 }, where 1 − 2ml gl (xl ) gl (xl ) ϒ0 g := , and ϒ1 g := 1 gr (xr ) g (x ) 2mr r r g = gl ⊕ gr ∈ dom(T ∗ ), is a boundary triplet for T ∗ . Note that T0 = T ∗ ker(ϒ0 ) is the restriction of T ∗ to the domain dom(T0 ) = g ∈ dom(T ∗ ) : gl (xl ) = gr (xr ) = 0 , that is, T0 corresponds to Dirichlet boundary conditions. It is not difficult to see that σ (T0 ) = [min{vl , vr }, ∞) and hence the Weyl function of T = {C2 , ϒ0 , ϒ1 } is holomorphic on C\[min{vl , vr }, ∞). Lemma 4.12 Let T ⊂ T ∗ and T = {C2 , ϒ0 , ϒ1 } be as above. Then the corresponding Weyl function coincides with τ (·) in (4.22).
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Proof A straightforward calculation shows that i hl,λ (x) := √ exp −i 2ml (λ − vl )(x − xl ) 2ml (λ − vl ) belongs to L2 ((−∞, xl )) for λ ∈ C\[vl , ∞) and satisfies −
1 d 1 d hl,λ (x) + vl hl,λ (x) = λhl,λ (x). 2 dx ml dx
Analogously the function i exp i 2mr (λ − vr )(x − xr ) kr,λ (x) := √ 2ml (λ − vr ) belongs to L2 ((xr , ∞)) for λ ∈ C\[vr , ∞) and satisfies −
1 d 1 d kr,λ (x) + vr kr,λ (x) = λkr,λ (x). 2 dx mr dx
Therefore the functions hλ := hl,λ ⊕ 0
and kλ := 0 ⊕ kr,λ
∗
belong to G and we have ker(T − λ) = sp{hλ , kλ }. As the Weyl function τ (·) associated with T and T = {C2 , ϒ0 , ϒ1 } is defined by ϒ1 gλ = τ (λ)ϒ0 gλ
for all
gλ ∈ ker(T ∗ − λ),
λ ∈ C\[min{vl , vr }, ∞), we conclude from i √ 1 − m1 2ml (λ−vl ) l ϒ1 hλ = and ϒ0 hλ = 0 0 2 and 1 ϒ1 kλ = 2
0 − m1r
and ϒ0 kλ =
that τ has the form (4.22), τ (·) = τ (·).
0 √
i 2mr (λ−vr )
Let A, = {C2 , 0 , 1 } and T, T = {C2 , ϒ0 , ϒ1 } be as above. Then according to Theorem 4.3 the operator
f − ϒ0 g = 0 L := A∗ ⊕ T ∗ f ⊕ g ∈ dom(A∗ ⊕ T ∗ ) : 0 (4.24) 1 f + ϒ1 g = 0 is a self-adjoint extension of A ⊕ T in H ⊕ G. We can identify H ⊕ G with L2 ((−∞, xl )) ⊕ L2 ((xl , xr )) ⊕ L2 ((xr , ∞)) and L2 (R). The elements f ⊕ g in H ⊕ G, f ∈ H, g = gl ⊕ gr ∈ G will be written in the form gl ⊕ f ⊕ gr . The conditions 0 f = ϒ0 g and 1 f = −ϒ1 g, f ∈ dom(A∗ ), g ∈ dom(T ∗ ), have the form 1 1 g (xl ) f (xl ) f (xl ) gl (xl ) 2ml 2m l = = and . 1 1
f (xr ) gr (xr ) − 2m g (x ) fr (xr ) − 2m r r
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Therefore an element gl ⊕ f ⊕ gr in the domain of (4.24) has the properties gl (xl ) = f (xl ) and as well as 1 g (xl ) = ml l
1 f (xl ) and m
f (xr ) = gr (xr )
1 1 f (xr ) = g (xr ) m mr r
and the self-adjoint operator L in (4.24) becomes L(gl ⊕ f ⊕ gr ) = ⎛ 1 d 1 d − 2 dx ml dx gl + vl gl ⎜ ⎜ 0 ⎝ 0
0
0
d 1 d − 21 dx f + Vf m dx
0
0
d 1 − 21 dx mr
d g dx r
⎞ ⎟ ⎟. ⎠ + vr gr
With the help of (4.20) and (4.21) we see that (4.24) can be regarded as the usual self-adjoint second order differential operator 1 d 1 d L=− +V dx 2 dx m on the maximal domain in L2 (R), that is, (4.24) coincides with the so-called Buslaev–Fomin operator from [14]. Denote by M(·) the Weyl function corresponding to A and the boundary triplet = {C2 , 0 , 1 }, cf. (3.33)–(3.34). Sinceσ (A0) consists of eigenvalues Corollary 4.6 implies that the scattering matrix S(λ) of the scattering system L, L0 , L0 = A0 ⊕ T0 , is given by −1 Im (τ (λ)) Hτ (λ) S(λ) = IHτ (λ) − 2iPτ (λ) Im (τ (λ)) M(λ) + τ (λ) for all λ ∈ ρ(A0 ) ∩ N , where
⎧ ⎪ ⎨{0} λ ∈ (−∞, min{vl , vr }], Hτ (λ) = ran (Im (τ (λ))) = C λ ∈ (min{vl , vr }, max{vl , vr }], ⎪ ⎩ 2 C λ ∈ (max{vl , vr }, ∞). The scattering system L, L0 was already investigated in [13, 14]. There it was in particular shown that the scattering matrix S(λ) and the characteristic function W A−τ (λ) (·) of the maximal dissipative extension A−τ (λ) from (4.23) are connected via S(λ) = W A−τ (λ) (λ − i0)∗ , which we here immediately obtain from Corollary 4.6. Acknowledgements The authors are grateful to Professor Peter Lax for helpful comments and fruitful discussions. Moreover, we would like to thank one of the referees for drawing our attention to further physical applications.
