Commun. Math. Phys. 300, 1–46 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1096-6
Communications in
Mathematical Physics
Congruence Subgroups and Generalized Frobenius-Schur Indicators Siu-Hung Ng1, , Peter Schauenburg2, 1 Department of Mathematics, Iowa State University, Ames, IA 50011, USA. E-mail:
[email protected] 2 Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany.
E-mail:
[email protected] Received: 8 October 2008 / Revised: 6 November 2009 / Accepted: 14 May 2010 Published online: 26 August 2010 – © Springer-Verlag 2010
Abstract: We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z (C) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay’s second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert. 0. Introduction The importance of the role of the modular group SL2 (Z) in conformal field theory has been known since the work of Cardy [Car86]. Associated to a 2D rational conformal field theory (RCFT) is a finite-dimensional representation of SL2 (Z) with a distinguished basis formed by the characters of the primary fields. This modular representation conceives some interesting algebraic and arithmetic properties. One notable example is the Verlinde formula (cf. [Ver88,MS89]). The kernel of the modular representation associated with a RCFT is of particular interest. It has been conjectured the kernel is always a congruence subgroup of SL2 (Z) (cf. [Moo87,Eho95,ES95,DM96,BCIR97]), i.e. it contains some principal congruence subgroup (N ) of SL2 (Z). The conjecture was first addressed by Coste and Gannon in [CG], and they showed that the conjecture 1 1 ∈ SL2 (Z) is represented by an odd order matrix under the modular holds if t = 0 1 The first author is supported by the NSA grant H98230-08-1-0078. The second author is supported by Deutsche Forschungsgemeinschaft through a Heisenberg fellowship.
2
S.-H. Ng, P. Schauenburg
representation. The conjecture was later established by Bantay in [Ban03] under certain assumptions. More recently, Xu also solved the conjecture for the modular representation associated to a local conformal net [Xu06]. The language of modular tensor categories, termed by I. Frenkel, constitutes a formalization of the chiral data of a conformal field theory (cf. [MS90,BK01]). A modular tensor category may be thought of as the representation category of some chiral algebra which corresponds to a conformal field theory. Huang has proved this for some vertex operator algebras [Hua05] (see also [Lep05]). The recent progress in representation theory has revealed that a modular tensor category over an algebraically closed field k of characteristic zero can always be realized as the representation category of some connected ribbon factorizable semisimple weak Hopf algebra over k (cf. [Szl01,NTV03]). Moreover, Müger has also shown in [Müg03] that the center (quantum double) Z (C) of a spherical fusion category C over k is naturally a modular tensor category. In particular, the representation category of a semisimple factorizable Hopf algebra and the representation category of the Drinfeld double D(H ) of a semisimple Hopf algebra H are modular tensor categories. Parallel to rational conformal field theory, each modular tensor category A over k is associated with a natural projective modular representation ρ A on K0 (A) ⊗Z k, where K0 (A) is the Grothendieck (fusion) ring of A. This projective modular representation is projectively equivalent to an ordinary representation, but such a lifting is not unique. However, if A = Z (C) for some spherical fusion category C, then there exists a canonical ordinary modular representation ρ Z (C ) which is a lifting of ρ Z (C ) . It is natural to ask whether the kernels of these canonical projective or ordinary modular representations are congruence subgroups of SL2 (Z). These questions were answered affirmatively by Sommerhäuser and Zhu in [SZ] for factorizable semisimple Hopf algebras and the Drinfeld doubles of semisimple Hopf algebras. In this paper, we generalize their results to spherical fusion categories, and prove the congruence subgroup conjectures in Theorems 6.7 and 6.8. Moreover, every lifting of the projective modular representation of a modular tensor category has a finite image. We approach these questions by studying the generalized Frobenius-Schur indicators for spherical fusion categories introduced in this paper. The classical notion of the second Frobenius-Schur (FS) indicators for the representations of a finite group has been generalized to many different contexts. A version for semisimple Hopf algebras was introduced by Linchenko and Montgomery [LM00]. A more general version for semisimple quasi-Hopf algebras was studied by Mason and Ng in [MN05], and Schauenburg in [Sch04]. Some categorical versions of the 2nd FS indicator were studied by Fuchs, Ganchev, Szlachányi, and Vescernyés in [FGSV99] and by Fuchs and Schweigert in [FS03]. Bantay also introduced another version of the 2nd FS indicator for RCFT as a formula in terms of the modular data. The less well-known higher FS indicators for the representations of a finite group were generalized to semisimple Hopf algebras in [LM00], and studied extensively by Kashina, Sommerhäuser and Zhu [KSZ06]. They were further generalized to semisimple quasi-Hopf algebras by Ng and Schauenburg [NS08]. All these FS indicators in different contexts are specializations of the higher FS indicators for pivotal categories introduced in [NS07b]. The main tool employed in [SZ] to prove the congruence subgroup theorems is the equivariant indicators for semisimple Hopf algebras, which are extensions of the higher FS indicators for semisimple Hopf algebras. Their discovery suggests a more general version of indicators for pivotal categories. In this paper, we introduce the generalized X (V ) for a pair (m, l) of integers, an object V of a Frobenius-Schur (GFS) indicator νm,l
Congruence Subgroups and Generalized Frobenius-Schur Indicators
3
pivotal category C and an object X in the center Z (C). For a pair of integers m, l ∈ Z, the X (V ) is defined as the trace of a linear endomorphism E (m,l) of the vector indicator νm,l X,V space C(X, V ⊗m ), where X is the underlying C-object of X. If X is the unit object of X (V ) coincides with the (m, l)th FS indicator ν Z (C) and m > 0, then νm,l m,l (V ) of V defined in [NS07b]. In a spherical fusion category C, one can extend the assignment X (V ) for each simple X ∈ Z (C) to a linear functional I ((m, l), −) on the X → νm,l V fusion algebra Kk (Z (C)) = K0 (Z (C)) ⊗Z k for each pair (m, l) of integers and V ∈ C; this extension is called the equivariant indicator in Sect. 6. It is equivalent to the corresponding notion introduced by Sommerhäuser and Zhu when C is the representation category of a semisimple Hopf algebra. The set of all equivariant indicators is closed under the SL2 (Z)-action on Kk (Z (C))∗ induced by the contragredient of the modular representation ρ Z (C ) . Moreover, the indicators are invariant under the action of the principal congruence subgroup (N ), which is the kernel of the epimorphism SL2 (Z) → SL2 (Z/N Z), where N is the Frobenius-Schur exponent of C. The study of the relation between the equivariant indicators and the modular representations associated with the center of a modular tensor category leads to our major theorems. These theorems imply that all the modular representations of a modular category have finite images, and a conjecture of Eholzer on these representations. In the course of studying the equivariant indicators, we obtain two formulae for X (V ) for the GFS indicators. The first formula, obtained in Corollary 5.6, expresses νm,1 a spherical fusion category C in terms of the modular data of Z (C); it contains the FS indicator formula discovered in [NS07a, Theorem 4.1] as a special case. The second formula, described in Proposition 6.1 which is a consequence of the first formula, expresses X (V ) for a modular category A in terms of its modular data. It implies Bantay’s indiνm,1 cator formula [Ban97] when we specify m = 2 and X to be the unit object of Z (A). More importantly, this formula suggests a close relationship between the GFS indicators c indexed by the primary fields a, b, c of a RCFT introduced in and a family of scalars Yab c are integers and it is further conjectured [PSS95]. It is conjectured in [PSS95] that Yab c in [BHS98] certain inequality holds for Yab . Gannon has proved these conjectures under the condition that the T matrix of the RCFT has odd order [Gan00]. More recently, Kac, Longo and Xu have proved these conjectures via Z2 -permutation orbifolds of conformal nets [KLX05]. As an application of GFS indicators, we prove these conjectures hold for all modular categories. The organization of this paper is as follows: In Sect. 1 we cover some basic definitions, notations, conventions and preliminary results on pivotal categories for the remaining discussion. In Sect. 2 we define the generalized FS indicators, discuss their basic properties and an alternative characterization. This continues in Sect. 3 under the additional assumption that the category is semisimple, and we give another characterization of the GFS indicators for spherical fusion categories. In Sect. 4, we show how this characterization recovers the equivariant indicators introduced in [SZ] when the underlying spherical category is the representation category of a semisimple Hopf algebra. We define the equivariant indicators for a spherical fusion category in Sect. 5. We show that the set of equivariant indicators admits a natural action of SL2 (Z), and derive some important consequences of this modular action. In Sect. 6, we study the equivariant indicators for a modular tensor category and its center, and prove the congruence subgroup theorems. We also provide an example for the congruence subgroups arising. The study of modular representations of a modular category continues in Sect. 7. We prove the images of these representations are finite, and a conjecture of Eholzer for modular categories holds. In Sect. 8, we prove a conjecture of Pradisi-Sagnotti-Stanev and a conjecture of
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S.-H. Ng, P. Schauenburg
Borisov-Halpern-Schweigert using a generalized Bantay’s formula for GFS indicators. In Sect. 9, we introduce the definition of generalized Frobenius-Schur endomorphisms in a pivotal fusion category C. For relatively prime positive integers m, l these turn out to be natural endomorphisms of the identity functor idC , and the corresponding GFS indicators can be expressed as their pivotal traces. This is a generalization of the formulas expressing higher indicators as character values on certain central elements of a quasi-Hopf algebra. 1. Preliminaries In this section, we will collect some conventions and facts on pivotal categories. Most of these are quite well-known, and the readers are referred to to [NS07b,NS08,NS07a] and the literature cited there. Additional key results on fusion categories and their centers are taken from Müger’s work [Müg03] 1.1. Pivotal and spherical monoidal categories. In a monoidal category C with tensor product ⊗, we denote : (U ⊗ V )⊗ W → U ⊗(V ⊗ W ) the associativity isomorphism. If X, Y ∈ C are obtained by tensoring together the same sequence of objects with two different arrangements of parentheses, one can obtain an isomorphism between them by composing several instances of the tensor products of , −1 and the identity. It is unique by the coherence theorem, and will be denoted by ? : X → Y . We will assume throughout that the unit object I ∈ C is strict. A left dual of an object V ∈ C is an object V ∨ ∈ C together with the morphisms ev : V ∨ ⊗ V → I and db : I → V ⊗ V ∨ such that db ⊗V V ⊗ev → V ⊗ (V ∨ ⊗ V ) −−−→ V , id V = V −−−→ (V ⊗ V ∨ ) ⊗ V − −1 V ∨ ⊗db ev ⊗V ∨ id V ∨ = V ∨ −−−−→ V ∨ ⊗ (V ⊗ V ∨ ) −−→ (V ∨ ⊗ V ) ⊗ V ∨ −−−−→ V ∨ . A right dual of an object can be defined similarly. A monoidal category C is called left (resp. right) rigid if every object of C admits a left (resp. right) dual. If C is a left rigid monoidal category, then taking duals can be extended to a monoidal functor (−)∨ : C → C op , and so (−)∨∨ : C → C is consequently a monoidal functor. Moreover, we can choose I ∨ = I and ev I = db I = id I . A pivotal category is a left rigid monoidal category equipped with an isomorphism j : I d → (−)∨∨ , called a pivotal structure, of monoidal functors. Let C be a pivotal category, and V ∈ C. Then V ∨ together with the morphisms db :=
V ∨ ⊗ jV−1
db
I − → V ∨ ⊗ V ∨∨ −−−−−→ V ∨ ⊗ V
and
ev := V ⊗ V
∨ ∨ jV ⊗V
−−−−→ V
∨∨
⊗V
, ∨ ev
− →I
becomes a right dual of V . In particular, C is also right rigid. Let f : V → V be a morphism in the pivotal category C. The left and right pivotal traces of f are respectively f ⊗id db ev → V ⊗ V ∨ −−−→ V ⊗ V ∨ − →I ptrr ( f ) = I − id ⊗ f db ev → V ∨ ⊗ V −−−→ V ∨ ⊗ V − →I . and ptr ( f ) = I −
Congruence Subgroups and Generalized Frobenius-Schur Indicators
5
The left and right pivotal dimensions of V ∈ C are d (V ) = ptr (id V ) and dr (V ) = ptrr (id V ). A spherical category is a pivotal category in which the left and right pivotal traces of every morphism are identical. In a spherical category, the pivotal traces and dimensions will be denoted by ptr( f ) and d(V ), respectively. A pivotal category is called strict if the associativity isomorphism , the pivotal structure j, and the canonical isomorphism (V ⊗ W )∨ → W ∨ ⊗ V ∨ are identities. It has been shown in [NS07b, Theorem 2.2] that every pivotal category C is equivalent to a strict pivotal category Cstr ; equivalence as pivotal categories means that the monoidal equivalence C → Cstr preserves pivotal structures in a suitable sense [NS07b]. If C is spherical, then so is Cstr . In a strict pivotal category, we make free use of graphical calculus. Our convention for a morphism is a diagram with the source at the top and the target at the bottom. For instance, the morphisms ev : V ∨ ⊗ V → I , db : I → V ⊗ V ∨ , ev : V ⊗ V ∨ → I and db : I → V ∨ ⊗ V are respectively the diagrams: V∨
V
,
V
V∨
,
V
V∨
, and
V∨
V
.
Notice that evV = evV ∨ and dbV = dbV ∨ in a strict pivotal category. The left and right pivotal traces of a morphism f : V → V are given by the diagrams: ptr ( f ) = V ∨
f
and ptrr ( f ) =
f
V ∨.
If the pivotal category is spherical, the two pivotal traces coincide. Now let C be a left rigid braided monoidal category. In the graphical calculus, we depict the braiding c and its inverse as cV W =
V W W V
and c−1 VW =
W V V W
.
Associated to c is the Drinfeld isomorphism u : I d → (−)∨∨ defined by db ⊗ id id ⊗c−1 u V := V −−−−→ (V ∨ ⊗ V ∨∨ ) ⊗ V − → V ∨ ⊗ (V ∨∨ ⊗ V ) −−−−→ V ∨ ⊗ (V ⊗ V ∨∨ ) −1 ∨ ∨∨ ev ⊗ id ∨∨ −−→ (V ⊗ V ) ⊗ V −−−→ V which also satisfies −1 u V ⊗W = (u V ⊗ u W )c−1 V W cW V for V, W ∈ C.
In particular, the equation θ = u −1 j describes a one-to-one correspondence between pivotal structures j and twists θ on C. Here, a twist is by definition an automorphism of the identity functor on C satisfying θV ⊗W = (θV ⊗ θW )cW V cV W and θ I = id I .
(1.1)
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S.-H. Ng, P. Schauenburg
For a strict pivotal category with a braiding c, the Drinfeld isomorphism and the associated twist θ are respectively given by V
V
and θV = u −1 V =
uV = V ∨ V
V ∨. V
A twist θ on C is called a ribbon structure if it satisfies θV∨ = θV ∨ . The triple (C, c, θ ) is called a ribbon category if θ is a ribbon structure on the braided monoidal category C with the braiding c. In a ribbon category, the associated pivotal structure on C is spherical. If C is strict, then the associated ribbon structure θ can be depicted as V
θV =
V V∨
V
=
.
V∨ V
1.2. The center construction. The (left) center Z (C) of a monoidal category C is a category whose objects are pairs X = (X, σ X ) in which X is an object of C and the half-braiding σ X (−) : X ⊗ (−) → (−) ⊗ X is a natural isomorphism satisfying the properties σ X (I ) = id X and (V ⊗ σ X (W )) ◦ V,X,W ◦ (σ X (V ) ⊗ W ) = V,W,X ◦ σ X (V ⊗ W ) ◦ X,V,W for all V, W ∈ C. We will often write σ X,V in place of σ X (V ). It is well-known that Z (C) is a braided monoidal category (cf. [Kas95]). The tensor product (X, σ X ) ⊗ (Y, σY ) := (X ⊗ Y, σ X ⊗Y ), of two objects (X, σ X ) and (Y, σY ), and the unit object (I, σ I ) are given by σ X ⊗Y (V ) = V,X,Y ◦ (σ X (V ) ⊗ Y ) ◦ −1 X,V,Y ◦ (X ⊗ σY (V )) ◦ X,Y,V and σ I (V ) = id V for any V ∈ C. The associativity isomorphisms are inherited from C, so that the forgetful functor Z (C) → C is a strict monoidal functor. The canonical braiding on Z (C) is given by cX,Y = σ X (Y ) for X = (X, σ X ), Y = (Y, σY ) ∈ Z (C). If there is no danger of confusion with a previously given braiding on C, we will depict the half-braiding of Z (C) by σ X (V ) =
X V V X
and σ X (V )−1 =
V X X V
.
If C is left rigid, then so is Z (C). If C is a pivotal (resp. spherical) category, then Z (C) is also a pivotal (resp. spherical) category with the pivotal structure inherited from C. Any equivalence F : C → D of monoidal categories naturally induces an equivalence Fˆ : Z (C) → Z (D) of braided monoidal categories. In addition, if C and D are pivotal categories and F preserves their pivotal structures, then Fˆ : Z (C) → Z (D) also preserves their pivotal structures and the twists associated with their pivotal structures. There is a one-to-one correspondence between braidings on a monoidal category and sections of the forgetful functor Z (C) → C, where the section C → Z (C) corresponding to a braiding c maps X ∈ C to (X, σ X ) ∈ Z (C) with σ X (V ) = c X,V . Since the inverse
Congruence Subgroups and Generalized Frobenius-Schur Indicators
7
of a braiding gives another braiding, we can combine the two resulting sections of the forgetful functor to yield a functor C × C → Z (C) which maps (X, Y ) ∈ C × C to (X ⊗ Y, σ X ⊗Y ) given in the strict case by X Y V
σ X ⊗Y (V ) = (c X V ⊗ Y )(X ⊗ cV Y )−1 =
.
(1.2)
V X Y
If C is left rigid, one can check that the dual of (X ⊗ Y, σ X ⊗Y ) is (X ⊗ Y, σ X ⊗Y )∨ = (X ∨ ⊗ Y ∨ , σ X ∨ ⊗Y ∨ )
(1.3)
with evaluation and coevaluation morphisms given by X∨ Y ∨
X
Y
.
and X
Y
X∨
Y∨
1.3. k-linear and semisimple monoidal categories. Almost all results obtained in this paper pertain to k-linear monoidal categories, where we assume throughout that k is an algebraically closed field of characteristic zero, although it may be worth to note that Sects. 2 and 3 do not require any additional assumptions on the field k. We also fix the convention that a k-linear monoidal category C is said to be semisimple if the underlying k-linear category is semisimple with finite-dimensional morphism spaces, and the unit object I is simple. Following [ENO05], a fusion category over k is a semisimple left rigid k-linear monoidal category with finitely many simple objects. Note that if C is a pivotal category over k and I is absolutely simple, then the pivotal traces ptr ( f ) and ptrr ( f ) of an endomorphism f , which were defined as endomorphisms of I , can be identified with scalars in k. In this case we use the pivotal traces to define bilinear pairings (·, ·) , (·, ·)r : C(U, V ) × C(V, U ) → k by ( f, g) := ptr ( f ◦ g), and ( f, g)r := ptrr ( f ◦ g).
(1.4)
Note that ( f, g) = (g, f ) holds (cf. [NS07b]), and these bilinear pairings coincide when C is spherical. In this case, we simply denote ( f, g)r or ( f, g) by ( f, g). If C is also semisimple, then the pairings in (1.4) are non-degenerate (cf. [GK96]). It follows that for a braided spherical semisimple k-linear category the twist associated to the pivotal structure is always a ribbon structure. For any object V in a pivotal fusion category C over k, we write [V ] for its isomorphism class. If {[Vi ] | i ∈ } denotes the finite set of isomorphism classes of simple objects in C, then the index set is always assumed to contain 0 by setting V0 to be the unit object of C. For i ∈ , we define i ∈ by the isomorphism Vi∨ ∼ = Vi . By [ENO05, Theorem 2.3 and Proposition 2.9], the pivotal dimension d (Vi ) of Vi is a non-zero algebraic integer in k, and
d (Vi )d (Vi∨ ) = 0. dim C = i∈
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S.-H. Ng, P. Schauenburg
In addition, if C is spherical, then d(Vi ) = d(Vi∨ ) (cf. [ENO05, Corollary 2.10]) and so
dim C = d(Vi )2 = 0. i∈
We denote by K0 (C) the Grothendieck ring of the fusion category C, and by Kk (C) := K0 (C) ⊗Z k its Grothendieck algebra. The set {[Vi ] | i ∈ } is a basis for the free Zmodule K0 (C), and (with the obvious identification) a k-basis of Kk (C); we will refer to it as the canonical basis. The center of a k-linear monoidal category is itself k-linear monoidal in the obvious way. By [Müg03, Theorem 3.16, Corollary 5.6] (see also [ENO05, Theorem 2.15]) the center of a pivotal (resp. spherical) fusion category over k is also a pivotal (resp. spherical) fusion category over k ([Müg03] assumes a spherical category, but note [Müg03, Remark 3.17]). Moreover, by [Müg03, Proposition 8.1] the forgetful functor F : Z (C) → C, which maps X = (X, σ X ) to X , admits a two-sided adjoint functor K : C → Z (C). Alternatively, [Szl01] and [NTV03] also imply the existence of a left adjoint K to F. Consider such an adjoint, and a natural isomorphism V,X : C(V, X ) → D(K (V ), X). Since C is a pivotal fusion category over k, so is D = Z (C) by [ENO05, Theorem 2.15]. Therefore, the bilinear forms (·, ·) defined in D are non-degenerate. Let X,V : D(X, K (V )) → C(X, V ) be the adjoint operator of V,X with respect to (·, ·) , i.e. ( ( f ), g) = ( f, (g)) for all f ∈ C(V, X ), g ∈ D(X, K (V )).
(1.5)
By the naturality of , the linear isomorphism is also natural in X and V . In particular, K is a right adjoint to F, and is an associated adjunction isomorphism. In fact if either of the adjunction isomorphisms for the right or for the left adjoint is given, we can define the other one by (1.5), which will define a natural isomorphism because (·, ·) is nondegenerate. 1.4. Modular categories. A modular tensor category over k (see [BK01, Chapter 3]), also simply called a modular category, is a ribbon fusion category A = (A, c, θ ) over k such that, for the set {[Ui ] | i ∈ } of isomorphism classes of simple objects, the matrix S = [Si j ] defined by (1.6) Si j = ptr cU j ,Ui ◦ cUi ,U j is non-singular. This matrix S is called the S-matrix of A. In the strict case, Si j can be depicted as Ui
Si j =
U∨ j
.
Let θUi = ωi idUi for some ωi ∈ k. The matrices T and C (charge conjugation matrix) of A are defined as T = [δi j ωi ] and C = [δi j ] . These matrices S, T and C satisfy the conditions: + 2 + − (ST )3 = pA S , S 2 = pA pA C, C T = T C, C 2 = id,
(1.7)
Congruence Subgroups and Generalized Frobenius-Schur Indicators
9
± ±1 ± 2 where pA = i∈ ωi d(Ui ) are called the Gauss sums of A. Note that pA are non-zero scalars and
+ − pA = d(Ui )2 = dim A. (1.8) pA i∈
By [Vaf88], ωi and the quotient
+ pA − pA
are roots of unity, and so T N = 1, where N = ord θ .
Recall that the modular group SL2 (Z) is the group generated by 0 −1 1 1 s= , t= with defining relations (st)3 = s2 and s4 = 1. 1 0 0 1 Therefore, the relations (1.7) imply that ρ A : SL2 (Z) → PGL(Kk (A)); s → S and t → T
(1.9)
defines a projective representation of SL2 (Z) on the Grothendieck algebra of A, where we identify the S and T matrices with automorphisms of Kk (A) using the latter’s canonical basis of simple objects. This projective representation will be called the projective modular representation of A. The projective representation (1.9) can be lifted to an ordinary representation λ,ζ
ρA : SL2 (Z) → GL(Kk (A)); s → s :=
1 1 S and t → t := T, λ ζ
(1.10)
by choosing scalars λ, ζ ∈ k such that λ2 = dim A and ζ 3 =
pA+ λ
.
(1.11)
It follows from (1.7) that (st)3 = s 2 , s 2 = C, and s 4 = 1.
(1.12)
The following well-known properties of the matrix s = [si j ] will be used frequently (cf. [BK01, Chapter 3]) : for i, j ∈ , s0i = d(Ui )/λ, si j = s ji = si j , and s −1 = [si j ] .
(1.13)
By Müger’s results [Müg03], the center of a spherical fusion category is a modular fusion category, whose Gauss sums are p ± Z (C ) = dim C. In this case, the projective representation (1.9) for A = Z (C) can be lifted in a canonical way to an ordinary representation by choosing λ = dim C and ζ = 1, which satisfy (1.11). We will call s = dim1 C S the normalized S-matrix of Z (C). This ordinary representation ρ Z (C ) : SL2 (Z) → GL(Kk (Z (C))); s →
1 S and t → T dim C
is called the canonical modular representation of Z (C). The center of a modular fusion category A = (A, c, θ ) can be described explicitly as follows: Let {Ui | i ∈ } be a complete set of non-isomorphic simple objects. Then by [Müg03], {(Ui ⊗ U j , σUi ⊗U j ) | i, j ∈ } is a complete set of non-isomorphic simple
10
S.-H. Ng, P. Schauenburg
objects of Z (A), where the half-braiding σUi ⊗U j is defined in (1.2). In other words, we have isomorphisms K0 (A) ⊗ K0 (A) → K0 (Z (A)), Kk (A) ⊗ Kk (A) → Kk (Z (A)); Z
(1.14)
[Ui ] ⊗ [U j ] → [Ui j ] := [(Ui ⊗ U j , σUi ⊗U j )]. Note that (1.3) implies Ui∨j ∼ = Ui j .
(1.15)
2. Generalized Frobenius-Schur Indicators In this section, we introduce the definition of generalized Frobenius-Schur indicators for each object in a pivotal category over the field k, and we derive some properties of these indicators from the definition. Let C be a pivotal category over k. For V ∈ C and m ∈ Z we define V m ∈ C by setting V 0 = I and V m = V ⊗ V m−1 if m > 0, and V m := (V ∨ )−m for m < 0. Duality (−)∨ is a contravariant monoidal functor with respect to a canonical isomorphism ξ : Y ∨ ⊗ X ∨ → (X ⊗ Y )∨ coherent with the associativity isomorphisms in C. For a non-negative integer m and V ∈ C, there exists, by the coherence theorem, a unique isomorphism tm : V −m → (V m )∨ which is a composition of instances of tensor products of id, ±1 and ξ ±1 . Combining with the pivotal structure j of C, we can extend the definition to negative m as follows: tm := V
−m
j
− → (V
−−→ (V ) .
∨ −m ∨∨ t−m
)
m ∨
Using tm we define, for any m ∈ Z: tm ⊗id ev evm := V −m ⊗ V m −−−→ (V m )∨ ⊗ V m − →I and dbm :=
db
I − →V
m
−1 m ∨ id ⊗tm
⊗ (V ) −−−−→ V
m
⊗V
−m
.
Note that if C is a strict pivotal category, then tm is the identity, evm = ev V m , and dbm = dbV m . Next, for any m, l ∈ Z, there is a canonical morphism Jm,l (V ) : V −l ⊗(V m ⊗ V l ) → V m defined using only evaluation and coherence. More precisely, ⎧ evl ⊗ id ? ⎨ −l V ⊗ (V m ⊗ V l ) −→ (V −l ⊗ V l ) ⊗ V m −−−−→ V m if ml ≥ 0, Jm,l (V ) := id ⊗ evl ? ⎩ −l V ⊗ (V m ⊗ V l ) −→ V m ⊗ (V −l ⊗ V l ) −−−−→ V m if ml ≤ 0. Note that there is no difference between these two expressions for Jm,l (V ) if ml = 0. We write Jm,l for Jm,l (V ) when the context is clear.
Congruence Subgroups and Generalized Frobenius-Schur Indicators
11
Now for X = (X, σ X ) ∈ Z (C), V ∈ C, and l ∈ Z set X ⊗db−l −1 DX,l := X −−−−→ X ⊗ (V −l ⊗ V l ) −−→ (X ⊗ V −l ) ⊗ V l σ X ⊗V l −−−−→ (V −l ⊗ X ) ⊗ V l − → V −l ⊗ (X ⊗ V l ) . Finally for m, l ∈ Z, X = (X, σ X ) ∈ Z (C), V ∈ C, we define the k-linear map (m,l) E X,V : C(X, V m ) → C(X, V m ) by DX,l Jm,l (V ) V −l ⊗ f ⊗V l (m,l) E X,V ( f ) := X −−→ V −l ⊗ (X ⊗ V l ) −−−−−−−→ V −l ⊗ (V m ⊗ V l ) −−−−→ V m (m)
(m,1)
for f ∈ C(X, V m ). It will sometimes be convenient to write E X,V := E X,V for m > 0 (0,1)
and FX,V := E X,V . Definition 2.1. Let C be a pivotal category over k. For X ∈ Z (C), V ∈ C and (m, l) ∈ Z × Z, we define the generalized Frobenius-Schur (GFS) indicator (m,l) X νm,l (2.1) (V ) = Tr E X,V . Remark 2.2. Let F : C → D be an equivalence of monoidal categories, with the induced equivalence Fˆ : Z (C) → Z (D) of braided monoidal categories. Consider V ∈ C and (m,l) X ∈ Z (C). Similar to the reasoning in [NS07b, Sect. 4], the endomorphisms E X,V
of C(X, V m ) and E (m,l) of D(F(X ), F(V )m ) are conjugate. More precisely, by Fˆ (X),F (V ) the general coherence results for monoidal functors there is a unique isomorphism ξ ? : F(V m ) → F(V )m composed from instances of the monoidal functor structures of F, and (if m < 0) the canonical isomorphism F(V ∨ ) → F(V )∨ . For f : X → V m we then have (m,l) (m,l) (ξ ? ◦ F( f )). ξ ? ◦ F E X,V ( f ) = E ˆ F (X),F (V )
As a consequence, monoidal category equivalences preserve generalized indicators: Fˆ (X)
X νm,l (F(V )) = νm,l (V ).
In particular, to deal with the theory of generalized indicators it will be sufficient to treat the case where the category C is strict pivotal. Remark 2.3. Assuming the pivotal category C is strict, we have the following diagrams in the graphical calculus: X
(m,l) E X,V (f) = V
−l
X Vl f
V −l
=
f
Jm,l (V )
Jm,l (V )
Vm
Vm
Vl
for f ∈ C(X, V m ),
(2.2)
12
S.-H. Ng, P. Schauenburg
where V −l V m Jm,l (V )
Vl
V −l
Vl
is
Vm
if ml ≥ 0, but equals
Vm
V −l
Vl
if ml ≤ 0.
Vm
Moreover, since Jm,l (V ∨ ) = J−m,−l (V ), (2.2) immediately implies (m,l)
(−m,−l)
E X,V = E X,V ∨
(2.3)
for V ∈ C, X ∈ Z (C), and m, l ∈ Z. (m,l)
Remark 2.4. It is immediate from the definition or (2.2) that E X,V is natural in X ∈ (m,l)
Z (C), i.e. for morphisms g : X → Y in Z (C) and f : Y → V in C we have E X,V ( f g) =
(m,l) X (V ) is additive in its paramE Y,V ( f )g. As a consequence, the generalized indicator νm,l eter X, that is X⊕Y X Y νm,l (V ) = νm,l (V ) + νm,l (V )
for X, Y ∈ Z (C), V ∈ C and (m, l) ∈ Z × Z. Lemma 2.5. Consider a pivotal monoidal category C. Then we have (m,k+l)
E X,V
(m,k)
(m,l)
= E X,V E X,V
(2.4)
for all V ∈ C, X ∈ Z (C), and m, k, l ∈ Z such that kl ≥ 0 or m = 0. In particular (m,l) (m,1) l E X,V = E X,V if m = 0. Proof. We can assume that C is strict pivotal. By (2.3) we may assume that m ≥ 0. If k, l ≥ 0, then V −k V −l V m
Vl
Vk
=
Jm,l
V −k V −l V l V m
Vm
Vk
=
V −k V −l V l V k V m
= Jm,k+l .
Jm,k
Jm,k
Vm
Vm
Vm
The same conclusion Jm,k (V ) ◦ (id V −k ⊗Jm,l (V ) ⊗ id V k ) = Jm,k+l (V ) holds for k, l ≤ 0 by a calculation which is the mirror image of that above. Thus, whenever kl ≥ 0, we have X X
(m,k)
(m,l)
E X,V E X,V ( f ) =
f
=
f
Jm,l
Jm,k+l
Jm,k
Vm
Vm
(m,k+l)
= E X,V
( f ).
Congruence Subgroups and Generalized Frobenius-Schur Indicators
13
l m,±l m,±1 Note that (2.4) for k, l of the same sign implies E X,V = E X,V for all non-negative integers l. Thus, to prove (2.4) for m = 0 and arbitrary k, l, it suffices to show that m,±1 E X,V are mutually inverse. It suffices to assume m > 0. Then X
X
(m,1) E X,V (f) =
,
f
V m−1
V∨
(m,−1) E X,V (f) =
(2.5)
f
V V m−1
V
and they are inverse of each other.
Remark 2.6. In particular, for m > 0 and X equal to the unit object of Z (C), the GFS X (V ) coincides with the (m, l)th Frobenius-Schur indicator ν indicator νm,l m,l (V ) defined in [NS07b]. Lemma 2.7. Consider a pivotal monoidal category C. Then we have (m,m+l)
E X,V
(m,l)
( f ) = E X,V ( f )θX−1
(2.6)
for all V ∈ C, X ∈ Z (C), and m, l ∈ Z with m = 0 and f ∈ C(X, V m ), where θ is the twist on Z (C) associated with the pivotal structure on C. In addition, if θXN = idX for (m,m N ) some positive integer N , then E X,V = id. Proof. We may assume that C is strict. By Lemma 2.5 it is enough to treat the case l = 0, (m,m) i.e. to show E X,V ( f ) = f ◦ θX−1 . But indeed Jm,m (V ) = evm ⊗V m , and thus X
X
(m,m)
E X,V ( f ) =
=
f
Vm
X
=
f∨
Vm
f
= f ◦ θX−1 .
Vm
If, in addition, θXN = idX , then (m,m N )
E X,V for all f ∈ C(X, V m ).
(m,m) N ( f ) = E X,V ( f ) = f ◦ θX−N = f
Proposition 2.8. Let C be a pivotal category over k, V ∈ C, X ∈ Z (C), and (m, l) ∈ Z × Z. Then X (I ) = dim C(X, I ). (i) νm,l X (V q ) = ν X (ii) νm,l qm,ql (V ) for q ∈ Z. X X (V ), where (V ) = ω−1 νm,l (iii) If X is (absolutely) simple and m > 0, then νm,m+l ω ∈ k is given by θX = ω idX . Moreover, ω = 1 or C(X, I ) = 0.
14
S.-H. Ng, P. Schauenburg
Proof. We can assume that C is strict. (m,l)
(i) For V = I the morphism Jm,l (V ) is the identity, and so is E X,V . (ii) From the graphical representation of E X,V displayed in Remark 2.3, it is straight(mq,lq) (m,l) forward to read off that E X,V . q = E X,V (iii) The first statement is a direct consequence of the preceding lemma. The second then follows by setting V = I and using (i), since we have assumed k to have characteristic zero. Without that assumption we could still set V = I and find (1,1)
(1,0)
idC (X,I ) = E X,I = ω−1 E X,I = ω−1 idC (X,I ) , and so ω = 1 or C(X, I ) = 0.
Lemma 2.9. Let C be a pivotal category over k, V ∈ C, X ∈ Z (C), and m, l ∈ Z. Then, for all f ∈ C(X, V m ) and g ∈ C(V m , X ), we have ⎛ m −l ⎞ V
V
⎜ ⎜ ⎜ (m,l) (g, E X,V ( f ))r = ptrr ⎜ ⎜ ⎜ ⎝
⎟ ⎟ ⎟ ⎟ if lm ≤ 0 , ⎟ ⎟ ⎠
g
f V −l V m
⎛
Vm
Vl
⎜ ⎜ ⎜ (m,l) ⎜ (g, E X,V ( f )) = ptr ⎜ ⎜ ⎝
g
f Vm
⎞ ⎟ ⎟ ⎟ ⎟ if lm ≥ 0. ⎟ ⎟ ⎠
Vl
Proof. We may assume that C is strict pivotal. We treat the case ml ≥ 0 first. Note that in this case m m −m −m V
V −l V
Jm,l (V )
Vl
V −l V l
V
V
=
= evm+l .
Therefore ⎛
db−m−l g (m,l)
g
(m,l)
(g, E X,V ( f )) = ptr (E X,V ( f )g) =
f
= f
Jm,l (V )
⎜ ⎜ ⎜ ⎜ = ptr ⎜ ⎜ ⎝
The proof in the case lm ≤ 0 is similar but using
Jm,l (V )
Vl
Vm g
f Vm
evm+l V −l V m
Vl
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
Vl
V −m
= evl−m this time.
Congruence Subgroups and Generalized Frobenius-Schur Indicators
15
Lemma 2.10. Assume that C is a strict spherical monoidal category. Then for V ∈ C, X = (X, σ X ) ∈ Z (C), m, l ∈ Z, f ∈ C(X, V m ) and g ∈ C(V m , X ) we have: (m,l)
(m,l)
ptr(E X,V ( f )g) = ptr( f E X∨ ,V ∨ (g ∨ )∨ ). Proof. (m,l) ptr(g E X,V ( f )) = ptr((V −l ⊗ f )σ X,V −l (g ⊗ V −l )) = ptr ((V −l ⊗ f )σ X,V −l (g ⊗ V −l ))∨ = ptr (V l ⊗ g ∨ )σ X ∨ ,V l ( f ∨ ⊗ V l ) = ptr ((V ∨ )−l ⊗ g ∨ )σ X ∨ ,(V ∨ )−l ( f ∨ ⊗ (V ∨ )−l ) ∨ = ptr( f ∨ E X(m,l) ∨ ,V ∨ (g )) (m,l)
= ptr(E X∨ ,V ∨ (g ∨ )∨ f ). Remark 2.11. It is worthwhile to rewrite the last lemma slightly in the context of a strict spherical category over k. We define (m,l)
(m,l)
E X,V : C(V m , X ) → C(V m , X ); f → E X∨ ,V ∨ ( f ∨ )∨ .
(2.7)
Thus, the definition of the E maps is obtained by turning that of the E maps upside down; we will return to this aspect later. Then the above lemma says that (m,l) (m,l) E X,V ( f ), g = f, E X,V (g) . (2.8) (m,l)
(m,l)
Note also that by definition the linear map E X,V is conjugate to E X∨ ,V ∨ , so that (m,l) X∨ (2.9) νm,l (V ∨ ) = Tr E X,V . 3. The Case of a Semisimple Pivotal Category In this section, we continue to study the GFS indicators for semisimple pivotal categories over k. In such a category, Lemma 2.9 allows us to express the GFS indicators of V as the pivotal traces of certain endomorphisms of tensor powers of V in the category. In the case where the category is spherical, we will obtain additional properties as well as another expression for the indicators in terms of pivotal traces of certain endomorphisms in the center Z (C). The latter expression will be used in the following section to compare our indicators to those defined by Sommerhäuser and Zhu in the Hopf algebra case. Let C be a semisimple pivotal category over k. Recall that the pairings (·, ·) , (·, ·)r defined in (1.4) are always non-degenerate in the semisimple case. Suppose { pα } is a basis for C(V, W ). Then the non-degenerate pairing (·, ·) defines a dual basis { p α } for C(W, V ), where = or r . The two bases { p α }, { prα } may not be the same. However, when V or W = I , these two bases are identical because dr (I ) = d (I ) = 1. In addition, if C is spherical these two bases are always identical, and we will simply write { p α } for this dual basis in this case.
16
S.-H. Ng, P. Schauenburg
Proposition 3.1. Let C be a semisimple strict pivotal category over k, and let V ∈ C, X = (X, σ X ) ∈ Z (C) and (m, l) ∈ Z2 . Suppose { pα }α is a basis for C(V m , X ). Then ⎛ m −l ⎞ ⎧ V V ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎜ pα ⎟ ⎪ ⎪
⎜ ⎟ ⎪ r ⎪ ⎜ ⎟ if ml ≤ 0, ⎪ ptr ⎜ ⎪ ⎟ ⎪ ⎪ ⎜ ⎟ ⎪ α prα ⎠ ⎪ ⎝ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Vm V −l X (3.1) νm,l (V ) = ⎛ Vl Vm⎞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ pα ⎟ ⎪ ⎜ ⎪
⎪ ⎜ ⎟ ⎪ ⎪ ⎟ ⎪ if ml ≥ 0. ptr ⎜ ⎪ ⎜ ⎟ ⎪ ⎪ α ⎜ p ⎟ ⎪ ⎪ ⎝ α ⎠ ⎪ ⎪ ⎩ Vm Vl
Proof. This is a direct consequence of the definition of the indicators and Lemma 2.9. Remark 3.2. One may also see that ⎛ Vl X νm,l (V ) =
α
⎜ ⎜ ⎜ ptr ⎜ ⎜ ⎜ ⎝
Vm
⎞
⎛V l
⎟ ⎜ ⎜ ⎟ ⎜ ⎟= ⎜ ptr ⎟ ⎜ ⎟ ⎜ α ⎠ ⎝
pα ⎟ p α
Vm Vl Vm
⎛V l =
α
⎜ ⎜ ⎜ ⎜ ptr ⎜ ⎜ ⎝
Vm
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
pα ⎟ p α Vm Vl
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
pα p α Vm
⎞
Vl
for ml ≥ 0. Proposition 3.3. Let C be a semisimple spherical category over k. For V ∈ C and X ∈ Z (C), we have ∨
X X X ν−m,−l (V ) = νm,l (V ∨ ) = νm,l (V )
for all (m, l) ∈ Z × Z. Proof. The first equality follows immediately from Proposition 2.8 (ii) by setting q = (m,l) (m,l) −1. In the semisimple case (2.8) says that E X,V and E X,V are adjoint maps and have the same trace, so that, by (2.9), (m,l) (m,l) X X∨ νm,l (V ∨ ) = Tr E X∨ ,V = Tr E X∨ ,V = νm,l (V ).
Congruence Subgroups and Generalized Frobenius-Schur Indicators
17
Next, we will derive an expression for the GFS indicators as the pivotal traces of certain endomorphisms in Z (C). For this we will assume that the category C is spherical. We will use the two-sided adjoint K : C → Z (C) to the forgetful functor with the conventions at the end of Sect. 1.3. Associated with the adjunction , we define (m,l) −1
X,V m
X,V m
E X,V
(m,l) ϕX,V := D(X, K (V m )) −−−−→ C(X, V m ) −−−→ C(X, V m ) −−−−→ D(X, K (V m ))
(3.2) (m,l)
for m, l ∈ Z, where D simply denotes the center Z (C). Obviously, ϕX,V is natural in X. By Yoneda’s lemma, (m,l) ϕX,V ( f ) = κV(m,l) ◦ f
(3.3)
for f ∈ D(X, K (V m )), where (m,l)
κV
(m,l)
:= ϕ K (V m ),V (id) : K (V m ) → K (V m ).
Note that for f ∈ D(X, K (V m )) and g ∈ D(K (V m ), X) we have, abbreviating (m,l) κ = κV , −1 (gκ, f ) = ptr(gκ f ) = (g, κ f ) = g, −1 E ( f ) = E (g), f , (m,l)
where W,X : C(W, X ) → D(K (W ), X) and E X,V : C(V m , X ) → C(V m , X ) are (m,l) respectively the adjoint maps of X,W and E X,V with respect to the bilinear form (·, ·)
described in (1.5) and (2.8). Thus, if we define ϕ (m,l) V,X to be the composition: (m,l) ϕ V,X :=
(m,l)
−1
E X,V
D(K (V ), X) −−→ C(V , X ) −−−→ C(V , X ) − → D(K (V ), X) m
m
m
m
, (3.4)
then by the non-degeneracy of the pairing (·, ·) we have shown (m,l)
(m,l)
ϕ V,X (g) = g ◦ κV (m,l)
(3.5)
(m,l)
= ϕ V,K (V m ) (id K (V m ) ). for all g ∈ D(K (V m ), X). In particular, κV The morphisms κ defined above can be used to compute the GFS indicators with the following theorem. Theorem 3.4. Let C be a spherical fusion category over k, and X a simple object of D := Z (C). For m, l ∈ Z, we have X νm,l (V ) =
1 ptr κV(m,l) ◦ z X , d(X)
where z X is the natural projection of K (V m ) onto the isotypic component of X.
18
S.-H. Ng, P. Schauenburg
Proof. It follows from (3.2) that (m,l) (m,l) X νm,l (V ) = Tr E X,V = Tr ϕX,V
(3.6)
for all l, m ∈ Z. Let { f α }α be a basis for D(X, K (V m )) and { f α }α the dual basis for D(K (V m ), X) with respect to the pairing (·, ·). Then f α ◦ fα =
δα,α idX d(X)
and z X := d(X)
α
fα ◦ f α
is the idempotent corresponding to the isotypic component of X in K (V m ). Let us write (m,l) κ for κV . Then, by (3.3), (m,l)
X νm,l (V ) = Tr(ϕX,V ) =
=
α
(m,l) ( f α , ϕX,V ( f α )) = ( f α , κ fα ) α
ptr( f α κ f α ) = ptr(κ
α
α
fα f α ) =
1 ptr(κ ◦ z X ). d(X)
(m,l)
(m,l)
Since ϕX,V is conjugate to E X,V , Lemma 2.5 implies analogous rules for the ϕ maps, as well as for the κ morphisms. More precisely, we have (m,k+l)
κV
(m,k) (m,l) κV
= κV (m,1)
for kl ≥ 0 or m = 0. We write βV m := κV (m,l)
βVl m = κV
(3.7) (0,1)
for m = 0 and γV = κV (0,l)
, γVl = κV
. Then we have
for all l ≥ 0.
In view of Proposition 3.3 and Theorem 3.4, the GFS indicator for spherical fusion categories can be summarized in terms of β and γ : l ⎧ 1 for m > 0, ⎪ d(X) ptr βV m ◦ z X ⎪ ⎨ 1 X for m = 0 and l ≥ 0, ptr γVl ◦ z X (3.8) νm,l (V ) = d(X) ⎪ ⎪ ⎩ X∨ ν−m,−l (V ) for otherwise. X (V ) Thus, the values of the GFS indicators are completely determined by those values νm,l with m ≥ 0. This characterization will be useful in the following section.
4. Equivariant Frobenius-Schur Indicators for Semisimple Hopf Algebras We will use the results in the preceding section to compare our generalized indicators with the equivariant indicators defined by Sommerhäuser and Zhu [SZ]. Let C = H -modfin for a semisimple Hopf algebra H over k. We follow the conventions for the Drinfeld double D(H ) of H described in [Kas95] and [Mon93]. As a coalgebra, D(H ) = (H ∗ )cop ⊗ H . We abbreviate the element p ⊗ k in D(H ) as pk and
Congruence Subgroups and Generalized Frobenius-Schur Indicators
19
simply write p for p1 H and k for 1 H ∗ k. Recall that the multiplication in D(H ) is given by
pq(S(k3 )?k1 ) ⊗ k2 h, pk · qh = (k)
where (k) k1 ⊗ k2 ⊗ k3 is the Sweedler notation for ( ⊗ id)(k), and q(S(k3 )?k1 ) denotes the linear functional a → q(S(k3 )ak1 ) on H . The center Z (C) of H -modfin is equal to D(H )-modfin as a rigid monoidal category. For X ∈ D(H )-modfin , the half-braiding σ X (V ) : X ⊗ V → V ⊗ X for V ∈ C is given by
σ X (V )(x ⊗ v) := h i v ⊗ S ∗ (h i )x, i
where S denotes the antipode of H , {h i } is a basis for H and {h i } its dual basis for H ∗ . Note that S 2 = id H (cf. [LR87,LR88]). The Drinfeld isomorphism u X ∈ End D(H ) (X) is given by
u X (x) = hi hi x = hi hi x i
i
for all x ∈ X. The induction functor K (−) = D(H ) ⊗ H − is left adjoint to the forgetful functor D(H )-modfin → H -modfin with the adjunction isomorphisms V,X : Hom H (V, X ) → −1 Hom D(H ) (K (V ), X) and V,X : Hom D(H ) (K (V ), X) → Hom H (V, X ) given by
( f )( p ⊗ v) = p f (v) and
−1
(g)(v) = g(1 D(H ) ⊗ v)
for v ∈ V , p ∈ H ∗ . Note that D(H )⊗ H V is naturally isomorphic to H ∗ ⊗ V as k-linear spaces. Every element is a linear combination of the tensor products p⊗v ∈ D(H )⊗ H V with p ∈ H ∗ and v ∈ V . (m,l) As we have mentioned following (2.7), the definition of E X,V can be obtained in (m,l) the graphical calculus by turning the definition of E X,V upside down. Explicitly, this gives, for m > 0: V V m−1 (m,1)
E X,V (g) =
(0,1)
g
and E X,V (g) =
V
g
. X
X
Thus, for V ∈ C, X ∈ Z (C), m > 0, f ∈ Hom H (V m , X ) and g ∈ Hom H (k, X), it is straightforward to verify that (m,1)
E X,V ( f )(v ⊗ w) =
i, j
(h i v j )(v)S ∗ (h i ) f (w ⊗ v j ) =
i
h i f (w ⊗ h i v)
20
S.-H. Ng, P. Schauenburg
for v ∈ V and w ∈ V m−1 , and (0,1)
E X,V (g)(1k ) =
(h i v j )(v j )S ∗ (h i )g(1k ) = v j (h i v j )h i g(1k ) = χV g(1k ), i, j
i, j
where χV denotes the character afforded by V , {v j } a basis for V and {v j } its dual basis. Therefore, for p ∈ H ∗ , v ∈ V and w ∈ V m−1 , βV m ( p ⊗ (v ⊗ w)) =
ph i ⊗ (w ⊗ h i v) and γV ( p ⊗ 1k ) = pχV ⊗ 1k . (4.1)
i
The above formula for βV m is identical to the map βV,V m−1 defined in [SZ]. Let ρm : D(H ) → End(D(H ) ⊗ H V m ) be the corresponding representation of the D(H )th module D(H ) ⊗ H V m and z an element in the
center of D(H ). The (m, l) equivariant Frobenius-Schur indicator of V and z = i pi ki is defined in [SZ] as ⎧ Tr(βVl m ◦ ρm (z)) ⎪ ⎪ ⎪ ⎨
(ki )( pi χV l )() I VSZ ((m, l), z) := dim H ⎪ ⎪ i ⎪ ⎩ SZ I V ((−m, −l), S D (z))
if m > 0, if m = 0 and l ≥ 0,
(4.2)
otherwise,
where S D is the antipode of D(H ), the counit of H , χV l the character of H afforded by V l , and the normalized integral of H , i.e. the integral of H satisfying () = 1. The following corollary highlights the relationship between our GFS indicators and the equivariant FS indicators defined for semisimple Hopf algebras. Corollary 4.1. Let C = H -modfin for some finite-dimensional semisimple Hopf algebra H over k. For simple X ∈ D(H )-modfin , V ∈ H -modfin and (m, l) ∈ Z × Z, X νm,l (V ) =
1 I SZ ((m, l), eX ), dim X V
where eX is the central idempotent of D(H ) associated with X. Proof. We first consider m > 0. Since eX is the central idempotent of D(H ) associated with the simple D(H )-module X, ρm (eX ) is the central idempotent z X of End D(H ) (D(H ) ⊗ H V m ) corresponding to the isotypic component of X in D(H ) ⊗ H V m . The pivotal traces in H -modfin as well as D(H )-modfin are identical to the ordinary trace of linear operators. Therefore, by Theorem 3.4 or (3.8), 1 1 1 Tr(βVl m ◦ z X ) = Tr(βVl m ◦ ρm (eX )) = I SZ ((m, l), eX ). dim X dim X dim X V
Let eX = i pi ki for some pi ∈ H ∗ and ki ∈ H , {h j } a basis for H and {h j } its dual basis for H ∗ . Then {h j ⊗ 1k } is a basis for D(H ) ⊗ H k and X (V ) = νm,l
γV l ◦ ρ0 (eX )(h j ⊗ 1k ) = γV l (eX h j ⊗ 1k ) = γV l (h j eX ⊗ 1k ) =
i
(ki )h j pi χV l ⊗ 1k .
Congruence Subgroups and Generalized Frobenius-Schur Indicators
21
Let be the normalized integral, and χ H the regular character of H . Then χ H is a twosided integral of H ∗ and χ H () = 1 (cf. [LR88, Theorem 4.4]). By [Rad94, Proposition 2],
(ki )(h j pi χV l )(h j ) = χ H (1) (ki ) pi χV l (). Tr(γV l ◦ ρ0 (eX )) = i, j
i
It follows from (3.8) that X ν0,l (V ) =
1 dim H
Tr(γV l ◦ ρ0 (eX )) = (ki ) pi χV l () dim X dim X i
1 I SZ ((0, l), eX ) = dim X V for l ≥ 0. Thus, if (i) m < 0, or (ii) m = 0 and l < 0, then, by Proposition 3.3, we find ∨
X X νm,l (V ) = ν−m,−l (V ) =
1 1 I SZ ((−m, −l), eX∨ ) = I SZ ((m, l), eX ). dim X V dim X V
The last equality follows from the fact S D (eX ) = eX∨ and the definition of equivariant FS-indicators illustrated in (4.2). 5. SL2 (Z)-Equivariant Indicators for Spherical Fusion Categories Given a pair (m, l) of integers, and an object V in a pivotal fusion category, the values X (V ) for X ∈ Z (C) can be extended to a functional on the of the GFS indicators νm,l Grothendieck algebra Kk (Z (C)) = K0 (Z (C)) ⊗Z k. In this section, these functionals are introduced as the equivariant indicators, and studied in detail for a spherical fusion category C. In this case, the center Z (C) is a modular tensor category, and so Kk (Z (C)) admits a natural representation of SL2 (Z) as described in Sect. 1.4. We show that the set of all equivariant indicators for a spherical fusion category C is closed under the contragredient action of SL2 (Z) on Kk (Z (C))∗ , and this action on the equivariant indicators is compatible with the action of SL2 (Z) on Z2 . This property of equivariant indicators X (V ) in V , and that its values lie in the implies the additivity of the GFS indicator νm,l cyclotomic field Q N , where N is the Frobenius-Schur exponent of C. Moreover, a formula for the GFS indicators for spherical fusion categories is obtained in Corollary 5.6 as a consequence. This formula implies the FS indicator formula discovered in [NS07a, Theorem 4.1]. Throughout the section, we consider a spherical fusion category C, and we let ˆ with X j = (X j , σ X j ) be the set of isomorphism classes of the simple {[X j ]| j ∈ } objects in D := Z (C). The equivariant indicators for C are defined as follows. Definition 5.1. For m, l ∈ Z, the (m, l)th equivariant indicator of V ∈ C is defined as the functional I V ((m, l), −) ∈ (Kk (D))∗ determined by the assignment X I V ((m, l), [X]) := νm,l (V )
for X ∈ Z (C); this is well-defined in view of remark 2.4.
22
S.-H. Ng, P. Schauenburg
Remark 5.2. (i) Definition 5.1 obviously makes sense for pivotal fusion categories. However, there is no natural modular action on the Grothendieck algebras of the centers of these categories, so we will reserve the term for the spherical case. (ii) For C = H -modfin for some semisimple Hopf algebra H over k, it follows from Corollary 4.1 that I V ((m, l), z) = I VSZ ((m, l), ψ(z)), where ψ : Kk (Z (C)) → Center(D(H )) is the k-linear isomorphism given by ψ([X]) = dim1 X eX for every simple D(H )-module X. Therefore, the equivariant indicator defined in Definition 5.1 is a generalization of the corresponding notion introduced by Sommerhäuser and Zhu in [SZ] for spherical fusion categories. Recall from Sect. 1.4 the canonical modular representation ρ Z (C ) : SL2 (Z) → GL(Kk (Z (C))) of Z (C). The association action of SL2 (Z) on Kk (Z (C)) is given by s[X j ] =
1
Si j [Xi ] and t[X j ] = ω j [X j ], dim C
(5.1)
i∈ˆ
where [Si j ]ˆ and [δi j ω j ]ˆ are the S and T -matrices of Z (C). For the convenience of the remaining discussion, we summarize some properties of the equivariant indicators in the following lemma. Lemma 5.3. Let C be a spherical fusion category over k. For z ∈ Kk (Z (C)), V ∈ C, and q, m, l ∈ Z, we have (i) I V ((−m, −l), z) = I V ∨ ((m, l), z) = I V ((m, l), s2 z), (ii) I V ((qm, ql), z) = I V q ((m, l), z). ˆ Therefore, the statements follow Proof. By (1.12), s2 [X j ] = [X j ] = [X∨j ] for j ∈ . immediately from Propositions 2.8, 3.3 and Definition 5.1. We define an outer automorphism σ of SL2 (Z) by 1 −1 σ (g) = jgj where j = 0
0 −1
(5.2)
for g ∈ SL2 (Z). We will write g˜ for σ (g) in the sequel. In particular, s˜ = s−1 and ˜t = t−1 , and so g˜˜ = g for all g ∈ SL2 (Z). If ρ : SL2 (Z) → GL(V ) is a representation, then we denote by ρ˜ its twist by the automorphism σ , i.e. ρ(g) ˜ := ρ(˜g) for all g ∈ SL2 (Z).
(5.3)
We proceed to show the SL2 (Z)-equivariance of the equivariant indicators as stated in the following theorem. Theorem 5.4. Let C be a spherical fusion category over k, V ∈ C and m, l ∈ Z. Then I V ((m, l)g, z) = I V ((m, l), g˜ z) for all g ∈ SL2 (Z) and z ∈ Kk (Z (C)).
Congruence Subgroups and Generalized Frobenius-Schur Indicators
23
Proof. Since SL2 (Z) is generated by s and t, it suffices to prove the equality holds for g = s, t. Let {[Vi ] | i ∈ } denote the set of isomorphism classes of simple objects in C. ˆ we let { p n }α be a basis for C(V n , Vi ), and {ιi,k,β }β a For any n ∈ Z, i ∈ and k ∈ , i,α n } n n basis for C(Vi , X k ). Then {ιi,k,β ◦ pi,α α,β,i is a basis for C(V , X k ). Let {qi,α }α be the n basis for C(Vi , V ), and {πi,k,β }β the basis for C(X k , Vi ) such that n n pi,α ◦ qi,α and πi,k,β ◦ ιi,k,β = δβ,β id Vi . = δα,α id Vi ,
1 n } q n ◦ πi,k,β }α,β,i forms a basis for C(X k , V n ) dual to {ιi,k,β ◦ pi,α Then { d(V α,β,i i ) i,α relative to the non-degenerate bilinear form (·, ·). (i) s-equivariance. We first consider the case ml ≥ 0. Then, by Proposition 3.1 and Remark 3.2,
⎛
Vl
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
1 ⎜ ptr ⎜ I V ((m, l), [Xk ]) = ⎜ d(V j ) ⎜ α,β, j ⎜ ⎜ ⎜ ⎜ ⎝
=
α,α β,i, j
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
p mj,α Vj ι j,k,β
π j,k,β Vj qm j,α
⎛
⎞
Vm
Vl
⎜ pl ⎜ i,α ⎜ ⎜ ⎜ ⎜ Vi ⎜ 1 ptr ⎜ ⎜ d(V j ) ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
Vm Vm
Vl
⎞
p mj,α Vj ι j,k,β
π j,k,β Vj qm j,α
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ Vi ⎟ ⎟ ⎟ ⎟ ⎟ l qi,α ⎠
Vm
Vl
Vi
Vj
It follows from [Müg03, Lemma 5.9] that Vi
Vj
ιi,k ,β
ι j,k,β
1 d(V j )
β
=
β,k
π j,k,β Vj
Vi
Skk d(Vi ) dim C
πi,k ,β Vj
Vi
24
S.-H. Ng, P. Schauenburg
ˆ where [Sab ] ˆ is the S-matrix of Z (C). Thus, we have for all i, j ∈ and k ∈ , ⎛
I V ((m, l), [Xk ]) =
α,α β,i, j,k
Vm
Vl
⎜ pl ⎜ i,α ⎜ Vi ⎜ ⎜ ιi,k ,β ⎜ ⎜ Skk ptr ⎜ ⎜ d(Vi ) dim C ⎜ ⎜ V j ⎜ ⎜ ⎜ qm ⎝ j,α
⎞
⎟ ⎟ ⎟ ⎟ Vj ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ πi,k ,β ⎟ ⎟ Vi ⎟ ⎟ l qi,α ⎠
Vm
p mj,α
Vl
By Proposition 3.1, the expression on the right hand side is equal to
Skk X
Skk k (V ) = νl,−m I V ((l, −m), [Xk ]) = I V ((l, −m), s[Xk ]) dim C dim C k
k
= I V ((m, l)s, s[Xk ]), where the second to last equality holds by definition of the SL2 (Z)-action on Kk (Z (C)) ˆ and hence (cf. (5.1)). Therefore, I V ((m, l)s, s[Xk ])) = I V ((m, l), [Xk ])) for all k ∈ , I V ((m, l), z)) = I V ((m, l)s, sz)) for all z ∈ Kk (Z (C)). Replacing z by s˜ z, we obtain I V ((m, l), s˜ z)) = I V ((m, l)s, s˜sz)) = I V ((m, l)s, z)) for all z ∈ Kk (Z (C)) and (m, l) ∈ Z2 with ml ≥ 0. If ml ≤ 0, then m(−l) ≥ 0 and (m, l) = (−l, m)s. In view of the preceding discussion, we find I V ((m, l), s˜ z) = I V ((−l, m)s, s˜ z) = I V ((−l, m), s2 z) = I V ((l, −m), z) = I V ((m, l)s, z). Here the second to last equality follows from Proposition 3.3, and this completes the proof of s-equivariance. (ii) t-equivariance. For k ∈ ˆ and m, l ∈ Z with m > 0, it follows from Proposition 2.8 (iii) that Xk I V ((m, l)t, [Xk ]) = νm,m+l (V ) = ωk−1 νm,l (V ) = I V ((m, l), t˜[Xk ]),
where [δab ωa ]ˆ is the T -matrix of Z (C). If m < 0, then, by Lemma 5.3, we also have I V ((m, l)t, [Xk ]) = I V ∨ ((−m, −l)t, [Xk ]) = I V ∨ ((−m, −l), t˜[Xk ]) = I V ((m, l), t˜[Xk ]). Therefore, I V ((m, l)t, z) = I V ((m, l), t˜z) for all z ∈ Kk (Z (C)) whenever m = 0. By Lemma 5.3 (ii), I V ((0, 0), z) = I I ((1, 0), z) = I I ((1, 1), z). Thus, I V ((0, 0), t˜z) = I I ((1, 0), t˜z) = I I ((1, 1), z) = I V ((0, 0), z).
Congruence Subgroups and Generalized Frobenius-Schur Indicators
25
Note that (˜s˜t)3 = s˜ 2 . Applying what we have just obtained, we find I V ((0, l), z) = I V ((−l, 0)s, z) = I V ((−l, 0), s˜ z) = I V ((−l, 0), t˜s˜ t˜s˜ t˜z) = I V ((−l, −l), s˜ ˜ts˜ ˜tz) = I V ((−l, l), t˜s˜ ˜tz) = I V ((−l, 0), s˜ ˜tz) = I V ((−l, 0)s, t˜z) = I V ((0, l), t˜z) for l ∈ Z. In conclusion, we have I V ((m, l)t, z) = I V ((m, l), t˜z) for all (m, l) ∈ Z2 , and the proof of t-equivariance is complete. The theorem implies that the space of equivariant indicators is closed under the contragredient action of SL2 (Z) on Kk (Z (C))∗ , and g · I V ((m, l), −) = I V ((m, l)˜g−1 , −) for all g ∈ SL2 (Z). One consequence of the SL2 (Z)-equivariance of the indicators is the additivity of X (−) for any pair of relatively prime integers (m, l) and X ∈ Z (C). A different proof νm,l will be given in Sect. 9. Corollary 5.5. Let C be a spherical fusion category over k, (m, l) a pair of relatively prime integers, and z ∈ Kk (Z (C)). Then I V ⊕W ((m, l), z) = I V ((m, l), z) + IW ((m, l), z) for V, W ∈ C. X (V ) = dim C(X, V ). Therefore, ν X (V ) Proof. For V ∈ C and simple X ∈ Z (C), ν1,0 1,0 is additive in V , and so is I V ((1, 0), z) for all z ∈ Kk (Z (C)). Since m, l are relatively prime, there exists g ∈ SL2 (Z) such that (m, l) = (1, 0)g. By Theorem 5.4, we find
I V ⊕W ((m, l), z) = I V ⊕W ((1, 0)g, z) = I V ⊕W ((1, 0), g˜ z) = I V ((1, 0), g˜ z) + IW ((1, 0), g˜ z) = I V ((1, 0)g, z) + IW ((1, 0)g, z) = I V ((m, l), z) + IW ((m, l), z). The equivariant indicators shed new light on the relationship between the higher indicators for the spherical fusion category C and the modular data of the Z (C). The following corollary which generalizes [NS07a, Theorem 4.1] is one of the examples. Corollary 5.6. Let C be a spherical fusion category over k, and Z (C) the center of C ˆ is a complete set of with the ribbon structure θ and braiding c. Suppose {Xi | i ∈ } non-isomorphic simple objects of Z (C) and [Si j ]i j∈ˆ , [δi j ω j ]i j∈ˆ are the corresponding S and T matrices. Then, for m ∈ Z, i ∈ ˆ and V ∈ C, we have 1 m ωk Sik N VX k dim C k 1 ptr c K (V ),Xi ◦ cXi ,K (V ) ◦ (idXi ⊗θ Km(V ) ) , = dim C
Xi νm,1 (V ) =
(5.4)
where N VX k = dim C(X k , V ) and K is a left adjoint to the forgetful functor from Z (C) to C. In particular, if N = ord θ , then ν N (V ) = d(V ) for all V ∈ C.
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S.-H. Ng, P. Schauenburg
Proof. By Theorem 5.4, we find Xi (V ) = I V ((m, 1), [Xi ]) = I V ((−m, −1), [Xi ]) = I V ((1, 0)t−m s, [Xi ]) νm,1
= I V ((1, 0), tm s−1 [Xi ]) = I V ((1, 0), tm s[Xi ]) 1 m = ωk Sik I V ((1, 0), [Xk ]). dim C k∈ˆ
It follows from the definition that Xk I V ((1, 0), [Xk ]) = ν1,0 (V ) = dim C(X k , V ) = N VX k ,
and so the first equality follows. Since K (V ) ∼ = ⎛ X∨ i
⎜ ⎜ ⎜ ⎜ ⎜ 1 ptr ⎜ ⎜ dim C ⎜ ⎜ ⎜ ⎝ Xi∨
K (V ) θm
k∈ˆ
N VX k Xk , we can conclude that
⎞
⎛ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜
⎟ ⎜ m Xk ⎟= 1 ω N ptr ⎜ k V ⎟ dim C ⎜ ⎟ ⎜ ˆ k∈ ⎟ ⎝ ⎟ ⎠
K (V )
Xi
Xk
Xi
Xk
⎞ ⎟ ⎟ ⎟ 1 m ⎟ ωk Sik N VX k . ⎟= ⎟ dim C k ⎟ ⎠
In view of [ENO05, Proposition 5.4] (or [NS07a, Proposition 4.5]), d(K (V )) = (dim C)d(V ) for all V ∈ C. Therefore, ν N (V ) = ν NX0,1 (V ) =
d(K (V )) = d(V ). dim C
Following [NS07a, Definition 5.1], the Frobenius-Schur exponent FSexp(C) of a pivotal category C over k with simple unit object is defined as the minimum of the set {n ∈ N | νn (V ) = d (V ) for all V ∈ C}. It has been proved in [NS07a, Theorem 5.5] that if C is a spherical fusion category over C, then FSexp(C) = ord θ , where θ is the ribbon structure of Z (C). Indeed, the theorem holds for any spherical fusion category over k. If C is a spherical fusion over k, then we learn immediately from Corollary 5.6 that FSexp(C) ≤ N , where N is the order of the ribbon structure θ of Z (C). Let Q N be the subfield of k obtained by adjoining a primitive N th root of unity in k to Q. For V ∈ C, ν N (V ) is an algebraic integer in Q N (cf. [NS07b]), and so is d(V ). Obviously, if ω ∈ k such that θX = ω idX for some simple X ∈ Z (C), then ω ∈ Q N . The subfield Q N of k can be identified with the N th cyclotomic field contained in C. Under this identification, and by [ENO05, Corollary 2.10], d(V ) is totally real for simple V ∈ C. Using the same proof of [NS07a, Theorem 5.5], we have FSexp(C) = N .
Congruence Subgroups and Generalized Frobenius-Schur Indicators
27
Proposition 5.7. Let C be a spherical fusion category over k with Frobenius-Schur expoˆ form a complete set of non-isomorphic simple objects of Z (C). nent N , and let Xi , i ∈ , Then 1 Xi (V ) ∈ Q N Si j , νm,l dim C for all m, l ∈ Z, i, j ∈ ˆ and V ∈ C, where S = [Si j ] denotes the S-matrix of Z (C). Proof. From the above remark, we find d(V ) ∈ Q N for all V ∈ C. Hence ˆ dim C and d(Xk ) ∈ Q N for k ∈ . ˆ By Let θ be the ribbon structure of Z (C) and ωk ∈ k such that θXk = ωk idXk for k ∈ . [BK01, 3.1.2], we also have Si j = ωi−1 ω−1 j
k∈ˆ
Nikj ωk d(Xk ) ∈ Q N .
(5.5)
Therefore, dim1 C Si j ∈ Q N . Note that s = dim1 C S is the normalized matrix of Z(C). Since s 4 = 1, s −1 is also a matrix over Q N . X (V ) = For any non-zero (m, l) ∈ Z2 , it follows from Proposition 2.8 that νm,l X q νm ,l (V ), where q = gcd(l, m), l = l/q and m = m/q. By Corollary 5.5, it suffices to show that νmX ,l (V ) ∈ Q N for all V ∈ C. Let g ∈ SL2 (Z) such that (m , l ) = (1, 0)g. By Theorem 5.4, X
νm j,l (V ) = I V ((1, 0)g, [X j ]) = I V ((1, 0), g˜ [X j ])
Xi gi j I V ((1, 0), [Xi ]) = gi j ν1,0 (V ) = gi j N VX i , = i∈ˆ
i∈ˆ
(5.6)
i∈ˆ
where [gi j ]ˆ = ρ Z (C ) (˜g). Since s, t generate SL2 (Z), [gi j ] is a product of the matrices ρ Z (C ) (˜s)±1 = s ∓1 and ρ(˜t)±1 = T ∓1 , where T = [δi j ωi ]ˆ is the T -matrix of Z (C). These matrices have been shown to be matrices over Q N , and so is [gi j ]ˆ . Therefore, the last term in (5.6) is an element of Q N . (m,1)
X (V ) is a cycloRemark 5.8. Since E X,V has finite order dividing m N for m > 0, νm,l tomic integer in Q N for m = 0. It has been shown in [CG94,dBG91] and [ENO05, Theorem 10.1] that dim1 C Si j ∈ Q(ξ ) for some root of unity ξ . The above proposition proves ξ can be chosen as a primitive N th root of unity for the modular tensor category λ,ζ Z (C). It will be developed in Theorem 7.1 that the image ρA (s) of a modular repreλ,ζ λ,ζ sentation ρA of a modular category A is a matrix over Qm , where m = ord(ρA (t)).
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S.-H. Ng, P. Schauenburg
6. The Center of a Modular Tensor Category and Congruence Subgroups In this section, we study the GFS indicators for a modular tensor category. We obtain a generalization of Bantay’s formula in Proposition 6.1, and our Congruence Subgroup Theorems 6.7 and 6.8. We prove in Theorem 6.8 that the kernel of the projective modular representation ρ A associated with a modular category A is a level N congruence subgroup of SL2 (Z), where N is the Frobenius-Schur exponent of A. In the case that A is the center Z (C) of some spherical fusion category C, we know more precisely that the kernel of the canonical modular representation ρ Z (C ) of Z (C) is a level N congruence subgroup of SL2 (Z). In fact this is proved first, in Theorem 6.7, and used in the proof of Theorem 6.8. An example for the congruence subgroup theorem is provided at the end this section. We begin with the discussion of the center of a modular category. Let A = (A, c, θ ) be a modular category over k with a complete set of nonisomorphic simple objects {Ui | i ∈ }, the S-matrix S = [Si j ] , and the T -matrix T = [δi j ωi ] . Without loss of generality, we may further assume that the underlying spherical fusion category of A is strict. Let Ui j = (Ui ⊗U j , σUi ⊗U j ), (i, j) ∈ × , be the complete set of simple objects of Z (A) described in Sect. 1.4. We have noted Ui∨j ∼ = Ui j in (1.15), and so the (i j, kl)-entry of the S-matrix S = [S i j,kl ] × of Z (A) is given by ⎛U U
S i j,kl
⎜ ⎜ ⎜ ⎜ = ptr ⎜ ⎜ ⎜ ⎜ ⎝
i
j Uk Ul
⎞
⎛ ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ = ptr ⎜Ui ⎟ ⎜ ⎟ ⎝ ⎟ ⎠
U j Uk
⎞ ⎟ ⎟ ⎟ = Sik S jl . ⎠
Ul ⎟
(6.1)
U j Uk
U j Ui Uk Ul
Since θUi = ωi idUi , we have the equalities Ui
Ui
ωi
and ωi−1
= Ui
Ui
Ui
Ui
= Ui
. Ui
The Ui j component of the ribbon structure of Z (A) is given by Ui U j
=
ωi idUi ⊗U j . ωj
(6.2)
Ui U j
Thus, the T -matrix of Z (A) is
ωi T = δi j,kl . ω j ×
(6.3)
Using Corollary 5.6, we can prove the following generalization of [NS07a, Theorem 7.5] which is also a further generalization of Bantay’s formula [Ban97] to GFS indicators.
Congruence Subgroups and Generalized Frobenius-Schur Indicators
29
Proposition 6.1. Let A be a modular category over k with a complete set of nonisomorphic simple objects {Ui | i ∈ }. Then, for Ui j = (Ui ⊗ U j , σUi ⊗U j ) ∈ Z (A), Ui j (Ua ) νm,1
ωk m 1 = Sik S jl Nkla , dim A ωl k,l∈
where Nkla = dim A(Uk ⊗ Ul , Ua ) and [Si j ] , [δi j ωi ] are the S and T -matrices of A respectively. Proof. The S and T -matrices of Z (A) have been shown in (6.1) and (6.3). By Corollary 5.6, we find
ωk m
ωk m 1 1 Ui j νm,1 (Ua ) = S i j,kl Nkla = Sik S jl Nkla . dim A ωl dim A ωl k,l∈
k,l∈
Note that the (2, 1)st indicator for the unit object U00 given by U00 ν2,1 (Ua )
ωk 2
ωk 2 1 a = S0k S0l Nkl = s0k s0l Nkla dim A ωl ωl k,l∈
k,l∈
is identical to Bantay’s indicator formula for RCFT introduced in [Ban97]. The representation ρ Z (A) is determined by the actions s[Ui j ] =
1
1
S kl,i j [Ukl ] = Ski Sl j [Ukl ], dim A dim A k,l
and t[Ukl ] =
k,l
ωk ωl−1 [Ukl ],
(6.4)
and it is isomorphic to a tensor product of two representations as described in the following lemma. Lemma 6.2. Let A be a modular category over k with a complete set of non-isomorphic λ,ζ simple objects {Ui | i ∈ }, and let ρ denote the representation ρA for some λ, ζ ∈ k + 2 3 such that λ = dim A and ζ = pA /λ described in (1.10). Then: (i) The k-linear isomorphism φ : Kk (A) ⊗ Kk (A) → Kk (Z (A)), [Ui ] ⊗ [U j ] → [Ui j ] from (1.14) defines an isomorphism ρ ⊗ ρ˜ → ρ Z (A) of representations of SL2 (Z). (ii) The bilinear form ·, · : Kk (A) ⊗ Kk (A) → k defined by [Ui ], [U j ] = δi j is SL2 (Z)-invariant under the representation ρ ⊗ ρ. ˜ Proof. Note that the SL2 (Z)-action on Kk (A) associated with the representation ρ is given by s[U j ] =
ωj 1
Si j [Ui ], and t[U j ] = [U j ], λ ζ i∈
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S.-H. Ng, P. Schauenburg
where [Si j ] and [δi j ω j ] are the S and T matrices of A. Note that Sik = Ski and S jl = S jl . By (1.13) the representation ρ ⊗ ρ˜ satisfies s([Ui ] ⊗ [U j ]) = s[Ui ] ⊗ s˜ [U j ] = s[Ui ] ⊗ s−1 [U j ] 1
= 2 Ski Sl j [Uk ] ⊗ [Ul ] λ
(6.5)
k,l∈
and also t([Ui ] ⊗ [U j ]) = t[Ui ] ⊗ ˜t[U j ] = t[Ui ] ⊗ t−1 [U j ] =
ωi ζ
ωj ζ
−1
[Ui ] ⊗ [U j ]. (6.6)
Comparing with (6.4) we see that φ satisfies φ(g([Ui ] ⊗ [U j ])) = gφ([Ui ] ⊗ [U j ]) for g ∈ {s, t} which implies that φ is SL2 (Z)-equivariant. Now apply the bilinear form ·, · to the rightmost expressions in (6.5) and (6.6). It follows from (1.13) that they are both equal to δi j , and this proves the second statement. Definition 6.3. The kernel (n) of the natural group homomorphism SL2 (Z) → SL2 (Z/nZ) is called the principal congruence subgroup of level n. A finite index subgroup G of SL2 (Z) is called a congruence subgroup if G contains a principal congruence subgroup of SL2 (Z). If n is the least positive integer such that (n) ⊆ G, then G is called a congruence subgroup of level n. In view of (5.2), = (n) for all positive integers n. (n)
(6.7)
We proceed to show that the principal congruence subgroup (N ), where N is the Frobenius-Schur exponent, always fixes the equivariant indicators. Lemma 6.4. Let C be a spherical fusion category over k with FSexp(C) = N . Then I V ((m, l), g˜ z) = I V ((m, l), z) = I V ((m, l), gz) for all m, l ∈ Z, V ∈ C, z ∈ Kk (Z (C)) and g ∈ (N ). ) = (N ), the first and second equality are equivalent. It suffices Proof. Since (N to show one of these two equalities holds. Note that t N ∈ (N ) and ρ Z (C ) (t N ) = id. Therefore, ker ρ Z (C ) contains the normal closure of t N in SL2 (Z). For N = 2, it is wellknown that the normal closure of t2 in SL2 (Z) is (2) (cf. [Bre60]). Thus, gz = z for all g ∈ (2) and z ∈ Kk (Z (C)). In particular, we have I V ((m, l), gz) = I V ((m, l), z). Now, we may assume N > 2 and consider the relation ∼ on Z2 defined by (m, l) ∼ (m , l ) iff I V ((m, l), z) = I V ((m , l ), z) for all z ∈ Kk (Z (C)), V ∈ C. It is obvious that ∼ is an equivalence relation on Z2 . By Theorem 5.4, if (m, l) ∼ (m , l ), then (m, l)g ∼ (m , l )g for all g ∈ SL2 (Z).
Congruence Subgroups and Generalized Frobenius-Schur Indicators
31
We need to show that (m, l) ∼ (m, l)g for all (m, l) ∈ Z2 and g ∈ (N ). To prove this, we use a version of [SZ, Theorem 1.3] which requires to verify the following conditions for each (m, l) ∈ Z2 : (i) (m, l) ∼ (m, m N + l) and (ii) (m, l) ∼ (m, kl) whenever gcd(m, l) = gcd(m, kl) for some integer k ≡ 1 (mod N ). The first condition follows directly from Theorem 5.4 and the fact that ρ Z (C ) (t˜ N ) = id. For the second condition, we consider m, l, k ∈ Z such that k ≡ 1 (mod N ) and gcd(m, l) = gcd(m, kl) = q. Obviously, if l = 0, then (m, l) ∼ (m, kl). We may assume l = 0. In this case, q ≥ 1 and gcd(m/q, l/q) = gcd(m/q, kl/q) = 1. If (m/q, l/q) ∼ (m/q, kl/q), then I V ((m, l), z) = I V q ((m/q, l/q), z) = I V q ((m/q, kl/q), z) = I V ((m, kl), z) for all V ∈ C and z ∈ Kk (Z (C)). Hence (m, l) ∼ (m, kl). Therefore, it suffices to prove (m, l) ∼ (m, kl) for gcd(m, l) = gcd(m, kl) = 1. If m = 0, then this condition forces k = ±1. Since k ≡ 1 (mod N ) and N > 2, k = 1, and hence (0, l) ∼ (0, kl). So, we may further assume m = 0. Since gcd(m, kl) = 1, k and m N are relatively prime. Let ξ ∈ k be a primitive |m|N th root of unity and consider the automorphism σk ∈ Gal(Q|m|N /Q) defined by σk : ξ → ξ k . Since k ≡ 1 (mod N ), we have σk (ξ m ) = ξ m or equivalently σk |Q N = id. Since θXN = idX for X ∈ Z (C), by Lemmas 2.5 and 2.7, we (m) m N have E X,V = id. Therefore, (m) l (m) kl X X = Tr E X,V = νm,kl (V )) = σk Tr E X,V (V ). σk (νm,l X (V ) ∈ Q by Proposition 5.7, and so it is fixed by σ . Thus, On the other hand, νm,l N k
I V ((m, l), [X]) = I V ((m, kl), [X]) for all V ∈ C and simple X ∈ Z (C). Hence, (m, l) ∼ (m, kl).
Lemma 6.5. Let C be a spherical fusion category over k with FSexp(C) = N . Suppose X0 is the unit object of Z (C). Then [X0 ] ∈ Kk (Z (C)) is (N )-invariant. Proof. Let f ∈ Kk (Z (C))∗ be defined by 1
f (z) = d(Vk )I Vk ((0, 1), z) for z ∈ Kk (Z (C)), dim C k∈
where {Vk | k ∈ } is a complete set of non-isomorphic simple objects in C. As a consequence of Lemma 6.4, we have f (gz) = f (z) for all z ∈ Kk (Z (C)) and g ∈ (N ). By Theorem 5.4, we find 1
1
f (s[X j ]) = d(Vk )I Vk ((0, 1), s[X j ]) = d(Vk )I Vk ((0, 1)s−1 , [X j ]) dim C dim C k∈ k∈ 1
1 = d(Vk )I Vk ((1, 0), [X j ]) = d(X j ) = s0 j = [X0 ], s[X j ] dim C dim C k∈
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S.-H. Ng, P. Schauenburg
and thus f (z) = [X0 ], z for all z ∈ Kk (Z (C)). Now g[X0 ], z = [X0 ], g˜ −1 z = f (˜g−1 z) = f (z) = [X0 ], z for all z ∈ Kk (Z (C)) by Lemma 6.2 (ii), and the result follows. Lemma 6.6. Let A be a modular tensor category over k with FSexp(A) = N . Suppose + /λ and consider the representation ρ λ,ζ λ, ζ ∈ k such that λ2 = dim A and ζ 3 = pA A of SL2 (Z) on Kk (A). Then, for z, z ∈ Kk (A), (gz)(˜gz ) = zz for all g ∈ (N ). Proof. Consider the non-degenerate bilinear form ·, · on Kk (A) from Lemma 6.2. Let m : Kk (A) ⊗ Kk (A) → Kk (A) denote the multiplication in the Grothendieck algebra. Then, by Lemma 6.2, we have mφ −1 (Ui j ), [Uk ] = [Ui ][U j ], [Uk ] = dim A(Ui ⊗ U j , Uk ) = IUk ((1, 0), [Ui j ]) for all i, j ∈ . Therefore, mφ −1 (w), [Uk ] = IUk ((1, 0), w) for all w ∈ Kk (Z (A)). By Lemma 6.4, for g ∈ (N ), mφ −1 (gw), [Uk ] = IUk ((1, 0), gw) = IUk ((1, 0), w) = mφ −1 (w), [Uk ], and hence mφ −1 (gw) = mφ −1 (w). Now for w = φ(z⊗z ) we have φ −1 (gw) = gz⊗ g˜ z by Lemma 6.2 and the claim follows. Theorem 6.7. Let C be a spherical fusion category over k with FSexp(C) = N . The kernel of the canonical modular representation ρ Z (C ) : SL2 (Z) → GL(Kk (Z (C))) of Z (C) is a congruence subgroup of level N . In particular, Kk (Z (C)) is (N )-invariant, Proof. Recall that Kk (Z (C)) is a k-algebra with [X0 ] as the identity element. It follows from Lemmas 6.5 and 6.6 that z = z[X0 ] = (gz)(˜g[X0 ]) = (gz)[X0 ] = gz for all g ∈ (N ), z ∈ Kk (Z (C)). Therefore, (N ) ⊆ ker ρ Z (C ) . Suppose (N ) ⊆ ker ρ Z (C ) for some positive integer N ≤ N . Then t N ∈ ker ρ Z (C ) or t N z = z for all z ∈ Kk (Z (C)). Therefore, T N = 1, where T is the T -matrix of Z (C). Since ord(T ) = N (cf. [NS07a, Theorem 5.5]), N | N and so N = N . Theorem 6.8. Let A be a modular category over k with FSexp(A) = N . Then the kernel of the projective modular representation ρ A of A is a congruence subgroup of level N . λ,ζ
Proof. Let λ, ζ ∈ k such that the representation ρ := ρA is well-defined. In view of Lemma 6.2, ρ ⊗ ρ˜ ∼ = ρ Z (A) . Therefore, by Theorem 6.7, gz ⊗ g˜ z = z ⊗ z for all z, z ∈ Kk (A) and g ∈ (N ).
(6.8)
Pick z ∈ Kk (A) and ∈ Kk (A)∗ with (z ) = 1. Then (6.8) implies gz(˜gz ) = z for λ,ζ all z ∈ Kk (A). In particular, ρA (g) is a scalar multiple of idKk (A) and hence ρ A (g) is the unit of PGL(Kk (A)). Thus, (N ) ⊆ ker ρ A .
Congruence Subgroups and Generalized Frobenius-Schur Indicators
33 λ,ζ
Suppose (N ) ⊆ ker ρ A for some positive integer N ≤ N . Then ρA (t N ) = α id for some nonzero scalar α ∈ k. Therefore, 1 ζN
T N = α id,
where T = [δi j ωi ] is the T -matrix of A. Since ω0 = 1, we find
1 = ω0 = ζ N α,
and hence T N = id. This implies N | N .
Example 6.9. Let G be a finite abelian group and ω a normalized complex valued 3-cocycle on G such that quasi-Hopf algebra D ω (G), introduced in [DPR92], is commutative. By [MN01, Corollary 3.6], the function θz defined by θz (x, y) =
ω(z, x, y)ω(x, y, z) ω(x, z, y)
is a 2-coboundary for all z ∈ G. Let tz : G → C× be a normalized cochain such that θz = δtz , i.e. θz (x, y) = tz (x)tz (y)/tz (x y). In addition, we chose t1 = 1. Following Sect. 9 of [MN01], the irreducible characters χα,u of D ω (G) are indexed by the set Gˆ × G, where Gˆ is the character group of G. As a vector space D ω (G) = C[G]∗ ⊗ C[G]. Let {e(u) | u ∈ G} be the basis of C[G]∗ dual to G. Then χα,u (e(h) ⊗ y) = α(y)tu (y)δh,u
(6.9)
for α ∈ Gˆ and h, u, y ∈ G. The universal R-matrix of D ω (G) and the canonical ribbon structure (cf. [AC92] and [GMN07, p. 869]) are given respectively by
R= (e(g) ⊗ 1) ⊗ (e(h) ⊗ g), v = e(g) ⊗ g. g,h∈G
g∈G
Since the pivotal trace of the canonical pivotal structure of D ω (G) is equal to the ordinary trace, it follows from (6.9) that the ((α1 , u 1 ), (α2 , u 2 ))-entry of the S-matrix for D ω (G)-modfin is the complex conjugate of bω ((α1 , u 1 ), (α2 , u 2 )) = (χα1 ,u 1 ⊗ χα2 ,u 2 )(R 21 R) = α1 (u 2 )α2 (u 1 )tu 1 (u 2 )tu 2 (u 1 )
(6.10)
and the ((α, u), (α, u))-entry of the T -matrix is q(α, u) = χα,u (v) = α(u)tu (u).
(6.11)
It is worth to note that q is a quadratic form canonically defined on the group ω of group-like elements of D ω (G) and bω is the associated non-degenerate bicharacter on ω defined in [MN01, p. 3491]. By [NS07a, Theorem 9.2], the Frobenius-Schur exponent of D ω (G)-modfin is given by the formula FSexp(D ω (G)) = lcm |ωC ||C|,
(6.12)
where C runs through all the maximal cyclic subgroups of G and |ωC | denotes the order of the restriction of the cohomology class of ω to C. Moreover, D ω (G)-modfin = Z (H -modfin ) for a certain semisimple quasi-Hopf algebra H of dimension |G|, [Maj98].
34
S.-H. Ng, P. Schauenburg
Now we consider the order 2 multiplicative group G = {1, x}, and let α be the non-trivial character of G. Then H 3 (G, C× ) = Z2 , D ω (G) is commutative for all normalized 3-cocycles ω of G, and the irreducible characters of D ω (G) are indexed by {(1, 1), (α, 1), (1, x), (α, x)}. (i) For ω ≡ 1, we can choose tz = 1 for all z ∈ G. It follows from (6.10) and (6.11) that the normalized S-matrix and T -matrix are ⎤ ⎡ ⎤ ⎡ 1 0 0 0 1 1 1 1 1 ⎢1 0⎥ 1 −1 −1 ⎥ ⎢0 1 0 . s= ⎣ ⎦, T = ⎣0 0 1 0⎦ 1 −1 1 −1 2 0 0 0 −1 1 −1 −1 1 Since s 2 = T 2 = (sT )3 = id and s(sT )s = T s = (sT )−1 , the image of the canonical representation ρ of D(G)-modfin is isomorphic S3 and so ker ρ = (2). (ii) For the non-trivial class of H 3 (G, C× ), we consider ω : G 3 → C× defined by ω(a, b, c) = −1 if a = b = c = x and ω(a, b, c) = 1 otherwise. Then ω is a nontrivial normalized 3-cocycle on G. If one defines tz : G → C× as tz (a) = i whenever a = z = x and tz (a) = 1 otherwise, then θz = δtz (cf. [GMN07, p. 857]). Using the same index set for the irreducible characters as in case (i), the normalized S and T -matrices of D ω (G)-modfin are ⎤ ⎡ ⎤ ⎡ 1 0 0 0 1 1 1 1 1 ⎢1 0⎥ 1 −1 −1 ⎥ ⎢0 1 0 . s= ⎣ ⎦, T = ⎣0 0 i 0⎦ 1 −1 −1 1 2 0 0 0 −i 1 −1 1 −1 Then s 2 = T 4 = (sT 2 )4 = (sT )3 = id and FSexp(D ω (G)) = 4. Since ssT 2 s = T 2 s = (sT 2 )−1 , the subgroup generated by s, sT 2 is a dihedral group of order 8 and hence the image of the canonical representation ρ of D ω (G)-modfin contains at least 24 elements. Since s 2 = id and s2 = −1, we have (4)1, −1 ⊆ ker ρ, and thus Im ρ is a homomorphic image of SL2 (Z)/ (4)1, −1 = PSL(2, Z4 ) ∼ = S4 . Thus we have ker ρ = (4)1, −1 and Im ρ ∼ = S4 . More examples of small modular categories can be found in [RSW09]. 7. Modular Representations and a Conjecture of Eholzer A matrix representation ρ : SL2 (Z) → GL(n, k) which has finite image is called t-rational if Im ρ ⊆ GL(n, Qm ), where m = ord(ρ(t)). It is conjectured in [Eho95] that the representation ρ : SL2 (Z) → GL(n, C) associated with a RCFT satisfies the conditions: (i) The kernel of ρ is a congruence subgroup of SL2 (Z), and (ii) ρ is t-rational. In this section, we prove that every modular representation of a modular category has finite image and is t-rational. Let A be a modular category over k with a complete set of non-isomorphic simple objects {Ui | i ∈ }. We denote by M (R) the ring of square matrices indexed by over
Congruence Subgroups and Generalized Frobenius-Schur Indicators
35
a commutative ring R, and GL( , R) the group of invertible matrices in M (R). A modular representation of A is an ordinary group representation ξ : SL2 (Z) → GL( , k) such that ρ A (g) = π(ξ(g)) for all g ∈ SL2 (Z), where π : GL( , k) → PGL( , k) λ,ζ is the natural surjection. In particular, ρA , for λ, ζ ∈ k satisfying (1.11), is a modular representation of A. Suppose ξi : SL2 (Z) → GL( , k), i = 1, 2, are modular representations of A. Then there exist xs, xt ∈ k× such that ξ2 (s) = xsξ1 (s) and ξ2 (t) = xtξ1 (t). Using the relations of the s and t, we find xs4 = 1, xt3 xs3 = xs2 , and hence xt12 = 1 and xs = xt−3 . This implies there are 12 modular representations of A. Moreover, for g ∈ ker ξ1 , ξ2 (g) = α(g) id, where α(g) is some power of xt. Since xt12 = 1, ξ2 (ker ξ1 ) is isomorphic to a subgroup of Z12 . In particular, we find (ker ξ1 )(ker ξ2 ) ∼ ker ξ1 ≤ Z12 . = ker ξ2 ker ξ1 ∩ ker ξ2 By the same argument, (ker ξ1 )(ker ξ2 )/ ker ξ1 is a cyclic group of order dividing 12. Consequently, ker ξ1 is a finite index subgroup of SL2 (Z) if, and only if, ker ξ2 is of finite index. λ,ζ Now, we consider a modular representation ξ = ρA , where λ, ζ ∈ k satisfy (1.11). Suppose N = FSexp(A), [Si j ] and [δi j ωi ] are the S and T matrices of A respectively. We have pointed out in the paragraph preceding Proposition 5.7 that d(Ui ) ∈ Q N for i ∈ . Therefore, ± Si j , pA ∈ Q N, + / p − ∈ Q . Since ζ 6 ∈ Q is a root of unity, ord(ζ 6 ) divides 2N . and so ζ 6 = pA N N A 12N = 1 and hence Thus, ζ
ξ(t)12N = id.
(7.1)
Obviously, ξ(s)4 = id, and so we have (det ξ(g))12N = 1 for all g ∈ SL2 (Z). By Theorem 6.8, there exists a group homomorphism α : (N ) → k× such that ξ(g) = α(g) id for all g ∈ (N ). Therefore, α(g)12N | | = 1. In particular, the image of α is a finite cyclic group. Now we find (ker ξ )(N ) ∼ (N ) ∼ = = ξ((N )) ∼ = α((N )). ker ξ (N ) ∩ ker ξ Since (N ) is a finite index subgroup of SL2 (Z), so is ker ξ . Hence, all the modular representations of A have finite images. In view of the preceding remark, {ξx | x ∈ k, x 12 = 1} is the set of all modular representations of A, where ξx : SL2 (Z) → GL( , k) given by ξx (s) :=
1 1 1 ξ(s) = 3 S, ξx (t) := xξ(t) = T. 3 x x λ ζ /x
(7.2)
36
S.-H. Ng, P. Schauenburg
It follows from (7.1) that m := ord(ξx (t)) divides 12N . Since ( ζω/xi )m = 1 for all i ∈ and ω0 = 1, we find (ζ /x)m = 1 and so ωim = 1 for all i ∈ . Therefore, + ∈ Q , we also have N | m | 12N and ζ /x, ωi ∈ Qm for all i ∈ . Since λζ 3 = pA m p+
A x 3 λ = (ζ /x) 3 ∈ Qm . Since S ∈ GL( , Q N ), we have ξ x (s) and ξ x (t) ∈ GL( , Qm ). This completes the proof of
Theorem 7.1. Let A be a modular category over k. Then every modular representation ρ of A has finite image, and is t-rational. Moreover, FSexp(A) | ord(ρ(t)) | 12 · FSexp(A). λ,ζ
If one sets ξ = ρA for some λ, ζ ∈ k satisfying (1.11), then ξx , x ∈ k a 12th root of unity, are all the modular representations of A. In the proof of the above theorem, we have seen that (N )/((N ) ∩ ker ξ ) is always a finite cyclic group. However, there exist many linear characters α : (N ) → C× with finite images whose kernels are noncongruence subgroups of SL2 (Z) (cf. [KL08]). So, it is still unclear whether there always exists a modular representation of a modular tensor category whose kernel is a congruence subgroup. In view of Theorem 6.7, this is true when the modular category is the center of a spherical fusion category. 8. GFS Indicators and Integers in the Open String c indexed by the primary fields a, b, c of a RCFT was introduced A family of scalars Yab by Pradisi, Sagnotti and Stanev. It is conjectured in [PSS95] that c Yab ∈ Z.
(8.1)
Borisov, Halpern and Schweigert also considered these scalars, and they conjecture in [BHS98] that
s2 s s ad bd cd d
2 s0d
c ± Yab ≥0
(8.2)
for all primary fields a, b, c, where [sab ] denotes the S-matrix of the RCFT. By considering the Galois group actions, Gannon has shown these two conjectures under the assumption that the T -matrix of the RCFT has odd order [Gan00]. More recently, Kac, Longo and Xu have proved these conjectures via Z2 -permutation orbifolds of conformal nets [KLX05]. In this section, we use the GFS indicators to prove these two conjectures for all modular categories. Let A be a modular category with the set of simple objects {Ui | i ∈ }, and the S and T matrices [Si j ] and [δi j ωi ] . Recall from Sect. 1.4 that {Ui j = (Ui ⊗ U j , σUi ⊗U j ) | i, j ∈ } forms a complete set of simple objects of Z (A). In the remainder of this section, we consider the normalizations s, t of S and T respectively: s :=
1 1 S, t := T, λ ζ
(8.3)
where λ, ζ ∈ k satisfy (1.11). In particular, λ2 = dim A. The assignment s → s, t → t defines an ordinary representation of SL2 (Z).
Congruence Subgroups and Generalized Frobenius-Schur Indicators
37
c = dim A(U , U ⊗ U ) of A and s are related by The fusion coefficients Nab c a b Verlinde’s formula (cf. [BK01]):
sad sbd scd c = . (8.4) Nab s0d d∈
c , the assignment K (A) → Defining the matrix Na ∈ M (k) by (Na )bc = Nab 0 M (Z); [Ua ] → Na is the regular representation of K0 (A) in matrix form. The Verlinde formula (8.4) can also be rewritten in matrix form as
Na = s Da s −1 ,
(8.5)
s
where Da is the diagonal matrix [δi j s0a jj ] . Definition 8.1. Let s be the normalized S-matrix of the modular category A described in (8.3). For J, K ∈ GL( , k), we define Ya (J, K ) ∈ M (k) with the (b, c)-entry given by c Yab (J, K ) :=
sad Q bd Q cd , s0d
d∈
where Q = J s K s J = [Q i j ] and Q −1 = [Q i j ] . It is worth noting that Ya (J, K ) is independent of the choice of λ used in the normalization (8.3) of s. Moreover, for any non-zero scalars x, y ∈ k, c c Yab (x J, y K ) = Yab (J, K ).
For any ω ∈ k and positive integer m, we write ω1/m for an m th root of ω. Similarly, for any diagonal matrix D ∈ M (k), D 1/m abbreviates a diagonal matrix in M (k) m which satisfies the equation D 1/m = D. For any m th root t 1/m of t, there exists an m th root T 1/m of T which is a scalar multiple of t 1/m . For these, Ya (J, t m ) = Ya (J, T m ), Ya (t 1/m , K ) = Ya (T 1/m , K ) and Ya (t 1/m , t m ) = Ya (T 1/m , T m ). c (t 1/2 , t 2 ) are the scalars Y c considered in [PSS95 and BHS98]. The In particular, Yab ab c and the GFS indicators following lemma suggests a relation between these scalars Yab via Proposition 6.1.
Lemma 8.2. Let J, K ∈ GL( , k) such that K is symmetric and J is a diagonal matrix of the form [δi j ηi ] . Then, for a ∈ , Ya (J, K ) = (J s K )Na (J s K )−1 . In particular, the assignment K0 (A) → M (k); [Ua ] → Ya (J, K ) defines a matrix representation of K0 (A). Moreover, for any positive integer m and a, b, c ∈ , c Yab (J, T m ) =
ηb Ubc ν (Ua ). ηc m,1
38
S.-H. Ng, P. Schauenburg
Proof. Since J, K are symmetric, so is Q = J s K s J . Let Da be the diagonal matrix [δi j ssai0i ] . Then we have Ya (J, K ) = Q Da Q −1 = J s K s J Da J −1 s −1 (J s K )−1 = J s K s Da s −1 (J s K )−1 = J s K Na (J s K )−1 . Here, the third equality follows from the Verlinde formula (8.5). In particular, ωm ηb l ωkm ηb a ηb Ubc c Yab (J, T m ) = Nak m sbk scl = Nkl sbk scl km = νm,1 (Ua ). ηc ωl ηc ωl ηc k,l
k,l
Here the last equation is an immediate consequence of Proposition 6.1.
c (T 1/m , T m ) is an algeTheorem 8.3. For any positive integer m and a, b, c ∈ , Yab braic integer in Qm and
sm s s ad bd cd c = dim A(Uc , Ua⊗m ⊗ Ub ) ≥ |Yab (T 1/m , T m )|, (8.6) m s0d d∈
where the inequality is considered in any subfield of C isomorphic to Qm . In particular, c (T 1/2 , T 2 ) ∈ Z and Yab
s2 s s ad bd cd d∈
2 s0d
c ± Yab (T 1/2 , T 2 ) ≥ 0.
If, in addition, m is relatively prime to the Frobenius-Schur exponent N of A, then there c (T 1/m , T m ) is a rational integer for all exists an m th root T 1/m ∈ M (Q N ), and Yab 1/m ∈ M (Q N ). T ω−1 (m,1) m = b−1 id as operators on A(Ub ⊗ Proof. In view of Lemma 2.7 and (6.2), E U ,Ua ωc
bc
Uc , Ua⊗m ). If we set
E˜ =
1/m
ωb
1/m
ωc
(m,1) , bc ,Ua
EU
˜ is an algebraic integer in Qm . Note that then E˜ m = id and hence Tr( E) ˜ = Tr( E)
1/m
ωb
1/m
ωc
ω1/m U (m,1) c bc Tr E U ,Ua = b1/m νm,1 (Ua ) = Yab (T 1/m , T m ), bc ωc
c (T 1/m , T m ) is an algewhere the last equality follows from Lemma 8.2. Therefore Yab ˜ braic integer in Qm for all positive integers m. Since E is a finite order k-linear operator on A(Ub ⊗ Uc , Ua⊗m ), if one identifies Qm with a subfield of C, then c ˜ ≤ dim A(U ⊗ Uc , Ua⊗m ) = dim A(Uc , Ua⊗m ⊗ Ub ). |Yab (T 1/m , T m )| = | Tr( E)| b
By (8.5), Nam = s Dam s −1 . Note that the (b, c)-entries of Nam and s Dam s −1 are respectively given by
sm s s ad bd cd dim A(Uc , Ua⊗m ⊗ Ub ) and . m s0d d∈
Thus, the first equality of (8.6) follows.
Congruence Subgroups and Generalized Frobenius-Schur Indicators
39
Let ς N ∈ k be a primitive N th root of unity. Then ωa is a power of ς N for any 1/m a ∈ . If m, N are relatively prime, then there is an m th root ωa ∈ Q N , and hence th 1/m T has a diagonal m root T ∈ M (Q N ). In particular, these diagonal m th roots Ubc (Ua ) is an algebraic integer in Q N , and also have finite order. By Remark 5.8, νm,1 c (T 1/m , T m ). Therefore, Y c (T 1/m , T m ) is an algebraic integer in Q ∩ Q . so is Yab N m ab c (T 1/m , T m ) is a rational integer for any Since (m, N ) = 1, Q N ∩ Qm = Q and so Yab a, b, c ∈ . Remark 8.4. The specialization m = 2 of Theorem 8.3 implies the conjecture of Pradisi-Sagnotti-Stanev (8.1) and the conjecture of Borisov-Halpern-Schweigert (8.2). As a consequence of Lemma 8.2, for m = 2 or (m, N ) = 1 with T 1/m ∈ M (Q N ), the assignment K0 (A) → M (Z); [Ua ] → Ya (T 1/m , T m ) defines an integral representation of K0 (A). 9. Generalized Frobenius-Schur Endomorphisms It has been shown in [NS07b] that the Frobenius-Schur indicators of an object V in a pivotal fusion category C over k are the pivotal traces of certain endomorphisms, called the Frobenius-Schur (FS) endomorphisms. In this section, we introduce the definition of (m,l) a generalized Frobenius-Schur (GFS) endomorphism FSV,z for a pair (m, l) of positive integers, an object V ∈ C, and a natural endomorphism z of the identity functor of Z (C). These GFS endomorphisms reduce to the FS endomorphisms defined in [NS07b] when z is the projection onto the trivial component. For a simple object X ∈ Z (C), we show X (V ) is the left pivotal trace of FS(m,l) that the GFS indicator νm,l V,z X /d (X) , where z X is the natural projection onto the isotypic component of X. Moreover, if (m, l) is a pair of (m,l) relatively prime integers, the GFS endomorphism FSV,z is natural in V . This implies once again the additivity property X X X νm,l (U ⊕ V ) = νm,l (U ) + νm,l (V )
for a simple object X ∈ Z (C) and a pair (m, l) of relatively prime positive integers proved already in Corollary 5.5 above when C is spherical. Let C be a pivotal fusion category over k, D = Z (C), and F : D → C the natural forgetful functor. Note that F maps the morphisms of D injectively to the morphisms of C and the pivotal trace of a morphism f in D is identical to the pivotal trace of F( f ) in C. Therefore, we may simply use the same notations for a morphism (or an object) in D and its images in C under F. Now, we consider the two-sided adjoint K to the forgetful functor F with adjunction isomorphisms arranged as in (1.5). For W ∈ C, we define −1
uW := W,K (W ) (id K (W ) ) : W → K (W ), and cW := K (W ),W (id K (W ) ) : K (W ) → W. Then uW ◦g = K (g) ◦ u V for all g ∈ C(V, W ).
and cW ◦K (g) = g ◦ cV
40
S.-H. Ng, P. Schauenburg
Let X ∈ D, and let { pα }α be a basis for C(W, X ), and {qα }α its dual basis for C(X, W ) with respect to the pairing (·, ·) . Set Pα = ( pα ) and Q α = −1 (qα ). Then pα = Pα ◦ u , qα = c ◦Q α and (Pα , Q α ) = δα,α . Therefore, {Pα }α and {Q α }α are dual bases for D(K (W ), X) and D(X, K (W )) respectively. If X is simple, then
Q α ◦ Pα z X = d (X) α
is the natural projection of K (W ) onto its isotypic component of X. Note that z X /d (X) is a natural endomorphism of the identity functor on D. Assume C strict and set W = V m for some V ∈ C and positive integer m. Then V
X νm,1 (V )
=
qα
V
=
pα
α
pα
X
qα
α
=
α
pa
qα
X
=
α
θ −1 qα
⎛ ⎜ ⎜ ⎜ ⎜ 1 ⎜ = ptr ⎜ ⎜ d (X) ⎜ ⎜ ⎝
V
⎜ ⎜ ⎜ ⎜
⎜ = ptr ⎜ ⎜ θ −1 α ⎜ Qα ⎜ c ⎝
pα
⎛
V u θ −1 zX c
u Pα
⎞
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜
⎟ ⎜ ptr ⎜ ⎟ = ⎟ ⎜ α ⎟ ⎜ ⎟ ⎜ ⎠ ⎝
V
⎞
⎛
V u θ −1 Pα Qα c
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
V
⎟ ⎟ ⎟ ⎟ ⎟ (m) ⎟ = ptr (FSV,z X /d (X) ), ⎟ ⎟ ⎟ ⎠
(9.1)
V
where we use the following Definition 9.1. The (m, 1)st GFS endomorphism of V associated to an endomorphism z of the identity functor on Z (C) is V u θ −1
(m)
FSV,z =
z c V
.
Congruence Subgroups and Generalized Frobenius-Schur Indicators
41
Proposition 9.2. Each of the (m, 1)st GFS endomorphisms defines a natural endomorX (V ) is additive in V for any simple phism of the identity functor on C. In particular, νm,1 X ∈ Z (C). Proof. Consider f : V → W in C, and write f k = id W k−1 ⊗ f ⊗ id V m−k : W k−1 ⊗ V m−k+1 → W k ⊗ V m−k for all 1 ≤ k ≤ m. We have V
V
V
V fk
u
u
u
θ −1
θ −1
K ( fk )
=
z
=
z
u
=
θ −1
θ −1
c
K ( fk )
z
z
fk
c
c
c
W
W
W
(9.2)
W
and when k < m we can continue V
V
fk
u
=
z
z
c
c
W
=
θ −1
V
u
u
θ −1
θ −1
fk
u θ −1
V
=
z c
c fk
W
.
z
f k+1
W
W
We conclude by induction that V
(m)
f FSV,z =
z
V
u
u
u
θ −1
θ −1
θ −1
=
z
c f W
V
= ··· =
z
V
V
fm
f
u
u
θ −1 =
=
θ −1
c
c
z
z
f1
fm
c
c
W
W
W
(m)
= FSW,z f.
W
(m)
(m)
(m)
By the naturality of the (m, 1)st GFS endomorphisms, we find FSU ⊕V,z = FSU,z ⊕ FSV,z for U, V ∈ C. If X ∈ Z (C) is simple, then it follows from (9.1) that X X X (U ⊕ V ) = νm,1 (U ) + νm,1 (V ). νm,1
42
S.-H. Ng, P. Schauenburg
For C = H -modfin for some semisimple quasi-Hopf algebra over C, the natural (m) endomorphism FS−,z is associated to a central element μm,z (H ) of H . Moreover, (m)
ptr(FSV,z ) = χV (μm,z (H )), where χV is the character afforded by V ∈ C. This central element was determined in [NS08] for z = z I but, for a general z, is yet to be determined. We now turn to our more general version of GFS endomorphisms. Definition 9.3. For non-negative integers k, r, m with l := k + r + 1 ≤ m and a natural endomorphism z of the identity functor on D, we define V u
V −k (m,k,r )
FSV,z
θ −1
=
V l−m
z
V −r
c V
where the distribution of tensor factors over the legs of the graphical symbols for the unit and counit of adjunction is as follows: V m−l V k
V
K (V m )
Vr
and
u K (V m ) (m,l)
(m,l−1,0)
We abbreviate FSV,z = FSV,z
.
c Vk
V
V r V m−l
for 1 ≤ l ≤ m.
Lemma 9.4. Let m ∈ N, k, r ≥ 0 with 1 ≤ l := k + r + 1 ≤ m. X (V ) = 1 ptr FS(m,l) . (i) For simple X ∈ Z (C), νm,l V,z X d (X) ) −1) (ii) If r > 0, then ptr FS(m,k,r = ptrr FS(m,k+1,r . V,z V,z (iii) In addition, if C is spherical, then 1 (m,k,r ) (m,l) (m,k,r ) X ptr FSV,z X = ptr FSV,z and νm,l ptr FSV,z (V ) = d(X) (m,k,r )
for simple X ∈ D. Moreover, FSV,z
(m,l)
= FSV,z for simple V ∈ C.
Proof. The proof of (i) is similar to (9.1), and (ii) can be obtained directly from graphical calculus. If C is spherical, then by induction we have ) = ptr FS(m,l) . ptr FS(m,k,r V,z V,z X (V ) = 1 ptr FS(m,k,r ) for simple X ∈ Z (C). If V is simple, then Hence, by (i), νm,l V,z X d(X) (m,k,r )
FSV,z
(m,l)
and FSV,z are scalar multiples of id V . Since they have the same pivotal trace (m,k,r )
and d(V ) = 0, FSV,z
(m,l)
= FSV,z .
Congruence Subgroups and Generalized Frobenius-Schur Indicators
43
(m,k,r )
Proposition 9.5. FSV,z is natural in V provided m and l := k + r + 1 are relaX (V ) is additive in V . In addition, if C is spherical, then tively prime. In particular, νm,l (m,k,r )
FSV,z
(m,l)
= FSV,z for all V ∈ C.
Proof. Let s ∈ Sm be the permutation determined by requiring s(i) ∈ {1, . . . , m} to be congruent to i + l modulo m. Note that s is an m-cycle since m and l are relatively prime. Consider f : V → W . For any X i , Y j in C we will write f p: X 1 ⊗. . . ⊗ X p−1 ⊗ V ⊗Y1 ⊗. . . ⊗ Yu → X 1 ⊗ . . . ⊗ X p−1 ⊗ W ⊗ Y1 ⊗ . . . ⊗ Yu for the morphism that acts as f in the p th position and the identity elsewhere. Define a series of objects and morphisms f [1]
f [2]
f [n]
V [0] −−→ V [1] −−→ V [2] → . . . −−→ V [n] by V [0] = V ⊗n and f [i] = f s i−1 (k+1) ; this fixes V [i] which has to be the appropriate target. Note that the sequence is well-defined since s is transitive, and we have V [n] = W ⊗n . Let V
V
f [i] u
u θ −1
θ −1
G i :=
=
for any i ∈ {1, . . . , m}.
z
z
c
c
f [i]
W
W
The last equality follows from a similar argument as (9.2). If p := s i−1 (k + 1) ≤ m − l, then V
V fp u θ −1
Gi = z
=
fp
c W
u
u
θ −1
θ −1
z c
W
V
=
z c f [i+1] W
= G i+1
44
S.-H. Ng, P. Schauenburg
since p + l = s( p) = s i (k + 1). If m − l < p ≤ m − l + k, then for q := p − m + l, V
V fp u
θ −1
θ −1
=
=
z
z
f [i+1]
fp
W
W
since q = s( p) = and find
s i (k+1). If
W
p > m−l+k+1 = m−r +1, then we set q = p−m+l−k−1 V
V fp u θ −1
Gi = z
= G i+1
z c
c
c
=
V
u
u
θ −1
θ −1
=
z c
c W
u
u
θ −1
Gi =
V
z
= G i+1 ,
c f [i+1]
fp W
W
since now q + k + 1 = p + l − m = s( p) = + 1). Because s has order m and s(m − l + k − 1) = k + 1, the case p = m − l + k + 1 = s m−1 (k + 1) occurs when i = m. Thus, using induction, we have s i (k
) ) f ◦ FS(m,k,r = G 1 = · · · = G m = FS(m,k,r ◦ f. V,z V,z
The remaining statements are direct consequences of the naturality of FSm,k,r V,z and the previous lemma. Acknowledgement. The authors would like to thank Ling Long for her comments and suggestions on congruence subgroups.
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Communicated by Y. Kawahigashi
Commun. Math. Phys. 300, 47–64 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1106-8
Communications in
Mathematical Physics
Kontsevich Deformation Quantization and Flat Connections Anton Alekseev1 , Charles Torossian2 1 Section de Mathématiques, Université de Genève, 2-4 Rue du Lièvre, C.P. 64, 1211 Genève 4, Switzerland.
E-mail:
[email protected]
2 Institut Mathématiques de Jussieu, Université Paris 7, CNRS; Case 7012, 2 Place Jussieu, 75005 Paris,
France. E-mail:
[email protected] Received: 25 June 2009 / Accepted: 26 April 2010 Published online: 21 August 2010 – © Springer-Verlag 2010
Abstract: In Torossian (J Lie Theory 12(2):597–616, 2002), the second author used the Kontsevich deformation quantization technique to define a natural connection ωn on the compactified configuration spaces C n,0 of n points on the upper half-plane. Connections ωn take values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ωn is flat. The configuration space C n,0 contains a boundary stratum at infinity which coincides with the (compactified) configuration space of n points on the complex plane. When restricted to this stratum, ωn gives rise to a flat connection ωn∞ . We show that the parallel transport defined by the connection ω3∞ between configuration 1(23) and (12)3 verifies axioms of an associator. We conjecture that ωn∞ takes values in the Lie algebra tn of infinitesimal braids. If correct, this conjecture implies that ∈ exp(t3 ) is a Drinfeld’s associator. Furthermore, we prove = K Z showing that is a new explicit solution of associator axioms. 1. Introduction The Kontsevich proof of the formality conjecture and the construction of the star product on Rd equipped with a given Poisson structure make use of integrals of certain differential forms over compactified configuration spaces C n,m of points on the upper half-plane. Here n points are free to move in the upper half-plane, m points are bound to the real axis, and we quotient by the diagonal action of the group z → az + b with a ∈ R+ , b ∈ R. In this paper, we use the same ingredients to study a certain connection ωn on C n,0 with values in the Lie algebra of derivations of the free Lie algebra with n generators. This connection was introduced by the second author in [14]. One of our results is flatness of ωn . The compactified configuration space C n,0 contains a boundary stratum “at infinity” which coincides with the configuration space of n points on the complex plane (quotient
48
A. Alekseev, C. Torossian
by the diagonal action z → az + b with a ∈ R+ , b ∈ C). Over this boundary stratum, the connection ωn restricts to the connection ωn∞ with values in the Lie algebra krvn defined in [3]. We conjecture that in fact ωn∞ takes values in the Lie algebra tn ⊂ krvn defined by the infinitesimal braid relations. Let be the parallel transport defined by the connection ω3∞ for the straight path between configurations 1(23) and (12)3 of 3 points on the complex plane. We show that verifies axioms of an associator with values in the group K RV3 = exp(krv3 ). If the conjecture of the previous paragraph holds true, then ∈ exp(t3 ), and it becomes a Drinfeld’s associator. The key ingredient in the proof of the pentagon axiom is the flatness property of ωn∞ . The construction of is parallel to the construction of the Knizhnik-Zamolodchikov associator K Z in [7] with ω3∞ replacing the Knizhnik-Zamolodchikov connection. Furthermore, one can show that is even, and hence = K Z . While this paper was in preparation, we learnt of the work [12] proving our conjecture stated above. The plan of the paper is as follows. In Sect. 2, we review some standard facts about the Kontsevich déformation quantization technique and free Lie algebras. In Sect. 3, we prove flatness of the connection ωn . Section 4 contains the proof of associator axioms for the element . 2. Deformation Quantization and Free Lie Algebras Many sources are now available on the Kontsevich formula for quantization of Poisson brackets (see e.g. [5]). For the convenience of the reader, we briefly recall the main ingredients of [11] for Rd and the construction [14] of the connection ωn .
2.1. Free Lie algebras and their derivations. 2.1.1. Free Lie algebras and derivations. Let K be a field of characteristic zero, and let lien = lie(x1 , . . . , xn ) be the degree completion of the graded free Lie algebra over K with generators x1 , . . . , xn of degree one. We shall denote by dern the Lie algebra of derivations of lien . An element u ∈ dern is completely determined by its values on generators, u(x1 ), . . . , u(xn ) ∈ lien . The Lie algebra dern carries a grading induced by the one of lien . Definition 1. A derivation u ∈ dern is called tangential if there exist ai ∈ lien , i = 1, . . . , n such that u(xi ) = [xi , ai ]. Tangential derivations form a Lie subalgebra tdern ⊂ dern . Elements of tdern are in one-to-one correspondence with n-tuples of elements of lien , (a1 , . . . , an ), which verify the condition that ak has no linear term in xk for all k. By abuse of notations, we shall often write u = (a1 , . . . , an ). For two elements of tdern , u = (a1 , . . . , an ) and v = (b1 , . . . , bn ), we have [u, v]tder = (c1 , . . . , cn ) with ck = u(bk ) − v(ak ) + [ak , bk ]lie.
(1)
Definition 2. A derivation n u = (a1 , . . . , an ) ∈ tdern is called special if u(x) = [x , a ] = 0 for x = i i i i=1 x i .
Kontsevich Deformation Quantization and Flat Connections
49
We shall denote the space of special derivations by sdern . It is obvious that sdern ⊂ tdern is a Lie subalgebra. Both tdern and sdern integrate to prouniportent groups denoted by T Autn and S Autn , respectively. In more detail, T Autn consists of automorphisms of lien such that xi → Ad gi xi = gi xi gi−1 , where gi ∈ exp(lien ). Similarly, nelements of S Autn are tangential automorphisms of lien with an extra property x = i=1 xi → x. The family of Lie algebras tdern is equipped with simplicial Lie homomorphisms tdern → tdern+1 . For instance, for u = (a, b) ∈ tder2 we define u 1,2 = (a(x, y), b(x, y), 0), u 2,3 = (0, a(y, z), b(y, z)), u 12,3 = (a(x + y, z), a(x + y, z), b(x + y, z)), and similarly for other simplicial maps. These Lie homomorphisms integrate to group homomorphisms of T Autn and S Autn . k 2.1.2. Cyclic words. Let Assn+ = ∞ k=1 Ass (x 1 , . . . , x n ) be the graded free associative algebra (without unit) with generators x1 , . . . , xn . Every element a ∈ Assn+ admits n a unique decomposition of the form a = i=1 (∂i a)xi , where ∂i a ∈ Assn (Assn is a free associative algebra with unit). We define the graded vector space cyn as a quotient cyn = Assn+ /(ab − ba); a, b ∈ Assn . Here (ab − ba); a, b ∈ Assn is the subspace of Assn+ spanned by commutators. The multiplication map of Assn+ does not descend to cyn which only has a structure of a graded vector space. We shall denote by tr : Assn+ → cyn the natural projection. By definition, we have tr(ab) = tr(ba) for all a, b ∈ Assn imitating the defining property of trace. In general, graded components of cyn are spanned by words of a given length modulo cyclic permutations. Example 1. The space cy1 is isomorphic to the space of formal power series in one variable without constant term, cy1 ∼ = xk[[x]]. This isomorphism is given by the following formula: f (x) =
∞
fk x k →
k=1
∞
f k tr(x k ).
k=1
2.1.3. Divergence. Let u = (a1 , . . . , an ) ∈ tdern . We define the divergence as div(u) =
n
tr(xi (∂i ai )).
i=1
It is a 1-cocycle of tdern with values in cyn (see Proposition 3.6 in [3]). We define krvn ⊂ sdern ⊂ tdern as the Lie algebra of special derivation with vanishing n divergence. Hence, n u = (a1 , . . . , an ) ∈ krvn is a solution of two equations: i=1 [x i , ai ] = 0 and i=1 tr(x i (∂i ai )) = 0. We shall denote by K RVn = exp(krvn ) the corresponding prounipotent group.
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Iris
Fig. 1. Variety C 2,0
2.2. Kontsevich construction. 2.2.1. Configurations spaces. We denote by Cn,m the configuration space of n distinct points in the upper half plane and m points on the real line modulo the diagonal action of the group z → az + b (a ∈ R+ , b ∈ R). In [11], Kontsevich constructed compactifications of spaces Cn,m denoted by C n,m . These are manifolds with corners of dimension + 2n − 2 + m. We denote by C n,m the connected component of C n,m with real points in the standard order (id. 1 < 2 < · · · < m). The compactified configuration space C 2,0 (the “Kontsevich eye”) is shown on Fig. 1. The upper and lower eyelids correspond to one of the points (z 1 or z 2 ) on the real line, left and right corners of the eye are configurations with z 1 , z 2 ∈ R and z 1 > z 2 or z 1 < z 2 . The boundary of the iris takes into account configurations where z 1 and z 2 collapse inside the complex plane. The angle along the iris keeps track of the angle at which z 1 approaches z 2 . 2.2.2. Graphs. The Kontsevich graphical calculus (in the case of linear Poisson brackets) was studied in [4] and [10]. A graph is a collection of vertices V and oriented edges E . Vertices are ordered, and the edges are ordered in a way compatible with the order of the vertices. We denote by G n,2 the set of graphs with n + 2 vertices and 2n edges verifying the following properties: i - There are n vertices of the first type 1, 2, . . . , n and 2 vertices of the second type 1, 2. ii - Edges start from vertices of the first type, 2 edges per vertex. iii - Source and target of an edge are distinct. iv - There are no multiple edges (same source and target). We are interested in the case of linear graphs. That is, vertices of the first type admit at most one incoming edge. Such graphs are superpositions of simple graphs of two types, Lie type graphs (graphs with one root as on Fig. 2) and wheel type graphs (graph with one oriented loop, as on Fig. 3). 2.2.3. The angle map and Kontsevich weights. Let p and q be two points on the upper half plane. Consider the hyperbolic angle map on C2,0 : q−p φh ( p, q) = arg ∈ T1 . (2) q−p This function admits a continuous extension to the compactification C 2,0 . Consider a graph ∈ G n,2 , and draw it in the upper half plane with vertices of the + second type on the real line. By restriction, each edge e defines an angle map φe on C n,2 .
Kontsevich Deformation Quantization and Flat Connections
exp(X)
51
exp(Y)
Fig. 2. Lie type graph with symbol (x, y) = [[x, [x, y]], y]
exp(X)
exp(Y)
Fig. 3. Wheel type graph with symbol (x, y) = tr(y 2 [x, y]x)
The ordered product =
dφe
(3)
e∈E +
is a regular 2n-form on C n,2 (which is a 2n-dim compact space). Definition 3. The Kontsevich weight of is given by the following formula: 1 w = (2π )2n
+
C n,2
.
(4)
2.3. Campbell-Hausdorff and Duflo formulas. Lie type graphs in G n,2 are binary rooted trees. Hence, to each ∈ G n,2 a simple Lie type graph one can associate a Lie word (x, y) ∈ lie2 of degree 2n in variables x, y (see Fig. 2). Similarly, if is a wheel type graph, it corresponds to an element (x, y) ∈ cy2 (see Fig. 3).
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Recall the definition of the Duflo density function duf(x, y) =
1 ( j (x) + j (y) − j (ch(x, y))) ∈ cy2 , 2
where ch(x, y) = log(e x e y ) is the Campbell-Hausdorff series and j (x) =
bn tr(x n ) n · n! n≥2
with bn the Bernoulli numbers. The following theorem relates functions ch(x, y) and duf(x, y) to the Kontsevich graphical calculus. Theorem 1 ([10], [4]). The following identities hold true: ch(x, y) = x + y +
duf(x, y) =
n≥1
simple geometric Lie type (n,2)
n≥1
simple geometric wheel type (n,2)
w (x, y),
w (x, y), m
(5)
(6)
where m is the order of the symmetry group of the graph . Here geometric means that graphs are not labeled. Note that the definition of both (x, y) and w requires an order on the set of edges, but the product w (x, y) is independent of this order. Even though ch(x, y) and duf(x, y) are defined over rationals, some of the coefficients w are very probably irrational (see example of [8]). Remark 1. Note that the associativity of the Kontsevich star product implies that the right hand side of Eq. (5) is an associative Lie series. If we denote it by χ (x, y), we have χ (χ (x, y), z) = χ (x, χ (y, z)). Then, χ (x, y) coincides (up to rescaling of arguments) with the Campbell-Hausdorff series (see e.g. Proposition 2.1 in [3]). Remark 2. Denote the right hand side of Eq. (6) by (x, y). Similarly to the previous remark, the associativity of the star product implies
(x, y) + (ch(x, y), z) = (x, ch(y, z)) + (y, z). By Proposition 2.2 in [3], this gives (x, y) = f (x) + f (y) − f (ch(x, y)). Finally, by looking at degreeone in y contributions (see Remark 8.5.5 in [5], and also [6]) one n arrives at f (x) = n wn tr(x )/n with coefficients wn given by Kontsevich graphs presented on Fig. 4. To the best of our knowledge, there is no direct computation of these graphs available in literature.
Kontsevich Deformation Quantization and Flat Connections
1
2
3
53
4 n−1
n
Fig. 4. Coefficient wn
2.4. ξ -deformation. In [14], one studies the following deformation for the CampbellHausdorff formula. Let ξ ∈ C 2,0 , ∈ G n,2 , and let π be the natural projection from C n+2,0 onto C 2,0 . We define the coefficients w (ξ ) for ξ ∈ C 2,0 as 1 . w (ξ ) = (2π )2n π −1 (ξ ) Functions w (ξ ) are smooth over C2,0 , and they are continuous over the compactification C 2,0 . The ξ -deformation of the Campbell-Hausdorff series chξ (x, y) is defined as w (ξ )(x, y). (7) chξ (x, y) = x + y + n≥1
simple geometric Lie type (n,2)
In a similar fashion, we introduce a deformation of the Duflo function, duf ξ (x, y) =
n≥1
simple geometric wheel type (n,2)
w (ξ ) (x, y). m
For ξ = (0, 1) (the right corner of the eye on Fig. 1), the expression (7) is given by the standard Campbell-Hausdorff series, and for ξ in the position α on the iris, the Kontsevich Vanishing Lemma implies chα (x, y) = x +y. By the results of [4] and [13], for ξ = (0, 1) the Duflo function duf ξ (x, y) coincides with the standard Duflo function, and for ξ in the arbitrary position α on the iris one has duf α (x, y) = 0. 2.5. Connection ω2 . In [14], one defines a connection on C2,0 with values in tder2 ,
ω2 = Fξ (x, y), G ξ (x, y) .
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A. Alekseev, C. Torossian A
Z2 Z1
Fig. 5. Extended graph
Here Fξ and G ξ are 1-forms on C2,0 taking values in lie2 . They satisfy the following two (Kashiwara-Vergne type) equations (see Theorem 1 and Theorem 2 in [14]) d chξ (x, y) = ω2 (chξ (x, y)), d duf ξ (x, y) = ω2 (duf ξ (x, y)) + div(ω2 ),
(8) (9)
where d is the de Rham differential on C2,0 , and ω2 acts on chξ (x, y) and duf ξ (x, y) as a derivation of lie2 . Let us briefly recall the construction of ω2 . We will denote by A, B ∈ G n,2 simple graphs of Lie type, and we define an extended graph A (resp. B ) as a graph with an additional edge starting at 1 (resp. 2) and ending at the root of A (resp. B), see Fig. 5. Note that n = 0 is allowed, but since the source and the target of an edge must be distinct, will have a single edge starting at 1 and ending at 2, or starting at 2 and ending at 1. Draw the extended graph in the upper half plane (with vertices of the second type corresponding to ξ ∈ C 2,0 ). Then, dφe A = e∈E A
is a 2n +1-form on C n+2,0 (which is a 2n +2-dim compact space). The push forward along the natural projection π : C n+2,0 → C 2,0 yields a 1-form on C 2,0 , ω A = π∗ ( A ). The connection ω2 = (Fξ (x, y), G ξ (x, y)) is defined by the following formula,1 ⎧ Fξ (x, y) = ω A A(x, y) ⎪ ⎪ ⎪ n≥0 A simple ⎪ ⎪ ⎪ graph of ⎨ (n,2) Lie type (10) G (x, y) = ω B B(x, y) ⎪ ξ ⎪ ⎪ n≥0 B simple ⎪ ⎪ ⎪ graph of ⎩ Lie type (n,2)
2.6. Definition of ωn and ωn∞ . We now extend this construction to an arbitrary number of vertices of the second type. Consider ∈ G p,n a simple graph of Lie type with vertices of the second type labeled 1, . . . , n. Define the extended graph (i) by adding an edge from the vertex i to the root 1 For n = 0, A = y and B = x.
Kontsevich Deformation Quantization and Flat Connections
55
of . Consider the natural projection π : C p+n,0 → Cn,0 and take the pushforward 1-form ω (i) = π∗ ( (i) ). We define 1-forms with values in lien , Fi =
dφh (z i , z j ) xj + 2π j=i
p>0
simple graph of Lie type ( p,n)
ω (i) (x1 , . . . , xn ).
Here the first term gives an explicit expression of the p = 0 contribution. The expression ωn = (F1 , . . . , , Fn ) defines a connection with values in tdern . The connection ωn is smooth over Cn,0 . Over the compactification C n,0 , it belongs to the class L 1 when restricted to piece-wise differentiable curves. Hence, along such curves all iterated integrals converge, and there is a unique solution of the initial value problem dg = −gω with g(z 0 ) = 1 for the base point z 0 (e.g. by using Grönwall’s Lemma). Therefore, parallel transports are well defined. The same applies to restrictions of ωn to boundary strata of C n,0 of dimension at least one. For instance, in the case of C 2,0 one can consider a path along the eyelid, or a generic path from the corner of C 2,0 to the iris. We will need restrictions of ωn to various boundary strata of co-dimension one of C n,0 . First of all, there is a stratum “at infinity” equal to the configuration space Cn of n points on the complex plane (modulo the diagonal action of the group z → az + b for a ∈ R+ , b ∈ C). We denote the corresponding connection ωn∞ . It is given by the same formula as ωn with the configuration space C n,0 replaced by C n , and the hyperbolic angle is replaced with the Euclidean angle. In particular, we have dφe (z i , z j ) ωn∞ = + ··· , ti, j 2π i< j
where ti, j = (0, . . . , x j , 0, . . . , xi , . . . , 0) with x j placed at the position i and xi at the position j. Next, for the first q points collapsing inside the upper half plane, we have a stratum of the form Cq × Cn−q+1,0 . We denote the natural projections by π1 and π2 , and obtain an expression for the connection ωn |Cq ×Cn−q+1,0 = π1∗ (ωq∞ )1,2,...,q + π2∗ ωn−q+1
12...q,q+1,...,n
.
Other choices of points to collapse can be described by using the action of the symmetric group Sn . A similar property holds for the connection ωn∞ on the stratum Cq × Cn−q+1 corresponding to (the first) q points collapsing together, ∞ ωn∞ |Cq ×Cn−q+1 = π1∗ (ωq∞ )1,2,...,q + π2∗ (ωn−q+1 )12...q,q+1,...,n .
In the case when (the first) q points are collapsing to a point on the real axis, we obtain the stratum Cq,0 × Cn−q,1 , and for the connection we get ωn |Cq,0 ×Cn−q,1 = π1∗ ωq
1,2,...,q
+ π2∗ ωn−q+1
12...q,q+1,...,n
|Cn−q,1 .
Note that the restriction of the connection form ωn to the boundary stratum Cn−1,1 corresponds to configurations with the point z 1 on the real axis, and it has the following property: its first component (as an element of tdern ) vanishes since the 1-form dφe vanishes when the source of the edge e is bound to the real axis.
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3. Zero Curvature Equation and Applications One of our main results is flatness of the connection ωn . 3.1. The zero curvature equation. Theorem 2. The connection ωn is flat. That is, the following 2-form on Cn,0 vanishes: 1 dωn + [ωn , ωn ] = 0. 2
(11)
Proof. The argument is based on the Stokes formula, and we give details in the case of ω2 . The case of arbitrary n is treated in a similar fashion. Let Cξ be a small circle around ξ ∈ C2,0 , ξ be the corresponding disk, and consider π −1 ( ξ ). Since the forms A , B are closed, we have d A A(x, y), B B(x, y) = 0. π −1 ( ξ )
A
B
By applying Stokes formula and the definition of the connection ω2 , one obtains 0= ω2 + A A(x, y), B B(x, y) . (12)
Cξ
∂(π −1 (z))
A
B
z∈ ξ
By using again Stokes’s formula (on the disk ξ ), one can rewrite the first term in the form ξ dξ ω2 . For the second term, one obtains contributions from the boundary strata of codimension one. The usual arguments in Kontsevich theory rule out strata where more than two points collapse (by the Kontevich Vanishing Lemma), and strata corresponding to collapse of internal edges (by Jacobi identity). The remaining strata correspond to collapsing of a vertex of the first type and a vertex of the second type. Figures below illustrate different cases of such boundary strata (for the first component): here [x, B]·∂x A(x, y) = d d d A(x + [x, B], y)| =0 and [y, B] · ∂ y A(x, y) = d A(x, y + [y, B])| =0 . – Fig. 6 computes terms of the type [x, ωB B(x, y)] · ∂x (ω A A(x, y)) . – Fig. 7 represents terms of the type [y, ω B B(x, y)] · ∂ y (ω A A(x, y)) . A B
Z 1
Z2
Fig. 6. [x, ω B B(x, y)] · ∂x (ω A A(x, y))
Kontsevich Deformation Quantization and Flat Connections
57
A B
Z
1 Z2
Fig. 7. [y, ω B B(x, y)] · ∂ y (ω A A(x, y)) B
A
Z1 Z2
Fig. 8. [ω A A(x, y), ω B B(x, y)]
– Fig. 8 computes terms of the type [ω A A(x, y), ωB B(x, y)]. This term appears only once. It corresponds to the componentwise bracket in lie2 × lie2 , 1 [ω2 (x, y), ω2 (x, y)]lie2 . 2 These are exactly the three terms of the bracket 21 [ω2 , ω2 ]tder (see Eq. (1)). By Eq. (12), one gets 1 dω2 + [ω2 , ω2 ] = 0. 2
ξ Since the curvature dω2 + 21 [ω2 , ω2 ] is a continuous function of ξ , we conclude that it vanishes on C2,0 . 3.2. Parallel transport and symmetries. In this section we discuss various properties of the connection ω2 , including the induced holonomies and their symmetries. 3.2.1. Parallel transport. Since ω2 is flat, the equation d g = −gω2 ,
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α
Fig. 9. A simple path from Duflo-Kontsevich star product to standard product
has a local solution on C2,0 with values in T Aut2 = exp(tder2 ). By abuse of notations we write (u, v) ∈ T Aut2 for an element acting on generators by x → Adu x = uxu −1 , y → Adv y = vyv −1 . Take the initial data gα = 1 for α on the iris of C 2,0 (see Fig. 9), and consider a path from α to ξ . The value at ξ for the parallel transport is well-defined since the connection is integrable, and by the flatness property it only depends on the homotopy class of the path. Integrating Eq. (8), we obtain gξ (chξ (x, y)) = chα (x, y) = x + y. Recall [1] that for ξ = (0, 1) this parallel transport F defines a solution of the Kashiwara-Vergne conjecture [9]. We conclude that this solution is independent of the choice of a path in the trivial homotopy class (the straight line joining α = 0 and ξ = (0, 1)). 3.2.2. Holonomy. Solutions of equation d g = −gω2 are not globally defined on C 2,0 because of the holonomy around the iris. Lemma 1. The restriction of ω2 to the iris is equal to ωθ = d2πθ (y, x). The holonomy around the Iris H2π is given by the inner automorphism (exp(x+y), exp(x+y)) ∈ T Aut2 . Proof. When the point ξ ∈ C 2,0 reaches the iris, the Kontsevich angle map degenerates to the Euclidean angle map on the complex plane, and the connection ω2 is replaced by ω2∞ , ω2∞ =
w∞A A(x, y),
w∞ B B(x, y) .
Since the Euclidean angle is rotation invariant, so is the 1-form w∞A . Hence, it is sufficient ∞ to compute T1 w A . By the Kontsevich Vanishing Lemma (see [11] § 6.6), integrals of 3 and more angle 1-forms vanish. Therefore, for A a nontrivial graph one gets T1 w∞A = 0 which implies w∞A = 0. As a result, we obtain the connection ωθ by adding two trivial graph contributions, ωθ =
dθ dθ dθ (y, 0) + (0, x) = (y, x). 2π 2π 2π
Let’s integrate the equation dθ g = −gωθ over the boundary of the iris. Note that t = (y, x) is actually an inner derivation since t (x) = [x, y] = [x, x + y] and t (y) = [y, x] = [y, x + y]. We conclude that the parallel transport around the iris is given by Hθ = exp(θ t/2π ) = (exp(θ (x + y)/2π ), exp(θ (x + y)/2π )). In particular, for θ = 2π we obtain H2π = (exp(x + y), exp(x + y)), as required.
Kontsevich Deformation Quantization and Flat Connections
59
3.2.3. Symmetries of the connection. Consider the following involutions on C 2,0 : σ1 : (z 1 , z 2 ) → (z 2 , z 1 ) and σ2 : (z 1 , z 2 ) → (−¯z 1 , −¯z 2 ). Identifying C 2,0 with the Kontsevich eye (see Fig. 1), σ1 is the reflection with respect to the center of the eye, and σ2 is the reflection with respect to the vertical axis (see Fig. 1). We shall denote by τ1 and τ2 the following involutions of tder2 , τ1 : (F(x, y), G(x, y)) → (G(y, x), F(y, x)), τ2 : (F(x, y), G(x, y)) → (F(−x, −y), G(−x, −y)). They lift to involutions of T Aut2 . Proposition 1. The connection ω2 verifies σ1∗ (ω2 ) = τ1 (ω2 ),
σ2∗ (ω2 ) = τ2 (ω2 ).
Proof. The involution σ1 simply exchanges the colors x and y of all graphs which induces the involution τ1 on tder2 . The involution σ2 flips the sign of the one form dφe for each edge (since the reflection changes sign of the Euclidean angle), and changes the orientation of each integration over a complex variable. Hence, for a graph with n internal vertices we collect −1 to the power (2n +1)+n ≡ n +1 (mod 2). Corresponding rooted trees have exactly n +1 leaves. Hence, one should change a sign of each leaf which results in applying the involution τ2 . Let F ∈ T Aut2 be the parallel transport of the equation dg = −gω2 for the straight path between the position 0 on the iris to the right corner of the eye C 2,0 . Since the path is invariant under the composition σ1 ◦ σ2 = σ2 ◦ σ1 , the parallel transport is invariant under τ = τ1 τ2 = τ2 τ1 , τ (F) = F. In order to discuss the involution τ1 and τ2 separately, we need the following lemma. Lemma 2. The parallel transport along the lower eyelid in the counter-clockwise direction is equal to R = (exp(y), 1) ∈ T Aut2 . Proof. The connection restricted to the lower eyelid has a trivial second component because edges starting from the real line give rise to a vanishing 1-form. Write the corresponding parallel transport as R = (g(x, y), 1) ∈ T Aut2 . Integrating the equation d chξ (x, y) = ω2 (chξ (x, y)) along the lower eyelid, we obtain R(ch(x, y)) = ch(Ad g(x,y) x, y) = ch(y, x), and this equation implies g(x, y) = exp(y), as required.
Note that the path along the upper eyelid (oriented in the counter-clockwise direction) can be obtained by applying the involution σ1 to the lower eyelid. Hence, the corresponding parallel transport is given by τ1 (R) = R 2,1 . Proposition 2. The element F ∈ T Aut2 verifies the following identities: F = et/2 τ1 (F)τ1 (R −1 ) = e−t/2 τ1 (F)R. Proof. These equations express the flatness condition for two contractible paths shown on Fig.10. These equations can be re-interpreted as the property of the parallel transport under the involution τ1 , F 2,1 = τ1 (F) = e−t/2 F R 2,1 = et/2 F R −1 .
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A. Alekseev, C. Torossian
21 R
e
τ 1(F)
τ 1(F)
F −t/2 e
t/2
F
12 R
Fig. 10. The R matrix
s
0
1
Fig. 11. A simple contribution in ω3∞
4. Connection ω3 and Associators 4.1. Connection ω3∞ . An important element of our construction is the connection ω3∞ which is built using the Kontsevich technique applied to the complex plane equipped with the Euclidean angle form. Example 2. Consider ω3∞ for 3 points situated on the real line at the positions 0, s, 1 1 (see Fig. 11). The Euclidean angle form is dθ with tan(θ ) = xy22 −y −x1 . One of the simplest trees is shown on Fig. 11. The corresponding 3-form is given by the following expression: y 1 1 1 y y ∧ ∧ d d d y 2 y 2 x (x − s) (x − 1) 1 + ( xy )2 1 + ( x−s ) 1 + ( x−1 ) =−
y2 d x ∧ dy ∧ ds. (x 2 + y 2 )((x − s)2 + y 2 )((x − 1)2 + y 2 )
(13)
By [2] §I.1, the orientation is given by −d x ∧ dy ∧ ds, and one gets log(1 − s) log(s) 1 ω∞(1) = − 2 + ds. 8π s (1 − s) 1 1 This 1-form is integrable (semi-algebraic), and one has 0 ω∞(1) = 24 . Remark 3. Let α : z → z¯ be a the complex conjugation, and let κ be an involution of tder3 defined by formula (a(x, y, z), b(x, y, z), c(x, y, z)) → (a(−x, −y, −z), b(−x, −y, −z), c(−x,−y,−z)). Then, α ∗ ω3∞ = κ(ω3∞ ). The proof is similar to the one of Proposition 1.
Kontsevich Deformation Quantization and Flat Connections
61
Proposition 3. Connection ω3∞ is flat and takes values in krv3 . Proof. The flatness condition dω3∞ + 21 [ω3∞ , ω3∞ ] = 0 is obtained by replacing the hyperbolic angle form on the upper half-plane by the Euclidean angle form on the complex plane in the proof of Theorem 2. Define ch∞ w∞ (ξ )(x, y, z), ξ (x, y, z) = x + y + z + n≥1
simple geometric Lie type (n,3)
and duf ∞ ξ (x, y, z) =
n≥1
simple geometric wheel type (n,3)
w∞ (ξ ) (x, y, z). m
Similarly to (8) and (9), ω3∞ satisfies equations ∞ ∞ d ch∞ ξ (x, y, z) = ω3 (chξ (x, y, z)),
∞ d duf ∞ ξ (x, y, z) = ω3 (duf ξ (x, y, z)) + div(ω3 ).
By the Kontsevich Vanishing Lemma (Lemma 6.6 in [11]), w∞ (ξ ) = 0 for all non∞ trivial graphs. Hence, ch∞ ξ (x, y, z) = x + y + z and duf ξ (x, y, z) = 0. Therefore, the ∞ differential equations for ω3 yield ω3∞ (x + y + z) = 0, div(ω3∞ ) = 0. That is, ω3∞ takes values in krv3 as required.
Results of the previous proposition extend to connections ωn∞ . 4.2. Associator. We will use the following notation. Recall that T = (u, v) ∈ T Aut2 is an automorphism of lie2 acting by (x, y) → (Adu x, Adv y). We denote T 1,2 = (u(x, y), v(x, y), 1) ∈ T Aut3 , T 12,3 = (u(x +y, z), u(x +y, z), v(x + y, z) ∈ T Aut3 , etc. For F ∈ T Aut2 the parallel transport from the iris to the right corner of the eye C 2,0 , we define = F 1,23 F 23 (F 12,3 F 12 )−1 ∈ T Aut3 . This element is the main topic of study in this section. Proposition 4. The element coincides with the parallel transport for the equation dg = −gω3∞ between positions 1(23) to (12)3.
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Proof. Consider the following path in the configuration space C 3,0 : First, place z 1 , z 2 , z 3 on the stratum at infinity and move them along the horizontal line (the real axis of the complex plane at infinity) from the position 1(23) (z 2 and z 3 collapsed) to the position (12)3 (z 1 and z 2 collapsed). The connection at infinity is ω3∞ , and we denote the corresponding parallel transport by ∞ . Next, make z 3 descend from the stratum at infinity to plus infinity of the real axis of the upper half-plane (this corresponds to moving to the right corner of the eye for the points (12) and 3). On this stratum, the connection is ω212,3 , and the parallel transport is given by F 12,3 . Continue with descending both z 1 and z 2 to the real axis. The connection on this stratum is ω21,2 , and the parallel transport gives F 1,2 . Then, move z 2 to the vicinity of z 3 along the real axis of the upper half-plane. The parallel transport is trivial since the connection vanishes along the real axis. Finally, lift z 2 and z 3 from the real axis and make them collapse on each other (parallel transport (F 2,3 )−1 ), and lift z 1 from the real axis and make it collapse with z 2 = z 3 (parallel transport (F 1,23 )−1 ). Thus, we made a loop and returned to the position 1(23) at infinity. This loop is contractible, and the total parallel transport is trivial by the flatness property of the connection. Hence, ∞ F 12,3 F 1,2 (F 2,3 )−1 (F 1,23 )−1 = 1, and we obtain ∞ = F 1,23 F 2,3 (F 12,3 F 1,2 )−1 = , as required. We have = F 1,23 F 23 (F 12,3 F 12 )−1 , and the first term of is given by 1 1 ([y, z], −[x, z], [y, z]) + · · · = 1 + [t 1,2 , t 2,3 ] + · · · , 24 24 1 1 (see with t 1,2 = (y, x, 0) and t 2,3 = (0, z, y). Here we used the fact that 0 ω∞(1) = 24 the example considered above), and the minus sign is coming from the orientation of the boundary stratum C3 ⊂ ∂C3,0 . The main properties of the element are summarized in the following theorem. (x, y, z) = 1 −
Theorem 3. The element satisfies associator axioms. Proof. The axioms to verify are as follows: is a group like element and it is a solution of the following equations: 3,2,1 1,2,3 = 1,
(i)
= , 1 1 1 3,1,2 2,3,1 exp ± t12 exp ± t13 exp ± t23 1,2,3 2 2 2 1 = exp ± (t12 + t13 + t23 ) . 2
1,2,34
12,3,4
2,3,4
1,23,4
1,2,3
(ii)
(iii)
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s B s=t=1
s=t t=1
s=t=0 s=0 t A
Fig. 12. The compactification space of 4 positions on a line
(i) - (1,2,3 )−1 is the parallel transport between positions 1(23) to (12)3. Let β be the reflection with respect to the vertical axis. Then, similar to Proposition 1, we obtain β ∗ ω3∞ = (ω3∞ )3,2,1 . Hence, we get 3,2,1 = (1,2,3 )−1 , as required. (iii) - Consider the following path: 1(23) → (12)3 → (21)3 → 2(13) → 2(31) → (23)1 → 1(23). Here the last step is by moving the collapsed pair (23) around the point 1 along the iris of the corresponding C 2,0 stratum. By the flatness property, the total parallel transport is trivial. This gives exactly the pair of hexagonal equations (iii) by using the equation (i) and the fact that c = t12 + t13 + t23 is central in sder3 (see Proposition 3.4, [3]). The plus or minus sign in equation (iii) depends on the choice of the (clockwise or anti-clockwise) semi-circle for each exchange of two points (e.g. 1 moves above 2 or below 2 in the move (12)3 → (21)3). (ii) - Consider four points z 1 = 0, z 2 = s, z 3 = t, z 4 = 1 on the horizontal line (the real axis of the complex plane) representing a point of the configuration space of the complex plane placed at infinity of C 4,0 . The path ((12)3)4 → (1(23))4 → 1((23)4) → 1(2(34)) is contractible. Hence, the parallel transport defined by the flat connection ω4∞ is trivial. It is easy to see that it reproduces the pentagon equation (ii) (see Fig. 12).
Note that is an element of the group K RV3 which contains the subgroup T3 = exp(t3 ). If is actually an element of T3 , it becomes a Drinfeld associator. Since is even (κ() = , by Remark 3), it does not coincide with the Knizhnik-Zamolodchikov associator (the only known Drinfeld associator defined by an explicit formula). Conjecture 1. The element is a Drinfeld associator. That is, ∈ T3 ⊂ K RV3 .
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In a recent work [12], Severa and Willwacher prove this conjecture affirming that the element is indeed a new Drinfeld associator admitting a presentation as a parallel transport of the flat connection ω3∞ defined by explicit formulas. Acknowledgements. We thank D. Barlet and F. Brown for useful discussions and remarks. We are grateful to P. Severa and T. Willwacher for letting us know of their forthcoming work [12]. We are indebted to the referee of this paper for valuable remarks and suggestions. Research of A.A. was supported in part by the grants 200020-121675 and 200020-120042 of the Swiss National Science Foundation. Research of C.T. was supported by CNRS.
References 1. Alekseev, A., Meinrenken, E.: On the Kashiwara-Vergne conjecture. Invent. Math. 164(3), 615–634 (2006) 2. Arnal, D., Manchon, D., Masmoudi, M.: Choix des signes pour la formalité de Kontsevich. Pacific J. Math. 203, 23–66 (2002) 3. Alekseev, A., Torossian, C.: The Kashiwara-Vergne conjecture and Drinfeld’s associators. http://arxiv. org/abs/0802.4300v1[math.QA], 2008 4. Andler, M., Sahi, S., Torossian, C.: Convolution of invariant distributions: proof of the Kashiwara-Vergne conjecture. Lett. Math. Phys. 69, 177–203 (2004) 5. Cattaneo, A.S., Keller, B., Torossian, C., Bruguières, A.: Déformation, quantification, théorie de Lie. Collection Panoramas et Synthèse no. 20, SMF, 2005. 6. Dito, G.: Kontsevich star product on the dual of a Lie algebra. Lett. Math. Phys. 48(4), 307–322 (1999) 7. Drinfeld, V.G.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q). (Russian) Algebra i Analiz 2, no. 4, 149–181 (1990); translation in Leningrad Math. J. 2, no. 4, 829–860 (1991) 8. Felder, G., Willwacher, T.: On the (ir)rationality of Kontsevich weights. http://arxiv.org/abs/0808. 2762v2[math.QA], 2008 9. Kashiwara, M., Vergne, M.: The Campbell-Hausdorff formula and invariant hyperfunctions. Inventiones Math. 47, 249–272 (1978) 10. Kathotia, V.: Kontsevich’s universal formula for deformation quantization and the Campbell-BakerHausdorff formula. Internat. J. Math. 11(4), 523–551 (2000) 11. Kontsevich, M.: Deformation quantization of Poisson manifolds, I. Lett. Math.Phys. 66(3), 157–216 (2003) 12. Severa, P., Willwacher, T.: Equivalence of formalities of the little discs operad. http://arxiv.org/abs/0905. 1789v1[math.QA], 2009 13. Shoikhet, B.: Vanishing of the Kontsevich integrals of the wheels. EuroConférence Moshe Flato 2000, Part II (Dijon). Lett. Math. Phys. 56(2), 141–149 (2001) 14. Torossian, C.: Sur la conjecture combinatoire de Kashiwara-Vergne. J. Lie Theory 12(2), 597–616 (2002) Communicated by A. Kapustin
Commun. Math. Phys. 300, 65–94 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1112-x
Communications in
Mathematical Physics
Natural Equilibrium States for Multimodal Maps Godofredo Iommi1 , Mike Todd2, 1 Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna 4860,
Santiago, Chile. E-mail:
[email protected]
2 Departamento de Matemática Pura, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre,
687, 4169-007 Porto, Portugal. E-mail:
[email protected] Received: 22 July 2009 / Accepted: 18 May 2010 Published online: 21 August 2010 – © Springer-Verlag 2010
Abstract: This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials −t log |D f |, for the largest possible interval of parameters t. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained. 1. Introduction The class of dynamical systems whose ergodic theory is best understood is the class of hyperbolic dynamical systems, or, more generally, systems where the interesting dynamics behaves in a uniformly hyperbolic way: Axiom A maps. This is due to several reasons, one of them is the fact that these systems often have a compact symbolic model whose dynamics is well known [Bo,Ru2]. For real one-dimensional maps, Axiom A maps are defined to be the class of maps where all points are either uniformly expanded or map into an attracting basin. This class is large even within families of maps with critical points such as the quadratic family, in which case it is a dense set, see [Ly2,GS]. Note that these maps do have a compact symbolic model (see [KH, Chap. 16]). In the example of the quadratic family, maps which are not Axiom A are nowhere dense, but nevertheless have positive Lebesgue measure, see [J,BC]. Due to the rich dynamics of these systems, the expansion properties of such systems, can be very delicate. GI was partially supported by Proyecto Fondecyt 11070050 and by Research Network on Low Dimensional Systems, PBCT/CONICYT, Chile. MT is supported by FCT grant SFRH/BPD/26521/2006 and also by FCT through CMUP. Current address: Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 2215, USA.
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In recent years a great deal of attention has been paid to non-Axiom A systems which are expanding on most of the phase space, but not in all of it. The simplest example of these type of maps, namely non-uniformly hyperbolic dynamical systems, are interval maps with a parabolic fixed point (e.g. the Manneville-Pomeau map [MP]). The ergodic theory for these maps is fairly well understood [MP,PreSl,S3] and qualitatively different from the one observed in the hyperbolic case. We will study the ergodic theory of the class of maps for which the lack of hyperbolicity can be even stronger: interval maps with critical points. The techniques we develop are different from the ones used to study hyperbolic systems and systems with a parabolic fixed point. In this paper we will be devoted to study a particular branch of ergodic theory, namely thermodynamic formalism. This is a set of ideas and techniques which derive from statistical mechanics [Dobr,Si,Bo,K2,Ru2,Wa]. It can be thought of as the study of certain procedures for the choice of invariant measures. Let us stress that the dynamical systems we will consider have many invariant measures, hence the problem is to choose relevant ones. The main object in the theory is the topological pressure: Definition 1.1. Let f : X → X be a Borel function of a compact metric space X and denote by M f the set of f -invariant Borel probability measures. Let ϕ : X → [−∞, ∞] be a Borel potential. Assuming that M f = ∅, the topological pressure of ϕ with respect to f is defined, via the Variational Principle, by P f (ϕ) = P(ϕ) = sup h(μ) + ϕ dμ : μ ∈ M f and − ϕ dμ < ∞ , where h(μ) denotes the measure theoretic entropy of f with respect to μ. We refer to the quantity in the curly brackets as the free energy of μ with respect to (X, f, ϕ). Note that this is sometimes thought of as being minus the free energy; see for example [K2] for a discussion of this terminology. Note that we do not specify the regularity properties we require on the potential ϕ. If it is a continuous function, then the above definition coincides with classical notions of topological pressure (see [Wa, Chap. 9]). In this paper we will be interested in the geometric potential x → −t log |D f (x)| for some parameter t ∈ R. This function is continuous in the uniformly hyperbolic case, but is not upper/lower semicontinuous for t positive/negative for the class of dynamical systems that we will consider. A measure μϕ ∈ M f is called an equilibrium state for ϕ if it satisfies: h(μϕ ) + ϕ dμϕ = P(ϕ). In such a way, the topological pressure provides a natural way to pick up measures. Questions about existence, uniqueness and ergodic properties of equilibrium states are at the core of the theory. For instance, if the dynamical system f is transitive, uniformly hyperbolic and the potential ϕ is Hölder continuous then there exist a unique equilibrium state μϕ for ϕ and it has strong ergodic properties [Bo,Ru2]. Moreover, the hyperbolicity of the system is reflected on the regularity of the pressure function t → P(tϕ). Indeed, the function is real analytic. When the system is no longer hyperbolic, as in the case of the Manneville-Pomeau map, then uniqueness of equilibrium states may break down [PreSl] and the pressure function might exhibit points where it is not analytic, the so called phase transitions [S3].
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As mentioned above, we will consider maps for which the lack of hyperbolicity is strong: not only do the maps have critical points, but the orbit of these points can be dense. We consider a family of real multimodal maps, that is smooth interval maps with a finite number of critical points. More precisely, let F be the collection of C 2 multimodal interval maps f : I → I , where I = [0, 1], satisfying: a) the critical set Cr = Cr ( f ) consists of finitely many critical points c with critical order 1 < c < ∞, i.e., there exists a neighbourhood Uc of c and a C 2 diffeomorphism gc : Uc → gc (Uc ) with gc (c) = 0 f (x)√= f (c) ± |gc (x)|c ; b) f has negative Schwarzian derivative, i.e., 1/ |D f | is convex; c) f is topologically transitive on I ; d) f n (Cr ) ∩ f m (Cr) = ∅ for m = n. Conditions c) and d) are for ease of exposition, but not crucial. In particular, Condition c) excludes that f has any attracting cycles, or homtervals (a homterval is an interval U such that U, f (U ), f 2 (U ), . . . are disjoint and the omega limit set is not a periodic orbit). Condition d) in particular excludes that one critical point is mapped onto another. If that happened, it would be possible to consider these critical points as a ‘block’, but to simplify the exposition, we will not do that here. Condition d) also, in particular, excludes that critical points are preperiodic, a case which is easier to handle (for example by combining [KH, Chap. 16] and [Bo]) and does not require the theory we present here, see Sect. 10.3. Together c) and d) exclude the renormalisable case. Remark 1.1. General C 2 multimodal maps satisfying a) and b) have no homtervals and the non-wandering set (the set of points x ∈ I such that for arbitrarily small neighbourhoods U of x there exists n(U ) 1 such that f n (U ) ∩ U = ∅) can be broken down into finitely many elements k , on each of which f is topologically transitive, see [MvS, Sect. III.4]. However, for the maps we consider, assumption c) means this fact follows automatically without the C 2 assumption. We note that in the case where there is more than one transitive element in , for example the renormalisable case, the analysis presented in this paper can be applied to any one of the transitive elements consisting of a union of intervals permuted by f . Now let int denote the union of all elements of which consist of intervals permuted by f . If, contrary to the assumptions on F above, int did not cover I then there would be a (hyperbolic) Cantor set consisting of points which are always outside int . Dobbs [D3] showed that for renormalisable maps these hyperbolic Cantor sets can give rise to phase transitions in the pressure function not accounted for by the behaviour of critical points themselves. Remark 1.2. The smoothness of our maps is important for two further reasons: to allow us to bound distortion on iterates, and to guarantee the existence of ‘local unstable manifolds’. For the first, the tool we use is the Koebe Lemma, see [MvS, Sect. IV]. The negative Schwarzian condition we impose still allows us to use this for C 2 maps. For a detailed explanation of this issue see [C]. Given a measure μ ∈ M f , the existence of local unstable manifolds was used in [B1,BT1] to show the existence of some natural ‘inducing schemes’ (see Sect. 3). As shown by Ledrappier [L], and later generalised by Dobbs [D4] (see the Appendix), we only need a C 1+α condition on f to guarantee the existence of local unstable manifolds. Note that our class F includes transitive Collet-Eckmann maps, that is maps where |D f n ( f (c))| grows exponentially fast. Therefore the set of quadratic maps in F has positive Lebesgue measure in the parameter space of quadratic maps (see [J,BC]).
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In the Appendix we show that our theory can be extended to a slightly more general class of maps, similar to the above, but only piecewise continuous. As mentioned above, we will be particularly interested in the thermodynamic formalism for the geometric potentials x → −t log |D f |. The study of these potentials has various motivations, for example the relevant equilibrium states and the pressure function are related to the Lyapunov spectrum, see for example [T]. Moreover, important geometric features are captured by this potential. Indeed, in several settings, the equilibrium states for this family are associated to conformal measures on the interval. This allows the study of the fractal geometry of dynamically relevant subsets of the space. Moreover, by [L] any equilibrium state μ for the potential x → − log |D f | is an absolutely continuous invariant probability measure (acip) provided log |D f | dμ > 0. For μ ∈ M f , we define the Lyapunov exponent of μ as λ(μ) := log |D f | dμ. We let λ M := sup{λ(μ) : μ ∈ M f }, λm := inf{λ(μ) : μ ∈ M f }. Remark 1.3. Our assumptions on f ∈ F, particularly non-flatness of critical points and a lack of attracting periodic cycles, means that by [Pr], λm 0. We let p(t) := P(−t log |D f |) and define t − := inf{t : p(t) > −λ M t} and t + := sup{t : p(t) > −λm t}.
(1)
Note that if t − ∈ R (resp. t + ∈ R) then p is linear for all t t − (resp. t t + ). We will later prove that for maps in F, t − = −∞. We prove in Proposition 8.1 that t + > 0. In some cases t + = ∞. As we will show later, for non-Collet Eckmann maps with quadratic critical point, λm = 0 and t + = 1. [MS] suggests that there should also be Collet-Eckmann maps with t + ∈ (1, ∞). In Proposition 9.2 we prove that under certain assumptions t + 1: we expect that to be true for any map f ∈ F. The following is our main theorem. Theorem A. For f ∈ F and t ∈ (−∞, t + ) there exists a unique equilibrium measure μt for the potential −t log |D f |. Moreover, the measure μt has positive entropy. A classical way to show the existence of equilibrium states is to use upper semicontinuity of entropy and thepotential ϕ (see [K2, Chap. 4]), and in particular the upper semicontinuity of μ → ϕ dμ. However, in our setting even though, as noted in [BK], for f ∈ F the entropy map is upper semicontinuous, the existence of equilibrium measures in the above theorem is not guaranteed since the potential −t log |D f | is not upper semicontinuous for t > 0. So for example, by [BK, Prop. 2.8] for unimodal maps satisfying the Collet-Eckmann condition, μ → −λ(μ) is not upper semicontinuous. Theorem A generalises [BK] which applies to unimodal Collet-Eckmann maps for a small range of t near 1; [PS] which applies to a subset of Collet-Eckmann maps, but for all t in a neighbourhood of [0, 1]; and [BT2, Theorem 1] which applies to a class of non-Collet Eckmann multimodal maps with t in a left-sided neighbourhood of 1.
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In order to prove Theorem A we use the theory of inducing schemes developed in [B1,BT1,BT2,T]. Let us note that the thermodynamic formalism is understood for certain complex rational maps. For example, Przytycki and Rivera-Letelier [PrR] proved that if f : C → C is a rational map of degree at least two, is expanding away from the critical points and has ‘arbitrarily small nice couples’ then the pressure function p is real analytic in a certain interval. These conditions are met for a wide class of rational maps including topological Collet-Eckmann rational maps, any at most finitely renormalisable polynomial with no indifferent periodic orbits, as well as every real quadratic polynomial. Also see [DU], where they show the existence and uniqueness of equilibrium states for all rational maps with degree greater than or equal to two, for all Hölder potentials ϕ with sup ϕ < P(ϕ). Related to the above are the regularity properties of the pressure function. Definition 1.2. Let ϕ : [0, 1] → R be a Borel potential. The pressure function has a first order phase transition at t0 ∈ R if p is not differentiable at t = t0 . The pressure function, being the supremum of convex functions, is convex (see [Roc, p. 35]) and when finite is continuous (see [Roy, p. 113]). This implies that the left and right derivatives D − p(t) and D + p(t) at each t exist. Moreover, the pressure, when finite, can have at most a countable number of points ti where it is not differentiable (i.e, Dp − (ti ) = D + p(ti )), hence of first order phase transitions. The regularity of the pressure is related to several dynamical properties of the system. For example, it has deep connections to large deviations [E] and to different modes of recurrence [S3,S5]. In Sect. 8 we prove that the pressure function restricted to the interval (−∞, t + ) not only does not have first order phase transitions, but it is C 1 . Theorem B. For f ∈ F, the pressure function p is C 1 , strictly convex and strictly decreasing in t ∈ (−∞, t + ). First order phase transitions are also related to the existence of absolutely continuous invariant probability measures. If p(t) = 0 for all t 1 and there is an acip, then the pressure function is not differentiable at t = 1. This occurs for example if f ∈ F is unimodal and non-Collet Eckmann, but has an acip (see [NS]). The following proposition gives the converse result. Proposition 1.1. Let f ∈ F be such that p(1) = 0. If the pressure function has a first order phase transition at t = 1 then the map f has an acip. We summarise some of the other results we present here for the potential x → −t log |D f (x)| in the simpler case of unimodal maps with quadratic critical point in the following proposition. Proposition 1.2. If f ∈ F is unimodal, non-Collet Eckmann and c = 2 then p is C 1 , strictly convex and decreasing throughout (−∞, 1) and p(t) = 0 for all t 1. Moreover, (a) if f has no acip then p is C 1 throughout R; (b) if f has an acip then p has a first order phase transition at t = 1. The paper is organised as follows. In Sect. 2 we give an introduction to the theory of thermodynamic formalism for countable Markov shifts, which was developed by Mauldin and Urba´nki and by Sarig. In Sect. 3 we give some preliminary results on inducing
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schemes, which will allow us to code any of our systems by a countable Markov shift. In Sect. 4 we show that the inducing schemes in Sect. 3 have some of the properties which will allow us to produce equilibrium states for our systems. In Sect. 5 we prove the most technically complex part of our paper which gives us the existence of equilibrium states for our systems. Section 6 gives details of the uniqueness of these equilibrium states which then allows us to prove Theorem A in Sect. 7. In Sect. 8 we prove Theorem B and in Sect. 9 we prove Propositions 1.1 and 1.2. In Sect. 10 we discuss statistical properties of the measures constructed, the ergodic optimisation problem and the case in which the critical points are preperiodic. Finally in the Appendix we show how the results of this paper extend to a class of Lorenz-like maps, of the kind studied by Rovella [Rov] and Keller and St Pierre [KStP]. Note that many of the results we quote in this paper are proved using the theory of Markov extensions introduced by Hofbauer. To prove our main theorems it is not necessary to explain this theory in any detail since it is sufficient to quote results from elsewhere. However, for a short description of this construction, see the Appendix. 2. Preliminaries: Countable Markov Shifts In this section we present the theory of countable Markov shifts: an extension of the finite case, and the relevant model for many non-uniformly hyperbolic systems, including maps in F. Let σ : → be a one-sided Markov shift with a countable alphabet S. That is, there exists a matrix (ti j ) S×S of zeros and ones (with no row and no column made entirely of zeros) such that = {x ∈ S N0 : txi xi+1 = 1 for every i ∈ N0 }, and the shift map is defined by σ (x0 x1 · · · ) = (x1 x2 · · · ). We say that ( , σ ) is a countable Markov shift. We equip with the topology generated by the cylinder sets Ci0 ···in = {x ∈ : x j = i j for 0 j n}. Given a function ϕ : → R, for each n 1 we set Vn (ϕ) = sup {|ϕ(x) − ϕ(y)| : x, y ∈ , xi = yi for 0 i n − 1}. We say that ϕ has summable variations if ∞ n=2 Vn (ϕ) < ∞. We will sometimes refer to ∞ n=2 Vn (ϕ) as the distortion bound for ϕ. Clearly, if ϕ has summable variations then it is continuous. We say that ϕ is weakly Hölder continuous if Vn (ϕ) decays exponentially. If this is the case then it has summable variations. In what follows we assume ( , σ ) to be topologically mixing (see [S1, Sect. 2] for a precise definition). It is a subtle matter to define a notion of topological pressure for countable Markov shifts. Indeed, the classical definition for continuous maps on compact metric spaces is based on the notion of (n, ε)-separated sets (see [Wa, Chap. 9]). This notion depends upon the metric of the space. In the compact setting, since all metrics generating the same topology are uniformly equivalent, the value of the pressure does not depend upon the metric. However, in non-compact settings this is no longer the case. Based on work of Gurevich [Gu1,Gu2], Sarig [S1] introduced a notion of pressure for countable Markov shifts which does not depend upon the metric of the space and which satisfies a Variational Principle. Let ( , σ ) be a topologically mixing countable Markov shift, fix
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a symbol i 0 in the alphabet S and let ϕ : → R be a potential of summable variations. We let Z n (ϕ, Ci0 ) := exp (Sn ϕ(x)) χCi0 (x), (2) x:σ n x=x
where χCi0 is the characteristic function of the cylinder Ci0 ⊂ , and Sn ϕ(x) := ϕ(x) + · · · + ϕ ◦ σ n−1 (x). Moreover, the so-called Gurevich pressure of ϕ is defined by 1 P G (ϕ) := lim log Z n (ϕ, Ci0 ). n→∞ n Since σ is topologically mixing, one can show that P G (ϕ) does not depend on i 0 . We define Mσ (ϕ) := μ ∈ Mσ : − ϕ dμ < ∞ . If ( , σ ) is the full-shift on a countable alphabet then the Gurevich pressure coincides with the notion of pressure introduced by Mauldin and Urba´nski [MU1]. Furthermore, the following property holds (see [S1, Theorem 3]): Proposition 2.1 (Variational Principle). If ϕ : → R has summable variations and P G (ϕ) < ∞ then P G (ϕ) = sup h μ (σ ) + ϕ dμ : μ ∈ Mσ (ϕ) .
Let us stress that the right hand side of the above inequality only depends on the Borel structure of the space and not on the metric. Therefore, a notion of pressure which is to satisfy the Variational Principle need not depend upon the metric of the space. The Gurevich pressure also has the property that it can be approximated by its restriction to compact sets. More precisely [S1, Cor. 1]: Proposition 2.2 (Approximation property). If ϕ : → R has summable variations then P G (ϕ) = sup{Pσ |K (ϕ) : K ⊂ : K = ∅ compact and σ -invariant}, where Pσ |K (ϕ) is the classical topological pressure on K . We consider a special class of invariant measures. As in [MU2] (see also [S4]), we say that μ ∈ Mσ is a Gibbs measure for the function ϕ : → R if for some constants P, C > 0 and every n ∈ N and x ∈ Ci0 ···in we have μ(Ci0 ···in ) 1 C. C exp (−n P + Sn ϕ(x)) This definition is analogous to that in the finite Markov shift case considered by Bowen [Bo]. We refer to any such C as a distortion constant for the Gibbs measure. It was proved by Mauldin and Urba´nski [MU2] that if ( , σ ) is a full-shift and the function ϕ is of summable variations with finite Gurevich pressure P G (ϕ) then it has an invariant Gibbs G measure. Moreover P = P (ϕ), and if − ϕ dμ < ∞ then μ is an equilibrium state for ϕ. Furthermore, this is the unique equilibrium state for ϕ by [MU2, Theorem 3.5] (note that this was later generalised for any topologically mixing countable Markov shift in [BuS]).
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3. Inducing Schemes In order to prove Theorem A we will use the machinery of inducing schemes. We will use the fact that inducing schemes for the system (I, f ) can be coded by the full-shift on countably many symbols. Given f ∈ F, we say that (X, {X i }i , F, τ ) is an inducing scheme for (I, f ) if • X is an interval and {X i }i is a finite or countable collection of disjoint intervals such that F maps each X i diffeomorphically onto X , with bounded distortion on all iterates (i.e. there exists K > 0 so that if there exist i 0 , . . . , i n−1 and x, y such that F j (x), F j (y) ∈ X i j for j = 0, 1, . . . , n − 1 then 1/K D F n (x)/D F n (y) K ); • τ | X i = τi for some τi ∈ N and F| X i = f τi . If x ∈ / ∪i X i then τ (x) = ∞. The function τ : ∪i X i → N is called the inducing time. It may happen that τ (x) is the first return time of x to X , but that is certainly not the general case. For ease of notation, we will write (X, F, τ ) = (X, {X i }i , F, τ ) and moreover, frequently write (X, F) = (X, F, τ ). We denote the set of points x ∈ I for which there exists k ∈ N such that τ (F n ( f k (x))) < ∞ for all n ∈ N by (X, F)∞ . Given an inducing scheme (X, F, τ ), we say that a probability measure μ F is a lift of μ if for any μ-measurable subset A ⊂ I , μ(A) =
i −1 τ 1 μ F (X i ∩ f −k (A)). τ dμ F X
i
(3)
k=0
Conversely, given a measure μ F for (X, F), we say that μ F projects to μ if (3) holds. Note that if (3) holds then μ F is F-invariant if and only if μ is f -invariant. We call a measure μ compatible with the inducing scheme (X, F, τ ) if • μ(X ) > 0 and μ (X \(X, F)∞ ) = 0; and • there exists a measure μ F which projects to μ by (3): in particular X τ dμ F < ∞ (equivalently μ F ∈ M F (−τ )). Remark 3.1. Given f ∈ F and an ergodic measure μ ∈ M f with positive Lyapunov exponent, there exists an inducing scheme (X, F, τ ) with a corresponding F−invariant measure μ F , see for example [BT2, Theorem 3]. Definition 3.1. Let (X, F, τ ) be an inducing scheme for the map f . Then for a potential ϕ : I → R, the induced potential for (X, F, τ ) is given by
(x) = F (x) := Sτ (x) ϕ(x). Note that in particular for the potential log |D f |, the induced potential for a scheme (X, F) is log |D F|. Moreover, the map x → log |D F(x)| has summable variations (see for example [BT2, Lemma 8]). Note that if (X, F, τ ) is some inducing scheme for the map f ∈ F and if ∂ X ∈ / (X, F)∞ , then the system F : (X, F)∞ → (X, F)∞ is topologically conjugated to the full-shift on a countable alphabet. For an inducing scheme (X, F, τ ) and a potential ϕ : X → [−∞, ∞] with summable variations, we can define the Gurevich pressure as in Sect. 2, and denote it by PFG (ϕ), where we drop the subscript if the dynamics is clear.
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In fact the domains for the inducing schemes used above come from the natural cylinder structure of the map f ∈ F. More precisely, the domains X are n-cylinders f coming from the so-called branch partition: the set P1 consisting of maximal intervals j f on which f is monotone. So if two domains Ci1 , C1 ∈ P1 intersect, they do so only at f elements of Cr . The set of corresponding n-cylinders is denoted Pn := ∨nk=1 f −k P1 . f We let P0 := {I }. For an inducing scheme (X, F) we use the same notation for the corresponding n-cylinders PnF . Note the transitivity assumption on our maps f implies f that P1 is a generating partition for any Borel probability measure. 4. Zero Pressure Schemes For t ∈ R, we let ψt := −t log |D f | − p(t). Similarly, for an inducing scheme (X, F) the induced potential is t . As in [PS,BT1, BT2] in order to apply the theory developed by Mauldin and Urba´nski and later by Sarig, we need to find an inducing scheme (X, F, τ ) so that P G (t ) = 0. Then [MU2, Cor. 2.10] gives a Gibbs measure for (X, F, t ), which if it projects to a measure in M f by (3), must be an equilibrium state by the Abramov formula. The main purpose of this section is to show that there are inducing schemes with P G (t ) = 0. We note that a major difficulty when working with inducing schemes is that, in general, no single inducing scheme is compatible with all measures of positive Lyapunov exponent. As a direct consequence of work by Bruin and Todd [BT2, Remark 6] we obtain in Lemma 4.3 that for each > 0, there exists η > 0 and a finite number of inducing schemes for which any measure of entropy greater than η is compatible with one of them. This will allow us to prove that for each t ∈ (t − , t + ) there exists an inducing scheme for which P(t ) = 0 and such that the pressure, p(t), can be approximated with f -invariant measures of positive entropy compatible with the inducing scheme. Proposition 4.1. For each t ∈ (t − , t + ), there exist an inducing scheme (X, F) and a sequence (μn )n ⊂ M f all compatible with (X, F) and such that h(μn ) − tλ(μn ) → p(t) and inf h(μn ) > 0. n
Moreover, P G (t ) = 0. We need some lemmas and a definition for the proof. Lemma 4.1. For each t ∈ R and any inducing scheme (X, F), we have P G (t ) 0. N X Proof. We let (X N , FN , τ N ) denote the subsystem of (X, F, τ ) where X N = ∪n=1 n and FN , τ N are the restrictions of F, τ to X N . Similarly, PFGN (t ) is defined in the obvious way. By Proposition 2.2, PFG (t ) > 0 implies that for large enough N , PFGN (t ) > 0. Hence there is an equilibrium state μ FN for this system so that τ N dμ FN < ∞ and h(μ FN ) − t log |D F| dμ FN − p(t) τ N dμ FN > 0.
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Similarly to the use of the Abramov formula above, the corresponding projected measure μ f N as in (3) has h(μ f N ) − t log |D f | dμ f N > p(t). This contradiction to the Variational Principle proves the lemma.
Remark 4.1. By [BT2, Lemma 8], the potentials t we consider for the inducing schemes (X, F) in Lemma 4.3 are weakly Hölder continuous. Definition 4.1. Given a function g : [a, b] → R, for x0 ∈ R, as in [Roy, p115], we refer to s : [a, b] → R as a supporting line for g at x0 if s(x) = g(x0 ) + b(x − x0 ) for some b ∈ R, and g(x) s(x) for all x ∈ R. Lemma 4.2. For each t ∈ (t − , t + ), there exists η > 0 such that any measure μ with free energy with respect to ψt close enough to 0 has h(μ) > η. Proof. Let us first consider the case in which t0 > 0. Suppose that there exists a sequence of invariant measures (μn )n such that limn→∞ h(μn ) = 0 and p(t0 ) = −t0 a, where a := limn→∞ λ(μn ). We will show that t0 t + . Let L(t) := −at. Since all measures have non-negative Lyapunov exponent, we have a 0. Since we also know that t0 > 0, this implies that p(t0 ) = −at0 < 0. Claim. p(t) = −at for all t t0 . Proof of the claim. Suppose the opposite, i.e. p(t) > −at for some t t0 . Then since the pressure function p(t) can be found via a limit of supporting lines h(μ) − tλ(μ) for μ ∈ M, we must have some t1 > t0 and μ ∈ M such that h(μ) − t1 λ(μ) > −at1 .
(4)
˜ We will show that this leads to a contradiction. Let L(t) := h(μ) − tλ(μ). We may ˜ assume L = L. By definition, the pressure always satisfies ˜ p(t) L(t) and p(t) L(t).
(5)
Since L˜ is affine and L is linear, both with negative slope, and both lines distinct, either these lines cross at a unique t ∗ ∈ (0, t0 ) or there is no such t ∗ . In the first case, L˜ must ˜ L(0), for all t t ∗ we must start above L and then go below it after t ∗ : since L(0) ˜ ˜ have L(t) < L(t). This means that L(t1 ) > L(t1 ), contradicting (4). In the second case, ˜ 0 ) > L(t0 ), so by (5), L˜ must be above the pressure function at t0 : we must have L(t L(t0 ) cannot have been the pressure at t0 , a contradiction. If there is a measure μ ∈ M such that λ(μ) < a then for some, possibly very large, t > 0 we must have p(t) h(μ) − tλ(μ) > L(t) by the same arguments as in the claim. But this then contradicts the claim. Hence a = λm , the infimum of the Lyapunov exponents. Therefore, by definition of t + we have t0 t + . If t0 < 0 an analogous argument proves that we must have t0 t − . So in either case, t0 ∈ / (t − , t + ), as required.
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Lemma 4.3. For each > 0 there exists θ > 0 and a finite number of inducing schemes N such that any ergodic measure with h(μ) > is compatible with one {(X n , Fn , τn )}n=1 of these schemes (X n , Fn , τn ) and τn dμ Fn < θ . Proof. This follows from [BT2, Remark 6]. We give a brief sketch of the ideas there. N such that for each μ ∈ M with That remark gives, for > 0, a set {(X n , Fn , τn )}n=1 f n h(μ) > , μ must be compatible with some (X , Fn , τn ). These schemes are constructed from sets Xˆ n on the so-called Hofbauer extension (see the Appendix for details). The map F is derived from a first return map Fˆ in this tower. Measures μ ∈ M f with h(μ) > 0 can be lifted to the tower, and if they have h(μ) > they must give one of the sets Xˆ n mass greater than some η = η() > 0. Since Fˆ is a first return map with return time τˆn , we use Kac’s lemma to get ˆ Xˆ n )−1 < η−1 , τn dμ = τˆn d μˆ = μ( as required.
As in [BT2, Remark 6], we denote this set of inducing schemes by Cover (). Proof of Proposition 4.1. By Lemmas 4.3 and 4.2, we can take a sequence of ergodic measures μ p such that h(μ p ) + ψt dμ p = p where p → 0 as p → ∞, h(μ p ) > η (some η > 0), all μ p are compatible with some inducing scheme (X, F, τ ) ∈ Cover () and τ dμ p < θ for all p ∈ N. This implies that P G (t ) 0 since we have a sequence of measures μ F, p such that h(μ F, p ) + t dμ F, p = h(μ p ) + ψt dμ p θ p . τ dμ F, p On the other hand P G (t ) 0 by Lemma 4.1. So the proposition is proved.
Since the inducing scheme (X, F) can be coded by the full-shift on countably many symbols we have, as explained in Sect. 2, a Gibbs measure μt for t . We need to show that this measure has integrable inducing time and thus that it projects to a measure in M f . 5. The Gibbs Measure has Integrable Inducing Times This section is devoted to proving that the inducing time is integrable with respect to the Gibbs measure constructed in Sect. 4. In particular, this implies that the measure has finite entropy and that it is an equilibrium state for the induced potential. It also implies that it can be projected to a measure in M f . Proposition 5.1. Let t ∈ (t − , t + ) and ψ = ψt . Suppose that we have an inducing scheme ˜ Then there exists k ∈ N such that replacing (X, F) ˜ by (X, F), where F = F˜ k , (X, F).
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the following holds. There exist γ0 ∈ (0, 1) and, for any cylinder Cn ∈ PnF any n ∈ N, j a constant δn < 0 such that any measure μ F ∈ M F with j
j μ F (Cn )
j (1 − γ0 )m (Cn )
or
j μ F (Cn )
m (Cn ) , 1 − γ0
where m denotes the conformal measure for the system (X, F, ), must have h(μ F ) + j dμ F δn .
j j ∞ ˜ Note that δn → 0 as m (Cn ) → 0. Also note that if K = exp V ( ) is a k=1 k ˜ then it is also a ˜ for the inducing scheme (X, F) distortion constant for the potential distortion constant for the potential on (X, F). The following lemma will allow us to choose k in the proof of Proposition 5.1. It is true for = t , but also for more general potentials of summable variation. Lemma 5.1. Suppose that we have an inducing scheme (X, F) and potential = t
∞ G () = 0. We let m denote with distortion constant K = exp V () and P k k=1 the conformal measure for the system (X, F, ). Then for any Cn ∈ PnF and n ∈ N, m (Cn ) e−λn ,
where λ := − log K supC1 ∈P F m (C1 ) . 1 Proof. Since m is a conformal measure, for Cin ∈ PnF we have 1 = m (F n (Cin )) = e−Sn dm . Cin
So by the Intermediate Value Theorem we can choose x ∈ Cin so that e Sn (x) = m (Cin ). For future use we will write Sni := Sn (x). Therefore, m (Cin ) = e Sn en sup . i
By the Gibbs property, esup K sup m (C1 ). C1 ∈P1F
Therefore ⎛
⎞
sup log ⎝ K sup m (C1 )⎠ . C1 ∈P1F
We can choose this as our value for −λ.
In the following proof we use the notation A = θ ±C to mean θ −C A θ C .
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˜ for the scheme Proof of Proposition 5.1. Suppose that the distortion of the potential ˜ is bounded by K 1. We first prove that measures giving cylinders very small (X, F) mass compared to m must have low free energy. Note that for any k ∈ N, the potential for the scheme (X, F), where F = F˜ k also has distortion bounded by K . We will choose k later so that λ = λ(K , supi m (X i )) for (X, F), as defined in Lemma 5.1, is large enough to satisfy the conditions associated to (7), (8) and (10). Note that as in [S3, Lemma 3] we also have P G () = 0. In Lemma 5.3 below, we will use the Variational Principle to bound the free energy of measures for the scheme which, for some γ , have μ(Cin ) K m (Cin )(1 − γ )/ (1 − m (Cin ))n in terms of the Gurevich pressure. However, instead of using , which, in the computation of Gurevich pressure weights points x ∈ Cin by e(x) , we use a potential which weights points in Cin by (1 − γ )e (x). That is, we consider (X, F, ), where (x) + log(1 − γ ) if x ∈ Cin , (x) = j (x) if x ∈ Cn , for j = i. Firstly we will compute P G ( ).
Lemma 5.2. P G ( ) = log 1 − γ m (Cin ) . Proof. We prove the lemma assuming that n = 1 since the general case follows similarly. We will estimate Z j ( , Ci1 ), where Z j is defined in (2). The ideas we use are similar to those in the proof of Claim 2 in the proof of [BT2, Prop. 2]. As can be seen from the definition, Z j ( , Ci1 ) = e±
j−1 k=0
Vk ()
eS j
(x)
.
C j ∈P jF ∩Ci1 any x∈C j
As in the proof of Lemma 5.1, the conformality of m and the Intermediate Value Theorem imply that for each k there exists xCk ∈ Ck1 such that m (Ck1 ) = e 1
duration of this proof we write k := (xCk ). As above, we have e 1 Therefore,
i
(xCk ) 1
. For the
:= (1 − γ )ei .
ei = 1 − γ ei .
i
For each C j ∈ P jF and for any k ∈ N, there exists a unique C j+1 ⊂ C j such that F j (C j+1 ) = Ck1 . Moreover, there exists xC j+1 ∈ C j+1 such that F j (xC j+1 ) = xCk . Then 1
for C j ⊂ Ci1 ,
e
S j+1 (xC j+1)
=e
±V j+1 () S j (xC j )
e
C j+1 ⊂C j
e
i
i
= e±V j+1 () e
S j (xC j )
(1 − γ ei ).
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Therefore, Z j+1 ( , Ci1 ) = (1 − γ ei )e
j−1 ± V j+1 ()+ k=0 Vk ()
Z j ( , Ci1 ),
hence j
Z j+1 ( , Ci1 ) = (1 − γ ei ) j e± k=0 (k+1)Vk () . j As in Remark 4.1, is weakly Hölder, so k=0 (k + 1)Vk () < ∞. Therefore we have P G ( ) = log(1 − γ ei ) = log(1 − γ m (Ci1 )), proving the lemma. For the next step in the proof of the upper bound on the free energy of measures giving Cin small mass, we relate properties of (X, F, ) and (X, F, ). Lemma 5.3. M F () = M F ( ) and for any Cin ∈ PnF we have K (1 − γ ) i i sup h F (μ) + dμ : μ ∈ M F (), μ(Cn ) < m (Cn ) (1 − m (Cin ))n K (1 − γ ) i i m (Cn ) sup h F (μ) + dμ : μ ∈ M F ( ), μ(Cn ) < (1 − m (Cin ))n K (1 − γ ) log(1 − γ ) m (Cin ) − (1 − m (Cin ))n K (1 − γ ) log(1 − γ ) G m (Cin ). P ( ) − (1 − m (Cin ))n Note that we can prove that the final inequality is actually an equality, but since we don’t require this here we will not prove it. Proof. The fact that M F () = M F ( ) is clear from the definition. Suppose that μ ∈ M F () and μ(Cin ) m (Cin )K (1 − γ )/(1 − m (Cin ))n . Then h F (μ) + dμ − h F (μ) + dμ = − dμ K (1 − γ ) log(1 − γ ) m (Cin ), = μ(Cin )(− log(1 − γ )) − (1 − m (Cin ))n proving the first inequality in the lemma. The final inequality follows from the definition of pressure. Lemmas 5.2 and 5.3 imply that any measure μ F with μ F (Cin ) < K (1 − γ )m (Cin )/ (1 − m (Cin ))n must have K (1 − γ ) log(1 − γ ) G m (Cin ) h(μ F ) + dμ F P ( ) − (6) (1 − m (Cin ))n
K (1 − γ ) log(1 − γ ) log 1 − γ m (Cin ) − m (Cin ). (7) (1 − m (Cin ))n
If m (Cin ) is very small then log 1 − γ m (Cin ) ≈ −γ m (Cin ) and so choosing γ ∈ (0, 1) close enough to 1 the above is strictly negative. By Lemma 5.1, m (Cin ) < e−λn
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so Cin is small if λ large. Hence if λ is sufficiently large then we can set γ = γ˜ ∈ (0, 1) so that
K (1 − γ˜ ) log(1 − γ˜ ) −λn e log 1 − γ˜ e−λn − (1 − e−λn )n is strictly negative for all n ∈ N. This implies that (7) with γ = γ˜ is strictly negative i, for any Cin ∈ PnF and any n, so we set (7) to be the value δn . For the upper bound on the free energy of measures giving Cin relatively large mass, we follow a similar proof, but with (x) − log(1 − γ ) if x ∈ Cin , (x) = j (x) if x ∈ Cn , for j = i. Similarly to above, one can show that M F () = M F ( ) and ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ i m (Cn ) i sup h F (μ) + dμ : μ ∈ M F (), μ(Cn ) > n
γ ⎪ ⎪ ⎩ ⎭ K (1 − γ ) 1 + m (Cin ) 1−γ ⎧ ⎫ ⎪ ⎪ ⎨ ⎬ i m (Cn ) i sup h F (μ)+ dμ : μ ∈ M F ( ), μ(Cn ) >
n γ ⎪ ⎪ ⎩ ⎭ K (1−γ ) 1+m (Cin ) 1−γ +
log(1 − γ )m (Cin ) n
γ K (1 − γ ) 1 + m (Cin ) 1−γ
P G ( ) +
log(1 − γ )m (Cin ) n .