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Math Phys Anal Geom (2007) 10:359–373 DOI 10.1007/s11040-008-9036-9
The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators Christian Remling
Received: 22 October 2007 / Accepted: 5 February 2008 / Published online: 12 March 2008 © Springer Science + Business Media B.V. 2008
Abstract This paper deals with general structural properties of onedimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper (Remling, The absolutely continuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas. Keywords Absolutely continuous spectrum · Schrödinger operator · Reflectionless potential Mathematics Subject Classifications (2000) Primary 34L40 · 81Q10
1 Introduction This note discusses basic properties of one-dimensional Schrödinger operators d2 + V(x) dx2 with some absolutely continuous spectrum. It is a supplement to my recent paper [19]. In [19], I dealt with the discrete case exclusively. As one would H=−
C. Remling (B) Mathematics Department, University of Oklahoma, Norman, OK 73019, USA e-mail:
[email protected] URL: www.math.ou.edu/∼cremling
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expect, the basic ideas that were presented in [19] can also be used to analyze the continuous case. It is the purpose of this paper to give such a treatment; basically, this will be a matter of making the appropriate definitions. Therefore, my general philosophy will be to keep this note brief. I will assume that the reader is familiar with at least the general outline of the discussion of [19] and only focus on those aspects where the extension to the continuous case is perhaps not entirely obvious. By the same token, I will not say much about related work here; please see again [19] for a fuller discussion. Given a potential V, we will consider limit points W under the shift (Sx V)(t) = V(x + t), as x → ∞. We will thus need a suitable topology on a suitable space of potentials. This will naturally lead us to consider generalized Schrödinger operators, with measures as potentials. The basic result, from which everything else will follow, is Theorem 3 below. It says that the limits W are necessarily reflectionless (this notion will be defined later) on the support of the absolutely continuous part of the spectral measure. This is a very strong condition; it severely restricts the structure of potentials with some absolutely continuous spectrum. As in [19], this result crucially depends on earlier work of Breimesser and Pearson [7, 8]. We will present two applications of Theorem 3 here; both are analogs of results from [19]. The first application gives an easy and transparent proof of a continuous Denisov–Rakhmanov [11, 12, 17] type theorem. We denote by ac the essential support of the absolutely continuous part of the spectral measure; this is determined up to sets of (Lebesgue) measure zero. If we write ρ for the spectral measure, we can define (a representative of) ac as the set where dρ/dt > 0. The absolutely continuous spectrum, σac , may be obtained from ac by taking the essential closure. The essential spectrum, σess , can be defined as the set of accumulation points of the spectrum. Theorem 1 Let V be a uniformly locally integrable (half line) potential (that is, n+1 we assume that supn n |V(x)| dx < ∞). Suppose that σess = ac = [0, ∞). Then
lim
x→∞
V(x + t)ϕ(t) dt = 0
for every continuous ϕ of compact support. Denisov proved this earlier [11, Theorem 2], under the somewhat stronger assumption that V is bounded. The conclusion of Theorem 1 says that V(x) tends to zero as x → ∞ in weak ∗ sense (more precisely, it is the sequence of measures V(x + t) dt that converges). It will become clear later that this mode of convergence is natural here. Also, examples of the type V = U 2 + U with a rapidly decaying, but oscillating U show that stronger modes of convergence of V can not be expected.