γ K (1 − γ ) 1 + m (Cin ) 1−γ
Moreover, we can show that γ γ m (Cin ) . P G ( ) = log 1 + m (Cin ) 1−γ 1−γ Therefore, if μ(Cin ) >
m (Ci ) n
n , γ K (1−γ ) 1+m (Cin ) 1−γ
h F (μ) +
dμ
m (Cin )
γ 1−γ
+
we have
log(1 − γ )m (Cin ) n .
γ K (1 − γ ) 1 + m (Cin ) 1−γ
(8)
If λ is sufficiently large then we can choose γ = γ˜ ∈ (0, 1) so that this is strictly i, negative and can be fixed to be our value δn . This can be seen as follows: let and γ = p/( p + 1) for some p to be chosen later. Then the right hand side of (8) becomes p log( p + 1) m (Cin ) ( p + 1) . (9) − p+1 K (1 + pe−λn )n
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If λ is sufficiently large, then there exists some large λ ∈ (0, λ) such that (1+ pe−λn )n 1 + pe−λ n for all n ∈ N. Hence with this suitable choice of λ we can choose p so that i, the quantity in the square brackets in (9) is negative for all n. So we can choose δn < 0 to be (8) with γ = γ˜ . We let n γ˜ . (10) γ = 1 − (1 − γ˜ ) 1 + e−λn 1 − γ˜ For appropriately chosen λ this is in (0, 1). i, i, We set γ0 := max{γ , γ } and for each Cin ∈ PnF we let δni := max{δn , δn }. The proof of the proposition is completed by setting γ0 := 1 − K (1 − γ0 ), which we may assume is in (0, 1). Proposition 5.2. There exists an inducing scheme (X, F)such that for t ∈ (t − , t + ) and ψ = ψt , any sequence of measures (μn )n with h(μn ) − ψ dμn → 0 as n → ∞ has a limit measure μψ which is an equilibrium state for ψ. Note that (X, F) and (μn )n can be chosen as in Proposition 4.1. ˜ and a sequence Proof. By Proposition 4.1, we canfind θ > 0, an inducing scheme (X, F) ˜ and with of measures (μn )n with h(μn ) + ψ dμn → 0 each compatible with (X, F) G k ˜ ˜ τ˜ dμ F,n ˜ < θ . Proposition 4.1 also implies P (t ) = 0. Taking F = F for k as in Proposition 5.1, that proposition then implies that there exists K > 0 such that for any Ck ∈ PkF , for all large enough n, 1 μ F,n (Ck ) S (x) K K e k for all x ∈ Ck (note that as in Proposition 5.1, we can actually take K = K /(1 − γ0 ), ˜ t ). Note that (μ F,n )n is tight (see [Bi, Sect. 25] where K is the distortion bound for for a discussion of this notion) and that any limit of the sequence μ F,∞ must satisfy the Gibbs property with distortion constant K . By the uniqueness of Gibbs measures ([MU2, Theorem 3.5]), μ = μ . We now show that τ dμ < θ k. First note that F,∞ τ dμ F,n = τ˜ k dμ F,n ˜ < θ k. For the purposes of this proof we let τ N := min{τ, N }. By the Monotone Convergence Theorem, τ dμ = lim τ N dμ lim lim sup τ N dμ F,n θ k. N →∞
N →∞ n→∞
Thus we can project μ to μψ by (3). The fact that μψ is a weak∗ limit of (μn )n follows as in, for example [FT, Sect. 6]. The fact that we have a uniform bound μ F,n {τ j} θ k/j for all n ∈ N is again crucial in proving this. The Abramov formula implies that dμ = ψ dμψ = τ dμ (λ(μψ ) − p(t)). τ dμ Since λ(μ) ∈ [λm , λ M ] and both p(t) and τ dμ are finite, this implies that − dμ < ∞ and hence μ is an equilibrium state for . Using the Abramov formula again we have that μψ is an equilibrium state for ψ.
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Remark 5.1. Here we give an example of a way our setting can be changed so that the arguments in Proposition 5.1 and 5.2 fail. In the case where f is the (appropriately scaled) quadratic Chebyshev polynomial, t − ∈ (−∞, 0). In this case there is a periodic point p such that the Dirac measure δ p on the orbit of p has λ(δ p ) = λ M . The point p is the image of the critical point which means that our class of inducing schemes can not be compatible with δ0 (indeed the only inducing scheme for δ0 has only one domain and the only measure compatible to it is δ0 ). However, any measure μ ∈ M f orthogonal to δ0 must have h(μ)−tλ(μ) h(μ1 )− tλ(μ1 ) for all t ∈ R, where μ1 is the acip. In particular, h(μ)−tλ(μ) < p(t) for t < t − . If P G (t ) = 0, then arguments similar to those in the proofs of Lemma 4.1 and Proposition 4.1 imply that there are measures with free energy w.r.t. ψt is arbitrarily close to zero and positive entropy. This contradiction implies that for t < t − , P G (t ) < 0, so we cannot begin to apply the arguments above to that case. So it is important that t ∈ (t − , t + ). 6. Uniqueness of Equilibrium States The result in Proposition 5.2 gives the existence of equilibrium states for −t log |D f | for each t ∈ (t − , t + ). In this section we obtain uniqueness. To do this we will use more properties of the inducing schemes described in [BT2]. They were produced in as first return maps to an interval in the so-called Hofbauer tower. This theory was further developed in [BT1] and [T]. The following theorem gives some of their properties. Theorem 6.1. There exists a countable collection {(X n , Fn )}n of inducing schemes with ∂ Xn ∈ / (X n , Fn )∞ such that: a) any ergodic invariant probability measure μ with λ(μ) > 0 is compatible with one of the inducing schemes (X n , Fn ). In particular there exists an ergodic Fn -invariant probability measure μ Fn which projects to μ as in (3); b) any ergodic equilibrium state for −t log |D f |, where t ∈ R with λ(μ) > 0 is compatible with all inducing schemes (X n , Fn ). Remark 6.1. Note that it is crucial in our applications of Theorem 6.1, for example in the proofs of Proposition 6.1 and Proposition 7.1, that in b) we are able to weaken the condition h(μ) > 0 to λ(μ) > 0 when we wish to lift measures. This is why we need to use a countable number of inducing schemes in Theorem 6.1 rather than the finite number in [BT2, Remark 6]. Before proving Theorem 6.1, we prove the following easy lemma. Lemma 6.1. If t ∈ (t − , t + ) and an equilibrium state μt from Proposition 5.2 is compatible with an inducing scheme (X, F), then P G (t ) = 0. Moreover the lifted measure μt,F is a Gibbs measure and an equilibrium state for t . Proof. First note that by Lemma 4.1, P G (t ) 0. Denote by μt an equilibrium measure for the potential −t log |D f | of positive Lyapunov exponent and let μt,F be the lifted measure. Note that by Proposition 2.1 and by the Abramov formula, see for example [PS, Theorem 2.3], we have P G (t ) h(μt,F ) + t dμt,F = h(μt,F ) − t log |D F| dμt,F − p(t) τ dμt,F
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log |D F| dμt,F h(μt,F ) − p(t) −t τ dμt,F τ dμt,F h(μt ) − t log |D f | dμt − p(t) . = τ dμt,F
=
τ dμt,F
But recall that μt is an equilibrium measure: p(t) = h(μt ) − t log |D f | dμt . Therefore P G (t ) 0. Since P G (t ) = 0 there exists a unique Gibbs measure μ F corresponding to (X, F, t ). By the Abramov formula, h(μt,F ) + t dμt,F = 0, so μt,F is an equilibrium state for (X, F, t ). Since, in this setting, equilibrium states are unique (see [MU2, Theorem 3.5]) we have that μt,F = μ F . Proof of Theorem 6.1. Part (a) of the theorem follows from the proof of [BT2, Theorem 3]. Part (b) is proved similarly to [BT2, Prop. 2], but with added information from our Proposition 5.1. We sketch some details. Suppose that μ is compatible to (X n , Fn ). Then Lemma 6.1 implies that P G (n ) = 0. Claim 1 of the proof of [BT2, Prop. 2] implies that for any other inducing scheme (X n , Fn ) is ‘topologically connected’ to (X n , Fn ). Proposition 5.1, which is an improved version of Claim 2 in the proof of [BT2, Prop. 2], then can be used as in that proof to give a ‘metric connection’ which means that an equilibrium state compatible with (X n , Fn ) must be compatible with (X n , Fn ). Proposition 6.1. For any t ∈ (−∞, t + ) there is at most one equilibrium state for −t log |D f |. Moreover, if t + > 1 then for any t ∈ R there is at most one equilibrium state for −t log |D f |. Clearly the equilibrium states, when unique, must be ergodic. Proof. The idea here is first to show that any equilibrium state can be decomposed into a sum of countably many measures, each of which is an equilibrium state and is compatible with an inducing scheme as in Theorem 6.1. [MU2, Theorem 3.5] implies that there is only one equilibrium state per inducing scheme. Lemma 6.1 then implies that this equilibrium state must be unique. We suppose that μ is an equilibrium state for −t log |D f | for t ∈ (−∞, t + ). We first note that μ may be expressed in terms of its ergodic decomposition, see for example [K2, Sect. 2.3], μ(·) = μ y (·) dμ(y), where y ∈ I is a generic point of the ergodic measure μ y ∈ M f . Clearly, for any set A ⊂ I such that μ(A) > 0, the measure 1 μ A (·) := μ(A) A μ y (·) dμ(y) must have h(μ A ) − tλ(μ A ) = p(t), i.e. it must be an equilibrium state itself (otherwise, removing μ A from the integral for μ would increase h μ − tλ(μ)). As in the proof of Lemma 4.2, λ(μ A ) > 0. Theorem 6.1(a) implies that any such μ A must decompose into a sum μ = n αn μn , where μn is a probability measure compatible with the scheme (X n , Fn ) and αn ∈ (0, 1].
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Then there are Fn -invariant probability measures μ Fn , each of which projects to μn by (3). By Lemma 6.1 and [BuS], μ Fn must be the unique equilibrium state for the scheme (X n , Fn , τn ) with potential −t log |D Fn |− p(t)τn . Therefore, μn is the only equilibrium state for −t log |D f | which is compatible with (X n , Fn ). We finish the proof by using Theorem 6.1 b) which implies that any of these equilibrium states compatible with an inducing scheme (X n , Fn ) as above must be compatible with each of the other inducing schemes (X j , F j ). Hence μi = μ j for every i, j ∈ N. Since μ was an arbitrary equilibrium state, this argument implies that μ is ergodic and is the unique equilibrium state for −t log |D f |, as required. Suppose that t + > 1. Since λm 0 this means that t → p(t) must be strictly decreasing in the interval (1, t + ). Since Bowen’s formula implies that p(t) 0, this means that p(t) < 0. Ruelle’s formula [Ru1] then implies that we must have λm > 0. Therefore, if t + > 1 then λ(μ) > 0 for all μ ∈ M f and so we can apply Theorem 6.1 to the case t t + also. 7. Proof of Theorem A The previous sections give most of the information we need to prove Theorem A. In this section we prove the remaining part: that the critical parameter t − , defined in Eq. (1), is not finite. We then put the proof of Theorem A together. Lemma 7.1. There exists a measure μ M such that λ(μ M ) = λ M . Proof. This follows from the compactness of M f and the upper semicontinuity of x → log |D f (x)|. Proposition 7.1. t − = −∞. Proof. Suppose, for a contradiction, that t − > −∞. This implies that for t t − , the measure μ M in Lemma 7.1 also maximises h(μ) − tλ(μ) for μ ∈ M f , and must have h(μ M ) = 0. By Theorem 6.1, we can choose an inducing scheme (X, F) compatible with μ M . Claim 1. P G (t ) = 0 for all t t − . Proof. P G (t ) 0 follows by Lemma 4.1. P G (t ) 0 follows since μ M is compatible with our scheme. Since by construction, μ M is compatible with (X, F), the induced measure being denoted by μ F,M , and since h(μ M ) + ψt dμ M = 0, we have h(μ F,M ) + t μ F,M = 0, and so μ F,M is an equilibrium state for t . However, by Theorems 1.1 and 1.2 of [BuS] any equilibrium state of t must have positive entropy, a contradiction. Proof of Theorem A. The existence of the equilibrium state for −t log |D f | and t ∈ (t − , t + ) follows from Proposition 5.2. Uniqueness follows from Proposition 6.1. Positivity of the entropy of μt comes from Lemma 4.2. Finally the fact that t − = −∞ comes from Proposition 7.1.
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8. The Pressure is of Class C 1 and Strictly Convex in (−∞, t + ) As discussed in the Introduction, for general systems the pressure function t → p(t) is convex, therefore it can have at most a countable number of first order phase transitions. In [S2] an example is constructed with the property that the set of parameters at which the pressure function is not analytic has positive measure (in this case, there also exist higher order phase transitions, see [S5]). Nevertheless, for multimodal maps it has been shown that in certain intervals the pressure function is indeed real analytic, see [BT1,BT2]. Dobbs [D3, Prop. 9] proved that in the quadratic family x → γ x(1 − x), γ ∈ (3, 4) there exists uncountably many parameters for which the pressure function admits infinitely many phase transitions. However, these transitions are caused by the existence of an infinite sequence renormalisations of the map, so for these parameters the corresponding quadratic maps do not have a representative in the class F. He also notes [D3, Prop. 4] that in the quadratic family there is a always a phase transition for negative t caused by the repelling fixed point at 0. Since this fixed point is not in the transitive part of the system (which actually must be contained in [ f 2 (c), f (c)]), from our perspective this point is not dynamically relevant, so any representative of such a map in F would miss this part of the dynamics, and hence not exhibit this transition. Proposition 8.1. For f ∈ F, the pressure function p is C 1 in the interval (−∞, t + ). Proof. We first show that p is differentiable. By Theorem 6.1, we can choose an inducing scheme (X, F, τ ) which is compatible with μt for each t ∈ (−∞, t + ). Then we have the limits lim = log |D F| dμ and lim = τ dμt . log |D F| dμ τ dμ t t t t →t
t →t
We emphasise that these limits are the same if t are taken to the left or to the right of t. Hence λ(μψt ) is continuous in (t − , t + ). Since the derivative of p is −λ(μψt ), the derivative is continuous, proving the lemma. This standard fact can be seen as follows (see also, for example, [K2, Theorem 4.3.5]): given ε > 0, by the definition of pressure the free energy of μt with respect to ψt+ε is no more than p(t + ε). Similarly the free energy of μt+ε with respect to ψt is no more than p(t). Hence p(t + ε) − p(t) (−(t + ε) + t)λ(μt ) (−(t + ε) + t)λ(μt+ε ) . ε ε ε So whenever t → λ(μt ) is continuous, Dp(t) = −λ(μt ).
Proposition 8.2. For f ∈ F, t + > 0 and the pressure function p is strictly convex in (−∞, t + ). Before proving this proposition, we need two lemmas: the first guarantees that t + > 0, while the second will be used to obtain strict convexity of the pressure function (both these facts are in contrast with the quadratic Chebyshev case). Lemma 8.1. For f ∈ F, λ(μ0 ) > λm , where μ0 is the measure of maximal entropy for f . Proof. The existence of a (unique) measure of maximal entropy μ0 is guaranteed by [H]. Suppose for a contradiction that the lemma is false and hence λ(μ0 ) = λm . Since when the derivative of p exists at a point t, it is equal to −λ(μt ) (see [Ru2] as well as
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the computation in the proof of Proposition 8.1) and by convexity, the pressure function must be affine with constant slope −λm . i.e. p(t) = h top ( f ) − tλm for t ∈ [0, ∞). This implies that μ0 must be an equilibrium state for the potential −t log |D f | for every t ∈ R. In particular this applies when t = 1. Moreover, by Ruelle’s inequality [Ru1], we have λ(μ0 ) > 0, so μ0 must be an acip by [L]. By [D2, Prop. 3.1], this implies that f has finite postcritical set, which is a contradiction. Lemma 8.2. For any ε > 0 there exists an inducing scheme (X, F) and a sequence i k → ∞ such that the domains X ik have |X ik | e−(λm +ε)τik . Proof. It is standard to show that for any ε > 0, there exists a periodic point p with Lyapunov exponent λm +ε/3, see for example [D3, Lemma 19]. We can choose (X, F) as in Theorem 6.1 so that the orbit of p is disjoint from X . We may further assume that (X, F) has distortion bounded by eδ for some δ > 0, i.e. |D F(x)| eδ |D F(y)| for all x, y ∈ X i for any i ∈ N. In this case, by the transitivity of (I, f ), which is reflected in our inducing scheme, there must exist an infinite sequence of domains X n k of (X, F) which shadow the orbit of p for longer and longer. One can use standard distortion arguments to prove that for all large k, |X i | |X |e−δ e−(λm +ε/2)τi . Choosing δ > 0 appropriately completes the proof of the lemma. Proof of Proposition 8.2. For the first part of the proposition, t + > 0 is guaranteed by Lemma 8.1. For the second part of the proposition, since p is convex, we only have to rule out p being affine in some interval. Suppose first that p is affine in an interval [t1 , t2 ] ⊂ (−∞, t ∗ ), where t ∗ := inf{t : Dp(t) = −λm }. I.e. for some β > λm , t ∈ [t1 , t2 ] implies p(t) = p(−t1 ) − (t − t1 )β. We let ε > 0 be such that β > λm + ε. By Lemma 8.2, there exists i k → ∞ such that |X ik | e−(λm +ε)τik . The fact that the pressure function is affine in [t1 , t2 ] implies that the equilibrium state is the same μ for every t ∈ [t1 , t2 ]. Denote the induced version of μ by μ F . By the Gibbs property of our inducing schemes, μ F (X i ) |X i |t e−τi p(t) for all t ∈ [t1 , t2 ]. Therefore, |X i
|X i |t1 e−τi p(t1 ) t | e−τi ( p(t1 )−(t−t1 )β)
1,
which implies that |X i | e−τi β for all i. Since |X ik | e−(λm +ε)τik for an infinite sequence of domains X ik , and β > λm + ε, this yields a contradiction.
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We next want to prove that t + = t ∗ . We suppose not in order to get a contradiction. In the first case suppose that λm = 0. Then p(t) 0 for all t ∈ R. Coupled with Bowen’s formula this implies that p(1) = 0. So the convexity of p implies t + = t ∗ , as required. Now suppose that λm > 0. Since we assumed t + the graph of p(t) must be above, and parallel to t → −tλm on [t ∗ , ∞). This implies that t + = ∞ and so Theorem A gives equilibrium states for all t ∈ R. Hence we can mimic the argument above, with the inducing scheme as in Theorem 6.1 compatible with μt ∗ , but instead taking [t1 , t2 ] ⊂ [t ∗ , ∞) and β = λm . Noting that the argument of Lemma 8.2 ensures that we chose the scheme (X, F) so that there is a sequence of domains |X ik | e−(λ M −ε)τik , we can complete the argument. Proof of Theorem B. The convexity of p follows from Proposition 8.2, the smoothness from 8.1 and the fact that the pressure is decreasing from [Pr]. 9. Phase Transitions in the Positive Spectrum In this section we study the relation between the existence of first order phase transitions at the point t = 1 and the existence of an acip. The following proposition has Proposition 1.1 as a corollary. Proposition 9.1. Suppose that f ∈ F has λm = 0. Then f has an acip if and only if p has a first order phase transition at t = 1. Remark 9.1. Note that if λm > 0 then the situation is quite different. For example if f ∈ F satisfies the Collet-Eckmann condition (which by [BS] implies λm > 0), in which case the map also has an acip, then by [BT2, Theorem 3], p is real analytic in a neighbourhood of t = 1. The following lemma will be used to prove Proposition 9.1. Lemma 9.1. Suppose that t + ∈ (0, ∞) and there is a first order phase transition at t + . Then there exists an inducing scheme (X, F), an equilibrium state μ for = t + , and an equilibrium state μψ for ψ = ψt + with h(μψ ) > 0. Proof. The fact that there is a first order phase transition implies that the left derivative of p at t + has Dp − (t + ) < −λm . The convexity of the pressure function implies that the graph of the pressure lies above the line t → D − p(t + )t − t + (λm + D − p(t + )). This means that we can take a sequence of equilibrium states μψt for t arbitrarily close to, and less than, t + with free energy converging to p(t + ) with h(μψt ) −t + (Dp − (t + ) + λm ) > 0. Hence the arguments used to prove Proposition 5.2 give us an equilibrium state for μt + with positive entropy. Proof of Proposition 9.1. If there exists an acip μ then Dp − (1) = −λ(μ) < 0. Since λm = 0 implies p(t) = 0 for all t 1, the existence of an acip implies that there is a first order phase transition at t = 1. On the other hand, if there exists a first order phase transition at t = 1 then Lemma 9.1 implies that there is an equilibrium state μ1 for − log |D f |, with h(μ1 ) > 0. By [L] this must be an acip.
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Remark 9.2. If λm > 0 and there is a measure μm such that λ(μm ) = λm , then by Lemma 9.1 and the arguments in the proof of Proposition 7.1, we have that t + = ∞. Remark 9.3. There are examples of maps in F with {μ ∈ M f : λ(μ) = λm } = ∅, for example [BK, Lemma 5.5], a quadratic map in F is defined so that λm = 0, but there are no measures with zero Lyapunov exponent. There are also examples of maps f ∈ F with {μ ∈ M f : λ(μ) = λm } = ∅, for example in [B2] examples of quadratic maps in F are given for which the omega-limit set of the critical point supports (multiple) ergodic measures with zero Lyapunov exponent. Moreover, Cortez and Rivera-Letelier [CRL] proved that given E a non-empty, compact, metrisable and totally disconnected topological space then there exists a parameter γ ∈ (0, 4] such that the set of invariant probability measures of x → γ x(1 − x), supported on the omega-limit set of the critical point is homeomorphic to E. It is plausible that there are maps f ∈ F for which inf {t ∈ R : p(t) 0} < 1. However, the following argument shows that this is not true for unimodal maps with quadratic critical point in F. Given an interval map f : I → I , we say that A ⊂ I is a metric attractor if B(A) := {ω(x) ⊂ A} has positive Lebesgue measure and there is no proper subset of A with this property. On the other hand A is a topological attractor if B(A) is residual and there is no proper subset of A with this property. We say that f has a wild attractor if there is a set A which is a metric attractor, but not a topological one. Proposition 9.2. If f ∈ F is a unimodal map with no wild attractor then for t < 1, p(t) > 0. Remark 9.4. It was shown in [BKNS] that there are unimodal maps with wild attractors in F. However, if c = 2 then this is not possible by [Ly1]. Lemma 9.2. If f ∈ F is a unimodal map with no wild attractor then for each ε > 0 there exists a measure μ ∈ M f so that h(μ) > 1 − ε. λ(μ) Proof. By [MvS, Theorem V.1.4], originally proved by Martens, there must be an inducing scheme (X, F) such that Leb(X \ ∪i X i ) = 0. For any δ > 0 we can truncate (X, F) N X so that Leb(∪ N X ) > (1 − δ)|X |. to a finite scheme (X N , FN ), where X N = ∪i=1 i i=1 i We therefore have dim H {x : τ k (x) < ∞ for all k ∈ N} > 1 − δ , where δ depends on δ and the distortion of F (in particular → 0 as δ → 0). It follows from the Variational Principle and the Bowen formula (see [P, Chap. 7]) that there is an F-invariant measure, μ F , for this system with h(μ F ) > 1 − δ. λ(μ F )
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By the Abramov formula, for μ the projection of μ F , h(μ) > 1 − δ λ(μ) also. Choosing δ > 0 so small that δ ε completes the proof.
Proof of Proposition 9.2. Let t < 1 and choose ε = 1 − t > 0. Then the measure μ in Lemma 9.2 has h(μ) − tλ(μ) > 0. Hence by the definition of pressure, p(t) > 0. Proof of Proposition 1.2. By Proposition 9.2 and Remark 9.4 we can take t + = 1. Hence we can conclude that p is C 1 strictly convex decreasing in (−∞, t + ) by Theorem B. The fact that p(t) = 0 for all t 1 follows from [NS]. Part (a) follows from Proposition 9.1 since this implies that both left and right derivatives of p(t) at t = 1 are zero. Part (b) is the converse of this since the left derivative is strictly negative and the right derivative is zero. 10. Remarks on Statistical Properties and Chebyshev Polynomials In this section we collect some further comments on our results.
10.1. Statistical properties. Given f ∈ F and an equilibrium state μ as in Theorem A, one can ask about the statistical properties of the system (I, f, μ). For an equilibrium state μt from Theorem A, we expect that as described in [BT2, Sect. 6], it should be possible to prove exponential decay of correlations (as in [Y]) and large deviations (see [MN,RY]), along with many other statistical laws. These laws can be proved when an inducing scheme (X, F, τ ) compatible with μt has exponential decay in n of the induced measure of {τ > n}. However, we do not have sufficient information on this quantity here. Nevertheless, we can use [BT3] to show that the system (I, f, μ) has ‘exponential return time statistics’. We give a sketch of this theory here, but for more definitions see for example [BT3]. Given f ∈ F and A ⊂ I , we let r A (x) := min{ j ∈ N ∪ {+∞} : f j (x) ∈ A}. For μ ∈ M f , letting μ A be the conditional measure on A, by Kac’s Lemma, the expected value of r A with respect to μ A is A r A dμ A = 1/μ(A). Given a sequence of sets (Un )n so that μ(Un ) → 0, the system has exponential return time statistics for (Un )n if for all τ 0, μUn rUn ≥
τ μ(Un )
→ e−τ
as n → ∞.
(11)
Let t ∈ (−∞, t + ) and μt be the equilibrium state for −t log |D f | given by Theorem A. By [BT3, Theorem 3], for μt -a.e. x0 ∈ I , and any set of open intervals (Un )n such that Un → x0 as n → ∞, the system has exponential return time statistics for (Un )n .
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10.2. Ergodic Optimisation. Let f ∈ F and ϕ : [0, 1] → R a function. The study of invariant probability measures whose ergodic ϕ−average is as large (or as small) as possible is known as ergodic optimisation. A measure μ ∈ M f is called ϕ−minimising/maximising if ϕ dμ = sup ϕ dμ = inf ϕ dν : ν ∈ M f ϕ dν : ν ∈ M f or respectively. For a survey on the subject see [Je]. Let t ∈ (−∞, t + ) and denote by μt the unique equilibrium state corresponding to the potential −t log |D f |. A consequence of the results in this paper is that: any accumulation point μ of a sequence of measures μtn , given by Theorem A, where tn → −∞ is a log |D f |-maximising measure. This is because log |D f | is upper semicontinuous; Dp(t) = −λ(μt ); and this derivative is asymptotic to −λ M . Hence there is a subsequence of these measures (μtnk )k so that lim λ(μtnk ) = λ(μ) = λ M .
k→∞
Note that Lemma 7.1 guarantees the existence of a log |D f |−maximising measure. (We do not assert anything about the uniqueness of this measure.) Actually, any measure μ, which is an accumulation point of μtn as tn → −∞, is a measure maximising entropy among all measures which maximise log |D f |. Then in fact p(t) is asymptotic to the line h(μ) − tλ M as t → −∞. 10.3. The preperiodic critical point case. For our class of maps F we assumed that the orbit of points in Cr are infinite. Here we comment on an alternative case. In the case of the quadratic Chebychev polynomial x → 4x(1 − x) on I , it is well known that the two relevant measures are the acip μ1 , which has λ(μ1 ) = log 2 = λm , and the Dirac measure δ0 on the fixed point at 0, which has λ(δ0 ) = log 4 = λ M . So t − = −1 and (1 − t) log 2 if t −1, p(t) = −t log 4 if t −1. Note that the above piecewise affine form for the pressure function does not conflict with Theorem B, which might be expected to apply in the interval (t − , t + ), since t + = t ∗ = −1, where t ∗ is defined in the proof of Proposition 8.2. Acknowledgements. We would like to thank N. Dobbs for his comments on earlier versions of this paper which improved both the results and the exposition. We would also like to thank H. Bruin, J. Rivera-Letelier as well as the referees for their useful remarks. MT is grateful to the mathematics department at PUC, where some of this work was done, for their hospitality.
Appendix A. Cusp Maps In this section we outline how to extend the above results to some maps which are not smooth. This class includes the class of contracting Lorenz-like maps, see for example [Rov]. Definition A.1. f : ∪ j I j → I is a cusp map if there exist constants C, α > 1 and a set {I j } j is a finite collection of disjoint open subintervals of I such that
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(1) f j := f | I j is C 1+α on each I j =: (a j , b j ) and |D f j | ∈ (0, ∞). (2) D + f (a j ), D − f (b j ) exist and are equal to 0 or ±∞. (3) For all x, y ∈ I j such that 0 < |D f j (x)|, |D f j (y)| 2 we have |D f j (x) − D f j (y)| < C|x − y|α . (4) For all x, y ∈ I j such that |D f j (x)|, |D f j (y)| 2, we have |D f j−1 (x)−D f j−1 (y)| < C|x − y|α . We denote the set of points a j , b j by Cr. Remark A.1. Notice that if for some j, b j = a j+1 , i.e. I j ∩ I j+1 intersect, then f may not continuously extend to a well defined function at the intersection point b j , since the definition above would then allow f to take either one or two values there. So in the definition above, the value of f j (a j ) is taken to be lim xa j f j (x) and f j (b j ) = lim xb j f j (x), so for each j, f j is well defined on I j . Remark A.2. In contrast to the class of smooth maps F considered previously in this paper, for cusp maps we can have λ M = ∞ and/or λm = −∞. The first possibility follows since we allow singularities (points where the one-sided derivative is ∞). The second possibility follows from the presence of critical points (although it is avoided for smooth multimodal maps with non-flat critical points by [Pr]). Examples of both of these possibilities can be found in [D1, Sect. 3.4]. We will ultimately be interested in cusp maps without singular points with negative Schwarzian derivative (in fact the latter rules out the former). Note that since we are only interested in the transitive parts the system, transitive multimodal maps as in the rest of the paper can be considered to fit into this class. We show below that we can build a Hofbauer extension ( Iˆ, fˆ). We note that the possible issue of f not being well defined at the boundaries of I j , discussed in Remark A.1, does not change anything in the definition of the Hofbauer tower. We next define the Hofbauer extension. The setup we present here can be applied to general dynamical systems, since it only uses the structure of dynamically defined cylinders. An alternative way of thinking of the Hofbauer extension specifically for the case of multimodal interval maps, which explicitly makes use of the critical set, is presented in [BB]. We let Cn [x] denote the member of Pn , defined as above, containing x. If x ∈ ∪n 0 f −n (Cr) there may be more than one such interval, but this ambiguity will not cause us any problems here. The Hofbauer extension is defined as Iˆ :=
f k (Ck )/ ∼ , k 0 Ck ∈Pk
where f k (Ck ) ∼ f k (Ck ) as components of the disjoint union Iˆ if f k (Ck ) = f k (Ck ) as subsets in I . Let D be the collection of domains of Iˆ and π : Iˆ → I be the natural inclusion map. A point xˆ ∈ Iˆ can be represented by (x, D), where xˆ ∈ D for D ∈ D and x = π(x). ˆ Given xˆ ∈ Iˆ, we can denote the domain D ∈ D it belongs to by Dxˆ . The map fˆ : Iˆ → Iˆ is defined by
fˆ(x) ˆ = fˆ(x, D) = ( f (x), D )
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if there are cylinder sets Ck ⊃ Ck+1 such that x ∈ f k (Ck+1 ) ⊂ f k (Ck ) = D and D = f k+1 (Ck+1 ). In this case, we write D → D , giving (D, →) the structure of a directed graph. Therefore, the map π acts as a semiconjugacy between fˆ and f : π ◦ fˆ = f ◦ π. We denote the ‘base’ of Iˆ, the copy of I in Iˆ, by D0 . For D ∈ D, we define lev(D) to be the length of the shortest path D0 → · · · → D starting at the base D0 . For each R ∈ N, let IˆR be the compact part of the Hofbauer tower defined by IˆR := {D ∈ D : lev(D) R}. For maps in F, we can say more about the graph structure of (D, →) since Lemma 1 of [BT2] implies that if f ∈ F then there is a closed primitive subgraph DT of D. That is, for any D, D ∈ DT there is a path D → · · · → D ; and for any D ∈ DT , if there is a path D → D then D ∈ DT too. We can denote the disjoint union of these domains by IˆT . The same lemma says that if f ∈ F then π( IˆT ) = , the non-wandering set and fˆ is transitive on IˆT . Theorem A.1 gives these properties for transitive cusp maps. Given an ergodic measure μ ∈ M f , we say that μ lifts to Iˆ if there exists an ergodic fˆ-invariant probability measure μˆ on Iˆ such that μˆ ◦ π −1 = μ. For f ∈ F, if μ ∈ M f is ergodic and λ(μ) > 0, then μ lifts to Iˆ, see [K1,BK]. Property (∗) is that for any x, ˆ yˆ ∈ / ∂ Iˆ with π(x) = π(y) there exists n such that n n ˆ = fˆ ( yˆ ). This follows for cusp maps by the construction of Iˆ using the branch fˆ (x) partition. We will only use the following result in the context of equilibrium states for cusp maps with no singularities. However, for interest we state the theorem in greater generality. Theorem A.1. Suppose that f : I → I is a transitive cusp map with h top ( f ) > 0. Then: (1) there is a transitive part IˆT of the tower such that π( IˆT ) = I ; (2) any measure μ ∈ M f with 0 < λ(μ) < ∞ lifts to μˆ with μ = μˆ ◦ π −1 ; (3) for each ε > 0 there exists η > 0 and a compact set Kˆ ⊂ IˆT \∂ Iˆ such that any measure μ ∈ M f with h(μ) > ε and 0 < λ(μ) < ∞ has μ( ˆ Kˆ ) > η. Proof. Part (1): The first part can be shown as in [BT2, Lemma 2], but we argue as in [H] (see also [KStP, Theorem 6]). Theorem 11 of that paper gives a decomposition of Iˆ into a countable union of irreducible (maximal with these properties) closed (if there is a path D → D for D ∈ E then D ∈ E) primitive (there is a path between any D, D ∈ E) subgraphs E along with some sets which carry no entropy. Since h top ( fˆ) = h top ( f ) and we have positive topological entropy, this means that = ∅. Let E ∈ . Clearly π(E) is open, so by the transitivity of f , there must be a point x ∈ π(E) which has a dense orbit in I . By definition, ω(x) ⊂ π(E). By property (∗), π(E) ∩ π(E ) = ∅ for any E, E ∈ which implies that # = 1. That there is a dense orbit in (E, fˆ) follows from the Markov property of this subgraph, so we let IˆT = D∈E D. Part (2): Ledrappier, in [L, Prop. 3.2] proved the existence of non-trivial local unstable manifolds for a more general class of maps (so-called PC-maps) with an ergodic measure μ ∈ M f with λ(μ) > 0. However, he also required a non-degeneracy condition. For cusp maps, Dobbs [D4, Theorem 13] was able to do this but without the non-degeneracy requirement.
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Keller showed in [K1, Theorem 6] that the existence of such unstable manifolds means that any non-atomic ergodic measure μ ∈ M f with λ(μ) > 0 lifts to μˆ on ( Iˆ, fˆ) and that μ = μˆ ◦ π −1 . Using Dobbs and assuming that μ is not supported on ∪n 0 f n (Cr) we can drop the non-atomic assumption (see also [BK, Theorem 3.6]). Part (3): The third part follows exactly as in [BT2, Lemma 4]. Suppose now that f is a cusp map without singularities (i.e. |D f | is bounded above), with negative Schwarzian and such that the non-wandering set is an interval. We consider f : → . For each t ∈ (t − , t + ), we can find a finite number of inducing schemes as in Proposition 6.1 with which all measures with large enough free energy w.r.t. ψt will be compatible. It is important here that we assume the negative Schwarzian derivative since we need bounded distortion for our inducing schemes. This then allows us to prove Theorem A for this class of maps, but we may have t − > −∞. If we exclude maps with preperiodic critical points then we again have t − = −∞. Similarly we can prove Theorem B for this class of maps, although again we only get t − = −∞ if we exclude maps with preperiodic critical points. Note also that the fact that λm can be negative, and may even be −∞, implies that t + , which for the class F had to lie in [1, ∞], could be any value in the range [0, ∞] for cusp maps. Note that for the maps considered by Rovella in [Rov], the critical values are periodic and so the measure supported on them is not seen by our inducing schemes. This is like the Chebyshev case, so as in that situation, the pressure function could be piecewise affine. References Benedicks, M., Carleson, L.: On iterations of 1 − ax 2 on (−1, 1). Ann. of Math. 122, 1–25 (1985) Billingsley, P.: Probability and measure. Second edition. New York: John Wiley and Sons, 1986 Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Springer Lect. Notes in Math. 470, Berlin-Heidelberg-New York: Springer, 1975 [BB] Brucks, K.M., Bruin, H.: Topics from one-dimensional dynamics. London Mathematical Society Student Texts 62. Cambridge: Cambridge University Press, 2004 [B1] Bruin, H.: Induced maps, markov extensions and invariant measures in one–dimensional dynamics. Commun. Math. Phys. 168, 571–580 (1995) [B2] Bruin, H.: Minimal cantor systems and unimodal maps. J. Difference Eq. Appl. 9, 305–318 (2003) [BK] Bruin, H., Keller, G.: Equilibrium states for s-unimodal maps. Erg. Th. Dynam. Syst. 18, 765–789 (1998) [BKNS] Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild cantor attractors exist. Ann. of Math. 143(2), 97–130 (1996) [BS] Bruin, H., van Strien, S.: Expansion of derivatives in one–dimensional dynamics. Israel. J. Math. 137, 223–263 (2003) [BT1] Bruin, H., Todd, M.: Equilibrium states for potentials with sup ϕ − inf ϕ < h top ( f ). Commun. Math. Phys. 283, 579–611 (2008) [BT2] Bruin, H., Todd, M.: Equilibrium states for interval maps: the potential −t log |d f |. Ann. Sci. École Norm. Sup. 42(4), 559–600 (2009) [BT3] Bruin, H., Todd, M.: Return time statistics for invariant measures for interval maps with positive lyapunov exponent. Stoch. Dyn. 9, 81–100 (2009) [BuS] Buzzi, J., Sarig, O.: Uniqueness of equilibrium measures for countable markov shifts and multidimensional piecewise expanding maps. Erg. Th. Dynam. Systs. 23, 1383–1400 (2003) [C] Cederval, S.: Invariant measures and correlation decay for S-multimodal interval maps. PhD thesis, Imperial College, 2006 [CRL] Cortez, M.I., Rivera-Letelier, J.: Invariant measures of minimal post-critical sets of logistic maps. Israel J. Math. 176, 157–193 (2010) [DU] Denker, M., Urbanski, M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4, 103–134 (1991) [D1] Dobbs, N.: Critical points, cusps and induced expansion in dimension one. Thesis, Université ParisSud, Orsay, 2006 [BC] [Bi] [Bo]
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Dobbs, N.: Visible measures of maximal entropy in dimension one. Bull. Lond. Math. Soc. 39, 366–376 (2007) Dobbs, N.: Renormalisation induced phase transitions for unimodal maps. Commun. Math. Phys. 286, 377–387 (2009) Dobbs, N.: On cusps and flat tops. http://arXiv.org/abs/0801.3815v1[math.DS], 2008 Dobrušhin, R.L.: Description of a random field by means of conditional probabilities and conditions for its regularity. Teor. Verojatnost. i Primenen 13, 201–229 (1968) Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Classics in Mathematics, Berlin: Springer-Verlag, 2006 Frietas, J., Todd, M.: Statistical stability of equilibrium states for interval maps. Nonlinearity 22, 259–281 (2009) Graczyk, J., Swi¸atek, G.: Generic hyperbolicity in the logistic family. Ann. Math. 146, 1–52 (1997) Gureviˇc, B.M.: Topological entropy for denumerable markov chains. Dokl. Akad. Nauk SSSR 10, 911–915 (1969) Gureviˇc, B.M.: Shift entropy and markov measures in the path space of a denumerable graph. Dokl. Akad. Nauk SSSR 11, 744–747 (1970) Hofbauer, F.: Piecewise invertible dynamical systems. Probab. Theory Relat. Fields 72, 359–386 (1986) Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Commun. Math. Phys. 81, 39–88 (1981) Jenkinson, O.: Ergodic optimization. Discrete Contin. Dyn. Syst. 15, 197–224 (2006) Katok, A., Hasselblat, B.: Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge: Cambridge University Press, 1995 Keller, G.: Lifting measures to markov extensions. Monatsh. Math. 108, 183–200 (1989) Keller, G.: Equilibrium states in ergodic theory. London Mathematical Society Student Texts, 42. Cambridge: Cambridge University Press, 1998 Keller, G., Pierre, M. St.: Topological and measurable dynamics of Lorenz maps. In: Ergodic theory, analysis, and efficient simulation of dynamical systems, Berlin: Springer, 2001, pp. 333–361 Ledrappier, F.: Some properties of absolutely continuous invariant measures on an interval. Erg. Th. Dynam. Syst. 1, 77–93 (1981) Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. 140(2), 347–404 (1994) Lyubich, M.: Dynamics of quadratic polynomials i-ii. Acta Math. 178, 185–297 (1997) Makarov, N., Smirnov, S.: On thermodynamics of rational maps. ii. non-recurrent maps. J. London Math. Soc. 67(2), 417–432 (2003) Mauldin, R., Urba´nski, M.: Dimensions and measures in infinite iterated function systems. Proc. London Math. Soc. 73(3), 105–154 (1996) Mauldin, R., Urba´nski, M.: Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125, 93–130 (2001) Melbourne, I., Nicol, M.: Large deviations for nonuniformly hyperbolic systems. Trans. Amer. Math. Soc. 360, 6661–6676 (2008) de Melo, W., van Strien, S.: One dimensional dynamics. Ergebnisse Series 25, Berlin-HeidelbergNew York: Springer–Verlag, 1993 Nowicki, T., Sands, D.: Non-uniform hyperbolicity and universal bounds for s-unimodal maps. Invent. Math. 132, 633–680 (1998) Pesin, Y.: Dimension Theory in Dynamical Systems. Cambridge: Cambridge Univ. Press, 1997 Pesin, Y., Senti, S.: Equilibrium measures for maps with inducing schemes. J. Mod. Dyn. 2, 1–31 (2008) Pomeau, Y., Manneville, P.: Intermittent transition to turbulence in dissipative dynamical systems. Commun. Math. Phys. 74, 189–197 (1980) Prellberg, T., Slawny, J.: Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66, 503–514 (1992) Przytycki, F.: Lyapunov characteristic exponents are nonnegative. Proc. Amer. Math. Soc. 119, 309– 317 (1993) Przytycki, F., Rivera-Letelier, J.: Nice inducing schemes and the thermodynamics of rational maps. Preprint, http://arXiv.org/abs/0806.4385v2[math.DS], 2008 Rey-Bellet, L., Young, L.-S.: Large deviations in non-uniformly hyperbolic dynamical systems. Erg. Th. Dynam. Syst. 28, 587–612 (2008) Rockafellar, R.T.: Convex analysis. Princeton Mathematical Series, No. 28, Princeton, N.J.: Princeton University Press, 1970 Rovella, A.: The dynamics of perturbations of the contracting lorenz attractor. Bol. Soc. Brasil. Mat. (N.S.) 24, 233–259 (1993)
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Royden, H.L.: Real analysis. Third edition. New York: Macmillan Publishing Company, 1988 Ruelle, D.: An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat. 9, 83–87 (1978) Ruelle, D.: Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics. With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. Encyclopedia of Mathematics and its Applications, 5. Reading, MA: Addison-Wesley Publishing Co., 1978 Sarig, O.: Thermodynamic formalism for countable markov shifts. Erg. Th. Dynam. Syst. 19, 1565– 1593 (1999) Sarig, O.: On an example with topological pressure which is not analytic. C.R. Acad. Sci. Serie I: Math. 330, 311–315 (2000) Sarig, O.: Phase transitions for countable markov shifts. Commun. Math. Phys. 217, 555–577 (2001) Sarig, O.: Existence of gibbs measures for countable markov shifts. Proc. Amer. Math. Soc. 131, 1751–1758 (2003) Sarig, O.: Critical exponents for dynamical systems. Commun. Math. Phys. 267, 631–667 (2006) Sinai, J.G.: Gibbs measures in ergodic theory. Uspehi Mat. Nauk 27, 21–64 (1972) Todd, M.: Multifractal analysis for multimodal maps. Preprint http://arXiv.org/abs/0809. 1074v3[math-DS], 2009 Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79, Berlin-Heidelberg-New York: Springer, 1981 Young, L.S.: Recurrence times and rates of mixing. Israel J. Math. 110, 153–188 (1999)
Communicated by G. Gallavotti
Commun. Math. Phys. 300, 95–145 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1110-z
Communications in
Mathematical Physics
A Kinetic Flocking Model with Diffusion Renjun Duan1 , Massimo Fornasier2 , Giuseppe Toscani3 1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong.
E-mail:
[email protected]
2 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences,
Altenbergerstrasse 69, A-4040 Linz, Austria. E-mail:
[email protected]
3 Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy
Received: 10 September 2009 / Accepted: 11 March 2010 Published online: 22 August 2010 – © Springer-Verlag 2010
Abstract: We study the stability of the equilibrium states and the rate of convergence of solutions towards them for the continuous kinetic version of the Cucker-Smale flocking in presence of diffusion whose strength depends on the density. This kinetic equation describes the collective behavior of an ensemble of organisms, animals or devices which are forced to adapt their velocities according to a certain rule implying a final configuration in which the ensemble flies at the mean velocity of the initial configuration. Our analysis takes advantage both from the fact that the global equilibrium is a Maxwellian distribution function, and, on the contrary to what happens in the Cucker-Smale model (IEEE Trans Autom Control 52:852–862, 2007), the interaction potential is an integrable function. Precise conditions which guarantee polynomial rates of convergence towards the global equilibrium are found.
Contents 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . 1.1 Main results . . . . . . . . . . . . . . . 1.2 Formal derivation of diffusive model . . Preparations . . . . . . . . . . . . . . . . . 2.1 Coercivity of the linearized operator . . 2.2 Macro-micro decomposition . . . . . . Linearized Cauchy Problem . . . . . . . . . 3.1 Hypocoercivity . . . . . . . . . . . . . 3.2 Proof of hypocoercivity: Fourier analysis Nonlinear Cauchy Problem . . . . . . . . . 4.1 Uniform a priori estimates . . . . . . . 4.2 Proof of global existence and uniqueness 4.3 Proof of rates of convergence . . . . . .
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96 96 102 106 106 109 113 113 114 121 121 124 129
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A. Proofs of Uniform a Priori Estimates . . . . . . . . . . . . . . . . . . . . . 133 A.1 A priori estimates: Microscopic dissipation . . . . . . . . . . . . . . . 133 A.2 A priori estimates: Macroscopic dissipation . . . . . . . . . . . . . . . 140 1. Introduction 1.1. Main results. Description of the collective and interactive motion of multi-agents such as school of fish, flocking of birds or swarm of bacteria became recently a major research topic in population and behavioral biology and ecology [4–6,21,22,25]. Among them, the phenomenon of flocking can be regarded as a universal behavior of multi-agents systems, where consensus is reached at large times [4]. Both the numerical and theoretical studies of some related mathematical models which describe various self-organized patterns in the collective motion [3,18,28], have shown recently an increasing interest. In this paper we are concerned with a kinetic flocking model in the presence of diffusion. Denoting by f = f (t, x, ξ ) ≥ 0 the number density of particles (e.g. flying birds) which have position x = (x1 , x2 , . . . , xn ) ∈ Rn and velocity ξ = (ξ1 , ξ2 , . . . , ξn ) ∈ Rn at time t ≥ 0, n ≥ 1, the evolution of the density is described by the Fokker-Planck type equation ∂t f + ξ · ∇x f + U ∗ ρξ f · ∇ξ f = U ∗ ρ f ∇ξ · (∇ξ f + ξ f ), f (0, x, ξ ) = f 0 (x, ξ ), where
ρ f (t, x) =
Rn
(1.1) (1.2)
f (t, x, ξ )dξ, ρξ f (t, x) =
Rn
ξ f (t, x, ξ )dξ.
Both the interactive potential U = U (x) and the initial data f 0 = f 0 (x, ξ ) are given. The operator “*” denotes the convolution with respect to the spatial variable. Throughout this paper, it is supposed that U is continuous in x with U (x) = U (|x|) ≥ 0, U (x)d x = 1. (1.3) Rn
Let us briefly present the origin of the model equation (1.1). When there is no diffusion term in (1.1), the equation ∂t f + ξ · ∇x f + U ∗ ρξ f · ∇ξ f = U ∗ ρ f ∇ξ · (ξ f )
(1.4)
has been derived and analyzed by Ha-Tadmor [18] as the mean-field limit of the discrete and finite dimensional flocking model considered by Cucker-Smale [4]. Recently, (1.4) was also obtained as the grazing collision limit of a Boltzmann equation of Povzner type in [2]. Within this kinetic picture, the velocities of birds are modified through binary interactions, which dissipate energy according to their mutual distance. Consequently, both Eqs. (1.4) and (1.1) describe a system of particles (e.g. birds) which influence each other according to the potential function of their mutual spatial distance, in such a way that the difference between the respective velocities is diminishing. This can be seen also looking at the characteristic equations, which in the Ha-Tadmor model (1.4) read ⎧ dX ⎪ ⎪ ⎪ ⎨ dt = , ⎪ d ⎪ ⎪ U (|x − y|)(ξ − ) f (t, y, ξ )dydξ. = ⎩ dt R n ×R n
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97
Equation (1.1) differs from the Ha-Tadmor model (1.4) in two essential features. In (1.1) particles are subject to random fluctuations whose strength depends on the density, which implies that randomness increases as soon as particles are closer to each other. Resorting again to the collisional kinetic picture, in this new model the velocities of birds are modified through binary interactions, in which, in addition to dissipation of energy according to their mutual distance, also random fluctuations of bird velocities are introduced to mimic a more realistic behavior. The additional presence of random terms, which is reasonable from a physical point of view, is responsible for the existence of a global equilibrium configuration of Maxwellian type. This type of interactions induces a substantial difference in the asymptotic behavior of the solution to (1.1), with respect to model (1.4), where all particles tend exponentially fast to move with their global mean velocity whenever the mutual interaction was strong enough at far distance, independently of the initial conditions. This situation is called unconditional flocking, and the global equilibrium (due to the intrinsic dissipation) is represented by a Dirac delta function concentrated at the mean velocity. A second main difference between model (1.4) and the present one is that, on the contrary to what happens in the former, here the interaction potential U (·) is integrable. This corresponds to a weak interaction between birds, or, in other words, to a rapid decay of the interaction in terms of the mutual distance. In consequence of this choice, the relaxation towards equilibrium is not universal, but depends on the size of the perturbation. We remark that also this condition can be reasonably justified from a physical point of view, since it reflects the fact that birds mainly adapt their velocity to birds which are close enough to them. We remark however that the unconditional flocking phenomenon observed in the original Cucker and Smale discrete model [4] heavily depends on the fact that the interaction potential is not integrable. Otherwise, results can be recovered only for well-prepared initial configurations of birds. By direct inspection, one can easily check that the global Maxwellian function 1 M = M(ξ ) = exp −|ξ |2 /2 , (1.5) n/2 (2π ) is a steady state of (1.1). Notice that M has zero bulk velocity and unit density and temperature. The main goal of this paper is to study the stability of solutions near M and the rate of convergence of these solutions towards it for the Cauchy problem (1.1)–(1.2). For this purpose, introduce the perturbation u = u(t, x, ξ ) by setting √ f = M + Mu. Then, u satisfies √ 1 ∂t u + ξ · ∇x u + U ∗ ρξ √Mu · ∇ξ u − U ∗ ρξ √Mu · ξ u − U ∗ ρξ √Mu · ξ M 2
√ 1 √ 1 U d x∇ξ · M∇ξ u + ξ Mu =√ 2 M
√ 1 1 √ √ + √ U ∗ ρ Mu ∇ξ · M∇ξ u + ξ Mu . 2 M It is straightforward to verify that √ 1 1 √ 1 √ ∇ξ · ( M∇ξ u + ξ Mu) = ξ u + (2n − |ξ |2 )u. 2 4 M
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Thus, the Cauchy problem (1.1)–(1.2) is reformulated as ∂t u + ξ · ∇x u + U ∗ ρξ √Mu · ∇ξ u = Lu + (u, u),
(1.6)
u(0, x, ξ ) = u 0 (x, ξ ),
(1.7)
where u 0 takes the form of u 0 = M−1/2 ( f 0 − M), and the linear part Lu and the nonlinear part (u, u) are respectively given by √ 1 Lu = ξ u + (2n − |ξ |2 )u + U ∗ ρξ √Mu · ξ M, (1.8) 4 1 1 (u, u) = U ∗ ρ√Mu [ξ u + (2n − |ξ |2 )u] + U ∗ ρξ √Mu · ξ u. (1.9) 4 2 m , H m , H m to denote We introduce some notations. For any integer m ≥ 0, we use Hx,ξ x ξ the usual Hilbert spaces H m (Rnx × Rnξ ), H m (Rnx ), H m (Rnξ ), respectively, where L 2x,ξ , L 2x , L 2ξ are also used for m = 0. For a Banach space X , · X denotes the corresponding norm, while · always denotes the norm · L 2 for simplicity when X = L 2x,ξ . We x,ξ
use ·, · to denote the inner product over the Hilbert space L 2ξ , i.e. g, h = g(ξ )h(ξ )dξ, g, h ∈ L 2ξ . Rn
For q ≥ 1, we also define Zq =
q L 2ξ (L x )
= L 2 (Rnξ ; L q (Rnx )), g Z q =
Rn
Rn
1/2
2/q |g(x, ξ )|q d x
dξ
Let ν(ξ ) = 1 + |ξ |2 . Denote | · |ν and · ν by |g|2ν = |∇ξ g(ξ )|2 + ν(ξ )|g|2 dξ, g = g(ξ ), n R 2 gν = |∇ξ g(x, ξ )|2 + ν(ξ )|g(x, ξ )|2 dξ d xdξ, g = g(x, ξ ). R n ×R n
and · U by 2 φU =
.
(1.10) (1.11)
R n ×R n
U (|x − y|)|φ(x, y)|2 d xd y, φ = φ(x, y).
(1.12)
Define the linear operator T by T b(x, y) = b(x)−b(y) for b = b(x). For the multiple indices α = (α1 , α2 , . . . , αn ) and β = (β1 , β2 , . . . , βn ), we denote β
β
β
β
∂xα ∂ξ = ∂xα11 ∂xα22 · · · ∂xαnn ∂ξ11 ∂ξ22 · · · ∂ξnn . As usual, the length of α is |α| = α1 + α2 + · · · + αn , and α ≤ α means that αi ≤ αi for 1 ≤ i ≤ n, while α < α means α ≤ α and |α | < |α|. For simplicity, we also use ∂i to denote ∂xi for each i = 1, 2, . . . , n. In addition, C denotes a generic positive (generally large) constant and λ a generic positive (generally small) constant, where both of them may take different values at different places. When necessary, we write C0 , C1 , . . ., λ0 , λ1 , . . ., to distinguish them. Now, the main results of this paper are stated as follows.
Kinetic Flocking Model with Diffusion
99
Theorem √ 1.1. Let n ≥ 3 and N ≥ 2[n/2] + 2, and let (1.3) hold. Suppose that f 0 ≡ M + Mu 0 ≥ 0, and u 0 H N is small enough. Then, the Cauchy problem (1.6)–(1.7) x,ξ admits a unique global solution u(t, x, ξ ), satisfying √ u ∈ C([0, ∞); H N (Rn × Rn )), f ≡ M + Mu ≥ 0, (1.13) and u(t)2H N x,ξ
t
+λ
+λ
|α|+|β|≤N
t
0
∇x (a u , bu )(s)2
β ∂xα ∂ξ {I
HxN −1
0
− P}u(s)2ν ds
+λ
|α|≤N
t
0
2 T ∂xα bu (s)U ds
ds ≤ Cu 0 2H N ,
(1.14)
x,ξ
for any t ≥ 0, where P, I − P, a u , bu are defined in (2.10). Moreover, if u 0 Z 1 is bounded and u 0 H N + ξ ∇x u 0 is small enough, then the time-decay estimate x,ξ
n u(t) H N ≤ C u 0 H N + u 0 Z 1 (1 + t)− 4 , x,ξ
x,ξ
(1.15)
is valid for any t ≥ 0. It has to be outlined that (1.1) is a nonlinear Fokker-Planck equation where both the nonlocal drift term and the diffusion coefficient depend on the macroscopic momentum and density, respectively. This kind of nonlinear character leads to the fact that (1.1) does not have the same properties of the classical linear Fokker-Planck equation ∂t f + ξ · ∇x f + ∇x V · ∇ξ f = ∇ξ · (∇ξ f + ξ f ),
(1.16)
where V = V (x) is a confining force potential. In fact, whether or not V is present, (1.16) possesses only one total conservation law (the conservation of mass), while (1.1) conserves not only the total mass but also the total momentum. This difference would imply that the kinetic dissipation of (1.1) should be much weaker than that of (1.16). In addition, (1.16) has a natural Lyapunov functional |ξ |2 E FP ( f ) = V (x) + + log f f d xdξ, 2 R n ×R n nonincreasing in time d E FP ( f ) = −D FP ( f ) ≡ − dt
R n ×R n
1 |∇ξ f + ξ f |2 d xdξ ≤ 0. f
In the current case, denoting 2 |ξ | + log f f d xdξ, E( f ) = n n 2 R ×R U ∗ ρf |∇ξ f + ξ f |2 d xdξ − D( f ) = U ∗ ρξ f · ρξ f d x, f R n ×R n Rn
(1.17) (1.18)
solutions to (1.1) satisfy a similar equation d E( f ) = −D( f ), dt
(1.19)
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but it is presently unknown if D( f ) is non-negative and consequently E( f ) is decreasing in time. The eventual existence of a Lyapunov functional for (1.1) is an interesting problem to study. In the case without diffusion Ha-Liu [17] explicitly constructed such Lyapunov functional for (1.4), and used its decay to give a simple proof of the exponential convergence of solutions to the flocking state. Since the nonlinear equation (1.1) lacks such natural a priori bound (only total conservations of mass and momentum hold), we need to turn to the perturbation theory of equilibrium (cf. Theorem 1.1 of this paper) to recover convergence to equilibrium. We remark that it is straightforward to check that the functional D(·) denoted by (1.18) satisfies D(M) = 0,
d d2 φ D(M + φ)|=0 = 0, D(M + φ)|=0 = 2L( √ ). d d 2 M
Consequently M is a critical point of the nonlinear functional D(·). This makes it possible to apply the perturbation method to obtain the stability of the equilibrium state M if the linearized operator L satisfies certain coercivity inequalities. As stated in Theorem 2.1, it turns out that L is degenerately dissipative over the full phase space Rnx × Rnξ in the sense of (2.9) below. Then, the classical energy method together with suitable smallness assumptions produce some uniform a priori estimates in high-order Sobolev spaces, which together with the local existence and the continuum argument yield the global existence. The rate of convergence of solutions to the steady state M is the other issue under consideration in this paper. For the classical linear Fokker-Planck equation (1.16), thank to the existence of Lyapunov functionals, the hypocoercivity with almost exponential rate or exponential rate in time has been extensively studied by Desvillettes-Villani [7], Mouhot-Neumann [20], Dolbeault-Mouhot-Schmeiser [8] and Villani [27] in a general framework. In the case without diffusion, Ha-Tadmor [18] showed that the energy of solution to (1.4) tends exponentially fast in time to zero for a certain strong potential function U . This result has been recently improved in [2], where it has been shown that both the discrete model by Cucker and Smale [4] and its kinetic version (1.4) produce a flocking behavior under the same conditions on the interaction potential. In the case considered in this paper, as shown in Theorem 1.1, solutions to the Cauchy problem (1.1)–(1.2) which are near equilibrium M, converge to it with an explicit algebraic rate f (t) − M √ ≤ C f (1 + t)− n4 , 0 M for any t ≥ 0, where C f0 is a constant depending on the size of initial data. As pointed out in [27], the hypocoercivity, which produces the trend towards the equilibrium, essentially stems from the interplay between the conservative free transport operator and the kinetic relaxation. Here, only the algebraic rate is found because the particles move in the whole space Rn and also the number of the total conservation laws exceed two. Another issue of this paper is concerned with the direct velocity regularized equation ∂t f + ξ · ∇x f + U ∗ ρξ f · ∇ξ f = U ∗ ρ f ∇ξ · (ξ f ) + κξ f,
(1.20)
for κ > 0. Notice that in the above equation, the strength of noise is spatially homogeneous, and the steady state is Mκ =
|ξ |2 1 e− 2κ . n/2 (2π κ)
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However, for fixed κ > 0, it is unclear whether or not Mκ is uniformly stable in time under a certain topology, again due to the lack of a Lyapunov functional. Moreover, the stability of Mκ with κ > 0 is unknown even for small smooth perturbation of the type considered in Theorem 1.1, because the linearized operator of (1.20) has no coercivity properties similar to that of L defined in (1.8). Therefore, as far as Eq. (1.1) is concerned, it is fundamental for the stability of equilibrium that the strength of noise would depend non-locally on the density. As mentioned before, this dependence implies that randomness is weaker at position x around which the density is lower. A similar phenomenon has been observed in a recent paper [28], where it is argued that coherence in collective swarm motion is facilitated in the presence of randomness, which has to be weaker at some position around which mean velocity of particles is larger. Last, we discuss a variant of the model (1.1). When the potential function U reduces to the Dirac delta function concentrated on the origin, the nonlocal nonlinear FokkerPlanck equation (1.1) takes the form ∂t f + ξ · ∇x f = ρ f ∇ξ · [∇ξ f + (ξ −
ρξ f ) f ]. ρf
(1.21)
We refer to [26] for an exhaustive presentation and discussion on the above local nonlinear Fokker-Planck equation. We emphasize that various results including Theorem 1.1 and Theorem 3.1 respectively in the nonlinear and linear cases also hold for Eq. (1.21). Moreover, when U reduces to the Dirac delta function concentrated on the origin, (1.19) remains true for (1.21) with E( f ) in (1.17) unchanged and D( f ) in (1.18) reducing to D0 ( f ) given by ρf |∇ξ f + ξ f |2 d xdξ − D0 ( f ) = |ρξ f |2 d x n n n f R R ×R ρξ f ρf 2 ) f | d xdξ ≥ 0. = |∇ξ f + (ξ − ρf R n ×R n f Therefore, (1.21) possesses a natural Lyapunov functional, and the non-perturbation theory would be also possible for the study of well-posedness and large-time behavior of (1.21). This will be the object of a separate forthcoming paper. Finally, we should point out that the present study presents analogies with a recent paper by Guo [15], where the global well-posedness on the torus for the classical Landau equation in the absence of external forcing ∂t f + ξ · ∇x f = ∇ξ · (ξ − ξ )[ f (ξ )∇ξ f (ξ ) − f (ξ )∇ξ f (ξ )]dξ , (1.22) Rn
was studied. In the Landau equation (1.22) is the non-negative matrix given by
ξi ξ j (ξ ) = 0 δi j − |ξ |γ +2 , γ ≥ −3, 0 > 0. |ξ |2 We remark that the main difficulties in this paper rely both in reckoning the kinetic dissipation of the nonlocal linearized operator L defined in (1.8) and in the control of the nonlinear term in the process of energy estimates. The method used to prove Theorem 3.1, which gives the time-decay estimates on the linearized solution operator is general enough to deal with the time-decay estimates of some other kinetic equations
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with both the free transport operator and the kinetic relaxation in the full space Rn , whenever the classical spectral analysis is difficult to apply [23]. Actually, we can resort to a similar strategy to obtain results analogous to those of Theorem 3.1 for the Landau equation (1.22), linearized in the case of hard potentials γ ≥ 0, where the space domain is the whole Rn . The rest of this paper is organized as follows. We shall end this Introduction with the next subsection by presenting how Eq. (1.2) can be derived as a kinetic version of a particle system. In particular, we formally show that this equation arises naturally either as a mean-field limit or as a grazing collision limit of a Boltzmann type equation from the Cucker-Smale particle model of flocking in the presence of an additional stochastic term. In Sect. 2, for the later study of both the linearized and the original nonlinear equation (1.6), we make two preparations, one of which is to obtain the coercivity of the linearized operator L as in Theorem 2.1, and the other one to make a macro-micro decomposition of the perturbation u and Eq. (1.6) where a system of equations for the evolution of moments of u up to second order is derived (cf. (2.21)–(2.24)). In Sect. 3 we employ Fourier analysis methods to establish the hypocoercivity property for the linearized Cauchy problem with a non-homogeneous microscopic source, and we obtain the precise algebraic time-decay rates. The main idea in the proof of Theorem 3.1 is to construct a temporal-frequency free energy functional defined in (3.11) able to capture the macroscopic dissipation in the Fourier space. A similar approach has been recently used in [11] for the study of the Vlasov-Poisson-Boltzmann system. In Sect. 4 we devote ourselves to the proof of the main result for the fully nonlinear Cauchy problem (1.6)–(1.7) (Theorem 1.1 introduced before). To this extent, in Subsect. 4.1 we list a series of uniform a priori estimates on the solution, whose proofs are postponed to Appendices A.1 and A.2 for the sake of a simpler presentation. The dissipative property of L proven in Theorem 2.1 and the construction of the other temporal free energy in the phase space contained in (4.6) play a key role in the proof of those a priori estimates. We continue in Subsect. 4.2 the proof of the local existence and uniqueness stated in Theorem 4.1 by using an iterative scheme and the standard stability method, concluding with the global existence, which follows by combining the established uniform a priori estimates with the continuum argument. In the last Subsect. 4.3 we apply the time-decay properties of the linearized solution operator of Theorem 3.1 to obtain the optimal time-decay rates for the perturbation solution u in some smooth Sobolev space. Here, it should be remarked that the idea of proof in this subsection is similar to one used in [12,13].
1.2. Formal derivation of diffusive model. In this subsection, we shall give a formal derivation of the kinetic equation (1.1). There are at least two ways to do it which we will introduce in what follows. The first way to derive (1.1) is based on the discrete Cucker-Smale model with noise whose strength depends on the distance between particles. Consider the evolutions of m (m ≥ 1) particles (e.g. birds) with positions and velocities (xi , ξi ) = (xi (t), ξi (t)) (1 ≤ i ≤ m) at time t in the phase space Rn × Rn : ⎧ d xi = ξi dt, ⎪ ⎪ ⎨ m m
dξ = U (|x − x |)(ξ − ξ )dt + 2μ U (|x j − xi |)dWi . ⎪ i j i j i ⎪ ⎩ j=1
j=1
(1.23)
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Here, U denotes the distance potential (communication rate) function defined as in (1.3). A typical example goes back to the original Cucker-Smale model [4], where U (x) =
Cn,γ , x ∈ Rn . (1 + |x|2 )γ
In the random noise term, Wi = Wi (t) (1 ≤ i ≤ m) are m independent Wiener processes with values in Rn , and μ ≥ 0 is a constant denoting the coefficient of noise strength. Notice that the strength of noise for the i th particle is μ
m
U (|x j − xi |),
j=1
which is proportional to the summation of distance potentials of the i th particle with all particles. We remark that if there is only one particle, i.e. m = 1, then the system reduces to d x = ξ dt, dξ = 2μU (0)dW, which means that the motion of a single particle is just a random walk, and further that if μ = 0, then the system is the same as the Cucker-Smale model. We are interested in the so-called mean field limit of particle systems (1.23). Thus, set U=
κ U0 m
for some function U0 and some constant κ > 0 which are independent of m, and let f (m) (t, x, ξ ) =
m 1 δ(x − xi (t))δ(ξ − ξi (t)), m i=1
where δ(·) is the Dirac delta function. Since f (m) for each m ≥ 1 belongs to M(R2n ) which is the space of Radon measure on R2n and f (m) (t, x, v)d xdv ≡ 1, t ≥ 0, R n ×R n
then, up to a subsequence, there is a temporal measure f (t) ∈ M(R2n ) such that f (m) → f (t) in w ∗ -M(R2n ) as m → ∞. Moreover, formally it is a usual way to show (see for instance [17]) that f (t) is the measure-valued weak solution in M(R2n ) to the kinetic equation ∂t f + ξ · ∇x f + κU0 ∗ ρξ f · ∇ξ f = κU0 ∗ ρ f ∇ξ · (μ∇ξ f + ξ f ).
(1.24)
In Eq. (1.24) the nonlinear diffusion term follows from the so-called Ito’s formula. On the other hand, the nonlinear kinetic model equation (1.24) of Fokker-Planck type can be also obtained as the grazing limit of a certain kinetic equation of Boltzmann
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type. Let us assume that the post-interaction velocities (ξ ∗ , η∗ ) of two birds which have positions and velocities (x, ξ ) and (y, η) before interaction are determined by the law ξ ∗ = (1 − κU (|x − y|))ξ + κU (|x − y|)η + 2μκU (|x − y|) θξ , η∗ = κU (|x − y|)ξ + (1 − κU (|x − y|))η + 2μκU (|x − y|) θη , where U is defined as before, while κ > 0 and μ ≥ 0 are constants which will enter into the equation exactly in the same way as in (1.24), and θξ = (θξ,1 , θξ,2 , . . . , θξ,n ) ∈ Rn , θη = (θη,1 , θη,2 , . . . , θη,n ) ∈ Rn . θξ,i and θη,i , (1 ≤ i ≤ n) are identically distributed independent random variables of zero mean and unit variance. For this time, it is also supposed that supx U (|x|) is finite and κ sup U (|x|) < x
1 . 2
(1.25)
Notice that this assumption can be removed in the later grazing limit since U will be scaled up to a small parameter > 0. As in [2], the evolution of the bird density can be described at a kinetic level by the following integro-differential equation of Boltzmann type: ∂t f + ξ · ∇x f = Q( f, f ), with
Q( f, f ) = σ
R n ×R n
(1.26)
1 f (x, ξ∗ ) f (y, η∗ ) − f (x, ξ ) f (y, η) dydη, J (|x − y|)
where (ξ∗ , η∗ ) mean the pre-collisional velocities of particles that generate the pair velocities (ξ, η) after interaction, and J (|x − y|) = (1 − 2κU (|x − y|))n is the Jacobian of the transformation of (ξ, η) into (ξ ∗ , η∗ ). Notice that J is a well-defined nonnegative function due to the assumption (1.25). Definition 1.1. The function f (t, x, ξ ) is said to be a weak solution to the Cauchy problem of Eq. (1.26) with initial data f 0 (x, ξ ) provided that for any smooth function φ(x, ξ ) with compact support, it holds that d φ(x, ξ ) f (t, x, ξ )d xdξ = ξ · ∇x φ(x, ξ ) f (t, x, ξ )d xdξ dt R2n R2n (φ(x, ξ ∗ ) − φ(x, ξ )) f (t, x, ξ ) f (t, y, η)d xd ydξ dη (1.27) +σE R4n
for any t > 0 and lim
t→0+ R2n
φ(x, ξ ) f (t, x, ξ )d xdξ =
R2n
φ(x, ξ ) f 0 (x, ξ )d xdξ.
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To carry out the grazing limit, we scale U as U = U0 , where > 0 is a small parameter. Suppose that f = f (t, x, ξ ) satisfies Eq. (1.26), where f actually takes the form of f σ, which depends on parameters σ and but the superscripts are omitted for brevity. Let us begin with the weak form (1.27) and we consider the Taylor’s expansion φ(x, ξ ∗ ) − φ(x, ξ ) = ∇ξ φ(x, ξ ) · (ξ ∗ − ξ ) +
1 β ∂ξ φ(x, ξ )(ξ ∗ − ξ )β 2 |β|=2
1 β + ∂ξ φ(x, (ξ ∗ , ξ ))(ξ ∗ − ξ )β , 6 |β|=3
where (ξ ∗ , ξ ) is a vector between ξ ∗ and ξ . Recall also that ξ ∗ − ξ = κU0 (|x − y|)(η − ξ ) + 2μκU0 (|x − y|) θξ . Then, formally one has E[φ(x, ξ ∗ ) − φ(x, ξ )] = ∇ξ φ(x, ξ ) · κU0 (|x − y|)(η − ξ ) + ξ φ(x, ξ ) · μκU0 (|x − y|) + O( 2 ). Thus,
d φ(x, ξ ) f (t, x, ξ )d xdξ = ξ · ∇x φ(x, ξ ) f (t, x, ξ )d xdξ dt R2n R2n +σ ∇ξ φ(x, ξ ) · (η − ξ )κU0 (|x − y|) f (t, x, ξ ) f (t, y, η)d xd ydξ dη 4n R +σ μξ φ(x, ξ )κU0 (|x − y|) f (t, x, ξ ) f (t, y, η)d xd ydξ dη R4n
+ σ O(). Taking the so-called grazing limit, so that → 0, σ → 1, then the limit function, still denoted by f (t, x, ξ ), satisfies d φ(x, ξ ) f (t, x, ξ )d xdξ = ξ · ∇x φ(x, ξ ) f (t, x, ξ )d xdξ dt R2n R2n + ∇ξ φ(x, ξ ) · (η − ξ )κU0 (|x − y|) f (t, x, ξ ) f (t, y, η)d xd ydξ dη 4n R + μξ φ(x, ξ )κU0 (|x − y|) f (t, x, ξ ) f (t, y, η)d xd ydξ dη. R4n
This implies that f satisfies ∂t f + ξ · ∇x f = κ∇ξ · ( f U0 ∗ (ρξ f − ξρ f )) + κμξ ( f U0 ∗ ρ f ), which is in the same form as (1.24).
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2. Preparations 2.1. Coercivity of the linearized operator. In this subsection, we are concerned with some properties of the linearized operator L defined by (1.8), especially the coercivity estimate of L over the Hilbert space L 2x,ξ . Notice that L is the summation of the classical linearized Fokker-Planck operator L FP and the convolution-type operator A, i.e. L = L FP + A,
(2.1)
where L FP and A are defined by 1 L FP u = ξ u + (2n − |ξ |2 )u, 4 √ Au = U ∗ ρξ √Mu · ξ M.
(2.2) (2.3)
In (2.2) and (2.3) n ≥ 1 denotes the spatial dimension, U satisfies the condition (1.3) and M is the normalized global Maxwellian given by (1.5). Firstly, it is well-known from [1] that L FP enjoys some dissipative properties, stated in the following Proposition 2.1. L FP is a linear self-adjoint operator with respect to the duality induced by the L 2ξ -scalar product, and it is local in x. Furthermore, the following properties hold: (i) One has
u 2 L FP u, u = − ∇ξ √M Mdξ, Rn √ √ Ker L FP = Span{ M}, Range L FP = Span{ M}⊥ .
(ii) Define the projector P0 by √ √ P0 u = a u M, a u ≡ M, u . Then, one has the identity 1 |∇ξ {I − P0 }u|2 dξ − |ξ |2 |{I − P0 }u|2 dξ L FP u, u = − n n 4 R R n 2 + |{I − P0 }u| dξ. 2 Rn (iii) There exists a constant λ FP > 0 such that the Poincaré inequality holds: |{I − P0 }u|2 dξ. −L FP u, u ≥ λ FP Rn
(iv) More strongly, there is a constant λ0 > 0 such that the coercivity estimate holds: − L FP u, u ≥ λ0 |{I − P0 }u|2ν , where the norm | · |ν is defined in (1.10).
(2.4)
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Next, we shall obtain some coercivity estimate similar to (2.4) on the non-local linear operator L in the phase space Rnx ×Rnξ . Notice that it is straightforward to make estimates on A as ≤ Cn Au, u d x |{I − P0 }u|2 d xdξ, Rn
R n ×R n
where Cn = |ξ |2 , M depends only on n. On the other hand, from (2.4), L FP u, u d x ≥ λ0 {I − P0 }u2ν holds. − Rn
Since it is not clear presently whether λ0 is strictly larger than Cn , it is nontrivial to get a coercivity estimate on L directly from (2.4). It turns out that one has to extract part of dissipation of L FP corresponding to the momentum component of u in order to control the non-local operator A. To do that, let us decompose the Hilbert space L 2ξ as √ √ L 2ξ = N ⊕ N ⊥ , N = Span{ M, ξ M},
and define the projector P by √ P : L 2ξ → N , u → Pu ≡ {a u + bu · ξ } M. √ √ √ Notice that since M, ξ1 M, . . . , ξn M forms an orthonormal basis of N , then one has √ √ a u = M, u , bu = ξ M, u . We also introduce the projector P1 by √ √ √ P1 u = bu · ξ M = ξ M, u · ξ M. Then P can be written as P = P0 ⊕ P1 , in L 2ξ . The main result of this subsection concerning the coercivity estimate of L is stated as follows. Theorem 2.1. Let n ≥ 1 and (1.3) hold. The operators L, L FP , A are defined by (2.1), (2.2) and (2.3), respectively. Then, the following holds: (i) A and hence L are linear nonlocal operators which are self-adjoint with respect to the duality induced by the L 2x,ξ -scalar product; (ii) One has identities: √ Au = APu = PAu = AP1 u = P1 Au = U ∗ bu · ξ M, (2.5) A{I − P}u = {I − P}Au = 0, (2.6) L FP Pu = L FP P1 u = −P1 u, PL FP u = −P1 u, (2.7) √ (2.8) LPu = −[P1 , A]u = −(bu − U ∗ bu ) · ξ M, where [P1 , A] denotes the commutator P1 A − AP1 ;
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(iii) Let λ0 be defined in (2.4). Then, the coercivity inequality 1 2 − Lu, u d x ≥ λ0 {I − P}u2ν + T bu U , n 2 R
(2.9)
holds for any u = u(x, ξ ), where the norms · ν and · U are defined in (1.11) and (1.12), respectively. Proof. To prove (i), for any u = u(x, ξ ), v = v(x, ξ ), √ Au, v d x = U ∗ bu · ξ Mvd xdξ = U ∗ bu bv d x Rn R n ×R n Rn U (|x − y|)bu (y) · bv (x)d xd y = R n ×R n u v b U ∗ b dx = u, Av d x holds, = Rn
Rn
where the symmetry of U = U (|x|) was used. Then, A is self-adjoint on L 2x,ξ . Since L FP is also self-adjoint with respect to the duality induced by the L 2x,ξ -scalar product, so is L = L FP + A. To prove (ii), (2.5) and (2.6) directly follow from definitions of A, P1 and P. Notice L FP Pu = L FP P0 u + L FP P1 u = L FP P1 u holds from Proposition 2.1 (i). Then, one can compute √ √ √ 1 L FP Pu = ξ (bu · ξ M) + (2n − |ξ |2 )(bu · ξ M) = −bu · ξ M = −P1 u, 4 where we used
√ √ √ ξ (bu · ξ M) = ξ (bu · ξ ) + 2∇ξ (bu · ξ ) · ∇ξ M + bu · ξ ξ M,
and √ √ √ 1 √ n√ 1 ∇ξ M = − ξ M, ξ M = − M + |ξ |2 M. 2 2 4 Moreover, √ √ √ √ PL FP u = M, L FP u M + ξ M, L FP u · ξ M √ √ √ √ = L FP P0 M, u M + L FP P1 (ξ M), u · ξ M √ √ = −ξ M, u · ξ M = −P1 u. Then, Eq. (2.7) is proved. Equation (2.8) follows from L = L FP + A and (2.6)–(2.7). To prove (iii), for any u, one has L FP u, u = L FP Pu, Pu + L FP Pu, {I − P}u + L FP {I − P}u, Pu + L FP {I − P}u, {I − P}u = L FP Pu, Pu + 2L FP Pu, {I − P}u + L FP {I − P}u, {I − P}u ,
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where for the first two terms on the r.h.s., further from (2.7), L FP Pu, Pu = −P1 u, Pu = −P1 u, P1 u = −|bu |2 , L FP Pu, {I − P}u = −P1 u, {I − P}u = 0. Then, one has L FP u, u = L FP {I − P}u, {I − P}u − |bu |2 . On the other hand, for A, similarly one has Au, u d x = AP1 u, u d x = P1 u, AP1 u d x = Rn
Rn
Rn
Rn
U ∗ bu · bu d x.
Thus, combining the above estimates on L FP and A, it follows that Lu, u d x = L FP {I − P}u, {I − P}u d x Rn Rn − |bu |2 d x + U ∗ bu · bu d x. Rn
Rn
One can further compute |bu |2 d x − U ∗ bu · bu d x Rn Rn = U (|x − y|)bu (x)(bu (x) − bu (y))d xd y R n ×R n U (|y − x|)bu (y)(bu (y) − bu (x))d xd y = n n R ×R 1 1 2 U (|x − y|)|bu (x) − bu (y)|2 d xd y = T bu U . = 2 R n ×R n 2 Therefore, (2.9) follows from the coercivity inequality (2.4) for L FP and {I − P0 }{I − P} = {I − P}. Hence, also (iii) is proved. This completes the proof of Theorem 2.1.
2.2. Macro-micro decomposition. As usual, for fixed (t, x), u(t, x, ξ ) can be uniquely decomposed as ⎧ ⎨ u(t, x, ξ ) = Pu + {I √− P}u, u u Pu ≡ {a (2.10) √ + b · ξ } M, √ ⎩ u a = M, u , bu = ξ M, u , where Pu is called the macroscopic component of u while {I − P}u is called the corresponding microscopic component. Notice that by the definitions of a u and bu , Pu ⊥ {I − P}u holds in L 2ξ for any (t, x).
(2.11)
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In what follows, let us suppose that u satisfies the perturbation equation (1.6) and the spatial dimension n ≥ 1 holds. For later use, let us now derive some macroscopic balance laws satisfied by the macro components a u and bu . To do that, rewrite (1.6) as ∂t u + ξ · ∇x u + U ∗ bu · ∇ξ u = Lu + (u, u),
(2.12)
where by (1.9), (·, ·) is regarded as a bilinear operator defined by 1 (u, v) = U ∗ a u L FP v + U ∗ bu · ξ v. 2
(2.13)
After taking velocity integration from the unperturbed equation (1.1), one has the local conservation law of mass: ∂t f dξ + ∇x · ξ f dξ = 0, (2.14) Rn
Rn
and the local balance law of momentum: ∂t ξi f dξ +∇x · ξ ξi f dξ −U ∗ ρξi f Rn
Rn
Rn
f dξ = −U ∗ ρ f
Rn
ξi f dξ
(2.15)
for 1 ≤ i ≤ n. By using the macro-micro decomposition (2.10) and the property (2.11), one can compute the moments of f up to second order as follows: √ f dξ = (M + Mu)dξ = 1 + a u , n n R R √ ξi f dξ = ξi (M + Mu)dξ = biu , Rn
and
Rn
Rn
√ ξi ξ j (M + Mu)dξ Rn √ √ = δi j + ξi ξ j MPudξ + ξi ξ j M{I − P}udξ Rn Rn √ u = (1 + a )δi j + ξi ξ j M, {I − P}u ,
ξi ξ j f dξ =
for 1 ≤ i, j ≤ n, where δi j is the Kronecker delta. Thus, it follows from (2.14) and (2.15) that ∂t a u + ∇x · bu = 0, and ∂t biu + ∂i a u +
√ ∂ j ξi ξ j M, {I − P}u − U ∗ biu (1 + a u )
j
= −U ∗ (1 + a u )biu , 1 ≤ i ≤ n. Next, we need to derive the evolution of second-order moments of {I − P}u: √ ξ ⊗ ξ M, {I − P}u .
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By using L = L FP + A, L FP P = −P1 , and the macro-micro decomposition (2.10), one can further rewrite (2.12) as √ ∂t u + ξ · ∇x u + U ∗ bu · ∇ξ u = L FP {I − P}u + (U ∗ bu − bu ) · ξ M √ + U ∗ a u L FP {I − P}u − U ∗ a u bu · ξ M 1 1 + U ∗ bu · ξ Pu + U ∗ bu · ξ {I − P}u, 2 2 that is ∂t Pu + ξ · ∇x Pu + U ∗ bu · ∇ξ Pu √ √ 1 − (U ∗ bu − bu ) · ξ M + U ∗ a u bu · ξ M − U ∗ bu · ξ Pu 2 = −∂t {I − P}u + l + r.
(2.16)
In (2.16), the linear term l and the nonlinear term r , are given respectively by l = −ξ · ∇x {I − P}u + L FP {I − P}u, 1 r = U ∗ a u L FP {I − P}u + U ∗ bu · ξ {I − P}u − U ∗ bu · ∇ξ {I − P}u. 2
(2.17) (2.18)
By using the representation of Pu as in (2.10), one can expand the l.h.s. of (2.16) as √ u ∂t a + U ∗ bu · bu M +
+
n
u √ ∂t bi + ∂i a u − (U ∗ biu − biu ) + U ∗ a u biu − U ∗ biu a u ξi M i=1 n
√ ∂i buj − U ∗ biu buj ξi ξ j M
i j=1
= −∂t {I − P}u + l + r.
(2.19)
Let us define the moment function A = (Ai j (·))n×n by √ Ai j (u) = (ξi ξ j − 1) Mudξ. Rn
(2.20)
Then, applying Ai j (·) to both sides of (2.19) yields ∂i biu − U ∗ biu biu = −∂t Aii ({I − P}u) + Aii (l + r ), and ∂i buj + ∂ j biu − U ∗ biu buj − U ∗ buj biu = −∂t Ai j ({I − P}u) + Ai j (l + r ), i = j, where 1 ≤ i, j ≤ n.
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In summary, the macro components a u and bu satisfy the equations ∂t a u + ∇x · bu = 0,
(2.21)
∂t biu + ∂i a u − (U ∗ biu − biu ) + U ∗ a u biu n
−U ∗ biu a u + ∂ j Ai j ({I − P}u) = 0,
(2.22)
j=1
∂t Aii ({I − P}u) + ∂i biu − U ∗ biu biu = Aii (l + r ), ∂t Ai j ({I
− P}u) + ∂i buj
+ ∂ j biu
−U
∗ biu buj
−U
∗ buj biu
(2.23)
= Ai j (l + r ), i = j, (2.24)
for 1 ≤ i, j ≤ n, where l, r are defined by (2.17) and (2.18), respectively. Notice that (2.24) is symmetric in (i, j). The similar derivation of the system of Eqs. (2.21)–(2.24) is inspired by [14] and used recently in [10] and [11] for the study of the Boltzmann equation and the Vlasov-Poisson-Boltzmann system, respectively. The following important observation, which plays a key role in the estimates on the macroscopic dissipation pointed firstly in [16] and later in [14], is that from (2.23) and (2.24), bu satisfies the following Proposition 2.2. For fixed 1 ≤ j ≤ n, ⎤ ⎡
∂t ⎣ ∂ j Aii ({I − P}u) − ∂i Ai j ({I − P}u)⎦ − x buj i= j
=
∂ j (U
∗ biu biu ) −
i
+
i
∂i (U ∗ biu buj + U ∗ buj biu )
i
∂ j Aii (l + r ) −
i= j
∂i Ai j (l + r ),
(2.25)
i
holds for t ≥ 0 and x ∈ Rn . Proof. For simplicity, set R = −∂t {I − P}u + l + r. From (2.24), one can compute
−x buj = − ∂i (∂i buj ) − ∂ j ∂ j buj i= j
=−
∂i [−∂ j biu + U ∗ biu buj + U ∗ buj biu + Ai j (R)] − ∂ j ∂ j buj
i= j
⎡
= ∂j ⎣
⎤ ∂i biu − ∂ j buj ⎦ −
i= j
∂i [U ∗ biu buj + U ∗ buj biu + Ai j (R)].
i= j
Thanks to (2.23), one has
−x buj = ∂ j [U ∗ biu biu + Aii (R)] − ∂ j [U ∗ buj buj + A j j (R)] i= j
−
i= j
∂i [U ∗ biu buj + U ∗ buj biu + Ai j (R)].
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A further simplification gives
−x buj = ∂ j (U ∗ biu biu ) − ∂i (U ∗ biu buj + U ∗ buj biu ) i
+
∂ j Aii (R) −
i= j
i
∂i Ai j (R).
i
Then, (2.25) follows from the definition of R and the linearity of Ai j . This completes the proof of Proposition 2.2. 3. Linearized Cauchy Problem 3.1. Hypocoercivity. Let us now consider the Cauchy problem of the linearized equation with a nonhomogeneous source, namely ∂t u = Bu + h, t > 0, x ∈ Rn , (3.1) u|t=0 = u 0 , x ∈ Rn , where n ≥ 1 is the spatial dimension, h = h(t, x, ξ ) and u 0 = u 0 (x, ξ ) are given, and the linear operator B is defined by B = −ξ · ∇x + L, L = L FP + A, 1 L FP u = ξ u + (2n − |ξ |2 )u, 4 √ √ Au = U ∗ bu · ξ M, bu = ξ M, u . Formally, the solution to the Cauchy problem (3.1) can be written as the Duhamel formula t u(t) = eBt u 0 + eB(t−s) h(s)ds, 0
where eBt denotes the solution operator to the Cauchy problem of the linearized equation without source corresponding to (3.1) with h ≡ 0. In this section, we shall show that eBt has the algebraic decay as time tends to infinity as in the case of the Boltzmann equation [12,13,23,24]. The approach of proof is here based on the Fourier analysis. To this end, for 1 ≤ q ≤ 2 and m ≥ 0, set the rate index σq,m by
1 m n 1 − + . σq,m = 2 q 2 2 The main result of this section, whose proof is left to the next subsection, is stated as follows: Theorem 3.1. Let 1 ≤ q ≤ 2 and n ≥ 1, and let (1.3) hold.
(i) For any α, α with α ≤ α, and for any u 0 satisfying ∂xα u 0 ∈ L 2x,ξ and ∂xα u 0 ∈ Z q , one has
∂xα eBt u 0 ≤ C(1 + t)−σq,m (∂xα u 0 Z q + ∂xα u 0 ),
(3.2)
for t ≥ 0 with m = |α − α |, where C is a positive constant depending only on n, m, q.
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(ii) Similarly, for any α, α with α ≤ α, and for any h such that, for all t ≥ 0,
ν(ξ )−1/2 ∂xα h(t) ∈ L 2x,ξ , ν(ξ )−1/2 ∂xα h(t) ∈ Z q holds and further √ √ Mh(t, x, ξ )dξ = ξi Mh(t, x, ξ )dξ = 0, i = 1, 2, . . . , n, (3.3) Rn Rn ,
Rn
x∈ one has t 2 α B(t−s) ∂ e h(s)ds x 0 t
≤C (1 + t − s)−2σq,m (ν −1/2 ∂xα h(s)2Z q + ν −1/2 ∂xα h(s)2 )ds,
(3.4)
0
for t ≥ 0 with m = |α − α |, where C is a positive constant depending only on n, m, q. 3.2. Proof of hypocoercivity: Fourier analysis. In what follows we devote ourselves to the proof of Theorem 3.1. Let u = u(t, x, ξ ) be the solution to the Cauchy problem (3.1) with the nonhomogeneous source h(t, x, ξ ) and initial data u 0 (x, ξ ). Similarly as before, we decompose u as ⎧ ⎨ u(t, x, ξ ) = Pu + {I √− P}u, u u + b · ξ } M, √ Pu ≡ {a √ ⎩ u u a = M, u , b = ξ M, u . Then, from the same procedure as in Sect. 2.2, a suitable skipping of the nonlinear term (u, u) leads to the macroscopic balance laws satisfied by a u , bu : ∂t a u + ∇x · bu = 0, n
∂ j Ai j ({I − P}u) = 0, ∂t biu + ∂i a u − (U ∗ biu − biu ) + j=1 u ∂t Aii ({I − P}u) + ∂i bi = Aii (l + h), ∂t Ai j ({I − P}u) + ∂i buj + ∂ j biu = Ai j (l + h),
i = j,
(3.5) (3.6) (3.7) (3.8)
where 1 ≤ i, j ≤ n, the velocity moment function Ai j (·) is defined by (2.20), and l has the same form as before, given by l = −ξ · ∇x {I − P}u + L FP {I − P}u.
(3.9)
Here we notice that h does not appear in the first n + 1 Eqs. (3.5)–(3.6) because the assumption (3.3) implies that Ph(t, x) ≡ 0, t ≥ 0, x ∈ Rn . Furthermore, following a procedure similar to that used to derive (2.25) from (2.23) and (2.24), for fixed 1 ≤ j ≤ n, it follows from (3.7) and (3.8) that ⎤ ⎡
∂t ⎣ ∂ j Aii ({I − P}u) − ∂i Ai j ({I − P}u)⎦ − x buj i= j
=
i= j
∂ j Aii (l + h) −
i
i
∂i Ai j (l + h).
(3.10)
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Up to the end of this subsection, let us introduce some notations. For an integrable function g : Rn → R, its Fourier transform ! g = F g is defined by ! g (k) = F g(k) =
Rn
e−2π ix·k g(x)d x, x · k =:
n
xjkj.
j=1
√ Here, k ∈ Rn , and i = −1 ∈ C is the imaginary unit. For two complex vectors a, b ∈ Cn , (a | b) denotes the dot product a · b over the complex field, where b is the complex conjugate of b. Lemma 3.1. There is a temporal-frequency free energy functional E lf r ee (! u (t, k)) in the form of E lf r ee (! u (t, k)) = 3
i= j
j
−3
iki (Ai j ({I − P}! u ) | b!uj ) 1 + |k|2
ij
−
ik j (Aii ({I − P}! u ) | b!uj ) 1 + |k|2
ik · (b!u | a!u ) 1 + |k|2
(3.11)
such that ∂ |k|2 Re E lf r ee (! (|a!u |2 + |b!u |2 ) u (t, k)) + ∂t 4(1 + |k|2 ) ! 1 − Re U C ≤ |b!u |2 + ν −1/2! u 2L 2 h2L 2 + C{I − P}! 2 ξ ξ 1 + |k| 1 + |k|2
(3.12)
holds for t ≥ 0 and k ∈ Rn . Moreover, one has the estimate u (t, k))| ≤ C! u (t, k)2L 2 , |E lf r ee (!
(3.13)
ξ
for t ≥ 0 and k ∈ Rn . Proof. Let us first notice that F({I − P}u) = {I − P}Fu, F(Pu) = PFu. After taking the Fourier transform, (3.10) reads ⎡
∂t ⎣
ik j Aii ({I − P}! u) −
i= j
=
i= j
ik j Aii (! l +! h) −
i
⎤ iki Ai j ({I − P}! u )⎦ + |k|2 b!uj
i
iki Ai j (! l +! h),
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which by further taking the inner product with b!uj gives ⎛ ⎞
∂t ⎝ ik j Aii ({I − P}! u) − iki Ai j ({I − P}! u ) | b!uj ⎠ + |k|2 |b!uj |2 i= j
i
⎛ ⎞
=⎝ ik j Aii (! l +! h) − iki Ai j (! l +! h) | b!u ⎠ j
i= j
⎛ +⎝
i
ik j Aii ({I − P}! u) −
i= j
⎞ iki Ai j ({I − P}! u ) | ∂t b!uj ⎠ .
(3.14)
i
The first term on the r.h.s of (3.14) is estimated by ⎛ ⎞
u ! ⎝ ! ! ! ! ⎠ ik j Aii (l + h) − iki Ai j (l + h) | b j i= j i
1 ≤ |k|2 |b!uj |2 + C |Ai j (! l)|2 + |Ai j (! h)|2 . 4
(3.15)
ij
By (3.9), the Fourier transform ! l of l is given by ! l = −iξ · k{I − P}! u + L FP {I − P}! u, and thus one has √ ! (ξi ξ j − 1) M(−iξ · k{I − P}! u + L FP {I − P}! u )dξ |Ai j (l)| = n R √ √ = [−iξ · k(ξi ξ j − 1) M + L FP ((ξi ξ j − 1) M)]{I − P}! u dξ Rn √ √ ≤ − iξ · k(ξi ξ j − 1) M + L FP ((ξi ξ j − 1) M) L 2 {I − P}! uL 2 ξ
≤ C(1 + |k|){I − P}! uL 2 .
(3.16)
ξ
Similarly one has for ! h,
ξ
|Ai j (! h)| =
√ (ξi ξ j − 1) M! hdξ Rn √ ≤ ν 1/2 (ξi ξ j − 1) M L 2 ν −1/2! h L 2 ξ
ξ
≤ Cν −1/2! h L 2 . ξ
(3.17)
Therefore, (3.15) together with (3.16) and (3.17) imply ⎛ ⎞
u ⎠ ! ⎝ ! ! ! ! ik A ( l + h) − ik A ( l + h) | b j ii i ij j i= j i ≤
1 2 !u 2 u 2L 2 + Cν −1/2! h2L 2 , |k| |b j | + C(1 + |k|2 ){I − P}! ξ ξ 4
(3.18)
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which gives the estimate on the first term on the r.h.s. of (3.14). For the second term, one can use the Fourier transform of (3.6),
!b!u − b!u ) + ∂t b!iu + iki a!u − (U ik j Ai j ({I − P}! u) = 0 (3.19) i i j
to estimate
⎛ ⎝
ik j Aii ({I − P}! u) −
i= j
⎞ iki Ai j ({I − P}! u ) | ∂t b!uj ⎠
i
⎛
=⎝ ik j Aii ({I − P}! u) − iki Ai j ({I − P}! u) | i= j
i
!b!u − b!u ) − −ik j a!u + (U j j
ik A j ({I − P}! u)
1 ≤ δ|k|2 |a!u |2 + |k|2 |b!u |2 + Cδ (1 + |k|2 ){I − P}! u 2L 2 , ξ 4
(3.20)
!| ≤ U L 1 = 1 where the constant 0 < δ ≤ 1 is arbitrary. In (3.20), the property supk |U x and |Ai j ({I − P}! u )| ≤ C{I − P}! uL 2 ξ
have been used. From (3.14) as well as (3.18) and (3.20), one has ⎛ ⎞
1 ∂t ⎝ ik j Aii ({I − P}! u) − iki Ai j ({I − P}! u ) | b!uj ⎠ + |k|2 |b!uj |2 2 i= j
≤ δ|k|
i
2
|a!u |2
+ Cδ (1 + |k| ){I − P}! u 2L 2 + Cν −1/2! h2L 2 , 2
ξ
ξ
(3.21)
for 0 < δ ≤ 1 to be determined later. To get the dissipation |k|2 |a!u |2 as in (3.21), we take the inner product of (3.19) with −ik a!u : & ' ' & ! − 1)ik · b!u | a!u −∂t ik · b!u | a!u + |k|2 |a!u |2 + (U ⎞ ⎛
ki k j Ai j ({I − P}! u ) | a!u ⎠ = 0. (3.22) +⎝ ij
Let us write & ' & ' & ' −∂t ik · b!u | a!u = ∂t −ik · b!u | a!u + ik · b!u | ∂t a!u . From (3.5), which implies ∂t a!u + ik · b!u = 0,
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one has & ' & ' ik · b!u | ∂t a!u = ik · b!u | −ik · b!u = −|k · b!u |2 . Then, the first term on the l.h.s. of (3.22) reduces to & ' & ' −∂t ik · b!u | a!u = ∂t −ik · b!u | a!u − |k · b!u |2 . Notice that !(k) = Im U
Rn
U (x) sin(k · x)d x = 0,
since U is even. Then, ! = Re U ! U holds. Thus, one has the estimate on the third term on the l.h.s. of (3.22) as & ' (U ! − 1)ik · b!u | a!u ≤ 1 (1 − Re U !)|k|2 |a!u |2 + (1 − Re U !)|b!u |2 4 1 !)|b!u |2 . ≤ |k|2 |a!u |2 + (1 − Re U 4 !| ≤ |U !| ≤ 1. For the fourth term on the l.h.s. of (3.22), one finally Again we used |Re U has ⎛ ⎞
1 u ⎠ ≤ |k|2 |a!u |2 + C|k|2 ⎝ ! k k A ({I − P}! u ) | a |Ai j ({I − P}! u )|2 i j ij 4 ij ij ≤
1 2 u2 |k| |a! | + C|k|2 {I − P}! u 2L 2 . ξ 4
Thus, plugging all the above estimates into (3.22) yields ' 1 & ∂t Re −ik · b!u | a!u + |k|2 |a!u |2 2 !)|b!u |2 + C|k|2 {I − P}! ≤ |k|2 |b!u |2 + (1 − Re U u 2L 2 . ξ
(3.23)
Therefore, (3.12) follows by taking the proper linear combination of (3.21) and (3.23) with a fixed small constant 0 < δ ≤ 1 and then dividing it by 1 + |k|2 . This completes the proof of Lemma 3.1. Lemma 3.2. 1 ∂ !)|b!u |2 ≤ Cν −1/2! u |2ν + (1 − Re U h(t, k)2L 2 holds ! u (t, k) L 2 + λ|{I − P}! ξ ξ 2 ∂t (3.24) for any t ≥ 0 and k ∈ Rn .
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Proof. Since √ L FP Pu = −P1 u = −bu · ξ M, we can rewrite the first equation in (3.1) as √ ∂t u + ξ · ∇x u = L FP {I − P}u − (bu − U ∗ bu ) · ξ M + h. Taking the Fourier transform in x yields √ !)b!u · ξ M + ! ∂t ! u + iξ · k! u = L FP {I − P}! u − (1 − U h. u , integrating it in ξ over Rn and then using the By taking further the inner product with ! coercivity estimate on L FP (2.4), one has 1 ∂ !)|b!u |2 ≤ |! ! u (t)2L 2 + λ|{I − P}! u |2ν + (1 − Re U h, ! u |, ξ 2 ∂t
(3.25)
! = Re U ! and {I − P0 }{I − P} = {I − P}. For the r.h.s. term, since where we used U Ph = 0, it holds u = ! h, P! u + ! h, {I − P}! u = ! h, {I − P}! u , ! h, ! which implies that 1 |! h, ! u | ≤ δν 1/2 {I − P}! u 2L 2 + ν −1/2! h2L 2 ξ ξ 4δ 1 −1/2! 2 2 ≤ Cδ|{I − P}! u |ν + ν h L 2 , ξ 4δ
(3.26)
where δ > 0 is arbitrary. Therefore, (3.24) follows from (3.25) together with (3.26) by taking a properly small constant δ > 0. This completes the proof of Lemma 3.2. Proof of Theorem 3.1. Let u 0 and h be given as in Theorem 3.1, and u be the solution to the Cauchy problem (3.1). Then, by choosing M > 0 large enough, it follows from l (! (3.12) and (3.24) that there is an energy E M u (t, k)) with l EM (! u (t, k)) = M! u (t, k)2L 2 + Re E lf r ee (! u (t, k)) ξ
such that |k|2 ∂ l !u |2 + |b!u |2 ) E M (! u (t, k)) + λ |{I − P}! u |2ν + (| a ∂t 1 + |k|2 !)|b!u |2 ≤ Cν −1/2! h(t, k)2 2 , + λ(1 − Re U Lξ
(3.27)
for any t ≥ 0 and k ∈ Rn , where E lf r ee (! u (t, k)) is defined by (3.11). From (3.13) it follows that l EM (! u (t, k)) ∼ ! u (t, k)2L 2 , ξ
(3.28)
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if M > 0 is large enough. Notice that ) |k|2 |k|2 ( 2 u |2 +|b!u |2 . !u |2 + |b!u |2 ) ≥ ! |{I − P}! u | (| a +| a ν 1 + |k|2 1 + |k|2 On the other hand, one also has |{I−P}! u |2ν +
l EM (! u (t, k)) ≤ C! u (t, k)2L 2 ≤ CP! u (t, k)2L 2 + C{I − P}! u (t, k)2L 2 ξ ξ ξ ) ( 2 2 2 u u ! ! ≤ C |{I − P}! u |ν + |a | + |b | .
(3.29)
(3.30)
Thus, (3.27) together with (3.29) and (3.30) yield ∂ l λ|k|2 l !)|b!u |2 ≤ Cν −1/2! E M (! u (t, k)) + E (! u (t, k)) + λ(1 − Re U h(t, k)2L 2 , ξ ∂t 1 + |k|2 M which, by using the Gronwall inequality, gives t 2 2 − λ|k| t l − λ|k| (t−s) −1/2 l ! EM (! u (t, k)) ≤ e 1+|k|2 E M (u!0 (k)) + C e 1+|k|2 ν h(s, k)2L 2 ds. ξ
0
Thus, from (3.28) one obtains ! u (t, k)2L 2 ξ
≤ Ce
−
λ|k|2 t 1+|k|2
u!0 (k)2L 2 ξ
t
+C
e
−
λ|k|2 (t−s) 1+|k|2
ν −1/2! h(s, k)2L 2 ds, ξ
0
(3.31)
for any t ≥ 0 and k ∈ Rn . Now, in order to get the decay estimate (3.2), let h = 0 so that u(t) = eBt u 0 . Write α k = k1α1 k2α2 · · · knαn . Then, from (3.31), one has λ|k|2 α Bt 2 2α 2 2α − 1+|k|2 t ∂x e u 0 = |k | · ! u (t, k) L 2 dk ≤ C |k |e u!0 (k)2L 2 dk. (3.32) ξ
Rnk
ξ
Rnk
As in [19], one can further estimate it by 2 − λ|k| t |k 2α |e 1+|k|2 u!0 (k)2L 2 dk Rnk
≤
ξ
|k|≤1
|k 2(α−α ) |e
2 − λ|k| 2 t 1+|k|
|k 2α | · u!0 (k)2L 2 dk + ξ
| − qn + n−2|α−α 2
≤ C(1 + t)
α
∂x u 0 2Z q + Ce
− λ2 t
λ
|k|≥1
e− 2 t |k 2α | · u!0 (k)2L 2 dk ξ
∂xα u 0 2 ,
(3.33)
where the Hölder and Hausdorff-Young inequalities were used in the usual way. Hence, (3.2) follows from (3.32) and (3.33). On the other hand, to get the decay estimate (3.4), let u 0 = 0 so that t u(t) = eB(t−s) h(s)ds. 0
From (3.31), one obtains
! u (t, k)2L 2 ≤ C ξ
t
e 0
−
λ|k|2 (t−s) 1+|k|2
ν −1/2! h(s, k)2L 2 ds. ξ
(3.34)
Proceeding as in the derivation of (3.32) and (3.33), (3.4) follows from (3.34). This completes the proof of Theorem 3.1.
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4. Nonlinear Cauchy Problem 4.1. Uniform a priori estimates. From now on, we devote ourselves to the proof of the main result Theorem 1.1. Through this subsection, let u be the solution to the Cauchy problem (1.6) or equivalently (2.12) and (1.7), and let n ≥ 3, N ≥ 2[n/2] + 2.
(4.1)
Also we suppose that u is smooth enough to justify that all calculations can be carried out. By using the classical energy method, we shall obtain in this subsection some uniform a priori estimates on u on the basis of some energy and energy dissipation rate inequalities. By these a priori estimates one will obtain in the next subsection a proof of the global existence of solutions with the help of the local existence as well as the continuum argument, under the smallness and regularity conditions on initial data u 0 . For the time-decay rate of u, we shall apply in the last subsection the energy-spectrum method recently developed in [13] and later in [12], which combine the linearized spectral analysis given in Sect. 3 with the nonlinear high-order energy estimates. For the above purpose, we begin with the proof of uniform a priori estimates on u to obtain the microscopic dissipation rate
β 2 {I − P}∂xα ∂ξ u(t)2ν + T ∂xα bu (t)U , |α|+|β|≤N
|α|≤N
which corresponds to the total temporal energy. Firstly, from Eq. (1.6) or equivalently (2.12), one can obtain estimates on u and its space derivatives. The proof of these estimates will be postponed to Appendix A.1 for a simpler presentation. Here, we need to take care of the zero-order individually since the estimate on the nonlinear term (u, u) is a little subtle in the case of zero-order. Lemma 4.1 (Zero-order). 1 d 1 2 u(t)2 + λ{I − P}u2ν + T bu U 2 dt 2 2 ≤ C(a u , bu ) L 2x ∩L ∞ ({I − P}u2ν + T bu U ) x + C(a u , bu ) L 2x bu 2L ∞ x
(4.2)
holds for t ≥ 0, where λ > 0 and C are constants depending only on n. Lemma 4.2 (Space derivatives). 1 d 2 dt
∂xα u(t)2 + λ
1≤|α|≤N
⎛
≤ C∇x (a u , bu ) HxN −1 ⎝
+ C∇x b HxN −1
1≤|α|≤N
|α|≤N
u
2 ∂xα {I − P}u2ν + T ∂xα bu U
⎞
∂xα {I − P}u2ν + ∇x (a u , bu )2
∂xα ∇ξ {I − P}u2
1≤|α|≤N −1
holds for t ≥ 0, where λ > 0 and C are constants depending only on n.
HxN −1
⎠ (4.3)
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Next, we shall obtain estimates on the mixed space-velocity derivatives of u which appears on the r.h.s. of (4.3). Notice that by taking the velocity derivatives we do not affect L 2x,ξ -norms for the macroscopic component Pu. Thus, let us apply I − P to both sides of (2.12) to get ∂t {I − P}u + {I − P}(ξ · ∇x u + U ∗ bu · ∇ξ u) = {I − P}Lu + {I − P}(u, u).
(4.4)
One can make further simplifications on the r.h.s. terms. In fact, from Theorem 2.1 (ii) it follows that {I − P}Lu = {I − P}L FP u + {I − P}Au = {I − P}L FP u = L FP {I − P}u + L FP Pu − PL FP u = L FP {I − P}u − P1 u + P1 u = L FP {I − P}u. Similarly, 1 {I − P}(u, u) = {I − P}(U ∗ a u L FP u + U ∗ bu · ξ u) 2 1 = U ∗ a u L FP {I − P}u + U ∗ bu · ξ {I − P}u 2 1 + U ∗ bu · [ξ, {I − P}]u 2 1 = (u, {I − P}u) + U ∗ bu · [ξ, {I − P}]u 2 holds, where {I − P}L FP = L FP {I − P} was used. Moreover [ξ, {I − P}] denotes the commutator [ξi , {I − P}] = ξi {I − P} − {I − P}ξi = [ξi , P], 1 ≤ i ≤ n, with ξ regarded as the velocity multiplier operator. Therefore, (4.4) is simplified as ∂t {I − P}u + {I − P}(ξ · ∇x u + U ∗ bu · ∇ξ u) 1 = L FP {I − P}u + (u, {I − P}u) + U ∗ bu · [ξ, P]u, 2 which further can be rewritten as the evolution equation of {I − P}u: ∂t {I − P}u + ξ · ∇x {I − P}u + U ∗ bu · ∇ξ {I − P}u 1 = L FP {I − P}u + (u, {I − P}u) + U ∗ bu · [ξ, P]u 2 + P(ξ · ∇x {I − P}u + U ∗ bu · ∇ξ {I − P}u) − {I − P}(ξ · ∇x Pu + U ∗ bu · ∇ξ Pu).
(4.5)
Then, on the basis of the above equation, one can use the energy estimates to obtain the following technical lemma, which is proven in Appendix A.1.
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123
Lemma 4.3 (Mixed space-velocity derivatives). Let 1 ≤ k ≤ N .
1 d β β ∂xα ∂ξ {I − P}u2 + λ ∂xα ∂ξ {I − P}u2ν 2 dt |β|=k |α|+|β|≤N
|β|=k |α|+|β|≤N
⎛
≤ C(a u , bu ) HxN ⎝ ⎛ +C ⎝
|α|+|β|≤N
∂xα {I
|α|≤N −k+1
+ Cχ{2≤k≤N }
β ∂xα ∂ξ {I
− P}u2ν + ∇x (a u , bu )2
HxN −1
⎠
⎞
− P}u2ν
⎞
+ ∇x (a , b u
u
)2 N −k ⎠ Hx
β
∂xα ∂ξ {I − P}u2ν
1≤|β|≤k−1 |α|+|β|≤N
holds for t ≥ 0, where λ > 0 and C are constants depending only on n, and χ D denotes the characteristic function of a set D. Finally, in order to control the nonlinear term and close the a priori estimates under the smallness condition, we need to obtain the macroscopic dissipation rate:
∂xα ∇x Pu(t)2 ∼ ∂xα ∇x (a u , bu )2 |α|≤N −1
|α|≤N −1
which corresponds to a certain temporal free energy. Actually, the following lemma exactly gives the above dissipation for the macroscopic component Pu or equivalently the coefficients (a u , bu ). Here, the analysis is essentially based only on the macroscopic balance laws (2.21)–(2.24) satisfied by (a u , bu ) which have been derived in Subsect. 2.2. The proof will be carried out in the physical phase space by using a method close to the proof of Lemma 3.1 in the case of the linearized equation. Again, we postpone it to Appendix A.2. Lemma 4.4. There exists a temporal free energy E nf r ee (u(t)) of the form
E nf r ee (u(t)) = 3 Aii (∂xα ∂ j {I − P}u)∂xα buj d x |α|≤N −1
j
−3
i= j
|α|≤N −1 i j
+
n |α|≤N −1 R
Rn
Rn
Ai j (∂xα ∂i {I − P}u)∂xα buj d x
∂xα ∇x a u · ∂xα bu d x,
(4.6)
such that d n E (u(t)) + λ∇x (a u , bu )2 N −1 Hx dt f r ee
α u 2 α ≤C (T ∂x b U + ∂x {I − P}u2 ) |α|≤N
+ C(a u , bu )2H N (∇x (a u , bu )2 x
HxN −1
+
|α|≤N
∂xα {I − P}u2 )
(4.7)
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R. Duan, M. Fornasier, G. Toscani
holds for t ≥ 0, where λ > 0 and C are constants depending only on n. Moreover, |E nf r ee (u(t))| ≤ Cu(t)2L 2 (H N ) ξ
(4.8)
x
holds for t ≥ 0. We remark that an estimate similar to the one stated in Lemma 4.4 was firstly considered in [9] and recently developed in [10] in the study of the Boltzmann equation for the hard sphere model in Rn . In addition, the proofs of Lemma 4.4 and Lemma 3.1 at the level of linearization are in the same spirit even though the analysis of the latter is made pointwise both in time and frequency.
4.2. Proof of global existence and uniqueness. In this subsection, we are going to make a few preparations in order to prove Theorem 1.1 along the line mentioned at the beginning of Subsect. 4.1. Let us first consider the local existence of solutions to the Cauchy problem (1.6) or equivalently (2.12) and (1.7). We define iteratively the sequence ( f m (t, x, ξ ))∞ m=0 of solutions to the Cauchy problems ⎧ ∂t f m+1 + ξ · ∇x f m+1 + U ∗ ρξ f m · ∇ξ f m+1 ⎪ ⎪ ⎨ m+1 m ∇ξ · (∇ξ f = U ∗ ρf√ + ξ f m+1 ), m+1 m+1 ⎪ ≡ M + Mu ,√ f ⎪ ⎩ m+1 |t=0 = f 0 ≡ M + Mu 0 , f or equivalently in terms of u m (t, x, ξ ): ⎧ m+1 m + ξ · ∇x u m+1 + U ∗ bu · ∇ξ u m+1 ⎨ ∂t u = L u m+1 + (u m , u m+1 ) + Au m , ⎩ m+1 FP |t=0 = u 0 , u
(4.9)
(4.10)
where m ≥ 0, and u 0 ≡ 0 is set at the beginning of iteration. Let the solution space X (0, T ; M) be defined by ⎧ ⎫ ⎨ v ∈ C([0, T ]; H N (Rn × Rn )) : ⎬ √ X (0, T ; M) = . v(t) ≤ M, M + Mv ≥ 0 sup N H ⎩ ⎭ 0≤t≤T
x,ξ
We prove the following Theorem 4.1. Let n, N satisfy (4.1). There √ are constants T∗ > 0, 0 , M0 such that if u 0 ∈ H N (Rn × Rn ) with f 0 ≡ M + Mu 0 ≥ 0 and u 0 H N ≤ 0 , then for each x,ξ
m ≥ 1, u m is well-defined with u m ∈ X (0, T∗ ; M0 ).
(4.11)
Furthermore, (u m )m≥0 is a Cauchy sequence in the Banach space C([0, T∗ ]; H N −1 (Rn × Rn )), and the corresponding limit function denoted by u belongs to X (0, T∗ ; M0 ), and u is a solution to the Cauchy problem (1.6)–(1.7). Meanwhile, there exists at most one solution in X (0, T∗ ; M0 ) to the Cauchy problem (1.6)–(1.7).
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Proof. One can use induction to prove (4.11). Suppose that (4.11) holds true for m ≥ 0. Without loss of generality, one can also suppose that u m is smooth enough so that all the forthcoming calculations can be carried out. Otherwise, one can instead consider the Cauchy problem on the regularized iterative equation ∂t f m+1, + ξ · ∇x f m+1, + U ∗ ρξ f m, · ∇ξ f m+1, = U ∗ ρ f m, ∇ξ · (∇ξ f m+1, + ξ f m+1, ) + x f m+1, , f m+1 |t=0 = u 0 . for any > 0 with u 0 a smooth approximation of u 0 , prove the same for f m, and then pass to the limit by letting → 0. Thanks to the nonnegativity m
U ∗ ρ f m = 1 + U ∗ au , one has 1 − C1 Mm (T ) ≤ U ∗ ρ f m ≤ 1 + C1 Mm (T ), 0 ≤ t ≤ T, for some constant C1 > 0, where Mm (T ) = sup u m (t)2H N 0≤t≤T
x,ξ
for any 0 ≤ T ≤ T∗ . Note that 2C1 M0 ≤ 1 if M0 > 0, to be chosen later, is sufficiently small. If this is the case, from the induction hypothesis, the estimate 2C1 Mm (T ) ≤ 2C1 M0 ≤ 1 holds. Then 1/2 ≤ U ∗ ρ f m ≤ 3/2 follows. By the maximum principle for (4.9), one has √ f m+1 ≡ M + Mu m+1 ≥ 0. To obtain the bound on u m+1 , for any α with |α| ≤ N , it follows from (4.10) 1 d α m+1 ∂ u (t)2 + λ0 {I − P0 }∂xα u m+1 2ν 2 dt x
m
= Cαα
U ∗ ∂xα−α bu · ∇ξ ∂xα u m+1 , ∂xα u m+1 d x Rn
α <α
+
A∂xα u m , ∂xα u m+1 d x
Rn m
≤ Cu L 2 (H N ) ξ
x
|α |+|β |≤N
∂xα (u m , u m+1 ), ∂xα u m+1 d x Rn
β
∂xα ∂ξ u m+1 2ν +
+ Cu m L 2 (H N ) u m+1 L 2 (H N ) . ξ
x
ξ
x
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Taking the summation over |α| ≤ N implies 1 d m+1 λ0 u (t)2L 2 (H N ) + ∂xα u m+1 2ν x ξ 2 dt 2 |α|≤N
β m ≤ Cu L 2 (H N ) ∂xα ∂ξ u m+1 2ν ξ
x
|α|+|β|≤N
+ Cu L 2 (H N ) u m+1 L 2 (H N ) + Cu m+1 2L 2 (H N ) . m
ξ
ξ
x
x
ξ
x
Similarly, for any 0 ≤ t ≤ T ≤ T∗ ,
d m+1 β u (t)2H N + λ0 ∂xα ∂ξ u m+1 2ν dt x,ξ |α|+|β|≤N
β ≤ Cu m L 2 (H N ) ∂xα ∂ξ u m+1 2ν ξ
x
|α|+|β|≤N
+ Cu H N u m+1 H N + Cu m+1 2L 2 (H N ) x,ξ x,ξ x ξ
α β m+1 2 ≤ C2 Mm (T ) ∂x ∂ξ u ν + C3 Mm (T ) + C4 Mm+1 (T ) (4.12) m
|α|+|β|≤N
√ holds for some constant C2 > 0. By letting C2 Mm (T ) ≤ C2 M0 ≤ λ0 /2, from the induction hypothesis and taking time integration, the above inequality gives T λ0 β ∂xα ∂ξ u m+1 (s)2ν ds Mm+1 (T ) + 2 0 |α|+|β|≤N
≤ u 0 2H N + C3 Mm (T )T + C4 Mm+1 (T )T.
(4.13)
x,ξ
Now, one can choose M0 = min{
1 λ0 1 1 1 , }, T∗ = min{ , }, 0 = M0 2C1 2C2 4C3 2C4 2
so that Mm+1 (T∗ ) ≤ 202 + 2C3 T∗ Mm (T∗ ) ≤
1 2 1 2 M + M ≤ M02 , 2 0 2 0
that is, sup u m+1 (t)2H N = x,ξ
0≤t≤T∗
Mm+1 (T∗ ) ≤ M0 .
Finally, proceeding as in the proof of (4.12), for any 0 ≤ s ≤ t ≤ T∗ , we obtain t m+1 d 2 m+1 2 m+1 2 u u (t) H N − u (s) H N = (θ ) H N dθ dθ x,ξ x,ξ x,ξ s
t β ≤ C(M0 + 1) ∂xα ∂ξ u m+1 (θ )2ν dθ + C M02 |t − s|. (4.14) |α|+|β|≤N
s
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This implies that u m+1 (t)2
is continuous over 0 ≤ t ≤ T∗ since from (4.13),
N Hx,ξ
β
∂xα ∂ξ u m+1 2ν is integrable over [0, T∗ ]. Hence Eq. (4.11) holds true for m + 1 and so it does for any m ≥ 0. Next, the difference between two subsequent solutions of (4.10) satisfies ⎧ m ∂t (u m+1 − u m ) + ξ · ∇x (u m+1 − u m ) + U ∗ bu · ∇ξ (u m+1 − u m ) ⎪ ⎪ ⎨ = L (u m+1 − u m ) + (u m , u m+1 − u m ) + (u m − u m−1 , u m ) FP m m−1 m ⎪ + − u m−1 ) − U ∗ bu −u · ∇ξ u m , ⎪ ⎩ m+1 A(u m − u )|t=0 = 0. (u As for (4.12), it follows that
d m+1 β u (t) − u m (t)2 N −1 + λ0 ∂xα ∂ξ (u m+1 − u m )2ν Hx,ξ dt |α|+|β|≤N −1
β m ≤ Cu L 2 (HxN −1 ) ∂xα ∂ξ (u m+1 − u m )2ν ξ
|α|+|β|≤N −1
+ Cu − u
m−1
+ Cu − u
m−1
m
m
L 2 (HxN −1 ) ξ
|α|+|β|≤N
L 2 (HxN −1 ) u
m+1
ξ
+ Cu m+1 − u m 2 2
L ξ (HxN −1 )
β
∂xα ∂ξ u m ν
|α|+|β|≤N −1
β
∂xα ∂ξ (u m+1 − u m )ν
− u L 2 (HxN −1 ) m
ξ
,
where N ≥ 2[n/2] + 2 and the Sobolev embedding H [n/2]+1 (Rn ) → L ∞ (Rn ) were used. Since u 0 H N and hence 0 , T∗ , M0 can be small enough, and from (4.13), x,ξ
T∗
sup m
0
β
|α|+|β|≤N
∂xα ∂ξ u m (s)2ν ds
can be also small enough, it further follows that there is a constant μ < 1 such that sup u m+1 (t) − u m (t) H N −1 ≤ μ sup u m (t) − u m−1 (t) H N −1 .
0≤t≤T∗
x,ξ
x,ξ
0≤t≤T∗
(4.15)
It can be seen from (4.15) that (u m )m≥0 is a Cauchy sequence in the Banach space C([0, T∗ ]; H N −1 (Rn × Rn )), and thus the limit function u ∈ C([0, T∗ ]; H N −1 (Rn × Rn )) exists. By letting m → ∞ in (4.9) or (4.10), u is a solution to the Cauchy problem (1.6)– (1.7). From the pointwise convergence of u m to u by the Sobolev embedding theorem and the lower semicontinuity of the norms, u m ∈ X (0, T∗ ; M0 ) implies √ f ≡ M + Mu ≥ 0, sup u(t) H N ≤ M0 . 0≤t≤T∗
x,ξ
Similarly to the proof of (4.14), one can conclude that u ∈ C([0, T∗ ]; H N (Rn × Rn )). Thus, u ∈ X (0, T∗ ; M0 ) follows.
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Finally, let v ∈ X (0, T∗ ; M0 ) be another solution to the Cauchy problem (1.6)–(1.7). Proceeding as in the proof of (4.15) we obtain sup u(t) − v(t) ≤ μ sup u(t) − v(t), 0≤t≤T∗
0≤t≤T∗
for μ < 1. Then, u ≡ v, and uniqueness follows. This completes the proof of Theorem 4.1. Proof of global existence and uniqueness in Theorem 1.1. At this time it suffices to obtain the uniform a priori estimates. For a given T > 0, let u be the solution to the Cauchy problem (1.6)–(1.7) over [0, T ] which satisfies sup u(t) H N ≤
0≤t≤T
x,ξ
for 0 < ≤ 1 small enough. Now, one can apply Lemmas 4.1, 4.2, 4.3 and 4.4 to u. We claim that there are the equivalent energy E(u(t)) and energy dissipation rate D(u(t)), defined by E(u(t)) ∼ u(t)2H N , x,ξ
β 2 D(u(t)) = ∂xα ∂ξ {I − P}u(t)2ν + T ∂xα bu (t)U |α|+|β|≤N
(4.16)
|α|≤N
+ ∇x (a , b u
u
)(t)2 N −1 , Hx
(4.17)
such that d E(u(t)) + λD(u(t)) ≤ 0, dt
(4.18)
holds for any 0 ≤ t ≤ T . In fact, since 0 < ≤ 1 is small enough, the linear combination of (4.2), (4.3) and (4.7) gives the dissipation of the macroscopic component Pu or equivalently of its coefficients (a u , bu ), the microscopic component {I − P}u, and their space derivatives with remaining terms including L 2 -norms of the mixed space-velocity derivatives with small-coefficients, that is, d dt
Mu(t)2L 2 (H N ) + E nf r ee (u(t)) x ξ
α 2 ∂x {I − P}u2ν + λT ∂xα bu U + λ∇x (a u , bu )2 +λ
HxN −1
|α|≤N
≤ CD(u(t)) + C
∂xα ∇ξ {I − P}u2 ,
(4.19)
1≤|α|≤N −1
where E nf r ee (u(t)) is defined by (4.6), and M ≥ 1 is large enough, so that, by (4.8), Mu(t)2L 2 (H N ) + E nf r ee (u(t)) ∼ u(t)2L 2 (H N ) ξ
x
ξ
x
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129
holds. On the other hand, the linear combination of (4.6) over 1 ≤ k ≤ N gives the dissipation of all the space-velocity derivatives in L 2 -norm, that is,
d β β Ck ∂xα ∂ξ {I − P}u2 + λ ∂xα ∂ξ {I − P}u2ν dt |β|≥1 |α|+|β|≤N
|β|=k |α|+|β|≤N
1≤k≤N
≤ CD(u(t)) + C
|α|≤N
∂xα {I − P}u2ν + C∇x (a u , bu )2
HxN −1
,
(4.20)
for some properly chosen constants Ck . Thus, the further linear combination of (4.19) and (4.20) leads to (4.18) by letting > 0 small enough. Then, after taking time integration, t D(u(s))ds ≤ E(u 0 ) ≤ Cu 0 2H N sup u(t) H N + λ x,ξ
0≤t≤T
x,ξ
0
holds, where C is independent of T and u 0 . Thus, the global existence and uniqueness of solutions to the Cauchy problem (1.6)–(1.7) follows from the above uniform a priori estimate (4.20) together with the local existence obtained in Theorem 4.1 as well as the continuum argument, and moreover, (1.13) and (1.14) hold. Here, the details are omitted for simplicity. This completes the proof of global existence and uniqueness in Theorem 1.1. 4.3. Proof of rates of convergence. In this subsection, in order to prove (1.15) in Theorem 1.1, we are concerned with the time-decay rates of solutions. The main idea of the proof is based on the energy-spectrum method recently developed in [12,13]. To this end, let us suppose that all conditions in Theorem 1.1 hold, and let u be the solution to the Cauchy problem (1.6)–(1.7) satisfying (1.13) and (1.14). Firstly, the time-decay properties of the linearized solution operator eBt given in Sect. 3 will be applied in the following lemma to obtain some formal time-decay estimates for the solution u in terms of the total temporal energy E(u(t)). Lemma 4.5. If u 0 Z 1 is bounded, then n
u(t)2 ≤ C(E(u 0 ) + u 0 2Z 1 )(1 + t)− 2 t n +C (1 + t − s)− 2 E(u(s))[E(u(s)) + ξ {I − P}u(s)2 ]ds 0
+C
t
n
(1 + t − s)− 4 E(u(s))ds
2 (4.21)
0
holds for any t ≥ 0. Proof. From (2.12), u can be written in mild form as t Bt u(t) = e u 0 + eB(t−s) G(s)ds, 0
where the source term G is denoted by G = (u, u) − U ∗ bu · ∇ξ u.
(4.22)
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R. Duan, M. Fornasier, G. Toscani
By the definition of , given in (2.13), G can be rewritten as G = G1 + G2 + G3, with G 1 = U ∗ a u L FP {I − P}u, 1 G 2 = U ∗ bu · ξ {I − P}u − U ∗ bu · ∇ξ {I − P}u, 2 1 G 3 = −U ∗ a u P1 u + U ∗ bu · ξ Pu − U ∗ bu · ∇ξ Pu. 2 It is straightforward to check that both G 1 and G 2 satisfy condition (3.3). Then, one can apply both (i) and (ii) in Theorem 3.1 to (4.22) to obtain n
u(t)2 ≤ C(E(u 0 ) + u 0 2Z 1 )(1 + t)− 2 2 t
n +C (1 + t − s)− 2 (ν −1/2 G i (s)2Z 1 + ν −1/2 G i (s)2 )ds 0
i=1
t
+C
− n4
(1 + t − s)
0
2 (G 3 (s) Z 1 + G 3 (s))ds
.
(4.23)
By using the inequalities U ∗ (a u , bu ) L 2x ≤ U L 1x (a u , bu ) L 2x , ≤ U L 1x (a u , bu ) L ∞ ≤ CU L 1x ∇x (a u , bu ) H [n/2] , U ∗ (a u , bu ) L ∞ x x x
it is straightforward to get ν −1/2 G 1 2Z 1 + ν −1/2 G 1 2 ≤ CE(u)[E(u) + ξ {I − P}u2 ], ν −1/2 G 2 2Z 1 + ν −1/2 G 2 2 ≤ C[E(u)]2 , G 3 (s) Z 1 + G 3 (s) ≤ E(u). Plugging the above estimates into (4.23) leads to (4.21). This completes the proof of Lemma 4.5. The next lemma is devoted to obtain a uniform bound on the velocity-weighted norm ξ {I − P}u(s), under the additional condition on initial data which imply that the time integral term on the r.h.s. of (4.21) can be controlled. This, together with (4.18) can lead to the desired time-decay rates of the total temporal energy E(u(t)). Lemma 4.6. If u 0 H N is small enough and ξ u 0 is bounded, then x,ξ
{I − P}uν ≤ C(u 0 H N + ξ u 0 ) x,ξ
holds for any t ≥ 0.
(4.24)
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131
Proof. For simplicity, set wi = ξi {I − P}u. Thanks to (4.5), wi satisfies ∂t wi + ξ · ∇x wi + U ∗ bu · ∇ξ wi 1 = L FP wi + (u, wi ) + U ∗ bu · ξi [ξ, P]u 2 + ξ P(ξ · ∇x {I − P}u + U ∗ bu · ∇ξ {I − P}u) − ξ {I − P}(ξ · ∇x Pu + U ∗ bu · ∇ξ Pu) + U ∗ biu {I − P}u − 2(1 + U ∗ a u )∂ξi {I − P}u. The last line follows from the computation of commutators [∇ξ , ξi ] = ei , [L FP , ξi ] = [ξ , ξi ] = 2∂ξi . The zero-order energy estimate as before gives 1 d wi (t)2 + λ0 {I − P0 }wi 2ν ≤ 2 dt
λ0 + CE(u) wi 2ν + C(E(u) + 1)D(u), 8
where E(u), D(u) are defined in (4.16) and (4.17), respectively. From (4.18) and sup E(u(t)) ≤ E(u 0 ) ≤ Cu 0 2H N , x,ξ
t≥0
which is small enough, d wi (t)2 + λ0 wi 2ν ≤ CD(u) + CP0 wi 2ν ≤ CD(u) follows. dt Then, further taking time integration and using (4.18), one has wi (t)2 + λ0 0
t
wi (s)2ν ds ≤ wi (0)2 + C
t
D(u(s))ds
0 2
≤ Cξi {I − P}u 0 + CE(u 0 ) ≤ C(u 0 2H N + ξ u 0 2 ), x,ξ
which gives (4.24). This completes the proof of Lemma 4.6.
Proof of time-decay rates in Theorem 1.1. For simplicity, let us denote K 0 = u 0 H N + u 0 Z 1 , δ0 = u 0 H N + ξ u 0 . x,ξ
x,ξ
(4.25)
Notice that K 0 is finite and δ0 can be arbitrarily small thanks to the assumptions of Theorem 1.1. In order to get the time decay of the total temporal energy in (1.15), we define n
E∞ (t) = sup (1 + s) 2 E(u(s)). 0≤s≤t
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Thus, to prove (1.15), it suffices to prove that E∞ (t) is uniformly bounded in time. In fact, combining (4.21) and (4.24) gives t n n u(t)2 ≤ C K 02 (1 + t)− 2 + Cδ02 (1 + t − s)− 2 E(u(s))ds 2 3 −2
+ Cδ0
0
t
0 n
≤ C K 02 (1 + t)− 2 2 3 −2
+ Cδ0
(1 + t − s)
− n4
2 3 +
2
[E(u(s)] ds t n n + Cδ02 E∞ (t) (1 + t − s)− 2 (1 + s)− 2 ds
[E∞ (t)]
2 3 +
0
t
n
n
n
(1 + t − s)− 4 (1 + s)− 3 − 2 ds
2 , (4.26)
0
where 0 < < 1/3 is a constant. Since n ≥ 3 holds, one has t n n n (1 + t − s)− 2 (1 + s)− 2 ds ≤ C(1 + t)− 2 , 0 t n n n n (1 + t − s)− 4 (1 + s)− 3 − 2 ds ≤ C(1 + t)− 4 , 0
where n/2 > 1 and n/3 + n/2 > max{1, n/4} were used. Then, it follows from (4.26) that 2 2 n −2 u(t)2 ≤ C K 02 + δ02 E∞ (t) + Cδ03 [E∞ (t)] 3 + (1 + t)− 2 . (4.27) Notice that (4.18) implies d E(u(t)) + λE(u(t)) ≤ Cu(t)2 . dt By the Gronwall inequality, it follows from (4.27) and (4.28) that t E(u(t)) ≤ e−λt E(u 0 ) + C e−λ(t−s) u(s)2 ds 0 2 2 n −2 2 2 [E∞ (t)] 3 + (1 + t)− 2 , ≤ C K 0 + δ0 E∞ (t) + Cδ03
(4.28)
for any t ≥ 0. In fact E∞ (t) is nondecreasing in time. Then, it holds 2
E∞ (t) ≤ C K 02 + Cδ02 E∞ (t) + Cδ03
−2
2
[E∞ (t)] 3 + ,
for any t ≥ 0. Since δ0 in (4.25) is small enough, one has 2
E∞ (t) ≤ C K 02 + Cδ03
−2
2
[E∞ (t)] 3 + ,
for any t ≥ 0. Again using the smallness of δ0 and 2/3 − 2 > 0, one further has sup E∞ (t) ≤ C K 02 . t≥0
Thus, the uniform boundness of E∞ (t) is obtained and hence (1.15) is proved. This completes the proof of time decay rates in Theorem 1.1 and thus the proof of the whole Theorem 1.1.
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Acknowledgements. Renjun Duan and Massimo Fornasier acknowledge the financial support provided by the FWF project Y 432-N15 START-Preis “Sparse Approximation and Optimization in High Dimensions”. This work was partially supported by CBDif. Renjun Duan would like to acknowledge the support and continuous encouragement of Peter Markowich during his stay at RICAM. The authors would like to thank the referee for useful comments.
A. Proofs of Uniform a Priori Estimates A.1. A priori estimates: Microscopic dissipation. In the first part of this appendix, we shall prove Lemmas 4.1, 4.2 and 4.3 which are related to the microscopic dissipation rate. As a preparation, we first obtain a lemma about some estimate on the nonlinear term. Lemma A.1. |(u, v), w | & ' ≤ C|U ∗ (a u , bu )| |{I − P}v|ν + |(a v , bv )| (|{I − P}w|ν + |(a w , bw )|),
(A.1)
holds for some constant C depending only on n. Proof. Recall the definition (2.13) of (u, v). One has (u, v), w = U ∗ a u (L FP {I − P}v, {I − P}w − P1 v, P1 w ) 1 + U ∗ bu · ξ, ({I − P}v + Pv)({I − P}w + Pw) . 2
(A.2)
Then, (A.1) follows by applying integration by parts and Hölder inequality to (A.1). Proof of Lemma 4.1. From (2.12), the zero-order energy estimate over Rnx × Rnξ gives 1 d 2 u(t) − Lu, u d x = (u, u), u d x. 2 dt Rn R3
(A.3)
To estimate the nonlinear term on the r.h.s., let us define I (u; v, w) = (u, v), w , where (·, ·) means the bilinear operator given in (2.13). Notice that I (u; v, w) can be written as the summation of two terms: I (u; v, w) = I1 (u; v, w) + I2 (u; v, w), where as in (A.2), from the macro-micro decomposition (2.10), I1 , I2 are defined by I1 (u; v, w) = U ∗ a u L FP {I − P}v, {I − P}w 1 + U ∗ bu · ξ, {I − P}v{I − P}w 2 + {I − P}vPw + Pv{I − P}w , 1 I2 (u; v, w) = −U ∗ a u P1 v, P1 w + U ∗ bu · ξ, PvPw . 2
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One can further simplify the form of I2 (u; v, w) in terms of the coefficients of Pv and Pw. In fact, P1 v, P1 w = bv · ξ bw · ξ Mdξ = bv · bw holds, Rn
and
ξ, PvPw =
Rn
ξ(a v + bv ·
√
M)(a w + bw ·
√
M)dξ = a v bw + a w bv .
Then, it follows that 1 I2 (u; v, w) = −U ∗ a u bv · bw + U ∗ bu · (a v bw + a w bv ). 2 Next, we estimate the nonlinear term (u, u), u = I1 (u; u, u) + I2 (u; u, u). For I1 (u; u, u), one has I1 (u; u, u)d x = Rn
Rn
+
(A.4)
U ∗ a u L FP {I − P}u, {I − P}u d x
Rn
U ∗ bu · ξ, Pu{I − P}u d x
1 + U ∗ bu · ξ, |{I − P}u|2 d x. 2 Rn The terms on the r.h.s are estimated as U ∗ a u L FP {I − P}u, {I − P}u d x Rn u ∞ ≤ U ∗ a L x |L FP {I − P}u, {I − P}u |d x Rn
{I − P}u2ν , ≤ Ca L ∞ x u
U ∗ bu · ξ, Pu{I − P}u d x ≤ |U ∗ bu | · ξ Pu L 2 · {I − P}u L 2 d x ξ ξ Rn ≤C |U ∗ bu | · (|a u | + |bu |) · {I − P}u L 2 d x
Rn
Rn u
ξ
{I − P}uν , ≤ C(a , b ) L 2x b L ∞ x and
u
u
1 1 U ∗ bu · ξ, |{I − P}u|2 d x ≤ |U ∗ bu | · |{I − P}u|2ξ d x 2 Rn 2 Rn 1 ≤ bu L ∞ {I − P}u2ν . x 2
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135
Then, it follows that I1 (u; u, u)d x ≤ Ca u L ∞ {I − P}u2ν Rn
+ C(a u , bu ) L 2x bu L ∞ {I − P}uν x 1 + bu L ∞ {I − P}u2ν . x 2
(A.5)
For I2 (u; u, u) one has I2 (u; u, u)d x = −U ∗ a u |bu |2 + U ∗ bu · a u bu d x Rn Rn = U (|x − y|)a u (x)bu (y)(bu (x) − bu (y))d xd y R n ×R n U (|x − y|)a u (x)(bu (x) − bu (y))2 d xd y =− R n ×R n U (|x − y|)a u (x)bu (x)(bu (x) − bu (y))d xd y. + R n ×R n
Since −
R n ×R n
2 U (|x − y|)a u (x)(bu (x) − bu (y))2 d xd y ≤ a u L ∞ T bu U , x
and R n ×R n
U (|x − y|)a u (x)bu (x)(bu (x) − bu (y))d xd y
≤ ≤
1 U (|x − y|) |a (x)b (x)| d xd y 2
u
u
2
R n ×R n u u a b L 2x T bu U ,
2
T bu U
it follows that Rn
2 I2 (u; u, u)d x ≤ a u L ∞ T bu U + a u bu L 2x T bu U . x
(A.6)
Plugging estimates (A.5) and (A.6) into (A.3) and using the coercivity inequality (2.9) of −L, we obtain (4.2). This completes the proof of Lemma 4.1. Proof of Lemma 4.2. Let 1 ≤ |α| ≤ N . Notice that although A is nonlocal in x, √ √ ∂xα Au = ∂xα (U ∗ ρξ √Mu · ξ M) = U ∗ ρξ √M∂ α u · ξ M = A∂xα u, x
and hence ∂xα Lu = ∂xα L FP u + ∂xα Au = L FP ∂xα u + A∂xα u = L∂xα u
(A.7)
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holds. Then, from (2.12), the energy estimate on ∂xα u over Rnx × Rnξ gives 1 d α L∂xα u, ∂xα u = ∂xα (u, u), ∂xα u d x ∂x u(t)2 − n n 2 dt R R
α α−α u α
+ Cα
−U ∗ ∂x b · ∇ξ ∂x u, ∂xα u d x. Rn
α <α
(A.8)
Next we estimate each term on the r.h.s. of (A.8). Similar to (A.7), from Lemma A.1 we obtain
α α α ∂x (u, u), ∂x u d x = Cα
(∂xα u, ∂xα−α u), ∂xα u d x ≤ C Iα,α , Rn
Rn
α ≤α
where
I
α,α
=
Rn
|U ∗ ∂xα (a u , bu )| · (|{I − P}∂xα−α u|ν + |∂xα−α (a u , bu )|)
· (|{I − P}∂xα u|ν + |∂xα (a u , bu )|)d x. When |α | ≥ [n/2] + 1 or α = α, one has
Iα,α ≤ U ∗ ∂xα (a u , bu ) L 2x ({I − P}∂xα uν + ∂xα (a u , bu ) L 2x )
×(sup |{I − P}∂xα−α u|ν + sup |∂xα−α (a u , bu )|). x
x
≤ C∇x g H [n/2] for n ≥ 3 and g = g(x), it further By the Sobolev inequality g L ∞ x x holds that
sup |{I − P}∂xα−α u|2ν x
= sup |∇ξ {I − P}∂xα−α u|2 + ν(ξ )|{I − P}∂xα−α u|2 dξ x Rn
≤ ∇ξ {I − P}∂xα−α u2L ∞ + ν(ξ ){I − P}∂xα−α u2L ∞ dξ x x Rn
≤C ∇ξ ∇x {I − P}∂xα−α u2 [n/2] + ν(ξ ){I − P}∇x ∂xα−α u2 [n/2] dξ H Hx n x R
α
2 {I − P}∂x uν , ≤C 1≤|α |≤N
and similarly
sup |∂xα−α (a u , bu )| ≤ C∇x ∂xα−α (a u , bu ) H [n/2] ≤ C∇x (a u , bu ) HxN −1 . x
x
Hence it follows that
⎛
Iα,α ≤ C∇x (a u , bu ) HxN −1 ⎝
1≤|α |≤N
⎞ α
∂x {I − P}u2ν +∇x (a u , bu )2
HxN −1
⎠.
(A.9)
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When |α | ≤ [n/2] and α < α, one obtains similarly that
Iα,α ≤ sup |U ∗ ∂xα (a u , bu )| · ({I − P}∂xα−α uν + ∂xα−α (a u , bu ) L 2x ) x
· ({I − P}∂xα uν + ∂xα (a u , bu ) L 2x ). A further bound also holds for the r.h.s. of (A.9) since
sup |U ∗ ∂xα (a u , bu )| ≤ sup |∂xα (a u , bu )| ≤ C∇x (a u , bu ) HxN −1 . x
x
For the second term on the r.h.s. of (A.8), by using the same proof as for Iα,α , one has
Cαα
−U ∗ ∂xα−α bu · ∇ξ ∂xα u, ∂xα u d x Rn
α <α
≤C
n α <α R
|U ∗ ∂xα−α bu | · ∇ξ ∂xα u L 2 ∂xα u L 2 d x ξ
⎛
≤ C∇x b HxN −1 ⎝ ≤ C∇x bu HxN −1
⎞
u
ξ
∇ξ ∂xα u2
+ ∂xα u2 ⎠
1≤|α |≤N −1
∂xα ∇ξ {I − P}u2
1≤|α |≤N −1
⎛
+ C∇x bu HxN −1 ⎝
1≤|α |≤N
⎞ α
∂x {I − P}u2ν + ∇x (a u , bu )2
HxN −1
⎠.
Putting all the above estimates into (A.8), taking summation over 1 ≤ |α| ≤ N and using the coercivity inequality (2.9) of −L yields (4.3). This completes the proof of Lemma 4.2. Proof of Lemma 4.3. Fix k with 1 ≤ k ≤ N , and choose α, β with |β| = k and |α|+|β| ≤ β N . From (4.5), the energy estimate on ∂xα ∂ξ u over Rnx × Rnξ gives 7
1 d α β β β ∂x ∂ξ {I−P}u2 − L FP ∂xα ∂ξ {I−P}u, ∂xα ∂ξ {I−P}u d x = Ii , (A.10) 2 dt Rn i=1 where Ii (1 ≤ i ≤ 7) take the form of β β −∂xα [∂ξ , ξ · ∇x ]{I − P}u, ∂xα ∂ξ {I − P}u d x, I1 = n R β β I2 = ∂xα [∂ξ , −|ξ |2 ]{I − P}u, ∂xα ∂ξ {I − P}u d x, Rn
and I3 =
α <α
I4 =
Rn
I5 =
Cαα
Rn β
β
β
−U ∗ ∂xα−α bu · ∇ξ ∂xα ∂ξ {I − P}u, ∂xα ∂ξ {I − P}u d x, β
∂xα ∂ξ (u, {I − P}u), ∂xα ∂ξ {I − P}u d x,
1 β β ∂xα ∂ξ (U ∗ bu · [ξ, P]u), ∂xα ∂ξ {I − P}u d x, Rn 2
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R. Duan, M. Fornasier, G. Toscani
and I6 = I7 =
β
Rn
β
∂xα ∂ξ P(ξ · ∇x {I − P}u + U ∗ bu · ∇ξ {I − P}u), ∂xα ∂ξ {I − P}u d x, β
Rn
β
−∂xα ∂ξ {I − P}(ξ · ∇x Pu + U ∗ bu · ∇ξ Pu), ∂xα ∂ξ {I − P}u d x.
Here, the commutator in I2 follows from β β β 1 β [∂ξ , L FP ] = [∂ξ , ξ ] + [∂ξ , (2n − |ξ |2 )] = [∂ξ , −|ξ |2 ]. 4
(A.11)
Next, we estimate each term Ii (1 ≤ i ≤ 7). For I1 and I2 , one has β
β
I1 ≤ δ∂xα ∂ξ {I − P}u2 + Cδ [∂ξ , ξ · ∇x ]∂xα {I − P}u2
β ≤ δ∂xα ∂ξ {I − P}u2 + Cδ ∂xα ∇x {I − P}u2 |α|≤N −k
+χ{2≤k≤N } Cδ
β
∂xα ∂ξ {I − P}u2 ,
1≤|β|≤k−1 |α|+|β|≤N
and β
β
I2 ≤ δ∂xα ∂ξ {I − P}u2 + Cδ [∂ξ , −|ξ |2 ]∂xα {I − P}u2
β ≤ δ∂xα ∂ξ {I − P}u2 + Cδ ∂xα {I − P}u2ν |α|≤N −k
+χ{2≤k≤N } Cδ
β
∂xα ∂ξ {I − P}u2ν ,
1≤|β|≤k−1 |α|+|β|≤N
where δ > 0 is arbitrary, to be chosen later. For I3 and I5 , similar to the proof of (A.9), I3 ≤ C
n α <α R
β
β
|U ∗ ∂xα−α bu | · ∇ξ ∂xα ∂ξ {I − P}u L 2 ∂xα ∂ξ {I − P}u L 2 d x ξ
≤ C∇x bu HxN −1
|α |+|β |≤N
ξ
β
∂xα ∂ξ {I − P}u2 holds,
and I5 ≤ C ≤C
α ≤α
Rn
α ≤α
Rn
β
β
|U ∗ ∂xα−α bu | · ∂ξ ([ξ, P]∂xα u) L 2 ∂xα ∂ξ {I − P}u L 2 d x ξ
ξ
β
|U ∗ ∂xα−α bu | · |∂xα (a u , bu )| · ∂xα ∂ξ {I − P}u L 2 d x ξ
≤ C(a u , bu ) HxN (∇x (a u , bu )2
HxN −1
β
+ ∂xα ∂ξ {I − P}u2 ).
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For I6 and I7 , it follows in the same way that
β
I6 ≤ δ∂xα ∂ξ {I − P}u2 + Cδ + Cbu HxN
|α |≤N −k
|α |+|β |≤N
∇x ∂xα {I − P}u2
β
∂xα ∂ξ {I − P}u2 ,
and β
I7 ≤ δ∂xα ∂ξ {I − P}u2 + Cδ ∇x (a u , bu )2
HxN −k
+ C(a u , bu ) HxN (∇x (a u , bu )2
HxN −1
β
+ ∂xα ∂ξ {I − P}u2 ).
Now, it remains to estimate I4 . To this extent, one can use the identity 1 β β β β ∂ξ (u, v) = (u, ∂ξ v) + U ∗ a u [∂ξ , L FP ]v + U ∗ bu · [∂ξ , ξ ]v 2 1 β β β = (u, ∂ξ v) + U ∗ a u [∂ξ , −|ξ |2 ]v + U ∗ bu · [∂ξ , ξ ]v, 2 where (A.11) was used again. Then, I4 can be rewritten as I4 = I4,1 + I4,2 , with
I4,1 =
β
n R
I4,2 =
Rn
+
β
∂xα (u, ∂ξ {I − P}u), ∂xα ∂ξ {I − P}u d x, β
β
∂xα (U ∗ a u [∂ξ , −|ξ |2 ]{I − P}u), ∂xα ∂ξ {I − P}u d x
1 β β ∂xα ( U ∗ bu · [∂ξ , ξ ]{I − P}u), ∂xα ∂ξ {I − P}u d x. 2 Rn
Similarly to the bound on I3 , I4,2 can be controlled by
β
I4,2 ≤ C(a u , bu ) HxN ∂xα ∂ξ {I − P}u2ν . |α |+|β |≤N
For I4,1 , it follows from Lemma A.1 that
β β I4,1 = Cαα
(∂xα−α u, ∂xα ∂ξ {I − P}u), ∂xα ∂ξ {I − P}u d x α ≤α
≤C
α ≤α
Rn
Rn
|U ∗ (a(∂xα−α u), b(∂xα−α u))|
β
β
β
×[|{I − P}∂xα ∂ξ {I − P}u|ν + |(a(∂xα ∂ξ {I − P}u), b(∂xα ∂ξ {I − P}u))|] β
β
β
×[|{I − P}∂xα ∂ξ {I − P}u|ν + |(a(∂xα ∂ξ {I − P}u), b(∂xα ∂ξ {I − P}u))|]d x, where a(w) = a w , b(w) = bw for any w. For any α , β , one can use the properties
β
a(∂xα u) = ∂xα a u , |a(∂xα ∂ξ {I − P}u)| ≤ C∂xα {I − P}u L 2 , ξ
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and similarly for b. Then, it further holds that
|U ∗ ∂xα−α (a u , bu )| I4,2 ≤ C n α ≤α R
β
×(|{I − P}∂xα ∂ξ {I − P}u|ν + ∂xα {I − P}u L 2 ) ξ
×(|{I
β − P}∂xα ∂ξ {I
≤ C(a u , bu ) HxN
− P}u|ν + ∂xα {I
|α |+|β |≤N
− P}u L 2 )d x ξ
β
∂x ∂ξ {I α
− P}u2ν .
Finally, using the coercivity inequality (2.4) of −L FP , the second term on the l.h.s. of (A.10) satisfies the low bound β β − L FP ∂xα ∂ξ {I − P}u, ∂xα ∂ξ {I − P}u d x Rn
β
≥ λ0 {I − P0 }∂xα ∂ξ {I − P}u2ν
λ0 α β β ∂ ∂ {I − P}u2ν − λ0 P0 ∂xα ∂ξ {I − P}u2ν 2 x ξ λ0 β ≥ ∂xα ∂ξ {I − P}u2ν − C∂xα {I − P}u2 . 2 ≥
Finally, plugging all the above estimates into (A.10), taking summation over {|β| = k, |α| + |β| ≤ N } and then choosing a properly small δ > 0 yields (4.6). This completes the proof of Lemma 4.3. A.2. A priori estimates: Macroscopic dissipation. In this second part of the Appendix, we shall prove Lemma 4.4 for the macroscopic dissipation which plays a key role in the proof of global existence. We first prove estimates on the space derivatives of Ai j (l) and Ai j (r ) in the following Lemma A.2. Let l, r be given by (2.17) and (2.18) respectively.
∂xα Ai j (l) L 2x ≤ C ∂xα {I − P}u2 , |α|≤N −1
|α|≤N −1
|α|≤N
∂xα Ai j (r ) L 2x ≤ C∇x (a u , bu ) HxN −1
∂xα {I − P}u
(A.12) (A.13)
|α|≤N
holds for some constant C depending only on n, where the moment function Ai j (·), 1 ≤ i, j ≤ n, are defined by (2.20). Proof. For any α, integration by parts yields straightforwardly |∂xα Ai j (l)| = |Ai j (∂xα l)| √ = |(ξi ξ j − 1) M, −ξ · ∇x ∂xα {I − P}u + L FP ∂xα {I − P}u | √ √ ≤ |−ξ(ξi ξ j − 1) M, ∇x ∂xα {I − P}u | + |L FP ([ξi ξ j − 1] M), ∂xα {I − P}u |
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√ ≤ ξ(ξi ξ j − 1) M L 2 ∇x ∂xα {I − P}u L 2 ξ ξ √ +L FP ([ξi ξ j − 1] M) L 2 ∂xα {I − P}u L 2 ξ
ξ
≤ C(∇x ∂xα {I − P}u L 2 + ∂xα {I − P}u L 2 ). ξ
ξ
The exponential decay in ξ for M was used above. Thus, (A.12) follows by further taking the L 2x -norm and then summation over |α| ≤ N − 1. For Ai j (n), one has ∂xα Ai j (r ) = Ai j (∂xα r ) √ = (ξi ξ j − 1) M, ∂xα [U ∗ a u L FP {I − P}u] 1 + U ∗ bu · ξ {I − P}u − U ∗ bu · ∇ξ {I − P}u] 2
√
Cαα (ξi ξ j − 1) M, U ∗ ∂xα−α a u L FP ∂xα {I − P}u = α ≤α
1
+ U ∗ ∂xα−α bu · ξ ∂xα {I − P}u − U ∗ ∂xα−α bu · ∇ξ ∂xα {I − P}u , 2 which from integration by parts gives
√
∂xα Ai j (r ) = Cαα L FP ([ξi ξ j − 1] M), U ∗ ∂xα−α a u ∂xα {I − P}u α ≤α
√ 1
+ ξ(ξi ξ j − 1) M, U ∗ ∂xα−α bu ∂xα {I − P}u 2 √
+ ∇ξ ((ξi ξ j − 1) M) , U ∗ ∂xα−α bu ∂xα {I − P}u .
Thus, it follows from the Hölder inequality that
|∂xα Ai j (r )| ≤ C |U ∗ ∂xα −α (a u , bu )| · ∂xα {I − P}u L 2 ξ
α ≤α
for any α. Hence, similarly as before, (A.13) follows by taking further L 2x -norm, applying the Young and Sobolev inequalities and then taking summation over |α| ≤ N − 1. This completes the proof of Lemma A.2. Proof of Lemma 4.4. Recall Eqs. (2.21)–(2.24) satisfied by (a u , bu ) and the parabolictype equation (2.25) derived from (2.23)–(2.24). We begin with the estimate on bu from (2.22) and (2.25). Let |α| ≤ N − 1. It follows from (2.25) that
d α α u ∂ [ ∂ j Aii ({I − P}u) − ∂i Ai j ({I − P}u)]∂x b j d x + |∇x ∂xα buj |2 d x n dt Rn x R i= j i
= ∂xα [ ∂ j Aii ({I − P}u) − ∂i Ai j ({I − P}u)]∂xα ∂t buj d x Rn
+
+
Rn
Rn
i= j
∂xα [
i
∂ j (U ∗ biu biu ) −
i
∂xα [
∂i (U ∗ biu buj + U ∗ buj biu )]∂xα buj d x
i
∂ j Aii (l + n) −
i= j
=: I1 + I2 + I3 ,
∂i Ai j (l + n)]∂xα buj d x
i
(A.14)
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R. Duan, M. Fornasier, G. Toscani
where Ii , i = 1, 2, 3, denote the corresponding terms on the r.h.s., respectively. For I3 , I3 ≤ λ∇x ∂xα buj 2L 2 + x
C α (∂x Ai j (l)2L 2 + ∂xα Ai j (n)2L 2 ) holds, x x λ ij
where 0 < λ ≤ 1 is a constant to be chosen later. For I2 , similarly one has I2 ≤ λ∇x ∂xα buj 2L 2 + x
C α ∂x (U ∗ biu buj )2L 2 x λ ij
≤ λ∇x ∂xα buj 2L 2
C
+ U ∗ ∂xα−α biu ∂xα buj 2L 2 x λ
≤ λ∇x ∂xα buj 2L 2
C + bu 2H N ∇x bu 2 N −1 , Hx x λ
x
x
i j α ≤α
where the Young and Sobolev inequalities were used. For I1 , one can use (2.22) to rewrite it as
∂xα [ ∂ j Aii ({I − P}u) − ∂i Ai j ({I − P}u)] I1 = Rn
i= j α · ∂x [−∂i a u
+
+ (U
∗ biu
i u − bi ) − (U
∗ a u biu − U ∗ biu a u )
∂ j Ai j ({I − P}u)]d x.
j
Then, it holds I1 ≤ δ∇x ∂xα a uj 2L 2 + x
C Ai j (∇x ∂xα {I − P}u)2L 2 x δ ij
C + λ∇x ∂xα buj 2L 2 + Ai j (∂xα {I − P}u)2L 2 x x λ ij
+ ∂xα (U ∗ a u biu − U ∗ biu a u )2L 2 + C Ai j (∇x ∂xα {I − P}u)2L 2 , x
x
ij
where 0 < δ ≤ 1 is a constant to be also determined later, and again from the Young and Sobolev inequalities, one has ∂xα (U ∗ a u biu − U ∗ biu a u ) L 2x ≤ C(a u , bu ) HxN ∇x (a u , bu ) HxN −1 . Thus, it follows that I1 ≤ λ∇x ∂xα buj 2L 2 + δ∇x ∂xα a uj 2L 2 x
x
+ C(a u , bu )2H N ∇x (a u , bu )2 x
HxN −1
+ Cλ,δ
∂xα {I − P}u2 .
|α|≤N
By plugging the above estimates on I1 , I2 and I3 into (A.14), taking summation over |α| ≤ N − 1, 1 ≤ j ≤ n and then choosing 0 < λ ≤ 1 properly small, one has
Kinetic Flocking Model with Diffusion
143
d n,b 1 E f r ee (u(t)) + ∇x bu 2 N −1 Hx dt 2
u 2 ≤ δ∇x a N −1 + Cδ ∂xα {I − P}u2 Hx
|α|≤N
+ C(a u , bu )2H N ∇x (a u , bu )2 N −1 Hx x
α 2 α +C (∂x Ai j (l) L 2 + ∂x Ai j (n)2L 2 ), x
ij
(A.15)
x
u where the free energy E n,b f r ee (u(t)) corresponding to the dissipation of b is given by
(u(t)) = Aii (∂xα ∂ j {I − P}u)∂xα buj d x E n,b f r ee |α|≤N −1
j
−
i= j
|α|≤N −1 i j
Rn
Rn
Ai j (∂xα ∂i {I − P}u)∂xα buj d x.
Next, we shall obtain the dissipation of a u from (2.21) and (2.22). Let |α| ≤ N − 1. It follows from (2.22) that d ∂ α ∇x a u · ∂xα bu d x + ∇x ∂xα a u 2L 2 x dt Rn x = ∂xα ∇x ∂t a u · ∂xα bu d x + ∂xα ∇x a u · ∂xα (U ∗ bu − bu )d x Rn Rn + ∂xα a u · ∂xα (U ∗ bu a u − U ∗ a u bu )d x Rn
− ∂xα ∂i a u ∂xα ∂ j Ai j ({I − P}u)d x ij
Rn
= I4 + I5 + I6 + I7 ,
(A.16)
where Ii , 4 ≤ i ≤ 7, denote the corresponding terms on the r.h.s., respectively. From the conservation law of mass (2.21), I4 is rewritten as ∂xα ∂t a u ∂xα ∇x · bu d x = ∂xα ∇x · bu 2L 2 . I4 = − Rn
x
For I5 , one can use the general inequality a(U ∗ b − b)d x = a(x)U (|x − y|)(b(y) − b(x))d xd y Rn
R n ×R n
≤
R n ×R n
1/2
|a(x)|2 U (|x − y|)d xd y
≤ a L 2x T bU for a = a(x) and b = b(x) so that I5 ≤
1 2 ∇x ∂xα a u 2L 2 + CT ∂xα bu U holds. x 6
T bU
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For I6 and I7 , similarly as before, one has 1 ∇x ∂xα a u 2L 2 + C∂xα (U ∗ bu a u − U ∗ a u bu )2L 2 x x 6 1 ≤ ∇x ∂xα a u 2L 2 + C(a u , bu )2H N ∇x (a u , bu )2 N −1 , Hx x x 6
I6 ≤
and I7 ≤
1 Ai j (∂xα ∂ j {I − P}u)2L 2 ∇x ∂xα a u 2L 2 + C x x 6 ij
1 ≤ ∇x ∂xα a u 2L 2 + ∇x ∂xα {I − P}u2 . x 6 Putting the above estimates on Ii (4 ≤ i ≤ 7) into (A.16) and taking summation over |α| ≤ N − 1 gives d n,a 1 E f r ee (u(t)) + ∇x a u 2 N −1 Hx dt 2
u 2 2 ≤ ∇x b N −1 + C (T ∂xα bu U + ∇x ∂xα {I − P}u2 ) Hx
+ C(a , b u
|α|≤N −1 u
)2H N ∇x (a u , bu )2 N −1 , Hx x
(A.17)
u where the free energy E n,a f r ee (u(t)) corresponding to the dissipation of a is given by
E n,a (u(t)) = ∂xα ∇x a u · ∂xα bu d x. f r ee n |α|≤N −1 R
Therefore, (4.7) with E nf r ee (u(t)) defined by (4.6) follows from taking the proper linear combination of (A.15) and (A.17) and then choosing a properly small 0 < δ ≤ 1. Finally, (4.8) holds true since one has
|E nf r ee (u(t))| ≤ C (∂xα {I − P}2 + ∂xα (a u , bu )2L 2 ) x
|α|≤N
≤C
(∂xα {I − P}2 + ∂xα P2 )
|α|≤N
≤ Cu(t)2L 2 (H N ) . ξ
x
This completes the proof of Lemma 4.4. References 1. Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. Part. Diff. Eqs. 26, 43–100 (2001) 2. Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker-Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010) 3. Chuang, Y.-L., D’Orsogna, M.R., Marthaler, D., Bertozzi, A.L., Chayes, L.: State transitions and the continuum limit for a 2D interacting, self-propelled particle system. Physica D 232, 33–47 (2007)
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4. Cucker, F., Smale, S.: Emergent behavior in flocks. IEEE Transactions of Automatic Control 52, 852–862 (2007) 5. Degond, P., Motsch, S.: Continuum limit of self-driven particles with orientation interaction. Math. Models Meth. Appl. Sci. 18(Suppl), 1193–1215 (2008) 6. Degond, P., Motsch, S.: Large scale dynamics of the persistent turning walker model of fish behavior. J. Stat. Phys. 131, 989–1021 (2008) 7. Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54(1), 1–42 (2001) 8. Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for kinetic equations with linear relaxation terms. C.R. Acad. Sci. Paris 347, 511–516 (2009) 9. Duan, R.-J.: On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in L 2x (HxN ). J. Diff. Eqs. 244, 3204–3234 (2007) 10. Duan, R.-J.: Stability of the Boltzmann equation with potential forces on torus. Physica D: Nonlinear Phenomena 238, 1808–1820 (2009) 11. Duan, R.-J., Strain, R.M.: Optimal time decay of the Vlasov-Poisson-Boltzmann system in R3 . Arch. Rat. Mech. Anal. (2010). doi:10.1007/s00205-010-0318-6 12. Duan, R.-J., Ukai, S., Yang, T.: A combination of energy method and spectral analysis for studies on systems for gas motions. Front. Math. China 4(2), 253–282 (2009) 13. Duan, R.-J., Ukai, S., Yang, T., Zhao, H.-J.: Optimal decay estimates on the linearized Boltzmann equation with time-dependent forces and their applications. Commun. Math. Phys. 277(1), 189–236 (2008) 14. Guo, Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53, 1081–1094 (2004) 15. Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231, 391–434 (2002) 16. Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9), 1104–1135 (2002) 17. Ha, S.-Y., Liu, J.-G.: A simple proof of the Cucker-Smale flocking dynamics and mean-field limit. Commun. Math. Sci. 7, 297–325 (2009) 18. Ha, S.-Y., Tadmor, E.: From particle to kinetic and hydrodynamic descriptions of flocking. Kinetic Rel. Models 1(3), 415–435 (2008) 19. Kawashima, S.: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. Thesis, Kyoto University, 1983 20. Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19, 969–998 (2006) 21. Topaz, C., Bertozzi, A.L.: Swarming patterns in a two-dimensional kinematic model for biological groups. SIAM J. Appl. Math. 65, 152–174 (2004) 22. Topaz, C.M., Bertozzi, A.L., Lewis, M.A.: A nonlocal continuum model for biological aggregation. Bull. Math. Bio. 68, 1601–1623 (2006) 23. Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Japa. Acad. 50, 179–184 (1974) 24. Ukai, S., Yang, T.: The Boltzmann equation in the space L 2 ∩ L ∞ β : Global and time-periodic solutions. Anal. Appl. 4, 263–310 (2006) 25. Vicsek, T., Czirók, A., Ben-Jacob, E., Shochet, O.: Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226–1229 (1995) 26. Villani, C.: A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, Vol. I, Amsterdam: North-Holland, 2002, pp. 71–305 27. Villani, C.: Hypocoercivity. Memoirs Amer. Math. Soc. 202, Providence, RI: Amer. Math. Soc. 2009, iv+141 pp 28. Yates, C., Erban, R., Escudero, C., Couzin, L., Buhl, J., Kevrekidis, L., Maini, P., Sumpter, D.: Inherent noise can facilitate coherence in collective swarm motion. Proc. Nat. Acad. Sci. 106(14), 5464– 5469 (2009) Communicated by P. Constantin
Commun. Math. Phys. 300, 147–184 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1111-y
Communications in
Mathematical Physics
Superconductivity Near the Normal State Under the Action of Electric Currents and Induced Magnetic Fields in R2 Yaniv Almog1 , Bernard Helffer2 , Xing-Bin Pan3 1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA.
E-mail:
[email protected]
2 Laboratoire de Mathématiques, Univ. Paris-Sud et CNRS, Bât 425, 91 405 Orsay Cedex,
France. E-mail:
[email protected]
3 Department of Mathematics, East China Normal University, Shanghai 200062, P.R. China.
E-mail:
[email protected] Received: 10 September 2009 / Accepted: 16 March 2010 Published online: 27 August 2010 – © Springer-Verlag 2010
Abstract: We consider the linearization of the time-dependent Ginzburg-Landau system near the normal state. We assume that an electric current is applied through the sample, which captures the whole plane, inducing thereby, a magnetic field. We show that independently of the current, the normal state is always stable. Using Fourier analysis the detailed behaviour of solutions is obtained as well. Relying on semi-group theory we then obtain the spectral properties of the steady-state elliptic operator. Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Properties of A0,c . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Preliminaries and maximal accretiveness . . . . . . . . . . . . . 2.2 Proof of maximal accretiveness . . . . . . . . . . . . . . . . . . 2.3 Compactness of the resolvent . . . . . . . . . . . . . . . . . . . 2.4 L ∞ (Rn ) spectral pairs . . . . . . . . . . . . . . . . . . . . . . . 3. The Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 3.1 Large |β| asymptotics of the first eigenvalues . . . . . . . . . . 3.2 Auxiliary functions of E 0 . . . . . . . . . . . . . . . . . . . . . 4. Analysis of Time-Dependent Problems . . . . . . . . . . . . . . . . 4.1 Long time decay . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Upper bounds of the resolvent with large |λ| . . . . . . . . . . . 5. Long Time Asymptotics for the Fourier Transform and Applications . 5.1 Technical preliminaries . . . . . . . . . . . . . . . . . . . . . . 5.2 An evolution problem . . . . . . . . . . . . . . . . . . . . . . . 5.3 Lower bounds for the resolvent . . . . . . . . . . . . . . . . . . 5.4 Estimates for dense subspaces . . . . . . . . . . . . . . . . . . Appendix A. Proof of (5.12) . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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148 151 151 152 154 156 158 158 160 161 161 164 165 165 167 176 180 181 183
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1. Introduction Consider a superconductor placed at a temperature lower than the critical one. If an electric current is applied through the sample it will induce a magnetic field. It is wellunderstood that if the electric current is sufficiently strong, then the sample must be at a normal state. If the current is then lowered, the normal state would lose stability and the sample would become superconducting again. Such a pattern of behaviour has been observed in numerous experiments [28]. It has also been obtained theoretically by analyzing the stability of the normal state for the time dependent Ginzburg-Landau system, but with the induced magnetic field neglected [3,20]. Our interest here is the joint effect of the current and the magnetic field it induces on the stability of the normal state. From a mathematical point of view we thus need to consider the linearization of the Ginzburg-Landau system near the normal state, this time with the effect of induced magnetic field included. In contrast with the simplified model, for which the induced magnetic field is neglected, one cannot present the full model in a one-dimensional setting. In the present contribution we thus consider a two-dimensional superconducting sample capturing the entire x y-plane. For a complete analysis of the effect of induced magnetic fields, we need, of course, to include boundaries and three-dimensional effects. However, the present contribution appears to be a necessary first step before one moves on to include these effects. Assuming that a magnetic field of magnitude He is perpendicularly applied to the sample the time-dependent Ginzburg-Landau system can be written as follows (see for instance [4,6,9,10,20,27,29]):
2 ψ + κ 2 (1 − |ψ|2 )ψ ∂t ψ + iκψ = ∇κA
in (0, T ) × R2 ,
¯ κA ψ) + κ 2 curlHe in (0, T ) × R2 , κ 2 curl2 A + σ (∂t A + ∇) = κ Im (ψ∇
(1.1)
where ψ is the order parameter, A is the magnetic potential, is the electric potential, the Ginzburg-Landau parameter of the superconductor is denoted by κ and the normal conductivity of the sample by σ . The triplet (ψ, A, ) should also satisfy an initial condition at t = 0. A solution (ψ, A, ) is called a normal state solution if ψ ≡ 0. From (1.1) we see that if (0, A, ) is a time-independent normal state solution then (A, ) satisfies the equality κ 2 curl2 A + σ ∇ = κ 2 curlHe in R2 .
(1.2)
In the following we further assume that a current of constant magnitude J is flown through the sample in the y-axis direction, and that the applied magnetic field is of constant magnitude, hence He = hiz , throughout the entire sample. Here ix , i y and iz denote the canonical basis in R3 . Under these additional assumptions (1.2) admits the following solution, which is also a normal state solution of (1.1), A=
1 κ2 J (J x + h)2 i y , = y. 2J σ
Note that the magnetic field H = curlA = (J x + h)iz ,
(1.3)
Superconductivity in Current and Magnetic Fields
149
is the sum of the constant applied magnetic field hiz and a linear term produced by the electric current. The linearization of (1.1) near the normal state solution (1.3) is ∂t ψ +
κ 2 iκ iκ 3 J y ψ = ψ − (J x + h)2 ∂ y ψ − (J x + h)4 ψ +κ 2 ψ in (0, T ) × R2 . σ J 2J (1.4)
Applying the transformation
h (t, x, y) → t, x − , y , J
we obtain iκ 3 J y ∂t ψ + ψ = ψ − iκ J x 2 ∂ y ψ − σ
κJ 2
2 x −κ 4
2
ψ.
(1.5)
In this work we shall analyze the asymptotic behavior of the solutions of (1.5) for large t. We assume J > 0 in the sequel. Otherwise we may either consider the complex conjugate of (1.4) or apply the transformation y → −y. Hence, we can rescale x and t by applying t → (κ J )2/3 t, (x, y) → (κ J )1/3 (x, y),
(1.6)
∂t u = −(A0,c − λ)u,
(1.7)
yielding
where A0,c is the differential operator defined by 2 i A0,c := −∂x x − ∂ y + x 2 + icy, 2
(1.8)
and c=
κ 4/3 κ2 , λ = 2/3 , u(x, y, t) = ψ((κ J )−1/3 x, (κ J )−1/3 y, (κ J )−2/3 t). σ J
Our main objective in this work is to analyze the long time behavior of the semi-group associated with A0,c . If one assumes that the magnetic field induced by the current is negligible, the following simplified elliptic operator is obtained from (1.8): B = − + icy. In [3] it is shown that one can obtain the spectral properties of B in large two or threedimensional bounded domains, by analyzing first a pair of one-dimensional problems for functions which depend on y only and involve the so-called complex Airy operator D 2y + icy on R or R+ . Obviously, for the present operator (1.8), it is impossible to find a meaningful one-dimensional setting which could teach us anything about the properties of A0,c . Thus, we have to discuss two-dimensional settings as our basic problems, imposing a significant complication on the spectral analysis. We shall therefore confine the present
150
Y. Almog, B. Helffer, X.-B. Pan
discussion to samples capturing the whole plane. We leave the analysis of the effect of boundaries to future research. Applying to (1.7) the transformation u → u eicyt yields 2 1 ∂t u = ∂x x u + ∂ y − i( x 2 + ct) u + λu. 2 Note that by applying the partial Fourier transform in y, 1 u(x, ˆ ω, t) = √ e−iωy u(x, y, t) dy, 2π R
(1.9)
(1.10)
to (1.9) we obtain
∂t uˆ = ∂x x uˆ −
1 2 x + (ct − ω) 2
2
− λ u. ˆ
(1.11)
The above can in turn be rewritten as a family (depending on ω ∈ R) of time-dependent problems on R, ∂t uˆ = −Lβ(t,ω) uˆ + λu, ˆ
(1.12)
with Lβ being the well-known anharmonic oscillator [22]: 1 Lβ = −∂x x + ( x 2 + β)2 , 2
(1.13)
and β(t, ω) = ct − ω. From this point we may proceed by translating t by −ω/c, i.e. by setting τ =t−
ω , c
v(x, τ ) = u(x, ˆ t)
(1.14)
to obtain from (1.12) the following: ∂τ v(x, τ ) = −(Lcτ v)(x, τ ) + λv(x, τ ).
(1.15)
The initial condition at t = 0 is then prescribed at ω τ =− . c Hence the dependence on ω appears only through the time at which the initial condition is set up. The main result of this work is that all solutions of (1.5) decay exponentially fast as t → ∞, for every J = 0. This means that the normal state ψ ≡ 0 is stable even for very weak currents. The simplest form in which this fact is being displayed in this
Superconductivity in Current and Magnetic Fields
151
contribution is (4.2). We bring it here in terms of the physical variables and parameters in (1.5): √ 2 2 κ 2 J 3/2 2 3/4 ψ(t) L 2 (R2 ) ≤ ψ(t = 0) L 2 (R2 ) exp − t + κ t + Ct , (1.16) 3 σ 1/2 where C is an appropriately chosen positive constant. The rest of this contribution is arranged as follows. In Sect. 2, we derive some of the basic properties of the non-selfadjoint operator introduced in (1.8). In Sect. 3 we review some semi-classical properties of the anharmonic operator (1.13) for large β, which will play a key role in our analysis. In Sect. 4 we obtain the long-time asymptotic behaviour of solutions of (1.7) as shown in (1.16), and then apply the estimates obtained there to control the norm of the resolvent. In Sect. 5, we propose a finer analysis of (1.7) which allows us in particular to obtain a lower bound for the norm of the resolvent. 2. Basic Properties of A0,c 2.1. Preliminaries and maximal accretiveness. Since some of the methods applied in this section can deal with classes of operators more general than (1.8), we state our results more generally, and then apply them to A0,c . Denote then by L 2 (Rn ) the space of real valued functions with finite L 2 norm, and L 2 (Rn , C) the complex valued functions. To simplify notation we use u L 2 (Rn ) to denote the norms in both spaces and u, v = u vd ¯ x Rn
to denote the inner product in them. Let A = (A1 , . . . , An ) ∈ C ∞ (Rn , Rn ) and V ∈ C ∞ (Rn , C). Define ∇A2 =
n (∂x j − i A j )2 ,
PA,V = −∇A2 + V (x).
(2.1)
j=1
Then ∇A2 and PA,V are well-defined in Cc∞ (Rn , C). The operator PA,V with domain C0∞ (Rn ) is closable and hence we can define P = PA,V
(2.2)
as its closure. By the definition, the domain of P, D(P), is the closure of C0∞ (Rn , C) under the graph norm u → ( u 2L 2 (Rn ) + Pu 2L 2 (Rn ) )1/2 . We study here the mapping properties of this operator and its spectrum. The operator A0,c , introduced in (1.8), clearly belongs to the class (2.1) via the particular choice n = 2,
A1 (x, y) = 0,
A2 (x, y) = −
x2 , V (x, y) = icy, c ∈ R\{0}. (2.3) 2
Hence, we can derive from the spectral properties of P the spectral properties of A = A0,c .
(2.4)
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Semi-boundedness. Clearly, for all u ∈ C0∞ (Rn , C), we have PA,V u, u =
−∇A2 u, u + V u, u
=
∇A u 2L 2 (Rn )
+
Rn
V |u|2 d x.
(2.5)
Assuming on Re V ≥ 0, we can introduce the bottom of the numerical range E∗ =
inf
u∈Cc∞ (Rn ,C)\{0}
Re PA,V u, u . u 2L 2 (Rn )
(2.6)
Note that E ∗ is the bottom of the spectrum of 21 (P + P ∗ ) (where P is given in (2.2) and P ∗ is the conjugate of P), which is the bottom of the spectrum of the (unique) selfadjoint realization of −∇A2 + Re V : E ∗ := inf σ (PA,Re V ) ≥ 0.
(2.7)
In the particular case where (2.3) is satisfied, E ∗ is the bottom of the selfadjoint operator 2 Dx2 + (D y + x2 )2 . As we shall see in the next section, we have in this case x2 2 ∗ 2 (2.8) E = inf inf σ Dx + (β + ) . 2 β∈R Theorem 2.1. Consider the magnetic Schrödinger operator P defined in (2.2) on Rn with A ∈ C ∞ (Rn , Rn ) and V ∈ C ∞ (Rn , C) such that Re V (x) ≥ 0.
(2.9)
Then the operator P − E ∗ is maximally accretive. Moreover P = (PA,V¯ )∗ .
(2.10)
The proof of this theorem is given in the next subsection. 2.2. Proof of maximal accretiveness. By the definition of E ∗ , PA,V − E ∗ is positive on C0∞ (Rn ). Hence the main point for the proof of Theorem 2.1 is to apply the Lumer-Phillips theorem for closable operators (see [8, Theorem 8.3.5]) to PA,V − E ∗ and PA,V¯ − E ∗ . By this theorem, if the range of (PA,V − E ∗ + γ ) is dense in L 2 (Rn , C) for some γ > 0, then P − E ∗ is the generator of a one-parameter contracting semigroup, and hence also maximally accretive. This reduces the proof of Theorem 2.1 to the following proposition:
Proposition 2.2. Under the condition (2.9), for any γ > 0, (PA,V −E ∗ +γ ) C0∞ (Rn , C) is dense in L 2 (Rn , C). Proof. Changing V into V − E ∗ if necessary, we can assume that E ∗ = 0. We rely on a proof given in [12] to a similar statement appearing in the proof of essential selfadjointness of PA,V in the case when V is real. Suppose that f ∈ L 2 (Rn , C) is such that f, (PA,V + γ )u = 0 for all u ∈ C02 (Rn , C). Proving that f ≡ 0 would achieve our goal.
(2.11)
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We first observe that (2.11) implies that (−∇A2 + V¯ + γ ) f = 0 in the sense of distributions. Standard elliptic regularity theory for the Laplacian (with 2 (Rn , C). We now our assumptions on V and A in mind) implies then that f ∈ Hloc introduce a family of cut-off functions, ζk , by x ζk (x) := ζ ( ), for all k ∈ N, k where ζ ∈ C0∞ (Rn , C) satisfies 0 ≤ ζ ≤ 1 , ζ = 1 on the unit ball B1 (0) and Supp ζ ⊂ B2 (0) , where B R (x0 ) denotes the ball with center at x0 and radius R. For any u ∈ C0∞ (Rn , C) we have the identity {∇A (ζk f ) · ∇A (ζk u) + ζk2 (V¯ + γ ) f u}d ¯ x Rn = f, (PA,V + γ )(ζk2 u) + {|∇ζk |2 f u¯ + ζk ∇ζk · [ f ∇A u − u∇ ¯ A f ]}d x. (2.12) Rn
Since f satisfies (2.11) we obtain {∇A (ζk f ) · ∇A (ζk u) + ζk2 (V¯ + γ ) f u}d ¯ x n R = {|∇ζk |2 f u¯ + ζk ∇ζk · [ f ∇A u − u∇ ¯ A f ]}d x. Rn
1 (Rn , C) . In particThis formula can be extended by continuity to all functions u ∈ Hloc ular, letting u = f and taking the real part of the above identity we obtain ∇A (ζk f ) 2L 2 (Rn ) + ζk2 (Re V + γ )| f |2 d x = |∇ζk |2 | f |2 d x,
Rn
hence, by (2.9),
γ
Rn
Rn
ζk2 | f |2 d x ≤
Rn
|∇ζk |2 | f |2 d x.
Using this and the definition of ζk , and taking the limit k → ∞ , we obtain γ f 2L 2 (Rn ) = γ lim ζk f 2 ≤ lim sup |∇ζk |2 | f |2 d x = 0, k→∞
k→∞
Rn
furnishing, thereby, the density of the range of PA,V + γ in L 2 (Rn , Rn ).
Theorem 2.1 permits us to apply some results of semi-group theory and of the theory of maximally accretive operators. We refer to [8] for a recent presentation of the first theory (and particularly Theorem 8.3.5) and to [18, Theorem 5.4] for the second. The following proposition gives a simple description of the domain D(P) of P (see the definition of D(P) given at the beginning of Sect. 2). Proposition 2.3. Let P be the operator defined in (2.2) and D(P) be the domain of P. Then, D(P) = {u ∈ L 2 (Rn , C) : PA,V u ∈ L 2 (Rn , C)}.
(2.13)
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We now observe that (2.6) implies by Hille-Yosida Theorem (or Theorem 12.8 in [2]) that, for λ such that Re λ > −E ∗ , the operator P + λ : D(P) → L 2 (Rn , C) is an isomorphism and (P + λ)−1 ≤ (Re λ + E ∗ )−1 .
(2.14)
2.3. Compactness of the resolvent. Although more general results have been obtained for example in [17] for the selfadjoint case, in [19] for the polynomial case, or in [18] for the case of Fokker-Planck operators, the next criterion is sufficient for all purposes we have in mind. Proposition 2.4. Let A ∈ C ∞ (Rn , Rn ) and let V ∈ C ∞ (Rn , C) satisfy (2.9). Let B = curlA and (2.15) m B,V (x) := |B(x)|2 + |V (x)|2 + 1. Suppose that lim m B,V (x) = +∞,
|x|→∞
(2.16)
and that either (i) Im V , and Bk = ∂xk A − ∂x Ak do not change sign in Rn for all 1 ≤ k, ≤ n, or (ii) there exists a constant C0 such that for all x ∈ Rn , |∇V (x)| + |∂i Bk (x)| ≤ C0 m B,V (x). (2.17) i,k,
Then, P has a compact resolvent. Proof. It is enough to show the existence of a constant C such that, for all u ∈ C0∞ (Rn , C), m B,V (x)|u(x)|2 d x ≤ C{ PA,V u 2L 2 (Rn ) + u 2L 2 (Rn ) }. (2.18) Rn
In the following we write m(x) for m B,V (x) and write Re V = V1 , Im V = V2 in order to simplify the notation. Using the condition (2.9) we have, for all u ∈ C0∞ (Rn , C), 1 {|∇A u|2 +|V1 ||u|2 }d x = Re PA,V u, u ≤ { PA,V u 2L 2 (Rn ) + u 2L 2 (Rn ) }. (2.19) 2 Rn In the following we estimate the integrals |V2 ||u|2 d x and |Bk ||u|2 d x, ∀(k, ) ∈ {1, . . . , n} × {1, . . . , n}. Rn
Rn
Suppose first that V2 has constant sign. In this case we immediately have, for all u ∈ C0∞ (Rn , C), 1 |V2 ||u|2 d x ≤ PA,V u, u ≤ { PA,V u 2L 2 (Rn ) + u 2L 2 (Rn ) }. (2.20) 2 Rn
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Without this sign assumption, we use integration by parts to obtain Im PA,V u, m −1 V2 u = Im (∇ − iA)u · ∇(m −1 V2 )u¯ d x + m −1 V22 |u|2 d x, Rn
Rn
and hence Im PA,V u, m −1 V2 u ≥ m −1 V22 |u|2 d x − sup |∇(m −1 V2 )| u L 2 (Rn ) ∇A u L 2 (Rn ) . Rn
x∈Rn
Using (2.17) and (2.19), we then obtain, for all u ∈ C0∞ (Rn , C), 2 |V2 ||u| d x ≤ m −1 V22 |u|2 d x ≤ C{ PA,V u 2L 2 (Rn ) + u 2L 2 (Rn ) }, (2.21) Rn
Rn
where C depends on the constant C0 in (2.17). To control the magnetic field part, suppose again first that a component Bk has a constant sign. In this case we use the operator identity ([·, ·] being the Poisson bracket) Bk = i[∂xk − i Ak , ∂x − i A ],
(2.22)
to obtain that Rn
|Bk | |u|2 d x ≤ 2 (∂x − i A )u L 2 (Rn ) · (∂xk − i Ak )u L 2 (Rn ) 1 ≤ | PA,V u, u | ≤ ( PA,V u 2L 2 (Rn ) + u 2L 2 (Rn ) ). 2
(2.23)
In case (ii), we use (2.22) once again to obtain 2 m −1 Bk |u|2 d x = i {[∂xk − i Ak , ∂x − i A ]u} · (m −1 Bk u) ¯ dx Rn Rn =i m −1 Bkl {[(∂xl − i Al )u][(∂xk − i Ak )u] − [(∂xk − i Ak )u][(∂xl − i Al )u]}d x n R −1 +i u{[(∂ ¯ Bkl ) − [(∂xk − i Ak )u]∂xl (m −1 Bkl )}d x xl − i Al )u]∂xk (m Rn ≤ C (∂x − i A )u L 2 (Rn ) (∂xk − i Ak )u L 2 (Rn ) + ( (∂x − i A )u L 2 (Rn ) + (∂xk − i Ak )u L 2 (Rn ) ) u L 2 (Rn ) sup |∇(m −1 Bk )| . x∈Rn
As before, the above inequality, in conjunction with (2.17), lead to 2 2 |Bk ||u| d x ≤ m −1 Bk |u|2 d x ≤ C{ PA,V u 2L 2 (Rn ) + u 2L 2 (Rn ) ), (2.24) Rn
Rn
for all u ∈ C0∞ (Rn ). Combining (2.19), (2.20), (2.21), (2.23), (2.24), we get (2.18).
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Note that the same proof gives Proposition 2.5. Assume A ∈ C ∞ (Rn , Rn ) and V ∈ C ∞ (Rn , C) and the conditions (2.9) and (2.17) hold. Let P = PA,V . Then we have 1 D(P) ⊂ HA,V (Rn , C) := {u ∈ L 2 (Rn , C), ∇A u ∈ L 2 (Rn , Cn ),
|V |1/2 u ∈ L 2 (Rn , C)},
(2.25)
and there exists a constant C depending on the constant C0 in (2.17) such that |V (x)||u(x)|2 d x ≤ C{ PA,V u 2L 2 (Rn ) + u 2L 2 (Rn ) }, ∀u ∈ D(P). ∇A u 2L 2 (Rn ) + Rn
(2.26) Corollary 2.6. Let A = A0,c be the operator on R2 defined in (2.4) where c = 0. Then σ (A) = ∅. Proof. From Proposition 2.4 we see that A has a compact resolvent. Hence it has a discrete spectrum. For some a ∈ R, we now introduce the translation operator Ta defined by (Ta u)(x, y) = u(x, y − a), to obtain: Ta A = (A + ia)Ta . Consequently, if σ (A) = ∅ then σ (A) is not discrete.
(2.27)
Note that the same result holds for the complex Airy operator Dx2 +i x on R, for which the emptiness of the spectrum has been established, using various techniques, in several other contributions including [3,8,21]. 2.4. L ∞ (Rn ) spectral pairs. We now move to consider bounded generalized eigenfunctions of the operator PA,V . We say that (ψ, λ) is an L ∞ -spectral pair for the operator PA,V if λ ∈ C and ψ ∈ L ∞ (Rn , C)\{0} is a solution, in the sense of distributions, of (PA,V − λ)ψ = 0.
(2.28)
Theorem 2.7. Assume A ∈ C ∞ (Rn , Rn ) and V ∈ C ∞ (Rn , C) and the conditions (2.9) and (2.17) hold. If (ψ, λ) is an L ∞ -spectral pair of the operator PA,V , then λ ∈ σ (P), where P = PA,V . Proof. Let (ψ, λ) denote an L ∞ spectral pair of the operator PA,V , and suppose λ ∈ / σ (P). We shall derive a contradiction. As the operator PA,V is elliptic and A and V are of class of C ∞ , it is clear that ψ is a ∞ C -function. The proof is reminiscent of the so-called Schnol’s theorem [7]. Consider, for R ≥ 1, a family of cut-off functions x , χR = χ R
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with χ being a non-negative smooth function satisfying χ = 1 on the ball of radius 1, and with compact support in a ball of radius 2. Set ψ R := χ R ψ. It is clear from the assumption that ψ R belongs to L 2 (Rn , C), with ψ R L 2 (Rn ) ≤ C R n/2 ψ L ∞ (Rn ) .