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Theorem 1 will be proved in Section 4. As in [19, Theorem 1.8], it should be possible to use the same technique to establish an analogous result for finite gap potentials and spectra (and beyond), but we will not pursue this theme here. Let us now discuss a second structural consequence of Theorem 3; in [19], I introduced the designation Oracle Theorem for statements of this type. The Oracle Theorem says that for operators with absolutely continuous spectrum, it is possible to approximately predict future values of the potential, with arbitrarily high accuracy, based on information about past values. The precise formulation will involve measures μ as potentials and some additional technical devices; these will of course be explained in more detail later. To get a preliminary impression of what the Oracle Theorem is saying, it is possible to replace μ by a (uniformly locally integrable) potential V in Theorem 2 below. We will work with spaces V JC of signed Borel measures μ on intervals J. For now, we can pretend that a measure μ is in V JC if |μ|(J) C|J|, but, for inessential technical reasons, the actual definition will be slightly different. If endowed with the weak ∗ topology, these spaces V JC are compact and in fact metrizable. The metric d that is used below arises in this way. We will also use a similarly defined space V C of measures on R. See Section 2 for the precise definitions. Finally, Sx μ will denote the shift by x of the measure μ, that is,
f (t) d(Sx μ)(t) =
f (t − x) dμ(t).
(1.1)
If dμ = V dt is a locally integrable potential V, then this reduces to the shift map (Sx V)(t) = V(x + t) that was introduced above. Theorem 2 (The Oracle Theorem) Let A ⊂ R be a Borel set of positive (Lebesgue) measure, and let > 0, a, b ∈ R (a < b), C > 0. Then there exist L > 0 and a continuous function (the oracle) C C : V(−L,0) → V(a,b)
so that the following holds. If μ ∈ V C and the half line operator associated with μ satisfies ac ⊃ A, then there exists an x0 > 0 so that for all x x0 , we have that d χ(−L,0) Sx μ , χ(a,b) Sx μ < . In other words, for large enough x, we can approximately determine the potential on (x + a, x + b) from its values on (x − L, x), and the function (oracle) that does this prediction is in fact independent of the potential. Moreover, by adjusting a, b , we can also specify in advance how far the oracle should look into the future.
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2 Topologies on Spaces of Potentials We need a topology on a suitable set of potentials that makes this space compact and also interacts well with other basic objects such as m functions. This is easy to do if we are satisfied with working with potentials that obey a local L p condition with p > 1. Indeed, for every p > 1 (and C > 0), we can define n+1 p p VC = V : R → R : |V(x)| dx C for all n ∈ Z . p n
Closed balls in L p are compact in the weak ∗ topology if p > 1; in fact, these compact topological spaces are metrizable. Pick such metrics dn ; in other words, if Wj, W ∈ L p (n, n + 1), Wj p , W p ≤ C, then dn (Wj, W) → 0 precisely if Wj → W in the weak ∗ topology, that is, precisely if n+1 n+1 Wj(x)g(x) dx → W(x)g(x) dx ( j → ∞) n
n
for all g ∈ Lq (n, n + 1), where 1/ p + 1/q = 1. Then, using these metrics, define, for V, W ∈ V C p d(V, W) =
∞
2−|n|
n=−∞
dn (Vn , Wn ) ; 1 + dn (Vn , Wn )
here Vn , Wn denote the restrictions of V, W to (n, n + 1). This metric generates the product topology on V C p , where this space is now viewed as the product of the closed balls of radius C in L p (n, n + 1). In particular, (V C p , d) is a compact metric space. This simple device allows us to establish continuous analogs of the results of [19] without much difficulty at all, but it is unsatisfactory because the most natural and general local condition on the potentials is an L1 condition. Since L1 is not a dual space, we will then need to consider measures to make an analogous approach work. Thus we define V C = {μ ∈ M(R) : |μ|(I) C max{|I|, 1} for all intervals I ⊂ R} .