(2.29)
We shall now show that if λ ∈ σ (P), then there exist k > 0 and Ck > 0 depending on λ such that, for all R ≥ 1, we have ψ R L 2 (Rn ) ≤ C R −k .
(2.30)
Once the above inequality is proved, letting R → +∞ will lead to a contradiction. Proof of (2.30). From (2.29) we see that there exist k0 ∈ R and C0 > 0 such that (2.30) holds for k = k0 and C = C0 . We now show that there exists C1 such that (2.30) holds for k = k0 + 1 and C = C1 . To this end we first observe that χ R ψ satisfies the following equation in the sense of distributions: (PA,V − λ)(χ R ψ) = −2(∇χ R ) · (∇A ψ) − ( χ R )ψ.
(2.31)
It is immediately seen (using the local regularity of PA,V ) that the right-hand side is in L 2 (Rn , C) and consequently that ψ R ∈ D(P). To obtain an effective estimate for the L 2 -norm of this right-hand side we take the scalar product of (2.31) with χ R ψ, integrating by parts, and then taking the real part, to obtain Re (PA,V − λ)(χ R ψ), χ R ψ = (∇χ R )ψ 2L 2 (Rn ) .
(2.32)
Using the assumption that Re V ≥ 0, and the fact that ∇χ R is supported in a ball with radius 2R, we have |∇A ψ R |2 d x ≤ (∇χ R )ψ 2L 2 (Rn ) + |Re λ| ψ R 2L 2 (Rn ) Rn
≤ C R −2 ψ2R 2L 2 (Rn ) + |Re λ| ψ R 2L 2 (Rn ) ≤ C R −2 C0 (2R)−2k0 + |Re λ|C02 R −2k0 .
Hence we can find positive constants C and Cˆ depending on λ, such that for any R ≥ 1, ∇A (χ R ψ) L 2 (Rn ) ≤ C0 C R −(k0 +1) + |Re λ|C0 R −k0 ≤ Cˆ R −k0 .
(2.33)
Then, we have from (2.31) and (2.33) (with R replaced by 3R) (P − λ)(χ R ψ) L 2 (Rn ) ≤ C R −(k0 +1) .
(2.34)
By the assumption λ is not in the spectrum of P, so there exists C1 > 0 depending on λ such that (2.30) is satisfied for k = k0 + 1 and C = C1 . Thus, we can repeat the above argument to show that there exists a constant C2 depending on λ such that (2.30) holds for k = k0 + 2 and C = C2 . After a finite number of iterations we reach the conclusion that (2.30) holds for some k > 0. Corollary 2.8. For any c = 0, there is no L ∞ spectral pair (ψ, λ) for A0,c . Proof. We have indeed σ (A) = ∅ for A = A0,c as proven in Corollary 2.6. Combining this with Theorem 2.7 we see that no L ∞ spectral pair for A0,c exists.
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3. The Anharmonic Oscillator In this section we consider the operator d2 Lβ = − 2 + dx
1 2 x +β 2
2 ,
(3.1)
which is the well-known anharmonic oscillator [22,26]. We consider its selfadjoint realization on L 2 (R). It is well known that this operator has a compact resolvent, and we are particularly interested in the limiting behaviour as β → ±∞ of its discrete spectrum. 3.1. Large |β| asymptotics of the first eigenvalues. The next proposition lists a few properties of the eigenvalues and the eigenfunctions of Lβ in L 2 (R) in the limit β → −∞. (2) ∞ Proposition 3.1. Let {E k(1) (β)}∞ k=0 (respectively {E k (β)}k=0 ) denote the eigenvalues of Lβ corresponding to the even (respectively odd) spectrum and {φk(1) (x, β)}∞ k=0 (respec(2) ∞ 2 tively {φk (x, β)}k=0 ) denote their corresponding eigenmodes in L (R), i.e., ()
() ()
Lβ φk = E k φk
in R,
(3.2)
()
with φk L 2 (R) = 1, = 1, 2. Then we have the following conclusions: √
(i) For all S ∈ (0, 4 3 2 ) and k ≥ 0, there exist Ck > 0 and βk < 0 such that, for β ≤ βk , (2)
(1)
0 ≤ E k (β) − E k (β) ≤ Ck e−S|β|
3/2
holds.
(ii) For any k ≥ 0, we have, for sufficiently large −β and for = 1, 2, Ck , |E k() (β) − (2k + 1) −2β| ≤ |β|
(3.3)
Ck () . (E k ) (β) ≤ √ −β
(3.4)
and
()
(iii) For sufficiently large −β and for all k ≥ 0, = 1, 2, we can choose the φk such that () Cˆ k φ (x, β) − √1 |2β|1/8 h k |2β|1/4 (x − −2β) ≤ , k |β|3/4 2 2 L (R )
(3.5) where h k (x) = Hk (x)e−x
2 /2
,
and Hk is the k th Hermite polynomial, normalized by the condition h k L 2 (R) = 1, and Cˆ k depends only on k.
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Proof. We omit from now on the superscript () to simplify the notation. The statements in the sequel (in this specific limit) are equally true for both the even and the odd cases. Except for (3.4) all the statements of the proposition have been proved in [13,14,26] (sometimes in a refined way, see [16, Chaps. 2–4]). Note that after an appropriate dilation we arrive at a standard semi-classical problem with the semiclassical parameter = |β|−3/2 . More precisely, by introducing the new coordinate x˜ = β −1/2 x, we obtain the operator β 2 (−2
d2 + W (x)), ˜ d x˜ 2
with
1 W (x) ˜ = ( x˜ 2 − 1)2 . 2
To prove (3.4), we take the derivative of (3.2) with respect to β and obtain ∂φk (Lβ − E k ) = −(x 2 + 2β) + E k (β) φk . ∂β Taking the inner product with φk yields the following Feynman-Hellmann formula [11]: E k (β) = (x 2 + 2β)|φk (x, β)|2 d x. R
Then, (3.4) is √ readily verified with the aid of (3.5) and the decay properties of φk far from x = ± −2β. Similar approximations are valid in the limit β → +∞. () ∞ Proposition 3.2. For the above-defined {E k() (β)}∞ k=0 and {φk (x, β)}k=0 we have, as β → +∞, the asymptotics C k () , (3.6a) E k (β) − β 2 − [2(2k + ) − 1]β 1/2 ≤ β
and ()
φk (x, β) − |β|1/8 h 2k+−1 (|β|1/4 x) L 2 (R) ≤
Cˆ k . |β|3/2
(3.6b)
In contrast with the case β → −∞, the Schrödinger operator (3.1) with β > 0 has a single well potential. The proof is again a standard application of semi-classical analysis (see [16, Sect. 3.4] for instance). In the following we denote E 0 (β) = E 0(1) (β), which is the lowest eigenvalue of Lβ . By (3.3) and (3.6), and since Lβ is positive-definite, it is clear that E 0 (β) has a strictly positive infimum E ∗ , E ∗ = inf E 0 (β), β∈R
(3.7)
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and that there exists at least one β ∗ < 0 such that E ∗ = E 0 (β ∗ ). In [22] the values of E ∗ and β ∗ have been numerically computed. In [24] Pan and Kwek stated that β ∗ is unique. In a recent contribution Helffer [15] proved, using a different approach, that β ∗ is indeed unique and that in addition the minimum is non degenerate. 3.2. Auxiliary functions of E 0 . For later use, we define some auxiliary quantities depending on the behavior of E 0 (β) as |β| → ∞. For sufficiently large ρ the equation E 0 (β) = ρ,
(3.8)
has exactly two distinct solutions β− (ρ) and β+ (ρ) satisfying β− (ρ) < 0 < β+ (ρ). Moreover, we have the following asymptotics by (3.3) and (3.6): β− (ρ) ∼ −
ρ2 + O(ρ −1 ), 2
β+ (ρ) ∼
√ ρ + O(ρ −1/4 ) as ρ → +∞.
(3.9)
We then define the natural quantity (ρ) =
β+ (ρ)
β− (ρ)
E 0 (β) dβ.
(3.10)
By (3.3), (3.6), and (3.9) we have (cf. [23, Sect. 1.2] or [5, Sect. 3.8]), as ρ → +∞, 1 3 ρ + ρ 3/2 + O(ln ρ). (3.11) (ρ) = 3 We further define the quantity ρ(τ ) as the solution of β+ (ρ(τ )) − β− (ρ(τ )) = τ, which has the asymptotics (by (3.9)) √ ρ(τ ) ∼ 2τ + O(τ −1/4 ) as τ → +∞.
(3.12)
(3.13)
Then we set (τ ) = (ρ(τ )). We observe from (3.11) and (3.13) that √ 2 2 3/2 (τ ) = τ + O(τ 3/4 ) as τ → +∞. 3
(3.14)
(3.15)
Finally, we define the function 1 L( , c, μ) = sup μt − (ct) . c t∈R
(3.16)
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Observe first that , c, μ) = L(
1 , 1, μ). L( c
(3.17)
, 1, μ), Hence, in order to compute (3.16) as μ → ∞, we need to approximate only L( which is called the Legendre transform of (see [25, Chap. 11]). Using (3.10) and (3.14), together with a short computation, gives that for sufficiently large μ, when c = 1, the supremum in (3.16) is attained at t = t (μ) which is defined by t (μ) = β+ (μ) − β− (μ), and that , c, μ) = L(
1 c
β+ (μ) β− (μ)
(μ − E 0 (β)) dβ.
(3.18)
, c, μ) for large μ can be derived by using (3.9), (3.10) and The asymptotics of L( (3.11): , c, μ) = L(
μ3 + O(μ3/2 ) as μ → +∞. 6c
(3.19)
4. Analysis of Time-Dependent Problems 4.1. Long time decay. We start this section by considering the long time behaviour of the solutions of (1.9). Using the properties of the operator A0,c that were derived in Sect. 2, we can apply semigroup theory to prove the global existence of solutions [8]. Our goal here is to improve the general results of semi-group theory using the more particular structure of the operator A in (2.4). Let u(x, y, t) be a solution of (1.9) in R2 . We denote by u(t) the one-parameter family of functions (x, y) → u(x, y, t). Their norm is given by u(t) L 2 (R2 ) =
1/2 R2
|u(x, y, t)|2 d xd y
.
Proposition 4.1. Let u(x, y, t) be a solution of (1.9) defined for all t ≥ 0 such that u(x, y, 0) = u 0 (x, y) ∈ L 2 (R2 , C). Then there exists T0 > 0 such that, for any t ≥ T0 and any u 0 ∈ L 2 (R2 ), 1 (ct) + λt u 0 L 2 (R2 ) , u(t) L 2 (R2 ) ≤ exp − c
(4.1)
introduced in (3.10) and (3.14). As a result there exists a constant C > 0 such with that for t ≥ T0 and any u 0 ∈ L 2 (R2 , C), √ 2 2c 3/2 (4.2) u(t) L 2 (R2 ) ≤ exp(− t + λt + Ct 3/4 ) u 0 L 2 (R2 ) . 3
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Proof. We prove (4.1) first for u 0 ∈ S(R2 ) (where S(R2 ) denotes the Schwartz space of the rapidly decreasing functions in x, y). The extension to all u 0 ∈ L 2 (R2 , C) then follows by density. Thus, it is sufficient to prove (4.1) for the partial Fourier transform of u with respect to y which is denoted by (x, ω, t) → u(x, ˆ ω, t). For given ω, we multiply (1.11) by u(x, ˆ¯ ω, t) (the complex conjugate of u), ˆ and integrate the resulting equality over R with respect to x to obtain 1 d u(·, ˆ ω, t) 2L 2 (R ) = − u(·, ˆ ω, t), Lct−ω u(·, ˆ ω, t) + λ u(·, ˆ ω, t) 2L 2 (R ) , x x 2 dt where Lβ is defined by (3.1), and ·, · denotes the L 2 (Rx ) inner product. Clearly, u(·, ˆ ω, t), Lct−ω u(·, ˆ ω, t) ≥ E 0 (ct − ω) u(·, ˆ ω, t) 2L 2 (R ) , x
where E 0 (β) = E 0(1) (β) is defined in Sect. 3. Consequently, we have d ˆ ω, t) 2L 2 (R ) . u(·, ˆ ω, t) 2L 2 (R ) ≤ 2(λ − E 0 (ct − ω)) u(·, x x dt Hence, it readily follows that, for any t > t0 ≥ 0, ˆ ω, t0 ) 2L 2 (R ) exp 2 λ(t −t0 )− u(·, ˆ ω, t) 2L 2 (R )≤ u(·, x
x
t
E 0 (cs −ω)ds
. (4.3)
t0
Setting t0 = 0, integrating over ω, and making use of the Plancherel formula, we obtain u(t) L 2 (R2 ) ≤ exp λt − inf (ω, t) u 0 L 2 (R2 ) , (4.4) ω∈R
with (ω, t) :=
t−ω/c
−ω/c
E 0 (cs) ds =
1 c
ct−ω −ω
E 0 (β) dβ.
(4.5)
Hence, it remains necessary to estimate the quantity inf (ω, t).
ω∈R
(4.6)
In view of the asymptotic behaviour, as |β| → ∞, of E 0 (β) it follows that, for any given t, the infimum (4.6) exists and must be attained at a point ω = ω1 (t) ∈ R such that ∂ (ω, t) = 0, ∂ω ω=ω1 (t) implying that E 0 (−ω1 ) = E 0 (ct − ω1 ).
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Then we may use (3.8) to obtain for sufficiently large t that the number ρ = E 0 (−ω1 ) satisfies β− (ρ) = −ω1 , β+ (ρ) = ct − ω1 . By these equalities, (4.5) and (3.10) we then have (ω1 , t) =
1 c
β+ (ρ) β− (ρ)
E 0 (β)dβ =
(ρ) , c
ρ = E 0 (−ω1 ).
(4.7)
However, as β+ (ρ) − β− (ρ) = ct, ρ = E 0 (−ω1 ) is the solution of (3.12) for τ = ct, so E 0 (−ω1 ) = ρ = ρ(ct). So we obtain from (4.7) and (3.14) that (ω1 , t) =
(ct) (ρ(ct)) = , c c
which, together with (4.4), establishes (4.1). Using (3.15), we obtain (4.2) as well. Coming back to (4.3), one can define a one-parameter semigroup St = exp(−tA)
(4.8)
on L 2 (R2 , C) associated to the operator A = A0,c introduced in (2.4), such that the solution of (1.9) with initial data is given by u(x, y, t) = St u(x, y, 0).
(4.9)
As a direct consequence of Proposition 4.1 with λ = 0 we have: Theorem 4.2. Let A be the operator defined in (2.4). Then there exists T0 > 0 such that for any t ≥ T0 we have (ct) . exp(−tA) ≤ exp − c In particular, there exists a constant C > 0 such that, for t ≥ T0 , √ 2 2c 3/2 3/4 t + Ct exp(−tA) ≤ exp − . 3
(4.10)
(4.11)
Remark 4.3. It follows from Theorem 4.2 and [8, Theorem 8.2.1] that A has an empty spectrum. We provide another proof of this fact in Sect. 2.
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4.2. Upper bounds of the resolvent with large |λ|. We look for a bound for the norm of the resolvent (A − λ)−1 with large |λ| for the operator A defined in (2.4), where λ is a complex number. Let λ = μ + iν,
(4.12)
with both μ and ν being real. Without loss of generality we can set ν = 0, otherwise we translate y by ν/c (that is, mapping y → y + ν/c). For A0,c with real constant c we have E ∗ ≥ 0 (see the definition of E ∗ for A0,c in (2.6)). So from (2.14) we see that for λ < 0, (A − λ)−1 ≤
1 . |λ|
(4.13)
On the other hand, as λ → +∞ we have the following estimate. Lemma 4.4. For the operator A defined in (2.4), there exist positive constants λ0 and C such that, for all λ > λ0 , 1 3 −1 3/2 . (4.14) λ + Cλ (A − λ) ≤ exp 6c Proof. We use the formula relating the semi-group and the resolvent given in the book [8]: +∞ −1 (A − λ) = exp(λt)St dt, (4.15) 0
with St = exp(−tA). Using (4.10), we get the universal upper bound +∞ (ct) −1 dt. exp λt − (A − λ) ≤ c 0
(4.16)
(4.17)
Using (4.11) and the substitution t = λ2 τ we obtain −1
(A − λ)
≤λ
2 0
+∞
exp λ
3
√ 2 2c 3/2 −3/2 3/4 τ + Cλ τ τ− dτ. (4.18) 3
The integral on the right-hand side can be estimated using the Laplace method (see [23, Chap. 2] for instance) to obtain (4.14). Remark 4.5. One can improve the upper bound of the norm of the resolvent in (4.14) by using (3.15)-(3.19) and an estimate with greater accuracy for the Legendre transform of (ct). The Laplace integral method can then provide better estimates for the right-hand side of (4.18).
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5. Long Time Asymptotics for the Fourier Transform and Applications 5.1. Technical preliminaries. In this section we look for a finer estimate of the norm of the resolvent of A. We approximate solutions of Eq. (1.15) which is obtained by first taking a partial Fourier transform of Eq. (1.9) and then translating the time while fixing the Fourier variable ω as a parameter. To obtain with greater accuracy the asymptotics of solutions for (1.15), we need to obtain some additional spectral properties of the anharmonic oscillator (1.13). We use semi-classical analysis to obtain these properties, which all involve asymptotics in the limit |β| → ∞. () Let E k (β), = 1, 2, k = 0, 1, 2, . . . be the eigenvalues of the operator Lβ intro() duced in Sect. 3, and φk be the associated eigenfunctions of the unit L 2 norm. Since the following discussion is valid for both = 1 and 2, and we omit the superscript in () () the sequel, namely, we replace E k (β) and φk by E k (β) and φk . Then, we set k (β) = E k (β) − E 0 (β). We have seen from Proposition 3.1 that 1 (β) = 2 −2β + O(|β|−1 ) as β → −∞.
(5.1)
(5.2)
We further introduce
∞ 2 1/2 1 x 2 φk (x, β)φm (x, β)d x , k = 0, 1, (5.3) gk (β) = E (β) − E (β) m k R m=2
and
1 f (β) = x 2 φ1 (x, β)φ0 (x, β)d x. 1 (β) R
(5.4)
g(β) = [g02 (β) + f 2 (β)]1/2 .
(5.5)
Finally, let
The following results will be useful in the next subsection. Lemma 5.1. There exist α = 0, C > 0 and β0 > 0 such that f (β) − α|β|−1/4 ≤ C|β|−1 , ∀β < −β0 , | f (β)| ≤ C |β|−5/4 , ∀β < −β0 , |g0 (β)| ≤ C |β|−1 , ∀β < −β0 , |g1 (β)| ≤ C |β|−1/4 , ∀β < −β0 , g(β) < Cβ −3/2 ∀β > β0 .
(5.6a) (5.6b) (5.6c) (5.6d) (5.6e)
Note that (5.6e) is related to the asymptotic behaviour of the anharmonic oscillator in the limit β → +∞, in contrast with the rest of the statements that deal with the limit β → −∞. Proof. Proof of (5.6a). We observe that, since {φk (·, β)}∞ k=0 are orthogonal to each other,
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R
x 2 φ j (x, β)φk (x, β)d x =
R
(x 2 + 2β)φ j (x, β)φk (x, β)d x,
j = k.
(5.7)
√ By (3.5), and the exponential rate of decay of φ0 and φ1 away from x = ± −2β (see for instance in [16, Chap. 3]) we have (x 2 +2β)φk L 2 (R) = (x + −2β)(x − −2β)φk L 2 (R) ≤ Cβ 1/4 , for k = 0, 1. (5.8) (5.6a) is easily verified using (3.5) and (5.2).
Proof of (5.6b). We first write, for i = 0, 1, ∞ ∂φ1−i ∂φ1−i x 2 φi , = x 2 φi , φk , φk . ∂β ∂β
(5.9)
k=0
Here ·, · denotes the inner product in L 2 (R). We then use the fact that φ j L 2 (R) = 1 and differentiate (3.2) with k = j in β to obtain 1 − E k −E x 2 φ j , φk if k = j, ∂φ j j , φk = (5.10) ∂β 0 if k = j. Substituting (5.10) with j = 1 − i into (5.9) yields ∞ (β) 1 1 1 x 2 φ0 , φk x 2 φ1 , φk . + f (β) = − 21 f (β)− 1 (β) (β) E − E E − E k 0 k 1 1 k=2
(5.11) The first term on the right side of (5.11) can be easily estimated. To control the second term we recall that the sequence {E k } is monotonically increasing, and hence for any k ≥ 2, 1 1 2 1 1 ≤ + . E − E E − E E − 1 (β) 1 2 E 1 k 0 k 1 Using (3.3) we see that the right-hand side of the above inequality is controlled by C/|β| as β → −∞. Hence the second term on the right side of (5.11) is bounded by 1/2 ∞ 1/2
∞ C 2 2 2 2 | x φ0 , φk | | x φ1 , φk | . |β| k=2
k=2
Using (5.7) with β replaced by 2β and then using the Parseval equality we have, for k ≥ 2, ∞
| x 2 φ0 , φk |2 =
k=2
∞
| (x 2 + 2β)φ0 , φk |2
k=2
=
∞
| (x 2 +2β)φ0 , φk |2 −| (x 2 +2β)φ0 , φ0 |2 −| (x 2 +2β)φ0 , φ1 |2
k=0
= (x 2 + 2β)φ0 2L 2 (R) −| (x 2 + 2β)φ0 , φ0 |2 −| (x 2 + 2β)φ0 , φ1 |2 .
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Now we claim that (x 2 + 2β)φ0 2L 2 (R) − | (x 2 + 2β)φ0 , φ0 |2 − | (x 2 + 2β)φ1 , φ0 |2 ≤
C . |β|
(5.12)
This is true due to the orthogonality of all Hermite functions h k for k ≥ 2 with xh 0 . The complete proof of (5.12) is given in Appendix A. Using (5.12) we get ∞
| x 2 φ0 , φk |2 ≤
k=2
C . |β|
(5.13)
Similarly we have ∞
| x 2 φ1 , φk |2 =
k=2
∞
| (x 2 +2β)φ1 , φk |2 ≤ (x 2 +2β)φ1 2L 2 (R) ≤ C|β|1/2 .
(5.14)
k=2
Here we have used the inequality (5.8). Combining (5.11), (5.13) and (5.14) yields (5.6b). The proof of (5.6c) and (5.6d) follows in exactly the same manner from (5.8) and (5.12). Finally we prove (5.6e) by first observing that g(β) ≤ From (3.6b) we now obtain (5.6e).
1 x 2 φ0 2L 2 (R) . ˆ1
5.2. An evolution problem. More detailed information on the solutions u of (1.7) can be obtained by deriving the large t asymptotic behavior of their (partial) Fourier transform which satisfies (1.11) or the large τ asymptotic behavior of the solutions of (1.15), with the relationship between t and τ given by (1.14). Note that in (1.15) the initial value corresponding to t = 0 is τ = −ω/c, and that −τ might be very large. Hence we need to separately consider two different regions of the variable τ : Case 1. 1 −τ ≤
ω c;
Case 2. 1 τ . The next proposition deals with the first case. We assume the initial data to be either even or odd, thus saving the need for marking each eigenvalue and eigenfunction by an appropriate superscript as in Sect. 3. For a given T =
ω >0 c
(5.15)
and some even L 2 -normalized function v0 ∈ L 2 (R), we analyze the properties of the unique solution v of (1.15) in the region −T < τ < +∞ such that v(x, −T ) = v0 (x),
v0 L 2 (R) = 1.
(5.16)
We denote by C(T ) the union set of all such solutions with initial data v0 satisfying (5.16).
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Remark 5.2. We note that this problem is an evolution problem attached to a τ -dependent problem. We can no more use the semi-group theory but fortunately our case enters in the theory developed by Kato for extending the Hille-Yosida theorem to time-dependent problems. We refer to [30, Chap. XIV] for a presentation of the theory. We observe that the domain of Lcτ is independent of τ and contains C0∞ (R) as a dense subspace. Moreover if the initial condition at τ = −T is in S(R) then the solution is in C ∞ ([−T, +∞), S(R)). Proposition 5.3. For every 0 < δ < 1/2, there exist T0 ≥ 1, T1 ≥ 1, Cδ > 0, Cˆ δ > 0 and Cδ −δ T (δ) = T − T , (5.17) 2 such that, if T > T0 ,
−T (δ) ≤ τ ≤ −T1 ,
(5.18)
and if v ∈ C(T ), then there exists C1 such that τ τ γ v(·, τ ) exp −λ(τ + T ) + E 0 (cs) ds − C1 φ0 (·, cτ ) 2 T −T L (R ) ≤ Cˆ δ |τ |−3/4 ,
(5.19)
and |C1 − v0 , φ0 (·, −cT ) | ≤ Cˆ δ T −1/2 ,
(5.20)
where γ =
|α|2 c √ , 2 2
(5.21)
and α is the positive number given in Lemma 5.1. Proof. We shall use (4.3) to derive the estimates for v(x, τ ). Step 1. We first transform (1.15) into an equivalent equation whose solution is both bounded and independent of λ. From (1.14), (5.15), (5.16), and using (4.3) (with t0 = 0) we find that, for −T < τ < −T1 , τ E 0 (cs) ds . (5.22) v(·, τ ) L 2 (R) ≤ exp λ(τ + T ) − −T
Set then
w(x, τ ) = v(x, τ ) exp −λ(τ + T ) +
τ −T
E 0 (cs) ds .
(5.23)
Clearly, by (5.22) we have w(·, τ ) L 2 (R) ≤ 1. Substituting (5.23) into (1.15), we obtain
⎧ 2 1 2 ⎪ ⎨ ∂ w(x, τ ) = ∂ w(x, τ ) − x + cτ − E 0 (cτ ) w(x, τ ), τ xx 2 ⎪ ⎩ w(x, −T ) = v0 (x).
(5.24)
(5.25)
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Step 2. We now define an expansion using the eigenfunctions {φk (x, cτ )} of Lcτ , and derive the equations for the first two coefficients and the remainder. Set then ak (τ ) = w(·, τ ), φk (·, cτ ) , k ∈ N,
(5.26)
where ·, · is the inner product in L 2 (R, C). Taking the inner product of (5.25) with φk (x, cτ ) and integrating over Rx , we obtain dak ∂φk (τ ) + k (τ )ak (τ ) = w(·, τ ), (·, cτ ) , (5.27) dτ ∂τ in which k (cτ ) = E k (cτ ) − E 0 (cτ ). k (τ ) ≡
(5.28)
Since w(·, τ ) =
∞
am (τ )φm (·, cτ ),
m=0
we have
∞ ∂φk ∂φk w(·, τ ), (·, cτ ) = (x, cτ )φm (x, cτ ) d x. am ∂τ R ∂τ m=0
From (5.10) it now follows that
∞ ∂φk am w(·, τ ), (·, cτ ) = −c x 2 φk (x, cτ )φm (x, cτ ) d x. ∂τ E m (cτ ) − E k (cτ ) R m=0 m=k
(5.29) Next we set w(x, ˇ τ ) = w(x, τ ) − a0 (τ )φ0 (x, cτ ) − a1 (τ )φ1 (x, cτ ).
(5.30)
From (5.26) and (5.30) we have w(·, ˇ τ ) 2L 2 (R) =
∞
|am |2 .
(5.31)
m=2
By (5.27) and (5.29) we have ∞ da0 (τ ) am (τ ) = −c x 2 φ0 (x, cτ )φm (x, cτ ) d x dτ E m (cτ ) − E 0 (cτ ) R m=1 ∞ am (τ ) = −ca1 (τ ) f (cτ ) − c x 2 φ0 (x, cτ )φm (x, cτ ) d x. E m (cτ ) − E 0 (cτ ) R m=2
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The second term can be controlled by
∞ 1/2 ∞ 2 1/2 1 2 c am x 2 φ0 (x, cτ )φm (x, cτ ) d x E (cτ ) − E (cτ ) m 0 R m=2
m=2
= cg0 (cτ ) w(·, ˇ cτ ) L 2 (R) , where g0 (cτ ) and f (cτ ) are given in (5.3) and (5.4). Hence da0 ˇ τ ) L 2 (R) . dτ (τ ) + c f (cτ )a1 (τ ) ≤ Cg0 (cτ ) w(·, Similarly we have da1 ≤ Cg1 (cτ ) w(·, (τ )a (τ ) + c f (cτ )a (τ ) ˇ τ ) L 2 (R) , (τ ) + 1 1 0 dτ
(5.32)
(5.33)
where g1 (cτ ) is given by (5.3). Note for later use that, since ak (τ ) is bounded for τ −1 for each k, (5.32) together with (5.6a) and (5.6c) imply that da0 −1/4 . (5.34) dτ (τ ) ≤ C|τ | From the definition of wˇ it follows that ∂ w(·, ˇ τ) + (Lcτ − E 0 (cτ )) w(·, ˇ τ) ∂τ 1 = − 1 (τ )a1 (τ )φ1 (·, cτ ) − ∂τ ak (τ )φk (·, cτ ).
(5.35)
k=0
Recall that w(·, ˇ τ ) is orthogonal to φi (·, cτ ) for i = 0, 1. Taking the inner product of (5.35) with w(·, ˇ τ ) and using this orthogonality yield that 1 d w(·, ˇ τ ) 2L 2 (R) + w(·, ˇ τ ), (Lcτ − E 0 (cτ )) w(·, ˇ τ) 2 dτ 1 ∂φk (·, cτ ) . = −c ak (τ ) w(·, ˇ τ ), ∂τ
(5.36)
k=0
Using (5.36) it is easy to show that d w(·, ˇ τ ) L 2 (R) + 2 (τ ) w(·, ˇ τ ) L 2 (R) ≤ c |ak (τ )| gk (cτ ). dτ 1
(5.37)
k=0
Step 3. Now we establish a preliminary upper bound of a1 (τ ). Set r1 (τ ) =
da1 (τ ) + 1 (τ )a1 (τ ). dτ
(5.38)
By (5.33) we have | r1 (τ )| ≤ Cg1 (cτ ) w(·, ˇ τ ) L 2 (R) + c| f (cτ )| |a0 (τ )|.
(5.39)
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Since
we have
171
τ τ d exp 1 (s)ds a1 (τ ) = exp 1 (s)ds r1 (τ ), dτ −T −T
exp
τ −T
1 (s)ds
a1 (τ ) = a1 (−T ) +
τ −T
exp
s −T
1 (η)dη r1 (s) ds.
Hence, by (5.39) τ a1 (τ ) − a1 (−T ) exp − 1 (s)ds −T τ τ # $ ≤ exp − 1 (η) dη Cg1 (cs) w(·, ˇ s) L 2 (R) + c| f (cs)||a0 (s)| ds. −T
s
(5.40) In view of (5.24) we have that ˇ τ ) 2L 2 (R) ≤ 1. a02 (τ ) + a12 (τ ) + w(·,
(5.41)
Hence, by (5.6) and (5.40),
τ |a1 (τ )| ≤ |a1 (−T )| exp − 1 (s)ds −T τ τ exp − 1 (η)dη |s|−1/4 ds. +C −T
(5.42)
s
By (3.3) we have that 1 (s) > C 1 (τ ) for s < τ < 0. Therefore,
τ 1 (s)ds |a1 (τ )| ≤ |a1 (−T )| exp − −T τ τ C exp − 1 (η)dη 1 (s) ds. + 1/4 |τ | 1 (τ ) −T s
(5.43)
We now estimate the right-hand side of (5.43). From (5.2) and (5.28), there exists C > 0 such that for all τ < 0, C . (5.44) 1 (τ ) − 2 2c|τ | ≤ |τ | Consequently, we obtain, for −T < τ ≤ −1, τ C T exp − 1 (s)ds ≤ exp −2 2c|τ |(τ + T ) . τ −T
(5.45)
From the above inequality and the fact that |a1 (−T )| ≤ 1, it is readily verified that the first term on the right-hand side of (5.43) is bounded from above by the right side of
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(5.45). The second term on the right-hand-side of (5.43) can be immediately integrated. So we obtain, for −T < τ ≤ −1, % C −3/4 T |a1 (τ )| ≤ C max |τ | , exp −2 2c|τ |(τ + T ) . (5.46) τ Let now 0 < δ < 1/2. Then for sufficiently large T0 , there exists Cδ > 0 such that Cδ −δ T and T ≥ T0 T − T1 > τ + T > 2 C T ⇒ exp −2 2c|τ |(τ + T ) ≤ |τ |−5/2 . τ
(5.47)
Let T (δ) be given by (5.17). For fixed δ, if T is large then T (δ) > T /2 > 0, and τ ∈ [−T (δ), −T1 ]
⇒
|a1 (τ )| ≤ C |τ |−3/4 .
(5.48)
Step 4. Now we establish some differential inequalities for w(·, ˇ τ ) L 2 (R) , a0 (τ ) and a1 (τ ). We deduce from (5.37), (5.41), (5.46), (5.6c) and (5.6d) that there exists C > 0 such that % C −3/2 −3/4 T w(·, ˇ τ ) L 2 (R) ≤ C max |τ | , |τ | τ exp −2 2c|τ |(τ + T ) . (5.49) From the above and (5.47), for τ ∈ [−T (δ), −T1 ] we have w(·, ˇ τ ) L 2 (R) ≤ C|τ |−3/2 .
(5.50)
By(5.32), (5.6a), (5.6c), (5.46) and (5.49) we now have for −T ≤ τ ≤ −T1 , % C da0 −1 −1/4 T τ exp −2 2c|τ |(τ + T ) . (5.51) dτ (τ ) ≤ C max |τ | , |τ | We now derive a differential inequality for a1 (τ ). To this end we use (5.33), (5.49) and (5.6d) to obtain for τ ∈ [−T (δ), −T1 ] that da1 ≤ C|τ |−7/4 . (τ ) + (τ )a (τ ) + c f (cτ )a (τ ) (5.52) 1 1 0 dτ Step 5. Now we establish an asymptotic expansion for a1 (τ ) for τ ∈ [−T (δ), −T1 ]. Integrating (5.52) from −T (δ) to τ we obtain τ a1 (τ ) − a1 (−T (δ)) exp − 1 (s)ds −T (δ) τ τ exp − 1 (η)dη f (cs)a0 (s) ds ≤ C|τ |−9/4 . +c −T (δ)
(5.53)
s
In the following we obtain the asymptotic expansion, in the large T1 limit, for the second and the third terms on the left-hand-side of (5.53). The second term can be easily
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estimated. In fact, from (5.41) we see that |a1 (−T (δ))| ≤ 1, and using (5.47) we obtain that, for τ ∈ [−T (δ), −T1 ], τ a1 (−T (δ)) exp − ≤ C|τ |−5/2 . (s)ds (5.54) 1 −T (δ)
To approximate the third term on the left-hand-side of (5.53) we represent it in the form τ ca0 (τ ) f (cτ ) τ exp − 1 (η) dη 1 (s)ds 1 (τ ) −T (δ) s τ τ a0 (s) f (cs) a0 (τ ) f (cτ ) − 1 (s) ds. (5.55) exp − 1 (η)dη +c 1 (s) 1 (τ ) −T (δ) s The first term in (5.55) can be readily integrated, to obtain for large T (δ) + τ : τ ca0 (τ ) f (cτ ) τ exp − 1 (η) dη 1 (s)ds 1 (τ ) −T (δ) s & ' ca0 (τ ) f (cτ ) + O exp −βˆ |τ |(T (δ) + τ ) , = (5.56) 1 (τ ) √ for some 0 < βˆ ≤ 2c. To estimate the second term in (5.55), we observe that, for −T (δ) ≤ s ≤ τ , d a0 (ξ ) f (cξ ) a0 (s) f (cs) a0 (τ ) f (cτ ) (τ − s). ≤ sup − (s) 1 (τ ) ξ ∈(s,τ ) dξ 1 (ξ ) 1 In view of (5.6a), (5.6b), (3.4), (5.41) and (5.51), for −T (δ) ≤ s ≤ ξ < τ ≤ −T1 , we also have
Hence,
τ −T (δ)
d a0 (ξ ) f (cξ ) ≤ C|ξ |−7/4 ≤ C|τ |−7/4 . dξ 1 (ξ ) exp −
≤ C|τ |−7/4
s τ
τ
a0 (s) f (cs) a0 (τ ) f (cτ ) − 1 (s) ds 1 (η)dη 1 (s) 1 (τ ) τ exp − 1 (η) dη (τ − s) 1 (s)ds. (5.57)
−T (δ)
s
Integration by parts then yields for τ ∈ [−T (δ), −T1 ], τ τ 0≤ exp − 1 (η) dη (τ − s) 1 (s)ds −T (δ) s τ τ τ exp − 1 (η) dη ds − (τ + T (δ)) exp − 1 (η) dη = −T (δ) s −T (δ) τ τ C exp − 1 (η) dη 1 (s)ds ≤ C|τ |−1/2 , ≤ 1 (τ ) −T (δ) s
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which together with (5.57) yields τ τ a0 (s) f (cs) a0 (τ ) f (cτ ) exp − (η)dη (s) ds − 1 1 1 (s) 1 (τ ) −T (δ) s ≤ C|τ |−7/4 |τ |−1/2 = C|τ |−9/4 .
(5.58)
Substituting (5.56) and (5.58) into (5.55) we obtain that the third term on the lefthand-side of (5.53) admits the expansion ca0 (τ ) f (cτ ) + O(1) exp −βˆ |τ |(T (δ) + τ ) + O(|τ |−9/4 ) 1 (τ ) ca0 (τ ) f (cτ ) + O(|τ |−9/4 ). = 1 (τ ) Inserting the above and (5.54) into (5.53) yields, for τ ∈ [−T (δ), −T1 ] with T1 large, a1 (τ ) =
ca0 (τ ) f (cτ ) + O(|τ |−9/4 ). 1 (τ )
(5.59)
Step 6. We now estimate a0 (τ ). We substitute (5.59) into (5.32), and use (5.6c), (5.47) and (5.49) to obtain that for all τ ∈ [−T (δ), −T1 ] we have da0 c2 | f (cτ )|2 ≤ C|τ |−9/4 . (τ ) + a (τ ) 0 dτ 1 (τ ) From (5.6a) and (5.44) it follows that for τ ∈ [−T (δ), −T1 ] we have da0 γ −7/4 , dτ (τ ) + |τ | a0 (τ ) ≤ C|τ | where γ is given in (5.21). Consequently, τ γ + O(|τ |−3/4 ). a0 (τ ) = a0 (−T (δ)) T (δ)
(5.60)
Step 7. Denote C1 = a0 (−T (δ)). From (5.30), (5.48), (5.50) and (5.60) we have, for τ ∈ [−T (δ), −T1 ], γ w(·, τ ) − C1 τ φ0 (·, cτ ) T (δ) 2 L (R ) ≤ w(·, ˇ τ ) L 2 (R) + a1 φ1 (·, cτ ) L 2 (R) + (a0 (τ ) − a0 (−T (δ)))φ0 (·, cτ ) L 2 (R) ≤ C|τ |−3/2 + C|τ |−3/4 + C|τ |−3/4 ≤ C|τ |−3/4 . Thus (5.19) is proved.
(5.61)
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To obtain (5.20) we use (5.34), which is valid also for −T ≤ τ ≤ −T (δ) = −T + Cδ T −δ , to obtain a0 (−T + Cδ T −δ ) − a0 (−T ) ≤ C T −δ−1/4 . Hence, (5.20) follows for δ ≥ 1/4. Note that the proposition follows for all δ < 1/2, since by decreasing δ we impose a weaker constraint on the τ domain where (5.19) is valid. We conclude this subsection by stating the asymptotic behaviour of u in the limit τ → +∞. Proposition 5.4. Given T > 0 there exist positive numbers T2 and C such that, for any v ∈ C(T ) and τ > T2 , there exists C0 such that τ v exp −λ(τ + T ) + E 0 (cs) ds − C0 φ0 (·, cτ ) ≤ C|T2 |−5/2 . (5.62) −T
L 2 (R )
Proof. We define w once again according to (5.23) to obtain (5.25). Set again a0 (τ ) = φ0 (·, cτ ), w(·, τ ) . Then, following the steps of the proof of Proposition 5.3 we obtain the inequality da0 ˜ τ ) L 2 (R) g(cτ ), i = 1, 2, (5.63) dτ (τ ) ≤ c w(·, where w(x, ˜ τ ) = w(x, τ ) − a0 (τ )φ0 (x, cτ ), and g is given by (5.5). Furthermore, we obtain that d w(·, ˜ τ ) L 2 (R) + 1 w(·, ˜ τ ) L 2 (R) ≤ g(cτ ). dτ By (5.64) and (5.6e) we obtain that w(·, ˜ τ ) L 2 (R) ≤
C . τ2
Hence, by (5.63) we obtain that −5/2
|a0 (τ ) − a0 (T2 )| ≤ C T2
.
Let C0 = a0 (T2 ) and write w(x, τ ) = w(x, ˜ τ ) + C0 φ0 (x, cτ ) + (a0 (τ ) − a0 (T2 ))φ0 (x, cτ ). An integration yields (5.62).
(5.64)
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5.3. Lower bounds for the resolvent. We now provide a lower bound for the norm of the resolvent of the operator A defined in (2.4). Proposition 5.5. Let θ=
4γ , 4γ + 3
(5.65)
where γ is given by (5.21). Then there exists a positive constant μ1 such that, for all μ > μ1 and θ˜ > θ , 3 μ −1 1+2θ˜ , (5.66) − 2μ (A − μ) ≥ exp 3c where c is the constant appearing in the definition of A0,c . Proof. To prove (5.66) we will construct a pair (u, f ) with u ∈ D(A) such that (A − μ)u = f,
(5.67)
f L 2 (R2 ) = 1,
(5.68)
where
and the norm of u will be estimated from below. To simplify notation we set c = 1. For general values of c the proof is almost identical. Step 1. We continue by applying a partial Fourier transform u(x, y) → u(x, ˆ ω) to (5.67) to obtain − μ)uˆ = fˆ, (A
(5.69)
in which uˆ and fˆ are the respective partial Fourier transforms of u and f and = ∂ω + Lω , A
(5.70)
where Lω is the operator in (3.1) with β replaced by ω. Choose f such that (1) fˆ(x, ω) = μ1/4 1 I (ω)φ0 (x, ω),
where I is the interval
1 μ2 1 μ2 − √ ,− + √ , I = − 2 2 μ 2 2 μ
(5.71)
(5.72)
1 I (ω) is the characteristic function of the interval I , and φ0(1) (·, ω) is the eigenfunction (1) of the operator Lω associated to E 0 (ω). It is immediately seen, using the Fubini Theorem, that the condition (5.68) is satisfied. Since A has empty spectrum in L 2 (R2 , C) (see cannot have a non-empty spectrum in L 2 (R2 , C), and hence (A − μ) Corollary 2.6), A is invertible. Therefore (5.69) has a unique solution u. ˆ
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Step 2. We next consider (5.69) locally as an evolution equation associated to the ωdependent unbounded operator (Lω − μ), where the time is the parameter ω. To find the solution we follow Duhamel’s principle for solving non-homogeneous equations. We thus introduce an additional parameter ξ ∈ I and consider the following problem: $ # ∂ω vξ (·, ω) = − (Lω − μ)vξ (·, ω) for ω ≥ ξ, (5.73) (1) vξ (·, ξ ) = φ0 (·, ξ ). This problem has a unique solution vξ which is well defined for ω > ξ using Kato’s theorem (as recalled in Remark 5.2). We extend vξ into the region ω < ξ by letting vξ (·, ω) = 0
for ω < ξ.
Once vξ is obtained, we claim that the following equality holds: u(x, ˆ ω) = μ1/4 vξ (x, ω) dξ.
(5.74)
I
We can indeed verify that the function in the right side of (5.74) belongs to L 2 (R2 ), and is a distribution solution of (5.69). ˆ For simplicity we denote the L 2 (R, C) Step 3. We now estimate the L 2 (R2 , C)-norm of u. norm by · 2 . We set μ to be so large that we can make use of Proposition 5.3 for any ξ ∈ I . We first state the obvious identity (recall that we have taken c = 1) γ ( ω (1) (1) 1/4 ω vξ (x, ω) = μ φ0 (x, ξ )eμ(ω−ξ ) e− ξ E 0 (τ ) dτ ξ γ ( ω (1) (1) −μ(ω−ξ ) ξ E 0 (τ ) dτ 1/4 ω + vξ (x, ω)e e − μ φ0 (x, ξ ) eμ(ω−ξ ) ξ × e− Hence
(ω ξ
(1)
E 0 (τ ) dτ
.
(5.75)
vξ (·, ω) dξ I 2γ ( ω (1) ω (1) 1/4 μ(ω−ξ ) − ξ E 0 (τ ) dτ ≥ μ φ0 (·, ξ )e e dξ I ξ 2 γ ( ω (1) ω (1) E (τ ) dτ −μ(ω−ξ ) 1/4 0 ξ v − (·, ω)e e − μ φ (·, ξ ) ξ ξ 0 I ( ω (1) ×eμ(ω−ξ ) e− ξ E 0 (τ ) dτ dξ .
(5.76)
2
Using (5.76) and (5.19) we obtain that, if μ ≥ 1, then for all ω > −μ2 /2 + 1 (hence ω ≥ ξ for all ξ ∈ I ), we have γ ( ω (1) vξ (·, ω) dξ ≥ ω φ (1) (·, ξ )eμ(ω−ξ ) e− ξ E 0 (τ ) dτ dξ ξ 0 I I 2 2 ( ω (1) C2 μ(ω−ξ ) − ξ E 0 (τ ) dτ − 3/4 e e dξ. |ω| I
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We now observe that by (3.5) and the expression of h k (x) that (1) (1) φ0 (x, ξ1 )φ0 (x, ξ2 ) d x ≥ C3 > 0, inf (ξ1 ,ξ2 )∈I ×I
R
(5.77)
where C3 is independent of μ. Consequently, for all ω > −μ2 /2 + 1 we have γ ( ω (1) ω μ(ω−ξ ) − ξ E 0 (τ ) dτ vξ (·, ω) dξ ≥ C4 − 1 e e dξ. (5.78) ξ |ω|3/4 I I 2 Step 4. This step involves an estimate of the right-hand side in (5.78). Let θ be the number defined in (5.65), and define μ2 I˜ = − + 1, −μ2θ . 2 If μ > 2 and ξ ∈ I , then |ξ | ≤
μ2 1 2μ2 + √ ≤ . 2 2 μ 3
Hence for all ω ∈ I˜ we have γ γ ω ≥ 3 μ−2γ (1−θ) , ξ 2
1 ≤ μ−3θ/2 . |ω|3/4
By our choice of θ we have 2γ (1 − θ ) =
3θ 6γ 3 = ≤ . 2 4γ + 3 2
Hence for μ > 2, ξ ∈ I and ω ∈ I˜ we have γ ω − 1 ≥ d , ξ |ω|3/4 μ3/2 where
(5.79)
γ 3 − 1 > 0. d= 2
Consequently for μ > 2 and ω ∈ I˜ we have by (5.78) and (5.79) that ( ω (1) μ(ω−ξ ) − ξ E 0 (τ ) dτ vξ (·, ω) dξ ≥ C5 e e dξ μ3/2 I I 2 ω C5 (1) = 3/2 exp [μ − E 0 (τ )] dτ dξ, μ I ξ where C5 = C4 d. Note that by (3.3) and (3.4), C6 (1) sup E 0 (ξ ) − μ ≤ 3/2 . μ ξ ∈I
(5.80)
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Set z(μ) = − Then
ω ξ
[μ −
(1) E 0 (τ )] dτ
−2
≥ −C6 μ
In view of (3.3) we have ω (1) E 0 (τ ) dτ = z(μ)
Hence
exp I
z(μ)
√ ω μ μ3 (1) − − + μω + E 0 (τ ) dτ. 2 2 z(μ)
√ 1 −2τ + O( ) dτ |τ |
√ z(μ) 2 2 3/2 3/2 . (|z(μ)| − |ω| ) + O log = 3 ω
ω
μ2 1 + √ . 2 2 μ
ξ
ω
[μ − E 0(1) (τ )] dτ
dξ
√ √ μ 2 2 μ3 1 −2 − − (|z(μ)|3/2 − |ω|3/2 ) ≥ 1/2 exp − C6 μ + μω + μ 2 2 3 z(μ) . (5.81) − O log ω
Note that for ω ∈ I˜ we have
z(μ) 2(1−θ) . ω ≤μ
Therefore we plug (5.81) into (5.80) and find that there exists a positive constant C7 such that, for sufficiently large μ and for all ω ∈ I˜, % √ 3 μ 2 2 C 7 3/2 vξ (·, ω) dξ ≥ + |ω| + μω . exp (5.82) μ2 6 3 I 2 Step 5. Now we recall (see (5.74)) that u 2L 2 (R2 )
=
2 =μ vξ (·, ω) dξ dω R I 2 2 vξ (·, ω) dξ dω.
u ˆ 2L 2 (R2 ) x,ω
≥ μ1/2
ω∈ I˜
1/2
I
2
It follows from (5.82) and (5.83) that % √ −μ2θ C72 μ3 4 2 3/2 2 + |ω| + 2μω dω exp u L 2 (R2 ,C) ≥ 7/2 2 μ 3 3 − μ2 +1 % √ μ2 −1 2 C72 μ3 4 2 3/2 + s − 2μs ds. = 7/2 exp μ 3 3 μ2θ
(5.83)
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If μ is sufficiently large (for instance μ ≥ 23/(2(1−θ)) ), then for any s ∈ [μ2θ , μ2 − 1], √ we have μ − 2s > 0, and √ √ √ μ − 2s 2μ − 2 2s μ − 2s . 1= ≥ √ ≥ 2μ μ − 2s μ − 2μ2θ 2
Hence for large μ we have u 2L 2 (R2 ,C)
% √ μ2 −1 √ 2 C72 μ3 4 2 3/2 + s − 2μs (2μ − 2 2s)ds ≥ 9/2 exp μ 3 3 μ2θ 3 μ ˜ − 2μ1+2θ , ≥ exp 3
for all θ˜ > θ . Hence (5.66) is true for sufficiently large μ.
5.4. Estimates for dense subspaces. Let A be the operator defined in (2.4). In view of the different asymptotic behaviors (3.3) for β → −∞ and (3.6) for β → +∞, one may expect the asymptotic dependence on μ of the norm (A − λ)−1 f L 2 (R2 ) for fixed f to be different from that of the norm of the resolvent (A − λ)−1 . This is indeed true. We now derive this different dependence if some additional conditions on the support of fˆ are assumed. As an example consider, for a ≥ 0, the space & ' (5.84) L a2 (R2 , C) = u ∈ L 2 (R2 , C) : Supp uˆ ⊂ {(x, ω) : ω ≤ a} , where, as above, we denote by uˆ the partial Fourier transform with respect to the y variable of u. Denote by a the orthogonal projector a : L 2 (R2 , C) → L a2 (R2 , C). The next theorem shows that exp(−tA) ◦ a and (A − λ)−1 ◦ a have a different respective behavior as t → +∞ or λ → +∞, than that of exp(−tA) and (A − λ)−1 . Theorem 5.6. For any a ≥ 0, there exists a constant T0 (a) such that, for any t ≥ T0 (a) we have a (ct) , (5.85) exp(−tA) ◦ a ≤ exp − c where a (t) = inf (ω, t), ω≤a
(5.86)
where is given by (4.5). In particular, for any a ≥ 0, there exists C(a) > 0, such that for t ≥ T0 (a) we have 2 c 3 3/2 . (5.87) exp(−tA) ◦ a ≤ C(a) exp − t + C(a)t 3
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The proof is the same as that for Theorem 4.2 and Lemma 4.4 except that now we take into consideration the information on Supp u. ˆ Clearly, (4.3) is still valid, but as uˆ ≡ 0 (ct). To for ω ≥ a we obtain an estimate similar to (4.10) but with a (ct) instead of prove (5.87) we note that, for sufficiently large t we have a (t) = (a, t), and hence (5.87) easily follows from (3.3). As in Subsect. 4.1, Theorem 5.6 implies: Corollary 5.7. For any a ≥ 0, there exists λ0 (a) and C(a), such that, for λ ≥ λ0 (a), 2 3/2 −1 3/4 . (5.88) (A − λ) ◦ a ≤ exp λ + C(a)λ 3c Observing that the subspace V =
)
L a2 (R2 , C)
a≥0
is dense in L 2 (R2 , C), we also obtain Corollary 5.8. There exists a dense set V in L 2 (R2 , C), such that for each f ∈ V there exist positive constants α = α( f ) and μ0 = μ0 ( f ) such that for all λ with Re λ = μ > μ0 ( f ), (A − λ)−1 f L 2 (R2 ) ≤ exp(αμ3/2 ) f L 2 (R2 ) holds.
(5.89)
Acknowledgements. Y. Almog was supported by NSF grant DMS 0604467. The work of X. B. Pan was partially supported by the National Natural Science Foundation of China grant no. 10871071, and the National Basic Research Program of China grant no. 2006CB805902.
Appendix A. Proof of (5.12) By (3.5) we may write φk = φk,0 + |β|−3/4 φk,1 , k = 1, 2, in which φk,1 2 ≤ C, where C is independent of β. Note that since φk 2 = 1 and φk,0 2 = 1 + O(e−Sβ ) for some S > 0, we have 1 = φk,0 + |β|−3/4 φk,1 22 = 1 + 2|β|−3/4 φk,0 , φk,1 + |β|−3/2 φk,1 22 + O(e−Sβ ), which leads to | φk,0 , φk,1 | ≤
C , k = 1, 2. |β|3/4
(A.1)
Furthermore, by (3.5) and (5.8) we have that (x 2 + 2β)φk 2 ≤ C|β|1/4 , k = 1, 2.
(A.2)
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We can then write (x 2 + 2β)φ0 22 = (x 2 + 2β)φ0,0 22 + 2|β|−3/4 (x 2 + 2β)φ0,0 , (x 2 + 2β)φ0,1 + O(|β|−1 ), (A.3a) | (x 2 + 2β)φ0 , φ0 |2 = | (x 2 + 2β)φ0,0 , φ0,0 |2 + 4|β|−3/4 (x 2 + 2β)φ0,0 , φ0,0 (x 2 + 2β)φ0,0 , φ0,1 + O(|β|−1 ),
(A.3b)
and | (x 2 + 2β)φ0 , φ1 |2 = | (x 2 + 2β)φ0,0 , φ1,0 |2 + 2|β|−3/4 (x 2 + 2β)φ0,0 , φ1,1 + (x 2 + 2β)φ1,0 , φ0,1 × (x 2 + 2β)φ0,0 , φ1,0 + O(|β|−1 ).
(A.3c)
Furthermore, orthogonality properties of Hermite functions yield C 2 (x + 2β)φ0,0 22 − | (x 2 + 2β)φ0,0 , φ0,0 |2 − | (x 2 + 2β)φ1,0 , φ0,0 |2 ≤ . β We thus obtain from (A.3) that (x 2 + 2β)φ0 22 − | (x 2 + 2β)φ0 , φ0 |2 − | (x 2 + 2β)φ1 , φ0 |2 & = |β|−3/4 2 (x 2 + 2β)φ0,0 , (x 2 + 2β)φ0,1 −4 (x 2 + 2β)φ0,0 , φ0,0 (x 2 + 2β)φ0,0 , φ0,1 ' − 2 (x 2 + 2β)φ0,0 , φ1,1 + (x 2 + 2β)φ1,0 , φ0,1 × (x 2 + 2β)φ0,0 , φ1,0 + O(|β|−1 ).
(A.4)
By the standard theory of orthogonal polynomials and Hermite functions in particular (see [1] for instance) we have that 2 (A.5) (x + 2β)φ0,0 , φ0,0 ≤ C|β|−1/2 . Furthermore, 2 (x + 2β)φ0,0 , φ1,1 ≤ |2β|1/4 φ1,0 , φ1,1 + (x 2 + 2β)φ0,0 − |2β|1/4 φ1,0 2 φ1,1 2 . By (A.1) and by evaluating the integral of Hermite functions on the right-hand-side of the above inequality we obtain 2 (x + 2β)φ0,0 , φ1,1 ≤ C|β|−1/2 . Substituting the above together with (A.5) into (A.4) yields (x 2 + 2β)φ0 22 − | (x 2 + 2β)φ0 , φ0 |2 − | (x 2 + 2β)φ1 , φ0 |2 & = |β|−3/4 2 (x 2 + 2β)φ0,0 , (x 2 + 2β)φ0,1 − 2 (x 2 + 2β)φ1,0 , φ0,1 ' × (x 2 + 2β)φ0,0 , φ1,0 + O(|β|−1 ). (A.6)
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To complete the proof we expand (x 2 + 2β)φ0,0 into a series of Hermite functions to obtain (x 2 + 2β)φ0,0 , (x 2 + 2β)φ0,1 = (x 2 + 2β)φ0,0 , φ0,0 (x 2 + 2β)φ0,0 , (x 2 + 2β)φ0,1 + (x 2 + 2β)φ0,0 , φ1,0 (x 2 + 2β)φ0,0 , (x 2 + 2β)φ0,1 + O(β −1/4 ). Substituting into (A.6) we obtain (5.12) with the aid of (A.5). References 1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. New York: Dover, 1972 2. Agmon, S.: Lectures on Elliptic Boundary Value Problems. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr., Van Nostrand Mathematical Studies, No. 2, Princeton, NJ-Toronto-London: D. Van Nostrand Co., Inc., 1965 3. Almog, Y.: The stability of the normal state of superconductors in the presence of electric currents. SIAM J. Math. Anal. 40, 824–850 (2008) 4. Bauman, P., Jadallah, H., Phillips, D.: Classical solutions to the time-dependent Ginzburg-Landau equations for a bounded superconducting body in a vacuum. J. Math. Phys. 46, 095104, (2005) 5. Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. International Series in Pure and Applied Mathematics, New York: McGraw-Hill Book Co., 1978 6. Chapman, S.J., Howison, S.D., Ockendon, J.R.: Macroscopic models for superconductivity. SIAM Review 34, 529–560 (1992) 7. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry. Texts and Monographs in Physics, Study ed., Berlin: Springer-Verlag, 1987 8. Davies, E.B.: Linear Operators and Their Spectra. Cambridge Studies in Advanced Mathematics, 106 Cambridge: Cambridge University Press, 2007 9. Dolgert, A., Blum, T., Dorsey, A., Fowler, M.: Nucleation and growth of the superconducting phase in the presence of a current. Phys. Rev. B 57, 5432–5443 (1998) 10. Du, Q., Gunzburger, M.D., Peterson, J.S.: Analysis and approximation of the Ginzburg-Landau model of superconductivity. SIAM Rev. 34, 54–81 (1992) 11. Feynman, R.P.: Forces in molecules. Phys. Rev. 56, 340–343 (1939) 12. Fournais, S., Helffer, B.: Spectral Methods in Surface Superconductivity. Basel-Boston: Birkhäuser, 2010 13. Harrell, E.M.: On the rate of asymptotic eigenvalue degeneracy. Commun. Math. Phys. 60, 73–95 (1978) 14. Harrell, E.M.: Double wells. Commun. Math. Phys. 75, 239–261 (1980) 15. Helffer, B.: The Montgomery model revisited. Colloq. Math. 118, 391–400 (2010) 16. Helffer, B.: Semi-Classical Analysis for the Schrödinger Operator and Applications. Lecture Notes in Mathematics, 1336, Berlin-Heidelberg-New York: Springer-Verlag, 1988 17. Helffer, B., Mohamed, A.: Caractérisation du spectre essentiel de l’opérateur de Schrödinger avec un champ magnétique. Ann. Inst. Fourier (Grenoble) 38, 95–112 (1988) 18. Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Lectures Notes in Mathematics 1862, Berlin-Heidelberg-New York: Springer, 2005 19. Helffer, B., Nourrigat, J.: Hypoellipticité Maximale pour des Opérateurs Polynômes de Champs de Vecteurs. Basel-Boston: Birkhäuser, 1985 20. Ivlev, B.I., Kopnin, N.B.: Electric currents and resistive states in thin superconductors. Adv. in Phys. 33, 47–114 (1984) 21. Martinet, J.: Sur les Propriétés Spectrales d’Opérateurs Non auto-adjoints Provenant de la Mécanique des Fluides. PHD Thesis, University of Paris-Sud, 2009 22. Montgomery, R.: Hearing the zero locus of a magnetic field. Commun. Math. Phys. 168, 651–675 (1995) 23. Murray, J.D.: Asymptotic Analysis. Springer, 1984 24. Pan, X.-B., Kwek, K.-H.: Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains. Trans. Amer. Math. Soc. 354, 4201–4227 (2002) 25. Rockafellar, R.T.: Convex Analysis. Princeton Mathematical Series, No. 28, Princeton, NJ: Princeton University Press, 1970 26. Simon, B.: Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38, 295–308 (1983) 27. Sivakov, A.G., Glukhov, A.M., Omelyanchouk, A.N., Koval, Y., Müller, P., Ustinov, A.V.: Josephson behavior of phase-slip lines in wide superconducting strips. Phys. Rev. Lett. 91, 267001 (2003)
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28. Tinkham, M.: Introduction to Superconductivity, New York: McGraw-Hill, 1996 29. Vodolazov, D.Y., Peeters, F.M., Piraux, L., Matefi-Tempfli, S., Michotte, S.: Current-voltage characteristics of quasi-one-dimensional superconductors: An s-shaped curve in the constant voltage regime. Phys. Rev. Lett. 91, 157001 (2003) 30. Yosida, K.: Functional Analysis. Classics in Mathematics, Berlin: Springer-Verlag, 1980 Communicated by P. Constantin
Commun. Math. Phys. 300, 185–204 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1115-7
Communications in
Mathematical Physics
Yang-Mills Flows on Nearly Kähler Manifolds and G 2 -Instantons Derek Harland1 , Tatiana A. Ivanova2 , Olaf Lechtenfeld1 , Alexander D. Popov2 1 Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany.
E-mail:
[email protected];
[email protected];
[email protected]
2 Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia.
E-mail:
[email protected];
[email protected] Received: 23 September 2009 / Accepted: 8 April 2010 Published online: 21 August 2010 – © Springer-Verlag 2010
Abstract: We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H , R × G/H and R2 × G/H , where G/H is a compact nearly Kähler sixdimensional homogeneous space, and the manifolds R × G/H and R2 × G/H carry G 2 - and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on R×G/H is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G 2 -structures on R × G/H . It is shown that both G 2 -instanton equations can be obtained from a single Spin(7)-instanton equation on R2 × G/H . 1. Introduction and Summary The Yang-Mills equations in two, three and four dimensions have been intensively studied both in physics and mathematics. In mathematics, this study (i.e. projectively flat unitary connections and stable bundles in d = 2 [1], the Chern-Simons model and knot theory in d = 3, instantons and Donaldson invariants [2] in d = 4) has yielded a lot of new results in differential and algebraic geometry. In particular, a crucial role in d = 4 gauge theory is played by the first-order anti-self-duality equations, which on manifolds R × X 3 are precisely the Chern-Simons gradient flow equations. The program of extending familiar constructions in gauge theory, associated to problems in lowdimensional topology, to higher dimensions, was proposed in [3] and developed in [4–11].1 An important role in this investigation is played by first-order gauge equations which are a generalisation of the anti-self-duality equations in d = 4 to higher-dimensional manifolds with special holonomy (or, more generally, with G-structure [12,13]). Such equations in d > 4 dimensions were first introduced in [14] and further considered e.g. in [6,7,11,15–23]. Some of their solutions were found e.g. in [24–31]. 1 For more literature see references therein.
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In physics, interest in Yang-Mills theories in dimensions greater than four grew essentially after the discovery of superstring theory, which contains supersymmetric Yang-Mills in the low-energy limit in the presence of D-branes as well as in the heterotic case. In particular, heterotic strings yield d = 10 heterotic supergravity, which contains the N = 1 supersymmetric Yang-Mills model as a subsector [32]. Supersymmetry-preserving compactifications on spacetimes M10−d × X d with further reduction to M10−d impose the above-mentioned first-order BPS-type gauge equations on X d [14,32]. Initial choices for the internal manifold X 6 were Kähler coset spaces and Calabi-Yau manifolds, as well as manifolds with an exceptional holonomy group G 2 for d = 7 and Spin(7) for d = 8. However, it was realised that Calabi-Yau compactifications suffer from the presence of many massless moduli fields in the resulting four-dimensional effective theories.2 This problem can be cured (at least partially) by allowing for non-trivial p-form fluxes on X d . String vacua with p-form fields along the extra dimensions (‘flux compactifications’) have been intensively studied in recent years (see e.g. [33–35] for reviews, and also the references therein). Compactifications in the presence of fluxes can be described in the language of G-structures on d-dimensional manifolds X d : SU(3)-structure for dimension d = 6, G 2 -structure for d = 7 and Spin(7)-structure for d = 8. In the definition of all these G-structures there enters a (d−4)-form on X d . Thus, we deal with internal manifolds of special geometry and consider the three-form field H = ∗d as torsion, where ∗ denotes the Hodge star operator. In particular, in six dimensions these manifolds may be non-Kähler and sometimes even non-complex. Flux compactifications have been investigated primarily for type II strings and to a lesser extent in the heterotic theories, despite their long history [36–39]. The number of torsionful geometries that can serve as a background for heterotic string compactifications seems rather limited. Among them there are six-dimensional nilmanifolds, solvmanifolds, nearly Kähler and nearly Calabi-Yau coset spaces. The last two kinds of manifolds carry a natural almost complex structure which is not integrable (for a discussion of their geometry see e.g. [40–47] and references therein). In heterotic string compactifications one has the freedom to choose a gauge bundle since the simple embedding of the spin connection into the gauge connection is ruled out for compactifications with dH = 0. For the torsionful backgrounds, the allowed gauge bundle is restricted by the Bianchi identity for the torsion field (anomaly cancellation) and by the Donaldson-Uhlenbeck-Yau equations [16–19] for d = 6 or the G 2 -instanton equations [3] for d = 7. The construction of such vector bundles over G 2 -manifolds of topology R × X 6 is the subject of the present paper. The only known examples of compact nearly Kähler six-manifolds are the four coset spaces SU(3)/U(1)×U(1), Sp(2)/Sp(1)×U(1), G 2 /SU(3) = S 6 and SU(2)3 /SU(2) = S 3 × S 3 . On all four cosets G/H we have a torsion H = ∗dω for an almost Kähler form ω. We describe some solutions of the Donaldson-Uhlenbeck-Yau equations for the gauge group G on these cosets. Our ansatz for a G-invariant connection is parameterised by a complex number φ, and the solutions show the 3-symmetry characteristic of all nearly Kähler spaces. Next, we step up to seven dimensions, extending G/H by a real line Rτ , so that φ → φ(τ ) ∈ C in our G-invariant ansatz. For the torsion H = − 13 κ1 ∗(dτ ∧dω)+ 13 κ2 dω with κ1 , κ2 ∈ R, our ansatz reduces the Yang-Mills equations to Newton’s equations φ¨ = f (φ) for a particle in the complex φ plane, subject to a 3-symmetric cubic force f . For κ2 = 0, there exists a potential of φ 4 type, so f ∼ ∂∂Vφ¯ , and an action can be 2 Kähler cosets also lead to non-realistic effective theories.
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formulated, which surprisingly agrees with the torsionful Yang-Mills action on our ansatz. Yet, even for κ2 = 0, we construct an explicit solution. In special instances, φ¨ ∼ ∂∂Vφ¯ is implied by a flow equation φ˙ ∼ ∂∂W . This flow is gradiφ¯ ent or hamiltonian, depending on whether the proportionality is real or imaginary. Among the complex φ(τ ) trajectories, finite-action kinks occur when (κ1 , κ2 ) = (±3, 0) and (−1, 0), as solutions to the gradient and hamiltonian flow, respectively. The corresponding connections are finite-action solutions of the Yang-Mills equations on R × G/H . By a duality transformation, which relates solutions for different values of κ1 , infinite-action solutions to the Yang-Mills equations are presented as well. The cases (κ1 , κ2 ) = (3, 0) and (−1, 0) mentioned above have a clear geometrical meaning. The corresponding gradient and hamiltonian flow equations for φ follow from seven-dimensional anti-self-duality conditions based on one of two G 2 -structures, called the G 2 -instanton equations. Now, these both descend from anti-self-duality equations based on the Spin(7)-structure of the eight-dimensional space Rτ × Rσ × G/H . For the gradient case one reduces over Rσ , while the hamiltonian case arises upon reduction over Rτ . The G 2 -instanton equations can themselves be interpreted as gradient and hamiltonian flows for a certain action functional on the space of all connections. We do not know of a similar geometrical interpretation for any other special value of the torsion. 2. Nearly Kähler Coset Spaces 2.1. Basic definitions. An SU(3)-structure on a six-manifold is by definition a reduction of the structure group of the tangent bundle to SU(3). Manifolds of dimension six with SU(3)-structure admit a set of canonical objects fixed by SU(3), consisting of an almost complex structure J , a Riemannian metric g, a real two-form ω and a complex threeform . With respect to J , the forms ω and are of type (1,1) and (3,0), respectively, and there is a compatibility condition, g(J ·, ·) = ω(·, ·). With respect to the volume form Vg of g, ω and are normalised so that ω ∧ ω ∧ ω = 6Vg
and
¯ = −8iVg . ∧
(2.1)
A nearly Kähler six-manifold is an SU(3)-structure manifold such that dω = 3ρ Im
and
d = 2ρ ω ∧ ω
(2.2)
for some real non-zero constant ρ, proportional to the square of the scalar curvature (if ρ was zero, the manifold would be Calabi-Yau). Nearly Kähler manifolds were first studied by Gray [40], and they solve the Einstein equations with positive cosmological constant. More generally, six-manifolds with SU(3)-structure are classified by their intrinsic torsion, and nearly Kähler manifolds form one particular intrinsic torsion class. There are only four known examples of compact nearly Kähler six-manifolds, and they are all coset spaces: SU(3)/U(1) × U(1),
Sp(2)/Sp(1) × U(1),
G 2 /SU(3) = S 6 ,
SU(2)3 /SU(2) = S 3 × S 3 .
(2.3)
Here Sp(1) × U(1) is chosen to be a non-maximal subgroup of Sp(2): if elements of Sp(2) are written as 2 × 2 quaternionic matrices, then elements of Sp(1) × U(1) are written diag( p, q), with p ∈ Sp(1) and q ∈ U(1). Also, SU(2) is the diagonal subgroup
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of SU(2) × SU(2) × SU(2). These coset spaces G/H were named 3-symmetric by Wolf and Gray, because the subgroup H is the fixed point set of an automorphism s of G satisfying s 3 = Id [41,42,44]. The 3-symmetry actually plays a fundamental role in defining the canonical structures on the coset spaces. The automorphism s induces an automorphism S of the Lie algebra g of G, that is S : g → g is linear and satisfies [S X, SY ] = S[X, Y ]
∀X, Y ∈ g.
(2.4)
The cosets under consideration are all reductive, which means that there is a decomposition g = h ⊕ m, where h is the Lie algebra of H and m satisfies [h, m] ⊂ m. Actually, on SU(2)3 /SU(2) there is a choice of subspaces m; we choose m so that it is orthogonal to h with respect to the Cartan-Killing form in this case. The map S acts trivially on h and non-trivially on m; one can define a map J : m → m by √ S|m = − 21 + 23 J = exp 2π (2.5) 3 J . The map J satisfies J 2 = −1 and provides the almost complex structure on G/H . A natural quadratic form on m is given by the Cartan-Killing form of g, X, Y g = −Tr g(ad(X ) ◦ ad(Y )).
(2.6)
This extends to a G-invariant metric g on G/H . The (1,1)-form ω is fixed by its compatibility with g and J , and is the unique suitably normalised G-invariant (3,0)-form. 2.2. Lie algebra identities. In calculations, it is useful to choose a basis {I A } for the Lie algebra g. We do so in such a way that Ia for a = 1, . . . , 6 form a basis for m and Ii for C are defined by i = 7, . . . , dim(G) yield a basis for h. The structure constants f AB C [I A , I B ] = f AB IC
with
D C f AC f D B = δ AB ,
(2.7)
where we have chosen the basis so that it is orthonormal with respect to the Cartan-Killing D δ form. Then f ABC := f AB DC is totally antisymmetric. The reductive property of the coset means that the structure constants f ai j vanish. The components Jab of the almost complex structure J are defined via J (Ia ) = Jab Ib . Then the 3-symmetry property (2.4) implies useful identities involving J : notably, the tensor f˜abc := f abd Jdc
(2.8)
Jcd f adi = Jad f cdi .
(2.9)
Jab f abi = 0.
(2.10)
is totally antisymmetric; also
Another useful identity is
We do not have a general proof of this identity, but we have verified it on each of the four coset spaces. It has the following interpretation: the action of H on m defines an embedding of H in GL(6, R). It is easy to show that H fixes the quadratic form ·, · g and almost complex structure J ; hence H is contained in U(3) ⊂ GL(6, R). The above
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identity merely asserts that H ⊂ SU(3). Geometrically, this means that the natural H -structure on G/H is contained within the SU(3)-structure. Apart from ·, · g, there are two other natural quadratic forms on m: X, Y m := −Tr m(Pm ◦ ad(X ) ◦ Pm ◦ ad(Y )), X, Y h := −Tr h (Ph ◦ ad(X ) ◦ Pm ◦ ad(Y )),
(2.11) (2.12)
where Pm and Ph denote the projections onto m and h, respectively. It is easy to show that ·, · g = ·, · m + 2 ·, · h .
(2.13)
Furthermore, on the coset spaces in question, one also has ·, · m =
1 3
·, · g .
(2.14)
Hence, in terms of the structure constants, f aci f bci = f acd f bcd = 13 δab .
(2.15)
The proof of this identity will be deferred until the end of this section. Note that for three of the four coset spaces this identity has been verified directly in [48]. 2.3. Orthonormal frame for the coset. The metric and almost complex structure on m lift to a G-invariant metric and almost complex structure on G/H . Local expressions for these can be obtained by introducing an orthonormal frame as follows. The basis elements I A of the Lie algebra g can be represented by left-invariant vector fields Eˆ A on the Lie group G, and the dual basis eˆ A is a set of left-invariant one-forms. The space G/H consists of left cosets g H and the natural projection g → g H is denoted π : G → G/H . Over a contractible open subset U of G/H , one can choose a map L : U → G such that π ◦ L is the identity (in other words, L is a local section of the principal bundle G → G/H ). The pull-backs of eˆ A by L are denoted e A . In particular, ea form an orthonormal frame for T ∗ (G/H ) over U (where again a = 1, . . . 6), and we can write ei = eai ea with real functions eai . The dual frame for T (G/H ) will be denoted E a . The forms e A obey the Maurer-Cartan equations, dea = − f iba ei ∧ eb −
c 1 a b 2 f bc e ∧ e , i b i e ∧ ec − 21 f jk e j ∧ ek . dei = − 21 f bc
(2.16)
Since all the connections we consider will be invariant under some action of G, it will suffice to do calculations just over the subset U . Local expressions for the G-invariant metric, almost complex structure, and nearly Kähler form on G/H are then g = δab ea eb ,
J = Jab ea E b ,
and
ω = 21 Jab ea ∧ eb .
(2.17)
One can also obtain a local expression for the (3,0)-form . From (2.16) one can compute dω and hence ∗dω: dω = − 21 f˜abc ea ∧ eb ∧ ec
and
∗ dω =
a 1 2 f abc e
∧ eb ∧ ec .
(2.18)
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We have that dω = 3ρ Im , and should be normalised so that Im 2 = 4.√ On the other hand, from (2.15) we compute that dω2 = 3. So it must be that ρ = 1/2 3 and Im = − √1 f˜abc ea ∧ eb ∧ ec ,
Re = − √1 f abc ea ∧ eb ∧ ec .
3
3
(2.19)
Given a pair of differential forms u, v such that the degree of u is less than or equal to the degree of v, their contraction is defined to be uv := ∗(u ∧ ∗v).
(2.20)
If u and v have the same degree, uv coincides with the usual inner product of forms induced by the metric. We are now in a position to prove (2.15): from (2.19), it is equivalent to g(u Re , v Re ) = 2g(u, v) ∀u, v ∈ 1 .
(2.21)
This identity holds on any 6-manifold with SU(3)-structure, as can be verified by direct calculation in an orthonormal basis. 3. Instantons in Six Dimensions 3.1. ω-anti-self-duality. Let be a (d−4)-form on a d-dimensional Riemannian manifold. A natural generalisation of the d = 4 anti-self-duality equations is the so-called -anti-self-duality equation, ∧ F = − ∗ F.
(3.1)
If is closed, this equation implies the Yang-Mills equation, D ∗ F = 0. Equations of this sort were first written down in [14], using the language of tensors rather than differential forms. They often have an interpretation as BPS equations, in particular all of the -anti-self-duality equations considered in this paper are BPS equations. On a nearly Kähler six-manifold, a natural choice for is the (1,1)-form ω, giving ω∧F =−∗F
⇔
∗(ω ∧ F) = −F.
(3.2)
Of course, ω is not closed, so (3.2) does not imply the Yang-Mills equation, but rather the Yang-Mills equation with torsion, D ∗ F + dω ∧ F = 0.
(3.3)
The ω-anti-self-duality equation (3.2) means that we are looking for eigen-two-forms F of the operator ∗(ω ∧ ·), with eigenvalue λ = −1. The space 2 of two-forms decomposes into three eigenspaces 2λ , with the following properties: λ dim 2λ F-type
2 1 ∼ω
1 6 (2, 0), (0, 2)
−1 8 (1, 1) ⊥ ω
(3.4)
Hence, (3.2) is equivalent to the so-called Donaldson-Uhlenbeck-Yau, or HermitianYang-Mills, equations [16–19]: F 0,2 = F 2,0 = 0
and
ω F = 0.
(3.5)
It is interesting to note that when F solves the ω-anti-self-duality equation (3.2), the torsional term in the Yang-Mills equation (3.3) vanishes, as was pointed out by Xu [49]. This is because F is a (1, 1)-form and dω is a sum of (3, 0)- and (0, 3)-forms: their wedge product then has to vanish.
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3.2. Gauge group H . First we consider H -instantons on G/H . The natural projection G → G/H defines a principal bundle with structure group H , on which G acts from the left. There is a unique G-invariant connection on this bundle, the so-called canonical connection [50–52]. On S 2 = SU(2)/U(1) the canonical connection is the Dirac monopole and on S 4 = Sp(2)/Sp(1)×Sp(1) it is the sum of an instanton and an anti-instanton, so it seems a good candidate solution to (3.2) on a nearly Kähler coset space. In local coordinates, the canonical connection is written A = ei Ii = ea eai Ii .
(3.6)
Its curvature F = dA + A ∧ A is given in [50], Chap. II, Theorem 11.1, and is also easily computed using (2.16): i a F = − 21 f ab e ∧ e b Ii .
(3.7)
The identity (2.9) implies that this F is a (1,1)-form, since (1,1)-forms θ are defined by the property θ (J ·, J ·) = θ (·, ·). The identity (2.10) tells us that ω F = 0, where ω = 21 Jab ea ∧ eb . So on each of the four nearly Kähler coset spaces, the canonical connection satisfies the Donaldson-Uhlenbeck-Yau equation (3.5), or equivalently the ω-anti-self-duality equation (3.2). The case G/H = G 2 /SU(3) was considered by Xu [49], who also showed that the canonical connection admits no continuous deformation preserving (3.2). In other words, this connected component of the moduli space of solutions consists of just a point. 3.3. Gauge group G. Next, we consider G-instantons on G/H . According to [50] and [53], G-invariant connections with gauge group G are determined by linear maps : m → g which commute with the adjoint action of H : (Ad(h)X ) = Ad(h) (X ), ∀h ∈ H, X ∈ m. Such a linear map is represented by a matrix (a B ), such that (Ia ) = a B I B , and in local coordinates the connection is written A = ei Ii + ea a B I B .
(3.8)
We make the simple choice ab = φ1 δab + φ2 Jab
and
ai = 0,
(3.9)
for real numbers φ1 and φ2 , which is more general than the choice considered in [31]. On the space G 2 /SU(3), (3.8) with (3.9) is the most general G 2 -invariant connection, but on the other coset spaces it is not – we will briefly discuss more general choices in the next section. The curvature F = 21 Fab ea ∧ eb is given in [50], Chap. II, Theorem 11.7, and can also be computed using (2.16): i Fab = f ac ( − Id)cb Ii + f abc (− + ( )2 )cd Id ,
(3.10)
where (Id, J ) = (Id, −J ). Note that F remembers the 3-symmetry S: → exp( 23 π J )
⇒
(− + ( )2 ) → exp( 23 π J )(− + ( )2 ). (3.11)
The two-forms ec ∗ dω =
a 3 2 f abc e
∧ eb
and
ec dω = − 23 f˜abc ea ∧ eb
(3.12)
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are clearly of type (2, 0) + (0, 2), since dω is of type (3, 0) + (0, 3). So this connection solves the ω-anti-self-duality equation (3.2) if and only if − + ( )2 = 0.
(3.13)
Apart from the canonical connection = 0, the other solutions to this equation are = Id, exp( 23 π J ), exp( 43 π J ).
(3.14)
Note that these connections in fact all have zero curvature. 4. Yang-Mills Equations in Seven Dimensions 4.1. From Yang-Mills theory to a 4 model. On a d-dimensional Riemannian manifold, the Yang-Mills equation with torsion is D ∗ F + ∗H ∧ F = 0,
(4.1)
with H a three-form. Equation (3.3) is a special case in d = 6. We will study solutions of this equation on the seven-dimensional manifolds R × G/H , with G/H a nearly Kähler coset space. We choose the metric and volume form, g7 = (e0 )2 + g6
and
V7 = e0 ∧ V6 ,
(4.2)
where e0 = dτ and τ is a coordinate on R, while g6 and V6 are the metric and volume form on G/H . For H we make the choice ∗ H = − 13 κ1 dτ ∧ dω + 13 κ2 ∗ dω.
(4.3)
This choice for H is clearly invariant under the action of G, and under translations in and reversals of τ – in fact, it is the most general possible choice satisfying these conditions. If one does not require invariance under τ -reversals, then a term proportional to dτ ∧ ω could be added to H and possibly others, depending on the choice of coset space. For the connection one-form A = A0 e0 + Aa ea , we copy from the previous section the G-invariant ansatz (3.8) and (3.9), A(τ ) = ei Ii + ea ab (τ ) Ib with = φ1 Id + φ2 J,
(4.4)
where φ1 and φ2 are now functions of τ . This ansatz has A0 = 0, but no generality is lost here since such a gauge can always be chosen. The curvature F = F0a e0 ∧ ea + 1 a b 2 Fab e ∧ e of this connection has the components (see (3.10)) i ( − Id)cb Ii + f abc (− + ( )2 )cd Id Fab = f ac
and
˙ ab Ib , F0a = (4.5)
where a dot denotes a derivative with respect to τ . In order to write the Yang-Mills equation in components, it is necessary to introduce the torsionful spin connection on G/H [53]. Recall that a linear connection is a matrix of a . The connection is metric compatible if ωc g is anti-symmetric, one-forms ωba = ec ωcb a cb a eb ∧ ec determined by the structure and its torsion is a vector of two-forms T a = 21 Tbc equation dea + ωba ∧ eb = T a .
(4.6)
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Our choice is T a = −ea H
⇔
a Tbc = κad f dbc
with
κ := κ1 Id − κ2 J.
(4.7)
We take ωba to be the unique metric-compatible linear connection with this torsion. Explicitly, a ωcb = eci f iba + 21 (κ + Id)ad f dcb .
(4.8)
The torsionful spin connection on R×G/H is given by a similar formula, with additional components vanishing: 0 a 0 = ω0b = ωcb = 0. ω0b
(4.9)
Using (4.6), one can show that the Yang-Mills equation with torsion (4.1) is equivalent to a F b0 + [Aa , F a0 ] = 0, E a F a0 + ωab
E0 F
0b
+ Ea F
ab
d + ωda F ab
b + ωcd F cd
+ [Aa , F ] = 0. ab
(4.10) (4.11)
It is now a matter of computation to substitute the ansatz (4.4) into (4.10) and (4.11), making use of structure constant identities introduced above. One finds that (4.10) is identically satisfied, while (4.11) is equivalent to ¨ = (κ−1) − (κ +3)( )2 + 4 2 . 6
(4.12)
For more general choices of connection, Eq. (4.10) is not automatically solved. For example, in the cases G/H = SU(3)/U(1) × U(1) and SU(2)3 /SU(2), the most general G-invariant connection is parametrised by three complex scalars ψ1 , ψ2 , ψ3 .3 Equation (4.10) then reads ψ˙ 1 ψ¯ 1 − ψ¯˙ 1 ψ1 = ψ˙ 2 ψ¯ 2 − ψ˙¯ 2 ψ2 = ψ˙ 3 ψ¯ 3 − ψ˙¯ 3 ψ3 .
(4.13)
The general solution to these equations is difficult to find, but one obvious solution is ψ1 = ψ2 = ψ3 . This returns us to our original ansatz (4.4). More general ansätze will be discussed in a future publication. 4.2. Action. There is an alternative method of deriving (4.12), which involves working with actions rather than equations of motion. Notice that, with κ2 = 0, the Yang-Mills equation with torsion (4.1) is the equation of motion for the action S= Tr F ∧ ∗F + 13 κ1 dτ ∧ ω ∧ F ∧ F . (4.14) R×G/H
In the Calabi-Yau (ρ = 0) limit this action agrees with the standard Yang-Mills action up to a boundary term. Substituting the ansatz (4.4) into this action gives () ˙ +V ˙ S = Vol(G/H ) dτ Tr 2 with (4.15) R
() = (1− κ1 ) Id + (κ1 −1) − (1+ κ1 ) 3 + ( )3 + 2 ( )2 . (4.16) 3V 3 3 3 For G/H = Sp(2)/Sp(1) × U(1) one has ψ = ψ ; for S 6 one has ψ = ψ = ψ = φ. 1 2 1 2 3
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The Euler-Lagrange equation for this integral is again (4.12). The matrix-valued poten is invariant under the action of the S3 permutation group generated by the 3-symtial V metry and the conjugation. Let us review what we have done. Starting from the action (4.14), we first substituted an ansatz and then derived an equation of motion. Previously, we derived the equation of motion (4.1) for the action and then substituted the ansatz. There is no reason to expect these two procedures to lead to the same differential equation, unless the ansatz chosen is the most general ansatz invariant under a given symmetry (this is called the principle of symmetry criticality [54]). In our case, the two procedures did lead to the same differential equation. However, our ansatz, while G-invariant, is certainly not the most general G-invariant ansatz, except in the case of S 6 . That the two procedures lead to the same differential equation on the other coset spaces could perhaps be attributed to the algebraic similarities between all four coset spaces. 5. Solutions of the Yang-Mills Equation 5.1. Critical points of the potential. Throughout this section, we identify the matrixvalued function = φ1 Id + φ2 J with the complex-valued function φ = φ1 + iφ2 and, () = V ( ) = V () in (4.16), likewise, interpret κ as a complex number. Since V we also define () =: V (φ) Id V
⇒
3 V (φ) = (1− κ31 ) + (κ1 −1) |φ|2 − (1+ κ31 ) 2 Re φ 3 + 2 |φ|4 . (5.1)
If κ2 = 0, the equation of motion (4.12) can be written in terms of the real function V , ¯ 2=3 6 φ¨ = (κ − 1) φ − (κ + 3)φ¯ 2 + 4 φφ
∂V . ∂ φ¯
(5.2)
This is the equation of motion of a particle moving in the complex plane under the influence of a potential −V . Equation (5.2) admits this mechanical interpretation only when κ2 = 0 since, if κ was complex, the potential function V could not be chosen real. In this section, we study solutions of (5.2) with κ2 = 0, using this mechanical analogy. We briefly discuss solutions of (4.12) with κ2 = 0 at the end of the section. Figure 1 displays some equipotential lines of V (φ) for two special cases. We are particularly interested in instantons, which correspond to particle trajectories interpolating between critical points of V , attained at τ = ±∞. By conservation of energy, such a trajectory can exist only if V |+∞ = V |−∞ . The critical points φ 0 of V along the real axis are φ0 V (φ 0 )
0 2 9 (1−2ν)
1 0
ν
2 3 9 (1+ν)(1 − ν)
with ν :=
1 4 (κ1 −1).
(5.3)
Since V is invariant under φ → exp( 23 π i)φ, for nonzero φ 0 there are further critical points exp( 23 π i)φ 0 and exp( 43 π i)φ 0 , degenerate in energy with φ 0 . At any value of κ1 , one may therefore search for trajectories connecting two critical points related by 3-symmetry, which we shall call “transverse”. We will show below that transverse trajectories exist for κ1 = −7, −1 (i.e. ν = −2, − 21 ). If κ1 = −3, 3, 9 (i.e. ν = −1, 21 , 2), two of the critical points on the real axis are degenerate in energy, and one may in addition look for “radial” trajectories connecting them.
Yang-Mills Flows on Nearly Kähler Manifolds and G 2 -Instantons
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
−0.5
−0.5
−1.0
−1.0
−1.5 −1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5 −1.5
195
−1.0
−0.5
0.0
0.5
1.0
1.5
Fig. 1. Contour plots of the potential V (φ) for κ = +3 (left) and for κ = −1 (right)
The sought-for instanton configurations have finite action only when V |±∞ = 0. Among the five special cases just mentioned, this occurs for κ1 = −3, −1, 3 (i.e. ν = −1, − 21 , 21 ). Finite-energy bounce solutions, connecting φ 0 = 0 to itself, may exist for κ1 < −3 and for 3 < κ1 < 5. κ1 −7 −3 −1 3 9 1 ν −2 −1 − 21 2 2 degeneration none V (exp(iα)) none V (0) = V (1) V (0) = V (2) instanton transverse radial transverse radial radial 1 1 φ 0 (±∞) exp(± 23 π i)(−2) exp(iα)(±1) exp(± 23 π i)(+1) ± 1±1 2 2 action infinite finite finite finite infinite
(5.4) 5.2. Duality. When κ2 = 0, there is a surprising duality that relates pairs of values of κ1 . This is best seen when the equation of motion (5.2) and the potential (5.1) are rewritten in terms of ν: ¯ 2 6 φ¨ = 4ν φ − 4(ν+1) φ¯ 2 + 4 φφ 3 V (φ) =
2 2 3 (1−2ν) + 4ν |φ|
It is straightforward to check that ( ν , φ(τ ) ) →
− 1 ν
and
3 8 3 (1+ν) Re φ
,
1 ν
φ( τν )
+ 2 |φ|4 .
(5.5)
(5.6)
maps solutions of (5.2) to other solutions. We do not know the origin of this duality. 5.3. Gradient flow. We discuss here the case (κ1 , κ2 ) = (3, 0). Our discussion applies also to (κ1 , κ2 ) = (9, 0), via the duality transformation. The potential V can be written in terms of a real “superpotential” W : ∂ W 2 3V = 2 with W = 13 (φ 3 + φ¯ 3 ) − |φ|2 . (5.7) ∂ φ¯
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2
1
0
−1
−2
−2
−1
0
1
2
Fig. 2. Contour plot of the superpotential W (φ) for κ = +3 (and for κ = −1)
So (5.2) is implied by the gradient flow equation ±
√ ∂W 3 φ˙ = φ¯ 2 − φ = . ∂ φ¯
Finite-action kink solutions are φ(τ ) =
1 2
√0 1 ± tanh τ −τ , 2 3
(5.8)
(5.9)
with τ0 being the collective coordinate. Further solutions are obtained by applying the 3-symmetry. Since W is 3-symmetric, it is clear that the gradient flow (which reduces the value of W along a path) does not have any transverse solutions. We have also found explicit infinite-action solutions of the gradient flow equations: if we define two real functions r (τ ) and ϕ(τ ) by the polar decomposition φ = r exp(iϕ), then (5.8) is equivalent to √ √ 3 r˙ = r − r 2 cos 3ϕ and 3 ϕ˙ = r sin 3ϕ. (5.10) It follows that dr = cosec 3ϕ − r cot 3ϕ, dϕ assuming ϕ˙ = 0. This can be integrated using a standard formula to give
r (ϕ) = − 13 (sin 3ϕ)−1/3 cos 3ϕ 2 F1 21 , 56 , 23 , cos2 3ϕ + C
(5.11)
(5.12)
for some real integration constant C. The hypergeometric function 2 F1 arises from the antiderivative of (sin 3ϕ)−2/3 . Since r (ϕ) diverges for 3ϕ = nπ , the trajectories are unbounded. However, this solution does not capture the special case of radial motion: √ ϕ˙ = 0 ⇔ 3ϕ = nπ and 3 r˙ = r (1−r ), (5.13) which yields our previous kinks, moving radially in the special directions.
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5.4. Continuous symmetry. The case (κ1 , κ2 ) = (−3, 0) is special: firstly, because it is fixed by the duality transformation; and secondly, because the potential function V is invariant under not only the 3-symmetry, but also under U(1) rotations of φ. Again, we can find a superpotential W , ∂ W 2 3 V = 2 with W = 23 |φ|3 − 2|φ|. (5.14) ∂ φ¯ Like before, solutions of the gradient flow equation
∂W √ φ ± 3 φ˙ = |φ| 1 − |φ|2 = ∂ φ¯
(5.15)
solve (5.2). However, care should be taken near the origin, where W is not differentiable. The finite-action solutions are
−τ0 (5.16) φ(τ ) = ± tanh τ√ 3
and U(1) rotations of these. The only other solutions are infinite-action trajectories connecting the critical circle |φ| = 1 with φ = ∞:
−τ0 φ(τ ) = ∓ coth τ√ , (5.17) 3
modulo U(1) rotations. In particular, there are no transverse solutions of the gradient flow equation. 5.5. Hamiltonian flow. Now we consider transverse trajectories. Without loss of generality, we look for transverse trajectories connecting exp(− 23 π i)φ 0 and exp( 23 π i)φ 0 with φ 0 a non-zero real critical point, i.e. φ 0 = 1 or φ 0 = ν. For simplicity, we assume that φ1 is constant along these trajectories. Then φ1 and κ should be chosen so that φ¨ 1 = 0 for all φ2 . Since 6 φ¨1 = (κ1 − 1)φ1 − (κ1 + 3)φ12 + 4φ13 + κ2 (2φ1 + 1) φ2 + [(κ1 + 3) + 4φ1 ] φ22 , (5.18) we get exactly three solutions for κ2 = 0: (φ1 , κ1 ) = (0, −3) , (− 21 , −1) , (1, −7).
(5.19)
We will return to κ2 = 0 solutions momentarily. The case κ1 = −3 was treated above and yields radial trajectories. The cases κ1 = −1 and κ1 = −7 are related by the duality transformation, so we consider here just (κ1 , κ2 ) = (−1, 0). Then (5.2) is implied by a first-order hamiltonian flow,
√ ∂W ± 3 φ˙ = i(φ¯ 2 − φ) = i with W = 13 φ 3 + φ¯ 3 − |φ|2 , (5.20) ∂ φ¯ i.e. the hamiltonian is exactly the superpotential of (5.7). There are finite-action solutions φ(τ ) = − 21 ± i
√
3 2
0 tanh( τ −τ 2 ).
Two further solutions are obtained on application of the 3-symmetry.
(5.21)
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It is straightforward to write down infinite-action solutions of (5.20), at least in implicit form. The value of W is conserved by the flow, so solutions φ(τ ) of the flow obey W (φ(τ )) = C for some constant C. In polar coordinates φ = r exp(iϕ), this reads 3 r2 + C , 2 r3
cos 3ϕ =
(5.22)
which yields r (ϕ) as a solution of a cubic equation. In particular, for −1/3 < C < 0 there are periodic trajectories. It is amusing to note that the gradient-flow and hamiltonian-flow cases are related by flipping the sign of ν. 5.6. Solutions without action. Referring again to Eq. (5.18) we see that, if κ2 = 0, then φ¨ 1 = 0 enforces φ1 = − 21 , hence the case (φ1 , κ1 ) = (− 21 , −1)
(5.23)
of (5.19) allows us to turn on κ2 . With these values fixed, (4.12) is equivalent to 6 φ¨2 = 4 (φ2 −
√ 3 2 ) (φ2
√
+
3 2 ) (φ2
−
κ2 4 ).
(5.24)
This equation has kink-type solutions whenever the roots of the polynomial on the righthand side are evenly spaced. This occurs not only in the case κ2 = 0 discussed above, √ but also when κ2 = ±6 3. The corresponding kink solutions are
√ √ φ = − 21 + i 23 ± 3 tanh(τ −τ0 ) with = sgn(κ2 ). (5.25) 6. Instanton Equations in Seven and Eight Dimensions 6.1. Anti-self-duality in eight dimensions. In the previous section we constructed solutions of the Yang-Mills equation on G/H × R for special values of κ. We found that the second-order Yang-Mills equations actually reduced to first-order equations for these special values. In this section we will show that those first-order equations which admit finite-energy instantons have a natural geometrical interpretation: they take the anti-selfduality form (3.1) with a suitably chosen three-form . It is most convenient to start in eight dimensions rather than in seven. Let x 7 and x 8 denote coordinates on R2 and let e7 = dx 7 and e8 = dx 8 ; then the forms
ω = ω + e7 ∧ e8
and
= ∧ (e7 + ie8 )
(6.1)
define an SU(4)-structure on G/H × R2 . The associated metric and volume form are g8 = g6 + (e7 )2 + (e8 )2
and
V8 = V6 ∧ e7 ∧ e8 .
(6.2)
The four-form =
1 2
ω∧ ω − Re
(6.3)
defines a Spin(7)-structure. The operator ∗8 ( ∧·) on two-forms has eigenvalues −1 and 3, with eigenspaces of dimensions 21 and 7, respectively. So it makes sense to consider the -anti-self-duality equation ∧ F = − ∗8 F.
(6.4)
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This equation has been studied in [3,4,14,11]. With respect to a complex basis {α } :
1 = e1 + ie2 , 2 = e3 + ie4 , 3 = e5 + ie6 , 4 = e7 + ie8 (6.5)
it reads
ωF =0
and
¯
Fα¯ β¯ = − 21 α¯ β¯ γ¯ δ¯ F γ¯ δ (6 real equations), ¯
(6.6) ¯
where we have raised indices using the almost Hermitian metric: F α¯ β = Fαβ g α α¯ g β β with g α α¯ = δ α α¯ . For A7 = A8 = 0, Eqs. (6.6) reduce to (∂7 + i ∂8 ) A p¯ = p¯ q¯ r¯ F q¯ r¯ for p, q, r = 1, 2, 3, whose stable points satisfy F 0,2 = 0. In the real basis, (6.6) read √ ∂7 Aa − Jab ∂8 Ab = 3 f abc Fbc .
(6.7)
(6.8)
6.2. Gradient flow. Now we step down to seven dimensions. Consider the seven-manifold G/H × R, with R parametrised by x 7 , the metric g7 induced from g8 and the volume form V7 = V6 ∧ e7 . Then our four-form descends as follows: = ∧ e8 + ∗7 ,
(6.9)
where = ω ∧ e7 + Im
and
∗7 =
1 2
ω ∧ ω − Re ∧ e7
(6.10)
live on G/H × R. The three-form defines a G 2 -structure, which is compatible with the metric in the sense that ∗7 (i u ∧ i v ∧ ) = 6 g7 (u, v) for all tangent vectors u, v [55]. Associated to is an anti-self-duality equation (3.1). The eigenvalue problem for the operator ∗7 ( ∧ ·) on two-forms F is characterised as follows: λ dim 2λ F-type
2 7 iu
−1 14 ∗7 ∧ F = 0
(6.11)
∼ so(7). The space 2−1 maps to the Lie algebra of G 2 under the isomorphism 2 = 2 Now suppose that F is a connection on G/H × R pulled back from G/H × R or, equivalently, that A8 = 0 and A1 , . . . , A7 are independent of x 8 in some gauge. Then it is easy to show that ∧ F = ∧ F ∧ e 8 + ∗7 ∧ F
and
∗8 F = ∗ 7 F ∧ e 8 .
(6.12)
Hence, the -anti-self-duality (6.4) in eight dimensions descends to ∧ F = − ∗7 F.
(6.13)
This equation was studied in detail in [9]. Differentiating, one sees that this equation implies the Yang-Mills equation (4.1) with torsion given by (κ1 , κ2 ) = (3, 0), on identifying τ = x 7 . Further, a solution of the anti-self-duality equation (3.2) on G/H pulls back to a τ -independent solution of (6.13).
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Let us rewrite (6.13) in components. This is easily done using the fact that (6.13) is equivalent to (i u ) F = 0 for tangent vectors u = E 1 , . . . , E 7 . With √ i E 7 = − 21 Jab ea ∧ eb and i Ea = −Jab eb ∧ e7 + 3 f˜abc eb ∧ ec , (6.14) (6.13) is equivalent to Jab Fab = 0
and
F7a =
√
3 f abc Fbc .
(6.15)
The second relation is the flow equation introduced in [31]. We have shown in Sect. 3 that the first relation is satisfied by our ansatz (4.4); substituting this ansatz into the second relation yields precisely the gradient flow equation (5.8) for κ = 3. 6.3. Hamiltonian flow. We now repeat the discussion of the previous subsection, but with the roles of x 7 and x 8 reversed. We regard x 8 as a coordinate on the factor R of G/H × R, denote by g7 the metric induced from g8 and choose the volume form V7 = V6 ∧ e8 . Then = − ∧ e7 + ∗7 ,
(6.16)
where = ω ∧ e8 + Re
and
∗7 =
1 2
ω ∧ ω + Im ∧ e8 .
(6.17)
Again, defines a G 2 -structure on G/H × R, and the action of ∗7 ( ∧ ·) exactly mirrors that of ∗7 ( ∧ ·). In particular, if F is a connection on G/H × R pulled back to G/H × R2 , then (6.4) is equivalent to ∧ F = − ∗7 F.
(6.18)
Which second-order equation is implied by (6.18)? Differentiating, one obtains D ∗7 F + (3ρ Im ∧ e8 + 2ρ ω ∧ ω) ∧ F = 0.
(6.19)
∗7
∧ F = 0, one arrives at the Taking into account the equivalence of (6.18) and Yang-Mills equation (4.1) with torsion for (κ1 , κ2 ) = (−1, 0). The component form of (6.18) is √ Jab Fab = 0 and F8a = − 3 f˜abc Fbc . (6.20) With the ansatz (4.4), the first equation is again automatically satisfied, while the second one is exactly equivalent to the hamiltonian flow equation (5.20) with κ = −1. 6.4. Continuous symmetry. The fixed points of the gradient flow or hamiltonian flow equations are the critical points of the superpotential W . For the special U(1)-symmetric case (κ1 , κ2 ) = (−3, 0), the superpotential (5.14) yields the fixed points |φ|2 = 1. From the discussion in Sect. 3 it is clear that these are the solutions of the ω-self-duality equation, ∗6 F = ω ∧ F,
(6.21)
for a G-invariant connection on G/H . Indeed, differentiating this equation gives the Yang-Mills equation (4.1) with torsion via (κ1 , κ2 ) = (−3, 0). This simple geometrical interpretation for the static solutions does not seem to extend to the general solutions of the flow equation (5.8) in this situation. Note that the ω-self-duality equation is not a BPS equation.
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6.5. First-order flows. We close with a second interpretation of the G 2 -instanton equations (6.13) and (6.18), which accounts for the appearance of a superpotential with gradient and hamiltonian flows in Sect. 5. In his thesis [49], Xu noted that the ω-anti-self-duality equation (3.2) is equivalent to dω ∧ F = 0.
(6.22)
We have already seen that (3.2) implies (6.22). To show the converse, we first observe that (6.22) implies F 0,2 = F 2,0 = 0. It follows that Re ∧ F = 0.
(6.23)
Second, differentiating and applying the Bianchi identity yields ω∧ω∧F =0
⇔
ω F = 0.
(6.24)
Thus, F contains only a (1,1) part orthogonal to ω. Since the Donaldson-Uhlenbeck-Yau equations (3.5) are equivalent to (3.2), this proves the assertion. Xu then introduced an action Tr(ω ∧ F ∧ F) (6.25) X6
for a connection on a nearly Kähler six-manifold X 6 , whose equation of motion is (6.22) and whose gradient flow equation is ∂A = ∗(F ∧ dω). ∂τ
(6.26)
Given a local orthonormal frame ea for the cotangent bundle of X 6 , we contract both sides with the ea . Employing the identity ∗ (F ∧ dω)ea = (∗dω)(F ∧ ea ),
(6.27)
one sees that the flow equation is equivalent to ∂Aa = ∂τ
3 2 f abc Fbc ,
(6.28)
which coincides with the second equation of (6.15) after a rescaling in τ . Thus, the G 2 -instanton equation (6.13) implies the gradient flow for the action (6.25). This explains why (6.13) reduces to a gradient flow equation for φ. Substituting the ansatz (4.4) into (6.25) should give something proportional to the superpotential (5.7), and this is easily verified: 3 Tr(ω ∧ F ∧ F) = −Vol(G/H ) Tr 1 − 3 + 3 + ( )3 G/H
= −Vol(G/H ) (1 + 3W ).
(6.29)
Similarly, the hamiltonian flow equation for (6.25) is ∂Aa Jab eb = ∗(F ∧ dω). ∂σ
(6.30)
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This is equivalent to the second equation of (6.20), and reduces to the hamiltonian flow for W . Finally, we comment on the relation between the gradient and hamiltonian flow equations and the remaining part of the G 2 -instanton equations, ω F = 0 (the first equation in (6.15) or (6.20)). By employing the identities J ∗ ∂τ A = 21 ∂τ A ∧ ω ∧ ω and J (Im ∧ F) = Re ∧ F, the gradient flow (6.26) can be rearranged to read 1 ∂A ∧ ω ∧ ω = 3ρ Re ∧ F. 2 ∂τ
(6.31)
By taking the exterior derivative, and using the fact that D(∂τ A) = ∂τ F, this equation implies ∂ (ω F) = 12ρ 2 ω F. ∂τ
(6.32)
A similar argument shows that, for the hamiltonian flow (6.30), ∂τ (ω F) = 0. So, we should not be surprised that ω F = 0 holds for our gradient and hamiltonian flows: if one requires this to hold at τ = ±∞, it will hold everywhere. It is a curious fact that a nearly Kähler structure can itself be regarded as a critical point of a hamiltonian flow [56] – perhaps there is some connection between this and the gradient and hamiltonian flows described above. Acknowledgements. We thank Christoph Nölle for collaboration at an early stage. This work was supported in part by the cluster of excellence EXC 201 “Quantum Engineering and Space-Time Research”, by the Deutsche Forschungsgemeinschaft (DFG) and by the Heisenberg-Landau program. The work of T.A.I. and A.D.P. was partially supported by the Russian Foundation for Basic Research (grant RFBR 09-02-91347). The work of D.H. is supported by Graduiertenkolleg GRK 1463 “Analysis, Geometry and String Theory”.
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Commun. Math. Phys. 300, 205–242 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1116-6
Communications in
Mathematical Physics
Asymptotic Stability, Concentration, and Oscillation in Harmonic Map Heat-Flow, Landau-Lifshitz, and Schrödinger Maps on R2 Stephen Gustafson1 , Kenji Nakanishi2 , Tai-Peng Tsai1 1 Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada.
E-mail:
[email protected];
[email protected]
2 Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan. E-mail:
[email protected]
Received: 5 October 2009 / Accepted: 24 May 2010 Published online: 21 August 2010 – © Springer-Verlag 2010
Abstract: We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schrödinger flow as special cases) for degree m equivariant maps from R2 to S2 . If m ≥ 3, we prove that near-minimal energy solutions converge to a harmonic map as t → ∞ (asymptotic stability), extending previous work (Gustafson et al., Duke Math J 145(3), 537–583, 2008) down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m = 3, involving (among other tools) a “normal form” for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schrödinger operators with sufficiently repulsive potentials (which may be of some independent interest). When m = 2 this asymptotic stability may fail: in the case of heat-flow with a further symmetry restriction, we show that more exotic asymptotics are possible, including infinite-time concentration (blow-up), and even “eternal oscillation”. Contents 1.
2. 3. 4. 5.
6.
Introduction and Results . . . . . . . . . . 1.1 The main difficulty and the main idea 1.2 Organization of the paper . . . . . . 1.3 Some further notation . . . . . . . . Generalized Hasimoto Transform . . . . . Decomposition and Orthogonality . . . . . Coordinate Change . . . . . . . . . . . . . Decay Estimates for the Remainder . . . . 5.1 Dissipative L 2t estimate . . . . . . . 5.2 Dissipative decay . . . . . . . . . . . 5.3 Dispersive L 2t estimate . . . . . . . . 5.4 Dispersive decay . . . . . . . . . . . Parameter Evolution . . . . . . . . . . . .
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7. 8. 9. 10.
Partial Integration for the Parameter Dynamics Special Estimates for m = 3 . . . . . . . . . . Special Estimates for m = 2, a > 0, v2 = 0 . . Proofs of the Key Linear Estimates . . . . . . 10.1 Uniform bound on the right inverse Rϕ . 10.2 Double endpoint Strichartz estimate . . . Acknowledgements. . . . . . . . . . . . . . . . . . Appendix A. Landau-Lifshitz Maps from S2 . . . . References . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction and Results The Landau-Lifshitz (sometimes Landau-Lifshitz-Gilbert) equation describing the dynamics of an 2D isotropic ferromagnet is (eg. [13]) ut = a1 ( u + |∇ u|2 u) + a2 u × u,
a1 ≥ 0, a2 ∈ R,
(1.1)
where the magnetization vector u = u(t, x) = (u 1 , u 2 , u 2 ) is a 3-vector with normalized length, so can be considered a map into the 2-sphere S2 : u : [0, T ) × R2 → S2 := { u ∈ R3 | | u | = 1}.
(1.2)
The special case a2 = 0 of (1.1) is the very well-studied harmonic map heat-flow into S2 , while the special case a1 = 0 is known as the Schrödinger flow (or Schrödinger map) equation, the geometric generalization of the linear Schrödinger equation for maps into the Kähler manifold S2 . In order to exhibit the simple geometry of (1.1) more clearly, we introduce, for u ∈ S2 , the tangent space Tu S2 := u⊥ = {ξ ∈ R3 | u · ξ = 0}
(1.3)
to the sphere S2 at u. For any vector v ∈ R3 , we define two operations on vectors: J v := v×, S2 ,
P v := −J v J v .
P u
For u ∈ projects vectors orthogonally onto Tu (complex structure) on Tu S2 . Denoting
S2 ,
(1.4) while
J u
a = a1 + ia2 ∈ C,
is a π/2 rotation (1.5)
the Landau-Lifshitz equation (1.1) may be written u, ut = Pau
Pau := a1 P u + a2 J u .
The energy associated to (1.1) is simply the Dirichlet functional 1 |∇ u|2 d x, E( u) = 2 R2 and (1.6) formally yields the energy identity t E( u (t)) + 2a1 |P u u (s, x)|2 d xds = E( u (0)) 0
R2
(1.6)
(1.7)
(1.8)
implying, in particular, energy non-increase if a1 > 0, and energy conservation if a1 = 0 (Schrödinger map).
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207
To a finite-energy map u : R2 → S2 is associated the degree 1 deg( u ) := u x · J u u x2 d x. 4π R2 1
(1.9)
If lim|x|→∞ u(x) exists (which will be the case below), we may identify u with a map S2 → S2 , and if the map is smooth, deg( u ) is the usual Brouwer degree (in particular, an integer). It follows immediately from expression (1.9) that the energy is bounded from below by the degree: 1 | u x − J u u x2 |2 + 4π deg( u ) ≥ 4π deg( u ), (1.10) E( u) = 2 R2 1 and equality here is achieved exactly at harmonic maps solving the first-order equations u x1 = J u u x2
(1.11)
which, in stereographic coordinates S2 u ←→
u 1 + iu 2 ∈ C ∪ {∞} 1 − u3
(1.12)
are the Cauchy-Riemann equations, and the solutions are rational functions. These harmonic maps are critical points of the energy E and, in particular, static solutions of the Landau-Lifshitz equation (1.1). In this paper we specialize to the class of m-equivariant maps, for some m ∈ Z+ : u(t, x) = emθ R v(t, r ),
v : [0, T ) × [0, ∞) → S2
(1.13)
with notations k = (0, 0, 1), R := J k = k×,
(1.14)
x1 + i x2 = r eiθ .
(1.15)
and polar coordinates
In terms of the radial profile map v = (v1 , v2 , v3 ), the energy is
∞
| vr |2 +
E( u) = π 0
m2 2 2 (v + v ) r dr. 2 r2 1
(1.16)
limr →∞ v = ±k (see [11] Finite energy implies v is continuous in r and limr →0 v = ±k, for details). We force non-trivial topology by working in the class of maps v(∞) = k}. m := { u = emθ R v(r ) | E( u ) < ∞, v(0) = −k,
(1.17)
It is easy to check that the degree of such maps is m: deg M ≡ m.
(1.18)
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The harmonic maps saturating inequality (1.10) which also lie in m are those corresponding to βz m (β ∈ C× = C\{0}) in stereographic coordinates (1.12). In the representation S2 ⊂ R3 , the harmonic map corresponding to z m is given by ), h = (h 1 , 0, h 3 ), h 1 = emθ R h(r
2 r m − r −m , h = . 3 r m + r −m r m + r −m
(1.19)
The full two-dimensional family of m-equivariant harmonic maps in m is then generated by rotation and scaling, so for s > 0 and α ∈ R, we denote μ = m log s + iα, h[μ] = eα R hs , hs = h(r/s).
(1.20)
The harmonic map emθ R h[μ] corresponds under stereographic projection to e−μ z m . We are concerned here with basic global properties of solutions of the Landau-Lifshitz equations (1.1), especially the possible formation of singularities, and the long-time asymptotics. For finite-energy solutions of (1.1) in 2 space dimensions, finite-time singularity formation is only known to occur in the case of the 1-equivariant harmonic map heat-flow (a2 = 0) – the first such result [5] was for the problem on a disk with Dirichlet boundary conditions (this was extended to 1 on R2 in [10]). Examples of finite-time blow-up for different target manifolds (not the physical case S2 ) are also known (eg. [22]). For the Schrödinger case (a1 = 0), it is known that small-energy solutions remain regular (this was proved first in [6] for equivariant maps, and then in [2] without symmetry restriction). In the present setting, the energy is not small – indeed by (1.10) and (1.18), E m ≥ 4π m.
(1.21)
A self-similar blow-up solution, which however carries infinite energy, is constructed in [7]. In the recent works [9,10,12], it was shown that when m ≥ 4, solutions of (1.1) in m with near minimal energy (E( u ) ≈ 4π m) are globally regular, and converge asymp of the harmonic map family. In particular, the harmonic totically to a member emθ R h[μ] maps are asymptotically stable. The analysis there fails to extend to m ≤ 3, due to the d h[μ] (a point which we hope to clarify below). With a new slower spatial decay of dμ approach, we can now handle the case m = 3 as well: Theorem 1.1. Let m ≥ 3, a = a1 + ia2 ∈ C\{0}, and a1 ≥ 0. Then there exists δ > 0 such that for any u(0, x) ∈ m with E( u (0)) ≤ 4mπ + δ 2 , we have a unique global solution u ∈ C([0, ∞); m ) of (1.1), satisfying ∇ u ∈ L 2t,loc ([0, ∞); L ∞ x ). Moreover, for some μ ∈ C we have u (t) − emθ R h[μ] + a1 E( u (t) − emθ R h[μ]) → 0 as t → ∞. L∞ x
(1.22)
In short, every solution with energy close to the minimum converges to one of the harmonic maps uniformly in x as t → ∞. Even for the higher degrees m ≥ 4, this result is stronger than the previous ones [12,10,9], where the convergence was given only in time average.1 Note that in the dissipative case (a1 > 0), solutions converge to 1 The statements in the previous papers do not follow directly from Theorem 1.1, but are implied by the proof in this paper.
Stability and Oscillation of Harmonic Maps
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a harmonic map also in the energy norm, while this is impossible for the conservative Schrödinger flow (a1 = 0). The analysis for the case m = 2 seems trickier still, and we have results only in the special case of the harmonic map heat-flow (a2 = 0) with the further restriction that the image of the radial profile map v(r ) remain on a great circle: v2 ≡ 0 (though of course the map u(x) itself covers the full sphere m times) – this is a condition which is preserved by the evolution only for the heat-flow. These results show, in particular, that the strong asymptotic stability result of Theorem 1.1 for m ≥ 3 is no longer valid; instead, more exotic asymptotics are possible, including infinite-time concentration (blow-up) and “eternal oscillation”: Theorem 1.2. Let m = 2 and a > 0. Then there exists δ > 0 such that for any u(0, x) = e2θ R v(0, r ) ∈ 2 with E( u (0)) ≤ 8π + δ 2 , and v2 (0, r ) ≡ 0, we have a unique global solution u ∈ C([0, ∞); 2 ) satisfying ∇ u ∈ L 2t,loc ([0, ∞); L ∞ x ). Moreover, for some continuously differentiable s : [0, ∞) → (0, ∞) we have u (t) − emθ R h(r/s(t)) + E( u (t) − emθ R h(r/s(t))) → 0 as t → ∞. L∞ x In addition, we have the following asymptotic formula for s(t): √at v1 (0, r ) 2 dr + Oc (1), (1 + o(1)) log(s(t)) = π 1 r
(1.23)
(1.24)
where as t → ∞, o(1) → 0 and Oc (1) converges to some finite value. In particular there are initial data yielding each of the following types of asymptotic behavior: (1) (2) (3) (4) (5) (6)
s(t) → ∃ s∞ ∈ (0, ∞). s(t) → 0. s(t) → ∞. 0 = lim inf s(t) < lim sup s(t) < ∞. 0 < lim inf s(t) < lim sup s(t) = ∞. 0 = lim inf s(t) < lim sup s(t) = ∞.
Estimate (1.23) shows that these solutions do converge asymptotically to the family of harmonic maps. However, the evolution along this family, described by the parameter s(t), does not necessarily approach a particular map in 2 (although it might – case (1)). The solution may in fact converge pointwise (but not uniformly) to a constant map ±k (which has zero energy, zero degree, and lies outside 2 ) as in (2)–(3) (this is infinitetime blow-up or concentration), or it may asymptotically “oscillate” along the harmonic map family, as in (4)–(6). Note that the above classification (1)–(6) is stable against initial “local” perturbation. Namely, if two initial data v 1 (0) and v 2 (0) satisfy ∞ 1 |v1 (0, r ) − v12 (0, r )| dr < ∞, (1.25) r 1 the corresponding solutions have the same asymptotic type among (1)–(6). More precisely, the difference of their scaling parameters converges in (0, ∞). The point is that the energy just barely fails to control the above integral. In particular, the oscillatory behavior in (4)–(6) is driven solely by the distribution around spatial infinity. In fact, if we replace the domain R2 by the disk D = {x ∈ R2 | |x| < 1} with the same symmetry restriction with m = 2 and the same boundary
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S. Gustafson, K. Nakanishi, T.-P. Tsai
then it is known [1] (see also [8]) that all the conditions v(t, 0) = −k and v(t, 1) = k, solutions behave like (2), namely they concentrate at x = 0 as t → ∞, provided that v3 (0, r ) has only one zero. The formula (1.24) suggests that we should always have (2) on D without the additional condition. Also, if we replace the domain R2 by S2 , then we can rather easily show in the dissipative case a1 > 0 that the solution converges to one harmonic map for all m ∈ N, by the argument in this paper, or even those in the previous papers. We state the result on S2 in Appendix A with a sketch of the proof. We should mention that existence of eternal oscillation of the same type was first shown in [18] for the semilinear heat equation of u(t, x) : [0, ∞) × R N → R, u t − u = |u| p u,
√
(1.26)
for very high dimensions and power2 (N ≥ 11 and p > 4/(N − 4 − 2 N − 1)), by using the comparison principle, but they did not obtain an asymptotic formula valid for all solutions, nor the asymptotic stability of the family of stationary solutions in a solution class containing the eternal oscillations. There is another example in [21, Sect. 5] with less similarity to ours, but for the harmonic map heat flow, which shows existence of “eternal winding” around a compact 1-parameter family of harmonic maps from S2 to S2 × R2 with some warped metric, where the analysis is reduced to an ODE on the target by the special choice of initial data. In this case, the weird behavior of the solutions is entirely due to the artificial choice of the metric on the target. Compared with those results, we have the following advantages: (1) The setting is very simple and physically natural. (2) The asymptotic formula is explicit in terms of the initial data, and valid for all general solutions under the symmetry condition. We want to emphasize also that our analysis works in the same way in the dissipative (a1 > 0) and the dispersive (a1 = 0) cases. We need a2 = 0 in Theorem 1.2 only because the angular parameter α(t) gets beyond our control (hence we remove it by the constraint), but the rest of our arguments could work in the general case.3 1.1. The main difficulty and the main idea. The standard approach for asymptotic stability is to decompose the solution into a leading part with finite dimensional parameters varying in time, and the rest decaying in time either by dissipation or by dispersion. In our context, we want to decompose the solution in the form v(t) = h[μ(t)] + v(t) ˇ
(1.27)
such that the remainder v(t) ˇ decays, and the parameter μ(t) ∈ C converges as t → ∞ (at least for Theorem 1.1). In favorable cases (the higher m, in our context), we can choose μ(t) such that all secular modes for v(t) ˇ are absorbed into the time evolution of the main part h[μ(t)]. This means that the kernel of the linearized operator for v(t) ˇ is spanned by the parameter derivatives of h[μ], and hence we can put that component of ∂t v(t) into ∂t h[μ(t)]. This is good both for v(t) ˇ and μ(t), because 2 The power is bigger at least than the H 5 scaling critical exponent. 3 We will use the parameter convergence in the proof of Theorem 1.1 in the dispersive case a = 0 to 1
fix our linearized operator. However it is possible to treat the linearized operator even with non-convergent parameter and a1 = 0, if we assume one more regularity on the initial data. We do not pursue it here since the wild behavior of α(t) prevents us from using it.
Stability and Oscillation of Harmonic Maps
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(1) v(t) ˇ will be free from secular modes, and so we can expect it to decay by dissipation or dispersion, at least at the linearized level. (2) The decomposition is preserved by the linearized equation. Hence μ(t) ˙ is affected by vˇ only superlinearly, i.e. at most in quadratic terms. In particular, if we can get L 2 decay of vˇ in time, then μ(t) ˙ becomes integrable in time, and so converges as t → ∞. This is indeed the case for m > 3. However, the above naive argument does not take into account the space-time behavior of each component. The problem comes from the fact that the decomposition and the decay estimate must be implemented in different function spaces, and they may be incompatible if the eigenfunctions decay too slowly at the spatial infinity. In fact, the parameter derivative of h[μ] is given by d h[μ] = h s1 eα R [(h s3 , 0, −h s1 )dμ1 + (0, 1, 0)dμ2 ],
(1.28)
and hence the eigenfunctions are O(r −m ) for r → ∞, i.e. slower for lower m. On the other hand, the spatial decay property in the function space for the time decay estimate is essentially determined by the invariance of our problem under the scaling v(t, x) → v(λ2 t, λx),
(1.29)
which maps solutions into solutions, preserving the energy. If we want L 2 decay in time (so that we can integrate quadratic terms in μ), ˙ then a function space with the right scaling is given by v/r ˇ ∈ L 2t L ∞ x .
(1.30)
To preserve such norms in x under the orthogonal projection, the eigenfunction must be in the dual space, for which m > 3 is necessary. Indeed, this is the essential reason for the restriction m ≥ 4 in the previous works [10–12]. We emphasize that the above difficulty is common for the dissipative and dispersive cases, since they share the same scaling property. That is, the dissipation does not help with this issue, even though it gives us more flexibility in the form of decay estimates. The main novelty of the present approach is the non-orthogonal decomposition L 2x = (h s1 ) ⊕ (ϕ s )⊥ ,
(1.31)
where ϕ s (r ) is smooth and supported away from r = 0 and from r = ∞, so that the (non-orthogonal) projection may preserve the decay estimates. This is good for the remainder v, ˇ but not for the parameter μ —the decomposition is no longer preserved by the linearized evolution, since they have no particular relation. This implies that we get a new error term in μ(t) ˙ which is linear in v(t) ˇ (see Sect. 6). This contribution is handled by including it in a sort of “normal form” for the dynamics of the parameters μ(t), explained in Sect. 7. In particular, it is this new term which drives the non-trivial dynamics for the m = 2 heat-flow given in Theorem 1.2. For the purely dispersive (Schrödinger map) case, one tool we use should be of some independent interest: the 2D radial “double-endpoint Strichartz estimate” for Schrödinger operators with sufficiently “repulsive” potentials (in the absence of a potential, the estimate is false). The proof is given in Sect. 10.2.
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S. Gustafson, K. Nakanishi, T.-P. Tsai
1.2. Organization of the paper. In Sect. 2, we use the “generalized Hasimoto transform” to derive the main equation used to obtain time-decay estimates of the remainder term. Section 3 gives the details of the solution decomposition described above, and addresses the inversion of the Hasimoto transform. The estimates for going back and forth between the different coordinate systems (the “Hasimoto” one of Sect. 2 and the decomposition of Sect. 3) are given in Sect. 4. Section 5 is devoted to establishing the time-decay (dispersive if a1 = 0, diffusive if a1 > 0) of the remainder term, using energy-, Strichartz-, and scattering-type estimates. The dynamics of the parameters μ(t) are derived and estimated in Sect. 6. The leading term in the equation for μ˙ is not integrable in time, and so Sect. 7 gives an integration by parts in time to identify (and estimate) a kind of “normal form” correction to μ(t), whose time derivative is integrable. At this stage, the proof of Theorem 1.1 for m > 3 is complete. A more subtle estimate of an error term for m = 3 is done in Sect. 8, completing the proof in that case. Finally, in Sect. 9, the normal form correction is analyzed in the case m = 2, a2 = 0, v2 = 0, in order to prove Theorem 1.2. Proofs of certain linear estimates (including the double-endpoint Strichartz) are relegated to Sect. 10. Appendix A states the analogous theorems for domain S2 and sketches the proofs. At the end of each of the main sections, we will put a proposition summarizing the main contents of that section. 1.3. Some further notation. We distinguish inner products in R3 and C by a · b =
3
ak bk , a ◦ b = Re a Re b + Im a Im b.
(1.32)
k=1
Both will be used for C3 vectors too. The L 2x inner-product is denoted ( f | g) = f (x)g(x)d x, R2
(1.33)
while ( f, g) just denotes a pair of functions. For any radial function f (r ) and any parameter s > 0, we denote rescaled functions by f s (r ) := f (r/s),
f s (r ) := f (r/s)s −2 .
(1.34)
We denote the Fourier transform on R2 by F, and, for radial functions, the Fourier-Bessel transform of order m by Fm : ∞ 1 −i x·ξ (F f )(ξ ) = f (x)e d x, (Fm ) f (ρ) = Jm (rρ) f (r )r dr, (1.35) 2π R2 0 where Jm is the Bessel function of order m. For m ∈ Z we have π 1 Jm (r ) = eimθ−ir sin θ dθ, F[ f (r )eimθ ] = i m (Fm f )eimθ . (1.36) 2π −π We denote the Laplacian x on the subspace spanned by d-dimensional spherical harmonics of order m by (m)
d Finally, the space
:= ∂r2 + (d − 1)r −1 ∂r − m(m + d − 2)r −2 .
(1.37)
p Lq
is the dyadic version of L p (r dr ) defined by the norm f L qp = f (r ){2 j < r < 2 j+1 } L p (r dr ) q . j (Z)
(1.38)
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213
2. Generalized Hasimoto Transform In this section, we recall from the previous papers [9–11] the equation for the remainder part, which is written in terms of a derivative vanishing exactly on the harmonic maps, and so independent of the decomposition. The equation was originally derived in [6] in the case of small energy solutions (hence with no harmonic map component), and called there the generalized Hasimoto transform. Under the m-equivariance assumption (1.13), the Landau-Lifshitz equation (1.6) is equivalent to the following reduced equation for v(r, t): ∂r m 2 2 (2.1) vt = Pav ∂r2 + + 2 R v. r r Define the operator ∂v on vector-valued functions by ∂v := ∂r −
m v J R. r
(2.2)
v · b) − (k · v)b, we have J v R = −v3 on the tan J v R b = k( Since for any vector b, gent space Tv S2 = v⊥ . For future use, we denote the corresponding operator on scalar functions by mv3 . r
(2.3)
vt = −Pav Dv∗ ∂v v,
(2.4)
D := P v ∂ P v
(2.5)
L v := ∂r + Then Eq. (2.1) can be factored as
where
will always denote a covariant derivative (which acts on Tv S2 -valued functions), and ∗ denotes the adjoint in L 2 (R2 ). Denote the right-most factor in (2.4) by w := ∂v v = vr −
m v P k. r
(2.6)
and applying Dv to both sides yields Then (2.4) becomes vt = −Pav Dv∗ w, = −Pav Dv Dv∗ w. Dt w
(2.7)
Now we rewrite the equation for w by choosing an appropriate orthonormal frame field on Tv S2 , realized in C3 . Let e = e(t, r ) satisfy Re e ∈ v⊥ , | Re e| = 1, Im e = J v Re e.
(2.8)
Let S, T be real scalar, and let q, ν be complex scalar, defined by w = q ◦ e,
P v k = ν ◦ e,
Dt e = −i Se,
Dr e = −i T e.
(2.9)
Then we have the general curvature relation
[Dr , Dt ]e = i(Tt − Sr )e = i det v vr vt e.
(2.10)
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S. Gustafson, K. Nakanishi, T.-P. Tsai
Using Eq. (2.4) for v, we get m v v ∗ P k) · Pia Dv w. r
(2.11)
Dr e = 0, e(r = ∞) = (1, i, 0).
(2.12)
+ Tt − Sr = (w Now we fix e by imposing
(The unique existence of such e will be guaranteed by Lemma 4.1.) Then (2.11) yields m (2.13) − Sr = (q + ν) ◦ (ia L ∗v q). r A key observation is that in the Schrödinger (non-dissipative) case a = i, we can pull out the derivative on q: Sr = (∂r + r2 )( 21 |q|2 + mr ν ◦ q), and so ∞ 1 dr m 1 2 mw3 Q := |q|2 + ν ◦ q = |w| + . (2.14) 2Q , S = −Q + r 2 r 2 r r The evolution equation (2.7) for w yields our equation for q: ∞ m ∗ S= (q + ν) ◦ (ia L ∗v q)dr. (∂t + i S)q = −a L v L v q, r r
(2.15)
This is the basic equation used to establish diffusive (a1 > 0) or dispersive (a1 = 0) decay estimates. The operator acting on q can be expanded as (m − 1)2 2m(1 − v3 ) m + + w3 . r2 r2 r Following is a summary of this section: L v L ∗v = ∂r∗ ∂r +
(2.16)
Proposition 2.1. Let m ∈ N and u(t, x) = emθ R v(t, r ) be a (local) solution of the Landau-Lifshitz equation (1.1), and let e(t, r ) be a complex orthonormal frame field on Tv S2 satisfying Dr e = 0, e(r = ∞) = (1, i, 0), where D denotes the covariant derivative (2.5). Define w, q and ν by m w = vr − P v k, q=w · e, ν = P v k · e. r Then they solve equations ∞ m (∂t + i S)q = −a L v L ∗v q, S = (q + ν) ◦ (ia L ∗v q)dr, r r
(2.17)
(2.18)
(2.19)
where L v = ∂r + mv3 /r and L ∗v is its adjoint. If a = i, the equation of S can be rewritten as ∞ 1 dr m S = −Q + 2Q , Q = |q|2 + ν ◦ q. (2.20) r 2 r r We will use the above equations to derive decay estimates on the remainder v − h[μ] via q. The following two sections are devoted to the correspondence between q and the remainder (including the existence of e), and then in Sect. 5 we derive the decay estimates.
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3. Decomposition and Orthogonality In this section, we investigate the interplay between the decay estimates and the orthogonality condition for the decomposition into the harmonic map and the remainder, illuminating the difference between the higher and the lower degrees. We introduce coordinates for the decomposition of the original map v = h[μ] + v, ˇ
(3.1)
or more precisely for the remainder v, ˇ and a localized orthogonality condition which determines the decomposition. The choice of coordinates is the same as in the previous works [9,10,12], while the decomposition itself is different. For each harmonic map profile h[μ], μ = m log s + iα, we introduce an orthonormal frame field f = f[μ] := eα R (−hs × j + i j)
(3.2)
2 on the tangent space Th[μ] S , such that the parameter derivative of h[μ] is given by
d h[μ] = h s1 dμ ◦ f.
(3.3)
We express the difference from the harmonic map in this frame by z := vˇ · f.
(3.4)
In other words P h[μ] v = z ◦ f, or vˇ = z ◦ f + γ h[μ], where we denote γ := 1 − |z|2 − 1 = −O(|z|2 ).
(3.5)
As explained in the Introduction, the orthogonality condition in the previous works (z | h s1 ) = 0
(3.6)
would not work for m ≤ 3 due to the slow decay of h s1 for r → ∞. Hence instead we determine the parameter μ by imposing localized orthogonality (z | ϕ s ) = 0,
ϕ s = ϕ(r/s),
(3.7)
with some smooth localized function ϕ(r ) ∈ C0∞ ((0, ∞); R), satisfying (h 1 | ϕ) = 1. solves (1.11) means that The fact that emθ R h[μ] ∂h h = 0,
(3.8)
and so we have w = ∂v v = vˇr +
m s m (h 3 vˇ + vˇ3 v) = L s vˇ + vˇ3 v. r r
(3.9)
m m vˇ3 z + h s1 γ . r r
(3.10)
Hence L s z = L s vˇ · f + vˇ · fr = w ·f −
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In order to estimate z by w (or equivalently q), we introduce a right inverse of the operator L s = ∂r + mr h s3 , defined by ∞ r h s1 (r )−1 g(r )dr ϕ s (r )h s1 (r )r dr . (3.11) Rϕs g := 2π h s1 (r ) 0
r
Then we have L s Rϕs g = g,
Rϕs L s g = g − h s1 (g | ϕ s ),
(3.12)
hence Rϕs = (L s )−1 on (ϕ s )⊥ . Moreover we have the following uniform bounds Lemma 3.1. For all p ∈ [1, ∞] and |θ | < m, we have ϕr −θ L 1 gr θ+1 L 1p , Rϕs gr θ L ∞ p
(3.13)
p
p
where the L q norm is defined in (1.38). Moreover, the condition on ϕ is optimal in the following sense: if ϕ ≥ 0, then ϕ ∈ r −1 L 1p is necessary for Rϕs to be bounded r θ−1 L ∞ p → D (0, ∞). We give a proof in Sect. 10. Note that the above bounds are scaling invariant: denoting Ds f := f (r/s), we have Rϕs = s Ds Rϕ1 Ds−1 ,
L s = s −1 Ds L 1 Ds−1 .
(3.14)
We can combine the estimates of the lemma with the embedding r θ1 L q11 ⊂ r θ2 L q22 ⇐⇒ p
p
2 2 − θ1 = − θ2 , p1 ≥ p2 , q1 ≤ q2 . p1 p2
(3.15)
The above lemma is used as follows. First note that the orthogonality (ϕ s | z) = 0 implies that z = Rϕs L s z because of (3.12). For the energy norm, we choose θ = 0 and p = 2 in Lemma 3.1. Then ϕ L 1 L s zr L 1 L s z L 2x . z/r L 2x z L ∞ 2 2
2
(3.16)
Since |L s − ∂r | 1/r , we further obtain Rϕs : r θ L 2 → r θ X, (|θ | < m),
(3.17)
where the space X is defined by the norm z X := z/r L 2x + zr L 2x .
(3.18)
The Sobolev embedding X ⊂ L ∞ is trivial by Schwarz: ≤ z/r L 2x zr L 2x . z2L ∞ x
(3.19)
Hence we get by using (3.10), z X L s z L 2 q L 2 + z L ∞ z X .
(3.20)
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217
For L 2t estimates of z, we use Lemma 3.1 with θ = 1 and p = ∞. Then we have ϕr −1 L 1 L s zr 2 L 1p L s z L ∞ , z/r L ∞ p p
(3.21)
1
for any p ∈ [1, ∞], and so by using (3.10), z L 2 L ∞ . z/r L 2 L ∞ q L 2 L ∞ + z L ∞ t,x t
t
p
p
t
(3.22)
p
If we were to use h 1 instead of ϕ, then we would need m > 3 for the Strichartz-type bound (3.22), and m > 2 for the energy bound (3.16), by the last statement of the lemma. As a summary of this section, we have Proposition 3.2. Let m ≥ 2, v(r ) ∈ m and, for some μ = m log s + iα ∈ C, v = h[μ] + v, ˇ z = vˇ ◦ f,
(3.23)
2 where f is the orthonormal frame on Th[μ] S defined in (3.2). Suppose that
(z|ϕ s ) = 0, ϕ s := ϕ(r/s)
(3.24)
for a fixed ϕ ∈ C0 (0, ∞) satisfying (h 1 |ϕ) = 1. Then we have the estimates z X , z X := z/r L 2x + zr L 2x q L 2x + z L ∞ x z/r L ∞ q L ∞ + z L ∞ z L ∞ (1 ≤ p ≤ ∞), p x x p
(3.25)
z L ∞ z X , x where q = w · e is the same as in Proposition 2.1. In the next section, we see that such an orthogonal decomposition uniquely exists for v ∈ m with energy close to the ground one, with small norms for q and z, so that we can dispose of the quadratic terms in the above estimates. 4. Coordinate Change Before beginning the estimates for the evolution, we establish in this section the bi-Lipschitz correspondence between the different coordinate systems: v and (μ, q), including unique existence of the decomposition. It is valid for any map in our class m with energy close to the ground states. For that purpose, we need to translate between the different frames e and f. At each point (t, r ), we define M = f ⊗ e ∈ G L R (C), a real-linear map C → C, by Mz := f · (e ◦ z).
(4.1)
Its transpose tM = e ⊗ f, defined by tMz = e · (f ◦ z), is the adjoint in the sense that (Mz) ◦ w = z ◦ (tMw). For any b, c, d ∈ C3 we have b · (c ◦ d) = (Re b · c) ◦ d + (Im b · c) ◦ d.
(4.2)
we have Since f(∞) = e−iα e(∞), and f ⊥ h[μ], ˇ · vr ) − M(∞) = e−iα , Mr = f ⊗ er + fr ⊗ e = −f ⊗ v(e
m s h vˇ ⊗ e. r 1
(4.3)
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Then e can be recovered from M by
e = P h[μ] e + (h[μ] · e)h[μ] = tMf − (1 + γ )−1 (tMz)h[μ],
(4.4)
provided that |γ | < 1. We further introduce some spaces with (pseudo-)norms: v ) = E(emθ R v(r )), |μ|C = min(| Re μ|, 1) + dist (Im μ, 2π Z), Em ( z X = z/r L 2x + zr L 2x , MY = Mr L 1 (dr ) + M L r∞ , v(∞) = k, Em ( m (δ) = { v (r ) : [0, ∞] → S2 | v(0) = −k, v ) ≤ 4mπ + δ 2 }, √ L 2 (δ) = {q(r ) : [0, ∞) → C | q L 2x ≤ 2δ}, C = C/2πiZ.
(4.5)
The metric on C is defined such that 1 ] − h[μ 2 ] X ∼ h[μ 1 ] − h[μ 2 ] L ∞ ∼ |μ1 − μ2 |C . h[μ
(4.6)
The following lemma is the goal of this section. Lemma 4.1. Let m ∈ N and ϕ ∈ C01 (0, ∞) satisfy (ϕ | h 1 ) = 1. Then there exists δ > 0 such that the system of equations v = h[μ] + v, ˇ z = v · f[μ], (z | ϕ s ) = 0, γ = 1 − |z|2 − 1, (4.7) m vr − P v k), q = e ◦ ( Dr e = 0, e(∞) = (1, i, 0), r defines a bijection from v ∈ m (δ) to (μ, q) ∈ C × L 2 (δ), which is unique under the condition z L ∞ δ. v, ˇ z and e are also uniquely determined. Moreover, if ( v j, . . . , ej) x with j = 1, 2 are such tuples given in this way, then we have vˇ 1 − vˇ 2 X + z 1 − z 2 X + e1 − e2 L ∞ + M1 − M2 Y v 1 − v2 X ∼ |μ1 − μ2 |C + q 1 − q 2 L 2 ,
(4.8)
where M j := f j ⊗ e j . In particular, we have pointwise smallness, ∼ z L ∞ δ 1, v ˇ L∞ x x
(4.9)
so that we can neglect higher order terms in z or v. ˇ Proof. We always assume (3.9), (3.4) and (3.5), which define the maps v , μ) → vˇ ↔ z → γ , (v, ˇ μ) → v, v → w = q ◦ e, (
(4.10)
with the Lipschitz continuity 2 L 2 v 1 − v2 X , γ 1 − γ 2 X z 1 − z 2 X ∼ vˇ 1 − vˇ 2 X , w 1 − w 1 v − v2 X − vˇ 1 − vˇ 2 X |μ1 − μ2 |C . (4.11)
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The energy can be written as m 2 m 2 2Em ( v ) = vr 2L 2 + R v = vr 2L 2 + P v k x x r r L 2x L 2x ) = w = w 2 2 + 2m( vr | P v k/r 2 2 + 4π [v3 (∞) − v3 (0)]. Lx
Lx
(4.12)
v )1/2 , X ⊂ L ∞ v | = 1, the boundary conditions v3 (0) = −1 Since P k v X Em ( x and | and v3 (∞) = 1 make sense in the energy norm. Next we consider a point orthogonality. Let v ∈ m (δ). Since v3 (0) < 0 < v3 (∞) for some μ0 = m log s0 + iα0 , so and v3 (r ) is continuous, we have v(s0 ) = eiα0 R h(1) that v = h[μ0 ] + vˇ is a decomposition satisfying (z | ϕ s0 ) = 0 if ϕ(r ) = δ(r − 1). In this case v is recovered from (w, μ0 ) by solving the ODE: m m · f[μ0 ] − vˇ3 z + h s10 γ , z(s0 ) = 0, (4.13) Ls z = w r r or the equivalent integral equation
m m s0 s0 w · f[μ ] − z + (4.14) v ˇ h γ . z = Rδ(r 0 3 −1) r r 1 s The uniform bound on Rδ(r −1) can be localized onto any interval I s0 , because z is the solution of the above initial value problem. Hence we get, in the same way as in (3.16),
q L 2x (I ) + z L ∞ zr L 2x (I ) . zr L 2x ∩L ∞ x (I ) x (I )
(4.15)
Since z(s0 ) = 0 and q L 2x ≤ δ 1, we get by continuity in r for I → (0, ∞), q L 2x δ. z X ∩L ∞ x
(4.16)
0 ], and we have v1 − v2 ∈ X by Thus every v ∈ m (δ) is close at least to some h[μ (4.11). m (δ) is a complete metric space with this distance. Now we take any ϕ ∈ C01 (0, ∞) satisfying (ϕ | h 1 ) = 1, and look for μ around μ0 solving the orthogonality F(μ) := ( v · f[μ] | ϕ s ) = (vˇ · f[μ] | ϕ s ) = (z | ϕ s ) = 0.
(4.17)
Its derivative in μ is given by v · h s3 f[μ] | ϕ s )dα − ( v · f[μ] | (r ∂r + 2)ϕ s ) | ϕ s )dμ − i( d F = −( v · h s1 h[μ]
ds s
= −dμ − (vˇ · h[μ] | h s1 ϕ s )dμ − (vˇ · f[μ] | (r ∂r + 2)ϕ s ds/s + i h s3 ϕ s dα) = −dμ + O(δ|dμ|). (4.18) In particular we have |F(μ0 )| δ,
∂F (μ0 ) = −I + O(δ). ∂μ
(4.19)
In addition, both F(μ) and ∂μ F are Lipschitz in v. Therefore by the implicit mapping theorem, if δ > 0 is small enough, there exists a unique μ ∈ C for each v such that F(μ) = 0 and |μ − μ0 | δ, and v → μ is Lipschitz. Then 0 ] L ∞ + |μ0 − μ| δ 1, v − h[μ z L ∞ x x
(4.20)
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S. Gustafson, K. Nakanishi, T.-P. Tsai
and so by the same argument as for (4.16), we get z X δ, and in addition, z 1 − z 2 X |μ1 − μ2 |C + w 1 − w 2L 2 .
(4.21)
If we have two such μ = μ1 , μ2 with z j L ∞ δ, then 1 ]− h[μ 2 ] L ∞ 1 ] L ∞ + 2 ] L ∞ δ, v − h[μ v − h[μ |μ1 −μ2 |C ∼ h[μ x x x
(4.22)
and so the implicit mapping theorem implies that μ1 = μ2 . Thus we get a bijection v → (μ, w) with the Lipschitz continuity 1 − w 2 L 2x . v 1 − v2 X ∼ |μ1 − μ2 |C + w
(4.23)
For the frame field e, we consider the matrix M = f ⊗ e, together with the equivalent set of Eqs. (4.3) and (4.4). Integrating (4.3) from r = ∞, we get M − eiα Y v/r ˇ L 2x vr L 2x + v/r ˇ L 2x h s1 /r L 2x δ, M1 − M2 Y |μ1 − μ2 |C + δ v 1 − v2 X + δe1 − e2 L ∞ + vˇ 1 − vˇ 2 X ,
(4.24)
while (4.4) provides e1 − e2 L ∞ M1 − M2 L ∞ + |μ1 − μ2 |C + z 1 − z 2 L ∞ . x x x
(4.25)
Hence for fixed v ∈ m (δ) (and μ), we can get (M, e) ∈ Y × L ∞ by the contraction mapping principle for the system of (4.3) and (4.4). Moreover we get v 1 − v2 X . M1 − M2 Y + e1 − e2 L ∞ x
(4.26)
If (μ, q) ∈ C × L 2 (δ) is given, we consider the system of Eqs. (4.3), (4.4) and
m m z = Rϕs Mq − vˇ3 z + h s1 γ , r r
(4.27)
which is equivalent to the q equation in (4.7) under the orthogonality (z | ϕ s ) = 0. The last equation provides, through the uniform bound on Rϕs , z 1 −z 2 X |μ1 − μ2 |C + q 1 − q 2 L 2x +δM1 − M2 Y + δz 1 − z 2 X . (4.28) Combining this with (4.24) and (4.25), we get (z, M, e) for any fixed (μ, q) by the contraction mapping, and moreover they satisfy z 1 − z 2 X + M1 − M2 Y + e1 − e2 L ∞ |μ1 − μ2 |C + q 1 − q 2 L 2x . (4.29) # " So far we have derived estimates at each fixed t, for the energy norms in the above lemma, and for the dispersive norms in Proposition 3.2. Now we turn to the main part of this paper, the analysis of the global dynamics.
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221
5. Decay Estimates for the Remainder In this section, we derive dissipative or dispersive space-time estimates of the remainder vˇ in terms of z, from Eq. (2.15) for q. First by the smallness of z, we obtain from (3.20) and (3.22), z(t) X q(t) L 2 δ, z/r L ∞ q L ∞ , p p
(5.1)
for all p ∈ [1, ∞]. Next we estimate the factor S, by using ∞ r f gdr L ∞ 2 j−k f g L 1 (r ∼2k ) ∼ f g L 1 ≤ f L 2 g L 2 . (5.2) 1 r
j∈Z k≥ j
Then from the expression in (2.15) for S, we have S(t) L 2 S(t)r −1 L ∞ (q L 2x + zr L 2x + 1)L ∗v q L 2x L ∗v q L 2x . 1
1
(5.3)
In the dispersive case a1 = 0, we avoid the derivative by using expression (2.14) S(t) L 2 S(t)r −1 L ∞ (q L 2x + zr L 2x + 1)q L ∞ q L ∞ (a1 = 0). 2 2 1
1
(5.4) For the time decay estimates, we treat the dissipative and the dispersive cases separately. 5.1. Dissipative L 2t estimate. Here we assume a1 > 0. By Eq. (2.15) of q, we have ∂t q2L 2 = −2a1 L ∗v q2L 2 ,
(5.5)
∗ q L ∞ 2 + L v q L 2 L 2 q(0) L 2x ∼ δ. t Lx
(5.6)
hence t
x
Since Rϕs∗ L s∗ = I and Rϕs∗ : L 2 → r L 2 by Lemma 3.1 and duality, we have ˇ L ∞ qr L 2x . qr L 2x L s∗ q L 2x L ∗v q L 2x + v
(5.7)
Since the last term can be absorbed by (4.9) smallness of v, ˇ we get q X q/r L 2x + L s∗ q L 2x L ∗v q L 2x .
(5.8)
So by using the bound (3.17) on Rϕs , we obtain zr L 2 X q L 2 X L ∗v q L 2 q(0) L 2x ∼ δ, t
t
t,x
(5.9)
and also from (5.3), S L 2 L 2 δ. t
x
(5.10)
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5.2. Dissipative decay. Next we show the convergence q → 0 as t → ∞, by comparing it with the free evolution. For T > 0, let (m−1)
q T := q − e(t−T )a2
q(T ).
(5.11)
Then we have qtT − a2(m−1) q T = (i S − aV )q, q T (T ) = 0,
(5.12)
where the potential V (t, x) is given by V =
2m(1 − v3 ) m + w3 . r2 r
(5.13)
Multiplying the equation with q T , we get the energy identity t m−1 T 2 1 T 2 q L 2 + q L 2 )dt a1 (qrT 2L 2 + x x x 2 r T t = Re (−aV q + i Sq | q T )dt,
(5.14)
T
and hence by Schwarz, and using estimate (5.3) to put S ∈ L 2t L 2x , q T L ∞
t>T
L 2x ∩L 2t>T X
q/r L 2
L 2x
+ Sq L 2
q/r L 2
L 2x
+ q L ∞ 2 q L 2 t Lx
t>T t>T
t>T
L 1x
+ q2L 4
t>T
t>T
L 4x
→0
X
(T → ∞). (5.15) (m−1)
Hence q(t) L 2x can not converge to a positive number, since e(t−T )a2 as t → ∞ for all T > 0. Thus we obtain z(t) X q(t) L 2x → 0 (t → ∞).
q(T ) → 0 (5.16)
5.3. Dispersive L 2t estimate. Next we consider the case a1 = 0 (and a2 = 0). We set (with no loss of generality) a = i. Since the energy identity provides only L 2x bound on q, we have to work with the Strichartz estimate in a perturbative way. Denoting H s := L s L s ∗ , the equation of q is given by qt + i H s(0) q = N1 + N2 ,
(5.17)
where s(0)
N1 := −2am
h3
s(t)
− h3 q, r2
N2 := i Sq − 2am
vˇ3 w3 q − am q, 2 r r
(5.18)
and S is given by (2.14). We have |N1 | |h 3 (s(t)/s(0))||q|/r 2 ,
(5.19)
and so N1 L 2 L 1 h 3 (s(t)/s(0)) L ∞ q L 2 L ∞ . t t
2
t
2
(5.20)
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223
Using (5.4), we have Sq L 1 L 2 ≤ S L 2 L 2 q L 2 L ∞ q2L 2 L ∞ . t
t
x
t
x
x
t
(5.21)
2
The other terms in N2 are bounded in L 1t L 2x by q2L 2 L ∞ + z/r L 2 L ∞ q L 2 L ∞ q2L 2 L ∞ . t
t
2
t
2
2
t
(5.22)
2
Now we need the endpoint Strichartz estimate for H s with fixed scaling s: (m−1)
Lemma 5.1. Let H s = L s L s ∗ = −2
+ 2mr −2 (1 − h s3 ) and m > 1. Then we have
e−i H t ϕ L ∞ L 2 ∩L 2 L ∞ ϕ L 2x s
t
t −∞
e
−i H s (t−t )
x
t
2
f (t )dt L ∞ L 2 ∩L 2 L ∞ f L 1 L 2 +L 2 L 1 , t
x
t
t
2
x
t
(5.23)
2
uniformly for any fixed s > 0. This lemma will be proved in Sect. 10.2. Hence if | log(s(t)/s(0))| 1 for all t, then we have q L ∞ L 2 ∩L 2 L ∞ q(0) L 2 ∼ δ,
(5.24)
S L 2 L 2 δ.
(5.25)
t
x
t
2
and also from (5.4) t
x
5.4. Dispersive decay. Next we prove the following asymptotics of scattering type for q and z: (m−1)
e−it2
q(t) → ∃q+ in L 2x , z → 0 in L ∞ (t → ∞). x
(5.26)
For the scattering of q, we further expand the equation (m−1)
qt − i2
q = N0 + N2 ,
(5.27)
1 − h s3 q. r2
(5.28)
where N2 is as in (5.18), and N0 := −2am
Then the global Strichartz bound implies that N0 L 2 L 1 (T,∞) → 0, N2 L 1 L 2 (T,∞) → 0 t
t
2
x
(5.29)
(m−1)
as T → ∞. By Strichartz (for 2 ) once again, we get the scattering of q. For the vanishing of z, we use the inversion formula z = Rϕs g, g = Mq + r −1 m(h s1 γ − vˇ3 z).
(5.30)
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Since Rϕs is bounded L 2x → L ∞ , the latter two terms contribute at most with z L ∞ z/r x L 2x z L ∞ , hence we may drop them. Also we may replace q by its asymptotic free x (m−1)
solution q ∞ := eit2 q+ . Moreover we may approximate q+ by nicer functions. Hence we assume that q := Fm−1 q+ ∈ C0∞ (0, ∞). Then we may further replace the free solution with the stationary phase part: ∞ 2 2 q ∞ (t, r ) = Cm t −1 eir /(4t) Jm−1 (rρ/(2t))eiρ /(4t) q+ (ρ)ρdρ 0 −1 ir 2 /(4t)
= Cm t
q (r/(2t)) + R,
e
(5.31)
where the error is bounded by Plancherel R L 2x ∼ (1 − eir
2 /(4t)
)q+ L 2x t −1 r 2 q+ L 2x → 0.
(5.32)
Now that spatially local vanishing is clear (eg. it follows from Rϕs Mq(t)r L ∞ q ∞ (t) L ∞ → 0), we may extract the leading term of Rϕs for large x. We assume s that s(t) ∈ L ∞ t and supp ϕ ⊂ (0, b) for a fixed b ∈ (0, ∞). Then for r > b we have r s h 1 (r ) (Rϕs g)(r ) = o(1) + s g(r )dr b h 1 (r ) r = o(1) + (r /r )m g(r )dr as r → ∞. (5.33) b
Thus we are reduced to showing that r 2 Gχ := (ρ/r )m M(t, ρ)t −1 eiρ /(4t) χ (ρ/t)dρ → 0 in L r∞
(5.34)
b
for any χ ∈ C0∞ (0, ∞). By partial integration on (ρ/t)eiρ
2 /(4t)
, we have
r m Gχ = (i/2)[ρ m−1 M(ρ)eiρ /(4t) χ (ρ/t)]rb r (m − 1)M(ρ)χ (ρ/t)/ρ + M(ρ)χ (ρ/t)/t − 2
b
+ Mr (ρ)χ (ρ/t)] ρ m−1 eiρ
2 /(4t)
dρ.
(5.35)
by r m /t,
using |χ (ρ/t)| ρ/t for the first, second and The right-hand side is bounded 1 fourth terms, |χ (ρ/t)| 1 for the third, and Mr ∈ L ∞ t L (dr ) for the fourth term. Thus we obtain z(t) L ∞ → 0. x Thus we have obtained the following a priori estimates in this section Proposition 5.2. Let m ≥ 2 and u(t, x) = emθ R v(t, r ) be a solution of (1.1) on 0 < t < T with u(0) ∈ m and E( u (0)) ≤ 4mπ + δ 2 for some small δ > 0. Let q, z, S be as in Proposition 2.1, and let μ(t) be given by Lemma 4.1. (I) If a1 > 0, then we have z L ∞ (0,T ;X )∩L 2 (0,T ;r X ) q L ∞ (0,T ;L 2 )∩L 2 (0,T ;X ) q(0) L 2x ∼ δ. (5.36) t
t
t
x
t
Moreover, if T = ∞ then z(t) X q(t) L 2x → 0 (t → ∞).
(5.37)
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225
(II) If a1 = 0 and s(t) = s(0) + O(δ), then we have z L ∞ (0,T ;X )∩L 2 (0,T ;r L ∞ ) q L ∞ (0,T ;L 2 )∩L 2 (0,T ;L ∞ ) q(0) L 2x ∼ δ. t
t
2
t
t
x
2
(5.38) Moreover, if T = ∞ and s(t) converges as t → ∞, then (m−1)
→ 0, q(t) − e−it2 z(t) L ∞ x (m−1)
for some radial q+ ∈ L 2x . 2
q+ L 2x → 0, (t → ∞)
(5.39)
is the (m − 1)-equivariant Laplacian, see (1.37).
Note that the decay of z is transferred to the remainder vˇ by Lemma 4.1. By using the above arguments and Lemma 4.1, it is easy to see that the solution is global unless s(t) → 0 in finite time (for a detailed proof, see [11, Sect. 3]). The remaining sections are therefore devoted to the analysis of the parameter dynamics, which is the most novel part of this paper. 6. Parameter Evolution It remains to control the asymptotic behavior of the parameter μ(t) of the harmonic map part of the solution. Its evolution is determined by differentiating the localized orthogonality condition 0 = ∂t (z | ϕ s ) = (vˇt · f | ϕ s ) + (vˇ · ft | ϕ s ) + (z | ∂t ϕ s ),
(6.1)
and each term on the right is expanded by using t = −(a L ∗ q) ◦ e − h s1 μ˙ ◦ f, vˇt = vt − h[μ] v s˙ s s ˙ − h 1 μh[μ], ˙ ∂t ϕ s = − r ∂r ϕ s . ft = −i h 3 αf s
(6.2)
Plugging this into the above and then dividing it by s 2 , we get μ˙ = −(Ma L ∗v q | ϕ s ) − (h s1 μγ ˙ | ϕ s ) − (z | (
μ˙ 1 r ∂r − i μ˙ 2 h s3 )ϕ s ). m
(6.3)
The last two terms are bounded by |μ|z ˙ L ∞ (ϕ L 1 + r ∂r ϕ L 1 ),
(6.4)
and so absorbed by the left-hand side since z L ∞ δ 1. h s + |v| Since |ν| = |P v k| ˇ and hence 1 | vr | |q| + |z|/r + h s1 /r,
(6.5)
|Mr | |qz| + |z|2 /r + |zh s1 |/r.
(6.6)
we get from (4.3), The leading (first in the r.h.s) term in (6.3) can be estimated, using [Ma, L ∗v ]
= Mr a,
as follows:
)qr L 2x |(Ma L ∗v q | ϕ s )| s −1 (Mr L 2x + M L ∞ x s −1 (q L 2x + z/r L 2x + 1)qr L 2x .
(6.7)
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S. Gustafson, K. Nakanishi, T.-P. Tsai
Hence using that z L ∞ z X q L 2 δ 1, we get x s μ ˙ L 2 q/r L 2 q L 2 L ∞ . t
t,x
t
(6.8)
2
Then the last two terms of (6.3) are bounded in L 1t by s
s μ ˙ L 2 z/r L 2 L ∞ (r |ϕ| + r 2 |ϕr |) L 1x q2L 2 L ∞ , t
t
x
t
(6.9)
2
where we used (5.1). Thus we have obtained Proposition 6.1. Let v, q, μ, ϕ and M as in Proposition 5.2 and Lemma 4.1. Then μ(t) satisfies μ˙ = −(Ma L ∗v q | ϕ s ) + err or,
(6.10)
where L ∗v = −∂r − 1/r + mv3 /r , and s μ ˙ L 2 (0,T ) q L 2 (0,T ;L ∞ ) , err or L 1 (0,T ) q2L 2 (0,T ;L ∞ ) . t
t
2
t
t
(6.11)
2
Thus our problem is reduced to the global behavior of the above term on the right, which is linear in q. 7. Partial Integration for the Parameter Dynamics Now we want to integrate in t the right-hand side of (6.3), which is not bounded in L 1t . The key idea is to employ the q Eq. (2.15), by identifying a factor of L v L ∗v q, through a partial integration in space. For the spatial integration, we first freeze the phase factor M. Since h[μ] = v = −k i α i α at r = 0, we have M(t, 0) = e , i.e. f(t, 0) = e e(t, 0) for some real α (t). Then Dt f(t, 0) = i α (t)f(t, 0) − i S(t, 0)f(t, 0), and so α (t) = S(t, 0) + α (t).
(7.1)
α ˇ M = ei + M,
(7.2)
We decompose
and rewrite the leading term of (6.3) as follows. Let c = h 1 −2 . Since L v = L s +m vˇ3 /r L2 s and L s h 1 = 0, we have α ˇ L ∗ q | ϕ s ) (Ma L ∗v q | ϕ s ) = aei (L ∗v q | ϕ s ) + (Ma v
i α ∗ s = ae (L v q | (ϕ − ch 1 ) ) + (mq vˇ3 /r | ch 1 s )
ˇ r aq | ϕ s ). ˇ + (Maq | L v ϕ s ) + (M
(7.3)
The second term is bounded by q vr ˇ −3 L 1x q/r L 2x z/r 2 L 2x , and the last two terms are bounded by ˇ L ∞ ), q/r L 2x (Mr /r L 2x + M/r x
(7.4)
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227
ˇ = 0 at r = 0, where the last factor is further bounded by using that M ˇ L ∞ Mr /r L 2 q/r L 2 + z/r 2 L 2 q L ∞ . M/r x 2 x x x
(7.5)
We further rewrite the remaining (main) term. By the definition of c, we have (ϕ − ch 1 | h 1 ) = 1 − ch 1 2L 2 = 0,
(7.6)
ϕ s − ch s1 = L s∗ Rϕs∗ (ϕ s − ch s1 ),
(7.7)
and so we have
where the operator Rϕs was defined in (3.11). Let ψ := Rϕ∗ (ϕ − ch 1 ) = −
c r 1−m + O(r 1−3m ) (r → ∞), m−1
(7.8)
where the asymptotic form easily follows from the fact that ∞ −1 ψ(r ) = −c(h 1 (r )r ) h 1 (r )2 r dr (r $ 1).
(7.9)
r
Then we have, by using Eq. (2.15) for q, (−a L ∗v q|(ϕ − ch 1 )s ) = (−a L s L ∗v q | ψ s /s)
= (qt − i Sq | ψ s /s) + (amq | L v vˇ3 r −1 ψ s /s),
(7.10)
and, using (3.9), the last term is bounded by q(|q| + |v/r ˇ |)r −2 L 1x (q/r L 2x + z/r 2 L 2x )2 q2L ∞ .
(7.11)
2
For m ≥ 2, ψ ∈ L 2∞ , and so (Sq | ψ s /s) L 1 S L 2 L 2 q L 2 L ∞ δq L 2 L ∞ , t
t
t
x
t
2
(7.12)
2
either by (5.10) or (5.25). Thus we have obtained α (qt | ψ s /s) L 1 δq L 2 L ∞ . μ˙ − ei t
t
(7.13)
2
Integrating by parts in t, the leading term is rewritten as α α α (qt | ψ s /s) = ∂t (ei q | ψ s /s) − i(s α˙ + s S(t, 0))(ei q | ψ s ) ei α +˙s (ei q | (r ∂r + 1)ψ s ).
(7.14)
The last term can be bounded in L 1t by using (6.8), s μ ˙ L 2 (q | (r ∂r + 1)ψ s ) L 2 q L 2 L ∞ (q | (r ∂r + 1)ψ s ) L 2 . t
t
t
2
t
(7.15)
If m = 2 or m > 3, then (r ∂r + 1)ψ ∈ L 1 , and so the above is further bounded by q2L 2 L ∞ . When m = 3, we need some extra effort to bound the last factor in L 2t – this t
2
is done in the next section.
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S. Gustafson, K. Nakanishi, T.-P. Tsai
If m > 2, we have for the leading term α |(ei q | ψ s /s)| q L 2x ψ L 2x q(0) L 2x ,
(7.16)
while for m = 2 this term can be infinite from the beginning. We will show in Sect. 9 that α q | ψ s /s)]t can be controlled for finite t, but still may become the time difference [(ei 0 unbounded as t → ∞ for some initial data. This also means that the second to last term of (7.14) is beyond our control when m = 2, and so in this case we force it to vanish by making the assumptions a2 = 0 and v2 = 0. For the other cases (m > 2), we should estimate S(t, 0), for which we use in the Schrödinger case (a = i) that ∞ 1 2Q m dr, Q = |q|2 + w3 = O(|q|2 + |z/r |2 + qh s1 /r ), (7.17) S = −Q + r 2 r r since |w3 | |q||ν| and |ν| h s1 + |z|. Thus we get at each t, using (5.1), S L ∞ q2L ∞ + s −1 q L ∞ . x
(7.18)
2
Then the second term in (7.14) is bounded in L 1t , + (s α ˙ L 2 + q L 2 L ∞ )(q | ψ s ) L 2 q2L 2 L ∞ (q | ψ s /s) L ∞ t t
t
2
t
t
x
q L 2 L ∞ (δ + (q | ψ s ) L 2 ), t
(7.19)
t
2
where we used (6.8). If m > 3, then ψ ∈ L 1 and hence the last factor (q | ψ s ) L 2 is t bounded by q L 2 L ∞ δ. Its estimate for m = 3 is deferred to the next section. t 2 In the dissipative case a1 > 0, we estimate simply by (2.15) at each t, S L ∞ (q/r L 2x + z/r 2 L 2x + h s1 /r 2 L 2x )L ∗v q L 2x , x
(7.20)
and hence the second term in L 1t is bounded by ˙ L 2 + q L 2 X )(q | ψ s ) L 2 q L 2 L ∞ q L 2 X + (s μ t
t
2
t
t
t
s
q L 2 L ∞ (δ + (q | ψ ) L 2 ), t
(7.21)
t
2
where we used (6.8) and (5.9). Thus we have obtained all the necessary estimates to prove Theorem 1.1 when m > 3. In summary, we have Proposition 7.1. Under the same assumptions as for Proposition 6.1, we have α α α μ˙ = ∂t (ei q|ψ s /s) − is α(e ˙ i q|ψ s ) + s˙ (ei q|(r ∂r + 1)ψ s ) + err or,
(7.22)
where err or L 1 (0,T ) δq L 2 (0,T ;L ∞ ) . Moreover, if m = 2 or m > 3, then the second t t 2 to last term can be included in the error. The proof of Theorem 1.1 for m = 3 will be complete once we show (q | ψ s ) L 2 + (q | (r ∂r + 1)ψ s ) L 2 δ, t
t
(7.23)
which will be done in Sect. 8. This estimate together with the above proposition implies the convergence of μ(t) = μ(0) + O(δ), closing all the estimates and the assumptions in the previous sections. For Theorem 1.2, it remains to derive the asymptotic formula (1.24) from the leading term (q | ψ s /s), and to show that all of the asymptotic behavior (1)–(6) can be realized by the choice of the initial data u(0, x) – this is done in Sect. 9.
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8. Special Estimates for m = 3 In this section we finish the proof of Theorem 1.1 by showing (7.23). It suffices to estimate the leading term for r → ∞: (r −2 χ (r ) | q) L 2 δ,
(8.1)
t
with χ ∈ C ∞ satisfying χ (r ) = 0 for r < 1 and χ (r ) = 1 for r > 2, since the rest decays at slowest O(r −8 ) ∈ L 1x , for which we can simply use q ∈ L 2t L ∞ x . Once the above is proved, we can conclude that μ L ∞ |μ(0)| + q L ∞ L 2 ∩L 2 L ∞ . t t
x
t
(8.2)
2
The boundedness of μ and the scattering of q imply that the “normal form” correction (ei α˜ q | ψ s /s) converges to zero, and so μ(t) is convergent as t → ∞. (2) To estimate (8.1), we use perturbation from the free evolution eat2 : (2)
q˙ − a2 q = N0 + N2 ,
(8.3)
where N0 and N2 are as in (5.28) and (5.18), satisfying N0 ∈ r −2 &r/s'−4 L 2t L ∞ x ,
N2 ∈ L 1t L 2x .
(8.4)
For the contribution of N2 as well as the initial data, we use the following estimate. Lemma 8.1. For any l > 0, any a ∈ C× with Re a ≥ 0, and any functions g(r ), f (r ), and F(t, r ), we have (l)
(g |
(g | eat2 f ) L 2 (0,∞) r 2 g L ∞ f L 2x , x t
t −∞
e
(l) a(t−s)2
F(s)ds) L 2 (R) r 2 g L ∞ F L 1 L 2 . x t
t
(8.5)
x
f = Fl f . The Proof. We start with the estimate for the free part. Let g = Fl g and 2 above L t norm equals by Plancherel in space, ∞ 2 ( g | e−atr e−atσ G(σ )dσ L 2 (0,∞) , (8.6) f ) L 2 (0,∞) ∼ t
t
0
where we put G(σ ) := g (σ 2 ) f (σ 2 ).
(8.7)
If a1 > 0, then (8.6) is bounded by Minkowski, ∞ ∞ ≤ e−a1 tσ |G(σ )|dσ L 2 (0,∞) ≤ e−a1 σ t −1 G(σ/t) L 2 (0,∞) dσ t t 0 0 ∞ σ −1/2 e−a1 σ dσ G L 2σ (0,∞) . (8.8) ≤ G L 2σ (0,∞) 0
If a1 = 0, then a2 = 0 and (8.6) is bounded by Plancherel in t, ∞ ≤ e−ia2 tσ G(σ )ds L 2 (R) ∼ G L 2σ (0,∞) . 0
t
(8.9)
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Thus in both cases we obtain (l)
(g | eat2 f ) L 2 (0,∞) G(σ ) L 2σ (R) gL ∞ f L 2 ∼ g L ∞ f L 2x . t
Then the first desired estimate follows from ∞ | g (ρ)| ≤ |Jl (rρ)||g(r )|r dr ≤ r 2 g L ∞ x 0
∞
|Jl (r )|
0
(8.10)
dr ∼ r 2 g L ∞ , (8.11) x r
since |Jl (r )| min(r l , r −1/2 ) for r > 0. By duality, the estimate on the Duhamel term is equivalent to ∞ (l) 2 λ(s + t)eas2 g(x)ds L ∞ 2 r g L ∞ λ L 2 , x t Lx
(8.12)
which is equivalent to
(8.13)
t
0
∞ 0
(l)
λ(t)eat2 g(x)dt L 2x r 2 g L ∞ λ L 2 , x t
which is dual to the first estimate.
# "
For the potential part N0 , we transfer the equation to R6 by u = r −2 q and consider −2 u t − a(0) N0 . 6 u =r
(8.14)
Then thanks to the decay of the potential, we have r −2 N0 ∈ L 2t L x
10/7
−1/5
(R6 ) ⊂ L 2t H˙ 3/2 (R6 ),
(8.15)
as long as s(t) is away from 0 and ∞. Then by the endpoint Strichartz or the energy −1/5 estimate on R6 , the corresponding Duhamel term is bounded in L 2t H˙ 3 (R6 ), and since 1/5 1 −4 −5 −4 6 6 |∇x r χ (r )| r , we have r χ ∈ H˙ 5/4 (R ) ⊂ H˙ 3/2 (R ). Thus to summarize, we have (r −2 χ | q) L 2 q(0) L 2x + q L 2 L ∞ . t
t
2,x
(8.16)
Completion of the proof of Theorem 1.1. Let initial data u(0) be specified as in Theorem 1.1. The existence of a unique local-in-time solution u(t) in the given spaces can be deduced by working in the (μ, q) variables (using the bijection of Lemma 4.1) and using estimates similar to those of Sects. 5 and 6. The details are carried out in the Schrödinger case (a = i) in [12], and carry over to the general case in a straightforward way (in fact, there are well-established methods for energy-space local existence in the dissipative case, starting with the pioneering work [19] on the heat-flow). It follows from this local theory that the solution continues as long as μ(t) is bounded and q is bounded 2 2 ∞ in L ∞ t Lx ∩ Lt L2 . For m > 3, the estimates of the previous four sections give the boundedness of q and μ which ensure the solution is global, as well as the convergence of μ(t). The convergence to a harmonic map then follows from the estimates of Sect. 5. " #
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9. Special Estimates for m = 2, a > 0, v2 = 0 Let m = 2 and (with no further loss of generality) a = 1. By the bijective correspondence v ↔ (μ, q), it is clear that v2 = 0 is equivalent to μ, q ∈ R. It remains to control the leading term for the parameter dynamics (q | ψ s /s).
(9.1)
In particular, we will show that this can diverge to ±∞, or oscillate between them for certain initial data. −1 ∈ L 2x , where we denote First by the asymptotics for r → ∞, we have ψ + cr1< −1 ra<
r −1 (r > a) = 0 (r ≤ a)
−1 r
r −1 (r < b) = 0 (r ≥ b)
−1 ra
r −1 (a < r < b) = . (9.2) 0 (otherwise)
−1 modulo o(1)L ∞ . Hence we may replace s −1 ψ s by −crs< t
(1)
Next we want to replace q by the free solution q 0 := et2 q(0). For that we use the following pointwise estimate to bound q − q0 : Lemma 9.1. Let Re a > 0 and l ≥ 1. Then for any function g(r ) satisfying |g(r )| ≤ &r '−1 , we have (l)
|eat2 g(r )| min(1, 1/r, r/t).
(9.3)
Proof. Let a1 = Re a. By using the explicit kernel we have (l)
e2at2 g(r ) =
C t
∞ π
0
−π
e−a
r 2 −2rq cos θ+q 2 +ilθ t
g(q)qdθ dq,
(9.4)
and the integral in θ can be rewritten by partial integration on eiθ as
π −π
r 2 −2rq cos θ+q 2 rq +ilθ t sin θ e−a g(q)qdθ dq. ilt
(9.5)
The double integral for |q − r | > r/2 is bounded by using the second form by
∞ 0
r2 rq −a1 r 2 +q 2 4t e min(q, 1)dq r e−a1 4t min(t −1/2 , t −1 ) min(1, r/t, 1/r ), 2 t
(9.6)
and that for |q − r | < r/2 is bounded by using the first form by t −1
∞ π 0
−π
e−a1
(r −q)2 t
e−a1
r2θ2 8t
dθ min(r, 1)dq t −1 t 1/2 (t/r 2 )1/2 min(r, 1)
and by the second form by r 2 /t × 1/r = r/t.
min(1, 1/r ), # "
(9.7)
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The nonlinear part of q contributes as t (m−1) −1 −1 | q − q 0 ) = − (e2 (t−t )rs(t)< | V (t )q(t ))dt , (rs<
(9.8)
0
where the potential term is given by V =
2m(1 − h s3 ) 2m(1 − v3 ) m + = + O(vˇ3 /r 2 ) + O(q/r ). w 3 r2 r r2
(9.9)
The contribution from the last two parts is estimated with the r −1 bound from the above lemma, thus bounded by (vˇ3 /r 2 L 2 + q/r L 2 )q/r L 2 q(0)2L 2 . t,x
t,x
t,x
x
(9.10)
We need to be more careful to estimate the other term q(1 − h s3 )/r 2 . First by Schwarz and the pointwise estimate, we have t 2 (m−1) (ea2 (t−t )r −1 | r −2 q(1 − h s ))dt 3 s(t)< 0 t ∞ −4m dr dt , (9.11) ≤ q/r 2L 2 min(s(t)−1 , r −1 , r/(t − t ))2 &r/s(t )' t,x 0 r 0 where we also used that |1 − h 3 (r )| &r '−2m . It suffices to bound the last double integral. Let τ = t − t . For 0 < τ < s(t)2 , the r integral is bounded by s(t) ∞ τ/s(t) 2 r dr 1 dr 1 dr + + s(t)−2 (1 + log(s(t)2 /τ )), (9.12) 2 r 2 r 2 r τ s(t) r 0 τ/s(t) s(t) hence its τ integral is bounded by s(t)2 −2 2 s(t) (1 + log(s(t) /τ ))dτ = 1 + 0
1
| log θ |dθ < ∞.
(9.13)
0
For s(t)2 < τ , the r integral is bounded by ∞ 2 4m s(t ) 2 s(t )2 r dr r s(t ) dr + , 2 4m τ2 r r τ2 0 s(t ) τ r
(9.14)
and its τ integral is bounded by the square of s(t )/τ L 2τ (s(t)2 ,t) s(t)/τ L 2τ (s(t)2 ,t) + (s(t) − s(t − τ ))/τ L 2τ (s(t)2 ,t) 1 1+ ˙s (t − θ τ ) L 2τ (s(t)2 ,t) dθ 0
1+ 0
1
˙s L 2 θ −1/2 dθ 1.
(9.15)
Thus we obtain −1 |(rs(t)< | q − q 0 )| q/r L 2 (vˇ3 /r 2 L 2 + ˙s L 2 + 1) q(0) L 2 , t,x
t,x
t
namely we may replace q by the free solution q 0 in the leading asymptotic term.
(9.16)
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233
Furthermore, we can freeze the scaling parameter because −1 −1 −1 −1 − rs(0)< | q 0 )| rs(t)< − rs(0)< L 2 q 0 (t) L 2 |(rs(t)<
|[log s]t0 |1/2 o(1) o(1)(|[log s]t0 | + 1).
(9.17)
Thus we obtain −1 (1 + o(1))[2 log s]t0 = −c[(rs(0)< | q 0 )]t0 + O(1), (t → ∞),
(9.18)
where O(1) is convergent. The leading term is further rewritten in the Fourier space by using that ∞ −1 F1 [rs(0)< ](ρ) = ρ −1 J1 (r )dr s(0)ρ
=ρ
−1
−1 J0 (s(0)ρ) = ρ<1/s(0) + s(0)R(s(0)ρ), ∃R ∈ L 2x .
(9.19)
q0 L 2x and Let q0 := F1 q(0). By Plancherel we have q(0) L 2x = −1 −1 | q 0 )]t0 = ((1 − e−tr )r<1/s(0) | q0 ) + O(1) [−(rs(0)< 2
−1√ = (r1/ | q0 ) + O(1) t<1/s(0) −1√ = (F1r1/ | q(0)) + O(1) t<1/s(0) √ t = 2π q(0, r )dr + O(1).
(9.20)
s(0)
Thus we obtain (using that c = h 1 −2 = π −2 ), L2 x
1 (1 + o(1))[log s]t0 = π
√
t
q(0, r )dr + O(1),
(9.21)
s(0)
and the error term O(1) converges to a finite value as t → ∞. Completion of the proof of Theorem 1.2. As in the proof of Theorem 1.1, we now have all the estimates to conclude the solution is global (in particular, μ(t) remains finite by the above formula and estimates), and the convergence to the harmonic map family follows from the estimates of Sect. 5. It remains to consider the asymptotics of s(t). ∞ Since q(0) ∈ L 2x does not require 1 |q(0, r )|dr < ∞, it is easy to make up q(0) ∈ L 2 , for any given s(0) ∈ (0, ∞), such that the first term on the right of (9.21) attains arbitrarily given lim sup ≥ lim inf ∈ [−∞, ∞] as t → ∞. In particular, all of the asymptotic behaviors (1)-(6) in Theorem 1.2 can be realized by appropriate choices of (q(0), s(0)), for which Lemma 4.1 ensures existence of corresponding initial data u(0) ∈ 2 . Using that v2 = 0, we can further rewrite the leading term in terms of v. Since e = (v3 , i, −v1 ), we have q=w · e = v1 v3r − v3 v1r +
2v1 2v1 = −βr + , r r
(9.22)
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S. Gustafson, K. Nakanishi, T.-P. Tsai
where β is defined by v = (cos β, 0, sin β). Hence we have (1 + o(1))[log s]t0 where O(1) converges as t → ∞.
2 = π
√
t
s(0)
v1 (0, r ) dr + O(1), r
(9.23)
# "
10. Proofs of the Key Linear Estimates 10.1. Uniform bound on the right inverse Rϕ . Proof of Lemma 3.1. Let s = 1 and omit it. It suffices to prove Rϕ gr θ L ∞ ϕr −θ L 1 gr θ+1 L 1∞ ,
Rϕ∗
f r −θ−1 L ∞ ϕr −θ L 1∞ f r −θ L 1∞ .
(10.1)
From this we get by duality, Rϕ gr θ L ∞ ϕr −θ L 1∞ gr θ+1 L 1 ,
Rϕ∗
1
f r −θ−1 L ∞ ϕr −θ L 1 f r −θ L 1 ,
(10.2)
1
and the bilinear complex interpolation covers the intermediate cases. It remains to prove (10.1). We rewrite the kernel of Rϕ , h 1 (r ) χ (r, r , r )ϕ(r )h 1 (r )r g(r )dr dr , Rϕ g = h 1 (r )
(10.3)
where χ (r ) is defined by
⎧ ⎪ (r < r < r ), ⎨1 χ (r, r , r ) = −1 (r < r < r ), ⎪ ⎩0 (otherwise).
(10.4)
We decompose the double integral dyadically such that r ∼ 2 j , r ∼ 2k and r ∼ 2l , and let A j = 2−θ j Rϕ g L ∞ (r ∼2 j ) ,
B j = 2(θ+1) j Rϕ∗ f L ∞ (r ∼2 j ) ,
ϕl = 2θl ϕ L 1 (r ∼2l ) , gk = 2(−θ−1)k g L 1 (r ∼2k ) , For Rϕ , we have Aj
f k = 2θk f L 1 (r ∼2k ) .
2−m| j|−θ j+m|k|+θk−m|l|−θl ϕl gk .
(10.5)
(10.6)
j−1≤k≤l+1 l−1≤k≤ j+1
The sums over k are bounded for j − 1 ≤ k ≤ l + 1 and for l − 1 ≤ k ≤ j + 1 respectively by max(2(−m+θ) j , 1) max(2(m+θ)l , 1), −m| j|−θ j−m|l|−θl 2 ϕl sup gk × (10.7) max(2(−m+θ)l , 1) max(2(m+θ) j , 1), k
Stability and Oscillation of Harmonic Maps
235
and since the exponential factors are bounded, after summation over l we get Rϕ gr θ L ∞ sup ϕl gk , l
(10.8)
k
as desired. For Rϕ∗ in (10.1), we have
Bj
2m| j|+θ j−m|k|−θk−m|l|−θl ϕl f k .
(10.9)
k−1≤ j≤l+1 l−1≤ j≤k+1
Then the sums over k and l are bounded in both cases by 2m| j|+θ j sup ϕl f k min(2m j−θ j , 1) min(2−m j−θ j , 1),
(10.10)
k,l
and hence Rϕ∗ f r −θ−1 L ∞ supl,k ϕl gk , as desired. Next we show the optimality. Let b ∈ Z, and choose any g which is piecewise constant on each dyadic interval (2 j , 2 j+1 ), supp g ⊂ [2b , ∞), and g ≥ 0. Then for 0 < r ≤ 2b we have ∞ a h(s)−1 ϕ(a)h(a)ag(s)dsda Rϕ g(r ) = h(r ) 2b
h(r )
2b j−1
j≥b k=b
j≥b
2m|k|+θk gk 2−m| j|−θ j ϕ j h(r )
g j−1 ϕ j ,
(10.11)
where we denote gk = gr θ−1 L ∞ (r ∼2k ) and ϕ j = ϕr −θ L 1 (r ∼2 j ) . Choosing a test function ψ ∈ C0∞ (0, ∞) satisfying ψ ≥ 0, supp ψ ⊂ (0, 2b ) and (h | ψ) = 1, we p
see that ϕ j ∈ j ( j > b) is necessary since we can choose arbitrary non-negative p gk ∈ k (k > b). Similarly by choosing supp g ⊂ (0, 2b ] and supp ψ ⊂ (2b , ∞), we see p
that ϕ j ∈ j ( j < b) is also necessary.
# "
10.2. Double endpoint Strichartz estimate. Lemma 5.1 holds for more general radial potentials. We call −1 t i(t−s)H r e f (s)ds r f L 2 (10.12) 0
t,x
L 2t,x
the Kato estimate for the operator H , and u L 2 (L ∞ ) u(0) L 2x + iu t + H u L 2 (L 1 ) t
2
t
2
(10.13)
the double endpoint Strichartz estimate for H . Lemma 5.1 is a consequence of the following. Theorem 10.1. For any m > 0, the double endpoint Strichartz (10.13) holds for radially (m) symmetric u(t, x) = u(t, |x|) and H = 2 = ∂r2 + r −1 ∂r − m 2 r −2 .
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Lemma 10.2. Suppose H0 and H = H0 + V are both self-adjoint on L 2 (R2 ) and |x|2 V (x) ∈ L ∞ (R2 ). Assume that the Kato estimate (10.12) holds for H , and that the double endpoint Strichartz estimate (10.13) holds for H0 . Then we have the double endpoint Strichartz also for H . The same is true when we restrict all functions to radially symmetric ones, if V is also symmetric. Corollary 10.3. Let V = V (|x|) ∈ C 1 (R2 \{0}) be a radially-symmetric function with |x|2 V ∈ L ∞ (R2 ), and suppose H = − + V is self-adjoint on L 2 (R2 ). Let f (t, x) = f (t, |x|) be radial. Then (1) the Kato estimate (10.12) holds for H if and only if the double-endpoint estimate (10.13) holds for H , (2) both estimates hold provided inf r 2 V (r ) > 0, inf −r 2 (r V (r ))r > 0.
r >0
r >0
(10.14)
Since our linearized operator H s satisfies (10.14), the above implies Lemma 5.1. Proof of Corollary 10.3. The first statement follows directly from Theorem 10.1 and Lemma 10.2. For the second statement: the methods of [4], adapted to the 2-dimensional radial setting (detailed in [12]), imply that conditions (10.14) yield the resolvent estimate (10.15), hence the Kato estimate, and the double endpoint estimate. " # Remark 1. While the double-endpoint estimate (10.13) always implies the Kato estimate (10.12), the reverse implication does not hold in general. For example, consider H = − acting on 2D functions with zero angular average. The Kato estimate in this case can be verified, for example, by using the methods of [4] to establish the resolvent estimate sup (H − λ)−1 ϕ L 2,−1 ϕ L 2,1 ,
λ=0
x
x
(10.15)
from which the Kato estimate follows by Plancherel in t (see [12] for details). On the other hand, if the double-endpoint estimate were to hold for zero-angular-average functions, so would the endpoint homogeneous estimate. Since the latter is known to hold for radial functions (see Tao [20]), it would therefore hold for all 2D functions, which is false (see Montgomery-Smith [16], also see [20]). Alternatively, a constructive counterexample is given by placing delta functions of the same mass but opposite sign at (1, 0) and (0, 1) in the plane. Proof of Theorem 10.1. Following [15], we use the identity ∞ a+3r dr da a−r F(s, t)dsdt = C ds dt F(s, t), r R r a−3r s
0 is some explicit positive constant. Define the bilinear operators I j for j ∈ Z by I j ( f, g) :=
2 j+1 2j
a+3r (m) dr da a−r ds dt ei(t−s)2 f (s) g(t) x , r R r a−3r a+r
where f and g are radial (i.e. f (s) = f (|x|, s), etc.).
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237
The desired estimate follows from |I j ( f, g)| f L 2 L 1 g L 2 L 1 . t
j∈Z
x
t
x
(m)
Using the 1/t decay for eit2 L 1 →L ∞ , we can easily bound the supremum of the summand. To get summability, we need decay both faster and slower than 1/t. In fact we have, for ϕ = ϕ(|x|) radial, (m)
eit2 ϕ L ∞,μ |t|−1+μ ϕ L 1,−μ , −m ≤ μ ≤ 1/2. x x
(10.16)
This follows easily from the explicit fundamental solution (m) cm ∞ i(r 2 +ρ 2 )/4t rρ it2 φ(ρ)ρdρ (e ϕ)(r ) = e Jm t 0 2t (cm a constant) in terms of the Bessel function Jm of the first kind, for which sup s μ Jm (s) < ∞, −m ≤ μ ≤ 1/2. s>0
Next, when the decay is slower than 1/t, namely if we choose μ > 0 in (10.16), then we get a non-endpoint Strichartz estimate using the Hardy-Littlewood-Sobolev inequality in time: (m) e−is2 f (s)ds L 2x f p 1,−α , (10.17) Lt Lx
R
for 0 < α ≤ 1/2 and 1/ p = 1/2 − α/2. Now the rest of the proof follows along the lines of Keel-Tao [14]. We will prove that |I j ( f, g)| 2 j (α+β)/2 f L 2 L 1,−α g L 2 L 1,−β ,
(10.18)
−m ≤α =β <0
(10.19)
0 < α, β ≤ 1/2.
(10.20)
t
x
t
x
for
and for For the first exponents (10.19), we use the decay estimate (10.16) and the L ∞,α − L 1,−α duality at each (s, t). Then we get 2 j+1 a+3r f (s) 1,−α g(t) 1,−α dr da a−r Lx Lx ds dt |I j ( f, g)| r R r a−3r |t − s|1−α 2j a+r 2 j+1 dr da jα 2 f L 2 (a−3r,a−r ;L 1,−α ) g L 2 (a+r,a+3r ;L 1,−α ) x x t t r R r 2j 2 j+1 dr jα 1 2 f L 2 (−3r
x
where we used Hölder for s, t, a.
t
x
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For the second exponents (10.20), we use the non-endpoint Strichartz (10.17) for both integrals in s and t, after applying the Schwartz inequality in x. Then we get
2 j+1
2j 2 j+1
|I j ( f, g)|
2j
dr r R dr r R
da f p 1,−β g q L t (a−3r,a−r ;L x ) L t (a+r,a+3r ;L 1,−α ) x r da j (α+β)/2 f L 2 (a−3r,a−r ;L 1,−β ) g L 2 (a+r,a+3r ;L 1,−α ) , 2 x x t t r
and the rest is the same as above, where 1/ p = 1/2 + α/2 and 1/q = 1/2 + β/2. Thus we get (10.18) both for (10.19) and (10.20). By bilinear complex interpolation (cf. [3]), we can extend the region (α, β) to the convex hull: 1 1 m 1 m β − α − , β > , α, β < . α> (10.21) 2 2 2 m + 21 m + 21 The only property we need is that this set includes a neighborhood of (0, 0), where we are looking for the summability. Now we use bilinear interpolation (see [3, Exercise 3.13.5(b)] and [17]) T : X i × X j → Yi+ j (i, j, i + j ∈ {0, 1}) (⇒ T : X θ1 ,r1 × X θ2 ,r2 → Yθ1 +θ2 ,r0 1/r0 = 1/r1 + 1/r2 , where X θ,r := (X 0 , X 1 )θ,r denotes the real interpolation space. The above bound (10.18) can be written as I ( f, g)−(α+β)/2 f L 2 L 1,−α g L 2 L 1,−β , ∞
t
x
t
x
where αp denotes the weighted space over Z: aαp := 2 jα a j p (Z) . j
Hence the bilinear interpolation implies that I ( f, g)−(α+β)/2 f L 2 L 1,−α g L 2 L 1,−β , t
1
2
t
2
for all (α, β) in (10.21), where q
ϕ L p,s := q
k∈Z
q
2ks ϕ L p (|x|∼2k ) ,
and we used the interpolation property of weighted spaces (cf. [3]): β
(α∞ , ∞ )θ,q = q(1−θ)α+θβ , α = β, p,(1−θ)α+θβ
(L p,α , L p,β )θ,q = L q
, α = β.
By choosing α = β = 0 in (10.22), we get the desired result.