Here, M(R) denotes the set of (signed) Borel measures on R. We can now proceed as above to define a metric on V C : Pick a countable dense (with respect to · ∞ ) subset { fn : n ∈ N} ⊂ Cc (R), the continuous functions of compact support, and put ρn (μ, ν) = fn (x) d(μ − ν)(x) . Then define the metric d as d(μ, ν) =
∞ n=1
2−n
ρn (μ, ν) . 1 + ρn (μ, ν)
(2.1)
Absolutely continuous spectrum of one-dimensional Schrödinger operators
Clearly, d(μ j, μ) → 0 if and only if f (x) dμ j(x) → f (x) dμ(x)
363
( j → ∞)
for all f ∈ Cc (R). Moreover, (V C , d) is a compact space. To prove this, let μn ∈ V C . By the Banach-Alaoglu Theorem, closed balls in M([−R, R]) are compact. Use this and a diagonal process to find a subsequence μn j with the property that f dμn j → f dμ for all f ∈ Cc (R), for some μ ∈ M(R). The proof can now be completed by noting that a measure ν ∈ M(R) is in V C if and only if f (x) dν(x) C max{diam(supp f ), 1} f ∞ for all f ∈ Cc (R). The same construction can be run if R is replaced by an interval J, and these spaces, which we will denote by V JC , will also play an important role later on. 3 Schrödinger Operators with Measures We are thus led to consider Schrödinger operators with measures as potentials; therefore, we must now clarify what the precise meaning of this object is. There is, of course, a considerable amount of previous work on these issues; see, for example, [1, 3, 5, 6] and the references cited therein. Here, we will follow the approach of [3]. Actually, Schrödinger operators will not play a central role in this paper, at least not explicitly. Therefore, we will only indicate how to make sense out of the Schrödinger equations − f + μf = zf . We can then use these to define Titchmarsh-Weyl m functions, spectral measures etc., and we refer the reader to [3] for the (straightforward) definition of domains that yield self-adjoint operators. There are two obvious attempts, and these conveniently lead to the same result: If I ⊂ R is an open interval and f ∈ C(I), we can call f a solution to the Schrödinger equation − f + f μ = zf
(3.1)
if (3.1) holds in the sense of distributions on I. Alternatively, one can work with the quasi-derivative (A f )(x) = f (x) − f (t) dμ(t);
[0,x]
if x < 0, then [0,x] needs to be replaced with − (x,0) here. We now say that f solves (3.1) on I if both f and A f are (locally) absolutely continuous and −(A f ) = zf on I. This new definition is motivated by the observation that, at least formally, (A f ) = f − f μ. A slight modification of the argument from the proof of [3, Theorem 2.4] then shows that this latter interpretation of (3.1) is equivalent to the equation
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holding in D (I). The basic observation here is that if f is continuous, then (A f ) = f − f μ in D , not only formally. Note that if f solves (3.1), then f is of bounded variation and the jumps can only occur at the atoms of μ. If μ ∈ V C , we have limit point case at both endpoints. This means that for z ∈ C+ (the upper half plane in C), there exist unique (up to a factor) solutions f± (x, z) of (3.1) on R satisfying f− ∈ L2 (−∞, 0), f+ ∈ L2 (0, ∞). The Titchmarsh-Weyl m functions of the problems on (−∞, x) and (x, ∞), with Dirichlet boundary conditions at x (u(x) = 0), are now defined as follows: m± (x, z) = ±
f± (x, z) f± (x, z)
(3.2)
We will use this formula only for points x with μ({x}) = 0 so that the possible discontinuities of f cannot cause any problems here. Definition 1 Let A ⊂ R be a Borel set. We call a potential μ ∈ V C reflectionless on A if m+ (x, t) = −m− (x, t)
for almost every t ∈ A
(3.3)
for some x ∈ R with μ({x}) = 0.
The set of reflectionless potentials μ ∈ C>0 V C on A is denoted by R(A). This is a key notion for everything that follows. If we have (3.3) for some x, then we automatically get this equation at all points of continuity of μ. Moreover, the exceptional set implicit in (3.3) can be taken to be independent of x. To prove these remarks, observe that if m± (x, t) ≡ lim y→0+ m± (x, t + iy) exists for some x, t ∈ R, then this limit exists for all x (and the same t). Moreover, as a function of x, the m functions are of bounded variation and (using distributional derivatives) ±
d m± = μ − z − m2± . dx
The claim now follows by considering (d/dx)(m+ + m− )(x, t). The other key notion is that of the ω limit set of a potential μ ∈ V C under the shift map. This was already mentioned in the introduction, and we can now give the more precise definition ω(μ) = ν ∈ V C : There exist xn → ∞ so that d(Sxn μ, ν) → 0 . For the definition of the shifted measures Sx μ, see (1.1). Typically, μ will be given as a half line potential V, but it is of course easy to interpret V as an element dμ = V dx of V C (such a μ automatically gives zero weight to (−∞, 0]). The compactness of V C ensures that ω(μ) is non-empty, compact, and invariant under {Sx : x ∈ R}. Moreover, and in contrast to the discrete case, ω(μ) is also connected because we now have a flow Sx .
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4 Main Results and their Proofs It will be useful to introduce V+C as the set of all μ ∈ V C with |μ|((−∞, 0]) = 0. These measures will serve as the potentials of half line problems on (0, ∞). We can think of such a μ as a measure on (0, ∞) or on R. Theorem 3 Let μ ∈ V+C . Then ω(μ) ⊂ R(ac ). Here, ac denotes an essential support of the absolutely continuous part of the spectral measure of the half line problem on (0, ∞) (say). Theorem 3 is proved in the same way as the analogous result (Theorem 1.4) from [19]. Therefore, we will only make a few quick remarks and then leave the matter at that. First of all, note that although the original result of Breimesser and Pearson [7, Theorem 1] is formulated for Schrödinger operators with locally integrable potentials, the same proof also establishes the result for operators with measures as potentials. Indeed, one never works with the potential itself but only with solutions to the Schrödinger equation (3.1) or with transfer matrices. See also [19, Appendix A]. As a second ingredient, we need continuous dependence of the (half line) m functions m± on the potential. Lemma 1 Let μn , μ ∈ V C and suppose that d(μn , μ) → 0. Fix x ∈ R with μn ({x}) = μ({x}) = 0. Then m± (x, z; μn ) → m± (x, z; μ), uniformly on compact subsets of C+ . This follows because convergence in V C implies weak ∗ convergence of the restrictions of the measures to compact intervals, at least if the endpoints of these intervals are not atoms of μ. It then follows that the solutions to the Schrödinger equation converge, locally uniformly in z. This is most conveniently established by rewriting the Schrödinger equation as an integral equation. See, for example, [3, Lemma 6.3] for more details. One can now use (3.2) to obtain the Lemma. In fact, it is also helpful to approximate m± by m functions of problems on bounded intervals. This allows us to work with solutions that satisfy a fixed initial condition. To prove Theorem 3, fix ν ∈ ω(μ). By definition of the ω limit set, there exists a sequence x j → ∞ so that Sx j μ → ν in (V C , d). Fix x ∈ R with μ({x + x j}) = ν({x}) = 0. By Lemma 1, m± (x + x j, z; μ) → m± (x, z; ν)
( j → ∞),
locally uniformly in z. The Breimesser-Pearson Theorem [7, Theorem 1] (see also [19, Theorem 3.1]) together with [19, Theorem 2.1] then yield a relation between m+ (x, z; ν) and m− (x, z; ν) which turns out to be equivalent to the
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condition from Definition 1, with A = ac . This last part of the argument is identical with the corresponding treatment of [19]. Honesty demands that I briefly comment on a technical (and relatively insignificant point) here: To run the argument in precisely this form, one needs a slight modification of either Lemma 1 or the original BreimesserPearson Theorem. The easiest solution would be to prove the BreimesserPearson Theorem for two half line m functions m± (in the original version from [7, 8], m− refers to a bounded interval). Alternatively, one can use a variant of Lemma 1 where the approximating m functions may be associated with bounded (but growing) intervals. Let us now show how Theorem 3 can be used to produce DenisovRakhmanov type theorems. We will automatically obtain the following slightly more general version of Theorem 1. Theorem 4 Let μ ∈ V+C , and suppose that the half line operator generated by μ satisfies σess = ac = [0, ∞). Then d(Sx μ, 0) → 0 as x → ∞. The proof will also depend on the following observation (whose discrete analog was pointed out in [15]; see also [16]). Proposition 1 Let μ ∈ V+C and assume that ν ∈ ω(μ). Then + (μ). σ (ν) ⊂ σess + Here, σ (ν) is the spectrum of −d2 /dx2 + ν on L2 (R), while σess (μ) denotes 2 2 the essential spectrum of the half line operator −d /dx + μ on L2 (0, ∞) (say).
Proof In the discrete case, this followed from a quick argument using Weyl sequences. In the continuous case, this device is not as easily implemented because of domain questions. The following alternative argument avoids these issues and thus seems simpler: Suppose that d(Sx j μ, ν) → 0. Then the whole line (!) operators associated with Sx j μ converge in strong resolvent sense to the (whole line) operator generated by ν. To prove this fact, one can argue as in Lemma 1 above. Since the operators with shifted potentials are unitarily equivalent to the operator generated by μ itself, it follows from [18, Theorem VIII.24(a)] that σ (ν) ⊂ σ (μ). The ω limit set does not change if μ is modified on a left half line; any discrete eigenvalue, however, can be moved (or removed) by such a + − − modification. Similarly, σess = σess ∪ σess , and σess is completely at our disposal, so we actually obtain the stronger claim of the Proposition. Proof of Theorem 4 We will show that ω(μ) consists of the zero potential only. This will imply the claim because the distance between Sx μ and ω(μ) must go to zero as x → ∞.
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By Theorem 3 and Proposition 1, any ν ∈ ω(μ) must satisfy σ (ν) = [0, ∞),
ν ∈ R((0, ∞)).