# "
(10.22)
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239
Proof of Lemma 10.2. By time translation, we can replace the interval of integration in (10.12) and (10.13) by (−∞, t). Then by taking the dual, we can also replace it by (t, ∞). Adding those two, we can replace it by R. Then the standard T T ∗ argument implies that ei H t ϕ L 2 (L 2,−1 ) ϕ L 2 , ei H0 t ϕ L 2 (L ∞ ) ϕ L 2 . t
t
2
Now let u=
t −∞
ei(t−s)H f (s)ds.
Then the Duhamel formula for the equation iu t + H0 u = f − V u implies that u=
t
ei(t−s)H0 ( f − V u)(s)ds.
−∞
Applying (10.12) for H and (10.13) for H0 , and using L 2,1 ⊂ L 12 , we get u L 2 (L ∞ ) f − V u L 2 (L 1 ) t
t
2
2
f L 2 (L 1 ) + r 2 V L ∞ u L 2 (L 2,−1 ) f L 2 (L 2,1 ) . x t
t
2
t
(10.23)
Then by duality we also get u L 2 (L 2,−1 ) f L 2 (L 1 ) . t
t
2
Feeding this back into (10.23), we get u L 2 (L ∞ ) f L 2 (L 1 ) . t
2
t
2
The estimate on ei H t ϕ is simpler, or can be derived from the above by the T T ∗ argument. # "
Acknowledgements. The authors thank Kyungkeun Kang for joining our discussion in the early stage for getting Theorem 1.2. They also thank the referee for suggesting improvements of the presentation and the question in Remark 2. This work was conducted while the second author was visiting the University of British Columbia, by the support of the 21st century COE program, and also by the support of the Kyoto University Foundation. The research of Gustafson and Tsai is partly supported by NSERC grants no. 251124-07 and 261356-08. The research of Nakanishi was partly supported by the JSPS grant no. 15740086.
Appendix A. Landau-Lifshitz Maps from S2 The same stability problem on S2 , instead of R2 , is much easier in the dissipative case, because the eigenfunctions get additional decay from the curved metric on S2 . Indeed we have convergence for all m ≥ 1:
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Theorem A.1. Let m ≥ 2, a ∈ C and Re a > 0. Then there exists δ > 0 such that for any u(0, x) ∈ m with E( u (0)) ≤ 4mπ + δ 2 , we have a unique global solution u ∈ C([0, ∞); m ) satisfying ∇ u ∈ L 2t,loc ([0, ∞); L ∞ x ). Moreover, for some μ∞ ∈ C we have ∞ ] L ∞ + E( ∞ ]) → 0 (t → ∞). u (t) − emθ R h[μ u (t) − emθ R h[μ x
(A.1)
Our proof does not give a uniform bound on δ if m = 1, but we have Theorem A.2. Let m = 1, a ∈ C, Re a > 0 and μ0 ∈ C. Then there exists δ > 0 such 0 ]) ≤ δ 2 , we have a unique global solution that for any u(0, x) ∈ 1 with E( u (0) − h[μ 2 u ∈ C([0, ∞); 1 ) satisfying ∇ u ∈ L t,loc ([0, ∞); L ∞ x ). Moreover, for some μ∞ ∈ C we have ∞ ] L ∞ + E( ∞ ]) → 0 (t → ∞). u (t) − emθ R h[μ u (t) − emθ R h[μ x
(A.2)
The proof is essentially a small subset of that in the R2 case, so we just indicate necessary modifications. Outline of Proof. By the stereographic projection, we can translate the problem to R2 with the metric g(x)d x 2 , where g(x) = (1 + r 2 /4)−2 . The harmonic maps are the same, while the evolution equation is changed to r m −1 ∗ g −1 (q + ν) ◦ ia L ∗v qdr. (A.3) qt = i Sq − a L v g L v q, S = r ∞ In this setting we can use the “standard" orthogonality to decide μ: 0 = (z | gh s1 ),
(A.4)
since gh 1 ∈ &r '−1 L 1 . The energy identity ∂t q2L 2 = −2a1 (g −1 L ∗v q | L v∗ q) x
(A.5)
implies the a priori bound on q: −1/2 ∗ q L ∞ L v q L 2 L 2 q(0) L 2x ∼ δ. 2 + g t Lx t
x
(A.6)
Since g −1/2 r , we get (by using q = Rϕs∗ L s∗ q as on R2 ), q L 2x r L ∗v q L 2x g −1/2 L ∗v q L 2x ∈ L 2t .
(A.7)
Then by the orthogonality (A.4) we have z = Rφs s L s z with φs := s 2 g(r s)h 1 /(gh s1 | h s1 ),
(A.8)
z/r L 2x L s z L 2x φs L 1 .
(A.9)
and so 2
Since
∞ 0
g(r s) min(r, 1/r )
−m
min(1, s −2 ) (m > 2) r dr ∼ , min(s −1 , s −2 ) (m = 1)
(A.10)
Stability and Oscillation of Harmonic Maps
we have
241
φs L 1 ∼ 2
1 (m ≥ 2) . max(s −1 , 1) (m = 1)
(A.11)
Anyway, if δ is small enough (depending on s), we get by the same argument as on R2 , z X q L 2x ∈ L 2t ∩ L ∞ t .
(A.12)
Differentiating the orthogonality, we get μ(h ˙ s1 | gh s1 ) = −(Ma L ∗v q | h s1 ) − (gz | ( = −((Mr +
μ˙ 1 r ∂r + i μ˙ 2 h s3 )h s1 ) m
m vˇ3 M)aq | h s1 ) r
μ˙ 1 (r ∂r + m) + i μ˙ 2 (h s3 − 1)}h s1 ), (A.13) m where on the second equality we used that L s h s1 = 0 and (gz | h s1 ) = 0. Using that −(gz | {
|(r ∂r + m)h 1 | + |(h 3 − 1)h 1 | min(r m−1 , r −3m−1 ) &r '−4 ,
(A.14)
we can bound the last term in (A.13) by |μ|z ˙ min(s 2 , 1), L∞ x
(A.15)
which is much smaller than the term on the left. The second to last term in (A.13) is bounded at each t by q2L 2 h s1 L ∞ q2L 2 . x x
x
(A.16)
If m > 1, we can improve this for s < 1 as follows. By the same argument as on R2 , we have q L ∞ L ∗v q L 2x g −1/2 L ∗v q L 2x , 2,x
(A.17)
1 s∗ where we need m > 1 for the boundedness of Rφs s : r 2 L 12 → r L ∞ 2 and Rϕ : r L 2 → L∞ 2 . Then we can replace the above estimate in the region r < 1 by
q2L 2 min(s 2 , 1). q L ∞ ∩L 2x h s1 L 1∞ +L ∞ x 2
x
Thus we obtain
(A.18)
μ ˙ L1 t
δ2 (m ≥ 2) , C(s)δ 2 (m = 1)
and hence if δ > 0 is small enough, we get the desired convergence as on R2 .
(A.19) # "
Remark 2. It is a natural question whether one can prove a weaker asymptotic stability as in Theorem A.2 also on R2 . It is impossible in the energy space, at least in the heat flow case (a > 0), because of the presence of blow-up solutions arbitrarily close to the ground state, together with the scaling invariance of the energy space. It is however quite likely that the stability holds for sufficiently localized initial perturbation. This requires weighted estimates on the linearized evolution, which will be pursued in a forthcoming paper.
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References 1. Angenent, S., Hulshof, J.: Singularities at t = ∞ in equivariant harmonic map flow. Contemp. Math. 367, Geometric evolution equations, Providence, RI: Amer. Math. Soc., 2005, pp. 1–15 2. Bejenaru, I., Ionescu, A., Kenig, C., Tataru, D.: Global Schrödinger maps in dimensions d ≥ 2: small data in the critical Sobolev spaces. http://arxiv.org/abs/0807.0265v1 [math.AP], 2008 3. Bergh, J., Löfström, J.: Interpolation spaces. Berlin-Heidelberg-New York: Springer-Verlag, 1976 4. Burq, N., Planchon, F., Stalker, J., Tahvildar-Zadeh, S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Ind. U. Math. J 53(6), 519–549 (2004) 5. Chang, K.-C., Ding, W.Y., Ye, R.: Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Diff. Geom. 36(2), 507–515 (1992) 6. Chang, N.-H., Shatah, J., Uhlenbeck, K.: Schrödinger maps. Comm. Pure Appl. Math. 53(5), 590–602 (2000) 7. Germain, P., Shatah, J., Zeng, C.: Self-similar solutions for the Schrödinger map equation. Math. Z. 264(3), 697–707 (2010) 8. Grotowski, J.F., Shatah, J.: Geometric evolution equations in critical dimensions. Calc. Var. Part. Diff. Eqs. 30(4), 499–512 (2007) 9. Guan, M., Gustafson, S., Kang, K., Tsai, T.-P.: Global Questions for Map Evolution Equations. CRM Proc. Lec. Notes 44, Providence, RI: Amer. Math. Soc., 2008, pp. 61–73 10. Guan, M., Gustafson, S., Tsai, T.-P.: Global existence and blow-up for harmonic map heat flow. J. Diff. Eq. 246, 1–20 (2009) 11. Gustafson, S., Kang, K., Tsai, T.-P.: Schrödinger flow near harmonic maps. Comm. Pure Appl. Math. 60(4), 463–499 (2007) 12. Gustafson, S., Kang, K., Tsai, T.-P.: Asymptotic stability of harmonic maps under the Schrödinger flow. Duke Math. J. 145(3), 537–583 (2008) 13. Kosevich, A., Ivanov, B., Kovalev, A.: Magnetic Solitons. Phys. Rep. 194, 117–238 (1990) 14. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998) 15. Machihara, S., Nakanishi, K., Ozawa, T.: Nonrelativistic limit in the energy space for the nonlinear Klein-Gordon equations. Math. Ann. 322(3), 603–621 (2002) 16. Montgomery-Smith, S.J.: Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math. J. 91(2), 393–408 (1998) 17. O’Neil, R.: Convolution operators and L( p, q) spaces. Duke Math. J. 30, 129–142 (1963) 18. Poláˇcik, P., Yanagida, E.: On bounded and unbounded global solutions of a supercritical semilinear heat equation. Math. Ann. 327, 745–771 (2003) 19. Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60, 558–581 (1985) 20. Tao, T.: Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm. PDE 25(7–8), 1471–1485 (2000) 21. Topping, P.M.: Rigidity in the harmonic map heat flow. J. Diff. Geom. 45, 593–610 (1997) 22. Topping, P.M.: Winding behaviour of finite-time singularities of the harmonic map heat flow. Math. Z. 247, 279–302 (2004) Communicated by P. Constantin
Commun. Math. Phys. 300, 243–271 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1119-3
Communications in
Mathematical Physics
Differing Averaged and Quenched Large Deviations for Random Walks in Random Environments in Dimensions Two and Three Atilla Yilmaz1, , Ofer Zeitouni1,2 1 Faculty of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel 2 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
E-mail: [email protected] Received: 8 October 2009 / Accepted: 6 April 2010 Published online: 29 August 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract: We consider the quenched and the averaged (or annealed) large deviation rate functions Iq and Ia for space-time and (the usual) space-only RWRE on Zd . By Jensen’s inequality, Ia ≤ Iq . In the space-time case, when d ≥ 3 + 1, Iq and Ia are known to be equal on an open set containing the typical velocity ξo . When d = 1 + 1, we prove that Iq and Ia are equal only at ξo . Similarly, when d = 2 + 1, we show that Ia < Iq on a punctured neighborhood of ξo . In the space-only case, we provide a class of non-nestling walks on Zd with d = 2 or 3, and prove that Iq and Ia are not identically equal on any open set containing ξo whenever the walk is in that class. This is very different from the known results for non-nestling walks on Zd with d ≥ 4. 1. Introduction 1.1. The models. Consider a discrete time Markov chain on the d-dimensional integer lattice Zd with d ≥ 1. For any x, z ∈ Zd , denote the transition probability from x to x + z by π(x, x + z). Refer to the transition vector ωx := (π(x, x + z))z∈Zd as the environment at x. If the environment ω := (ωx )x∈Zd is sampled from a probability space (, B, P), then this process is called random walk in a random environment (RWRE). Here, B is the Borel σ -algebra corresponding to the product topology. For every y ∈ Zd , define the shift Ty on by Ty ω x := ωx+y . In order to have some statistical homogeneity in the environment, P is generally assumed to be stationary and ergodic with respect to (Ty ) y∈Zd . In this paper, we will make the stronger assumption that P is a product measure with equal marginals.
(1.1)
In other words, ω = (ωx )x∈Zd is a collection of independent and identically distributed (i.i.d.) random vectors. Current address: Department of Mathematics, University of California, Berkeley, CA 94720, USA. E-mail: [email protected]
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A. Yilmaz, O. Zeitouni
The set R := {z ∈ Zd : P(π(0, z) > 0) > 0} is the range of allowed steps of the walk (here and throughout, we often use 0 to denote the origin in Zd when no confusion d denote the canonical basis for Zd . The walk is said to be space-time occurs). Let (ei )i=1 if R = Rst := {(z 1 , . . . , z d ) ∈ Zd : |z 1 | + · · · + |z d−1 | = 1, z d = 1},
(1.2)
and it is said to be space-only if d . R = Rso := {±ei }i=1
(1.3)
In either case, we will assume throughout the paper that there exists a κ > 0 such that P(π(0, z) ≥ κ) = 1 for every z ∈ R. This condition is known as uniform ellipticity. Space-time is a natural term for the case (1.2) since then, the walk decomposes into two parts. Its projection on the ed -axis is deterministic and can be identified with time. d−1 The motion in the span of (ei )i=1 can be thought of as a variation of space-only RWRE where the environment is freshly sampled at each time step. To emphasize this decomposition, we will write the dimension as d = (d − 1) + 1. For example, when d = 3, we will say that the dimension is 2 + 1. For every x ∈ Zd and ω ∈ , the Markov chain with environment ω induces a probability measure Pxω on the space of paths starting at x. Statements about Pxω that hold for P-a.e. ω are referred to as quenched. Statements about the semi-direct product Px := P × Pxω are referred to as averaged (or annealed). Expectations under P, Pxω and Px are denoted by E, E xω and E x , respectively. See [25] for a survey of results and open problems on RWRE. It is clear that no model satisfies both (1.2) and (1.3). Nevertheless, it turns out that many of the results that hold for space-only RWRE are valid also under the space-time assumption, and it is fair to say that space-time RWRE is easier to analyze than spaceonly RWRE because (1.2) ensures that the walk never visits the same point more than once. 1.2. Regeneration times. In the next subsection, we will give a brief survey of the previous results on large deviations for RWRE in order to put the present work in context. Some of these results involve certain random times which are introduced below for convenience. Let (X n )n≥0 denote the path of a space-only RWRE. Consider a unit vector uˆ ∈ S d−1 . Define a sequence (τm )m≥0 of random times, which are referred to as regeneration times (relative to u), ˆ by τo := 0 and τm := inf j > τm−1 : X i , u ˆ < X j , u ˆ ≤ X k , u ˆ for all i, k with i < j < k for every m ≥ 1. (Regeneration times first appeared in the work of Kesten [9] on onedimensional RWRE. They were adapted to the multidimensional setting by Sznitman and Zerner, cf. [18].) Because we assumed the environment ω = (ωx )x∈Zd to be an i.i.d. collection, if the walk is directionally transient relative to u, ˆ i.e., if Po (limn→∞ X n , u ˆ = ∞) = 1, then Po (τm < ∞) = 1 for every m ≥ 1. In this setup, as noted in [18], the significance of (τm )m≥1 is due to the fact that X τm +1 − X τm , X τm +2 − X τm , . . . , X τm+1 − X τm , τm+1 − τm m≥1 is an i.i.d. sequence under Po .
Inequality of Quenched and Averaged Rate Functions
245
The walk is said to satisfy Sznitman’s transience condition (T) if Eo
sup exp {c1 |X i |} < ∞
for some c1 > 0.
1≤i≤τ1
(Here and throughout, the norm | · | denotes the 2 norm.) When d ≥ 2, Sznitman [17] proves that (1.1), (1.3) and (T) imply a ballistic law of large numbers (LLN), an averaged central limit theorem and certain large deviation estimates. Condition (T) holds as soon as the walk is non-nestling relative to u, ˆ i.e., when the random drift vector v(ω) := π(0, z)z satisfies ess inf v(·), u ˆ > 0. (1.4) P
z∈R
The walk is said to be non-nestling if it is non-nestling relative to some unit vector. Otherwise, it is referred to as nestling. In the latter case, the convex hull of the support of the law of v(·) contains the origin. In the case of space-time RWRE, regeneration times are defined naturally by taking uˆ = ed and τm = m for every m ≥ 1. Clearly, the space-time walk is always non-nestling relative to uˆ = ed . 1.3. Previous results on large deviations for RWRE. Recall that a sequence (Q n )n≥1 of probability measures on a topological space X is said to satisfy the large deviation principle (LDP) with a rate function I : X → R+ ∪ {∞} if I is lower semicontinuous and for any measurable set G, − inf o I (x) ≤ lim inf x∈G
n→∞
1 1 log Q n (G) ≤ lim sup log Q n (G) ≤ − inf I (x). n n n→∞ x∈G¯
Here, G o is the interior of G, and G¯ its closure. See [4] for general background regarding large deviations. We will focus on the following large deviation principles for walks in uniformly elliptic environments.
Theorem 1.1 (Quenched LDP). For P-a.e. ω, Poω Xnn ∈ · satisfies the LDP with n≥1
a deterministic and convex rate function Iq .
Theorem 1.2 (Averaged LDP). Po Xnn ∈ ·
n≥1
satisfies the LDP with a convex rate
function Ia . There are many works on large deviations for space-only RWRE. We briefly mention them in chronological order. Greven and den Hollander [7] prove Theorem 1.1 for walks on Z under the i.i.d. environment assumption. They provide a formula for Iq and show that its graph typically has flat pieces. Zerner [26] establishes Theorem 1.1 for nestling walks on Zd in i.i.d. environments. Comets, Gantert and Zeitouni [3] generalize the result of [7] to walks on Z in stationary and ergodic environments. Also, they prove Theorem 1.2 for walks on Z in i.i.d. environments and give a formula that links Ia to Iq . Varadhan [20] generalizes Zerner’s result to stationary and ergodic environments without any nestling assumption. He also proves Theorem 1.2 for walks on Zd in i.i.d.
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environments and gives a variational formula for Ia . Rassoul-Agha [12] generalizes the latter result of [20] to certain mixing environments. Rosenbluth [15] gives an alternative proof of Theorem 1.1 for walks on Zd in stationary and ergodic environments, and provides a variational formula for Iq . Yilmaz [23] generalizes the result of [15] to a so-called level-2 LDP. Berger [1], Peterson and Zeitouni [11], and Yilmaz [21] obtain certain qualitative properties of Ia . Rassoul-Agha and Seppäläinen [14] generalize the result of [15] to a so-called level-3 LDP. In the case of space-time RWRE, Rassoul-Agha and Seppäläinen [13] prove Theorem 1.1 by adapting the quenched argument in [20]. Theorem 1.2 does not require any work. Indeed, Assumption (1.2) implies that the walk under Po is a sum of i.i.d. increments. The common distribution of these increments is (q(z))z∈R , where q(z) := E[π(0, z)] for every z ∈ R. Therefore, Theorem 1.2 in the space-time setup is simply Cramér’s Theorem, cf. [4]. In addition to the works mentioned in the last two paragraphs, there are two more results on large deviations for RWRE that are relevant to this paper. We state them in detail. Theorem 1.3 (Yilmaz [22]). Assume (1.1) and (1.2). If d ≥ 3 + 1, then Iq = Ia on a set Ast × {ed } containing the LLN velocity ξo , where Ast is an open subset of Rd−1 . Theorem 1.4 (Yilmaz [24]). Assume (1.1), (1.3), d ≥ 4, and that Sznitman’s (T) condition holds for some uˆ ∈ S d−1 . (a) If the walk is non-nestling, then Iq = Ia on an open set Aso containing the LLN velocity ξo . (b) If the walk is nestling, then (i) Iq = Ia on an open set A+so , (ii) there exists a (d − 1)-dimensional smooth surface patch Abso such that ξo ∈ Abso ⊂ ∂A+so , (iii) the unit vector ηo normal to Abso (and pointing inside A+so ) at ξo satisfies ηo , ξo > 0, and (iv) Iq (tξ ) = t Iq (ξ ) = t Ia (ξ ) = Ia (tξ ) for every ξ ∈ Abso and t ∈ [0, 1]. It is worthwhile to emphasize that the equality Iq = Ia does not extend, in the setup of Theorems 1.3 and 1.4, to the whole space. Indeed, for any d ≥ 1, Ia < Iq at the extremal points of the domain of Ia .
(1.5)
By continuity, this inequality holds also at some interior points. See Proposition 4 of [24] for details.
1.4. Our results. For space-time RWRE, it is natural to ask whether Theorem 1.3 can be generalized to d ≥ 1 + 1 or 2 + 1. The answer turns out to be no. Theorem 1.5. Assume (1.1) and (1.2). If d = 1 + 1, then Iq (ξ ) = Ia (ξ ) < ∞ if and only if ξ = ξo , the LLN velocity. Theorem 1.6. Assume (1.1) and (1.2). If d = 2+1, then Ia < Iq on a set (Gst ×{e3 })\{ξo }, where Gst ⊂ R2 is open and Gst × {e3 } contains ξo .
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247
In the case of space-only RWRE on Z, a consequence of Comets et al. [3], Proposition 5, is that Iq (ξ ) = Ia (ξ ) < ∞ if and only if ξ = 0 or Ia (ξ ) = 0. In particular, Theorem 1.4 cannot be generalized to d ≥ 1. Our next result shows that the conclusion of Theorem 1.4 is false for a class of space-only RWRE’s in dimensions d = 2, 3. Definition 1.7. Assume d ≥ 2, and fix a triple p = ( p + , p o , p − ) of positive real numbers such that p − < p + and p + + p o + p − = 1. For any > 0, a probability measure P on (, B) is said to be in class M (d, p) if (a) (b) (c) (d)
(1.1) and (1.3) hold, P(π(0, ed ) = p + , π(0, −ed ) = p − ) = 1, po P( /2 < |π(0, e1 ) − 2(d−1) | < ) = 1, and P is invariant under the rotations of Zd that preserve ed . (We will refer to this as isotropy.)
Theorem 1.8. Assume d = 2 or 3. Fix a triple p = ( p + , p o , p − ) as in Definition 1.7. Then there exists an o = o ( p) such that if < o and P is in class M (d, p), then the quenched and the averaged rate functions Iq and Ia are not identically equal on any open set containing the LLN velocity ξo . The proofs of our results are based on a technique that combines the so-called fractional moment method with a certain change of measure (which we will refer to as tilting the environment). This technique has been developed for analyzing the so-called polymer pinning model, cf. [5,6,19], and it has been recently refined by Lacoin [10] for obtaining certain lower bounds for the free energy of directed polymers in random environments. Comparing with the polymer setup, an extra complication occurs in the RWRE model due to the dependence of the transition probabilities of the walk on the environment. (In the polymer model discussed above, the walk is a simple random walk, and the environment only appears in the evaluation of exponential moments with respect to the random walk.) The difficulty in the RWRE setup, and much of our work, lies in overcoming this dependency. For space-time RWRE, this task is greatly simplified because each site is visited at most once. For space-only RWRE, where this is not true, we employ a perturbative approach that unfortunately restricts the class of models considered, see Sect. 4 for further comments. Here is how the rest of the paper is organized: In Sect. 2, we consider space-time RWRE and prove Theorems 1.5 and 1.6 by adapting the relevant arguments given in [10]. In Sect. 3, we focus on space-only walks that are non-nestling relative to ed , and modify the previous proofs by making use of regeneration times. This way, we establish a result (see Theorem 3.4) analogous to Theorems 1.5 and 1.6. The only difference is that Theorem 3.4 is valid under a certain correlation condition, cf. (3.17). Finally, we prove Theorem 1.8 by checking that (3.17) holds whenever P is in class M (d, p) with some triple p (as in Definition 1.7) and a sufficiently small > 0. 2. Inequality of the Rate Functions for Space-Time RWRE 2.1. Reducing to a fractional moment estimate. Assume d ≥ 1 + 1. Recall (1.2). Consider a space-time random walk on Zd in a uniformly elliptic and i.i.d. environment. For every θ ∈ Rd , define φ(θ ) := eθ,z q(z), z∈R
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where q(z) := E[π(0, z)]. Since the walk visits every point at most once, E o [exp {θ, X N }] = φ(θ ) N for every N ≥ 1. Define the logarithmic moment generating functions 1 log E oω exp{θ, X N } and N →∞ N 1 log E o exp{θ, X N } = log φ(θ ). a (θ ) := lim N →∞ N By Varadhan’s Lemma, cf. [4], q (θ ) = supξ ∈Rd θ, ξ − Iq (ξ ) = Iq∗ (θ ), the convex conjugate of Iq at θ . Similarly, a (θ ) = log φ(θ ) = Ia∗ (θ ). For every N ≥ 1, θ ∈ Rd and ω ∈ , define q (θ ) := lim
W N (θ, ω) := E oω [exp{θ, X N − N log φ(θ )}]. Given any α ∈ (0, 1), Jensen’s inequality and the bounded convergence theorem imply that
1 1 log W N (θ, ·) = E lim log W N (θ, ·) q (θ ) − log φ(θ ) = lim N →∞ N N →∞ N 1 1 E log W N (θ, ·) = lim E log W N (θ, ·)α = lim N →∞ N N →∞ N α 1 log E W N (θ, ·)α ≤ lim sup (2.1) N →∞ N α 1 log (E [W N (θ, ·)])α = 0. ≤ lim N →∞ N α Lemma 2.1. Assume (1.1) and (1.2). Fix any α ∈ (0, 1). If d = 1 + 1, then lim sup N →∞
1 log E W N (θ, ·)α < 0 N
(2.2)
whenever θ ∈ / sp{e2 }, the one-dimensional vector space spanned by e2 . Lemma 2.2. Assume (1.1) and (1.2). Fix any α ∈ (0, 1). If d = 2 + 1, then there exists a β > 0 such that (2.2) holds whenever dist(θ, sp{e3 }) ∈ (0, β). Remark 2.3. For every θ ∈ sp{ed }, (1.2) implies that W N (θ, ·) = 1 and q (θ ) = log φ(θ ). When d = 1 + 1, it follows from (2.1) and Lemma 2.1 that q (·) < log φ(·) on {θ ∈ R2 : θ ∈ / sp{e2 }}. By convex duality, Ia < Iq on {∇ log φ(θ ) : θ ∈ / sp{e2 }}. It is easy to see that the latter set is equal to ((−1, 1) × {e2 })\{ξo }. In combination with (1.5), this proves Theorem 1.5. Similarly, when d = 2 + 1, Lemma 2.2 implies that Ia < Iq on {∇ log φ(θ ) : dist(θ, sp{e3 }) ∈ (0, β)}. One can check that this set is of the form (Gst × {e3 })\{ξo }, where Gst ⊂ R2 is open and Gst × {e3 } contains ξo . This proves Theorem 1.6. The rest of this section is devoted to proving Lemmas 2.1 and 2.2.
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249
2.2. Decomposing into paths. Assume d = 1 + 1 or 2 + 1. Let Vd := Zd−1 × {0} ⊂ Zd . Fix an n of the form k 2 , with k an integer to be determined later (e.g., for d = 1 + 1, this n is chosen so that the conclusion of Lemma 2.4 below holds). When d = 1 + 1, let
1 √ 1 √ Jy := (y − ) n, (y + ) n × {0} ⊂ R2 (2.3) 2 2 for every y = (y , 0) ∈ V2 . Similarly, when d = 2 + 1, let
1 √ 1 √ 1 √ 1 √
Jy := (y − ) n, (y + ) n × (y − ) n, (y + ) n × {0} ⊂ R3 2 2 2 2 for every y = (y , y
, 0) ∈ V3 . Take N = nm for some m ≥ 1. For every θ ∈ Rd , ω ∈ and Y = (y1 , . . . , ym ) ∈ (Vd )m , define W¯ N (θ, ω, Y ) := E oω [exp{θ, X N − N log φ(θ )}, X jn − jnξ(θ ) ∈ Jy j for every j ≤ m],
(2.4)
where ξ(θ ) = ∇ log φ(θ ). (For u ∈ Rd , u denotes the closest element of Zd to u. If there is more than one closest element, then take the one whose index is the smallest with respect to the lexicographic order.) Note that ξ(θ ), ed = 1 because z, ed = 1 for every z ∈ Rst . Since Vd is contained in the disjoint union ∪ y∈Vd Jy , we see that W N (θ, ω) = α α ¯ ¯ Y W N (θ, ω, Y ). Hence, W N (θ, ω) ≤ Y W N (θ, ω, Y ) by subadditivity, and E[W N (θ, ·)α ] ≤ (2.5) E W¯ N (θ, ·, Y )α . Y
In the rest of this section, we will treat the cases d = 1 + 1 and d = 2 + 1 separately. 2.3. Tilting along a path (d = 1 + 1). Our aim is to prove Lemma 2.1 which states that E[W N (θ, ·)α ] decays exponentially in N . Let us say a few words about our strategy. For any function g(θ, ·) on , E[W N (θ, ·)α ] = E (W N (θ, ·)g(θ, ·))α g(θ, ·)−α 1−α α (2.6) ≤ E [W N (θ, ·)g(θ, ·)]α E g(θ, ·)− 1−α by Hölder’s inequality. For every i ≥ 1, E ωX i [exp{θ, X i+1 − X i − log φ(θ )}] and θ, v(TX i ω) − ξo are correlated, cf. (2.23), where v(·) denotes the random drift vector. We could try to exploit this fact by tilting the environment at the points on the path in a clever way, e.g., by choosing a g(θ, ·) that penalizes the environments for which 1 N i=1 θ, v(TX i ω) − ξo deviates from zero. This way, we could make the first expecN tation in (2.6) small. However, there is a problem: we do not know where the path is, and if we naively tilt the environment everywhere, then the second expectation in (2.6) might become too large. Fortunately, it is possible to resolve this issue by first decomposing E[W N (θ, ·)α ] as in (2.5) (so that we know roughly where the path is), and then tilting the environment on a tube which contains most of the path with a high probability.
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Given m ≥ 1, θ ∈ / sp{e2 }, C1 ≥ 1 and Y = (y1 , . . . , ym ) ∈ (V2 )m , let √ √ B j := {(s, i) ∈ Z2 : ( j − 1)n ≤ i < jn, (s, i) − iξ(θ ) − n y j−1 ≤ C1 n} (2.7) for every j ∈ {1, . . . , m}. Here, yo = (0, 0). Recall that n = k 2 for some integer k. Fix a large K and a small δn , both to be determined later (depending on the choice of α, see (2.12), (2.13) and Lemma 2.4). Define f K (u) := −K 1u≥e K 2 and g(θ, ω, Y ) := exp
m
f K δn D(B j ) > 0 ,
(2.8)
j=1
where D(B j ) :=
a(θ, (s, i))
for every j ∈ {1, . . . , m},
(2.9)
(s,i)∈B j
and a(θ, x) := θ, v(Tx ω) − ξo for every x ∈ Z2 , cf. (1.4). Note that E[a(θ, x)] = 0. As before, take N = nm. By Hölder’s inequality, E[W¯ N (θ, ·, Y )α ] = E (W¯ N (θ, ·, Y )g(θ, ·, Y ))α g(θ, ·, Y )−α 1−α α α . (2.10) ≤ E W¯ N (θ, ·, Y )g(θ, ·, Y ) E g(θ, ·, Y )− 1−α Let us control the second term in (2.10). B j ’s are pairwise disjoint and they each √ have n(2C1 n + 1) elements. Since the environment is i.i.d., ⎡ ⎛ ⎞⎤ m α α f K δn D(B j ) ⎠⎦ E g(θ, ·, Y )− 1−α = E ⎣exp ⎝− 1−α j=1
α f K δn D(B j ) = E exp − 1−α j=1 m
α f K (δn D(B1 )) = E exp − 1−α
m α 2 ≤ 1 + e 1−α K P δn D(B1 ) ≥ e K . m
Note that, by Chebyshev’s inequality,
P δn D(B1 ) ≥ e K
2
⎡
≤ e−2K δn2 E D(B1 )2 = e−2K δn2 E ⎣ 2
2
(2.11) ⎤
a(θ, (s, i))2 ⎦
(s,i)∈B1
√ =e δn2 n(2C1 n + 1)E a(θ, (0, 0))2 2 ≤ e−2K δn2 3C1 n 3/2 E a(θ, (0, 0))2 −2K 2
since, by the i.i.d. assumption on the environment, only the diagonal terms survive. Take −1/2 −3/4
δn = C1
n
,
(2.12)
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251
where C1 is still to be defined (and will be chosen as in Lemma 2.4). Then, the RHS of (2.11) is bounded from above by α
α 2 m 2 m 1 + 3E a(θ, (0, 0))2 e 1−α K −2K ≤ 1 + 12e 1−α K −2K ≤ 2m as soon as α
12e 1−α K −2K ≤ 1. 2
(2.13)
Recalling (2.5) and (2.10), we see that α E[W N (θ, ·)α ] ≤ 2m E W¯ N (θ, ·, Y )g(θ, ·, Y ) .
(2.14)
Y
2.4. Estimating the expectation under the tilt (d = 1 + 1). For every m ≥ 1, θ ∈ / sp{e2 }, ω ∈ and Y ∈ (V2 )m , let N = nm as before. By the Markov property, W¯ N (θ, ω, Y ) = E oω [exp{θ, X N − N log φ(θ )}, X jn − jnξ(θ ) x1 ,...,xm ∈Z2
=
x1 ,...,xm
= x j ∈ Jy j ∀ j ≤ m] E oω [exp{θ,
X n − n log φ(θ )}, X n − nξ(θ ) = x1 ∈ Jy1 ]
∈Z2
×E xω1 +nξ(θ) [exp{θ, X n − (x1 + nξ(θ )) − n log φ(θ )}, X n − 2nξ(θ ) = x2 ∈ Jy2 ] =
×···
E oω [exp{θ, X n − n log φ(θ )}, X n − nξ(θ ) = x1 ∈ Jy1 ]
x1 ,...,xm ∈Z2 Tnξ(θ)+√n y ω 1 √ [exp{θ, 1 − n y1
×E x
X n − (x1 −
√
n y1 ) − n log φ(θ )}, √ √ X n − nξ(θ ) = x2 − n y1 ∈ Jy2 − n y1 ]
×··· . Recall (2.8) and (2.9). It follows from the i.i.d. environment assumption that E W¯ N (θ, ·, Y )g(θ, ·, Y ) = E[E oω [exp{θ, X n −n log φ(θ )+ f K (δn D(B1 ))}, X n −nξ(θ ) = x1 ∈ Jy1 ] x1 ,...,xm Tnξ(θ)+√n y ω 1 √ [exp{θ, 1 − n y1
×E x
X n − (x1 −
√
n y1 ) − n log φ(θ ) + f K (δn D(B1 ))}, √ √ X n − nξ(θ ) = x2 − n y1 ∈ Jy2 − n y1 ]
×···] = E o [exp{θ, X n − n log φ(θ ) + f K (δn D(B1 ))}, X n − nξ(θ ) = x1 ∈ Jy1 ] x1 ,...,xm
√ n y1 ) − n log φ(θ ) + f K (δn D(B1 ))}, √ √ X n − nξ(θ ) = x2 − n y1 ∈ Jy2 − n y1 ]
×E x1 −√n y1 [exp{θ, X n − (x1 −
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×··· ≤ E o [exp{θ, X n − n log φ(θ ) + f K (δn D(B1 ))}, X n − nξ(θ ) ∈ Jy1 ] √ × max E x1 −√n y1 [exp{θ, X n − (x1 − n y1 ) − n log φ(θ ) + f K (δn D(B1 ))}, x1 ∈J y1
X n − nξ(θ ) ∈ Jy2 −
√ n y1 ]
×··· = E o [exp{θ, X n − n log φ(θ ) + f K (δn D(B1 ))}, X n − nξ(θ ) ∈ Jy1 ] × max E x1 [exp{θ, X n −x1 −n log φ(θ )+ f K (δn D(B1 ))}, X n −nξ(θ ) ∈ Jy2 −y1 ] x1 ∈Jo
×··· .
Plugging this in (2.14), we conclude that E[W N (θ, ·)α ]
⎛
≤ ⎝2
y∈V2
max E x x∈Jo
⎞m α exp{θ, X n −x−n log φ(θ )+ f K (δn D(B1 ))}, X n −nξ(θ ) ∈ Jy ⎠ .
The RHS of this inequality decays exponentially in m if the term in the parentheses is strictly less than 1. Since N = nm and n was fixed, this proves Lemma 2.1 (and hence Theorem 1.5), provided that we have Lemma 2.4. Assume (1.1) and (1.2). If d = 1 + 1, α ∈ (0, 1), θ ∈ / sp{e2 } and δn = −1/2 C1 n −3/4 , then α max E x exp{θ, X n −x−n log φ(θ )+ f K (δn D(B1 ))}, X n −nξ(θ ) ∈ Jy < 1/2 y∈V2
x∈Jo
(2.15) whenever n, K and C1 are sufficiently large. (The proof is valid with the constant 1/2 replaced by any arbitrarily small positive number.) 2.5. Finishing the proof of Theorem 1.5. It remains to give the Proof of Lemma 2.4. We write the sum in (2.15) as max E x [· · · ]α = max E x [· · · ]α + max E x [· · · ]α y∈V2
x∈Jo
y∈V2 : |y|>R
x∈Jo
y∈V2 : |y|≤R
x∈Jo
(2.16)
with some large constant R, to be determined. Since f K (u) = −K 1u≥e K 2 ≤ 0, the first sum on the RHS of (2.16) is bounded from above by √ α
√ n max E x exp{θ, X n − x − n log φ(θ )}, |X n − nξ(θ ) − n y| ≤ x∈Jo 2 y∈V : 2 |y|>R
≤
y∈V2 : |y|>R
α
X n − nξ(θ ) E o exp{θ, X n − n log φ(θ )}, − y ≤ 1 . √ n
(2.17)
Inequality of Quenched and Averaged Rate Functions
253
Consider a tilted space-time walk on Z2 (in a deterministic environment) with transition probabilities q θ (z) := q(z) exp{θ, z − log φ(θ )} for z ∈ Rst . Let Pˆoθ denote the probability measure it induces on paths. Note that the LLN velocity under Pˆoθ is
zq(z) exp{θ, z − log φ(θ )} = ∇ log φ(θ ) = ξ(θ ).
z∈Rst
With this notation, (2.17) is equal to y∈V2 : |y|>R
α α θ X n − nξ(θ ) θ X n − nξ(θ ) ˆ ˆ Po Po − y ≤ 1 ≤ √ √ ≥ |y| − 1 n n y∈V : 2 |y|>R
which, by Chebyshev’s inequality, can be made arbitrarily small (uniformly in large n) by choosing R sufficiently large. The second sum on the RHS of (2.16) is bounded from above by α (2R + 1) max E x exp{θ, X n − x − n log φ(θ ) + f K (δn D(B1 ))} . x∈Jo
Therefore, to conclude the proof of Lemma 2.4, it suffices to show that E o exp{θ, X n − n log φ(θ ) + f K (δn D(B1 − x))} ≤
1 8R
α −1 (2.18)
for every x ∈ Jo . Similar to B1 defined in (2.7), introduce a new set √ B¯ 1 := {(s, i) ∈ Z2 : 0 ≤ i < n, |(s, i) − iξ(θ )| ≤ (C1 − 1/2) n}. Note that B¯ 1 ⊂ B1 − x for every x ∈ Jo since |x| ≤
√
n/2. We have
E o exp{θ, X n − n log φ(θ ) + f K (δn D(B1 − x))} 2 = e−K E o exp{θ, X n − n log φ(θ )}, δn D(B1 − x) ≥ e K 2 +E o exp{θ, X n − n log φ(θ )}, {X i : 0 ≤ i < n} ⊂ B¯ 1 , δn D(B1 − x) < e K 2 +E o exp{θ, X n − n log φ(θ )}, {X i : 0 ≤ i < n} ⊂ B¯ 1 , δn D(B1 − x) < e K ≤ e−K + Pˆoθ {X i : 0 ≤ i < n} ⊂ B¯ 1 2 +E o exp{θ, X n − n log φ(θ )}, {X i : 0 ≤ i < n} ⊂ B¯ 1 , δn D(B1 − x) < e K . (2.19) The first term in (2.19) is small when K is large. Donsker’s invariance principle ensures that the second term can be made arbitrarily small (uniformly in n) by choosing C1 sufficiently large.
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Let us focus on the third term in (2.19). For any sequence (An )n≥1 of natural numbers, 2 E o [exp{θ, X n − n log φ(θ )}, {X i : 0 ≤ i < n} ⊂ B¯ 1 , δn D(B1 − x) < e K ] ≤ E o [exp{θ, X n −n log φ(θ )}, {X i : 0 ≤ i < n} ⊂ B¯ 1 , δn a(θ, (s, i)) < −An ] (s,i)∈B1 −x (s,i) = X i
+E o [exp{θ, X n −n log φ(θ )}, {X i : 0 ≤ i < n} ⊂ B¯ 1 , δn
2
a(θ, X i ) < e K + An ]
i=0
≤
n−1
E[E oω [exp{θ, X n −n log φ(θ )}, X i = xi ∀i < n],
x1 ,...,xn−1
δn
a(θ, (s, i)) < −An ]
(s,i)∈B1 −x (s,i) =xi
+ E o [exp{θ, X n − n log φ(θ )}, δn =
n−1
2
a(θ, X i ) < e K + An ]
i=0
E o [exp{θ, X n − n log φ(θ )}, X i = xi ∀i < n]
x1 ,...,xn−1
× P(δn
a(θ, (s, i)) < −An )
(2.20)
(s,i)∈B1 −x (s,i) =xi
+E o [exp{θ, X n − n log φ(θ )}, δn ≤
max
x1 ,...,xn−1
P(δn
n−1
2
a(θ, X i ) < e K + An ]
i=0
a(θ, (s, i)) < −An )
(s,i)∈B1 −x (s,i) =xi
+E o [exp{θ, X n − n log φ(θ )}, δn
n−1
2
a(θ, X i ) < e K + An ]
i=0 2 3/2 ≤ A−2 E[a(θ, (0, 0))2 ] n δn 2C 1 n
+ E o [exp{θ, X n − n log φ(θ )}, δn
(2.21) n−1
2
a(θ, X i ) < e K + An ].
i=0
Here, (2.20) follows from the independence assumption on the environment, and (2.21) −1/2 is an application of Chebyshev’s inequality. Since δn = C1 n −3/4 , the first term in (2.21) goes to zero as n → ∞ if An → ∞.
Inequality of Quenched and Averaged Rate Functions
255
Choose An such that An → ∞ and An = o(n 1/4 ) as n → ∞. For any μ ∈ R+ , the second term in (2.21) is equal to n−1 2 (a(θ, X i ) − μ) < e K + An − μnδn ] E o [exp{θ, X n − n log φ(θ )}, δn i=0
≤ Mn E o [exp{θ, X n − n log φ(θ )}
n−1
(a(θ, X i ) − μ)2 ]
i=0
+ Mn E o [exp{θ, X n − n log φ(θ )}
(a(θ, X i ) − μ)(a(θ, X j ) − μ)] (2.22) i= j
by Chebyshev’s inequality, where Mn =
δn 2 μnδn −An −e K
2
= O(n −2 ).
By the FKG inequality (cf. [8]), E o exp{θ, X 1 − log φ(θ )}a(θ, (0, 0)) = E E oω exp{θ, X 1 − log φ(θ )} a(θ, (0, 0)) (2.23) > E E oω exp{θ, X 1 − log φ(θ )} E [a(θ, (0, 0))] = 0, since E oω exp{θ, X 1 − log φ(θ )} and a(θ, (0, 0)) are easily checked to be either both strictly increasing functions (when θ, e1 > 0) or both strictly decreasing functions (when θ, e1 < 0) of the random variable π((0, 0), (1, 1)). If we choose (2.24) μ = E o exp{θ, X 1 − log φ(θ )}a(θ, (0, 0)) , then the second term in (2.22) vanishes by the independence assumption on the environment. Finally, observe that the first term in (2.22) is equal to n Mn E o exp{θ, X 1 − log φ(θ )}(a(θ, (0, 0)) − μ)2 = O(n −1 ). 2.6. Proof of Theorem 1.6. Let us recall a few points regarding the arguments in Subsects. 2.3–2.5. There, since d = 1 + 1, the volume of B1 (defined in (2.7)) is O(n 3/2 ). The variance of D(B1 ) (cf. (2.9)) scales like that volume. We take δn = O(n −3/4 ) so that the variance of δn D(B1 ) is O(1). With this choice, nδn → ∞ as n → ∞. As we saw, this fact is crucial in the proof of Theorem 1.5. In this subsection, we will assume that d = 2 + 1. For every m ≥ 1, 1 ≤ j ≤ m, θ∈ / sp{e3 }, C1 ≥ 1 and Y = (y1 , . . . , ym ) ∈ (V3 )m , we define √ √ B j := {(r, k) : r ∈ Z2 , ( j − 1)n ≤ k < jn, (r, k) − kξ(θ ) − n y j−1 ≤ C1 n}, (2.25) similar to (2.7). Note that the volume of this new set is O(n 2 ). If we were to define D(B1 ) analogously to (2.9), then we would have to take δn ≤ O(n −1 ) in order to make the variance of δn D(B1 ) not grow with n, in which case nδn remains bounded. Hence, the proof for d = 1 + 1 does not directly carry over to the case d = 2 + 1.
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To resolve this issue, following [10], we will modify the proof by redefining D(B1 ) and δn . (We will continue using these names so that we can refer to the parts of Subsects. 2.3–2.5 that carry over word by word.) The modification amounts essentially to using a tilting that is quadratic, instead of linear, in the local drift, as follows. For every (r, k) and (s, l) with r, s ∈ Z2 and k, l ≥ 1, let V ((r, k), (s, l)) :=
1 √ 1 |k − l| {|(s,l)−(r,k)−(l−k)ξ(θ)|
(2.26)
if k = l, and set it to be equal to zero if k = l. Here, the constant C2 ≥ 1 will be determined later. Given any n integer and x1 , . . . , xn ∈ Z3 with xk , e3 = k, it follows easily from (2.26) that for any s ∈ Z2 , l ∈ {1, . . . , n},
n
V (xk , (s, l)) ≤ 2 log n,
k=1 n
k=1 (s,l)∈B1
(s,l)∈B1
n
2 1 2C2 |k − l| ≤ 4C22 n 2 , |k − l|
V (xk , (s, l)) ≤
1≤k,l≤n k =l
2
V (xk , (s, l))
=
max
(s ,l )
k=1
n
V (xk , (s , l ))
n
V (xk , (s, l))
(s,l)∈B1 k=1
k=1
≤ (2 log n)(4C22 n 2 ) = 8C22 n 2 log n, and n n √ 1{k=l} V ((r, k), (s, l))2 ≤ (2C1 n)2 (2C2 |k − l|)2 2 |k − l| (r,k)∈B , 1 (s,l)∈B1
k=1
(2.27)
l=1
= 16C12 C22 n
1≤k,l≤n k =l
1 ≤ 32C12 C22 n 2 log n. (2.28) |k − l|
Recall the tilted law Pˆoθ introduced in the proof of Lemma 2.4. Lemma 2.5. For any δ > 0, there exists a C2 ≥ 1 such that ν(n, X ) := X j ) satisfies Pˆoθ (ν(n, X ) < n log(n − 1)/2) ≤ δ for every n ≥ 2. Proof. For any realization of X = (X i )i≥1 , ν(n, X ) ≤
1≤i, j≤n i = j
1 =: H (n). |i − j|
1≤i, j≤n
V (X i ,
Inequality of Quenched and Averaged Rate Functions
Observe that
Eˆ oθ [ν(n, X )] =
257
Eˆ oθ [V (X i , X j )]
1≤i, j≤n
=
1≤i, j≤n i = j
1 Pˆoθ (|X i − X j − (i − j)ξ(θ )| < C2 |i − j|). |i − j|
When C2 is sufficiently large, the CLT implies that Pˆoθ (|X i − X j − (i − j)ξ(θ )| < C2 |i − j|) ≥ (1 − δ/2) for any i = j. Therefore, Eˆ oθ [ν(n, X )] ≥ (1−δ/2)H (n). Applying Markov’s inequality, we see that Pˆoθ (ν(n, X ) < H (n)/2) = Pˆoθ (H (n) − ν(n, X ) > H (n)/2) ≤ δ. This implies the desired result since H (n) ≥ n log(n − 1).
For any θ ∈ R3 and x ∈ Z3 , define a(θ, x) := θ, v(Tx ω) − ξo as before, where v(ω) = z∈R π(0, z)z. Lemma 2.6. There exists a β > 0 such that μ := E o exp{θ, X 1 − log φ(θ )}a(θ, (0, 0, 0)) > 0 whenever dist(θ, sp{e3 }) ∈ (0, β). Proof. For every θ ∈ / sp{e3 }, let F(θ ) := E{E oω [eθ,X 1 ]E oω [θ, X 1 ]} and G(θ ) := E o [eθ,X 1 ]E o [θ, X 1 ] = φ(θ )θ, ξo . Our aim is to show that F(θ ) > G(θ ). Write θ = ce3 + θ for some c ∈ R and θ ∈ R3 such that θ , e3 = 0. Then, F(θ ) = ec F(θ ) + cec φ(θ ) and G(θ ) = ec G(θ ) + cec φ(θ ). Therefore, it suffices to show that F(θ ) > G(θ ). Clearly, we have ∇ F(θ )|θ=0 = ∇G(θ )|θ=0 = E o [X 1 ] = ξo . Also, for any
u, u
∈ R3 , with D 2 F denoting the Hessian of F, = 2E{E oω [X 1 , u]E oω [X 1 , u ]} u, D 2 F(θ )u θ=0
and
u, D 2 G(θ )u
θ=0
= 2E o [X 1 , u]E o [X 1 , u ] = 2ξo , uξo , u .
By Schwarz’ inequality (which is strict since the walk is uniformly elliptic in the directions other than e3 ),
inf u, D 2 F(θ )u > 0. − u, D 2 G(θ )u |u|=1 u,e3 =0
θ=0
θ=0
Finally, Taylor’s theorem implies the existence of a β > 0 such that F(θ ) > G(θ ) whenever |θ | ∈ (0, β).
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Now, we are ready to give the new definition of D(B1 ) which is suitable for d = 2+1. For any θ ∈ R3 such that dist(θ, sp{e3 }) ∈ (0, β) (with β as in Lemma 2.6), let V ((r, k), (s, l))a(θ, (r, k))a(θ, (s, l)). (2.29) D(B1 ) := (r,k)∈B1 , (s,l)∈B1
Note that V ((·, k), (·, k)) = 0 for every 1 ≤ k ≤ n. Since E[a(θ, 0)] = 0, it follows from the independence of the environment that E[D(B1 )] = 0. Also, E[D(B1 )2 ] ≤ 1024|θ |4 C12 C22 n 2 log n by (2.28) and the fact that |a(θ, 0)| ≤ 2|θ |. If we choose δn := n −1 (log n)−1/2 , then the variance of δn D(B1 ) is O(1). Once we have this fact, the arguments in Subsects. 2.3–2.5 carry over until (2.18). So, it suffices to show that E o [exp{θ, X n − 2 n log φ(θ )}, δn D(B1 − x) < e K ] is small for all x ∈ Jo when n and K are large. In the estimate below, we will (WLOG) take x = 0. Let γ = 1/2, and observe that 2
E o [exp{θ, X n − n log φ(θ )}, δn D(B1 ) < e K ] 2
≤ E o [exp{θ, X n − n log φ(θ )}, ν(n, X ) ≥ γ n log(n − 1), δn D(B1 ) < e K ] + E o [exp{θ, X n − n log φ(θ )}, ν(n, X ) < γ n log(n − 1)] = E o [exp{θ, X n − n log φ(θ )}, ν(n, X ) ≥ γ n log(n − 1), 2 δn (D(B1 ) − μ2 ν(n, X )) < e K − μ2 δn ν(n, X )] + Pˆoθ (ν(n, X ) < γ n log(n − 1)) ≤ Mn E o [exp{θ, X n − n log φ(θ )}(D(B1 ) − μ2 ν(n, X ))2 , ν(n, X ) ≥ γ n log(n − 1)] (2.30) + Pˆoθ (ν(n, X ) < γ n log(n − 1)) ≤ Mn E o [exp{θ, X n − n log φ(θ )}(D(B1 ) − μ2 ν(n, X ))2 ] + Pˆoθ (ν(n, X ) < γ n log(n − 1)).
(2.31)
Here, (2.30) follows from the elementary inequality 1a
2
δn μ2 δn γ n log(n − 1) − e K
2
.
(2.32)
Choose C2 sufficiently large so that the second term in (2.31) is small for all n ≥ 2 by Lemma 2.5. It remains to control the first term in (2.31). Note that D(B1 ) − μ2 ν(n, X ) n = 2μ V (X k , (s, l))(a(θ, (s, l)) − μ1{X l =(s,l)} ) k=1 (s,l)∈B1
+
(r,k)∈B1 , (s,l)∈B1
V ((r, k), (s, l))(a(θ, (r, k)) − μ1{X k =(r,k)} )(a(θ, (s, l)) − μ1{X l =(s,l)} ),
Inequality of Quenched and Averaged Rate Functions
259
and E o [exp{θ, X n − n log φ(θ )}(D(B1 ) − μ2 ν(n, X ))2 ] ⎡ ⎛ ⎞2 ⎤ n ⎢ ⎥ V (X k , (s, l))(a(θ, (s, l)) − μ1{X l =(s,l)} )⎠ ⎦ ≤ 8μ2 E o ⎣exp{· · · } ⎝ ⎡
⎛
k=1 (s,l)∈B1
⎢ ⎜ ⎜ +2E o ⎢ V ((r, k), (s, l))(a(θ, (r, k)) − μ1{X k =(r,k)} )(a(θ, (s, l)) ⎣exp{· · · } ⎝ (r,k)∈B1 , (s,l)∈B1
⎞2 ⎤
⎟ ⎥ ⎥ −μ1{X l =(s,l)} )⎟ ⎠ ⎦ =: 8μ2 E1 + 2E2
(2.33)
by the inequality (a + b)2 ≤ 2(a 2 + b2 ). One should note at this stage that in fact, even though μ was chosen to equal the mean under the tilted measure of a(θ, 0), it is not necessarily the case that the mean of D(B1 ) − μ2 ν(n, X ) under that tilted measure vanishes. This makes the control of Ei somewhat messy, involving a local CLT (Lemma 2.7). We turn to the details of the computation. E1 can be written as a double sum over pairs (s, l), (s , l ) ∈ B1 . If (s, l) = (s , l ), then it is clear from independence that this pair does not contribute to E1 on the event {X l = (s, l)} ∪ {X l = (s , l )}. Therefore, ⎡ ⎤ n 2 E1 = E o ⎣exp{· · · } V (X k , (s, l))(a(θ, (s, l)) − μ1{X l =(s,l)} ) ⎦ +
(s,l)∈B1
k=1
E o [exp{θ, X n − n log φ(θ )}V (X k , X l )V (X k , X l )
k,k ,l,l : l =l
×(a(θ, X l ) − μ)(a(θ, X l ) − μ)] ⎡ n V (X k , (s, l))(a(θ, (s, l)) − μ1{X l =(s,l)} ) =: E o ⎣exp{· · · } +
k,k ,l,l : l =l
(s,l)∈B1
2
⎤ ⎦
k=1
E1 (k, k , l, l ) ⎡
≤ (2|θ | + μ)2 E o ⎣exp{· · · }
(s,l)∈B1
≤ (2|θ | + μ)2 8C22 n 2 log n +
n
2
V (X k , (s, l))
k=1
E1 (k, k , l, l )
k,k ,l,l : l =l
by (2.27) and the fact that |a(θ, ·)| ≤ 2|θ |.
⎤ ⎦+
E1 (k, k , l, l )
k,k ,l,l : l =l
(2.34)
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If l > max(k, k , l), then E1 (k, k , l, l ) is equal to zero since we can condition on
the path up to l and use the fact that, for any (xi )l1 ,
E o exp{θ, X n − X l − (n − l ) log φ(θ )}(a(θ, X l ) − μ) (X i )l1 = (xi )l1 = 0 by the definition of μ, cf. Lemma 2.6. If l < l < k < k, then V (X k , X l ) and V (X k , X l ) create a slight complication since X k and X k are not independent of X l +1 − X l . Indeed, E1 (k, k , l, l )
= E o exp{· · · }(a(θ, X l )−μ)(a(θ, X l )−μ), (X i )l1 = (xi )l1 , X l +1 − X l = z x1 ,...,xl z∈R
× Eˆ oθ [V (X k , xl )V (X k , xl ) | X l +1 = xl + z], and the latter expectation depends on z. (If it were independent of z, we could simply take the sum over z ∈ R and conclude that E1 (k, k , l, l ) = 0.) However, for any z, z ∈ R, ˆθ E o [V (X k , xl )V (X k , xl ) | X l +1 = xl + z]− Eˆ oθ [V (X k , xl )V (X k , xl ) | X l +1 = xl +z ] ≤ (k − l )−1 Pˆoθ (X k = xk |X l +1 = xl + z)− Pˆoθ (X k = xk |X l +1 = xl +z ) xk : V (xk ,xl )>0
× Eˆ oθ [V (X k , xl ) | X k = xk ] ≤ 4C22 (k − l )(k − l )−1 O((k − l )−3/2 )(k − l)−1 = O((k − l)−1 (k − l )−3/2 )
uniformly in (xi )l1 , cf. Lemma 2.7 (given below). Hence,
E1 (k, k , l, l ) ≤ O(n 2 log n).
l
It is easy to see that this technique works for E1 (k, k , l, l ) in all other cases, and we get E1 ≤ O(n 2 log n) by (2.34). E2 is a quadruple sum over (r, k), (r , k ), (s, l), (s , l ) ∈ B1 that is symmetric in (r, k) and (s, l) (as well as in (r , k ) and (s , l )). Recall that V ((r, k), (s, l)) = 0 when (r, k) = (s, l). If (r, k) ∈ / {(r , k ), (s , l )}, then it is clear from independence that there is no contribution to E2 on the event {X k = (r, k)}. The contribution from the complementary event can be estimated using Lemma 2.7, just like in the case of E1 .
Inequality of Quenched and Averaged Rate Functions
261
Putting everything together and recalling (2.28), we see that ⎡ ⎢ E2 ≤ 2E o ⎢ (V ((r, k), (s, l))(a(θ, (r, k)) ⎣exp{· · · } (r,k)∈B1 , (s,l)∈B1
⎤
−μ1{X k =(r,k)} )(a(θ, (s, l)) − μ1{X l =(s,l)} )
2 ⎥ ⎥ + O(n 2 log n) ⎦
⎡
⎤
⎢ ⎥ 2 ≤ 2(2|θ |+μ)4 E o ⎢ V ((r, k), (s, l))2 ⎥ ⎣exp{θ, X n −n log φ(θ )} ⎦ + O(n log n) (r,k)∈B1 , (s,l)∈B1
≤ 2(2|θ | + μ)4 32C12 C22 n 2 log n + O(n 2 log n) ≤ O(n 2 log n). Finally, Mn E o [exp{θ, X n − n log φ(θ )}(D(B1 ) − μ2 ν(n, X ))2 ] ≤ Mn (8μ2 E1 + 2E2 ) ≤ O((log n)−1 ) by (2.32) and (2.33). This concludes the proof of Theorem 1.6, apart from Lemma 2.7. For any z, z ∈ R, sup | Pˆzθ (X m = x) − Pˆzθ (X m = x)| ≤ O(m −3/2 ) as m → ∞.
x∈Z3
Proof. Let G θ be the centered Gaussian density on R2 that has the same covariance with (X 1 , e1 , X 1 , e2 ) under Pˆoθ . For any z ∈ R, it is shown in Theorem 22.1 of [2] that x − z − mξ(θ ), e1 x − z − mξ(θ ), e2 2 sup Pˆzθ (X m = x) − G θ , √ √ m m m x ≤ O(m −3/2 ) as m → ∞. Here, the supremum is taken over all x = (x1 , x2 , x3 ) ∈ Z3 such that x1 +x2 +m+1 is even and x3 = m + 1. (Otherwise, Pˆzθ (X m = x) is equal to zero.) Since sup y∈R2 |∇ y G θ (y)| < ∞, the desired result follows from the triangle inequality. 3. Inequality of the Rate Functions for Space-Only RWRE 3.1. Reducing to a fractional moment estimate. Consider space-only RWRE on Zd with d ≥ 1. Assume that the walk is non-nestling relative to the canonical basis vector ed . By Jensen’s inequality, the quenched and the averaged logarithmic moment generating functions
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A. Yilmaz, O. Zeitouni
1 log E oω exp{θ, X N } and N →∞ N 1 a (θ ) := lim log E o exp{θ, X N } N →∞ N
q (θ ) := lim
satisfy q (θ ) ≤ a (θ ) ≤ |θ | for every θ ∈ Rd . Recall the definition of regeneration times (τn )n≥0 (relative to ed ) given in Subsect. 1.2. Let β := inf{i ≥ 0 : X i , ed < X o , ed } ∈ [1, ∞]. By the non-nestling assumption, there exist constants c2 , c3 > 0 such that ess inf Poω (β = ∞) ≥ c2 and ess sup Poω (τ1 > n) ≤ e−c3 n P
P
(3.1)
for every n ≥ 1, cf. [16]. These bounds clearly imply that ess sup E oω [ exp{cτ1 }| β = ∞] ≤ c2−1 ess sup E oω [exp{cτ1 }] =: H (c) < ∞ P
P
(3.2)
whenever c < c3 . For every c ∈ (0, c3 ], introduce the set C(c) := {θ ∈ Rd : 2|θ | < c}.
(3.3)
Lemma 3.1. For every θ ∈ C(c3 ), E o [ exp{θ, X τ1 − a (θ )τ1 } β = ∞] = 1.
(3.4)
a is analytic on C(c3 ). ∇a (0) = ξo . The Hessian Ha of a is positive definite on C(c3 ). For every c < c3 and θ ∈ C(c), the smallest eigenvalue of Ha (θ ) is bounded from below by a positive constant that depends only on c and the ellipticity constant κ of the walk. Proof. See the proofs of Lemmas 6 and 12 of [21]. In particular, the desired lower bound for the smallest eigenvalue of Ha is evident from Eq. (2.10) of that paper. Given any N ≥ 1, θ ∈ C(c3 ) and ω ∈ , define and Wˆ N (θ, ω) := E oω [exp{θ, X τ N − a (θ )τ N }] W N (θ, ω) := E oω [ exp{θ, X τ N − a (θ )τ N } β = ∞]. Lemma 3.2. For every θ ∈ C(c3 ), if lim sup N →∞
holds P-a.s., then q (θ ) < a (θ ).
1 log Wˆ N (θ, ·) < 0 N
Inequality of Quenched and Averaged Rate Functions
263
Proof. Let θ ∈ C(c3 ). Then, θ ∈ C(c) for some c < c3 . By hypothesis, for P-a.e. ω, there exist C3 ≥ 1 and c4 > 0 (both depending on ω) such that Wˆ N (θ, ω) ≤ C3 e−c4 N for every N ≥ 1. Given any n ≥ 1 and K ≥ 1, it follows from Chebyshev’s inequality and (3.2) that E oω [exp{θ, X n − a (θ )n}] = E oω [exp{θ, X n − a (θ )n}, n < τ Kn ] +
n j= Kn
E oω [exp{θ, X n − a (θ )n}, τ j ≤ n < τ j+1 ]
≤ e2|θ|n Poω (n < τ Kn ) +
n
E oω [exp{θ, X τ j − a (θ )τ j }]
j= Kn
ω
× ess sup E o [ exp{2|θ |τ1 }| β = ∞] P
≤ e(2|θ|−c)n E oω [exp{cτ Kn }] +
n
Wˆ j (θ, ω)ess sup E oω [ exp{cτ1 }| β = ∞]
j= Kn
P
n −1 K
≤ e(2|θ|−c)n E oω [exp{cτ1 }] ess sup E oω [ exp{cτ1 }| β = ∞] P
+
n
Wˆ j (θ, ω)ess sup E oω [ exp{cτ1 }| β = ∞]
P
j= Kn
n
n
≤ e(2|θ|−c)n H (c) K + H (c)
j= Kn
C3 e−c4 j .
Take K sufficiently large, and conclude that q (θ ) − a (θ ) = lim
n→∞
1 log E oω [exp{θ, X n − a (θ )n}] < 0. n
Lemma 3.3. For every θ ∈ C(c3 ), if lim sup N →∞
1 log E W N (θ, ·)α < 0 N
(3.5)
for some α ∈ (0, 1), then q (θ ) < a (θ ). Hence, by convex duality, Ia < Iq at ξ = ∇a (θ ). Proof. For any N ≥ 1 and θ ∈ C(c3 ), it follows from the renewal structure and (3.4) that E[Poω (β = ∞)W N (θ, ·)] = Po (β = ∞)E o [ exp{θ, X τ N − a (θ )τ N } β = ∞] N = Po (β = ∞) E o [ exp{θ, X τ1 − a (θ )τ1 } β = ∞] = Po (β = ∞).
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A. Yilmaz, O. Zeitouni
Given any α ∈ (0, 1), by the same reasoning as in (2.1), 1 1 lim sup log Wˆ N (θ, ·) ≤ lim sup log E Wˆ N (θ, ·)α , P-a.s. N →∞ N N →∞ N α
(3.6)
On the other hand, if 2|θ | < c < c3 , then we see by subadditivity, Chebyshev’s inequality, and (3.2) that ⎡⎛ ⎞α ⎤ E Wˆ N +1 (θ, ·)α = E ⎣⎝ E oω [exp{θ, X τ1 − a (θ )τ1 }, X τ1 = x]W N (θ, Tx ·)⎠ ⎦ ⎡ ≤ E⎣ ⎡ ≤ E⎣ ⎡
x∈Zd
E oω [exp{θ, X τ1 − a (θ )τ1 }, X τ1 = x]
x∈Zd
E oω [exp{2|θ |τ1 }, τ1 ≥ |x|1 ]
α
α
⎤ W N (θ, Tx ·)α ⎦ ⎤
W N (θ, Tx ·)α ⎦
x∈Zd
⎤
α ≤ E⎣ e(2|θ|−c)|x|1 E oω [exp{cτ1 }] W N (θ, Tx ·)α ⎦ x∈Zd
(2|θ|−c)α|x| 1. ≤ H (c)α E W N (θ, ·)α e
(3.7)
x∈Zd
The desired result follows immediately from (3.6), (3.7) and Lemma 3.2.
3.2. The correlation condition. In this subsection, we will consider space-only RWRE on Zd with d = 2, 3, assume that the walk is non-nestling relative to ed , and outline how one can modify the arguments given in Sect. 2 in order to reduce (3.5) to a simpler inequality. We start with d = 2. For every n ≥ 1 of the form k 2 , and for every y = (y , y
) ∈ Z2 , let 1 √ 1 √ 1 √ 1 √ Jy := [(y − ) n, (y + ) n) × [(y
− ) n, (y
+ ) n) ⊂ R2 , 2 2 2 2 cf. (2.3). Take N = nm for some m ≥ 1. For every θ ∈ C(c3 ), ω ∈ and Y = (y1 , . . . , ym ) ∈ (Z2 )m , define W¯ N (θ, ω, Y ) := E oω [exp{θ, X τ N − a (θ )τ N }, X τ jn − jnζ (θ ) ∈ Jy j for every j ≤ m| β = ∞], cf. (2.4), where
ζ (θ ) := E o [ X τ1 exp{θ, X τ1 − a (θ )τ1 } β = ∞].
By subadditivity, E[W N (θ, ·)α ] ≤
E W¯ N (θ, ·, Y )α , Y
(3.8)
Inequality of Quenched and Averaged Rate Functions
265
cf. (2.5). Given any C1 ≥ 1, Y = (y1 , . . . , ym ) ∈ (Z2 )m and j ∈ {1, . . . , m}, let √ B j = B j (y j−1 , y j ) := {(s, i) ∈ Z2 : ( j − 1)nζ (θ ), e2 + n(y
j−1 + 1/2) √ ≤ i < jnζ (θ ), e2 + n(y
j − 1/2), √ √ √ ζ (θ ), e1 |(s − n y j−1 ) − (i − n y
j−1 )| ≤ C1 n}, ζ (θ ), e2 cf. (2.7). Also, redefine a(θ, ·) by setting a(θ, x) := θ, v(Tx ω) − E[θ, v(·)] for every x ∈ Z2 , where v(ω) = z∈R π(0, z)z as before. Note that, under the assumptions stated in Definition 1.7, we have E[θ, v(·)] = θ, ξo . However, this equality does not necessarily hold in general. With these modified definitions, the arguments in Subsects. 2.3 and 2.4 easily carry over, once one replaces the i.i.d. random variables TX i ω
Eo
[exp{θ, X 1 − log φ(θ )}]
by the variables TX τ ω
Eo
i
[exp{θ, X τ1 − a (θ )τ1 }|β = ∞].
Therefore, in order to prove (3.5), it suffices to show that max E x exp{θ, X τn − x − a (θ )τn + f K (δn D(B1 ))}, y∈Z2
x∈Jo
α X τn − nζ (θ ) ∈ Jy | β = ∞ < 1/2
(3.9)
when ζ (θ ) is as in (3.8) and n, K , C1 are sufficiently large, cf. Lemma 2.4. Here, −1/2 α ∈ (0, 1) is fixed, f K (u) := −K 1u≥e K 2 and δn = C1 n −3/4 , as before. We imitate (2.16), and write the sum in (3.9) as max E x [· · · ]α = max E x [· · · ]α + max E x [· · · ]α (3.10) y∈Z2
x∈Jo
y∈Z2 : |y|>R
x∈Jo
y∈Z2 : |y|≤R
x∈Jo
with some large constant R, to be determined. Just like in the space-time case, the first sum on the RHS of (3.10) is bounded from above by α X n − nζ (θ ) ≥ |y| − 1 . (3.11) Pˆoθ √ n 2 y∈Z : |y|>R
Here, Pˆoθ is redefined to be the probability measure on paths induced by the random walk (in a deterministic environment) whose transition probabilities are given by q θ (x) := E o [ exp{θ, X τ1 − a (θ )τ1 }, X τ1 = x β = ∞], x ∈ Z2 .
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(Note that x∈Z2 q θ (x) = 1 by (3.4).) If Eˆ oθ denotes the corresponding expectation, it is clear that Eˆ oθ [exp{c|X 1 |}] < ∞
for every c ∈ (0, c3 − 2|θ |).
(3.12)
Therefore, by Chebyshev’s inequality, (3.11) can be made arbitrarily small (uniformly in large n) by choosing R sufficiently large. The second sum on the RHS of (3.10) can be controlled by showing that max max E x exp{θ, X τn − x − a (θ )τn + f K (δn D(B1 ))}, y∈Z2 : |y|≤R
x∈Jo
X τn − nζ (θ ) ∈ Jy | β = ∞
(3.13)
is small when n, K and C1 are sufficiently large. In the space-time case, the verification of the analogous statement, i.e., (2.18), relied on the fact that E o [exp{θ, X n − n log φ(θ )}
n−1
a(θ, X i )]
(3.14)
i=0
grows linearly in n, cf. (2.21) and (2.22). In the space-only case, the drift vectors at the points off the path do not contribute to the mean of D(B1 ) under the tilted measure, and the drift vector at any point on the path contributes only once even if it is visited multiple times. Therefore, the statement concerning (3.14) needs to be replaced by the statement that E o [exp{θ, X τn − a (θ )τn } a(θ, x)|β = ∞] (3.15) x∈S(X,τn )
grows linearly in n. Here, for any j ≥ 1, S(X, j) := {X i : 0 ≤ i < j}.
(3.16)
In the space-time case, the variance of D(B1 ) under the tilted measure was shown to be O(n 3/2 ) since the only non-vanishing terms were those corresponding to points x, y ∈ Z2 such that x = y. In the space-only case, steps of the walk between consecutive regeneration times are not independent, and we therefore need to also consider terms corresponding to x and y that are both on the path in the same regeneration block. However, since regeneration times have exponentially decaying tails, the total contribution of such terms is O(n), and the variance of D(B1 ) under the tilted measure is still O(n 3/2 ). With these modifications, the argument in Subsect. 2.5 enables us to deduce (3.5) provided that (3.15) grows linearly in n. By the renewal structure, the latter is equivalent to the following correlation condition: μ := E o [exp{θ, X τ1 − a (θ )τ1 } a(θ, x)|β = ∞] > 0. (3.17) x∈S(X,τ1 )
(This replaces the choice of μ for the space-time case, see (2.24).) For d = 3, after modifying (3.13) by (i) taking the first maximum over {y ∈ Z3 : |y| ≤ R}, (ii) replacing the sets Jy and B1 by their three dimensional analogs, and (iii) redefining D(B1 ) as in (2.29), one can employ the reasoning above in order to reduce (3.5) to showing that (3.13) is small when n, K and C1 are sufficiently large. After that,
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one can set δn := n −1 (log n)−1/2 , apply the same kind of modifications to the argument given in Subsect. 2.6, and further reduce (3.5) to (3.17). In particular, note that Lemma 2.7 continues to hold under the new definition of Pˆoθ , thanks to (3.12). We omit the (routine) details. We have arrived at the following theorem. Theorem 3.4. Consider space-only RWRE on Zd with d = 2, 3. Assume (1.1), (1.3) and that the walk is non-nestling relative to ed . Then, there exists an open set Aso ⊂ Rd with the following properties: (i) Ia is strictly convex and analytic on Aso , (ii) ξo ∈ Aso , and (iii) for every ξ ∈ Aso , the strict inequality Ia (ξ ) < Iq (ξ ) holds if (3.17) is satisfied at θ := ∇ Ia (ξ ). Proof. Recall (3.3), and define Aso := {∇a (θ ) : θ ∈ C(c3 )}. It follows from Lemma 3.1 and the inverse function theorem that Ia is strictly convex and analytic on Aso which is an open set containing ξo . Take any ξ ∈ Aso . Note that θ := ∇ Ia (ξ ) satisfies ξ = ∇a (θ ) by convex duality. As outlined above, (3.17) implies (3.5). Hence, the desired result follows from Lemma 3.3. 3.3. Proof of Theorem 1.8. Consider space-only RWRE on Zd with d = 2, 3. Fix a triple p = ( p + , p o , p − ) of positive real numbers such that p − < p + and p + + p o + p − = 1. Assume that P is in class M (d, p) for some small > 0, cf. Definition 1.7. Assume po that ≤ 4(d−1) so that the ellipticity constant κ of the walk satisfies
po κ ≥ min p , p , . 4(d − 1) +
−
(3.18)
Lemma 3.5. There exist C4 ≥ 1 and c5 > 0 (depending only on p) such that |a (θ ) − θ, ξo | ≤ C4 |θ |2 holds for every θ ∈ C(c5 ). Proof. Recall (3.1). Note that c3 depends only on the law of the regeneration times which, in turn, is determined by the fixed triple p. Moreover, the ellipticity constant κ of the walk satisfies (3.18). Fix any c5 < c3 . The desired result follows immediately from Lemma 3.1. Consider the set Ct (c5 ) := {θ ∈ C(c5 ) : θ, ed = 0}. (Here, the subscript stands for transversal.) Take any θ ∈ Ct (c5 ). Recall the notation in (3.16). Since P is in class M (d, p), it is easy to see that ξo = ( p + − p − )ed , θ, ξo = 0, and |a(θ, x)| = |θ, v(Tx ω)| ≤ 2 (d − 1)|θ | (3.19)
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for every x ∈ Zd . Similarly, the isotropy assumption ensures that Z (θ ) = Z (θ, X, τ1 , ω) := a(θ, x) x∈S(X,τ1 )
satisfies E o [Z (θ )|β = ∞] = E o [τ1 Z (θ )|β = ∞] = 0.
(3.20)
Our aim is to show that E o [exp{θ, X τ1 − a (θ )τ1 }Z (θ )|β = ∞] > 0
(3.21)
for certain choices of θ , to be determined later. Expanding the exponential on the LHS of (3.21), we see that E o [exp{θ, X τ1 − a (θ )τ1 }Z (θ )|β = ∞] ≥ E o [(1 + θ, X τ1 − a (θ )τ1 )Z (θ )|β = ∞] − C5 |θ |3
(3.22)
= E o [θ, X τ1 Z (θ )|β = ∞] − C5 |θ | .
(3.23)
3
Indeed, (3.22) follows from |Z (θ )| ≤ 2 (d − 1)|θ |τ1 and 1 + θ, X τ1 − a (θ )τ1 ≤ exp{θ, X τ1 − a (θ )τ1 } ≤ 1 + θ, X τ1 − a (θ )τ1 + 2|θ |2 τ12 exp{2|θ |τ1 }, C5 is some constant that depends only on p and c5 and finally, (3.20) implies (3.23). In order to estimate (3.23), we first provide a more convenient representation of the RWRE. Let (bi )i≥0 be an i.i.d. sequence of random variables taking values in {ed , 0, −ed }, with P(b1 = ed ) = p + ,
P(b1 = 0) = p o , and P(b1 = −ed ) = p − .
Let ( f i )i≥0 be another i.i.d. sequence of random variables (independent of (bi )i≥0 ) taking values in the set {±e j : 1 ≤ j < d} ∪ {0}, with P( f 1 = 0) =
2 (d − 1) 1 − o if 1 ≤ j < d. and P( f 1 = ±e j ) = po 2(d − 1) p
For any ω ∈ , the walk (X i )i≥0 under Poω can be constructed by setting X i+1 − X i := bi + (1 − |bi |) f i + (1 − |bi |)(1 − | f i |)Ui , where (Ui )i≥0 is a sequence of independent random variables taking values in {±e j : 1 ≤ j < d}, with o
ω
P (Ui = ±e j |Fi ) =
p π(X i , X i ± e j ) − ( 2(d−1) − )
2 (d − 1)
.
Here, Fi = σ (X 1 , . . . , X i ). Note that the laws of the sequences (bi )i≥0 and ( f i )i≥0 do not depend on the environment, and that τ1 is a function of (bi )i≥0 only.
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Let Ni :=
i−1
11=(1−|bi |)(1−| fi |) .
j=0
Introduce the events L 0 := {Nτ1 = 0}, L 1 := {Nτ1 = 1}, and L 2 := {Nτ1 ≥ 2}. Let G := σ ((bi , f i )i≥0 ). Note that the events L 0 , L 1 and L 2 are G-measurable, and so is the event {β = ∞}. On the event L 0 , the walker never sees the environment until τ1 , and thus X τ1 is G-measurable. Also, for any i ≥ 0, on the event {X i ∈ / S(X, i)} (i.e., when X i is a fresh point), a(θ, X i ) is independent of Fi and G under Po . Therefore, by isotropy, τ −1 1 a(θ, X i )1 X i ∈S(X,i) E o [Z (θ )|G] = E o G = 0. / i=0
Putting these observations together, we see that E o [θ, X τ1 Z (θ ), L 0 , β = ∞] = E o [θ, X τ1 E o [Z (θ )|G], L 0 , β = ∞] = 0.
(3.24)
On the other hand, it is easy to check that Po (L 2 ) ≤ c6 2 for some c6 = c6 ( p). By Hölder’s inequality, |E o [θ, X τ1 Z (θ ), L 2 , β = ∞]| ≤ Po (L 2 )2/3 E o [|θ, X τ1 Z (θ )|3 , β = ∞]1/3 ≤ c7 7/3 |θ |2
(3.25)
for some c7 = c7 ( p) > 0. (Recall that a(θ, ·) ≤ 2 (d − 1)|θ |, cf. (3.19).) Finally, let L 1 = L 1 ∩ {(1 − |b |)(1 − | f |) = 1, < τ1 }. Then, E o [θ, X τ1 Z (θ ), L 1 , β = ∞] =
∞
E o [θ, X τ1 Z (θ ), L 1 , β = ∞].
(3.26)
=0
For every ≥ 0, E o [θ, X τ1 Z (θ ), L 1 , β = ∞] = E o [θ, X Z (θ ), L 1 , β = ∞] +E o [θ, X +1 − X Z (θ ), L 1 , β = ∞] +E o [θ, X τ1 − X +1 Z (θ ), L 1 , β = ∞].
(3.27)
By computations similar to the one involving L 0 , the first and the third terms on the RHS of (3.27) are zero. The second term is equal to E o [θ, X +1 − X θ, v(TX ω), L 1 , β = ∞]
= Po (L 1 , β = ∞)E[E ω [θ, Uo ]θ, v(ω)] ⎞ ⎤ ⎡⎛ po π(0, z) − ( 2(d−1) −
) θ, z⎠ θ, v(ω)⎦ = Po (L 1 , β = ∞)E ⎣⎝ 2 (d − 1) z=±ed
=
Po (L 1 , β = ∞) E θ, v(ω)2 . 2 (d − 1)
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Therefore, by (3.26), E o [θ, X τ1 Z (θ ), L 1 , β = ∞] =
Po (L 1 , β = ∞) E θ, v(ω)2 . 2 (d − 1)
It is easy to see that Po (L 1 , β = ∞) ≥ c8 for some c8 = c8 ( p) > 0 if is small enough. Also, part (c) of Definition 1.7 ensures that E θ, v(ω)2 ≥ c9 2 |θ |2 for some c9 = c9 ( p) > 0. Hence, E o [θ, X τ1 Z (θ ), L 1 , β = ∞] ≥ c10 2 |θ |2
(3.28)
for some c10 = c10 ( p) > 0. Combining (3.24), (3.25) and (3.28) gives E o [θ, X τ1 Z (θ )|β = ∞] − C5 |θ |3 ≥ c10 2 |θ |2 − c7 7/3 |θ |2 − C5 |θ |3
= (c10 − c7 1/3 ) 2 − C5 |θ | |θ |2 . (3.29) If < (c10 /c7 )3 , then, for every θ ∈ Ct (c5 ) such that 0 < |θ | < (c10 − c7 1/3 ) 2 /C5 , E o [exp{θ, X τ1 − a (θ )τ1 }Z (θ )|β = ∞] > 0 by (3.23) and (3.29). Finally, Theorem 3.4 implies that Ia < Iq on the set {∇a (θ ) : θ ∈ Ct (c5 ), 0 < |θ | < (c10 − c7 1/3 ) 2 /C5 } whose closure contains the LLN velocity ξo = ∇a (0). We have proved Theorem 1.8. 4. Open Problems Our technique of proof puts several restrictions on the class of models treated. The following are natural questions we have not addressed. (1) Does Theorem 1.8 extend to all space-only RWRE in dimension d = 2, 3, or at least to those satisfying Sznitman’s condition (T)? Note that, for non-nestling walks, it suffices to show that the correlation condition (3.17) is satisfied on a sequence (θn )n≥1 that converges to zero, cf. Theorem 3.4. (2) In case π(0, z)z, e is random for any e ∈ Rso , is it true that Iq (ξ ) = Ia (ξ ) only when ξ = 0 or Ia (ξ ) = 0, as is the case in dimension d = 1? In our proof of Theorem 1.8 (specifically, in the proof of the correlation condition (3.17)), we used the isotropy assumption in order to get rid of a centering term under the (untilted) measure; this does not seem essential and probably, the lack of isotropy could be handled in the perturbative regime. However, getting rid of the perturbative restriction, or of the non-randomness in the ed direction, requires additional arguments. Acknowledgements. This research was supported partially by a grant from the Israeli Science Foundation, and by the Alhadeff Fund at the Weizmann Institute. We thank Francis Comets for providing us with an update on polymer models and bringing the work of Lacoin [10] to our attention. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Commun. Math. Phys. 300, 273–299 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1091-y
Communications in
Mathematical Physics
Continuous Spectrum of Automorphism Groups and the Infraparticle Problem Wojciech Dybalski Zentrum Mathematik, Technische Universität München, D-85747 Garching, Germany. E-mail: [email protected] Received: 12 December 2009 / Accepted: 24 February 2010 Published online: 11 July 2010 – © Springer-Verlag 2010
Abstract: This paper presents a general framework for a refined spectral analysis of a group of isometries acting on a Banach space, which extends the spectral theory of Arveson. The concept of a continuous Arveson spectrum is introduced and the corresponding spectral subspace is defined. The absolutely continuous and singular-continuous parts of this spectrum are specified. Conditions are given, in terms of the transposed action of the group of isometries, which guarantee that the pure-point and continuous subspaces span the entire Banach space. In the case of a unitarily implemented group of automorphisms, acting on a C ∗ -algebra, relations between the continuous spectrum of the automorphisms and the spectrum of the implementing group of unitaries are found. The group of spacetime translation automorphisms in quantum field theory is analyzed in detail. In particular, it is shown that the structure of its continuous spectrum is relevant to the problem of existence of (infra-)particles in a given theory. 1. Introduction In the familiar case of a strongly continuous group of unitaries U (t) = ei H t , acting on a Hilbert space H, spectral theory is well understood. In particular, H can be decomposed into the pure-point, absolutely continuous and singular-continuous subspaces, which reflects the decomposition of the spectral measure of H into measure classes. On the other hand, for a strongly continuous group of isometries αt = ei Dt acting on a Banach space A, spectral theory is much less developed. While the Arveson spectral theory provides subspaces associated with closed subsets of the spectrum of D [Ar82,Ev76,Lo77], there does not seem to exist any general definition of the continuous spectral subspace, not to speak of its absolutely continuous or singular continuous parts. It is the main goal of the present paper to introduce such notions and demonstrate their relevance to the problem of particle interpretation in quantum field theory (QFT). In the Hilbert space setting these detailed spectral concepts provided a natural framework for the formulation and resolution of the problem of asymptotic completeness
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in quantum mechanics [En78,SiSo87,Gr90,De93]. The absence of such structures on the side of Banach spaces impedes the study of particle aspects in quantum field theory, where the time evolution is governed by a group of automorphisms α acting on a C ∗ -algebra of observables A. Asymptotic completeness is an open problem in all known models of interacting quantum fields, except for a recently constructed class of two-dimensional theories with factorizing S-matrices [Le08]. Since pairs of charged particles may be produced in collisions of neutral particles, it is not even a priori clear what particle types a given theory describes. Finally, the possible presence of charges with weak localization properties, like the electric charge in quantum electrodynamics, forces one to depart from the conventional Wigner concept of a particle as a state in some irreducible representation space of the Poincaré group [Wi39]. In spite of decades of research [Sch63,FMS79,Bu86], this infraparticle problem is still a largely open issue. One approach to the problems listed above is to study simplified models which capture some relevant features of quantum field theories. Over the last decade there has been significant progress along these lines [DG00,FGS04,Sp,Pi05,CFP07,He07,Re09]. A complementary approach, pursued in algebraic quantum field theory, aims at a development of a model-independent concept of a particle which is sufficiently general to encompass all the particle-like structures appearing in quantum field theory [Bu94]. Substantial steps in this direction were made by Buchholz, Porrmann and Stein [BPS91]. In order to clarify the relation between the particle aspects of quantum field theory and the (Arveson) spectrum of the group of automorphisms Rs+1 (t, x) → α(t,x ) , which describes spacetime translations of observables, we recall the main steps of this analysis: To extract the particle content of a physical state ω ∈ A∗ , one has to compensate for dispersive effects. To this end, one paves the whole space with observables and sums up the results. This amounts to studying the time evolution of the integrals σω(t) (A) := d s x ω(α(t,x ) (A)), (1.1) (t)
where A ∈ A is a suitable observable. The limit points of σω as t → ∞, called the (+) asymptotic functionals σω , carry information about all the particle types appearing in the theory. In fact, for theories of Wigner particles it was shown in [AH67] that each asymptotic functional can be represented as a mixture of plane wave configurations of all the particle types contributing to the state ω, σω(+) (A) = (1.2) d s p ρλ ( p) p, λ|A| p, λ . λ
This mixture is labeled by λ = [m, s, q] (i.e. mass, spin and charge), possibly including pairs of charged particles. The functions ρλ stand for the asymptotic densities of the respective particle types. In the general case, including both Wigner particles and infraparticles, a similar expression was derived by Porrmann [Po04.2] σω(+) (A) = dμ(λ) σλ(+) (A). (1.3) Here the analogues of the plane wave configurations are the so-called pure particle (+) weights σλ labeled by λ = [ p, γ ], where p is the four-momentum of the particle and γ carries information about the other quantum numbers, like spin or charge. Each pure particle weight gives rise to an irreducible representation of A. In contrast to the
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previous case, the representations corresponding to different four-momenta p may be inequivalent. Thus the infraparticle situation, e.g. the electron whose velocity gives a superselection rule [FMS79,Bu82,Bu86], can be treated in this framework. A necessary prerequisite for this approach is the existence of non-zero asymptotic functionals. It is well known from the study of generalized free fields that this property does not follow from the general postulates of quantum field theory. However, it has been established for theories of Wigner particles [AH67] and in a non-interacting model of an infraparticle introduced by Schroer [Sch63,Joh91]. Moreover, in [Dy08.2] we supplied a model-independent argument, ensuring the existence of non-zero asymptotic functionals in a certain class of theories containing a stress-energy tensor.1 In view of this recent progress, one can hope for a general classification of quantum field theories w.r.t. their particle structure. It is the main goal of the present work to develop a natural language for such a classification. As a first step in this direction, we infer from formula (1.2) that the asymptotic functional is non-trivial, only if its domain contains sufficiently many observables, whose energy-momentum transfer includes zero. We recall that the energy-momentum transfer of an observable A coincides with the Arveson spectrum of the group of automorphisms α restricted to the subspace spanned by the orbit of A (cf. definition (2.3)). Therefore, essential information about the particle content of a given theory should be encoded in the properties of the Arveson spectrum of α in a neighborhood of zero. As it is evident from definition (1.1) that the joint eigenvectors of the generators of α do not belong to the domains of the asymptotic functionals, the relevant part of the spectrum is the continuous one. However, the existing spectral theory is not yet sufficiently developed to test such fine features of the Arveson spectrum. It provides a functional calculus and spectral mapping theorems for some classes of functions as well as a definition and properties of spectral subspaces associated with closed subsets of the spectrum. (See [Ar82] for a review.) This allows e.g. for a study of the pure-point spectrum which has attracted much attention from various perspectives [Ba78,Jo82,AB97,Hu99]. To our knowledge, there is no systematic analysis of the continuous Arveson spectrum in the literature. Nevertheless, there exist some interesting results pertaining to decay properties of the functions Rd x → ω(αx (A)) and regularity properties of their Fourier transforms p)). We mention the analysis of Jorgensen [Jo92], inspired by the Stone Rd p → ω( A( formula, and a result of Buchholz [Bu90], concerning space translations in QFT, which · )) appear will be used in Sect. 5 of the present paper. The Fourier transforms ω( A( also as a tool in the literature related to the Rieffel project of extending the notions of proper action and orbit space from the setting of group actions on locally compact spaces to the context of C ∗ -dynamical systems [Ri90,Ex99,Ex00,Me01]. This recent revival of interest in the subject is an additional motivation for the general analysis of the continuous Arveson spectrum which we undertake in this work. Let (α, A) be a strongly continuous group of isometries acting on a Banach space A and let (α ∗ , A∗ ) be the transposed action of α on a suitable closed, invariant subspace A∗ ⊂ A∗ . In this framework we define the pure-point and continuous spectral subspaces of α and α ∗ , denoted by App , Ac and A∗,pp , A∗,c , respectively. Certainly, App , (resp. A∗,pp ), is spanned by the joint eigenvectors of the generators of α, (resp. α ∗ ). In the absence of orthogonality, our definition of the continuous subspace is motivated by the
1 The relevance of the stress-energy tensor to particle aspects was first pointed out in [Bu94].
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Ergodic Theorem from the setting of groups of unitaries: 1 e−iq x ϕ(αx (A)) d d x = 0 }. Ac := { A ∈ A | ∀ lim q∈Rd K Rd |K | K
(1.4)
ϕ∈A∗
The subspace A∗,c is obtained by exchanging the roles of A and A∗ above. In analogy with the Hilbert space setting, A∗,pp , Ac = A∗,c , App = 0,
(1.5)
where · , · denotes the evaluation of functionals from A∗ on elements of A. Exploiting these facts, we find necessary and sufficient conditions for the following decompositions to hold: A = App ⊕ Ac , A∗ = A∗,pp ⊕ A∗,c ,
(1.6) (1.7)
as well as examples, where these equalities fail. More refined analysis of the continuous ˆ ⊂ A and A ˆ ∗ ⊂ A∗ . By spectrum requires the choice of some norm dense subspaces A definition, the absolutely continuous part Aac is generated by all A ∈ Ac s.t. the Fourier transforms of the corresponding functions Rd x → ϕ(αx (A)) are integrable for all ˆ ∗ . As for the singular-continuous space, in the Banach space setting the canonical ϕ∈A choice is the quotient Asc := Ac /Aac
(1.8)
on which there acts the reduced group of isometries α, defined naturally on the equivalence classes. The point-spectrum Sppp α consists of joint eigenvalues of the generators of α. The continuous spectrum of α, denoted by Spc α, is defined as the Arveson spectrum of α restricted to the subspace Ac . The absolutely continuous spectrum Spac α is constructed analogously. The singular-continuous part Spsc α is specified as the Arveson spectrum of α. The corresponding spectral concepts on the side of α ∗ are defined in obvious analogy. We remark that in the Hilbert space setting the above spectra and the subspaces App , Ac and Aac coincide with the standard ones. The space Asc is isomorphic, as a Banach space, to the conventional singular-continuous subspace. We are primarily interested in the case of a group of automorphisms α acting on a C ∗ -algebra A. We assume the existence of a pure state ω0 on A s.t. ker ω0 ⊂ Ac ,
(1.9)
if A∗ is chosen as the predual of the GNS representation (π, H, ) induced by ω0 . Then there follow decompositions (1.6) and (1.7) as well as the invariance of ω0 under the action of α ∗ . Hence, there acts in H a strongly-continuous unitary representation U of Rd which implements α, i.e. π(αx (A)) = U (x)π(A)U (x)−1 ,
A ∈ A, x ∈ Rd .
(1.10)
We establish new relations between the spectral concepts on the side of α and U . In particular, we find the following continuity transfer relations: π(Ac ) ⊂ Hc , π(Aac ) ⊂ Hac ,
(1.11) (1.12)
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akin to the spectrum transfer property (4.23) from the standard Arveson theory. We also verify the inclusions ±Spsc U ⊂ Spsc α, ±Spac U ⊂ Spac α ∗ ,
(1.13) (1.14)
which provide means to estimate the shapes of the spectra introduced above. This analysis applies, in particular, to the group of spacetime translation automorphisms α in any local, relativistic quantum field theory equipped with a normal vacuum state ω0 . We obtain from inclusions (1.13), (1.14) that in a theory of Wigner particles the singular-continuous spectrum of α contains the mass hyperboloids of these particles, whereas the multiparticle spectrum contributes to Spac α ∗ . The subgroup of space translation automorphisms βx = α(0,x ) allows for a more detailed analysis. Relying on results from [Bu90], we obtain that Spac β = Spac β ∗ = Rs , whereas Spsc β = Spsc β ∗ are either empty or consist of {0}. These spectra turn out to be empty in theories satisfying certain timelike asymptotic abelianess condition introduced in [BWa92] or complying with a regularity criterion L (1) which restricts the continuous spectrum of α in a neighborhood of zero. We show, following [Dy08.2], that this latter condition implies the existence of particles in theories containing a stress-energy tensor. This paper is organized as follows: In Sect. 2 we recall the basics of the Arveson spectral theory, introduce the continuous Arveson spectrum and decompose it into the absolutely continuous and singular-continuous parts. Section 3 focuses on relations between the spectra of α and α ∗ . Unitarily implemented groups of automorphisms acting on C ∗ -algebras are studied in Sect. 4. In Sect. 5 we establish general properties of the continuous spectrum of spacetime translation automorphisms valid in any local relativistic quantum field theory admitting a normal vacuum state. In Sect. 6, which contains some results from the author’s PhD thesis [Dy08.2], we restrict attention to models complying with the regularity condition L (1) . We strengthen our spectral results in this setting and show that such theories describe particles, if they contain a stress-energy tensor. Section 7 summarizes our results and outlines future directions. In the Appendix we consider spectral theory of space translation automorphisms in the absence of normal vacuum states. In particular, we provide examples which violate relations (1.6), (1.7). 2. Continuous Spectrum of a Group of Isometries In this section we consider a strongly continuous group of isometries Rd x → αx acting on a Banach space A. We choose a subspace A∗ ⊂ A∗ which satisfies the following: Condition S: The subspace A∗ is norm closed in A∗ and invariant under the action of α ∗ . Moreover, for any A ∈ A, A = sup |ϕ(A)|. ϕ∈A∗,1
(2.1)
The foundations of spectral theory in this setting were laid by Arveson [Ar74,Ar82]. We recall below the familiar concepts of spectral subspaces and the pure-point spectrum. Next, we propose a new notion of the continuous Arveson spectrum and decompose it into the absolutely continuous and singular-continuous parts. For this latter purpose we ˆ ∗ ⊂ A∗ . choose a norm dense subspace A
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For any ϕ ∈ A∗ and A ∈ A we consider the Fourier transforms of bounded, continuous functions Rd x → ϕ(αx (A)), p)) := 1 ϕ( A( (2.2) d d x e−i px ϕ(αx (A)), d (2π ) 2 which are tempered distributions. Here px stands for some non-degenerate inner product in Rd and we adhere to the Fourier transform convention which omits the minus sign in front of px in the case of test-functions. The Arveson spectrum of the group of isometries α is defined as follows: · )). Sp α := supp ϕ( A( (2.3) A∈A ϕ∈A∗
We also define the Arveson spectrum of an individual element A ∈ A as Sp A α := Sp α| X , where X = Span{ αx (A) | x ∈ Rd }n-cl and n-cl denotes the norm closure. (We mention as an aside that for a strongly continuous one-parameter group of isometries, Sp α coind cides with the operator-theoretic spectrum of the infinitesimal generator D = 1i dt αt |t=0 [Ev76,Lo77]). Similarly as in the Hilbert space setting, for any closed set ⊂ Rd one defines the spectral subspace · )) ⊂ }. A( ) := { A ∈ A | ∀ supp ϕ( A( ϕ∈A∗
(2.4)
In particular, for a single point q ∈ Rd there holds A({q}) = { A ∈ A | αx (A) = eiq x A for all x ∈ Rd }.
(2.5)
This leads us to natural definitions of the pure-point subspace and the pure-point spectrum, analogous to those from the Hilbert space setting: App := Span{ A({q}) | q ∈ Sp α }n-cl , A({q}) = {0} }. Sppp α := { q ∈ Sp α |
(2.6) (2.7)
We note that these objects are independent of the choice of A∗ , as long as it satisfies Condition S. The spectral subspaces (2.4) and the pure-point spectrum have been thoroughly studied in the literature. (See [Ar82,Pe] for a review of the former subject and [Ba78,Hu99] for interesting results on the latter.) However, there does not seem to exist any generally accepted definition of the continuous Arveson spectrum. Motivated by the Ergodic Theorem from the setting of groups of unitaries acting on Hilbert spaces [RS1], we propose the following norm closed, invariant subspace Ac and the corresponding spectrum 1 Ac := { A ∈ A | ∀ lim e−iq x ϕ(αx (A)) d d x = 0 }, (2.8) q∈Rd K Rd |K | K ϕ∈A∗
Spc α := Sp α|Ac .
(2.9)
Here K Rd denotes a family of cuboids, centered at zero, whose edge lengths tend to infinity.2 In contrast to the pure-point part, the continuous subspace depends on the 2 In some cases we impose further restrictions on this family. See the discussion preceding Theorem 5.1.
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choice of A∗ . However, for any such choice App ∩ Ac = {0}, if Sppp α is a finite set. On the other hand, the equality A = App ⊕ Ac , expected from the Hilbert space setting, fails in some cases, as we show in the Appendix. Necessary and sufficient conditions for this equality, which we establish in Sect. 3, will suggest judicious choices of A∗ . Let us now introduce more refined spectral concepts which are sensitive to regular · )). With the help of the norm dense subspace ity properties of the distributions ϕ( A( ˆ ∗ in A∗ we define the absolutely continuous subspace and the corresponding spectrum: A · )) ∈ L 1 (Rd , d d p) }n-cl , Aac := { A ∈ Ac | ∀ ϕ( A( ˆ∗ ϕ∈A
Spac α := Sp α|Aac .
(2.10) (2.11)
Let us recall that the singular-continuous part of a Hilbert space is the orthogonal complement of the absolutely continuous subspace in the continuous one. In the case of a Banach space there may not exist any direct sum complement of Aac which is invariant under the action of α. (As a matter of fact, there may not exist any direct sum complement at all [Ru]). Therefore, we define the singular continuous space as a quotient Asc := Ac /Aac .
(2.12)
We denote the equivalence class of an element A ∈ Ac by [A] and introduce the strongly continuous group of isometries Rd x → α x of Asc given by α x [A] := [αx (A)]. We define the singular continuous spectrum as the Arveson spectrum of α, · )), Spsc α := Sp α = supp ϕ([A](
(2.13)
(2.14)
A∈Ac ϕ∈A∗sc
p)) = (2π )− d2 d d x e−i px ϕ(α x ([A])). Clearly, Spsc α = ∅, if and only if where ϕ([A]( Aac = Ac . To conclude this section, let us consider briefly the case of a group of unitaries α ˆ ∗ = A∗ = A∗ . Noting that the distributions ϕ( A( · )) acting on a Hilbert space A with A are then just the spectral measures, we obtain that the above concepts of the pure-point, continuous and absolutely continuous spectral subspaces, and the corresponding spectra coincide with the standard ones. Moreover, Spsc α is equal to the conventional singular-continuous spectrum and the space (2.12) is isomorphic, as a Banach space, to the singular-continuous subspace. 3. Spectral Analysis of the Transposed Action In the setting of groups of unitaries, acting on a Hilbert space, there always holds A = App ⊕ Ac .
(3.15)
However, as shown in the Appendix, in the Banach space setting the above equality fails, if the space A∗ is excessively large. In order to characterize the choices of A∗ which entail relation (3.15), we develop the spectral theory for the transposed action α ∗ . Thus we define the spectra Sp α ∗ , Sppp α ∗ , Spc α ∗ , as well as the subspaces A∗ ( ),
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A∗,pp , A∗,c analogously as in the previous section, by exchanging the roles of A∗ and ˆ ⊂ A, we also define A∗,ac , A∗,sc and the A. After selecting a norm dense subspace A corresponding spectra Spac α ∗ , Spsc α ∗ .3 The evaluation of functionals from A∗ on elements of A provides a natural substitute for the scalar product from the Hilbert space setting. We exploit this observation in the following lemma. To stress the analogy with the Hilbert space case, we denote this evaluation by · , · . Moreover, for any subset A0 ⊂ A we define the annihilator A⊥ 0 = { ϕ ∈ A∗ | ∀ A∈A0 ϕ(A) = 0 }, and analogously for subsets of A∗ . Lemma 3.1. There holds A( ) = 0, if , ⊂ Rd are closed and ∩ = ∅, (a) A∗ ( ), (b) A∗,pp , Ac = 0 and A∗,c , App = 0. Proof. As for part (a), let χ , (resp. χ ), be a bounded, smooth function on Rd which is equal to one on , (resp. on ). Moreover, suppose that supp χ ∩ supp χ = ∅. Then there holds for any ϕ ∈ A∗ ( ), A ∈ A( ) and f ∈ S(Rd ), p)) f˜( p) d d x ϕ(αx (A)) f (x) = d d p ϕ( A( p)) f˜( p)χ ( p) χ ( p) = 0. (3.16) = d d p ϕ( A( Therefore Rd x → ϕ(αx (A)) vanishes as a distribution. Since it is a continuous function, it vanishes pointwise. To prove (b), suppose that ϕ ∈ A∗ is an eigenvector, i.e. αx∗ ϕ = eiq x ϕ for some q ∈ Rd . Then there holds for any A ∈ Ac , 1 1 d d x e−iq x eiq x ϕ(A) = d d x e−iq x ϕ(αx (A)) → 0. (3.17) ϕ(A) = |K | K |K | K K R d Since eigenvectors form a total set in A∗,pp , the proof of the first statement in (b) is complete. The proof of the second statement is analogous. With the help of the above observation we easily obtain the following list of necessary conditions for the decomposition (3.15). Theorem 3.2. Suppose that A = App ⊕ Ac . Then: (a) For any q ∈ Rd there holds dim A({q}) ≥ dim A∗ ({q}), (b) A⊥ = A , ∗,c pp (c) Sppp α ⊃ Sppp α ∗ and Spc α ⊃ Spc α ∗ . Remark. This result also holds with the roles of A and A∗ exchanged. Proof. In part (a) it suffices to consider the case dim A({q}) = m < ∞. Suppose there exist m + 1 linearly independent functionals {ϕk }m+1 in A∗ ({q}). Then, according to 1 Lemma 3.1 (b), their restrictions ϕˆk to the subspace ˚ pp := Span{ A A({q }) | q ∈ Sppp α }
(3.18)
3 If α ∗ does not act norm continuously on A ∗ ∗,sc , we refrain from defining Spsc α . All the other definitions remain meaningful, if the action of α ∗ is only weakly∗ continuous.
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which consists of finite, linear combinations of eigenvectors, still form a linearly independent family. In view of Lemma 3.1 (a), ker ϕˆ1 ∩ · · · ∩ ker ϕˆ m+1 ⊃ Span{ A({q }) | q ∈ Sppp α, q = q }, (3.19) which is a contradiction, since the subspace on the r.h.s. of this relation has codimension ˚ pp . m in A As for part (b), we note that A⊥ pp ⊃ A∗,c , by Lemma 3.1 (b). In order to prove the opposite inclusion, we pick ϕ ∈ A⊥ pp and arbitrary A ∈ A. According to our assumption, A = App + Ac , where App ∈ App and Ac ∈ Ac . There holds for any q ∈ Rd , 1 1 d −iq x d xe ϕ(αx (A)) = d d x e−iq x ϕ(αx (Ac )) → 0, (3.20) |K | K |K | K K R d and therefore ϕ ∈ A∗,c . The statement concerning the point spectrum in (c) follows immediately from part (a). As for the continuous spectrum, we note that · )) ⊃ · )) = Spc α ∗ , Spc α = supp ϕ( A( supp ϕ( A( (3.21) A∈Ac ϕ∈A∗
A∈Ac ϕ∈A∗,c
where the last equality relies on the fact that A⊥ pp ⊃ A∗,c and on the assumption.
In the Appendix we find counterexamples to relation (3.15). To this end, we exploit part (a) of the above theorem as follows: First, we show that for the group of space translation automorphisms acting on the algebra of observables A in quantum field theory dim A({0}) = 1 always holds. Next, we note that A∗ ({0}) is spanned by the functionals which are invariant under the transposed action. Hence A = App ⊕ Ac , if A∗ contains more than one vacuum state4 . Coming back to the general setting of groups of isometries, we provide sufficient conditions for relation (3.15) to hold. The following theorem accounts for the important role of the point spectrum of α ∗ in the study of the continuous spectrum of α. Its assumptions have a natural formulation in the framework of automorphism groups of C ∗ -algebras which we explore in the next section. Theorem 3.3. Suppose there exists a functional ω0 ∈ A∗ s.t. ker ω0 ⊂ Ac and a non-zero element I ∈ A({q}) for some q ∈ Rd . Then: (a) A = App ⊕ Ac , where App = Span {I }, Ac = ker ω0 , A∗ ({q}), (b) ω0 ∈ (c) A∗ = A∗,pp ⊕ A∗,c , where A∗,pp = Span {ω0 }, A∗,c = ker I . Moreover, Spc α = Spc α ∗ . Proof. By assumption, Ac is a subspace of codimension at most one in A. Since App ⊃ Span {I }, part (a) follows. Hence any A ∈ A can be expressed as A = cI + Ac , where Ac ∈ Ac . Since Ac is invariant under the action of automorphisms, there holds αx∗ ω0 (A) = cω0 (αx (I )) = eiq x ω0 (A), which entails (b). From Theorem 3.2 (b) we obtain that A∗,c = ker I . Since it is a subspace of codimension one, part (c) follows from part (b). The last statement is a consequence of Theorem 3.2 (c) and parts (a), (c) of the present theorem. 4 That is a translationally invariant state on the algebra of observables s.t. the relativistic spectrum condition holds in its GNS representation (cf. Sect. 5).
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4. Spectral Analysis of a Group of Automorphisms In this section we consider a group of automorphisms α acting on a C ∗ -algebra A containing a unity I . We assume that there exists a pure state ω0 on A which satisfies ker ω0 ⊂ Ac ,
(4.22)
where A∗ , entering the definition of Ac , is chosen as the predual of the GNS representation (π, H, ) induced by the state ω0 . (As we will see in Sect. 5, in the case of spacetime translation automorphisms in QFT any pure vacuum state satisfies this inclusion as a ˆ ∗ ⊂ A∗ we impose the following condition: consequence of locality.) On the subspace A ˆ ∗ is self-adjoint5 , norm dense in A∗ and contains all the ˆ The subspace A Condition S: functionals of the form (|π( · ) ), where belongs to some dense subspace in H. Having specified the framework, we proceed to the spectral analysis of α. Exploiting Theorem 3.3 and the fact that A is unital, we obtain: Theorem 4.1. Let A, ω0 and A∗ be specified as above. Then: (a) A = App ⊕ Ac , where App = Span {I }, Ac = ker ω0 , A∗ ({0}), (b) ω0 ∈ (c) A∗ = A∗,pp ⊕ A∗,c , where A∗,pp = Span {ω0 }, A∗,c = ker I . Moreover, Spc α = Spc α ∗ . In view of part (b) of the above theorem, the state ω0 is invariant under the action of α ∗ . Hence there exists a strongly continuous group of unitaries Rd x → U (x), acting on H, s.t. π(αx (A)) = U (x)π(A)U (x)−1 and the vector ∈ H is invariant under its action. We denote by H( ), Hpp , Hc , Hac , Hsc the spectral subspaces of H w.r.t. the action of U . One of the central problems in the present setting is to find relations between the Arveson spectrum of α and the spectrum of the implementing group of unitaries. An important and well known property is the spectrum transfer: If A ∈ A( 1 ) then 2 ) ⊂ H( 1 + 2 ), π(A)H(
(4.23)
where 1 , 2 ⊂ Rd are closed sets [Ar82]. It turns out that similar properties, which can be called continuity transfer relations, hold at the level of more detailed spectral theory: Proposition 4.2. Let A, ω0 and A∗ be specified as above and suppose that Condition Sˆ holds. Then: (a) Hpp = Span{ }, (b) Hc = { π(Ac ) }n-cl , (c) Hac ⊃ π(Aac ) . Proof. First, we show that π(Ac ) ⊂ Hc : Let A ∈ Ac and ∈ H be arbitrary. Then for any q ∈ Rd , 1 d d x e−iq x (|U (x)π(A) ) |K | K 1 = d d x e−iq x ϕ(αx (A)) → 0, (4.24) |K | K K R d 5 That is if ϕ ∈ A ˆ ∗ then ϕ ∈ A ˆ ∗ , where ϕ(A) = ϕ(A∗ ), A ∈ A.
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where in the last step we made use of the fact that ϕ( · ) = (|π( · ) ) ∈ A∗ . Thus, by the Ergodic Theorem, π(A) ∈ Hc . Next, we verify that { π(Ac ) }n-cl = { }⊥ . Suppose that ∈ H and (| ) = 0. By cyclicity of , there exists a sequence {An }n∈N of elements of A s.t. −π(An ) → 0. Then Bn := An − ω0 (An )I is a sequence of elements of Ac which satisfies − π(Bn ) → 0. This completes the proof of (a) and (b). To prove (c), let A ∈ Aac and let {An }n∈N be a sequence of elements of Aac which converges to A in norm and s.t. n ( p))| < ∞ (4.25) d d p |ϕ( A ˆ ∗ . Then, for any from the dense set appearing in Condition S, ˆ for all n ∈ N and ϕ ∈ A n ( p)) )| < ∞. · )π(An ) )( p)| = d d p |(|π( A (4.26) d d p |(|U ( Hence the vectors π(An ) belong to Hac and so does their norm limit π(A) .
After this preparation we establish relations between the spectra of U and α. We use these facts in the next section, in a quantum field theoretic context. Theorem 4.3. Under the assumptions of Proposition 4.2 the following relations hold: (a) (b) (c) (d)
Sppp U = Sppp α = Sppp α ∗ = {0}, Spc U − Spc U ⊂ Spc α, ±Spac U ⊂ Spac α ∗ , ±Spsc U ⊂ Spsc α.
Proof. Part (a) follows immediately from Proposition 4.2 (a) and Theorem 4.1. To prove (b), let q = p1 − p2 , where p1 , p2 ∈ Spc U . We choose an open neighborhood Vq of q and bounded neighborhoods 1 , 2 of p1 and p2 , respectively, s.t. 1 − 2 ⊂ Vq . We pick 1 ∈ Ran P( 1 ) and 2 ∈ Ran P( 2 ). Since the continuous spectrum of U is a subset of the essential spectrum, we can find 1 , 2 such that (1 |2 ) = 0. By purity of the state ω0 , π(A) acts irreducibly on H and we can find such A ∈ A that (1 |π(A)2 ) = 0.
(4.27)
Replacing A with A−ω0 (A)I , if necessary, we can assume that A ∈ Ac . Now we choose f ∈ S(Rd ) s.t. f˜( p) = (2π )−d/2 for p ∈ 1 − 2 and f˜( p) = 0 for p outside of Vq . · ))2 ) is contained Making use of the fact that the support of the distribution (1 |π( A( in 1 − 2 , which follows from relation (4.23), we obtain d p))2 ) d x f (x)(1 |π(αx (A))2 ) = d d p f˜( p)(1 |π( A( = (1 |π(A)2 ) = 0,
(4.28)
which implies that q ∈ Spc α. As for (c), let q ∈ Spac U and let Vq be any open neighborhood of q. Then there exists f ∈ S(Rd ) s.t. supp f˜ ⊂ Vq and f := d d x f (x)U (x) = 0 for some ∈ Hac . We note that the functional ϕ ( · ) := (|π( · ) ) belongs to A∗,ac , since for any A ∈ A, p))| = d d p |(|U ( d d p |ϕ ( A( · )π(A) )( p)| < ∞, (4.29)
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where in the last step we made use of the fact that ∈ Hac . Next, since π(A) is dense in H, we find such B ∈ A that ( f |π(B) ) = 0. Hence (4.30) d d x f¯− (x) αx∗ ϕ (B) = 0, where f − (x) = f (−x). Since supp f˜¯− ⊂ Vq , we obtain that q ∈ Spac α ∗ . To show that −q ∈ Spac α ∗ , we repeat the argument using the functional ϕ ( · ) = ( |π( · )) instead of ϕ . To prove (d), let q ∈ Spsc U and let Vq be an open neighborhood of q. Then there exists f ∈ S(Rd ) s.t. supp f˜ ⊂ Vq and f := d d x f (x)U (x) = 0 for some ∈ Hsc . By Proposition 4.2 (b), π(Ac ) is dense in Hc , so we can find A ∈ Ac s.t. ( f |π(A) ) = 0.
(4.31)
We note that the functional ϕ ( · ) = (|π( · ) ) is an element of A∗c which, by Proposition 4.2 (c), contains Aac in its kernel. Therefore, ϕ induces a well defined, bounded functional ϕ on Asc = Ac /Aac s.t. ϕ ([B]) = ϕ (B) for B ∈ Ac . By relation (4.31), (4.32) d d x f¯− (x)ϕ (α x [A]) = 0, which proves that q ∈ Spsc α. To show that −q ∈ Spsc α, we repeat the argument using the functional ϕ ( · ) = ( |π( · )) instead of ϕ . 5. Spectral Theory of Automorphism Groups in QFT In this section we analyze the spectrum of the group of spacetime translation automorphisms acting on the algebra of observables in quantum field theory. To keep our investigation general, we rely on the Haag-Kastler framework of algebraic quantum field theory [Ha]: The theory is based on a net O → A(O) ⊂ B(H) of unital C ∗ -algebras, attached to open, bounded regions of spacetime O ⊂ Rs+1 , which satisfies isotony and locality. The ∗-algebra of local operators is given by Aloc := A(O) (5.1) O⊂Rs+1
and its norm closure A is irreducibly represented on the infinitely dimensional Hilbert space H. Moreover, H carries a strongly continuous unitary representation of translations Rs+1 x → U (x) which satisfies the relativistic spectral condition, i.e. the joint spectrum of the generators H, P1 , . . . , Ps is contained in the closed forward lightcone V + . It is also assumed that the translation automorphisms αx ( · ) = U (x) · U (x)∗ act geometrically on the net, i.e. αx (A(O)) = A(O + x),
(5.2)
and are strongly continuous, that is any function Rs+1 x → αx (A), A ∈ A, is continuous in the norm topology of A. Suppose that the Hilbert space contains a vacuum vector
, which is invariant under the action of U . Then we set ω0 ( · ) = ( | · ) and say that the theory admits a normal vacuum state ω0 . This state belongs to the general class of functionals of bounded energy, defined as follows: Let PE be the spectral projection of
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H (the Hamiltonian) on the subspace spanned by vectors of energy lower than or equal to E. We identify B(H)∗ with the space of trace-class operators on H and denote by S E the set of states from PE B(H)∗ PE . The following norm dense subspace of B(H)∗ : SE } (5.3) B(H)∗,bd := Span{ E≥0
is called the space of functionals of bounded energy. It is easy to incorporate the triple (α, A, ω0 ), introduced above, into the algebraic setting of the previous section: We note that the GNS representation of A, induced by ω0 , coincides with the defining representation, up to unitary equivalence. We set ˆ ∗ = B(H)∗,bd , A ˆ = Aloc A∗ = B(H)∗ , A
(5.4)
and check that Condition Sˆ is satisfied. It remains to verify that there holds the key assumption (4.22), i.e. ker ω0 ⊂ Ac . To this end, we choose the family of sets K Rs+1 , appearing in definition (2.8), so as to exploit the local structure of the theory. In the case of spacetime translations we set K L = [−L ε , L ε ] × [−L , L]×s for some 0 < ε < 1 and any L ∈ R+ , while for the subgroup of space translations we choose K L = [−L , L]×s . Clearly, K L Rs+1 and K L Rs as L → ∞. Since we work with the Minkow ski space Rs+1 , we set px = p0 x0 − p x, p x = sj=1 p j x j , in definitions (2.2) and (2.8). Accordingly, px = p0 x0 for the time axis and px = − p x for the spacelike hyperplane Rs . After this preparation, we obtain: Theorem 5.1. Let Rs+1 x → αx be the group of spacetime translation automorphisms acting on the algebra of observables A in a quantum field theory admitting a normal vacuum state ω0 and let A∗ = B(H)∗ . Then ker ω0 ⊂ Ac and, consequently, (a) A = App ⊕ Ac , where App = Span {I }, Ac = ker ω0 , (b) A∗ = A∗,pp ⊕ A∗,c , where A∗,pp = Span {ω0 }, A∗,c = ker I , (c) Spc α = Spc α ∗ . The above statements are also true for the subgroup Rs x → βx := α(0,x ) of space translation automorphisms. Proof. To verify that ker ω0 ⊂ Ac , we have to show that for any A ∈ ker ω0 and q ∈ Rs+1 , 1 ∗ w - lim d s+1 x e−iq x αx (A) = 0. (5.5) L→∞ |K L | K L This can be proven by a rather standard argument: We note that for any normal functional ϕ ∈ B(H)∗ the function Rs+1 x → αx∗ ϕ is continuous w.r.t. the norm topology in B(H)∗ so the following Bochner integrals: 1 d s+1 x e−iq x αx∗ ϕ (5.6) ϕL = |K L | K L define functionals from B(H)∗ . Now we fix a state ω ∈ B(H)∗ and obtain, from the Banach-Alaoglu theorem, a net { K L β ⊂ Rs+1 | β ∈ I } and a functional ωI ∈ A∗ s.t. w∗ - lim ω L β = ωI . β
(5.7)
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Next, for any A ∈ A, we consider the following net of elements from B(H): 1 d s+1 x e−iq x αx (A). PL (A) := |K L | K L
(5.8)
By locality and the slow growth of the timelike dimension of K L , the net {PL (A)} L>0 satisfies lim [PL (A), B] = 0
L→∞
(5.9)
for any B ∈ A. Therefore, all its limit points w.r.t. the weak∗ topology on B(H) are multiples of the identity by the assumed irreducibility of A. It follows that for any ϕ ∈ B(H)∗ , A ∈ A, lim (ϕ L (A) − ω L (A)ϕ(I )) = 0.
L→∞
Consequently w∗ - lim β
1 |K L β |
d s+1 x e−iq x αx (A) = ωI (A)I.
(5.10)
(5.11)
K Lβ
By evaluating this relation on the translationally invariant state ω0 , we obtain that ωI = ω0 for q = 0 and ωI = 0 otherwise, which entails equality (5.5). Now parts (a), (b) and (c) follow from Theorem 4.1. By an obvious modification of the above argument we obtain the statement concerning the subgroup of space translations. Proceeding to more detailed analysis of the spectrum of α, we obtain, with the help of Theorem 4.3, the following facts: Theorem 5.2. Let Rs+1 x → αx be the group of spacetime translation automorphisms acting on the algebra of observables A in a quantum field theory admitting a normal vacuum state ω0 . Let m > 0 and suppose that Sp U = {0} ∪ hm ∪ gm , where hm = { p ∈ Rs+1 | p 2 = m 2 , p 0 > 0 } is the mass hyperboloid and gm = { p ∈ Rs+1 | p 2 ≥ (2m)2 , p 0 > 0 } is the multiparticle spectrum. Then there holds: (a) (b) (c) (d)
Sppp α = Sppp α ∗ = {0}, Spc α = Spc α ∗ = Rs+1 , Spac α ∗ ⊃ ±gm , Spsc α ⊃ ±hm .
ˆ ∗ and A ˆ are chosen as in (5.4). Here the subspaces A∗ , A Proof. Parts (a) and (b) follow directly from the corresponding statements in Theorem 4.3 and from Theorem 5.1 (c). Part (c) is a consequence of Theorem 4.3 (c) under the premise that gm ⊂ Spac U . To prove this fact, we proceed as follows: Let Hm be the spectral subspace corresponding to hm and let H0 = (Hm ) be the symmetric Fock space over Hm . Moreover, let Um be the restriction of U to Hm and let U0 = (Um ) be its second quantization acting on H0 . Then, by the Haag-Ruelle scattering theory [BF82,Ar,Dy05], there exists the isometric wave-operator + : H0 → H which satisfies
+ U0 (x) = U (x) + , x ∈ Rs+1 .
(5.12)
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It is a simple exercise to show that gm = Spac U0 . Hence, if q ∈ gm , then for any open neighborhood Vq of q there exists f ∈ S(Rs+1 ) s.t. supp f˜ ⊂ Vq and (5.13) d s+1 x U0 (x) f (x)0 = 0 for some 0 ∈ H0,ac . Thus, making use of property (5.12) and the fact that + is an isometry, we obtain that + 0 ∈ Hac and d s+1 x U (x) f (x) + 0 = 0, (5.14) which proves that q ∈ Spac U . To justify (d), we note that hm is a set of Lebesgue measure zero in Rs+1 . Hence, if q ∈ hm , then it either belongs to the singular-continuous spectrum or to the pure-point spectrum of U . The latter possibility is excluded by Proposition 4.2 (a) (or Lemma 3.2.5 of [Ha]). Now the statement follows from Theorem 4.3 (d). This theorem exhibits an interplay between the spectral properties of α and the particle aspects of quantum field theory: The mass hyperboloids of Wigner particles contribute to the singular-continuous spectrum of α. Thereby, this result provides a large class of physically relevant examples of automorphism groups with non-empty singular-continuous spectrum. However, it leaves open the question of non-triviality of Spac α. Since we do not have sufficient control over the full group of spacetime translations, we will study this problem in the case of the subgroup Rs x → βx := α(0,x ) of space translation automorphisms. Theorem 5.1 fixes the decomposition of A and A∗ into the pure-point and continuous parts w.r.t. the action of β. To facilitate further analysis, we introduce the following subspaces: ˆ c := { A ∈ A ˆ | ω0 (A) = 0 }, A ˆ ∗ | ϕ(I ) = 0 }, ˆ ∗,c := { ϕ ∈ A A
(5.15) (5.16)
which are norm dense in Ac and A∗,c , respectively. Our study of the absolutely continuous and singular-continuous spectrum of β is based on two ingredients: The first of them is the following fact, mentioned in [Bu90]: ˆ c, Lemma 5.3. For any non-zero A ∈ A · )) = Rs , supp ϕ( A(
(5.17)
ˆ ∗,c ϕ∈A
where the Fourier transform is taken w.r.t. the group of space translations β. ˆ ∗,c Proof. Let X be the set on the l.h.s. of relation (5.17) and X 0 its counterpart with A replaced with its norm closure A∗,c . First, we prove that X 0 = Rs . In fact, suppose that ˆ and the complement of X 0 is a non-empty (open) set. Then, for any operator B ∈ A, ϕ ∈ B(H)∗ , the functional ϕ( · B) − ϕ(B · ) is in A∗,c . Hence the following analytic function of p ∈ Rs 1 p), B]) = ϕ([ A( (5.18) d s x ei px ϕ([βx (A), B]) (2π )s/2
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is identically equal to zero. Thus A belongs to the commutant of A which consists of multiples of the identity. Since ω0 (A) = 0, we obtain that A = 0, which is a contradiction. Now suppose that X has a non-empty (open) complement O in Rs . Then, for any ˆ ∗,c , ϕ(A( f )) = 0 holds, where f ∈ S(Rs ) s.t. supp f˜ ⊂ O and for any ϕ ∈ A x )βx (A) ∈ A. (5.19) A( f ) := d s x f ( ˆ ∗,c is norm dense in A∗,c , the same holds for ϕ ∈ A∗,c , contradicting the fact Since A that X 0 = Rs . The second ingredient is the following estimate, due to Buchholz [Bu90]: p))|2 < ∞, sup d s p | p|s+1+ε |ω( A( ω∈S E
(5.20)
valid for any E ≥ 0, any local observable A ∈ Aloc and ε > 0. With these two facts at hand, we are ready to analyze the spectrum of β. Theorem 5.4. Let Rs x → βx be the group of space translation automorphisms acting on the algebra of observables A in a quantum field theory admitting a normal vacuum state ω0 . Then there holds: (a) Sppp β = Sppp β ∗ = {0}, (b) Spac β = Spac β ∗ = Rs , (c) Spsc β ⊂ {0}, Spsc β ∗ ⊂ {0}. ˆ ∗ and A ˆ are given by (5.4). The subspaces A∗ , A Proof. Part (a) follows directly from Theorem 5.1. To prove (b) and (c), we proceed as follows: For any function f ∈ C0∞ (Rs ) and n ∈ N we introduce f n ∈ C0∞ (Rs ) given ˆ c and ϕ ∈ A ˆ ∗,c we set by f˜n ( p) = f˜( p)| p|2n . Next, for any A ∈ A x ) βx (A), (5.21) A( f n ) := d s x f n ( ϕ fn := d s x f n ( x ) βx∗ ϕ. (5.22) ˆ c , since each local algebra A(O) is a norm closed subspace We note that A( f n ) ∈ A ˆ ∗,c , since β ∗ acts of A and the action of β is strongly continuous. Similarly, ϕ fn ∈ A ˆ strongly continuously on B(H)∗ and each ϕ ∈ A∗,c belongs to the closed subspace Span S E ⊂ B(H)∗ for E sufficiently large. Setting 4n > s+1 and noting that A( f n )( p) = p), we obtain from estimate (5.20), (2π )s/2 f˜n ( p) A( A( f n )( p))|2 < ∞. (5.23) d s p |ϕ( ∀ ˆ ∗,c ϕ∈A ˆc A∈A
Hence, recalling definition (2.10) and noting that the distributions ϕ( A( f n )( · )) are compactly supported, we conclude that A( f n ) ∈ Aac and ϕ fn ∈ A∗,ac . Now we show that
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ˆ c , one some of these elements are different from zero: Clearly, for any non-zero A ∈ A can choose such f ∈ C0∞ (Rs ) that A( f ) = 0. Thus we conclude from Lemma 5.3 ˆ ∗,c . Hence, ϕ(A( f n )) = 0 or, that supp ϕ( A( f )( · )) contains p = 0 for some ϕ ∈ A equivalently, ϕ fn (A) = 0. Now part (b) of the theorem follows from Lemma 5.3 and the inclusions supp ϕ( A( f n )( · )) ⊂ Spac β, (5.24) ˆc A∈A ϕ∈A∗
· )) ⊂ Spac β ∗ . supp ϕ fn ( A(
(5.25)
A∈A ˆ ∗,c ϕ∈A
To verify the first statement in part (c), we have to show that for any A ∈ Ac the corresponding element [A] ∈ Asc = Ac /Aac satisfies [A]( f ) = [A( f )] = 0 for any f ∈ S(Rs ) s.t. supp f˜ ∩ {0} = ∅. To this end, we pick a sequence {Am }m∈N of elements ˆ c s.t. {Am ( f )}m∈N tends to A( f ) in norm. From estimate (5.20) we obtain of A d s p |ϕ( A ))|2 < ∞ (5.26) m ( f )( p ˆ ∗ . This implies that A( f ) ∈ Aac , i.e. [A( f )] = 0. for any ϕ ∈ A To prove the second part of (c), one shows that for any ϕ ∈ A∗,c the corresponding element [ϕ] ∈ A∗,sc satisfies [ϕ f ] = 0 for any f ∈ S(Rs ) s.t. supp f˜ ∩ {0} = ∅. The argument is analogous as above. Part (c) of the above theorem states that Spsc β and Spsc β ∗ are either empty or consist only of {0}. It is an interesting question, whether this latter possibility can be excluded in general. As a step in this direction, we show in the Appendix that in theories complying with a timelike asymptotic abelianess condition, introduced in [BWa92], Spsc β = Spsc β ∗ = ∅
(5.27)
ˆ slightly smaller than Aloc chosen here. This includes, in particular, the theory of for A scalar, non-interacting massive and massless particles.6 In the next section we provide further evidence for triviality of the singular-continuous spectra of β and β ∗ : We propose a regularity condition, suitable for massive theories, which implies (5.27). We also show that this condition guarantees the existence of particles, if the theory contains a stress-energy tensor. 6. Structure of the Continuous Spectrum and the Particle Content in QFT In the present section, which is based on Sect. 2.3 of [Dy08.2], we augment the general postulates of quantum field theory, adopted in the previous section, by Conditions L (1) and T stated below. The former is a regularity condition, restricting the structure of the continuous spectrum of α near zero, while the latter encodes the presence of a stress-energy tensor among the pointlike-localized fields of the theory. We will show 6 In the massless case for s ≥ 3.
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that Spsc β = Spsc β ∗ = ∅ in theories complying with Condition L (1) . If, in addition, Condition T is satisfied, we demonstrate that the theory describes particles in the sense of non-zero asymptotic functionals. In order to formulate Condition L (1) , we have to introduce some terminology: We define, for any E ≥ 0 and C ∈ A, the (possibly infinite) quantity C E,1 := sup d s x|ω(βx (C))|, (6.28) ω∈S E
and introduce the following subspace of A A(1) := { C ∈ A | ∀ C E,1 < ∞ }, E≥0
(6.29)
(+)
which is a natural domain for the asymptotic functionals σω mentioned in the Introduction and defined precisely in (6.36) below. To study the properties of this subspace, we introduce two useful concepts: First, an operator B ∈ A is called energy-decreasing, if its Arveson spectrum w.r.t. the group of spacetime translation automorphisms does not intersect with the closed forward lightcone, i.e. Sp B α ∩ V + = ∅. Second, an observable B ∈ A is called almost local, if there exists a net of local operators { Br ∈ A(O(r )) | r > 0 }, s.t. for any k ∈ N0 , lim r k B − Br = 0,
r →∞
(6.30)
where O(r ) is a double cone of radius r , centered at the origin. After this preparation we state a result, due to Buchholz [Bu90], which guarantees non-triviality of A(1) in any local, relativistic quantum field theory. Theorem 6.1. [Bu90] Let B ∈ A be almost local and energy-decreasing. Then, for any E ≥ 0, there holds B ∗ B E,1 < ∞. Our regularity condition specifies another class of observables from A(1) . These opera ˆ c (see definition (5.15)) and tors are of the form A(g) = dt g(t)αt (A), where A ∈ A g˜ is supported in a small neighborhood of zero. More precisely: Condition L (1) : There exists μ > 0 s.t. for any g ∈ S(R) with supp g˜ ⊂] − μ, μ[ and ˆ c, A∈A (a) A(g) ∈ A(1) , (b) A(g) E,1 ≤ cl,E,r R l A R l g1 , for all E ≥ 0, l ≥ 0, where R = (1 + H )−1 and r > 0 is s.t. A ∈ A(O(r )). This condition has been verified in massive scalar free field theory in Appendix D of [Dy08.2], so it is consistent with the basic postulates of quantum field theory. The quantitative part (b) of this criterion is needed in Theorem 6.3 below to prove the existence of non-trivial asymptotic functionals in theories admitting a stress-energy tensor. On the other hand, the qualitative part (a) suffices to conclude that the singular-continuous spectrum of the space translation automorphisms is empty. Theorem 6.2. Let Rs x → βx be the group of space translation automorphisms acting on the algebra of observables A in a quantum field theory admitting a normal ˆ ∗ and A ˆ be given by (5.4). vacuum state ω0 and satisfying Condition L (1) (a). Let A∗ , A ∗ Then Spsc β = Spsc β = ∅.
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ˆ c . To show that A ∈ Aac , it suffices to verify that for any Proof. Suppose that A ∈ A E ≥ 0 and ω ∈ S E , d s x |ω(βx (A))|2 < ∞. (6.31) (This follows from the Plancherel theorem and the fact that the distributions p)) are compactly supported.) We fix E ≥ μ, where μ appeared Rs p → ω( A( 1 (1) in Condition L , and choose a function f ∈ S(R) s.t. f˜ = (2π )− 2 on [−E, E] and supp f˜ ⊂ [−2E, 2E]. With the help of a smooth partition of unity we can decompose f as follows: f = f − + f + + f 0 , where supp f˜− ⊂ [−2E, −μ/2], supp f˜+ ⊂ [μ/2, 2E], and supp f˜0 ⊂] − μ, μ[. Then PE A PE = PE A( f )PE = PE A( f − )PE + PE A( f + )PE +PE A( f 0 )PE ,
(6.32)
where the first equality is a consequence of relation (4.23). By Condition L (1) (a), A( f 0 ) satisfies the bound (6.31). To the remaining terms we can apply Theorem 6.1, since both A( f − ) and A( f + )∗ are almost local and energy-decreasing. This latter fact follows from the equality 1 p 0 , p), A( f − )( p) = (2π ) 2 f˜− ( p 0 ) A(
(6.33)
which implies that the support of A( f − ) does not intersect with the closed forward lightcone. (An analogous argument applies to A( f + )∗ .) We obtain for any ω ∈ S E ,
s 2 d s x ω βx (A( f − )∗ A( f − )) d x |ω(βx (A( f − )))| ≤ sup ω ∈S E
= A( f − )∗ A( f − ) E,1 ,
(6.34)
where the last expression is finite by Theorem 6.1. Since an analogous estimate holds ˆ c ⊂ Aac and therefore Ac = Aac , i.e. Spsc β = ∅. Now for A( f + ), we conclude that A ˆ ˆ suppose that ϕ ∈ A∗,c . Then, by (6.31), for any A ∈ A, d s x |βx∗ ϕ(A)|2 < ∞, (6.35) which implies that ϕ ∈ A∗,ac . We conclude that Spsc β ∗ = ∅.
Proceeding to particle aspects of the theory, we note that the space A(1) , equipped with the family of seminorms { · E,1 | E ≥ 0 }, is a locally convex Hausdorff space and we call the corresponding topology T (1) . (This is established as in Sect. 2.2 of [Po04.1].) (t) We define, for any ω ∈ S E , a net {σω }t∈R+ of functionals on A(1) given by (t) σω (C) := d s x ω(α(t,x ) (C)), C ∈ A(1) . (6.36) (t)
This net satisfies the uniform bound |σω (C)| ≤ C E,1 . Therefore, by the (+) Alaoglu-Bourbaki theorem (see [Ja], Sect. 8.5), it has weak limit points σω in the
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topological dual of (A(1) , T (1) ) which we call the asymptotic functionals. The set of such functionals P := { σω(+) | ω ∈ S E for some E ≥ 0 }
(6.37)
will be called the particle content of the theory. This terminology was justified in the Introduction, where we argued that the asymptotic functionals should carry information about all the (infra-)particle types appearing in the theory. A general argument for the existence of non-zero asymptotic functionals has been given to date only for theories of Wigner particles [AH67]. It is now our goal to show that P = {0} not relying on the Wigner concept of a particle. Since our argument is based on the existence of a stress-energy tensor, which is postulated in Condition T below, we recall the definition and simple properties of pointlike-localized fields: We set R = (1+ H )−1 and introduce the space of normal functionals with polynomially damped energy R l B(H)∗ R l . (6.38) B(H)∗,∞ := l≥0
We equip this space with the locally convex topology given by the norms · l = R −l · R −l for l ≥ 0. The field content of the theory is defined as follows [FH81]: {R l A(O(r ))R l }w-cl for some l ≥ 0 }, FH := { φ ∈ (B(H)∗,∞ )∗ | R l φ R l ∈ r >0
(6.39) where w-cl denotes the weak closure in B(H). Since the normal vacuum state ω0 is an element of B(H)∗,∞ , we can define FH,c := { φ ∈ FH | ω0 (φ) = 0 }.
(6.40)
There holds the following useful approximation property for the pointlike-localized fields which is due to Bostelmann [Bos05]: For any φ ∈ FH,c there exists l ≥ 0 and a net Ar ∈ A(O(r )), r > 0, ω0 (Ar ) = 0, s.t. lim R l (Ar − φ)R l = 0.
r →0
(6.41)
Making use of Condition L (1) (b), we also obtain, for any time-smearing function g ∈ S(R) s.t. supp g˜ ⊂] − μ, μ[, lim Ar (g) − φ(g) E,1 = 0.
r →0
(6.42)
This implies, in particular, that φ(g) E,1 < ∞ for any φ ∈ FH,c , which prepares the ground for our next assumption: Condition T : There exists a field T 00 ∈ FH,c which satisfies ω ∈ SE , d s x ω(βx (T 00 (g))) = ω(H ),
(6.43) 1
for any E ≥ 0 and any time-smearing function g ∈ S(R) s.t. g(0) ˜ = (2π )− 2 and (1) supp g˜ ⊂] − μ, μ[, where μ appeared in Condition L .
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This condition holds, in particular, in massive scalar free field theory as shown in Sect. B.2 of [Dy08.2]. With Conditions L (1) and T at hand, we are ready to prove the existence of non-zero asymptotic functionals. Theorem 6.3. Suppose that a quantum field theory, admitting a normal vacuum state ω0 , satisfies Conditions L (1) and T and let ω ∈ S E be s.t. ω(H ) > 0. Then all the limit (+) points σω are non-zero. Proof. We choose g ∈ S(R) as in Condition T and 0 < ε ≤ 21 |ω(H )|. Making use of Condition L (1) (b) and relation (6.42), we can find C ∈ A(1) s.t. T 00 (g) − C E,1 ≤ ε. Then, exploiting Condition T and invariance of H under time translations, we obtain
s 00 |ω(H )| = | d x ω α(t,x ) (T (g)) | ≤ ε + | d s x ω(α(t,x ) (C))|. (6.44) Thus we arrive at a positive lower bound ω(H ) ≤ 2|σω(t) (C)| which is uniform in t.
We emphasize that we have proven more than non-triviality of the particle content we have verified that every physical state, with non-zero mean energy, gives rise to a non-trivial asymptotic functional. On the other hand, we did not touch upon the problem (t) of convergence of the nets {σω }t∈R+ which is essential for their physical interpretation in terms of particle measurements. The question, if the energy of the state ω can be reconstructed from the four-momenta characterizing the pure particle weights, appearing in the decomposition (1.3) of σω(+) , is another important open problem. Such a result would be an essential step towards a model-independent understanding of the problem of asymptotic completeness in quantum field theory (cf. discussion in [Bu94]). Regularity properties of the continuous spectrum of α should be of relevance to the study of these issues. 7. Conclusions and Outlook In this paper we defined and analyzed the continuous Arveson spectrum of a group of isometries α acting on a Banach space A. We introduced new notions of the absolutely continuous and singular continuous spectra of α and defined the corresponding spectral spaces. By studying relations between the spectral concepts on the side of α and α ∗ we found necessary and sufficient conditions for the pure-point and continuous subspaces to span the entire Banach space. The sufficient conditions have a natural formulation, if A is a unital C ∗ -algebra equipped with a distinguished, invariant state ω0 . In this setting we established relations between the continuous spectrum of α and the spectrum of the implementing group of unitaries in the GNS representation induced by ω0 . We verified that in any quantum field theory, admitting a normal vacuum state, the group of spacetime translation automorphisms fits into this algebraic framework. We concluded that in a theory of Wigner particles the singular-continuous spectrum of α contains the corresponding mass hyperboloid, while the multiparticle spectrum of the energy-momentum operators is included in the absolutely continuous spectrum of α ∗ . Moreover, we found conditions on the continuous spectrum of α in a neighborhood of zero which, on the one hand, imply triviality of the singular-continuous spectrum of the space translation automorphisms, on the other hand entail the existence of particles, if the theory contains a stress-energy tensor.
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While this latter assumption is physically reasonable, we feel that the presence of (infra-)particles, which is a large-scale phenomenon, should not depend on short-distance properties, like the existence of certain pointlike-localized fields. It should be possible to find general necessary and sufficient conditions for non-triviality of the particle content in terms of some spectral properties of the group of translation automorphisms. In the second step of the analysis these criteria should be related to physical properties of the theory (e.g. the phase space structure or the existence of constants of motion). Since pure particle weights corresponding to different momenta of an infraparticle can give rise to inequivalent representations of the algebra of observables, it may be necessary to look for more general spectral concepts than these introduced in the present work. Such notions should not depend on the choice of a specific vacuum state, but rather carry information about some large class of positive energy representations. First steps in this direction are taken in the Appendix, where we choose as A∗ the space of energetically accessible functionals (see definition (A.5)), rather than the predual in some vacuum representation. We hope to return to these problems in a future publication. Acknowledgements. I would like to thank Prof. D. Buchholz for pointing out to me the absence of essential spectral concepts in the theory of automorphism groups and the relevance of this problem to particle aspects of QFT. I am also grateful to him for numerous valuable discussions in the course of this work, in particular, for indicating the role of the Ergodic Theorem and for pointing out the importance of the singular-continuous spectrum. Financial support from Deutsche Forschungsgemeinschaft and Graduiertenkolleg ’Mathematische Strukturen in der modernen Quantenphysik’ of the University of Göttingen is gratefully acknowledged. This work was completed at the TU-München, where it was supported by the DFG grant SP181/25-1. Travel grants from Wilhelm und Else Heraeus-Stiftung are also acknowledged as well as a fellowship supported by the Austrian Federal Ministry of Science and Research, the High Energy Physics Institute of the Austrian Academy of Sciences and the Erwin Schrödinger International Institute of Mathematical Physics, which supported the author’s participation in the 4th Vienna Central European Seminar on Particle Physics and Quantum Field Theory.
Appendix: Spectral Theory of Automorphism Groups in QFT in the Absence of Normal Vacuum States Theorem 3.3 provides sufficient conditions for the following equalities: A = App ⊕ Ac , A∗ = A∗,pp ⊕ A∗,c ,
(A.1) (A.2)
which we verified in Sect. 5 in a large class of examples. In the first part of this Appendix we describe situations where decompositions (A.1), (A.2) fail. In particular, we show that in the case of the group of space translation automorphisms β, acting on the algebra of observables in quantum field theory, equality (A.1) fails, if A∗ contains more than one vacuum state. This occurs in some low-dimensional massless theories, if A∗ is chosen as the space of energetically accessible functionals, defined in (A.5) below. On the other hand, for theories complying with Condition A, stated below, equalities (A.1), and (A.2) hold for this choice of A∗ . In the second part of the Appendix, which relies on results from [BWa92], we analyze briefly the continuous spectra of β and β ∗ . It turns out that their singular-continuous parts are empty in this setting. We start from the following simple observation: Proposition 7.1. Let Rs x → βx be the group of space translation automorphisms acting on the algebra of observables A in a quantum field theory (possibly without normal vacuum states). Then App = Span {I } for any A∗ satisfying Condition S.
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Proof. Suppose that A ∈ A is an eigenvector, i.e. βx (A) = e−i q x A, x ∈ Rs
(A.3)
for some q ∈ Rs . Then A belongs to the center of A, since locality gives [A, B] = lim [αx (A), B] = 0, | x |→∞
B ∈ A.
(A.4)
The irreducibility assumption ensures that the center of A consists only of multiples of the identity. In view of Theorem 3.2 (a), equality (A.2) fails, in particular, if dim A∗ ({0}) < dim A({0}). Let us consider the group of space translation automorphisms β acting on the algebra of observables A in a quantum field theoretic model which does not admit normal vacuum states, (see [BHS63] for an example). Thus, choosing A∗ = B(H)∗ , we obtain dim A∗ ({0}) = 0, whereas Proposition 7.1 gives dim A({0}) = 1. On the other hand, equality (A.1) fails when dim A∗ ({0}) > dim A({0}). To exhibit an example, let us choose as A∗ the space of energetically accessible functionals ⎧ ⎫n-cl ⎨ ∗ ⎬ B(H)(a) S Ew -cl , (A.5) ∗ := Span ⎩ ⎭ E≥0
where w∗ -cl denotes the closure in the weak∗ topology of A∗ . (Clearly, this space satisfies Condition S.) It is well known that massless free field theory in s = 2 dimensional space (a) has an infinite family of vacuum states in B(H)∗ [BWa92]. Hence, by Theorem 7.1, A = App ⊕ Ac in this situation. (a) However, there exists a large class of theories, in which the choice A∗ = B(H)∗ entails equalities (A.1) and (A.2). These are, in particular, models which satisfy the following asymptotic abelianess assumption, proposed by Buchholz and Wanzenberg [BWa92]. Condition A: There exists a norm dense subspace D ⊂ Aloc s.t. for any A ∈ D there exists some positive number 1 ≤ r < s, s.t. for all ∈ H, sup d s x[A∗ , αx0 ,x (A)]r < ∞. (A.6) x0
These authors have shown that Condition A holds in massive (for s ≥ 1) and massless (for s ≥ 3) free field theory. Moreover, it was verified in [BWa92] that in theories complying with Condition A there exists a distinguished vacuum state ω0 in B(H)(a) ∗ s.t. for A ∈ A, w∗ - lim βx (A) = ω0 (A)I. | x |→∞
(A.7)
(Other conditions which imply this property can be found in [Dy08.1,Dy09].) By a slight modification of the discussion from Sect. 4 of [BWa92], we obtain that for any (a) A ∈ ker ω0 , ϕ ∈ B(H)∗ and q ∈ Rs , 1 lim ei q x ϕ(βx (A)) d s x = 0. (A.8) K Rs |K | K
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W. Dybalski (a)
Hence ker ω0 ⊂ Ac and, consequently, ω0 is the unique element of B(H)∗ invariant under the action of β ∗ . Thus the decomposition of A and A∗ into the pure-point and continuous subspaces is given by Theorem 3.3. We summarize: Theorem 7.2. Let Rs x → βx be the group of space translation automorphisms acting on the algebra of observables A in a quantum field theory satisfying Condition A. (a) (a) Let A∗ = B(H)∗ and let ω0 be the unique vacuum state in B(H)∗ . Then ker ω0 ⊂ Ac and, consequently, (a) A = App ⊕ Ac , where App = Span {I }, Ac = ker ω0 , (b) A∗ = A∗,pp ⊕ A∗,c , where A∗,pp = Span {ω0 }, A∗,c = ker I . The analysis of the absolutely continuous and singular-continuous spectrum is perˆ ⊂ A and formed similarly as in Theorem 5.4. However, the norm dense subspaces A ˆ ∗ ⊂ A∗ , are now chosen as follows: A ˆ = D, A ˆ ∗ = B(H)(a) := Span A ∗,bd
⎧ ⎨ ⎩
∗ -cl
S Ew
E≥0
⎫ ⎬ ⎭
(A.9) ,
(A.10)
where D appeared in Condition A. In the present case we are able to show that the singular-continuous spectra of β and β ∗ are empty. Theorem 7.3. Let Rs x → βx be the group of space translation automorphisms acting on the algebra of observables A in a quantum field theory satisfying Condition A. ˆ and A ˆ ∗ be given by (A.5), (A.9) and (A.10), respectively. Then the following Let A∗ , A holds: (a) Sppp β = Sppp β ∗ = {0}, (b) Spac β = Spac β ∗ = Rs , (c) Spsc β = Spsc β ∗ = ∅. Proof. Part (a) follows from Theorem 7.2. To prove statements (b) and (c), we define the subspaces ˆ (a) := { A ∈ A ˆ | ω0 (A) = 0}, A c (a) ˆ ˆ A ∗,c := { ϕ ∈ A∗ | ϕ(I ) = 0},
(A.11) (A.12)
s ˆ (a) which are norm dense in Ac and A∗,c , respectively. We choose As ∈ Ac , f ∈ S(R ) s.t. f˜ vanishes in some neighborhood of zero and set A( f ) := d x f ( x )βx (A). Then we obtain from Lemma 2.1 of [BWa92] the bound 1/2 PE A( f )PE ≤ c (A.13) d s p | p|−(s−ε) | f˜( p)|2 ∗
for some constants c > 0, ε > 0 independent of f . Noting that for any ω ∈ S Ew -cl , |ω(A( f ))| ≤ PE A( f )PE holds, making use of the assumption that β acts strongly continuously on A to exchange the action of the state ω with integration and proceeding as in [BWa92], p.581, we obtain x ) + ω0 (A), ω(βx (A)) = l(
(A.14)
Continuous Spectrum of Automorphism Groups and the Infraparticle Problem
297 ∗
where l˜ ∈ L 1 (Rs , d s p) and ω0 = w∗ - lim|x |→∞ βx∗ ω is an element of S Ew -cl , invariant under the action of β ∗ . So, by Theorem 7.2 (b), ω0 = ω0 and consequently p))| < ∞. ∀ (A.15) d s p |ϕ( A( (a)
ˆc A∈A ˆ (a) ϕ∈A ∗,c
We conclude which that Ac = Aac and A∗,c = A∗,ac , which proves (c). Part (b) follows from the inclusions · )) ⊂ Spac β, supp ϕ( A( (A.16) (a)
ˆc A∈A ϕ∈A∗
· )) ⊂ Spac β ∗ , supp ϕ( A(
(A.17)
A∈A ˆ (a) ϕ∈A ∗,c
ˆ (a) ˆ ˆ (a) ˆ and from Lemma 5.3. To apply this latter fact, we note that A c ⊂ Ac and A∗,c ⊃ A∗,c , ˆ ˆ where Ac and A∗,c are given by (5.15) and (5.16), respectively. The above result has the following immediate corollary: After adding Condition A to ˆ = D, part (c) of this theorem can be the assumptions of Theorem 5.4 and choosing A strengthened to Spsc β = Spsc β ∗ = ∅. References [Ar] [AH67] [AB97] [Ar74] [Ar82] [Ba78] [BHS63] [Bos05] [Bu82] [Bu86] [Bu90] [Bu94] [BF82] [BPS91]
Araki, H.: Mathematical Theory of Quantum Fields. Oxford: Oxford University Press, 1999 Araki, H., Haag, R.: Collision cross sections in terms of local observables. Commun. Math. Phys. 4, 77–91 (1967) Arendt, W., Batty, C.J.K.: Almost periodic solutions of first and second order Cauchy problems. J. Diff. Eqs. 137, 363–383 (1997) Arveson, W.: On groups of automorphisms of operator algebras. J. Funct. Anal. 15, 217–243 (1974) Arveson, W.: The harmonic analysis of automorphism groups. In: Operator Algebras and Applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math. 38, Providence, RI: Amer. Math. Soc., 1982, pp. 199–269 Baskakov, A.G.: Spectral criteria for almost periodicity of solutions of functional equations. Math. Notes 24, 606–612 (1978) Borchers, H.J., Haag, R., Schroer, B.: The vacuum state in quantum field theory. Nuovo Cimento 29, 148–162 (1963) Bostelmann, H.: Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 46, 052301–052318 (2005) Buchholz, D.: The physical state space of quantum electrodynamics. Commun. Math. Phys. 85, 49–71 (1982) Buchholz, D.: Gauss’ law and the infraparticle problem. Phys. Lett. B 174, 331–334 (1986) Buchholz, D.: Harmonic analysis of local operators. Commun. Math. Phys. 129, 631–641 (1990) Buchholz, D.: On the manifestations of particles. In: Mathematical Physics Towards the 21st Century. Proceedings Beer-Sheva 1993, Sen, R.N., Gersten, A. Eds., Ben-Gurion University of the Negev Press 1994, pp. 177–202 Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982) Buchholz, D., Porrmann, M., Stein, U.: Dirac versus Wigner: towards a universal particle concept in quantum field theory. Phys. Lett. B 267, 377–381 (1991)
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Communicated by Y. Kawahigashi
Commun. Math. Phys. 300, 301–315 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1098-4
Communications in
Mathematical Physics
Universal Bounds for the Littlewood-Paley First-Order Moments of the 3D Navier-Stokes Equations Felix Otto1 , Fabio Ramos1,2 1 Institute of Applied Mathematics, University of Bonn, Bonn 53115, Germany.
E-mail: [email protected]
2 Department of Computer Science and Applied Mathematics, Weizmann Institute of Science,
Rehovot 76100, Israel Received: 21 April 2009 / Accepted: 27 April 2010 Published online: 15 September 2010 – © Springer-Verlag 2010
Abstract: We derive upper bounds for the infinite-time and space average of the L 1 -norm of the Littlewood-Paley decomposition of weak solutions of the 3D periodic Navier-Stokes equations. The result suggests that the Kolmogorov characteristic velocity scaling, Uκ ∼ 1/3 κ −1/3 , holds as an upper bound for a region of wavenumbers near the dissipative cutoff. 1. Introduction Consider the three-dimensional incompressible Navier-Stokes equations in the box = [0, L]3 : ∂t u + u · ∇u − νu + ∇ p = f in × [0, ∞), ∇ · u = 0 in × [0, ∞), u(x, 0) = u0 in ,
(1) (2) (3)
where u, the velocity field, and p, the pressure field, are the unknowns. The constant ν > 0 is the kinematic viscosity, f, the body forcing, is a smooth time-independent, divergence-free vector field with zero spatial average, and the initial velocity u0 is a divergence-free vector field belonging to (L 2 ())3 . We impose -periodic boundary conditions on u, p, f and u0 . Clearly, because of the periodicity assumption, and the conditions imposed on the d forcing term, f, (1) preserves the spatial average of u, i. e. dt u = 0, where for any -periodic vector field v(x), we use the abbreviation v := L −3 v d x. Because (1) is invariant under the Galilean transformation t = tˆ, x = xˆ + U t, u = uˆ + U,
(4)
one may restrict oneself to the study of solutions with vanishing spatial average, that is ∀t ∈ [0, ∞), u = 0.
(5)
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F. Otto, F. Ramos
We denote by H˙ 1 () and L˙ 2 () the space of functions belonging, respectively, to H 1 () and L 2 (), with zero spatial average, and satisfying -periodic boundary conditions. In this work, we consider weak solutions of the Navier-Stokes equations, a notion first introduced by J. Leray in [9], and further developed by E. Hopf in [8]. Definition 1. We say that (u, p) is a Leray-Hopf weak solution of the Navier-Stokes equations if (i) (u, p) satisfies (1) and (2) in the sense of distributions, and u(t) → u0 weakly in (L 2 ())3 as t → 0. (ii) 2 u ∈ L loc ([0, ∞), ( H˙ 1 ())3 ) ∩ L ∞ ([0, ∞), ( L˙ 2 ())3 ),
(6)
and if, for a.e. T > 0, u satisfies the following energy inequality: (iii) 2 3 T ∂u i 1 2 u(·, T ) L 2 + (·, s) 2 ds 2 ∂x j L i, j=1 0 T 1 ≤ (f, u(s)) L 2 ds + u(·, 0) 2L 2 , 2 0
(7)
where (·, ·) L 2 stands for the inner product in L 2 (), and · L 2 is the norm in L 2 (). The classical results concerning existence of Leray-Hopf weak solutions can be found in [4,8,9,13,14]. Notice that our definition of weak solutions is different from the usual one found, for example, in [4,13,14]. There, the authors consider only test functions ϕ satisfying ∇ · ϕ = 0, which eliminate the pressure term in the weak formulation. However, we remark that both formulations are equivalent since one can always recover the pressure term, see, e.g., [14, Theorem IV.2.2]. Moreover, one can also obtain from [14, Theorem IV.2.2] that Leray-Hopf weak solutions satisfy 1 ∇ p ∈ L loc ([0, T ), (L 1 ())3 ).
(8)
The reason we chose to work with this definition of weak solutions comes from the fact that, in the course of this work, we may perform energy estimates with a non-divergencefree test function. Furthermore, because we are interested in L 1 estimates, one may not circumvent this problem by using the Leray-Helmoltz orthogonal projector, since this is not a bounded operator in (L 1 ())3 . 1.1. Main result. We begin with some definitions Definition 2. (i) For any suitable scalar -periodic function u(x), we define the spatial average by 1 u := 3 u(x)d x. L
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(ii) For any suitable scalar periodic function u(x, t), defined for x ∈ and t ≥ 0, which is -periodic, and for any p ∈ [1, ∞), we denote the finite time and space average of the p th power by 1 T |u| p T := |u(t, .)| p dt. T 0 (iii) Similarly, for any vector field u(x, t) = (u 1 , u 2 , u 3 ), defined in and for all t ≥ 0, which is -periodic, and for any p ∈ [1, ∞), we define 3 3 p p |u| p := |u i | p , |u| p T := |u i | p T , i=1
i=1
and ∇u 2 T :=
3 ∂u i 2 | | T . ∂x j
i, j=1
(iv) For any vector field u(x, t) = (u 1 , u 2 , u 3 ), defined in and for all t ≥ 0, which is -periodic, and for any p ∈ [1, ∞), we define the infinite-time and space average by 3 p |u| p := lim sup |u i | p T , T →∞
i=1
and ∇u 2 := lim sup T →∞
3 ∂u i 2 ∂ x T . j
i, j=1
Definition 3. (i) For any suitable -periodic function u(x) we define the Fourier series (Fu)(q) via (Fu)(q) = L −3 exp(iq · x) u(x) d x, for q ∈ 2π L −1 Z3 . [0,L]3
(ii) For a Schwartz function φ(x), x ∈ R3 , we define its Fourier transform (Fφ)(q), q ∈ R3 via exp(iq · x) φ(x) d x. (9) (Fφ)(q) = R3
(iii) We select a family of Schwartz functions {φk (x)}k∈Z defined in R3 such that their Fourier transforms, {(Fφk )(q)}k∈Z , satisfy (Fφ0 )(q) = 0, only for q ∈ A2−1 ,2 , k∈Z
(Fφk )(q) = (Fφ0 )(2
−k
q), for all k and q,
(Fφk )(q) = 1, for all q,
(10) (11) (12)
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F. Otto, F. Ramos
where A2−1 ,2 is the annulus defined as A2−1 ,2 := q ∈ R3 ; |q| ∈ (2−1 , 2) .
(13)
(iv) For any suitable -periodic function u(x), we define the Littlewood-Paley decomposition {u k }k∈Z by u k := φk ∗ u, where ∗ denotes convolution in the x variable in R3 . Remark 1.1. It is easy to prove, see [1,11], that there exist universal constants c > 0 and C > 0 such that c 22k |u k |2 ≤ |∇u k |22 ≤ C 22k |u k |2 ,
(14)
and so that c
∞
∞
22k |u k |2 ≤ |∇u|22 ≤ C
k=−∞
22k |u k |2 .
(15)
k=−∞
Remark 1.2. Throughout this work, when a convolution product is taken between a scalar function and a vector or a tensor field, it is meant to be a convolution coordinate by coordinate. In particular, the Littlewood-Paley decomposition, {uk }k∈Z , of a -periodic vector field u(x) = (u 1 , u 2 , u 3 ) is defined by uk := φk ∗ u := (φk ∗ u 1 , φk ∗ u 2 , φk ∗ u 3 ). Definition 4. Let u be a weak solution of the Navier-Stokes equations. We denote by the mean rate of dissipation of kinetic energy defined by := ν ∇u 2 . Remark 1.3. We remark that Definition 4 makes sense because of energy inequality (7), see [4] for details. The main result of this work states the following: Theorem 1. Let u(x, t) be a weak solution of the Navier-Stokes equations, and assume that the forcing term f in (1) satisfies fk = 0,
for every k ≥ k0 .
Then, |uk |1 ≤ Cν −2 (2k )−3 ,
(16)
for every k ≥ k0 , where C := C(φ0 ) is a universal constant depending only on the choice of the partition function φ0 in the definition of Littlewood-Paley decomposition.
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1.2. Physical interpretation. Let us now comment on the significance of Theorem 1. Our arguments are partially similar to the ones used by P. Constantin in [2,3]. It is believed that the Navier-Stokes equations describe a wide range of incompressible Newtonian fluid flows, including the ones in the turbulent regime. In 1941, in a celebrated series of papers, see [7] for details, Kolmogorov argued that there exists a range of wavenumbers, called the inertial range, such that the energy spectrum of a homogeneous turbulent flow, described by a velocity field u(x, t), satisfies E(m) ∼ 2/3 m −5/3 ,
(17)
where the energy spectrum E(m) is a function with dimensions of energy per wavenumber defined by 1 1 T ˆ t)|22 d S(ξ ) dt, lim E(m) = |u(ξ, (18) 2 T →∞ T 0 |ξ |=m where uˆ denotes the spatial Fourier transform. The asymptotic law (17) is expected to hold in the range of wavenumbers [m 0 , m d ], as the Reynolds number, Re, tends energy input scale, m d is the Kolmogorov dissipation waveto ∞, where m −1 0 is an
1/4 , and the Reynolds number, Re, can be defined by Re := number, m d := /ν 3 −1 2 1/2 (|u|2 m 0 )/ν. Using (18), one may define a characteristic velocity at length scale 1/m by 1/2 Um := 21/2
2m
m 2
E(κ) dκ
.
(19)
Now, from (17), it is clear that Um satisfies Um ∼ C 1/3 m −1/3 .
(20)
Kolmogorov’s heuristic laws are obtained by a universality hypothesis of the statistical properties of small scales in a homogeneous turbulent flow, i. e., that in the limit of infinite Reynolds number, all the small scale statistical properties are uniquely and universally determined by the scale m −1 , and by the mean energy dissipation rate , see [7] for details. The definition of the characteristic velocity at scale m −1 given by (19), although natural, is arbitrary. Nonetheless, arguing by dimensional analysis, we have by the Kolmogorov’s universality hypothesis that any working definition of characteristic veloc˜ m , must satisfy the scaling law (20), i. e., ity at scale m −1 , U ˜ m ∼ C 1/3 m −1/3 . U
(21)
We remark, however, that intermittent dynamics observed in many turbulent flows is inconsistent with the universality hypothesis in K 41, and several modifications of this assumption have been proposed, see [7] for more details. The nature of the intermittent phenomena in turbulent flows is presently not well understood, and the question about the universality of the scaling law (21) for a given characteristic velocity at scale 1/m remains elusive. We refer the reader to [7, Ch. 7] for a phenomenological derivation of (21) using a second-order correlation function of the velocity field as the characteristic velocity, and with the universality hypothesis replaced by a self-similarity assumption.
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Now, we interpret the result obtained in Theorem 1 by following the arguments used in [2,3]. For any weak solution u(x, t) of the Navier-Stokes equations, let us define the Littlewood-Paley first-order characteristic velocity at length scale 1/m by UmL P := |uk |1 ,
(22)
for m ∈ [2k−1 , 2k ). With this definition, Theorem 1 implies that for m ≥ m 0 , UmL P ≤ Cν −2 (2k )−3 ≤ Cν −2 m −3 . Arguing as in [2,3], we can rewrite the inequality above as
m 8/3 d 1/3 m −1/3 , UmL P ≤ C m
(23)
(24)
and, again, arguing as in [2], this implies that UmL P ≤ C 1/3 m −1/3 ,
(25)
m0 holds for every m ∈ [βm d , m d ], with m ≤ β ≤ 1, and C := Cφ0 β −8/3 , where Cφ0 d is a universal constant depending only on the choice of the partition function φ0 in the definition of Littlewood-Paley decomposition. Therefore, (25) states that the scaling 1/3 m −1/3 is obtained in the last decades before the dissipative cutoff, if we define the characteristic velocity as in (22). The results discussed above are in the same vein as the ones obtained in [2] and [3]. There, similar bounds were obtained for the Littlewood-Paley second-order moments, which are somehow equivalent to the Fourier second-order moments by using the isometry of the Fourier transform in L 2 , see [2] for details. However, we remark an important difference in our work. The equivalent of our constant C displayed in the last inequality is not universal in [2] and [3]. Besides that, in [2], the author works with solutions of a regularized Navier-Stokes equations, and uses a different definition of , which involves a Besov norm. We conclude this introduction by mentioning that the analysis of Littewood- Paley components has also been used in [12], where the authors obtained upper bounds for the L 2 -norms of the Littewood-Paley decomposition as a necessary condition for the inertial energy transfer to take place. We also would like to mention that our bounds is only relevant for a range near the dissipative cut-off, and for the dissipative range our result is far from being sharp. Several results deal with the dissipative range, and we refer the reader to [5,6] and the references therein for relevant results concerning this range. Now, we proceed to the proof of Theorem 1.
2. Proof Let Br (z) denote the ball in R3 of radius r > 0, and centered in z. Lemma 1 (Narrow-bandedness in Fourier space). Let u : → R be a smooth -periodic function, satisfying F(u)(q) = 0, for all q ∈ / Bδ (η),
(26)
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where 0 < δ < 1,
1 < |η| ≤ 2. 2
and
We claim that | − u − |η|2 u| ≤ Cδ|u|,
(27)
for some universal constant C > 0. Proof. The proof is a direct modification of Step 1 in the proof of Proposition 2 in [10]. Select a Schwartz function φ : R3 → R, such that its Fourier transform (Fφ)(q) = 3 R3 exp(iq · z) φ(z) dz, q ∈ R , satisfies (Fφ)(q) = 1 for |q| ≤ 1.
(28)
η
Consider the rescaled and modulated version φδ of φ: η
φδ (z) := exp (i η · z) δ 3 φ(δz).
(29)
An easy calculation shows that (Fφ)(q) = 1,
for q ∈ Bδ (η).
η
(30)
η
Therefore, because (F(φδ ∗ u))(q) = (Fφδ )(q)(Fu)(q), q ∈ 2π L −1 Z3 , the η narrow-bandedness assumption (26) implies that (F(φδ ∗ u))(q) = (Fu)(q), which η means that φδ leaves u invariant under convolution, i. e., η
u = φδ ∗ u.
(31)
Now (27) follows easily because (31) implies the representation η
η
−u − |η|2 u = (−φδ − |η|2 φδ ) ∗ u. Indeed, we obtain on the one hand
| − u − |η| u| ≤
η
2
R3
η
| − φδ − |η|2 φδ |dz |u|.
(32)
On the other hand, because of − exp (i η · z) = |η|2 exp (i η · z), we obtain η
η
(−φδ − |η|2 φδ )(z) = −2i exp (i η · z)δ 4 (∇φ)(δz) · η + exp (i η · z)δ 5 φ(δz), so that, since φ is a Schwartz function, |η| < 2 and δ < 1, we have η η | − φδ − |η|2 φδ |dz ≤ C δ. R3
Inserting (33) into (32) yields (27).
(33)
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Lemma 2 (Energy estimate). There exist universal δ > 0 and C < ∞ with the following property: Assume that u ∈ L ∞ ([0, ∞), L 2 ()), 1 g ∈ L loc ([0, ∞), L 1 ()),
(34) (35)
and 2 ∞ v ∈ L loc ([0, ∞), (H 1 ())3 ) ∩ L loc ([0, ∞), (L 2 ())3 ).
Suppose that the following equation holds in the sense of distributions: ⎧ ∂t u + v · ∇u − νu = g, ⎨ ∇ · v = 0, ⎩ u, v periodic, with periodic box = [0, L]3 ,
(36)
(37)
and u(t) → u 0 weakly in (L 2 ()) as t → 0. Assume that u is narrow-banded in Fourier space, i.e., F(u(·, t))(q) = 0, for all q ∈ / Bδ (η), and t ≥ 0,
(38)
where 0 < δ < 1, and
1 ≤ |η| ≤ 2. 2
We claim that under these assumptions, for a.e. T > 0, we have 1 ν 1 |u(., T )| + |u|T ≤ |g|T + |u(., 0)|. T 8 T
(39)
Proof. We first observe that because of (38), we have |u(t, ·)|22 + ∇u(t, ·) 2 |u(t, ·)|2 , for a.e. t ∈ [0, ∞), so that (34) improves to u ∈ L ∞ ([0, ∞), H 2 ()). In particular, 1 1 −v · ∇u + νu + g ∈ L ∞ (L 1 ) + L ∞ (L 2 ) + L loc (L 1 ) ⊂ L loc (L 1 ),
so that by (37) we obtain that du 1 ∈ L loc (L 1 ), dt and that (37) also holds in the a.e. sense. Now, let A(z) be a smooth approximation of A(z) = |z|.
(40)
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By the chain rule for weak derivatives, we obtain from (37) in the a.e. sense that d A(u) + v · ∇ A(u) − νu A (u) = g A (u). dt
(41)
Since ∇ · v = 0 in the weak sense, this yields d A(u) − νu A (u) = g A (u). dt At this stage, we may carry out our approximation argument in A so that (41) holds for (40). Now, we will show with help of Lemma 1 that the narrow-bandedness (26) implies − A (u) u = −signu u ≥
1 |u|, 8
(42)
for δ small enough. Indeed, by Lemma 1 we have for δ small enough: | − u − |η|2 u| ≤
1 |u|. 8
(43)
Therefore, −sign u u = (sign u) |η|2 u + sign u (−u − |η|2 u) 1 1 ≥ |η|2 |u| − |u| ≥ |u|, 8 8 where we have used the fact that |η| ≥ 21 in the last inequality above. We now return to (41) with A given by (40), in which we insert (42), yielding ν ∂t |u| + |u| ≤ |g|, 8
(44)
for a.e. t > 0. By (34) and (35), we may take the time average from 0 to T in the equation above, yielding (39). Lemma 3 (Commutator Estimates). Let φ : R3 → R be a Schwartz function. Let v be a smooth -periodic function. Consider the commutator [v, φ∗] of the operation “multiplication with v” and the operation “convolution with φ”, that is, [v, φ∗]w := v (φ ∗ w) − φ ∗ (v w),
(45)
for any smooth -periodic function w. We claim that the two estimates |[v, φ∗]∂xi w| ≤ C |∇v|22 1/2 |w|2 1/2 ,
(46)
|∇v|22 1/2 |w|2 1/2 ,
(47)
|[v, φ∗]w| ≤ C
hold for i = 1, 2, 3, with a constant C(φ) > 0.
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Proof. Both estimates rely on the elementary inequality, which proof we omit, 1/2 1 2 |v(x − z) − v(x)| d x ≤ |z| |∇v|22 1/2 , L3
(48)
for every z ∈ . Indeed, in order to prove (47), we apply Fubini’s Theorem, Cauchy-Schwarz’ Inequality and inequality (48), to obtain |([v, φ∗]w)| = v(·) φ(z) w(· − z) dz − φ(z) v(· − z) w(· − z) dz R3 R3 1 ≤ 3 |φ(z)| |v(x) − v(x − z)| |w(x − z)| d x dz L R3 ≤C |φ(z)||z| dz|∇v|22 1/2 |w|2 1/2 , (49) R3
and (47) follows because φ is a Schwartz function. For (46) we again use Fubini’s Theorem, Cauchy-Schwarz’ Inequality and inequality (48) to write |([v, φ∗]∂xi w)| = φ(z) (v(·) − v(· − z)) ∂xi w(· − z) dz R3 1 ≤ 3 |∂zi φ(z)| |v(x) − v(x − z)||w(x − z)| d x dz L R3 1 |φ(z)| |∂xi v(x − z)||w(x − z)| d x dz + 3 L R3 ≤C |∂zi φ(z)||z| dz |∇v|22 1/2 |w|2 1/2 R3 +C |φ(z)| dz|∇v|22 1/2 |w|2 1/2 , R3
and (46) follows because φ is a Schwartz function.
Now, our aim is to estimate the component u0 , and to extend this estimate for every uk by using the scaling of the NSE as described in Lemma 6. In order to establish it, we first estimate a microlocal decomposition of u around a point η ∈ A2−1 ,2 by using Lemma 3, and, then, we recover the estimates for u0 by covering A2−1 ,2 with finite open balls centered at points belonging to A2−1 ,2 . We split this strategy in two lemmas. Let us first make a couple of definitions. Let δ > 0 and η ∈ A2−1 ,2 . For any h ∈ (L 2 ())3 , we define η
η
η
hη := ((h 1 )η , (h 2 )η , (h 3 )η ) := (φδ ∗ h 1 , φδ ∗ h 2 , φδ ∗ h 3 ),
(50)
η
where φδ is a Schwartz function satisfying η
/ Bδ (η). F(φδ )(q) = 0, for all q ∈ We also define v0 :=
k≤−1
uk , and w0 :=
k≥0
uk .
(51)
(52)
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Lemma 4 (Micro-local estimates). There exist universal constants δ > 0 and C = η C(φδ ) < ∞ with the following property: Let u be a weak solution of the Navier-Stokes equations (1). Let v0 , w0 be as defined in (52). Let uη be as defined in (50). Then, for a.e. T > 0, we have 1 ν |uη (·, T )|1 + |uη |1 T ≤ C(|uη |22 T + |w0 |22 T + ∇v0 2 T ) T 8 1 1 + |uη (·, 0)|1 + |fη |1 . (53) T T Proof. We begin by treating the pressure gradient term in the Navier-Stokes equations. We remind the reader that the Leray-Helmholtz orthogonal projector (54) PL H : L 2 () → H = u ∈ L 2 () ; u is -periodic, and ∇ · u = 0 can be explicitly described in Fourier domain by the tensor k⊗k F(u)(k), ∀k ∈ 2π L −1 Z3 . F(PL H (u))(k) = I d − |k|2
(55)
η
Now, let ψδ be the tensor defined componentwise by ki k η η F((ψδ )i )(k) = δi − 2 F(φδ )(k), ∀k ∈ R3 |k|
(56)
η
for all k ∈ R3 , and i, ∈ {1, 2, 3}, where φδ is as described in (51), and δi is the η Kronecker delta tensor. It is easy to see that each component of the tensor ψδ is a Schwartz function. Moreover, because ∇ · u = 0, and ∇ · f = 0, the following identities hold: η
η
η
η
(u i )η = φδ ∗ u i = (ψδ )i ∗ u , ( f i )η = φδ ∗ f i = (ψδ )i ∗ f ,
(57)
where, throughout the work, repeated indices are summed over (Einstein summation). Moreover, because of (8), one easily obtains ki k η η F((∂ p) ∗ (ψδ )i )(k) = δi − 2 F(φδ )(k)F(∂ p)(k) |k| ki k η (58) = δi − 2 F(∂ ( p ∗ φδ ))(k) = 0. |k| Now, we decompose the nonlinear term as u · ∇u = v0 · ∇u + w0 · ∇v0 + w0 · ∇w0 , and rewrite the Navier-Stokes equations (1) as ∂t u + v0 · ∇u − νu + ∇ p = f − w0 · ∇v0 − ∇ · (w0 ⊗ w0 ).
(59)
η
Taking the convolution product of the equation above with the tensor ψδ , we obtain by (57) and (58), that the weak solution u satisfies the following equations in the distributional sense: η
∂t (u i )η + (v0 ) j ∂ j (u i )η − ν(u i )η = gi , for i = 1, 2, 3,
(60)
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where, for each i = 1, 2, 3 fixed, we define η
η
gi := ( f i )η − (ψδ )i ∗ ((w0 ) j ∂ j (v0 ) ) η
η
− (ψδ )i ∗ ((w0 ) j ∂ j (w0 ) ) + [(v0 ) j , (ψδ )i ∗]∂ j u .
(61)
Now, we prove that there exists a universal δ > 0, such that each coordinate of uη satisfies the hypothesis of the function u in Lemma 2, with v replaced by v0 , and g replaced by the correspondent coordinate of gη . η Indeed, by Definition 1 of weak solutions, u satisfies (6), and because φδ satisfies (51), it clearly implies that each coordinate of uη satisfies (34) and (38). It is also clear that (6) implies that v0 satisfies (36). It remains only to prove that each coordinate (g η )i satisfies (35). First, it is immediate to see that for each set of indices i, j, fixed, we have
η |(ψδ )i ∗ ((w0 ) j ∂ j (v0 ) )| ≤ C|((w0 ) j ∂ j (v0 ) )| ≤ C |w0 |22 + ∇v0 2 . (62) The following inequality is also straightforward: η
η
|(ψδ )i ∗ (∂ j ((w0 ) j (w0 ) ))| = |∂ j (ψδ )i ∗ ((w0 ) j (w0 ) )| ≤ C|w0 |22 .
(63)
For the commutator term, by splitting u i = (vi )0 + (wi )0 , we have for each set of indices i, j, fixed,
η η η |[(v0 ) j , (ψδ )i ∗]∂ j u | ≤ |[(v0 ) j , ψδ ∗]∂ j (v0 ) | + |[(v0 ) j , (ψδ )i ∗]∂ j (w0 ) | . Applying (47) to the first term on the right-hand side of the equation above, and, similarly, applying (46) to the second term on the right-hand side of the equation above yields
η η |[(v0 ) j , (ψδ )i ∗]∂ j u | ≤ C(φδ ) |w0 |22 + ∇v0 2 . (64) Thus, from (62), (63) and (64), we obtain that each coordinate (g η )i satisfies (35). This concludes the proof that each coordinate (u i )η of uη satisfies the hypothesis of Lemma 2, with v replaced by v0 , and g replaced by the correspondent coordinate of g η . Therefore, by Lemma 2, 1 ν 1 η |(u i )η (., T )| + |(u i )η |T ≤ |gi |T + |(u i )η (., 0)|. (65) T 8 T Taking the time-average of (62), (63), (64), summing in i, j, , and substituting into (65), yields (53). Now, we recover the result obtained in Lemma 4 to the annulus A2−1 ,2 . From now on, throughout the paper, we fix δ > 0 such that δ is small enough to satisfy the hypothesis of Lemma 2 and Lemma 4. Lemma 5 (Local dyadic estimates). There exist universal δ > 0 and C < ∞ with the following property: Let u be a weak solution of the Navier-Stokes equations (1). Let v0 , w0 be defined as in (52). Then, for every T > 0, we have 1 ν |u0 (·, T )|1 + |u0 |1 T ≤ C(|u0 |22 T + |w0 |22 T + ∇v0 2 T T 8 1 + |u0 (·, 0)|1 ) + |f0 |1 . (66) T
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Proof. Let A2−1 ,2 be as defined in (13). Consider a finite family of functions χ j j such that F(χ j ) j is a partition of unity of the annulus A2−1 ,2 , subordinated to an open covering of A2−1 ,2 given by a finite family of open balls Bδ (η j ) j , with η j ∈ A2−1 ,2 . As already mentioned, we consider that δ > 0 is small enough to satisfy the hypothesis of Lemma 2 and Lemma 4. Now, because F(u0 ) is supported on the annulus A2−1 ,2 , and because F(χ j ) j is a partition of unity of A2−1 ,2 , we have that u0 = χ j ∗ u0 = (χ j ∗ φ0 ) ∗ u. (67) j
j
Now, notice that because χ j ∗ u0 = (χ j ∗ φ0 ) ∗ u, and F(χ j ∗ φ0 )(q) = 0, for all q ∈ / Bδ (η j ),
(68)
we have that each function (χ j ∗ φ0 ) ∗ u satisfies the hypothesis of uη in Lemma 4, with η χ j ∗ φ0 playing the role of φδ . Therefore, there are constants C(χ j , φ0 ) > 0, such that 1 1 ν ν |u0 (·, T )|1 + |u0 |1 T ≤ |χ j ∗ u0 (·, T )|1 + |χ j ∗ u0 |1 T T 8 T 8 j C(χ j , φ0 ) |χ j ∗ u0 |22 T + |w0 |22 T + ∇v0 2 T ≤ j
1 + |χ j ∗ u0 (·, 0)|1 + |χ j ∗ f0 |1 T 1 |u0 (·, 0)|1 + |f0 |1 ). (69) T Thus, because the choice of the finite set of partition functions χ j j , and of the function φ is independent of the particular setting of the Navier-Stokes equations, and its weak solutions, we conclude that (66) indeed holds with a universal C > 0. ≤ C(|u0 |22 T + |w0 |22 T + ∇v0 2 T +
Lemma 6 (Scaling). For any ∈ Z, ν 22 |u (·, T )|1 + 22 |u |1 T ≤ C(2 |u |22 T + 2 |w |22 T T 8 22 |u (·, 0)|1 + |f |1 , + 2− ∇v 2 T ) + T (70) where v and w are defined analogously to v0 and w0 in (52), that is, uk , w := uk . v := k≤−1
(71)
k≥
Proof. Indeed, we notice that the change of variables ˆ x = 2− x, ˆ t = 2−2 tˆ, u = 2 u,
ˆ p = 22 p,
f = 23 fˆ
(72)
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leaves (1) invariant. Notice that (11) translates into φk (z) = 23k φ0 (2k z),
(73)
so that 2−3 φk (2− zˆ ) = φk− (ˆz ). Hence we deduce from (72) the following relation between the Littlewood-Paley decompositions: uk = 2 uk− ,
f k = 23 f k− .
In particular, we have u0 , ∂x v = 22 ∂xˆ j v0 , w = 2 w0 . u = 2 ˆ fˆ) yields in terms of (t, x, u, f ): Hence (66), applied to (tˆ, x, ˆ u, 2− ν |u (·, T )|1 + 2− |u |1 T ≤ C(2−2 |u |22 T + 2−2 |w |22 T T 8 2− |u (·, 0)|1 + 2−4 ∇v 2 T ) + T + 2−3 |f |1 T . (74)
Multiplication with 23 yields (70).
Proof of Theorem 1. We now proceed to the proof of our main result. From now on, we consider k ≥ k0 , so that fk = 0. From Lemma 6 and Lemma 7, if u is a weak solution of (1), then 22k ν {|uk (., T )|1 − |uk (., 0)|1 } + 22k |uk |1 T T 8 ≤ C(2k |uk |22 T + 2k |wk |22 T + 2−k | ∇vk 2 T ).
(75)
Now, taking the lim sup in T of both sides of the expression above, multiplying the result by 2k , and using (14) and (15), we obtain ν23k |uk |1 ≤ C(22k |uk |22 + 22k |wk |22 + ∇vk 2 ) ∞
≤ C(22k
|um |22 +
m=k
≤ C(
∞ m=k −1
≤ Cν
22m |um |22 +
k
∇um 2 )
m=−∞ k
22m |um |22 )
m=−∞
.
(76)
Therefore, |uk |1 ≤ Cν −2 (2k )−3 .
(77)
Acknowledgement. This research has been partially supported by the German Science Foundation through the Hausdorff Center for Mathematics. F.R. has also been partially supported by the Koshland Center for Basic Research at the Weizmann Institute of Science.
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References 1. Bergh, J., Löfström, J.: Interpolation spaces, an introduction. Berlin-Heidelberg-New York: Springer, 1976 2. Constantin, P.: Euler equations, Navier-Stokes equations and turbulence. In: Mathematical foundation of turbulent viscous flows, Lecture Notes in Math. Vol. 1871, Berlin: Springer, 2006, pp. 1-43 3. Constantin, P.: The Littlewood-Paley spectrum in two-dimensional turbulence. Theor. Comput. Fluid Dyn. 9, 183–189 (1997) 4. Constantin, P., Foias, C.: Navier-Stokes Equations. Chicago, IL: The University of Chicago Press, 1988 5. Doering, C.R., Gibbon, J.D.: Bounds on moments of the energy spectrum of weak solutions of the three-dimensional Navier-Stokes equations. Physica D. 165, 163–175 (2002) 6. Doering, C.R., Titi, E.S.: Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations. Phys. Fluids. 7, 1384–1390 (1995) 7. Frisch, U.: Turbulence. Cambridge: Cambridge University Press, 1995 8. Hopf, E.: Uber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951) 9. Leray, J.: Essai sur le mouvement dun fluide visqueux emplissant lespace. Acta Math. 63, 193–248 (1934) 10. Otto, F.: Optimal bounds on the Kuramoto-Sivashinsky Equation. J. Funct. Anal. 257(7), 2188–2245 (2009) 11. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton, NJ: Princeton University Press, 1970 12. Sulem, P.-L., Frisch, U.: Bounds on energy flux for finite energy turbulence. J. Fluid. Mech. 72, 417–424 (1975) 13. Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. In: Studies in Mathematics and its Applications, 3rd edition, Amsterdam-New York: North-Holland Publishing Co., 1984, Reedition AMS Chelsea Series, Providence, RI: Amer. Math. Soc., 2001 14. von Wahl, W.: The Equations of Navier-Stokes and Abstract Parabolic Equations. Aspects of Mathematics, Braunschweig-Weisbaden: Friedrich Vieweg & Sahn, 1988 Communicated by A. Kupiainen
Commun. Math. Phys. 300, 317–329 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1122-8
Communications in
Mathematical Physics
Quantum Random Walks and Thermalisation Alexander C. R. Belton Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK. E-mail: [email protected] Received: 28 April 2009 / Accepted: 17 May 2010 Published online: 6 September 2010 – © Springer-Verlag 2010
Abstract: It is shown how to construct quantum random walks with particles in an arbitrary faithful normal state. A convergence theorem is obtained for such walks, which demonstrates a thermalisation effect: the limit cocycle obeys a quantum stochastic differential equation without gauge terms. Examples are presented which generalise that of Attal and Joye (J Funct Anal 247:253–288, 2007). 1. Introduction A quantum random walk [3] may be interpreted as the dynamics of a quantum system that interacts periodically with a stream of identical particles, each of which lies in the vacuum state. As observed in [1], a Gelfand–Naimark–Segal construction may be employed to consider particles in a more general state (see also Franz and Skalski [7]); for a particular choice of evolution, one generated by a Hamiltonian which describes dipole-type interaction, Attal and Joye demonstrated that the limit flow involves fewer quantum noises than might naïvely be expected, a so-called ‘thermalisation’ effect [1, Theorem 7]. Inspired by this work, techniques from [3] are used herein to show that thermalisation occurs for a large class of quantum random walks. Section 2 is a rapid introduction to quantum random walks and presents the main convergence theorem from [3]. The formulation adopted uses matrix spaces over operator spaces, the Lindsay–Wills approach to quantum stochastics. This setting is both natural and fruitful, allowing the consideration of walks on C ∗ algebras [3] and quantum groups [8], for example, as well as on von Neumann algebras. The key to this approach is a matrix-space tensor product which is, in general, strictly intermediate between the spatial and ultraweak tensor products. The main result, Theorem 3, which has a remarkably simple proof, is obtained in Sect. 3. Let ρ be a faithful normal state on the particle algebra B(K), where the Hilbert space K may be infinite dimensional, and let δ be a ρ-preserving conditional expectation on B(K). Suppose the maps Φ(τ ) : B(h) → B(h ⊗ K) are linear, appropriately bounded
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and such that, in a suitable sense, τ −1 d(Φ(τ )(a) − a ⊗ IK ) + τ −1/2 d ⊥ (Φ(τ )(a)) → Ψ (a)
as τ → 0+
for all a ∈ B(h), where d := I B(h) ⊗ δ and d ⊥ := I B(h⊗K) − d. By Theorem 3, the random walk with generator Φ(τ ) and particle state ρ converges to a quantum stochastic cocycle j on h ⊗ F, where F is the Boson Fock space over L 2 (R+ ; k), which satisfies the following quantum Langevin equation: t jt (a) = a ⊗ IF + js (ψβα (a)) dΛβα (s) ∀ t ∈ R+ , a ∈ B(h). (1) α,β
0
(Here the walk takes place on a type I factor, for simplicity; this is generalised below to any operator space.) The generator of the cocycle j is a linear mapping ψ : B(h) → B(h ⊗ k), where k is the Hilbert space in the GNS representation corresponding to ρ, the vector ω gives the associated state and k := k Cω. The generator ψ, given explicitly in terms of δ and Ψ , is such that ψβα = 0 unless α = 0 or β = 0: there is no contribution in (1) from the gauge integrals. If n := dim K is finite then there are n 2 − 1 independent quantum noises when working with particles in the vacuum state and at most twice that in the situation described above, not n 4 − 1 as might first appear necessary: this is the thermalisation phenomenon. In Sect. 4 two classes of examples are presented, which include the Attal–Joye model (Examples 4 and 5). The problem of extending these results to the case of walks with a normal but not necessarily faithful state ρ has recently been solved completely [4]. 1.1. Conventions and notation. The symbol := is to be read as ‘is defined to equal’ (or similarly). The indicator function of a set A is denoted by 1 A . The sets of non-negative integers and non-negative real numbers are denoted by Z+ := {0, 1, 2, . . .} and R+ := [0, ∞[. The identity operator on a vector space V is denoted by I V . Algebraic, Hilbert-space and ultraweak tensor products are denoted by , ⊗ and ⊗, respectively, with the symbol ⊗ also denoting the spatial tensor product of operator spaces. The Dirac dyad |u v| denotes the linear transformation on an inner-product space V such that w → v, w V u. All vector spaces have complex scalar field; all inner products are linear in the second argument. An empty sum or product equals the appropriate additive or multiplicative unit, respectively. Throughout, upper-case Greek letters are used for the generators of walks and lower-case Greek letters for the generators of cocycles. 2. Preliminaries 2.1. Quantum random walks. Definition 1. Let V be a concrete operator space acting on the Hilbert space h, i.e., V is a closed subspace of B(h). Recall that the matrix space V ⊗M B(H) := {T ∈ B(h ⊗ H) : E x T E y ∈ V ∀ x, y ∈ H} is an operator space for any Hilbert space H, where E x ∈ B(h ⊗ H; h) is the adjoint of E x : u → u ⊗ x.
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The inclusions V⊗ B(H) ⊆ V ⊗M B(H) ⊆ V ⊗ B(H) hold, with the latter an equality if and only if V is ultraweakly closed; in general, both inclusions are proper. There is a natural identification (V ⊗M B(H1 )) ⊗M B(H2 ) = V ⊗M B(H1 ⊗ H2 ). Definition 2. Suppose H = {0} and let W be another operator space. A linear map Φ : V → W is H bounded if it is completely bounded, or it is bounded and the space H is finite dimensional. The Banach space of such H-bounded maps is denoted by HB(V; W) and is equipped with the Hb norm (dim H) Φ if dim H < ∞, · Hb : Φ → ΦHb := Φcb if dim H = ∞, where · cb denotes the completely bounded norm. Remark 1. Note that T → E x T E y equals the slice map I B(h) ⊗ ωx,y , where ωx,y is the ultraweakly continuous functional X → x, X y H . As {ωx,y : x, y ∈ H} is total in the predual B(H)∗ , it follows that (I B(h) ⊗ ω)(T ) ∈ V
∀ ω ∈ B(H)∗ , T ∈ V ⊗M B(H).