(4.1)
Here, we use the (well known) fact that Im m± > 0 almost everywhere on A if the corresponding potential is reflectionless on this set. Indeed, (3.3) shows that otherwise we would have m+ + m− = 0 on a set of positive measure, hence everywhere, but this is clearly impossible. What we will actually prove now is that only the zero potential, ν = 0, satisfies (4.1). This argument follows a familiar pattern; see, for example, [9] or [21] (especially Lemma 4.6 and the discussion that follows) for similar arguments in somewhat different situations. Suppose that ν ∈ V C obeys (4.1), and fix x ∈ R with ν({x}) = 0. Let m± be the m functions of Hν = −d2 /dt2 + ν on L2 (x, ∞) and L2 (−∞, x), respectively, and consider the function H(z) = m+ (x, z) + m− (x, z) = −
W( f+ , f− ) . f+ (x, z) f− (x, z)
Here, W(u, v) = uv − u v denotes the Wronskian. This last expression identifies H as H(z) = −
1 , G(x, x; z)
the negative reciprocal of the (diagonal of the) Green function of Hν . Compare, for example, [10, Section 9.5]. (Of course, this reference does not discuss Schrödinger operators with measures, but the rather elementary argument based on the variation of constants formula generalizes without any difficulty.) The defining property of G is given by ∞ −1 (Hν − z) ϕ (x) = G(x, y; z)ϕ(y) dy. −∞
This holds for z ∈ / σ (ν) = [0, ∞), ϕ ∈ L2 (R). The spectral theorem shows that if z = −t < 0, then
d E(s)ϕ 2 −1 > 0. ϕ, (Hν + t) ϕ = s+t [0,∞) Since G is continuous in x, y, this implies that G(x, x, t) 0, thus H(t) < 0 for t < 0. Furthermore, the fact that ν ∈ R((0, ∞)) implies that Re H(t) = 0 for almost every t > 0. So we know the phases of the boundary values of the Herglotz function H almost everywhere. By the exponential Herglotz representation (or, synonymously, the Herglotz representation of ln H(z)), this determines H up to a (positive) multiplicative constant. Since H0 (z) = (−z)1/2 has the properties described above, this says that for suitable c > 0, √ t c c ∞ 1 H(z) = c −z = − √ + − 2 t1/2 dt. (4.2) π t − z t + 1 2 0
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We will now need some information on the large z asymptotics of m functions. This subject has been analyzed in considerable depth; see, for example, [2, 13, 14, 20]. Of course, the treatment of these references needs to be adjusted here to cover the case of Schrödinger operators with measures, but this is easy to do, especially since we will only need the rather unsophisticated estimate (κ → ∞). m± x, −κ 2 = −κ + o(1) Here, it is important that we assumed that ν({x}) = 0. Since H = m+ + m− , it now follows that c = 2 in (4.2). Furthermore, the measures from the Herglotz representations of m± are absolutely continuous with respect to the measure dρ(t) =
2 χ(0,∞) (t)t1/2 dt π
from the Herglotz representation (4.2) of H. In fact, we can write 1 t m± (x, z) = A± + − g± (t) dρ(t), t − z t2 + 1 √ with A± ∈ R, A+ + A− = − 2, and, more importantly, 0 g± 1 and g+ +g− = 1. More can be said here: Since ν is reflectionless on (0, ∞), we can use (3.3) to deduce that Im m+ (x, t) = Im m− (x, t) for almost every t > 0. But for almost every t > 0, we have that Im m± (x, t) = g± (t)(2/π )t1/2 , thus g+ = g− = 1/2 almost everywhere. It now follows that √ m± (x, z) = −z. √ But m0 = −z is the m function for zero potential, thus ν = 0, as desired. This last step is a basic result in inverse spectral theory for potentials (m determines V); here, we of course need a version for measures, but this extension poses no difficulties. See, for instance, [3, Theorem 6.2(b)] (this needs to be combined with the fact that m determines φ, but this is also discussed in [3]). It remains to prove the Oracle Theorem. We prepare for this by making a couple of new definitions. First of all, put RC (A) = R(A) ∩ V C .