If θ : B(H1 ) → B(H2 ) is linear, h bounded and ultraweakly continuous then so is I B(h) ⊗ θ : B(h ⊗ H1 ) → B(h ⊗ H2 ), by [6, Lemma 1.5(b)], and the previous observation shows that this ampliation respects the matrix-space structure: (I B(h) ⊗ θ )(V ⊗M B(H1 )) ⊆ V ⊗M B(H2 ). The following type of ampliation was introduced by Lindsay and Wills [10]. The map to be lifted need not be ultraweakly continuous, and this ampliation extends that on the minimal operator-space product which is a consequence of complete boundedness. Proposition 1. If Φ ∈ HB(V; W) then the H lifting of Φ is the unique map Φ ⊗M I B(H) : V ⊗M B(H) → W ⊗M B(H) such that E x (Φ ⊗M I B(H) (T ))E y = Φ(E x T E y )
∀ x, y ∈ H, T ∈ V ⊗M B(H).
The lifting is linear, H bounded and is completely bounded if Φ is. It satisfies the inequalities Φ ⊗M I B(H) ΦHb and Φ ⊗M I B(H) cb Φcb . Proof. See [3, Theorem 2.5]. Proposition 2. For any Φ ∈ HB(V; V ⊗M B(H)) there exists a unique family of maps Φ (n) : V → V ⊗M B(H⊗n ) indexed by n ∈ Z+ , the quantum random walk with generator Φ, such that Φ (0) = IV and E x Φ (n+1) (a)E y = Φ (n) (E x Φ(a)E y )
∀ x, y ∈ H, a ∈ V, n ∈ Z+ .
These maps are linear, H bounded and completely bounded if Φ is; if n 1 then Φ (n) Hb ΦnHb and Φ (n) cb Φncb . Proof. Given Φ (n) , use Proposition 1 to let Φ (n+1) := (Φ (n) ⊗M I B(H) ) ◦ Φ. For the first inequality, see [3, Theorem 2.7]; the second is immediate.
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2.2. Toy and Boson Fock space. Notation 1. Let k be a Hilbert space containing the distinguished unit vector ω and let k := k Cω, the orthogonal complement of Cω in k. Define x := ω + x ∈ k for any x ∈ k. Definition 3. The toy Fock space over k is Γ := ∞ n=0 k(n) , where k(n) := k for all n ∈ Z+ , with respect to the stabilising sequence (ω(n) := ω)∞ ; the suffix (n) is used n=0 n−1 to indicate the relevant copy of k. Note that Γ = Γn[ ⊗ Γ[n , where Γn[ := m=0 k(m) and Γ[n := ∞ m=n k(m) , for all n ∈ Z+ . Definition 4. Let F be the Boson Fock space over L 2 (R+ ; k), the Hilbert space of squareintegrable k-valued functions on the half line. Recall that F may be considered as the completion of E, the linear span of exponential vectors ε( f ) labelled by f ∈ L 2 (R+ ; k), with respect to the inner product ∞ f (t), g(t) k dt ∀ f, g ∈ L 2 (R+ ; k). ε( f ), ε(g) F := exp 0
The following gives sense to the idea that the toy space Γ approximates F. Proposition 3. For all τ > 0 there is a unique co-isometry Dτ : F → Γ such that (n+1)τ ∞ Dτ ε( f ) = f (t) dt, f (n; τ ), where f (n; τ ) := τ −1/2 nτ
n=0
for all f ∈ L 2 (R+ ; k). Furthermore, Dτ∗ Dτ → IF strongly as τ → 0+. (See [2].) 2.3. QS cocycles. Definition 5. An h process X is a family {X t }t∈R+ of linear operators in h ⊗ F, such that the domain of each operator contains h E and the map t → X t uε( f ) is weakly measurable for all u ∈ h and f ∈ L 2 (R+ ; k); this process is adapted if uε( f ), X t vε(g) h⊗F = uε(1[0,t[ f ), X t vε(1[0,t[ g) h⊗F ε(1[t,∞[ f ), ε(1[t,∞[ g) F for all u, v ∈ h, f, g ∈ L 2 (R+ ; k) and t ∈ R+ . (As is conventional, the tensor-product sign is omitted between elements of h and exponential vectors.) A mapping process j is a family { j· (a)}a∈V of h processes such that the map a → jt (a) is linear for all t ∈ R+ ; this process is adapted if each j· (a) is, it is strongly regular if jt (·)E ε( f ) ∈ B(V; B(h; h ⊗ F))
∀ f ∈ L 2 (R+ ; k), t ∈ R+ ,
with norm locally uniformly bounded as a function of t, and it is cb regular if these two conditions hold with “bounded” replaced by “completely bounded”. Theorem 1. Given any φ ∈ k B(V; V ⊗M B( k)) there exists a unique strongly regular φ adapted mapping process j , the QS cocycle generated by φ, such that t φ uε( f ), ( jt (a) − a ⊗ IF )vε(g) = uε( f ), jsφ (E f (s) φ(a)E )vε(g) ds (2) g(s) 0
for all u, v ∈ h, f, g ∈ L 2 (R+ ; k), a ∈ V and t ∈ R+ . If φ is completely bounded then j φ is cb regular. Proof. See [10]; the proof contained there is valid for any operator space.
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Remark 2. Let {eα } be an orthonormal basis for k, with e0 = ω. The QS cocycle j ψ generated by ψ satisfies the quantum stochastic differential equation (1) if ψβα : a → E eα ψ(a)E eβ ; Eq. (2) is a coordinate-free version of (1). QS cocycles are the correct limit objects for quantum random walks, as the next section makes clear. 2.4. Random-walk convergence. Definition 6. If τ > 0 and Φ ∈ k B(V; V ⊗M B( k)) then the embedded walk with generator Φ and step size τ is the mapping process J Φ,τ such that JtΦ,τ (a) := (Ih ⊗ Dτ )∗ (Φ (n) (a) ⊗ IΓ[n )(Ih ⊗ Dτ )
if t ∈ [nτ, (n + 1)τ [
for all a ∈ V and t ∈ R+ . Definition 7. If τ > 0 and Φ ∈ k B(V; V ⊗M B( k)) then the scaled modification m(Φ, τ ) ∈ k B(V; V ⊗M B(k)) is defined by setting m(Φ, τ )(a) := (τ −1/2 Δ⊥ + Δ)(Φ(a) − a ⊗ Ik )(τ −1/2 Δ⊥ + Δ)
∀ a ∈ V,
where Δ is the orthogonal projection from h ⊗ k onto h ⊗ k and Δ⊥ := Ih⊗k − Δ. Note that m(Φ, τ ) is completely bounded if Φ is. In block-matrix form,
0 0 (a) τ −1 (Φ00 (a) − a) τ −1/2 Φ× Φ00 Φ× then m(Φ, τ )(a) = . if Φ = × × Φ0× Φ× (a) − a ⊗ Ik τ −1/2 Φ0× (a) Φ× Remark 3. For a sequence (Φn ) in HB(V; W), recall that Φn ⊗M I B(H) → 0 strongly if and only if
Φn → 0 strongly when dim H < ∞, Φn → 0 in cb norm when dim H = ∞,
by [3, Prop. 2.11 and Lemma 2.13]. The following result is a quantum analogue of Donsker’s invariance principle. Theorem 2. Let τn > 0 and Φn , φ ∈ k B(V; V ⊗M B( k)) be such that τn → 0+
and
m(Φn , τn ) ⊗M I B(k) → φ ⊗M I B(k) strongly
(i.e., pointwise in norm) as n → ∞. If f ∈ L 2 (R+ ; k) and T ∈ R+ then lim
φ
sup JtΦn ,τn (a)E ε( f ) − jt (a)E ε( f ) = 0
n→∞ t∈[0,T ]
∀ a ∈ V.
(3)
If, further, m(Φn , τn ) − φkb → 0 as n → ∞ then lim
φ
sup JtΦn ,τn (·)E ε( f ) − jt (·)E ε( f ) kb = 0;
n→∞ t∈[0,T ]
(4)
when Φn and φ are completely bounded, the same implication holds if · kb is replaced by · cb . Proof. See [3, Theorem 7.6].
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Notation 2. The conclusion (3) will be abbreviated to J Φn ,τn → j φ and the stronger conclusion (4) will be denoted by J Φn ,τn →kb j φ , or by J Φn ,τn →cb j φ if the completely bounded version holds. Example 1 (Dipole interaction in the Schrödinger picture). Choose self-adjoint operators Hsys ∈ B(h) and Hpar ∈ B( k), and let V ∈ B(h; h ⊗ k). Define Htot (τ ) := Hsys ⊗ Ik + Ih ⊗ Hpar + Hint (τ ) for all τ > 0, where
0 V∗ −1/2 ω ∗ ∗ −1/2 Hint (τ ) := τ (QV E + E ω V Q ) = τ V 0 and Q : h ⊗ k → h ⊗ k is the natural embedding map. If W (τ ) := exp(−iτ Htot (τ )) then (τ −1/2 Δ⊥ + Δ)(W (τ ) − Ih⊗k )(τ −1/2 Δ⊥ + Δ)
Hsys + μIh V ∗ 1 V ∗V 0 = −i − + O(τ ) 2 V 0 0 0 as τ → 0+, where μ := ω, Hpar ω k . Thus if τn → 0+ and Φ(τ ) : B(h) → B(h ⊗ k); a → (a ⊗ Ik )W (τ ) then, by Theorem 2, J Φ(τn ),τn →cb j φ , where φ(a) := (a ⊗ Ik )F for all a ∈ B(h) and
−i(Hsys + μIh ) − 21 V ∗ V −iV ∗ F := ∈ B(h ⊗ k). −iV 0 The cocycle j φ obtained in the limit is an evolution of Hudson–Parthasarathy type: if φ Ut := jt (Ih ) for all t ∈ R+ then Ut is unitary, by [9, Proof of Theorem 7.1 and Theorem 7.5], and the adapted h process U satisfies the quantum Langevin equation U0 = Ih⊗F ,
dUt = dΛ F (t)Ut .
(5)
φ
(The cocycle j φ may be recovered from U by setting jt (a) := (a ⊗ IF )Ut .) Example 2 (Dipole interaction in the Heisenberg picture). Let τn and W (τ ) be as in Example 1 and let Φ(τ ) : B(h) → B(h ⊗ k); a → W (τ )∗ (a ⊗ Ik )W (τ ). Then J Φ(τn ),τn →cb j φ , by Theorem 2, where
−i[a, Hsys ] + V ∗ (a ⊗ Ik )V − 21 {a, V ∗ V } φ(a) := −i(a ⊗ Ik )V + iV a
−iaV ∗ + iV ∗ (a ⊗ Ik )
0
for all a ∈ B(h), with the commutator [x, y] := x y − yx and the anticommutator {x, y} := x y + yx. Here, the limit cocycle is an inner Evans–Hudson flow: if U is the unitary adapted h process which satisfies (5) then, by [9, Theorem 7.4], φ
jt (a) = Ut∗ (a ⊗ IF )Ut
∀ a ∈ B(h), t ∈ R+ .
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Remark 4. In Example 1, if k has orthonormal basis {e j } Nj=1 , where N may be infinite, then Hint (τ ) = τ −1/2
N (V j ⊗ |e j ω| + V j∗ ⊗ |ω e j |), j=1
where V j := E e j V for all j; the series converges strongly if N = ∞. For N < ∞, this is the dipole-interaction Hamiltonian used by Attal and Joye in [1]. With the convention that e0 = ω, the quantum Langevin equation (5) takes the form U0 = Ih⊗F ,
N
dUt =
α,β=0
(Fβα ⊗ IF )Ut dΛβα (t),
where Fβα := E eα F E eβ for all α, β. 3. Thermal Walks Notation 3. Let be a density matrix which acts on the Hilbert space K, i.e., a positive operator with unit trace, and suppose that the corresponding normal state ρ : X → tr(X ) on the von Neumann algebra B(K) is faithful (so K is separable). Fix an orthonormal basis {e j } Nj=0 of K, where N ∈ Z+ or N = ∞, such that =
N
λ j |e j e j |.
j=0
The eigenvalues λ j are positive and sum to 1, so this series is norm convergent. Definition 8. Let ( k, π, ω) be the GNS representation corresponding to ρ and let X → [X ] denote the induced mapping from B(K) into k, so that ω := [IK ]. Note that the representation π : B(K) → B( k) is ultraweakly continuous, injective and unital [5, Theorems 2.3.16 and 2.4.24], and [ker ρ] is dense in k := k Cω. Lemma 1. The slice map ρ := I B(h) ⊗ ρ : B(h ⊗ K) → B(h) is completely positive and such that ρ (T ) =
N
λ j E e j T E e j = tr K ((Ih ⊗ )T )
∀ T ∈ B(h ⊗ K),
j=0
where tr K is the partial trace over K. The unital ∗-homomorphism π := I B(h) ⊗ π : B(h ⊗ K) → B(h ⊗ k) is injective and such that E [X ] π (T )E [Y ] = ρ ((Ih ⊗ X )∗ T (Ih ⊗ Y )) ∀ X, Y ∈ B(K), T ∈ B(h ⊗ K).
(6)
Proof. The existence of ρ and π as claimed follows from [11, Prop. IV.5.13 and Theorem IV.5.2]. The identities are immediate if T ∈ B(h) B(K), so hold everywhere by ultraweak continuity.
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Definition 9. Let Φ ∈ K B(V; V ⊗M B(K)). The map π ◦Φ ∈ k B(V; V ⊗M B( k)) is the GNS generator of the quantum random walk with generator Φ and particle state ρ. (The vector state on B( k) given by ω corresponds to the state ρ on B(K).) Definition 10. Choose a conditional expectation δ on B(K), i.e., a linear idempotent onto a C ∗ subalgebra D such that δ(X ∗ X ) 0 and δ(δ(X )Y ) = δ(X )δ(Y ) = δ(X δ(Y )) ∀ X, Y ∈ B(K).
(7)
Recall that δ is completely positive, by [11, Cor. IV.3.4]. Suppose further that δ preserves the state ρ, i.e., ρ ◦ δ = ρ. Then δ is ultraweakly continuous, so D is a von Neumann algebra, and δ(IK ) = IK , since IK − δ(IK ) is an orthogonal projection with ρ(IK − δ(IK )) = 0. -preserving Notation 4. Letting d := I B(h) ⊗ δ, where δ is as above, the lifted map d is a ρ conditional expectation onto B(h) ⊗ D which leaves V ⊗M B(K) invariant. Furthermore, d(a ⊗ IK ) = a ⊗ IK and d(T1 d(T2 )) = d(T1 )d(T2 ) = d(d(T1 )T2 )
(8)
◦ d = ρ , for all a ∈ B(h) and T1 , T2 ∈ B(h ⊗ K). These, together with the identity ρ imply that (T1 d(T2 )) ρ (d(T1 )T2 ) = ρ ρ (T ) ρ ((a ⊗ IK )T ) = a
∀ T1 , T2 ∈ B(h ⊗ K)
and
∀ a ∈ B(h), T ∈ B(h ⊗ K).
(9) (10)
The conditional expectation d plays a vital rôle in scaling walks in order to obtain convergence. The following example is a natural choice for this map, that given by the eigenbasis {e j } Nj=0 . Example 3. Define the diagonal map de : B(h ⊗ K) → B(h ⊗ K); T → (Ih ⊗ S)∗ (T ⊗ IK )(Ih ⊗ S), where the Schur isometry S ∈ B(K; K ⊗ K) is such that Se j = e j ⊗ e j for all j. If De is the maximal Abelian subalgebra of B(K) generated by {|e j e j |} Nj=0 then de is the lifting of the unique ρ-preserving conditional expectation δe onto De . Definition 11. If τ > 0 and Φ ∈ K B(V; V ⊗M B(K)) then the scaled modification m δ (Φ, τ ) ∈ K B(V; V ⊗M B(K)) is defined by setting m δ (Φ, τ ) := τ −1 d ◦ Φ + τ −1/2 d ⊥ ◦ Φ = (τ −1 d + τ −1/2 d ⊥ ) ◦ Φ , where Φ ∈ K B(V; V ⊗M B(K)) is such that Φ (a) := Φ(a) − a ⊗ IK for all a ∈ V and d ⊥ := I B(h⊗K) − d. If Φ is completely bounded then so is m δ (Φ, τ ). Theorem 3. Let τn > 0 and Φn , Ψ ∈ K B(V; V ⊗M B(K)) be such that τn → 0+
and
m δ (Φn , τn ) ⊗M I B(K) → Ψ ⊗M I B(K) strongly
as n → ∞. If ψ ∈ k B(V; V ⊗M B( k)) is defined by setting π ◦ Ψ )(a)Δ⊥ + Δ( π ◦ d ⊥ ◦ Ψ )(a)Δ⊥ + Δ⊥ ( π ◦ d ⊥ ◦ Ψ )(a)Δ ψ(a) := Δ⊥ ( then ψ is completely bounded whenever Ψ is and
π ◦Φn ,τn J
if m δ (Φn , τn ) − Ψ Kb → 0 then J
→
π ◦Φn ,τn
jψ.
Furthermore,
→kb j ψ
and, when Φn and Ψ are completely bounded, π ◦Φn ,τn →cb j ψ . if m δ (Φn , τn ) − Ψ cb → 0 then J
(11)
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Proof. Note first that Φ = (τ d + τ 1/2 d ⊥ ) ◦ m δ (Φ, τ ) for all τ > 0. If X, Y ∈ ker ρ and a ∈ V then (6) and the identity ρ ◦ d = ρ imply that E ω m( π ◦ Φ, τ )(a)E ω = τ −1 E ω π (Φ (a))E ω = τ −1 ρ (Φ (a)) =ρ (m δ (Φ, τ )(a)) = E ω ( π ◦ m δ (Φ, τ ))(a)E ω , π ◦ Φ, τ )(a)E ω = τ −1/2 E [X ] π (Φ (a))E ω E [X ] m( [X ] = E ( π ◦ (τ 1/2 d + d ⊥ ) ◦ m δ (Φ, τ ))(a)E ω , π ◦ Φ, τ )(a)E [Y ] = E ω ( π ◦ (τ 1/2 d + d ⊥ ) ◦ m δ (Φ, τ ))(a)E [Y ] E ω m( and E [X ] m( π ◦ Φ, τ )(a)E [Y ] = E [X ] π (Φ (a))E [Y ] = E [X ] ( π ◦ (τ d + τ 1/2 d ⊥ ) ◦ m δ (Φ, τ ))(a)E [Y ] . Letting Θ := m δ (Φ, τ ) − Ψ , this working shows that (m( π ◦ Φ, τ ) − ψ)(a) = Δ⊥ ( π ◦ Θ)(a)Δ⊥ + Δ( π ◦ d ⊥ ◦ Θ)(a)Δ⊥ +Δ⊥ ( π ◦ d ⊥ ◦ Θ)(a)Δ + τ 1/2 R(a), where R(a) := Δ( π ◦ d ◦ m δ (Φ, τ ))(a)Δ⊥ + Δ⊥ ( π ◦ d ◦ m δ (Φ, τ ))(a)Δ +Δ( π ◦ (τ 1/2 d + d ⊥ ) ◦ m δ (Φ, τ ))(a)Δ. The result follows, by Theorem 2. Remark 5. The presence of thermalisation in the conclusions of Theorem 3 is apparent from the appearance of the generator ψ. The lack of a term in (11) of the form Δ⊥ T Δ⊥ indicates the absence of gauge integrals in Eq. (1) satisfied by the limit cocycle j ψ . If Ψ ∈ K B(V; V ⊗M B(K)) and ψ is defined by (11) then the identities (6) and (9) imply that E ω ψ(a)E ω = ρ (Ψ (a)) = E ω π (Ψ (a))E ω ,
(12)
E [X ] ψ(a)E ω = ρ ((Ih ⊗ X )∗ (d ⊥ ◦ Ψ )(a)) = E ω
⊥
[δ ⊥ (X )]
ω
π (Ψ (a))E ω , (13)
((d ◦ Ψ )(a)(Ih ⊗ Y )) = E π (Ψ (a))E [δ ⊥ (Y )] E ψ(a)E [Y ] = ρ and E [X ] ψ(a)E [Y ] = 0
(14) (15)
for all X, Y ∈ ker ρ and a ∈ V, where δ ⊥ := I B(K) − δ. In the quantum Langevin equation (1), the quantities (12–15) appear in the time, creation, annihilation and gauge integrals, respectively; it follows that no gauge terms are present. If n := dim K < ∞ then at most 2 dim{[δ ⊥ (X )] : X ∈ ker ρ} 2(n 2 − 1) independent noises are required to drive the cocycle j ψ ; equality holds in the above if δ(X ) = ρ(X )IK for all X ∈ B(K).
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Remark 6. Although the construction in Example 3 may appear to depend on the choice of orthonormal basis which diagonalises , this is not really so. To see this, suppose { f j } Nj=0 is another eigenbasis for , labelled so that e j and f j have the same eigenvalue for all j, let δe and δf be the ρ-preserving conditional expectations from B(K) onto the subalgebras generated by {|e j e j |} Nj=0 and {| f j f j |} Nj=0 , and let de and df be their lifts to B(h ⊗ K). If U ∈ B(K) is the unique unitary operator such that U e j = f j for all j then ρ(U ∗ XU ) = ρ(X ) and U ∗ δf (X )U = δe (U ∗ XU )
∀ X ∈ B(K).
:= Ih ⊗ U , Hence, if Φ, Ψ ∈ K B(V; V ⊗M B(K)) and U ∗ (m δf (Φ, ˇ τ ) − Ψˇ )(a)U (m δe (Φ, τ ) − Ψ )(a) = U
∀ a ∈ V,
Φ(a)U ∗ et cetera. Let ψe and ψf be defined by (11), but with d where Φˇ : a → U equal to de and df , respectively, and Ψ replaced by Ψˇ in the latter case; if a ∈ V and X, Y ∈ ker ρ then (Ψˇ (a)) = ρ (Ψ (a)) = E ω ψe (a)E ω , E ω ψf (a)E ω = ρ E [X ] ψf (a)E ω = ρ ((Ih ⊗ X )∗ (df⊥ ◦ Ψˇ )(a)) U ∗ df⊥ (Ψˇ (a))U ) ∗ (Ih ⊗ X ∗ )U =ρ (U =ρ ((Ih ⊗ U ∗ XU )∗ de⊥ (Ψ (a))) = E [U ω
∗ XU ]
ψe (a)E ω ,
ω
E ψf (a)E [Y ] = E ψe (a)E [U ∗ Y U ] and E [X ] ψf (a)E [Y ] = 0 = E [U
∗ XU ]
ψe (a)E [U ∗ Y U ] .
Thus if W ∈ B( k) is the unique unitary operator such that W [X ] = [U ∗ XU ] for all X ∈ B(K) then ψf (a) = (Ih ⊗ W )∗ ψe (a)(Ih ⊗ W )
∀ a ∈ V;
the change of orthonormal basis used to define the diagonal map is manifest as a change of coordinates (an isometric isomorphism of k which preserves ω) and unitary conjugation of the map Ψ . 4. Examples Henceforth V will be a von Neumann algebra A and A ⊗M B(K) = A ⊗ B(K). As above, δ is a conditional expectation on B(K) which preserves the faithful normal state ρ and d = I B(h) ⊗ δ. 4.1. Hudson–Parthasarathy evolutions. Remark 7. Suppose F ∈ A ⊗ B(K) and let Ψ : A → A ⊗ B(K) be such that Ψ (a) = (a ⊗ IK )F for all a ∈ A. If ψ is defined by (11) then ψ(a) = (a ⊗ Ik )G for all a ∈ A, by (8), where π (F)Δ⊥ + Δ π (d ⊥ (F))Δ⊥ + Δ⊥ π (d ⊥ (F))Δ. G = Δ⊥
(16)
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327 ψ
Furthermore, the cocycle j ψ is such that jt (a) = (a ⊗ IF )X t for all a ∈ A and t ∈ R+ , ψ where the adapted h process X = {X t := jt (Ih )}t∈R+ satisfies the Hudson–Parthasarathy equation X 0 = Ih⊗F ,
dX t = dΛG (t)X t .
(17)
(See [9, Proof of Theorem 7.1].) Theorem 4. Let Hd and Ho be self-adjoint elements of A ⊗ B(K) such that Hd = d(Hd ) and Ho = d ⊥ (Ho ). If τ > 0 and Htot (τ ) := Hd +τ −1/2 Ho then the completely isometric map Φ(τ ) : A → A ⊗ B(K); a → (a ⊗ IK ) exp(−iτ Htot (τ ))
(18)
is such that m δ (Φ(τ ), τ ) − Ψ cb → 0 as τ → 0+, where Ψ : A → A ⊗ B(K); a → (a ⊗ IK )(−i(Hd + Ho ) − 21 d(Ho2 )). π ◦Φ(τn ),τn → ψ Thus J cb j if τn → 0+, where ψ : A → A ⊗ B(k) is such that
E ω ψ(a)E ω = −ia ρ (Hd ) − 21 a ρ (Ho2 ), ρ ((Ih ⊗ X )∗ Ho ), E [X ] ψ(a)E ω = −ia ρ (Ho (Ih ⊗ Y )) E ω ψ(a)E [Y ] = −ia and E [X ] ψ(a)E [Y ] = 0 ψ
for all X, Y ∈ ker ρ and a ∈ A, and Ut := jt (Ih ) is unitary for all t ∈ R+ . Proof. If a ∈ A then d(Φ(τ ) (a)) = (a ⊗ IK )d(exp(−iτ Htot (τ )) − Ih⊗K ), by (8), and the same holds with d replaced by d ⊥ . As τ → 0+, τ −1 d(exp(−iτ Htot (τ )) − Ih⊗K ) = −iHd − 21 d(Ho2 ) + O(τ 1/2 ) and τ −1/2 d ⊥ (exp(−iτ Htot (τ )) − Ih⊗K ) = −iHo + O(τ 1/2 ), which gives the first claim. Theorem 3 and Remark 5, simplified with use of the identities (8), (10) and ρ = ρ ◦ d, complete the result; unitarity holds for the adapted h process U = {Ut }t∈R+ by [9, Theorem 7.5]. The following is the appropriate version of Example 1 for thermal walks. Example 4 (Dipole interaction in the Schrödinger picture). Let d = de be the diagonal map of Example 3 and suppose Hd := Hsys ⊗ IK + Ih ⊗ Hpar , where the self-adjoint operators Hsys ∈ A and Hpar =
N
μ j |e j e j | ∈ B(K),
j=0
with this series strongly convergent when N = ∞. Let K× := K Ce0 , choose V ∈ A ⊗ B(C; K× ) and define
0 V∗ e0 ∗ ∗ Ho := QV E + E e0 V Q = , V 0
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where Q : h ⊗ K× → h ⊗ K is the natural embedding. If τn → 0+ and Φ(τ ) is defined π ◦Φ(τn ),τn → ψ by (18) then Theorem 4 implies that J cb j , where E ω ψ(a)E ω = −ia(Hsys + ρ(Hpar )Ih ) − 21 a ρ (Ho2 ), E [X ] ψ(a)E ω = −ia(E X e0 QV + V ∗ Q ∗ E X ∗ e0 ), E ω ψ(a)E [Y ] = −ia(E Y
∗e
0
QV + V ∗ Q ∗ E Y e0 )
and E [X ] ψ(a)E [Y ] = 0 for all X, Y ∈ ker ρ and a ∈ A. If dim K < ∞ then this is in agreement with [1, Theorem 7]. Letting λ0 = 1 and λ j = 0 for all j > 0, the above is also in formal agreement with Example 1; note that the GNS space is spanned by {[|e j e0 |]} Nj=0 in this case. For rigorous results when ρ is not necessarily faithful, see [4]. 4.2. Evans–Hudson evolutions. Definition 12. An Evans–Hudson flow is a solution j of the quantum stochastic differential equation (1) which is ∗-homomorphic, i.e., each jt (a) extends to a bounded operator on h⊗F and the mapping a → jt (a) is a ∗-homomorphism from A to B(h⊗F) for all t ∈ R+ . Remark 8. If F ∈ A ⊗ B(K) and Ψ : A → A ⊗ B(K) is such that Ψ (a) = (a ⊗ IK )F + F ∗ (a ⊗ IK ) + d(d ⊥ (F)∗ (a ⊗ IK )d ⊥ (F))
∀ a ∈ A,
then a short calculation shows that ψ defined by (11) is such that ψ(a) = (a ⊗ Ik )G + G ∗ (a ⊗ Ik ) + G ∗ Δ(a ⊗ Ik )ΔG
∀ a ∈ A,
ψ
with G as in (16). It follows [9, Theorem 7.4] that jt (a) = X t∗ (a ⊗ Ik )X t for all a ∈ A and t ∈ R+ , where X is the solution of (17); if X is co-isometric then j ψ is an inner Evans–Hudson flow. Recall that [x, y] := x y − yx and {x, y} := x y + yx are the commutator and the anticommutator, respectively. Theorem 5. If Hd , Ho and Htot are as in Theorem 4 then the ultraweakly continuous unital ∗-homomorphism Φ(τ ) : A → A ⊗ B(K); a → exp(iτ Htot )(a ⊗ IK ) exp(−iτ Htot )
(19)
is such that m δ (Φ(τ ), τ ) − Ψ cb → 0 as τ → 0+, where Ψ : A → A ⊗ B(K); a → −i[a ⊗ IK , Hd + Ho ] + d(Ho (a ⊗ IK )Ho ) − 21 {a ⊗ IK , d(Ho2 )}. π ◦Φ(τn ),τn → ψ Hence J cb j if τn → 0+, where ψ : A → A ⊗ B(k) is completely bounded and
E ω ψ(a)E ω = −i[a, ρ (Hd )] + ρ (Ho (a ⊗ IK )Ho ) − 21 {a, ρ (Ho2 )}, ((Ih ⊗ X )∗ Ho )], E [X ] ψ(a)E ω = −i[a, ρ (Ho (Ih ⊗ Y ))] E ω ψ(a)E [Y ] = −i[a, ρ and E [X ] ψ(a)E [Y ] = 0 for all X, Y ∈ ker ρ and a ∈ A, and j ψ is an inner Evans–Hudson flow.
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Proof. This follows in the same manner as Theorem 4. Remark 9. Since Φ(τ ) in (19) is an ultraweakly continuous ∗-homomorphism, so are π ◦Φn (τn ),τn . It follows, by strong Φ(τ )(n) , for all n ∈ Z+ , as is the embedded walk J convergence, that the limit cocycle j ψ of Theorem 5 is ∗-homomorphic; this shows directly that j ψ is an Evans–Hudson flow. The following example is analogous to Example 2, but in the thermal setting. Example 5 (Dipole interaction in the Heisenberg picture). If the maps d, Hd , Ho and Q π ◦Φ(τn ),τn → ψ are as in Example 4 and the generator Φ(τ ) is defined by (19) then J cb j if τn → 0+, where ψ is such that E ω ψ(a)E ω = −i[a, Hsys ] + ρ (Ho (a ⊗ IK )Ho ) − 21 {a, ρ (Ho2 )}, E [X ] ψ(a)E ω = −i[a, E X e0 QV + V ∗ Q ∗ E X ∗ e0 ], E ω ψ(a)E [Y ] = −i[a, E Y
∗e
0
QV + V ∗ Q ∗ E Y e0 ]
and E [X ] ψ(a)E [Y ] = 0 for all X, Y ∈ ker ρ and a ∈ A. When dim K < ∞, the map a → E ω ψ(a)E ω is the Lindblad generator of [1, Cor. 13]. Formally, the above agrees with Example 2 when λ0 = 1 and λ j = 0 for all j > 0; see [4]. Acknowledgements. The author completed part of this work while an Embark Postdoctoral Fellow at University College Cork, funded by the Irish Research Council for Science, Engineering and Technology. It was begun at the International Workshop on Quantum Probability and its Applications held in Ferrazzano, near Campobasso, Italy, thanks to the hospitality of the meeting’s organiser, Professor Michael Skeide. Earlier drafts of this work received generous comments from Professor Martin Lindsay and Dr. Adam Skalski; an observation of Dr. Stephen Wills led to the significant improvement of Sect. 3. Helpful suggestions from the referee have hopefully given rise to a clearer exposition, and those obscurities which remain are solely the responsibility of the author.
References 1. Attal, S., Joye, A.: The Langevin equation for a quantum heat bath. J. Funct. Anal. 247, 253–288 (2007) 2. Belton, A.C.R.: Approximation via toy Fock space – the vacuum-adapted viewpoint. In: Quantum Stochastics and Information, eds. V.P. Belavkin, M. Gu¸ta˘ , Singapore: World Scientific, 2008, pp. 3–22 3. Belton, A.C.R.: Random-walk approximation to vacuum cocycles. J. London Math. Soc. 81(2), 412– 434 (2010) 4. Belton, A.C.R.: Quantum random walks and thermalisation II. In preparation 5. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics 1. Second edition, Berlin: Springer, 1987 6. de Canniere, J., Haagerup, U.: Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107, 455–500 (1985) 7. Franz, U., Skalski, A.: Approximation of quantum Lévy processes by quantum random walks. Proc. Indian Acad. Sci. Math. Sci. 118, 281–288 (2008) 8. Lindsay, J.M., Skalski, A.G.: Quantum stochastic convolution cocycles II. Commun. Math. Phys. 280, 575–610 (2008) 9. Lindsay, J.M., Wills, S.J.: Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise. Probab. Theory Related Fields 116, 505–543 (2000) 10. Lindsay, J.M., Wills, S.J.: Existence of Feller cocycles on a C ∗ -algebra. Bull. London Math. Soc. 33, 613–621 (2001) 11. Takesaki, M.: Theory of operator algebras I. Springer, New York (1979) Communicated by A. Connes
Commun. Math. Phys. 300, 331–373 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1100-1
Communications in
Mathematical Physics
Foliations by Stable Spheres with Constant Mean Curvature for Isolated Systems with General Asymptotics Lan-Hsuan Huang1,2 1 Department of Mathematics, Stanford University, Stanford, CA 94305, USA.
E-mail: [email protected]
2 Current address: Department of Mathematics, Columbia University, 2990 Broadway,
New York, NY 10027, USA. E-mail: [email protected] Received: 15 May 2009 / Accepted: 29 April 2010 Published online: 31 August 2010 – © Springer-Verlag 2010
Abstract: We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge–Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliation is asymptotically concentric, and its geometric center is the center of mass. The construction of the foliation generalizes the results of Huisken–Yau, Ye, and Metzger, where strongly asymptotically flat manifolds and their small perturbations were studied. 1. Introduction Whether a foliation of constant mean curvature surfaces uniquely exists in an exterior region of an asymptotically flat manifold is a fundamental problem in general relativity. The significance of this problem is that the foliation provides an intrinsic geometric structure near infinity, supplies a definition of the center of mass in general relativity, and has a relation to quasi-local mass. Currently, a widely-used definition of asymptotic flat manifolds at infinity is expressed in terms of Cartesian coordinates outside a compact set and requires suitable decay rates on the data. The definition is convenient for calculation purposes, but it may obscure interesting geometry [16, p. 697]. In order to understand the canonical structure of asymptotically flat manifolds, Yau suggests that the constant mean curvature foliation is a promising description of asymptotic flat manifolds near infinity. Moreover, once the foliation exists and is unique, one can develop polar coordinates analogous to the polar coordinates in Euclidean space, and a canonical concept of center of mass can be defined. Also, the Hawking mass is a quantity introduced to capture the energy content of the region bounded by a two-surface N which is defined as follows: 1
m H (N ) =
|N | 2
3
(16π ) 2
2 16π − H dσ . N
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Christodoulou and Yau [4] proved that the Hawking mass is non-negative on a stable surface with constant mean curvature for initial data sets satisfying the dominant energy condition. Bray [3] showed that the Hawking mass is monotonically increasing along the isoperimetric constant mean curvature surfaces and converges to the ADM mass at infinity. For the existence and uniqueness of constant mean curvature foliation, some results have been achieved for strongly asymptotically flat manifolds whose metrics, in some asymptotically flat coordinate chart, are of the form: 2m gi j (x) = 1 + δi j + pi j , |x| pi j (x) = O(|x|
−2
α
), ∂ pi j (x) = O(|x|
(1.1) −2−|α|
),
where m is the ADM mass. Huisken and Yau [10] proved the existence of constant mean curvature foliations for strongly asymptotically flat manifolds, if m > 0. They also showed that the foliation is unique if each leaf is stable, and if it lies outside a suitable compact set. Using the unique foliation, they defined a geometric center of the foliation. Corvino and Wu [7] proved that the geometric center of the foliation is the center of mass if the metric is conformally flat near infinity. The condition that the metric is conformally flat near infinity is later removed by the author [9]. Ye [17] used a different approach to prove the existence of the foliation under the same assumption that the metric is strongly asymptotically flat, and the uniqueness of the foliation under slightly different conditions. A more general uniqueness result was proven by Qing and Tian [13]. Metzger [12] generalized the previous results to manifolds whose metrics are small perturbations of strongly asymptotically flat metrics. However, these results have been limited to asymptotically flat manifolds with special restrictions on the |x|−1 -term of the metrics. Especially, the metric being strongly asymptotically is not coordinate invariant; namely, it no longer has the expression (1.1) if the metric is written in a boosted coordinate chart. Furthermore, center of mass is defined for asymptotically flat manifolds satisfying a more general condition: the Regge–Teitelboim condition (see Definition 1.2) [2,9], so it is desirable to generalize the previous results to this setting. In this paper, we show that the foliation exists in the exterior region of an asymptotically flat manifold satisfying the Regge–Teitelboim condition, when the ADM mass is strictly positive. We not only remove the condition on the |x|−1 -term of the metrics, but also allow metrics to have the most general decay rates q > 1/2. Also, we prove that the foliation is unique under certain assumptions analogous to those in [10,12]. From our construction, the geometric center of the foliation is equal to the center of mass. To clearly state the results, we first provide some definitions. A three-dimensional manifold M with a Riemannian metric g and a symmetric (0, 2)-tensor K is called an initial data set if g and K satisfy the constraint equations Rg − |K |2g + (trg (K ))2 = 16πρ, divg (K − trg (K )g) = 8π J,
(1.2)
where Rg is the scalar curvature of M, trg (K ) = g i j K i j , ρ is the observed energy density, and J is the observed momentum density. We use the Einstein summation convention
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and sum over repeated indices; though, sometimes we employ summation symbols for clarity. Definition 1.1. (M, g, K ) is asymptotically flat (AF) at the decay rate q ∈ (1/2, 1] if it is an initial data set, and there exist coordinates {x} outside a compact set, say B R0 , such that gi j (x) = δi j + O5 (|x|−q ),
K i j (x) = O1 (|x|−1−q ).
Also, ρ and J satisfy ρ(x) = O(|x|−2−2q ),
J (x) = O(|x|−2−2q ).
Here, the subscript in the big O notation denotes the order of the derivatives which possess the corresponding decay rates. For example, if f = O2 (|x|−q ), then f ∈ C 2 and | f (x)| ≤ c|x|−q , |D f (x)| ≤ c|x|−1−q , |D 2 f (x)| ≤ c|x|−2−q pointwisely for |x| large, where c is a constant depending only on g and K . Remark. The condition on the regularity of g up to the fifth order of derivatives is used in the proof of uniqueness: Theorem 2 and Theorem 3. For the existence of the constant mean curvature foliation (Theorem 1), we only need gi j = δi j + O2 (|x|−q ). For AF manifolds, the ADM mass m is defined by m=
1 lim 16π r →∞
|x|=r i, j
gi j,i − gii, j
xj dσe , |x|
(1.3)
i 2 where |x| = i (x ) , and dσe is the induced area form with respect to the Euclidean metric. The ADM mass is well-defined when the decay rate q is greater than 1/2 (see [1,5]). Another equivalent definition of ADM mass is 1 lim m= 16π r →∞
|x|=r
RiciMj
1 xj dσe , − Rg gi j (−2x i ) 2 |x|
(1.4)
where Ric M is the Ricci curvature of g. Definition 1.2. (M, g, K ) is asymptotically flat satisfying the Regge–Teitelboim condition (AF–RT) at the decay rate q ∈ (1/2, 1] if (M, g, K ) is asymptotically flat, and g, K satisfy these asymptotically even/odd conditions gi j (x) − gi j (−x) = O2 (|x|−1−q ),
K i j (x) + K i j (−x) = O1 (|x|−2−q ).
Also, ρ and J satisfy ρ(x) − ρ(−x) = O(|x|−3−2q ),
J (x) − J (−x) = O(|x|−3−2q ).
Remark. The RT condition on the data is preserved under coordinate translations, rotations, and boost.
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Assume that (M, g, K ) is AF–RT. Then, the center of mass C is defined by, for α = 1, 2, 3, ⎡ 1 xj α ⎣ C = x α (gi j,i − gii, j ) dσe lim 16π m r →∞ |x|=r |x| i, j xi xα giα − gii dσe . − (1.5) |x| |x| |x|=r i
The above notion is well-defined [2,6,9] for AF–RT manifolds. It is noted that another notion of center of mass analogous to (1.4) has been studied and proven to be equivalent to C in [9]. For the purpose of this paper, we use the above definition (1.5). We denote S R (C) = {x : |x − C| = R} and νg as the outward unit normal vector on S R (C) with respect to g. If ψ ∈ C 2,α (S R (C)), then ψ ∗ (y) := ψ(Ry + C) and ψ ∗ ∈ C 2,α (S1 (0)). (ψ ∗ )odd denotes ψ ∗ (y) − ψ ∗ (−y). For the definitions of strictly stable and stable, please refer to Definition 3.6. Also, throughout this article, c and ci denote constants independent of R. Our main theorems are the following: Theorem 1. Assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1]. If m = 0, then there exist surfaces { R } with constant mean curvature H R in the exterior region of M, and H R = (2/R) + O(R −1−q ). Moreover, R is a c0 R 1−q -graph over S R (C), i.e.
R = x + ψ0 (x)νg : ψ0 ∈ C 2,α (S R (C)) with ψ0∗ C 2,α (S1 (0)) ≤ c0 R 1−q , and (ψ0∗ )odd C 2,α (S1 (0)) ≤ c0 R −q . Therefore, the geometric center of { R } is the center of mass C. Additionally, if m > 0, then each R is strictly stable, and { R } form a foliation. For one single surface N , we have the following uniqueness result where the minimal radius is denoted by r = min{|x| : x ∈ N }. Theorem 2. Assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1] and m > 0. Then there exists σ1 so that if N has the following properties: (1) (2) (3) (4)
N is topologically a sphere, N has constant mean curvature H = H R for some R ≥ σ1 , N is stable, 5−q r ≥ H −a for some a satisfying 2(2+q) < a ≤ 1,
then N = R . Notice that the topological condition (1) is used in Lemma 4.4. In Theorem 2, we do not assume that N is a leaf of the foliation. Thus, in the region M\B H −a (0), R is the only stable surface with constant mean curvature H R . In particular, { R } is the only foliation by stable surfaces of constant mean curvature so that each leaf with mean curvature H lies in the region M\B H −a (0). It is noted that when the decay rate q = 1,
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335
a > 2/3, which is exactly the restriction imposed in [12] to derive the a priori estimates, but the radius H −a increases as q approaches 1/2. If we replace the condition on r by the condition that r and the maximal radius r = max{|x| : x ∈ N } are comparable, we derive a uniqueness result which holds outside a fixed compact set. Theorem 3. Assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1] and m > 0. There exist σ2 and c2 so that if N has the following properties: (1) N is topologically a sphere, (2) N has constant mean curvature H = H R for some R ≥ σ2 , (3) N is stable, 5−q (4) r ≤ c2 (r )1/a for some a satisfying 2(2+q) < a ≤ 1, then N = R . An ingredient used in Sect. 2 (Lemma 2.1) and hence in Theorem 1 is the density theorem for (M, g, K ) satisfying the AF–RT condition. Denote the momentum tensor π = K − (trg K )g below and denote the modified Lie derivative, for any metric g, Lg X := L X g − divg (X )g, where L X g is the Lie derivative. Definition 1.3. (M, g, π ) is said to have harmonic asymptotics if (M, g, π ) is asymptotically flat and g = u 4 δ, π = u 2 (Lδ X )
(1.6)
outside a compact set for some function u and vector field X tending to 1 and 0 at infinity respectively. k, p
Definition 1.4. We denote W−q (M) the weighted Sobolev spaces. We say that f ∈ k, p
k, p
W−q (M), if f ∈ Wloc (M) and, in addition, when p < ∞, f W k, p (M) −q
⎛ ⎝ :=
⎞1 p p D α f ρ |α|+q ρ −3 dvol g ⎠ < ∞, M |α|≤k
where α is a multi-index and ρ is a continuous function with ρ = |x| on M\B R0 ; when p = ∞, f W k,∞ (M) := ess sup |D α f |ρ |α|+q < ∞. −q
|α|≤k
M
Theorem 4 (Density Theorem [9]). Assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1). Then, there is a sequence of data (g k , π k ) of harmonic asymptotics satisfying (1.2) (with the same ρ and J ) such that: Given any > 0 and q0 ∈ (0, q), there exist R and k0 = k0 (R) so that, for any p > 3/2, (g k , π k ) is within an -neighborhood 2, p 1, p of (g, π ) in W−q (M) × W−1−q (M) and g k (x) − g k (−x)W 2, p
≤ ,
π k (x) + π k (−x)W 1, p
≤ ,
−1−q0 (M\B R ) −2−q0 (M\B R )
for all k ≥ k0 .
Moreover, mass, linear momentum, center of mass, angular momentum of (g k , π k ) are within of those of (g, π ).
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Remark. The density theorem stated in [9] is for vacuum initial data, i.e. ρ = 0 and J = 0. A slight modification of the proof generalizes to the current situation. Also, notice that as in [9], the theorem holds more generally for (g, π ) satisfying weaker regularity (in weighted Sobolev spaces). Here, we only need the version that (g, π ) satisfies the pointwise regularity at the suitable decay rates defined by Definition 1.2. The article is organized as follows. In Sect. 2, an important identity relating the mean curvature to center of mass (2.2) is derived using the density theorem. In Sect. 3, we prove the existence of the foliation (Theorem 3.1 and Theorem 3.9) and show its geometric center is equal to the center of mass (Corollary 3.4). In Sect. 4, Theorem 2 and Theorem 3 are proven after certain a priori estimates are established. 2. Estimates on Surfaces Close to Euclidean Spheres This section contains three technical lemmas. Throughout this section, we assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1]. Denote S R ( p) := {x : |x − p| = R}. We can view S R ( p) as a submanifold in M with respect to either the physical metric g or the Euclidean metric ge . Because g is asymptotic to Euclidean metric near infinity, the induced metric on S R ( p) is close to the standard spherical metric, for R large. Hence, the geometric quantities on S R ( p) are close to those on the standard sphere, up to the error terms. In order to construct constant mean curvature surfaces, we need to compute explicitly the leading order terms in the error terms and also estimate the rest of the terms. In the first lemma, the mean curvature of S R ( p) with respect to g is derived. Its mean curvature after integration with x α − p α gives the difference of p and center of mass C. The estimates on the second fundamental form, Laplacian, and Ric(νg , νg ) on S R ( p) are obtained in the second lemma. The analogous estimates for surfaces close to S R ( p) are derived in the third lemma. If f is a function defined on S R ( p), we define f odd (x) = f (x) − f (−x + 2 p) and f even (x) = f (x) + f (−x + 2 p), where x and −x + 2 p are antipodal points on S R ( p). Also, h i j denotes gi j − δi j . Lemma 2.1. Let HS be the mean curvature of S R ( p) and dσe be the area form of the standard spherical metric. Then HS (x) =
2 1 (x i − pi )(x j − p j )(x k − p k ) + h i j,k (x) R 2 R3 i, j,k
+2
i, j
h i j (x)
(x i − pi )(x j − p j ) x j − pj − h (x) i j,i R3 R i, j
1 x j − p j h ii (x) − + E 0 (x), + h ii, j (x) 2 R R i, j
(2.1)
i
where E 0 (x) = O(R −1−2q ) and E 0odd (x) = O(R −2−2q ). For α = 1, 2, 3, 2 α α dσe = 8π m( p α − C α ) + O(R 1−2q ). (x − p ) HS − R S R ( p)
(2.2)
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Proof. Let ∇ be the covariant derivative of g. HS = divg νg , where νg is the outward unit normal vector field on S R ( p) with respect to g and νg =
∇|x − p| . |∇|x − p||g
Computing directly, we have νg = 1 +
s,t
−
k,l
(x s − p s )(x t − p t ) x l − pl ∂ h st (x) 2 R2 R ∂ xl
1
l
x k − pk ∂ h kl (x) + E(x), R ∂ xl
(2.3)
where E(x) = O(R −2q ) and E even (x) = O(R −1−2q ). Then a straightforward computation gives (2.1). To prove (2.2), we let f (x) = HS − 2/R. First we notice that the leading order term of f (x) is even and vanishes after integration with the odd function x α − p α . Moreover, the error term E 0 after integration with (x α − p α ) is of lower order O(R 1−2q ). We define, for α = 1, 2, 3, ⎡ ⎤ i − p i )(x j − p j )(x k − p k ) 1 (x ⎦ dσe . Igα (R) = (x α − p α ) ⎣ h i j,k (x) 2 R3 S R ( p) i, j,k
Because the asymptotically flat coordinates are not globally defined in the interior, we use the Euclidean divergence theorem in the annulus A = {R ≤ |x − p| ≤ R1 }: (x j − p j )(x k − p k )(x α − p α ) dx |x − p|2 A i, j,k ,i j (x − p j )(x k − p k )(x α − p α ) 1 h i j,k (x) dx = 2 A |x − p|2 ,i i, j,k (x j − p j )(x k − p k )(x α − p α ) 1 h i j,i (x) dx + 2 A |x − p|2 ,k i, j,k j (x − p j )(x k − p k )(x α − p α ) 1 h i j,i (x) d x. − 2 A |x − p|2 ,k
Igα (R1 ) − Igα (R) =
1 2
h i j,k (x)
i, j,k
Using integration by parts and simplifying the expression, we obtain an identity containing purely the boundary terms Igα (R1 ) − Igα (R) = Bgα (R1 ) − Bgα (R)
for all R1 ≥ R,
(2.4)
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where Bgα (R) equals the boundary integral: 1 x j − pj (x i − pi )(x j − p j ) α α dσe (x − p ) h i j,i (x) − 2h i j (x) 2 R R3 S R ( p) i, j 1 x α − pα x i − pi h ii (x) + h iα (x) dσe . + R R S R ( p) 2 i
Claim: Igα (R) = Bgα (R). Proof. First notice that if g = u 4 δ outside a compact set, then by direct computation and (2.4), for any R1 large (so that g = u 4 δ outside B R1 ( p)), Igα (R) − Bgα (R) = Igα (R1 ) − Bgα (R1 ) = 0
for α = 1, 2, 3.
To prove the identity for general metrics, we apply Theorem 4 and would like to show that, given 0 > 0, there exists g so that, for some R1 , |Igα (R1 ) − Bgα (R1 )| ≤ |Igα (R1 ) − Bgα (R1 )| + 0 = 0 .
(2.5)
We denote symbolically |D(g − g)|r dσe = Igα (r ) − Bgα (r ) − Igα (r ) − Bgα (r ) . Sr ( p)
Then by Hölder’s inequality, 2R |D(g − g)|r dσe dr ≤ C(g, q, p)g − gW 2, p (M) R 3−q . −q
Sr ( p)
R
That means, for a.e. r ∈ (R, 2R), say r = R1 , that |D(g − g)|R1 dσe ≤ C(g, q, p)g − gW 2, p (M) R 3−q . −q
S R1 ( p)
Given = 0 /(C(g, q, p)R 3−q ), there exists g so that g − gW 2, p (M) ≤ by Theo−q
rem 4. Hence
S R1 ( p)
|D(g − g)|R1 dσe ≤ 0 ,
and then (2.5) holds. Because 0 is arbitrary, we prove the claim. Igα (R)
Bgα (R)
Then, substituting by into (2.1) and (2.2), and simplifying the expression, we have 2 α α dσe (x − p ) HS − R S R ( p) ⎡ 1⎣ x j − pj dσe =− (x α − p α ) (h i j,i − h ii, j ) 2 R S R ( p) i, j ⎤ x i − pi x α − pα h iα − h ii dσe ⎦ + O(R 1−2q ). − R R S R ( p) i
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Using the definitions of the ADM mass (1.3) and center of mass (1.5), we derive (2.2). In the following lemmas, c denotes a constant independent of R. Also, we denote f ∗ to be the pullback of f defined by f ∗ (y) = f (Ry + p), so f ∗ is a function on S1 (0). Also, define ( f ∗ )odd = f ∗ (y) − f ∗ (−y), ( f ∗ )even = f ∗ (y) + f ∗ (−y). Lemma 2.2. Let A S be the second fundamental form on (S R ( p), g S ) where g S is the induced metric on S R ( p) from g, S be the Laplacian on (S R ( p), g S ), and νg be the outward unit normal vector. Let eS be the standard spherical Laplacian on S R ( p). Then (i) |A S |2 =
2 R2
+ E 1 , where |E 1 | ≤ c R −2−q and |E 1odd | ≤ c R −3−q .
(ii) For any f ∈ C 2,α (S R ( p)), S f = eS f + E 2 , where |E 2 | ≤ c R −2−q f ∗ C 2 (S1 (0)) and |E 2odd | ≤ c R −3−q f ∗ C 2 (S1 (0)) + R −2−q ( f ∗ )odd C 2 (S1 (0)) . (iii) Ric M (νg , νg ) = E 3 , where |E 3 | ≤ c R −2−q and |E 3odd | ≤ c R −3−q . Proof. Let {u 1 , u 2 } be local coordinates on S R ( p) and ∇ be the covariant derivative of (M, g). In the rest of the section, we temporarily denote gab = g (∂a , ∂b ) for a, b ∈ {1, 2, 3}, where ∂a = ∂u∂ a if a ∈ {1, 2} and ∂3 = νg (instead of the original meaning of {gi j } on the asymptotically flat coordinates in Definition 1.1). Therefore, the second fundamental form A S is (A S )ab
= −g ∇
∂ ∂u a
∂ , νg ∂u b
3 = − ab .
(2.6)
Because g is asymptotically flat, g(x) = ge + h and h = O(|x|−q ). Locally, we have 3
ab =
∂ 1 ga3,b + gb3,a − gab,3 = ge ∇ e ∂ , νe + |h∂h| + |∂h|, 2 ∂u a ∂u b
3 and g (∇ e where we denote the difference of ab e ∂
∂u a
(2.7)
∂ ∂u b , νe ) symbolically by |h∂h|+|∂h|,
where ∇ e is the covariant derivative and the Christoffel symbols of (M\B R0 , ge ) and ∂ denotes the derivative in either tangential or normal directions on S R ( p). Remark. More precisely, writing f = |∂h| symbolically means | f | ≤ c|∂h|, | f even | ≤ c|(∂h)even |, | f odd | ≤ c|(∂h)odd |. The constant c is independent of R. Notice that the derivatives in the tangential and normal directions do not affect the asymptotic even/odd property, but only improve the decay rate. For example, if h = O(|x|−q ) and h odd = O(|x|−1−q ), then ∂h = O(|x|−1−q ) and ∂h is still asymptotically even at the decay rate (∂h)odd = O(|x|−2−q ). In the following arguments, we will use similar notations to bound lower order terms for simplicity.
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The second fundamental forms are (A S )ab = Ae ab + |h∂h| + |∂h|. Therefore, if the principal curvature of (S R ( p), g S ) are denoted by (λ S )i , the above identity says: (λ S )i =
1 + |h∂h| + |∂h|, R
(2.8)
where 1/R is the principal curvature of the spheres in Euclidean space. Then |A S |2 = (λ S )21 + (λ S )22 =
2 1 + (|h∂h| + |∂h|) + (|h∂h| + |∂h|)2 . R2 R
We could conclude (i) by analyzing the error terms on the right-hand side and by using the AF–RT condition. Using g = ge + h, the Laplacian in the local coordinates is ∂ √ ij ∂ gg f S f = g ∂u i ∂u j = eS f + |h||∂g||∂ f | + |h||∂ 2 f | + |∂h||∂ f | . √
−1
(2.9)
By the definition of f ∗ , |∂ f (x)| = R −1 |∂ f ∗ (y)| and |∂ 2 f (x)| = R −2 |∂ 2 f ∗ (y)|, and then (ii) follows. For (iii), notice that Ric M (νg , νg ) = |D 2 g|, where Dg denotes the usual deriva tives of g in { ∂∂xi } directions as in Definition 1.1. Therefore, |D 2 g| = O |x|−2−q and 2 odd 2 odd = O(|x|−3−q ). (D g) = D g In the following lemma, we generalize the above results and prove that similar estimates also hold for surfaces which are c R 1−q -graphs over S R ( p) for some constant c (recall q ∈ (1/2, 1], the decay rate of the AF metrics). Notice that when R is large, the unit normal vector νg is close to the Euclidean normal vector, so the normal graphs over S R ( p) are well-defined. Let N be a normal graph over S R ( p) defined by
N = (x) = x + ψνg : ψ ∈ C 2 (S R ( p)) . For any f ∈ C 2 (N ), we let f (x) := f ((x)) and f ∗ := ( f )∗ , the pull-back function defined on S1 (0). Let μg be the outward unit normal vector field on N , A N be the second fundamental form, and N be the Laplacian on (N , g N ), where g N is the induced metric on N by g. Lemma 2.3. Assume that ψ ∗ C 2 (S1 (0)) ≤ c R 1−q and (ψ ∗ )odd C 2 (S1 (0)) ≤ c R −q .
(2.10)
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Then 2 + E 1 where |E 1 | ≤ c R −2−q and |(E 1 )odd | ≤ c R −3−q . R2 f (x) + E 2 , For f ∈ C 2 (N ), ( N f )((x)) = eS where |E 2 | ≤ c R −2−q f ∗ C 2 (S1 (0)) and |(E 2 )odd | ≤ c R −2−2q f ∗ C 2 (S1 (0)) + R −2−q ( f ∗ )odd C 2 (S1 (0)) . Ric M (μg , μg ) ((x)) = E 3 , where |E 3 | ≤ c R −2−q and |(E 3 )odd | ≤ c R −3−q .
(i) |A N |2 = (ii)
(iii)
Proof. Similarly as in the proof of Lemma 2.2, let {u 1 , u 2 } be local coordinates on an open set U of x ∈ S R ( p). Moreover, without loss of generality, we assume { ∂u∂ 1 , ∂u∂ 2 , νg } are orthonormal at x with respect to the metric g. Let {v1 , v2 } be the corresponding local coordinates on V = (U ) ⊂ N and μg be the outward unit normal vector field on N with respect to g. Because M is AF, up to lower order terms, we have ∂ ∂ ∂ ∂ψ = + (A S )i j ψ + νg , ∂vi ∂u i ∂u j ∂u i ∂ψ ∂ μg = νg + ψ HS νg − , ∂u i ∂u i
(2.11) (2.12)
i=1,2
where we parallel transport ∂v∂ 1 , ∂v∂ 2 , μg to x along the unique geodesic connecting x and (x). In this proof, we denote g ab = g(ea , eb ) gab = g(ea , eb )
∂ if a ∈ {1, 2} and e3 = μg , ∂va ∂ where ea = if a ∈ {1, 2} and e3 = νg , ∂u a where ea =
where gab is defined the same as in the proof of the previous lemma. By (2.11) and (2.12), we have for i ∈ {1, 2}, a, b ∈ {1, 2, 3}, g ia = gia + |ψ||A S ||g| + |∂ψ||g|, g ia,b = gia,b + |∂ψ||A S ||g| + |ψ||∂ A S ||g| + |ψ||A||∂g| +|∂ 2 ψ||g| + |∂ψ|2 |∂g|.
(2.13)
To prove (i), notice that (A N )i j = −g ∇
∂ ∂vi
∂ , μg ∂v j
3
= − i j
and 3
1 g a3,b + g b3,a − g ab,3 2 ∂ , νg =g ∇ ∂ ∂u a ∂u b
ab =
+|∂ψ||A S ||g| + |ψ||∂ A S ||g| + |ψ||A S ||∂g| + |∂ 2 ψ||g| + |∂ψ|2 |∂g|.
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Therefore, by (2.8) and the previous two identities, we get |A N |2 = |A S |2 +
1 |∂ψ||A S ||g| + |ψ||∂ A S ||g| R
+|ψ||A||∂g| + |∂ 2 ψ||g| + |∂ψ|2 |∂g| .
Above, the terms of the weakest decay rate in the error terms are, for instance, 1 2 |∂ ψ||g| = O(R −2−q ). R Similarly, we could compute (E 1 )odd and use Lemma 2.2(i) to conclude (i). Moreover, we can derive from the above two identities to conclude that the trace-free second fundamental form is | A˚ N | = O(R −1−q ),
(2.14)
and the mean curvature of N is HN =
2 + O(R −1−q ). R
(2.15)
For (ii), the Laplacian in local coordinates is ∂ ij ∂ g¯ g¯ f ((x)) ∂vi ∂v j ∂ √ ij ∂ √ = g −1 gg f ((x)) + |∂ψ||A S ||g||∂ f | ∂u i ∂u j
( N f )((x)) =
g¯
−1
+ |ψ||∂ A S ||g||∂ f | + |ψ||A S ||g||∂g||∂ f | + |ψ||A S ||g||∂ 2 f |. Then ( N f )((x)) = S f| f (x) + |∂ψ||A S ||g||∂ + |ψ||∂ A S ||g||∂ f | + |ψ||A S ||g||∂g||∂ f | + |ψ||A S ||g||∂ 2 f |, where the terms at the weakest decay rate of the error terms are, for instance, |∂ψ||A S ||g||∂ f (x)| ≤ R −1 |∂ψ||A S ||g||∂ f ∗ (x)| ≤ C R −2−q f ∗ C 2,α . Then, (ii) follows from Lemma 2.2 (ii). Using Lemma 2.2(iii) and the identity Ric M (μg , μg )((x)) = Ric M (νg , νg ) + |D 2 g||ψ||A S | + |D 2 g||∂ψ|, we can conclude (iii).
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3. Existence of the Foliation In this section, we prove the existence of the foliation of constant mean curvature surfaces, assuming the ADM mass m > 0. An idea similar to [17] is employed in which normal perturbations of Euclidean spheres are considered. However, our construction is more subtle because we have to perturb a Euclidean sphere S R ( p) twice to construct a constant mean curvature surface. Roughly speaking, the first perturbation is of the order O(R 1−q ) and the second one is of the order O(R 1−2q ). Geometrically, it reflects the fact that, under weaker asymptotics, constant mean curvature surfaces are too far away from some S R ( p) to apply the implicit function theorem directly. Therefore, we have to construct a family of approximate spheres S( p, R) from S R ( p) using a PDE construction. Then by carefully choosing the center p, we find the nearby constant mean curvature surfaces from S( p, R). While we only require m = 0 in proving Theorem 3.1, assuming m > 0 is used to prove the stability of the surfaces and then to show that they form a foliation. From our construction, each leaf of the foliation is a graph over the Euclidean sphere centered at some p = p(R). We also show that p converges to the center of mass C as R → ∞. Throughout this section, c = c(α, g, ∂g) or ci = ci (α, g, ∂g) denote constants independent of R. Recall that if ψ ∈ C 2,α (S R ( p)), then ψ ∗ (y) := ψ(Ry + p) and ψ ∗ ∈ C 2,α (S1 (0)), and define (ψ ∗ )odd = ψ ∗ (y) − ψ ∗ (−y), (ψ ∗ )even = ψ ∗ (y) + ψ ∗ (−y). The first theorem states the existence of a surface with the given constant mean curvature. Theorem 3.1. Assume that (M, g, K ) is AF–RT with q ∈ (1/2, 1] and m = 0. There exist constants σ0 and c0 so that, for all R > σ0 , there is R with constant mean curvature H R =
2 + O(R −1−q ). R
R is a c0 R 1−q -graph over S R ( p), i.e.
R = x + ψ0 νg : ψ0 ∈ C 2,α (S R ( p)) and ψ0 satisfies ψ0∗ C 2,α (S1 (0)) ≤ c0 R 1−q , and (ψ0∗ )odd C 2,α (S1 (0)) ≤ c0 R −q .
(3.1)
Because the mean curvature of S R ( p) is equal to 2/R up to O(R −1−q )-terms (2.1), we would like to construct a constant mean curvature surface by perturbing S R ( p) in the normal direction. However, in contrast to the case that (M, g) is strongly asymptotically flat, the mean curvature of S R ( p) is not close to some constant enough to apply the implicit function theorem. Therefore, we first construct the unique approximate spheres S( p, R) associated to S R ( p) whose mean curvature is closer to some constant up to O(R −1−2q )-terms. Recall that f denotes HS − 2/R, and HS is the mean curvature of S R ( p).
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Lemma 3.2. There exists c independent of R so that, for R large, there is an approximate sphere
S( p, R) = x + φ(x)νg : φ ∈ C 2,α (S R ( p)) , where φ satisfies φ ∗ C 2,α (S1 (0)) ≤ c R 1−q , (φ ∗ )odd C 2,α (S1 (0)) ≤ c R −q .
(3.2)
Moreover, the mean curvature of S( p, R) is HS = where f := (4π R 2 )−1
S R ( p)
2 + f + O(R −1−2q ), R
(3.3)
f dσe .
Remark. When q = 1, φ is bounded by a constant. However, when q < 1, the size of φ may increase as R increases. Proof. L 0 = − eS − 2/R 2 denotes the linearized mean curvature operator on the standard sphere S R ( p) in Euclidean space , where eS is the standard spherical Laplacian. It is known that because mean curvature is preserved by translations in the Euclidean space, L 0 has the kernel K = span{x 1 − p 1 , x 2 − p 2 , x 3 − p 3 }. Also notice that, by the self-adjointness of L 0 , the L 2 orthogonal complement K⊥ = RangeL 0 . Let L 0 : C 2,α (S R ( p)) → C 0,α (S R ( p)). Consider L 0 φ = f − R −3−q Ai (x i − pi ) − f .
(3.4)
i
We choose the constants Ai to satisfy 3 −1+q R A = 4π
i
S R ( p)
(x i − pi ) f (x) dσe ,
(3.5)
so the right-hand side of (3.4) is in RangeL 0 and then (3.4) is solvable. Notice that because of the AF–RT condition, Ai = O(1). We let φ be the unique solution in K⊥ to Eq. (3.4). To estimate φ ∗ , note that it satisfies (− 0 − 2)φ ∗ = R 2 ( f ∗ (y) − R −2−q Ai y i − f ), i
where 0 is the standard spherical Laplacian of the unit sphere in Euclidean space. Because φ ∗ ∈ (Ker(− 0 − 2))⊥ , by the Schauder estimate and because f = |Dh| = O(R −1−q ), φ ∗ C 2,α (S1 (0)) ≤ cR 2 f ∗ (y) − R −q Ai y i − R 2 f C 0,α (S1 (0)) ≤ c R 1−q . i
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Moreover, (φ ∗ )odd satisfies the following equation (− 0 − 2)(φ ∗ )odd = R 2 ( f ∗ )odd − 2R −q
Ai y i .
i
Then, because (φ ∗ )odd ∈ (Ker(− 0 − 2))⊥ , by the Schauder estimates and the fact that f is asymptotically even with f odd = O(R −2−q ), we have (φ ∗ )odd C 2,α (S1 (0)) ≤ cR 2 ( f ∗ )odd − R −q i Ai y i C 0,α (S1 (0)) ≤ c R −q . (3.6) Then we define ! S( p, R) = x + φνg . In particular, S( p, R) is a graph over S R ( p) which satisfies the conditions for N in Lemma 2.3. We compute the mean curvature of S( p, R). Denoting HS the mean curvature map HS : C 2,α (S R ( p)) → C 0,α (S R ( p)) which maps a function φ to the mean curvature of the normal graph of φ over S R ( p). Then the mean curvature of S( p, R) is HS (φ). By Taylor’s theorem, 1 HS (φ) = HS (0) − L S φ + (d HS (sφ) − d HS (0)) φ ds, 0
where d HS is the first Fréchet derivative in the φ-component, and L S is the linearized mean curvature operator on S R ( p) defined by L S = − S − |A S |2 − Ric M (νg , νg ), where S , A S , and Ric M (νg , νg ) are defined by Lemma 2.2. The integral term above can be bounded by sups∈[0,1] d 2 HS (sφ)φφ by the mean value inequality, and ∂2 d 2 HS (sφ)φφ = 2 HS (tφ) . ∂t t=s The left-hand side is the second Fréchet derivative and the right-hand side is the second derivative of the mean curvature of the surface ! Ns := x + sφ(x)νg : y ∈ S R ( p) . For R large, the unit outward normal vector field on Ns is close to νg , and a straightforward calculation gives us 2 ∂ ≤ c Ri jkl A N |φ|2 + |A N ||∂φ|2 + A N |φ||∂ 2 φ| + A N 3 |φ|2 H (tφ) S s s s s ∂t 2 2 ≤ c R −3 φ ∗ C 2 (S
1 (0))
.
(3.7)
In the last inequality, we use that |Ri jkl | = O(R −2−q ) and |A Ns | = O(R −1 ) from Lemma 2.3.
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Noticing that HS (0) is the mean curvature of S R ( p), so, by Lemma 2.1, we have HS (φ) =
2 + f (x) − L 0 φ + (L 0 − L S )φ + R
1
(d HS (sφ) − d HS (0) φ ds.
0
By (3.4), HS (φ) =
2 + f + R −3−q Ai (x i − pi ) + E 4 , R
(3.8)
i
where
1
E 4 = (L 0 − L S )φ +
(d HS (sφ) − d HS (0)) φ ds.
0
By Lemma 2.2, (3.2), and (3.7), the error term E 4 is bounded by 2 E 4∗ C 0,α ≤ c R −q φ ∗ C 2,α + R −1 φ ∗ C ≤ c R 1−2q , 2,α (E 4∗ )odd C 0,α ≤ c R −1−q φ ∗ C 2,α + R −q (φ ∗ )odd C 2,α 2 −1 + R −2 φ ∗ C (φ ∗ )odd C 2,α φC 2,α 2,α + R ≤ c R −2q . Therefore, we derive (3.3).
(3.9)
Proof of Theorem 3.1. To construct a surface R with constant mean curvature, we consider the normal perturbations on S( p, R) := {(x) = x + φνg }. We denote the mean curvature of the normal graph ψ over S( p, R) by HS (ψ). By Taylor’s theorem, for any ψ ∈ C 2,α (S( p, R)), HS (ψ) = HS (0) + S ψ + |AS |2 + Ric M (μg , μg ) ψ 1 + (3.10) (d HS (sψ) − d HS (0)) ψ ds, 0
where S , AS , and μg are defined as in Lemma 2.3 for which we let N = S( p, R), and ψ ∗ denote the pull-back functions on S R ( p) and S1 (0) respectively. By (3.8) and ψ and (3.10), solving HS (ψ) =
2 + f R
is equivalent to solving ψ to the following equation: Ai (x i − pi ) + E 4 + S ψ + |AS |2 + Ric M (μg , μg ) ψ 0 = R −3−q i
1
+ 0
(d HS (sψ) − d HS (0)) ψ ds.
(3.11)
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That is, to solve = R −3−q L 0ψ
Ai (x i − pi ) + E 4 + E 5 ,
(3.12)
i
where E 5 (x) = ( S ψ) ◦ (x) − eS ψ 2 2 M + |AS | ((x)) − 2 + (Ric (μg , μg )) ◦ (x) ψ R 1 + [(d HS (sψ) − d HS (0)) ψ] ◦ (x) ds. 0
Using Lemma 2.3 and (3.7), we have 2 E 5∗ C 0,α ≤ c R −q ψ ∗ C 2,α + R −1 ψ ∗ C 2,α " " " " (E 5∗ )odd C 0,α ≤ c R −2q ψ ∗ C 2,α + R −q "(ψ ∗ )odd " 2,α " C " " " −2 ∗ 2 −1 ∗ + R ψ C 2,α + R ψ C 2,α "(ψ ∗ )odd "
C 2,α
We pull back (3.12) on S1 (0), #
(− 0 − 2)ψ ∗ = R 2 R −2−q
(3.13) .
$ Ai y i + E 4∗ + E 5∗
=: F( p, R, ψ ∗ ).
(3.14)
i
If ψ ∗ C 2,α ≤ 1, then by (3.9) and (3.13), F( p, R, ψ ∗ )C 0,α (S1 (0)) ≤ c R 1−2q , (F( p, R, ψ ∗ ))odd C 0,α (S1 (0)) ≤ c R −2q .
(3.15)
In order to find a solution ψ ∗ to the above equation, a necessary condition is that F( p, R, ψ ∗ ) lies inside Range(− 0 − 2). Using m = 0, we show this can be achieved by correctly choosing p = p(R, ψ ∗ ). By the definition of Ai (3.5), we have y α R −2 F( p, R, ψ ∗ ) dσe S1 (0) # $ α −2−q i i ∗ ∗ = y A y + E 4 + E 5 dσe R = =
S1 (0)
S1 (0) S R ( p)
i
y α E 4∗ + E 5∗ dσe S1 (0) x α − pα f (x)R −2 dσe + y α E 4∗ + E 5∗ dσe . R S1 (0)
y α f ∗ (y) dσe +
Using (2.2) in Lemma 2.1, the first integral is equal to 8π m p α − C α R −3 + O(R −2−2q ).
(3.16)
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Therefore, because m = 0, we can choose R3 y α E 4∗ + E 5∗ dσe + O(R 1−2q ), p α (R, ψ ∗ ) = C α − 8π m S1 (0)
(3.17)
such that (3.16) is zero; that is, F( p(R, ψ ∗ ), R, ψ ∗ ) ∈ Range(− 0 − 2). To complete the proof, we apply the Schauder fixed point theorem. Although F( p(R, ψ ∗ ), R, ψ ∗ ) contains also the second order derivatives of ψ ∗ from the error term E 5 , those second derivatives are quasi-linear and have small coefficients. We can rewrite (3.14) as, for R large, (− 0 − 2)ψ ∗ + ai j (x, ψ ∗ )∂i2j ψ ∗ = F( p(R, ψ ∗ ), R, ψ ∗ ), where ai j = O(R −q ) if ψ ∗ C 0 ≤ 1. Therefore, Lψ ∗ = (− 0 − 2)ψ ∗ + ai j (y, ψ ∗ )∂i2j ψ ∗ is a quasi-linear elliptic operator for R large. Define B = C 2,α (S1 (0)) ∩ {v : vC 2,α (S1 (0)) ≤ 1}. Let T : B → C 2,α (S1 (0)) be T (v) = u, where u is the unique solution in (KerL)⊥ to the linear equation (− 0 − 2)u + ai j (y, v)∂i2j u = F( p(R, v), R, v) By the Schauder estimates and (3.15), uC 2,α (S1 (0)) ≤ cFC 0,α (S1 (0)) ≤ c R 1−2q . For R large enough, the right-hand side is less than 1, so T is a map from B to itself. It is easy to check that T is compact and continuous by the standard linear theory. Therefore, the Schauder fixed point theorem applies, and there is a fixed point ψ ∗ to (3.14). Using the Schauder estimates and (3.14) to (ψ ∗ )odd , L 0 (ψ ∗ )odd = F odd . Therefore, ψ ∗ C 2,α (S1 (0)) ≤ ∗ odd
(ψ )
1 c0 R 1−2q , 2
C 2,α (S1 (0)) ≤ cF
odd
C 0,α (S1 (0))
(3.18)
1 ≤ c0 R −q . 2
(x) = ψ ∗ x− p , ψ is a solution to the identity (3.12). We let ψ((x)) = By letting ψ R ψ (x), then the graph of ψ over S( p, R) has constant mean curvature (2/R)+ f . Because μg is close to νg by (2.12), we can rearrange and write R as a graph over S R ( p),
R = x + ψ0 νg : ψ0 ∈ C 2,α (S R ( p)) . + O(R −q ), by (3.2) and (3.18), we derive (3.1). Because ψ0 = φ + ψ
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In [10], the geometric center of a constant mean curvature foliation is defined: Definition 3.3. Let { R } be the family of surfaces constructed in the previous theorem and X be the position vector. The geometric center of mass of (M, g, K ) is defined by, for α = 1, 2, 3, α
X dσe α C H Y = lim R . R→∞
R dσe From our construction, we not only prove that the geometric center converges, but we also show that it is equal to center of mass C. The following corollary generalizes the results in [7,9]. Corollary 3.4. Assume (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1] and m = 0. Then C H Y converges and is equal to C. Proof. Let be the diffeomorphism from S R ( p) to R defined by (x) = x + ψ0 νg . Then by the definition and the area formula, α + ψ ν α J dσ α dσ x 0 e X g e S R ( p)
R =
R dσe S ( p) J dσe R 1−2q ) dσ e S ( p) O(R = pα + R . S R ( p) dσe In the second identity, we use (3.1) and J = 1 + O(R −q ), so the second term in the last line is of lower order and vanishes after taking limits. We only need to study the limit of p. By (3.17), we estimate the error terms E 4∗ and E 5∗ in (3.17). By the asymptotically even/odd properties of E 4∗ in (3.9), α ∗ α ∗ odd ≤ y E dσ y (E ) dσ e e 4 4 S1 (0) S1 (0) (x α − p α ) ≤ (E 4 )odd R −2 dσe R S R ( p) ≤ c R −2 sup (E 4 )odd ≤ c R −2−2q . S R ( p)
Similarly, by (3.13) and (3.18), α ∗ y E 5 dσe ≤ c R −2 sup S R ( p) E 5odd ≤ c R −2−2q . S1 (0)
From (3.17), we derive p α = C α + O(R 1−2q ). After taking limits, we prove the corollary.
Because p is asymptotic to C, we can rearrange R to be graphs over S R (C).
(3.19)
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Corollary 3.5. The constant mean curvature surfaces R constructed in Theorem 3.1 are c0 R 1−q -graph over S R (C), i.e.
R = x + ψ0 νg : ψ0 ∈ C 2,α (S R (C)) , and ψ0 satisfies ψ0∗ C 2,α (S1 (0)) ≤ c0 R 1−q , (ψ0∗ )odd C 2,α (S1 (0)) ≤ c0 R −q . After constructing the family of surfaces with constant mean curvature { R }, we prove that they form a smooth foliation. We first estimate the eigenvalues of the linearized mean curvature operator. Definition 3.6. A smooth hypersurface N in M is called stable if the linearized mean curvature operator L N := − N − |A N |2 + Ric M (μg , μg ) has the lowest eigenvalue μ0 ≥ 0 among functions with zero mean value, i.e. % & μ0 := inf u L N u dσ : u L 2 (N ) = 1, u dσ = 0, and u ≡ 0 ≥ 0. N
(3.20)
N
If μ0 is strictly positive, N is called strictly stable. Remark. If N has constant mean curvature, N being stable means that N locally minimizes area among surfaces containing the same volume. The following two lemmas hold for more general surfaces. Lemma 3.7. Assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1] and m > 0. Let N be a normal graph of ψ over S R ( p):
N = (x) = x + ψνg : ψ ∈ C 2 (S R ( p)) , where ψ satisfies (2.10) in Lemma 2.3. For R large, N is strictly stable and the lowest eigenvalue μ0 ≥
6m + O(R −2−2q ). R3
Proof. Let L 0 = − eS − R22 be the linearized mean curvature operator of standard spheres of radius R in Euclidean space. L 0 has kernel K: & % 1 x − p1 x 2 − p2 x 3 − p3 . K = span , , R2 R2 R2 By Lemma 2.3 and recalling that u (x) = u((x)) ∈ C 2 (S R ( p)), for any u ∈ C 2 (N ), 2 (L N u) ((x)) − − e ≤ c R −2−q u ∗ C 2 (S (0)) . u − u S 1 2 R
Foliations by Stable Spheres with Constant Mean Curvature
351
Now, we normalize u to satisfy u L 2 (N ) = 1, and it implies u = O(R −1 ) because the area |N | = 4π R 2 + O(R 2−q ). Then by the area formula and (2.10), u L N u dσ = u L N u dσe + O(R −2−2q ) N N u (L N u)((x))J dσe + O(R −2−2q ) = S R ( p) = u L 0 u dσe + O(R −2−2q ). S R ( p)
Therefore, the infimum of (3.20) is achieved by u satisfying u ∈ K, up to lower order terms. We claim that, if u satisfies that u ∈ K and u L 2 (N ) = 1, then 3 u L N u dσ ≥ − Ric M (μg , μg ) dσ + O(R −2−2q ). (3.21) 4π R 2 N N Let u satisfy the assumption of the claim. By Lemma 2.3, − N u =
2 u + E 2 , R2
(3.22)
where ' E 2 = O(R −3−q ),
and
odd ' E 2 = O(R −3−q ).
(3.23)
Multiply (3.22) by u and integrate over N . Because u is an odd function with respect to the center p, 2 |∇ N u|2 dσ = 2 + O(R −2−q ). (3.24) R N Then notice that, by (2.14) and (2.15), H2 2 2 |A N | = N + | A˚ N |2 = 2 + 2 R R 2
2 HN − R
+ O(R −2−2q ).
By the definition of L N , 2 2 2 HN − u 2 dσ u L N u dσ = |∇ N u|2 dσ − 2 − R R N n N R − Ric M (μg , μg )u 2 dσ + O(R −2−2q ).
(3.25)
N
If we substitute the gradient term in the right-hand side by (3.24), it eliminates the second term 2/R 2 . However, we do not know the sign of the remainders which are still of higher order O(R −2−q ). Therefore, we have to derive a better estimate on the gradient term to cancel out the third term. Recall the Bochner–Lichnerowicz identity: 1 N |∇ N u|2 = (Hess N u)2 + ∇ N u, ∇ N N u + K(∇ N u, ∇ N u) 2 ( N u)2 + ∇ N u, ∇ N N u + K(∇ N u, ∇ N u), ≥ 2
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L.-H. Huang
where K is the Gauss curvature of N . After integrating the above inequality, the left-hand side vanishes because N is a compact manifold without boundary. Then using (3.22) and (3.23), 1 R2 N 2 |∇ u| dσ ≥ 2 + K|∇ N u|2 dσ + O(R −2−2q ). R 2 N N Using the Gauss equation, (2.14) and (2.15), 1 1 2 (H − |A|2 ) − Ric M (μg , μg ) + Rg 2 N 2 1 1 2 HN − − Ric M (μg , μg ) + O(R −2−2q ), = 2+ R R R
K=
where we use that Rg = O(R −2−2q ) by the constraint equations (1.2). Hence, 2 2 R N 2 N 2 HN − |∇ u| dσ |∇ u| dσ ≥ 2 + R R 2 N N R2 − Ric M (μg , μg ) |∇ N u|2 dσ + O(R −2−2q ). 2 N Substituting the above inequality back to (3.25), we have, for any u satisfying that u∈K and u L 2 (N ) = 1, R N 2 2 2 2 HN − |∇ u| − u dσ u L N u dσ ≥ R 2 R N N 2 R M N 2 2 − |∇ u| + u dσ + O(R −2−2q ). Ric (μg , μg ) 2 N In particular, we choose vi , for i = 1, 2, 3, to satisfy ( 3 x i − pi . v i = 4π R 2 Then, for each i, because |∇ e v i |2 =
3 v i 2 − , 4π R 4 R2
we get |∇ N vi |2 = Hence,
vi2 3 − + O(R −4−q ). 4π R 4 R2
vi L N vi dσ ≥ N
3 3 2 dσ v − 8π R 3 2R i N 3 1 2 dσ + O(R −2−2q ). v − Ric M (μg , μg ) + 8π R 2 2 i N HN −
2 R
Foliations by Stable Spheres with Constant Mean Curvature
Let u =
i
353
vi . Then, because i vi2 = 3/(4π R 2 ) + O(R −2−q ), u L N u dσ = vi L N vi dσ + O(R −2−2q ) N
i
N
3 ≥− 4π R 2
Ric M (μg , μg ) dσ + O(R −2−2q ). N
We prove the claim. To complete the proof, we use the alternative definition of the ADM mass (1.4) and obtain, Ric M (μg , μg ) dσ = Ric M (νe , νe ) dσe + O(R −1−q ) S R ( p)
N
=−
8π m + O(R −1−q ). R
In order to apply the inverse function theorem, we prove that L N is invertible. We show that the lowest eigenvalue of L N without any constraints is negative, and the next eigenvalue is strictly positive. Lemma 3.8. Assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1] and m > 0. Let N be a normal graph of ψ over S R ( p):
N = (x) = x + ψνg : ψ ∈ C 2 (S R ( p)) , where ψ satisfies (2.10) in Lemma 2.3. For R large, L N is invertible, and L −1 N : −1 R 3 . C 0,α (N ) → C 2,α (N ) satisfies |L −1 | ≤ cm N Proof. Let η0 be the lowest eigenvalue of L N without constraints. By Lemma 2.3, |∇ N u|2 − |A N |2 + Ric M (μg , μg ) u 2 dσ η0 = inf ! u L 2 =1
N
2 ≥ − 2 + O(R −2−q ). R On the other hand, if we replace u by a constant, we obtain the reverse inequality. Hence, η0 = −
2 + O(R −2−q ). R2
(3.26)
Let h 0 be the corresponding eigenfunction L N h 0 = η0 h 0 . 2 We show that h 0 is close to a constant and derive an L –estimate on the difference of −1 h 0 and its mean value h 0 := |N | N h 0 dσ , L N (h 0 − h 0 ) = η0 (h 0 − h 0 ) + η0 + |A N |2 + Ric M (μg , μg ) h 0 . (3.27)
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L.-H. Huang
Multiplying the above identity by (h 0 − h 0 ) and integrating it over N : η0 + |A N |2 + Ric M (μg , μg ) (h 0 − h 0 )2 dσ |∇ N (h 0 − h 0 )|2 dσ = N N + (η0 + |A N |2 + Ric M (μg , μg ))(h 0 − h 0 )h 0 dσ. N
Similarly as shown in the previous lemma, because h 0 − h 0 has zero mean value, the left-hand side is bounded below by −2−q 2 |∇ N (h 0 − h 0 )|2 dσ ≥ + O R |h 0 − h 0 |2 dσ. R2 N N Also, pointwisely η0 + |A N |2 + Ric M (μg , μg ) = O(R −2−q ).
(3.28)
Therefore, 2 2 −2−q 2 −2−q |h − h | dσ ≤ c R |h − h | dσ + c R |h 0 − h 0 ||h 0 | dσ. 0 0 0 0 R2 N N N Using the AM–GM inequality to the last integrand: 1 q R |h 0 − h 0 |2 + c R −q |h 0 |2 . 4c
|h 0 − h 0 ||h 0 | ≤ We obtain, when R is large,
h 0 − h 0 L 2 (N ) ≤ c R −q |h 0 ||N |1/2 .
(3.29)
In particular, h 0 = 0. Let η1 be the next eigenvalue with the corresponding eigenfunction h 1 . We show that η1 is positive and, moreover, η1 ≥ Note that
6m + O(R −2−2q ). R3
0=
h 0 h 1 dσ = N
(h 0 − h 0 )(h 1 − h 1 ) dσ + N
h 0 h 1 dσ. N
Then, by Hölder’s inequality, h 1 dσ ≤ |h 0 |−1 h 0 − h 0 L 2 (N ) h 1 − h 1 L 2 (N ) . N
Substituting (3.29) into the above inequality, we get h 1 ≤ c R −q |N |−1/2 h 1 − h 1 L 2 (N ) .
(3.30)
Because L N h 1 = η1 h 1 , (h 1 − h 1 )L N (h 1 − h 1 ) dσ = η1 (h 1 − h 1 )2 dσ N N h 1 (h 1 − h 1 ) η1 + |A N |2 + Ric M (μg , μg ) dσ. + N
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355
Because η1 + |A N |2 + Ric M (μg , μg ) = constant + (R −2−q ), and by Hölder’s inequality, the last integral is bounded above by c R −2−q |N |1/2 |h 1 |h 1 − h 1 L 2 (N ) .
(3.31)
By Lemma 3.7, (3.30), and (3.31), μ0 h 1 − h 1 2L 2 (N ) ≤ (η1 + c R −2−2q )h 1 − h 1 2L 2 (N ) . Therefore, η1 ≥ μ0 + c R −2−2q ≥ This finishes the proof.
6m + O(R −2−2q ). R3
The family of constant mean curvature surfaces { R } constructed in Theorem 3.1 satisfies the assumptions of N in the previous two lemmas. They imply that, in particular,
R is strictly stable and L R is invertible. In the next theorem, we use the invertibility of L R and the inverse function theorem to show that { R } form a smooth foliation. Theorem 3.9. Assume that (M, g, K ) is AF–RT at the decay rate q ∈ (1/2, 1] and m > 0. Let { R } be the family of surfaces with constant mean curvature constructed in Theorem 3.1. Then { R } form a smooth foliation in the exterior region of M. Proof. Let H : C 2,α ( R1 ) → C 0,α ( R1 ) be the mean curvature map so that H(u) is the mean curvature of the normal graph of u over R1 . Because dH = −L R1 is a linear isomorphism by Lemma 3.8, H is a diffeomorphism from a neighborhood U of 0 ∈ C 2,α ( R1 ) to a neighborhood V of H(0) by the inverse function theorem. By our construction of { R }, for R close to R1 , { R } are the unique constant mean curvature surfaces in a neighborhood of R . Moreover, { R } vary smoothly in R. To show that { R } form a foliation, we need to prove that R and R1 have no intersection for any R = R1 . First, when R is close to R1 and R is the graph of u for u ∈ U , we show that u has a sign; in particular, u cannot be zero. In the following, we denote R1 by . By the Taylor theorem, for any u ∈ U , 1 H(u) = H(0) − L u + (dH(su) − dH(0)) u ds, 0
where H(u) and H(0) are constants. By integrating the above identity over , H(0) − H(u) = −
2 u + E8, R2
(3.32)
and |E 8 | ≤ c R −2−q |u| + c R −3 u ∗ C 2 (S1 (0)) . We decompose u = h 0 +u 0 , where h 0 is the lowest eigenfunction of L and 0. Then |u − h 0 | ≤ |h 0 − h 0 | + |u 0 |.
h 0 u 0 dσ =
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L.-H. Huang
Claim. The right-hand side of the above inequality is small compared to h 0 . Moreover precisely, sup |h 0 − h 0 | ≤ c R −q |h 0 |,
sup |u 0 | ≤ c R −q |h 0 |.
Assuming the claim, we obtain, by choosing R large enough, 1 1 h 0 − h 0 ≤ u ≤ h 0 + h 0 . 2 2 Because h 0 is nonzero by (3.29), u has a sign, and the theorem follows. Proof of the Claim. Recall that h 0 − h 0 satisfies (3.27). On a coordinate chart, (3.27) is a second order elliptic equation. We choose the coordinate chart to be a ball of radius R on . Then the number of charts to cover is independent of R. Using the De Giorgi– Nash–Moser theory[8, Theorem 8.17] on each chart and summing over the charts, we obtain sup |h 0 − h 0 |
" " " " ≤ c R −1 h 0 − h 0 L 2 ( ) + c|h 0 | "η0 + |A |2 + Ric M (μg , μg )"
L 2 ( )
≤ c R −q |h 0 |,
(3.33)
where we use (3.28) and (3.29). To prove the second inequality in the Claim, we need the Hölder estimate on h 0 − h 0 . By [8, Theorem 8.22] and (3.33), ) * h 0 − h 0 0,α " " " " −α 2 M ≤ c R sup |h 0 − h 0 | + |h 0 | "η0 + |A | + Ric (μg , μg )" 2 L ( )
≤ cR
−α−q
|h 0 |.
(3.34)
To estimate sup |u 0 |, by the definition of u 0 and h 0 , 1 2 L u 0 = L u − η0 h 0 = − 2 u − η0 h 0 + E 8 + (dH(su) − dH(0)u)u ds R 0 2 2 = 2 (h 0 − h 0 ) − 2 u 0 + O(R −2−q u ∗ C 2 (S1 (0)) ), R R where we use (3.32) in the second equality and η0 = −2/R 2 + O(R −2−q ) in the third equality. Because L has no kernel, by pulling back the equation to unit spheres and using the Schauder estimates, u ∗0 C 2,α (S1 (0)) ≤ c h ∗0 − h 0 C 0,α (S1 (0)) + |u 0 | + R −q u ∗ C 2,α (S1 (0)) . (3.35) Because h 0 satisfies L N h 0 = η0 h 0 and η0 = O(R −2 ), using the Schauder estimate on h 0 in the second inequality below, we have u ∗ C 2,α (S1 (0)) ≤ cu ∗0 C 2,α (S1 (0)) + ch ∗0 C 2,α (S1 (0))
≤ cu ∗0 C 2,α (S1 (0)) + ch ∗0 C 0,α (S1 (0)) ≤ cu ∗0 C 2,α (S1 (0)) + ch ∗0 − h 0 C 0,α (S1 (0)) + c|h 0 |.
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357
Therefore, combining the above identities and absorbing the term c R −q u ∗0 C 2,α (S1 (0)) to the left of (3.35) for R large, we have u ∗0 C 2,α (S1 (0)) ≤ c h ∗0 − h 0 C 0,α (S1 (0)) + |u 0 | + R −q |h 0 | ≤ c R −q |h 0 | + c|u 0 |,
(3.36)
where we use (3.33) and (3.34) in the second inequality. It remains to estimate |u 0 |. Because h 0 u 0 dσ = 0, similarly as in (3.30), we have u 0 dσ ≤ 2|N ||h 0 |−1 sup |h 0 − h 0 | sup |u 0 |,
and then by (3.33), |u 0 | ≤ 2|h 0 |−1 sup |h 0 − h 0 | sup |u 0 | ≤ c R −q sup |u 0 |.
Then |u 0 | could be absorbed into the left-hand side of (3.36) for R large. We prove that in the neighborhood U where the inverse function theorem holds, two surfaces in the family of { R } have no intersection. Because the size of U is independent of R by the uniform bounds of |d 2 H| and |L −1
| (cf. [11, Prop. 2.5.6]), we could inductively proceed the argument toward infinity of M and conclude that { R } form a foliation in the exterior region. 4. Uniqueness of the Foliation In this section, we assume that (M, g, K ) is AF–RT with q ∈ (1/2, 1] and m > 0. R is the surface with constant mean curvature constructed in Theorem 3.1, and R is a c0 R 1−q -graph over S R (C) as in Corollary 3.5. 4.1. Local uniqueness. Theorem 4.1. Assume that N has constant mean curvature equal to H R . Given any c1 ≥ 2c0 , there exists σ1 = σ1 (c1 ) so that, for R ≥ σ1 , if N is a c1 R 1−q -graph over S R (C), i.e. N = {x + uνg : u ∈ C 2,α (S R (C))} with u ∗ C 2,α (S1 (0)) ≤ c1 R 1−q , then N = R . Remark. Notice that we do not impose any condition on (u ∗ )odd . Lemma 4.2. There exists a constant c1 so that N is a c1 -graph over S(C, R) and N has constant mean curvature equal to H R , then N = R .
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Proof of the lemma. Assume that N is the graph of v over R . By using the invertibility of L R , we first prove that there is a constant c1 so that if vC 2,α ( R ) ≤ 2c1 , then v ≡ 0. By Taylor’s Theorem, and because N and R have the same mean curvature, L R v =
1
d H R (sv) − d H R (0) v ds.
0
−1 R 3 by Lemma 3.8, and by (3.7), Because |L −1
R | ≤ cm
" " vC 2,α ( R ) ≤ cm −1 R 3 " " ≤ cm
−1
3
R R
" " d H R (sv) − d H R (0) v dσ " "
1
0 −3
2 vC 2,α ( ) R
≤ cm
−1
C 0,α ( R ) 2 vC 2,α ( ) . R
This implies c−1 m ≤ vC 2,α ( R ) . Choose any c1 < (2c)−1 m. If vC 2,α ( R ) ≤ 2c1 , then v ≡ 0. By the construction in Theorem 3.1 and (3.18), R is a 2−1 c0 R 1−2q –graph over S( p, R), and p = C + O(R 1−2q ) by Corollary 3.4. For R ≥ σ1 = σ1 (g, c0 , | p − C|, c1 ) large, S(C, R) is within c1 -distance of R . Also, because the normal vectors of R and S(C, R) are close for R large, if N is a c1 -graph over S(C, R), then N is a 2c1 -graph over R . Therefore, by the above analysis, N = R . Proof of Theorem 4.1. By the assumption, N is the graph of u over S R (C) with u ∗ C 2,α ≤ c1 R 1−q . Because p = C + O(R 1−2q ), for R large, we can assume that N is the graph of u over S R ( p) with u ∗ C 2,α ≤ 2c1 R 1−q . Recall that L S denotes the linearized mean curvature operator on S R ( p) with respect to g. By Taylor’s Theorem,
1
HS (u) = HS (0) − L S u +
(d HS (su) − d HS (0)) u ds.
(4.1)
0
Also, recall that L 0 = − eS − (2/R 2 ) and K = KerL 0 . Let φ be the function defined as in Lemma 3.2; that is, S( p, R) is the graph of φ ∈ K⊥ over S R ( p) and Ai (x i − pi ) − f , L 0 φ = f − R −3−q i
where f = HS (0) − (2/R). Then we show that u − φ is small. Because N and R have the same mean curvature, HS (u) = 2/R + f by the construction of R in Theorem 3.1. Therefore, L 0 (u − φ) = R −3−q Ai (x i − pi ) + (L 0 − L S )u i
1
+
(d HS (su) − d HS (0)) u ds.
0
We decompose u into u = u ⊥ + R −q
i
B i (x i − pi ),
(4.2)
Foliations by Stable Spheres with Constant Mean Curvature
where u ⊥ ∈ K⊥ and, for i = 1, 2, 3, 3R −4+q B = 4π
359
i
S R ( p)
(x i − pi )u dσe .
Notice that we only use |u| ≤ 2c1 R 1−q to guarantee B i = O(1), and we do not assume any condition on u odd . Applying the Schauder estimates on to (4.2), because u ⊥ − φ ∈ K⊥ , (u ⊥ − φ)∗ C 2,α (S1 (0)) ≤ c R −q 1 + u ∗ C 2,α (S1 (0)) ≤ c(1 + 2c1 )R 1−2q . (4.3) To estimate the part inside the kernel, we first rewrite (4.2): −q i i i LS R B (x − p ) = −L 0 (u − φ0 ) + R −3−q Ai (x i − pi ) i
+ (L 0 − L S )u ⊥ +
i
1
(d HS (su)−d HS (0)) u ds. (4.4)
0
Then we multiply the above identity by i B i (x i − pi ) and integrate it over S R ( p) with respect to the area measure dσ . First notice that, by (3.4), (3.5), and Lemma 2.1, for a = 1, 2, 3, (x a − pa )R −3−q Ai (x i − pi ) dσe = 8π m( pa − C a ) + O(R −q ) S R ( p)
i
= O(R 1−2q ). Also, S R ( p)
(x a − pa )(L 0 − L S )u ⊥ dσ
=
S R ( p)
⊥
(x − p )(L 0 − L S )(u − φ) dσ + a
a
S R ( p)
(x a − pa )(L 0 − L S )φ dσ.
Combining (4.3) and the fact that (φ ∗ )odd C 2,α ≤ c R −q , the above term is O(R 1−2q ). For other terms side of (4.4), they are of order O(R 1−2q ) after inte in ithei right-hand i grating with i B (x − p ) as well. Concluding the above estimates and using the eigenvalue estimate on μ0 in Lemma 3.7 (for the operator L S on S R ( p)), 6m −3−q −q + O(R ) R B i (x i − pi ) L 2 (S R ( p)) ≤ c R 1−2q . R3 i
That is, the bound on B i , i = 1, 2, 3, is improved: |B i | ≤ c R −q . Therefore, using (4.3) and (4.5), ∗
⊥
∗
(u − φ) C 2,α ≤ (u − φ) C 2,α
(4.5)
" " " " " 1−q i i " + "R B y" " " i
C 2,α
≤ 2c(1 + c1 )R 1−2q .
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By choosing R ≥ σ1 = σ1 (μ0 , φ ∗ C 2,α , (φ ∗ )odd C 2,α , c1 ), we have c1 . 2 Because the normal vectors of S R ( p) and of S(C, R) are close enough, we could arrange N to be a c1 –graph over S(C, R). Then by Lemma 4.2, N = R . (u − φ)∗ C 2,α ≤
The above theorem says that, among surfaces which are spherical and close to the Euclidean sphere centered at C, R is the only one with the constant mean curvature H R . In particular, we can generalize the above results to the spherical constant mean curvature surfaces. Corollary 4.3. Assume | p − C| ≤ c3 R 1−q . Given any c4 ≥ 2(c0 + c3 ), there exists σ1 = σ1 (c0 , c3 , c4 ) so that, for R ≥ σ1 , if N has constant mean curvature equal to H R , and if N is a c4 R 1−q -graph over S R ( p), then N = R . Proof. Assume that N is a c4 R 1−q -graph over S R ( p). Because the normal vectors on S R ( p) and S R (C) are close and | p − C| ≤ c3 R 1−q , N is a (c0 + c3 )R 1−q -graph over S R (C) for R large. Then we can apply Theorem 4.1 (by letting c1 = c0 + c3 ) and derive that N = R . 4.2. A Priori estimates. In this subsection, we assume (M, g, K ) is AF at the decay rate q ∈ (1/2, 1] (note that the RT condition is not assumed). For general surfaces N in M with constant mean curvature, we would like to derive a priori estimates and show that they are spherical under the condition that N is stable. Let N be a smooth surface with constant mean curvature H and N be topologically a sphere. Assume that N is stable, i.e. u L N u dσ ≥ 0, for all u satisfying u dσ = 0. N
N
Let the minimum radius and the maximal radius of N be defined by r = min{|z| : z ∈ N } and r = max{|z| : z ∈ N } respectively. A denotes the second fundamental form of N , and A˚ = A − 21 H g N denotes the trace-free part of A. μg is the outward unit normal vector field on N , and and ∇ are the Laplacian and the covariant derivative on N with respect to the induced metric g N . Moreover, we denote Ri jkl or Riem the Riemannian curvature tensor and Ric the Ricci curvature tensor of (M, g, K ) respectively. The following Sobolev inequality can be found in, for example, [10, Prop. 5.4]. Sobolev Inequality. For r large, there is a constant csob so that for any Lipschitz functions v on N , 1 2 v 2 dσ ≤ csob (|∇v| + H |v|) dσ. (4.6) N
N
Lemma 4.4. Assume that N is a smooth surface in M with constant mean curvature H . Also, assume that N is topologically a sphere and stable. Then there is some constant c so that the following estimates hold for r large: −s 2−s (1) For " s > 2, q N |x| dσ ≤ cr , " any " ˚" (2) "| A|" ≤ cr − 2 , L2
−1 (3) csob ≤ H 2 |N | ≤ c.
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Proof. Using the first variation formula as in [10, Lemma 5.2], for any s > 2, |x|−s dσ ≤ cr 2−s H 2 |N |. N
Because N is topologically a sphere, by the stability condition as in [10, Prop. 5.3] and the fact that the Ricci curvature is bounded by |x|−2−q , we have ˚ 2 dσ ≤ cr −q H 2 |N |. | A| N
If (3) holds, especially the upper bound, then both (1) and (2) directly follow. The lower bound in (3) can be derived by letting |v| = H in the Sobolev inequality (4.6). Let K be the Gauss curvature of N . For the upper bound, the Gauss equation implies 1 2 ˚ 2 − Rg + 2Ric(μg , μg ) dσ H dσ = 2K + | A| N 2 N ˚ 2 + |x|−2−q ) dσ ≤ c + c (| A| N
≤ c + cr −q H 2 |N |. For r large, the last term is absorbed to the left-hand side, and (3) is proved.
Assume that the Greek letters range over {1, 2}, and the Latin letters range over {1, 2, 3}. For any surface N in M, the Simons identity [15] states Aαβ = ∇α ∇β H + H Aδα Aδβ − |A|2 Aαβ + Aδα R β δ + Aδ Rδαβ +∇β Ricαk ν k + ∇ δ Rkαβδ ν k . ˚ We show that A˚ Because H is a constant, the Simons identity gives an equation on A. is small in the following lemma. Lemma 4.5.
" " " ˚ 2" "| A| "
L2
" " " ˚" + "∇| A| "
L2
" " " ˚" + "|∇ A| "
L2
" " " ˚" + "H | A| "
L2
≤ cr −1−q .
2 ˚ . Then by direct com˚ 2 ≥ ∇| A| Proof. First by the Cauchy–Schwarz inequality |∇ A| putations and the Codazzi equation (see [12, Cor. 3.5] or [14, p. 237]): 2 ˚ ≥ 1 |∇ A| ˚ 2 − ∇| A| ˚ 2 − 16 |ω|2 + |∇ H |2 |∇ A| 17 17 2 1 2 ˚ − 16 |ω|2 + |∇ H |2 , ˚ + 1 ∇| A| |∇ A| ≥ 34 34 17 where ω = Ric(·, μg )T denotes the projection of Ric(·, μg ) onto the tangent space of N . Substitute the above inequality into the following identity: 2 ˚ = | A| ˚ ˚ + 2 ∇| A| ˚ 2 = 2 A˚ αβ A˚ αβ + 2|∇ A| ˚ 2. 2| A| | A| (4.7)
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L.-H. Huang
Then we have
2 ˚ − 16 |ω|2 + |∇ H |2 . ˚ 2 + 1 ∇| A| ˚ ˚ ≥ A˚ αβ A˚ αβ + 1 |∇ A| | A| | A| 34 34 17
Because H is a constant, we use the Simons identity in the above inequality and have ˚ ˚ ≥ H A˚ αβ Aδα Aδβ − |A|2 A˚ αβ Aαβ + A˚ αβ Aδα R β δ + A˚ αβ Aδ Rδαβ | A| | A| + A˚ αβ ∇β Ricαk ν k + A˚ αβ ∇ δ Rkαβδ ν k 2 1 ˚ − 16 |ω|2 . ˚ 2 + 1 ∇| A| + |∇ A| (4.8) 34 34 17 A direct calculation shows that the first two terms on the right-hand side are ˚ 2 + H A˚ αβ A˚ δα A˚ δβ . H A˚ αβ Aδα Aδβ − |A|2 A˚ αβ Aαβ = −(|A|2 − H 2 )| A| ˚ which vanishes because A˚ The last term is the sum of the cubic of the eigenvalues of A, ˚ ˚ over N yields is trace-free and N is two-dimensional. Then integrating −| A| | A| 35 ˚ 2 + 1 |∇ A| ˚ 2 dσ |∇| A|| 34 N 34 2 2 ˚ 2 ≤ A˚ αβ Aδα R β δ + A˚ αβ Aδ Rδαβ dσ (|A| − H )| A| dσ − N N 16 2 αβ k αβ δ k ˚ ˚ |ω| dσ. A ∇β Ricαk ν + A ∇ Rkαβδ ν dσ + − (4.9) N N 17 The last term in the second line can be bounded by ˚ 2 |Riem| + H | A||Riem| ˚ ˚ 2 + H | A|) ˚ dσ. | A| dσ ≤ c |x|−2−q (| A| c N
N
Using integration by parts and the Codazzi equation, the first integral in the third line can be bounded by c N |ω|2 dσ . To estimate the first integral in the second line of (4.9), we use the stability condition. Because N is stable, for any u with mean value u, by the stability equation for u − u: |A|2 u 2 dσ N |∇u|2 dσ + |A|2 (2uu − u 2 ) dσ − Ric(μg , μg )(u − u)2 dσ ≤ N N N 1 2 2 2 ˚ + H (2uu − u 2 ) dσ + 2 | A| ≤ |∇u| dσ + |Ric(x)|(u 2 + u 2 ) dσ. 2 N N N ˚ and rewrite the above Because 2uu − u 2 ≤ u 2 and |Ric(x)| ≤ c|x|−2−q , we let u = | A| inequality as follows: 1 ˚ 2 dσ ≤ ˚ 2 dσ + 2u ˚ 3 dσ |A|2 − H 2 | A| |∇| A|| | A| 2 N N N −2−q 2 ˚ + u 2 dσ. + 2c | A| |x| N
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Multiplying the above inequality by 69/68, and adding it to (4.9),
˚ 2 ˚ 4 dσ + ˚ 2 dσ + H 2 ˚ 2 dσ | A| |∇ A| | A| ∇| A| dσ + N N N N 3 −2−q 2 2 ˚ ˚ ˚ −2−q dσ | A| + u dσ + c | A| dσ + c |x| H | A||x| ≤ cu N N N |x|−4−2q dσ. +c N
" " q " ˚" Because "| A| " 2 ≤ cr − 2 by Lemma 4.4 (2), by the Hölder inequality and Lemma 4.4 L (3), u 2 := |N |−2
2
˚ dσ | A|
≤ |N |−1
N
˚ 2 dσ ≤ c|N |−1r −q ≤ cr −q H 2 . | A| N
By the AM–GM inequality and the above identity, ˚ 3 dσ ≤ | A|
cu N
1 4
˚ 4 dσ + cr −q H 2 | A|
N
˚ 2 dσ. | A| N
For r large enough, these two terms could be absorbed to the left-hand side. Similarly, we estimate the rest of the terms: c
−2−q
2 2 ˚ | A| + u dσ + c |x|−4−2q dσ
|x| N 1 4 −2−q 2 ˚ dσ + r ≤ | A| u |N | + cr −2−2q , 4 N N
and c N
˚ −2−q dσ ≤ 1 H 2 H | A||x| 2
˚ 2 dσ + cr −2−2q . | A| N
We then derive " " " ˚ 2" "| A| "
L2
" " " ˚" + "∇| A| "
L2
" " " ˚" + "|∇ A| "
L2
" " " ˚" + "H | A| "
L2
≤ cr −1−q .
˚ is improved: Remark. In particular, comparing to Lemma 4.4 (2), the L 2 bound of | A| " " " ˚" "| A|"
L2
≤ cH −1 r −1−q .
(4.10)
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4.3. The position estimate. In this subsection, we assume that (M, g, K ) is AF at the decay rate q ∈ (1/2, 1] (note that the RT condition is not assumed). Assume that N has constant mean curvature H and that N is stable. In order to prove that N is spherical, ˚ in the previous sub˚ by the L p -estimates on | A| we derive the pointwise estimate of | A| ˚ Inspired by [13], we apply the Moser iteration section and the Simons identity for | A|. to functions satisfying this type of the differential equation below. Lemma 4.6. For any functions u ≥ 0, f ≥ 0, and h on N satisfying − u ≤ f u + h,
(4.11)
we have the pointwise control on u as follows: sup u ≤ c( f L 2 + H + r −1 )(u L 2 + r H −1 h L 2 ). N
Proof. Replacing v by v 2 in the Sobolev inequality (4.6) and using the Hölder inequality, we derive a variant of the Sobolev inequality
1 v 4 dσ
2
N
≤c
|v||∇v| dσ +
N
≤c
H v 2 dσ N
1 v dσ 2
1 1 2 2 2 2 |∇v| dσ + H v dσ .
2
2
N
N
(4.12)
N
Let k be a positive constant and uˆ = u + k. Then multiplying uˆ p−1 on both sides of (4.11), − uˆ p−1 uˆ ≤ f uˆ p − k f uˆ p−1 +
h p h uˆ ≤ f uˆ p + uˆ p = fˆuˆ p , uˆ k
(4.13)
where fˆ = f + k −1 h. Integrating (4.13) and using p
|∇(uˆ 2 )|2 = we have, for p ≥ 2, p 2 ∇(uˆ 2 ) dσ = N
p2 ( p − 1)uˆ p−2 |∇ u| ˆ 2, 4( p − 1)
p2 4( p − 1)
−uˆ
p−1
fˆuˆ p dσ.
uˆ dσ ≤ p
N
N
p
We let v be uˆ 2 in (4.12) and substitute the gradient term by the above inequality. Then, 1 1 1 1 2 2 2 2 2p p p ˆ 2 p p uˆ dσ ≤c uˆ dσ uˆ f dσ + H uˆ dσ . N
N
N
N
By the Hölder inequality, the last two terms can be bounded by 1 1 2 4 1 p ˆ 2 p p uˆ f dσ ≤ p2 uˆ dσ N
1
H uˆ dσ 2 p
N
2
N
≤
4
4
uˆ
H dσ N
N
4
f dσ N
1
1
ˆ2
2p
1 4
dσ
.
,
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Therefore, using the above inequalities and the AM–GM inequality,
1 2
uˆ 2 p dσ
≤
N
1 2
1 uˆ 2 p dσ N
2
uˆ p dσ
1 uˆ
2p
2
dσ
2
1 2
.
N
fˆ2 dσ
≤ cp
N
fˆ2 dσ N
1
H 4 dσ
N
Therefore,
N
+c
uˆ p dσ
+ cp
N
1
1
2
+
2
4
uˆ dσ . p
H dσ N
N
Then, using Lemma 4.4 (3) to bound H 2 |N |1/2 ≤ cH , we obtain
uˆ 2 p dσ N
1 2p
1 1 p 1 1 p ≤ c p p p fˆ L 2 + H uˆ p dσ . N
Now letting p = 2i , i = 1, 2, 3, . . ., we then have 2−l−1 li=1 2−i l −i 2l+1 ˆ uˆ dσ ≤ c f L 2 + H 2 i=1 (i2 ) u ˆ L2 . N
Let l → ∞,
sup uˆ ≤ c fˆ L 2 + H u ˆ L2 N ≤ c f L 2 + k −1 h L 2 + H u L 2 + k H −1 ,
where we use |N |1/2 ≤ cH −1 . Let k = r h L 2 . Then the proof is completed. Corollary 4.7. ˚ ≤ c(r −1−q + H −1r −2−q ). sup | A| Furthermore, if r ≥ H −a for some fixed a ≤ 1, then ˚ ≤ cH 1+ , sup | A| where = (2 + q)a − 2, and > 0 if
2 < a ≤ 1. 2+q
Proof. Notice that by (4.7) and the Cauchy–Schwarz inequality, ˚ ˚ ≤ − A˚ αβ A˚ αβ . −| A| | A| Using the Simons identity and the estimates in Lemma 4.5, we have ˚ 2 − A˚ αβ Aδα R β δ − A˚ αβ Aδ Rδαβ ˚ ˚ ≤ (|A|2 − H 2 )| A| −| A| | A| − A˚ αβ ∇β Ricαk ν k − A˚ αβ ∇ δ Rkαβδ ν k ˚ 4 + | A| ˚ 2 |x|−2−q + H | A||x| ˚ −2−q + | A||x| ˚ −3−q , ≤ c | A|
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L.-H. Huang
where we have used that |Ri jkl | ≤ c|x|−2−q and |∇ Ri jkl | ≤ c|x|−3−q . Set ˚ u = | A|, ˚ 2 + |x|−2−q ), f = c(| A|
h = c(H |x|−2−q + |x|−3−q ).
By Lemma 4.5, u L 2 ≤ cH −1 r −1−q ,
f L 2 ≤ cr −1−q ,
The corollary follows by Lemma 4.6.
h L 2 ≤ cr −2−q .
˚ yields the estimates on | A˚ e | when N is treated Because M is AF, the estimates on | A| as an embedded surface in Euclidean space. We prove that N is a graph over the sphere Sr0 ( p). The following lemma is a generalization of [10, Prop. 2.1] where M was assumed strongly asymptotically flat. A similar argument allows us to generalize to AF manifolds ˚ and r . We include the at the decay rate q > 1/2 and to remove the conditions on |∇ A| proof for completeness. Lemma 4.8. Let N satisfy the assumptions as in Theorem 2. Then, there exists the center p so that for all z ∈ N , |λe i − r0−1 | ≤ cH 1+ , νe (z) − z − p ≤ cH , r0
(4.14) (4.15)
where r0 = 2/H , λe i and νe (z) are the principal curvature and the outward unit normal vector at z with respect to the Euclidean metric. Moreover, N is a graph over Sr0 ( p) so that
N = z = x + νg v : x ∈ Sr0 ( p), v ∈ C 1 (Sr0 ( p)) and v ∗ C 1 ≤ cH −1+ . ˚ ≤ cH 1+ . Because M is AF and r ≥ H −a , for r Proof. By Corollary 4.7, sup N | A| large, ˚ + cr −1−q ≤ cH 1+ , sup | A˚ e | ≤ sup | A| N e
N
|H − H | ≤ cr −1−q ≤ cH 1+ . We would like to use the bound of these Euclidean quantities to show that N is close to some sphere in the Euclidean space. To derive (4.14), e λ i − 1 H ≤ λe i − 1 H e + 1 H e − 1 H 2 2 2 2 ≤ | A˚ e | + cH 1+ ≤ cH 1+ . Let r0−1 = (1/2)H , and then (4.14) follows. To prove (4.15), we first derive the upper bound on the diameter of N which is defined by the intrinsic distance on N equipped
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with its induced metric from the Euclidean space. Let K be the Gauss curvature of N . Using the Gauss equation on N in Euclidean space, K − 1 (H e )2 ≤ | A˚ e |2 ≤ cH 2+2 . 2 Hence, |K| ≥ 18 H 2 , for H small. The Bonnet–Myers theorem says that diam(N ) ≤ cH −1 . Then, let z be the position vector and g N be the induced metric on N from Euclidean space. By the Gauss–Weingarten relation 1 jk jk ∂i νe = Aiej g N ∂k z = A˚ iej − H e (g N ) jk g N ∂k z. 2 Then, 1 1 e jk e ˚ ∂i νe − H z = Ai j − (H − H )(g N ) jk g N ∂k z. 2 2 We integrate the above identity along a geodesic, and derive, for some p, |νe − r0−1 (z − p)| ≤ c sup | A˚ e | + |H e − H | diam(N ) ≤ cH . N
To prove that N is a graph over Sr0 ( p), we define v(x) = |z − x|, where x ∈ Sr0 ( p) is the intersection of the ray z − p and Sr0 ( p). By (4.15) , for H small, 1 z − p − νe ≤ . r 2 0 p In particular, νe never becomes perpendicular to the radial direction, so N = {x + v x− r0 : x ∈ Sr0 ( p)} is well-defined. To obtain the C 1 bound on v, we have ||z − p| − r0 | ≤ cH −1+ by (4.15), and then
vC 0 = sup |z − x| ≤ sup ||z − p| − r0 | ≤ cH −1+ . z∈N
z∈N
Moreover, |∂v| = |z − x|−1 ∇ e (z − x), z − x ≤ |∇ e (z − p) − ∇ e (x − p)|. Using (4.15) and that |∇ e (ν e −
x− p r0 )|
≤ | A˚ e |, we obtain
|∂v| ≤ cH . Therefore, we conclude v ∗ C 1 (S1 (0)) ≤ cH −1+ . Moreover, because νe and νg on N are close in C 2,α , N = {x + νg v : v ∈ C 1 (Sr0 ( p))} for some v satisfying v ∗ C 1 (S1 (0) ≤ cH −1+ .
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In order to use the Taylor theorem to the mean curvature map, N should be a graph whose C 2,α –norm is under control. Therefore, we have to derive the pointwise estimate ˚ A modified Moser iteration which involves the special choice on the C 1,α -norm of A. of the cut-off functions is employed. Lemma 4.9. For any functions u ≥ 0, f ≥ 0, and h on N satisfying − u ≤ f u + h,
(4.16)
we have the pointwise control on u as follows: sup u ≤ c f L 2 + H + r −1 (u L 2 + r 2 h L 2 ). N
Remark. Comparing this lemma with Lemma 4.6, the term H −1 h L 2 = (r −1 ) (r H −1 h L 2 ) is replaced by r h L 2 = (r −1 )(r 2 h L 2 ). The term H −1 h L 2 is unfavorable because if this term appeared in Corollary 4.10, it is bounded by H −1r −3−q which may not be bounded by H 2+ for > 0, when 2/(2 + q) < a ≤ 1. Proof. Let k be a positive constant. As in the proof of Lemma 4.6, we define uˆ = u + k and fˆ = f +k −1 h. Let χ be a cut-off function on N . The same calculations in Lemma 4.6 give p 2 χ 2 fˆuˆ p dσ + |∇χ |2 uˆ p dσ. ∇(χ uˆ 2 ) dσ ≤ p N
N
N
χ uˆ p/2 ,
By (4.12) and letting v = 1 2 χ 4 uˆ 2 p dσ ≤ c(H + sup |∇χ |) uˆ p dσ supp(χ ) N N 1/2 1/2 p χ 2 fˆuˆ p dσ χ 2 uˆ p dσ . +c N
Using the AM–GM inequality to the second line and absorbing the term χ 4 uˆ 2 p to the left, we obtain 1 2 4 2p ˆ χ uˆ dσ ≤ cp f L 2 + H + sup |∇χ | uˆ p dσ. N supp(χ ) N Let pi = 2i , i = 1, 2, 3, . . .. Fix z 0 ∈ N . The cut-off functions supported on N are defined by, for z ∈ N , % 1 if z ∈ B(1+2−i )r (z 0 ) χi (z) = , 0 if z outside B(1+2−i+1 )r (z 0 ) and |∇χi | ≤ 2i r −1 . Then 2−1−l B(1+2−l )r (z 0 )
uˆ
21+l
dσ
≤c
l i=1
+H
2−i
l
2
l
−i i=1 2
i=1
i2−i
l
2 fˆ L 2i=1
−i
li=1 2−i i −1 + 2r u ˆ L 2 (B2r (z 0 )) .
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Let l → ∞, sup uˆ ≤ c( fˆ L 2 + H + r −1 )u ˆ L 2 (B2r (z 0 )) .
Br (z 0 )
Let k = r h L 2 . Then 1/2 sup u ≤ c( f L 2 + H + r −1 )(u L 2 + r h L 2 B2r (z 0 ) ).
Br (z 0 )
By the area formula, because g is AF, and N is a graph of v over Sr0 ( p) satisfying |∂v| ≤ cH by Lemma 4.8, |B2r (z 0 )| = dσ ≤ (1 + cH ) dσe B2r (z 0 )
B2r (z 0 )
≤2
B2r (z 0 )
dσe ≤ cr 2 .
Corollary 4.10. Assume that N satisfies the assumptions in Theorem 2. Then ˚ ≤ cr −1−q (r −1 + H ). sup |∇ A| N
Moreover, if r ≥ H −a for some fixed a ≤ 1, then ˚ ≤ cH 2+ , sup |∇ A| N
where = (2 + q)a − 2 > 0, if
2 2+q
< a ≤ 1.
Proof. Let Tγ αβ = ∇γ A˚ αβ , 2|T | |T | + 2 |∇|T ||2 = |T |2 = 2T γ αβ Tγ αβ + 2|∇T |2 . Because |∇T |2 ≥ |∇|T ||2 by the Cauchy–Schwarz inequality, −|T | |T | ≤ −T γ αβ Tγ αβ . Changing the order of differentiation in the Laplacian term, (∇γ A˚ αβ ) = ∇γ A˚ αβ + g ρδ (∇ A˚ αβ )Rδ γρ + g ρδ (∇δ A˚ β )Rα γρ + g ρδ (∇δ A˚ α )Rβ γρ + g ρδ ∇ρ A˚ β Rα γ δ + A˚ α Rβ γ δ . By the Simons identity, ∇ γ A˚ αβ ∇γ A˚ αβ = H ∇ γ A˚ αβ ∇γ Aδα Aδβ − ∇ γ A˚ αβ ∇γ |A|2 Aαβ M M k + ∇ γ A˚ αβ ∇γ Aδα R β δ + Aδ Rδαβ + ∇β Ricαk ν + ∇ δ Rkαβδ ν k .
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L.-H. Huang
Then, ˚ 2 |∇ A| ˚2 ∇ γ A˚ αβ ∇γ A˚ αβ ≥ −| A| ˚ 2 |A| + |∇ A| ˚ 2 |Riem| + |∇ A|| ˚ A||∇Riem| ˚ −c H |∇ A|
˚ ||∇Riem| + |∇ A||∇ ˚ 2 Riem| + |∇ A| ˚ 2 |∇Riem| . +|∇ A|H
Using |Riem| ≤ c|x|−2−q , |∇Riem| ≤ c|x|−3−q , |∇ 2 Riem| ≤ c|x|−4−q , and combining the above estimates, ˚ ˚ ≤ −∇ γ A˚ αβ ∇γ A˚ αβ −|∇ A| |∇ A| ˚ 2 |∇ A| ˚ 2 +H 2 |∇ A| ˚ 2 +H | A||∇ ˚ ˚ 2 +|∇ A| ˚ 2 |x|−2−q +|∇ A| ˚ 2 |x|−3−q ≤ c | A| A| ˚ A||x| ˚ −3−q + H |∇ A||x| ˚ −3−q + |∇ A||x| ˚ −4−q . +|∇ A|| ˚ and By Lemma 4.9, we set u = |∇ A| ˚ + |x|−2−q + |x|−3−q ), ˚ 2 + H 2 + H | A| f = c(| A| ˚ −3−q + H |x|−3−q + |x|−4−q ). h = c(| A||x| By Lemma 4.4 and Lemma 4.5, u L 2 ≤ cr −1−q , f L 2 ≤ c(r −1−q + H ), and h L 2 ≤ cr −3−q . Then the proof follows directly from Lemma 4.9. ˚ ≤ cH 2+α+ , Similarly, we can derive that, if r ≥ H −a , then the Hölder norm |∇ A| α where = (2 + q)a − 2 and > 0 if 2/(2 + q) < a ≤ 1. Using the same argument in Lemma 4.8, we prove the following: Corollary 4.11. Assume that N satisfies the assumptions in Theorem 2. If r ≥ cH −a for some 2/(2 + q) < a ≤ 1, then N is a graph defined by N = {x + νg v : x ∈ Sr0 ( p)} with v ∗ C 2,α (S1 (0)) ≤ cH −1+ ≤ cr01− , where = (2 + q)a − 2 > 0. 4.4. Global uniqueness. Proof of Theorem 2. Assume that N has mean curvature equal to H = H R for some R. By Corollary 4.3, we only need to prove that N is a graph of v over S R ( p), where | p − C| ≤ c3 R 1−q and v ∗ C 2,α ≤ c4 R 1−q for some c3 and c4 . By Corollary 4.11, N is a graph over Sr0 ( p) and H = 2/r0 . The idea of the proof is to show that the center p does not drift away too much as H goes to zero. More pre cisely, we show that | p − C| ≤ cr01− for some > 0. Hence, r0 and r are comparable; consequently, H and r are comparable. By the Taylor theorem, because N is the graph of v over Sr0 ( p), 1 H = HS − L S v + (d H (sv) − d H (0)) v ds. 0
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Recall that L S = − S − (|A S |2 + Ric M (νg , νg )), L 0 = − eS − Also recall that φ in Lemma 3.2, and φ satisfies −3−q L 0 φ = f − r0 Ai (x i − pi ) − f ,
2 , R2
and K = KerL 0 .
i
where f = HS − 2/r0 . Therefore, −3−q
L 0 (v − φ) = r0
Ai (x i − pi ) + f + (L 0 − L S )v
i
1
+
(d H (sv) − d H (0)) v ds.
(4.17)
0
We decompose v = v ⊥ + r0− vK, where vK = i B i (x i − pi ) ∈ K and 3r −4+ (x i − pi )v dσe = O(1). Bi = 0 4π Sr0 ( p) Because v ⊥ − φ ∈ K⊥ , we apply the Schauder estimate to (4.17), v ⊥ − φC 2,α ≤ c(r01−2 + r0
1−q
).
Without loss of generality, we assume < q/2. The right-hand side of the above identity is dominated by cr01−2 . Consider (v ⊥ − φ)odd : −3−q Ai (x i − pi ) + (L 0 − L S )(v ⊥ )odd L 0 ((v ⊥ − φ)odd ) = 2r0 i − −2L S (r0 u K) + (L S 1
− L 0 )odd v odd v (d H (sv) − d H (0)) ds
+ 0 1
+
(d H (sv) − d H (0)) v odd ds.
(4.18)
0
Then by the Schauder estimate, and using Lemma 2.2 and Lemma 2.3 to estimate the last three terms, −q −q 1−q− ((v ⊥ )∗ )odd C 2,α ≤ (φ ∗ )odd C 2,α + c r0 + r0 (v ∗ )odd C 2,α + r0 −1−q ∗ + r0 v C 2,α + r0−1 ((v ⊥ )∗ )odd C 2,α vC 2,α . Bootstrapping the term ((v ⊥ )∗ )odd C 2,α yields ((v ⊥ )∗ )odd C 2,α (S1 (0)) ≤ cr 1−q− . We then integrate both sides of (4.18) with x a − pa on Sr0 ( p) with respect to the area measure dσ . By the definition of Ai and Lemma 2.1, −3−q 1−2q (x a − pa )r0 Ai (x i − pi ) dσ = 8π m( pa − C a ) + O(r0 ). Sr0 ( p)
i
372
L.-H. Huang −2−2q
Also, by Lemma 3.7 (there, the equality that μ0 = 6π m/r03 + O(r0 ) is achieved by the coordinate functions x i − pi ), 6π m 2−2q− r0− (x a − pa )L S B i (x i − pi ) dσ = r0− 3 B a r04 + O(r0 ). r Sr0 ( p) 0 i 2−2q−
). Therefore, we have 1−(q+2 −1) . | pa − C a | ≤ c r01− + r0
The rest terms are of order O(r0
Recall that = (2 + q)a − 2. By the assumption that a > (5 − q)/(4 + 2q), we have := q + 2 − 1 > 0. Then | pa | ≤ cr01− , so the center p may drift away but at a controlled rate. Let z 0 be a point so that r = |z 0 |,
r = |z 0 | ≥ |z 0 − p| − | p| ≥ r0 − cH −1+ − cr01− . For r0 large, r ≥ cr0 . Therefore, we can replace the assumption r ≥ H −a by r ≥ cr0 ≥ 1−q 1−q cH −1 in Corollary 4.11. Therefore, N is a cr0 -graph over Sr0 ( p) and | p−C| ≤ cr0 . Although H may not be exactly equal to H r0 , we can choose R so that H = H R with −q
R = r0 + O(r0 ). Then we can apply the local uniqueness result of Corollary 4.3 by viewing N as a graph over S R ( p) and conclude N = R .
To prove a result of the uniqueness outside a fixed compact set, we replace the condi−1 tion on r by the condition that r and r satisfy r ≤ c2 r a for any (5−q)/(4+2q) < a ≤ 1. Proof of Theorem 3. If N lies completely outside B H −a (0) for some a satisfying (5 − q)/(4 + 2q) < a ≤ 1, by Theorem 2, N = R . We assume that N = R . Therefore N ∩ B H −a (0) = φ for any (5 − q)/(4 + 2q) < a ≤ 1. Then r ≤ H −a ≤ 3R a if R large enough because H = (2/R) + O(R −1−q ). On the other hand, for any z ∈ N , 2 4 ≤ H e (z) ≤ 2H ≤ + c R −1−q . r R For R large, 2 6 ≤ , r R and then R/3 ≤ r . Therefore, 1 (3) Choosing any c2 < fore, N = R .
√1 , 3
1 a −1
1
1
(r ) a ≤
(3)
1 a −1
1
(3R a ) a ≤
R ≤ r. 3
1
we obtain c2 r a < r which contradicts the assumption. There-
Acknowledgements. I would like to thank my advisor Professor Rick Schoen for suggesting this problem and providing useful comments. I also would like to thank Professors Brian White, Leon Simon, Damin Wu, Justin Corvino, Jan Metzger, and Gerhard Huisken for discussions. Also, I thank the referee for suggestions and for pointing out several typographical errors. The global uniqueness result was completed at Institut Mittag-Leffler in Autumn 2008 when the author participated the program Geometry, Analysis, and General Relativity. The author is grateful for their hospitality and the generous support.
Foliations by Stable Spheres with Constant Mean Curvature
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References 1. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39(5), 661–693 (1986) 2. Beig, R., Ó Murchadha, N.: The Poincaré group as the symmetry group of canonical general relativity. Ann. Phys. 174(2), 463–498 (1987) 3. Bray, H.L.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Diff. Geom. 59(2), 177–267 (2001) 4. Christodoulou, D., Yau, S.-T.: Some remarks on the quasi-local mass. In: Mathematics and general relativity (Santa Cruz, CA, 1986), Volume 71 of Contemp. Math., Providence, RI: Amer. Math. Soc., 1986, pp. 9–14 5. Chru´sciel, P.T.: On the invariant mass conjecture in general relativity. Commun. Math. Phys. 120(2), 233– 248 (1988) 6. Chru´sciel, P.T., Delay, E.: On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mém. Soc. Math. Fr. (N.S.) 94, vi+103 (2003) 7. Corvino, J., Wu, H.: On the center of mass of isolated systems. Class. Quant. Grav. 25(8), 085008, 18, (2008) 8. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Berlin: Springer-Verlag, 2001, reprint of the 1998 edition 9. Huang, L.-H.: On the center of mass of isolated systems with general asymptotics. Class. Quant. Grav. 26(1), 015012, 25 (2009) 10. Huisken, G., Yau, S.-T.: Definition of center of mass for isolated physical systems and unique foliations by stable spheres with constant mean curvature. Invent. Math. 124(1–3), 281–311 (1996) 11. Ratiu, T., Marsden, J.E., Abraham, R.: Manifolds, tensor analysis, and applications. Volume 75 of Applied Mathematical Sciences. New York: Springer-Verlag, second edition, 1988 12. Metzger, J.: Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature. J. Diff. Geom. 77(2), 201–236 (2007) 13. Qing, J., Tian, G.: On the uniqueness of the foliation of spheres of constant mean curvature in asymptotically flat 3-manifolds. J. Amer. Math. Soc. 20(4), 1091–1110 (electronic) (2007) 14. Schoen, R., Yau, S.T.: Proof of the positive mass theorem. II. Commun. Math. Phys. 79(2), 231–260 (1981) 15. Simons, J.: Minimal varieties in Riemannian manifolds. Ann. of Math. 88(2), 62–105 (1968) 16. Yau, S.T.: Problem section. In: Seminar on Differential Geometry, Volume 102 of Ann. of Math. Stud. Princeton, NJ: Princeton Univ. Press, 1982, pp. 669–706 17. Ye, R.: Foliation by constant mean curvature spheres on asymptotically flat manifolds. In: Geometric analysis and the calculus of variations, Cambridge, MA: Int. Press, 1996, pp. 369–383 Communicated by P.T. Chru´sciel
Commun. Math. Phys. 300, 375–410 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1105-9
Communications in
Mathematical Physics
Energy Decay for the Damped Wave Equation Under a Pressure Condition Emmanuel Schenck Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France. E-mail: [email protected] Received: 29 July 2009 / Accepted: 14 March 2010 Published online: 9 September 2010 – © Springer-Verlag 2010
Abstract: We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This result holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by the imaginary part of a finite number of eigenvalues close to the real axis. 1. Introduction One of the standard questions in geometric control theory concerns the so-called stabilization problem: given a dissipative wave equation on a manifold, one is interested in the behaviour of the solutions and their energies for long times. The answers that can be given to this problem are closely related to the underlying manifold and the geometry of the control (or damping) region. In this paper, we shall study these questions in the particular case of the damped wave equation on a compact Riemannian manifold (M, g) with negative curvature and dimension d ≥ 2. For simplicity, we will assume that M has no boundary. If a ∈ C ∞ (M) is a real valued function on M, this equation reads (∂t2 − g + 2a(x)∂t )u = 0, (t, x) ∈ R × M,
(1.1)
with initial conditions u(0, x) = ω0 (x) ∈ H 1 , i ∂t u(0, x) = ω1 (x) ∈ H 0 . Here H s ≡ H s (M) are the usual Sobolev spaces on M. The Laplace-Beltrami operator g ≡ is expressed in local coordinates by 1 g = √ ∂i (g i j g∂ ¯ j ), g¯ = det g. (1.2) g¯
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E. Schenck
√ We ¯ x the natural Riemannian density, and u, v = will also denote by dvol = gd u vdvol ¯ the associated scalar product. M In all the following, we will consider only the case where the waves are damped, which corresponds to take a ≥ 0 with a non identically zero. We can reformulate the above problem into an equivalent one by considering the unbounded operator 0 Id B= : H1 × H0 → H1 × H0 −g −2 i a with domain D(B) = H 2 × H 1 , and the following evolution equation: (∂t + i B)u = 0, u = (u 0 , u 1 ) ∈ H 1 × H 0 .
(1.3)
From the Hille-Yosida theorem, one can show that B generates a uniformly bounded, strongly continuous semigroup e− i t B for t ≥ 0, mapping any (u 0 , u 1 ) ∈ H 1 × H 0 to a solution (u(t, x), i ∂t u(t, x)) of (1.3). Since B has compact resolvent, its spectrum Spec B consists in a discrete sequence of eigenvalues {τn }n∈N . The eigenspaces E n cor responding to the eigenvalues τn are all finite dimensional, and the sum n E n is dense in H 1 × H 0 , see [GoKr]. If τ ∈ Spec B, there is v ∈ H 1 such that u(t, x) = e− i tτ v(x),
(1.4)
P(τ )u = 0, where P(τ ) = − − τ 2 − 2 i aτ.
(1.5)
and the function u then satisfies
From (1.5), it can be shown that the spectrum is symmetric with respect to the imaginary axis, and satisfies −2 a ∞ ≤ Im τn ≤ 0 while | Re τn | → ∞ as n → ∞. Furthermore, if Re τ = 0, we have Im τ ∈ [− a ∞ , 0], and the only real eigenvalue is τ = 0, associated to the constant solutions of (1.1). The question of an asymptotic density of modes has been adressed by Markus and Matsaev in [MaMa], where they proved the following Weyl-type law, also found later independently by Sjöstrand in [Sjö]: d λ Card{n : 0 ≤ Re τn ≤ λ} = d xdξ + O(λd−1 ). 2π p −1 ([0,1]) Here p = 21 gx (ξ, ξ ) is the principal symbol of −g and d xdξ denotes the Liouville measure on T ∗ M coming from its symplectic structure. Under the assumption of ergodicity for the geodesic flow with respect to the Liouville measure, Sjöstrand also showed that most of the eigenvalues concentrate on a line in the high-frequency limit. More precisely, he proved that given any ε > 0, Card{n : τn ∈ [λ, λ + 1] + i(R\[−a¯ − ε, −a¯ + ε])} = o(λd−1 ).
(1.6)
The real number a¯ is the ergodic mean of a on the unit cotangent bundle S ∗ M = {(x, ξ ) ∈ T ∗ M, gx (ξ, ξ ) = 1}. It is given by T a¯ = lim T −1 a ◦ t dt, well defined d xdξ − almost everywhere on S ∗ M. T →+∞
0
Energy Decay for the Damped Wave Equation Under a Pressure Condition
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Hence the eigenvalues close to the real axis, say with imaginary parts in [α, 0], 0 > α > −a, ¯ can be considered as “exceptional”. In this direction, the first result we will present in this paper shows that a spectral gap of finite width can exist below the real axis under some dynamical hypotheses, see Theorem 1 below. The second object studied in this work is the energy of the waves. From now on, we call H = H 0 × H 1 the space of Cauchy data. Let u be a solution of (1.1) with initial data ω ∈ H. The energy of u is defined by E(u, t) =
1 ( ∂t u 2L 2 + ∇u 2L 2 ). 2 t→∞
As a well known fact, E is decreasing in time, and E(u, t) −→ 0. It is then natural to ask if a particular rate of decay of the energy can be identified. Let s > 0 be a positive number, and define the Hibert space Hs = H 1+s × H s ⊂ H. Generalizing slightly a definition of Lebeau, we introduce the best exponential rate of decay with respect to · Hs as ρ(s) = sup{β ∈ R+ : ∃C > 0 such that ∀ω ∈ Hs , E(u, t) ≤ C e−βt ω Hs },
(1.7)
where the solutions u of (1.1) have been identified with the Cauchy data ω ∈ Hs . It is shown in [Leb] that ρ(0) = 2 min(G, C(∞)), where G = inf{− Im τ ; τ ∈ Spec B\{0}} is the spectral gap, and 1 t ∗ π a( s ρ)ds ≥ 0. C(∞) = lim inf∗ t→∞ ρ∈T M t 0 Here t : T ∗ M → T ∗ M is the geodesic flow, and π : T ∗ M → M is the canonical projection along the fibers. It follows that the presence of a spectral gap below the real axis is of significant importance in the study of the energy decay. However, an explicit example is given in [Leb], where G > 0 while C(∞) = 0, and then ρ(0) = 0. This particular situation is due to the failure of the geometric control, namely, the existence of orbits of the geodesic flow not meeting supp a (which implies C(∞) = 0). Hence, the spectrum of B may not always control the energy decay, and some dynamical assumptions on the geodesic flow are required if we want to solve positively the stabilization problem. In the case where geometric control holds [RaTa], it has been shown in various settings that ρ(0) > 0, see for instance [BLR,Leb,Hit]. In [Chr], a particular situation is analyzed where the geometric control does not hold near a closed hyperbolic orbit of the geodesic flow: in this case, there is a sub-exponential decay of the energy with respect to · Hε for some ε > 0. Dynamical assumptions. In this paper, we first assume that (M, g) has strictly negative sectional curvatures. This implies that the geodesic flow has the Anosov property on every energy layer, see Sect. 2.1 below. Without loss of generality, we suppose that the injectivity radius satisfies r ≥ 2. Then, we drop the geometric control assumption, and replace it with a dynamical hypothesis involving the topological pressure of the geodesic flow on S ∗ M, which we define now. For every ε > 0 and T > 0, a set S ⊂ S ∗ M is (ε, T )−separated if ρ, θ ∈ S implies that d( t ρ, t θ ) > ε for some t ∈ [0, T ], where
378
E. Schenck
d is the distance induced from the adapted metric on T ∗ M. For f continuous on S ∗ M, set ⎫ ⎧ T −1 ⎬ ⎨ exp f ◦ k (x) . Z ( f, T, ε) = sup ⎭ S ⎩ ρ∈S
k=0
The topological pressure Pr( f ) of the function f with respect to the geodesic flow is defined by Pr( f ) = lim lim sup ε→0 T →∞
1 log Z ( f, T, ε). T
The pressure Pr( f ) contains useful information on the Birkhoff averages of f and the complexity of the geodesic flow, see for instance [Wal] for a general introduction and further properties. The particular function we will deal with is given by 1 1 π ∗ a ◦ s (ρ) ds + log J u (ρ) ∈ R, (1.8) a u : ρ ∈ S ∗ M → a u (ρ) = − 2 0 where J u (ρ) is the unstable Jacobian at ρ for time 1, see Sect. 2.1. In this paper, we will always assume that Pr(a u ) < 0.
(1.9)
Main results. Under the condition Pr(a u ) < 0, we will see that a spectral gap of finite width exists below the real axis. As a consequence, there is an exponential decay of the energy of the waves with respect to · Hκ for any κ > d/2, and if G < | Pr(a u )|, we have ρ(κ) = 2G. We begin by stating the result concerning the spectral gap. Theorem 1 (Spectral gap). Suppose that the topological pressure of a u with respect to the geodesic flow on S ∗ M satisfies Pr(a u ) < 0, and let ε > 0 be such that Pr(a u ) + ε < 0. Then, there exists e0 (ε) > 0 such that for any τ ∈ Spec B with | Re τ | ≥ e0 (ε), we have Im τ ≤ Pr(a u ) + ε. The presence of a spectral gap of finite width below the real axis is not obvious a priori if the geometric control does not hold, since there may be a possibility for | Im τn | to become arbitrary small as n → ∞ : see for instance [Hit], Theorem 1.3. However, this accumulation on the real axis can not occur faster than a fixed exponential rate, as it was shown in [Leb] that ∃C > 0 such that ∀τ ∈ Spec B, Im τ ≤ −
1 −C| Re τ | . e C
Let us mention a result comparable to Theorem 1 in the framework of chaotic scattering obtained recently by Nonnenmacher and Zworski [NoZw], in the semiclassical setting. For a large class of Hamiltonians, including P() = − + V on Rd with V compactly supported, they were able to show a resonance-free region near the energy E: ∃δ, γ > 0 such that Res(P()) ∩ ([E − δ, E + δ] − i[0, γ ]) = ∅ for 0 < ≤ δ,γ .
Energy Decay for the Damped Wave Equation Under a Pressure Condition
379
This holds provided that the hamiltonian flow t on the trapped set K E at energy E is hyperbolic, and that the pressure of the unstable Jacobian with respect to the geodesic flow on K E is strictly negative. We will adapt several techniques of [NoZw] to prove Theorem 1, some of them coming back to [Ana1]. In a recent paper, Anantharaman [Ana2] studied the spectral deviations of Spec B with respect to the line of accumulation Im z = −a¯ appearing in (1.6). In the case of constant negative curvature, she obtained an upper bound for the number of modes with imaginary parts above −a, ¯ and showed that for α ∈ [−a, ¯ 0[, there exists a function H (α) such that ∀c > 0, lim sup λ→∞
log Card{τn : Re τn ∈ [λ − c, λ + c], Im τn ≥ α} ≤ H (α). log λ (1.10)
H (α) is a dynamical quantity defined by
H (α) = sup{h K S (μ), μ ∈ M 1 , 2
adμ = −α},
where M 1 denotes the set of t −invariant measures on S ∗ M, and h K S stands for 2 the Kolmogorov–Sinai entropy of μ. As a consequence of Theorem 1, the result of Anantharaman is not always optimal : H (α) = 0 for α ∈ [−a, ¯ 0[, but if Pr(a u ) < 0, there is no spectrum in a strip of finite width below the real axis, i.e. the lim sup in (1.10) vanishes for some α = α(a) = 0. The operator B being non-selfadjoint, its eigenfunctions may fail to form a Riesz basis of H. However, if a solution u of (1.3) has initial data sufficiently regular, it is still possible to expand it on eigenfunctions which eigenmodes are close to the real axis, up to an exponentially small error in time: Theorem 2 (Eigenvalues expansion). Let ε > 0 such that Pr(a u ) + ε < 0, and κ > d2 . There exists e0 (ε) > 0, n = n(ε) ∈ N and a (finite) sequence τ0 , . . . , τn−1 of eigenvalues of B with τ j ∈ [−e0 (ε), e0 (ε)] + i[Pr(a u ) + ε, 0],
j ∈ 0, n − 1,
such that for any solution u(t, x) of (1.3) with initial data ω ∈ Hκ , we have u(t, x) =
n−1
e− i tτ j u j (t, x) + rn (t, x), t > 0.
j=0
The functions u j , r n satisfy
u j (t, ·) H ≤ Ct m j ω H and rn (t, ·) H ≤ Cε et (Pr(a
u )+ε)
ω Hκ ,
where m j denotes the multiplicity of τ j , the constant C > 0 depends only on M and a, while Cε > 0 depends on M, a and ε. A similar eigenvalues expansion can be found in [Hit], where no particular assumption on the curvature of M is made, however the geometric control must hold. Our last result deals with an exponential decay of the energy, which will be derived as a consequence of the preceding theorem:
380
E. Schenck
Theorem 3 (Exponential energy decay). Let ε > 0, (τ j )0≤ j≤n(ε)−1 , κ and u as in Theorem 2. Set by convention τ0 = 0. The energy E(u, t) satisfies ⎛ ⎞2 n−1 u E(u, t) ≤ ⎝ et Im τ j t m j C ω H + Cε et (Pr(a )+ε) ω Hκ ⎠ , j=1
where m j denotes the multiplicity of τ j . The constant C > 0 depends only on M and a, while Cε > 0 depends on M, a and ε. In particular, ρ(κ) = 2 min(G, | Pr(a u ) + ε|) > 0. Remark. In our setting, it may happen that geometric control does not hold, while Pr(a u ) < 0. In this particular situation, it follows from [BLR] that we can not have an exponential energy decay uniformly for all Cauchy data in H, where by uniform we mean that the constant C appearing in (1.7) does not depend on u. However, if for κ > d2 we look at ρ(κ) instead of ρ(0), our results show that we still have uniform exponential decay, namely ρ(κ) > 0 while ρ(0) = 0. 1.1. Semiclassical reduction. The main step yielding to Theorem (1) is more easily achieved when working in a semiclassical setting. From the eigenvalue equation (1.5), we are lead to study the equation P(τ )u = 0, where Im τ = O(1). To obtain a spectral gap below the real axis, we are lead to study eigenvalues with arbitrary large real parts since Spec B is discrete. For this purpose, we introduce a semiclassical parameter ∈ ]0, 1], and write the eigenvalues as τ=
1 + ν, ν ∈ C, |ν| = O(1). →0
If we let go to 0, the eigenvalues τ we are interested in then satisfy τ −−−→ 1. Putting τ = λ and z = λ2 /2, we rewrite the stationary equation 2 √ − − z − i qz u = 0, qz (x) = 2za(x). 2 Equivalently, we write (P(z, ) − z)u = 0,
(1.11)
where P(z, ) = − 2 − i qz . The parameter z plays the role of a complex eigenvalue of the non-selfadjoint quantum Hamiltonian P. In the semiclassical limit, it is close to the “energy” E = 1/2, while Im z is of order and represents the “decay rate” of the mode. In order to recall these properties, we will also often write 2
z=
1 + ζ, ζ ∈ C and |ζ | = O(1). 2
(1.12)
In most of the following, we will deal with the semiclassical analysis of the nonselfadjoint Schrödinger operator P(z, ) and the associated Schrödinger equation i ∂t = P(z, )
with L 2 = 1.
(1.13)
Energy Decay for the Damped Wave Equation Under a Pressure Condition
381
The basic facts and notations we will use from semiclassical analysis are recalled in Appendix A. The operator P has a principal symbol equal to p(x, ξ ) = 21 gx (ξ, ξ ), and a subprincipal symbol given by − i qz . Note that the classical Hamiltonian p(x, ξ ) generates the geodesic flow on the energy surface p −1 ( 21 ) = S ∗ M. The properties of the geodesic flow on S ∗ M which will be useful to us are summarized in the next section, where is also given an alternative definition of the topological pressure more adapted to our purposes. We will denote the quantum propagator by U t ≡ e− P , it
so that if ∈ L 2 (M) satisfies (1.13), we have (t) = U t (0). Using standard methods of semiclassical analysis, one can show that U t is a Fourier integral operator (see [EvZw], Chap. 10) associated with the symplectic diffeomorphism given by the geodesic flow
t . Since we assumed that a ≥ 0, it is true that U t L 2 →L 2 ≤ 1, ∀t ≥ 0. Denote 1 = 1 () = {z = 2
2
1 + O() ∈ C, ∃ ∈ L 2 (M), (P(z, ) − z) = 0}. 2
If z ∈ 1 and is such that (1.11) holds, the semiclassical wave front set of satisfies 2
WF() ⊂ S ∗ M. This comes from the fact that is an eigenfunction associated with the eigenvalue 21 of a pseudodifferential operator with principal symbol p(x, ξ ) = 21 gx (ξ, ξ ). Using these semiclassical settings, we will show the following key result : Theorem 4. Let ε > 0 be such that Pr(a u ) + ε < 0, and z ∈ 1 (). There exists 2 0 = 0 (ε) such that ≤ 0 ⇒
Im z ≤ Pr(a u ) + ε.
From (1.12), equation implies Im τ ≤ Pr(a u ) + ε +O() √we also notice that the above −1 −1 since τ = 2z, and then Im τ = Im z + O(). It follows by rescaling that Theorem 4 is equivalent to Theorem 1. 2. Quantum Dynamics and Spectral Gap 2.1. Hyperbolic flow and topological pressure. We call
t = et H p : T ∗ M → T ∗ M the geodesic flow, where H p is the Hamilton vector field of p. In local coordinates, def
Hp =
d ∂p ∂p ∂x − ∂ξ = { p, ·}, ∂ξi i ∂ xi i i=1
where the last equality refers to the Poisson bracket with respect to the canonical symd plectic form ω = i=1 dξi ∧ d xi . Since M has strictly negative curvature, the flow
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E. Schenck
generated by H p on constant energy layers E = p −1 (E) ⊂ T ∗ M, E > 0 has the Anosov property: for any ρ ∈ E, the tangent space Tρ E splits into flow, stable and unstable subspaces Tρ E = RH p ⊕ E s (ρ) ⊕ E u (ρ). The spaces E s (ρ) and E u (ρ) are d − 1 dimensional, and are preserved under the flow map: ∀t ∈ R, d tρ (E s (ρ)) = E s ( t (ρ)), d tρ (E u (ρ)) = E u ( t (ρ)). Moreover, there exists C, λ > 0 such that i) d tρ (v) ≤ C e−λt v , for all v ∈ E s (ρ), t ≥ 0,
(2.1)
−λt
v , for all v ∈ E u (ρ), t ≥ 0. ii) d −t ρ (v) ≤ C e
One can show that there exists a metric on T ∗ M called the adapted metric, for which one can take C = 1 in the preceding equations. At each point ρ, the spaces E u (ρ) are tangent to the unstable manifold W u (ρ), the set of points ρ u ∈ E such that d( t (ρ u ), t→−∞
t (ρ)) −−−−→ 0, where d is the distance induced from the adapted metric. Simis larly, E (ρ) is tangent to the stable manifold W s (ρ), the set of points ρ s such that t→+∞ d( t (ρ s ), t (ρ)) −−−−→ 0. The adapted metric induces the volume form ρ on any d dimensional subspace of T (Tρ∗ M). Using ρ , we now define the unstable Jacobian at ρ for time t. Let us define the weak-stable and weak-unstable subspaces at ρ by E s,0 (ρ) = E s (ρ) ⊕ RH p ,
E u,0 (ρ) = E u (ρ) ⊕ RH p .
We set Jtu (ρ) = det d −t | E u,0 ( t (ρ)) =
ρ (d −t v1 ∧ · · · ∧ d −t vd ) , t (ρ) (v1 ∧ · · · ∧ vd )
def
J u (ρ) = J1u (ρ),
where (v1 , . . . , vd ) can be any basis of E u,0 (ρ). While we do not necessarily have J u (ρ) < 1, it is true that Jtu (ρ) decays exponentially as t → +∞. The definition of the topological pressure of the geodesic flow given in the Introduction, although quite straightforward to state, is not really suitable for our purposes. The alternative definition of the pressure we will work with is based on refined covers of S ∗ M, and can be stated as follows. For δ > 0, let E δ = p −1 ] 21 − δ, 21 + δ[ be a thin neighbourhood of the constant energy surface p −1 ( 21 ), and V = {Vα }α∈I a finite number of open sets of E δ such that V is an open cover of p −1 [ 21 − 43 δ, 21 + 43 δ]. In what follows, we shall always choose δ < 1/2. For T ∈ N∗ , we define the refined cover V (T ) , made of the sets Vβ =
T −1
−k (Vbk ), β = b0 b1 . . . bT −1 ∈ I T .
k=0
It will be useful to coarse-grain any continuous function f on E δ with respect to V (T ) by setting f T,β = sup
T −1
ρ∈Vβ i=0
f ◦ i (ρ).
Energy Decay for the Damped Wave Equation Under a Pressure Condition
One then defines Z T (V, f ) = inf BT
⎧ ⎨ ⎩
exp( f T,β ) : BT ⊂ I T , E 3δ/4 ⊂
β∈BT
383
β∈BT
⎫ ⎬ Vβ
⎭
.
The topological pressure of f with respect to the geodesic flow on E 3δ/4 is defined by : Pr δ ( f ) =
lim
lim
diam V →0 T →∞
1 log Z T (V, f ). T
The pressure on the unit tangent bundle S ∗ M is simply obtained by continuity, taking the limit Pr( f ) = limδ→0 Pr δ ( f ). To make the above limits easier to work with, we now take f = a u and fix ε > 0 such that Pr(a u ) + ε < 0. Then, we choose the width of the energy layer δ ∈ ]0, 1[ sufficiently small such that | Pr(a u ) − Pr δ (a u )| ≤ ε/2. Given a cover V = {Vα }α∈A of E 3δ/4 (with arbitrary small diameter), there exists a time t0 depending on the cover V such that 1 log Z t (V, a u ) − Pr δ (a u ) ≤ ε . 0 2 t 0 Hence there is a subset of t0 −strings Bt0 ⊂ At0 such that {Vα }α∈Bt0 is an open cover of E 3δ/4 and satisfies ε exp(a u t0 ,β ) ≤ exp t0 (Pr δ (a u ) + ) ≤ exp t0 (Pr(a u ) + ε) . (2.2) 2 β∈Bt0
For convenience, we denote by {Wβ }β∈Bt0 ≡ {Vβ }β∈Bt0 the sub-cover of V (t0 ) such that (2.2) holds. Note that in this case, the diameter of V, t0 and then W depends on ε. 2.2. Discrete time evolution. Let {ϕβ }β∈Bt0 be a partition of unity adapted to W, so that def
its Weyl quantization ϕβw = β (see Appendix A) satisfies WF(β ) ⊂ E δ , ∗β = β , β = 1 microlocally near E δ/2 . β
We will also consider a partition of unity {ϕ˜α }α∈A adapted to the cover V, and its Weyl ˜ def quantization = ϕ˜ w . In what follows, we will be interested in the propagator U N t0 +1 , and N = T log −1 , T > 0. It is important to note that T can be arbitrary large, but is fixed with respect to . The propagator U N t0 is decomposed by inserting β∈Bt β at each time step of length t0 . 0 Setting first Uβ = U t0 β , we have (microlocally near E δ/2 ) the equality U t0 = β Uβ , and then U N t0 = Uβ N . . . Uβ1 , near E δ/2 . (2.3) β1 ,β2 ,...,β N ∈Bt0
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E. Schenck
2.3. Proof of Theorem 4. We begin by choosing χ ∈ C0∞ (T ∗ M) such that supp χ E δ and χ ≡ 1 on E δ/4 , and considering Op(χ ). Applying the triangle inequality, we get immediately
U N t0 +1 Op(χ ) ≤
Uβ N . . . Uβ1 U 1 Op(χ ) + O L 2 →L 2 (∞ ). (2.4) β1 ,β2 ,...,β N ∈BtN0
Unless otherwise stated, the norms · always refer to · L 2 →L 2 or · L 2 , according to the context. The proof of Theorem 4 relies on the following intermediate result, proven much later in Sect. 4. Proposition 5 (Hyperbolic dispersion estimate). Let ε > 0, and δ, V, t0 be as in Sect. 2.1. For N = T log −1 , T > 0, take a sequence β1 , . . . β N and W1 , . . . , Wβ N the associated open sets of the refined cover W. Finally , let Op(χ ) be as above. There exists a constant C > 0 and 0 (ε) ∈ ]0, 1[ such that ≤ 0 ⇒ U t0 β N . . . U t0 β1 U 1 Op(χ ) ≤ C−d/2
N
e
a u t0 ,β j
j=1
where a u t0 ,β = supρ∈Wβ manifold M.
t0 −1 j=0
a u ◦ j (ρ). The constant C only depends on the
We also state the following crucial consequence: Corollary 6. Take ε > 0 such that Pr(a u )+ε < 0. There exists C > 0 and 0 (ε) ∈ ]0, 1[ such that d
≤ 0 ⇒ U N t0 +1 Op(χ ) ≤ C− 2 e N t0 (Pr(a
u )+ε)
.
The constant C only depends on M. Proof. Given ε > 0, we choose δ, V, t0 , W as in the preceding proposition. Using (2.4), we then have ⎞ ⎛ N u d a ⎝ e t0 ,β j +O(∞ )⎠
U N t0 +1 Op(χ ) ≤ C− 2 β1 ...β N ∈BtN0
⎛ ≤ C
− d2
⎝
e
a u
j=1
⎞N t0 ,β
⎠ + O(∞ ).
β∈Bt0
To get the second line, notice that the number of terms in the sum is of order (Card Bt0 ) N = −T log Card Bt0 . From our choice of ε and δ, we can use (2.2), and for small enough, rewrite this equation as d
U N t0 +1 Op(χ ) ≤ C− 2 e N t0 (Pr d
δ (a u )+ε/2)
≤ C− 2 e N t0 (Pr(a where C > 0 only depends on the manifold M.
u )+ε)
,
Energy Decay for the Damped Wave Equation Under a Pressure Condition
385
Let us show how this result implies Theorem 4. We assume that satisfies (1.11), and therefore U N t0 +1 = e
(N t0 +1) Im z
. Notice also that we have
Op(χ ) = + O(∞ ), since WF() ⊂ S ∗ M, and then
U N t0 +1 Op(χ ) = U N t0 +1 + O(∞ ) = e(N t0 +1) Im z + O(∞ ). It follows from the corollary that e
N t0 +1
Im z
d
≤ C− 2 e N t0 (Pr(a
u )+ε)
+ O(m ),
where m can be arbitrary large. Taking the logarithm, this yields to Im z d 1 log C − log + Pr(a u ) + ε + O( ). ≤ N t0 2N t0 N t0 But given ε > 0, we can take N = T log −1 with T arbitrary. Hence there is 0 (ε) ∈ ]0, 1[ and T sufficiently large, such that ≤ 0 (ε) ⇒
Im z ≤ Pr(a u ) + 2ε.
Since the parameter ε can be chosen as small as wished, this proves Theorem 4. 3. Eigenvalues Expansion and Energy Decay 3.1. Resolvent estimates. To show the exponential decay of the energy, we follow a standard route from resolvent estimates in a strip around the real axis. Let us denote Q(z, ) = −
√ 2 − z − i 2za(x) = P(z, ) − z. 2
The following proposition establishes a resolvent estimate in a strip of width | Pr(a u )+ε| below the real axis in the semiclassical limit. This is the main step toward Theorems 2 and 3, see also [NoZw2] for comparable resolvent estimates in the chaotic scattering situation: Proposition 7. Let ε > 0, and z =
1 2
+ ζ, with |ζ | = O(1) satisfying
Pr(a u ) + ε ≤ Im ζ ≤ 0.
(3.1)
There exists 0 (ε) > 0, Cε > 0 depending on M, a and ε such that d
≤ 0 (ε) ⇒ Q(z, )−1 L 2 →L 2 ≤ Cε −1− 2 log −1 .
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E. Schenck
Proof. Given ε > 0, we choose ε = ε/4, and fix 0 (ε ) so that Corollary 6 holds with ε . Note that ≤ 0 implies that any z ∈ 1 satisfies −1 Im z ≤ Pr(a u )+2ε < Pr(a u )+ε. 2
This implies that for ≤ 0 , any z = 21 +ζ with ζ satisfying (3.1) is not in the spectrum. Hence we fix from now on ≤ 0 (ε ), and Q(z, )−1 is well defined. Finally, we set Pr(a u )+ = Pr(a u ) + ε . In order to bound Q(z, )−1 , we proceed in two steps, by finding two operators which approximate Q −1 : one on the energy surface E δ , the other outside E δ . Let χ be as in Sect. 2, and choose also Let χ˜ ∈ C0∞ (T ∗ M) with supp χ˜ supp χ , such that we also have χ˜ = 1 near S ∗ M. We first look for an operator A0 = A0 (z, ) such that Q A0 = (1 − Op(χ )) + O L 2 →L 2 (∞ ). def
˜ = Q 0 (z, ). Because of the property of χ˜ , the For this, consider Q(z, ) + i Op(χ) operator Q 0 is elliptic. Hence, there is an operator A˜ 0 , uniformly bounded in L 2 (M), such that Q 0 A˜ 0 = 1 + O L 2 →L 2 (∞ ). The operator A0 we are looking for is obtained by taking A0 = A˜ 0 (1−Op(χ )). Indeed, Q(z, )A0 (z, ) = 1 − Op(χ ) − i Op(χ˜ ) A˜ 0 (1 − Op(χ )) + O L 2 →L 2 (∞ ) = 1 − Op(χ ) + O L 2 →L 2 (∞ ), since χ˜ and 1 − χ have disjoint supports by construction. We now look for the solution on E δ . From Corollary 6, we have an exponential decay of the propagator U N t0 if N becomes large. To use this information, we set i A1 (z, , T) =
0
T
it
U t e z Op(χ ) dt,
where T has to be adjusted. Hence, it
Q(z, )A1 (z, , T) = Op(χ ) − U t e z |t=T Op(χ ) = Op(χ ) + R1 . Since Im z/ = Im ζ = O(1), we have
R1 = e−T Im ζ U T Op(χ ) .
(3.2)
From Corollary 6, we know that for N = T log −1 with ≤ 0 , d
U N t0 +1 Op(χ ) ≤ C− 2 e N t0 Pr(a
u )+
.
We observe that this bound is useful only if does not diverge as → 0, which is the case if T ≥
d def = T0 . 2t0 | Pr(a u )+ |
Energy Decay for the Damped Wave Equation Under a Pressure Condition
387
Let us define T0 = T0 t0 log −1 + 1 =
d log −1 + 1. 2| Pr(a u )+ |
(3.3)
If T = T log −1 t0 + 1, with T > T0 chosen large enough, we find
R1 ≤ CT t0 (Pr(a
u )+ε)
d
− 2 −T t0 Pr(a
u )+
= O(m )
with m = m(T) ≥ 0, since ε − ε > 0. Consequently, there is T1 = T1 (ε) > 0 such that T1 = T1 t0 log −1 + 1 satisfies m(T1 ) = 0. This means that for T ≥ T1 , we have Q(z, )(A0 (z, ) + A1 (z, , Th )) = 1 + O L 2 →L 2 (1), in other words, A0 + A1 is “close” to the resolvent Q −1 . Hence, we impose now T ≥ T1 , and evaluate the norms of A0 and A1 . By construction, A0 = O(1). For A1 , we have to estimate an integral of the form T e−t Im ζ U t Op(χ ) dt. IT = 0
Let us split the integral according to T0 , and use the decay of U t Op(χ ) for t ≥ T0 : ∞ d 0 u + |IT | ≤ T0 e−T Im ζ +C− 2 e−t Im ζ e(t−1) Pr(a ) dt T0
≤ T0 e
−T0 Im ζ
0 −T0 Im ζ
≤ Cε T e
d
(1 + Cε − 2 e(Th −1) Pr(a 0
u )+
)
.
Using (3.3) and (3.1), this gives d
A1 (z, , T) ≤ Cε −1− 2 log −1 , where Cε > 0 depends now on M, a and ε. We now translate these results obtained in the semiclassical settings in terms of τ . Recall 1 P(τ ) = − − τ 2 − 2 i aτ ≡ 2 Q(z, ), def
and set R(τ ) = P(τ )−1 . The operator R(τ ) is directly related to the resolvent (τ −B)−1 : a straightforward computation shows that R(τ )(−2 i a − τ ) −R(τ ) −1 . (τ − B) = R(τ )(2 i aτ − τ 2 ) −R(τ )τ Proposition 8. Let ε > 0 be such that Pr(a u ) + ε < 0. Let τ ∈ C\ Spec B be such that 1 Pr(a u ) + ε ≤ Im τ < 0. Set also τ = (1 + |τ |2 ) 2 . For |τ | large enough, there exists a constant C > 0 depending on M, a and ε such that for any κ > d/2, we have d
(i) R(τ ) L 2 →L 2 ≤ Cε τ −1+ 2 logτ , d
(ii) R(τ ) L 2 →H 2 ≤ Cε τ 1+ 2 logτ , (iii) R(τ ) H κ →H 1 ≤ Cε , (iv) τ R(τ ) H κ →H 0 ≤ Cε .
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E. Schenck
Proof. (i) follows directly from rescaling the statements of the preceding proposition. For (ii), observe that
R(τ )u H 2 ≤ C( R(τ )u L 2 + R(τ )u L 2 ), C > 0. But
R(τ )u L 2 ≤ u L 2 + |τ 2 + 2τ i a| R(τ )u L 2 , so using (i), we get d
R(τ )u H 2 ≤ C (1 + |τ 2 + 2 i aτ |) R(τ )u L 2 + u L 2 ≤ Cε τ 1+ 2 logτ u L 2 . To arrive at (iii), we start from the following classical consequence of the Hölder inequality:
R(τ )u 2H 1−s ≤ R(τ )u 1−s
R(τ )u 1+s , L2 H2
s > 0.
(3.4)
From (i) and (ii), we obtain d
R(τ )u H 1−s ≤ Cε τ 2 −s logτ u L 2 . If we choose s > d2 , we get R(τ ) H 0 →H 1−s ≤ Cε . Hence, for any s ≥ 0 we have
R(τ ) H s →H s +1−s ≤ Cε . Taking s = s shows (iii), where we must have κ > d2 . The last equation (iv) is derived as (iii), by considering
τ R(τ )u 2H 1−s ≤ |τ |2 R(τ )u 1−s
R(τ )u 1+s , s > 0, L2 H2 and choosing s so that τ R(τ ) 2H 0 →H 1−s ≤ Cε . 3.2. Eigenvalues expansion . We now prove Theorem 2. Let us fix ε > 0 so that Pr(a u ) + ε < 0. From Theorem 1 we know that def Card Spec B ∩ (R + i[Pr(a u ) + ε, 0]) = n(ε) < ∞. Hence there is e0 (ε) > 0 such that Spec B ∩ (R + i[Pr(a u ) + ε]) ⊂ , where = (ε) = [−e0 , e0 ] + i[Pr(a u ) + ε, 0]. We then call {τ0 , . . . , τn(ε)−1 } = Spec B ∩ , and set by convention τ0 = 0. We define as above Pr(a u )+ = Pr(a u ) + ε. Since we look at the eigenvalues τ ∈ , let us introduce the spectral projectors on the generalized eigenspace E j for j ∈ 0, n − 1: 1 j = (τ − B)−1 dτ, j ∈ L(H, D(B ∞ )), 2 i π γj where γ j are small circles centered in τ j . We also denote by =
n j=0
j
Energy Decay for the Damped Wave Equation Under a Pressure Condition
389
the spectral projection onto nj=0 E j . We call E 0 the eigenspace corresponding to the eigenvalue τ0 = 0. It can be shown [Leb] that E 0 is one dimensional over C and spanned by (1, 0), so 0 ω = (c(ω), 0) with c(ω) ∈ C. Let now ω = (ω0 , ω1 ) be in Hκ . Near a pole τ j of (τ − B)−1 , we have mj
−1
(τ − B)
(B − τ j )k−1 j j = + + H j (τ ), τ − τj (τ − τ j )k k=2
where H j is an operator depending holomorphically on τ in a neighbourhood of τ j , and m j is the multiplicity of τ j . Since ∈ L(H, D(B ∞ )), we have the following integral representation of e− i t B ω, with absolute convergence in H: +∞+i α 1 e− i t B ω = e− i tτ (τ − B)−1 ωdτ, t > 0, α > 0. (3.5) 2 i π −∞+i α The integrand in the right hand side has poles located at τ j , j ∈ 0, n − 1, so that mj 1 (B − τ j )k−1 j e− i tτ ωdτ 2 i π γj (τ − τ j )k j k=1 def − i tτ j = e− i tτ j pτ j (t)ω = e u j (t),
e− i t B ω =
j
j
where pτ j (t) = j +
mj (− i t)k−1 k=2
(k − 1)!
(B − τ j )k−1 j .
The operators pτ j (t) are polynomials in t, with degree at most m j , taking their values in L(H, D(B ∞ )). It follows that for some C > 0 depending only on M and a,
u j (t) H ≤ Ct m j ω H . The remainder term appearing in Theorem 2 is now identified: rn (t) = e− i t B (1 − )ω. To conclude the proof, we have therefore to evaluate rn H . To do so, we will use in a crucial way the resolvent bounds below the real axis that we have obtained in the preceding section. We consider the solution u(t, x) of (1.1) with initial data u = (u 0 , u 1 ) = (1 − )ω, with ω ∈ Hκ , κ > d/2. Let us define χ ∈ C ∞ (R), 0 ≤ χ ≤ 1, such that χ = 0 for t ≤ 0 and χ = 1 for t ≥ 1. If we set v = χ u, we have (∂t2 − + 2a∂t )v = g1 ,
(3.6)
g1 = χ u + 2χ ∂t u + 2aχ u.
(3.7)
where
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E. Schenck
Note also that supp g1 ⊂ [0, 1] × M, and v(t) = 0 for t ≤ 0. Let us denote the inverse Fourier transform in time by ˇ )= ei tτ u(t)dt. Ft→−τ : u → u(τ R
Applying Ft→−τ (in the distributional sense) to both sides of (3.6) yields to P(τ )v(τ, ˇ x) = gˇ 1 (τ, x). We then remark that R(τ )gˇ 1 (τ, x) is the first component of i(τ − B)−1 Ft→−τ χ (t)(u, i ∂t u) . From the properties of , it is clear that the operator (τ − B)−1 (1 − ) depends holou + − i t B (1 − )ω, morphically on τ in the half-plane Im τ ≥ Pr(a ) . From (u, i ∂t u) = e we then conclude that i(τ − B)−1 Ft→−τ χ (t)(u, i ∂t u) depends also holomorphically on τ in the half plane Im τ ≥ Pr(a u )+ . Hence v(τ, ˇ x) = R(τ )gˇ 1 (τ, x) and an application of the Parseval formula yields to
e−t Pr(a
u )+
v L 2 (R+ ,H 1 ) = v(· ˇ + i Pr(a u )+ L 2 (R,H 1 ) = R(· + i Pr(a u )+ )gˇ 1 (· + i Pr(a u )+ ) L 2 (R,H 1 ) ≤ Cε gˇ 1 (· + i Pr(a u )+ ) L 2 (R,H κ ) ≤ Cε g1 L 2 (R+ ,H κ ) ,
where we have used Proposition 8. The term appearing in the last line can in fact be controlled by the initial data. From (3.7), we have
g1 L 2 (R+ ;H κ ) ≤ C u L 2 ([0,1];H κ ) + ∂t u L 2 ([0,1];H κ ) . (3.8) A direct computation shows ∂t u 2Hκ ≤ C( u 2Hκ + ∂t u 2H κ + ∇u 2H κ ). The Gronwall inequality for t ∈ [0, 1] gives t ( ∂s u(s) 2H κ + ∇u(s) 2H κ )ds
u(t, ·) 2H κ ≤ C u(0, ·) 2H κ + 0
≤ C ω 2Hκ , since the κ−energy E κ (t, u) =
1 ( ∂t u 2H κ + ∇u 2H κ ) 2
is also decreasing in t. Coming back to (3.8), we see that g1 L 2 (R+ ;H κ ) ≤ C ω Hκ and then,
e−t (Pr(a
u )+ε)
v(t, x) L 2 (R+ ,H 1 ) ≤ Cε ω Hκ .
This is the exponential decay we are looking for, but in the integrated form. It is now easy to see that
u(t, ·) H 1 ≤ Cε et (Pr(a
u )+ε)
ω Hκ .
Energy Decay for the Damped Wave Equation Under a Pressure Condition
391
We have to check that the same property is valid for ∂t u. Using the same methods as above, we also have P(τ )Ft→−τ (∂t v) = −τ gˇ 1 (τ ), and then, Ft→−τ (∂t v) = −τ R(τ )gˇ 1 (τ ). It follows that
e−t Pr(a
u )+
∂t v(t, x) L 2 (R+ ,H 0 ) = v(τ ˇ + i Pr(a u )+ L 2 (R,H 0 ) = τ R(τ + i Pr(a u )+ )gˇ 1 (τ + i Pr(a u )+ , x) L 2 (R,H 0 ) ≤ Cε gˇ 1 (τ + i Pr(a u )+ , x) L 2 (R,H κ ) ≤ Cε g1 (t, x) L 2 (R+ ,H κ ) .
Grouping the results, we see that
u H ≤ Cε et (Pr(a
u )+ε)
ω Hκ ,
and this concludes the proof of Theorem 2. 3.3. Energy decay. We end this section with the proof of Theorem 3, which gives the exponential energy decay. This is an immediate consequence of the following lemma, that tells us that the energy can be controlled by the H 1 norm of u, for t ≥ 2: Lemma 9. There exists C > 0 such that for any solution u of (1.1) and E(u, t) the associated energy functional, we have E(u, T ) ≤ C u 2L 2 ([T −2,T +1];H 1 ) , T ≥ 2. Proof. This is a standard result, we borrow the proof from [EvZw]. For T > 2, we choose χ2 ∈ C ∞ (R), 0 ≤ χ2 ≤ 1 such that χ2 (t) = 1 for t ≥ T and χ2 (t) = 0 if t ≤ T − 1. Setting u 2 (t, x) = χ2 (t)u(t, x), we have (∂t2 − + 2a∂t )u 2 = g2 for g2 = χ2 u + 2χ2 ∂t u + 2aχ2 u. Note that g2 is compactly supported in t. Define now 1 E 2 (u, t) = (|∂t u 2 |2 + |∇u 2 |2 )dvol, 2 M and compute E 2 (u, t) = ∂t2 u 2 , ∂t u 2 − u 2 , ∂t u 2 = −2a∂t u 2 , ∂t u 2 + g2 , ∂t u 2 ≤C |∂t u 2 |(|∂t u| + |u|)dvol M (|∂t u|2 + |u|2 )dvol . ≤ C E 2 (u, t) + M
We remark that E 2 (u, T − 1) = 0 and E 2 (u, T ) = E(u, T ), so the Gronwall inequality on the interval [T − 1, T ] gives E(u, T ) ≤ C ∂t u 2L 2 ([T −1,T ];L 2 ) + u 2L 2 ([T −1,T ];L 2 ) . (3.9)
392
E. Schenck
To complete the proof, we need to bound the term ∂t u 2L 2 ([T −1,T ];L 2 ) . For this purpose, we choose χ3 ∈ C ∞ (R), 0 ≤ χ3 ≤ 1 such that χ3 (t) = 1 for t ∈ [T − 1, T ] and χ3 (t) = 0 if t ≤ T − 2 and t ≥ T + 1. From (1.1), we get T +1 χ32 u, ∂t2 u − u + 2a∂t udt 0= =
T −2 T +1 T −2
−χ32 ∂t u, ∂t u − 2χ3 χ3 u, ∂t u + 2χ32 u, a∂t u + χ32 u, −udt,
whence
∂t u L 2 ([T −1,T ];L 2 ) ≤ C u L 2 ([T −2,T +1];H 1 ) . Substituting this bound in (3.9) yields the result. Theorem 3 follows now from the preceding lemma. Let us denote by u j (t, x) and rn (t, x) the first component of pτ j (t)ω and e− i t B (1 − )ω respectively. We learned above that u(t, x) =
n
e− i tτ j u j (t, x) + rn (t, x)
j=0
with u j (t, ·) H 1 ≤ Ct m j ω Hκ , and rn (t, ·) H 1 ≤ Cε et (Pr(a )+ε) ω Hκ . Suppose first that the projection of ω on E 0 vanishes, i.e. 0 ω = 0. Then, from the preceding lemma we clearly have u
1
E(u, t) 2 ≤
n
et Im τ j C u j (t, x) H 1 + Cε et (Pr(a
u )+ε)
ω Hκ .
j=1
This shows Theorem 3 when 0 ω = 0. But the general case follows easily: we can write u(t, ˜ x) = u(t, x) − 0 ω for which we have the expected exponential decay, and notice that E(u, ˜ t) = E(u, t) since 0 ω is constant. 4. Hyperbolic Dispersion Estimate This last section is devoted to the proof of Proposition 5. Let ε, δ, V, W and Op(χ ) be as in Sect. 2. We also set N = T log −1 , T > 0. 4.1. Decomposition into elementary Lagrangian states. Recall that each set Wβ ≡ Wb0 ...bt0 −1 in the cover W has the property
k (Wβ ) ⊂ Vbk , k ∈ 0, t0 − 1
(4.1)
for some sequence b0 , b1 , . . . , bt0 −1 . To every sequence β1 , . . . , β N of open sets Wβk , we associate a sequence γ1 , . . . γ N t0 of open sets Vγk ⊂ V such that
k (Wβi ) ⊂ Vγ(i−1)t0 +k+1 k ∈ 0, t0 − 1. Hence, any point of Wi visits the sets Vγ(i−1)t0 +1 , . . . , Vγ(i−1)t0 +t0 at the times 0, . . . , t0 −1.
Energy Decay for the Damped Wave Equation Under a Pressure Condition
393
We now decompose further each evolution of length t0 in (2.3) by inserting additional quantum projectors. To unify the notations, we define for j ∈ 1, N t0 the following projectors and the corresponding open sets in T ∗ M: βk if j − 1 = kt0 , k ∈ N Wβk if j − 1 = kt0 , k ∈ N Pγ j = ˜ , Vγ j = γ j if j − 1 = 0 mod t0 Vγ j if j − 1 = 0 mod t0 . (4.2) C0∞ (T ∗ M)
We will also denote by Fγ ∈ the function such that supp Fγ ⊂ Vγ and Pγ = Fγw . Let us set up also a notation concerning the constants appearing in the various estimates we will deal with. Let , K ∈ N be two parameters (independent of ), and e1 , e2 , e3 > 0 some fixed numbers. For a constant C depending on M and derivatives of χ , a, t (for t bounded) up to order e1 + e2 K + e3 , we will write C (,K ) (M, χ ), or simply C (K ) (M, χ ) if only one parameter is involved. If the constant C depends also on the cutoff functions Fγ and their derivatives, we will write C = C (,K ) (M, χ , V), in order to recall the dependence on the cutoff function χ supported inside E δ , and the refined cover V. We will sometimes use the notation C (,K ) (M, V) when no dependence on χ is assumed. Note that V depends implicity on ε since its diameter was chosen such that (2.2) holds. Using (4.1), standard propagation estimates give U t0 β1 = UPγt0 . . . UPγ1 + O L 2 →L 2 (∞ ), U ≡ U 1 , and similar properties for U t0 βk , k > 1. Finally, Uβ N t0 . . . Uβ1 U Op(χ ) = UPγ N t0 . . . UPγ1 U Op(χ ) + O L 2 →L 2 (∞ ).
(4.3)
Take now ∈ L 2 (M). In order to show Proposition 5, we will write Op(χ ) as a linear decomposition over some elementary Lagrangian states, and study the individual evolution of such elementary states by U N t0 +1 . This type of method comes back to [Ana1] and is the key tool to prove Proposition 5. The decomposition of Op(χ ) is obtained by making explicit the action of Op(χ ) in local coordinates (see Appendix A). When applying Op(χ ) to using local charts labelled by , we get η,x−z 0 1 x + z0 , η)ϕ (z 0 )φ (x)(z 0 )dη dz 0 [Op(χ )](x) = ei χ ( d (2π ) 2 = δχ ,z 0 (x)(z 0 )dz 0 ,
where we have defined def δχ ,z 0 (x) =
1 (2π )d
ei
η,x−z 0
χ(
x + z0 , η)ϕ (z 0 )φ (x)dη. 2
This is a Lagrangian state, for which the Lagrangian manifold is given by 0 = Tz∗0 M ∩ E δ ⊂ T ∗ M. def
394
E. Schenck
Geometrically, 0 corresponds to a small, connected piece taken out of the union of spheres {Tz∗0 M ∩ p −1 ( 21 + ν) , |ν| ≤ δ}. If we project and evolve according to the operator appearing in the right-hand side of (4.3), we get: sup U t Pγ N t0 . . . Pγ1 Uδχ ,z 0
|(x)|d x
U t Pγ N t0 . . . UPγ1 U Op(χ ) ≤
≤C
z
M
sup U t Pγ N t0 . . . Pγ1 Uδχ ,z 0
,
(4.4)
z
where C > 0 depends only on the manifold M. Hence we are lead by this superposition principle to study in detail states of the form U t Pγn . . . UPγ1 Uδχ ,z 0 , for n ∈ 1, N t0 and t ∈ [0, 1]. For simplicity, because the local charts will not play any role in the following, we will omit them in the formulæ. 4.2. Evolution of Lagrangian states and their Lagrangian manifolds. 4.2.1. Ansatz for short times In this section we investigate the first step of the sequence of projection–evolution given in (4.3): our goal is to describe the state U t δχ ,z 0 with t ∈ [0, 1]. Since U t is a Fourier integral operator, we know that U t δχ ,z 0 is a Lagrangian state, supported on the Lagrangian manifold def
0 (t) = t ( 0 ), t ∈ [0, 1]. Because of our assumptions on the injectivity radius, the flow t : 0 (s) → 0 (t) for 1 ≥ t ≥ s > 0, induces on M a bijection from π 0 (s) to π 0 (t). In other words, 0 (t) projects diffeomorphically on M for t ∈ ]0, 1], i.e. ker dπ | 0 (t) = 0: in this case, we will say that 0 (t) is projectible. This is the reason for introducing a first step of propagation during a time 1: the Lagrangian manifold 0 (0) is not projectible, but as soon as t ∈ ]0, 1], 0 (t) projects diffeomorphically. Treating separately this evolution for times t ∈ [0, 1] avoids some unnecessary technical complications. The remark above implies that the Lagrangian manifold 0 (t), t ∈ ]0, 1] is generated by the graph of the differential of a smooth, well defined function S0 : 0 (t) = {(x, dx S0 (t, x, z 0 )) : 1 ≥ t > 0, x ∈ π t ( 0 )}. This means that for t ∈ ]0, 1], we have the Lagrangian Ansatz: def
v 0 (t, x, z 0 ) = U t δχ ,z 0 (x) K −1 S (t,x,z ) 1 i 0 0 k 0 K 0 = bk (t, x, z 0 ) + B K (t, x, z 0 ) . e d (2π ) 2 k=0
(4.5)
The functions bk0 (t, x, z 0 ) are smooth, and x ∈ π 0 (t). Furthermore, given any multi index , they satisfy
∂x bk0 (t, ·, z 0 ) ≤ C,k ,
(4.6)
where the constants C,k depend only on M (via the Hamiltonian flow of p), the damping a, the cutoff function χ and their derivatives up to order 2k + . However, note that C0,0 only depends on M. The remainder satisfies B K0 ≤ C K , where the constant C K also depends on M, a, χ and is uniformly bounded with respect to x, z 0 . The base point z 0 will be fixed until Sect. 4.5, so it will be ommited in the following to simplify the notations.
Energy Decay for the Damped Wave Equation Under a Pressure Condition
395
4.2.2. Further evolution. In the sequence of projection–evolution (4.3), we then have performed the first step, and obtained an Ansatz for U t δχ , t ∈ ]0, 1] up to terms of order K −d/2 , for any K ≥ 0. The main goal of the next paragraphs consists in finding an Ansatz for the full state def
v n (t, x) = U t Pγn UPγn−1 . . . UPγ1 Uδχ , t ∈ [0, 1], n ≥ 1.
(4.7)
The β j are defined according to j − 1 mod t0 as in the preceding section, but here n is arbitrary in the interval 1, N t0 . Because the operator U t P is a Fourier integral operator, v j (t, x), j ≥ 1 is a Lagrangian state, with a Lagrangian manifold which will be denoted by j (t). This manifold consists in a small piece of j+t ( 0 ), because of the successive applications of the projectors Pγ between the evolution operator U. If j = 1, the Lagrangian manifold 1 (0) is given by 1 (0) = 0 (1) ∩ Vγ1 , and for t ∈ [0, 1] we have 1 (t) = t ( 1 (0)). For j ≥ 1, j (t) can be obtained by a similar procedure: knowing j−1 (1), we take for j (t), t ∈ [0, 1] the Lagrangian manifold def
j (0) = j−1 (1) ∩ Vγ j , and j (t) = t ( j (0)). Of course, if the intersection j−1 (1) ∩ Vγ j is empty, the construction has to be stopped, since by standard propagation estimates, v j is of order O(∞ ) and Proposition 5 is true. We will then assume that ∀ j ∈ 1, n, j (0) = ∅. One can show (see [AnNo], Sect. 3.4.1 for an argument) that the Lagrangian manifolds j (t) are projectible for all j ≥ 1. This is mainly because M has no conjugate points. In particular, any j (t) can be parametrized as a graph on M of a differential, which means that there is a generating function S j (t, x) such that j (t) = {x, dx S j (t, x)}. By extension, we will call a Lagrangian state projectible if its Lagrangian manifold is. Let us introduce now some notations that will be often used later. Suppose that x ∈ π j (t), j ≥ 1. Then, there is a unique y = y(x) ∈ π j (0) such that π ◦ t (y, d y S j (0, y)) = x. If we denote for t ∈ [0, s] the (inverse) induced flow on M by φ S−tj (s) : x ∈ π j (s) → π −t x, dx S j (s, x) ∈ π j (s − t), we have y(x) = φ S−tj (t) (x). If x ∈ π j (t), then by construction
−t−k (x, dx S j (t, x)) ∈ j−k (0) ⊂ j−k−1 (1), k ∈ 0, j − 1. By definition, we will write (x) = π −t−k (x, dx S j (t, x)) and φ S−kj (x) = π −k (x, dx S j (1, x)). φ S−t−k j (t)
396
E. Schenck
To summarize, our sequence of projections and evolutions can be cast into the following way: δχ
0
U1
/ v 0 (1, ·)
P1
/ v 1 (0, ·)
U
/ v 1 (1, ·)
P2
/ ...
Pn
/ v n (0, ·)
Ut
/ v n (t, ·),
1
/ 0 (1)
|V1
/ 1 (0)
1
/ 1 (1)
|V2
/ ...
|Vn
/ n (0)
t
/ n (t). (4.8)
On the top line are written the successive evolutions of the Lagrangian states, while the evolution of their respective Lagrangian manifolds is written below (the notation |V denotes a restriction to the set V ⊂ T ∗ M). 4.3. Evolution of a projectible Lagrangian state. Let Vγ and Pγ be as in (4.2). The next proposition contains an explicit description of the action of the Fourier integral operators U t P on projectible Lagrangian states localized inside Vγ . i
Proposition 10. Let Vγ and Pγ = Fγw be as in Sect. 4.1. Let w(x) = w(x) e ψ(x) be a projectible Lagrangian state, supported on a projectible Lagrangian manifold = {x, dx ψ(x)} ⊂ Vγ . def
Assume also that (t) = t is projectible for t ∈ [0, 1]. We have the following asymptotic development: i
[U t Pγ w](x) = e ψ(t,x)
K −1
k wk (t, x) + K r K (t, x),
(4.9)
k=0
where ψ(t, ·) is a generating function for (t). The amplitudes wk can be computed from the geodesic flow (via the function ϕγ ), the damping q and the function Fγ . Moreover, the following bounds hold:
wk C ≤ C,k w C +2k ,
r K C ≤ C,K w C +2K +d , where the constants depend on ϕγ , a, Fγ and their derivatives up to order + 2K + d, namely C,k = C (,k) (M, V). An explicit expression for wk will be given in the proof. Proof. The steps we will encounter below are very standard in the non-damping case, i.e. q = 0. If the diameter of the partition V of E δ is chosen small enough, we can assume without loss of generality the existence of a function ϕγ ∈ C ∞ ([0, 1] × Rd × Rd ) which generates the canonical transformation given by the geodesic flow on Vγ for times t ∈ [0, 1], in other words: ∀(y, η) ∈ Vγ , t (y, η) = (x, ξ ) ⇔ ξ = ∂x ϕγ (t, x, η) and y = ∂η ϕγ (t, x, η). (4.10)
Energy Decay for the Damped Wave Equation Under a Pressure Condition
397
2 ϕ = 0, and solves the following Hamilton-Jacobi Furthermore, ϕγ satisfies det ∂x,η γ equation: ∂t ϕγ + p(x, dx ϕγ ) = 0 ϕγ (0, x, η) = η, x.
We first look for an oscillatory integral representation: U t Pγ w(x) =
1 (2π )d
i
e (ϕγ (t,x,η)−y,η+ψ(y))
K −1
γ
k ak (t, x, y, η)w(y)dydη
k=0
+ O L 2 ( ) K
def
= b(t, x) + K r˜K (t, x), ˜r K = O(1),
(4.11)
with (y, η) ∈ Vγ . For simplicity, we will omit the dependence on γ in the formulæ. We have to determine the amplitudes ak . For this, we want b to solve i g ∂b =( − q)b ∂t 2 up to order K . Direct computations using (1.2) show that the functions ϕ and ak must satisfy the following equations: ⎧ ∂t ϕ + p(x, dx ϕ) = 0 (Hamilton-Jacobi equation) ⎪ ⎪ ⎨ (0th transport equation) (4.12) ∂t a0 + qa0 + X [a0 ] + 21 a0 divg X = 0 ⎪ ⎪ ⎩ ∂t ak + qak + X [ak ] + 21 ak divg X = 2i g ak−1 (k th transport equation) with initial conditions
⎧ ⎪ ⎨ ϕ(0, x, η) = x, η a0 (0, x, y, η) = F( x+y 2 , η) ⎪ ⎩ a (0, x, y, η) = 0 for k ≥ 1. k
The variables y and η are fixed in these equations, so they will play the role of parameters for the moment and will sometimes be skipped in the formulæ. X is a vector field on M depending on t, and divg X its Riemannian divergence. In local coordinates, X = g i j (x)∂x j ϕ(t, x) ∂xi = ∂ξi p(x, ∂x ϕ(t, x))∂xi
1 and divg X = √ ∂i ( g¯ X i ). g¯
The Hamilton-Jacobi equation is satisfied by construction. To deal with the transport equations, we notice that X corresponds to the projection on M of the Hamiltonian vector field H p at (x, dx ϕ(t, x, η)) ∈ T ∗ M. Let us call first t,η = {(x, dx ϕ(t, x, η)), x ∈ π t }, η fixed. This Lagrangian manifold is the image of the Lagrangian manifold 0,η = {(y, η) : y ∈ π } by the geodesic flow t . The flow κst on M generated by X can be now identified with the geodesic flow restricted to s,η : κst : π s,η x → π t (x, ∂x ϕ(t, x, η)) ∈ π t+s,η .
398
E. Schenck
−t The inverse flow (κst )−1 will be denoted by κs+t . Let us extend now the flow κst of X on t M to the flow K generated by the vector field X = ∂t + X on R × M: R×M →R×M t K : (s, x) → (s + t, κst (x)).
We then identify the functions ak with Riemannian half-densities on R × M, see [Dui,EvZw]: 1 1 ¯ 2 |dtd x| 2 ∈ C ∞ (R × M, 1 ). ak (t, x) ≡ ak (t, x) dtdvol(x) = ak (t, x) g(x) 2
Since we have √ √ 1 LX (ak dtdvol) = (X [ak ] + ak divg X ) dtdvol, 2 the 0th transport equation takes the simple form of an ordinary differential equation: √ √ LX (a0 dtdvol) + qa0 dtdvol = 0. This is the same as √ √ d (Kt )∗ a0 dtdvol = −qa0 dtdvol, dt which is solved by t √ √ s−t a0 dtdvol = e− 0 q◦K ds (K−t )∗ a0 dtdvol.
We now have to make explicit the coordinates dependence, which yields to t 1 s−t 1 a0 (t, x) g(x) ¯ 2 |d xdt| 2 = e− 0 q◦κt (x)ds a0 (0, κt−t (x)) ! 1 1 ¯ t−t (x))| det dx κt−t | 2 |d xdt| 2 . × g(κ
Consequently, ! a0 (t, x) = e−
t
s−t (x)ds 0 q◦κt
a0 (0, κt−t (x))
g(κ ¯ t−t (x)) 1 | det dx κt−t | 2 . √ g(x) ¯
Since κt−t : x → π −t (x, ∂x ϕ(t, x, η)) = ∂η ϕ(t, x, η), 2 ϕ(t, x, η)|. For convenience, we introduce the it is clear that | det dx κt−t (x)| = | det ∂xη following operator Tst transporting functions f on M with support inside π s,η to functions on π t+s,η while damping them along the trajectory: ! −t t g(κ ¯ t+s (x)) σ −t 1 −t −t (x)) √ (x)| 2 . Tst ( f )(x) = e− 0 q◦κt+s dσ f (κt+s | det dx κt+s g(x) ¯
Energy Decay for the Damped Wave Equation Under a Pressure Condition
399
This operator plays a crucial role, since we have a0 (t, ·) = T0t (a0 (0, ·)) = T0t F,
(4.13)
from which we see that a0 (t, ·) is supported inside π t,η . By the Duhamel formula, the higher order terms can now be computed, they are given by t i t−s ak (t, ·) = Ts g ak−1 (s) ds. 2 0 The ansatz b(t, x) constructed so far satisfies the approximate equation i ∂b i = (i g − q)b − K e S(t,x,η,y) w(y)g a K −1 (t, x, y, η) dydη. ∂t 2 The difference with the actual solution U t P is bounded by K t g a K −1 ≤ Ct K , where C = C (2K ) (M, V), so (4.11) is satisfied. As noticed above, for time t > 0, the state U t Pw is a Lagrangian state, supported on the Lagrangian manifold (t) = t . By hypothesis, (t) is projectible, so we expect an asymptotic expansion for b(t, x), exactly as in (4.5). To this end, we now proceed to the stationary phase development of the oscillatory integral in (4.11). We set i 1 Ik (x) = e (ϕ(t,x,η)−y,η+ψ(y)) ak (t, x, y, η)w(y)dydη. d (2π ) The stationary points of the phase are given by ψ (y) = η ∂η ϕ(t, x, η) = y, for which there exists a solution (yc , ηc ) ∈ (0) in view of (4.10). Moreover, this solution is unique since (t) is projectible: yc = yc (x) ∈ π (0) is the unique point in π (0) such that x = π t (yc , ψ (yc )), and then ηc = ψ (yc ) is the unique vector allowing the point yc to reach x in time t. The generating function for (t) we are looking for is then given by ψ(t, x) = S(t, x, yc, (x), ηc (x)). Applying now the stationary phase theorem for each Ik (see for instance [Hör], Theorem 7.7.6 or [NoZw], Lemma 4.1 for a similar computation), summing up the results and ordering the different terms according to their associated power of , we see that (4.14) holds with a0 (t, x, yc , ηc )
i
w0 (t, x) = e β(t)
1
2 ϕ(t, x, η ) ◦ ψ (y ))| 2 | det(1 − ∂ηη c c
w(yc ),
β ∈ C ∞ (R),
and wk (t, x) =
k i=0
A2i (x, Dx,η )(ak−i (t, x, y, η)w(y))|(y,η)=(yc ,ηc ) .
(4.14)
400
E. Schenck
A2i denotes a differential operator of order 2i, with coefficients depending smoothly on ϕ, ψ and their derivatives up to order 2i + 2. This yields to the following bounds:
wk C ≤ C,k w C +2k , where C,k = C (,k) (M, V). The remainder term r K (t, x) is the sum of the remainders coming from the stationary phase development of Ik up to order K − k. Each remainder of order K − k has a C norm bounded by C,K −k K −k w C +2(K −k)+d , so we see that
r K C ≤ C,K w C +2K +d , C = C (,K ) (M, V). The principal symbol w0 can also be interpreted more geometrically. As in Sect. 4.2, −t denote by φψ(t) the following map −t φψ(t)
:
π (t) → π (0) x → π −t (x, dx ψ(t, x)).
Let us write the differential of t : (y, η) → (x, ξ ) as d t (δy, δη) = (δx, δξ ). Using (4.10), we have 2 2 δy = ∂xη ϕδx + ∂ηη ϕδη 2 ϕδη, δξ = ∂x2x ϕδx + ∂xη 2 ϕ is invertible, and then, since ∂xη
δx δξ
=
2 ϕ −1 ∂xη 2 ϕ −1 ∂x2x ϕ∂xη
2 ϕ −1 ∂ 2 ϕ −∂xη ηη 2 ϕ − ∂ ϕ∂ 2 ϕ −1 ∂ 2 ϕ ∂xη xx xη ηη
δy δη
.
If we restrict t to (0), we have δη = ψ (y)δy, which means that for x ∈ π (t), −t 2 2 (x) = ∂xη ϕ(t, x, ηc )(1 − ∂ηη ϕ(t, x, ηc )ψ (yc ))−1 . dφψ(t)
It follows from (4.14) that
w0 (t, x) = e
i β(t)
=e
i β(t)
w(yc )F(yc , ηc ) e w(yc )F(yc , ηc ) e
−
−
1
−t+s 0 q(φψ(t) (x))ds
1
−t+s 0 q(φψ(t) (x))ds
" −t | det dφψ(t) (x)|
1 2
−t g(φ ¯ ψ(t) (x))
1 2 −t (x)) , Jac(dφψ(t)
1 2
g(x) ¯
where Jac( f ) denotes the Jacobian of f : M → M measured with respect to the Riemannian volume.
Energy Decay for the Damped Wave Equation Under a Pressure Condition
401
4.4. Ansatz for n > 1. In this paragraph, we construct by induction on n a Lagrangian state bn (t, x) supported on n (t), in order to approximate v n (t, x) up to order K −d/2 . Proposition 11. There exists a sequence of functions {bkn (t, x), Sn (t, x) : n ≥ 1, k < K , x ∈ M, t ∈ [0, 1]} such that Sn (t, x) is a generating function for n (t) and supp bkn (t, ·) ⊂ π n (t). Furthermore, we have v n (t, x) =
1 (2π )
d 2
ei
Sn (t,x)
K −1
d
k bkn (t, x) + K − 2 R nK (t, x),
(4.15)
k=0
where R nK satisfies
R nK
≤ C K (1 + C)
n
n K −1
bki−1 (1, ·) C 2(K −k)+d
+C
.
(4.16)
i=2 k=0
The constants satisfy C = C (K ) (M, χ ), C K = C (K ) (M, χ , V) and C > 0 is fixed. Proof. The construction of the amplitudes bkn for all k ≥ 0 is done by induction on n, following step by step the sequence (4.8). In Sect. 4.2 we obtained U 1 δχ as a projectible Lagrangian state: K −1 S (1,x) 1 i 0 v (1, x) = e k bk0 (1, x) + K −d/2 B K0 (1, x) (2π )d/2 0
k=0
1 = b0 (1, x) + K −d/2 R 0K (1, x), (2π )d/2
def
and we know that b0 (1, ·) satisfies the hypotheses of Proposition 10, which will be used to describe U t Pγ1 v 0 (1, ·). More generally, suppose that the preceding step has lead for some n ≥ 1 to v n−1 (t, x) = =
1 (2π ) 1
d 2
(2π )
d 2
ei
Sn−1 (t,x)
K −1
k bkn−1 (t, x) + K −d/2 R n−1 K (t, x)
k=0
b
n−1
(t, x) + K −d/2 R n−1 K (t, x),
where bn−1 (t, ·) is a Lagrangian state, supported on the Lagrangian manifold n−1 (t), is some remainder in L 2 (M). We now apply Proposition 10 to each and R n−1 K i
Lagrangian state e Sn−1 (1,x) k bkn−1 (1, x) appearing in the definition of bn−1 . Because of the term k , if we want an Ansatz as in (4.15), it is enough to describe U t Pγn vkn−1 (1, ·) up to order K − k, which gives a remainder of order C K −k K −k bkn−1 (1, ·) C 2(K −k)+d . Grouping the terms corresponding to the same power of when applying Proposition
402
E. Schenck
10 to each (vkn−1 )0≤k
n−1
](x) = e
i Sn (t,x)
K −1
def
k bkn (t, x) + K B Kn (t, x) = bn (t, x) + K B Kn (t, x),
k=0
where Sn (t, x) is a generating function of the Lagrangian manifold n (t) = t ( n−1 (1) ∩ Vγn ). The coefficients bkn are given by bkn (t, x) =
k k−i
γ
n A2l (ak−i−l (t, x, y, η)bin−1 (1, y))|(y,η)=(yc ,ηc ) ,
(4.17)
i=0 l=0
where yc = φ S−tn (t) (x), ηc = d y Sn−1 (1, yc ). In particular, b0n (t, x) = Dn (t, x)b0n−1 (1, yc ), with 1 i 1 s−t 2 Dn (t, x) = e− 0 q(φ Sn (t) (x))ds Jac(dφ S−tn (t) (x)) e βn (t) Fγn (yc , ηc ) (4.18) for some βn (t) ∈ C ∞ (R). The remainder B Kn satisfies K −1
B Kn (t) ≤ C K
bkn−1 (1, ·) C 2(K −k)+d , C K = C (K ) (M, χ , V).
(4.19)
k=0
Hence, we end up with v (1, x) = n
def
=
1 d
(2π ) 2 1 d
(2π ) 2
e
i
Sn (1,x)
K −1
k bkn (1, x) + K (B Kn (1, x) + U 1 Pγ1 B Kn−1 (1))
k=0
bn (1, x) + K −d/2 R nK (1, x), d
n where R nK (1, ·) = (2π )− 2 (B Kn (1, ·) + U 1 Pγn R n−1 K (1, ·)). Again, b satisfies the hypotheses of Proposition 10, so we can continue iteratively. To complete the proof, we now have to take all the remainders into account. From the discussion above, we get: d
U t Pγn v n−1 (1, ·) = (2π )− 2 U t Pγn bn−1 (1, ·) + K −d/2 U t Pγn (R n−1 K (1, ·)) d = (2π )− 2 bn (t, ·) + K B Kn (t.·) + K −d/2 U t Pγn (R n−1 K (1, ·)) d
= (2π )− 2 bn (t, ·) + K −d/2 R nK (t, ·), d
where we defined R nK = (2π )− 2 B Kn + U t Pγn (R n−1 K ). Since |Fγn | ≤ 1, we have
U t Pγn L 2 →L 2 ≤ 1 + C, C > 0. n−1 n This implies that U t Pγn (R n−1 K (1, ·)) ≤ (1 + C) R K (1, ·) , and finally R K satisfies
R nK ≤ (1 + C)n B Kn + B Kn−1 + · · · B K1 + B K0 . (4.20)
In view of (4.19) and (4.5), this concludes the proof.
Energy Decay for the Damped Wave Equation Under a Pressure Condition
403
Given v n−1 (1, ·), we have then constructed v n (t, x) as in (4.15), but it remains to control the remainder R nK in the L 2 norm: from (4.19) and (4.20), we see that it is crucial for this j to estimate properly the C norms of the coefficients bk for j ≥ 1 and k ∈ 0, K − 1. Lemma 12. Let n ≥ 1, and define n−1 Dn−i (1, φ S−in (x)) , D 0 = 1. D n = sup x∈π n (1) i=0
If x ∈ π n (1), the principal symbol b0n is given by ⎛ ⎞ n−1 −j Dn− j (1, φ Sn (x))⎠ b00 (1, φ S−n (x)). b0n (1, x) = ⎝ n
(4.21)
j=0
For k ∈ 0, K − 1, the functions bkn satisfy
bkn (1, ·) C ≤ Ck, (n + 1)3k+ D n ,
(4.22)
where Ck, = C (,k) (M, χ , V). It follows that
B Kn (1, ·) ≤ C K n 3K +d D n−1 , n n n
R K (1) ≤ C K (1 + C) j 3K +d D j−1 ,
(4.23) (4.24)
j=1
where C > 0 and C K = C (K ) (M, χ , V). On the other hand, if x ∈ / π n (1), we have n bk (x) = 0 for k ∈ 0, K − 1. Proof. First, if x ∈ / π n (1), then there is no ρ ∈ Vγn such that π 1 (ρ) = x, and then n ∞ v (1, x) = O( ). In what follows, we then consider the case x ∈ π n (1). We first see that (4.21) simply follows from (4.18) applied recursively. If ρn = (xn , ξn ) ∈ n (1), we call ρ j = (x j , ξ j ) = j−n (ρn ) ∈ j (1) if j ≥ 0. In other words, ∀ j ∈ 1, n, x j−1 = φ S−1 (x j ). j It will be useful to keep in mind the following sequence, which illustrates the backward trajectory of ρn ∈ n (1) under −k , k ∈ 1, n and its projection on M: ρ0 ∈ 0 (1) o π
x0 o
−1
φ S−1 1
ρ1 ∈ 1 (1) o π
x1 o
−1
φ S−1 2
... o
... o
−1
ρn−1 ∈ n−1 (1) o
φ S−1
n−1
We denote schematically the Jacobian matrix dφ S−ij =
π
xn−1 o ∂ x j−i ∂x j
−1
φ S−1 n
ρn ∈ n (1) π
xn
for 1 ≤ i ≤ j ≤ n. Since
for any E > 0, the sphere bundle Tz∗ M ∩ p −1 (E) is transverse to the stable direction [Kli], the Lagrangians n ⊂ n 0 converge exponentially fast to the weak unstable foliation as n → ∞. This implies that t | 0 is asymptotically expanding as t → ∞,
404
E. Schenck
except in the flow direction. Hence, the inverse flow −t | n acting on n and its projection φ S−tn on M have a tangent map uniformly bounded with respect to n, t. As a result, the Jacobian matrices ∂ x j−i /∂ x j are uniformly bounded from above: for 1 ≤ i ≤ j ≤ n there exists C = C(M) independent of n such that # # # ∂ x j−i # # # (4.25) # ∂ x # ≤ C. j It follows that if we denote D j = supx j D j (1, x j ), there exists C = C(M) > 0 such that C −1 ≤ D j ≤ C. Note also that
(4.26)
n−1 n−1 − j sup Dn− j (1, φ Sn (x)) = Dn− j = D n . x∈π n (1) j=0 j=0
We first establish the following crucial estimate: Lemma 13. Let n ≥ 1, and k ∈ 1, n. For every multi index α of length |α| ≥ 2, there exists a constant Cα > 0 depending on M such that # α # # ∂ xn−k # α−1 # # . (4.27) # ∂ x α # ≤ Cα k n Proof. We proceed by induction on k, from k = 1 to k = n. The case k = 1 is clear. Let us assume now that # # α # ∂ xn−k # α−1 # # , k ∈ 1, k − 1 # ∂ x α # ≤ Cα k n
and show the bound for
k
= k. For simplicity, we will denote ∂ αj =
def
α ∂α def ∂ x j α , ∂ x = . j ∂ x αj ∂ x αj+1
In particular, ∂ α x j ≤ Cα . We also recall the Faà di Bruno formula: let be the set of partitions of the ensemble {1, . . . , |α|}, and for π ∈ , write π = {B1 , . . . Bk }, where Bi is some subset of {1, . . . , |α|}. Here |α| ≥ k ≥ 1, and we denote |π | = k. For two smooth functions g : Rd → Rd and f : Rd → Rd such that f ◦ g is well defined, one has ∂α f ◦ g ≡ ∂ |π | f (g) ∂ B g. (4.28) π ∈
B∈π
The term in the right hand side is written schematically, to indicate a sum of derivatives of f of order |π |, times a product of |π | terms, each of them corresponding to derivatives of g of order |B|. It is important for our purpose to note that |B| = |α|. Continuing from these remarks, we compute X k = ∂nα xn−k = ∂ xn−k ∂nα xn−k+1 def + ∂ |π | xn−k ∂nB xn−k+1 = ∂ xn−k X k−1 + Yk−1 . def
π ∈,|π |>1
B∈π
Energy Decay for the Damped Wave Equation Under a Pressure Condition
405
By the induction hypothesis,
Yi ≤ Cα i α−2 ,
(4.29)
since the partitions π involved in the sum contains at least two elements. Setting Mk−1 = ∂ xn−k , we have X k = Mk−1 . . . M1 X 1 + Mk−2 . . . M1 Y1 + Mk−3 . . . M1 Y2 + · · · + M1 Yk−1 . From the chain rule we have ∂ x j−i ∂ x j−i ∂ x j−1 = ··· , ∂x j ∂ x j−i+1 ∂x j and (4.25) yields to Mi−1 . . . M1 = O(1) for 2 ≤ i ≤ k. Adding up all the terms contributing to X k and taking (4.29) into account yields to
X k ≤ Cα (1 + 1α−2 + 2α−2 + · · · + (k − 1)α−2 ) ≤ Cα k α−1 and the lemma is proved. We now prove (4.22). For this, we will proceed in two steps. First, we show the bounds for the principal symbol b0n . Then, we treat the higher order terms bkn , k ≥ 1 using the bounds on b0n C for any . For b0n , the C 0 norm estimate follows directly from (4.21). From now on, we denote for convenience def
D0 (x0 ) = b00 (1, x0 ). Computing ∂n b0n (xn ) = ∂n (Dn (xn ) . . . D1 (x1 )D0 (x0 )), we will obtain a sum of terms, each of them of the form α
Mαn ...α0 = ∂nαn Dn ∂n n−1 Dn−1 . . . ∂nα1 D1 ∂nα0 D0 , with αn + · · · + α0 = . Note that if is fixed with respect to n, most of the multi-indices αi vanish when n becomes large: actually, at most || are non-zero, and we will denote them by αi1 , . . . , αik , k ≤ ||. Hence the above expression is made of long strings of Di , alternating with some derivative terms ∂nαi Di which number depends only on . We can then write αi
Mαn ...α0 C 0
αi
αi
∂n 1 Di1 . . . ∂n k−1 Dik−1 ∂n k Dik C 0 ≤ Dn × . Di1 . . . Dik
(4.30)
Let us examinate each term ∂nα Di appearing in the right-hand side individually. By the Faà di Bruno formula and Lemma 13, we have for i = 0, |π | ∂nα Di (xi ) = ∂i Di ∂nB xi ≤ Cπ n α−|π | ≤ Cα n α−1 , (4.31) π
B∈π
where Cα = C (,K ) (M, χ , V). Of course, if i = 0, ∂0α D0 (x0 ) C 0 ≤ Cα ∂0α b00 C 0 for some constant Cα > 0. Now, for a fixed configuration of derivatives {α} = {αi1 , . . . αik }
406
E. Schenck
we have to choose i 1 , . . . , i k indices among n + 1 to form the right-hand side in (4.30), and the number of such choices is at most of order O((n + 1)k ). Hence, α
∂n b0n C 0
∂nαi1 Di1 . . . ∂n ik−1 Dik−1 ∂nαik Dik C 0 ≤ Dn Di1 . . . Dik {α} i i ,...,i k ≤ Dn Cα (n + 1)k (n + 1)α1 −1 . . . (n + 1)αk −1 {α}
≤ C D n (n + 1) ,
(4.32)
where C = C () (M, χ , V). For higher order terms (bkn , k > 0), we remark from (4.17) that we can write bkn (xn )
=
Dn (xn )bkn−1 (xn−1 ) +
k
n−1 n α jα (x n )∂n−1 bk− j (x n−1 ).
(4.33)
j=1 |α|≤2 j
The function njα can be expressed with the flow, the damping and the cutoff function Fγn . It follows that the norms njα C are uniformly bounded with respect to n:
n j,α C
= C (,K ) (M, χ , V).
In order to show the bounds (4.22) for k > 0, we will proceed by induction on the index k. The case k = 0 has been treated above. Suppose now that for any and k ∈ 0, k − 1 we have proven
∂n bkn C 0 ≤ C (n + 1)3k + D n , C = C (,K ) (M, χ , V). As above, to treat the case k = k, we begin by the situation where = 0. To shorten the formulæ, we introduce for 1 ≤ i ≤ j ≤ n the functions i,k (xn ) =
k
i−1 i α jα (x i )∂i−1 bk− j (x i−1 ),
j=1 |α|≤2 j j J i (xn )
= D j (x j )D j−1 (x j−1 ) . . . Di (xi ),
(xn ). where the xi , i ≤ n have to be considered as functions of xn , namely xi = φ S−n+i n Iterating (4.33) further, we have: bkn (xn ) = J nn bkn−1 (xn−1 ) + n,k n−2 n−1,k = J nn ( J n−1 ) + n,k n−1 bk (x n−2 ) +
= J nn−1 bkn−2 (xn−1 ) + J nn n−1,k + n,k = J n1 bk0 (x0 ) + J n2 1,k + J n3 2,k + · · · + J nn n−1,k + n,k . By the induction hypothesis and (4.26), each term i,k , i > 0 satisfies
n−i,k C 0 ≤ Ck (n − i)3k−1 D n−i ,
(4.34)
Energy Decay for the Damped Wave Equation Under a Pressure Condition
407
hence adding up all the terms we get
bkn C 0 ≤ Ck D n (bk0 (x0 ) +
n−1
(n − i)3k−1 ) ≤ Ck D n (n + 1)3k ,
i=0
and we obtain the bounds (4.22) for = 0. To evaluate ∂ bkn , > 1, we start from the expression (4.34). We notice first that k
∂nβ n−i,k =
(∂nβ1
n−i β2 α n−i−1 jα (x n−i ))(∂n ∂ bk− j (x n−i−1 )).
β1 +β2 =β j=1 |α|≤2 j
Using the Faà di Bruno formula and Lemma 13, we get
∂nβ1
n−i,k jα (x n−i ) C 0
n−i−1 3k−1+β2 ≤ Cβ1 i β1 −1 and ∂nβ2 ∂ α bk− , j (x n−i−1 ) C 0 ≤ C β2 i
and this implies
∂nβ n−i,k C 0 ≤ Cβ i 3k−1+β . Then, exactly the same strategy used to derive (4.32) shows that n
∂n J i+1 i,k C 0 ≤ C n 3k−1+ .
Using these estimates and (4.34) yields to
∂n bkn C 0 ≤ C (n + 1)n 3k−1+ ≤ C (n + 1)3k+ , where the constant C is such that C = C (,K ) (M, χ , V). 4.5. The main estimate: Proof of Proposition 5. As noted before, the Lagrangians n converge exponentially fast as n → ∞ to the weak unstable foliation. This implies that for x ∈ π j (1), the Jacobians JS j (x) = | det φ S−1 (x)| satisfy j (1) def
JS j (x) − 1 ≤ C e− j/C , C = C(M) > 0. ∀ j ≥ 2, ∀(x, ξ ) ∈ j (1), J u (x) S (x,ξ )
Here, generates the (Lagrangian) local weak instable manifold at point (x, ξ ). Moreover, these Jacobians decay exponentially with j as j → ∞. This means that uniformly with respect to n, Su
n−1 j=0
−j
JSn− j (φ Sn (x)) ≤ C(M)
n−1
−j
JS u ( − j (x,ξ )) (φ Sn (x)).
j=0
The Jacobian JS u (x,ξ ) (x) measures the contraction of −1 along the unstable subspace E u ( 1 (ρ)), where 1 (ρ) = (x, ξ ), and x ∈ M serves as coordinates to compute this def Jacobian (via the projection π ). The unstable Jacobian J u (ρ) = | det d −1 | E u,0 ( (ρ)) |
408
E. Schenck
defined in Sect. 2.1 expresses also this contraction, but in different coordinates: for n large enough, the above inequality can then be extended to n−1
−j
JSn− j (φ Sn (x)) ≤ C
j=0
n−1
−j JS u ( − j (x,ξ )) (φ Sn (x)) ≤ C˜
n−1
j=0
J u ( − j (ρ)), (4.35)
j=0
where C, C˜ only depends on M. As noted above, because of the Anosov property of the geodesic flow, the above products decay exponentially with n. Together with the fact that the damping function is positive, it follows that the right-hand side in (4.23) also decays exponentially with n. Recall now that 1 ≤ n ≤ N t0 and N = T log −1 . Using (4.24), we then see that the remainders R nK in (4.15) are uniformly bounded: they satisfy
R nK ≤ C K , C K = C (K ) (M, χ , V) uniformly in n and z 0 , the point on which δχ ,z 0 was based. From the very construction of bn (t, x), we then have
U 1 Pγn . . . U 1 Pγ1 U 1 δχ −
1 bn (1, ·) ≤ C K K −d/2 . (2π )d/2
(4.36)
But the bounds on the symbols bkn , k > 0 given in Lemma 12 tells us that (4.36) also holds if we replace the full symbol bn by the principal symbol b0n , provided is chosen small enough, say ≤ 0 (ε). Hence, for ≤ 0 , d
U 1 Pγn . . . U 1 Pγ1 U 1 δχ ≤ (2π )− 2 b0n (1, ·) + C K K −d/2 . Now, using (4.21), (4.35) and the fact that |Fγ | ≤ 1, we conclude that for a u as in (1.8),
b0n (1, x)
≤Ce
n O ()
sup
x∈π n (1)
exp
n
a u ◦ − j (x, dx Sn (1, x)).
j=1
Here, C = C(M) depends only on the manifold M. Let us consider now the particular case n = N t0 with N = T log −1 . It follows immediately that ⎞ ⎛ N t0 t N 0 −1 exp a u ◦ − j (x, dx Sn (1, x)) ≤ sup ⎝exp a u ◦ j (ρ)⎠ . sup x∈π N t0 (1)
k=1 ρ∈Wβk
j=1
j=0
By the superposition principle already mentionned in (4.4), we then obtain for some C = C(M) > 0 depending only on M:
UPγ N t0 . . . UPγ1 U 1 Op(χ ) ≤ C sup UPγ N t0 . . . Pγ1 U 1 δz,α
0 ≤
z0 C−d/2 b0n + C K K −d/2
≤ C
− d2
N
⎛
sup ⎝exp
k=1 ρ∈Wβk
t 0 −1
⎞ a u ◦ j (ρ)⎠ .
j=0
To get the last line, we have noticed that K can be chosen arbitrary large: since n ≤ T t0 log −1 , we see that for small enough, the main term in the right-hand side of the second line is larger than the remainder C K K −d/2 , and e N t0 O() = O(1). This completes the proof of Proposition 5.
Energy Decay for the Damped Wave Equation Under a Pressure Condition
409
Acknowledgements. I am very grateful to Stéphane Nonnenmacher for numerous helpful discussions on the subject of the present paper. I would also like to thank sincerely Maciej Zworski for interesting suggestions, and the UC Berkeley for its hospitality during March and April 2008.
Appendix A. Semiclassical Analysis on Compact Manifolds In this Appendix we gather standard notions of pseudodifferential calculus on a compact, d dimensional manifold M endowed with a Riemannian structure coming from a metric g. As usual, M is equipped with an atlas { f , V }, where {V } is an open cover of M and each f is a diffeomorphism form V to a bounded open set W ⊂ Rd . Functions on Rd can be pulled back via f ∗ : C ∞ (W ) → C ∞ (V ). The canonical lift of f between T ∗ V and T ∗ W is denoted by f˜ : (x, ξ ) ∈ T ∗ V → f˜ (x, ξ ) = ( f (x), (D f (x)−1 )T ξ ) ∈ T ∗ W , where A T denotes the transpose of A. Its corresponding pull-back will be denoted by of unity adapted to the cover {V } f˜∗ : C ∞ (T ∗ W ) → C ∞ (T ∗ V ). A smooth partition is a set of functions φ ∈ Cc∞ (V ) such that φ = 1 on M. Any observable (i.e. a function a ∈ C ∞ (T ∗ M)) can now be split into a = a where a = φ a, and each term pushed to a˜ = ( f˜−1 )∗ a ∈ C ∞ (T ∗ W ). If a belongs to a standard class of symbols, for instance % $ def β a ∈ S m,k = S k (ξ m ) = a = a ∈ C ∞ (M), |∂xα ∂ξ a| ≤ Cα,β −k ξ m−|β| , each a can be be Weyl-quantized into a pseudodifferential operator on S(R) via the formula x + y i 1 ∀u ∈ S(Rd ), Opw , ξ ; u(y) dy dξ ( a ˜ )u(x) = e x−y,ξ a˜ d (2π ) 2 To pull-back this operator on C ∞ (V ), one first takes another smooth cutoff ψ ∈ Cc∞ (V ) such that ψ = 1 in a neighbourhood of supp φ . The quantization of a ∈ S m,k is finally defined by gluing local quantizations together, yielding to ∀u ∈ C ∞ (M), Op(a)u =
−1 ∗ ψ × f ∗ ◦ Opw (a˜ ) ◦ ( f ) (ψ u)
The space of pseudodifferential operators obtained from S k,m by this quantization will be denoted by m,k . Although this quantization depends on the cutoffs, the principal symbol map σ : m,k → S m,k /S m,,k−1 is intrinsically defined and do not depend on the choice of coordinates. The residual class is made of operators in the space m,−∞ . As an example, the (semiclassical) Laplacian −2 g ∈ 0,2 is a pseudodifferential operator, and its principal symbol is given by σ (−2 g ) = ξ 2g = gx (ξ, ξ ) ∈ S 2,0 . In this article, we are concerned with a purely semiclassical theory and then deal only with compact subsets of T ∗ M. If A ∈ m,k , we will denote by WF(A) the semiclassical wave front set of A. A point ρ ∈ T ∗ M belongs to WF (A) if for some choice of local coordinates near the projection of ρ, the full symbol of A is in the class S m,−∞ . WF(A) is a closed subset of T ∗ M, and WF(AB) ⊂ WF(A)∩WF(B). In particular,
410
E. Schenck
if WF(A) = ∅, then A is a negligible operator, i.e. A ∈ m,−∞ . If ∈ L 2 (M), we also define the semiclassical wave front set of by: $ %c WF() = (x, ξ ) : ∃a ∈ S m,0 , a(x, ξ ) = 0, Op(a) L 2 (M) = O(∞ ) where the superscript c indicates the complementary set. We will often make use of the following fundamental propagation property: if U t is a Fourier integral operator associated to a symplectic diffeomorphism t : T ∗ M → T ∗ M, then WF(U t ) = t (WF()). References [Ana1] [Ana2] [AnNo] [Ano] [AsLe] [BLR] [Chr] [CoZu] [Dui] [EvZw] [GoKr] [Hit] [Hör] [Kli] [Leb] [MaMa] [NoZw] [NoZw2] [RaTa] [Sjö] [Wal]
Anantharaman, N.: Entropy and the localization of eigenfunctions. Ann. of Math. 168(2), 435–475 (2008) Anantharaman, N.: http://arXiv.org/abs/0904.1736v1[math.DG], to appear in G.A.F.A. Anantharaman, N., Nonnenmacher, S.: Half delocalization of eigenfunctions of the laplacian on an anosov manifold. Ann. Inst. Fourier 57(7), 2465–2523 (2007) Anosov, D.V.: Geodesic flows on closed riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov. 90, 1–235 (1967) Asch, M., Lebeau, G.: The spectrum of the damped wave operator for a bounded domain in R2 . Exp. Math. 12, 227–241 (2003) Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions fot the observation, control and stabilization of waves from the boundary. SIAM J. Control and Optimization 30(5), 1024–1065 (1992) Christianson, H.: Semiclassical non-concentration near hyperbolic orbits, corrigendum. J. Funct. Anal. 246(2), 145–195 (2007) Cox, S., Zuazua, E.: The rate at which the energy decays in a dumped string. C.P.D.E.; estimations sur le taux de décroissance exponentielle de l’énergie dans l’équation d’ondes. Note C.R.A.S. Paris, 317, 249–254 (1993) Duistermaat, J.J.: Fourier Integral Operators. Progress in Mathematics 130, Basel: Birkhäuser, 1996 Evans, L.C., Zworski, M.: http://math.berkeley.edu/~zworski/semiclassical.pdf Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear non selfadjoint operators. Trans. Math. Monograph 18, Providence, RI: Amer. Math. Soc., 1969 Hitrik, M.: Eigenfrequencies and expansions for damped wave equations. Meth. Appl. Anal. 10(4), 1–22 (2003) Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. I, II, Berlin: Springer Verlag, 1983 Klingenberg, W.P.A.: Riemannian manifolds with geodesic flows of anosov type. Ann. of Math. 99(2), 1–13 (1974) Lebeau, G.: Équation des ondes amorties. In: Algebraic and geometric methods in mathematical physics (Kaciveli, 1993), Math. Phys. Stud., 19, Dordrecht: Kluwer Acad. Publ., 1996, pp. 73–109 Markus, A.S., Matsaev, V.I.: Comparison theorems for spectra of linear operators and spectral asymptotics. Trudy. Moskov. Mat. Obshch. 45, 133–181 (1982) Nonnenmacher, S., Zworski, M.: Quantum decay rates in chaotic scattering. Acta Math. 203(2), 149–304 (2009) Nonnenmacher, S., Zworski, M.: Semiclassical resolvent estimates in chaotic scattering, http:// arXiv.org/abs/0904.2986v2[math-ph], 2009 Rauch, J., Taylor, M.: Decay of solutions to nondissipative hyperbolic systems on compact manifolds. C.P.A.M. 28, 501–523 (1975) Sjöstrand, J.: Asymptotic distribution of eigenfrequencies for damped wave equations. Publ. Res. Inst. Math. Sci. 36(5), 573–611 (2000) Walters, P.: An Introduction to Egodic Theory. Graduate Texts in Mathematics 79, BerlinHeidelberg-New York: Springer, 1975
Communicated by P. Constantin
Commun. Math. Phys. 300, 411–433 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1120-x
Communications in
Mathematical Physics
(Non)Invariance of Dynamical Quantities for Orbit Equivalent Flows Katrin Gelfert1 , Adilson E. Motter2 1 Instituto de Matemática, UFRJ, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil.
E-mail: [email protected]
2 Department of Physics and Astronomy & Northwestern Institute on Complex Systems,
Northwestern University, Evanston, Illinois 60208-3112, USA. E-mail: [email protected] Received: 21 September 2009 / Accepted: 27 April 2010 Published online: 1 September 2010 – © Springer-Verlag 2010
Abstract: We study how dynamical quantities such as Lyapunov exponents, metric entropy, topological pressure, recurrence rates, and dimension-like characteristics change under a time reparameterization of a dynamical system. These quantities are shown to either remain invariant, transform according to a multiplicative factor or transform through a convoluted dependence that may take the form of an integral over the initial local values. We discuss the significance of these results for the apparent noninvariance of chaos in general relativity and explore applications to the synchronization of equilibrium states and the elimination of expansions. 1. Introduction In the context of dynamical systems theory one may argue that the existence of chaotic behavior together with its quantifiers should be invariant under changes of coordinates. Depending on the quantifier of chaos, such changes could be either measurable, continuous, or differentiable. The topological entropy, for example, stands as a fundamental topological conjugacy invariant. The Hausdorff dimension, on the other hand, is invariant under bi-Lipschitz transformations of the phase space only. In the case of continuous-time dynamical systems, as generated by flows of vector fields, the problem is more involved when one considers that any change of coordinates allows in addition for a change of time parameterization. Some properties, such as ergodicity and the finiteness of the entropy, are preserved by any measurable time transformation of the flow. Other dynamical quantities or spectral invariants can exhibit surprising transformation properties. For example, suitable changes of time can strengthen or simulate chaotic properties in the flow. In particular, for any ergodic flow there exists a transformation of time that yields a mixing flow and for any ergodic flow with positive entropy there exists a time change that yields a K-flow (see [24] and references therein). In this paper, we offer a complete characterization of how quantifiers of chaotic behavior transform under a smooth time reparameterization. The importance of this
412
K. Gelfert, A. E. Motter
work is two-fold. First, from a strictly mathematical perspective, it provides a scheme of relationships for the classification and study of dynamical invariants across orbit equivalent flows. For example, it is useful to consider the time transformation of Lyapunov exponents and dimension quantities if one wants to study the generality of conjectures that relate these quantities, such as the Kaplan-Yorke relation [27]. Second, this work provides a mathematical foundation for the study of chaos in general relativity and other physical theories that do not have an absolute time parameter. The foundations of chaos in relativistic dynamics became a topic of intense research after the observation, first reported in [15], that space-time transformations commonly used in the study of (relativistic) cosmology could transform positive Lyapunov exponents into zero Lyapunov exponents. This finding, which was confirmed in numerous subsequent studies [7,8,20,39] and was even the topic of a conference in the early nineties [21], led to the tacit assumption that Lyapunov exponents are unsuitable indicators of chaos in relativistic dynamics. Many alternatives have been proposed, including the use of methods based on symbolic dynamics [12,37], Painlevé analysis [10], local curvature [40], fractals [13,28], and system-specific analyses such as those based on Misner-Chitré-like variables [6]. More recently, it has been shown that Lyapunov exponents can be restated as reliable quantifiers of chaos if the transformation is such that it does not create singularities in the invariant measure [26,30]. However, the conditions under which different approaches can be used to provide an invariant characterization of chaos under time reparameterizations remain debatable. In particular, it is currently unclear whether the fractal-based characterization of chaos by means of dimensions of measures, which has stood out as the most promising alternative, is any more invariant than the Lyapunov exponents themselves. This is an important motivation for establishing a mathematical basis for the study of chaos under time reparameterizations, which is one of the goals of this paper. Among the main results in the paper, we anticipate that for sufficiently regular transformations: (i) the Lyapunov exponents remain essentially invariant, except for a multiplicative factor that accounts for the change of the time units; (ii) a time reparameterization can be used not only to synchronize the measure of maximal entropy and the SRB measure, but also to synchronize expansion rates and equilibrium states of smooth potentials; (iii) the generalized dimensions of measures Dq are invariant for q = 1, whereas the information dimension D1 and entropies may change. These and accompanying results address a longstanding problem in relativistic dynamics, namely, they identify the conditions under which different quantities can (or cannot) be used as invariant indicators of chaos. The paper is organized as follows. In Sect. 2, we introduce definitions and recall basic results on time transformations, such as the transformation of an invariant measure. In Sect. 3, we establish a fundamental result on the time transformation of the solutions of the variational equations. In Sect. 4, we present the main results about the transformation of dynamical quantities under time reparameterizations. Results on the time transformation of dimension-like characteristics of measures are presented in Sect. 5. In the last section we discuss implications of our results and future directions. 2. Orbit Equivalent Flows and Time Reparameterizations Throughout the paper, M is a manifold of smoothness C m , m ≥ 2, g is a Riemannian metric of class C m−1 on M, and f is a C vector field on M, 1 ≤ < m. We write ρ for the distance induced by the Riemannian structure on M.
(Non)Invariance of Dynamical Quantities for Orbit Equivalent Flows
413
We assume that the differential equation dx = f (x) dt
(1) def
has a C flow ϕ : R × M → M, and we write ϕ t (·) = ϕ(t, ·). The curve t → ϕ t (x0 ), t ∈ R, is the unique solution of (1) with the initial condition x(0) = x0 . For the flow Φ = {ϕ t }t∈R we denote by M(Φ) the set of all Φ-invariant Borel probability measures on M. We endow M(Φ) with the weak∗ topology. Moreover, we denote by Me (Φ) ⊂ M(Φ) the subset of ergodic measures. A C flow Φr is a time reparameterization of the flow Φ if for every x ∈ M the orbits τ {ϕr (x)}τ ∈R and {ϕ t (x)}t∈R coincide and if they have the same orientation given by the change of t and τ in the positive direction.1 We study a C flow Φr = {ϕrτ }τ ∈R on M that is C k -orbit equivalent to Φ, k ≤ . This means that there exists a C k diffeomorphism τ }τ ∈R given by ψ τ def h : M → M such that the flow {ψ = h −1 ◦ ϕrτ ◦ h is a time reparamt eterization of the flow {ϕ }t∈R . Without loss of generality for the results of this paper, we assume that h is the identity, meaning that the orbits themselves are not deformed. See [32,11,22,41] for treatments of the classical theory of time reparameterization of flows. If Φr is a time transformation of Φ, then we have ϕrτ (x) = ϕ t (x,τ ) (x)
(2)
for every x ∈ M, where (x, τ ) → t (x, τ ) is a real-valued function which satisfies t (x, τ1 + τ2 ) = t (x, τ1 ) + t (ϕ τ1 (x), τ2 ), t (x, −τ ) = −t (ϕ −τ (x), τ ), and t (x, τ ) ≥ 0 if τ ≥ 0. For every point x that is not a fixed point, (x, τ ) → t (x, τ ) is a C function satisfying t (x, τ ) > 0 whenever τ > 0. One can view the flow Φr as being obtained from Φ by means of a change of time parameterization determined by dτ = r (x) dt,
(3)
where def
r (x) =
∂ t (x, τ )|τ =0 ∂τ
−1
at every non-fixed point x. In particular, at every non-fixed x the function r is C −1 and positive, and τ → ϕrτ (x) solves the differential equation f (x) dx = , dτ r (x)
(4)
with initial condition x(0) = x. Of particular interest are orbit equivalent flows that are a priori obtained by means of a time change determined by (3). Clearly, to accomplish orbit equivalence, such a function r is non-negative everywhere, and positive and sufficiently regular at any non-fixed point. 1 Throughout the paper, the index r is used to indicate time reparameterized quantities. The use of this notation will become clear shortly.
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As indicated above, the description of the behavior of the flows close to fixed points requires particular care. To describe this in a general setting, assume that r is a nonnegative measurable function and denote by def M = {x ∈ M : f (x) = 0}
(5)
the regular set of Φ (and hence of Φr ). In the particular case in which r is positive and def and t ∈ R, of class C −1 everywhere on M, we set M = M instead of (5). Given x ∈ M we define t def τ (x, t) = r (ϕ s (x)) ds. (6) 0
If μ ∈ M(Φ) and r is μ-integrable and positive on some set of positive measure, then the function t → τ (x, t) is strictly increasing and satisfies for almost every x ∈ M τ (x, 0) = 0 and lim τ (x, t) = ±∞.
t→±∞
(7)
where B def B), is the restriction Then Φr = {ϕrτ }τ ∈R is a measurable flow on ( M, = B∩ M of the σ -field B to M. Condition (7) provides an important criterion for the selection of eligible time reparameterizations for the study of chaos in relativistic systems exhibiting cosmological singularities [29] or event horizons [30]. The time-transformed flow Φr in general evolves at a different speed and possesses different invariant probability distributions. Given some μ ∈ M(Φ), assume that r is a non-negative function that is positive on some positive measure set and satisfies 1 def s is μr = r dμ on M 0 r (ϕ (x))ds < ∞ at μ-almost every x. This measure given by d Φr -invariant and σ -finite. The measure is finite, and hence normalizable, if and only if r is μ-integrable, in which case we introduce the Φr -invariant probability measure μr by def dμr =
r dμ. r M dμ
(8)
Note that for every Φ-invariant measurable A we have μr (A) = 0 if μ(A) = 0. The measure μr is ergodic if μ is. If r is μ-integrable, from (8) we conclude that for every continuous function ξ : M → R we have ξ dμ ξ . (9) dμr = M M r M r dμ For a proof of the above stated properties in the context of abstract metrically isomorphic flows, see [11,41] and references therein. We continue to denote by M(Φ ≡ Φ1 ) (by M(Φr )) the set of Φ-invariant (Φr -invariant) probability measures. Remark 1. If the function r is such that 1/r ∈ L 1 (M, μ), then the original flow Φ is obtained by the time reparameterization of the flow Φr determined by the function 1/r . In this case the map μ → μr defined by (8) is a bijection from Me (Φ) to Me (Φr ). The decomposition of a measure into its ergodic components is preserved under this transformation, although the particular weight associated to each component is not necessarily the same.
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Remark 2. Given two orbit equivalent flows Φ and Φr without fixed points, for every μ ∈ Me (Φ) the sets of generic points with respect to the measures μ and μr coincide [31]. This means that we have G(μ, Φ) = G(μr , Φr ), where def
G(μ, Φ) =
1 t→∞ t
x : lim
t
ξ(ϕ s (x)) ds =
ξ dμ for every ξ ∈ C 0 (M, R) M
0
and G(μr , Φr ) is defined similarly. 3. Variational Equations Given a point x ∈ M and the trajectory {ϕ t (x) : t ∈ R} that passes through x, we introduce the variational differential equation Dy = ∇ f (ϕ t (x)) y, dt
(10)
where ∇ f is the covariant derivative of the vector field f . The absolute derivative is taken def along the curve t → ϕ t (x). For x ∈ M and v ∈ Tx M, the linearization y(t) = y(t, x, v) t of the flow map ϕ given by def
y(t) = Dx ϕ t (v)
(11)
the linearization is the solution of Eq. (10) with y(0) = v. Analogously, given x ∈ M, def
z(τ ) = z(τ, x, v) of the flow map ϕrτ given by z(τ ) = Dx ϕrτ (v) def
is the solution of the variational differential equation 1 Dz 1 τ τ = ∇ f (ϕ (ϕ (x)) z + D (x)) · z f (ϕrτ (x)) r dτ r (ϕrτ (x)) r r
(12)
with the initial condition z(0) = v. and v ∈ Tx M, the maps t → y(t) and τ → z(τ ) given Proposition 1. For x ∈ M above satisfy f (ϕ t (x)) z(τ (x, t)) = y(t) + κ(t) r
(13)
for every t, where the function κ : R → R is given by
t
κ(t) = − 0
Dr (ϕ s (x)) · y(s) ds.
(14)
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K. Gelfert, A. E. Motter def
Proof. By transforming the time in (10) for γ (τ ) = y(t (x, τ )) and using relation (2) and dt (x, τ )/dτ = r (ϕ t (x,τ ) (x))−1 , we obtain Dγ dt Dγ f = =∇ (ϕrτ (x)) γ (τ ) − (Dr −1 (ϕrτ (x)) · γ (τ )) f (ϕrτ (x)). dτ dτ dt r Making the ansatz z(τ ) = y(t (x, τ )) + κ(τ ) ( f /r ) (ϕ t (x,τ ) ) for some function κ : R → R, a short calculation gives f Dγ f τ =∇ (ϕr (x)) γ (τ ) − κ(τ (ϕrτ (x)). ˙ ) (15) dτ r r From the above and (15) we obtain that κ(τ ) satisfying κ(0) = 0 is given by τ D log r (ϕrs (x)) · y(t (x, s)) ds κ(τ ) = − 0
for every τ . After a change of variable, we obtain (13). Example 1 (Hyperbolic splitting). Let us consider the example of a vector field f with a C 1 Anosov flow Φ on a compact smooth manifold M. This means that there exists λ > 0 and a Riemannian metric such that for every x ∈ M there is a splitting Tx M = u/s u/s E u ⊕ E c ⊕ E s with f (x) ∈ E xc \{0}, dim E xc = 1, Dx ϕ t E x = E ϕ t (x) for all t, and
||Dx ϕ ∓t (v u/s )|| ≤ e−tλ ||v u/s || for every v u/s ∈ E x \{0} and t > 0. By the AnosovSinai theorem [2], a change of the time parameterization by means of a positive C 1 funcu ⊕ Ec ⊕ Es tion r produces again an Anosov flow. Moreover, the splitting Tx M = E x,r x,r x,r that is invariant and hyperbolic with respect to the flow Φr can be described explicitly as follows (see [2,33]). Given x ∈ M, we have
u/s u/s E x,r = v + κ u/s (v) f (x) : v ∈ E x , u/s
where κ u/s (v) = −
1 r (x)
∓∞
Dr (ϕ t (x)) · Dx ϕ t (v) dt.
0 u/s
This integral converges uniformly since v u/s is in the unstable/stable subbundle E x and thus κ u/s is continuous. Note that this is in agreement with the proposition above, where (13) describes how an arbitrary vector v ∈ Tx M is stretched under Dϕrt . Recall u/s that Dx ϕrt (v) converges towards the direction of E ϕ t (x),r as t → ±∞ whenever r
s/u
c . v∈ / E x,r ∪ E x,r
4. Transformation of Dynamical Quantities 4.1. Lyapunov exponents. Given x ∈ M and v ∈ Tx M, the forward Lyapunov exponent of v at x with respect to the flow Φ is defined by def
λ+ (x, v) = lim sup t→∞
1 log ||Dx ϕ t (v)||, t
(16)
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with the convention that log 0 = −∞. Here the norm || · || is the one derived from the scalar-product structure on the tangent bundle induced by the Riemannian metric. Analogously, the backward Lyapunov exponent λ− (x, v) is defined by replacing the map ϕ t in (16) with ϕ −t . For every point x ∈ M that is Lyapunov regular with respect to Φ, there exists a positive integer k(x) ≤ dim M and a unique splitting Tx M =
k(x)
E xi
(17)
i=1
of the tangent space, where each linear subspace E xi depends measurably on x and satisfies Dx ϕ t (E xi ) = E ϕi t (x) for every t ∈ R. Moreover, there are numbers λ1 (x) < · · · < λk(x) (x) such that for each v ∈ E xi we have lim
t→±∞
1 log ||Dx ϕ t (v)|| = λ+ (x, v) = −λ− (x, v) = λi (x). t j
In addition, for every v ∈ E xi and w ∈ E x with i = j, we have lim
t→±∞
1 log | sin ∠(Dx ϕ t (v), Dx ϕ t (w))| = 0, t
(18)
j
i.e., the angles between the two subspaces E xi and E x can go to zero at most subexponentially along the orbit of x (see [4] for details on Lyapunov regularity). By the multiplicative ergodic theorem, the set of Lyapunov regular points has full measure with respect to any μ ∈ M(Φ). Furthermore, the functions x → λi (x), dim E xi , and k(x) are μ-measurable and Φ-invariant μ-almost everywhere, and hence are constant μ-almost everywhere if μ is ergodic. Here invariant means that we have λi (x) = λi (ϕ s (x)) for every s. The numbers λi (x) are called the Lyapunov exponents of Φ at x and dim E xi is their multiplicity. For the remainder of the paper we use the same notation for the Lyapunov exponents to account for multiplicities: λ1 (x) ≤ λ2 (x) ≤ · · · ≤ λdim M (x). Given μ ∈ M(Φ), we also consider def
λi (μ) =
λi (x) dμ(x).
(19)
M
the set of Lyapunov regular points such that there exists Let us denote by R ⊂ M j (x) = 1 and λ j (x) (x) = a positive integer number j (x) ≤ dim M for which dim E x λ(x, f (x)) is the exponent in the direction of f (x). We now consider the case where at every Lyapunov regular point x ∈ M all Lyapunov exponents except the one related to the direction of the vector field are non-zero. More precisely, we assume that for every j (x) x ∈ R there exists a number j (x) such that dim E x = 1, the exponent λ j (x) (x) is associated to the direction of the flow, and λ1 (x) ≤ · · · ≤ λ j (x)−1 (x) < λ j (x) (x) = 0 < λ j (x)+1 (x) ≤ · · · ≤ λdim M (x). (20) This also includes the case that all exponents except the one related to the flow direction are positive (or negative). A Φ-invariant measure μ is called hyperbolic if (20) holds for μ-almost every point x, in which case R is necessarily of full measure.
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we define Given x ∈ M, def
1 t→±∞ t
r± (x) = lim
t
r (ϕ s (x)) ds
(21)
0
whenever the limit exists. that is Lyapunov regular with respect to Φ Proposition 2. For every point x ∈ R ⊂ M and Φr and satisfies 0 < r± (x) < ∞ and every v ∈ Tx M\{0} we have 1 1 log ||Dx ϕrτ (x,t) (v)|| = lim log ||Dx ϕ t (v)|| = ± λ± (x, v). t→±∞ t t→±∞ t lim
(22)
Proof. Take x ∈ R to be a point that is Lyapunov regular with respect to Φ and Φr ,
and let v ∈ Tx M\{0}. Denote y(t) = Dx ϕ t (v) and let ζ (t) = Dx ϕrτ (x,t) (v). From the Lyapunov regularity, it follows that the limits def
1 log ||y(t)|| t→±∞ t
λ+ (x, v) = −λ− (x, v) = lim
(23)
exist. We first consider the case in which v is in one of the remaining subspaces E xi , i = j (x), which implies ∠(v, f (x)) = 0. From relation (13), it follows that ||ζ (t)|| =
| sin ∠(y(t), f (ϕ t (x)))| ||y(t)|| | sin ∠(ζ (t), f (ϕ t (x)))|
(24)
for every t ∈ R. Using (18), we get 1 log | sin ∠(y(t), f (ϕ t (x)))| = 0. t→±∞ t lim
(25)
we have ϕ t (x) = ϕrτ (x,t) (x). Since Note that for every t ∈ R and every x ∈ M s s Dx ϕr (( f /r )(x)) = ( f /r )(ϕr (x)) for every s ∈ R, we obtain | sin ∠(Dx ϕrτ (x,t) (v), Dx ϕrτ (x,t) ( f (x)))| = | sin ∠(ζ (t), f (ϕ t (x)))|. From property (18) applied to the flow Φr we obtain lim
t→±∞
(note that τ (x, −t) = −
1 log | sin ∠(ζ (t), f (ϕ t (x)))| = 0 τ (x, t)
t 0
r (ϕ −s (x))ds). Using (21) and 0 < r± (x) < ∞, this leads to
1 log | sin ∠(ζ (t), f (ϕ t (x)))| = 0. t→±∞ t lim
(26)
Using (25) and (26) in (24), we obtain lim
t→±∞
1 1 log ||ζ (t)|| = lim log ||y(t)||. t→±∞ t t τ (x,t)
In the other case, v = f (x), we have ||Dx ϕr (v)|| = r (x)/r (ϕ t (x))|| f (ϕ t (x))||. 1 From 0 < r± (x) < ∞, it follows that limt→±∞ t log r (ϕ t (x)) = 0. Hence, we obtain τ (x,t) (v)|| for every vector v ∈ Tx M\{0} parallel λ± (x, v) = ± limt→±∞ 1t log ||Dx ϕr to f (x). The observation that an arbitrary v can be split into components in subspaces E xi completes the proof.
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Remark 3. If condition 0 < r± (x) < ∞ in Proposition 2 is relaxed, the statement remains valid for any v ∈ E xi and any i = j (x), while for v = f (x) the right-hand side of (22) has to be replaced by ± λ± (x, f (x)) − R± (x), where def
R± (x) = lim
t→±∞
1 log r (ϕ t (x)). t
Note that R± (x) ≤ 0 if r± (x) < ∞ and R± (x) ≥ 0 if r± (x) > 0. Moreover, if log r ∈ L 1 (μ) then R± (x) = 0 for μ-almost every x. Given a point x that is Lyapunov regular with respect to the flow Φr , we use the notation λri (x) to distinguish the values of the Lyapunov exponents from those corresponding to Φ. that is Lyapunov regular with respect to Φ Theorem 1. For every point x ∈ R ⊂ M and Φr and satisfies 0 < r+ (x) < ∞, we have λri (x) r+ (x) = λi (x), i = 1, . . . , dim M,
(27)
and hence the number of positive, null, and negative Lyapunov exponents at x remains the same. def
τ (x,t)
Proof. Let v ∈ Tx M and denote ζ (t) = Dx ϕr t → ∞ and hence lim
t→∞
(v). Note that τ (x, t) → ∞ as
1 1 log ||ζ (t)|| = r+ (x) lim log ||Dx ϕrτ (v)||. τ →∞ τ t
Lyapunov regularity and Proposition 2 imply that the Lyapunov exponents at x with respect to Φr satisfy λri (x) r+ (x) = λi (x). Note that the joint Lyapunov regularity automatically implies that r+ (x) = r− (x). This proves the theorem. Remark 4. Let us consider the case in which the condition 0 < r+ (x) < ∞ in Theorem 1 is relaxed. It follows from the proof that for every Lyapunov regular point, r+ (x) = 0 implies that we must have λi (x) = 0, i = 1,. . ., dim M (see also Example 2). Analogously, if r+ (x) = ∞ then we have λri (x) = 0, i = 1, . . . , dim M (compare with Example 3 below). The following is a consequence of Theorem 1. Corollary 1. Let μ ∈ M(Φ) be hyperbolic and let r ∈ L 1 (M, μ). For the measure μr given by (8) we have r dμ = λi (μ), i = 1, . . . , dim M. (28) λri (μr ) M
Therefore, μr ∈ M(Φr ) is hyperbolic if r is positive μ-almost everywhere. Proof. In order to prove (28), what remains to be shown is that for every i = 1, . . . , dim M we have r dμ λri (x) dμr (x) = λi (μ). (29) M
M
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By the Birkhoff theorem, the limit (21) exists μ-almost everywhere and is finite, and r+ ∈ L 1 (M, μ). By the multiplicative ergodic theorem, the set of Lyapunov regular points with respect to any of the flows Φ and Φr has full measure. By (8) we have r dμ λri (x) dμr (x) = λri (x) r (x) dμ(x). M
M
M
Recall that the function x → r+ (x) is μ-measurable and invariant according to the Birkhoff theorem, and that x → λi (x) is μ-measurable and invariant. Here invariance implies that we have r+ (x) = r+ (ϕ s (x)) and λi (x) = λi (ϕ s (x)) for every s, and thus r r dμ λi (x) dμr (x) = λri (x) r (ϕ s (x)) dμ(x). M
M
M
Further, by the first claim of the theorem, we can conclude that
λri (x) r (ϕ s (x)) dμ(x)
λi (μ) − M
r s
= λi (x) dμ(x) − λi (x) r (ϕ (x)) dμ(x)
M
M
r r s
= λi (x) r+ (x) dμ(x) − λi (x) r (ϕ (x)) dμ(x)
M M
t
1 r s
≤ λi (x) r+ (x) − r (ϕ (x)) ds
dμ(x) → 0 as t → ∞. t 0 M Thus it follows that (29) holds true, and this proves the theorem. We illustrate the above results with some explicit constructions. Example 2 (Lyapunov exponents for singular transformations). We consider the flow Φ = {ϕ t }t≥0 generated by the vector field f (x) = −x|x| for every x ∈ R and the time transformed flow Φr with the associated function r (x) = |x|, which has a stable fixed point x = 0. For every x ∈ R we have λ(x) = 0 and λr (x) = −1. Note that r+ (x) = 0 and R+ (x) = 0 (this is not in conflict with Remark 3 since in (22) the 1/t-factor is not time-transformed). This demonstrates that a singular time reparameterization with r+ (x) = 0 may create non-zero Lyapunov exponents in the time-transformed flow. Example 3 (Elimination of non-zero exponents). Let us consider the hyperbolic toral automorphism h : T2 → T2 of the two-torus given by h(x, y) = (2x + y, x + y) mod 1. This flow is ergodic with respect to the Lebesgue measure m. Consider the mapping torus def
M = T2 × [0, 1]/ ∼, where ∼ is the identification of (x, y, 1) with (h(x, y), 0), and consider the standard susdef pension flow of h on M that is defined by ηt (x, y, s) = (x, y, s + t) for 0 ≤ t + s < 1. t The flow {η }t∈R preserves the measure μ defined by 1 def ξ dμ = ξ(x, y, t) dt dm(x, y) M
T2 0
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for any continuous function ξ : M → R, and μ is ergodic since m is. Note that the Lyapunov exponents (with respect to the flow {ηt }t∈R ) are given by √ √ λ1 ( p) = (3 − 5)/2, λ2 (x) = 0, and λ3 ( p) = (3 + 5)/2 for every point p ∈ M. Denote by (x0 , y0 ) the fixed point of h and let p0 = (x0 , y0 , 0). Following the construction in [38], given some δ > 0 one can construct two C ∞ functions, ω1 , ω2 : M → [0, 1], that satisfy 1. ω1 ( p) = ω2 ( p) = 0 if and only if p = p0 , 2. ω1 ( p) = ω2 ( p) = 1 for every p ∈ M\B( p0 , δ), 3. M ω1−1 dμ = 1, M ω2−1 dμ = ∞, where B( p0 , δ) is a ball of radius δ around p0 . Let Φ1 and Φ2 be the time transformations def of {ηt }t∈R with reparameterization functions rk given by rk ( p) = ωk ( p)−1 for every p = p0 , k = 1, 2, respectively. The flow Φ1 preserves the probability measure μ1 given by dμ1 = ω1−1 dμ, and we have r1+ ( p) = 1 for μ−almost every p and μ1 −almost every p.
(30)
The flow Φ2 preserves the σ -finite measure μ2 given by dμ2 = ω2−1 dμ, and we have r2+ ( p) = ∞ for μ−almost every p and μ2 −almost every p.
(31)
We sketch how property (31) can be shown. Choose some monotonically decreasing sequence of continuous functions ηn : M → (0, 1] satisfying ηn ( p) = ω2 ( p) for every p ∈ M\B( p0 , 2−n ) and ηn ( p0 ) ≤ 2−n , and ηn ( p) ≥ ω2 ( p) othdef
In particular, ηn converges pointwise to ω2 . Clearly, we have ξn (μ) = erwise. −1 dμ < ∞ for every n ≥ 1 and lim η n→∞ ξn (μ) = ∞. By the Birkhoff n M ergodic theorem, for each n there is a set An ⊂ M of full μ-measure for which (ηn−1 )+ ( p) = ξn (μ) and hence r2+ ( p) ≥ ξn (μ) for every p ∈ An . Using a Borel-Cantelli argument, we conclude that r2+ ( p) = ∞ for μ-almost every p and hence for μ2 -almost every p. From the arguments above, we obtain λrk1 ( p) = λk ( p) for μ1 −almost every p, and λrk2 ( p) = 0 for μ2 −almost every p. This construction demonstrates that an appropriate time transformation (here by means of r2 ) can eliminate any non-zero Lyapunov exponents almost everywhere. Recall that a set A ⊂ M is of total probability with respect to Me (Φ) if for all μ ∈ Me (Φ) we have μ(A) = 1. Example 4 (Synchronized expansions). In the context of Parry’s analysis of the so-called “synchronization” of canonical measures [33], let us consider the example of a vector field with a topologically transitive C 1 Anosov flow Φ on a compact smooth manifold M. Extending the result in [33], we show that not only the measure of maximal entropy and the SRB measure can be synchronized (see Example 5 below), but also the expansion rates for almost every point. If Φ is C 2 , then x → E xu is Hölder continuous so that Dx ϕ t | E xu and the function defined by def
r (x) = ξ u (x) = − lim
t→0
1 log |det Dx ϕ t | E xu | t
(32)
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K. Gelfert, A. E. Motter
is Hölder continuous. If we assume that Φ is C 2 and also assume2 that the distribution x → E xu is C 1 , then ξ u is C 1 smooth and the so obtained reparameterized flow Φr is a C 1 flow and orbit equivalent to Φ. It follows from the Birkhoff theorem and the multiplicative ergodic theorem that for any μ ∈ Me (Φ), a μ-full measure set of points satisfies
1 r+ (x) = lim log det Dx ϕ t | E xu = λi (x). t→∞ t λi (x)>0
An immediate consequence of Theorem 1 is that we have λr (x)>0 λri (x) = 1 for points i x in a set of total probability. Therefore, for a transformation defined by (32) there is synchronization in the sense that μ λi (x)>0 λi (x) dμ (before the time transformation) is μ-dependent while μr λr (x)>0 λri (x) dμr (after the time transformation) is the same i for all μr ∈ Me (Φr ). In the particular case in which E u is one-dimensional, the set of points of total probability satisfies 1 log ||Dx ϕ t | E xu || = λdim M (x), t→∞ t
r+ (x) = lim
and hence λrdim M (x) = 1 for points x in a set of total probability. 4.2. Topological pressure and entropy. In this section we focus on the dynamics on some compact invariant set Λ ⊂ M. We continue to use M(Φ) to now denote the set of Φ-invariant probability measure supported on Λ. In order to establish our main results, we first briefly recall some concepts from the thermodynamic formalism. Given t > 0 and ε > 0, we say that a set E ⊂ Λ of points is (t, ε)-separated if x1 , x2 ∈ E, x1 = x2 implies that ρ(ϕ s (x1 ), ϕ s (x2 )) > ε for some s ∈ [0, t]. Given a continuous function ξ : Λ → R, the topological pressure of ξ with respect to Φ|Λ is defined by t 1 def s Ptop (Φ, ξ ) = lim lim sup log sup exp ξ(ϕ (x)) ds , (33) ε→0 t→∞ t 0 x∈E
def
where the supremum is taken over all (t, ε)-separated sets E ⊂ Λ. We call h top (Φ) = Ptop (Φ, 0) the topological entropy of Φ|Λ . Note that Ptop (Φ, ξ ) = Ptop (ϕ 1 , ξ ), where the latter is the classically defined topological pressure of ξ with respect to the map ϕ 1 [42]. We have the following variational principle: h μ (ϕ 1 ) + ξ dμ , (34) Ptop (Φ, ξ ) = sup μ∈Me (Φ)
Λ
where h μ (ϕ 1 ) denotes the metric entropy of μ with respect to the map ϕ 1 . A measure that realizes the supremum in (34) is called an equilibrium state for ξ with respect to Φ. By the Abramov formula [1], for every t ∈ R, we have h μ (ϕ t ) = |t| h μ (ϕ 1 ).
(35)
2 This is a significant restriction made in order to obtain an orbit equivalent C 1 flow. For a related discussion, see [33, Sect. 4].
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423
def
One calls h μ (Φ) = h μ (ϕ 1 ) the metric entropy of μ with respect to the flow Φ. For μ ∈ M(Φ) and a flow Φr that is a time reparameterization of Φ by means of a function r ∈ L 1 (M, μ) with Λ r dμ > 0, it is well known [1,41] that the metric entropy of the measure μr ∈ M(Φr ) satisfies r dμ ≤ h μ (Φ). (36) h μr (Φr ) Λ
The equality holds if μ is ergodic. As the pressure is defined for continuous potentials only, we now specialize to time transformations with the same property. The following relation is an immediate consequence of the variational principle (34) and Eqs. (36) and (9). Theorem 2. Given a flow Φr that is a time reparameterization of Φ with an associated positive continuous function r : Λ → R>0 , for every continuous function ξ : Λ → R we have ξ − Ptop (Φ, ξ ) = 0. (37) Ptop Φr , r The following discussion sheds new light on synchronizations of Lyapunov exponents discussed in Example 4. Example 5 (Synchronized equilibrium states). Let Φ be a C 2 topologically transitive Anosov flow on M. Recall that for every q ∈ R, there exists a unique equilibrium state μq for the potential x → q ξ u (x). In particular, μ0 is the unique measure of maximal entropy and μ1 is the SRB measure for Φ. Let us consider ξ u (x, t) = − log |det Dx ϕ t | E u | and denote by ξru (x, τ ) the corresponding function for the flow Φr with respect to the unstable distribution Eru of Φr . It can be shown that ξru (x, τ ) = ξ u (x, t (x, τ )) + η(ϕrτ (x)) − η(x) for some continuous function η (see [2] or [33, Theorem 1]). With the particular choice of time reparameterization r (x) = ξ u (x) in (32), we obtain ξru (x, τ ) = τ + η(ϕrτ (x)) − η(x), which implies that ξru defined by ξru (x) = limτ →0 ξru (x, τ )/τ is Φr -cohomologous to 1.3 This means that for any q ∈ R, the potential q ξru is cohomologous to a constant, and hence there is a unique measure that is the equilibrium state for every q ξru . More important, this measure coincides with the measure of maximal entropy and the SRB measure for Φr . It is an immediate consequence that Ptop (Φr , q ξru ) = 1 − q for every q ∈ R. For a comparison, consider the nontrivial case in which ξ u is not Φ-cohomologous to a constant or, equivalently, that μ0 = μ1 . A change of time parameterization transforms the Φ-SRB measure μ1 into the Φr -SRB measure. Observe that the measure μ0 of maximal entropy with respect to Φ then must satisfy h μ0 (Φ) < M r dμ0 . This implies that the measure μ0r satisfies h μ0r (Φr ) < 1 = h top (Φr ), that is, after the change of time the transformed measure is no longer of maximal entropy. 3 Recall that ξ : M → R is Φ-cohomologous to a constant c ∈ R if there exists a bounded measurable function η : M → R such that ξ(x) − c = limt→0 (η(ϕ t (x)) − η(x))/t for every x ∈ M.
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In the following theorem we put the above example in a more general framework, which holds true for arbitrary potentials. Recall that a set Λ ⊂ M is said to be a basic set of an axiom A flow Φ if Λ contains a dense set of periodic orbits and Φ|Λ is hyperbolic, locally maximal, and topologically transitive. Theorem 3. Let Λ be a basic set of a C 1 axiom A flow Φ and ξ : M → R>0 be a C 1 function. Given the flow Φr that is the time reparameterization of Φ with the associated function r = ξ , we have Ptop (Φξ , q ξ ) = q0 − q for every q ∈ R,
(38)
where q0 is the unique number satisfying Ptop (Φ, q0 ξ ) = 0. Moreover, the equilibrium state for q ξ with respect to Φξ coincides with the measure of maximal entropy with respect to Φξ for every q ∈ R. Proof. By a well-known result on the equidistribution of closed orbits for Φ we know that the unique equilibrium state μ(ξ ) of ξ with respect to Φ is obtained as the weak∗ limit of the weighted orbital measures: given any ε > 0,
γ
(γ )eξ(γ ) m γ γ
def
→ μ(ξ ) for T → ∞, where ξ(γ ) =
(γ )eξ(γ )
Λ
ξ dm γ ,
with summation taken over closed orbits γ with period (γ ) between T and T + ε and m γ denoting the invariant probability measure supported on γ (see [36]). Considering the change of time parameterization given by r = ξ , any closed orbit γr = {ϕrτ (x) : 0 ≤ (γ ) τ ≤ (γr )} with respect to Φr satisfies 0 r ξ(ϕrτ (x)) dτ = (γ ). Hence, for any real parameter q, the equilibrium state μr (q ξ ) of q ξ with respect to Φr is obtained as the limiting measure
γ
r
(γr ) m γr γr
(γr )
→ μr (q ξ ) for T → ∞.
In particular, this equilibrium state coincides with the measure of maximal entropy for Φr . Combined with (37), this leads to h top (Φr ) = q0 , which implies (38) and proves the theorem. Now we present some results on topological entropy. Theorem 2 leads to the following characterization of the topological pressures in terms of the topological entropy. Theorem 4. Given a flow Φr that is a time reparameterization of Φ with an associated positive continuous function r : Λ → R>0 , we have h top (Φ) = 0. Ptop Φr , − r Proof. It follows from the definition (33) that Ptop (Φ, −h top (Φ)) = 0 and that the number h top (Φ) is the only zero of the function s → Ptop (Φ, −s) on the real line. The statement then follows from Theorem 2 with ξ = 0.
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If the reparameterization function r : Λ → R>0 is positive, by the variational principle (34) and by means of (36) we can characterize the topological entropy of the flow Φr as h top (Φr ) =
h (Φ) μ . μ∈Me (Φ) Λ r dμ sup
The topological entropy is then bounded as h top (Φ)
supμ∈M(Φ)
Λr
dμ
≤ h top (Φr ) ≤ h top (Φ)
sup
ν∈M(Φr ) Λ
1 dν. r
In general the topological entropy can change with the time transformation of the flow, as the following examples demonstrate. Example 6 (Topological entropy). Let us consider the example of a vector field f with a topologically transitive Anosov flow Φ on a compact smooth manifold M. Considering a time transformation by means of a C 1 function r : M → R>0 , we obtain h (Φ) 1 top dμΦr , ≤ h top (Φr ) ≤ h top (Φ) M r M r dμΦ where μΦ and μΦr denote the Margulis measures [23] with respect to Φ and to Φr (i.e., the unique and hence ergodic measure of maximal entropy). In particular, the topological entropy changes when r (x) ≡ r0 = 1. Zero topological entropy is an invariant for orbit equivalent flows without fixed points [31]. In equivalent flows with fixed points, this property can change with a time reparameterization if we consider a more general, not integrable, reparameterization function. Example 7 (Entropies for non-integrable r ). Sun et al. [38] constructed C ∞ flows Φ1 and Φ2 on a compact Riemannian manifold M of dimension greater than 2 that possess one fixed point x0 ∈ M and that are time transformations of each other. The flow Φ1 preserves an ergodic probability measure μ satisfying h μ (Φ1 ) > 0 and hence we have h top (Φ1 ) > 0. The only ergodic probability measure ν preserved by Φ2 is supported on the fixed point, leading to h ν (Φ2 ) = h top (Φ2 ) = 0. In view of (36) and h μ (Φ1 ) > 0 it can be concluded that the orbit equivalent flows Φ1 and Φ2 are related to each other by a time transformation with a function that is not μ-integrable. We refer to [38] for details on the construction of the flow.
4.3. Recurrences and waiting times. The Poincaré recurrence theorem says that, given a measure μ ∈ M(Φ), almost all orbits starting from a set of positive measure will come back to it infinitely many times (see, e.g., [11,42]). Quantitative indicators of such recurrent behavior are given by the lower and the upper recurrence rates with respect to the flow Φ. Given x, y ∈ M, let us define the lower and upper waiting time indicators by def
R(Φ, x, y) = lim inf ε→0
log E B(x,ε) (Φ, y) , − log ε
def
R(Φ, x, y) = lim sup ε→0
log E B(x,ε) (Φ, y) , − log ε
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K. Gelfert, A. E. Motter def
where B(x, ε) = {z : ρ(x, z) < ε} and def E A (Φ, y) = inf t > e A (Φ, y) : ϕ t (y) ∈ A def
denotes the time of first (re-)entrance of the trajectory of y ∈ Λ into A. Here e A (Φ, y) = inf{t > 0 : ϕ t (y) ∈ / A} denotes the escape time of y from A. If E B(x,ε) (Φ, y) is infinite for some ε, then R(Φ, x, y) and R(Φ, x, y) are set to be infinite. For x = y we call def
def
R(Φ, x) = R(Φ, x, x) and R(Φ, x) = R(Φ, x, x) the lower and the upper recurrence rates at x with respect to Φ, respectively. Given a measure μ ∈ M(Φ) and a positive measure set A, it follows from the Poincaré recurrence theorem that E A (Φ, x) is finite for μ-almost every x ∈ A. We study transformations of recurrence and waiting rates under a change of time parameterization by studying their local behavior. Notice that for x ∈ M, E B(x,ε) (Φ,y) E B(x,ε) (Φr , y) = r (ϕ s (y)) ds, (39) 0
which follows immediately from the time transformation (6). we have Lemma 1. Given μ ∈ M(Φ), for μ-almost every x ∈ M lim
ε→0
E B(x,ε) (Φr , x) = r+ (x). E B(x,ε) (Φ, x)
(40)
Proof. A well-known recurrence result [16, p. 61] states that for μ-almost every x ∈ M we have lim inf t→∞ ρ(x, ϕ t (x)) = 0. If x is non-periodic, this implies that E B(x,ε) (Φ, x) → ∞ as ε → 0, and (40) follows from the Birkhoff ergodic theorem. If x is periodic with period T > 0, then E B(x,ε) (Φ, x) = T , and (39) implies (40). We call μ ∈ M(Φ) non-atomic if it is not supported on a periodic orbit. Lemma 2. Let μ ∈ M(Φ) be non-atomic. Then given x ∈ M, for μ-almost every we have y∈M lim
ε→0
E B(x,ε) (Φr , y) = r+ (y). E B(x,ε) (Φ, y)
(41)
of positive measure such that Proof. Let us assume that there is a set A ⊂ M E B(x,ε) (Φ, y) → ∞ when ε → 0 and is bounded, that is, there exists a positive number C such that for every y ∈ A we have lim supε→0 E B(x,ε) (Φ, y) < C. Since the measure is nonatomic, let us consider a number ε small enough such that μ(B(x, ε)) < C −1 μ(A). Since every y ∈ A must enter B(x, ε) in less than C time units, there is a t ≤ C and a subset A ⊂ A such that μ(A ) ≥ C −1 μ(A) and ϕ t (A ) ⊂ B(x, ε). Because ϕ −t (B(x, ε)) will have larger measure than B(x, ε), this contradicts the invariance of the measure μ and hence proves that A has zero measure. Accordingly, for μ-almost every y we have E B(x,ε) (Φ, y) → ∞ as ε → 0 and, in view of (39), this implies (41). Thus, we obtain the following result. Theorem 5. Given μ ∈ M(Φ) and a flow Φr that is a time reparameterization of Φ with an associated function r ∈ L 1 (M, μ), for μ-almost every x we have R(Φr , x) = R(Φ, x) and R(Φr , x) = R(Φ, x). If μ is non-atomic, then for μ-almost every y we have R(Φr , x, y) = R(Φ, x, y) and R(Φr , x, y) = R(Φ, x, y).
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Waiting times may change after a change of time parameterization if for a “typical point” y we have r+ (y) = ∞. A construction of such a flow and reparameterization function can be obtained following Example 3 or the model in [38]. We illustrate this fact instead with a simple example in which the measure is atomic. Example 8 (Waiting times for singular transformations). Consider the flow Φ generated by the vector field f (x) = −x, for x ∈ R. In the case of the time reparameterized flow Φr1 with the associated function r1 (x) = − log |x|, for x = 0 and every y ∈ R\{0} we have r1+ (y) = ∞ and R(Φr1 , 0, y) = 0 < R(Φ, 0, y) = 1. In the case of the time reparameterized semi-flow Φr2 = {ϕrt 2 }t≥0 with the associated function r2 (x) = |x|−c , c ≥ 1, for x = 0 and every y ∈ R\{0} we have r2+ (y) = ∞ and R(Φ, 0, y) = 1 ≤ R(Φr2 , 0, y) = c. This shows that r+ (y) < ∞ is necessary for the invariance of the waiting time statistics, which is satisfied for μ-almost every y provided μ ∈ M(Φ) and r ∈ L 1 (M, μ). 5. Transformation of Dimensions of Measures We now consider concepts from dimension theory for dynamical systems. We refer to [35] and references therein for the proofs of the known properties used below. 5.1. Hausdorff dimension and local dimension. Given a σ -finite Borel measure μ and x ∈ M, we define the lower local dimension of μ at x by def
d μ (x) = lim inf ε→0
log μ(B(x, ε)) . log ε
(42)
Analogously, the upper local dimension d μ (x) of μ at x is defined by replacing the lower def
limit with the upper limit. If d μ (x) = d μ (x) = dμ (x), we call the common value the local dimension of μ at x. If μ is a Borel probability measure, we consider the Hausdorff dimension of μ defined by def
dimH μ = inf{dimH A : A ⊂ M and μ(A) = 1}, where dimH A denotes the Hausdorff dimension of a set A. Note that dimH μ = ess sup d μ (x),
(43)
x∈M
where ess sup denotes the essential supremum with respect to μ. Theorem 6. For any μ ∈ M(Φ) and μr defined by d μr = r dμ, we have d μr (x) = 1 d μ (x), d μr (x) = d μ (x) for all x ∈ M. Moreover, if r ∈ L (M, μ) and μr is given by (8), we have dimH μr ≤ dimH μ.
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the function r is C −1 and positive on Proof. Recall that Φr is C . For every x ∈ M, B(x, ε) for sufficiently small ε, and hence the first two equalities are immediate. Note that for every Φ-invariant A ⊂ M we have μr (A) = 0 whenever μ(A) = 0. This implies dimH μr ≤ dimH μ. Example 9 (Hausdorff dimension for non-integrable r ). In the context of Example 3, we have dμ (y) = 3 = dimH μ for every y ∈ M. The measure μ2 defined by dμ2 = ω2−1 dμ (induced by the time reparameterization in that example) is not normalizable. It satisfies It can be shown that the (only ergodic) Φ2 -invariant dμ2 (y) = 3 for every y ∈ M. probability measure ν is supported on the fixed point p0 and hence satisfies dν ( p0 ) = dν ( p) = 0 = dimH ν for every p ∈ M. 5.2. Information dimension. Given a Borel probability measure μ, the lower information dimension of μ is defined by 1 def D 1 (μ) = lim inf log μ(B(x, ε)) dμ(x) ε→0 log ε M (this definition is equivalent to the standard one using grids [3]). Analogously, the upper information dimension D 1 (μ) is defined by replacing the lower limit with the upper limit. Theorem 7. Given a flow Φ and a measure μ ∈ M(Φ) satisfying def
d μ (·) = d μ (·) = dμ (·)
(44)
μ-almost everywhere, and given a flow Φr that is a time reparameterization of Φ with an associated function r ∈ C −1 and μr given by (8), we have r (x) + D1 (μr ) = dμ (x) dμ(x). (45) M M r dμ Moreover, if additionally 1. μ (and hence μr ) is ergodic, or if 2. d μ = d μ = d almost everywhere for some constant d, then we have D1 (μr ) = D1 (μ). Proof. As a consequence of the Fatou Lemma, we obtain d μ (x) dμ(x) ≤ D 1 (μ) ≤ D 1 (μ) ≤ d μ (x) dμ(x). M
(46)
M
Under the assumptions of the theorem, from (46), (8), and Theorem 6, we have r (x) r (x) D1 (μr ) = dμ(x) = dμ(x). dμr (x) dμ (x) r dμ M M M M r dμ Since ϕ s : M → M, s ∈ (0, 1], is a Lipschitz map with Lipschitz inverse, one can verify that dμ (x) = dμ (ϕ s (x)). Thus for arbitrary s > 0, 1 dμ (x)r (x) dμ(x) = dμ (x) τ (x, s) dμ(x), s M M
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from which we obtain (45). If μ is ergodic, then r+ (x) = M r dμ for almost every x, and hence D1 (μr ) = D1 (μ). If d μ = d μ = d almost everywhere, then we can use the relation M r+ dμ = M r dμ, which follows from the Birkhoff ergodic theorem, to conclude that D1 (μr ) = D1 (μ) = d. Theorem 7 implies that for a non-ergodic measure μ with a local dimension that is not constant on a set of positive measure, we can find a suitable change of the time parameterization, and hence a “redistribution” of the invariant measure, such that the Hausdorff dimension and the information dimension of the resulting measure μr differ. Thus, the information dimension can sensitively depend on the time parameterization and as such cannot be regarded as an invariant. We conclude this section by giving illustrative examples. Example 10 a (Noninvariance of D1 ). Let Φ = {ϕ t }t∈R be a flow possessing a fixed point x0 and a periodic point y = ϕ T (y) of period T > 0. Let μ = δx0 be the Dirac-δ measure supported on x0 and let ν be the Φ-invariant Borel probability measure that is supported def
on the periodic orbit O through y. Then, D1 (μ) = 0 = d μ (x0 ) = d μ (x0 ) = dμ (x0 ) def
and D1 (ν) = 1 = d μ (x) = d μ (x) = dμ (x) for every x ∈ O. Given α ∈ (0, 1), the def
(non-ergodic) measure μ = αν + (1 − α)μ satisfies (44), and hence μ) = α D1 (ν) + (1 − α)D1 (μ) = 1 − α > D1 (
(1 − α)r+ (y) = D1 ( μr ) r (x0 )α + (1 − α)r+ (y)
whenever Φr is a flow obtained from Φ by means of a smooth change of the time parameterization satisfying r (x0 ) < r+ (y). Consider a flow on a compact Riemannian manifold that can be represented as a suspension flow over a two-sided topological Markov chain with Hölder continuous roof function. Such flows naturally occur as models for flows that possess strong hyperbolic behavior. Primary examples are geodesic flows on compact Riemannian manifolds with negative sectional curvature and their time reparameterizations. Note that the property (44) is satisfied for any invariant Borel probability measure μ [5, Theorem 1], and that dμ is constant almost everywhere if μ is ergodic. This is the starting point for the following (more involved) variation of Example 10 a. Example 10 b (Noninvariance of D1 ). Let in particular M be a compact orientable Riemannian surface of class C 4 with negative curvature K and consider the geodesic flow Φ = {ϕ t }t∈R on the unit tangent bundle S M. Note that the normalized Liouville measure μ S M on S M, which is induced by the volume μ M on M, is ergodic (see [4] for details). Hence, for μ S M -almost every x ∈ S M we have 1 t def K (ϕ s (x)) ds = K dμ M = 2π χ M , K + (x) = lim t→∞ t 0 M where χ M denotes the Euler characteristic of M and where the last equality follows from the Gauß-Bonnet theorem. Let us further assume that K is not Φ-cohomologous to a constant. It follows from the Livshitz theorem for flows (see, for example, [22, Theorem 19.2.4]) that there exist periodic points x1 = ϕ t1 (x1 ) and x2 = ϕ t2 (x2 ) such that 1 t1 1 t2 K (ϕ s (x1 )) ds = K (ϕ s (x2 )) ds, t1 0 t2 0
430
K. Gelfert, A. E. Motter
and, in particular, there exists a periodic point x0 ∈ S M such that K + (x0 ) = 2π χ M . def
We assume that k0 = K + (x0 ) < 2π χ M . We consider the Φ-invariant Borel probability measure ν that is supported on the periodic orbit O through x0 and satisfies d ν (x) = d ν (x) = D1 (ν) = 1 for every point x on this orbit. Analogously, for μ S M -almost every y ∈ S M we have d μ S M (y) = d μ S M (y) = D1 (μ S M ) = 3. In addition, we have K + (x) = k0 < 2π χ M = K + (y) for ν-almost every x and μ S M -almost every y. Consider the positive function r : M → R>0 given by r (x) = −K (x) for every x ∈ S M. (Note that we can, in fact, take any other Hölder continuous function ξ : S M → R that is not Φ-cohomologous to a constant and consider the function r = ξ −min x∈S M ξ(x)+1.) def
Given α ∈ (0, 1), the measure μ = αν + (1 − α)μ S M satisfies (44) and hence μ) = α D1 (ν) + (1 − α)D1 (μ S M ) = 3 − 2α. D1 ( If we now consider the flow Φr obtained from Φ after a change of time parameterization with the function r = −K , we obtain D1 ( μr ) =
αk0 (1 − α)2π χ M D1 (ν) + D1 (μ S M ), αk0 + (1 − α)2π χ M αk0 + (1 − α)2π χ M
μ) < D1 ( μr ). where k0 < 2π χ M implies αk0 /[αk0 +(1−α)2π χ M ] > α, and hence D1 ( 5.3. Generalized dimensions. Given q > 0, q = 1, we define the lower generalized dimension of order q of a Borel probability measure μ on M by 1 1 def lim inf log D q (μ) = μ(B(x, ε))q−1 dμ(x). q − 1 ε→0 log ε M The functions D q (μ) are defined analogously with the upper limit and are called the upper generalized dimension of order q of μ. Although the generalized dimensions a priori do not involve any dynamics, they were introduced to obtain dynamical information by observing individual trajectories [18], thus admitting dynamical interpretations when μ is an invariant measure of the system. Of particular dynamical interest is the so-called lower (upper) correlation dimension D 2 (μ) (D 2 (μ)), which is the most accessible one in numerical computations based on time series analysis (see [34,35] and references therein). The discrete analogs of the above defined dimensions, often considered in the physics literature, are called generalized lower (upper) Rényi dimensions of order q. Given ε > 0, cover the support of μ with boxes Bk of a uniform grid of size ε and denote by N (ε) the number of boxes needed to cover this set. For q > 0, q = 1, let N (ε)
1 1 lim inf log R D q (μ) = μ(Bk )q , q − 1 ε→0 log ε def
(47)
k=1
and define R D q (μ) analogously by replacing the lower limit with the upper limit. These quantities are invariant under smooth transformations of the phase space and their definitions are independent of the choice of the grid. Moreover, R D q (μ) = D q (μ) and R D q (μ) = D q (μ) (see [3,35] and references therein). Furthermore, we now show that the generalized dimensions of flow-invariant measures do not depend on a particular choice of time parameterization.
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431
Theorem 8. Given a flow Φr that is a time reparameterization of Φ with an associated uniformly bounded positive function r : M → R>0 , for μ ∈ M(Φ) and μr given by (8), we have D q (μr ) = D q (μ) and D q (μr ) = D q (μ) for every q > 0, q = 1. Proof. It suffices to notice that, by Eq. (8), measures μ and μr are absolutely continuous with respect to each other. Since r is positive and bounded, there exist positive constants c1 , c2 such that c1 μ(B(x, ε)) ≤ μr (B(x, ε)) ≤ c2 μ(B(x, ε)) for every x and ε > 0. This applied to the definitions of D q and D q proves the statement. Remark 5. We emphasize that the Examples 10a,b illustrate the non-invariance of the information dimension already in the case of orbit equivalent flows with a differentiable reparameterization function r . In contrast, note that, by Theorem 8, in each of these examples we have Dq (μ) = Dq (μr ) for every q > 0, q = 1. The implications of the non-invariance of the information dimension for the Kaplan-Yorke relation are discussed in [27]. 6. Concluding Remarks The transformation of dynamical quantities established in this paper addresses a longstanding problem in general relativity and cosmology, namely, of whether chaos is a property of the physical system or a property of the coordinate system [21]. At the center of this discussion is the mixmaster cosmological model [25], a spatially homogeneous anisotropic solution of Einstein’s equations that has been conjectured to describe the dynamics of the early universe. This model can be described as a geodesic flow on a Riemannian manifold with negative curvature [9], which nevertheless has been shown to have positive or vanishing largest Lyapunov exponent depending on the time coordinate adopted [15]. This problem has resisted rigorous solution because, as shown in [26] and [30], the transformations used in cosmology are not guaranteed to preserve the normalization property of the invariant measure and because the identification of truly invariant indicators of chaos is often elusive. To this regard, our results show once and for all that Lyapunov exponents, entropies, and dimension-like characteristics can be used to make invariant assertions about chaos. However, the same results also show that the values of some quantities that have been previously conjectured to be invariant, such as the information dimension and topological entropy (see [28] for a related discussion), are not invariant in general. Finally, we observe that there are several directions along which we expect this work to be extended. One concerns applications, such as in the above mentioned study of relativistic dynamics and to probe the invariance of dynamical quantities and identities, as exemplified by our synchronization analysis. Another direction concerns the study of different dynamical quantities and the relations between them. The spectrum of return time dimension [19], for example, which relates the recurrence times to the multifractal properties of strange attractors [17], can be shown to be invariant under uniformly bounded time transformations, whereas an exponential distribution of first return times can change within exponential bounds. Such transformation properties can stimulate further investigation on how robust the properties of the return time statistics are, particularly when the system gains or loses mixing properties after a time reparameterization. Note that the mixing properties are in general not preserved [14], since there are analytic time reparameterizations of a completely non-chaotic flow on T3 that are mixing (see [24] for additional references). Other extensions can be envisioned in the joint characterization of flows and their discretizations, particularly through the consideration of
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K. Gelfert, A. E. Motter
Poincaré maps and suspension flows, for which many of the necessary techniques have been developed. Acknowledgements. The authors thank Aysa Sahin for stimulating discussions and Stefano Galatolo for hints on Lemma 2. A.E.M. acknowledges support from the Alfred P. Sloan Foundation in the form of a Sloan Research Fellowship.
References 1. Abramov, L.: On the entropy of a flow. Amer. Math. Soc. Transl. Ser. 2 49, 167–170 (1966) 2. Anosov, D.V., Sinai, Ya.G.: Certain smooth ergodic systems. Russ. Math. Surv. 22, 103–167 (1967) 3. Barbaroux, J.-M., Germinet, F., Tcheremchnatsev, S.: Generalized fractal dimensions: Equivalences and basic properties. J. Math. Pures Appl. 80, 977–1012 (2001) 4. Barreira, L., Pesin, Y.: Smooth ergodic theory and nonuniformly hyperbolic dynamics, with an appendix by O. Sarig. In: Handbook of Dynamical Systems 1B, B. Hasselblatt, A. Katok eds., Amsterdam: Elsevier, 2006 5. Barreira, L., Radu, L., Wolf, C.: Dimension of measures for suspension flows. Dyn. Syst. 19, 89–107 (2004) 6. Benini, R., Montani, G.: Frame independence of the inhomogeneous mixmaster chaos via MisnerChitré-like variables. Phys. Rev. D 70, 103527 (2004) 7. Berger, B.K.: Comments on the computation of Liapunov exponents for the Mixmaster universe. Gen. Relativ. Gravit. 23, 1385–1402 (1991) 8. Burd, A.B., Buric, N., Ellis, G.F.R.: A numerical analysis of chaotic behaviour in Bianchi IX models. Gen. Relat. Gravit. 22, 349–363 (1990) 9. Chitré, D.M.: Investigations of Vanishing of a Horizon for Bianchy Type X (the Mixmaster) Universe. Ph.D. Thesis, University of Maryland, 1972 10. Contopoulos, G., Grammaticos, B., Ramani, A.: The mixmaster universe model, revisited. J. Phys. A 27, 5357–5361 (1994) 11. Cornfeld, I., Fomin, S., Sinai, Y.: Ergodic Theory. Berlin-Heidelberg-NewYork: Springer, 1982 12. Cornish, N.J., Levin, J.J.: Mixmaster universe: A chaotic Farey tale. Phys. Rev. D 55, 7489–7510 (1997) 13. Cornish, N.J., Levin, J.J.: The mixmaster universe is chaotic. Phys. Rev. Lett. 78, 998–1001 (1997) 14. Fayad, B.: Analytic mixing reparametrizations of irrational flows. Erg. Th. Dynam. Syst. 22, 437–468 (2002) 15. Francisco, G., Matsas, G.E.A.: Qualitative and numerical study of Bianchi IX models. Gen. Relat. Grav. 20, 1047–1054 (1988) 16. Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton, NJ: Princeton University Press, 1981 17. Galatolo, S.: Dimension via waiting time and recurrence. Math. Res. Lett. 12, 377–386 (2005) 18. Grassberger, P., Procaccia, I.: Characterization of Strange Attractors. Phys. Rev. Lett. 50, 346–349 (1993) 19. Haydn, N., Luevano, J., Mantica, G., Vaienti, S.: Multifractal properties of return time statistics. Phys. Rev. Lett. 88, 224502 (2002) 20. Hobill, D., Bernstein, D., Welge, M., Simkins, D.: The Mixmaster cosmology as a dynamical system. Class. Quant. Grav. 8, 1155–1171 (1991) 21. Hobill, D., Burd, A.B., Coley, A.A. (eds.): Deterministic chaos in general relativity. NATO ASI Series B, Vol. 332, London: Plenum Press, 1994 22. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. In: Encyclopedia of Mathematics and Its Applications 54, Cambridge: Cambridge University Press, 1995 23. Katok, A., Knieper, G., Weiss, H.: Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows. Commun. Math. Phys. 138, 19–31 (1991) 24. Katok, A., Thouvenot, J.-P.: Spectral properties and combinatorial constructions in ergodic theory. In: Handbook of Dynamical Systems 1B, B. Hasselblatt, A. Katok, eds., Amsterdam: Elsevier, 2006 25. Misner, C.W.: Mixmaster universe. Phys. Rev. Lett. 22, 1071–1074 (1969) 26. Motter, A.E.: Relativistic chaos is coordinate invariant. Phys. Rev. Lett. 91, 231101 (2003) 27. Motter, A.E., Gelfert, K.: Time-metric equivalence and dimension change under time reparatererizations. Phys. Rev. E 79, 065202(R) (2009) 28. Motter, A.E., Letelier, P.S.: Mixmaster chaos. Phys. Lett. A 285, 127–131 (2001) 29. Motter, A.E., Letelier, P.S.: FRW cosmologies between chaos and integrability. Phys. Rev. D 65, 068502 (2002) 30. Motter, A.E., Saa, A.: Relativistic invariance of Lyapunov exponents in bounded and unbounded systems. Phys. Rev. Lett. 102, 184101 (2009)
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31. Ohno, T.: A weak equivalence and topological entropy. Publ. Res. Inst. Math. Sci. 16, 289–298 (1980) 32. Parry, W.: Topics in Ergodic Theory. Cambridge University Press, Cambridge (1981) 33. Parry, W.: Synchronisation of canonical measures for hyperbolic attractors. Commun. Math. Phys. 106, 267–275 (1986) 34. Pesin, Y.: On rigorous mathematical definitions of correlation dimension and generalized spectrum for dimensions. J. Stat. Phys. 71, 529–547 (1993) 35. Pesin, Y.: Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Chicago Lectures in Mathematics, Chicago: Chicago University Press, 1998 36. Pollicott, M., Sharp, R., Tuncel, S., Walters, P.: The mathematical research of William Parry. Erg. Th. Dynam. Syst. 28, 321–337 (2008) 37. Rugh, S.E.: In Ref. [21], p. 359 38. Sun, W., Young, T., Zhou, Y.: Topological entropies of equivalent smooth flows. Trans. Amer. Math. Soc. 361, 3071–3082 (2009) 39. Szydlowski, M.: Chaos hidden behind time parametrization in the Mixmaster cosmology. Gen. Relativ. Gravit. 29, 185–203 (1997) 40. Szydlowski, M., Krawiec, A.: Description of chaos in simple relativistic systems. Phys. Rev. D 53, 6893–6901 (1996) 41. Totoki, H.: Time changes of flows. Mem. Fac. Sci. Kyushu Univ. Ser. A 20, 27–55 (1966) 42. Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79, Berlin-HeidelbergNew York: Springer, 1981 Communicated by G. Gallavotti
Commun. Math. Phys. 300, 435–486 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1117-5
Communications in
Mathematical Physics
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model M. Disertori1 , T. Spencer2 , M. R. Zirnbauer3 1 Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Université de Rouen,
76801 Saint-Étienne-du-Rouvray, France. E-mail: [email protected]
2 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA 3 Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany
Received: 2 October 2009 / Accepted: 5 April 2010 Published online: 31 August 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract: We study a lattice field model which qualitatively reflects the phenomenon of Anderson localization and delocalization for real symmetric band matrices. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. Correlations in this model may be described in terms of a random walk in a highly correlated random environment. We prove that in three or more dimensions the model has a ‘diffusive’ phase at low temperatures. Localization is expected at high temperatures. Our analysis uses estimates on non-uniformly elliptic Green’s functions and a family of Ward identities coming from internal supersymmetry.
1. Introduction 1.1. Some history and motivation. It has been known since the pioneering work of Wegner [19,20] that information about the spectral and transport properties of random band matrices and random Schrödinger operators can be inferred from the correlation functions of statistical mechanical models of a certain kind. These models have a hyperbolic symmetry, typically a noncompact group such as O( p, q) or U( p, q), and were originally studied in the limit of p = q = 0 replicas. The connection between random Schrödinger operators and statistical mechanics models was made more precise by Efetov [7], who introduced the so-called supersymmetry method to avoid the use of replicas. In Efetov’s formulation one employs both commuting (or bosonic) and anticommuting (or Grassmann) integration variables, and these are related by a natural symmetry that makes the emerging statistical mechanics system supersymmetric (SUSY). The simplest class of these models has a U(1, 1|2) symmetry. This means that for the bosonic variables there exists a hyperbolic symmetry U(1, 1) preserving an indefinite Hermitian form on C2 , and the Grassmann variables are governed by a compact U(2) symmetry. Moreover, there exist odd symmetries mixing Grassmann and bosonic variables.
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The fields Zd j → Q j of the supersymmetric models introduced by Efetov are 4 by 4 supermatrices built from bosonic as well as Grassmann entries. In the physics literature one usually assumes the sigma model approximation, which is believed to capture the essential features of the energy correlations and transport properties of the underlying quantum system. The sigma model approximation constrains the matrix field Q by Q 2j = Id for all j. This constraint is similar to the constraints appearing in the Ising or Heisenberg models, where S j · S j = 1. We refer the reader to [3,9,11,14] for an introduction to these ideas. The models described above are difficult to analyse with mathematical rigor in more than one dimension. In this paper we study a simpler SUSY model. Our exposition will be essentially self-contained and the full supersymmetric formalism alluded to here will serve primarily as a source of motivation. 1.2. Probabilistic representation of our model. In this paper we analyze a lattice field model which may be thought of as a simplified version of one of Efetov’s nonlinear sigma models. More precisely, it is related to the model that derives from real symmetric matrices, see Sect. 3. In this statistical mechanics model the field at site j has four degrees of freedom. Two of these, t j and s j , parametrize a hyperboloid and the other two, ψ¯ j and ψ j , are Grassmann (i.e., anticommuting) variables. Technically speaking, the field takes values in a target space denoted by H2|2 , which is a supermanifold extension of the hyperbolic plane H2 ; see Sect. 2. This model was introduced by one of us in [5,21], and localization was established in one dimension (1D) in the sense that the conductance was proven to decay exponentially in the system size [21]. The model is expected to reflect the qualitative behavior of random band matrices – namely localization and diffusion – in any dimension. This paper establishes the existence of a quasi-diffusive phase in three dimensions at low temperature.1 Our supersymmetric hyperbolic nonlinear sigma model, called the H2|2 model for short, will be formulated on a lattice cube ⊂ Zd of side L. We shall see (in Sect. 2.2) ¯ and s. This feature is special to that the action of the field variables is quadratic in ψ, ψ, the horospherical coordinate system that we use. It enables us to reduce the H2|2 model to the statistical mechanics of a single field t : → R, j → t j . Its free energy or effective action, F(t), is real, so the resulting statistical mechanical model has a probabilistic interpretation. In order to specify F(t), first consider the finite-difference elliptic operator Dβ,ε (t) defined by the quadratic form [v ; Dβ,ε (t) v] = β eti +t j (vi − v j )2 + ε etk vk2 . (1.1) (i j)
k∈
This operator plays a central role in our analysis. The first sum is over nearest neighbor pairs in , and [ ; ] denotes the usual scalar product in 2 (). We see that D1,0 (0) is the finite-difference Laplacian. The regularization parameter ε > 0 will serve to make the theory well-defined. One may interpret Dβ,ε (t) as the generator of a random walk in an environment given by the fluctuating field t, with a death rate of ε et j at site j. Note that the operator D is elliptic but not uniformly so, as t j ∈ R has unbounded range. The free energy or effective action Fβ,ε (t) is now expressed by Fβ,ε (t) = β (cosh(ti − t j ) − 1) − ln Det 1/2 Dβ,ε (t) + (tk − ε + ε cosh tk ). (1.2) (i j)
k∈
1 Localization has recently been established at high temperature in any dimension.
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model
If dμ denotes the product measure dμ = on R|| , then the partition function is Z (β, ε) =
dtk √ 2π k∈
R||
e−Fβ,ε dμ = 1.
437
(1.3)
(1.4)
The partition function is identically equal to unity independent of β, ε even when β depends on the edge (i j) and ε depends on the lattice point k; see (5.1). This is a reflection of an internal supersymmetry which will be explained in later sections. There exist many variants of this identity. One of them gives us easy control of nearest neighbor fluctuations of the field t (cf. Sect. 6). The expectation of an observable function t → f (t) is defined by f ,β,ε = f e−Fβ,ε dμ . (1.5) Let us make a few comments on these expository definitions. 1. The action or free energy Fβ,ε (t) is nonlocal due to the presence of the term − ln Det 1/2 D(t). This nonlocality arises from integrating out three massless free ¯ ψ) of Grassmann type. fields, one (s) of bosonic and two (ψ, 2. Fβ,ε (t) is not convex as a function of t and therefore the Brascamp-Lieb estimates used in earlier work on a related model [18] do not apply. The lack of convexity is an important feature and opens the possibility for a localization-delocalization transition to occur. 3. When ε = 0, Fβ,0 (t) is invariant under shifts t j → t j + c by any constant c ∈ R. To see this, note that for ε = 0 we have Dβ,0 (t + c) = e2c Dβ,0 (t) by (1.1). The result1/2 ing additional term −||c from − ln Det Dβ,0 (t) in (1.2) is canceled by another such term, which arises from shifting k∈ tk . This symmetry (which is a formal one, since the integral is ill-defined for ε = 0) is associated with the presence of a massless mode. The importance of the regularization ε, which was omitted from the present argument, becomes evident from the saddle point discussed below. 4. The model at hand describes a disordered quantum system at zero temperature. Nevertheless, adopting the familiar language of statistical mechanics and thermodynamics, we refer to the field stiffness β as the inverse ‘temperature’. (β is actually the dimensionless conductance for an Ohmic system of size L = 1 as measured in lattice units.) 1.3. Main result. The main goal of this paper is to estimate the fluctuations of the field t for large values of the parameter β and dimension d = 3. This will enable us to prove that the random walk in the random environment drawn from F(t) is transient. More precisely, we will prove the following. (Similar estimates hold for all dimensions d ≥ 3.) ¯ the fluctuations of the field Theorem 1. For d = 3, there is a β¯ ≥ 1 such that if β ≥ β, t are uniformly bounded in x, y, and : coshm (tx − t y ) ,β, ε ≤ 2, provided that m ≤
β 1/8 .
(1.6)
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This theorem implies that for any x and y, |tx −t y | is very unlikely to be large. A stronger version of (1.6) is given in (9.3). We will use this result to prove Theorem 2. Under the hypothesis of Theorem 1 the average field is bounded: cosh p (tx ) ,β, ε ≤
5 , 2
(1.7)
provided p ≤ 10 and ||1−α/3 ε ≥ 1 with α ≥ 1/ ln β. Thus in the thermodynamic limit || → ∞ we may send ε → 0 while maintaining the bound on cosh p tx . To investigate the localized or extended nature of the energy eigenstates of a disordered quantum system with Hamiltonian H , one looks at the average square of the quantum Green’s function, |(H − E + iε)−1 (x, y)|2 . The analog of this Green’s function in the H2|2 model is the two-point correlation function (1.8) C x y = et x s x et y s y , where the expectation is given by the full functional integral defined in Sects. 2.1, 2.2. ¯ ψ, and s, we have After integration over the fields ψ, ≡ D˜ β,ε (t)−1 (x, y) , (1.9) C x y = etx +t y Dβ,ε (t)−1 (x, y) ,β,ε
,β,ε
where D˜ = e−t D ◦ e−t . Note that C x y is positive both pointwise and as a quadratic form. A simple calculation shows that D˜ β,ε (t) = −β + βV (t) + ε e−t ,
(1.10)
where V (t) is a diagonal matrix (or ‘potential’) given by (eti −t j − 1) V j j (t) = |i− j|=1 −t −t j (sum over nearest neighbors) and e−t is the diagonal matrix with (e ) j j = e . In Appendix B we establish the sum rule ε y∈ C x y = 1, reflecting conservation of probability for the quantum dynamics generated by a Hamiltonian H . Note that if t were bounded, then D(t) (given by (1.1)) would be uniformly elliptic and we could establish good diffusive bounds on the two-point function C (1.9). However, Theorems 1 and 2 only say that large field values are unlikely. To get optimal bounds on C we would need to prove uniform ellipticity on a percolating set. The set on which |t j + t j | < M, is presumably a percolating set but this does not readily follow from our estimates. Our next theorem states a quasi-diffusive estimate on C. More precisely let G 0 = (−β + ε)−1 be the Green’s function for the discrete Laplacian (with a regularization term ε) and G˜ 0 = (−β + ε/2)−1 . In 3 dimensions G 0 (x, y) ≤ β −1 (1 + |x − y|)−1 (and the same is true for G˜ 0 ). Then we have
Theorem 3. Let f : → R be non-negative. Then assuming the hypotheses of Theorems 1 and 2 we have 1 ˜ ˜] ≤ [ f ; C f ] = [ f ; G Ci j f (i) f ( j) ≤ K [ f ; G˜ 0 f ], (1.11) f 0 K ij
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where f˜( j) = (1 + | j − x|α )−1 f ( j), x ∈ is any fixed point, and K and K are constants independent of f . The parameter α ≈ 1/ ln (β) was introduced in Theorem 2 and is small for large β. Remark. In this paper we always use periodic boundary conditions on ⊂ Z3 . The distance |x − y| between two points is always the distance on with periodic boundary conditions. 1.4. Saddle point. One may try to gain a crude understanding of the behavior of the H2|2 sigma model via a simple saddle-point analysis. Let t (0) be the configuration of t = {t j } which minimizes the effective action Fβ,ε (t) defined in (1.2). In Appendix A we prove (0) that t (0) is unique and t j = t ∗ independent of j. For large β we find ∗
1D: ε e−t β −1 ,
∗
2D: ε e−t e−β ,
(1.12)
in one and two dimensions, respectively. Thus in 1D or 2D the saddle point depends sensitively on the regularization parameter ε. The value of t ∗ suggests a strong asymmetry of the field favoring negative values of t. On the other hand, in 3D at low temperatures, we find t ∗ = 0 independent of ε. Our estimates (1.7) confirm this value by controlling fluctuations about the saddle. For β small, in 3D, the saddle t ∗ is again strongly ε-sensitive, suggesting localization. The bias to negative values of the field t is expected to be closely related to localization. Note that since − + V (t) ≥ 0, the additional term ε e−t makes D˜ β,ε strictly positive at the saddle suggesting that C x y decays roughly like e−m|x−y| with m 2 = ∗ ε e−t /β = β −2 and e−β in 1D and 2D respectively. There are important fluctuations away from this saddle but we do not expect them to spoil the exponential decay. For the 1D chain this has been proved [21]. 1.5. Edge reinforced random walk. A number of mathematicians (Kozma, Heydenreich, Sznitmann) have noted that our random walk looks similar to a linearly edge reinforced random walk (ERRW). ERRW is a history-dependent walk which prefers to visit edges it has visited in the past. Let n(e) denote the number of times the walk has visited the edge e. Then the probability that the walk at vertex v will visit a neighboring edge e equals (a + n(e))/Sa (v), where S is the sum of a + n(e ) over all the edges e touching v. The parameter a is analogous to our β. Coppersmith and Diaconis [2] proved that this history-dependent walk can be expressed as a random walk in a random environment; see also more recent work by Merkl and Rolles [13] in which recurrence of the walk is established on a 2D lattice for small β. This is analogous to localization in our model. The environment of ERRW is very similar to the environment in H2|2 . In fact, both environments have nonlocal actions arising from the square root of a determinant. Although the two models do not seem to be identical, they may have similar properties. 1.6. Outline of the paper. The remainder of this paper is organized as follows. In the next section we give a precise definition of the full H2|2 model and introduce the horospherical coordinate system. The effective action defined in (1.2) is then derived by integration of the field s and the Grassmann fields ψ¯ and ψ. Section 3 provides a physical motivation for the study of this model. In Sect. 4 we explain the symmetries of the model and briefly discuss its perturbative renormalization group flow. The basic Ward identities we shall need are given in Sect. 5 and are derived in Appendix C. Section 5 ends with a rough outline of our proof and a description of the remaining sections of this paper.
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2. Definition of the Model We now fill in the details of the definition the H2|2 model and derive the free energy Fβ,ε (t) given above. 2.1. Full supersymmetric model. As in Sect. 1.2, let ⊂ Zd be a cube of size L. For each lattice site j ∈ we introduce a supervector u j ∈ R3|2 , u j = (z j , x j , y j , ξ j , η j ),
(2.1)
with 3 real components x j , y j , z j and 2 Grassmann variable components ξ j , η j . We then define an inner product on R3|2 by (u, u ) = −zz + x x + yy + ξ η − ηξ ,
(2.2)
and constrain u j by the quadratic equation ∀j ∈ :
(u j , u j ) = −1,
(2.3)
which is solved by
z j = ± 1 + x 2j + y 2j + 2ξ j η j .
(2.4)
Here z j is an even element in the Grassmann algebra (defined as a terminating power series in ξ j η j ) and the sign ± refers to the bosonic part (the ξ η = 0 contribution). In the following we take the positive square root for all j ∈ . This singles out a choice of connected subspace, H2|2 , parametrized by two bosonic variables x j , y j and two fermionic variables ξ j , η j . On the product space (H2|2 )|| we introduce a ‘measure’ (more accurately, a Berezin superintegration form) Dμ =
(2π )−1 d xk dyk ∂ξk ∂ηk ◦ (1 + xk2 + yk2 + 2ξk ηk )−1/2 .
(2.5)
k∈
We use the notation ∂ξ ≡ ∂/∂ξ for the partial derivative. The statistical measure then is of the Gibbs form Dμ e−Aβ,ε with action β Ji j (u i − u j , u i − u j ) + ε (z k − 1) 2 i, j k∈ =β Ji j z i z j − (1 + xi x j + yi y j + ξi η j − ηi ξ j ) + ε (z k − 1).
Aβ,ε =
i, j
(2.6)
k∈
Here Ji j = 1 if i, j are nearest neighbors (NN) and Ji j = 0 otherwise. As will be discussed later, the action Aβ,0 is SO(1,
2)-invariant. The ε-term breaks this noncompact symmetry and makes the integral Dμ e−Aβ,ε converge.
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2.2. Horospherical coordinates. As with [18], it is very helpful to switch to horospherical coordinates — it is only in this coordinate system that we can obtain the probabilistic interpretation of Sect. 1.2. We thus use the following parametrization of the supermanifold: ¯ η = et ψ, ¯ x = sinh t − et 21 s 2 + ψψ , y = et s, ξ = et ψ, (2.7) ¯ ψ) are globally defined where t and s range over the real numbers. Note that (t, s; ψ, coordinates and ¯ ψ) = (0, 0; 0, 0) ⇔ (x, y; ξ, η) = (0, 0; 0, 0). (t, s; ψ, The expression for the action in them is Aβ,ε = β (Si j − 1) + ε (z k − 1), (i j)
(2.8)
k∈
where (i j) are NN pairs and Si j = Bi j + (ψ¯ i − ψ¯ j )(ψi − ψ j ) eti +t j , 1 2 (si
− sj) e Bi j = cosh(ti − t j ) + 1 2 z k = cosh tk + 2 sk + ψ¯ k ψk etk .
2 ti +t j
,
(2.9) (2.10) (2.11)
We also need the expression for the measure Dμ in horospherical coordinates. By applying Berezin’s transformation formula [1] for changing variables in a (super-)integral, one finds that Dμ = (2π )−1 e−t j dt j ds j ∂ψ¯ j ∂ψ j . (2.12) j∈
For any function f of the lattice field variables {t j , s j , ψ¯ j , ψ j } j∈ we now define its expectation as f β,ε = Dμ e−Aβ,ε f, (2.13) whenever this integral exists. ¯ ψ, 2.3. Effective bosonic field theory. Since the action (2.8) is quadratic in the fields ψ, and s, each with covariance Dβ,ε (t)−1 , we know from standard free-field calculus that ¯ ψ yields integration over s yields a factor of Det −1/2 (Dβ,ε (t)) while integration over ψ, Det Dβ,ε (t). By performing these integrations, we arrive at the nonlocal free energy functional Fβ,ε (t) given by (1.2). Moreover, the basic two-point functions are s(v)2 = + [v ; Dβ,ε (t)−1 v] , (2.14) ¯ ψ(v)ψ(v) = − [v ; Dβ,ε (t)−1 v] ,
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where s(v) =
s j v( j), ψ(v) =
j∈
ψ j v( j),
j∈
and the expectations on the left-hand and right-hand side are defined by (2.13) and (1.5), respectively. We will often use the formula (2.14) as well as its generalization n ¯ e λ=1 ψ(vλ )ψ(vλ ) = Det(1 − A) , (2.15) where A is the n × n matrix given by Aλλ (t) = [vλ ; Dβ,ε (t)−1 vλ ].
(2.16)
¯ ψ were absent, then Det 1/2 in (1.2) would be Remark 2.1. If the Grassmann ψ, fields −1/2 −t k replaced by Det (and k e dtk by k etk dtk ) and Z would be the partition function of the hyperbolic sigma model studied in [18]. ¯ ψ (but not over s) we produce a Remark 2.2. If we integrate only over the fields ψ, positive integrand depending on t and s. The square root of the determinant is then replaced by Det Dβ,ε (t) > 0. Remark 2.3. The logarithm of Det Dβ,ε (t) is convex in t. Proof (D. Brydges). By the matrix tree theorem we have t j +t Det Dβ,ε (t) = β ||−|R| ε|R| e j e tk , F
∈F
(2.17)
k∈R
where F denotes the spanning rooted forests, R the set of roots, |R| the cardinality of this set, and = ( j , j ) denotes an edge in the forest. The proof is now immediate since any positive sum of exponentials in t is log convex. Note that the logarithm of Det Dβ,ε (t) competes with the other factor, e−β (i j) cosh(ti −t j ) , which is log concave. 3. Microscopic Origin of the Model In this subsection we use the language and heuristic ideas of physics to sketch the origin of our field theory model from a microscopic model of disorder. Consider real symmetric random band matrices, H , say with independent Gaussian distributed entries, of band width W in d dimensions. (Such a band matrix model possesses a time-reversal symmetry and belongs to symmetry class AI – traditionally referred to as the WignerDyson class of orthogonal symmetry – of the 10-way classification of disordered fermion systems [10]). Now suppose that we wish to compute the disorder average of 2 Det(E + iη − H )/Det(E + iε − H ) × (E + iε − H )−1 (x, y) (3.1) for real energy parameters E and ε, η > 0. The unconventional feature here is that the square |(E + iε − H )−1 (x, y)|2 of the Green’s function is weighted by the square root
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of a ratio of one determinant taken at energy E + iε and another one at energy E + iη. Although one might think that the presence of these extra factors complicates the problem, quite the opposite is true; it will actually lead to simplifications when η is taken to be large. First of all, the combination (E + iε − H )−1 (x, y) Det −1/2 (E + iε − H ) can be generated by Gaussian integration over a single real boson field, φ1+ . Second, writing the complex conjugate (E + iε − H )−1 (x, y) of the Green’s function as a Gaussian integral requires two real boson fields φα− and two anticommuting fields ψα− (α = 1, 2). Third, to express the square root of Det(E + iη − H ) as a Gaussian integral, we need another real boson φ2+ and two more anticommuting fields ψα+ . Altogether, we then have four bosonic fields φασ and four fermionic fields ψασ (σ = ±, α = 1, 2). Now assume for the moment that η = ε, in which case the two determinants in (3.1) cancel each other. If the band width W is large enough, then the standard steps of disorder averaging followed by Hubbard-Stratonovich transformation and elimination of the massive modes, take us to Efetov’s nonlinear sigma model for systems with orthogonal symmetry (class AI). Physically speaking, the order parameter fields of retarded (+) and advanced (−) type acquire different expectation values: φασ φβσ = δαβ G σ , ψασ ψβσ = αβ G σ
(σ = ±; α, β = 1, 2),
where we are using the abbreviations φα+ (x)φβ+ (x) = φα+ φβ+ , G + = (E + iε − H )−1 (x, x), G − = G + , and αβ = −βα is the antisymmetric tensor for two degrees of freedom. In the region of nonzero average density of states, where G + = G − , these expectation values break a continuous symmetry of the Gaussian integrand at ε = 0. The components of Efetov’s sigma model field have the physical meaning of being the Goldstone modes associated with this broken symmetry. There are 4 bosonic Goldstone modes due to the symmetry breaking φα+ φα+ = φβ− φβ− and four more such modes due to ψ1+ ψ2+ = −ψ2+ ψ1+ not being equal to ψ1− ψ2− = −ψ2− ψ1− . There also exist 8 fermionic Goldstone modes due to the breaking of the odd symmetries connecting the boson-boson sector φ1σ φ1σ = φ2σ φ2σ with the fermion-fermion sector ψ1τ ψ2τ = −ψ2τ ψ1τ of opposite type τ = −σ . All these modes organize into a supermanifold with tangent space R8|8 over a symmetric space (H2 × H2 ) × S4 . Now let η ε > 0, so that the two determinants in the expression (3.1) no longer cancel. The difference η−ε ≈ η then acts as a mass term for the Goldstone modes connecting the advanced sector (−) with the η retarded sector φ2+ φ2+ = ψ1+ ψ2+ = −ψ2+ ψ1+ . By a Thouless-type argument, these massive Goldstone modes do not affect the renormalized physics at length scales much greater than the length L determined by the equation η = 2π D/L , 2
where D ∝ W 2 is the bare diffusion constant of the system. Thus at large length scales L L we may simply drop the massive Goldstone modes from the theory or, in a more careful treatment, integrate them out perturbatively.
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What we are left with, then, are the 2 + 2 = 4 massless bosonic and fermionic Goldstone modes connecting the retarded component φ1+ φ1+ of the order parameter with its four components φ1− φ1− = φ2− φ2− = ψ1− ψ2− = −ψ2− ψ1− in the advanced sector. These four residual Goldstone modes organize into a supermanifold with tangent space R2|2 and base manifold H2 — we thus arrive at the field space H2|2 of the model we are going to study. 4. Symmetries and Their Consequences As an effective theory derived by reduction from an underlying sigma model, the statistical mechanics problem posed by (1.1)–(1.5) enjoys a number of symmetries. First among these is a hidden supersymmetry which ensures that the partition function is always equal to unity, Z (β, ε) = 1, independent of the inverse temperature β and regularization parameter ε. Thus the reduced statistical measure e−Fβ,ε dμ can be regarded as a probability measure, and the physical observables of the model are given as expectations f = f e−Fβ,ε dμ . In the following subsection we provide some background to the normalization property Z (β, ε) = 1. 4.1. Q-symmetry. We start by observing that, for any ε, the full action Aβ,ε defined in (2.6) is invariant under transformations that preserve the short inner product xi x j + yi y j + ξi η j − ηi ξ j
(4.1)
for all i, j ∈ . Such transformations are given, at the infinitesimal level, by even and odd derivations (i.e., first-order differential operators) with the property that they annihilate the expression (4.1) for all i, j and their coefficients are linear functions of the coordinates xk , yk , ξk , ηk . These differential operators form a representation of the orthosymplectic Lie superalgebra osp2|2 . An important example of an odd operator Q ∈ osp2|2 is Q= x j ∂η j − y j ∂ξ j + ξ j ∂ x j + η j ∂ y j . (4.2) j∈
Since j d x j dy j ∂ξ j ∂η j is the Berezin superintegration form given by the inner product
(4.1), it is immediate that Dμ is osp2|2 -invariant, which implies that Dμ Q f = 0 whenever the function f is differentiable and Q f is integrable. For present use, let us record here the explicit expression for the osp2|2 generator Q in horospherical coordinates: a straightforward computation starting from (4.2) gives Q = j∈ q j with single-site generator (index j omitted) ¯ t + (ψ − s ψ)∂ ¯ s − s∂ψ¯ + 1 (1 − e−2t − s 2 − 4ψψ)∂ ¯ q = ψ∂ ψ. 2
(4.3)
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Now consider any differentiable integrand f which is invariant by Q, i.e., Q f = 0. This invariance property has strong consequences for the integral of f (if it exists): in Appendix C, Proposition 2, we prove that the integral of such f equals f evaluated on the zero-field configuration (i.e., on t j = s j = ψ¯ j = ψ j = 0 or equivalently, x j = y j = ξ j = η j = 0, for all j ∈ ): (4.4) Dμ f = f (o). The idea of the proof is easy to state: one shows that the integral of f remains unchanged by the replacement f → e−τ h f with h = j∈ (x 2j + y 2j + 2ξ j η j ) and τ ≥ 0, and then deduces the result (4.4) by sending the deformation parameter τ → +∞ to localize the integral at the zero-field configuration. Using the explicit expression (4.3) it is easy to check that the action Aβ,ε is Q-invariant. Since the differential operator Q is of first order, one directly infers the relation Q e−Aβ,ε = 0. Therefore, as a particular consequence of (4.4) and Aβ,ε (o) = 0 it follows that the partition function equals unity, Z (β, ε) = Dμ e−Aβ,ε = e−Aβ,ε (o) = 1, (4.5) for all values of β ≥ 0 and ε > 0. Further consequences of (4.4) will be elaborated below. 4.2. Hyperbolic symmetry. While Q is a symmetry of our action Aβ,ε for all values of ε, further symmetries emerge in the limit of vanishing regularization ε → 0+. Relegating a more detailed discussion to Appendix B, we here gather the crucial facts. The model (2.5), (2.6) for ε → 0+ acquires a global symmetry by the Lorentz group SO(1, 2) – the isometry group of the hyperbolic plane H2 viewed as a noncompact symmetric space H2 SO(1, 2)/SO(2). This global symmetry entails a number of conserved currents and associated Ward identities. Of these let us mention here the most important one, 1 (4.6) etx +t y Dβ, ε (t)−1 (x, y) = , ε y∈
which is the sigma model version of the quantum sum rule 2 1 π −1 Im (E − iε − H )−1 (x, x) = ρ(E), (E − iε − H ) (x, y) = ε ε y∈
where ρ(E) is the mean local density of states. In the sigma model approximation one sets πρ(E) = 1. The above relation reflects the unitarity of the quantum theory. Its classical interpretation is conservation of probability. Notice that the right-hand side of (4.6) diverges in the limit of vanishing regularization ε → 0. For an infinite lattice there exist two principal scenarios [12] by which to real−1 ize this divergence. In the first one, the correlation function C x y = etx +t y Dβ,ε (t)(x, y) , while bounded in ε, becomes of long range and thus ceases to be summable in the limit ε → 0. In this case the SO(1, 2) symmetry is spontaneously broken and the system is in a phase of extended states. On the other hand, C x y may already diverge for any
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fixed pair of lattice sites x, y, signaling strong field fluctuations and restoration of the noncompact symmetry SO(1, 2) as ε → 0. Exponential decay of C x y with distance |x − y| then corresponds to exponential localization of the energy eigenstates. Thus the question of extended versus localized states of the disordered quantum system translates to the question of the Lorentzian symmetry SO(1, 2) of the statistical mechanical model with free energy (1.1) being spontaneously broken or not. At this stage, a remark is called for: Niedermaier and Seiler have recently shown [15,16] for a large class of sigma models that if the symmetry group of the sigma model is non-amenable – this includes in particular the case of the Lorentz group SO(1, 2) – then spontaneous symmetry breaking occurs in all dimensions d ≥ 1 and for all β > 0. It must therefore be emphasized that, although our sigma model does acquire the nonamenable symmetry SO(1, 2) in the limit ε → 0, it does not belong to the class of models where the arguments of [15,16] apply. The reason is that the SO(1, 2) symmetry of the full action (2.6) is explicitly broken by the step of integrating over the Grassmann variables. More precisely, it is the choice of splitting between even and odd variables implied by the horospherical coordinate system (2.7) that fails to be SO(1, 2) invariant. (There exist other choices of splitting which do preserve the SO(1, 2) symmetry. However, these do not yield an effective bosonic field theory with a probabilistic interpretation.) Thus SO(1, 2) is present as a symmetry at the level of correlation functions or Ward identities such as (4.6), but there exists no group action of SO(1, 2) on the reduced free energy (1.2). More generally speaking, we do not expect the results of Niedermaier and Seiler ever to apply to any of the noncompact nonlinear sigma models of Anderson localization. Indeed, if they did there would be a contradiction with the field-theoretic interpretation of Anderson localization as a phase of unbroken noncompact symmetry. 4.3. Perturbative renormalization group. We now sketch a perturbative result from Wilsonian renormalization theory by which our model is expected to be in a symmetryunbroken phase also for d = 2 and all values of the inverse temperature β, and thus to exhibit Anderson localization of all electronic states. This result follows from Friedan’s work [8] on renormalization for the general class of nonlinear sigma models. According to it, the RG flow of the temperature T = β −1 with increasing renormalization scale a is given by dT = (2 − d)T + R T 2 + O(T 3 ), (4.7) da where R is the target space curvature – more precisely, the multiplicative constant R by which the Ricci tensor of the target space differs from its metric tensor. For both the H2|2 model and Efetov’s sigma model of class AI a quick computation shows the curvature R to be positive. In contrast, R = 0 and R < 0 for Efetov’s sigma models of class A (broken time-reversal symmetry) and class AII (spin-orbit scattering), respectively. According to (4.7), a positive value of R implies that a small initial value of the temperature T increases under renormalization in dimension d = 2. By extrapolation, one therefore expects the existence of a mass gap (or, equivalently, localization of all states) in this case. For the localization length ξ = ξ(a, T (a)), which is a physical observable and hence a renormalization group invariant, one obtains the formula a
ξ ∝ a e1/(R T ) (d = 2) by direct integration of the RG equation (4.7).
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In dimension d = 3, Eq. (4.7) predicts the localizing tendency of positive target space curvature to become irrelevant at small enough temperatures and hence the RG flow to be attracted to the fixed point T = 0 corresponding to extended states. As was remarked above, the Lorentzian symmetry SO(1, 2) is spontaneously broken at this fixed point. With increasing temperature T (or decreasing field stiffness β) the model in d = 3 is expected to undergo an Anderson-type transition to the phase of unbroken symmetry. This phase transition was studied numerically in [4], where the critical value of β was found to be βc ≈ 0.04. The transition has also been investigated in detail using the Migdal-Kadanoff renormalization scheme [5]. 5. Ward Identities and Outline of Proof In order to control fluctuations of the field t at low temperatures T = β −1 we rely on a family of Ward identities due to the internal supersymmetries of the model. These Ward identities are naturally expressed in terms of both the real variables t j , s j and the Grassmann variables ψ¯ j , ψ j . In order to obtain probabilistic information we integrate out the Grassmann variables using (2.14) and (2.15), thereby producing a Green’s function. As was already mentioned, our partition function always equals unity even when the temperature varies in space. By using this fact, we show that gradients of the field t between neighboring sites are strongly suppressed for small T . There also exist Ward identities at larger scales, and information may be extracted from them by using information on previous length scales. In addition to Ward identities, there are two other crucial ingredients of our proof. The first one is a basic estimate on Green’s functions which are non-uniformly elliptic. The second one is the use of SUSY characteristic functions, which help to control large-scale field fluctuations. A more detailed outline of our proof is given below, where the notation and the needed Ward identities are explained. Once the Ward identities are established, most of our proof is very classical. 5.1. Ward identities due to Q-symmetry. We recall the formula (4.4) for the integral of a Q-invariant function. It is easy to check that the functions Si j and z k given in (2.9) and (2.11) satisfy the invariance conditions Q Si j = 0 and Qz k = 0. Therefore, using Si j (o) = 1 and z k (o) = 1 we have the identity Dμ e−β x,y Jx y (Sx y −1)− x∈ εx (z x −1) = 1 (5.1) for all values of β ≥ 0, Jx y ≥ 0, and εx > 0. Note that in order for this statement to be true, Jx y does not have to be nearest neighbor. Moreover, for m ∈ R and any pair x, y ∈ we have tx +t y ¯ ¯ , (5.2) 1 = Sxmy β,ε = Bxmy + m Bxm−1 y (ψx − ψ y )(ψx − ψ y ) e β,ε
where the expectation · β,ε was defined in (2.13), and we used the nilpotency (ψx − ψ y )2 = (ψ¯ x − ψ¯ y )2 = 0. By integrating over the Grassmann fields ψ¯ and ψ as in (2.14) we obtain our basic identity, 1 = Sxmy β,ε = Bxmy 1 − mG x y . (5.3)
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The last expectation is taken with respect to the effective action for the fields t and s, and the Green’s function G x y is Gxy =
etx +t y (δx − δ y ); Dβ,ε (t)−1 (δx − δ y ) . Bx y
More generally if (xi , yi ) are n pairs of points, then n n 1= Sxmi yi = Bxmi yi Det (1 − mG) , i=1
β,ε
i=1
(5.4)
(5.5)
where G is an n × n matrix of Green’s functions Gi j = [gi ; Dβ,ε (t)−1 g j ]
(5.6)
and −1/2
gi = Bxi yi e(txi +t yi )/2 (δxi − δ yi ).
(5.7)
The matrix G is real symmetric and positive. It will be important later that we are choosing gi to be orthogonal to the zero mode (i.e., the constant functions). 5.2. Outline of proof. Our proof of Theorem 1 relies on the Ward identity (5.3), (5.5) and an induction on length scales. The basic idea is quite simple: suppose m > 0 and we had a uniform bound |G x y | ≤ C/β < 1/m on the Green’s function (5.4), for all configurations of t. Then we could conclude from (5.3) that coshm (tx − t y ) ≤ Bxmy ≤ (1 − mC/β)−1 ,
(5.8)
and this would imply Theorem 1. In Sect. 6 we prove that if |x − y| = 1 then indeed 0 ≤ G x y ≤ 1/β, and we establish an even stronger version of (5.8). This proves that nearest neighbor fluctuations of the field t are very unlikely for large β (see Lemma 4). For distances |x − y| > 1, however, there is no uniform bound on G x y . In Sect. 7 we study the Green’s function (5.4) and establish sufficient conditions on the field t to obtain the desired bound on G x y . In 3D these conditions are roughly given as follows (where | j − x| ≥ 1): cosh(t j − tx ) ≤ B j x ≤ a | j − x|α , 0 < α < 1/2,
(5.9)
and the same for cosh(t j − t y ). The number a is a constant. It will turn out that these estimates are needed only for the sites j in a 3D diamond-type region, Rx y , containing x and y; see Fig. 1. Notice that since the exponent α is positive, we are allowing larger fluctuations at larger scales. The probability that such a condition is violated will be shown to be small by induction. Section 8 uses the conditions described above to prove conditional estimates on the fluctuations of the field t at all scales. These conditions are initially expressed in terms of Q-invariant characteristic functions χ . Later we show that the nilpotent (or Grassmann) part of χ is not important, so we may think of χ in the usual classical sense. The remaining problem is to obtain unconditional estimates on the fluctuations and thereby prove Theorem 1. This is first done for short scales in Sect. 10. For larger scales
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y
Fig. 1. “Diamond” region: a double cone in 3 dimensions
we use induction. Our induction hypothesis is formulated in Sect. 11. Roughly speaking it asserts that n m Bxi yi ≤ 2n (5.10) i=1
holds under the assumption that the diamond-type regions Rxi yi (Fig. 1) associated with i = 1, . . . , n have disjoint interiors. The induction is in , defined as the maximal separation |xi − yi | in the product over i = 1, . . . , n. For = 1 this hypothesis was verified in Sect. 6. Section 12 contains the technical core of our paper. There we prove unconditional estimates on the fluctuations and thus obtain Theorem 1. The main idea is to consider a site b in Rx y closest to x or y such that condition (5.9) is violated for j = b. We shall then prove by induction that the probability for such an event to occur is small. The m B m (see Lemma 2 below) is used for a point c near b. Since the inequality Bxmy < 2m Bxc cy distances |x − c| and |c − y| are less than |x − y|, induction can be applied. The factor 2m is offset by the small probability of the event when β is large. Theorem 2 is proved in Sect. 13. Here we must estimate the contribution of the zero mode and at this stage ε > 0 plays a key role. Finally, Theorem 3 follows from the estimates of Theorem 2; its proof is given in Sect. 14. 5.3. Two simple lemmas. We conclude this section with two simple lemmas which will be frequently used below. The first lemma is useful for estimates on Green’s functions. To state it, let V be a finite-dimensional Euclidean vector space with scalar product [; ]V . Lemma 1. Let M : V → V be a positive real symmetric operator. Then for any set of n vectors vi ∈ V we have M−
n
vi [vi ; · ]V ≥ 0
(5.11)
i=1
if and only if the symmetric n × n matrix K with matrix elements K i j = [vi ; M −1 v j ]V satisfies 0 ≤ K ≤ Id.
(5.12)
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Proof. Letting wi = M −1/2 vi ∈ V we observe that K i j = [wi ; w j ]V and (5.11) is equivalent to Id −
n
wi [wi ; · ]V ≥ 0.
i=1
By evaluating this quadratic form at w = λi wi for any real numbers λi (since any function in the orthogonal complement of w satisfies (5.11) automatically) we see that (5.11) is equivalent to n
λi λ j (K − K 2 )i j ≥ 0,
(5.13)
i, j=1
or 0 ≤ K 2 ≤ K , from which our assertion follows.
The second lemma will be used in our induction process of Sect. 12. Lemma 2. If Bi j is defined as in (2.10) then for all distinct x, y, c ∈ , Bx y < 2Bxc Bcy .
(5.14)
The inequality (5.14) can be verified by direct computation (proof omitted). Remark. The raison d’etre behind (5.14) is easy to state: Bx y has an interpretation as the hyperbolic cosine of the geodesic distance on H2 . Therefore, if x, y, c are three points on H2 , then since the geodesic distance dist(x, y) is the minimal length of any curve connecting x and y, the triple of geodesic distances satisfy the triangle inequality dist(x, y) ≤ dist(x, c) + dist(c, y). Given this, the inequality (5.14) follows by taking the hyperbolic cosine of both sides and using that cosh(a + b) < 2 cosh(a) cosh(b) holds for any two real numbers a, b. 6. Bounds on NN Fluctuations As was already mentioned, for nearest neighbor (NN) pairs we can obtain a result stronger than (5.8). Recall that we have now fixed Jx y = 1 for all x y that are NN pairs, and Jx y = 0 otherwise. This fact is essential in the next lemma. Lemma 3. Let x, y be an NN pair and suppose that 0 < γ < 1. Then eβγ (Bx y −1) ≤ (1 − γ )−1 .
(6.1)
More generally, if (x j , y j ), j = 1, . . . , n is a set of n different NN pairs, then βγ nj=1 (Bx j y j −1) e ≤ (1 − γ )−n .
(6.2)
This shows that NN fluctuations are strongly suppressed. Remark. Since Jx j y j = 1 and γ < 1 the integrals in (6.1)–(6.2) are well defined. This would not be true if x j , y j were not NN, or if γ > 1, or if two or more NN pairs were allowed to be identical without further restrictions on the value of γ .
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Proof. For x, y an NN pair let Fx y ( j) = e(tx +t y )/2 (δx ( j) − δ y ( j)),
(6.3)
and introduce the Green’s function −1 G 0x y (t) = [Fx y ; Dβ,ε (t) Fx y ].
(6.4)
Since Sx y is Q-invariant, Proposition 2 of Appendix C implies ¯ eβγ = eβγ Sx y = eβγ (Bx y +ψ(Fx y )ψ(Fx y )) ¯ x y )ψ(Fx y )) = eβγ Bx y (1 + βγ ψ(F = eβγ Bx y (1 − βγ G 0x y ) ,
(6.5)
where we have used ψ 2 = ψ¯ 2 = 0 and (2.14). Now from (1.1) we have [v ; Dβ, ε (t) v] = β [v ; Fi j ]2 + ε etk vk2 ≥ β [v ; Fx y ]2 .
(6.6)
(i j)
k
Therefore Lemma 1 implies that 0 ≤ β G 0x y (t) ≤ 1 for all t, and (6.1) follows. Similarly, for n > 1 we have ¯ x y )ψ(Fx y ) βγ nj=1 Sx j y j βγ nj=1 Bx j y j βγ nj=1 ψ(F j j j j enβγ = e = e e βγ nj=1 Bx j y j = e Det(1 − γ K ) ,
(6.7)
where K is the n × n matrix K i j = β [Fx j y j ; Dβ,ε (t)−1 Fxi yi ]
(6.8)
given by n different NN pairs xi , yi . From (6.6) and Lemma 1 it follows that K ≤ 1. This implies |Det(1 − γ K )| ≥ (1 − γ )n and the lemma follows. As a corollary, since 1 ≤ Bxmy ≤ em(Bx y −1) for m ≥ 0, we have the bound
n
Bxmj y j
≤ (1 − m/β)−n ≤ 2n
(6.9)
j=1
for any m in the range m ≤ β/2. A first important consequence of Lemma 3 is the following statement. Lemma 4. Let x j , y j be a set of n different nearest neighbor pairs. Then Prob ∀ j = 1, . . . , n : Bx j y j > 1 + δ ≤ (1 − γ )−n e−n(βγ )δ for any 0 < γ < 1.
(6.10)
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Proof. Let n = 1. By the Chebyshev inequality,2 Prob Bx y > 1 + δ = χ (Bx y > 1 + δ) ≤ e−βγ (1+δ) eβγ
Bx y
,
(6.11)
where χ (Bx y > 1 + δ) is the characteristic function for Bx y > 1 + δ to hold. The desired inequality for n = 1 now follows directly from Lemma 3. The proof for n pairs is no different. 7. Conditional Estimates on Green’s Functions For general x, y (not NN) we do not have the option of considering eβγ Sx y , as the underlying integral need not exist. Nevertheless, Sxmy does exist and from (5.3) we have 1 = Bxmy 1 − mG x y , (7.1) with G x y defined by (5.4), Gxy =
etx +t y (δx − δ y ) ; Dβ,ε (t)−1 (δx − δ y ) . Bx y
Now, as was explained in Sect. 5.2, if we knew that G x y ≤ C/β for all configurations of t, then we could conclude that Bxmy ≤ (1 − mC/β)−1 .
(7.2)
While we have seen that this estimate is true for |x − y| = 1 (with C = 1), it is false in general, as there are rare configurations with large negative t surrounding x or y. Nonetheless, in 3D we can get an upper bound on G x y by estimating the local ‘conductance’ at an edge (i j) from below. This conductance is A x y (i j) ≡ Bx y e−tx −t y eti +t j ≥ 21 max eti +t j −2tx , eti +t j −2t y ,
(7.3)
where we have used Bx y ≥ cosh(tx − t y ). This ‘conductance’ appears as an explicit factor in G x y since etx +t y (x y) Dβ,ε (t)−1 = (Dβ,ε (t))−1 , Bx y (x y)
where for each pair x y, Dβ,ε (t) is the finite-difference elliptic operator defined by the quadratic form (x y) A x y (i j) (vi − v j )2 + ε Bx y etk −tx −t y vk2 . [v ; Dβ,ε (t) v] = β (i j)
k∈
It will suffice to estimate the expression (7.3) for NN pairs (i j) in a region Rx y which is like a 3D double cone with vertices at x and y. Note that Neumann boundary conditions increase G x y and δx − δ y is orthogonal to the zero mode. We will have to require that Rx y be essentially three-dimensional in the following sense: 2 Actually, the Chebyshev inequality states that for any random variable X with average X we have 0 Prob[(X − X 0 )2 > a 2 ] ≤ a −2 (X − X 0 )2 . Here we are using the same principle.
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x2
y
θ
x (a)
x1 (b)
Fig. 2. a The region R x y in the continuum limit. b On the lattice, the points x1 and x2 must be added to ensure connectedness
Definition 1. A region Rx y ⊂ containing x and y is called δ-admissible if it is connected by nearest-neighbor bonds and the two one-parameter families of intersections Rz (r ) ≡ Rx y ∩ Bzr with the ball Bzr of radius r centered at z = x, y satisfy √ vol{Rz (r )} ≥ r 3 δ for r ≤ |x − y|/ 2 (z = x, y). In addition we require that the following Poincaré inequality: f ( j)2 ≤ C0 r 2 (∇ f )2 ( j), j∈Rz (r )
j∈Rz (r )
holds for all functions f : → R subject to the condition fixed constant.
j∈Rz (r )
f ( j) = 0. C0 is a
√ We observe that by the choice of maximal radius r = |x − y|/ 2 the scaling of volume is monitored up to the full side length of a rectangular diamond Rx y (or a double cone Rx y , see Fig. 1) with opposite corners placed at x and y. In the continuum limit this definition is satisfied by a double cone obtained by rotating (around the line x y connecting x and y) a 2D diamond with vertices on x and y and angle θ ≥ θ0 (δ) ≥ π/10 (see Fig. 2a). Since we are on a lattice we may have to add a few lattice points near x and y to ensure connectedness (see Fig. 2b). The Poincaré inequality is straightforward to prove in such convex regions. Definition 2. Given a δ-admissible region Rx y , we define the regions Rxz y for z = x and z = y by √ Rxz y = { j ∈ Rx y | 1 ≤ | j − z| ≤ |x − y|/ 2}. (7.4) y
For the case of a diamond, Rxx y ∪ Rx y = Rx y \{x, y}. Remark 7.1. The values of the field t outside the region Rx y are not important, as we can use Neumann boundary conditions to eliminate the exterior of Rx y . Indeed, in the subspace orthogonal to the constant functions the Laplacian on Rx y with Neumann boundary conditions is bounded (by the Poincaré inequality) from below by some number, say c, times the inverse square of the linear size L of Rx y . By this token, since the vector δx − δ y used in the definition of G x y lies in that subspace, we may utilize the bound on the inverse of the Neumann Laplacian by c−1 L 2 and in this way eventually obtain an upper bound on G x y (see Lemma 5).
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Remark 7.2. From (7.3) we have A x y (i j)−1 ≤ 8 cosh(ti − tz ) cosh(t j − tz )
(7.5)
for both z = x and z = y. The main result of this section is that we can get an upper bound on G x y even without imposing a uniform upper bound on A x y (i j)−1 , as long as some growth restriction on the fluctuations of t is met for a δ-admissible region Rx y : Lemma 5. Fix two constants a > 1 and 1/2 > α > 0. If Rx y is a δ-admissible region in the sense of Def. 1 and the statement ∀ j ∈ Rxz y :
cosh(t j − tz ) ≤ a | j − z|α
(7.6)
holds for both z = x and z = y, then we have 0 ≤ G x y ≤ G xNy ≤ C/β,
(7.7)
where C(a, α, δ, C0 ) is some constant depending on the parameters a, α and the geometry of the region Rx y (encoded in the parameters δ and C0 ). G x y was introduced in (5.4) and we defined G xNy =
etx +t y N (δx − δ y ) ; Dβ,ε (t)−1 (δx − δ y ) , Bx y
(7.8)
where the notation D N means Neumann boundary conditions on ∂ Rx y . Proof. The following is a variation on an argument presented in [18]. For each k ∈ N consider two cubes of side 2k centered at x and y. (For concreteness, imagine the edges of the two cubes to be parallel to the vector x − y). Let Rxk , R ky denote the corresponding intersections with Rx y and let Ik , I˜k be the indicator functions of Rxk and R ky , respectively, normalized so that for each k, Ik ( j) = 1 = (7.9) I˜k ( j). j
j
We observe that ∀k ≥ km :
Rxk = R ky = Rx y ,
(7.10)
where km is the smallest number k ∈ N such that 2k−1 ≥ |x − y|. Since Rx y is δ-admissible, Rzk has the same properties as Rz (r = 2k ) in Def. 1 and we therefore have vol Rxk ≥ 2kd δ
(k ≤ km − 1, d = 3)
(7.11)
for all 2k > 10. For 2k ≤ 10 this is not true (see Def. 1) but the corresponding volume is no less than unity, as Rx y is connected. Now we express δx − δ y as a telescopic sum: δx − δ y =
km k=1
(ρk − ρ˜k ) ,
(7.12)
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455
where ρk = Ik−1 − Ik , ρ˜k = I˜k−1 − I˜k , I0 = δx , and I˜0 = δ y . This sum terminates at ˜ km because by (7.10) we have Ik = Ik for k≥ km . Note that ρk , ρ˜k are orthogonal to the constant functions: j∈ ρk ( j) = 0 = j∈ ρ˜k ( j). Next we put the telescopic sum to use by the following computation: km N N δx − δ y ; Dβ,ε ρk − ρ˜k ; Dβ,ε (t)−1 (δx − δ y ) = (t)−1 (ρl − ρ˜l ) k,l=1
k 2 m 1/2 1/2 N −1 N −1 ρk ; Dβ,ε (t) ρk + ρ˜k ; Dβ,ε (t) ρ˜k , ≤
(7.13)
k=1
where the Cauchy-Schwarz inequality was employed. Hence we need to estimate [ρk ; Dβ,ε (t)−1 ρk ] and [ρ˜k ; Dβ,ε (t)−1 ρ˜k ]. This is done, say for the former, by the inequality N [ρk ; Dβ,ε (t)−1 ρk ] ≤ D −1 ρk 22 , Rk
(7.14)
x
where D R (for a region R) stands for the operator (1.3) with Neumann boundary conditions on R. In view of ρk ( j) = 0 the operator norm is to be taken on the orthogonal complement of the constant functions. The square of the L 2 -norm of ρk+1 is bounded by (vol Rxk )−1 . Thus by (7.11) ρk+1 22 ≤ (vol Rxk )−1 ≤ 2−kd δ −1 .
(7.15)
The corresponding inequality also holds for ρ˜k+1 . . For this we observe that the conditions We must still bound the operator norm D −1 R xk (7.5) ensure that A x y ( j j )−1 ≤ 23 (a | j − x|α )2 ≤ 8a 2 22kα for all j, j ∈ Rxk and k ≤ km − 1, since in that case j j ∈ Rxx y and we apply (7.6) for y z = x. For k = km we are looking at pairs that belong to Rx y but not to Rxx y . In that case we apply (7.6) for z = y and still have A x y ( j j )−1 ≤ 8a 2 22kα . Therefore, since Rx y is δ-admissible and by the Poicaré inequality (see Def. 1) the lowest nonzero eigenvalue of the Neumann Laplacian on Rxk is of the order of (2k )−2 , we obtain −1 2 −1 2k 2kα etx +t y Bx−1 y D R k ≤ c(δ) a β 2 2
(7.16)
x
for some c(δ) and all k ≤ km . For 2k ≤ 10 the connectedness of Rx y ensures that D −1 ≤ const. The same bounds apply for D −1 . Rk Rk x
y
Finally, by combining (7.13) with (7.14), (7.15), and (7.16), we arrive at G xNy
etx +t y a2 N δx − δ y ; Dβ,ε = (t)−1 (δx − δ y ) ≤ c(δ) Bx y β
2
km √
2 2
k(2α+2−d)
k=1
For 2α < d − 2 = 1 the value of this sum is bounded uniformly in km .
.
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M. Disertori, T. Spencer, M. R. Zirnbauer y
Remark. The bound (7.7) also applies when the definition of Rxx y and Rx y is modified in the following way (for z = x, y as before): Rxz y = { j ∈ Rx y : | j − z| ≤ |x − y| f z },
(7.17)
where f x , f y are a pair of positive numbers which add up to (at least) unity and neither of which is too small. If these regions become too asymmetric or the angles too small the Poincaré inequality and (7.11) may no longer hold. It is easy to see that the relevant scales involved are the ones for k near km and we can get the same bound but with a change of overall factor. This remark will become important in Sect. 12, Lemma 11, where we will need this estimate with f y 1/5. 8. Conditional Estimates on Fluctuations In this section we establish bounds on the fluctuations of the field t by bounding Bxmy χ¯ x y where χ¯ x y has the property that χ¯ x y = 0 whenever the hypothesis (7.6) of Lemma 5 fails. Definition 3 (Characteristic function). As before, fix two constants a > 1 and 1/2 > α > 0, and let r j−k := (a | j − k|α )−1 for j, k ∈ , j = k. Let χ : R+ → R be the characteristic function of the interval [0, 1], i.e., χ (t) = 1 for 0 ≤ t ≤ 1 and χ (t) = 0 y for t > 1. Moreover, let Rx y be δ-admissible and choose the regions R xx y , Rx y as in (7.4). In this setting we define χx j χy j , χz j = χ (r j−z Bz j ) (z = x, y). (8.1) χ¯ x y = j∈R xx y
y
j∈R x y
Here the constants a, α are taken to coincide with those in Lemma 5. With these definitions we have Lemma 6. Let Rx y be a δ-admissible region, and let C = C(a, α, δ) be the constant that appears in Lemma 5. Then for 0 ≤ m < β/C we have Bxmy χ¯ x y ≤ (1 − mC/β)−1 . (8.2) Proof. Our proof uses the identity Sxmy χ¯ xSy = 1, where χ¯ xSy is a supersymmetric version of χ¯ x y defined above. After integrating out the Grassmann fields we shall show that this identity implies Bxmy χ¯ x y (1 − mG x y ) ≤ 1. Lemma 5 and the presence of χ¯ x y then yield (8.2). d More precisely, let χγ ∈ C∞ (R+ ) with dt χγ (t) ≤ 0 and 1 t ≤ 1 − γ, χγ (t) = 0 t ≥ 1, be a smooth regularization of χ = limγ →0 χγ . We fix a small value of γ > 0 and write χ˜ ≡ χγ for short. Then, recalling the definition (6.3) of Fx y we introduce ¯ x j )ψ(Fx j ), χxSj = χ˜ (r j−x Sx j ) = χ˜ x j + r j−x χ˜ x j ψ(F
(8.3)
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where χ˜ x j = χ˜ (r j−x Bx j ). Since χxSj is Q-invariant and χxSj (0) = 1 we have 1=
Sxmy
χxSj
χ ySj
≡ Sxmy χ¯ xSy .
(8.4)
y j∈R x y
j∈R xx y
Now, we express
Sxmy χ¯ xSy = Sxmy χˆ x y exp −[ψ¯ ; Aψ] ,
(8.5)
where
χˆ x y =
χ˜ x j
χ˜ y j ,
χ¯ x y = lim χˆ x y , γ →0
y j∈R x y
j∈R xx y
and the symmetric operator A is given by [ f ; Af ] = −
r j−x χ˜ x j χ˜ x j
j∈R xx y
[ f ; Fx j ]2 −
r j−y χ˜ y j y j∈R x y
χ˜ y j
[ f ; F y j ]2 .
(8.6)
¯ contribution to (8.5) Clearly A ≥ 0 as a quadratic form since χ˜ ≤ 0. The total ψψ including the fermionic part of the action is ¯ [ψ¯ ; (Dβ,ε (t) + A)ψ] − m Bx−1 y ψ(Fx y ) ψ(Fx y ), where the second summand stems from Sxmy , see (5.2)-(5.3). Thus, integration over the ¯ ψ gives Det(Q + A), where Grassmann fields ψ, Q = Dβ,ε (t) − m Bx−1 y Fx y [Fx y ; · ]. Since we are taking m to be less than β/C, the presence of the factor χˆ x y in (8.5) ensures (by Lemma 5) that mG x y < 1. Now by Lemma 1 the inequality 1 ≥ mG x y = −1 m Bx−1 y [Fx y ; Dβ,ε (t) Fx y ] is equivalent to Q ≥ 0. Therefore the result Det(Q + A) of ¯ ψ is bounded from below by integrating over ψ, Det(Q + A) ≥ Det(Q) = Det(Dβ,ε (t)) (1 − mG x y ) ≥ 0, and we obtain the estimate ¯ 1 = Bxmy χˆ x y e−[ψ ;(Q+A)ψ] ≥ Bxmy χˆ x y (1 − mG x y ) . We finally take the limit γ → 0. The smooth function χˆ x y then converges to the characteristic function χ¯ x y . Hence 1 ≥ Bxmy χ¯ x y (1 − mG x y ) ≥ Bxmy χ¯ x y (1 − mC/β), which is the desired result.
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Lemma 7. If all of the regions Rx1 y1 , Rx2 y2 , . . . , Rxn yn are δ-admissible and disjoint (meaning they have disjoint interiors), then we still have n m Bx j y j χ¯ x j y j ≤ (1 − mC/β)−n . (8.7) j=1
Proof. As before we use the fact that the supersymmetrized observable, which here results from replacing Bx j y j by Sx j y j , has expectation one. first the simpler problem of computing the expectation of the product Consider Sxmi yi χ¯ xi yi . After integrating over ψ and ψ¯ we see that n n m m Sxi yi χ¯ xi yi = Bxi yi χ¯ xi yi Det(1 − mG) , (8.8) i=1
i=1
where G is an n × n matrix of Green’s functions Gi j = β −1 [gi ; Dβ,ε (t)−1 g j ],
−1/2
gi = Bxi yi e(txi +t yi )/2 (δxi − δ yi ).
(8.9)
The matrix G is positive as a quadratic form. In order to reduce the problem to the previous case (of just a single region) note that G ≤ G N , where the subscript denotes Neumann boundary conditions on the boundaries of the disjoint regions Rxi yi . The presence of the factors χ¯ xi yi implies 1 − G N > 0, so that Det(1 − mG) ≥ Det(1 − mG N ) =
n
(1 − mG xNi yi ) ≥ (1 − mC/β)n ,
(8.10)
i=1
where G xNy was defined in (7.8). Note that since the regions Rxi yi have no common edge (but they may have one common vertex), the presence of Neumann boundary conditions implies that only the diagonal terms contribute in (8.10). The proof of the lemma is completed by introducing the effects of χ˜ as before. Since there are no new aspects to this argument, we omit it. Remark. From this lemma one obtains estimates for conditional probabilities only. Yet, in order to bound C x y in Theorem 3 we need probability estimates without any conditions, which is why we now have to develop an inductive argument. 9. Unconditional Estimates on Fluctuations We are now going to remove the constraints enforced by insertion of χ¯ . In order to do so, we have to consider χxc j = 1 − χx j for χx j defined by (8.1). Short scales (given by 0 < | j − x| < β 1/4 ) will be treated separately by monitoring, in Sect. 10, only the size of nearest neighbor gradients inside the region Rx y . At the very large scales of | j − x| ≥ β 1/4 , however, looking only at NN fluctuations is not enough. There, in order to remove the χ¯ constraints we will show by induction on the distance | j − x| that the corresponding contribution is small. We will distinguish between two types of geometry: diamonds and deformed diamonds. For deformed diamonds we will quantify the bounds given by (7.7) and call such regions C-admissible.
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y y
x
(a)
x
(b)
Fig. 3. a diamond region, b C-admissible region (deformed diamond)
Definition 4. Let Rx y ⊂ be δ-admissible in the sense of Def. 1. 1. We call Rx y a diamond if it is the set of lattice points which is contained in a 3-dimensional double cone obtained in the following way: we take a 2-dimensional rectangular √ diamond with opposite vertices placed on x and y and edges of length |x − y|/ 2 (see Fig. 3) and rotate it around the line x y. In order to ensure connectedness we may have to add a few lattice points near x and y (see Fig. 2 a, b). 2. We call Rx y a C-admissible region (or deformed diamond) if 0 ≤ G xNy χ¯ x y ≤ C/β for |x − y| > β 1/4 , 0 ≤ G xNy χ pq ≤ C/β for |x − y| ≤ β 1/4 ,
(9.1) (9.2)
pq
where χ¯ x y is defined in (8.1), the superscript N stands for Neumann boundary con ditions on Rx y and denotes the product over all nearest neighbor pairs in Rx y . Note that for short scales, dealt with in (9.2), instead of using χ¯ x y we impose constraints on all NN pairs in the region Rx y . With these definitions we can state the main result of this paper. Theorem 4. Let m = β 1/8 , and let Rxi yi for i = 1, . . . , n 1 be diamonds with disjoint interiors and |xi − yi | > β 1/4 . Then we have n 1 Bxmi yi ≤ 2n 1 (9.3) i=1
for all n 1 ≥ 0. Moreover if p j , q j for j = 1, . . . , n 2 are such that | p j − q j | > β 1/4 , the regions R p j q j are C-admissible, have disjoint interiors and do not overlap with any of the regions Rxi yi , then there exists a constant ρ ≤ 1/2 such that n n2 1 m 3m Bxi yi B p j q j χ¯ p j q j ≤ 2n 1 (1 + ρ)n 2 (9.4) i=1
j=1
for all n 1 ≥ 0 and n 2 ≥ 0. Finally, let rk , sk for k = 1, . . . , n 3 be such that |rk − sk | ≤ β 1/4 , Rrk sk are C-admissible, have disjoint interiors and do not overlap with any of the
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M. Disertori, T. Spencer, M. R. Zirnbauer
regions Rxi yi or R p j q j . Then for all n 1 ≥ 0, n 2 ≥ 0 and n 3 ≥ 0 we have n n2 n3 1 m 3m 3m Bxi yi B p j q j χ¯ p j q j Brk sk ≤ 2n 1 (1 + ρ)n 2 2n 3 i=1
j=1
(9.5)
k=1
with ρ ≤ 1/2. Let = max |x j − y j |. j
(9.6)
The proof of the theorem is carried out in Sects. 10–12 and will use an inductive argument on . We will need to distinguish between three situations, which we refer to as classes. Class 1. |x − y| > β 1/4 and the pair is not protected by a factor of χ¯ x y . In this case we need an inductive argument on scales to prove a bound on the expectation of Bxmy . The induction will be done on (defined above) and is carried out in Sects. 11 and 12. We will need to inductively select non-overlapping smaller diamonds inside the region Rx y while making sure that these remain δ-admissible. To arrange for all geometrical details to work out, we take Rx y to be a perfect diamond. Class 2. |x − y| > β 1/4 but the pair is protected by a factor of χ¯ x y . In this case we can apply the results of Sect. 8, thereby obviating the need for any induction. Rx y is then allowed to be a deformed diamond and the bound we can get is stronger than in Class 1 (power 3m instead of m). Class 3. |x − y| ≤ β 1/4 . This includes short scales and the NN case, which was already treated in Sect. 6. We will show in Sect. 10 that these scales do not require any factor of χ¯ x y to ensure a good bound. No induction is needed, and we can therefore take Rx y to be a deformed diamond. Note that the larger exponent 3m appearing in (9.4) and (9.5) is important for the inductive proof to go through. The enlarged exponent can be handled either because of the presence of χ¯ or because the pair is of Class 3. 9.1. Fixing the different parameters. We have introduced a certain number of parameters: m, a, ρ, δ, C0 , C, α. Before going on, we briefly review why they appeared and how to choose their values. 1. The parameter m is ubiquitous in this paper as the power of Bx y . Since the probability of large deviations will be bounded by K −m with K > 1, we want m to be as large as possible. On the other hand, to apply the SUSY argument of Sect. 8 we must have (1 − 3mC/β)−1 < 1, where the factor 3m in this inequality comes from the power of B in (9.4), (9.5). Therefore the magnitude of m is limited by β. To arrange for all the conditions to be met, we fix m = β 1/8 . The factor m will be kept fixed in the whole course of proof. 2. The constants C0 and δ appearing in the definition of the region Rx y (see Def. 1) are not subject to any special requirements, but their values do constrain the other parameters. They will be fixed throughout.
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model
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3. To prove the induction hypothesis we need 0 ≤ ρ ≤ 1. More precisely (see Eq. (12.6)) we need ρ + R(x, y) ≤ 1. Since we prove R(x, y) ≤ 1/2 we will take ρ ≤ 1/2. 4. The constant a in Lemma 5 in Sect. 7 plays a key role in bounding the entropy for small scales; see Sect. 12, Case 1, Eq. (12.16). It will become clear there that a > 10 is sufficient. 5. We need to take α > 0 in Lemma 5 in order to control entropy factors for large deviations (see Sect. 12). On the other hand, the result of Theorem 2 would be optimal for α = 0. Therefore we wish to make α as small as possible. We will see in Sect. 12 (Case 2b, Eq. (12.21) and Case 2c, Eq. (12.23)) that α ≥ O(1/ ln β) is a requirement for our analysis to go through. 10. Short-scale Fluctuations We now prove Theorem 4 for ≤ β 1/4 (see (9.6)), i.e., for Class 3 pairs (this is equivalent to take n 1 = 0 in (9.5)). These estimates will follow from the bounds on NN fluctuations established in Sect. 6. Lemma 8. There is a constant β0 such that for β ≥ β0 , |x − y| = ≤ β 1/4 and m = β 1/8 , we have (10.1) Bx3m y ≤ 2. More generally let (x1 , y1 ), . . . , (xn 1 , yn 1 ) be n 1 pairs with |x j − y j | ≤ for all j, and let the interiors of the corresponding C-admissible regions Rx1 y1 , Rx2 y2 , . . . , Rxn1 yn1 be disjoint. Moreover if p j , q j for j = 1, . . . , n 2 are such that | p j − q j | > β 1/4 , the regions R p j q j are C-admissible, have disjoint interiors and do not overlap with any of the regions Rxi yi , then there exists a constant ρ ≤ 1/2 such that n n2 1 n1 n2 Bx3m B 3m (10.2) p j q j χ¯ p j q j ≤ 2 (1 + ρ) i yi i=1
j=1
for all n 1 ≥ 0 and n 2 ≥ 0. Proof. As in Def. 3, let χ be the characteristic function of the interval [0, 1] and let (with a parameter δ to be defined shortly) χ pq = χ ((1 + δ)−1 B pq ), | p − q| = 1,
(10.3)
c = 1 − χ . Note that this parameter δ appears only in this section and has nothand χ pq pq ing to do withthe one controlling the geometry of the region in Def. 1. Using χ pq ≤ 1 c and we have 1 ≤ ( pq) χ pq + ( pq) χ pq
Bx3m y
≤
Bx3m y
( pq)
χ pq +
c Bx3m χ y pq ,
( pq)
where the product and the sum are over all nearest neighbor pairs ( pq) in Rx y .
(10.4)
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M. Disertori, T. Spencer, M. R. Zirnbauer
We estimate the first term on the right-hand side of (10.4) by applying the strategy of the proof of Lemma 6 to show that 3m χ pq ≤ 1. Bx y 1 − 3m G x y (10.5) ( pq)
To bound G x y , note that on the support of χ pq we have (10.6) 0 ≤ 21 (t p − tq )2 ≤ cosh(t p − tq ) − 1 ≤ δ. √ √ Thus |t p − tq | ≤ 2δ and |tz − t j | ≤ 2δ for z = x, y and all j ∈ Rx y . Now let us require √ 2δ = 1, or δ = 21 β −1/2 , (10.7) since ≤ β 1/4 . Thus we have a uniform lower bound on the conductance (7.5). It then follows that 0 ≤ G x y ≤ C/β with C independent of β, and (10.5) gives 3m Bx y χ pq ≤ (1 − 3m C/β)−1 ≤ 3/2 (10.8) ( pq)
for β large. For the second summand of (10.4) we use c χ pq
= 1 − χ (1 + δ)
−1
B pq ≤
B pq 1+δ
β/2
.
The factor Bx3m y is estimated by repeated application of (5.14): 2Bx y ≤ 2B p j q j ,
(10.9)
(10.10)
j
where the product ranges over a set of NN pairs connecting x and y. By combining (10.9) and (10.10) and then using the result (6.9) for NN pairs we have 23m(−1) β/2 3m c 3m Bx y χ pq ≤ Bpjqj B pq (1 + δ)β/2 j
≤
23m(−1) (1 + δ)β/2
1 2
− 3m/β
−1
≤ e3m e−βδ/3 .
(1 − 3m/β)− (10.11)
Since 3m ≤ 3β 1/8 β 1/4 by hypothesis, and δ = 21 β −1/2 by (10.7), we see that the expression (10.11) is less than exp(β 3/8 − β 1/2 /6). Combining our estimates on the two terms on the r.h.s. of (10.4) we have 3/2 + 33 eβ
3/8 −β 1/2 /6
≤2
for large enough β. The factor 33 ≤ 3β 3/4 comes from the sum over all NN pairs in Rx y .
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model
463
When several disjoint regions are present, the bounds over disjoint regions factor, and we can get the same result using the same argument. Each term B 3m p j q j is a Class 2 contribution and the corresponding bound (see Lemma 7) is (1 − mC/β)−1 = (1 − Cβ 1/8 /β)−1 ≤ (1 + ρ), for β > β0 . This concludes the proof of Lemma 8.
11. Induction Hypothesis and Some Preliminary Estimates The argument in the last section cannot be repeated for all values of . In order to control all scales we need an inductive argument. Induction Hypothesis. Let xi , yi (i = 1, . . . , n 1 ), p j , q j ( j = 1, . . . , n 2 ), and rk , sk (k = 1, . . . , n 3 ) be pairs of Class 1, 2, resp. 3, in the sense of Theorem 3. Then the bounds (9.3), (9.4), (9.5) hold when |xi − yi | ≤ for all i ≤ n 1 . The induction is on = maxi |xi − yi |. The said bounds were already established for = 1 (NN case, Sect. 6) and ≤ β 1/4 (Sect. 10). Assuming that the Induction Hypothesis holds up to scale , we shall prove (in Sect. 12) that it holds up to scale + 1. This will complete the proof of Theorem 4 and, as an immediate consequence, Theorem 1. The idea of the proof is the same as in Sect. 10. If the pair x y is protected by a χ¯ x y factor (Class 2), then we apply Lemma 6 in Sect. 8. To get the unconditional estimates we must study the situation when χ¯ x y is violated. This violation may happen at any scale from 1 up to . To quantify this we introduce the following definition. Definition 5. A point x ∈ is called n-good if Bx y ≤ a |x − y|α
(11.1)
for all y ∈ with distance 1 ≤ |x − y| ≤ 4n from x. Definition 6. For a cube Rn of side 4n we define χ Rc n to be the indicator function of the event that there exists no n-good point in Rn . Our goal in the present section is to bound the expectation of the indicator function χ Rc n . In brief we will achieve this by estimating χ Rc n by a sum of products of factors of Bx y and then using (9.5). The details are as follows. A 3D cube Rn of side 4n can be expressed as a union of 43 disjoint subcubes of side 4n−1 . It is clear by inspection of Fig. 4 that we can select 23 = 8 of these subcubes, j i i say Rn−1 (i = 1, . . . , 8), so that dist(Rn−1 , Rn−1 ) > 4n−1 (i = j). Our approach now rests on the following simple observation: if there is no n-good point in Rn , then there is i either no (n − 1)-good point in any of the 8 subcubes Rn−1 , or else there exists at least n−1 n one bad pair (x, y) ∈ Rn × at scale 4 < |x − y| ≤ 4 . Thus, χ Rc n is bounded by the inequality χ Rc n ≤ S cRn +
8 i=1
χ Rc i
n−1
,
(11.2)
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M. Disertori, T. Spencer, M. R. Zirnbauer
3
2’’ 4
6’
6’
R3
R2
4
3’
1’’ 3’
n
4’
4’ 2
2 4 R4
2
n−1
1’
1’
4
2’’ 1’’
R1
4
3
1 root
1
n−1
(b)
(a)
Fig. 4. a In a 3D cube of side 4n we select 8 cubes of side 4n−1 . b Here we see an example of a rooted tree (on a 2D square) with coordination number 5 or 1 at each vertex, and the corresponding set of subsquares. The root corresponds to the large square
where S cRn =
x∈Rn , y∈ 4n−1 <|x−y|≤4n
χxcy ≤
x∈Rn , y∈ 4n−1 <|x−y|≤4n
Bxmy a m |x − y|αm
.
(11.3)
In order to apply the induction hypothesis (9.3)-(9.5) we need to write χ Rc n as sums j
of products of Bx y factors. Therefore we iterate (11.2) inside each cube Rn−1 , thus selecting 82 subcubes of side 4n−2 , and we keep repeating this procedure until we reach cubes of side 40 = 1 (i.e. points). We denote by R˜ n−k (k = 0, . . . , n) the set of 8k cubes of side 4n−k obtained in this way. R˜ n = {Rn } is the starting cube. Moreover let R˜ = ∪nk=0 R˜ n−k . In this way we bound χ Rc n by a positive sum of products of χxcy , which in turn are bounded by Bxmy /(a |x − y|α )m . The resulting expression can be organized as a sum over rooted trees picturing the hierarchy of inclusion relations of the subcubes. The following set of definitions serves to prepare the statement of Lemma 9 below. ˜ We associate by a fixed bijective Let V be an abstract set of vertices such that |V | = | R|. ˜ We denote by kv the scale of the corresponding map each vertex v in V to a cube Rv ∈ R. cube: Rv ∈ R˜ n−kv . The vertex associated with the largest cube Rn is denoted by r (root). Finally let A(v) (ancestor of v) be the unique vertex in V such that Rv ⊂ RA(v) (see vertex 1 and 1’ in Fig. 4b). With these definitions we can introduce Tn the set of labelled rooted trees on some subset of V with root r , such that the root has coordination number dr = 8 or dr = 0 (in which case the tree is reduced to a single vertex) and the other vertices have coordination number dv = 9 or dv = 1. Moreover if v belongs to the tree then there must be a tree line connecting v to its ancestor A(v). The maximal distance of a vertex from the root is n. Let L T denote the set of vertices in T with dv = 1 (the leaves) or dr = 0 (then L T contains only the root). Let Vk be the set of vertices in T at distance k from the root. With these definitions the tree is completely fixed by the leaves L T (or equivalently by the choice of the coordination numbers for each vertex). See Fig. 4b for an example in the case of dv = 5 instead of 9.
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Lemma 9. With the definitions above we have the inequality S cRv , χ Rc n ≤
(11.4)
T ∈Tn v∈L T
where Rv is a cube of side 4n−kv and S cRv is defined as in (11.3), with n replaced by n − kv . Proof. Our trees T ∈ Tn are constructed by iterating (11.2). In each iteration we get to choose between the first and second term of the r.h.s. of (11.2). The construction starts with the root of the tree. In the first step of the iterative scheme, if we pick the first term of (11.2) then the construction ends and we have produced nothing but the trivial tree (the root). If we pick the second term, we have a product of 8 different indicator functions χ Rc , one for each subcube in R˜ n−1 . We represent them in the tree by attaching 8 vertices to the root. Each vertex is then associated to a subcube by lexicographical order (see Fig. 4b). Now we repeat this procedure at the end of each branch and, continuing in this way, construct a tree. Whenever we pick the first term in (11.2), the corresponding branch of the tree terminates and we produce a terminal vertex or leaf. If we pick the second term, we generate a vertex of coordination number 9. The iteration stops when we reach scale n. We are now in a position to state and prove the main result of this section. Proposition 1. Let d = 3 and let the parameters a, m, α be chosen such that mα ≥ 4d and a ≥ 23α+(d+2)/m . Assume that the induction hypothesis (9.5) holds up to scale = 4n . Then c χ Rn ≤ 2−(n+1)αm , (11.5) and if Rn(k) , k = 1, . . . , N , denotes a family of cubes of side 4n(k) ≤ such that
dist(Rn(k) , Rn(k ) ) ≥ max(4n(k) , 4n(k ) ) then
N
χ Rc n(k)
k=1
≤
N
(11.6)
2−(n(k)+1)αm .
(11.7)
k=1
Proof. Applying (11.4) we have χ Rc n ≤
S cRv ≤
T ∈Tn v∈L T
T ∈Tn
xv ∈Rv , yv ∈,v∈L T 4n−kv −1 <|xv −yv |≤4n−kv
v∈L T
Bxmv yv
a m |xv − yv |αm
(4n v )d (4n v 2)d 2 1 −(n−kv +1)αm ≤ ≤ , 2 2 a m 4(n v −1)αm T ∈Tn v∈L T
(11.8)
T ∈Tn v∈L T
where in the second line we applied (9.3) (holds by the induction hypothesis). Finally n v = n − kv when n − kv ≥ 1 and n v = 1 when n − kv = 0. Note that when n − kv = 0 or 1 the bound is ensured by the factor a m in the denominator only. To perform the
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remaining sum over trees note that each vertex appears either as a leaf (dv = 1) at scale kv or as a branch (dv = 9) connecting to 8 vertices at scale kv + 1. Then 8 (0) (0) (0) 8 8 1 −(n−kv +1)αm = C 2 = C + = C + C + = · · ·, (C (C ) ) 0 1 2 0 0 1 2 T ∈Tn v∈L T
(0)
where the contribution Ck of a vertex at scale 0 ≤ k ≤ n is the sum of Ck (when the vertex is a leaf) and (Ck+1 )8 (when it is connected to 8 vertices at scale k + 1). The value of Ck is defined by induction: (0)
Ck = Ck + (Ck+1 )8 , k = 0, . . . , n − 1, Cn = Cn(0) , (0)
Ck
=
1 2
2−(n−k+1)αm
k = 0, . . . , n.
With these definitions it is easy to see that Ck ≤ 2−(n−k+1)αm ∀0 ≤ k < n. Hence the result. To prove (11.7) we apply (11.4) to each term χ Rc n(k) . Then we can again apply (9.5) as long as the corresponding diamonds are disjoint – this is ensured by the procedure for choosing subcubes and by the constraint dist(Rn(k) , Rn(k ) ) ≥ max(4n(k) , 4n(k ) ). This concludes the proof of Proposition 1. 12. Proof of the Induction Hypothesis In this section we shall establish the induction hypothesis of Sect. 11 at scale assuming that it holds up to scale − 1. Since our regions Rxi yi are disjoint by assumption, we will be able to re-express each factor Bxi yi as a sum over non-overlapping regions where our induction hypothesis applies. To simplify the notation let xi = x and yi = y. We will assume that = |x − y| is large, i.e., || ≥ β 1/4 . (The case of small < β 1/4 was dealt with in Sect. 10.) For z = x, y we recall the meaning of the regions Rxz y from Def. 2, Sect. 7. To ensure that the new regions produced by the analysis below remain inside the original region Rx y we need to introduce the following subsets. Definition 7. Let Rx y be a diamond region as described in Def. 4. Then we define R˜ xz y for z = x, y as R˜ xz y = { j ∈ Rxz y : ∠( j z, x y) ≤ π/8 for | j − z| > 10},
(12.1)
where ∠( j z, x y) is the angle between the lines j z and x y. This definition roughly selects (at distances larger than 10) a double cone which is obtained by rotating around x y a 2D diamond with vertices on x and y and opening angle θ = π/8 (see Fig. 2a). The condition | j − k| > 10 ensures that R˜ xz y ∪ {z} is connected. We also define uxy = χx j χy j (12.2) j∈ R˜ xx y
j∈ R˜ x y y
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y for χx j , χ y j as defined in (8.1). Note that R˜ xx y ∪ R˜ x y ∪ {x, y} is a δ-admissible region in the sense of Def. 1 (Sect. 7), so Lemma 5 and Lemma 6 can be applied to give
Bxmy u x y ≤ (1 − mC/β)−1 .
(12.3)
χ yc j
= 1 − χx j and = 1 − χ y j . The next lemma is nothing but a Now let combinatorial identity based on the following partitions of unity: 1= (χx j + χxc j ), 1= (χ y j + χ yc j ), (12.4) χxc j
j∈ R˜ xx y
j∈ R˜ x y y
y where R˜ xx y and R˜ x y are the regions defined above.
Lemma 10. The identity function can be written as c c χxb χx j + χ yb χxi 1 = uxy + b∈ R˜ xx y
j,| j−x|<|b−x|
y b∈ R˜ x y
i∈ R˜ xx y
χ y j . (12.5)
j,| j−y|<|b−y|
Proof. We start from (12.4) and expand the first product over j, beginning with small | j − x|. For each factor χx j + χxc j we have two possibilities: either we pick χx j , in which case we proceed to the next factor and repeat, or else we pick χxc j and then we stop expanding and leave all the other factors (with larger | j − x|) in summed form χx j + χxc j = 1. In the resulting sum there is the term j∈ R˜ x χx j . This we multiply by xy y the other product (over j ∈ R˜ x y ) in (12.4), which we expand in the same way. y In total, we have either picked a factor χ for all j ∈ R˜ xx y and j ∈ R˜ x y (this results c c in the term u x y ), or we have picked a term χx j or χ y j somewhere during the course of the expansion process (this gives all the other terms). The point where we stopped is c > 0 or χ c > 0 means that there denoted by b (where b stands for ‘bad’) because χxb yb is a large deviation at that point. Using the equality (12.5) of Lemma 10 we can rewrite Bxmy as Bxmy = Bxmy u x y + R(x, y), where R(x, y) is defined as R(x, y) =
χx j
j: | j−x|<|b−x|
b∈ R˜ xx y
+
c Bxmy χxb
c Bxmy χ yb
y b∈ R˜ x y
χxi
i∈ R˜ xx y
(12.6)
χy j .
(12.7)
j: | j−y|<|b−y|
We have Bxmy u x y ≤ (1 − mC/β)−1 by (12.3), without any need for an inductive argument. Thus if we can prove that R(x, y) ≤ 2 − (1 − mC/β)−1 our proof of (9.3) will be complete. The desired statement is formulated in the next lemma. Lemma 11. For large β the remainder (12.7) is bounded in average by R(x, y) ≤ 1/2.
(12.8)
Moreover, R(x, y) can be written as a sum over products of Bx y with |x − y | ≤ − 1 in such a way that the corresponding regions Rx y are disjoint.
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Now, using this lemma and arranging for (1 − mC/β)−1 not to exceed 3/2, we have Bxmy ≤ 3/2 + 1/2 = 2,
(12.9)
thus completing the proof of the Induction Hypothesis for the case n 1 = 1 and n 2 , n 3 = 0. The general case – (9.5) – is done in the same way. Before starting the proof of Lemma 11 we provide some orientation and motivation. c we could try to use (5.14): To bound the expectation of Bxmy χxb m m Bxmy ≤ 2m Bxb Bby ,
(12.10)
c = 1 − χ can be bounded by while χxb xb c χxb ≤ Bxb a − p |b − x|−αp , p
(12.11)
where we used Def. 3 of Sect. 8. Since both |b − x| and |b − y| are smaller than it is natural to try to apply the Induction Hypothesis. However we face at least two problems: m+ p [A]. TheInduction Hypothesis does not cover Bxb when p > 0. Indeed, a factor χ¯ xb = χx j χbj must be present in order for (9.4) to apply in such a case with 0 < p ≤ 2m. Notice, however, that while we have no immediate control without the missing factors χbj , the factors χx j for j ∈ R˜ xx y and | j − x| < |b − x| are already in place. To overcome the problem, we shall introduce the needed factors χbj by the same partition of unity scheme that was used above. Before embarking on that scheme, let us quickly evaluate the situation which emerges after insertion of χ¯ xb . We can then choose p = 2m, the induction (9.4) applies, and we get a small contribution c m χ¯ xb ≤ 2m a − p |b − x|−αp Bxb χ¯ xb Bby Bxmy χxb m+ p
≤ 4 · 2m a −2m |b − x|−2αm
(12.12)
for |b − x| large. Here we used (9.4) since xb is of Class 2 and by is of Class 1. Note that the expression (12.12) is summable in b for mα large. Moreover, the factor a −2m ensures that also the contributions for |b − x| = O(1) are small. [B]. The second problem is that, in order for (9.4) to apply we must make sure that we can find inside Rx y two non-overlapping regions Rxb and Rby of which the former is C-admissible and the latter is of diamond type. Moreover, since we have χx j only y for j ∈ R˜ xx y we must ensure that Rxb is inside the reduced region R˜ xx y ∪ R˜ x y ∪ {x, y} in Rx y . Since this might have to be repeated many times at smaller and smaller scales, we must be sure that all regions remain δ-admissible (or, put differently, we do not want δ to be scale-dependent). We will see in the next lemma that this can be arranged. y
Lemma 12. For a diamond Rx y consider the subsets Rxx y and Rx y of (7.4). [1] Let w be any point in R˜ xx y . Then we can always find a point a ∈ Rx y and regions Rxw , Rwa , and Ray , such that Rwa and Ray are diamonds inside Rx y , Rxw lies inside R˜ xx y ∪ {x} and is C-admissible, and the three regions have disjoint interiors (see Fig. 5).
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w θ2 x
a
θ1
y
Fig. 5. We need one intermediate point a. The two angles θ1 and θ2 are never smaller than π/8
w2
w1 x
a1 a2
a4 a3
y
x
a1
w1 a2
y
w2
(a)
(b)
Fig. 6. a If the pair w1 w2 is right in the middle, then we need to add four intermediate points a1 , . . . , a4 in order to find a minimal connected path around w1 w2 paved with disjoint diamonds. b Even if the pair w1 w2 is located on the boundary of R˜ xx y , the region Rw1 w2 still lies inside R x y
[2] Let w be any point in R˜ xx y such that |w − x| > β 1/4 (xw not of Class 3). Let y w1 w2 be a pair in R˜ xx y ∪ R˜ x y ∪ {x, y} such that |w1 − w| ≤ |w − x|1/2 and |w − x|1/2 ≤ |w1 − w2 | < |w − x|/5 (see (12.18)). Let Rw1 w2 be the corresponding diamond region. Then we can always find 4 points ai ∈ Rx y (i = 1, . . . , 4), such that all of the regions R xa1 , Ra j a j+1 ( j = 1, . . . , 3), and Ra4 y , are diamonds with disjoint interiors and do not overlap with Rw1 w2 (see Fig. 6). The same can be done y for x ∈ Rx y . Proof. The most dangerous situations are shown in Figs. 5 and 6. It is a simple geometrical argument to see that the region Rxw in Fig. 5 is C-admissible, as the angles θ1 and θ2 are never smaller than π/8, see (12.1). In the cases shown in Figs. 6a and 6b one has to check that √ the diamonds do not transgress Rx y . This never happens since |w1 − x| ≤ |x − y|/ 2 and |w1 − w2 | ≤ |w1 − x|/5. Proof of Lemma 11. We split the sum over bad points b in (12.7) into several groups of terms. Case 1. The bad point b is located close to x, i.e., |b − x| ≤ β 1/4 . Then we can bound c by (12.11), which gives Bxmy by (5.14) and χxb
470
M. Disertori, T. Spencer, M. R. Zirnbauer c m Bxmy χxb ≤ 2m a − p |b − x|−αp Bxb Bby . m+ p
(12.13)
To apply the Induction Hypothesis we need to select inside the diamond Rx y two regions Rxb and Rby . The first one, Rxb , need only be C-admissible (since xb is of Class 3, see Sect. 9), so it may be a deformed diamond (Fig. 3b). On the other hand, Rby has to be diamond-shaped, since |b − y| > β 1/4 (Class 1). To make the requirement of diamond shape conform with our constrained geometry, we must add an intermediate point a as in Fig. 5 with w = b, and use m m m Bby ≤ 2m Bba Bay .
(12.14)
We have seen in Lemma 12 that we can always find such a point a, so the induction (9.4) does apply. Note that since |b − x| < β 1/4 there will be no additional induction on Rxb . Therefore there is no risk that the region might get more and more deformed by the induction steps and δ-admissibility might finally be lost. Thus we have c m m Bxmy χxb ≤ 22m a − p |b − x|−αp Bxb Bba Bay . m+ p
(12.15)
The situation for b near y is analogous. Summing the contributions from b near x or y we obtain m c χz j Bx y χzb z=x,y |b−z|≤β 1/4
≤
j: | j−z|<|b−z|
4m a −2m |b − x|−2αm 22 (1 + ρ)
z=x,y |b−z|≤β 1/4 β 4m ρ , |b − x|2−2αm ≤ (4/a 2 )m K 1 ≤ ≤ 2 (1 + ρ) 2m K 1 a 10 1/4
3
(12.16)
|b−x|=1
where in the second line we used (9.4) and p = 2m. We can accommodate m + p = 3m > m without any protection factor χ¯ since bx is Class 3. In the third line, K 1 |b − x|2 is the entropy factor for the 3D sum over bad points at distance |b − x|, the factor K 1 is a constant of order unity, and we used that 4/a 2 < 1 and m > 4d/α is large. We bounded the expression by ρ/10 for convenience; since both a and m are large, the factor (4/a 2 )m K 1 is in fact very small. Case 2. The first bad point b is far from x (i.e., |b − x| > β 1/4 ) and also far from y. Let y us consider the case b ∈ R˜ xx y for definiteness. (The other case, b ∈ R˜ x y , is treated in the same way.) Again, we have to estimate m c Bx y χxb χx j . (12.17) j: | j−x|<|b−x|
As was observed above, if we succeeded in promoting the last product in the average to a complete factor χ¯ xb , then we could apply the Induction Hypothesis as in (12.12). In order to satisfy the hypothesis √ of Lemma 5, Eq. (7.6), we should have a constraint χ jb for all | j − b| ≤ |b − x|/ 2. Actually, from the remark after the proof of that lemma we only require χ jb for | j − b| < |b − x|/5 since we know that all tx − t j fluctuations are good up to | j − x| ≈ |b − x|.
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Guided by the idea of partition of unity (cf. (12.4)–(12.5)), we will first check whether there is some large fluctuation χ c > 0 at large scale near b. If no such event occurs, we proceed to the step of checking fluctuations at intermediate distance scales. Then either all intermediate distance fluctuations are good too (and we have the desired factor χ), ¯ or there must be some bad event at intermediate scale. In this last case we will see that many bad events must happen. We will now make this more precise. Case 2a. The nearest bad point b is far from x (and y), |b−x| > β 1/4 , and there is a large y scale bad event near b. This means that B jk ≥ a | j − k|α for some pair j, k ∈ R˜ xx y ∪ R˜ x y such that | j − b| ≤ |b − x|1/2 and |b − x|1/2 ≤ | j − k| ≤ |b − x|/5.
(12.18)
Now, using (5.14) and (12.11), m Bxmy χ cjk ≤ 24m Bxa 1
3
Bami ai+1 Bam4 y B mjk | j − k|−αm a −m .