Next, we consider again spaces of half line potentials, and we now think of these as restrictions of measures μ ∈ V C : V+C = χ(0,∞) μ : μ ∈ V C , V−C = χ(−∞,0) μ : μ ∈ V C I emphasize that on V±C , we do not use the topology that is induced by V C ⊃ V±C . That would quite obviously be a bad idea because it would make the restriction map μ → χ(0,∞) μ discontinuous; consider, for example, the sequence μn = δ1/n . Instead, we just observe that in the notation from Section 2, we can identify V+C = V JC , where J = (0, ∞), and we use the topology and metric
Absolutely continuous spectrum of one-dimensional Schrödinger operators
369
described in Section 2. In other words, if we denote this metric by d+ , then d+ (μn , μ) → 0 if and only if f (x) dμn (x) → f (x) dμ(x) (n → ∞) for all continuous f whose support is a compact subset of (0, ∞). Similar remarks apply to V−C , of course. Now the restriction maps V C → V±C are continuous, and the spaces (V±C , d± ) are compact. Finally, we introduce C C RC + (A) = χ(0,∞) μ : μ ∈ R (A) ⊂ V+ , C C RC − (A) = χ(−∞,0) μ : μ ∈ R (A) ⊂ V− , and we use the same metrics d± on these spaces also. With this setup, we now obtain statements that are analogs of [19, Proposition 4.1]. Proposition 2 Let A ⊂ R be a Borel set of positive measure, and fix C > 0. Then: (a) RC (A), d and RC ± (A), d± are compact spaces; (b) The restriction maps RC (A) → RC + (A),
μ → χ(0,∞) μ;
RC (A) → RC − (A),
μ → χ(−∞,0) μ
are continuous and bijective (and thus homeomorphisms). (c) The inverse map C RC − (A) → R (A),
χ(−∞,0) μ → μ
is (well defined, by part (b), and ) uniformly continuous. Proof (a) Since RC (A) is a subspace of the compact space V C , it suffices to show that RC (A) is closed. This can be done exactly as in [19, Proof of Proposition 4.1(d)]; we make use of Lemma 1 of the present paper and Theorem 2.1, Lemma 3.2 of [19]. C The spaces RC ± (A) are the images of the compact space R (A) under the continuous restriction maps, so these spaces are compact, too. (b) Continuity of the restriction maps is clear (and was already used in the preceding paragraph). Moreover, these maps are surjective by the definition of the spaces RC ± (A). Injectivity follows from equation (3.3): μ on (0, ∞) determines m+ (x, ·) for all x > 0. Fix an x > 0 with μ({x}) = 0. Since μ is reflectionless on A and |A| > 0, we have condition (3.3) on a set of positive measure, and this lets us find m− (x, ·). This m function, in turn, determines μ on (−∞, x).
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Finally, recall that a continuous bijection between compact metric spaces automatically has a continuous inverse. (c) This is an immediate consequence of parts (a) and (b). As in [19], the Oracle Theorem will follow by combining Proposition 2 with Theorem 3. In fact, in rough outline, things are rather obvious now: Proposition 2 says that a reflectionless potential can be approximately predicted if it is known on a sufficiently large interval (recall how the topologies on the spaces RC (A), RC ± (A) were defined), and Theorem 3 makes sure that Sx μ is approximately reflectionless for sufficiently large x. Some care must be exercised, however, if a continuous oracle is desired. The following straightforward but technical considerations prepare for this part of the proof. We again consider the spaces V JC with metrics of the type described in Section 2; in the applications below, the interval J will be bounded and open, but this is not essential here. In a normed space, balls Br (x) = {y : x − y < r} are convex; Lemma 2 below says that balls with respect to the metric (2.1) enjoy the following weaker, but analogous property. Lemma 2 If w j 0, 1/4 (say), then
m j=1
w j = 1 and μ, ν j ∈ V JC satisfy d(μ, ν j) < with ⎛
d ⎝μ,
m
⎞ w jν j⎠ < 6 ln −1 .
j=1
Proof Let N = max{n ∈ N : 2n+1 1}, and abbreviate d(μ, ν j) < , it is clear from (2.1) that if n N, then ρn (μ, ν j) < The definition of ρn shows that ρn (μ, ν)
w jν j = ν. Since
2n 2n+1 . 1 − 2n
w jρn (μ, ν j),
so we obtain that ρn (μ, ν) < 2n+1
(n N).
This allows us to estimate N n=1
2 ln −1 ρn (μ, ν) 2−n · 2n+1 = 2N < < . 1 + ρn (μ, ν) ln 2 n=1 N
2−n
On the other hand, we of course have that ρn (μ, ν) 4 ln −1 2−n < 2−n = 2−N < 4 , 1 + ρn (μ, ν) n>N ln 4 n>N so we obtain the Lemma.
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Proof of Theorem 2 We begin by introducing some notation that will prove useful. Write J− = (−L, 0),
J+ = (a, b).
Our goal is to (approximately) predict the restriction of Sx μ to J+ , and we are given the restriction of Sx μ to J− . We will use subscripts + and −, respectively, for such restrictions. So, for example, ν+ = χ J+ ν, and this is now interpreted as an element of V JC+ . Next, note that although a metric is explicitly mentioned in Theorem 2, by compactness, it suffices to establish the assertion for some metric that generates the weak ∗ topology. We will of course want to work with the metric from (2.1) and Lemma 2. More specifically, denote this metric (on V JC+ ) by d+ . We use a similar metric d− on V JC− ; on V C , we also fix such a metric d, but, in addition, we demand, as we may, that d dominates d± in the following sense: If μ, ν ∈ V C , then d− (μ− , ν− ) d(μ, ν),
d+ (μ+ , ν+ ) d(μ, ν).
(4.3)
(The same notation, d± , was used for different purposes in Proposition 2; since we are not going to explicitly use those metrics here, that should not cause any confusion.) With these preparations out of the way, the proof can now be accomplished in four steps. Let A ⊂ R, |A| > 0, > 0, a, b ∈ R (a < b ), and C > 0 be given. Step 1: Use Proposition 2(c) and the definition of the topologies on RC (A), RC ν ∈ RC (A), − (A) to find L > 0 and δ > 0 such that the following holds: For ν, d− (ν− , ν− ) < 5δ
=⇒
ν+ ) < 2 . d+ (ν+ ,
(4.4)
We further assume that δ ≤ here. (The suspicious reader will have noticed that it is at this point only that we can define d− and d.) Step 2: The set RCJ− (A) := μ− : μ ∈ RC (A) is compact by Proposition 2 (again, this is a continuous image of a compact space). Since V JC− is compact, it follows that the closed δ neighborhood U δ = μ− ∈ V JC− : d− (μ− , ν− ) ≤ δ for some ν ∈ RC (A) is also compact. Therefore, there exist ν1 , . . . , ν N ∈ RC (A) so that the 2δ balls about the ν j,− cover U δ . At these points, we can define a preliminary version of the oracle in the obvious way as 0 ν j,− = ν j,+ .
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C. Remling
However, this will be modified in the next step. Step 3: We now define, for arbitrary σ ∈ U δ , (3δ − d− (σ, ν j,− ))0 (ν j,− ) . (σ ) = (3δ − d− (σ, ν j,− )) The sums are over those j for which d− (σ, ν j,− )< 3δ. It’s easy to see that : U δ → V JC+ is continuous. Moreover, if j0 ∈ {1,. . ., N} is such that d− (σ, ν j0 ,− ) < 2δ, then d− (ν j,− , ν j0 ,− ) < 5δ for all j contributing to the sum. Therefore, (4.4) shows that d+ (ν j,+ , ν j0 ,+ ) < 2 for these j. If > 0 was sufficiently small, then Lemma 2 now implies that d+ (σ ), ν j0 ,+ < 6 2 ln −2 < , (4.5) say. Recall that this holds for every j0 for which d− (σ, ν j0 ,− ) < 2δ. Moreover, for every σ ∈ U δ , there is at least one such index j0 . The oracle has now been defined on U δ , and this is all we need to do the prediction. However, if a (somewhat specious) continuous extension to all of V JC− is desired, one can proceed as above, by considering a suitable covering and taking convex combinations. It is also possible, somewhat more elegantly, to just refer to the extension theorem of Dugundji-Borsuk [4, Ch. II, Theorem 3.1]. Step 4: In this final step, we show that indeed predicts μ. Given a potential μ ∈ V C with ac ⊃ A, first of all take x0 so large that d (Sx μ, ω(μ)) < δ
for all x x0 .
In other words, if we fix x x0 , we then have that d(Sx μ, ν) < δ
(4.6)
for some (in general: x dependent) ν ∈ ω(μ). By Theorem 3, ν ∈ R (A). When we restrict to J± , then (4.3), (4.6) imply that C
d− ([Sx μ]− , ν− ) < δ,
d+ ([Sx μ]+ , ν+ ) < δ.
(4.7)
In particular, this ensures that [Sx μ]− ∈ U δ , and thus there exists a j ∈ {1, . . . , N} so that d− [Sx μ]− , ν j,− < 2δ. By (4.5),
d+ ([Sx μ]− ) , ν j,+ < .
(4.8)
But by the triangle inequality, we also have that d− (ν− , ν j,− ) < 3δ, so (4.4) shows that d+ ν+ , ν j,+ < 2 . If this is combined with (4.7), (4.8), we indeed obtain that d+ ( ([Sx μ]− ) , [Sx μ]+ ) < δ + + 2 < 3 (say), as desired.
Absolutely continuous spectrum of one-dimensional Schrödinger operators
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Acknowledgements I thank Sergey Denisov and Barry Simon for bringing [11] to my attention and Lenny Rubin for useful information on extension theorems.
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