(12.19)
i=1
To apply the Induction Hypothesis the corresponding regions must all be diamonds (all pairs are Class 1). By the assumptions made on the pair jk, Lemma 12 guarantees that we can choose the four intermediate points ai ∈ Rx y (i = 1, . . . , 4) so that all of the regions Rxa1 , Ra j a j+1 ( j = 1, . . . , 3), and Ra4 y , are diamonds with disjoint interiors and do not overlap with R jk (see Fig. 6). Since the regions are non-overlapping and > | j − k| ≥ |b − x|1/2 , our induction hypothesis yields Bxmy χ cjk ≤ 24m 26 |b − x|−αm/2 a −m . For large m the value of the sum over b is small. To estimate the entropy factor, note that there are less than |b − x|d+d/2 pairs jk satisfying (12.18). Altogether then, the present partial sum of contributions from r ≡ |b − x| ≥ β 1/4 is bounded by ρ (4/a)m K 2 . r (d−1)+d+d/2−αm/2 = O β −1/4 < 10 1/4 r >β
Note that there is nothing special or optimal about the exponent 1/4 of 1/β – it is just convenient. Case 2b. We now suppose that |b − x| ≥ β 1/4 and there is no large deviation near b, i.e., B jk ≤ a | j − k|α holds for all j, k subject to (12.18). This implies that at long scales | j − k| ≥ |b − x|1/2 we have χ jk = 1. It remains to check whether χ jk holds also at shorter scales | j − k| ≤ |b − x|1/2 . First we consider the case of there being a point g (g stands for good as in Def. 5 of Sect. 11) in Rx y with |g − b| ≤ |b − x|1/2 such that χgh = 1 holds for all h with |g − h| ≤ |b − x|1/2 . We then have in particular that χgb = 1, and so by Def. 3, Bgb ≤ a |b − g|α ≤ a (|b − x|1/2 )α = a |b − x|α/2 .
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M. Disertori, T. Spencer, M. R. Zirnbauer
c = 1 and (5.14) yields This inequality combined with the constraint χxb
2Bxg ≥
Bxb a |b − x|α ≥ = |b − x|α/2 . Bgb a |b − x|α/2
Thus we have c m m c 3m m χ¯ xg ≤ 2m (Bxg χ¯ xg )Bgy χxb ≤ 23m (Bxg χ¯ xg )Bgy |b − x|−αm . Bxmy χxb
(12.20)
Now we have to be somewhat careful about the choice of the regions Rxg and Rgy , as they may not have the canonical diamond shape. For Rxg this is not a problem, because of the presence of χ¯ xg (xg is of Class 2). All we need is that Rxg be C-admissible. On the other hand, Rgy is (as in Case 1) slightly more delicate. To be sure that we deal with diamond-shaped regions, we add an intermediate point a as in Fig. 5 and use m ≤ 2m B m B m . We have seen in Lemma 12 that it is always possible to find such a Bgy ga ay point a. The regions Rga and Ray are of diamond type, so induction applies. Note that Rxg comes with a χ¯ xg factor (xg is of Class 2) so no additional induction is required for it. Therefore, as in Case 1, there is no risk that the region might get more and more deformed by the induction steps. It should be emphasized, however, that χ¯ xg is not exactly the same as in (8.1), but rather is given by
χ¯ xg =
R˜ xx y j : | j−x|<|b−x|
χx j
χg j .
R˜ xx y j : | j−g|≤|b−x|/5
Since |b− g| ≤ |b− x|1/2 and |b− x| > β 1/4 we have |g − x| |b− x| up to a correction factor of order O(|b − x|−1/2 ) ≤ O(β −1/8 ) 1. Therefore χ¯ xg is equivalent to the following constraints: ∀ j ∈ Rxg , | j − x| ≤ |g − x| f 1 : and ∀ j ∈ Rxg , | j − g| ≤ |g − x| f 2 :
Bx j ≤ a | j − x|α , Bg j ≤ a | j − g|α ,
with f 2 = 1/5 and f 1 = 1 − O(β −1/8 ). From Remark 7.3 we know that Lemma 5 and hence Lemma 6 still hold, so we can apply the induction and
c Bxmy χxb χ¯ xg ≤ 23m (1 + ρ) 22 |b − x|−αm .
There are O(|b − x|d/2 ) choices for g, so the sum over these contributions is bounded by 23m K 3
r >β 1/4
r d−1+d/2−αm = O(β −1/4 ) <
ρ . 10
(12.21)
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473
Case 2c. The last case to consider is the situation where no such point g exists. In that y case we can always find a cube Rn ⊂ R˜ xx y ∪ R˜ x y which contains the point b and has side from the boundary of 4n = |b − x|1/2 such that Rn is at least at distance β 1/4 ≤ 4n Rx y and contains no n-good point (see Def. 5 in Sect. 11). Then by (11.4) and (12.11) with p = m, we have χ Rc n ≤
T ∈Tn { jv kv }v∈L T v∈L T
χ cjv kv ≤
T ∈Tn { jv kv }v∈L T v∈L T
B mjv kv
a m | jv − kv |αm
,
where according to (11.3) the sum over configurations of pairs ( jv , kv ) ∈ Rv × is constrained by 4n v −1 < | jv − kv | ≤ 4n v , with n v the scale of the leaf v. Since all cubes Rv are inside the small region Rn and all pairs jv , kv satisfy the conditions of Lemma 12 for the pair w1 w2 , we can proceed as in Case 2b and select 4 intermediate points a j , j = 1, . . . , 4 such that the corresponding regions are diamonds and do not overlap with any R jv kv (see Fig. 6). Then by (5.14) we have m · · · Bam4 y Bxmy χ Rc n ≤ 24m Bxa 1
T ∈Tn { jv kv }v∈L T v∈L T
B mjv kv
a m | jv − kv |αm
,
(12.22)
and we can apply the Induction Hypothesis. By Proposition 1 in Sect. 11 we have
Bxmy χ Rc n ≤ 24m 2−nαm , n ≈ ln |b − x| ≥ ln(β 1/4 ).
(12.23)
Therefore we have enough decay to control the entropy factors: 24m K 4
|b − x|−αm < O(β −1/4 ) <
r >β 1/4
ρ . 10
This concludes the proof of (12.8). From Eqs. (12.15) and (12.19)–(12.22), we see that R(x, y) can be written as a sum over products of such Bx y with |x − y | ≤ − 1 that the corresponding regions Rx y are disjoint. This concludes the proof of Lemma 11 and the Induction Hypothesis. 13. Proof of Theorem 2 Now that we have estimated Bxmy for all x, y we need to estimate cosh p (tx ) for moderate values of p ≤ 10. If we suppose that the field t is pinned at some point j0 , so that t j0 = 0, then Theorem 2 follows directly from Theorem 1: cosh p tx = cosh p (tx − t j0 ) ≤ 2, for any x in the lattice (since Theorem 1 does not require bounds on ε). When the field is not pinned, we need ε > 0 and some conditions on the volume. The rest of this section is devoted to this case. As in the proof of Theorem 1 we will first prove bounds on conditional expectations.
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Definition 8. A point x ∈ is called ‘good at all scales’ if ∀ j ∈ \{x} :
Bx j ≤ a | j − x|α
(see also Def. 5 in Sect. 11). The corresponding characteristic function is χx j , χ¯ x :=
(13.1)
j∈\{x}
where the factors χx j are those of Def. 3 (Sect. 8). Lemma 13. Let x be good at all scales, and let Bx = cosh tx + 21 etx sx2 .
(13.2)
If β 1 and ε ≥ 8 pa L −d+α , then for any 0 < p ≤ O(β) we have p
Bx χ¯ x ≤ 2.
(13.3)
Proof. The proof uses a combination of ideas already present in the proofs of Lemma 5 (Sect. 7) and Lemma 6 (Sect. 8). By supersymmetry (Proposition 2, Appendix C) we have p 1 = zx χxSj , (13.4) j∈\{x}
where z x is defined in (2.11) and χxSj in (8.3). Following exactly the same steps as in the proof of Lemma 6, we obtain the inequality p 1 ≥ Bx χ¯ x (1 − p G x ) ,
Gx =
et x [δx ; Dβ,ε (t)−1 δx ], Bx
(13.5)
if p G x < 1. We must now bound the Green’s function G x using the constraint χ¯ x (as we did in Lemma 5). For this purpose define D˜ = e−tx Bx Dβ,ε (t) by ˜ = et x B x β [v ; Dv] eti +t j −2tx (vi − v j )2 + ε Bx etk −tx vk2 , (i j)
k
and note that Bx j ≤ a | j − x|α implies the bound et j −tx ≥ (2a | j − x|α )−1 . We then follow the proof of Lemma 5 and introduce a telescopic sum δx = (δx − I1 ) + (I1 − I2 ) + · · · + (I N −1 − I N ) + I N =
N
ρn ,
(13.6)
n=0
where In is the (normalized) indicator function of a cube of center x and side 2n , and ρn = In − In+1 . There is no need to introduce I˜ as we did in the proof Lemma 5, as we are now working not on Rx y but on the whole volume. The sum terminates on reaching the system size 2 N . Note that for n < N we have j ρn ( j) = 0 and ρn 22 ≤ 2−nd = 2−3n . The function ρn for n = N is constant: ρ N ( j) = I N ( j) = ||−1 for all j ∈ .
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475
Now, by the Cauchy-Schwarz inequality, ˜ −1
[δx ; D
δx ] ≤
N
2 ˜ −1
[ρn ; D
ρn ]
1/2
.
(13.7)
n=0
For n < N we use the bound [ρn ; D˜ −1 ρn ] ≤ ( D˜ −1 )ρn ρn 22 , where ( D˜ −1 )ρn ≤ 22n+2nα c1 . Then for the sum of terms with n < N we have N −1
N −1 c1 √ −n(d−2−2α) γ1 [ρn ; D˜ −1 ρn ]1/2 ≤ √ 2 <√ β n=0 β n=0
(13.8)
uniformly in N since d = 3 and 2α 1. For n = N , on the other hand, we no longer have orthogonality to the constant functions (‘zero mode’) and therefore must ˜ take recourse to the ε-term in D: −1 Lα 1 . [ρ N ; D˜ −1 ρ N ] ≤ ε|| min et j −tx ≤ 2a d ≤ j∈ εL 4p
(13.9)
√ √ Hence G x ≤ (γ1 / β + 1/ 4 p )2 < 1/(2 p) for β 1. So, p 1 2 Bx
p
χ¯ x ≤ Bx χ¯ x (1 − p G x ) ≤ 1
by (13.5), and the lemma is proved.
(13.10)
With this lemma we can finally complete the proof of Theorem 2, i.e. the bound on the unconditional expectation of cosh p tx . Proof of Theorem 2. We recall from Def. 5 (Sect. 11) that a point x is said to be n-good if Bx j ≤ a | j − x|α for all j ∈ subject to 1 ≤ | j − x| ≤ 4n . A point x is good at all scales if Bx j ≤ a | j − x|α for all j ∈ \{x}; we then say that x is N -good. We proceed as in Lemma 11 (Sect. 12): cosh p tx = χ¯ x cosh p tx + χ¯ xc cosh p tx ,
(13.11)
where χ¯ x ensures that the point x is N -good. Then by Lemma 13 we have χ¯ x cosh p tx ≤ 2.
(13.12)
It remains to estimate the second term, χ¯ xc cosh p tx . We prove in Lemma 14 below that this term is bounded by a constant. Once this has been accomplished, the proof of Theorem 2 will be finished. Lemma 14. Let χ¯ xc = 1 − χ¯ x , with χ¯ x defined by (13.1). Let β 1 and ε ≥ 8 · 4 · 10 a L α−d . Then for any 0 ≤ p ≤ 10 we have χ¯ xc cosh p tx ≤ 1/2.
(13.13)
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M. Disertori, T. Spencer, M. R. Zirnbauer
Proof. While χ¯ xc means that x is not N -good, it is still possible for other points in to be N -good. If g = x is the nearest such point (as seen from x), then none of the points x inside the ball K |g−x| of radius |g − x| and center x is N -good. Denoting the indicator function for the latter event by χ Kc x we have the identity |g−x|
χ¯ xc =
χ¯ g χ Kc x
|g−x|
g=x
+
χ¯ cj ,
(13.14)
j∈
where the last term accounts for the possibility that there is no N -good point in at all. Thus we obtain the decomposition c p c p c p χ¯ x cosh tx = χ¯ g χ K x cosh tx + χ¯ j cosh tx . (13.15) g=x
|g−x|
j
We will prove that both of these two terms are bounded by 1/4. 1. We consider the first sum. Using cosh tx ≤ 2 cosh(tx − tg ) cosh tg and applying the Cauchy-Schwarz inequality twice, we obtain 1/2 1/2 χ Kc x χ¯ g cosh p tx ≤ χ Kc x χ¯ g χ¯ g cosh2 p tx |g−x| |g−x| 1/2 1/2 c ≤ χ K x χ¯ g 2 p χ¯ g cosh2 p (tx − tg ) cosh2 p tg |g−x| 1/2 1/4 1/4 p χ¯ g cosh4 p tg cosh4 p (tx − tg ) ≤ 2 χ Kc x χ¯ g |g−x| 1/2 ≤ 2 p c p χ Kc x χ¯ g , |g−x|
where in the last step we used Lemma 13 and Theorem 1, and we introduced c0 = 1 and c p = 21/2 for p ≥ 1. It remains to bound 1/2 1/2 2pcp χ Kc x χ¯ g χ Kc x χ¯ g = 2pcp . (13.16) |g−x|
g
n≥0 4n ≤|g−x|<4n+1
|g−x|
Let Rnx be the cube centered at x of side 4n . Now fixing a point g with 4n ≤ |g−x| < 4n+1 x we have Rnx ⊆ K |g−x| , and we distinguish between two cases: x 1a. The interior of K |g−x| is void not only of N -good points but also of n-good points. c Let χn denote the corresponding indicator function. Then for n ≥ 1, using Proposition 1 we have χ Kc x χ¯ g χnc ≤ χ Rc nx ≤ 2−(n+1)αm , (13.17) |g−x|
where χ Rc x n so
is given in Def. 6, Sect. 11. For n = 0 the cube R0x contains only the point x,
χ Rc x 0
≤
|z−x|=1
χx z ≤
|z−x|=1
Bxmz 2d ≤ 2 m < 2−αm . am a
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477
x 1b. There is at least one n-good point y inside K |g−x| . Let χn (y) be the corresponding indicator function. Because the point y cannot be N -good, there must be a first scale q > n so that y is q-bad. Thus there exists a first point b at distance |b − y| > 4n with B yb > a |b − y|α . It follows that
χ¯ g χ Kc x
x y∈K |g−x|
|g−x|
χn (y) ≤
m B yb
x b: |b−y|>4n y∈K |g−x|
≤
a m |b − y|αm
4nd k1 2 r d−1 k2 ≤ 2−nαm a −m ≤ 2−(n+1)αm , am r αm n r >4
where the factor 4nd k
1 comes from the sum over y and r
d−1 k
2 comes from the sum over b. Inserting these results into (13.16) we obtain 1/2 1/2 χ Kc x χ¯ g 2pcp ≤ 2pcp 4(n+1)d k3 2 · 2−(n+1)αm , n≥0 4n ≤|g−x|≤4n+1
|g−x|
n≥0
where 4(n+1)d k3 comes from the sum over g. This will be no greater than 1/4 provided that αm is large enough. 2. To complete the proof, we have to estimate the last term j χ¯ cj cosh p tx in (13.15). By Proposition 1 the probability for no N -good point to be found in a cube of side L = 4 N is bounded by 2−N αm = L −αm/2 . Hence 1/2 1/2 1/2 χ¯ cj cosh p tx ≤ cosh2 p tx χ¯ cj ≤ cosh2 p tx L −αm/4 . j
j
To get a bound on the expected value of cosh2 p tx we once again use supersymmetry (Proposition 2), as follows: eγ ε = eγ εz x = eγ ε Bx (1 − γ ε G x ) , where we choose 0 < γ < 1/2, and G x = etx [δx ; Dβ,ε (t)−1 δx ]. Since the operator Dβ,ε (t) − ε etx δx [δx ; ·] is non-negative, by Lemma 1 we have ε G x ≤ 1, so eγ ε(Bx −1) ≤ (1 − γ )−1 . Also, cosh2 p tx ≤ (2 p)! (γ ε)−2 p eγ ε Bx by an elementary computation, and hence cosh2 p tx ≤ (2 p)! (γ ε)−2 p eγ ε Bx ≤ O(ε−2 p ). We thus finally obtain c p χ¯ j cosh tx ≤ cosh2 p tx 1/2 L −αm/4 ≤ O(ε− p )L −αm/4 < 1/4,
(13.18)
j
since αm is large and ε ≥ L α−d . This concludes the proof of Lemma 14.
Remark. In the proof of Theorem 2 the ε term (zero mode) appears only in two places: (13.9) of Lemma 13 (the last term in the telescopic sum) and (13.18) (when no N good point is present). The inequality (13.9) is the reason why we cannot take ε = O(L −d ) but must take ε = O(L α−d ).
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14. Proof of Theorem 3 Finally we can prove the bound on the Green’s function C x y of (1.9). Let f be such that f ( j) ≥ 0 for all j ∈ . We need to estimate [ f ; C f ] = [et f ; Dβ,ε (t)−1 et f ] = [W ; G t W ] , (14.1) where Dβ,ε (t)−1 = G t was defined in (1.1), and W ( j) = et j f ( j). 14.1. Upper bound. Let L 0 = −β + ε and G 0 = L −1 0 (as defined in the statement of the theorem). Now [W ; G t W ] = [L 0 G 0 W ; G t W ] = β [∇(G 0 W ); ∇(G t W )] + ε [G 0 W ; G t W ] ∇ j j (G 0 W )∇ j j (G t W ) + ε (G 0 W )( j)(G t W )( j) =β ( j j )
=β
j
∇ j j (G 0 W ) ∇ j j (G t W )
( j j )
e(t j +t j )/2
e−(t j +t j )/2
+ε
(G 0 W )( j) (G t W )( j) e+t j /2
j
e−t j /2
.
Since |a · b + c · d| ≤ (a · a + c · c)1/2 (b · b + d · d)1/2 we have ⎛ [W ; G t W ] ≤ ⎝β
|∇ j j (G 0 W )|2 et j +t j
( j j )
+ε
|(G 0 W )( j)|2 et j
j
⎞1/2 ⎠
[W ; G t W ]1/2 .
Therefore3 [W ; G t W ] ≤ β
|∇ j j (G 0 W )|2 e
( j j )
Now |∇ j j (G 0 W )| ≤
t j +t j
+ε
|(G 0 W )( j)|2 j
et j
|(G 0 ( j, k) − G 0 ( j , k))| W (k) ≤ const
k
.
(14.2)
H jk W (k), (14.3)
k
where we defined H jk = β −1 (| j − k|2 + 1)−1 e−˜ε| j−k| , ε˜ = (ε/2β)1/2 , and we used |(G 0 ( j, k) − G 0 ( j , k))| ≤ const H jk . By inserting (14.3) into (14.2) we get [ f ; C f ] ≤ const β +ε
H jk H jl f (k) f (l) e(tk +tl −t j −t j )
( j, j ),k,l
G 0 ( j, k) G 0 ( j, l) f (k) f (l) etk +tl −t j .
j,k,l 3 We thank S.R.S. Varadhan for explaining the inequality (14.2) to us.
(14.4)
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479
By Theorems 1 and 2 the expectation over the field t is uniformly bounded. Now we can sum over j: H jk H jl ≤ const G˜ 0 (k, l), G 0 ( j, k) G 0 ( j, l) = G 20 (k, l), (14.5) j
j
where G˜ 0 = (−β + ε/2)−1 . Note that G 0 ≤ G˜ 0 . We finally obtain [ f ; C f ] ≤ const [ f ; G˜ 0 f ] + ε[ f ; G 20 f ] ≤ 2 const [ f ; G˜ 0 f ]. This completes our proof of the upper bound.
(14.6)
14.2. Lower bound. Let χ¯ x be the characteristic function ensuring that x ∈ is good at all scales (see (13.1)). Recall that if χ¯ x > 0 then et j −tx ≥ (2a(1 + | j − x|α ))−1
(14.7)
for all j ∈ . We have the inequality 1 = χ¯ x + χ¯ xc ≥ χ¯ x . Inserting it into (14.1) we obtain [W ; G t W ] ≥ χ¯ x [W ; G t W ] = χ¯ x W ( j)W (k) G t ( j, k) (14.8) jk
1 ˜ 1 ≥ 2 f ( j) f˜(k) χ¯ x D¯ t−1 ( j, k) = 2 χ¯ x [ f˜; D¯ t−1 f˜] , 4a 4a jk
where f˜( j) = (1 + | j − x|α )−1 f ( j) and D¯ t−1 = e2tx G t = (e−2tx Dβ,ε (t))−1 . In the first line we used the fact that G t is positive as a quadratic form for each configuration of t. In the second line we used the fact that this is a sum of positive terms since W ( j) ≥ 0 and G t is pointwise positive. Furthermore, we applied (14.7) to estimate W ( j). Now, χ¯ x [ f˜; D¯ t−1 f˜] = χ¯ x E [ f˜; D¯ t−1 f˜] ≥ χ¯ x [ f˜; E( D¯ t )−1 f˜], (14.9) where E(·) =
χ¯ x · χ¯ x
is a probability distribution and we used Jensen’s inequality. In order to complete the proof we need to estimate χ¯ x and χ¯ x D¯ t . From Lemma 14 (in the previous section) with p = 0 we know that χ¯ x = 1 − χ¯ xc ≥ 1/2. Moreover χ¯ x D¯ ≤ D¯ as a quadratic form and for any function u we have (u( j) − u(k))2 et j +tk −2tx + ε u( j)2 et j −2tx [u; D¯ u] = β ( jk)
≤ βc
( jk)
(u( j) − u(k))2 + εc
j
j
u( j)2 ≤ c1 [u; G 0 u],
(14.10)
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where we applied Theorems 1 and 2, and c1 = sup{c, c }. Thus D¯ ≤ c1 G 0 . By applying these relations we see that E( D¯ t ) ≤ 2c1 G 0 and hence 1 [ f˜; G 0 f˜]. 4a 2 c1
[ f;Cf] ≥ This concludes the proof of Theorem 3.
(14.11)
Remark. If W did not depend on t we would have the quadratic form estimate [W ; G 0 W ] with c2 = sup
1 ≤ [W ; G t W ] ≤ c2 [W ; G 0 W ] c1
( j j ),k
e−t j −t j , e−tk , c1 = sup et j +t j , etk . ( j j ),k
The upper bound follows directly from (14.2), the lower bound from Jensen’s inequality. Acknowledgements. It is our pleasure to thank S. Varadhan and J. Lebowitz for discussions and suggestions related to this paper. Very special thanks go to D. Brydges for sharing his many insights on the model and for many comments on an early version of this paper. We wish to thank the Newton Institute (Cambridge) for its support and hospitality during the completion of this article. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Appendices A. Minimum of the Effective Action Let j → t j ≡ t¯ ∈ R (for all j ∈ ) be a constant field configuration. Evaluating the statistical weight function on it we get e−ε||(cosh t¯−1) Det 1/2 (−β + ε e−t¯).
(A.1)
Let t ∗ be the number that maximizes this statistical weight. The condition for the first derivative to vanish at t ∗ is ∗
2 sinh t ∗ = −e−t G 0 (x, x)
(x ∈ ),
−t ∗
where G 0 ≡ (−β + ε e )−1 ≥ 0. Equivalently, 1 − e is non-negative, it follows that t ∗ ≤ 0. We thus infer that
2t ∗
(A.2)
= G 0 (x, x), and since G 0
∗
0 ≤ 1 − e2t = G 0 (x, x) ≤ 1.
(A.3)
Next, we show that the constant field t ∗ maximizes the integrand over the full set of all field configurations t = {t j }. For this, we recall the definition (1.2) of the effective action or free energy Fβ,ε in combination with (1.10): Fβ,ε (t) = β (cosh(ti − t j ) − 1) + ε (cosh tk − 1) (i j)
− ln Det
k
1/2
−β + βV (t) + ε e−t .
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model 1/2
Now we introduce A := G 0
βV (t) + ε (e−t − e−t
∗ Id
481
1/2 ) G 0 and write
Det −β + βV (t) + ε e−t = Det(G −1 0 ) Det(Id + A). Using ln Det(Id + A) ≤ Tr A we then obtain −1/2
Fβ,ε (t) ≥ − ln Det(G 0 ) + ε|| (cosh t ∗ − 1) +β (cosh(ti − t j ) − 1) − G 0 (x, x) 21 βVkk (i j)
k
∗ cosh tk − cosh t ∗ − 21 (e−tk − e−t ) G 0 (x, x) . +ε k
The second line of the r.h.s. is non-negative by 21 j V j j = (i j) (cosh(ti − t j ) − 1) ∗ and G 0 (x, x) ≤ 1, and so is the third line by the identity G 0 (x, x) = 1 − e2t and a trivial computation. This proves that Fβ,ε (t) is bounded from below by Fβ,ε (t ∗ ) = −1/2 − ln Det(G 0 ) + ε|| (cosh t ∗ − 1). B. Hyperbolic Symmetry In Sect. 4 we explained that the H2|2 nonlinear sigma model in the limit of vanishing regularization ε → 0+ acquires a global symmetry by the Lorentz group SO(1, 2). We will now exhibit the Ward identities due to this Lorentzian symmetry SO(1, 2). (Consequences due to the supersymmetries of model will be explored in Appendix C.) To prepare the discussion, the reader is invited to recall the expressions (2.7) for the functions ¯ x, y, ξ, η in horospherical coordinates. We also recall that z = cosh t + et ( 21 s 2 + ψψ). We now seek the first-order differential operator, L 1 , generating Lorentz boosts in the zx-plane, i.e., L 1 z = x,
L 1 x = z,
L 1 y = L 1 ξ = L 1 η = 0.
It is easy to verify that the unique operator with these properties is ¯ ψ¯ − ψ∂ψ − s∂s . L 1 = ∂t − ψ∂
(B.1)
Similarly, the generator L 2 of Lorentz boosts in the zy-plane and the generator L 0 of Euclidean rotations in the x y-plane, are expressed by ¯ ψ¯ − ψ∂ψ + 1 1 + e−2t − s 2 + 2ψψ ¯ ∂s , L 2 = s ∂t − ψ∂ 2 ¯ ψ¯ + ψ∂ψ + 1 1 − e−2t + s 2 − 2ψψ ¯ L 0 = s −∂t + ψ∂ ∂s . 2 Being the generators of the Lie algebra so1,2 of the Lorentz group, the operators L 0 , L 1 , L 2 satisfy the commutation relations: [L 0 , L 1 ] = −L 2 , [L 0 , L 2 ] = L 1 , [L 1 , L 2 ] = L 0 . In particular, the generator L 0 + L 2 = [L 1 , L 0 + L 2 ] is the generator of translations of the coordinate s.
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M. Disertori, T. Spencer, M. R. Zirnbauer
So far, we have been concerned with the case of a single site. To pass to a lattice with many sites, we take the sum La = L a ( j) (a = 0, 1, 2) j∈
of differential operators over all sites. L a ( j) are symmetries of the Berezin By construction, the so1,2 operators L a = measure Dμ . Therefore, they give rise to Ward identities: (B.2) 0= Dμ L a e−Aβ,ε F = L a F − F L a Aβ,ε (a = 0, 1, 2), which hold for any observable F as long as these expectations exist. By computing the symmetry-breaking terms from the formula L a Aβ,ε = ε j L a ( j) z j one obtains these Ward identities in the more explicit form L 1 F = ε (sinh t j − 21 et j s 2j − et j ψ¯ j ψ j )F , j
L 2 F = ε
et j s j F ,
L 0 F = 0.
j
The sum rule (4.6) now follows from the identity for L 2 F by taking F = eti si and ¯ and s. performing the Gaussian integrals over the fields ψ, ψ, Another important consequence results from making the choice F = et j s j . Since L 0 F = − sinh t j + et j ( 21 s 2j + ψ¯ j ψ j ), it follows from L 0 F = 0 that et j = cosh t j + sinh t j = cosh t j + et j ( 21 s 2j + ψ¯ j ψ j ) = z j = 1.
(B.3)
The last step, z j = 1, is by Proposition 2 of Appendix C. C. SUSY Ward Identities The action function of our H2|2 model has a global symmetry w.r.t. the Lie superalgebra g := osp2|2 (for any ε ≥ 0). As a result, there exist supersymmetric Ward identities for suitable (osp2|2 invariant) observables. Although such identities are standard material from the theory of localization of supersymmetric integrals [17], we nonetheless give their derivation for completeness here, as the said identities play a central role in our analysis. The essence of the argument can already be understood at the very special example of a lattice consisting of just a single site. For pedagogical reasons we first handle this simple situation and then, in a second step, give the generalization to arbitrary lattices. The treatment will be most transparent if we do all calculations using the coordinates x, y, ξ, η described at the beginning of Sect. 2. As stated there, for our purposes we may view osp2|2 as the space of first-order differential operators D with coefficients that are linear in the variables x, y, ξ, η and the property DH = 0
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model
483
of annihilating the quadratic polynomial H = x 2 + y 2 + 2ξ η. Let Q be the distinguished first-order differential operator defined by Q = x∂η − y∂ξ + ξ ∂x + η∂ y .
(C.1)
Clearly Q is odd, converting even coordinate generators x, y into odd generators ξ, η and vice versa. Q is also seen to annihilate H , and thus represents an element of osp2|2 . Notice that Q squares to Q 2 = x∂ y − y∂x + ξ ∂η − η∂ξ , which is a generator from the Lie algebra part o2 ⊕ sp2 of osp2|2 . Now recall from Sect. 2 that our Berezin superintegration form is Dμ = (2π )−1 d xd y ∂ξ ∂η ◦ (1 + H )−1/2 . Lemma 15. The Berezin superintegration form Dμ is Q-invariant, i.e., Dμ Q f = 0 R2
for any bounded smooth superfunction f = f (x, y, ξ, η). Proof. Since Q is a first-order differential operator, we have from Q H = 0 that Q(1 + H )−1/2 = 0. Therefore, Dμ Q f = Dμ (1 + H )1/2 Q(1 + H )−1/2 f and −1 Dμ Q f = (2π ) d xd y ∂ξ ∂η x∂η − y∂ξ + ξ ∂x + η∂ y (1 + H )−1/2 f. R2
R2
The desired result now follows because ∂ξ2 = ∂η2 = 0 and the integral over R2 of the total derivatives ∂x (1 + H 2 )−1/2 f and ∂ y (1 + H 2 )−1/2 f vanishes. An important property of the differential operator Q is that the joint zero locus of its coefficients is the origin x = y = 0 and ξ = η = 0. Denoting the origin by o we write f (x = 0, y = 0, ξ = 0, η = 0) ≡ f (o). Lemma 16. Let f = f (x, y, ξ, η) be a smooth superfunction which satisfies the invariance condition Q f = 0 and decreases sufficiently fast at infinity in order for the integral
R2 Dμ f to exist. Then Dμ f = f (o). R2
Proof. The idea is to ‘deform’ the integrand f (without changing the integral) by a factor that localizes the integral at o. We will do this deformation by multiplication with e−τ H for some positive real parameter τ . Thus we are going to show that (C.2) Dμ f = Dμ e−τ H f, independent of τ ≥ 0. The desired result will then follow by taking τ → +∞.
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We begin by observing that the localizing function H is Q-exact: it can be written as H = Qλ with λ := x η − y ξ an odd superfunction. Next, using the relation Q H = Q 2 λ = 0 we do the following calculation: e−τ (Qλ) − 1 −τ H −τ (Qλ) e =1+ e −1 =1+ Q λ . Qλ Here the term in parentheses stands for ∞
(−τ )n+1 e−τ (Qλ) − 1 := (Qλ)n . Qλ (n + 1)! n=0
Inserting this decomposition into the integral we obtain Dμ e
−τ H
f =
Dμ f +
e−τ (Qλ) − 1 Dμ f Q λ . Qλ
Since our integrand f is Q-invariant by assumption (Q f = 0), the second integral can also be written as e−τ (Qλ) − 1 e−τ (Qλ) − 1 Dμ f Q λ = Dμ Q f λ = 0, Qλ Qλ which vanishes by Lemma 15. This already proves (C.2). ∗ To complete√ the proof, we √ √ consider the √ effect of a scale transformation φτ : x → x/ τ , y → y/ τ , ξ → ξ/ τ , η → η/ τ . Note that φτ∗ H = H/τ and the Berezin superintegration form d xd y ∂ξ ∂η = Dμ ◦ (1 + H )1/2 is invariant by φτ∗ . The statement of the lemma now results from taking the limit Dμ f = lim Dμ e−τ H f = lim φτ∗ Dμ e−τ H f τ →∞ τ →∞ R2 Dμ (1 + H )1/2 (1 + H/τ )−1/2 e−H φτ∗ f = f (o), = lim τ →∞
where the last step is done by verifying the normalization integral 2 2 Dμ (1 + H )1/2 e−H = (2π )−1 d xd y ∂ξ ∂η e−x −y −2ξ η = 1, R2
R2
and observing that limτ →∞ φτ∗ f is the constant function of value f (o).
We finally turn to the setting of an arbitrary lattice . We have a first-order differential operator Q j for every site j ∈ and we now take the symmetry generator Q to be the sum of all of these:
Quasi-Diffusion in a 3D Supersymmetric Hyperbolic Sigma Model
Q=
j∈
Qj =
485
x j ∂η j − y j ∂ξ j + ξ j ∂ x j + η j ∂ y j .
j∈
By the same argument as before, one sees that Dμ is Q j -invariant for all j and hence Q-invariant. There still exists H = j∈ (x 2j +y 2j +2ξ j η j ) and λ = j∈ (x j η j −y j ξ j )
with Qλ = H . Hence we can still localize the integral Dμ F for any Q-invariant function F by deforming with e−τ H and sending τ → ∞. Thus we arrive at the following result which, though valid for any choice of coordinate system, will be stated in terms of the horospherical coordinates t j , s j , ψ¯ j , ψ j used in the body of the paper. Proposition 2. For any Q-invariant, smooth and integrable function F of the lattice variables t j , s j , ψ¯ j , ψ j the integral of F localizes at the zero-field configuration t j = s j = ψ¯ j = ψ j = 0 (for all j ∈ ): Dμ F = F(o). (R2 )||
In particular, for the partition function (4.5) we have Dμ e−Aβ,ε = 1. Z (β, ε) = (R2 )||
References 1. Berezin, F.A.: Introduction to Superanalysis. Dordrecht: Reidel Publishing Co., 1987 2. Coppersmith, D., Diaconis, P.: Random walk with reinforcement. Unpublished manuscript, 1986 3. Disertori, M.: Density of states for GUE through supersymmetric approach. Rev. Math. Phys. 16, 1191–1225 (2004) 4. Dupré, T.: Localization transition in three dimensions: Monte Carlo simulation of a nonlinear sigma model. Phys. Rev. B 54, 12763–12774 (1996) 5. Drunk, W., Fuchs, D., Zirnbauer, M.R.: Migdal-Kadanoff renormalization of a nonlinear supervector model with hyperbolic symmetry. Ann. Physik 1, 134–150 (1992) 6. Efetov, K.B.: Supersymmetry in Disorder and Chaos. Cambridge: Cambridge University Press, 1997 7. Efetov, K.B.: Supersymmetry and theory of disordered metals. Adv. Phys. 32, 874 (1983) 8. Friedan, D.H.: Nonlinear models in 2 + epsilon dimensions. Ann. Phys. 163, 318–419 (1985) 9. Fyodorov, Y.V.: Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation. Nucl. Phys. B 621, 643–674 (2002) 10. Heinzner, P., Huckleberry, A., Zirnbauer, M.R.: Symmetry classes of disordered fermions. Commun. Math. Phys. 257, 725–771 (2005) 11. Littelmann, P., Sommers, H.-J., Zirnbauer, M.R.: Superbosonization of invariant random matrix ensembles. Commun. Math. Phys. 283, 343–395 (2008) 12. McKane, A.J., Stone, M.: Localization as an alternative to Goldstone theorem. Ann. Phys. 131, 36–55 (1981) 13. Merkl, F., Rolles, S.W.W.: Asymptotic behavior of edge-reinforced random walks. Ann. Prob. 35, 115–140 (2007) 14. Mirlin, A.D.: Statistics of energy levels and eigenfunctions in disordered systems. Phys. Rep. 326, 260–382 (2000) 15. Niedermaier, M., Seiler, E.: Structure of the space of ground states in systems with non-amenable symmetries. Commun. Math. Phys. 270, 373–443 (2007) 16. Duncan, A., Niedermaier, M., Seiler, E.: Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models. Nucl. Phys. B 720, 235–288 (2005) 17. Schwarz, A., Zaboronsky, O.: Supersymmetry and localization. Commun. Math. Phys. 183, 463–476 (1997) 18. Spencer, T., Zirnbauer, M.R.: Spontaneous symmetry breaking of a hyperbolic sigma model in three dimensions. Commun. Math. Phys. 252, 167–187 (2004)
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19. Schäfer, L., Wegner, F.: Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes. Z. Phys. B 38, 113–126 (1980) 20. Wegner, F.: The mobility edge problem: continuous symmetry and a conjecture. Z. Phys. B 35, 207– 210 (1979) 21. Zirnbauer, M.R.: Fourier analysis on a hyperbolic supermanifold with constant curvature. Commun. Math. Phys. 141, 503–522 (1991) Communicated by M. Salmhofer
Commun. Math. Phys. 300, 487–528 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1118-4
Communications in
Mathematical Physics
A Construction of Blow up Solutions for Co-rotational Wave Maps C˘at˘alin I. Cârstea Department of Mathematics, University of Rochester, RC Box 270138, Rochester, NY 14627, USA. E-mail: [email protected] Received: 5 October 2009 / Accepted: 3 April 2010 Published online: 3 September 2010 – © Springer-Verlag 2010
Abstract: The existence of co-rotational finite time blow up solutions to the wave map problem from R2+1 → N , where N is a surface of revolution with metric dρ 2 +g(ρ)2 dθ 2 , g an entire function, is proven. These are of the form u(t, r ) = Q(λ(t)t)+R(t, r ), where Q is a time independent solution of the co-rotational wave map equation −u tt + u rr + r −1 u r = r −2 g(u)g (u), λ(t) = t −1−ν , ν > 1/2 is arbitrary, and R is a term whose local energy goes to zero as t → 0. 1. Introduction In the following wave maps (see [5]) from R2+1 into a surface of revolution N (with some restrictions on the metric that will be made explicit below), which are also co-rotational, will be considered. The wave map equation reduces in this case to: 1 1 − u tt + u rr + u r = 2 f (u), r r
(1.1)
with r > 0 and where the right-hand side is related to the metric of N (see below). This equation will be shown to have blow up solutions (solutions for which the ||u|| H˙ 1 norm goes to infinity in finite time) with initial data (u, u t ) in H 1+δ × H δ , for some δ > 0. The energy ∞ 1 g(u)2 2 2 E(u) = r dr (1.2) (∂t u) + (∂r u) + 2 r2 0 is preserved and the problem is energy critical in the sense that the scaling u → u(λt, λr ) leaves E(u) invariant. If the local energy with respect to the origin is defined to be 1 g(u)2 Eloc (u) = r dr, (1.3) (∂t u)2 + (∂r u)2 + r2 r
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then it is known (see [6]) that the solution u will blow up at the origin, as t → 0, iff lim inf Eloc (u)(t) > 0. t↓0
(1.4)
In [7] Struwe has shown that for solutions u with C ∞ data that have blow up at t0 there exist sequences ri ↓ 0 and ti ↑ t0 such that ri /ti → 0 and u i (t, r ) = u(ti + ri t, ri x) → u ∞ (x), where u ∞ is a non-constant time independent solution of the wave map equation. This motivates the construction detailed in this paper which produces a solution of the wave map equation which inside the light cone r < t is of the form u(t, r ) = Q(λ(t)r ) + u e (t, r ) + (t, r ),
(1.5)
where Q is a finite energy, non-trivial stationary solution of the wave map equation (a harmonic map) and λ(t) = t −1−ν , ν > 1/2. The first term is the one for which lim inf Eloc (Q(λ(t)r ))(t) > 0. t↓0
(1.6)
The second term is “large”, but does not cancel the energy concentration of the first term. The last term is “small”. The proof of the main result (Theorem 2.2 below) follows very closely the work of Krieger, Schlag, and Tataru ([2]) in the particular case when the surface of revolution N is the sphere. Indeed, certain portions of this paper are nearly identical to the ones in [2]. Section 3 here corresponds to Sect. 3 in [2] and it deals with iteratively constructing corrections to u 0 = Q(λ(t)r ), which will form the u e term. The procedure is split into four steps which alternate constructions of additive corrections (by two different methods) with estimations of the errors made. Here and in Appendix A is where most of the original contribution of the paper is concentrated. One of the differences from [2] is in the spaces introduced in Subsect. 3.1 below. Though only slightly changed, the definitions given here should also be used to replace the ones in [2] in order to make some of the computations there meaningful. The computations of the errors corresponding to each of the succesive approximate solutions are also new, as the right hand side term of the wave map equation is more general here. Section 4 corresponds to Sect. 4 in [2]. In it an equation satisfied by is derived. Section 5 (Sect. 6 in [2]) deals with rewriting this equation as a transport-like equation for the generalized Fourier transform of corresponding to a self adjoint operator L, which is a conjugate of the linearization of the spatial part of the wave map equation. The term is then obtained in Sects. 6 and 7 (Sects. 7, 8, and 9 in [2]) by means of a contraction principle argument. Finally, the proof of Theorem 2.2 is finalized in Sect. 8. Appendix A corresponds mostly to Sect. 5 in [2] and it contains an analysis of the spectral theory of the operator L mentioned above. This is based on results by Gesztesy and Zinchenko ([1]) on the spectral theory of Schrödinger operators with certain singular potentials. It is due to the fact that the same expansions (see Proposition A.4) can be derived for the generalized Fourier basis of L as in the particular case of N = S 2 that Sects. 5–8 are essentially identical to their correspondents in [2]. The original contribution here lies mostly in the proof of Proposition A.4. Lemma A.2 is also new as it deals with establishing the properties of a certain convenient system of fundamental solutions for L. This was not necessary in [2] since there explicit formulas for these solutions are available. The results of Lemma A.2 are essential for the first step of the iterative procedure of Sect. 3.
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See also the Introduction to [2] for a discussion of the history of the problem and a more in depth analysis of the motivation for the method. Krieger, Schlag and Tataru have also applied the same method to the H 1 (R3 ) critical focusing semilinear wave equation in [4] and to the critical Yang–Mills problem in [3]. 2. Setup 2.1. The manifold. Let N be a compact surface of revolution, with Riemannian metric ds 2 = dρ 2 + g(ρ)2 dθ 2 .
(2.1)
If N is produced by rotating the graph of the function y = y(x),
y(0) = 0,
y(x M ) = 0,
(2.2)
2 2 2 around the x-axis, then ρ is the arclength on the graph of the function, dρ = d x + dy . 2 2 Also, y(ρ) = g(ρ), hence d x = dρ 1 − g (ρ) and |g (ρ)| < 1 for any ρ ∈ (0, ρ M ). In order for the graph of y = y(x) to generate a surface of revolution, it has to be true that dy/d x → ∞, as x → 0+ , and dy/d x → −∞, as x → x M − . Since dy = g (ρ)/ 1 − g (ρ)2 , (2.3) dx
it follows that g (ρ) → 1, as ρ → 0+ , and g (ρ) → −1, as ρ → ρ M − . It also has to be true that g is an odd function of ρ and of (ρ M − ρ). Therefore it can be extended to a smooth periodic function of period 2ρ M . Throughout this paper, the function g is assumed to have the folowing properties: i) ii) iii) iv)
g : R → R is entire; g is an odd function of ρ and of ρ M − ρ; |g (ρ)| < 1 for all ρ ∈ (0, ρ M ); g (0) = 1, g (ρ M ) = −1.
(ρ M − ρ)2 , Note then that g can be written as g(ρ) = ρG(ρ 2 ), or as g(ρ) = (ρ M −ρ)G are entire functions, G(0) = 1, G(0) where G and G = −1. Let f (ρ) = g(ρ)g (ρ). This function is also entire, odd, and can be written as are entire (ρ M − ρ)2 , where F and F f (ρ) = ρ F(ρ 2 ), or as f (ρ) = (ρ M − ρ) F functions, F(0) = 1, F(0) = 1. 2.2. The equation. Co-rotational wave maps from R2+1 into N are of the form (t, r, θ ) → (u(t, r ), θ ), where u satisfies the following equation: 1 f (u) − ∂t2 u + ∂r2 u + ∂r u = 2 . r r
(2.4)
The energy of u is
∞
E(u) = 0
and it is constant in time.
1 g(u)2 2 2 r dr, (∂t u) + (∂r u) + 2 r2
(2.5)
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2.3. The harmonic map. Note that for any stationary solution u of (2.4), the following quantity is independent of r : (r ∂r u)2 − g(u)2 = C.
(2.6)
If such a solution is to have finite energy, then it is necessary that C = 0. It follows then that either r ∂r u = g(u), or r ∂r u = −g(u). A stationary solution of Eq. (2.4) is called a harmonic map. As seen just above, harmonic maps with finite energy are solutions of one of two first order ODE and therefore can be specified uniquely by a choice of sign in r ∂r u = ±g(u) and the value they take at r = 1 (for example). Let Q be the solution of r ∂r Q = g(Q),
Q(1) = 1.
(2.7)
It is clear that limr →0+ Q(r ) = 0, and limr →∞ Q(r ) = ρ M . With the ansatz Q(r ) = r Q(r 2 ), Eq. (2.7) becomes
2r 2 ∂r 2 Q(r 2 ) = Q(r 2 ) G(r 2 Q(r 2 )2 ) − 1 ,
(2.8)
therefore Q must be an analytic function of r 2 , Q(0) > 0. Similarly, notice that with the change of variable l = 1/r , Eq. (2.7) can be written as M − Q)2 ), l∂l (ρ M − Q) = (ρ M − Q)G((ρ
(2.9)
2 ), where Q is and, proceeding as above, it follows that Q(r ) = ρ M − (1/r )Q(1/r analytic, Q(0) > 0. From Eqs. (2.7) and (2.4) it follows that
Q = −
1 g(Q) 1 − g (Q) , 2 r
(2.10)
so Q (r ) is decreasing. Also,
r 2 Q = g(Q) 1 + g (Q) , so r 2 Q (r ) is increasing. To summarize: Lemma 2.1. The chosen harmonic map has the following properties: i) Q (r ) is decreasing and r 2 Q (r ) is increasing; ii) Q(r ) = r Q(r 2 ), with Q a real-analytic function, Q(0) > 0; 2 ), with Q a real-analytic function, Q(0) > 0. iii) Q(r ) = ρ M − (1/r )Q(1/r
(2.11)
Construction of Blow up Solutions for Co-rotational Wave Maps
491
2.4. The theorem. Define the local energy of a solution u of (2.4) with respect to the origin and at time t to be 1 2 g(u)2 2 r dr. (2.12) (u t + u r ) + Eloc (u)(t) = 2r 2 r 1/2 be arbitrary and t0 > 0 be suficiently small. Define λ(t) = t −1−ν and fix a large integer N . Then there exists a function u e satisfying u e ∈ C ν+1/2− ({t0 > t > 0, |x| < t}), Eloc (u e )(t) (tλ(t))−2 | log t|2 as t → 0,
(2.13) (2.14)
and a solution u of (2.4) in [0, t0 ] which is of the form u(t, r ) = Q(λ(t)r ) + u e (t, r ) + (t, r ), 0 ≤ r ≤ t,
(2.15)
where decays at t = 0. More precisely, 1+ν− ν− (R2 ), t ∈ t N −1 Hloc (R2 ), Eloc ()(t) t N ∈ t N Hloc
as t → 0, (2.16)
with spatial norms that are uniformly controlled as t → 0. Also, u(0, t) = 0 for all 0 < t < t0 . The solution u(t, r ) extends as an H 1+ν− solution to all R2 . 3. Approximate Solutions Let λ(t) = t −1−ν , ν > 1/2, R = λ(t)r . In this section a sequence u k of approximate solutions of (2.7) will be constructed. For each of these the corresponding error is defined to be 1 1 2 2 (3.1) ek = −∂t + ∂r + ∂r u k − 2 f (u k ). r r The first element of the sequence is u 0 (t, r ) = Q(R). For a large enough N , u N − u 0 will be the u e of Theorem 2.2. To motivate the particular construction, suppose that the sought solution of (2.4) is of the form: u = u k + , with small. Then
1 1 −∂t2 + ∂r2 + ∂r − 2 f (u k ) ≈ −ek . r r
(3.2)
(3.3)
Two different approximations of this linearized equation will be used. The first assumes the time derivative to be unimportant and also approximates u k ≈ u 0 , replacing f (u k ) by f (u 0 ). The second one retains the time derivative, but assumes that u k ≈ u 0 (∞) = ρ M , replacing f (u k ) by 1, as would be the case if r ≈ t and t would be close to zero. Successive corrections vk = u k − u k−1 to the approximate solutions will be constructed using these two ideas alternatively, that is the vk ’s will be required to solve 1 1 0 ∂r2 + ∂r − 2 f (u 0 ) v2k+1 = −e2k (3.4) r r
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and
1 1 2 2 0 −∂t + ∂r + ∂r − 2 v2k+2 = −e2k+1 , r r
(3.5)
with zero Cauchy data at r = 0 and where ek0 is the “principal part” of ek , in a sense that will be detailed below. The conclusion of this section requires the introduction of certain spaces of functions on the light cone. It can be found stated in Eqs. (3.24)–(3.27). This section mirrors Sect. 3 of [2]. Step 3, in particular is virtually identical to the reference as (3.5) does not depend on the particular geometry of the surface of revolution. The main difference lies in error estimates of Steps 2 and 4. It is in the course of these two steps that the assumption that g is entire is necessary. Note that in the definitions of the spaces of functions in the following subsection three “b” parameters are used (b, b1 , b2 ), instead of one as in [2]. The definitions given in [2] should be replaced by the ones below. Certain other typos have been fixed here. 3.1. Some spaces. Before proceeding with the construction, a few spaces of functions need to be introduced. Let C0 = {(t, r ) : 0 ≤ r ≤ t, 0 < t < t0 }
(3.6)
be a truncated forward light cone on which the u k ’s will be defined. Definition 3.1. For i ∈ N let j (i) = i if ν is irrational, and j (i) = 2i 2 if ν is rational. Q is the algebra of continuous functions q : [0, 1] → R with the following properties: i) q is analytic in [0, 1) with even expansion at 0; ii) near a = 1 there is an absolutely convergent expansion of the form: ⎛ j (2i−1) ∞ 1 ⎝(1 − a)(2i−1)ν+ 2 q = q0 (a) + q2i−1, j (a)(log(1 − a)) j i=1
+ (1 − a)2iν+1
j=0 j (2i)
⎞
q2i, j (a)(log(1 − a)) j ⎠
(3.7)
j=0
with analytic coefficients q0 , qi, j . Definition 3.2. With j (i) as above, Q is the space of functions q : [0, 1] → R with the following properties: i) q is analytic in [0, 1) with even expansion at 0; ii) near a = 1 there is an absolutely convergent expansion of the form: ⎛ j (2i−1) ∞ 1 ⎝(1 − a)(2i−1)ν− 2 q = q0 (a) + q2i−1, j (a)(log(1 − a)) j i=1
+ (1 − a)2iν+1
j=0 j (2i)
⎞
q2i, j (a)(log(1 − a)) j ⎠
j=0
with analytic coefficients q0 , qi, j .
(3.8)
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493
Definition 3.3. i) Qm is the sub-algebra of Q defined by the requirement that qi j (1) = 0 if i ≥ 2m + 1 and i is odd; ii) Qm is the sub-space of Q defined by the requirement that qi j (1) = 0 if i ≥ 2m + 1 and i is odd. Lemma 3.4. i) Q ⊂ Q and Qm ⊂ Qm ; ii) Qm ⊂ Qm+1 , Qm ⊂ Qm+1 . Proof. Note that the only difference between the definitions of the Q spaces and the Q spaces is that a power (1 − a)1/2 appears in the first term inside the bracket in (3.7), while in the same place in (3.8) there is a power (1 − a)−1/2 . i) follows from: (1 − a)1/2 = (1 − a)(1 − a)−1/2 .
(3.9)
ii) is obvious from Definition 3.3. Definition 3.5. S m R k (log R)l is the class of functions v : [0, ∞) → R with the following properties: 2 j for small R; i) v vanishes of order m at R = 0, and v(R) = R m ∞ j=0 c j R ii) v has a convergent expansion near R = ∞ of the form: v(R) = ci j R k−2i (log R) j . (3.10) 0≤ j≤l+i
Let B, B1 , B2 be positive constants to be specified shortly. Definition 3.6. S m R k (log R)l , Qn is the class of analytic functions v : [0, ∞) × [0, 1] × [0, B] × [0, B1 ] × [0, B2 ] → R with the following properties: i) v is analytic as a function of R, b, b1 , b2 , v : [0, ∞) × [0, B] × [0, B1 ] × [0, B2 ] → Qn ;
(3.11)
ii) v vanishes of order m at R = 0 and has a convergent expansion, v(R, a, b, b1 , b2 ) = R m
∞
c j (a, b, b1 , b2 )R 2 j ;
(3.12)
j=0
iii) v has a convergent expansion near R = ∞ of the form, v(R, ·, b, b1 , b2 ) = ci j (·, b, b1 , b2 )R k−2i (log R) j,
(3.13)
0≤ j≤l+i
where the coefficients ci j : [0, B] × [0, B1 ] × [0, B2 ] → Qn are analytic with respect to b, b1 , b2 . S m R k (log R)l , Qn is defined similarly. Here is a list of elementary, but useful, properties of these spaces: Lemma 3.7. i) S m+2 R k (log R)l , Qn ⊂ S m R k (log R)l , Qn ; ii) S m R k (log R)l , Qn ⊂ S m R k (log R)l+1 , Qn ;
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C. I. Cârstea
iii) S m R k (log R)l , Qn ⊂ S m R k+2 (log R)l−1 , Qn ; iv) S m R k (log R)l , Qn ⊂ S m R k (log R)l , Qn . All but the last one are also properties of S k R l (log R)m , Qn . With the notations R = λ(t)r , a = r/t = (tλ)−1 R, b = (tλ)−2 [log(2 + R 2 )]2 , b1 = (tλ)−2 [log(2 + R 2 )], b2 = (tλ)−2 , if (t, r ) ∈ C0 , then there are positive constants B, B1 , B2 such that b ∈ [0, B], b1 ∈ [0, B1 ], and b2 ∈ [0, B2 ]. Definition 3.8. I S m R k (log R)l , Qn is the class of analytic functions w defined on the cone C0 which can be represented as
(3.14) w(t, r ) = v(R, a, b, b1 , b2 ), v ∈ S m R k (log R)l , Qn . The definition of I S m R k (log R)l , Qn is similar. Note that the representations in the above definition are not at all unique. 3.2. Two useful lemmas. The following results will be useful throughout this section. Lemma 3.9. f (2k) (Q(R)) ∈ I S 1 (R −1 ) and f (2k+1) (Q(R)) ∈ I S 0 (1). Proof. f (2k) (ρ) has an odd expansion in ρ and also in (ρ M − ρ). Plugging in Q the first half of the result follows from Lemma 2.1. The case of f (2k+1) is similar, but with even expansions.
Lemma 3.10. If z∈
1 I S 1 (R(log R), Q) , (tλ)2
(3.15)
then f (2k) (Q(R) + z(R)) ∈
1 I S 1 (R(log R), Q) (tλ)2
(3.16)
and f (2k+1) (Q(R) + z(R)) ∈ I S 0 (1, Q).
(3.17)
Proof. First expand f (2k) (Q + z) =
1 f 2k+l (Q)z l . l!
(3.18)
l≥0
Note that
1 2 2 2 R I S (log R) , Q (tλ)4
1 1 2 2 0 2 2 0 R R bI S (log R) , Q + b I S (log R) , Q ⊂ 1 (tλ)2 (tλ)2
1 b2 I S 2 R 2 (log R)0 , Q + (tλ)2
1 ⊂ I S 2 R 2 (log R)0 , Q ⊂ a 2 I S 0 (1, Q). 2 (tλ)
z2 ∈
(3.19)
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495
Now f (2k+2m+1) (Q)z ∈
1 I S 1 (R(log R), Q) (tλ)2
(3.20)
and
1 1 3 2 R(log R) I S , Q ⊂ I S 1 (R(log R), Q). 4 (tλ) (tλ)2 Combining Eqs. (3.18)–(3.21) yields (3.16). To prove (3.17) proceed similarly by expanding 1 f (2k+1) (Q + z) = f 2k+l+1 (Q)z l . l! f (2k+2m) (Q)z 2 ∈
(3.21)
(3.22)
l≥0
Similar computations to the ones above give the result.
3.3. Step 0. As is mentioned above, the first element of the squence of approximate solutions is u 0 = Q(R). The corresponding error is then: 1 1 e0 = −∂t2 + ∂r2 + ∂r u 0 − 2 f (u 0 ) r r = −∂t2 Q(λ(t)r ) = −∂t [r λ (t)Q (λ(t)r )] = −r λ (t)Q (R) − r 2 λ (t)2 Q (R) 1
= − 2 (1 + ν)(2 + ν)R Q (R) + (1 + ν)2 R 2 Q (R) . t Therefore, t 2 e0 ∈ I S 1 R −1 .
(3.23)
3.4. Induction. The approximate solutions will be constructed by adding successive corrections to u 0 . With the notation vk = u k − u k−1 , it will be inductively shown that
1 3 2k−1 R(log R) , (3.24) v2k−1 ∈ I S , Q k−1 (tλ)2k
1 1 2k−1 R(log R) , (3.25) t 2 e2k−1 ∈ I S , Q k−1 (tλ)2k
1 v2k ∈ (3.26) I S 3 R 3 (log R)2k−1 , Qk , 2k+2 (tλ)
1 1 −1 2k I S R t 2 e2k ∈ (log R) , Q k (tλ)2k
+ bI S 1 R(log R)2k−1 , Qk
+ b1 I S 1 R(log R)2k−1 , Qk
+ b2 I S 1 R(log R)2k−1 , Qk . (3.27) The exact method for constructing the vk ’s will be described below. In the following, for a fixed k, it will be assumed that the above hold for k and for any smaller natural number.
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3.5. Step 1. It is assumed that t 2 e2k−2
⎡
1 ⎣ I S 1 R −1 (log R)2k−2 , Qk−1 ∈ (tλ)2k−2 ⎤
β I S 1 R(log R)2k−3 , Qk−1 ⎦ . +
(3.28)
β=b,b1 ,b2 0 Choose the “principal part” e2k−2 by setting b = b1 = b2 = 0 in a representation of e2k−2 (see Definition 3.8). Then
1 0 1 −1 2k−2 R , (3.29) t 2 e2k−2 ∈ I S (log R) , Q k−1 (tλ)2k−2
and
1 0 t 2 e2k−2 = t 2 e2k−2 − e2k−2
1 1 −1 2k−2 ∈ I S R β (log R) , Q k−1 (tλ)2k−2 β=b,b1 ,b2
+ I S 1 R(log R)2k−3 , Qk−1 .
Replacing the b, b1 , b2 by their definitions, it follows that
1 1 −1 1 t 2 e2k−2 I S R (log R)2k , Qk−1 ∈ 2k (tλ)
+ I S 1 R(log R)2k−1 , Qk−1 ,
(3.30)
(3.31)
so 1 t 2 e2k−2 ∈
1 I S 1 R(log R)2k−1 , Qk−1 . 2k (tλ)
(3.32)
1 f (u 0 ) ∂R − . R R2
(3.33)
This will be useful later. Let L = ∂ R2 +
Keeping a, b, b1 , b2 fixed, define v2k−1 to be the solution of 0 , (tλ)2 Lv2k−1 = −t 2 e2k−2
with vanishing Cauchy data at R = 0.
(3.34)
Lemma 3.11. The solution of Lv = ϕ ∈ S 1 R −1 (log R)2k−2 , with v(0) = v (0) = 0, has the regularity
(3.35) v ∈ S 3 R(log R)2k−1 .
Construction of Blow up Solutions for Co-rotational Wave Maps
497
Proof. Behavior at R ∼ 0. Close to zero, ϕ(R) =
∞
ϕk R 2k+1 ,
f (u 0 (R)) = 1 +
k=0
∞
f k R 2k .
(3.36)
k=1
Make the ansatz v(R) =
∞
Vk R 2k+1 .
(3.37)
k=1
Since ∂ R2 + R −1 ∂ R = R −1 ∂ R R∂ R , it follows that the Vk need to satisfy ((2k + 1)2 − 1)Vk = φk−1 +
k−1
fl Vk−l , ∀k ≥ 1.
(3.38)
l=1
This system can be solved to find Vk such that the sum in (3.37) converges absolutely in a neighborhood of zero. Such a v will vanish of order 3 at zero. √ √ Behavior at R ∼ ∞. Notice that v has to satisfy L Rv = − Rϕ, where L = −∂r2 +
3 + V (r ); 4r 2
V (r ) = −
1 1 − f (Q(r )) . 2 r
(3.39)
Using the fundamental system of L from Lemma A.2, v can be written as: 1 1 v(R) = cφ √ φ0 (R) + cθ √ θ0 (R) R R R √ 1 1 φ0 (S) Sϕ(S) d S + √ θ0 (R) 2 R 1 R √ 1 1 θ0 (S) Sϕ(S) d S. − √ φ0 (R) 2 R 1
(3.40)
By Lemma A.2, R −1/2 φ0 (R) ∈ S(R −1 ), R −1/2 θ0 (R) ∈ S(R −1 ),
R 1/2 φ0 (R)ϕ(R) ∈ S R −1 (log R)2k−2 ,
R 1/2 θ0 (R)ϕ(R) ∈ S R(log R)2k−2 .
(3.41) (3.42) (3.43) (3.44)
Therefore
R
1
1
R
√ φ0 (S) Sϕ(S) d S ∈ S (log R)2k−1
(3.45)
√ θ0 (S) Sϕ(S) d S ∈ S R 2 (log R)2k−2 + S (log R)2k−1
⊂ S R 2 (log R)2k−2 .
(3.46)
498
C. I. Cârstea
Since v is sought such that it has zero Cauchy data, then cφ = cθ = 0. Putting all these together, it follows that
v ∈ S R(log R)2k−1 .
(3.47)
An immediate consequence of the previous lemma is that v2k−1 ∈
1 I S 3 R(log R)2k−1 , Qk−1 . 2k (tλ)
(3.48)
3.6. Step 2. The error corresponding to v2k−1 is: 1 e2k−1 = e2k−2 + N2k−1 (v2k−1 ) + E t v2k−1 + E a v2k−1 ,
(3.49)
where N2k−1 (v) =
1 f (u 0 )v − f (u 2k−2 + v2k−1 ) − f (u 2k−1 ) , 2 r
(3.50)
E t v2k−1 designates the terms in ∂t2 v2k−1 with no derivatives on the a variable, and E a v2k−1 designates the terms in −∂t2 + ∂r2 + r1 ∂r v2k−1 with at least one derivative on the a variable. 3.6.1. The N2k−1 (v2k−1 ) term. First write t 2 N2k−1 (v2k−1 ) = −a −2 ( f (u 2k−2 + v2k−1 ) − f (u 2k−2 ) − f (u 2k−2 )v2k−1 + f (u 2k−2 ) − f (u 0 ) v2k−1 = −a −2 [I + I I ].
(3.51)
For l ≤ k,
1 3 2l−1 R(log R) I S , Q l−1 (tλ)2l 1 l−1 3 ⊂ b I S (R(log R), Ql−1 ) (tλ)2 1 l−2 b b2 I S 3 (R(log R), Ql−1 ) + (tλ)2 +····················· 1 l−1 3 b I S (R(log R), Ql−1 ) + (tλ)2 2 1 I S 3 (R(log R), Ql−1 ), ⊂ (tλ)2
v2l−1 ∈
(3.52)
Construction of Blow up Solutions for Co-rotational Wave Maps
499
and for l < k,
1 I S 3 R 3 (log R)2l−1 , Ql 2l+2 (tλ)
1 1 2l−1 R(log R) ⊂ a2 I S , Q l (tλ)2l 1 ⊂ I S 1 (R(log R), Ql ). (tλ)2
v2l ∈
(3.53)
Therefore, for any l < k, (u 2l+1 − u 0 ), (u 2l − u 0 ) ∈
1 I S 1 (R(log R), Qk−1 ). (tλ)2
(3.54)
Returning to (3.51), 1 l−2 f (l) (u 0 + (u 2k−2 − u 0 ))v2k−1 . l!
2 I = v2k−1
(3.55)
l≥2
1 2 1 0 Note that, since v2k−1 ∈ (tλ) 2 I S (R(log R), Q), v2k−1 ∈ I S (1, Q). For the even terms in the expansion above, using Lemma 3.10,
f (2m) (u 2k−2 )(z 2 )m−1 ∈
1 I S 1 (R(log R), Q) , (tλ)2
(3.56)
and for the odd ones f (2m+1) (u 2k−2 )z(z 2 )m−1 ∈
1 I S 1 (R(log R), Q) . (tλ)2
(3.57)
Therefore
1 7 3 4k−1 R I S (log R) , Q k−1 (tλ)4k+2
1 7 3 2k−1 R ⊂ I S (log R) , Q k−1 (tλ)2k+2
1 5 2k−1 R(log R) ⊂ a2 I S , Q k−1 . (tλ)2k
I ∈
(3.58)
So a −2 I ∈
1 5 2k−1 R(log R) I S , Q k−1 . (tλ)2k
(3.59)
Now I I = v2k−1
l≥2
1 f (l) (u 0 )(u 2k−2 − u 0 )l−1 . (l − 1)!
(3.60)
From computations above it follows that (u 2k−2 − u 0 )2 ∈ a 2 I S 0 (1, Q).
(3.61)
500
C. I. Cârstea
Then, using Lemma 3.9, 1 I S 2 (log R, Q) (tλ)2 1 ⊂ I S 2 (R 2 ) ⊂ a 2 I S 0 (1, Q) (tλ)2
f (2m) (u 0 )(u 2k−2 − u 0 )2m−1 ∈
(3.62)
and f (2m+1) (u 0 )(u 2k−2 − u 0 )2m ∈ a 2 I S 0 (1, Q).
(3.63)
1 3 2k−1 R(log R) I S , Q k−1 . (tλ)2k
(3.64)
Therefore a −2 I I ⊂
From (3.59) and (3.64) it follows that t 2 N2k−1 (v2k−1 ) ∈
1 3 2k−1 R(log R) I S , Q k−1 . (tλ)2k
(3.65)
3.6.2. The E t v2k−1 term. Recall that E t v2k−1 = ∂t2 v2k−1 with a fixed. Note that there 0 was obtained by setting these to zero. is no dependence on b, b1 , b2 in v2k−1 since e2k−2 v2k−1 can be written as 1 w(R, a) (tλ)2k
(3.66)
1 1 1 1+ν + (· · · ), = 2kν w+ R∂ R w − t (tλ)2k (tλ)2k t
(3.67)
v2k−1 = with w ∈ S 3 R(log R)2k−1 , Qk−1 ; ∂t v2k−1
where the terms left out are those that involve ∂a ; 1 1 1 1 w − 2kν 2 w 2 2k t (tλ) t (tλ)2k 1 1+ν 1 1+ν 1 + 2 R + 2(2kν) R ∂R w − ∂R w t (tλ)2k t t (tλ)2k 1 1+ν 2 + R∂ R∂ w − + (· · · ). R R (tλ)2k t
∂t2 v2k−1 = (2kν)2
Since
w, R∂ R w, (R∂ R )2 w ∈ S 3 R(log R)2k−1 , Qk−1 ,
(3.68)
(3.69)
it follows that t 2 E t v2k−1 ∈
1 3 2k−1 R(log R) I S , Q k−1 . (tλ)2k
(3.70)
Construction of Blow up Solutions for Co-rotational Wave Maps
501
3.6.3. The E a v2k−1 term. Using the same notation as above, remembering that there is no dependence on b, b1 , b2 in v2k−1 , and omitting to write explicitly the terms that will not become part of E a v2k−1 , 1 1 1 ∂r v2k−1 = wa 2 a −1 + (· · · ), (3.71) 2k r (tλ) t 1 1 ∂r2 v2k−1 = ∂ + λw w r a R (tλ)2k t 1 1 2λ (3.72) wa R + (· · · ), = waa + (tλ)2k t 2 t a 1 1 1 1 1+ν w + + − ∂t2 v2k−1 = ∂t 2kν w Rw R − a 2k 2k 2k (tλ) t (tλ) t (tλ) t a 2 2 1 1 2 a 1 wa − + = 2kν wa 2 a + waa − (tλ)2k t t (tλ)2k t (tλ)2k t a 1 1+ν + 2Rwa R − − + (· · · ). (3.73) (tλ)2k t t Putting these together t 2 E a v2k−1 =
1
(1 − a 2 )waa + [2a(2kν − 1) + a −1 ]wa (tλ)2k −2R[(1 + ν)a − a −1 ]wa R .
(3.74)
Since a∂a , a −1 ∂a , (1 − a 2 )∂a2 : Qk−1 → Qk−1 ,
(3.75)
it follows that t 2 E a v2k−1 ∈
1 3 2k−1 R(log R) I S , Q k−1 . (tλ)2k
To conclude, the results (3.32), (3.65), (3.70), and (3.76) imply that
1 1 2k−1 t 2 e2k−1 ∈ R(log R) . I S , Q k−1 (tλ)2k
(3.76)
(3.77)
3.7. Step 3. Let t 2 f 2k−1 =
2k−1 2k−1 R 1 j q (a)(log R) = aq j (a)(log R) j , (3.78) j 2k 2k−1 (tλ) (tλ) j=0
j=0
be the sum of the leading terms of the expansion of e2k−1 at R = ∞, with b = b1 = b2 = 0. By definition, q j ∈ Qk−1 for all j. Define w2k to be a solution of the equation 1 1 (3.79) t 2 −∂t2 + ∂r2 + ∂r − 2 w2k = −t 2 f 2k−1 . r r
502
C. I. Cârstea
Making the ansatz w2k =
2k−1 j 1 W2k (a)(log R) j , 2k−1 (tλ)
(3.80)
j=0
plugging into (3.79), and matching the corresponding powers of log R, it follows that j the W2k have to satisfy the equations 1 1 1 j 2 2 2 W (a) t −∂t + ∂r + ∂r − 2 r r (tλ)2k−1 2k 1 =− (aq j (a) + F j (a)), (3.81) (tλ)2k−1 j
where, with the convention that W2k = 0 when j ≥ 2k,
j+1 j+1 F j (a) = ( j + 1) (1 + ν)(2ν(2k − 1) − 1)W2k + 2(a −1 − (1 + ν)a)∂a W2k
j+2 + ( j + 1) ( j + 2)a −2 − j (1 + ν)2 w2k . (3.82) Conjugating by (tλ)−(2k−1) , the system of Eq. (3.81) becomes (2k − 1)ν 2 1 1 j 2 2 t − ∂t + + ∂r + ∂r − 2 W2k (a) t r r = −aq j (a) − F j (a).
(3.83)
With the notation L β = (1 − a 2 )∂a2 + (a −1 + 2aβ − 2a)∂a + (−β 2 + β − a −2 ), writing (3.83) in terms of derivatives in a yields: j L (2k−1)ν W2k = − aq j (a) + F j (a) .
(3.84)
(3.85)
Adding the requirement that the Cauchy data at a = 0 for this system is zero, the solutions will satisfy j
W2k ∈ a 3 Qk ,
j = 0, 2k − 1.
(3.86)
See [2] for a proof of this fact. The w2k constructed so far cannot be used as v2k as it is singular at zero. Instead, define j 2k−1 j 1 1 2 log(1 + R ) v2k = W2k (a) (tλ)2k−1 2 j=0
=
1 (tλ)2k+2
2k−1 j=0
a −3 W2k (a)R 3 j
1 log(1 + R 2 ) 2
j .
(3.87)
Then clearly v2k ∈
1 3 3 2k−1 R . I S (log R) , Q k (tλ)2k+2
(3.88)
Construction of Blow up Solutions for Co-rotational Wave Maps
503
3.8. Step 4. Define j 2k−1 R 1 2 log(1 + R q (a) ) j (tλ)2k 2 j=0
1 1 2k−1 R(log R) ∈ I S , Q k−1 . (tλ)2k
0 t 2 e2k−1 =
(3.89)
The error corresponding to v2k is 0 ) t 2 e2k = t 2 (e2k−1 − e2k−1 1 1 2 0 2 2 + t e2k−1 + −∂t + ∂r + ∂r − 2 v2k + t 2 N2k (v2k ), r r
(3.90)
where N2k (v) =
v 1 f (u 2k−1 + v) − f (u 2k−1 ) . − r2 r2
(3.91)
0 3.8.1. The first term of (3.90). Both t 2 e2k−1 and t 2 e2k−1 have the same leading order in their expansions at R = ∞. Therefore
1 0 1 −1 2k t 2 (e2k−1 − e2k−1 R (3.92) )∈ I S (log R) , Q k−1 . (tλ)2k
Suppose
w ∈ I S 1 R −1 (log R)2k , Qk−1 .
(3.93)
This can be written as w = (1 − a 2 )w +
R2 w. (tλ)2
(3.94)
The first term satisfies
(1 − a 2 )w ∈ I S 1 R −1 (log R)2k , Qk−1 .
(3.95)
In the case of the second term,
R2 1 3 2k R(log R) w ∈ I S , Q k−1 (tλ)2 (tλ)2
⊂ b1 I S 3 R(log R)2k−1 , Qk−1 + b2 I S 3 R, Qk−1 .
(3.96)
0 ), it follows that Applying this to t 2 (e2k−1 − e2k−1 ⎡
1 ⎣ 1 −1 0 t 2 (e2k−1 − e2k−1 )∈ I S R (log R)2k , Qk−1 2k (tλ)
+
β=b,b1 ,b2
β I S1
⎤
R(log R)2k−1 , Qk−1 ⎦ .
(3.97)
504
C. I. Cârstea
3.8.2. The second term of (3.90). The reason this term is not zero is the replacement of log R by 21 log(1 + R 2 ) made above. The second term of (3.90) consists of a sum of expressions of the type 2k−1
1 j −2 0 −2 2 j−1 a W 2k I S (R )(log(1 + R )) (tλ)2k−1 j=0 + I S 0 (R −2 )(log(1 + R 2 )) j−2 j + a −1 ∂a W2k I S 0 (R −2 )(log(1 + R 2 )) j−1 .
(3.98)
Using (3.86) it follows, using also the argument from Eqs. (3.93)–(3.96) as well as basic properties of the I S spaces, that 1 1 0 t 2 e2k−1 + −∂t2 + ∂r2 + ∂r − 2 v2k r r
1 1 −1 2k−2 R I S (log R) , Q ∈ k (tλ)2k ⎤ ⎡
1 ⎣ 1 β I S 1 R(log R)2k−1 , Qk ⎦ . (3.99) ⊂ I S R −1 (log R)2k , Qk + (tλ)2k β=b,b1 ,b2
3.8.3. The third term of (3.90). Write first − t 2 N2k (v2k ) = a −2 f (u 2k−1 + v2k ) − f (u 2k−1 ) − f (u 2k−1 )v2k + f (u 2k−1 ) − f (u 0 ) v2k + f (u 0 ) − 1 v2k = a −2 [I + I I + I I I ].
(3.100)
Now 2 I = v2k
1 l−2 f (l) (u 2k−1 )v2k . l!
(3.101)
l≥2
Remembering the computation (3.53), v2k ∈
1 2 I S 1 (R(log R), Q) , v2k ∈ I S 0 (1, Q). (tλ)2
(3.102)
By Lemma 3.10 2m−2 f (2m) (u 2k−1 )v2k ∈
1 I S 1 (R(log R), Q) (tλ)2
(3.103)
and 2m−2 f (2m+1) (u 2k−1 )v2k v2k ∈
1 I S 1 (R(log R), Q). (tλ)2
(3.104)
Construction of Blow up Solutions for Co-rotational Wave Maps
505
Therefore
1 1 1 7 7 4k−1 R I S I S (log R) , Q R), Q) ⊂ (R(log k (tλ)2 (tλ)4k+6
1 ⊂ β I S 7 R 7 (log R)2k−1 , Qk (tλ)2k+6 β=b,b1 ,b2
1 1 2k−1 ⊂ a6 R(log R) (3.105) β I S , Q k . (tλ)2k
2 I ∈ v2k
β=b,b1 ,b2
The second term in (3.100) can be written as I I = v2k
l≥2
1 f (l) (u 0 )(u 2k−1 − u 0 )l−1 . (l − 1)!
(3.106)
Recall that u 2k−2 − u 0 ∈
1 I S 1 (R(log R), Q) , (u 2k−2 − u 0 )2 ∈ I S 0 (1, Q). (tλ)2
(3.107)
Using Lemma 3.9, 1 I S 2 (log R, Q) (tλ)2 β I S 2 (R 2 ) ⊂ a 2 β I S 0 (1, Q)
f (2m) (u 0 )(u 2k−2 − u 0 )2m−1 ∈ ⊂
1 (tλ)2
β=b,b1 ,b2
(3.108)
β=b,b1 ,b2
and f (2m+1) (u 0 )(u 2k−2 − u 0 )2m ∈ a 2
β I S 0 (1, Q).
(3.109)
β=b,b1 ,b2
Then II ∈
1 (tλ)2k+2
1 ⊂ a2 (tλ)2k
β=b,b1 ,b2
β I S 3 R 3 (log R)2k−1 , Qk
β I S 1 R(log R)2k−1 , Qk .
(3.110)
β=b,b1 ,b2
The last term in (3.100) is I I I = v2k f (u 0 ) − 1 ∈ v2k I S 2 R −2 ,
(3.111)
therefore
1 5 2k−1 R(log R) I S , Q k (tλ)2k+2
1 ⊂ a2 I S 3 R −1 (log R)2k , Qk . 2k (tλ)
III ∈
(3.112)
506
C. I. Cârstea
Putting together the results of Eqs. (3.97), (3.99), (3.105), (3.110), and (3.112), it follows that
1 1 −1 2k 1 2k−1 t 2 e2k ∈ I S R + bI S R(log R) (log R) , Q , Q k k (tλ)2k
+ b1 I S 1 R(log R)2k−1 , Qk + b2 I S 1 R(log R)2k−1 , Qk . (3.113) By induction, (3.24), (3.25), (3.26), and (3.27) are now proved for any k. 4. The Perturbed Equation For a fixed k define (t, r ) to be such that u(t, r ) = u 2k−1 (t, r ) + (t, r ),
(4.1)
where u is the solution of (2.4) that is being constructed. Then needs to solve the following equation: 1 f (u 0 ) = −e2k−1 − N2k−1 (), − ∂t2 + ∂r2 + ∂r − r r2
(4.2)
1 f (u 0 ) − f (u 2k−1 + ) − f (u 2k−1 ) . r2
(4.3)
where N2k−1 () =
If the time variable is replaced by τ = ν1 t −ν , the space variable by R = λ(t)r , and with the notation v(τ, R) = (t, λ−1 R), then (4.2) becomes 2 ! λτ λτ λτ 1 f (Q(R)) v R∂ R + ∂τ + R∂ R − ∂τ + v + ∂ R2 + ∂ R − λ λ λ R R2 1 = − 2 N2k−1 () + e2k−1 . (4.4) λ After making the further change of function ˜ (τ, R) = R 1/2 v(τ, R), (4.2) becomes 2 ! λτ λτ 1 λτ 2 1 R∂ R + − ∂τ + + ∂τ ˜ − L˜ λ 4 λ 2 λ
(4.5) = −λ−2 R 1/2 N2k−1 (R −1/2 ˜ ) + e2k−1 , where L = −∂ R2 +
3 + V (R), 4R 2
V (R) = −
1 1 − f (Q(R)) . 2 R
(4.6)
This last change of function has the benefit that it produces L, which is a self-adjoint operator on L 2 (R+ , d R).
Construction of Blow up Solutions for Co-rotational Wave Maps
507
5. The Transference Identity The plan to deal with (4.5) is to expand ˜ in terms of the generalized Fourier basis φ(R, ξ ) of the operator L (see Theorem A.3): ∞ ˜ (τ, R) = x(τ, ξ )φ(R, ξ )ρ(ξ ) dξ. (5.1) 0
The coefficients x(τ, ξ ) would then hopefully satisfy a transport equation. However, R∂ R is not diagonal in this Fourier basis. To deal with this, R∂ R will be replaced by 2ξ ∂ξ and the error will be treated as a perturbation. This section follows closely Sect. 6 of [2], to the point of being identical. This is due to the fact that the estimates of Appendix A are identical to the ones in Sect. 5 of the reference. The main result of the section is Proposition 5.2, whose proof is omitted as it is identical to the proof of Proposition 6.2 in [2]. Let the operator K be defined by1 u + K" u, R∂ R u = −2ξ ∂ξ "
(5.2)
where " f = F f is the distorted Fourier transform defined in Theorem A.3. Using the definitions for this Fourier transform and its inverse, K can be written as # ∞ $ K f (η) = f (ξ )R∂ R φ(R, ξ )ρ(ξ ) dξ, φ(R, η) L 2R
0
# + 0
∞
$
2ξ ∂ξ f (ξ )φ(R, ξ )ρ(ξ ) dξ, φ(R, η)
L 2R
.
Integrating by parts with respect to ξ , # ∞ $ K f (η) = f (ξ ) R∂ R − 2ξ ∂ξ φ(R, ξ )ρ(ξ ) dξ, φ(R, η) 0
−2 1 +
ηρ (η) ρ(η)
f (η).
(5.3)
L 2R
(5.4)
The scalar product is interpreted in the principal value sense with f ∈ C0∞ (0, ∞). A priori K : C0∞ (0, ∞) → C ∞ (0, ∞), therefore there is a distribution valued function η → K (η, ξ ) such that ∞ K f (η) = k(η, ξ ) f (ξ ) dξ.
(5.5)
(5.6)
0
Theorem 5.1. The operator K can be written as 3 ηρ (η) + δ(ξ − η) + K0 , K=− 2 ρ(η) 1 This is what is referred to as a “transference identity”.
(5.7)
508
C. I. Cârstea
where the operator K0 has a kernel K 0 (η, ξ ) of the form (in the principal value sense): K 0 (η, ξ ) =
ρ(ξ ) F(ξ, η), ξ −η
(5.8)
with a symmetric function F(ξ, η) of class C 2 in (0, η) × (0, η) satisfying the bounds % ξ +η ξ +η ≤1 , (5.9) |F(ξ, η)| (ξ + η)−3/2 (1 + |ξ 1/2 − η1/2 |)−N ξ + η ≥ 1 % 1 ξ +η ≤1 , (5.10) |∂ξ F(ξ, η)| + |∂η F(ξ, η)| (ξ + η)−2 (1 + |ξ 1/2 − η1/2 |)−N ξ + η ≥ 1 % ξ +η ≤1 | log(ξ + η)|3 j sup |∂ξ ∂ηk F(ξ, η)| , (5.11) −5/2 (1 + |ξ 1/2 − η1/2 |)−N ξ + η ≥ 1 (ξ + η) j+k=2 where N is an arbitrary large integer. Proof. The off-diagonal behavior of K is addressed first. Let f ∈ C0∞ (0, ∞). Then ∞ u(R) = f (ξ )[R∂ R − 2ξ ∂ξ ]φ(R, ξ )ρ(ξ ) dξ (5.12) 0
R 3/2
at 0 and like a Schwartz function at infinity. The second factor in (5.4), behaves like φ(R, η), decays like R 3/2 at zero, but at infinity is bounded, with bounded derivatives. Using integration by parts: ηK f (η) = u, Lφ(R, η) L 2 = Lu, φ(R, η) L 2 . R
(5.13)
R
Moreover, ∞ ∞ Lu = f (ξ )[L, R∂ R ]φ(R, ξ )ρ(ξ ) dξ + f (ξ )(R∂ R − 2ξ ∂ξ )ξ φ(R, ξ )ρ(ξ ) dξ 0 ∞ 0 ∞ f (ξ )[L, R∂ R ]φ(R, ξ )ρ(ξ ) dξ + ξ f (ξ )(R∂ R − 2ξ ∂ξ )φ(R, ξ )ρ(ξ ) dξ = 0 0 ∞ ξ f (ξ )φ(R, ξ )ρ(ξ ) dξ, (5.14) −2 0
with the commutator [L, R∂ R ] = 2L − 2 V (R) + RV (R) = 2L + W (R). Thus
Lu = 0
(5.15)
∞ f (ξ )W (R)φ(R, ξ )ρ(ξ ) dξ + ξ f (ξ )(R∂ R − 2ξ ∂ξ )φ(R, ξ )ρ(ξ ) dξ.
∞
0
(5.16) Hence
#
∞
ηK f (η) − K(ξ f )(η) = 0
$ f (ξ )W (R)φ(R, ξ )ρ(ξ ) dξ, φ(R, η)
L 2R
. (5.17)
Construction of Blow up Solutions for Co-rotational Wave Maps
509
Changing the order of integration on the right hand side yields: (η − ξ )K (η, ξ ) = ρ(ξ ) W (R)φ(R, ξ ), φ(R, η) L 2 .
(5.18)
R
This gives the representation (5.8) when η = ξ , with F(ξ, η) = W (R)φ(R, ξ ), φ(R, η) L 2 .
(5.19)
R
It remains to study its size and regularity. By Proposition A.4, sup |φ(R, ξ )| < ξ >−3/4 ,
(5.20)
R≥0
|R∂ R φ(R, ξ )| min(Rξ −1/4 , R 3/2 ), ∀ξ > 1, |∂ξ φ(R, ξ )| min(Rξ
−5/4
,R
|∂ξ φ(R, ξ )| min(R
3/2
|∂ξ2 φ(R, ξ )| |∂ξ2 φ(R, ξ )|
2 −7/4
min(R ξ min(R
7/2
7/2
), ∀ξ > 1/2,
log(1 + R ), ξ 2
,R
(5.21)
11/2
−1/4
(5.22)
| log ξ |R), ∀0 < ξ < 1/2,
), ∀ξ > 1/2,
log(1 + R ), ξ 2
−3/4
(5.23) (5.24)
| log ξ |R ), ∀0 < ξ < 1/2, 2
(5.25)
therefore |F(ξ, η)| < ξ >−3/4 < η >−3/4 , −5/4
|∂ξ F(ξ, η)| < ξ >
−3/4
|∂η F(ξ, η)| < ξ > |∂ξ2η F(ξ, η)| |∂ξ2 F(ξ, η)| |∂η2 F(ξ, η)|
−5/4
<ξ >
(5.26)
−3/4
,
(5.27)
−5/4
,
(5.28)
−5/4
, ξ + η 1,
(5.29)
<η> <η> <η>
< ξ >−7/4 < η >−3/4 , ξ > 1, η > 1, −7/4
<ξ >
−3/4
<η>
(5.30)
, ξ > 1, η > 1.
(5.31)
To improve on these, two cases will be considered. Case 1. 1 ξ + η. By integration by parts: ηF(ξ, η) = W (R)φ(R, ξ ), Lφ(R, η) L 2
R
= [L, W (R)]φ(R, ξ ), φ(R, η) L 2 + ξ F(ξ, η). R
(5.32)
Evaluating the commutator: ' & (η − ξ )F(ξ, η) = − (2W ∂ R + W )φ(R, ξ ), φ(R, η) L 2 .
(5.33)
R
Since W (0) = 0 (it is odd), it follows that (2W ∂ R + W )φ(R, ξ ) has the same behavior as φ(R, ξ ) at R = 0. Then the argument can be repeated to obtain: & ' (η − ξ )2 F(ξ, η) = − [L, 2W ∂ R + W ]φ(R, ξ ), φ(R, η) L 2 . (5.34) R
This second commutator has the form: [L, 2W ∂ R + W ] = 4W L − 4W ∂ R − W (4) + 3R −2 (R −1 W − W ) − 2W V − 4W V.
(5.35)
510
C. I. Cârstea
Since R −1 W (R) − W (R) = O(R 2 ), this leads to & ' (η − ξ )2 F(ξ, η) = (W o (R)∂ R + W e (R) + ξ W e (R))φ(R, ξ ), φ(R, η) L 2 ,
(5.36)
R
where W o , respectively W e , are odd, respectively even, real-analytic functions with good decay at infinity. Inductively (η − ξ )2k F(ξ, η) ⎞ ) (⎛ k−1 k j o l e ξ Wk j (R)∂ R + ξ Wkl (R)⎠ φ(R, ξ ), φ(R, η) = ⎝ j=0
l=0
,
(5.37)
L 2R
where < R > |Wkoj (R)| + |Wkle | < R >−4−2k , ∀ j, l.
(5.38)
Using the pointwise bounds on φ and ∂ R φ from (5.20)–(5.25): |F(ξ, η)|
ξ k−3/4 < η >−3/4 , ∀ξ 1, η > 0. (η − ξ )2k
(5.39)
Combining this with (5.26)–(5.31), it yields, for arbitrary N , that |F(ξ, η)| (ξ + η)−3/2 (1 + |ξ 1/2 − η1/2 |)−N , if ξ + η 1.
(5.40)
For the derivatives of F a similar procedure can be used. If ξ and η are comparable, then from (5.26)–(5.31), |∂η F(ξ, η)| < ξ >−2 .
(5.41)
Otherwise, differentiating with respect to η in (5.37), (η − ξ )2k ∂η F(ξ, η) ⎞ ) (⎛ k−1 k j o l e ⎠ ⎝ ξ Wk j (R)∂ R + ξ Wkl (R) φ(R, ξ ), ∂η φ(R, η) = j=0
l=0
−2k(η − ξ )2k−1 F(ξ, η).
L 2R
(5.42)
Using also (5.39), it follows that |∂η F(ξ, η)|
ξ k−3/4 η−5/4 , 1 ξ, η, (η − ξ )2k
(5.43)
|∂η F(ξ, η)|
η−5/4 , ξ 1 η, (η − ξ )2k
(5.44)
|∂η F(ξ, η)|
ξ k−3/4 , η 1 ξ, (η − ξ )2k
(5.45)
respectively
and
which yield the desired bounds.
Construction of Blow up Solutions for Co-rotational Wave Maps
511
Finally, consider the second order derivatives with respect to ξ and η. For ξ and η close, (5.26)–(5.31) can be used. Otherwise, differentiate twice in (5.37) and continue as before. Note that it is important that the decay of Wkoj and Wkej improves with k. This is because the second order derivative bound at zero has a sizable growth at infinity which has to be canceled, |∂ξ2 φ(R, 0)| ≈ R 7/2 log R.
(5.46)
Case 2. ξ, η 1. First note that F(0, 0) = 0. This can be verified by direct computation. Also by direct computation it can be checked that |∂ξ F(ξ, η) 1.
(5.47)
To obtain the bound on the second derivatives, begin by observing that the following inequalities hold: ⎧ ⎨ R −1/2+2 j log(1 + R 2 ) R < ξ −1/2 j |∂ξ φ(R, ξ )| , j = 0, 1, 2. (5.48) ⎩ ξ 1/4− j/2 | log ξ |R j R ≥ ξ −1/2 If η < ξ < 1/2, then these bounds imply that ξ −1/2 |∂ξ2η F(ξ, η)| < R >−4 R 3 (log(1 + R 2 ))2 d R 0
+
η−1/2
ξ −1/2 ∞
+
η−1/2
< R >−4 R 5/2 ξ −1/4 | log ξ | log(1 + R 2 ) d R
< R >−2 ξ −1/4 η−1/4 | log ξ | | log η| d R | log ξ |3 .
(5.49)
The main contribution comes from the first term. When η < ξ < 1/2, a similar computation yields ξ −1/2 |∂ξ2 F(ξ, η)| < R >−4 R 3 (log(1 + R 2 ))2 d R 0
+ +
η−1/2
ξ −1/2 ∞ η−1/2
< R >−4 R 3/2 ξ −3/4 | log ξ | log(1 + R 2 ) d R
< R >−2 ξ −3/4 η1/4 | log ξ | | log η| d R | log ξ |3 .
(5.50)
It remains to consider ∂ξ2 F(ξ, η) when ξ η < 1/2. Differentiating (5.33), . (η − ξ )∂ξ2 F(ξ, η) = 2∂ξ F(ξ, η) − ∂ξ2 φ(R, ξ ), (2W ∂ R + W )φ(R, η) 2 . (5.51) LR
Differentiating and integrating with respect to η, (η − ξ )∂ξ2 F(ξ, η) η . = 2∂ξ2ζ F(ξ, ζ ) − ∂ξ2 φ(R, ξ ), (2W ∂ R + W )∂ζ φ(R, ζ ) ξ
L 2R
dζ.
(5.52)
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C. I. Cârstea
Using the bound
|∂ R ∂ζ φ(R, ζ )|
⎧ ⎨ R 1/2 log(1 + R 2 )
R < ζ −1/2
⎩
R ≥ ζ −1/2
ζ −1/4 | log ζ |
,
(5.53)
the inner product in (5.52) can be evaluated as follows: /. // / 2 / ∂ φ(R, ξ ), (2W ∂ R + W )∂ζ φ(R, ζ ) / / ξ L 2R / ζ −1/2 < R >−6 R 7/2 log(1 + R 2 )R 3/2 log(1 + R 2 ) d R 0
+ +
ξ −1/2
ζ −1/2 ∞ ξ −1/2
< R >−6 R 7/2 log(1 + R 2 )ζ −1/4 | log ζ |R d R
< R >−6 ξ −3/4 | log ξ |R 2 ζ −1/4 | log ζ |R d R | log ζ |3 .
(5.54)
Thus, (5.52) is controlled by / η / / / |(η − ξ )∂ξ2 F(ξ, η)| // (log ζ )3 dζ // η| log η|3 . ξ
(5.55)
Since ξ η, this yields |∂η2 F(ξ, η)| | log η|3 .
(5.56)
This concludes the analysis of the off-diagonal part of the kernel. All that is left now is to determine the δ measure that sits on the diagonal of the kernel K . To do so, first restrict ξ and η to a compact set of (0, ∞). Then the following asymptotics hold for Rξ 1/2 1: −1/4 i Rξ 1/2 φ(R, ξ ) = Re a(ξ )ξ 1+ e
3i 8Rξ 1/2
+ O(R −2 ),
(R∂ R − 2ξ ∂ξ ) φ(R, ξ )
1/2 1+ = −2Re ξ ∂ξ (a(ξ )ξ −1/4 )ei Rξ
3i 8Rξ 1/2
+ O(R −2 ),
(5.57)
(5.58)
where the O terms depend on the choice of compact subset. The R −2 terms are integrable, so they contribute a bounded kernel to the inner product in (5.4). The same applies to the contribution of a bounded R region. Therefore, the δ-measure contribution of the
Construction of Blow up Solutions for Co-rotational Wave Maps
inner product in (5.4) can only come from one of the following integrals: ∞ ∞
1/2 1/2 f (ξ )χ (R)Re ξ ∂ξ (a(ξ )ξ −1/4 )a(η)η−1/4 ei R(ξ +η ) − 0 0 3i 3i 1 + ρ(ξ )dξ d R, × 1+ 8Rξ 1/2 8Rη1/2 1 ∞ ∞ 1/2 1/2 − f (ξ )χ (R)ξ ∂ξ (a(ξ )ξ −1/4 )a(η)η−1/4 ei R(ξ −η ) 2 0 0 3i 3i × 1+ 1 − ρ(ξ )dξ d R, 8Rξ 1/2 8Rη1/2 1 ∞ ∞ 1/2 1/2 f (ξ )χ (R)ξ ∂ξ (a(ξ )ξ −1/4 )a(η)η−1/4 e−i R(ξ −η ) − 2 0 0 3i 3i 1+ ρ(ξ )dξ d R, × 1− 8Rξ 1/2 8Rη1/2
513
(5.59)
(5.60)
(5.61)
where ξ is a smooth cutoff function which equals 0 near R = 0 and 1 near R = ∞. In all of the above integrals it can be argued, as in the proof of the classical Fourier inversion formula, that the order of integration can be changed. Integration by parts in the first integral (5.59) reveals that it cannot contribute to the δ-measure. Discarding the O(R −2 ) terms in (5.60) and (5.61) reduces the two integrals to: ∞ ∞
1/2 1/2 f (ξ )χ (R)Re ξ ∂ξ (a(ξ )ξ −1/4 )a(η)η−1/4 ei R(ξ −η ) ρ(ξ )dξ d R, (5.62) − 0 0
3 ∞ ∞ 1/2 1/2 + f (ξ )χ (R)Im ξ ∂ξ (a(ξ )ξ −1/4 )a(η)η−1/4 ei R(ξ −η ) 8 0 0 ×R −1 (ξ −1/2 − η−1/2 )ρ(ξ )dξ d R.
(5.63)
Since (5.63) contains both an R −1 and a (ξ −1/2 − η−1/2 ) factor, its contribution to K is bounded. The integral (5.62) contributes both a Hilbert transform type kernel as well as a δ-measure to K . By inspection, the δ-measure contribution is:
1 ∞ 1/2 1/2 − Re ξ ∂ξ (a(ξ )ξ −1/4 )a(η)η−1/4 ei R(ξ −η ) ρ(ξ )d R 2 −∞
= −π Re ξ ∂ξ (a(ξ )ξ −1/4 )a(η)η−1/4 ρ(ξ )δ(ξ 1/2 − η1/2 )
= −2π ξ 1/2 ρ(ξ )Re ξ ∂ξ (a(ξ )ξ −1/4 )a(ξ )ξ −1/4 δ(ξ − η) 1 −1/2 1/2 2 1/2 = −2π ξ ρ(ξ )Re ξ |a(ξ )| + ξ a(ξ )a (ξ ) δ(ξ − η) 4 1 ξρ (ξ ) = + δ(ξ − η), (5.64) 2 ρ(ξ ) where the fact that ρ(ξ )−1 = π |a|2 was used in the last step. This finishes the proof.
The following proposition establishes some L 2 mapping properties of K. Since the conclusion of the preceding theorem and the results of Appendix A are the same as their
514
C. I. Cârstea
correspondents in [2], the proof of this result is omitted as it is identical to the proof of Proposition 6.2 in the reference. 2 First let L 2,α ρ be the L space with the norm f L 2,α = ρ
∞
1/2 | f (ξ )|2 < ξ >2α ρ(ξ ) dξ
.
(5.65)
0
Then Proposition 5.2.
i) The operator K0 maps 2,α+1/2 K0 : L 2,α ; ρ → Lρ
(5.66)
ii) In addition, the following commutator bound holds: 2,α [K0 , ξ ∂ξ ] : L 2,α ρ → Lρ .
(5.67)
Both statements hold for all α ∈ R. In particular, K and [K, ξ ∂ξ ] are bounded operators on L 2,α ρ . 6. The Final Equation To rewrite (4.5) in a final form, begin by expressing the operator R∂ R in terms of K. Therefore, with F as in Theorem A.3, λτ λτ F ∂τ + R∂ R = ∂τ + (−2ξ ∂ξ + K) F, (6.1) λ λ which gives 2 2 λτ λτ R∂ R = ∂τ + (−2ξ ∂ξ + K) F F ∂τ + λ λ 2 λτ λτ λτ = ∂τ − 2ξ ∂ξ F + 2 K ∂τ − 2ξ ∂ξ F λ λ λ
λ2τ 2 + 2 K + 2[ξ ∂ξ , K] F. λ This leads to a transport type equation for the Fourier transform x(τ, ξ ) of : 2 λτ λτ λτ − ∂τ − 2ξ ∂ξ x − ξ x = 2 K ∂τ − 2ξ ∂ξ x λ λ λ 2
2 1 λτ λτ λτ 1 2 + 2 K + 2[ξ ∂ξ , K] x − + ∂τ x λ 4 λ 2 λ
+ λ−2 F R 1/2 N2k−1 (R −1/2 F −1 x) + e2k−1 .
(6.2)
(6.3)
The aim is to obtain solutions of (6.3) which decay as τ → ∞. This means the equation will be solved backwards in time, with zero Cauchy data at τ = ∞. The problem will be treated iteratively, as a small perturbation of the linear equation governed by
Construction of Blow up Solutions for Co-rotational Wave Maps
515
the operator on the left-hand side. For this the following transport equation needs to be solved: ! 2 λτ ∂τ − 2ξ ∂ξ + ξ x(τ, ξ ) = b(τ, ξ ). − (6.4) λ Denote by H the backward fundamental solution of the operator 2 λτ ∂τ − 2ξ ∂ξ + ξ, λ and by H (τ, σ ) its kernel, i.e. (6.4) has solution ∞ x(τ ) = − H (τ, σ )b(σ ) dσ, τ
(6.5)
(6.6)
where the ξ variable has been suppressed. The mapping properties of H are described in the following result, which is proven in [2], Sect. 8. Proposition 6.1. For any α ≥ 0 there exists some (large) constant C = C(α) so that the operator H (τ, σ ) satisfies the bounds σ C H (τ, σ ) L 2,α →L 2,α+1/2 τ , (6.7) ρ ρ τ 0 0 σ C 0 0 0 ∂τ − λτ 2ξ ∂ξ H (τ, σ )0 τ , (6.8) 0 2,α 2,α 0 λ τ L ρ →L ρ
uniformly in σ ≥ τ . This leads to the introduction of the spaces L ∞,N L 2,α ρ with norm f L ∞,N L 2,α = sup τ N f (τ ) L 2,α . ρ
τ ≥1
ρ
(6.9)
Then an immediate consequence of the above proposition is the following Corollary 6.2. Given α ≥ 0, let N be large enough. Then 0 0 0 0 λτ 0 H b L ∞,N −2 L 2,α+1/2 + 0 ∂τ − 2ξ ∂ξ H b0 0 ∞,N −1 2,α ρ λ L Lρ ≤ C0 N −1 b L ∞,N L 2,α ,
(6.10)
ρ
with a constant C0 that depends on α but does not depend on N . The nonlinear operator N2k−1 from (6.3) has the following mapping properties (which are proved below): Proposition 6.3. Assuming that N is large enough and ν2 + 43 > α > 41 , then the map
(6.11) x → λ−2 F R 1/2 N2k−1 (R −1/2 F −1 x) is locally Lipschitz from L ∞,N −2 L ρ
2,α+1/2
to L ∞,N L 2,α ρ .
These two results above, combined with Proposition 5.2 allow for the use of a contraction argument to solve Eq. (6.3).
516
C. I. Cârstea
7. The Nonlinear Terms The aim of this section is to prove Proposition 6.3. First define Sobolev spaces Hρα , adapted to the operator L, such that u Hρα = " u L 2,α .
(7.1)
ρ
What needs to be shown is that the map → λ−2 R 1/2 N2k−1 (R −1/2 )
(7.2)
α+1/2 L ∞,N −2 Hρ
is locally Lipschitz from to L ∞,N Hρα . The following lemmas are proven in [2]: Lemma 7.1. Let q ∈ S(1, Q) and |α| <
ν 2
+ 34 . Then
q f Hρα f Hρα . Lemma 7.2. Let α > 41 . Then 0 0 0 −3/2 0 f g0 0R respectively
α+1/4
Hρ
0 0 0 −3/2 0 f g0 0R
Hρα
(7.3)
f H α+1/2 g H α+1/2 , ρ
ρ
f H α+1/4 g H α+1/2 , ρ
ρ
(7.4)
(7.5)
for all f , g such that the right-hand sides are finite. Lemma 7.3. Let α > 0. Then 0 0 0 −1 0 0R f gh 0
Hρα
f H α+1/2 g H α+1/2 h Hρα , ρ
ρ
(7.6)
for all f , g, h such that the right hand side is finite. Now
R 1/2 λ−2 N2k−1 () = −R −3/2 f (u 2k−1 ) − f (u 0 ) + f (u 2k−1 + ) − f (u 2k−1 ) − f (u 2k−1 ) = −R −3/2 [I + I I ].
(7.7)
For the first term write I () =
l≥2
Remember that (u 2k−1 − u 0 ) ∈
1 f (l) (u 2k−1 − u 0 )l−1 . (l − 1)! 1 (tλ)2
(7.8)
I S 1 (R(log R), Q) and that, by Lemma 3.9,
f (2m) (u 0 ) ∈ I S 1 (R −1 , Q), f (2m+1) (u 0 ) ∈ I S 0 (1, Q). Then
1 2m 2m−2 2m−1 f (2m) (u 0 )(u 2k−1 − u 0 )2m−1 ∈ R I S (log R) , Q (tλ)4m−2
1 1 2m 2m 2m−2 2m 2m 0 R R I S (log R) , Q ⊂ I S (log R) , Q ⊂ (tλ)4m−2 (tλ)2m 1 ⊂ I S 2 (R 2 , Q), (7.9) (tλ)2
Construction of Blow up Solutions for Co-rotational Wave Maps
517
and
1 2m 2m 2m R I S (log R) , Q (tλ)4m
1 1 I S 2m R 2m (log R)0 , Q ⊂ I S 2 (R 2 , Q). ⊂ 2m (tλ) (tλ)2
f (2m+1) (u 0 )(u 2k−1 − u 0 )2m ∈
(7.10)
Therefore ) ∈ R −3/2 I (R −1/2
1 I S 0 (1) ⊂ τ −2 I S 0 (1). (tλ)2
(7.11)
(The last step uses the fact that tλ τ .)2 So → R −3/2 I (R −1/2 )
(7.12)
has the desired mapping property. The second term can be split into two: I I () = I I1 () + I I2 (),
(7.13)
where I I1 =
1 f (2l) (u 2k−1 ) 2l (2l)!
(7.14)
1 f (2l+1) (u 2k−1 ) 2l+1 . (2l + 1)!
(7.15)
l≥1
and I I2 =
l≥1
By Lemma 3.10, f (2l) (u 2k−1 ) ∈
1 I S 1 (R(log R), Q) ⊂ I S 1 (R). (tλ)2
(7.16)
Then R −3/2 f (2l) (u 2k−1 )(R −1/2 )2l ∈ (R −1 2 )l−1 R −3/2 2 I S 0 (1).
(7.17)
Now, by Lemmas 7.1, 7.2, and (7.1), it follows that I I1 has the right mapping property in the space variable. More precisely, the claim follows from the fact that 0 0 0 −3/2 2 0 0 α 2 α+1/2 , (7.18) 0R Hρ
that, as an operator,
0 0 0 −1 2 0 0 0R
Hρα →Hρα
Hρ
2
α+1/2
Hρ
,
(7.19)
and from Lemma 7.1. The τ behavior follows from the fact that I I1 has no linear term in , only higher powers. 2 By a b it is meant that there is a positive constant C such that C −1 a < b < Ca.
518
C. I. Cârstea
Finally, note that R −3/2 f (2l+1) (u 2k−1 )(R −1/2 )2l+1 ∈ (R −1 2 )l−1 R −3 3 I S 0 (1). After noticing that 0 0 0 −3 3 0 0 0R
Hρα
0 0 0 0 0R −3/2 20
α+1/4
Hρ
H α+1/2 3 ρ
α+1/2
Hρ
,
(7.20)
(7.21)
the argument that I I2 has the right mapping property is the same as the one above for I I1 . 8. The Conclusion of the Argument To compare the Sobolev spaces Hρα with the usual ones H β (R2 ), define a map u(R) → (T u)(R, θ ) = eiθ R −1/2 u(R).
(8.1)
This is easily seen to be an isometry L 2 (R+ ) → L 2 (R2 ). Lemma 8.1. For any α ≥ 0, u H α/2 (R+ ) T u H α (R2 ) ρ
(8.2)
in the sense that if one side is finite then the other is also finite and they have comparable sizes. β
Proof. The spaces Hρ (R+ ) are defined using fractional powers of the operator L, but β since L − L0 is bounded in L 2 and in any Hρ , these spaces could be defined using L0 instead. The lemma follows from the identity T u = T L0 u, which holds whenever u ∈ L 2 and L0 u ∈ L 2 .
(8.3)
Fix now a ν > 1/2, and an index k sufficiently large (depending on ν). So far u 2k−1 and e2k−1 have only been defined inside the cone {r ≤ t}. They can be extended to be supported in the cone {r ≤ 2t} so that they have the same regularity and all relevant derivatives match on the boundary of the light-cone. Finally, choose α so that 1 ν <α< . 4 2
(8.4)
The error e2k−1 has a singularity of the type (1 − a)ν−1/2 logm (1 − a) on the cone a = 1, which means that e2k−1 is in H β locally around r = t, as long as β < ν. On the other hand, since e2k−1 has order one at R = 0, T (R 1/2 e2k−1 ) = eiθ R(c0 (τ ) + c1 (τ )R 2 + c2 (τ )R 4 + · · · ),
(8.5)
which is smooth around R = 0. Finally, taking into account the size of the error R(log(2 + R 2 ))2k−1 e2k−1 = O , (8.6) t 2 (tλ)2k
Construction of Blow up Solutions for Co-rotational Wave Maps
it follows that for all α < ν/2, 0 0 0 −2 1/2 0 0λ R e2k−1 (t (τ ), λ−1 R)0
Hρα
519
τ −2k+2 .
(8.7)
Using Propositions 5.2, 6.1, and 6.3, Eq. (6.3) can be solved through a contraction principle argument with respect to the norm 0 0 0 0 λτ 0 x L ∞,N −2 L 2,α+1/2 + 0 ∂ − 2 . (8.8) ξ ∂ τ ξ x0 0 ρ λ L ∞,N −1 L 2,α ρ The transference identity and Proposition 5.2 give that = F −1 x satisfies 0 0 0 0 λτ 2−N 1−N 0 (τ )0 R∂ R (τ ) H α+1/2 τ , 0 ∂τ + , N ≤ 2k. 0 α τ ρ λ Hρ
(8.9)
By Lemma 8.1, this construction yields a solution to (2.4) on the cone r ≤ t, 0 < t < t0 , which is of class H 1+ν− on the closure of the cone. To obtain a solution on all (0, t0 )×R2 extend the data u(t0 , ·), ∂t u(t0 , ·) to all of R2 with the same smoothness. The corresponding solution to (2.4) will coincide with the one constructed on the cone due to the finite speed of propagation. This solution satisfies the properties stated in Theorem 2.2. Acknowledgements. I would like thank Prof. C. Kenig and Prof. W. Schlag. Fruitful discussions with both have made this work possible.
A. An Analysis of the Operator L The material below mostly parallels Sect. 5 of [2]. The one main difference arises from the fact that a fundamental basis of solutions for L is not explicitly known here. This calls for a few changes in the proof of Proposition A.4, which coresponds to Proposition 5.4 in the reference. The asymptotic expansion for θ in that same proposition has been omitted here, as it is not needed for the main result. The rest is close to identical to [2]. Consider the operator on L 2 (0, ∞) L = L0 + V (r ) = −∂r2 +
3 + V (r ); 4r 2
V (r ) = −
1 1 − f (Q(r )) . (A.1) 2 r
As r → 0+ , V (r ) ∼ 1 (i.e. limr ↓0 V (r ) = C for some real number C). As r → ∞, V (r ) ∼ 1/r 4 . Therefore both L and L0 are self-adjoint with domains (A.2) D(L) = D(L0 ) = ϕ ∈ L 2 (0, ∞) : ϕ, ϕ ∈ ACloc , L0 ϕ ∈ L 2 . Notice that Lφ0 = 0,
φ0 (r ) = r 3/2 Q (r ).
(A.3)
From (2.7) it follows that φ0 is positive. As r → 0+ , φ0 (r ) ∼ r 3/2 , as r → ∞, φ0 (r ) ∼ r −1/2 , so φ0 ∈ L 2 (0, ∞). These two remarks together with the Sturm oscillation theorem give:
520
C. I. Cârstea
Lemma A.1. The spectrum of L is purely absolutely continuous and spec (L) = [0, ∞). To find a function θ0 such that together with φ0 it forms a fundamental system of L, it is enough to ask that it satisfies W (φ0 , θ0 ) = 1. This, together with the initial condition θ0 (1) = 0, yields: r 1 θ0 (r ) = −φ0 (r ) ds. (A.4) 2 1 φ0 (s) Lemma A.2. L has a fundamental system of solutions φ0 , θ0 with the following properties: i) φ0 (r ) = r 3/2 Q (r ); ii) φ0 (r ) ∼ r 3/2 , θ0 (r ) ∼ r −1/2 as r → 0+ ; iii) φ0 (r ) ∼ r −1/2 , θ0 (r ) ∼ r 3/2 as r → ∞; iv) r −3/2 φ0 (r ) is real-analytic, φ0 (R) ∈ R 3/2 S(R −2 ); v) θ0 (R) ∈ R 3/2 S(R 0 ). Proof. Only the last statement needs a proof. For large s, φ0 (s)−2 admits an absolutely convergent expansion of the form: ∞
1 = φ k s 3−2k . 2 φ0 (s)
(A.5)
k=1
Therefore, for large R,
θ0 (R) ∈ R 3/2 φ 0 + φ 1 R 2 + φ 2 log R + φ 3 R −2 + · · · S(R −2 ).
(A.6)
The following theorem will be useful: Theorem A.3 (Sect. 3 of [1], Theorem 5.3 of [2]). With the notation above: i) For each z ∈ C there exists a fundamental system φ(r, z), θ (r, z) for L − z which is analytic in z for each r > 0 and has the asymptotic behavior φ(r, z) ∼ r 3/2 ,
θ (r, z) ∼
1 −1/2 r as r → 0+ . 2
(A.7)
In particular, their Wronskian is W (θ (·, z), φ(·, z)) = 1 for all z ∈ C. By convention, φ(r, z), θ (r, z) are real-valued for z ∈ R. ii) For each z ∈ C, Im z > 0, let ψ + (r, z) denote the Weyl–Titchmarsh solution of L − z at r = ∞ normalized so that ψ + (r, z) ∼ z −1/4 ei z
1/2
r as r → ∞, Im z 1/2 > 0.
(A.8)
If ξ > 0, then the limit ψ + (r, ξ + i0) exists point-wise for all r > 0 and it will be denoted by ψ + (r, ξ ). Moreover, define ψ − (·, ξ ) := ψ + (·, ξ ). Then ψ + (r, ξ ), ψ − (r, ξ ) form a fundamental system of L − ξ with asymptotic behavior ψ ± (r, ξ ) ∼ ξ −1/4 e±iξ
1/2 r
as r → ∞.
(A.9)
Construction of Blow up Solutions for Co-rotational Wave Maps
521
iii) The spectral measure of L is absolutely continuous and its density is given by ρ(ξ ) =
1 Im m(ξ + i0)χ[ξ >0] , π
(A.10)
with the “generalized Weyl–Titchmarsh” function m(z) =
W (θ, ·, z), ψ + (·, z)) , Im z ≥ 0. W (ψ + (·, z), φ(·, z))
(A.11)
iv) The distorted Fourier transform defined as F: f → " f (ξ ) = lim
b
b→∞ 0
φ(r, ξ ) f (r ) dr
(A.12)
is a unitary operator from L 2 (R+ ) to L 2 (R+ , ρ) and its inverse is given by μ f → f (r ) = lim φ(r, ξ ) " f (ξ )ρ(ξ ) dξ. (A.13) F −1 : " μ→∞ 0
Here lim refers to the corresponding L 2 limit. Proposition A.4. The φ(r, z) in Theorem A.3 admits the absolutely convergent expansion: φ(r, z) = φ0 (r ) + r
−1/2
∞
(r 2 z) j φ j (r 2 ),
(A.14)
j=1
where the functions φ j are real-analytic on [0, ∞) and satisfy the bounds |φ j (u)| ≤
C2 C j log(1 + |u|), |φ1 (u)| > C log u if u 1, ( j − 1)!
(A.15)
where C, C2 are positive constants. In particular, φ j (0) = 0 and |φ j (0)| ≤ C2 C j / ( j − 1)! for j = 1, 2, . . .. Proof. First make the ansatz φ(r, z) = r −1/2
∞
z j f j (r ).
(A.16)
j=0
The functions f j will be constructed such that the series converges in a “reasonable” sense. They should solve L(r −1/2 f j ) = r −1/2 f j−1 ,
f 0 (r ) = r 1/2 φ0 (r ).
(A.17)
To obtain the f j ’s, the “forward fundamental solution” of L is used: H (r, s) =
1 [φ0 (r )θ0 (s) − φ0 (s)θ0 (r )] 1[r >s] , 2
(A.18)
522
C. I. Cârstea
therefore f j (r ) =
1 2
r
r 1/2 s −1/2 [φ0 (r )θ0 (s) − φ0 (s)θ0 (r )] f j−1 (s) ds.
(A.19)
0
Remembering that φ0 (r ) = r 3/2 Q (r ) and using the notation χ (r ) = r 2 that θ0 (r ) = −r −2 φ0 (r )χ (r )), the identity above becomes: 1 r Q (r )Q (s) 2 s χ (r ) − r 2 χ (s) f j−1 (s) ds. f j (r ) = 2 0 s Note now that χ (r ) can be written as: r Q(r ) 1 1 2 2 χ (r ) = r Q (s) ds = r dρ. 3 3 g(Q(s)) g(ρ) 1 Q(1)
1r
ds 1 φ 2 (s) 0
(so
(A.20)
(A.21)
Using the assumptions made on g this gives:
1 1 Q(1)−2 − Q(r )−2 − g (0) (log Q(r ) − log Q(1)) + · · · χ (r ) = r 2 (A.22) 2 2 1 = − g (0)r 2 log r + (terms analytic at 0). (A.23) 2 It follows then (by induction) that the singularity f j might have at zero is isolated and, in fact, removable. To see this, choose a branch of the logarithm which is holomorphic in C\R− . It is necessary to show that f j (r + i0) = f j (r − i0) for r < 0. Disregarding the terms not involving logarithms, it is enough to show that for any holomorphic function g r +i0 r −i0 [log s − log(r + i0)]g(s) ds = [log s − log(r − i0)]g(s) ds, (A.24) 0
0
which is obvious since for s < 0, log(s + i0) − log(r + i0) = log(s − i0) − log(r − i0).
(A.25)
Therefore, each f j is an even analytic function in a (uniform) neighborhood of the real line. Also, the assumption that f j−1 (r ) ∼ r 2 at zero implies f j (0) = 0, so f j (r ) ∼ r 2 at zero. Induction gives that f j (0) = 0 for all j. f j (u), Q (r ) = B(u), For the rest of this proof, let u = r 2 , v = s 2 , f j (r ) = χ (r ) = X (u). It is easy to see that there are positive constants C1 , C2 such that C1 ≤ (1 + u)B(u) ≤ C2 , ∀u. With this notation 1 f j (u) = 4
0
u
B(u)B(v) f j−1 (v) dv. [v X (u) − u X (v)] v
Also X (u) =
u 2
1
u
dv v 2 B(v)2
.
(A.26)
(A.27)
(A.28)
Construction of Blow up Solutions for Co-rotational Wave Maps
523
Therefore, if v ≤ u, v X (u) − u X (v) > 0,
(A.29)
which makes (by induction and the fact that f 0 > 0 and is increasing; see Lemma 2.1) each f j positive and increasing, uv u dw 1 1 u v X (u) − u X (v) = ≤ Cuv − + log + (u − v) 2 v w 2 B(w)2 v u v u−v u + + (u − v) ≤ Cu(1 + u + uv), (A.30) ≤ Cuv uv v and B(u)B(v) [v X (u) − u X (v)] v u u ≤ C [B(u)(1 + u)B(v) + u B(u)v B(v)] ≤ C . v v Note that f 0 (u) = u B(u). Then u / / u v / / dv = CC2 u log(1 + u). f 1 (u) ≤ CC2 0 v 1+v By induction, using the fact that u 1 x j−1 log(1 + x) d x ≤ u j log(1 + u), j 0
(A.31)
(A.32)
(A.33)
it follows that / / C2 C j j / u log(1 + u), f j (u)/ ≤ ( j − 1)! where C is the same constant from the last inequality in A.31. Finally, consider 1 u B(u)B(v) f 1 (u) = [v X (u) − u X (v)] v B(v) dv 4 0 v u dw 1 u dv B(u)B(v)2 uv . = 2 2 8 0 v w B(w) Using the fact that u B(u) is bounded and increasing (see Lemma 2.1), u 1 1 v 2 B(v)2 dv (u − v) f 1 (u) ≥ u B(u) 2 8 v u B(u)2 0 u
u 1 1 − 1 dv ≥ C B(1)2 u log u − (u − 1) . ≥ v 2 B(v)2 8 u B(u) 1 v
(A.34)
(A.35)
(A.36)
So, for u 1, f 1 (u) ≥ Cu log u.
(A.37)
524
C. I. Cârstea
Note that the logarithmic behavior of φ1 (u) for large u is inherited by φ(r, ξ ). If 1 ξ > 0 and r = δξ −1/2 , where δ > 0 is a small absolute constant, then φ(r, ξ ) r −1/2 log r.
(A.38)
The next proposition deals with ψ + : Proposition A.5. For any ξ > 0, the solution ψ + (·, ξ ) from Theorem A.3 is of the form ψ + (r, ξ ) = ξ −1/4 eir ξ
1/2
σ (r ξ 1/2 , r ), r 2 ξ 1,
(A.39)
where σ admits the asymptotic series approximation σ (q, r ) ≈
∞
q
−j
ψ +j (r ),
ψ0+
j=0
= 1,
ψ1+
3i 1 , = +O 8 1 + r2
(A.40)
with zero order symbols ψ +j that are analytic at infinity, / / / / sup /(r ∂r )k ψ +j (r )/ < ∞,
(A.41)
r >0
in the sense that for all large integers j0 , and all indices α, β, it holds that / ⎤/ ⎡ / / j0 / / sup //(r ∂r )α (q∂q )β ⎣σ (q, r ) − q − j ψ +j (r )⎦// ≤ cα,β, j0 q − j0 −1 r >0 / / j=0
(A.42)
for all q > 1. Proof. Let σ (q.r ) = ξ 1/4 ψ + (r, ξ )e−ir ξ
1/2
.
(A.43)
Since ψ + solves the equation (L − ξ )ψ + (r, ξ ) = 0,
(A.44)
it follows that σ has to solve 3 −∂r2 − 2iξ 1/2 ∂r + 2 + V (r ) σ (r ξ 1/2 , r ) = 0. 4r
(A.45)
First look for a formal power series solution to this equation: σ =
∞
ξ −1/2 f j (r ),
(A.46)
j=0
which would require that the f j satisfy 3 2i∂r f j = −∂r2 + 2 + V (r ) f j−1 , 4r
f 0 = 1.
(A.47)
Construction of Blow up Solutions for Co-rotational Wave Maps
Then i i f j (r ) = ∂r f j−1 + 2 2
r
∞
3 + V (s) 4s 2
525
f j−1 (s) ds.
(A.48)
Inductively, it is easy to see (recalling also that V (r ) ∼ r −4 at s = ∞) that all f j are analytic at infinity, with leading order term r − j . At zero however, the f j will be singular. Using (A.48) it is not hard to show, inductively, that / / / / (A.49) /(r ∂r )k f j / ≤ c j r − j , ∀k ∈ N, r > 0. Indeed, suppose the claim is true for f j−1 . Then
3 i i k k−1 + V (r ) f j−1 (r ) (r ∂r ) f j = (r ∂r ) ∂r f j−1 (r ) − (r ∂r ) 2 2 4r 2 i 1 i 1 (r ∂r )k+1 f j−1 (r ) + (r ∂r )k , (r ∂r ) f j−1 (r ) = 2r 2 r 3 i + V (r ) (r ∂r )k−1 f j−1 (r ) − 2 4r 2 i k−1 3 − (r ∂r ) , 2 + V (r ) f j−1 (r ). 2 4r k
Noting that
(A.50)
1 k = − (r ∂r )k−1 , r r 6(k − 1) 3 (r ∂r )k−1 , 2 = − (r ∂r )k−2 , 4r 4r 2
(r ∂r )k−1 , V (r ) = (k − 1)r V (r )(r ∂r )k−2 , (r ∂r )k ,
(A.51) (A.52) (A.53)
the induction is complete. Let then ψ +j (r ) = r j f j (r ). These will satisfy (A.41). It is known from symbol calculus that there exists a function σap (q, r ) which satisfies / ⎤/ ⎡ / / j0 / / α β⎣ −j + / ⎦ sup /(r ∂r ) (q∂q ) σap (q, r ) − q ψ j (r ) // ≤ cα,β, j0 q − j0 −1 (A.54) r >0 / / j=0 for all natural numbers α, β, and j0 . However, σap will not solve (A.45). Define the error 3 1/2 2 1/2 e(r ξ , r ) = −∂r − 2iξ ∂r + 2 + V (r ) σap (r ξ 1/2 , r ). (A.55) 4r It is easy to see that
/ / /(r ∂r )α (q∂q )β e(q, r )/ ≤ cα,β, j r −2 q − j ,
for all α, β, and j. Let σ1 = −σ + σap . This σ1 has to satisfy 3 −∂r2 − 2iξ 1/2 ∂r + 2 + V (r ) σ1 (r ξ 1/2 , r ) = e(r ξ 1/2 , r ). 4r
(A.56)
(A.57)
526
C. I. Cârstea
To obtain estimates on σ1 , first define v = (v1 , v2 ) = (σ1 , r ∂r σ1 ). Eq. (A.57) can be written in terms of v as 0 r −1 0 . (A.58) v = ∂r v − 3 −1 − 2iξ 1/2 −r e 4r + r V (r ) r From this it follows that
d v |2 + r | v | |e| , | v |2 ≥ −C r −1 | dr
(A.59)
d | v | ≥ −C r −1 | v | + r |e| . dr
(A.60)
so
By Gronwall’s inequality, | v (r )| ≤ r
∞ s C
r
s|e(s)| ds.
(A.61)
For large j, |e| < Cξ − j/2 r − j−2 ,
(A.62)
| v | < C j ξ − j/2 r − j = C j q j ,
(A.63)
which implies that
for large j. Entirely similar arguments can be applied to (r ∂r )α (q∂q )β v, to conclude in the end that / / /(r ∂r )α (q∂q )β σ1 (q, r )/ ≤ Cα,β, j q − j , (A.64) for large j and any α and β. Then σ = σap − σ1 is as desired.
The last result of this section deals with the spectral measure of L. Proposition A.6.
i) There is a function a(ξ ) such that φ(r, ξ ) = a(ξ )ψ + (r, ξ ) + a(ξ )ψ + (r, ξ ),
which is smooth, always nonzero, and has size3 % 1/2 −ξ log ξ ξ 1 |a(ξ )| . ξ −1/2 ξ 1 Moreover, this function satisfies the bounds / / / / /(ξ ∂ξ )k a(ξ )/ ≤ ck |a(ξ )|, ∀ξ > 0. 3 By a b it is meant that there is a positive constant C such that C −1 a < b < Ca.
(A.65)
(A.66)
(A.67)
Construction of Blow up Solutions for Co-rotational Wave Maps
527
ii) The spectral measure ρ(ξ )dξ has density 1 ρ(ξ ) = |a(ξ )|−2 , π and therefore satisfies % 1 ξ 1 ρ(ξ ) ξ(log ξ )2 . ξ ξ 1
(A.68)
(A.69)
Proof. i) Since φ is real valued and W (ψ + , ψ − ) = −2i, the function a must be i a(ξ ) = − W (φ(·, ξ ), ψ − (·, ξ )). (A.70) 2 By Proposition A.4 it follows that both φ(ξ −1/2 , ξ ) and (r ∂r φ)(ξ −1/2 , ξ ) can be written in the form ξ 1/4 f (ξ −1 ) with f (u) analytic and satisfying | f (u)| log(1 + |u|).
(A.71)
ψ + (ξ −1/2 , ξ )
and (r ∂r By Proposition A.5 it follows that both written in the form ξ −1/4 h(ξ −1/2 ) with h satisfying the bounds
ψ + )(ξ −1/2 , ξ )
|(r ∂r )k h(r )| ≤ ck . is a sum of terms of the form ξ 1/2
can be (A.72)
f (ξ −1 )h(ξ −1/2 ), with
Then the function a above. The bounds (A.67) and the upper bounds in (A.66) then follow. To prove the lower bounds, begin by noting that Im (ψ + (r, ξ )∂r ψ − (r, ξ )) = −1.
f and h as
(A.73)
Since φ is real-valued, this gives Im ∂r ψ + (r, ξ )W (φ(·, ξ ), ψ − (·, ξ )) = −∂r φ(r, ξ ),
(A.74)
which implies that for all r, |a(ξ )| ≥
|∂r φ(r, ξ )| . 2|∂r ψ + (r, ξ )|
(A.75)
There is a small constant δ such that, if r = δξ −1/2 , by Proposition A.4, |∂r φ(r, ξ )| r −3/2 log(1 + r 2 ),
(A.76)
|∂r ψ + (r, ξ )| ξ 1/4 (r 2 ξ )− j0 .
(A.77)
and by Proposition A.5,
These give the lower bounds in (A.66). ii) ψ + can be written in terms of φ and θ as ψ + = −φW (ψ + , θ ) + θ W (ψ + , φ). W (ψ + , ψ − )
Since both φ and θ are real-valued, inserting into Im W (ψ + , θ )W (ψ − , φ) = −1.
(A.78) = −2i, it follows that
Inserting into (A.10) and (A.11), this yields 1 1 1 Im W (ψ + , θ )W (ψ − , φ) = |W (ψ + , φ)|−2 = . ρ(ξ ) = + 2 π |W (ψ , φ)| π π |a(ξ )|2
(A.79)
(A.80)
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References 1. Gesztesy, F., Zinchenko, M.: On spectral theory for Schrödinger operators with strongly singular potentials. Math. Nachr. 279(9-10), 1041–1082 (2006) 2. Krieger, J., Schlag, W., Tataru, D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171(3), 543–615 (2008) 3. Krieger, J., Schlag, W., Tataru, D.: Renormalization and blow up for the critical Yang-Mills problem. Adv. Math. 221(5), 1445–1521 (2009) 4. Krieger, J., Schlag, W., Tataru, D.: Slow blow-up solutions for the H 1 (R3 ) critical focusing semilinear wave equation. Duke Math. J. 147(1), 1–53 (2009) 5. Shatah, J., Struwe, M.: Geometric wave equations. Volume 2 of Courant Lecture Notes in Mathematics. New York: New York University Courant Institute of Mathematical Sciences, 1998 6. Shatah, J., Shadi Tahvildar-Zadeh, A.: On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math. 47(5), 719–754 (1994) 7. Struwe, M.: Equivariant wave maps in two space dimensions. Comm. Pure Appl. Math. 56(7), 815–823 (2003) Communicated by P. Constantin
Commun. Math. Phys. 300, 529–597 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1129-1
Communications in
Mathematical Physics
Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials Robert M. Strain Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA. E-mail: [email protected]; URL: http://www.math.upenn.edu/∼strain/ Received: 8 October 2009 / Accepted: 2 June 2010 Published online: 3 September 2010 – © Springer-Verlag 2010
Abstract: In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in L ∞ . If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of Dudy´nski and Ekiel-Je˙zewska (Commun Math Phys 115(4):607–629, 1985); this resolves the open question of global existence for the soft potentials. Contents 1. Introduction . . . . . . . . . . . . . . . . 2. Statement of the Main Results . . . . . . . 3. Linear L 2 Bounds and Decay . . . . . . . 4. Linear L ∞ Bounds and Slow Decay . . . . 5. Nonlinear L ∞ Bounds and Slow Decay . . 6. Linear L ∞ Rapid Decay . . . . . . . . . . 7. Nonlinear L ∞ Rapid Decay . . . . . . . . Appendix: Derivation of the Compact Operator References . . . . . . . . . . . . . . . . . . . .
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1. Introduction The relativistic Boltzmann equation is a fundamental model for fast moving particles; it can be written with appropriate initial conditions as p μ ∂μ F = C(F, F). The authors research was partially supported by the NSF grant DMS-0901463.
530
R. M. Strain
The collision operator [4,9] is given by C( f, h) =
c dq dq dp W ( p, q| p , q )[ f ( p )h(q ) − f ( p)h(q)]. 2 R3 q0 R3 q0 R3 p0
The transition rate, W ( p, q| p , q ), can be expressed as W ( p, q| p , q ) = sσ (g, θ )δ (4) ( p μ + q μ − p μ − q μ ), where σ (g, θ ) is the differential cross-section or scattering kernel; it measures the interactions between particles. The speed of light is the physical constant denoted c. Standard references in relativistic Kinetic theory include [8,9,21,48,54]. The rest of the notation is defined in the sequel. 1.1. A brief history of relativistic kinetic theory. Early results include those on derivations [40], local existence [3], and linearized solutions [14,18]. DiPerna-Lions renormalized weak solutions [13] were shown to exist in 1992 by Dudy´nski and Ekiel-Je˙zewska [19] globally in time for large data, using the causality of the relativistic Boltzmann equation [16,17]. See also [1,57] and [37,38]. In particular [1] proves the strong L 1 convergence to a relativistic Maxwellian, after taking a subsequence, for weak solutions with large initial data that is not necessarily close to an equilibrium solution. There are also results in the context of local [7] and global [49] Newtonian limits, and near vacuum results [22,35,49] and blow-up [2] for the gain term only. We also mention a study of the collision map and the pre-post collisional change of variables in [23]. For more discussion of historical results, we refer to [49]. We review in more detail the results most closely related to those in this paper. In 1993, Glassey and Strauss [24] proved for the first time global existence and uniqueness of smooth solutions which are initially close to a relativistic Maxwellian and in a periodic box. They also established exponential convergence to the Maxwellian. Their assumptions on the differential cross-section, σ , fell into the regime of “hard potentials” as discussed below. In 1995, they extended that result to the whole space case [20] where the convergence rate is polynomial. More recent results with reduced restrictions on the cross-section were proven in [36], using the energy method from [29–32]; these results also apply to the hard potentials. For relativistic interactions–when particles are fast moving–an important physical regime is the “soft potentials”; see [15] for a physical discussion. Despite their importance, prior to the results in this paper there were no existence results for the soft potentials in the context of strong nearby relativistic Maxwellian global solutions. In 1988 a general physical assumption was given in [18]; see (2.7) and (2.8). In this paper we will prove global existence of unique L ∞ near equilibrium solutions to the relativistic Boltzmann equation and rapid time decay under the full physical assumption on the cross-section from [18]. Our main focus is of course the soft potentials; and we do not require any angular cut-off (although the angular singularity will not be worse than just barely integrable). 1.2. Notation. Prior to discussing our main results, we will now define the notation of the problem carefully. In special relativity the momentum of a particle is denoted by p μ , μ = 0, 1, 2, 3. Let the signature of the metric be (− + ++). Without loss of generality, we set the rest mass for each particle m = 1. The momentum for each particle is restricted to the mass shell p μ pμ = −c2 with p0 > 0.
Asymptotic Stability for Soft Potentials
531
Further with p ∈ R3 ,we may write p μ = ( p 0 , p)and similarly qμ = (q 0 , q). Then the energy of a relativistic particle with momentum p is p0 = c2 + | p|2 . We are def raising and lowering indices with the Minkowski metric pμ = gμν p ν , where (gμν ) = diag(−1 1 1 1). We use the Einstein convention of implicit summation over repeated indices. The Lorentz inner product is then given by p μ q μ = − p0 q 0 + def
3
pi qi .
i=1
Then also q0 = c2 + |q|2 > 0. Note our convention for raising and lowering indices; we only use it in this paragraph and in the Appendix. Now the streaming term of the relativistic Boltzmann equation is given by p μ ∂μ = p 0 ∂t + p · ∇ x . We thus write the relativistic Boltzmann equation as ∂t F + pˆ · ∇x F = Q(F, F).
(1.1)
Here Q(F, F) = C(F, F)/ p0 , with C defined at the top of this paper. Above we consider F = F(t, x, p) to be a function of time t ∈ [0, ∞), space x ∈ T3 and momentum p ∈ R3 . The normalized velocity of a particle is denoted pˆ = c
p p = . p0 1 + | p|2 /c2
(1.2)
Steady states of this model are the well known Jüttner solutions, also known as the relativistic Maxwellian. They are given by exp (−cp0 /(k B T )) , 4π ck B T K 2 (c2 /(k B T )) 2 ∞ def where K 2 (·) is the Bessel function K 2 (z) = z2 1 e−zt (t 2 − 1)3/2 dt, T is the temperature and k B is Boltzmann’s constant. In the rest of this paper, without loss of generality but for the sake of simplicity, we will now normalize all physical constants to one, including the speed of light to c = 1. So that in particular we denote the relativistic Maxwellian by J ( p) = def
J ( p) =
e − p0 . 4π
(1.3)
Henceforth we let C, and sometimes c denote generic positive inessential constants whose value may change from line to line. We will now define the quantity s, which is the square of the energy in the “center of momentum” system, p + q = 0, as def s = s( p μ , q μ ) = −( p μ + q μ )( pμ + qμ ) = 2 − p μ qμ + 1 ≥ 0. (1.4) The relative momentum is denoted def g = g( p μ , q μ ) = ( p μ − q μ )( pμ − qμ ) = 2(− p μ qμ − 1).
(1.5)
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R. M. Strain
Notice that s = g 2 + 4. We warn the reader that this notation, which is used in [9], may differ from other author’s notation by a constant factor. Conservation of momentum and energy for elastic collisions is expressed as p μ + q μ = p μ + q μ .
(1.6)
cos θ = ( p μ − q μ )( pμ − qμ )/g 2 .
(1.7)
The angle θ is defined by def
This angle is well defined under (1.6), see [21, Lemma 3.15.3]. We now consider the center of momentum expression for the collision operator below. An alternate expression for the collision operator was derived in [24]; see [49] for an explanation of the connection between the expression from [24] and the one we give just now. One may use Lorentz transformations as described in [9] and [50] to reduce the delta functions and obtain Q( f, h) = dq dω vø σ (g, θ ) [ f ( p )h(q ) − f ( p)h(q)], (1.8) R3
S2
where vø = vø ( p, q) is the Møller velocity given by √ p q 2 p q 2 g s def − − × = . vø = vø ( p, q) = p0 q0 p0 q0 p0 q 0 The post-collisional momentum in the expression (1.8) can be written:
( p + q) · ω p+q g , + ω + (γ − 1)( p + q) p = 2 2 | p + q|2
g ( p + q) · ω p+q , − ω + (γ − 1)( p + q) q = 2 2 | p + q|2 √ where γ = ( p0 + q0 )/ s. The energies are then g p0 + q 0 + √ ω · ( p + q), 2 2 s g p0 + q 0 − √ ω · ( p + q). q0 = 2 2 s
(1.9)
(1.10)
p0 =
(1.11)
These clearly satisfy (1.6). The angle further satisfies cos θ = k · ω with k = k( p, q) and |k| = 1. This is the expression for the collision operator that we will use. For a smooth function h( p) the collision operator satisfies ⎛ ⎞ 1 dp ⎝ p ⎠ Q(h, h)( p) = 0. R3 p0 By integrating the relativistic Boltzmann equation (1.1) and using this identity we obtain the conservation of mass, momentum and energy for solutions as ⎛ ⎞ 1 d dx dp ⎝ p ⎠ F(t) = 0. dt T3 R3 p 0
Asymptotic Stability for Soft Potentials
533
Furthermore the entropy of the relativistic Boltzmann equation is defined as def dx dp F(t, x, p) ln F(t, x, p). H(t) = − T3
R3
The celebrated Boltzmann H-Theorem is then formally d H(t) ≥ 0, dt which says that the entropy of solutions is increasing as time passes. Notice that the steady state relativistic Maxwellians (1.3) maximize the entropy which formally implies convergence to (1.3) in large time. It is this formal reasoning that our main results make mathematically rigorous in the context of perturbations of the relativistic Maxwellian for a general class of cross-sections. 2. Statement of the Main Results We are now ready to discuss in detail our main results. We define the standard perturbation f (t, x, p) to the relativistic Maxwellian (1.3) as √ def F = J + J f. With (1.6) we observe that the quadratic collision operator (1.8) satisfies Q(J, J ) = 0. Then the relativistic Boltzmann equation (1.1) for the perturbation f = f (t, x, p) takes the form ∂t f + pˆ · ∇x f + L( f ) = ( f, f ),
f (0, x, p) = f 0 (x, p).
(2.1)
The linear operator L( f ), as defined in (2.2), and the non-linear operator ( f, f ), defined in (2.5), are derived from an expansion of the relativistic Boltzmann collision operator (1.8). In particular, the linearized collision operator is given by √ √ def L(h) = −J −1/2 Q(J, J h) − J −1/2 Q( J h, J ) = ν( p)h − K (h). (2.2) Above the multiplication operator takes the form def ν( p) = dq dω vø σ (g, θ ) J (q). R3
S2
(2.3)
The remaining integral operator is def K (h) = dq dω vø σ (g, θ ) J (q) J (q ) h( p ) + J ( p ) h(q ) R3 S2 dq dω vø σ (g, θ ) J (q)J ( p) h(q) − R3
S2
= K 2 (h) − K 1 (h).
(2.4)
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R. M. Strain
The non-linear part of the collision operator is defined as √ √ def (h 1 , h 2 ) = J −1/2 Q( J h 1 , J h 2 ) = dq dω vø σ (g, θ ) J (q)[h 1 ( p )h 2 (q )−h 1 ( p)h 2 (q)]. (2.5) R3
S2
Without loss of generality we can assume that the mass, momentum, and energy conservation laws for the perturbation, f (t, x, p), hold for all t ≥ 0 as ⎛ ⎞ 1 dx dp ⎝ p ⎠ J ( p) f (t, x, p) = 0. (2.6) T3 R3 p0 We now state our conditions on the collisional cross-section. Hypothesis on the collision kernel. For soft potentials we assume the collision kernel in (1.8) satisfies the following growth/decay estimates: σ (g, ω) g −b σ0 (ω),
g σ (g, ω) √ g −b σ0 (ω). s
(2.7)
We also consider angular factors such that σ0 (ω) sinγ θ with γ > −2. Additionally σ0 (ω) ≥ 0 and σ0 (ω) should be non-zero on a set of positive measure. We suppose further that 0 < b < min(4, 4 + γ ). For hard potentials we make the assumption σ (g, ω) g a + g −b σ0 (ω),
(2.8) g σ (g, ω) √ g a σ0 (ω). s In addition to the previous parameter ranges we consider 0 ≤ a ≤ 2 + γ and also 0 ≤ b < min(4, 4 + γ ) (in this case we allow the possibility of b = 0). This hypothesis includes the full range of conditions which were introduced in [18] as a general physical assumption on the kernel (of course we add the corresponding necessary lower bound in each case); see also [15] for further discussions. Prior to stating our main theorem, we will need to introduce the following mostly standard notation. The notation A B will imply that a positive constant C exists such that A ≤ C B holds uniformly over the range of parameters which are present in the inequality and moreover that the precise magnitude of the constant is unimportant. The notation B A is equivalent to A B, and A ≈ B means that both A B and B A. We work with the L ∞ norm h ∞ = ess supx∈T3 , def
p∈R3 |h(x,
p)|.
If we only wish to take the supremum in the momentum variables we write |h|∞ = ess sup p∈R3 |h( p)|. def
Asymptotic Stability for Soft Potentials
535
We will additionally use the following standard L 2 spaces: def def 2 h 2 = dx dp |h(x, p)| , |h|2 = T3
R3
R3
dp |h( p)|2 .
Similarly in the sequel any norm represented by one set of lines instead of two only takes into account the momentum variables. Next we define the norm which measures the (very weak) “dissipation” of the linear operator h ν = def
T3
dx
R3
dp ν( p)|h(x, p)|2 .
The L 2 (Rn ) inner product is denoted ·, · . We use (·, ·) to denote the L 2 (Tn × Rn ) inner product. Now, for ∈ R, we define the following weight function: w = w ( p) = def
b/2
p0 , for the soft potentials: (2.7) p0 , for the hard potentials: (2.8).
(2.9)
For the soft potentials w1 ( p) ≈ 1/ν( p) (Lemma 3.1). We consider weighted spaces h ∞, = w h ∞ , h 2, = w h 2 , h ν, = w h ν . def
def
def
3 3 Here as usual L ∞ (T × R ) is the Banach space with norm · ∞, , etc. We will also use the momentum only counterparts of these spaces
|h|∞, = |w h|∞ , |h|2, = |w h|2 , |h|ν, = |w h|ν . def
def
def
We further need the following time decay norm: ||| f |||k, = sup (1 + s)k f (s) ∞, . def
(2.10)
s≥0
We are now ready to state our main results. We will first state our theorem for the soft potentials which is the main focus of this paper: 3 3 Theorem 2.1 (Soft Potential). Fix > 3/b. Given f 0 = f 0 (x, p) ∈ L ∞ (T × R ) which satisfies (2.6) initially, there is an η > 0 such that if f 0 ∞, ≤ η, then there exists a unique global mild solution, f = f (t, x, p), to Eq. (2.1) with soft potential kernel (2.7). For any k ≥ 0, there is a k = k (k) ≥ 0 such that
f ∞, (t) ≤ C,k (1 + t)−k f 0 ∞,+k . These √ solutions are continuous √ if it is so initially. We further have positivity, i.e. F = J + J f ≥ 0 if F0 = J + J f 0 ≥ 0. We point out that k (0) = 0 in the above theorem, and k (k) ≥ k can in general be computed explicitly from our proof. Our approach also applies to the hard potentials and in that case we state the following theorem which can be proven using the same methods.
536
R. M. Strain
3 3 Theorem 2.2 (Hard Potential). Fix > 3/2. Given f 0 = f 0 (x, p) ∈ L ∞ (T × R ) which satisfies (2.6) initially, there is an η > 0 such that if f 0 ∞, ≤ η, then there exists a unique global mild solution, f = f (t, x, p), to Eq. (2.1) with hard potential kernel (2.8) which further satisfies for some λ > 0 that
f ∞, (t) ≤ C e−λt f 0 ∞, . These √ solutions are continuous √ if it is so initially. We further have positivity, i.e. F = J + J f ≥ 0 if F0 = J + J f 0 ≥ 0. Previous results for the hard potentials are as follows. In 1993 Glassey and Strauss [24] proved asymptotic stability such as Theorem 2.2 in L ∞ with > 3/2. They consider collisional cross-sections which satisfy (2.8) for the parameters b ∈ [0, 1/2), a ∈ [0, 2 − 2b) and either γ ≥ 0 or 1 2 − 2b − a |γ | < min 2 − a, − b, , 2 3 which in particular implies γ > − 21 if b = 0 say. They further assume a related growth bound on the derivative of the cross-section ∂σ ∂g . In [36] this growth bound was removed while the rest of the assumptions on the cross-section from [24] remained the same. These results also sometimes work in smoother function spaces, and we note that we could also include space-time regularity to our solutions spaces. However for the relativistic Boltzmann equation in (1.1) the issue of adding momentum derivatives is more challenging. In recent years many new tools have been developed to solve these problems. A method was developed in non-relativistic kinetic theory to study the soft potential Boltzmann equation with angular cut-off by Guo in [30]. This approach makes crucial use of the momentum derivatives, and Sobolev embeddings to control the singular kernel of the collision operator. Yet in the context of relativistic interactions, high derivatives of the post-collisional variables (1.10) create additional high singularities which are hard to control. Worse in the more common relativistic variables from [24], derivatives of the post-collisional momentum exhibit enough momentum growth to preclude hope of applying the method from [30]; these growth estimates on the derivatives were known much earlier in [23]. Notice also that the methods for proving time decay, such as [10,52,53], require working in the context of smooth solutions. We would also like to mention recent developments on Landau Damping [44] proving exponential decay with analytic regularity. Furthermore we point out very recent results proving rapid time decay of smooth perturbative solutions to the Newtonian Boltzmann equation without the Grad angular cut-off assumption as in [25–27]. In this paper however we avoid smooth function spaces in particular because of the aforementioned problem created by the relativistic post-collisional momentum. Other recent work [33] developed a framework to study near Maxwellian boundary value problems for the hard potential Newtonian Boltzmann equation in L ∞ . In particular a key component of this analysis was to consider solutions to the Boltzmann equation (2.1) after linearization as ∂t + pˆ · ∇x + L f = 0, f (0, x, p) = f 0 (x, p), (2.11) with L defined in (2.2). The semi-group for this equation (relativistic or not) will satisfy a certain ‘A-Smoothing property’ which was pioneered by Vidav [56] about 40
Asymptotic Stability for Soft Potentials
537
years ago. This A-Smoothing property which appears at the level of the second iterate of the semi-group, as seen below in (4.4), has been employed effectively for instance in [21,24,33,34]. Related to this a new compactness connected to a similar iteration has been observed in the ‘mixing lemma’ of [41–43]. The key new step in [33] was to estimate the second iterate in L 2 rather than L ∞ and then use a linear decay theory in L 2 which does not require regularity and is exponential for the hard potentials case that paper considered. Note further that a method was developed in [52,53] to prove rapid polynomial decay for the soft potential Newtonian Boltzmann and Landau equations; this is related to the articles [5,6,10,55], all of which make use of smooth function spaces. In the present work we adapt the method from [53] to prove rapid L 2 polynomial decay of solutions to the linear equation (2.11) without regularity, and we further adapt the L 2 estimate from [33] to control the second iterate. This approach, for the soft potentials in particular, yields global bounds and slow decay, O(1/t), of solutions to (2.11) in L ∞ . The details and complexity of this program are however intricate in the relativistic setting. And fortunately, this slow decay is sufficient to just barely control the nonlinearity and prove global existence to the full non-linear problem as in Theorem 2.1. It is not clear how to apply the above methods to establish the rapid “almost exponential” polynomial decay from Theorem 2.1 in this low regularity L ∞ framework. To prove the rapid decay in Theorem 2.1 our key contribution is to perform a new high order expansion of the remainder term, R1 ( f ), from (4.15). This is contained in Proposition 6.1 and its proof. This term, R1 ( f ), is the crucial problematic term which only appears to exhibit first order decay. More generally, the main difficulty with proving rapid decay for the soft potentials is created by the high momentum values, where the time decay is diluted by the momentum decay. This results in the generation of additional weights, typically one weight for each order of time decay. At the same time the term R1 ( f ) only allows us to absorb one weight, w1 ( p), and therefore only appears to allow one order of time decay. In our proof of Proposition 6.1 we are able to overcome this apparent obstruction by performing a new high order expansion for k ≥ 2 as Rk ( f ) = G k ( f ) + Dk ( f ) + Nk ( f ) + L k ( f ) + Rk+1 ( f ). (The expansion from R1 ( f ) to R2 ( f ) requires a slightly different approach.) At every level of this expansion we can peel of each of the terms G k ( f ), Dk ( f ), Nk ( f ), and L k ( f ) which will for distinct reasons exhibit rapid polynomial decay to any order. In particular we use an L 2 estimate for L k ( f ) which crucially makes use of the bounded velocities that come with special relativity. On the other hand the last term Rk+1 ( f ) will be able to absorb k + 1 momentum weights, and therefore it will be able to produce time decay up to the order k + 1. By continuing this expansion to any finite order, we are able to prove rapid polynomial decay. We hope that this expansion will be useful in other relativistic contexts. The rest of this paper is organized as follows. In Sect. 3 we prove L 2 decay of solutions to the linearized relativistic Boltzmann equation (2.11). Then in Sect. 4 we prove global L ∞ bounds and slow decay of solutions to (2.11). Following that in Sect. 5 we prove nonlinear L ∞ bounds using the slow linear decay, and we thereby conclude global existence. In the remaining Sections 6 and 7 we prove linear and non-linear rapid decay respectively. Then in the Appendix we give an exposition of a derivation of the kernel of the compact part of the linear operator from (2.4).
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R. M. Strain
3. Linear L 2 Bounds and Decay It is our purpose in this section to prove global in time L 2 bounds for solutions to the linearized Boltzmann equation (2.11) with initial data in the same space L 2 . We will then prove high order, almost exponential decay for these solutions. We begin by stating a few important lemmas, and then we use them to prove the desired integral bounds (3.5) and the decay it implies in Theorem 3.7. We will prove these lemmas at the end of the section. Lemma 3.1. Consider (2.3) with the soft potential collision kernel (2.7). Then −b/2
ν( p) ≈ p0
.
−b/2 More generally, R3 dq S2 dω vø σ (g, θ ) J α (q) ≈ p0 for any α > 0. We will next look at the “compact” part of the linear operator K . The most difficult part is K 2 from (2.4). We will employ a splitting to cut out the singularity. The new element of this splitting is the Lorentz invariant argument: g. Given a small > 0, choose a smooth cut-off function χ = χ (g) satisfying 1 if g ≥ 2 χ (g) = (3.1) 0 if g ≤ . Now with (3.1) and (2.4) we define def 1−χ dq dω (1 − χ (g)) vø σ (g, θ ) J (q) J (q ) h( p ) K 2 (h) = R3 S2 + dq dω (1 − χ (g)) vø σ (g, θ ) J (q) J ( p ) h(q ). R3
S2
(3.2)
Define K 1 (h) similarly. We use the splitting K = K 1−χ + K χ . A splitting with the same goals has been previously used for the Newtonian Boltzmann equation in [53]. The advantage for soft potentials, on the singular region, is that one has exponential decay in all momentum variables. Then on the region away from the singularity we are able to extract a modicum of extra decay which is sufficient for the rest of the estimates in this paper, see Lemma 3.2 just below. In the sequel we will use the Hilbert-Schmidt form for the non-singular part. The following representation is derived in the Appendix: χ χ K i (h) = dq ki ( p, q) h(q), i = 1, 2. 1−χ
def
R3
χ
We will also record the kernel ki ( p, q) below in (3.9) and (3.10). We have Lemma 3.2. Consider the soft potentials (2.7). The kernel enjoys the estimate χ
0 ≤ k2 ( p, q) ≤ Cχ ( p0 q0 )−ζ ( p0 + q0 )−b/2 e−c| p−q| , Cχ , c > 0, χ
with ζ = min {2 − |γ |, 4 − b, 2} /4 > 0. This estimate also holds for k1 . def
Asymptotic Stability for Soft Potentials
539
We remark that for certain ranges of the parameters γ and b the decay in Lemma 3.2 χ could be improved somewhat. In particular the term k1 in (3.9) clearly yields exponential decay. However what is written above is sufficient for our purposes. We will use the decomposition given above and the decay of the kernels in Lemma 3.2 to establish the following lemma. Lemma 3.3. Fix any small η > 0, we may decompose K from (2.4) as K = Kc + Ks , where K c is a compact operator on L 2ν . In particular for any ≥ 0, and for some R = R(η) > 0 sufficiently large we have | w2 K c h 1 , h 2 | ≤ Cη |1≤R h 1 |2 |1≤R h 2 |2 . Above 1≤R is the indicator function of the ball of radius R. Furthermore | w2 K s h 1 , h 2 | ≤ η |h 1 |ν, |h 2 |ν, . This estimate will be important for proving the coercivity of the linearized collision operator, L, away from its null space. More generally, from the H-theorem L is nonnegative and for every fixed (t, x) the null space of L is given by the five dimensional space [21]: √ √ √ √ √ def J , p1 J , p2 J , p3 J , p0 J . (3.3) N = span We define the orthogonal projection from L 2 (R3 ) onto the null space N by P. Further expand Ph as a linear combination of the basis in (3.3): ⎧ ⎫ 3 ⎨ ⎬√ def Ph = a h (t, x) + bhj (t, x) p j + ch (t, x) p0 J. (3.4) ⎩ ⎭ j=1
We can then decompose f (t, x, p) as f = P f + {I − P} f. With this decomposition we have Lemma 3.4. L ≥ 0. Lh = 0 if and only if h = Ph, and ∃δ0 > 0 such that Lh, h ≥ δ0 |{I − P}h|2ν . This last statement on coercivity holds as an operator inequality at the functional level. The following lemma is a well-known statement of the linearized H-Theorem for solutions to (2.11) which was shown in the non-relativistic case in [33]. Lemma 3.5. Given initial data f 0 ∈ L 2 T3 × R3 for some ≥ 0, which satisfies (2.6) initially, consider the corresponding solution, f , to (2.11) in the sense of distributions. Then there is a universal constant δν > 0 such that 1 1 ds {I − P} f 2ν (s) ≥ δν ds P f 2ν (s). 0
0
540
R. M. Strain
We will give just one more operator level inequality. Lemma 3.6. Given δ ∈ (0, 1) and ≥ 0, there are constants C, R > 0 such that w2 Lh, h ≥ δ|h|2ν, − C|1≤R h|22 . Notice that Lemma 3.6 follows easily from Lemma 3.3. With these results, we can prove the following energy inequality for any ≥ 0: t f 22, (t) + δ ds f 2ν, (s) ≤ C f 0 22, , ∃δ , C > 0, (3.5) 0
as long as f 0 22, is finite. We will prove this first for = 0, and then for arbitrary > 0. In the first case we multiply (2.11) with f and integrate to obtain t ds (L f, f ) = f 0 22 . f 22 (t) + 0
First suppose that t ∈ {1, 2, . . .}. By Lemma 3.4 we have
t
ds (L f, f ) =
0
t−1
1
ds (L f, f )(s + j) ≥
j=0 0
=
δ0 2
t−1
j=0
t−1 1 j=0 0
1
δ0
ds {I − P} f 2ν (s + j)
0
ds {I − P} f 2ν (s + j)
t−1 δ0 1 + ds {I − P} f 2ν (s + j). 2 0 j=0
Then by Lemma 3.5 the second term above satisfies the lower bound t−1 t−1 δ0 1 δ 0 δν 1 ds {I − P} f 2ν (s + j) ≥ ds P f 2ν (s + j). 2 2 0 0 j=0
j=0
This follows in particular because f j (s, x, v) = f (s + j, x, p) satisfies the linearized Boltzmann equation (2.11) on the interval 0 ≤ s ≤ 1. Collecting the previous two estimates yields def
0
t
ds (L f, f ) ≥
t−1 δ0 1 ds {I − P} f 2ν (s + j) 2 0 j=0
+
t−1 δ0 δν 1 ds P f 2ν (s + j) 2 0 j=0
≥ δ˜
t−1 j=0 0
1
ds
f 2ν (s
+ j) = δ˜
0
t
ds f 2ν (s),
Asymptotic Stability for Soft Potentials
541
with δ˜ = 21 min δ02δν , δ20 > 0. Plugging this estimate into the last one establishes the energy inequality (3.5) for = 0 and t ∈ {1, 2, . . . }. For an arbitrary t > 0, we choose m ∈ {0, 1, 2, . . . } such that m ≤ t ≤ m + 1. We then split the time integral as [0, t] = [0, m] ∪ [m, t]. For the time interval [m, t] we have t f (t) 22 + ds (L f, f ) = f (m) 22 . (3.6) m
Since L ≥ 0 by Lemma 3.4, we see that f (t) 22 ≤ f (m) 22 . We then have t δ˜ m 2 ds (L f, f ) + ds f 2ν (s) ≤ Ck f 0 22 . f 2 (t) + 2 0 m Furthermore with Lemma 3.6 for Cδ > 0 independent of t we obtain t t t 2 ds (L f, f ) ≥ δ ds f ν (s) − Cδ ds 1≤R f 22 (s) m m m t ds f 2ν (s) − Cδ sup f 22 (s). ≥δ m
(3.7)
m≤s≤t
However we have already shown that supm≤s≤t f 22 (s) ≤ f 22 (m) ≤ C f 0 22 . Collecting the last few estimates for any t > 0 we have (3.5) for = 0. For > 0, we multiply Eq. (2.11) with w2 f and integrate to obtain t f 22, (t) + ds (w2 L f, f ) = f 0 22, . 0
In this case using Lemma 3.6 we have t t 2 2 ds (w L f, f ) ≥ δ ds f ν, (s) − Cδ,
t
Adding this estimate to the line above it, we obtain t f 22, (t) + δ ds f 2ν, (s) ≤ f 0 22, + Cδ,
t
0
0
0
0
0
ds f 2ν (s).
ds f 2ν (s).
Now the just proven integral inequality (3.5) in the case = 0 (as an upper bound for the integral on the right-hand side) establishes the claimed energy inequality (3.5) for any > 0. In fact we prove a more general time decay version of this inequality in (3.51). These will imply the following rapid decay theorem. Theorem 3.7. Consider a solution f (t, x, p) to the linear Boltzmann equation (2.11) with data f 0 2,+k < ∞ for some , k ≥ 0. Then f 2, (t) ≤ C,k (1 + t)−k f 0 2,+k . This allows “almost exponential” polynomial decay of any order. Theorem 3.7 is the main result of this section. We now proceed to prove each of Lemma 3.1, Lemma 3.2, Lemma 3.3, Lemma 3.4, Lemma 3.5, and then Theorem 3.7 in order. These proofs will complete this section on linear decay.
542
R. M. Strain
Proof of Lemma 3.1. We will use the soft potential hypothesis for the collision kernel (2.7) to estimate (2.3). For α > 0 we more generally consider π def α dq vø J (q) dθ σ (g, θ ) sin θ. να ( p) = R3
0
Initially we record the following pointwise estimates: | p − q| √ ≤ g ≤ | p − q|, and g ≤ 2 p0 q0 , √ p0 q 0
(3.8)
see [24, Lemma 3.1]. However notice that in [24] their “g” is actually defined to be 1/2 times our “g”. With the Møller velocity (1.9), we thus also have s = 4 + g 2 p0 q0 , vø 1. For b ∈ (1, 4), these estimates including (3.8) yield √ √ s 1−b s ( p0 q0 )(b−1)/2 ( p0 q0 )(b−2)/2 −b g . vø g = p0 q 0 p0 q0 | p − q|b−1 | p − q|b−1 We thus obtain
π ( p0 q0 )(b−2)/2 α J (q) dθ sin1+γ θ b−1 3 | p − q| R 0 π J α/2 (q) (b−2)/2 dq dθ sin1+γ θ p0 b−1 3 | p − q| R 0
να ( p)
dq
(b−2)/2 1−b p0
p0
−b/2
≈ p0
.
We note that the angular integral is finite since γ > −2: π dθ sin1+γ θ = Cγ < ∞. 0
In this case above and the cases below a key point is that J α/2 (q) −β dq ≈ p0 , ∀β < 3. | p − q|β R3 For b ∈ (0, 1), with (3.8) we alternatively have √ s 1−b ( p0 q0 )(1−b)/2 g ( p0 q0 )−b/2 . vø g −b = √ p0 q 0 p0 q 0 −b/2
Now in a slightly easier way than for the previous case we have να ( p) p0 For the lower bound with b ∈ (2, 4), we use (2.7), (3.8) and the estimate
g g 2−b ( p0 q0 )(2−b)/2 vø √ g −b = ≈ ( p0 q0 )−b/2 . p0 q 0 p0 q 0 s
.
Alternatively for b ∈ (0, 2), we use (3.8) to get the estimate
g | p − q| 2−b g 2−b 1 ≈ ( p0 q0 )(b−4)/2 | p − q|2−b . vø √ g −b = √ p0 q 0 p0 q 0 p0 q 0 s −b/2
We use these estimates to obtain να ( p) p0
as for the upper bound.
Asymptotic Stability for Soft Potentials
543
Now that the proof of Lemma 3.1 is complete, we develop the necessary formulation for the proof of Lemma 3.2. It is trivial to write the Hilbert-Schmidt form for the cut-off (3.1) part of K 1 from (2.4) as χ dω vø σ (g, θ ). (3.9) k1 ( p, q) = J (q)J ( p) χ (g) S2
Furthermore, we can write the Hilbert-Schmidt form for the cut-off (3.1) part of K 2 from (2.4) in the following somewhat complicated integral form
∞ √ g s 3/2 def χ −l 1+y 2 ,ψ k2 ( p, q) = c2 χ (g) dy e σ gp0 q0 sin (ψ/2) 0 y 1 + 1 + y2 × I0 ( j y). (3.10) 1 + y2 Here c2 > 0, and the modified Bessel function of index zero is defined by 2π 1 I0 (y) = e y cos ϕ dϕ. 2π 0
(3.11)
We are also using the simplifying notation sin (ψ/2) = def
√ 2g
. [g 2 − 4 + (g 2 + 4) 1 + y 2 ]1/2
(3.12)
Additionally l = ( p0 + q0 )/2,
j=
| p × q| . g
This derivation for a pure k2 operator appears to go back to [9,11], where it was done in the case of the alternate linearization F = J (1 + f ). The author gave a similar derivation with many details explained in full in [50], including the explicit form of the necessary Lorentz transformation; in particular Eq. (5.51) in this thesis. For the benefit χ of the reader, we have provided the derivation of k2 in the case of the linearization √ F = J + J f in the Appendix to this paper. Note that it is elementary to verify that (3.9) under (2.7) satisfies the estimate in Lemma 3.2. In the proof below we focus on the more involved estimate for (3.10). Proof of Lemma 3.2. We consider K 2 from (2.4) with Hilbert-Schmidt form represented by the kernel (3.10). From (2.7), we have the bound
sin (ψ/2) b γ g ,ψ σ sin ψ sin (ψ/2) g b+γ
γ sin (ψ/2) cosγ (ψ/2) . g (3.13) g We have just used the trigonometric identity sin ψ = 2 sin (ψ/2) cos (ψ/2) .
544
R. M. Strain
We estimate these angles in three cases. In each of the cases below we will repeatedly use the following known [24, p.317] estimates
y
2 1 + y2
≤ cos (ψ/2) ≤ 1.
(3.14)
The estimates above and below are proved for instance in [24, Lemma 3.1]: 1 1 sin (ψ/2) . √ 2 1/4 g g(1 + y 2 )1/4 s(1 + y )
(3.15)
Notice that in general from (3.10) and (3.13) we have the bound χ
k2 ( p, q)
∞ √ s 3/2 γ 2 g χ (g) dy e−l 1+y y I0 ( j y) gp0 q0 0
sin (ψ/2) b+γ × cosγ (ψ/2) . g
(3.16)
We will estimate this upper bound in three cases. Case 1. Take γ = − |γ | < 0 and b + γ = b − |γ | < 0. Then we have
sin (ψ/2) g
b+γ
γ
cos (ψ/2) s
−(b−|γ |)/2
1 (1 + y 2 )1/4
b−|γ |
y
−|γ |
1 + y2
s −(b−|γ |)/2 y −|γ | (1 + y 2 )−(b−3|γ |)/4 s −(b−|γ |)/2 y −|γ | (1 + y 2 )−(b+3γ )/4 . Above we have used (3.14) and (3.15). In this case from (3.16) we have χ k2 ( p, q)
∞ √ s 3/2 s −(b−|γ |)/2 −l 1+y 2 1+γ χ (g) dy e y I0 ( j y)(1 + y 2 )−(b+3γ )/4 gp0 q0 g |γ | 0 ∞ √ s (3+|γ |−b)/2 2 ≤ C χ (g) dy e−l 1+y y 1−|γ | I0 ( j y)(1 + y 2 )−(b−3|γ |)/4 . p0 q 0 0
Note that we have just used the > 0 from (3.1). From (2.7), b ∈ (0, 4 − |γ |) and γ ∈ (−2, 0). We have in this case b ∈ (0, |γ |). Hence b − 3 |γ | ∈ (−6, 0). We evaluate the relevant integral above as
∞
0
dy e−l
1
0
√
1+y 2 1−|γ |
dy e−l
y
√
I0 ( j y)(1 + y 2 )(3|γ |−b)/4 =
1+y 2 1−|γ |
y
∞
I0 ( j y) +
dy e−l
√
1
∞
dy + 0 1+y 2
dy 1
y I0 ( j y)(1 + y 2 )(|γ |−b)/4 .
1
(3.17) For the unbounded integral we have used the estimate y −|γ | (1 + y 2 )−2|γ |/4 . In this case |γ | − b ∈ (0, 2) since 0 < b < |γ |.
Asymptotic Stability for Soft Potentials
545
To estimate the remaining integrals above we use the precise theory of special functions, see e.g. [39,46]. We define ∞ √ 2 def dy e−l 1+y y I0 ( j y)(1 + y 2 )α/4 . K˜ α (l, j) = 0
Then for α ∈ [−2, 2] from [24, Cor. 1 and Cor. 2] it is known that K˜ α (l, j) ≤ Cl 1+α/2 e−c| p−q| .
(3.18)
We also define def I˜η (l, j) =
1
dy e−l
√
1+y 2 1−η
y
I0 ( j y).
0
Then for η ∈ [0, 2) from [24, Lemma 3.6] we have the asymptotic estimate: √ 2 2 I˜η (l, j) ≤ Ce−c l − j ≤ Ce−c| p−q|/2 .
(3.19)
We will use these estimates in each of the cases below. Thus in this Case 1 by (3.19) and (3.18) the integral in (3.17) is e−c| p−q|/2 + l 1+(|γ |−b)/2 e−| p−q|/2 l 1+(|γ |−b)/2 e−c| p−q|/2 . We may collect the last few estimates together to obtain χ
k2 ( p, q) ≤ C
s (3+|γ |−b)/2 χ (g) ( p0 + q0 )1+(|γ |−b)/2 e−c| p−q|/2 . p0 q 0
Note that for any ∈ R we have s e−c| p−q|/2 ≤ C e−c| p−q|/4 . This follows trivially from (3.8) and s = 4 + g 2 ≤ 4 + | p − q|2 . Furthermore, for any ∈ [0, 2], we claim the following estimate ( p0 + q0 ) −c| p−q|/4 e ( p0 q0 )/2−1 e−c| p−q|/8 . p0 q 0
(3.20)
Using (3.20) with = 1 + |γ |/2, in this Case 1, we have the general estimate √ |γ |/2−1 χ p0 q 0 k2 ( p, q) ≤ C ( p0 + q0 )−b/2 e−c| p−q| . This is the desired estimate in the current range of exponents for Case 1. Before moving on to the next case, we establish the claim. Suppose that 1 |q| ≤ | p| ≤ 2|q|. 2 In this case (3.20) is obvious. If 21 |q| ≥ | p|, then we have | p − q| ≥ |q| − | p| ≥
1 |q|. 2
(3.21)
546
R. M. Strain
Whence ( p0 + q0 ) −c| p−q|/4 e ≤ C ( p0 q0 )−1 e−c| p−q|/8 e−c|q|/64 . p0 q 0 In the last splitting | p| ≥ 2|q|, then we alternatively have | p − q| ≥ | p| − |q| ≥
1 | p|. 2
(3.22)
Similarly in this situation ( p0 + q0 ) −c| p−q|/4 e ≤ C ( p0 q0 )−1 e−c| p−q|/8 e−c| p|/64 . p0 q 0 These last two stronger estimates establish (3.20). We move on to the next case. Case 2. We still consider γ = − |γ | < 0 but now b + γ = b − |γ | ≥ 0. We have −|γ |
b−|γ | sin (ψ/2) b+γ y 1 γ −b+|γ | cos (ψ/2) g g (1 + y 2 )1/4 2 1 + y2 g −(b−|γ |) y −|γ | (1 + y 2 )−(b−3|γ |)/4 ≤ C y −|γ | (1 + y 2 )−(b−3|γ |)/4 . We have used g ≥ on the support of χ (g) in (3.1). We also used (3.14) and (3.15). Then again from (3.16) we have ∞ √ s 3/2 2 χ dy e−l 1+y y 1−|γ | I0 ( j y)(1 + y 2 )−(b−3|γ |)/4 . k2 ( p, q) ≤ C p0 q 0 0 From (2.7), we have in this case that γ ∈ (−2, 0) and b ∈ [|γ | , 4 − |γ |). Hence for the exponent above b − 3 |γ | ∈ (−4, 4). Then the relevant integral above is bounded by ∞ √ 2 dy e−l 1+y y 1−|γ | I0 ( j y)(1 + y 2 )−(b−3|γ |)/4 = 0
1
dy e
−l
√
1+y 2 1−|γ |
y
∞
I0 ( j y) +
0
dy e−l
√
0 1+y 2
1
∞
dy +
dy 1
y I0 ( j y)(1 + y 2 )(|γ |−b)/4 .
1
Here we used the same estimates as in Case 1. In this case |γ | − b ∈ (−4, 0). Let ζ2 = max(−2, |γ | − b). Then in this case by (3.19) and (3.18) the above is ≤ Ce−c| p−q|/2 + Cl 1+ζ2 /2 e−| p−q|/2 ≤ Cl 1+ζ2 /2 e−c| p−q|/2 . We may collect the last few estimates together to obtain χ
k2 ( p, q) ≤ C
s 3/2 ( p0 + q0 )1+ζ2 /2 −c| p−q|/4 e . ( p0 + q0 )1+ζ2 /2 e−c| p−q|/2 ≤ C p0 q 0 p0 q 0
We will further estimate the quotient.
Asymptotic Stability for Soft Potentials
547
If ζ2 = |γ | − b, then 1 + |γ |/2 ∈ [1, 2) and (3.20) implies ( p0 + q0 )1+|γ |/2 ( p0 + q0 )1+ζ2 /2 −c| p−q|/4 e = ( p0 + q0 )−b/2 e−c| p−q|/4 p0 q 0 p0 q 0 ≤ ( p0 q0 )(|γ |−2)/4 ( p0 + q0 )−b/2 e−c| p−q|/8 . Alternatively, if ζ2 = −2 then 1 + ζ2 /2 = 0 and (3.20) implies ( p0 + q0 )b/2 ( p0 + q0 )1+ζ2 /2 −c| p−q|/4 e = ( p0 + q0 )−b/2 e−c| p−q|/4 p0 q 0 p0 q 0 ( p0 q0 )(b−4)/4 ( p0 + q0 )−b/2 e−c| p−q|/8 . In either situation χ
k2 ( p, q) ≤ C ( p0 q0 )−ζ ( p0 + q0 )−b/2 e−c| p−q|/8 , with ζ = min {2 − |γ |, 4 − b} /4 > 0. def
Case 3. In this last case γ = |γ | ≥ 0 and b + γ ≥ 0. From (3.14) and (3.15):
b+|γ |
1 sin (ψ/2) b+γ γ −b−|γ | cos (ψ/2) g g (1 + y 2 )1/4 g −(b+|γ |) (1 + y 2 )−(b+|γ |)/4 ≤ C (1 + y 2 )−(b+|γ |)/4 . We have again used g ≥ on the support of χ (g) in (3.1). In this case from (3.16) we have √ s 3/2 g |γ | ∞ 2 χ k2 ( p, q) ≤ C dy e−l 1+y y I0 ( j y)(1 + y 2 )−(b+|γ |)/4 . p0 q 0 0 From (2.7), b ∈ [0, 4) and b + |γ | ∈ [0, 4 + |γ |). Let ζ3 = min(2, b + |γ |) ≥ 0. Again using (3.18) we have ∞ √ 2 dy e−l 1+y y I0 ( j y)(1 + y 2 )−(b+|γ |)/4 ≤ Cl 1−ζ3 /2 e−c| p−q|/2 . 0
Hence χ
k2 ( p, q) ≤ C
s 3/2 g |γ | ( p0 + q0 )1−ζ3 /2 −c| p−q|/4 e . ( p0 + q0 )1−ζ3 /2 e−c| p−q|/2 ≤ C p0 q 0 p0 q 0
If ζ3 = 2 this estimate can be handled exactly as in Case 2. If ζ3 = b + |γ |: ( p0 + q0 )1−|γ |/2 ( p0 + q0 )1−ζ3 /2 −c| p−q|/4 e = ( p0 + q0 )−b/2 e−c| p−q|/4 p0 q 0 p0 q 0 ( p0 + q 0 ) ≤ ( p0 + q0 )−b/2 e−c| p−q|/4 p0 q 0 ( p0 q0 )−1/2 ( p0 + q0 )−b/2 e−c| p−q|/16 . χ
We have again used the estimate (3.20). In all of the cases we see that k2 ( p, q) satisfies the claimed bound from Lemma 3.2 with ζ = min {2 − |γ |, 4 − b, 2} /4 > 0. We could obtain a larger ζ in Case 3 if it was needed.
548
R. M. Strain
With the estimate for the Hilbert-Schmidt form just proven in Lemma 3.2, we will now prove the decomposition from Lemma 3.3. Proof of Lemma 3.3. We recall the splitting K = K 1−χ + K χ from (3.2). For K χ we χ χ have the kernel k χ = k2 − k1 from (3.9) and (3.10). For a given R ≥ 1, choose another smooth cut-off function φ R = φ R ( p, q) satisfying φ R ≡ 1, if | p| + |q| ≤ R/2, |φ R | ≤ 1, supp(φ R ) ⊂ {( p, q) | | p| + |q| ≤ R } .
(3.23)
We will use this cut-off with several different R’s in the cases below. Now we split the kernels k χ ( p, q) of the operator K χ into k χ ( p, q) = k χ ( p, q)φ R ( p, q) + k χ (1 − φ R ) = kcχ ( p, q) + ksχ ( p, q). We further define K s = K 1−χ + K sχ , def
def χ χ where K s (h) = R3 dq ks ( p, q) h(q). Then the compact part is given by
K c = K cχ , def
def χ χ where K c (h) = R3 dq kc ( p, q) h(q). Note that the compactness of K c (h) is evident from the integrability of the kernel. In the following we will show that the operators K c and K s satisfy the estimates claimed in Lemma 3.3. First off, for K c , from the Cauchy-Schwartz inequality we have 2 dq dp w2 ( p) kcχ ( p, q) |h 1 (q)h 2 ( p)| w K c (h 1 ), h 2 ≤
R3
≤
R3
dqdp w2 ( p) kcχ ( p, q) |h 1 (q)|2
×
dqdp
1/2
w2 ( p) kcχ ( p, q) |h 2 ( p)|2
1/2 .
χ
From the definition of kc ( p, q) and Lemma 3.2, we see that w2 ( p) kcχ ( p, q) ≤ C R e−c| p−q| 1≤R ( p) 1≤R (q), where 1≤R is the indicator function of the ball of radius R centered at the origin as defined in Lemma 3.3. By combining the last few estimates we clearly have the claimed estimate for K c from Lemma 3.3. χ χ In the remainder of this proof we estimate K s = K 1−χ + K s . For K s we have 2 χ ≤ K (h ), h
dq dp w2 ( p) ksχ ( p, q) |h 1 (q)h 2 ( p)| . w s 1 2 R3
R3
χ
With the definition of ks ( p, q) and Lemma 3.2, we obtain w 2 ( p) w2 ( p) ksχ ( p, q) = w2 ( p) k χ ( p, q) (1 − φ R ) ζ ( p0 + q0 )−b/2 e−c| p−q| . R
Asymptotic Stability for Soft Potentials
549
Furthermore, we claim that w2 ( p)e−c| p−q| w ( p)w (q)e−c| p−q|/2 .
(3.24)
By combining the last few estimates including (3.24), with Lemma 3.1, we have 1 e−c| p−q| 2 χ |h 1 (q)h 2 ( p)| dq dp w ( p)w (q) w K s (h 1 ), h 2 ζ R R3 ( p0 q0 )b/4 R3 |h 1 |ν, |h 2 |ν, . Rζ Since ζ > 0 we conclude our estimate here by choosing R > 0 sufficiently large. Notice that the size of this R above clearly depends upon > 0 from (3.1). To prove the claim in (3.24), we use the same general strategy which was used to prove (3.20). Indeed, if 21 |q| ≥ | p| then because of (3.21) we have w2 ( p)e−c| p−q|/2 e−c| p−q|/2 ≤ w2 (q)e−c|q|/4 e−c| p−q|/2 ≤ C e−c| p−q|/2 , which is better than (3.24). Alternatively if | p| ≥ 2|q|, then with (3.22) we have w2 ( p)e−c| p−q|/2 e−c| p−q|/2 ≤ w2 ( p)e−c| p|/4 e−c| p−q|/2 ≤ C e−c| p−q|/2 . The only remaining case is 21 |q| ≤ | p| ≤ 2|q| for which the estimate (3.24) is obvious. 1−χ 1−χ The last term to estimate is K 1−χ = K 2 − K 1 . Notice that 1−χ 1−χ K 1 (h) = dq k1 ( p, q) h(q), R3
where from (2.4) and (3.1), 1−χ
k1
( p, q) = (1 − χ (g))
J (q)J ( p)
S2
dω vø σ (g, θ ).
1−χ
To estimate K 1 we apply Cauchy-Schwartz to obtain 2 1−χ 1−χ dq dp w2 ( p) k1 ( p, q) |h 1 (q)h 2 ( p)| w K 1 (h 1 ), h 2 ≤ R3
R3
1−χ
dpdq w2 ( p)k1
≤
1/2 ( p, q)|h 1 (q)|2
×
dpdq
1−χ w2 ( p)k1 ( p, q)|h 2 ( p)|2
1/2 .
We will estimate the kernel of each term above. We further split 1−χ
k1
1−χ
( p, q) = k1
1−χ
( p, q)φ R ( p, q) + k1
1−χ
(1 − φ R ) = k1S
1−χ
+ k1L .
The value of R ≥ 1 used here is independent of the case considered previously. The R here will be independent of . From (2.7) and (1.9), in general we have √ √ s 1−b π s 1−b dω vø σ (g, θ ) g dθ sin1+γ θ g . (3.25) p0 q 0 p0 q 0 0 S2
550
R. M. Strain
For g ≤ 2 as in (3.1), with (3.8) we conclude that √ | p − q| ≤ 2 p0 q0 .
(3.26)
Furthermore, on the support of φ R we notice that additionally | p − q| ≤ 4 R. Then if 1−χ b ∈ (1, 4), with the formula for k1 ( p, q), (3.25) and then (3.8), we obtain √ s 1−b 1−χ 2 dp g J (q)J ( p) dp w ( p) k1S ( p, q) ≤ C R p0 q 0 | p−q|≤4 R ≤ CR dp g 1−b J 1/4 (q)J 1/4 ( p) ≤ CR
| p−q|≤4 R
| p−q|≤4 R
√ dp
p0 q 0 | p − q|
b−1 J 1/4 (q)J 1/4 ( p)
≤ C R 4−b 4−b J 1/8 (q). √ b−1 1/8 The last inequality above follows easily from p0 q 0 J (q)J 1/8 ( p) ≤ C and also J 1/8 ( p) dp ≤ C R 4−b 4−b . | p − q|b−1 | p−q|≤4 R Thus when b ∈ (1, 4) we have 2 1−χ w K 1S (h 1 ), h 2 ≤ C R 4−b |J 1/16 h 1 |2 |J 1/16 h 2 |2 .
(3.27)
This is much stronger than the desired estimate for = (R) > 0 chosen sufficiently small. Alternatively if b ∈ [0, 1] then with (3.25) we have 1−χ dp w2 ( p) k1S ( p, q) ≤ C dp J 1/4 (q)J 1/4 ( p) ≤ C R 3 3 J 1/4 (q). R3
| p−q|≤4 R
1−χ Thus when b ∈ [0, 1] we have w2 K 1 (h 1 ), h 2 ≤ C R 3 |J 1/8 h 1 |2 |J 1/8 h 2 |2 . This 1−χ
concludes our estimate for the part containing k1S ( p, q) for any fixed R after choosing = (R) > 0 small enough (depending on the size of R). 1−χ For the term involving k1L ( p, q) the estimate is much easier. In this case 1−χ 2 dp w ( p) k1L ( p, q) dp g 1−b (1−φ R ( p, q)) J 1/4 (q)J 1/4 ( p) R3
R3 −R/16
e
.
(3.28)
The same estimates hold for the other term in the inner product above. These estimates are independent of . We thus obtain the desired estimate for this term in the same way as for the last term; here we first choose R > 0 sufficiently large. 1−χ The last term to estimate is K 2 from (3.2). With (3.26) we see that √ p0 ≤ | p − q| + q0 ≤ 2 p0 q0 + q0 ≤ p0 + (1 + ) q0 . The first inequality in this chain can be found in [18, Ineq. A.1]. We conclude p0 q0 and similarly q0 p0 . For 0 < < 1/4 say the constant in these inequalities can be
Asymptotic Stability for Soft Potentials
551
chosen to not depend upon . Furthermore from (1.11), if g ≤ 2 and is small (say less than 1/8), then it is easy to show that p0 ≥
p0 + q 0 p0 + q 0 , q0 ≥ . 4 4
(3.29)
These post-collisional energies are also clearly bounded from above by p0 and q0 , so that all of these variables are comparable on (3.2). We thus have 1−χ K 2 (h 1 ), h 2
dωdqdp (1 − χ (g)) vø σ (g, θ )e−cq0 −cp0
R3 ×R3 ×S2 × h 1 ( p ) + h 1 (q ) |h 2 ( p)| .
With Cauchy-Schwartz we obtain
2 dωdqdp vø σ (g, θ )e−cq0 −cp0 h 1 ( p )
1/2
g≤2
dωdqdp vø σ (g, θ )e−cq0 −cp0 |h 2 ( p)|2
× g≤2
dωdqdp vø σ (g, θ )e
+
−cq0 −cp0
h 1 (q ) 2
−cq0 −cp0
|h 2 ( p)|
g≤2
×
dωdqdp vø σ (g, θ )e
1/2 1/2 1/2
2
.
g≤2
From (3.25) and the arguments just below it, for any small η > 0 we can estimate
dωdq vø σ (g, θ )e−cq0 −cp0 ≤ ηe−cp0 /2 . g≤2
Above of course we have η = η() → 0 as → 0, and by symmetry the same estimate holds if the roles of p and q are reversed. Since the kernels of the integrals above are invariant with respect to the relativistic pre-post collisional change of variables [23], which is justified for (1.10), we may apply it as dpdq =
p0 q0 dp dq . p0 q 0
Putting all of this together with (1.6) we have
1/2 1/2 1−χ −cp0 2 −cp0 2 |h 1 ( p)| |h 2 ( p)| dp e . dp e K 2 (h 1 ), h 2 ≤ η() For > 0 small enough, this is stronger than the estimate which we wanted to prove. We have now completed the proof of this lemma. We will at this time use the prior lemma to prove the next lemma.
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R. M. Strain
Proof of Lemma 3.4. Most of this lemma is standard, see e.g. [21]. We only prove the coercive lower bound for the linear operator. Assuming the converse grants a sequence of functions h n ( p) satisfying Ph n = 0, |h n |2ν = νh n , h n = 1 and Lh n , h n = |h n |2ν − K h n , h n ≤
1 . n
Thus {h n } is weakly compact in | · |ν with limit point h 0 . By weak lower-semi continuity |h 0 |ν ≤ 1. Furthermore, Lh n , h n = 1 − K h n , h n . We claim that lim K h n , h n = K h 0 , h 0 .
n→∞
The claim will follow from the prior Lemma 3.3. This claim implies 0 = 1 − K h 0 , h 0 . Or equivalently Lh 0 , h 0 = |h 0 |2ν − 1. Since L ≥ 0, we have |h 0 |2ν = 1 which implies h 0 = Ph 0 . On the other hand since h n = {I − P}h n the weak convergence implies h 0 = {I − P}h 0 . This is a contradiction to |h 0 |2ν = 1. We now establish the claim. For any small η > 0, we split K = K c + K s as in Lemma 3.3. Then | K s h n , h n | ≤ η. Also K c is a compact operator in L 2ν so that lim |K c h n − K c h 0 |ν = 0.
n→∞
We conclude by first choosing η small and then sending n → ∞.
We are now ready to prove Lemma 3.5. We point out that similar estimates, but with strong Sobolev norms, have been established in recent years [28,29,31,51] via the macroscopic equations for the coefficients a, b and c. We will use the approach from [33], which exploits the hyperbolic nature of the transport operator, to prove our Lemma 3.5 in the low regularity L 2 setting. Proof of Lemma 3.5. We use the method of contradiction, if Lemma 3.5 is not valid then for any k ≥ 1 we can find a sequence of normalized solutions to (2.11) which we denote by f k that satisfy 1 1 1 ds {I − P} f k 2ν (s) ≤ ds P f k 2ν (s). k 0 0 Equivalently the normalized function Z k (t, x, p) = 1 def
0
f k (t, x, p) ds P f k 2ν (s)
,
Asymptotic Stability for Soft Potentials
553
satisfies
1
0
ds PZ k 2ν (s) = 1,
and
1 0
ds {I − P}Z k 2ν (s) ≤
1 . k
Moreover, from (2.6) the following integrated conservation laws hold: ⎛ ⎞ 1 1 ds dx dp ⎝ p ⎠ J ( p) Z k (s, x, p) = 0. T3 R3 0 p0
(3.30)
(3.31)
Furthermore, since f k satisfies (2.11), so does Z k . Clearly sup k≥1 0
1
ds Z k 2ν (s) 1.
(3.32)
Hence there exists Z (t, x, p) such that Z k (t, x, p) Z (t, x, p), as k → ∞, 1 1 weakly with respect to the inner product 0 ds (·, ·)ν of the norm 0 ds · 2ν . Furthermore, from (3.30) we know that 1 ds {I − P}Z k 2ν (s) → 0. (3.33) 0
We conclude that {I − P}Z k → {I − P}Z and {I − P}Z = 0 from (3.33). It is then straightforward to verify that
1
PZ k → PZ weakly in 0
ds · 2ν .
Hence √ Z (t, x, p) = PZ = {a(t, x) + p · b(t, x) + p0 c(t, x)} J .
(3.34)
At the same time notice that L Z k = L{I − P}Z k and we have (3.33). Send k → ∞ in (2.11) for Z k to obtain, in the sense of distributions, that ∂t Z + pˆ · ∇x Z = 0.
(3.35)
At this point our main strategy is to show, on the one hand, Z has to be zero from (3.33), the periodic boundary conditions, and the hyperbolic transport equation (3.35), 1 and (3.31). On the other hand, Z k will be shown to converge strongly to Z in 0 ds · 2ν with the help of the averaging lemma [12] in the relativistic formulation [47] and 1 2 0 ds Z ν (s) > 0. This would be a contradiction.
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R. M. Strain
Strong convergence. We begin by proving the strong convergence, and then later we will prove that the limit is zero. Split Z k (t, x, p) as Z k (t, x, p) = PZ k + {I − P}Z k =
5
Z k (t, x, ·), e j e j ( p) + {I − P}Z k ,
j=1
where e j ( p) are an orthonormal basis for (3.3) in · ν . 1 To prove the strong convergence in 0 ds · 2ν , recalling (3.33), we will show
1
1≤ j≤5 0
ds Z k , e j e j − Z , e j e j 2ν (s) → 0.
Since e j ( p) are smooth with exponential decay when p → ∞, it suffices to prove
1
ds 0
T3
d x | Z k , e j − Z , e j |2 → 0.
(3.36)
We will now establish (3.36) using the averaging lemma. 3 3 Choose any small η > 0 and a smooth cut off function χ 1 (t, x, p) in (0, 1) × T × R 1 such that χ1 (t, x, p) ≡ 1 in [η, 1 − η] × T3 × | p| ≤ η and χ1 (t, x, p) ≡ 0 outside [η/2, 1 − η/2] × T3 × | p| ≤ η2 . Split Z k (t, x, ·), e j = (1 − χ1 ) Z k (t, x, ·), e j + χ1 Z k (t, x, ·), e j .
(3.37)
For the first term above, notice that 1 ds d x | (1 − χ1 ) |Z k − Z | , e j |2 0
T3
1
0
(1 − χ1 ) |Z k | |e j | + + + . 2
2
T3 ×R3
0≤s≤η
1−η≤s≤1
1
2
0
T3 ×R3
(1 − χ1 )2 |Z |2 |e j |2
| p|≥1/η
Since e j = e j ( p) has exponential decay in | p| we have the crude bound |e j ( p)| ≤ C η, for | p| ≥ 1/η. Thus all three integrals above can be bounded by Cη sup Z k 22 (s) + Z 22 (s) ≤ C Z k (0) 22 + Z (0) 22 η ≤ Cη,
(3.38)
0≤s≤1
which will hold for Z k uniformly in k. These bounds follow from (2.11) and (3.35). The second term in (3.37), χ1 Z k (t, x, ·), e j , is actually uniformly bounded in H 1/4 ([0, 1] × T3 ). To prove this, notice that (2.11) implies that χ1 Z k satisfies [∂t + pˆ · ∇x ] (χ1 Z k ) = −χ1 L[Z k ] + Z k [∂t + pˆ · ∇x ]χ1 .
(3.39)
Asymptotic Stability for Soft Potentials
555
The goal is to show that each term on the right-hand side of (3.39) is uniformly bounded in L 2 ([0, 1] × T3 × R3 ). This would imply the H 1/4 bound by the averaging lemma. It clearly follows from (3.32) that Z k [∂t + pˆ · ∇x ]χ1 ∈ L 2 ([0, 1] × T3 × R3 ). Furthermore, it follows from Lemma 3.1 and Lemma 3.3 with = 0 that dp |χ1 L[Z k ]|2 ≤ dp |χ1 ν( p)Z k |2 + dp |χ1 K (Z k )|2 |Z k (t, x)|2ν . R3
R3
R3
Thus the right-hand side of (3.39) is uniformly bounded in L 2 ([0, 1] × T3 × R3 ). By the averaging lemma [12,47] it follows that χ1 Z k (t, x, ·), e j = dp χ1 (t, x, p)Z k (t, x, p)e j ( p) ∈ H 1/4 ([0, 1] × T3 ). R3
This holds uniformly in k, which implies up to a subsequence that χ1 Z k (t, x, ·), e j → χ1 Z (t, x, ·), e j in L 2 ([0, 1] × T3 ). Combining this last convergence with (3.38) concludes the proof of (3.36). As a consequence of this strong convergence we have 1 ds Z k − Z 2ν (s) → 0, 0
which implies that
1 0
ds PZ 2ν (s) = 1.
Now if we can show that at the same time PZ = 0, then we have a contradiction. The limit function Z (t, x, p) = 0. By analysing the equations satisfied by Z , we will show that Z must be trivial. We will now derive the macroscopic equations for PZ ’s coefficients a, b and c. Since {I − P}Z = 0, we see that PZ solves (3.35). We plug the expression for PZ in (3.34) into Eq. (3.35), and expand in the basis (3.3) to obtain p j j p j pi i 0 0 j ∂ a+ ∂ a + ∂ b j + p j ∂ b j + ∂ c + p0 ∂ 0 c J 1/2 ( p) = 0, p0 p0 where ∂ 0 = ∂t and ∂ j = ∂x j . By a comparison of coefficients, we obtain the important relativistic macroscopic equations for a(t, x), bi (t, x) and c(t, x): ∂ 0 c = 0, ∂ c + ∂ 0 bi = 0, (1 − δi j )∂ i b j + ∂ j bi = 0,
(3.40) (3.41) (3.42)
∂ i a = 0, ∂ 0 a = 0,
(3.43) (3.44)
i
which hold in the sense of distributions.
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R. M. Strain
We will show that these Eqs. (3.40)–(3.44), combined with the periodic boundary conditions imply that any solution to (3.40)–(3.44) is a constant. Then the conservation laws (3.31) will imply that the constant can only be zero. We deduce from (3.44) and (3.43) that a(t, x) = a(0, x), a.e. x, t, a(s, x1 ) = a(s, x2 ), a.e. s, x1 , x2 . Thus a is a constant for almost every (t, x). From (3.40), we have c(t, x) = c(x) for a.e. t. Then from (3.41) for some spatially dependent function b˜i (x) we have bi (t, x) = ∂ i c(x)t + b˜i (x). From (3.42) and the above 0 = ∂ i bi (t, x) = ∂ i ∂ i c(x)t + ∂ i b˜i (x), which implies ∂ i ∂ i c(x) = 0, ∂ i b˜i (x) = 0. Similarly if i = j we have
0 = ∂ j bi (t, x) + ∂ i b j (t, x) = ∂ j ∂ i c(x) + ∂ i ∂ j c(x) t + ∂ j b˜i (x) + ∂ i b˜ j (x),
so that ∂ j ∂ i c(x) = −∂ i ∂ j c(x), ∂ j b˜i (x) = −∂ i b˜ j (x), which implies ∂ i c(x) = ci , and c(x) is a polynomial. By the periodic boundary conditions c(x) = c˜ ∈ R. We further observe that bi (t, x) = bi (x) is a constant in time a.e. from (3.41) and the above. From (3.42) again ∂ i bi = 0 so that trivially ∂ i ∂ i bi = 0. Moreover (3.42) further implies that ∂ j ∂ j bi = 0. Thus for each i, bi (x) is a periodic polynomial, which must be a constant: bi (x) = bi ∈ R. We compute from (3.34) that 1/2 dp pi J ( p) Z (t, x, p) = bi dp pi2 J ( p), i = 1, 2, 3, R3 R3 1/2 dp J ( p) Z (t, x, p) = a dp J ( p) + c dp p0 J ( p), 3 3 3 R R R dp p0 J 1/2 ( p) Z (t, x, p) = a dp p0 J ( p) + c dp p02 J ( p). R3
R3
As in [51] we define J ( p)dp = 1, ρ0 = ρ1 = R3
R3
R3
p0 J ( p)dp, ρ2 =
R3
p02 J ( p)dp.
ρ1 ρ0 is invertible because ρ02 < ρ1 ρ2 . It then follows ρ0 ρ2 from the conservation law (3.31) which is satisfied by the limit function Z (t, x, p) that the constants a, bi , c must indeed be zero.
Now the matrix given by
Asymptotic Stability for Soft Potentials
557
We have now completed all of the L 2 energy estimates for the linearized relativistic Boltzmann equation (2.11). We will now use (3.5), Lemma 3.4, Lemma 3.5, and Lemma 3.6 to prove Theorem 3.7. This will be the final proof in this section. Proof of Theorem 3.7. For k ≥ 0, we define the time weight function by Pk (t) = (1 + t)k . def
(3.45)
For a solution f (t, x, p) to the linear Boltzmann equation (2.11), with Pk (t) from (3.45), Pk (t) f (t) satisfies the equation (3.46) ∂t + pˆ · ∇x + L (Pk (t) f (t)) − k Pk−1 (t) f (t) = 0. For the moment, suppose that t = m and m ∈ {1, 2, 3, . . .}. For the time interval [0, m], we multiply Pk (t) f (t) with (3.46) and take the L 2 energy estimate over 0 ≤ s ≤ m to obtain m 2 P2k (m) f 2 (m) + ds P2k (s)(L f, f ) 0 m −k ds P2k−1 (s) f 22 (s) = f 0 22 . (3.47) 0
We divide the time interval into ∪m−1 j=0 [ j, j + 1) and also f j (s, x, p) = f ( j + s, x, p) for j ∈ {0, 1, 2, . . . , m − 1}. We have def
P2k (m) f (m) 22 +
m−1 1 j=0
ds
0
P2k ( j + s)(L f j , f j ) − k P2k−1 ( j + s) f j 22 (s)
= f 0 22 . Clearly f j (s, x, p) satisfies the same linearized Boltzmann equation (2.11) on the interval 0 ≤ s ≤ 1. Notice that on this time interval P2k ( j + s) ≥ P2k ( j),
P2k−1 ( j + s) ≤ C˜ k P2k−1 ( j), ∀k ≥ 1/2, s ∈ [0, 1]. (3.48)
These estimates are uniform in j, so that P2k (m) f (m) 22
+
m−1 1 j=0
ds
0
P2k ( j)(L f j , f j )(s) − C˜ k k P2k−1 ( j) f j 22 (s)
≤ f 0 22 . Moreover, by Lemma 3.4, we have (L f j , f j ) ≥ δ0 {I − P} f j 2ν =
δ0 δ0 {I − P} f j 2ν + {I − P} f j 2ν . 2 2
Furthermore, with Lemma 3.5 applied to each f j (s, x, p) we obtain 1 1 m−1 m−1 δ 0 δν δ0 2 P2k ( j) ds {I − P} f j ν ≥ P2k ( j) ds P f j 2ν (s). 2 2 0 0 j=0
j=0
(3.49)
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R. M. Strain
We combine (3.49) with the estimate above it to conclude m−1
1
P2k ( j)
ds (L f j , f j )(s) ≥
0
j=0
1 m−1 δ0 P2k ( j) ds {I − P} f j 2ν (s) 2 0 j=0
+
1 m−1 δ0 δν P2k ( j) ds P f j 2ν (s) 2 0 j=0
≥ δ˜
m−1
j=0
where δ˜ =
1 2
min
δ0 δν δ0 2 , 2
P2k (m) f 22 (m) +
0
0
ds f j 2ν (s),
. Define Ck = C˜ k k. With this lower bound
m−1 1 j=0
1
P2k ( j)
ds δ˜ P2k ( j) f j 2ν − Ck P2k−1 ( j) f j 22 (s) ≤ f 0 22 .
Next, for λ > 0 sufficiently small we introduce the following splitting: b/2 b/2 c p E λ, j = p p0 < λ (1 + j) , E λ, = p ≥ λ + j) . (1 j 0
(3.50)
We incorporate this splitting into our energy inequality as follows: 1 m−1 δ˜ P2k ( j) ds f j 2ν (s) 2 0 j=0 δ˜ P2k ( j) f j 1 E λ, j 2ν (s) − Ck P2k−1 ( j) f j 1 E λ, j 22 (s) ds 2
P2k (m) f 22 (m) +
+
m−1 1 j=0
≤
f 0 22
0
+
m−1 1 j=0
0
2 c (s), ds Ck P2k−1 ( j) f j 1 E λ, j 2
where 1 E λ, j is the usual indicator function of the set E λ, j . On E λ, j , with the help of Lemma 3.1, we have −b/2
−1 ≥ −λ (1 + j) p0
≥ −Cν λ (1 + j) ν( p), Cν > 0.
Hence δ˜ P2k ( j) f j 1 E λ, j 2ν (s) − Ck P2k−1 ( j) f j 1 E λ, j 22 (s) 2 δ˜ − Ck Cν λ P2k ( j) f j 1 E λ, j 2ν (s). ≥ 2 We choose λ > 0 small enough such that Cλ = def
δ˜ − Ck Cν λ > 0. 2
Asymptotic Stability for Soft Potentials
559
Then we have the following useful energy inequality: 1 m−1 δ˜ 2 P2k (m) f 2 (m) + P2k ( j) ds f j 2ν (s) 2 0 j=0
+ Cλ
m−1 1 j=0
≤
f 0 22
+
0
ds P2k ( j) f j 1 E λ, j 2ν (s)
m−1 1 j=0
0
2 c (s). ds Ck P2k−1 ( j) f j 1 E λ, j 2
The last term on the left side of the inequality is positive and we discard it from the energy inequality. For the right side of the energy inequality, on the complementary set c , using Lemma 3.1 again, we have E λ, j b/2 2k−1 p0 Cν P2k−1 ( j) ≤ ≤ 2k−1 ν( p) w2k ( p). λ λ Thus we bound the time weights with velocity weights and the dissipation norm m−1 m−1 1 1 2 2 c (s) ≤ C c ds Ck P2k−1 ( j) f j 1 E λ, ds f j 1 E λ, (s). j 2 j ν,k j=0
0
j=0
0
We switch back to f j (t, x, p) = f (t + j, x, p) and use (3.48) to deduce m m 2 2 2 P2k (m) f 2 (m) + δk ds P2k (s) f ν (s) ≤ f 0 2 + C ds f 2ν,k (s). 0
0
We can obtain an upper bound for the right side above using the regular energy inequality in (3.5) to achieve m 2 ds P2k (s) f 2ν (s) ≤ Ck f 0 22,k . P2k (m) f 2 (m) + δk 0
We have thus established our desired energy inequality from Theorem 3.7 for any m ∈ {0, 1, 2, . . . } and = 0. For an arbitrary t > 0, we choose m ∈ {0, 1, 2, . . . } such that m ≤ t ≤ m + 1. We then split the time integral as [0, t] = [0, m] ∪ [m, t]. For the time interval [m, t], we have the L 2 energy estimate as in (3.6). Since L ≥ 0 by Lemma 3.4, we may use (3.48) and (3.6) to see that P2k (t) f (t) 22 ≤ Ck P2k (m) f (m) 22 , ∀t ∈ [m, m + 1]. Since (3.47) holds for any time t (not necessarily an integer), we can use the estimate above together with (3.47), as in (3.7), using Lemma 3.6, for any t > 0 to obtain t ds P2k (s) f 2ν (s) ≤ Ck f 0 22,k . (3.51) P2k (t) f 22 (t) + δk 0
This proves our time decay theorem for = 0. For general > 0 this estimate can be proven in exactly the same way, except in this case we use Lemma 3.6 in the place of Lemma 3.4 and Lemma 3.5 as we did in the proof of (3.5). This concludes our discussion of L 2 estimates for the linear Boltzmann equation. In the next section we use these L 2 estimates to prove L ∞ estimates.
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4. Linear L ∞ Bounds and Slow Decay In this section we will prove global in time uniform bounds for solutions to the linearized equation (2.11) in L ∞ ([0, ∞) × T3 × R3 ), and slow polynomial decay in time. We express solutions, f (t, x, p), to (2.11) with the semigroup U (t) as f (t, x, p) = {U (t) f 0 }(x, p),
(4.1)
with initial data given by {U (0) f 0 }(x, p) = f 0 (x, p). Our goal in this section will be to prove the following. 3 3 Theorem 4.1. Given > 3/b and k ∈ [0, 1]. Suppose that f 0 ∈ L ∞ +k (T × R ) satisfies (2.6) initially, then under (2.7) the semi-group satisfies
{U (t) f 0 } ∞, ≤ C(1 + t)−k f 0 ∞,+k . Above the positive constant C = C,k only depends on and k. The first step towards proving Theorem 4.1 is an appropriate decomposition. Initially we consider solutions to the linearization of (2.1) with the compact operator K removed from (2.11). This equation is given by (4.2) ∂t + pˆ · ∇x + ν( p) f = 0, f (0, x, p) = f 0 (x, p). Let the semigroup G(t) f 0 denote the solution to this system (4.2). Explicitly G(t) f 0 (x, p) = e−ν( p)t f 0 (x − pt, ˆ p). def
For soft potentials (2.7), with Lemma 3.1, this formula does not imply exponential decay in L ∞ for high momentum values. However, as we will see in Lemma 4.2 below, this formula does imply that one can trade between arbitrarily high polynomial decay rates and additional polynomial momentum weights on the initial data. More generally we consider solutions to the full linearized system (2.11), which are expressed with the semi-group (4.1). By the Duhamel formula t {U (t) f 0 } (x, p) = G(t) f 0 (x, p) + ds1 G(t − s1 )K {U (s1 ) f 0 } (x, p). 0
Kχ
We employ the splitting K = + which is defined with the cut-off function (3.1) and (3.2). We then further expand out t {U (t) f 0 } (x, p) = G(t) f 0 (x, p) + ds1 G(t − s1 )K 1−χ {U (s1 ) f 0 } (x, p) 0 t + ds1 G(t − s1 )K χ {U (s1 ) f 0 } (x, p). K 1−χ
0
We further iterate the Duhamel formula of the last term, as did Vidav [56]: s1 U (s1 ) = G(s1 ) + ds2 G(s1 − s2 )K U (s2 ). 0
(4.3)
Asymptotic Stability for Soft Potentials
561
This will grant the so-called A-Smoothing property. Notice below that we only iterate on the K χ term, which is different from Vidav. Plugging this Duhamel formula into the previous expression yields a more elaborate formula t {U (t) f 0 } (x, p) = G(t) f 0 (x, p) + ds1 G(t − s1 )K 1−χ {U (s1 ) f 0 } (x, p) 0 t + ds1 G(t − s1 )K χ G(s1 ) f 0 (x, p) 0 t s1 + ds1 ds2 G(t − s1 )K χ G(s1 − s2 )K {U (s2 ) f 0 } (x, p). 0
0
However this is not quite yet in the form we want. To get the final form, we once again split the compact operator K = K 1−χ + K χ in the last term to obtain t {U (t) f 0 } (x, p) = G(t) f 0 (x, p) + ds1 G(t − s1 )K 1−χ {U (s1 ) f 0 } (x, p) 0 t + ds1 G(t − s1 )K χ G(s1 ) f 0 (x, p) 0 t s1 + ds1 ds2 G(t −s1 )K χ G(s1 −s2 )K 1−χ {U (s2 ) f 0 } (x, p) 0 0 t s1 + ds1 ds2 G(t − s1 )K χ G(s1 − s2 )K χ {U (s2 ) f 0 } (x, p) 0
0
= H1 (t, x, p)+ H2 (t, x, p)+ H3 (t, x, p)+ H4 (t, x, p)+ H5 (t, x, p), (4.4) def
where H1 (t, x, p) = e−ν( p)t f 0 (x − pt, ˆ p), t def H2 (t, x, p) = ds1 e−ν( p)(t−s1 ) K 1−χ {U (s1 ) f 0 } (y1 , p), 0 t def −ν( p)(t−s1 ) H3 (t, x, p) = ds1 e dq1 k χ ( p, q1 ) e−ν(q1 )s1 f 0 (y1 − qˆ1 s1 , q1 ). def
R3
0
Just above and below we will be using the following short-hand notation: y1 = x − p(t ˆ − s1 ), def
ˆ − s1 ) − qˆ1 (s1 − s2 ). y2 = y1 − qˆ1 (s1 − s2 ) = x − p(t def
(4.5)
We are also using the notation q10 = 1 + |q1 |2 and q1 = (q11 , q12 , q13 ) ∈ R3 with qˆ1 = q1 /q10 . Furthermore the next term is H4 (t, x, p) = def
R3
χ
t
dq1 k ( p, q1 )
ds1 0
×K 1−χ {U (s2 ) f 0 } (y2 , q1 ).
0
s1
ds2 e−ν( p)(t−s1 ) e−ν(q1 )(s1 −s2 )
562
R. M. Strain
Lastly, we may also expand out the fifth component as t H5 (t, x, p) = dq1 k χ ( p, q1 ) dq2 k χ (q1 , q2 ) ds1 e−ν( p)(t−s1 ) R3 R3 0 s1 × ds2 e−ν(q1 )(s1 −s2 ) {U (s2 ) f 0 } (y2 , q2 ).
(4.6)
0
We will estimate each of these five terms individually. In Lemma 4.2 below we will show that the first and third term exhibit rapid polynomial decay. Then after that, in Lemma 4.3, we show that the second and fourth terms can be bounded by the time decay norm (2.10) multiplied by an arbitrarily small constant. For the last term, in Lemma 4.4, we will show that H5 can be estimated by (2.10) times a small constant plus the L 2− j norm of the semi-group (for any j > 0) multiplied by a large constant. After stating each of these lemmas, we will put these estimates together to prove our key decay estimate on the semi-group for solutions to (2.11) in L ∞ as stated above in Theorem 4.1. Once this is complete we give the proofs of the three key Lemmas 4.2, 4.3 and 4.4 at the end of this section. Lemma 4.2. Given ≥ 0, for any k ≥ 0 we have |w ( p)H1 (t, x, p)| + |w ( p)H3 (t, x, p)| ≤ C,k (1 + t)−k f 0 ∞,+k . Next Lemma 4.3. Fix ≥ 0. For any small η > 0, which relies upon the small > 0 from (3.1), and any k ≥ 0 we have |w ( p)H2 (t, x, p)| + |w ( p)H4 (t, x, p)| ≤ η(1 + t)−k ||| f |||k, . The estimates in the lemma above will be used to obtain upper bounds for the the L∞ norm of the semi-group. The final lemma in this series is below Lemma 4.4. Fix ≥ 0, choose any (possibly large) j > 0. For any small η > 0, which depends upon (3.1), and any k ≥ 0 we have the estimate t |w ( p)H5 (t, x, p)| ≤ η(1 + t)−k ||| f |||k, + Cη ds e−η(t−s) f 2,− j (s) + w ( p) |R1 ( f )(t)| .
0
By the L 2 decay theory from Theorem 3.7, and also Proposition 4.5, we have t ds e−η(t−s) f 2,− j (s) ≤ Cη (1 + t)−k f 0 2,k ≤ Cη (1 + t)−k f 0 ∞,+k . 0
The above estimates hold for any k ≥ 0 and > 3/b (as in (4.7) just below). On the other hand, for the last term if we restrict k ∈ [0, 1] then ∀η > 0 we have w ( p) |R1 ( f )| ≤ η(1 + t)−k ||| f |||k, . Above R1 is defined in (4.15) during the course of the proof.
Asymptotic Stability for Soft Potentials
563
These estimates would imply almost exponential decay except for the problematic term R1 ( f )(t), which only appears to decay to first order. This will be discussed in more detail in Sect. 6, where it is shown that this term can decay to any polynomial order by performing a new high order expansion. We now show that the above lemmas grant a uniform bound and slow decay for solutions to (2.11) in L ∞ . We are using the semi-group notation f (t) = {U (t) f 0 }. Lemmas 4.2, 4.3, and 4.4 together imply that for any η > 0 and k ∈ [0, 1] we have f ∞, (t) ≤ C,k (1 + t)−k f 0 ∞,+k + η(1 + t)−k ||| f |||k, + Cη (1 + t)−k f 0 2,k . Equivalently ||| f |||k, ≤ C,k f 0 ∞,+k + C1/2 f 0 2,k ≤ C,k f 0 ∞,+k . The last estimate holds when we choose > 3/b, with (2.9), as follows 2 ( p) f 0 2,k = dx dp wk2 ( p)| f 0 ( p)|2 ≤ f 0 ∞,+k dp w− T3
R3
R3
f 0 ∞,+k .
(4.7)
With this inequality, we have the desired decay rate for the L ∞ norm of solutions to the linear equation (2.11), which proves Theorem 4.1 subject to Lemmas 4.2, 4.3, and 4.4. We now prove those lemmas. Along this course we will repeatedly use the following basic decay estimate Proposition 4.5. Suppose without loss of generality that λ ≥ μ ≥ 0. Then t Cλ,μ (t) ds ≤ , λ (1 + s)μ (1 + t − s) (1 + t)ρ 0 where ρ = ρ(λ, μ) = min{λ + μ − 1, μ} and 1 if λ = 1, Cλ,μ (t) = C log(2 + t) if λ = 1. Furthermore, we will use the following basic estimate from the Calculus: e−ay (1 + y)k ≤ max{1, ea−k k k a −k }, a, y, k ≥ 0.
(4.8)
We will now write an elementary proof of this basic time decay estimate in Proposition 4.5. This result is not difficult and known, however we provide a short proof for the sake of completeness and because we have not seen a proof in the literature. Proof of Proposition 4.5. We will consider the cases μ = 0 and μ = λ separately. Then the general result will then be established by interpolation. Case 1. μ = 0. If λ = 1 we have t t ds ds 1 1−λ 1 − (1 + t) ≤ C(1 + t)−ρ(λ,0) . = = λ λ λ−1 0 (1 + t − s) 0 (1 + s) Note ρ = 0 if λ > 1 and ρ = λ − 1 < 0 otherwise. Alternatively if λ = 1, t ds = log(1 + t). 1 +s 0 This completes our study of the first case μ = 0.
564
R. M. Strain
Case 2. μ = λ. We split the integral as t t/2 t ds = + . λ λ 0 (1 + t − s) (1 + s) 0 t/2 For the first integral t/2 0
ds ≤ (1 + t/2)−λ (1 + t − s)λ (1 + s)λ
0
t/2
ds . (1 + s)λ
Now from Case 1, we can estimate the remaining integral as t/2 ds ≤ Cλ,λ (t)(1 + t)max{0,1−λ} , λ (1 + s) 0 which conforms with the claimed decay. The second half of the integral can be estimated in exactly the same way as the first. Case 3. 0 < μ < λ. By Hölder’s inequality, we have t t ds ds = λ λ λ μ 0 (1 + t − s) (1 + s) 0 (1 + t − s) p + q (1 + s)μ 1/ p t 1/q
t ds ds ≤ . λ λ qμ 0 (1 + t − s) 0 (1 + t − s) (1 + s) Above q =
λ μ
and p =
t
≤C 0
λ λ−μ
with
ds (1 + t − s)λ
1 p
+
1 q
= 1. Therefore the above is
(λ−μ)/λ 0
t
ds (1 + t − s)λ (1 + s)λ
μ/λ
.
By the previous two cases, this is (λ−μ)/λ μ/λ Cλ,λ (t)(1 + t)−ρ(λ,λ) ≤ Cλ,0 (t)(1 + t)−ρ(λ,0) = Cλ,0 (t)(λ−μ)/λ Cλ,λ (t)μ/λ (1 + t)−ρ(λ,0)(λ−μ)/λ−ρ(λ,λ)μ/λ . The proposition follows by adding the exponents.
We are now ready to proceed to the Proof of Lemma 4.2. We start with H1 . From Lemma 3.1 and (4.8) we have e−ν( p)t ≤ Ck p0
kb/2
(1 + t)−k ≤ Ck wk ( p)(1 + t)−k , ∀t, k > 0.
(4.9)
Here we use the notation from (2.9). This procedure grants high polynomial time decay on the solution if we admit similar high polynomial momentum decay on the initial data. In particular we have shown |w ( p)H1 (t, x, p)| = w ( p) e−ν( p)t f 0 (x − pt, ˆ p) ≤ C(1 + t)−k f 0 ∞,+k , which is the desired estimate for H1 .
Asymptotic Stability for Soft Potentials
565
We finish off this lemma by estimating H3 . Notice that we trivially have e−ν( p)(t−s1 ) e−ν(q)s1 ≤ e−ν(max{| p|,|q|})t , where ν(max{| p|, |q|}) is ν evaluated at max{| p|, |q|}. We have |w ( p)H3 (t, x, p)| ≤ w ( p)
t
ds1 e−ν( p)t
0
+ w ( p)
t
ds1 0
| p|≥|q1 |
dq1 k χ ( p, q1 ) sup | f 0 (y, q1 )| y∈T3
dq1 k χ ( p, q1 ) e−ν(q1 )t sup | f 0 (y, q1 )| .
| p|<|q1 |
y∈T3
We will estimate the second term, and we remark that the first term can be handled in exactly the same way. As in the previous estimate for H1 , we use (4.9) to obtain e−ν(q1 )t sup | f 0 (y, q1 )| ≤ Ck (1 + t)−k−1 wk+1 (q1 ) sup | f 0 (y, q1 )|. y∈T3
y∈T3
Next we use the estimate for k χ ( p, q) from Lemma 3.2. When using this estimate we may suppose |q1 | ≤ 2| p|. For otherwise, if say |q1 | ≥ 2| p|, then as in (3.21) we have | p − q1 | ≥ |q1 |/2 which leads directly to wk+1 (q1 )w ( p)e−c| p−q1 |/2 ≤ Cwk+1+ (q1 )e−c|q1 |/4 ≤ C.
(4.10)
In this case we easily obtain an estimate better than (4.11) below. In particular
t
0
ds1 w ( p) −k
(1 + t)
| p|<|q1 |
dq1 1|q1 |≥2| p| k χ ( p, q1 ) e−ν(q1 )t sup | f 0 (y, q1 )| y∈T3
f 0 ∞,− j , ∀k > 0, j > 0.
Thus in the following we assume | p| < |q1 | ≤ 2| p|. On this region we may plug in the last few estimates including (4.9) to obtain
t
0
ds1 w ( p)
≤ Ck
| p|<|q1 |≤2| p|
t (1 + t)k+1
dq1 k χ ( p, q1 ) e−ν(q1 )t sup | f 0 (y, q1 )|
| p|<|q1 |≤2| p|
y∈T3
dq1 w (q1 ) wk+1 (q1 ) k χ ( p, q1 ) sup | f 0 (y, q1 )|
≤ Ck (1 + t)−k f 0 ∞,+k w1 ( p)
| p|<|q1 |≤2| p|
dq1 k χ ( p, q1 ) .
y∈T3
(4.11)
Then from Lemma 3.2 we clearly have the following bound: w1 ( p) dq1 k χ ( p, q1 ) ≤ C. | p|<|q1 |≤2| p|
This completes the time decay estimate for H3 and our proof of the lemma.
566
R. M. Strain
Our next aim is to prove Lemma 4.3. To do this we will use the following Lemma 4.6. Fix any ≥ 0 and any j > 0. Then given any small η > 0, which depends upon χ in (3.1), the following estimate holds: w ( p)K 1−χ (h)( p) ≤ ηe−cp0 h ∞,− j . Above the constant c > 0 is independent of η. Proof of Lemma 4.6. We consider K 1−χ (h) as defined in (3.2). From (3.29), for in (3.1) chosen sufficiently small ( smaller than 1/4 is sufficient), we see that
J (q)J ( p ) +
J (q)J (q ) e−cq0 −cp0 , ∃c > 0.
With that estimate, and additionally the conservation of energy (1.6), we have 1−χ dωdq (1 − χ (g)) vø σ (g, θ )e−cq0 −cp0 w ( p)K 2 (h)( p) R3 ×S2 × h( p ) + h(q ) c c dωdq (1 − χ (g)) vø σ e− 2 q0 − 2 p0 3 2 R ×S c c × e− 2 p0 h( p ) + e− 2 q0 h(q ) c
≤ ηe− 4 p0 h ∞,− j . In the last inequality we have used the following estimate for any small η > 0: c c c dωdq (1 − χ (g)) vø σ (g, θ ) e− 2 q0 − 2 p0 ≤ η e− 4 p0 . (4.12) R3 ×S2
The proof of this bound uses (3.25), but also the exact strategy used in the paragraph containing (3.25) and the paragraph just below it. The idea is to use the splitting (3.23) inside the integral in (4.12): 1 = φ R ( p, q) + (1 − φ R ( p, q)). First when | p| + |q| is large, this is the term containing (1 − φ R ( p, q)), we have a bound for the integral in (4.12) which is of the form Ce−c R , where the constant C, c > 0 are independent of both R and similar to (3.28). On the other hand, when | p| + |q| ≤ R, this is the term containing φ R ( p, q), we have a bound which is of the form C R p . Here p = p(b) can be chosen to be p = 3 if b ∈ [0, 1] and p = 4 − b if b ∈ (1, 4). This second estimate is similar to (3.27). Since this strategy is already performed in detail nearby (3.25), we will not re-write the details. Since p > 0 we can first choose R >> 1 sufficiently large, and then choose > 0 sufficiently small so that the constant η > 0 in (4.12) can be chosen arbitrarily small. 1−χ This yields the desired estimate. We remark that the estimate for K 1 can be shown in the same exact way; it is in fact slightly easier.
Asymptotic Stability for Soft Potentials
567
With Lemma 4.6 and Proposition 4.5 in hand, we proceed to the Proof of Lemma 4.3. We begin with the estimate for H2 . Using Lemma 4.6, for any small η > 0 we have t |w ( p)H2 (t, x, p)| = w ( p) ds1 e−ν( p)(t−s1 ) K 1−χ {U (s1 ) f 0 } (y1 , p) 0 t −cp0 ≤η e ds1 e−ν( p)(t−s1 ) {U (s1 ) f 0 } ∞,− j , ∀ j > 0 0 t −cp0 ≤η e ||| f |||k, ds1 e−ν( p)(t−s1 ) (1 + s1 )−k . 0
The norm is from (2.10) for k ≥ 0. As in (4.9) for any λ > max{1, k} we have t |w ( p)H2 (t, x, p)| ≤ η wλ ( p)e−cp0 ||| f |||k, ds (1 + t − s)−λ (1 + s)−k 0
≤ η (1 + t)
−k
||| f |||k, ,
which follows from Proposition 4.5. This is the desired estimate for H2 . For H4 we once again use Lemma 4.6, for any small η > 0, to obtain s1 t χ |w ( p)H4 (t, x, p)| ≤ dq1 k ( p, q1 ) ds1 ds2 e−ν( p)(t−s1 ) R3 0 0 × e−ν(q1 )(s1 −s2 ) w ( p) K 1−χ ({U (s2 ) f 0 }) (y2 , q1 ) w ( p) −cq ≤ η ||| f |||k, e 10 dq1 k χ ( p, q1 ) w (q1 ) R3 t s1 × ds1 ds2 e−ν( p)(t−s1 ) e−ν(q1 )(s1 −s2 ) (1 + s2 )−k . 0
0
We recall that q10 = 1 + |q1 |2 . For the time decay, from Proposition 4.5 with (4.8) as in (4.9) we notice that t s1 ds1 ds2 e−ν( p)(t−s1 ) e−ν(q1 )(s1 −s2 ) (1 + s2 )−k 0 0 t s1 wλ ( p)wλ (q1 ) ds1 (1 + t − s1 )−λ ds2 (1 + s1 − s2 )−λ (1 + s2 )−k 0
0
wλ ( p)wλ (q1 )(1 + t)
−k
.
Above we have taken λ > max{1, k}. Combining these estimates yields −k |w ( p)H4 | η (1 + t) ||| f |||k, dq1 k χ ( p, q1 ) w+λ ( p)w−+λ (q1 )e−cq10 . R3
To estimate the remaining integral and weights we split into three cases. If either 2|q1 | ≤ | p|, or |q1 | ≥ 2| p|, then we bound all the weights and the remaining momentum integral by a constant as in (4.10). Alternatively if 21 |q1 | ≤ | p| ≤ 2|q1 |, then the desired estimate is obvious since we have strong exponential decay in both p and q1 . In either of these cases we have the estimate for H4 .
568
R. M. Strain
We will finish this section with a proof of the crucial Lemma 4.4. Proof of Lemma 4.4. We now turn to the proof of our estimate for H5 . Recall the definition of H5 from (4.6) with y2 defined in (4.5). We will utilize rather extensively the estimate for k χ from Lemma 3.2. We now further split high
H5 (t, x, p) = H5
(t, x, p) + H5low (t, x, p),
(4.13)
and estimate each term on the right individually. For M >> 1 we define 1high = 1| p|>M 1|q1 |≤M + 1|q1 |>M . def
(4.14)
Notice 1high + 1| p|≤M 1|q1 |≤M = 1. Now the first term in the expansion is t def high χ χ dq1 k ( p, q1 ) dq2 k (q1 , q2 ) 1high ds1 e−ν( p)(t−s1 ) H5 (t, x, p) = R3 R3 0 s1 × ds2 e−ν(q1 )(s1 −s2 ) {U (s2 ) f 0 } (y2 , q2 ). 0
We use (4.8), as in (4.9), and Lemma 3.1 to see that for any λ ≥ 0 we have s1 t ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) 0 0 t s1 ≤ Cλ wλ ( p)wλ (q1 ) ds1 ds2 (1 + (t − s1 ))−λ (1 + (s1 − s2 ))−λ . 0
0
When either | p| > M or |q1 | > M, by Lemma 3.2, we have the bound χ k ( p, q1 ) ≤ C M −ζ ( p0 + q10 )−b/2 e−c| p−q1 | . If either | p| ≥ 2|q1 | or |q1 | ≥ 2| p| then as in (4.10) we have w+λ ( p)wλ (q1 )e−c| p−q1 | ≤ C. Thus by combining the last few estimates we have χ dq1 k ( p, q1 ) dq2 k χ (q1 , q2 ) 1high 1| p|≥2|q1 | + 1| p|≤ 1 |q1 | w ( p) 2 R3 R3 t s1 × ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) |{U (s2 ) f 0 } (y2 , q2 )| 0 0 Cλ dq1 e−c| p−q1 | dq2 e−c|q1 −q2 | ≤ ζ +b/2 ||| f |||k, M R3 R3 t s1 (1 + s2 )−k × ds1 ds2 . (1 + (t − s1 ))λ (1 + (s1 − s2 ))λ 0 0 With Proposition 4.5, for any k ≥ 0 and λ > max{k, 1} the previous term is bounded from above by ≤
Ck,λ (1 + t)−k ||| f |||k, . M ζ +b/2
This is the desired estimate for M >> 1 chosen sufficiently large.
Asymptotic Stability for Soft Potentials
569 high
We now consider the remaining part of H5 . As in the previous estimates and (4.10), if either |q2 | ≥ 2|q1 | or |q1 | ≥ 2|q2 | then for any k ≥ 0 we have dq1 k χ ( p, q1 ) dq2 k χ (q1 , q2 ) 1|q2 |≥2|q1 | + 1|q2 |≤ 1 |q1 | w ( p) 2 R3 R3 t s1 ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) ×1 1 | p|≤|q1 |≤2| p| 1high 2 0 0 Ck,λ × |{U (s2 ) f 0 } (y2 , q2 )| ≤ 2ζ +b ||| f |||k, dq1 e−c| p−q1 | dq2 e−c|q1 −q2 | M R3 R3 t s1 (1 + s2 )−k × ds1 ds2 (1 + (t − s1 ))λ (1 + (s1 − s2 ))λ 0 0 Ck,λ ≤ 2ζ +b/2 (1 + t)−k ||| f |||k, . M Above we have used exactly the same estimates as in the prior case. Both of the last two terms have a suitably small constant in front if M is sufficiently large. high Thus the remaining part of H5 to estimate is def R1 ( f )(t) = dq1 k χ ( p, q1 ) dq2 k χ (q1 , q2 ) 1 1 | p|≤|q1 |≤2| p| 1 1 |q1 |≤|q2 |≤2|q1 | 2 2 R3 R3 t s1 × 1high ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) {U (s2 ) f 0 } (y2 , q2 ). 0
0
(4.15) It is only this term which slows down the time decay rate. In this current proof we will only make a basic argument to show that this term can naively exhibit first order decay. Since all the momentum variables are comparable, we have χ |R1 ( f )(t)| ≤ dq1 k ( p, q1 ) dq2 k χ (q1 , q2 ) R3
R3
×1 1 | p|≤|q1 |≤2| p| 1 1 |q1 |≤|q2 |≤2|q1 | 1high 2 2 t s1 −cν(q1 )(t−s1 ) × ds1 e ds2 e−cν(q1 )(s1 −s2 ) |{U (s2 ) f 0 } (y2 , q2 )|. 0
0
Next using similar techniques as in the previous two estimates, including Proposition 4.5 twice, we obtain the following upper bound for any k ∈ [0, 1]: Ck,λ −b/2−ζ −c| p−q1 | w ( p) |R1 ( f )(t)| ≤ 2ζ ||| f |||k, dq1 q10 e M t ds1 −b/2−ζ −c|q1 −q2 | × dq2 q10 e w2+2δ (q1 ) (1 + (t − s1 ))1+δ 0 s1 ds2 Ck,λ × ≤ 2ζ (1 + t)−k ||| f |||k, . 1+δ k (1 + (s1 − s2 )) (1 + s2 ) M 0 In the last line we used the fact that we have chosen δ > 0 to satisfy δ < 2ζ /b, where ζ > 0 is defined in the statement of Lemma 3.2. In Sect. 6 we will examine this term at length to show that (4.15) actually decays “almost exponentially.”
570
R. M. Strain
We are ready to define the second term in our splitting of H5 . It must be def low χ dq1 k ( p, q1 ) dq2 k χ (q1 , q2 ) H5 (t, x, p) = 1| p|≤M |q1 |≤M
t
×
ds1 e−ν( p)(t−s1 )
R3
s1
ds2 e−ν(q1 )(s1 −s2 ) {U (s2 ) f 0 } (y2 , q2 ).
0
0
For any small κ > 0, we further split this term into two terms, one of which is κ def H5low,κ (t, x, p) = dq1 k χ ( p, q1 ) dq2 k χ (q1 , q2 ) ds1 e−ν( p)(t−s1 ) |q1 |≤M
R3
s1
s1
0
×1| p|≤M ds2 e−ν(q1 )(s1 −s2 ) {U (s2 ) f 0 } (y2 , q2 ) 0 t + dq1 k χ ( p, q1 ) dq2 k χ (q1 , q2 ) ds1 e−ν( p)(t−s1 ) |q1 |≤M
×1| p|≤M
R3
ds2 e
s1 −κ
−ν(q1 )(s1 −s2 )
κ
{U (s2 ) f 0 } (y2 , q2 ).
The other term in this latest splitting is defined just below as H5low,2 . On this temporal integration domain s1 − s2 ≤ κ. Since we are proving uniform bounds, it is safe to assume when proving decay that t ≥ 1 for instance. Since p and q1 are both bounded by M, from Lemma 3.1 we have 1| p|≤M 1|q1 |≤M e−ν( p)(t−s1 )−ν(q1 )(s1 −s2 ) ≤ e−C(t−s2 )/M
b/2
.
(4.16)
Then for the first term in H5low,κ above multiplied by w ( p) we have the bound κ dq1 k χ ( p, q1 ) dq2 k χ (q1 , q2 ) ds1 e−ν( p)(t−s1 ) w ( p) R3
|q1 |≤M
×1| p|≤M
s1
−ν(q1 )(s1 −s2 )
ds2 e κ ds1 ≤ C M ||| f |||0,
0
|{U (s2 ) f 0 } (y2 , q2 )|
0
0
≤ C M κ ||| f |||0, e 2
s1
ds2 e−C(t−s2 )/M
0 −Ct/M b/2 Cκ/M b/2
e
b/2
≤ C M κ 2 (1 + t)−k ||| f |||0, .
We have just used (4.8). We obtain the desired estimate for the above terms by first choosing M large, and second choosing κ = κ(M) > 0 sufficiently small. For the second term in H5low,κ multiplied by w ( p) for any k ≥ 0 we have w ( p)
|q1 |≤M
dq1 k χ ( p, q1 )
×1| p|≤M
s1
R3
dq2 k χ (q1 , q2 )
t
κ
ds1 e−ν( p)(t−s1 )
ds2 e−ν(q1 )(s1 −s2 ) |{U (s2 ) f 0 } (y2 , q2 )|
s1 −κ
≤ C M ||| f |||k,
t κ
ds1
s1 s1 −κ
ds2 e−C(t−s1 )/M
b/2
e−C(s1 −s2 )/M
b/2
(1 + s2 )−k .
Asymptotic Stability for Soft Potentials
571
Since s2 ∈ [s1 − κ, s1 ] and κ ∈ (0, 1/2), then (1 + s2 ) ≥ 21 + s1 . We have t b/2 ds1 e−C(t−s1 )/M (1 + s1 )−k ≤ C M κ||| f |||k, κ
≤ C M κ(1 + t)−k ||| f |||k, . In the last step we have used (4.8) and Proposition 4.5. We conclude the desired estimate for H5low,κ by first choosing M large, and then κ > 0 sufficiently small. The only remaining part of H5low (t, x, p) to be estimated is given by t def H5low,2 (t, x, p) = dq1 k χ ( p, q1 ) dq2 k χ (q1 , q2 ) ds1 e−ν( p)(t−s1 ) |q1 |≤M
×1| p|≤M
R3
s1 −κ
κ
ds2 e−ν(q1 )(s1 −s2 ) {U (s2 ) f 0 } (y2 , q2 ).
0
high
and H5low terms defined above, we remark that (4.13) holds. With all of the H5 We now estimate H5low,2 . Since p and q1 are both bounded by M, from Lemma 3.1 we still have (4.16). For any j ≥ 0, Lemma 3.2 implies the following bound: 2 dq1 dq2 w j (q2 )k χ ( p, q1 )k χ (q1 , q2 ) ≤ C M . R3
|q1 |≤M
Indeed if |q2 | ≥ 2|q1 | then as in (4.10) we can prove this bound. Alternatively if |q2 | ≤ 2|q1 | then w j (q2 ) ≤ Cw j (q1 ) ≤ C M jb/2 and the bound above also holds. We use the above and Cauchy-Schwartz to estimate the momentum integrals: χ χ dq1 k ( p, q1 ) dq2 k (q1 , q2 ) {U (s2 ) f 0 } (y2 , q2 ) |q1 |≤M
|q1 |≤M
dq1
R3
×
R3
|q1 |≤M
dq1
≤ CM
2 dq2 w j (q2 )k χ ( p, q1 )k χ (q1 , q2 )
R3
2 dq2 w− j (q2 ) {U (s2 ) f 0 } (y2 , q2 )
|q1 |≤M
dq1
1/2
R3
1/2
2 dq2 w− j (q2 ) {U (s2 ) f 0 } (y2 , q2 )
1/2 .
That step was significant to yield rapid polynomial momentum decay. We change variables q1 → y2 on the dq1 integration with y2 given by (4.5). Then
2 −q q δmn q10 dy2 1m 1n = −(s1 − s2 ) . (4.17) 3 dq1 mn q10 2) , and a third eigenvalue This is a 3 × 3 matrix with two eigenvalues equal to − (s1q−s 10
given by −(s1 − s2 )
2 −|q |2 q10 1 3 q10
=
−(s1 −s2 ) . 3 q10
Thus the Jacobian is
dy2 |(s1 − s2 )|3 κ3 = ≥ C . dq 5 M5 q10 1
572
R. M. Strain
This lower bound holds on the set |q1 | ≤ M, s2 ∈ [0, s1 − κ] and s1 ∈ [κ, t] so that s1 − s2 ≥ κ. After application of this change of variables we have
2 1/2 dq1 dq2 w− j (q2 ) {U (s2 ) f 0 } (y2 , q2 ) |q1 |≤M
R3
1/2
2 1/2 M5 } {U w (y dy dq (q ) (s ) f , q ) 2 2 −j 2 2 0 2 2 κ3 |y2 −x|≤C(t−s2 ) R3 1/2
2 1/2 M5 3/2 } {U w (y {1+(t −s ) } dy dq (q ) (s ) f , q ) 2 2 2 −j 2 2 0 2 2 κ3 T3 R3
= C(M, κ){1 + (t − s2 )3/2 } {U (s2 ) f 0 } 2,− j . Putting together all of these estimates, in particular using (4.16), we have shown t s1 −κ b/2 b/2 low,2 ds1 ds2 e−C(t−s1 )/M e−C(s1 −s2 )/M ≤ Cκ,M w ( p)H5 κ
0
×{1 + (t − s2 )3/2 } {U (s2 ) f 0 } 2,− j t t C C b/2 b/2 − C2 (t−s1 )/M b/2 ≤ Cκ,M ds1 e ds2 e− 2 (t−s1 )/M e− 2 (s1 −s2 )/M 0
0
×{1 + (t − s2 )3/2 } {U (s2 ) f 0 } 2,− j . Notice that the first exponential controls the s1 time integral, and the second and third exponential control the remaining time integral as follows: t C b/2 low,2 ds2 e− 2 (t−s2 )/M {1 + (t − s2 )3/2 } {U (s2 ) f 0 } 2,− j w ( p)H5 ≤ Cκ,M 0 t C b/2 ≤ Cκ,M ds2 e− 4 (t−s2 )/M {U (s2 ) f 0 } 2,− j . 0
high
That was the last case. Adding up the individual estimates for H5 and H5low in (4.13) completes our proof after first choosing M large enough and then second choosing κ sufficiently small. 5. Nonlinear L ∞ Bounds and Slow Decay Suppose f = f (t, x, p) solves (2.1) with initial condition f (0, x, p) = f 0 (x, p). We may express mild solutions to this problem (2.1) in the form f (t, x, p) = {U (t) f 0 }(x, p) + N [ f, f ](t, x, p), where we have used the notation
t
N [ f 1 , f 2 ](t, x, p) = def
(5.1)
ds {U (t − s) [ f 1 (s), f 2 (s)]}(x, p).
0
Here as usual U (t) is the semi-group (4.1) which represents solutions to the linear problem (2.11). The main result of this section is to prove the following
Asymptotic Stability for Soft Potentials
573
Theorem 5.1. Choose > 3/b, k ∈ (1/2, 1]. Consider the following initial data f 0 = 3 3 f 0 (x, p) ∈ L ∞ +k (T × R ) which satisfies (2.6) initially. There is an η > 0 such that if f 0 ∞,+k ≤ η, then there exists a unique global in time mild solution (5.1), f = f (t, x, p), to Eq. (2.1) which satisfies f ∞, (t) ≤ C,k (1 + t)−k f 0 ∞,+k . These √ solutions are continuous √ if it is so initially. We further have positivity, i.e. F = μ + μ f ≥ 0, if F0 = μ + μ f 0 ≥ 0. In Theorem 7.1, which is proven in Sect. 7, we will show that these solutions exhibit rapid polynomial decay to any order. Notice that our main Theorem 2.1 will follow directly from Theorem 5.1 and Theorem 7.1. To prove this current Theorem 5.1 we will use the following non-linear estimate. Lemma 5.2. Considering the non-linear operator defined in (2.5) with (2.7), we have the following pointwise estimates: |w ( p) (h 1 , h 2 )( p)| ν( p) h 1 ∞, h 2 ∞, . These hold for any ≥ 0. Furthermore, (h 1 , h 2 ) ∞,+1 h 1 ∞, h 2 ∞, . The lemma above combined with Proposition 4.5 will be important tools in our proof of Theorem 5.1. We now give a simple proof. Proof of Lemma 5.2. We recall (2.3), (2.5), and (2.9). For ≥ 0, it follows from (1.6) that b/2
w ( p) p0
( p0 )b/2 (q0 )b/2 w ( p )w (q ).
A proof of this estimate above was given in [24, Lemma 2.2]. Thus w ( p) | (h 1 , h 2 )| dωdq vø σ (g, θ ) J (q) w ( p )w (q ) h 1 ( p )h 2 (q ) R3 ×S2 + dωdq vø σ (g, θ ) J (q) w ( p) |h 1 ( p)h 2 (q)| R3 ×S2 dωdq vø σ (g, θ ) J 1/2 (q) h 1 ∞, h 2 ∞, R3 ×S2
ν( p) h 1 ∞, h 2 ∞, .
The last inequality above follows directly from Lemma 3.1 since both the integral and ν( p) have the same asymptotic behavior at infinity. That yields the first estimate. For the second estimate we notice from the first estimate that w+1 ( p) | (h 1 , h 2 )| w1 ( p)ν( p) h 1 ∞, h 2 ∞, . But w1 ( p)ν( p) 1 from Lemma 3.1 and (2.9). This completes the proof. We now proceed to the Proof of Theorem 5.1. We will prove Theorem 5.1 in three steps. The first step gives existence, uniqueness and slow decay via the contraction mapping argument. The second step will establish continuity, and the last step shows positivity.
574
R. M. Strain
Step 1. Existence and uniqueness. When proving existence of mild solutions to (5.1) it is natural to consider the mapping M[ f ] = {U (t) f 0 }(x, p) + N [ f, f ](t, x, p). def
With the norm (2.10), we will show that this is a contraction mapping on the space R Mk, = { f ∈ L ∞ ([0, ∞) × T3 × R3 ) : ||| f |||k, ≤ R}, def
R > 0.
We first estimate the non-linear term N [ f, f ] defined in the equation display below (5.1). We apply Theorem 4.1, with > 3/b, and k ∈ (1/2, 1] to obtain t w ( p) |N [ f 1 , f 2 ](t, x, p)| ds w ( p) |{U (t − s) [ f 1 (s), f 2 (s)]}(x, p)| 0 t ds [ f 1 (s), f 2 (s)] ∞,+k . k 0 (1 + t − s) Next Lemma 5.2 allows us to bound the above by t ds f 1 (s) ∞,+k−1 f 2 (s) ∞,+k−1 . k 0 (1 + t − s) From Proposition 4.5 and the decay norm (2.10) we see that the last line is t ds ||| f 1 |||k, ||| f 2 |||k, k (1 + s)2k (1 + t − s) 0 (1 + t)−k ||| f 1 |||k, ||| f 2 |||k, . We have shown |||N [ f 1 , f 2 ]|||k, ||| f 1 |||k, ||| f 2 |||k, . To handle the linear semigroup, U (t), we again use Theorem 4.1 to obtain |||M[ f ]|||k, ≤ C,k f 0 ∞,+k + ||| f |||2k, . R into itself for 0 < R chosen sufficiently small and We conclude that M[·] maps Mk, R e.g. f 0 ∞,+k ≤ 2C,k . To obtain a contraction, we consider the difference
M[ f 1 ] − M[ f 2 ] = N [ f 1 − f 2 , f 1 ] + N [ f 2 , f 1 − f 2 ]. Then as in the previous estimates we have ∗ |||M[ f 1 ] − M[ f 2 ]|||k, ≤ C,k ||| f 1 |||k, + ||| f 2 |||k, ||| f 1 − f 2 |||k, . With these estimates, the existence and uniqueness of solutions to (2.1) follows from R when R > 0 is suitably small. the contraction mapping principle on Mk, Step 2. Continuity. We perform the estimates from Step 1 on the space R,0 Mk, = { f ∈ C 0 ([0, ∞) × T3 × R3 ) : ||| f |||k, ≤ R}, def
R > 0.
Asymptotic Stability for Soft Potentials
575
R,0 As in Step 1, we have a uniform in time contraction mapping on Mk, for suitable R. R,0 and f 0 is continuous. Since the converFurthermore M[ f ] is continuous if f ∈ Mk, gence is uniform, the limit will be continuous globally in time. This argument is standard and we refer for instance to [21,24,33] for full details.
Step 3. Positivity. We use the standard alternative approximating formula ∂t + pˆ · ∇x F n+1 + R(F n )F n+1 = Q+ (F n , F n ), √ with the same initial conditions F n+1 t=0 = F0 = J + J f 0 , for n ≥ 1 and for instance √ def F 1 = J + J f 0 . Here we have used the standard decomposition of Q = Q+ − Q− into gain and loss terms with Q− (F n+1 , F n ) = R(F n )F n+1 , √ def and R(F n ) = Q− (1, F n ). If we consider F n+1 (t, x, p) = J + J f n+1 (t, x, p), then related to Step 1 we may show that f n+1 (t, x, p) is convergent in L ∞ on a local time interval [0, T ], where T will generally depend upon the size of the initial data. In parn+1 ticular f n+1 (t, x, p) = F √ −J satisfies the equation J
∂t + pˆ · ∇x + ν( p) f n+1 = K ( f n ) + + ( f n , f n ) − − ( f n+1 , f n ). We rewrite this equation using the solution formula to the system (4.2) as f n+1 = G(t) f 0 + L( f n+1 , f n ). This solution formula G(t) is defined just below (4.2). Furthermore L( f
n+1
t
,f )= n
def
ds G(t − s)K ( f n )
0
t
+ 0
ds G(t − s) + ( f n , f n ) − G(t − s) − ( f n+1 , f n ).
R ([0, T ]) defined by For given T > 0 and R > 0 we consider the space Mk,
f ∈ L ∞ ([0, T ] × T3 × R3 ) : ess sup0≤t≤T (1 + t)k f (t) ∞, ≤ R .
R ([0, T ]) and f Now given f n ∈ Mk, 0 ∞,+k ≤
R 2Ck, f n+1
with R > 0 chosen sufficiently
R ([0, T ]). ∈ Mk, small, as in Step 1, we can prove the existence of With the estimates established in this paper, it is now not hard to show that
sup f n+1 − f n ∞, (t) ≤ C T sup f n − f n−1 ∞, (t).
0≤t≤T
0≤t≤T
Here T > 0 is sufficiently small, and the constant C > 0 can be chosen independent of any small T . Therefore there exists a T ∗ > 0 such that f n → f uniformly in L ∞ on [0, T ∗ ]. This will be sufficient to prove the positivity globally in time.
576
R. M. Strain
Indeed if F n ≥ 0, then so is Q+ (F n , F n ) ≥ 0. With the representation formula t
ˆ p) F0 (x − pt, ˆ p) F n+1 (t, x, p) = e− 0 ds R(F )(s,x− p(t−s), t t n ˆ ), p) + ds e− s dτ R(F )(τ,x− p(t−τ Q+ (F n , F n )(s, x − p(t ˆ − s), p). n
0
≥ 0 for all n ≥ 0 if F0 ≥ 0, which implies in the limit Induction shows F n+1 (t, x, p)√ n → ∞ that F(t, x, p) = J + J f (t, x, p) ≥ 0. Using our L ∞ uniqueness, this is the same F as the one from Step 1 on the time interval [0, T ∗ ]. We extend this positivity for all time intervals [0, T ∗ ] + T ∗ k for any k ≥ 1 by repeating this procedure and using the global uniform bound in L ∞ from Step 1. 6. Linear L ∞ Rapid Decay In this section we prove that the linear semi-group (4.1) exhibits rapid polynomial decay in L ∞ . For any k ≥ 1 we will discover a k = k (k) ≥ k such that {U (t) f 0 } ∞, ≤ C,k (1 + t)−k f 0 ∞,+k .
(6.1)
The main obstruction to proving such rapid decay in this low regularity L ∞ framework was the term (4.15) which came up during the course of the proof of Lemma 4.4. In this section we perform a new high-order expansion of this remainder which allows one to prove rapid decay as follows. Proposition 6.1. Consider R1 ( f )(t) defined in (4.15). Choose > 3/b. For any small η > 0, and any k ≥ 1, there exists a k = k (k) ≥ k such that w ( p) |R1 ( f )(t)| ≤ η(1 + t)−k ||| f |||k, + C,k ,η (1 + t)−k f 0 ∞,+k . The power k can be explicitly computed from the proof. The crucial difficulty with proving rapid decay for the soft potentials is caused by the high momentum values, for which the time decay is diluted by the momentum decay. This causes the generation of weights on the initial data, typically one weight for each order of time decay. In the proof below we are able to overcome this apparent obstruction by performing a new high order expansion for R1 ( f ) which is explained in detail at the beginning of the proof. We will first show that Proposition 6.1 implies (6.1). We use the expansion (4.4) and the semi-group notation f (t) = {U (t) f 0 }. We now see that Lemmas 4.2, 4.3, and 4.4 together imply that for any η > 0 and k ≥ 1/2 we have η f ∞, (t) ≤ C,k (1 + t)−k f 0 ∞,+k + (1 + t)−k ||| f |||k, + w ( p) |R1 ( f )(t)| . 2 Here we use (2.10). Then Proposition 6.1 further implies for some k ≥ k that η w ( p) |R1 ( f )(t)| ≤ C,k,η (1 + t)−k f 0 ∞,+k + (1 + t)−k ||| f |||k, . 2 Equivalently ||| f |||k, ≤ C,k f 0 ∞,+k . This is the desired decay rate for the L ∞ norm of mild solutions to the linear equation (2.11), which proves (6.1) subject to Proposition 6.1. In the rest of this section we prove this crucial new proposition.
Asymptotic Stability for Soft Potentials
577
Proof of Proposition 6.1. We will prove this proposition with a new high order expansion of (4.15) by iterating the semi-group (4.3). For ease of exposition we write χ
k≈ ( p, q1 ) = k χ ( p, q1 )1 1 | p|≤|q1 |≤2| p| . def
2
Recall that 1 1 | p|≤|q1 |≤2| p| is the function which is one when 21 | p| ≤ |q1 | ≤ 2| p| and zero 2
χ
elsewhere. We will use similar expressions for k≈ with different arguments. Then we may split (4.15) as R1 ( f ) = S1 ( f ) + L 1 ( f ) + R2 ( f ). For any small κ > 0 we choose κ1 = κ and κ2 = κ1 /2 so that κ1 s1 def χ ds1 e−ν( p)(t−s1 ) dq1 k≈ ( p, q1 ) 1high ds2 e−ν(q1 )(s1 −s2 ) S1 ( f ) = 0 R3 0 χ × dq2 k≈ (q1 , q2 ) {U (s2 ) f 0 } (y2 , q2 ), R3 t s1 −κ2 def χ −ν( p)(t−s1 ) L 1( f ) = ds1 e dq1 k≈ ( p, q1 ) 1high ds2 e−ν(q1 )(s1 −s2 ) R3 0 κ1 χ × dq2 k≈ (q1 , q2 ) {U (s2 ) f 0 } (y2 , q2 ). R3
Then we may define the remainder term as t s1 def χ −ν( p)(t−s1 ) ds1 e dq1 k≈ ( p, q1 ) 1high ds2 e−ν(q1 )(s1 −s2 ) R2 ( f ) = 3 R s1 −κ2 κ1 χ × dq2 k≈ (q1 , q2 ) {U (s2 ) f 0 } (y2 , q2 ). R3
Our notation above is from the proof of Lemma 4.4, in particular (4.14). We will show that the first term S1 ( f ) exhibits rapid decay in L ∞ . The last term L 1 ( f ) is bounded in 2 L which further has rapid decay as in Theorem 3.7. We will notice first of all that the term R2 ( f ) naively exhibits second order polynomial decay. However if we continue the expansion then we can obtain higher and higher order decay rates as follows. We may expand R2 ( f ) = G 2 ( f ) + D2 ( f ) + N2 ( f ) + L 2 ( f ) + R3 ( f ). Now each of the terms G 2 ( f ), D2 ( f ), N2 ( f ), and L 2 ( f )– to be defined below–will exhibit (for different reasons) high order polynomial decay right away again at a cost of momentum weights on the initial data. The term R3 ( f ) will clearly exhibit third order polynomial decay, however we may continue this expansion at each level so that at level k we can again expand Rk ( f ) = G k ( f ) + Dk ( f ) + Nk ( f ) + L k ( f ) + Rk+1 ( f ). As in the initial case each of the terms G k ( f ), Dk ( f ), Nk ( f ), and L k ( f )–which are defined recursively–will exhibit high order polynomial decay. The last term Rk+1 ( f ) will have k + 1 order polynomial decay. This expansion is well defined and can be continued to any order, which yields rapid decay.
578
R. M. Strain
We define the 2nd order terms by plugging the iteration (4.3) into R2 ( f ), using the expansion of K = K 1−χ + K χ with (3.1), and splitting the remaining time and momentum integrals in the following useful way. For κ3 = κ2 /2 = κ1 /22 , we define G2( f ) =
t
def
ds1 e−ν( p)(t−s1 )
κ1
χ
dq1 k≈ ( p, q1 ) 1high
R3
s1
s1 −κ2
ds2 e−ν(q1 )(s1 −s2 )
χ
× dq2 k≈ (q1 , q2 ) {G(s2 ) f 0 } (y2 , q2 ), R3 t s1 def χ −ν( p)(t−s1 ) D2 ( f ) = ds1 e dq1 k≈ ( p, q1 ) 1high ds2 e−ν(q1 )(s1 −s2 ) R3 s1 −κ2 κ1 s2 χ × dq2 k≈ (q1 , q2 ) ds3 e−ν(q2 )(s2 −s3 ) K 1−χ {U (s3 ) f 0 } (y3 , q2 ). R3
0
Furthermore N2 ( f ) =
t
def
κ1
ds1 e−ν( p)(t−s1 )
×
R3
χ
R3
χ dq2 k≈ (q1 , q2 )
dq1 k≈ ( p, q1 ) 1high
s2
ds3 e
s1 −κ2
×
0
R3
×
χ
R3
ds2 e−ν(q1 )(s1 −s2 )
−ν(q2 )(s2 −s3 )
χ × dq3 k= (q2 , q3 ) {U (s3 ) f 0 } (y3 , q3 ), R3 t def χ L 2( f ) = ds1 e−ν( p)(t−s1 ) dq1 k≈ ( p, q1 ) 1high κ1
s1
dq2 k≈ (q1 , q2 )
s2 −κ3
s1 s1 −κ2
ds2 e−ν(q1 )(s1 −s2 )
ds3 e−ν(q2 )(s2 −s3 )
0
χ
R3
dq3 k≈ (q2 , q3 ) {U (s3 ) f 0 } (y3 , q3 ).
And the remainder is given by R3 ( f ) = def
t
κ1
ds1 e
× ×
−ν( p)(t−s1 )
R3
χ
R3
χ dq1 k≈ ( p, q1 ) 1high
dq2 k≈ (q1 , q2 )
s2
s2 −κ3
s1
s1 −κ2
ds2 e−ν(q1 )(s1 −s2 )
ds3 e−ν(q2 )(s2 −s3 )
χ
R3
dq3 k≈ (q2 , q3 ) {U (s3 ) f 0 } (y3 , q3 ),
where above y1 and y2 are defined in (4.5), and more generally yi+1 = y1 − qˆ1 (s1 − s2 ) − · · · − qˆi (si − si+1 ) = x − p(t ˆ − s1 ) − qˆ1 (s1 − s2 ) − · · · − qˆi (si − si+1 ). def
So that in general for i ≥ 1 we have yi+1 = yi − qˆi (si − si+1 ).
(6.2)
Asymptotic Stability for Soft Potentials
579
χ
χ
Furthermore k χ ( p, q) = k= ( p, q) + k≈ ( p, q) with the notation def χ k= ( p, q) = k χ ( p, q) 1| p|≥2|q| + 1|q|≥2| p| . Now we will develop a collection of notations in order to put this expansion into a general framework and appropriately define the high order terms. We consider the sequence {κ}, where for i ≥ 1 we define κi+1 = κi /2 with a small κ1 = κ > 0 as above so that κi = κ/2i−1 . For i ≥ 1 we can now define t def χ −ν( p)(t−s1 ) ds1 e dq1 k≈ ( p, q1 ) 1high f (s1 , y1 , q1 ), A( f )(t, x, p, {κ}) = R3 κ1 def χ χ dq2 k≈ (q1 , q2 ) · · · dqi+1 k≈ (qi , qi+1 ) Bi+1 ( f )(s1 , y1 , q1 , {κ}) = 3 3 R R s1 −ν(q1 )(s1 −s2 ) × ds2 e ··· s1 −κ2 si dsi+1 e−ν(qi )(si −si+1 ) f (si+1 , yi+1 , qi+1 ). × si −κi+1
To finish off our expansion we further define def ˜ f )(s, y, q) = G(s) f 0 (y, q), G( si+1 def D( f )(si+1 , yi+1 , qi+1 ) = dsi+2 e−ν(qi+1 )(si+1 −si+2 )
0
×K 1−χ {U (si+2 ) f 0 } (yi+2 , qi+1 ). Above we recall that G(s) is defined just below (4.2). Additionally si+1 def N ( f )(si+1 , yi+1 , qi+1 ) = dsi+2 e−ν(qi+1 )(si+1 −si+2 ) 0 χ × dqi+2 k= (qi+1 , qi+2 ) {U (si+2 ) f 0 } (yi+2 , qi+2 ), R3 si+1 −κi+2 def ˜ f )(si+1 , yi+1 , qi+1 , {κ}) = L( dsi+2 e−ν(qi+1 )(si+1 −si+2 ) 0 χ × dqi+2 k≈ (qi+1 , qi+2 ) {U (si+2 ) f 0 } (yi+2 , qi+2 ). R3
Then we may use this notation to write ˜ f )))(t, x, p, {κ}), G 2 ( f ) = A(B2 (G(
D2 ( f ) = A(B2 (D( f )))(t, x, p, {κ}), def def ˜ f )))(t, x, p, {κ}), N2 ( f ) = A(B2 (N ( f )))(t, x, p, {κ}), L 2 ( f ) = A(B2 ( L( def
def
R3 ( f ) = A(B3 ( f ))(t, x, p, {κ}). def
We iteratively define the higher order terms of this expansion for i ≥ 2 as ˜ f )))(t, x, p, {κ}), G i ( f ) = A(Bi (G(
Di ( f ) = A(Bi (D( f )))(t, x, p, {κ}), def def ˜ f )))(t, x, p, {κ}), Ni ( f ) = A(Bi (N ( f )))(t, x, p, {κ}), L i ( f ) = A(Bi ( L( def
def
Ri+1 ( f ) = A(Bi+1 ( f ))(t, x, p, {κ}). def
580
R. M. Strain
This expansion works to high order using (4.3) which implies Bi = Bi G˜ + Bi D + Bi N + Bi L˜ + Bi+1 , i ≥ 2. This completes our general discussion of the expansion formula, and our strategy for obtaining the desired decay. In the following we prove the claimed time decay estimates for each term in a general framework. We initially estimate the main term, Rk+1 , with k ≥ 1. We claim that w (q1 ) |Bk+1 ( f )(s1 , y1 , q1 , {κ})| s1 w−k(1+δ) (q1 ) ||| f |||k+1,
s1 −κ2
ds2 e−cν(q1 )(s1 −s2 ) (1 + s2 )−k−1 .
(6.3)
Here we have chosen δ > 0 to satisfy δ < 2ζ /b, for ζ from Lemma 3.2. This claim (6.3) would imply with Lemma 3.2 that w ( p) |Rk+1 ( f )(t, x, p, {κ})| = w ( p) |A(Bk+1 ( f ))(t, x, p, {κ})| t χ −ν( p)(t−s1 ) ds1 e dq1 k≈ ( p, q1 ) 1high w−k(1+δ) (q1 ) ||| f |||k+1, 3 R κ1 s1 × ds2 e−cν(q1 )(s1 −s2 ) (1 + s2 )−k−1 s1 −κ2
w−(k+1)(1+δ) ( p) ||| f |||k+1, Mζ
t
ds1 e−cν( p)(t−s1 ) (1 + s1 )−k−1 .
0
We have used that s1 ≈ s2 for κ sufficiently small, and also e−cν(q1 )(s1 −s2 ) 1 since s1 − s2 ≥ 0. We have additionally used the fact that the momentum variables are comχ parable because of the support condition for k≈ . The large M comes from the support of 1high in (4.14) and Lemma 3.2. Furthermore t 1 ζ ||| f |||k+1, ds1 (1 + t − s1 )−k−1−δ (1 + s1 )−k−1 M 0 1 ζ (1 + t)−k−1 ||| f |||k+1, . M We have additionally used (4.8) and then Proposition 4.5. For M 1 chosen sufficiently large, this is the desired estimate for Rk+1 . We now prove the claim from (6.3). Since all of the momentum variables are comparable in this operator we have the following iterated estimate: e−ν(q1 )(s1 −s2 ) · · · e−ν(qk )(sk −sk+1 ) ≤ e−Cν(q1 )(s1 −sk+1 ) .
(6.4)
This uses in particular Lemma 3.1. We use (6.4) to obtain w (q1 ) |Bk+1 ( f )(s1 , y1 , q1 , {κ})| χ χ dq2 k≈ (q1 , q2 ) · · · dqk+1 k≈ (qk , qk+1 ) ||| f |||k+1, 3 R3 s1 sk R × ds2 e−ν(q1 )(s1 −s2 ) · · · dsk+1 e−ν(qk )(sk −sk+1 ) (1 + sk+1 )−k−1 s1 −κ2 sk −κk+1 χ χ ||| f |||k+1, dq2 k≈ (q1 , q2 ) · · · dqk+1 k≈ (qk , qk+1 )
×
s1 s1 −κ2
R3
ds2 · · ·
sk sk −κk+1
R3
dsk+1 e−Cν(q1 )(s1 −sk+1 ) (1 + sk+1 )−k−1 .
Asymptotic Stability for Soft Potentials
581
We sub-claim that Lemma 3.2 can be used to control the momentum integrals as χ χ dq2 k≈ (q1 , q2 ) · · · dqk+1 k≈ (qk , qk+1 ) w−k(1+δ) (q1 ). (6.5) R3
R3
The trick used in this estimate is Lemma 3.2 combined with q1 − q2 → q2 to obtain χ dq2 k≈ (q1 , q2 ) w−(1+δ) (q1 ) dq2 e−c|q2 | . R3
R3
We can do that k times starting with the dqk+1 integral and iterating backwards (using the essential point that all the momentum variables are comparable) to obtain the sub-claim. Now by the definition of the sequence {κ} we can say that sk+1 ≤ s2 for k ≥ 1 and more generally (on the integration region of Bk+1 ) sk+1 ≥ sk − κk+1 ≥ s2 − κ3 − · · · − κk+1
1 1 κ 1 + · · · + k ≥ s2 − ≥ s2 − . = s2 − κ 4 2 2 4 Thus for κ ≤ 1/2 we use these estimates above to obtain s1 sk ds2 · · · dsk+1 e−cν(q1 )(s1 −sk+1 ) (1 + sk+1 )−k−1 s1 −κ2
sk −κk+1 s1
κk+1 · · · κ3
s1 −κ2
ds2 e−cν(q1 )(s1 −s2 )
1 + s2 2
−k−1
.
Collecting the estimates above proves the claim (6.3). To estimate the first term above, S1 , we obtain χ χ w ( p) |S1 ( f )(t)| w ( p) dq1 k≈ ( p, q1 ) dq2 k≈ (q1 , q2 ) 1high 3 3 κ1 R κ1 R × ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) |{U (s2 ) f 0 } (y2 , q2 )| . 0
0
Now we use (4.8), and Lemma 3.1, to observe that e−ν( p)t wk ( p)(1 + t)−k . κ κ Furthermore 0 1 ds1 eν( p)s1 0 1 ds2 e−ν(q1 )(s1 −s2 ) κ12 . We thus have χ w ( p) |S1 ( f )(t)| κ12 (1 + t)−k ||| f |||0,+k dq1 k≈ ( p, q1 ) R3 wk ( p)1high χ × dq2 k≈ (q1 , q2 ) wk (q2 ) R3 κ12 (1 + t)−k f 0 ∞,+k . M 2ζ
(6.6)
We have used the uniform bound from Theorem 4.1 with no decay and the bound for |k χ | from Lemma 3.2 and (4.14). This is the desired estimate for S1 . We continue with an estimate for L i ( f ) with i ≥ 1. For all of the terms below we switch from the notation of k to the notation of i to indicate that the decay of each of
582
R. M. Strain
these terms will not depend upon the index of the term, which is contrary to the decay of the Rk terms above. We estimate from above χ χ dq1 k≈ ( p, q1 ) · · · dqi+1 k≈ (qi , qi+1 ) w ( p) |L i ( f )| w ( p) R3 R3
s1 t si−1 −ν( p)(t−s1 ) −ν(q1 )(s1 −s2 ) −ν(qi−1 )(si−1 −si ) × ds1 e ds2 e ··· dsi e κ1
s1 −κ2
si −κi+1
×1high
si−1 −κi
dsi+1 e−ν(qi )(si −si+1 ) |{U (si+1 ) f 0 } (yi+1 , qi+1 )| .
0
The term in parenthesis above would be simply unity in the case of L 1 . Since all the momentum variables are comparable, we control the time integrals as t si−1 s1 ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) · · · dsi e−ν(qi−1 )(si−1 −si ) κ1
s1 −κ2
si−1 −κi
si −κi+1
dsi+1 e−ν(qi )(si −si+1 ) 0 t−κi+1 t (κ2 · · · κi ) ds1 dsi+1 e−Cν( p)(t−si+1 ) . ×
κ1
0
−b/2 Cq j0
−b/2
−b/2
−b/2
We have used ν(q j ) ≥ from Lemma 3.1, ν( p) ≈ p0 , and q j0 ≥ C p0 ; these estimates hold for any j ∈ {1, . . . , i}. We have then used an estimate analogous to (6.4). Furthermore t−κi+1 t ds1 dsi+1 e−Cν( p)(t−si+1 ) (κ2 · · · κi ) κ1
κ i−1 (1 + t)
0 t−κi+1
0
κ
i−1
dsi+1 e−Cν( p)(t−si+1 )
t−κi+1
(1 + t) w j ( p)
dsi+1 (1 + t − si+1 )− j .
0
These estimates follow from the definition of the sequence {κ} as well as Lemma 3.1 together with (4.8) in the form (6.6) for any j ≥ 0. Next we use Cauchy-Schwartz to estimate the following two integrals: χ w+ j (qi−1 ) dqi k≈ (qi−1 , qi ) R3 χ × dqi+1 k≈ (qi , qi+1 ) |{U (si+1 ) f 0 } (yi+1 , qi+1 )| R3
R3
dqi
×
R3
R3
2 χ χ dqi+1 w2 (qi+1 )k≈ (qi−1 , qi )k≈ (qi , qi+1 )
dqi Zi
1/2
2 dqi+1 w+ j−2 (qi+1 ) {U (si+1 ) f 0 } (yi+1 , qi+1 )
1/2 . (6.7)
Above Z i = {qi+1 : 21 |qi | ≤ |qi+1 | ≤ 2|qi |}. Also in the case i = 1 we consider qi−1 = p. For now we focus on the second set of integrals involving the semi-group. def
Asymptotic Stability for Soft Potentials
583
We apply the change of variables qi → yi+1 on the dqi integration with yi+1 given by (6.2). Notice that similar to (4.17) the 3 × 3 matrix Jacobian is
2 −q q δmn qi0 dyi+1 im in = −(si − si+1 ) . 3 dqi mn qi0 We recall qi = (qi1 , qi2 , qi3 ) with qi0 = 1 + |qi |2 . This Jacobian matrix has two eigeni+1 ) , and a third eigenvalue which is given by −(si −si+1 ) values equal to − (si −s qi0
−(si − si+1 ) q13 . i0
2 −|q |2 qi0 i 3 qi0
=
Therefore the Jacobian determinant is
3 dyi+1 |(si − si+1 )|3 κi+1 κ3 ≥ = . dq = 5 5 5 qi0 25 q(i+1)0 25+3i q(i+1)0 i
This lower bound holds on the region qi0 ≤ 2q(i+1)0 , si+1 ∈ [0, si − κi+1 ]. Furthermore we have used that si − si+1 ≥ κi+1 ; this temporal estimate holds on the integration region of L i . These estimates explain the lower bound for the Jacobian. Notice while the old variable qi occupies the whole space, the new variable yi+1 satisfies the estimate |yi+1 − x| ≤ p(t ˆ − s1 ) + qˆ1 (s1 − s2 ) + · · · + qˆi (si − si+1 ) ≤ C ((t − s1 ) + (s1 − s2 ) + · · · + (si − si+1 )) ≤ C (t − si+1 ) . We remark that this procedure would not hold in the non-relativistic situation, since in that case we do not have bounded velocities. In particular, because of relativity, the mapping qi → yi+1 sends R3 into a bounded domain (for any finite t). After application of this change of variables, denoting yi+1 = y, we have 2 dqi dqi+1 w+ j−2 (qi+1 ) {U (si+1 ) f 0 } (yi+1 , qi+1 ) R3 Zi dqi 2 dy dqi+1 w+ j−2 (qi+1 ) {U (si+1 ) f 0 } (y, qi+1 ) dy Z i |y−x|≤C(t−si+1 ) 3 (1 + t − si+1 ) dy dqi+1 w2+2 j−4+10/b (qi+1 ) |{U (si+1 ) f 0 } (y, qi+1 )|2 κ3 T3 R3 = C(κ) (1 + t − si+1 )3 {U (si+1 ) f 0 } 22,+ j−2+5/b . 5 = w10/b (qi+1 ). This estimate above is the main one for the We have used that q(i+1)0 L i ( f ) terms which allows us to deduce high order decay. Since we have used (6.7), for the momentum integrals in L i , we are left to control the iteration of kernels. We claim the following estimate:
χ χ dq1 k≈ ( p, q1 ) · · · dqi−1 k≈ (qi−2 , qi−1 )
R3
×
R3
dqi
R3
R3
2 χ χ dqi+1 w2 (qi+1 )k≈ (qi−1 , qi )k≈ (qi , qi+1 ) 1high
1/2
1 . Mζ
Note that if i = 1 then the first term in parenthesis above is simply unity. Firstly, from Lemma 3.2 we have 1 χ χ w2 (qi+1 )k≈ (qi−1 , qi )k≈ (qi , qi+1 )1high ζ e−c|qi−1 −qi | e−c|qi −qi+1 | . M
584
R. M. Strain
This also uses (4.14) and the fact that all the momentum variables are comparable. They key point is then to employ the following series of changes of variables which begins with qi − qi+1 → qi+1 on dqi+1 , qi−1 − qi → qi on dqi , and ends with p − q1 → q1 on dq1 . The end result, with Lemma 3.2, is that
χ χ dq1 k≈ ( p, q1 ) · · · dqi−1 k≈ (qi−2 , qi−1 ) R3
×
R3
R3
R3
dqi
R3
1/2 −c|qi−1 −qi | −c|qi −qi+1 | 2 dqi+1 e e
dq1 e−c|q1 | · · ·
R3
1.
dqi−1 e−c|qi−1 |
R3
dqi
R3
dqi+1 e−2c|qi | e−2c|qi+1 |
1/2
Collecting the estimates in this paragraph establishes the claim. We gather all of the estimates for L i ( f ) to obtain t w ( p) |L i ( f )| (1 + t) dsi+1 (1 + t − si+1 )− j+3/2 0
× {U (si+1 ) f 0 } 2,+ j−2+5/b t f 0 2,+ j−2+5/b+k (1 + t) dsi+1 (1+t −si+1 )− j+3/2 (1+si+1 )−k−1 . 0
Above we have used the decay of the linear solutions to (2.11) from Theorem 3.7; these solutions are represented by (4.1). Then for j ≥ k + 1 + 3/2 and k = j − 2 + 5/b + k + (for any > 3/b) we use Proposition 4.5 to show that w ( p) |L i ( f )| f 0 2,+k − (1 + t)−k f 0 ∞,+k (1 + t)−k . The last inequality above follows as in (4.7). This is the desired estimate for L i ( f )(t) which holds for any i ≥ 1 and k ≥ 0. It remains to estimate G i+1 ( f ), Di+1 ( f ), and Ni+1 ( f ) for i ≥ 1. First t χ dq1 k≈ ( p, q1 ) 1high ds1 e−ν( p)(t−s1 ) w ( p) |G i+1 ( f )| R3 κ1 χ χ × dq2 k≈ (q1 , q2 ) · · · dqi+1 k≈ (qi , qi+1 ) 3 R3 Rs1 si −ν(q1 )(s1 −s2 ) × ds2 e ··· dsi+1 e−ν(qi )(si −si+1 ) s1 −κ2
×w (qi+1 )e
si −κi+1
−ν(qi+1 )si+1
f 0 (yi+1 − qˆi+1 si+1 , qi+1 ).
We have used that all the momentum variables are comparable and the trick from (6.4) to conclude that the upper bound above is further bounded as χ χ χ dq1 k≈ ( p, q1 ) 1high dq2 k≈ (q1 , q2 ) · · · dqi+1 k≈ (qi , qi+1 ) R3 R3 R3 si s1 t f 0 ∞,+k ×e−Cν( p)t ds1 ds2 · · · dsi+1 . wk (qi+1 ) κ1 s1 −κ2 si −κi+1
Asymptotic Stability for Soft Potentials
585
Notice that using the definition of κi = κ/2i−1 we have si s1 t ds1 ds2 · · · dsi+1 = κ i 2−i(i+1)/2 s1 −κ2
κ1
si −κi+1
t
κ1
ds1
(1 + t). Furthermore, as in (6.5) and Lemma 3.2 with (4.14) we have χ dq1 k≈ ( p, q1 ) 1high R3 χ χ × dq2 k≈ (q1 , q2 ) · · · dqi+1 k≈ (qi , qi+1 ) R3
R3
1 wk (qi+1 )
1 1 1 . 2ζ M wk+i+1 ( p) wk+2 ( p)
Moreover, for any k ≥ 0, by (4.8) as in (6.6) we have e−Cν( p)t wk+2 ( p)(1 + t)−k−2 . Collecting the last few estimates we obtain w ( p) |G i+1 ( f )| (1 + t)−k−1 f 0 ∞,+k . This is the desired estimate for G i+1 ( f ). We will now study Di+1 ( f ), which satisfies the following general estimate: χ χ w ( p) |Di+1 ( f )| dq1 k≈ ( p, q1 ) · · · dqi+1 k≈ (qi , qi+1 )
R3
R3
si ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) · · · dsi+1 e−ν(qi )(si −si+1 ) s1 −κ2 si −κi+1 κ1 si+1 −ν(qi+1 )(si+1 −si+2 ) 1−χ ×1high w ( p) dsi+2 e ({U (si+2 ) f 0 }) (yi+2 , qi+1 ) . K ×
t
s1
0
Since all the momentum variables are comparable, with Lemma 4.6, we have χ χ χ dq1 k≈ ( p, q1 ) 1high dq2 k≈ (q1 , q2 ) · · · dqi+1 k≈ (qi , qi+1 ) R3 R3 R3 t si s1 × ds1 e−ν( p)(t−s1 ) ds2 e−ν(q1 )(s1 −s2 ) · · · dsi+1 e−ν(qi )(si −si+1 ) s1 −κ2 si −κi+1 κ1 si+1 ×e−cp0 ||| f |||k, dsi+2 e−ν(qi+1 )(si+1 −si+2 ) (1 + si+2 )−k . 0
For this term Lemma 4.6 would allow a better estimate for the momentum weight on ||| f |||k, . As in the estimate for G i+1 above, with (4.14), we have 1 w ( p) |Di+1 ( f )| 2ζ e−cp0 ||| f |||k, M t si s1 × ds1 e−Cν( p)(t−s1 ) ds2 e−Cν( p)(s1 −s2 ) · · · dsi+1 e−Cν( p)(si −si+1 ) s1 −κ2 si −κi+1 κ1 si+1 × dsi+2 e−Cν( p)(si+1 −si+2 ) (1 + si+2 )−k . 0
586
R. M. Strain
We have again used the crucial fact that all the momentum variables are comparable. Since we have exponential decay, we can iterate the estimates from (4.8) and Proposition 4.5 as in (6.6) to obtain t si s1 ds1 e−Cν( p)(t−s1 ) ds2 e−Cν( p)(s1 −s2 ) · · · dsi+1 e−Cν( p)(si −si+1 ) s1 −κ2 si −κi+1 κ1 si+1 × dsi+2 e−Cν( p)(si+1 −si+2 ) (1 + si+2 )−k 0 t si s1 ≤ ds1 e−Cν( p)(t−s1 ) ds2 e−Cν( p)(s1 −s2 ) · · · dsi+1 e−Cν( p)(si −si+1 ) 0 0 0 si+1 −Cν( p)(si+1 −si+2 ) −k × dsi+2 e (1 + si+2 )
0 t
ds1 (1 + (t − s1 ))−k−1
0
sj
ds j+1 (1 + (s j − s j+1 ))−k−1
j=1 0
×w(k+1)(i+2) ( p)
i !
si+1
dsi+2 (1 + (si+1 − si+2 ))−k−1 (1 + si+2 )−k .
(6.8)
0
After iteratively applying Proposition 4.5 we obtain an upper bound of w(k+1)(i+2) ( p)(1 + t)−k . Plugging this into the previous estimate we have C w ( p) |Di+1 ( f )| ≤ 2ζ w(k+1)(i+2) ( p)e−cp0 (1 + t)−k ||| f |||k, M C ≤ 2ζ (1 + t)−k ||| f |||k, . M This is the desired estimate for Di+1 ( f ) when M is chosen sufficiently large. The final term to estimate is Ni+1 ( f ). In this case we have the upper bound t χ |N dq1 k≈ ( p, q1 ) 1high ds1 e−ν( p)(t−s1 ) w ( p) i+1 ( f )| w ( p) 3 R κ1 χ χ × dq2 k≈ (q1 , q2 ) · · · dqi+1 k≈ (qi , qi+1 ) Rs1
R3
3
× ×
s −κ 1si+12
ds2 e−ν(q1 )(s1 −s2 ) · · ·
dsi+2 e−ν(qi+1 )(si+1 −si+2 )
0
× |{U (si+2 ) f 0 }(yi+2 , qi+2 )| .
si
si −κi+1
R3
dsi+1 e−ν(qi )(si −si+1 )
χ dqi+2 k= (qi+1 , qi+2 )
We first estimate the time integrals above. Since the relevant momentum variables are all comparable, as in (6.8) and (6.6) we have si t s1 −ν( p)(t−s1 ) −ν(q1 )(s1 −s2 ) ds1 e ds2 e ··· dsi+1 e−ν(qi )(si −si+1 ) s1 −κ2 si −κi+1 κ1 si+1 × dsi+2 e−ν(qi+1 )(si+1 −si+2 ) (1 + si+2 )−k 0
w(k+1)(i+2) ( p)(1 + t)−k .
Asymptotic Stability for Soft Potentials
587
Now for the momentum integrals, from Lemma 3.2 and (4.14) we have χ χ χ w ( p)w(k+1)(i+2) ( p) dq1 k≈ ( p, q1 ) dq2 k≈ (q1 , q2 ) · · · dqi+1 k≈ (qi , qi+1 ) R3 R3 R3 χ × 1high dqi+2 k= (qi+1 , qi+2 ) R3 C ≤ (i+2)ζ dq1 e−c| p−q1 | 1high dq2 e−c|q1 −q2 | · · · dqi+1 e−c|qi −qi+1 | 3 3 3 M R R R −c|qi+1 −qi+2 | 1|qi+2 |≥2|qi+1 | + 1|qi+1 |≥2|qi+2 | w+k(i+2) (qi+1 ) × dqi+2 e R3
≤
C . M (i+2)ζ
In the last step we have used the following estimate: c 1|qi+2 |≥2|qi+1 | + 1|qi+1 |≥2|qi+2 | e− 2 |qi+1 −qi+2 | w+k(i+2) (qi+1 ) ≤ C. Indeed in either of the regions |qi+1 | ≥ 2|qi+2 | or |qi+2 | ≥ 2|qi+1 | we can use estimates such as those in (3.21) or (3.22) to directly establish this bound. Now by collecting the estimates in this paragraph, we have shown w ( p) |Ni+1 ( f )| ≤
C M (i+2)ζ
(1 + t)−k ||| f |||k,0 .
Since k ≥ 0 is arbitrary, we conclude our estimate and our proposition after choosing M sufficiently large in this last upper bound. This concludes our proof of rapid linear decay. 7. Nonlinear L ∞ Rapid Decay In Sect. 5, we have proven the existence of mild solutions (5.1) to the non-linear relativistic Boltzmann equation (2.1) with the soft potentials. For > 3/b and k ∈ (1/2, 1] we have shown in Theorem 5.1 that these solutions, f = f (t, x, p), satisfy f ∞, (t) ≤ C,k (1 + t)−k f 0 ∞,+k . Then in Sect. 6 we prove high order “almost exponential” decay for the linear semigroup as in (6.1). From these estimates and the solution formula, (5.1), we can prove the following non-linear almost exponential decay. Theorem 7.1. Given any > 3/b and k ≥ 0, there is a k = k (k) ≥ 0 such that the solutions which were proven to exist in Thereom 5.1 further satisfy f ∞, (t) ≤ C,k (1 + t)−k f 0 ∞,+k . Proof of Theorem 7.1. We use an induction which allows one to continually improve the decay. The main point is to bound the non-linear term, since we already know this kind of rapid decay for the linear part of (5.1) from (6.1) which follows from the crucial
588
R. M. Strain
Proposition 6.1. In the first step we note that by Thereom 5.1, Theorem 7.1 is true for k ∈ (1/2, 1]. Then given any j ≥ 0, from (6.1) we have t w ( p) |N [ f, f ](t, x, p)| ds w ( p) |{U (t − s) [ f (s), f (s)]}(x, p)| 0 t ds (1 + t − s)− j [ f (s), f (s)] ∞,+ j . 0
Above j ≥ j is the number corresponding to j in (6.1). From Lemma 5.2 we have t ds (1 + t − s)− j f (s) ∞,+ j −1 f (s) ∞,+ j −1 . 0
Next we use the non-linear decay from Theorem 5.1 to see t f 0 2∞,+ j +i−1 ds (1 + t − s)− j (1 + s)−2i (1 + t)
−ρ
0 2 f 0 ∞,+ j +i−1 .
The last estimate follows from Proposition 4.5 with ρ = min{ j + 2i − 1, min{ j, 2i}}. In the above estimates we can choose j ∈ (1, 2] and then i ∈ (1/2, 1] such that j = 2i > 1. Then we have shown Theorem 7.1 for k ∈ (1, 2] by choosing ρ = j = k and k = max{ j , j + i − 1} = j . Next suppose the theorem is correct for some k > 2; we will show that we may go beyond this k. Indeed similar to the initial case we have w ( p) |N [ f, f ](t, x, p)| ≤ C(1 + t)−ρ f 0 2∞,+ j +i −1 , with ρ = min{ j + 2i − 1, min{ j, 2i}} = min{ j, 2i}. Above j corresponds to the power of the weight coming from decay level j in (6.1) and i corresponds to the power of the weight generated by decay level i ∈ (0, k] in this Theorem 7.1. Choose i ∈ (k/2, k] and j = 2i ∈ (k, 2k]. This is always possible. Then we have ρ = j so that we have proven Theorem 7.1 for any k˜ ∈ (k, 2k] and the corresponding k˜ = max{ j , j + i − 1}. We conclude by induction. Acknowledgements. The author gratefully thanks the anonymous referees for their lengthy detailed comments which helped to substantially improve the presentation of this research paper.
Appendix: Derivation of the Compact Operator In this section, we give a complete exposition of the derivation of the Hilbert-Schmidt form (3.10) for the compact operator from (2.4). The linearized collision operator takes the form (2.2). In that formulation we have the multiplication operator as in (2.3). The remaining “compact” part of the linearized operator is given by (2.4) with K = K 2 − K 1 and in particular dq dq dp 1 def )h( p ) K 2 (h) = W ( p, q| p , q ) J (q) J (q p0 R3 q0 R3 q0 R3 p0 dq dq dp 1 )h(q ) . W ( p, q| p , q ) J (q) J ( p + p0 R3 q0 R3 q0 R3 p0
Asymptotic Stability for Soft Potentials
589
We are using the original notation from the top of this paper, which includes the delta functions. We will outline in detail a procedure which is sketched in [9, p. 277] (see also [11]), that allows a nice reduction of the term K 2 as in (3.10). In particular we give the exact form of the Lorentz transformation. This reduction for the K 1 term can be reduced to the form (3.9) using much simpler methods than the ones we use below, see e.g. [9,49,50]. We recall the definition of the transition rate, W , from the top of this paper. We plug the definition of W into K 2 above to obtain dq dq dp 1 def K 2 (h) = sσ (g, θ )δ (4) ( p μ + q μ − p μ − q μ ) p0 R3 q0 R3 q0 R3 p0 × J (q) J (q )h( p ) + J ( p )h(q ) . We will first reduce this to a Hilbert-Schmidt form and second carry out the delta function integrations in the kernel. Recall the discussion at the beginning of Sect. 1.2 regarding our convention for raising and lowering indices and the Lorentz inner product. In preparation, we write down some invariant quantities. By (1.6) and (1.4) we obtain ( p μ − q μ )( pμ − qμ ) = 2 p μ pμ + 2q μ qμ − ( p μ + q μ )( pμ + qμ ) = 2 p μ pμ + 2q μ qμ + s. Further notice that (1.6) implies ( p μ − p μ )( pμ − pμ ) = (q μ − q μ )(qμ − qμ ). Expanding this we have −2 − 2 p μ pμ = −2 − 2q μ qμ . We thus have another invariant p μ pμ = q μ qμ . Define g¯ = g( p μ , p μ ) as in (1.5). We will always use g without the bar to exclusively denote g = g( p μ , q μ ). From (1.4) and (1.5) we know s = g 2 + 4. We may re-express θ from (1.7) as
2 g¯ μ μ 2 . cos θ = ( p − q )( pμ − qμ )/g = 1 − 2 g This follows from the invariant calculations in the previous paragraph and ( p μ − q μ )( pμ − qμ ) = g 2 + 4 + 4 p μ pμ = g 2 − 2g¯ 2 . We further claim that 1 g 2 = g¯ 2 − ( p μ + p μ )(qμ + qμ − pμ − pμ ). 2 Let s¯ = s( p μ , p μ ) = g¯ 2 + 4. Then (7.1) is equivalent to 1 1 g 2 = g¯ 2 − s¯ − ( p μ + p μ )(qμ + qμ ) 2 2 1 1 = g¯ 2 − 2 − ( p μ + p μ )(qμ + qμ ) 2 2 1 1 = g¯ 2 + g 2 + 2 p μ qμ − ( p μ + p μ )(qμ + qμ ). 2 2
(7.1)
590
R. M. Strain
We thus prove (7.1) by showing that 1 2 1 g¯ + 2 p μ qμ − ( p μ + p μ )(qμ + qμ ) = 0. 2 2 Expanding this expression we obtain − p μ pμ − 1 + 2 p μ qμ −
1 μ 1 1 1 p qμ − p μ qμ − p μ qμ − p μ qμ . 2 2 2 2
Notice further that p μ qμ = p μ qμ and p μ pμ = q μ qμ as a result of (1.6) and (1.4). We thus obtain p μ qμ − 1 −
1 μ 1 1 1 p pμ − q μ qμ − p μ qμ − p μ qμ , 2 2 2 2
which by (1.6) is 1 1 p μ qμ − 1 − ( p μ + q μ )( pμ + qμ ) = p μ qμ − 1 + s = 0. 2 2 This establishes the claim (7.1). Now we establish the Hilbert-Schmidt form. First consider 1 dq dq dp sσ (g, θ )δ (4) ( p μ + q μ − p μ − q μ ) J (q) J (q )h( p ). p 0 R3 q 0 R3 q 0 R3 p 0 Exchanging q with p the integral above is equal to 1 dq dp dq (4) μ μ μ μ ) J (q ) , h(q) s ¯ σ ( g, ¯ θ )δ ( p + p − q − q ) J ( p p0 q0 q0 p0 where θ is now defined by
2 g cos θ = 1 − 2 , g¯
(7.2)
and from (7.1), with the new argument in the delta function above, we have 1 g¯ 2 = g 2 + ( p μ + q μ )( pμ + qμ − pμ − qμ ), 2
(7.3)
and further s¯ is defined by s¯ = g¯ 2 + 4. We do a similar calculation for the second term in K 2 h, e.g. exchange q with q and then swap the q and p notation. The result is that we can define 2 dq dp def k2 ( p, q) = s¯ σ (g, ¯ θ )δ (4) ( p μ + p μ −q μ −q μ ) J ( p ) J (q ). p0 q 0 q0 p0 (7.4) We now write the Hilbert-Schmidt form K 2 (h) = k2 ( p, q)h(q)dq. We will carry out the delta function integrations in k2 ( p, q) using a special Lorentz matrix.
Asymptotic Stability for Soft Potentials
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We first translate (7.4) into an expression involving the total and relative momentum variables, p μ + q μ and p μ − q μ respectively. Define u by u(r ) = 0 if r < 0 and u(r ) = 1 if r ≥ 0. Let g = g( p μ , q μ ) and s = s( p μ , q μ ). We claim that
dq dp 1 μ μ μ μ G( p , q , p , q ) = d( p μ , q μ )G( p μ , q μ , p μ , q μ ), 16 R4 ×R4 R3 q 0 R3 p 0 (7.5)
where we are now integrating over the eight vector ( p μ , q μ ) and d( p μ , q μ ) = dp μ dq μ u( p0 + q0 )u(s − 4)δ(s − g 2 − 4)δ(( p μ + q μ )( pμ − qμ )). To establish the claim, first notice that −( p μ + q μ )( pμ − qμ ) = − p μ pμ + q μ qμ = ( p0 )2 − | p |2 − (q0 )2 + |q |2 = A p − Aq , where now p 0 and q 0 are integration variables and we have defined A p = ( p0 )2 − (| p |2 + 1),
Aq = (q0 )2 − (|q |2 + 1).
Integrating first over dp μ , we see that alternatively −( p μ + q μ )( pμ − qμ ) = ( p0 )2 − (| p |2 + 1 + Aq ) 2 2 p0 + | p | + 1 + A q . = p0 − | p | + 1 + A q Furthermore, by (1.4) and (1.5) we have s − g 2 − 4 = −( p μ + q μ )( pμ + qμ ) − ( p μ − q μ )( pμ − qμ ) − 4 = −2 p μ pμ − 2q μ qμ − 4 = 2 A p + 2 Aq .
Then similarly s − g 2 − 4 = 2(q0 )2 − 2[|q |2 + 1 − A p ] 2 2 = 2 q0 − |q | + 1 − A p q0 + |q | + 1 − A p . Further note that p0 + q0 ≥ 0 and s − 4 ≥ 0 together imply p0 ≥ 0 and q0 ≥ 0. With these expressions and standard calculations we establish (7.5). We thus conclude that 1 1 k2 ( p, q) = d( p μ , q μ )¯s σ (g, ¯ θ )δ (4) ( p μ + p μ −q μ −q μ ) J (q )J ( p ). p 0 q 0 8 R 4 ×R 4 Now apply the change of variables p¯ μ = p μ + q μ , q¯ μ = p μ − q μ .
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R. M. Strain
This transformation has Jacobian = 16 and inverse tranformation as p μ =
1 μ 1 μ 1 1 p¯ + q¯ , q μ = p¯ μ − q¯ μ . 2 2 2 2
With this change of variable, for some c > 0, the integral becomes c k2 ( p, q) = d( p¯ μ , q¯ μ )¯s σ (g, ¯ θ )δ (4) ( p μ − q μ + q¯ μ ) J ( p), ¯ p 0 q 0 R 4 ×R 4 √ with J ( p) ¯ = e− p¯0 /2 (ignoring constants). Above the measure is now d( p¯ μ , q¯ μ ) = d p¯ μ d q¯ μ u( p¯ 0 )u(− p¯ μ p¯ μ − 4)δ(− p¯ μ p¯ μ − q¯ μ q¯μ − 4)δ( p¯ μ q¯μ ). Also g¯ ≥ 0 from (7.3) is now given by 1 g¯ 2 = g 2 + ( p μ + q μ )( pμ + qμ − p¯ μ ), 2 and θ and s¯ can be defined through the new g¯ with (7.2). We next carry out the delta function argument for δ (4) ( p μ − q μ + q¯ μ ) to obtain c k2 ( p, q) = d( p¯ μ )¯s σ (g, ¯ θ )e− p¯0 /2 , ∃c > 0, p 0 q 0 R4 where the measure is d( p¯ μ ) = d p¯ μ u( p¯ 0 ) u(− p¯ μ p¯ μ − 4) δ(− p¯ μ p¯ μ − g 2 − 4)δ( p¯ μ (qμ − pμ )). Since s = g 2 + 4 we have u( p¯ 0 )δ(− p¯ μ p¯ μ − g 2 − 4) = u( p¯ 0 )δ(− p¯ μ p¯ μ − s) = u( p¯ 0 )δ(( p¯ 0 )2 − | p| ¯ 2 − s) δ( p¯ 0 − | p| ¯ 2 + s) = . 2 | p| ¯ 2+s We then carry out one integration using the delta function to get d p¯ c k2 ( p, q) = u(− p¯ μ p¯ μ − 4)δ( p¯ μ (qμ − pμ ))¯s σ (g, ¯ θ )e− p¯0 /2 , 3 p0 q0 R p¯ 0 def with p¯ 0 = | p| ¯ 2 + s. Using s = g 2 + 4 we have − p¯ μ p¯ μ − 4 = s − 4 = g 2 ≥ 0. So always u(− p¯ μ p¯ μ − 4) = 1 and the integral reduces to d p¯ c μ ¯ δ( p¯ μ (qμ − pμ ))¯s σ (g, ¯ θ )e− p¯ Uμ /2 , k2 ( p, q) = 3 p0 q0 R p¯ 0 where U¯ μ = (1, 0, 0, 0), U¯ μ = (−1, 0, 0, 0) and e− p¯0 /2 = e− p¯
μU ¯
μ /2
.
Asymptotic Stability for Soft Potentials
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We finish off our reduction by moving to a new Lorentz frame. We consider a Lorentz transformation which maps into the center-of-momentum system as √ def Aν = μν ( pμ + qμ ) = ( s, 0, 0, 0),
B ν = − μν ( pμ − qμ ) = (0, 0, 0, g). def
We recall our notation for raising and lowering indices from the beginning of Sect. 1.2 def as pμ = gμν p ν , where gμν = diag(−1 1 1 1). Also recall p μ = ( p 0 , p) with p0 > 0. " Then we use the Einstein summation convention as μν pμ = 3μ=0 μν pμ . From this information, we have derived in [50] and exposited in [49] that ⎛ ⎞ p0√+q0 p1√+q1 p2√+q2 p3√+q3 − − − s s s ⎟ ⎜ s 01 11 21 31 ⎟ μν ⎜ ⎜ ⎟ =⎜ = ⎟, ( p×q)1 ( p×q)2 ( p×q)3 ⎟ ⎜ 0 ⎝ | p×q| | p×q| | p×q| ⎠ p0 −q0 1 2 3 − p1 −q − p2 −q − p3 −q g g g g with the second row given by 01 = and
2| p × q| √ , g s
& % & % 2 pi p0 + q0 p μ qμ + qi q0 + p0 p μ qμ = (i = 1, 2, 3). √ g s| p × q| i1
We have a complete description of this Lorentz transformation in terms of p, q. Define Uμ = νμ U¯ ν , notice that
p0 + q0 2| p × q| p0 − q 0 . Uμ = √ , √ , 0, g s g s Then 1 μ d p¯ d p¯ s¯ σ (g, ¯ θ )e− p¯0 /2 δ( p¯ μ (qμ − pμ )) = s¯ σ (g¯ , θ )e− 2 p¯ Uμ δ p¯ μ Bμ , p¯ 0 p¯ 0 where g¯ , s¯ ≥ 0 are now given by √ 1 μ 1√ A Aμ − p¯ μ = g 2 + s p¯ 0 − s , 2 2 2 , s¯ = 4 + g¯
2 g . cos θ = 1 − 2 g¯ 2 = g2 + g¯
The equality of the two integrals holds because d p/ ¯ p¯ 0 is a Lorentz invariant. We work with the integral on the right-hand side above. Now p¯ μ Bμ = p¯ 3 g.
(7.6)
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We switch to polar coordinates in the form d p¯ = | p| ¯ 2 d| p| ¯ sin ψdψdϕ,
p¯ ≡ | p|(sin ¯ ψ cos ϕ, sin ψ sin ϕ, cos ψ).
Then we can write k2 ( p, q) as π ∞ 2π | p| ¯ 2 d| p| ¯ c μ dϕ sin ψdψ s¯ σ (g¯ , θ ) e− p¯ Uμ /2 δ(| p|g ¯ cos ψ). p0 q 0 0 p¯ 0 0 0 We evaluate the last delta function at ψ = π/2 to write k2 ( p, q) as ∞ 2π p +q | p|d| ¯ p| ¯ c √ | p| − p¯ 0√ 0 | p×q| ¯ cos ϕ dϕ s¯ σ (g¯ , θ )e 0 2 s e g s . gp0 q0 0 p ¯ 0 0
(7.7)
This is already a useful reduced form for k2 ( p, q). √ | p| ¯ , the modified Bessel function of index zero given in We recall, for I0 | gp×q| s (3.11). We further re-label the integration as | p| ¯ = y. Notice that (7.6) implies ' 1 − cos θ θ g = g¯ = g¯ sin , 2 2 with √ θ g 2g g sin = = = . √ 2 g¯ g 2 − 4 + s y 2 /s + 1 g 2 − s/2 + 2s y 2 + s We may rewrite (7.7) as
∞ √ p +q g | p × q| ydy c − 20√s0 y 2 +s , θ e I0 k2 ( p, q) = s¯ σ √ y . gp0 q0 0 g s sin θ2 y2 + s From (7.6) we have
√ 1 1√ 1 s{ y 2 + s − s} = s + s y 2 /s + 1. 2 2 2 √ We apply the change of variables y → y/ s to obtain that k2 ( p, q) is given by
2 √ p +q g | p × q| c s 3/2 ∞ y 1 + y + 1 dy − 0 2 0 y 2 +1 y , ,ψ e I0 σ gp0 q0 0 g sin ψ2 y2 + 1 s¯ = 4 + g 2 +
where sin ψ2 is given by (3.12). This is the expression from (3.10) once we incorporate the cut-off function (3.1) which is insignificant for the purposes of this calculation. Significant simplifications can be performed in the case of the “hard ball” cross section where σ = constant. The relevant integral is then a Laplace transform and a known integral, which can be calculated exactly via a taylor expansion [39, p. 134]. For instance, it is well known that for any R > r ≥ 0, √ ∞ −R √1+y 2 2 2 e y I0 (r y) e− R −r dy = √ , R2 − r 2 1 + y2 0 √ ∞ √ 1 R 2 2 2 1 + e− R −r . e−R 1+y y I0 (r y)dy = 2 √ 2 2 2 R − r R −r 0 See for instance [46,45, p. 383, or 24, p. 322].
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Using these formulas we can express the integral as k2 ( p, q) =
c s 3/2 ˜ U1 ( p, q) exp −U˜ 2 ( p, q) , gp0 q0
def where U˜ 2 ( p, q) = {( p0 + q0 )/2}2 − (| p × q|/g)2 and
−1 p + q −2 −1 p0 + q 0 ˜ def 0 0 U˜ 2 ( p, q) U˜ 1 ( p, q) = 1 + U2 ( p, q) U˜ 2 ( p, q) + . 2 2 Further, s g2 + 4 | p − q| = | p − q| . U˜ 2 ( p, q) = 2g 4g 2 √
Therefore, U˜ 2 ( p, q) ≥ 21 | p − q| + 1. This completes our discussion of the HilbertSchmidt form for the linearized collision operator. References 1. Andréasson, H.: Regularity of the gain term and strong L 1 convergence to equilibrium for the relativistic Boltzmann equation. SIAM J. Math. Anal. 27(5), 1386–1405 (1996) 2. Andréasson, H., Calogero, S., Illner, R.: On blowup for gain-term-only classical and relativistic Boltzmann equations. Math. Meth. Appl. Sci. 27(18), 2231–2240 (2004) 3. Bichteler, K.: On the Cauchy problem of the relativistic Boltzmann equation. Commun. Math. Phys. 4, 352–364 (1967) 4. Boisseau, B., van Leeuwen, W.A.: Relativistic Boltzmann theory in D + 1 spacetime dimensions. Ann. Physics 195(2), 376–419 (1989) 5. Caflisch, R.E.: The Boltzmann equation with a soft potential. I. Linear, spatially- homogeneous. Commun. Math. Phys. 74(1), 71–95 (1980) 6. Caflisch, R.E.: The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic. Commun. Math. Phys. 74(2), 97–109 (1980) 7. Calogero, S.: The Newtonian limit of the relativistic Boltzmann equation. J. Math. Phys. 45(11), 4042– 4052 (2004) 8. Cercignani, C., Medeiros Kremer, G.: The relativistic Boltzmann equation: theory and applications. Progress in Mathematical Physics, Vol. 22, Basel: Birkhäuser Verlag, 2002 9. de Groot, S.R., van Leeuwen, W.A., van Weert, Ch.G.: Relativistic kinetic theory. Amsterdam: NorthHolland Publishing Co., 1980 10. Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomoge- neous kinetic systems: the Boltzmann equation. Invent. Math. 159(2), 245–316 (2005) 11. Dijkstra, J.J., van Leeuwen, W.A.: Mathematical aspects of relativistic kinetic theory. Phys. A 90(3–4), 450–486 (1978) 12. DiPerna, R.J., Lions, P.-L.: Global weak solutions of Vlasov-Maxwell systems. Comm. Pure Appl. Math. 42(6), 729–757 (1989) 13. DiPerna, R.J., Lions, P.-L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2) 130(2), 321–366 (1989) 14. Dudy´nski, M.: On the linearized relativistic Boltzmann equation. II. Existence of hydro-dynamics. J. Stat. Phys. 57(1–2), 199–245 (1989) 15. Dudy´nski, M., Ekiel-Je˙zewska, M.L.: The relativistic Boltzmann equation - mathematical and physical aspects. J. Tech. Phys. 48, 39–47 (2007) 16. Dudy´nski, M., Ekiel-Je˙zewska, M.L.: Causality of the linearized relativistic Boltzmann equation. Phys. Rev. Lett. 55(26), 2831–2834 (1985) 17. Dudy´nski, M., Ekiel-Je˙zewska, M.L.: Errata: Causality of the linearized relativistic Boltzmann equation. Investigación Oper. 6(1), 2228 (1985) 18. Dudy´nski, M., Ekiel-Je˙zewska, M.L.: On the linearized relativistic Boltzmann equation. I. Existence of solutions. Commun. Math. Phys. 115(4), 607–629 (1985)
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19. Dudy´nski, M., Ekiel-Je˙zewska, M.L.: Global existence proof for relativistic Boltzmann equation. J. Stat. Phys. 66(3–4), 991–1001 (1992) 20. Glassey, R.T., Strauss, W.A.: Asymptotic stability of the relativistic Maxwellian via fourteen moments. Trans. Th. Stat. Phys. 24(4–5), 657–678 (1995) 21. Glassey, R.T.: The Cauchy problem in kinetic theory. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1996 22. Glassey, R.T.: Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data. Commun. Math. Phys. 264(3), 705–724 (2006) 23. Glassey, R.T., Strauss, W.A.: On the derivatives of the collision map of relativistic particles. Trans. Th. Stat. Phys. 20(1), 55–68 (1991) 24. Glassey, R.T., Strauss, W.A.: Asymptotic stability of the relativistic Maxwellian. Publ. Res. Inst. Math. Sci. 29(2), 301–347 (1993) 25. Gressman, P.T., Strain, R.M.: Global Strong Solutions of the Boltzmann Equation without Angular Cut-off. Preprint, available at http://arXiv.org/abs/0912.0888v1 [math.AP], 2009 26. Gressman, P.T., Strain, R.M.: Global classical solutions of the Boltzmann equation with long-range interactions and soft potentials. Preprint, available at http://arXiv.org/abs/1002.3639v1 [math.AP], 2010 27. Gressman, P.T., Strain, R.M.: Global classical solutions of the Boltzmann equation with long-range interactions. Proc. Nat. Acad. Sci. U. S. A. 107(13), 5744–5749 (2010) 28. Guo, Y.: The Landau equation in a periodic box. Commun. Math. Phys. 231(3), 391–434 (2002) 29. Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55(9), 1104–1135 (2002) 30. Guo, Y.: Classical solutions to the Boltzmann equation for molecules with an angular cutoff. Arch. Rat. Mech. Anal. 169(4), 305–353 (2003) 31. Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153(3), 593–630 (2003) 32. Guo, Y.: The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53(4), 1081–1094 (2004) 33. Guo, Y.: Decay and continuity of Boltzmann equation in bounded domains. Arch. Rat. Mech. Anal. 197(3), 713–809 (2010) 34. Guo, Y., Strauss, W.A.: Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48(8), 861–894 (1995) 35. Ha, S.-Y., Kim, Y.D., Lee, H., Noh, S.E.: Asymptotic completeness for relativistic kinetic equations with short-range interaction forces. Meth. Appl. Anal. 14(3), 251–262 (2007) 36. Hsiao, L., Yu, H.: Asymptotic stability of the relativistic Maxwellian. Math. Meth. Appl. Sci. 29(13), 1481–1499 (2006) 37. Jiang, Z.: On the relativistic Boltzmann equation. Acta Math. Sci. (English Ed.) 18(3), 348–360 (1998) 38. Jiang, Z.: On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: global existence. Transport Theory Statist. Phys. 28(6), 617–628 (1999) 39. Lebedev, N.N.: Special functions and their applications. New York: Dover Publications Inc., 1972, revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication 40. Lichnerowicz, A., Marrot, R.: Propriétés statistiques des ensembles de particules en relativité restreinte. C. R. Acad. Sci. Paris 210, 759–761 (1940) 41. Liu, T.-P., Yu, S.-H.: The Green’s function and large-time behavior of solutions for the one-dimensional Boltzmann equation. Comm. Pure Appl. Math. 57(12), 1543–1608 (2004) 42. Liu, T.-P., Yu, S.-H.: Green’s function of Boltzmann equation, 3-D waves. Bull. Inst. Math. Acad. Sin. (N.S.) 1(1), 1–78 (2006) 43. Liu, T.-P., Yu, S.-H.: Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation. Comm. Pure Appl. Math. 60(3), 295–356 (2007) 44. Mouhot, C., Villani, C.: On the Landau damping. http://arXiv.org/abs/0904.2760v1 [math.AP], 2009 45. Poularikas, A.D. (ed.): The transforms and applications handbook. The Electrical Engineering Handbook Series, Boca Raton, FL: CRC Press, 1996 46. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and series. Vol. 2, New York: Gordon & Breach Science Publishers, 1988, translated from the Russian by N. M. Queen 47. Rein, G.: Global weak solutions to the relativistic Vlasov-Maxwell system revisited. Commun. Math. Sci. 2(2), 145–158 (2004) 48. Stewart, J.M.: Non-equilibrium relativistic kinetic theory. Berlin, New York: Springer-Verlag, 1971 49. Strain, R.M.: Global Newtonian limit for the relativistic Boltzmann equation near vacuum. SIAM J. Math. Anal. 42(4), 1568–1601 (2010) 50. Strain, R.M.: An energy method in collisional kinetic theory. Ph.D. dissertation, Division of Applied Mathematics, Brown University, May 2005 51. Strain, R.M., Guo, Y.: Stability of the relativistic Maxwellian in a collisional plasma. Commun. Math. Phys. 251(2), 263–320 (2004)
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52. Strain, R.M., Guo, Y.: Almost exponential decay near Maxwellian. Comm. Part. Diff. Eqs. 31(1–3), 417–429 (2006) 53. Strain, R.M., Guo, Y.: Exponential decay for soft potentials near Maxwellian. Arch. Rat. Mech. Anal. 187(2), 287–339 (2008) 54. Synge, J.L.: The relativistic gas. Amsterdam: North-Holland Publishing Company, 1957 55. Ukai, S., Asano, K.: On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Inst. Math. Sci. 18(2), 477–519 (1982) 56. Vidav, I.: Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30, 264–279 (1970) 57. Wennberg, B.: The geometry of binary collisions and generalized Radon transforms. Arch. Rat. Mech. Anal. 139(3), 291–302 (1997) Communicated by H. Spohn
Commun. Math. Phys. 300, 599–613 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1123-7
Communications in
Mathematical Physics
Vertex Operators, Grassmannians, and Hilbert Schemes Erik Carlsson Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA. E-mail: [email protected] Received: 28 October 2009 / Accepted: 1 June 2010 Published online: 12 September 2010 – © Springer-Verlag 2010
Abstract: We approximate the infinite Grassmannian by finite-dimensional cutoffs, and define a family of fermionic vertex operators as the limit of geometric correspondences on the equivariant cohomology groups, with respect to a one-dimensional torus action. We prove that in the localization basis, these are the well-known fermionic vertex operators on the infinite wedge representation. Furthermore, the boson-fermion correspondence, locality, and intertwining properties with the Virasoro algebra are the limits of relations on the finite-dimensional cutoff spaces, which are true for geometric reasons. We then show that these operators are also, almost by definition, the vertex operators defined by Okounkov and the author in Carlsson and Okounkov (http://arXiv.org/abs/ 0801.2565v2 [math.AG], 2009), on the equivariant cohomology groups of the Hilbert scheme of points on C2 , with respect to a special torus action. 1. Introduction The infinite wedge representation, ∞/2 (F · Z) =
∞/2
m
(F · Z),
m∈Z
is the infinite-dimensional vector space over F with basis eμ,m = em+μ1 ∧ em+μ2 −1 ∧ em+μ3 −2 ∧ · · · ,
(1)
where e j , j ∈ Z is a basis of F · Z, and μ is a partition. It realizes representations of the Virasoro algebra, affine Kaç-Moody algebras, and other infinite-dimensional Lie algebras [3,13]. These representations, and their relations with each other, can be explicitly constructed by formal generating functions of operators known as vertex operators. These operators include the operators of “wedging and contracting” with e j , and a larger family with interesting intertwining properties with the above Lie algebras.
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Surprisingly, these operators also appear in the cohomology groups of Hilbert schemes of points on a smooth complex surface, and more general moduli spaces of sheaves. If S is a smooth surface, let Hilbk S denote the Hilbert scheme of zero-dimensional subschemes Z ⊂ S, such that dim(H 0 (O Z )) = k. There are many interesting and valuable correspondences acting on the direct sum U S,k , U S,k = H ∗ (Hilbk S, C), (2) U S = k≥0 ∞/2
and when S = C2 , U S ∼ = 0 (C · Z), and we obtain geometric constructions of the above Lie algebras. For instance, consider the Nakajima subvariety [14,19,21], Z n = (Z , x, Z ) ∈ Hilbk S × S × Hilbk+n S Z ⊂ Z , supp(O Z /O Z ) = {x} , (3) which is a singular variety with a well-defined fundamental class [Z n ] ∈ H∗ (Hilbk S) ⊗ S ⊗ H∗ (Hilbk+n S). If S is a compact variety, then by Poincaré duality, we obtain an operator , αn (x)(γ ) = p3∗ p1∗ (γ ) ∪ p2∗ (x) ∩ [Z −n ] αn (x) : Uk → Uk−n
(4)
for n < 0, and dual operators for n > 0, under the inner product x ∪ y.
(5)
Nakajima proved that these operators satisfy the commutation relations of an infinitedimensional Heisenberg algebra. Also using correspondences, Grojnowski [9] described a family of vertex operators on the cohomology groups. These comprise a larger family of operators, but they are formally related to Nakajima’s operators. In the case of the Hilbert scheme on C2 , this relationship is just the boson-fermion correspondence [6,21], and so one might say that the two collections of operators are equivalent. In a similar direction, Lehn constructed a corresponding action of the Virasoro algebra using characteristic classes [14]. In [5], Okounkov and the author defined a new family of vertex operators in terms of characteristic classes of universal bundles on the Hilbert scheme, and proved a “bosonization” formula in terms of the Nakajima operators. These operators were new, but in special cases, such as (6) below, they agree with the vertex operators of the last paragraph. Furthermore, if S carries a torus action, then characteristic classes have an explicit expression in the fixed-point basis of equivariant cohomology. In fact, the primary motivation was to calculate certain equivariant integrals of characteristic classes on the Hilbert scheme of S = C2 , with respect to a torus action [4,23]. In this situation, Hilbk S is not proper, but one may simply define integration by the localization formula for equivariant cohomology. In this paper, we define a family of vertex operators as limits of correspondences on the equivariant cohomology groups of certain finite-dimensional Grassmannians. These are the Grassmannians of half-dimensional subspaces of the vector space of formal Laurent series with cutoffs in high and low degree (essentially the infinite Sato Grassmannian). The circle acts on this space by rotation of functions. We define operators
Vertex Operators, Grassmannians, and Hilbert Schemes
601
in the same way as in (4), where the correspondence is the two-step flag variety, and x is replaced by a characteristic class. We use the relationship between localization and geometry to deduce their (super) commutation relations, the boson-fermion correspondence, and their intertwining properties with the Virasoro Lie algebra. We then conclude that a subset of these operators coincide with the fermionic vertex operators discussed, for instance, in [13], or [6] Chap. 5, by identifying the fixed-point basis with eμ,m , up to a normalization. We then connect our results to the Hilbert scheme. Let S = C2 , and let the onedimensional torus act by z · (x, y) = (zx, z −1 y), z ∈ T.
(6)
The vertex operator in this paper equals the operator W defined in [5], in respective fixedpoint bases of the two moduli spaces. We show that this isomorphism is represented by a characteristic class on the product of the two moduli spaces, and that it intertwines the two vertex operators. Thus, this paper serves two purposes. First, it connects a family of vertex operators on the cohomology of the Hilbert scheme to a simpler moduli space. Second, many difficult identities follow from the equality of an integral over a complex variety with its localization expression. We recover the aforementioned relationships between infinitedimensional Lie algebras and vertex operators as limits of such identities as the variety tends to the infinite Grassmannian. 2. Grassmannians Let
H M,N = x M · C[[x]] /(x N ),
M ≤ N , dim(H M,N ) = N − M.
There is an action of the circle on H M,N by translation, (z · f )(x) = f (z −1 x), and an invariant Hermitian metric, x m , x n = δm,n . These spaces have an obvious system of commuting inclusion and projection maps
i M,M : H M,N → H M ,N
p N ,N : H M,N → H M,N ,
for M ≤ M, N ≤ N . Let GmM,N = Gr (m + N , H M,N ), the Grassmannian of m + N -dimensional subspaces of H M,N , and define
f mM,M : GmM,N → GmM ,N , gmN ,N : GmM,N → GnM,N by
f mM,M (V ) = i M,M (V ), gmN ,N (V ) = p −1 N ,N (V ).
(7)
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Their pullbacks form an inverse system on Vm,M,N = HT∗ (GmM,N , C) ⊗C[t] C(t), t ∈ Lie(T ),
with the induced action of T on GmM,N . The dual maps form a direct system
f ∗M,M : VmM,N → VmM ,N , g∗N ,N : VmM,N → VmM,N
on VmM,N , the dual vector space of Vm,M,N over C(t). Let Vm be the direct limit over
M, N , and let h mM,N : VmM,N → Vm be the induced maps. By the localization theorem, vU = iU∗ constitute a basis of VmM,N , where iU : pt → M,N Gm is the inclusion map of a fixed subspace, U . Therefore Vm ∼ =
C(t) · vμ,m ,
μ
where μ is a partition, M,N M,N M,N vμ,m = h mM,N (vμ,m ), vμ,m = vU , U = Vμ,m ,
and M,N Vμ,m =
C · x −m−(μi −i+1) ∈ GmM,N , −M, N 0.
1≤i≤m+N Vm,M,N inherits an inner-product on cohomology,
(u, v)mM,N
=
GmM,N
u · v,
which by localization is equal to
T U ∈ GmM,N
vU ⊗ vU ∈ VmM,N ⊗ VmM,N , α(U, V ) = e (Hom(U, V )) . α(U, U ⊥ )−1
Here e is the Euler class, and Hom(U, U ⊥ ) is the tangent space to GmM,N at U , viewed as an equivariant bundle over a point. Explicitly, if Ck is the representation of T by z → z k , and U∼ Ck , V ∼ Ck , = = k
k
then α(U, V ) = t dim(U ) dim(V )
k,k
(k − k) ∈ HT∗ ( pt) ∼ = C[t].
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This inner-product is not compatible with f ∗M,M or g∗N ,N , but the normalized innerproduct cm,m,M,N
cμ,m,M = t
|μ|
cμ,m,M · cμ,m,N
U
vU ⊗ vU , α(U, U ⊥ )−1
(−M − m + i − j), cν,n,N = t |ν|
(i, j)∈μ
cm,n,M,N
=t
m
(m+N )(−n−M)
−M
(N + n − i + j),
(i, j)∈ν
( j − i),
i=−(N −1) j=n+1
is compatible, and so defines an element of Vm ⊗ Vm . It is given in the dual basis by h()2 , (8) (vμ,m , vν,m ) = δμ,ν t 2|μ| (−1)|μ| ∈μ
where h() is the hook length. There is a unique Hermitian inner-product with vμ,m , vν,n = (vμ,m , vν,n ) with respect to the conjugation f (t) = f (−t). Then h()−1 vμ,m (9) eμ,m = t −|μ| ∈μ
is an orthonormal basis. At t = ∞/2 isomorphism Vm ∼ = m (Z · C).
√
−1, identifying eμ,m with the basis (1) gives an
3. Vertex Operators Consider the flag variety
M,N Fm,n = (U, V ) ∈ GmM,N × GnM,N U ⊂ V , m ≤ n.
(10)
M,N Fm,n has a canonical bundle with fiber V /U over each point (U, V ), denoted also by V /U , and an associated determinant line bundle L. Given a symmetric polynomial, f , let c f be the characteristic class such that c is a homomorphism, and
cen = cn , where en is the elementary symmetric polynomial. Define an inner-product by M,N (c1 , c2 ) f,z = p1∗ (c1 ) p2∗ (c2 ) ch(L)c f (V /U ), M,N Fm,n
c1 ∈
Vm,M,N ,
c2 ∈
Vn,M,N ,
p1 × p2 :
M,N Fm,n
→
GmM,N
(11)
× GnM,N
for m ≤ n. This is an inner-product on Vm,M,N ×Vn,M,N with values in C(z, t), where we have substituted et = z in the Chern character. We prefer to think of z, t as independent variables. When m = n, this becomes the inner-product of the previous section.
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Using the formula for the tangent bundle to the flag variety, we obtain the dual inner product in coordinates, ⊥ (vU , vV ) M,N f,z = δ(U ⊂ V )α(U, V )c f (V /U ) ch(det(V /U )).
Normalizing as before, (vU , vV ) f,z =
cμ,m,M cν,n,N cm,n,M,N
(vU , vV ) M,N f,z ,
(12)
we obtain an inner-product on Vm × Vn , given in coordinates by vμ,m , vν,n f,z = z k t |μ|+|ν| δ(Vμ,m ⊂ Vν,n ) (a + aν () + μ () + 1) ·
∈μ
(a − aμ () − ν () − 1)c f (Vν,n /Vμ,m ),
(13)
∈ν
a = n − m, k = |ν| +
m(m + 1) n(n + 1) − |μ| − , 2 2
where aμ , μ are the (possibly negative) arm and leg-lengths in μ. If n < m, let ma , (v, u) M,N (u, v) M,N f,z = (−1) f,z −1
where u and v are homogeneous classes of degrees m and n, and (Y (a, f, ±z) · u, v) = z −a(a+1)/2 (u, v) f,z .
(14)
The expression above vanishes if a = n − m, and ± means sgn(a). The signs are designed so that we always arrive at the expression in (13), which isdefined for all a. The coefficient of z k of Y (a, f, z) is an honest operator on V = m Vm , though not for any particular value of z. Instead, Y (a, f, z) is a formal power series with coefficients in End(V), Y (a, f, z) ∈ End(V)[[z ±1 ]], V = C(t) · vμ,m , μ,m
which is sometimes called a field. 4. Locality The commutation relations between the vertex operators are collectively called locality, and are described by the supercommutator, {Y (a, f, z), Y (b, g, w)} = Y (a, f, z)Y (b, g, w) − (−1)ab Y (b, g, w)Y (a, f, z).
(15)
We deduce the commutation relations among the Y (a, f, z) using the finite-dimensional approximations.
Vertex Operators, Grassmannians, and Hilbert Schemes
605
T
T Lemma 1. Let U ∈ GlM,N , W ∈ GnM,N , f = eκ , g = eκ , where eκ = i eκi , and let
A=
A =
M,N (vU , vV ) M,N f,z (vW , vV )g,w
T V ∈ GmM,N
(vV , vV ) M,N
M,N (vU , vV )g,w (vV , vW ) M,N f,z
(vV , vV ) M,N
T V ∈ GmM,N
,
, B = A − (−1)ab A ,
where m = l + n − m, a = m − l, b = n − m. a. If a, b ≥ 0, or a, b ≤ 0, then B = 0. b. If a > 0, b < 0, then B = h 1 (z, w)(z + w) K 1 + · · · + h d (z, w)(z + w) K d ,
(16)
where d = m − dim(Y ), K j = j (dim(Z ) + j − 2d − (κ) − (κ )), X = U ∩ V, Y = U + V, Z = X ⊕ Y ⊥ . Proof. Assume for simplicity that w = −1. To prove the first part, suppose a, b ≥ 0. Then A=
T V ∈ GmM,N
δ(U ⊂ V )δ(V ⊂ W )
α(U, V ⊥ )α(V, W ⊥ ) · α(V, V ⊥ )
ch(det(V /U ))c f (V /U )cg (W/V ) ⊥ = α(U, W ) ch(det(V ))c f (V )cg (V ⊥ ), G r (a,W/U )
where in the second line, V is the universal bundle on the Grassmannian, and V ⊥ is perpendicular within W/U . Similarly A = α(U, W ⊥ )
G r (b,W/U )
ch(det(V ⊥ ))cg (V )c f (V ⊥ ).
The map ⊥: Gr (a, W/U ) → Gr (b, W/U ) is a T -equivariant isomorphism of real manifolds of degree (−1)ab , which interchanges the integrands, proving that A = (−1)ab A . The case a, b < 0 is similar.
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For the second part, let dim(X ) = p, dim(Y ) = q: A = (−1)la ch(det(V /U ))c f (V /U )cg (V /W ) ·
T V ∈ GmM,N
α(U, V ⊥ )α(W, V ⊥ ) α(V, V ⊥ ) = (−1)la+ pd ch(det(Y/U ))α(X, Y ⊥ )c f (Y/U )cg (Y/W ) · c f g (V /Y ) ch(det(V /Y )) δ(V ⊃ Y ) α(V /Y, X ⊕ V ⊥ )
T
δ(V ⊃ Y )
V ∈ GmM,N
= B0 · F(Y ⊥ , X, c f g , d), where
F(Y ⊥ , X, c f , d) =
V ∈G r (d,Y ⊥ )T
ch(V )c f (V ) . α(V, X ⊕ Y ⊥ V )
(17)
In the second line, we used the fact that α(X, V /Y ) = (−1) pd α(V /Y, X ). Doing a similar calculation for A , we find that B = B0 · H (Y ⊥ , X, c f g , d), where H (Y ⊥ , X, c f , d) = F(Y ⊥ , X, c f , d) − (−1)d F(X, Y ⊥ , c f , d). We must prove that this has the form (16). Consider first the base case X = {0}. Then H (Y ⊥ , {0}, c f , d) = F(Y ⊥ , {0}, c f , d) =
h(z) . t Kd
At z = et , (17) represents a localization integral over a compact Grassmannian, so it is an element C[[t]], the closure of HT∗ ( pt). Therefore h(z) vanishes at z = 1 to degree K d . Now suppose c f (V ) = c S (V ) = α(V, S), where S is a representation of the circle. The remaining cases follow from the recursion relation H (Y ⊥ , X, c S , d) − (−1)d H (Y ⊥ Ck , X ⊕ Ck , c S , d) α(Ck , S)z k H (Y ⊥ Ck , X, c S⊕Ck , d − 1), = α(X ⊕ Y ⊥ Ck , Ck ) which is straightforward to verify.
(18)
Theorem 1. If a, b ≥ 0, or a, b ≤ 0, then {Y (a, f, z), Y (b, g, w)} = 0. Otherwise, there exists K > 0 such that (z − w) K {Y (a, f, z), Y (b, g, w)} = 0.
Vertex Operators, Grassmannians, and Hilbert Schemes
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Proof. For simplicity, we may assume w = 1. We substitute M = −N , and express the commutator as a limit as N approaches ∞. Since
cμ,m,−N cμ,m,N = lim = 1, |μ| N →∞ N N →∞ N |μ| lim
the first part follows immediately from the lemma. For the second, we see that {Y (a, f, z), Y (b, g, 1)} · vμ,m , vν,n = lim C N · H (Y ⊥ , X, c f g , d), N →∞
−N ,N −N ,N where X and Y are as above, with U = Vμ,m , W = Vν,n . This vanishes if dim(Y/ X ) > |a − b|. It follows from this fact and the lemma that there are overall bounds on the upper and lower degrees of z j N h j (z, 1) for all j, μ, m, ν, n. It is straightforward to check that if
h 1,N (z)z −N (1 + z)2N +c1 + · · · + h d,N (z)z −d N (1 + z)2d N +cd converges to h (z) as a formal distribution, and the upper and lower degrees of h j,N (z) are bounded, then h (z) is a linear combination of derivatives of the delta function, so that (z − 1) K h (z) = 0, for some overall K , independent of U, W .
5. Bosonization and the Virasoro Algebra We now define the Heisenberg and Virasoro actions on Vm , and use the previous section to obtain the intertwining properties of Y (a, z) = Y (a, c0 , z). As a result, we obtain a simple proof of the boson-fermion correspondence. The recursion relation shows that when a = −1, b > 0, f = g = 1, t = 1, B = α(U, W ⊥ )
k∈W/U
(k + z∂z ) ·
z M w −(N −1) (z + w) N −M−1 . (N − M − 1)!
The expression on the right-hand side comes from the formula ⊥ ⊥ H (Y , X, c0 , 1) = H (Y ⊕ X, {0}, c0 , 1) =
P(Y / X )
(19)
ch(L),
where L is the universal bundle on P(Y/ X ). Taking the limit normalized as in (12) at b = 1, we get the Clifford algebra relations ψ(z), ψ ∗ (w) = δ(z − w), (20) where ψ(z) = Y (1, z), ψ ∗ (z) = Y (−1, z), δ(z − w) =
j
z j w − j−1 .
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E. Carlsson
Let α0 =
ψ j ψ ∗j −
j>0
where ψ j and
ψ ∗j
ψ ∗j ψ j , αn =
j≤0
ψ j−n ψ ∗j , n = 0,
j
are defined by ψ(z) = ψ j z − j−1 , ψ ∗ (z) = ψ ∗j z − j . j
j
It is easy to see that these operators act by “wedging and contracting” in the basis (9), and that α0 is multiplication by m on Vm . By (20), these operators satisfy the Heisenberg commutation relations [αi , α j ] = iδi,− j . We also define the Virasoro generators Ln =
∗ kψk ψk+n .
(21)
k
Theorem 2. With the above notations, (1)
Y (a, z) = Q z
a aα0
exp a
α−n z n n>0
n
exp −a
αn z −n
n
n>0
,
where Q · vμ,m = vμ,m+1 . (2)
[L n , Y (a, z)] = z
n+1
a(a − 1) n (n + 1)z · Y (a, z). ∂z + 2
(3) V is a conformal vertex algebra, isomorphic to the free fermionic vertex superalgebra. Under this isomorphism, part (1) is the boson-fermion correspondence (see [6,13]). Proof. Let us compute the commutator
[α(z), Y (a, w)] = ψ(z) ψ ∗ (z), Y (a, w) .
As in Theorem 1, we deduce the expression on the right-hand side by taking a limit of the matrix elements ⎛ V0 ∈Gm 0
(U, V0 )zM,N (V0 , V0 ) M,N
⎜ ⎜ ⎜ ⎜ ⎝
T V1 ∈ GmM,N 1
M,N (V0 , V1 )zM,N (V1 , W )w M,N (V1 , V1 )
⎞ −(−1)a
T V1 ∈ G M,N m1
M,N (V , W ) M,N ⎟ ⎟ (V0 , V1 )w z 1 ⎟, ⎟ M,N ω (V1 , V1 ) ⎠
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where
T
T U ∈ GlM,N , W ∈ GnM,N , m 0 = l + 1, m 1 = l, m 1 = n + 1, a = n − l. Inserting (19), setting M = −N , and taking the normalized limit as N → ∞ gives
α(U, W ⊥ )
V ∈P(W/U )T
Extracting the coefficient of
ch z (V ) chw (V ⊥ ) (k + z∂z ) · δ(zw −1 − 1). α(V, V ⊥ ) ⊥ k ∈V
zn ,
we get
w n α(U, W ⊥ ) chw (W/U )
k − k − n , k − k k k =k
where k, k are the torus weights of W/U . The sum is the localization expression for the Euler characteristic of P(W/U ) which equals a for all n, U, W . The case a < 0 is similar with Gr (−a − 1, W/U ) replacing projective space. It follows that [αn , Y (a, z)] = az n Y (a, z). This determines Y (a, z), and in fact it equals the expression in part (1). Part (2) is similar, but uses the alternate formula k − k − n = k c (L) ci (T P(W/U ))(−n)a−i 1 k − k P(W/U ) k
k =k
i
a(a − 1) n. = c1 (W/U ) + 2 In particular, [T, Y (a, z)] = ∂z Y (a, z),
(22)
where T = L −1 is the time translation operator. Therefore part (3) follows from Theorem 1 and the reconstruction theorem [6]. It would be interesting if the additional operators Y (a, f, z) could be included in the vertex algebra. Unfortunately, while they are mutually local, they do not satisfy (22), and so the reconstruction theorem does not apply. 6. The Hilbert Scheme Let S be a smooth surface with an action of a torus, T . Then Hilbk S parametrizes subschemes Z ⊂ S with dimC H 0 (O Z ) = n, or equivalently, their ideal sheaves I Z = ker(O S → O Z ). By definition, there is a universal subscheme, Z ⊂ Hilb S × S,
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E. Carlsson
and its corresponding ideal sheaf I. Let U S = HT∗ (Hilbk S) ⊗C[t∗ ] C(t∗ ), k
and let U denote its dual vector space over C(t∗ ). In [5], Okounkov and the author defined a class
∨ ∨ ∗ · p E = p12 ∗ OZ + O − O · O (L) ∈ K T (Hilbk S × Hilbl S), Z2 Z2 3 Z1 1 ∗ O . This is a virtual vector bundle of rank where L is a line bundle on S, and OZi = pi3 Z k + l, and so we may consider its Euler class,
ck+l (E(L)) ∈ HT∗ (Hilbk × Hilbl ) → U S ⊗ U S . If S has compact fixed loci under T , then the vector space U S has an inner-product over C(t∗ ) given as above. The above characteristic class defines an operator W(L, z) ∈ End(U S )[[z, z −1 ]], given by (W(L) · η, ξ ) = z
l−k Hilbk × Hilbl
p1∗ (η) ∪ p2∗ (ξ ) ∪ e(E)
(23)
which we gave a formula for in terms of Nakajima’s Heisenberg operators, and the first Chern class of the canonical bundle of S. An important case is S = C2 with the torus action (z 1 , z 2 ) · (x, y) = (z 1 x, z 2 y), and where L is a character of T viewed as an equivariant bundle on C2 . In this case the fixed-points of the action of T on Hilbk C2 correspond to the monomial ideals Iμ = (x μ1 , x μ2 y, x μ3 y 2 , . . . , y μ ),
(24)
where μ is a partition. By the localization formula, U∼ C(t1 , t2 ) · u μ , = μ
where u μ is the pullback of the inclusion of Iμ . For a reference on localization and the equivariant cohomology of the Hilbert scheme, see [27,29]. Let us calculate the matrix elements of W(L) in this basis. To do this we need the character of the fiber of the bundle E(L), over a pair of fixed-points (Iμ , Iν ) ∈ Hilbk C2 × Hilbl C2 . Since ch H 0 (O Z ) = ch H 0 (OC2 ) − ch H 0 (I Z ) ∈ C(z 1−1 , z 2−1 ), we find that ch E(L)(I
μ ,Iν )
= ch χ (O, O) − ch χ (Iμ , Iν ) ch(L),
(25)
Vertex Operators, Grassmannians, and Hilbert Schemes
611
where χ S (F, G) = Ext 0S (F, G) − Ext 1S (F, G) + Ext 2S (F, G), the derived push-forward of H om S (F, G) to a point. By Riemann-Roch, ch χC2 (Iμ , Iν ) = e(C2 ) td(C2 )−1 ch H 0 (Iμ∨ ) ch H 0 (Iν ) = (1 − z 1 )(1 − z 2 )ch H 0 (Iμ ) ch H 0 (Iν ) ∈ C((z 1−1 , z 2−1 )), (26) where F ∨ is the dual in equivariant K -theory, td is the Todd class, 1− j ch H 0 (Iμ ) = z 11−i z 2 , e(C2 ) td(C2 )−1 = (1 − z 1−1 )(1 − z 2−1 ), i, j≥1, (i, j)∈μ /
ch(H 0 (F ∨ )) = z 1−1 z 2−1 ch(H 0 (F)),
f (z 1 , z 2 ) = f (z 1−1 , z 2−1 ).
The order of the multiplication matters in (26). For instance, the product of the last two terms is divergent. On the other hand, multiplication by (1 − z 1 )(1 − z 2 ) of either term yields an element of C(z 1 , z 2 ). We have assumed that multiplication happens from left to right, producing an element of C((z 1−1 , z 2−1 )). However, choosing a different order of multiplication does not affect the difference (25), so long as we choose the order consistently for each term. We may therefore write ch E(L)(I ,I ) = ch H 0 (I∅ )(1 − z 2−1 ) ch H 0 (I∅ )(1 − z 1 ) μ ν
− ch H 0 (Iμ )(1 − z 2−1 ) ch H 0 (Iν )(1 − z 1 ) ch L t 1−νi 1−i 1−i −μi 1−i 1−i z1 z2 z1 z2 − z1 z1 z2 ch L = ⎛ =⎝
i
i
i
a ()+1 −lν () z2
z 1μ
∈μ
+
i
⎞
−aν () lμ ()+1 ⎠ z2 ch L,
z1
(27)
∈ν
where μt is the transposed partition. Furthermore, the restriction of E = E(O) along the diagonal is the tangent bundle of the Hilbert scheme, and the above expression reduces to the well-known formulas for its character. It follows that in the dual coordinates, W(L) · u μ , u ν = z |ν|−|μ| t |μ|+|ν| c1 (L) + aμ ()t1 − ν ()t2 + t1 ∈μ
· c1 (L) − aν ()t1 + μ ()t2 + t2 .
(28)
∈ν
We may now explain how this relates to our results in this paper. Let U = UC2 , with the simpler torus action (6). This amounts to setting (t1 , t2 ) = (t, −t), c1 (L) = at, a ∈ Z in (28). Comparing with (13), the two operators are identical, with the partitions transposed. We now explain why.
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Given a vector space U ⊂ C(x, y) and an integer N , let
U N = U ∩ span x i y j | i, j < N ⊂ C(x, y).
(29)
For each k, m, and large N , there is a T -equivariant subbundle of the trivial bundle C(x, y) on Hilbk C2 whose fiber is given by Jm,n,N Z = (x m y n I Z ) N , dim(Jm,n,N ) = (N − m)(N − n) − k. There are no jumps in dimension by the construction of the Hilbert scheme on P2 , and the fact that the N th filtered subspace of C[x, y] is isomorphic to the N th graded subspace of C[x, y, z]. Now consider the bundle on Hilbk C2 × GmM,N given by EmM,N = H om C (J−m,0,N /J−m,1,N , V ⊥ ),
(30)
which has dimension (N + m)(N − m) for k, m, and sufficiently large −M, N . The Euler class of this bundle defines a map U → VmM,N whose composition with h mM,N is the isomorphism ϕm : U → Vm , ϕm (u μ ) = vμt ,m of inner-product spaces over C(t). Substituting t ⊥ z m+μi −i+1 , ch Vν,n = z n−νi +i , ch L = z a ch Vμ,m = i
i
in (27), we have deduced the following theorem: Theorem 3. When S = C2 with the torus action (6), W(L, z) = z −ma ϕn−1 Y (a, z)ϕm , where a = n − m, and L is the trivial bundle with character z a . Note that z is merely a place holder for the number of points on the Hilbert scheme side, but varies within each Grassmannian. Acknowledgements. The author would like to thank Andrei Okounkov and David Nadler for valuable discussions. Okounkov suggested connecting the Hilbert scheme to the Sato Grassmannian.
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7. Göttsche, L.: Hilbert schemes of points on surfaces. ICM Proceedings, Vol. II (Beijing, 2002), Beijing: Higher Ed. Press, 2002, pp. 483–494 8. Göttsche, L.: The Betti numbers of the Hilbert scheme of points on a smooth projective surface. Math. Ann. 286(1–3), 193–207 (1990) 9. Grojnowski, I.: Instantons and affine algebras I: the Hilbert scheme and vertex operators. Math. Res. Lett. 3, 275–291 (1996) 10. Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Amer. Math. Soc. 14(4), 941–1006 (2001) 11. Haiman, M.: Combinatorics, symmetric functions and Hilbert schemes. In: Current developments in mathematics, 2002, Somerville, MA: Int. Press, 2003, pp. 39–111 12. Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Braunschweig: Vieweg & Sohn, 1997 13. Kaç, V.: Infinite dimensional Lie algebras, third edition. Cambridge: Cambridge University Press, 1990 14. Lehn, M.: Geometry of Hilbert schemes. In: CRM Proceedings and Lecture Notes, Volume 38, Providence, RI: Amer. Math. Soc., 2004, pp. 1–30 15. Lehn, M.: Chern classes of tautological bundles on Hilbert schemes of points on surfaces. Invent. Math. 136(1), 157–207 (1999) 16. Licata, A., Savage, A.: Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves. http://arXiv.org/abs/0809.4010v3 [math.RT], 2009 17. Lehn, M., Sorger, C.: Symmetric groups and the cup product on the cohomology of Hilbert schemes. Duke Math. J. 110, 345–357 (2001) 18. Macdonald, I.: Symmetric functions and Hall polynomials. New York: The Clarendon Press/Oxford University Press, 1995 19. Nakajima, H.: Heisenberg algebra and Hilbert schemes of points on projective surfaces. Ann. of Math. (2) 145(2), 379–388 (1997) 20. Nakajima, H.: Jack polynomials and Hilbert schemes of points on surfaces. http://arXiv.org/abs/alggeom/9610021v1, 1996 21. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. Providence, RI: Amer. Math. Soc., 1999 22. Nekrasov, N.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831–864 (2003) 23. Nekrasov, N., Okounkov, A.: Seiberg-Witten Theory and Random Partitions. In: The Unity of Mathematics, ed. by Etingof, P., Retakh, V., Singer, I.M., Progress in Mathematics, Vol. 244, Basel-Boston: Birkhäuser, 2006 24. Okounkov, A.: Random partitions and instanton counting. International Congress of Mathematicians, Vol. III, Zürich: Eur. Math. Soc., 2006, pp. 687–711 25. Okounkov, A., Olshanski, G.: Shifted Jack polynomials, binomial formula, and applications. Math. Res. Lett. 4(1), 69–78 (1997) 26. Okounkov, A., Pandharipande, R.: The quantum differential equation of the Hilbert scheme of points in the plane. http://arXiv.org/abs/0906.3587v1 [math.AG], 2009 27. Vasserot, E.: Sur lanneau de cohomologie du schema de Hilbert de C2. C. R. Acad. Sci. Paris, Ser. I Math. 332, 7–12 (2001) 28. Li, W., Qin, Z., Wang, W.: Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces. Math. Ann. 324, 105–133 (2002) 29. Li, W., Qin, Z., Wang, W.: The cohomology rings of Hilbert schemes via Jack polynomials. CRM Proceedings and Lecture Notes, Vol. 38, Providence, RI: Amer. Math. Soc., 2004, pp. 249–258 Communicated by N.A. Nekrasov
Commun. Math. Phys. 300, 615–639 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1121-9
Communications in
Mathematical Physics
On the Atomic Photoeffect in Non-relativistic QED Marcel Griesemer1 , Heribert Zenk2 1 Fachbereich Mathematik, Universität Stuttgart, D-70569 Stuttgart, Germany.
E-mail: [email protected]
2 Mathematisches Institut, Ludwig-Maximilians-Universität München, D-80333 München, Germany
Received: 13 November 2009 / Accepted: 28 March 2010 Published online: 2 September 2010 – © Springer-Verlag 2010
Abstract: In this paper we present a mathematical analysis of the photoelectric effect for one-electron atoms in the framework of non-relativistic QED. We treat photo-ionization as a scattering process where in the remote past an atom in its ground state is targeted by one or several photons, while in the distant future the atom is ionized and the electron escapes to spacial infinity. Our main result shows that the ionization probability, to leading order in the fine-structure constant, α, is correctly given by formal time-dependent perturbation theory, and, moreover, that the dipole approximation produces an error of only sub-leading order in α. In this sense, the dipole approximation is rigorously justified.
1. Introduction Even today, more than 100 years after its discovery by Hertz, Hallwachs and Lenard, the phenomenon of photoionization is still investigated, both experimentally and theoretically [1,4]. This research is driven by novel experimental techniques that allow for the production of very strong and ultrashort laser pulses. In contrast, the photoelectric effect in the early experiments is produced by weak, non-coherent radiation of high frequency. There is a third physical regime, where the radiation is weak, of high frequency, and coherent. This regime is the subject of the present paper. We consider one-electron atoms within the standard model of non-relativistic QED, and we present a mathematically rigorous analysis of the ionization process caused by the impact of finitely many photons. Improving on earlier results concerning more simplified models, we show that the probability of ionization, to leading order in the fine-structure constant, is proportional to the number of photons, and, in the case of a single photon, it is given correctly by the rules of formal (time-dependent) perturbation theory. It turns out that the dipole approximation produces an error of subleading order, which provides a rigorous justification of this popular approximation.
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Let’s briefly recall the standard model of one-electron atoms within non-relativistic QED. More elaborate descriptions may be found elsewhere [20,31]. States of arbitrarily many transversal photons are described by vectors in the symmetric Fock space F := Sn ⊗n L 2 (R3 × {1, 2}) n≥0
over L 2 (R3 × {1, 2}). Here Sn denotes the projection of L 2 (R3 × {1, 2})n onto the subspace of all symmetric functions of (k1 , λ1 ), . . . , (kn , λn ) ∈ R3 ×{1, 2}, and S0 L 2 (R3 × {1, 2}) := C. We shall use to denote the vacuum vector (1, 0, . . .) ∈ F. N f is the number operator in F, and H f = d(ω) denotes the second quantization of multiplication with ω(k) = |k| in L 2 (R3 × {1, 2}). See [27], X.7, for the notation d(·) and for an introduction to second quantization. The creation and annihilation operators a ∗ (h) and a(h), for h ∈ L 2 (R3 × {1, 2}), are densely defined, closed operators with a ∗ (h) = a(h)∗ and with √ [a ∗ (h)](n) = nSn (h ⊗ (n−1) ) for vectors = ( (0) , (1) , . . .) from the subspace D(N f ). Here, (n) denotes the n-photon component of . The system studied in this paper is composed of a non-relativistic, (spinless) quantum mechanical, charged particle (the electron), and the quantized radiation field which is coupled to the electron by minimal substitution. In addition, there is an external potential V , which may be due to a static nucleus. The Hilbert space is thus the tensor product 1/2
H := L 2 (R3 ) ⊗ F, and the Hamiltonian is of the form 3
Hα = (p + α 2 A(αx))2 + V + H f = H0 + W,
(1.1)
where H0 = Hel + H f , Hel = − + V , and W = Hα − H0 . The quantized vector potential A(αx), for each x ∈ R3 , is a triple of self-adjoint operators, each of which is a sum of a creation and an annihilation operator. Explicitly, A(αx) = a(Gx ) + a ∗ (Gx ),
κ(k) Gx (k, λ) := √ ε(k, λ)e−iαk·x , 2|k|
(1.2)
where ε(k, λ) ∈ R3 , λ = 1, 2, are orthonormal polarization vectors perpendicular to k, and κ is an ultraviolet cutoff chosen from the space S(R3 ) of rapidly decreasing functions. No infrared cutoff is needed. Here and henceforth, the position of the electron, x ∈ R3 , and the wave vector of a photon, k ∈ R3 , are dimensionless and related to the corresponding dimension-full quantities X, K by X = (a0 /2)x and K = (2α/a0 )k, where a0 := 2 /me2 is the Bohr-radius, m > 0 is the mass of the particle, e its charge, and α = e2 /c is the fine structure constant. It follows that X · K = αx · k, and in units where , c, and four times the Rydberg energy 2mα 2 are equal to unity, the Hamiltonian of a one-electron atom with static nucleus at the origin takes the form (1.1) with V (x) = −Z /|x|, Z being the atomic number of the nucleus. For simplicity, we confine ourselves, in this Introduction, to this particular potential V . In nature, α ≈ 1/137, but in this paper α is treated as a free parameter that can assume any non-negative value.
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For all α ≥ 0, the Hamiltonian Hα is self-adjoint on D(H0 ) and its spectrum σ (Hα ) is a half-axis [E α , ∞) [23,24]. Moreover, E α := inf σ (Hα ) is an eigenvalue of Hα , and, at least for α sufficiently small, this eigenvalue is simple [2,21]. We use α to denote a normalized eigenvector associated with E α . Another important point in the spectrum of Hα is the ionization threshold α , which, for our system, is given by α = inf σ (Hα − V ). In a state vector from the spectral subspace Ran1(−∞, α ) (Hα ), the electron is exponentially localized in the sense that eβ|x| 1(−∞, α −ε] (Hα ) < ∞
(1.3)
for all β with β 2 < ε [19]. The phenomenon of photo-ionization can be considered as a scattering process, where in the limit t → −∞, the atom in its ground state is targeted by a (finite) number of asymptotically free photons, while in the limit t → ∞ the atom is ionized in a sense to be made precise. We begin by discussing incoming scattering states and their properties. To this end it is convenient to introduce the space L 2ω (R3 × {1, 2}) of all those f ∈ L 2 (R3 × {1, 2}) for which f 2ω := (1.4) | f (k, λ)|2 1 + ω(k)−1 d 3 k < ∞. λ=1,2
∗ ( f ) is defined by Given f ∈ L 2ω (R3 × {1, 2}), the asymptotic creation operator a− ∗ ( f ) := lim ei Hα t a ∗ ( f t )e−i Hα t , a− t→−∞
f t := e−iωt f,
(1.5)
and its domain is the space of all vectors ∈ D(|Hα |1/2 ) for which the limit (1.5) exists. This is known to be the case, e.g., for the ground state = α . Moreover, it is ∗ ( f ) · · · a ∗ ( f ) is well defined and that known that a− 1 α − n ∗ ∗ e−i Hα t a− ( f 1 ) · · · a− ( f n ) α
= a ∗ ( f 1,t ) · · · a ∗ ( f n,t )e−i Hα t α + o(1),
(t → −∞),
(1.6)
∗( f )··· whenever f i , ω f i ∈ L 2ω (R3 × {1, 2}) for all i = 1, . . . , n [22]. By (1.6), a− 1 ∗ a− ( f n ) α describes a scattering state, which, in the limit t → −∞ is composed of the atom in its ground state and n asymptotically free photons with wave functions ∗ ( f ) hold true for the asymptotic annihilaf 1 , . . . , f n . Results analogous to those on a− tion operators a− ( f ) [22]. The asymptotic annihilation and creation operators satisfy the usual canonical commutation relations: e.g. ∗ [a− ( f ), a− (g)] = f, g
(1.7)
for all f, g ∈ L 2ω (R3 × {1, 2}). Moreover, the ground state α is a vacuum vector for asymptotic annihilation operators in the sense that a− ( f ) α = 0
for all f ∈ L 2ω (R3 ).
(1.8)
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n Hence, if f = ( f 1 , m 1 , . . . , f n , m n ) ∈ L 2 (R3 × {1, 2}) × N with f i , f j = δi j , then it follows from (1.7) and (1.8) that ∗ a− ( f ) α :=
n
k=1
√
1 ∗ ( f k ) m k α a− mk !
(1.9)
∗ ( f ) hold mutatis mutandis is a normalized vector in H. All these properties of a− ( f ), a− ∗ for the asymptotic operators a+ (g), a+ (g) defined in terms of the limit t → +∞. ∗ ( f ) describes an ionized atom We are interested in the probability that e−i Hα t a− α in the distant future, but we are not interested in the asymptotic state of the electron or the radiation field in the limit t → +∞. We therefore shall not attempt to construct outgoing scattering states describing an ionized atom, which is a difficult open problem. Instead we base our definition of the probability of ionization on the following reasonable ∗ ( f ) is either ionized in the limit t → ∞, assumption: the atom described by e−i Hα t a− α ∗ ( f ) , for or else, in that limit, it relaxes to the ground state in the sense that e−i Hα t a− α t large enough, is well approximated by a linear combination of vectors of the form
a ∗ (g1,t ) . . . a ∗ (gn,t )e−i E α t α .
(1.10)
∗ ( f ) belongs to the closure More precisely, relaxation to the ground state occurs if a− α of the span of all vectors of the form
a+∗ (g1 ) . . . a+∗ (gn ) α = lim ei Hα t a ∗ (g1,t ) . . . a ∗ (gn,t )e−i E α t α , t→+∞
with gi , ωgi ∈ L 2ω (R3 × {1, 2}). Let H+α denote this space and let P+α be the orthogonal ∗ ( f ) 2 is the probability for relaxation to the ground projection onto H+α . Then P+α a− α state and ∗ ∗ 1 − P+α a− ( f ) α 2 = (1 − P+α )a− ( f ) α 2
(1.11)
is the probability of ionization. The assumption that relaxation to the ground state is the only alternative to ionization, is motivated by the conjecture of asymptotic completeness for Rayleigh scattering, which is the property, that every vector ∈ H describing a bound state in the sense that supt eε|x| e−i Hα t < ∞ for some ε > 0, will relax the ground state in the limit t → ∞. In view of (1.3), asymptotic completeness for Rayleigh scattering implies that H+α ⊇ 1(−∞, α ) (Hα ), which can be proven for simplified models of atoms [8,14,18,30]. The following two theorems will allow us to compute (1.11). Theorem 1.1. Suppose that f 1 , . . . , f n ∈ L 2 (R3 × {1, 2}), where 2λ=1 ε(·, λ) f i (·, λ) belongs to C02 (R3 \{0}, C3 ) for each i, and let f = ( f 1 , . . . , f n ). Then: ∗ a− ( f ) α = a+∗ ( f ) α − iα 3/2
∞ −∞
2p(s)ϕel ⊗ [A(0, s), a ∗ ( f )] ds + O(α 5/2 ), (1.12)
where p(s) = ei Hel s pe−i Hel s and A(0, s) = ei H f s A(0)e−i H f s .
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The first term of (1.12) gives no contribution to the ionization probability (1.11) because a+∗ ( f ) α ∈ H+α . The second term is proportional to α 3/2 and it is due to scattering processes where one of the n photons f 1 , . . . , f n is absorbed. The remainder terms are of order O(α 5/2 ) and stem from the dipole approximation A(αx) → A(0), from dropping α 3 A(αx)2 and from ignoring processes of higher order in α 3/2 . To isolate the contribution of order α 3 from (1.11) using (1.12), we need: Theorem 1.2. Suppose that H+α ⊇ 1(−∞, α ) (Hα ) for α in a neighborhood of 0, and suppose that Hel has only negative eigenvalues. Then lim P+α = 1 pp (Hel )
α→0
(1.13)
in the strong operator topology. Combining Theorem 1.1 and Theorem 1.2 we see that ∗ ∗ (1 − P+α )a− ( f ) α 2 = (1 − P+α ) a− ( f ) − a+∗ ( f ) α 2 ∗ = 1c (Hel ) a− ( f ) − a+∗ ( f ) α 2 + o(α 3 ) ∞ = α 3 1c (Hel ) p(s)ϕel ⊗ [A(0, s), a ∗ ( f )] ds 2 + o(α 3 ), −∞
where 1c (Hel ) = 1 − 1 pp (Hel ), and where the second equation is justified by the α ∗ ( f ) − a ∗ ( f ) as given by (1.12). We are now going to express dependence of a− α α + the coefficient of α 3 in terms of generalized eigenfunctions of Hel , which makes it explicitly computable in simple cases. A general and sufficient condition for the existence of a complete set of generalized eigenfunctions is the existence and completeness of a (modified) wave operator + associated with Hel . This condition is satisfied for our choice of V . It means that there exists an isometric operator + ∈ L(Hel ) with Ran+ = 1c (Hel )Hel and Hel + = + (−). In particular, the singular continuous spectrum of Hel is empty. Given the wave operator + and the fact that (Hel − i)−1 x −2 is a Hilbert-Schmidt operator, it is easy to establish existence of generalized eigenfunctions ϕq , q ∈ R3 , of Hel with the following properties [26]: (i) The function (x, q) → q −2 x −2 ϕq (x) is square integrable on R3 × R3 , in particular x −2 ϕq ∈ L 2 (R3 ) for almost every q ∈ R3 . x := (1 + |x|2 )1/2 . (ii) If ψ ∈ D(|x|2 ) then | ϕq , ψ |2 d 3 q. (1.14) 1ac (Hel )ψ 2 = R3
(iii) If F : R → C is a Borel function, ψ ∈ D(|x|2 ) ∩ D(F(Hel )), and F(Hel )ψ ∈ D(|x|2 ), then (1.15) ϕq , F(Hel )ψ = F(q2 ) ϕq , ψ for almost every q ∈ R3 . In (ii) and (iii) we use ϕq , ψ to denote the integral ϕq (x)ψ(x) d 3 x, which is well defined by (i) and by the assumption ψ ∈ D(|x|2 ). Theorem 1.1 in conjunction with (i)–(iii) implies the following theorem, which is our main result specialized to the case of only one asymptotic photon in the incident scattering state.
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Theorem 1.3. For all f ∈ L 2 (R3 ×{1, 2}) with 2λ=1 ε(·, λ) f (·, λ) ∈ C02 (R3 \{0}, C3 ), 1ac (Hel ) a ∗ ( f ) α − a ∗ ( f ) α 2 − + ∞ 2 3 3 i(q2 −E 0 )t ∗ , E(0, t)a ( f ) dt +O(α 4 ) (1.16) = α d q ϕq , xϕel · e R3
−∞
as α → 0. Here, E(0, t) = −i[H f , A(0, t)], ϕel is a normalized ground state of Hel and ϕq , q ∈ R3 , is any family of generalized eigenfunction of Hel with properties (i)–(iii) above. The expression (1.16) for the ionization probability can be understood, on a formal level, by first order, time-dependent perturbation theory. To this end one considers the transitions ϕel ⊗ f → ϕq ⊗ , for fixed q ∈ R3 , in the interaction picture defined by H0 . Then the time-evolution of state vectors is generated by the time-dependent interaction operator W (t) = ei H0 t W e−i H0 t = 2α 3/2 p(t) · A(αx, t) + α 3 A(αx, t)2 with p(t) = ei Hel t pe−i Hel t and A(αx, t) = ei H0 t A(αx)e−i H0 t . In the computation of the transition amplitude to the order α 3/2 one drops α 3 A(αx, t)2 and one replaces A(αx, t) by A(0, t), which is known as the dipole approximation. Then, an integration by parts using that 2p(t) =
d x(t), dt
−
∂ A(0, t) = E(0, t), ∂t
leads to a result for the transition amplitude which agrees with the expression in (1.16) whose modulus squared is integrated over q ∈ R3 . Theorem 1.3 and its proof justify this formal derivation and the use of the dipole approximation. Note that αx = X, hence the ionization probability is of order α 3 rather than of order α, as a formal computation, similar to the one above, in dimension-full quantities would suggest. We prove a more general result than Theorem 1.3, where the incoming scattering state may contain several asymptotic photons, and where the external potential V is taken from a large class of long range potentials. In the case where the asymptotic state 2 3 at t = −∞ is of the form (1.9) and each of the photons f 1 , . . . , f n ∈ L (R × {1, 2}) satisfies the hypotheses of Theorem 1.3, in addition to f i , f j = δi j , our result says that n 2 ∗ ∗ 3 m l P (3) ( fl ) + O(α 4 ) 1ac (Hel ) a− ( f ) α − a+ ( f ) α = α
(1.17)
l=1
with P (3) ( fl ) :=
d 3 q ϕq , xϕel ·
∞
−∞
R3
2 2 , E(0, t)a ∗ ( fl ) ei(q −E 0 )t dt . (1.18)
The integral with respect to t in (1.18) can be computed explicitly in terms of fl and G0 , and it gives ∞ 2 ei(q −E 0 )t , E(0, t)a ∗ ( fl ) dt −∞ κ(k) 2|k| ελ (k, λ) fl (k, λ)dσ (k), = iπ |k|=q2 −E 0
λ=1,2
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where dσ (k) is the surface measure of the sphere {k ∈ R3 : |k| = q2 − E 0 } in R3 . The integration over the spheres with |k| = q2 − E 0 expresses the conservation of energy in the scattering process, and the additivity (1.17) of the ionization probability with respect to the incoming photons corresponds to the experimental fact that the number of photo-electrons is proportional to the intensity of the incoming radiation. In Sect. 5 we give a second derivation of α 3 P3 ( f ) based on a space-time analysis of the ionization process. This approach, in a slightly different form, was introduced in the papers [3,33], and does not assume asymptotic completeness of Rayleigh scattering. The construction of outgoing scattering states describing an ionized atom and an electron escaping to spacial infinity is a difficult open problem in the model described above. Only for V = 0 such states have been constructed so far [5,25]. Hence it is not possible yet to study the ionization probability based on transition probabilities between asymptotic states. Previously ionization by quantized fields was investigated in [3,16,17,33]. [3] and [33] are precursors of the present paper on simpler models of atoms and the ionization probability defined in a different, but equivalent way. In [16,17] it is shown that a thermal quantized field leads to ionization in the sense of absence of an equilibrium state of atom and field. There is a large host of mathematical results on ionization by classical electric fields: Schrader and various coauthors study the phenomenon of stabilization by providing upper and lower bounds on the ionization probability, see [10–12] and the references therein. They use the Stark-Hamiltonian with a time dependent electric field E(t) that vanishes unless 0 ≤ t ≤ τ < ∞. Lebowitz and various coauthors compute the probability of ionization by an electric field that is periodic in time; see [7,29] and references therein. Most of these papers study one-dimensional Schrödinger operators with a single bound state that is produced by a δ-potential. Ionization in a three-dimensional model with a δ-potential is studied in [6]. 2. Notations and Hypotheses For easy reference, we collect in this section the definitions, our notations and all hypotheses. As usual, L 2 (R3 × {1, 2}) denotes the space of square integrable functions f : R3 × {1, 2} → C with inner product f, g := f (k, λ)g(k, λ)d 3 k. λ=1,2
R3
We recall from the introduction that L 2ω (R3 × {1, 2}) consists of those functions f ∈ L 2 (R3 × {1, 2}) for which the norm f ω defined in (1.4) is finite. Regularity assumptions will be imposed on the vector-valued function (ε f )(k) :=
2
ε(k, λ) f (k, λ),
(2.1)
λ=1
rather than on f (·, 1) and f (·, 2). It is useless to impose smoothness conditions on f (·, λ) because it is (2.1) that matters and because the polarization vectors ε(k, 1) and ε(k, 2) are necessarily discontinuous. On the other hand, every square integrable function f : R3 → C3 with k · f (k) = 0, for a.e. k ∈ R3 , can be approximated, in the L 2 -sense, by smooth functions of the form (2.1).
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It is convenient to collect a family f 1 , . . . , f N ∈ L 2 (R3 × {1, 2}) of photon wave functions in an N -tupel f = ( f 1 , . . . , f N ). We define a( f ) := a( f 1 ) · · · a( f N ), a ∗ ( f ) := a ∗ ( f 1 ) · · · a ∗ ( f N ). This should not lead to confusion with (1.9), where f also includes occupation numbers. For the various parts of the interaction operator W = Hα − H0 , we use the notations W dip := 2p · A(0), W (1) := 2p · A(αx), W (2) := A(αx)2 . It follows that 3
3
5
W = α 2 W (1) + α 3 W (2) = α 2 W dip + O(α 2 ), where the last equation is purely formal, but we shall give it a rigorous meaning in this paper. The Hamiltonian Hα = H0 + W is self-adjoint on the domain of − + H f provided that V is infinitesimally operator bounded with respect to −, [23,24]. This is the case, e.g., if V is the sum of Coulomb potentials due to static nuclei; all our results are valid for such V . Nonetheless, it is useful to identify the properties of V that are essential for our analysis. From now on, we shall only assume the following hypotheses on V : 2 (R3 ), lim Hypotheses. Both V and x · ∇V belong to ∈ L loc |x|→∞ V (x) = 0, and there exist constants μ > 0 and R > 0 such that for |β| = 1, 2 we have β
|∂x V (x)| ≤ |x|−|β|−μ ,
if |x| > R.
Moreover, E 0 := inf σ (Hel ) < 0. We define e1 := inf(σ (Hel )\{E 0 }). From these Hypotheses it follows that σess (Hel ) = [0, ∞), that σsc (Hel ) = ∅ and that E 0 is a simple eigenvalue. In fact, the decay assumptions on V imply long-range asymptotic completeness [9], which is what we use to infer the existence of a complete set of generalized eigenfunctions. All this remains true if a singular short-range potential is added to Hel . The time evolution of an operator B in the interaction picture will be denoted by B(t), that is, B(t) := ei H0 t Be−i H0 t , and Bt := B(−t). Note that p(t) = ei Hel t pe−i Hel t , A(0, t) = ei H f t A(0)e−i H f t and that a # ( f t ) = e−i H0 t a # ( f )ei H0 t = a # ( f )t .
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3. Commutator Estimates and Scattering States The main purpose of this section is to establish bounds on the commutators [W ( j) , a ∗ ( f t )] applied to α for W ( j) ∈ {W (1) , W (2) , W dip }. We are interested in the decay as |t| → ∞ and in the dependence on α. Typically, our estimates are valid for α ≤ α, ˜ where α˜ is defined in Proposition A.3. As a simple application of our decay estimates in t, we will obtain existence of the scattering states ∗ a± ( f ) α = lim eit Hα a ∗ ( f t )e−it Hα α , t→±
which was already established in [22] in larger generality. Here f t = ( f 1,t , . . . , f N ,t ) and f j,t := e−itω f j . Given l ∈ {1, . . . , N }, we write a ∗ ( f [l],t ) := a ∗ ( f 1,t ) · · · a ∗ ( fl−1,t )A(αx)a ∗ ( fl+1,t ) · · · a ∗ ( f N ,t ),
a ∗ ( f (l),t ) := a ∗ ( f 1,t ) · · · a ∗ ( fl−1,t )a ∗ ( fl+1,t ) · · · a ∗ ( f N ,t ). For x ∈ R3 , x := (1 + |x|2 )1/2 .
Lemma 3.1. Suppose that f ∈ L 2 (R3 × {1, 2}) with ε f ∈ C0n (R3 \{0}, C3 ) for a given n ∈ N. Then there exists a constant c1,n = c1,n ( f ) such that 1 , 1 + |t|n 1 + (α|x|)n | Gx , f t | ≤ c1,n 1 + |t|n α|x| αx n | Gx − G0 , f t | ≤ c1,n 1 + |t|n | G0 , f t | ≤ c1,n
(3.1) for all x ∈ R3 , for all x ∈ R3 .
(3.2) (3.3)
Proof. Estimate (3.1) follows from (3.2). We next prove (3.2). By a stationary phase analysis of κ(k) iαk·x−itω(k) Gx , f t = e d 3k √ (ε f )(k) (3.4) 3 2ω(k) R we obtain | Gx , f t | ≤ Cn |t|−n for α|x| ≤ |t|/2, [28] Theorem XI.14. It follows that Cn , |t|n 2α|x| n ≤C , |t|
| Gx , f t |1{2α|x|≤|t|} ≤ | Gx , f t |1{2α|x|>|t|}
where C := supt∈R, x∈R3 | Gx , f t | < ∞. This proves (3.2). To prove (3.3) we write e−itω(k) Fx (k)d 3 k, Gx − G0 , f t = R3
where κ(k) (ε f )(k)g(αk · x) Fx (k) = iαk · x √ 2ω(k)
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and g : R → C denotes the real-analytic function given by g(s) = (eis −1)/(is) for s = 0. g and all its derivatives are bounded, and by assumption on f , Fx ∈ C0∞ (R3 \{0}, C3 ) for each x. It follows that β sup ∂k Fx (k) |x|−1 αx −|β| < ∞, x,k∈R3 , x=0
which implies (3.3), again by stationary phase arguments.
Lemma 3.2. Suppose that ε f 1 , . . . , ε f N ∈ C0n (R3 \{0}, C3 ) for a given n ∈ N, and let α˜ be defined by Proposition A.3. Then there exists a constant c2,n = c2,n ( f ), such that for all α ≤ α, ˜ t ∈ R, and W ( j) ∈ {W (1) , W (2) , W dip }, ( j) ∗ W , a ( f t ) α ≤
c2,n . 1 + |t|n
(3.5)
Proof. By definition of a ∗ ( f t ), [W ( j) , a ∗ ( f t )] α =
N
a ∗ ( f 1,t ) · · · a ∗ ( fl−1,t ) W ( j) , a ∗ ( fl,t ) a ∗ ( fl+1,t ) · · · a ∗ ( f N ,t ) α ,
(3.6)
l=1
and by definition of W (1) and W (2) ,
W (1) , a ∗ ( fl,t ) = 2 Gx , fl,t · p, W (2) , a ∗ ( fl,t ) = 2 Gx , fl,t · A(αx).
(3.7) (3.8)
From (3.2), (3.6), (3.7), (3.8) and Lemma A.1 it follows that N −1 N cn (H f + 1) 2 αx n p α , n 1 + |t| N N cn (H f + 1) 2 αx n α [W (2) , a ∗ ( f t )] α ≤ n 1 + |t|
[W (1) , a ∗ ( f t )] α ≤
(3.9) (3.10)
with some constant cn . Thanks to Lemma A.4, these upper bounds are bounded uniformly in α ≤ α, ˜ α˜ being defined by Proposition A.3. This proves (3.5) for j = 1, 2. The assertion for W dip now follows from W dip = W (1) |x=0 , which leads to a bound for [W dip , a ∗ ( f t )] α of the form (3.9) with x = 0. Proposition 3.3. For all ε f 1 , . . . , ε f N ∈ C02 (R3 \{0}, C3 ) there exists a constant c3 = c3 ( f ), such that for all α ≤ α˜ and for all s ∈ R, c3 α (1) . W − W dip , a ∗ ( f s ) α ≤ 1 + s2
(3.11)
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Proof. By (3.6) for j = 1, (3.7), and the corresponding equations for W dip ,
N W (1) − W dip , a ∗ ( f s ) α = 2 Gx − G0 , fl,s · pa ∗ ( f (l),s ) α , l=1
where Gx − G0 , c ≤α 1 + s2 c ≤α 1 + s2
fl,s · pa ∗ ( f (l),s ) α |x| αx 2 a ∗ ( f (l),s )p α 1/2 1/2 2 |x| αx 4 p α a( f (l),s )a ∗ ( f (l),s )p α
by (3.3) and the Cauchy-Schwarz inequality. The norms in the last expression are bounded uniformly in α ≤ α˜ by Lemma A.1 and Lemma A.4. Lemma 3.4. For all ε f 1 , ..., ε f N ∈ C02 (R3 \{0}, C3 ), there exists a constant c4 = c4 ( f ) < ∞, such that for all α ≤ α˜ and s, t ∈ R, 3 c4 α 2 dip ∗ , (3.12) Ws , a ( f t ) ( α − 0 ) ≤ 1 + |t − s|2 c4 dip ∗ , (3.13) W s , a ( f t ) 0 ≤ 1 + |t − s|2 3 c4 α 2 dip ∗ . (3.14) W, Ws , a ( f t ) α ≤ 1 + |t − s|2 dip Proof. Since Ws , a ∗ ( fl,t ) = 2 G0 , fl,t−s · ps , which commutes with the creation operators a ∗ ( f i,t ),
dip Ws , a ∗ ( f t )
=
N
dip a ∗ ( f 1,t ) · · · a ∗ ( fl−1,t ) Ws , a ∗ ( fl,t ) a ∗ ( fl+1,t ) · · · a ∗ ( f N ,t )
l=1
=2
N l=1
a ∗ ( f (l),t ) G0 , fl,t−s · ps ,
(3.15)
where | G0 , fl,t−s | ≤ cl (1+(t −s)2 )−1 by (3.1). In view of Lemma A.1 and Lemma A.5, this proves (3.12). The proof of (3.13) is similar. From (3.15) we obtain, that
N 3 3 dip W, Ws , a ∗ ( f t ) α = 2α 2 G0 , fl,t−s · W (1) + α 2 W (2) , a ∗ ( f (l),t )ps α . l=1
Hence, by (3.1), it suffices to show that W ( j) a ∗( f (l),t )ps α and a ∗ ( f (l),t )ps W ( j) α
are bounded uniformly in t, s and α ≤ α. ˜ We shall do this for a ∗ ( f (l),t )ps W (1) α only,
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M. Griesemer, H. Zenk
the proofs in the other cases being similar. Let m ≥ (N − 1)/2. Then ∗ a ( f (l),t )ps W (1) α ≤
3 ∗ a ( f (l),t )(H f + 1)−m ps p j (Hel + i)−1 (Hel + i)(H f + 1)m A j (αx) α j=1
≤C
3
(Hel + i)(H f + 1)m A j (αx) α
j=1
with a constant C, that is finite by Lemma A.1. We now want to compare (Hel +i)(H f + 1)m A j (αx) α with A j (αx)(Hel +i)(H f +1)m α , because the latter norm is bounded uniformly in α ≤ α, ˜ by Lemma A.1 and by (A.17). Thus we compute the commutator of (Hel + i)(H f + 1)m and A j (αx) = a ∗ (G x, j ) + a(G x, j ) applied to α . Using
3
2a ∗ (km G x, j ) pm , Hel , a ∗ (G x, j ) = α 2 a ∗ (ω2 G x, j ) − α m=1
m
m ∗ l m ∗ a (ω G x, j )(H f + 1)m−l , (H f + 1) , a (G x, j ) = l l=1
and similar commutator equations for a(G x, j ), we see that all resulting terms have norms that are bounded, uniformly in α ≤ α, ˜ thanks to (A.17) and Lemma A.1. For completeness of this paper we now use Lemma 3.2 to prove existence of the asymptotic creation and annihilation operators on α . More general results can be found in [13,22]. Proposition 3.5. Suppose f = ( f 1 , . . . , f N ) ∈ [L 2ω (R3 ×{1, 2})] N . Then, for all α ≤ α, ˜ ∗ ( f ) α := lim ei Hα t a ∗ ( f t )e−i Hα t α a±
(3.16)
∗ a± ( f ) α ≤ c5 f 1 ω · · · f N ω ,
(3.17)
t→±∞
exists, and
with a constant c5 that is independent of α and f . If ε fl ∈ C0n+1 (R3 \{0}, C3 ) for l = 1, . . . , N , then there exists a constant cn ( f ), such that cn ( f ) ∗ . a± ( f ) α − ei Hα t a ∗ ( f t )e−i Hα t α ≤ α 3/2 1 + |t|n Proof. Suppose first that ε f 1 , . . . , ε f N ∈ C0n+1 (R3 \{0}, C3 ). Then d it Hα ∗ e a ( f t )e−it Hα α = iei(Hα −E α )t [W, a ∗ ( f t )] α , dt and, by Lemma 3.2,
±∞
± t
cn ( f ) . [W, a ∗ ( f s )] α ds ≤ α 3/2 1 + |t|n
(3.18)
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627
∗ ( f ), by Cook’s argument, and then it implies This estimate first proves existence of a± ∗ ( f ) in the case where f ∈ L 2 (R3 × {1, 2}) now follows (3.18). The existence of a± α j ω from the approximation argument given in [22], Prop. 2.1. By Lemmas A.1 and A.4, N
N
eit Hα a ∗ ( f t )e−it Hα α ≤ a ∗ ( f t )(H f + 1)− 2 (H f + 1) 2 α ≤ c5 f 1 ω · · · f N ω ,
uniformly in t ∈ R and α ∈ [0, α]. ˜ Letting t → ±∞ in this estimate, we obtain (3.17). 4. Proofs of the Main Theorems 4.1. A reduction formula. In this section we first prove Theorem 4.1 below, which is a generalization of Theorem 1.1, the latter corresponding to the choice τ = 0. The generalization to arbitrary τ ∈ R will be needed in Sect. 5. Theorem 4.1. Let ε f 1 , . . . , ε f N ∈ C02 (R3 \{0}, C3 ). Then ∗ ( f τ ) α a+∗ ( f τ ) α − a− ∞ 3 = iα 2 e−i(H0 −E 0 )τ 2p(s)ϕel ⊗ [A(0, s), a ∗ ( f )] ds + R(τ, α), −∞
where R(τ, α) = O(α 5/2 ) + O(α 3 |τ |) as α → 0. Remark. Part of the error O(α 5/2 ) stems from passing to the dipole-approximation W (1) → W dip . Hence its order 5/2 = 3/2 + 1 cannot be improved. Proof. Recall that Bt = B(−t) = e−it H0 Beit H0 . To compare the time-evolutions generated by Hα and H0 we will use that t ei(Hα −E α )t Bt α = B α + ei(Hα −E α )s [i W, Bs ] α ds. (4.1) 0
This equation may be iterated because [i W, Bs ] = [i W−s , B]s . From ∗ ( f ) α = lim ei(Hα −E α )t a ∗ ( f t ) α a± t→±∞
and (4.1) it follows that ∗ a+∗ ( f ) α − a− ( f ) α =
∞
−∞
ei(Hα −E α )s [i W, a ∗ ( f s )] α ds.
Only terms contributing to this integral of order α 3/2 need to be kept. Since W = 3 α 2 W (1) + α 3 W (2) , we may drop W (2) , W (1) − W dip and restrict the interval of integration to |s| ≤ α −1 by Lemma 3.2 and Proposition 3.3. We obtain ∗ ( f ) α a+∗ ( f ) α − a− ∞ = iα 3/2 ei(Hα −E α )s [W dip , a ∗ ( f s )] α ds + O(α 5/2 ) −∞ dip 3/2 ei(Hα −E α )s [W−s , a ∗ ( f )]s α ds + O(α 5/2 ). = iα |s|≤α −1
(4.2)
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M. Griesemer, H. Zenk
Applying now (4.1) to the integrand in (4.2) and the time interval [τ, s], rather than [0, s], we find dip ei(Hα −E α )s [W−s , a ∗ ( f )]s α ds −1 |s|≤α dip [W−s , a ∗ ( f )]τ α ds = ei(Hα −E α )τ −1 |s|≤α s dip ds ei(Hα −E α )r [i W, [Wr −s , a ∗ ( f r )]] α dr. (4.3) + |s|≤α −1
τ
By (3.14) in Lemma 3.4, the norm of the double integral is bounded by |τ | + |s| 3/2 const α ds = O(α 3/2 |τ |) + O(α 3/2 ln(α)). 2 |s|≤α −1 1 + |s|
(4.4)
In the integral (4.3) we use Lemma 3.4 to replace α by 0 and to extend the integration over all s ∈ R. We find that dip [W−s , a ∗ ( f )]τ α ds |s|≤α −1 ∞
=
=
−∞ ∞ −∞
[Wτ −s , a ∗ ( f τ )] 0 ds + O(α) dip
e−i(H0 −E 0 )τ [W−s , a ∗ ( f )] 0 ds + O(α). dip
(4.5)
∗ ( f ) = Equations (4.2), (4.3), (4.4) and (4.5) prove the theorem because e−i(Hα −E α )τ a± α ∗ a± ( f τ ) α and because 0 = ϕel ⊗ .
Theorem 4.1 in the case τ = 0 becomes Theorem 1.1, which implies that 2 ∗ ( f ) α − a+∗ ( f ) α = α 3 P (3) ( f ) + O(α 4 ), 1ac (Hel ) a− where P
(3)
( f ) := 1ac (Hel )
∞ −∞
2 ∗ 2p(s)ϕel ⊗ A(0, s), a ( f ) ds .
(4.6)
We next show that P (3) ( f ) is additive in its one-photon contributions.
n Proposition 4.2. Suppose that f = ( f 1 , m 1 , . . . , f n , m n ) ∈ L 2 (R3 ) × N with n f i , f j = δi j and ε fl ∈ C02 (R3 \{0}, C3 ). Then P (3) ( f ) = l=1 m l P (3) ( fl ) with 1 (H ) P (3) ( fl ) = ac el
2 2p(s)ϕel · , A(0, s)a ∗ ( fl ) ds −∞ 2 ∞ = x(s)ϕel · , E(0, s)a ∗ ( fl ) ds 1ac (Hel ) . ∞
−∞
(4.7)
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Proof. Since a ∗ ( f ) is a product of creation operators a ∗ ( fl ) and since [A(0, s), a ∗ ( fl )] = , A(0, s)a ∗ ( fl ) , a scalar multiple of the identity operator, we have
n √ A(0, s), a ∗ ( f ) = m l , A(0, s)a ∗ ( fl ) a ∗ ( f (l) ), l=1
where f (l) = ( f 1 , m 1 , . . . , fl , (m l − 1), . . . , f n , m n ). The vectors a ∗ ( f (l) ) are ortho-
normal by construction. Hence by definition of P (3) ( f ) and by the Pythagoras identity, P (3) ( f ) =
n
m l P (3) ( fl )
l=1
P (3) ( f
with l ) given by the first equation in the statement of the proposition. The second equation in the proposition follows from d d x(s)ϕel , , A(0, s)a ∗ ( fl ) = − , E(0, s)a ∗ ( fl ) 2p(s)ϕel = ds ds by an integration by parts. The differentiability of s → x(s)ϕel and the expression for its derivative are established in Lemma A.6. 4.2. Expansion in generalized eigenfunctions. In this section we prove Theorem 1.3 and the stronger statement expressed by Eqs. (1.17) and (1.18). The ingredients are Theorem 1.1, Proposition 4.2, and a set of generalized eigenfunctions ϕq with the properties (i)-(iii) in the Introduction. Concerning the existence of ϕq , we recall from [9], Theorem 4.7.1, that our hypotheses on V imply existence and completeness of a (modified) wave operator + associated with Hel . Moreover, (Hel − i)−1 x −2 is a HilbertSchmidt operator. Lemma 4.3. Suppose that ϕ : R → Hel ∩ D(|x|2 ) is such that s → ϕ(s) and s → 2 |x| and absolutely integrable with respect to the norm of Hel . Then ∞ ϕ(s) are continuous 2 ) and ϕ(s)ds ∈ D(|x| −∞ 2 ∞ ∞ 2 3 1ac (Hel ) = d q. ϕ ϕ(s)ds , ϕ(s) ds q R3
−∞
−∞
∞ Proof. From the existence of the improper Riemann integrals −∞ ϕ(s)ds and ∞ 2 and the fact that multiplication with |x|2 is a closed operator, it fol−∞ |x| ϕ(s)ds ∞ lows that −∞ ϕ(s)ds ∈ D(|x|2 ) and that ∞ ∞ 2 ϕ(s)ds = |x|2 ϕ(s)ds. |x| −∞
−∞
This equation and property (i) of ϕq imply that ∞ ∞ ϕq , ϕ(s)ds = |x|−2 ϕq , |x|2 ϕ(s)ds −∞ −∞ ∞ −2 |x| ϕq , |x|2 ϕ(s) ds = = −∞
In view of (1.14), this proves the assertion.
∞ −∞
ϕq , ϕ(s) ds.
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Proposition 4.4. Suppose that ε f ∈ C02 (R3 \{0}, C3 ). Then P
(3)
∞ 2 i(q2 −E 0 )s ∗ , E(0, s)a ( f ) ds (f) = d q ϕq , xϕel · e R3 −∞ 2 2 3 = 4π d q ϕq , xϕel · |k| G 0 (k, λ) f (k, λ)dσ (k) , R3 |k|=q 2 −E 0 λ=1,2
3
where dσ (k) denotes the surface measure of the sphere |k| = q 2 − E 0 in R3 . Proof. We start with the expression (4.7) for P (3) ( f ) and we shall apply Lemma 4.3 to ϕ(s) = x(s)ϕel · , E(0, s)a ∗ ( f ) . (4.8) By Lemma A.6, x(s)ϕel = ei(Hel −E 0 )s xϕel belongs to D(|x|2 ) and |x|2 x(s)ϕel ≤ C(1 + s 2 ). On the other hand ∗ , E(0, s)a ( f ) = iω(k)e−iω(k)s G 0 (k, λ) f (k, λ)d 3 k =
λ=1,2 ∞
dωe−iωs
0
|k|=ω
iω
G 0 (k, λ) f (k, λ)dσ (k) (4.9)
λ=1,2
is the Fourier transform of a function from C0∞ (R+ ), and hence rapidly decreasing as s → ∞. It follows that (4.8) satisfies the hypotheses of Lemma 4.3. Hence Lemma 4.3 proves the first asserted equation because
2 ϕq , x(s)ϕel = ei(q −E 0 )s ϕq , xϕel .
The second equation follows from the first one and from (4.9) by an application of the Fourier inversion theorem. 4.3. Proof of Theorem 1.2. Lemma 4.5. If f = ( f 1 , . . . , f n ) with ε f 1 , . . . , ε f n ∈ C0∞ (R3 \{0}, C3 ) and F ∈ C0∞ ((−∞, 0)), then 3
a+∗ ( f )F(Hα ) − a ∗ ( f )F(H0 ) = O(α 2 ). Proof. Choose R ∈ R, such that supp( f 1 ), . . . , supp( f n ) ⊂ {|k| < R} and then choose G ∈ C0∞ (R) with G = 1 on supp(F) + [0, n R]. Then, by the pull through formula for a ∗ ( f ) and by [13], Theorem 4 (iv), a ∗ ( f )F(H0 ) = G(H0 )a ∗ ( f )F(H0 ), 3
a+∗ ( f )F(Hα ) = G(Hα )a+∗ ( f )F(Hα ).
Using that F(H0 ) − F(Hα ) = O(α 2 ), by the Helffer-Sjöstrand functional calculus, that (a ∗ ( f ) − a+∗ ( f ))F(Hα ) = O(α 3/2 ), by the proof of Proposition 3.5, and that
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G(H0 )a ∗ ( f ), a+∗ ( f )F(Hα ) are bounded by Lemma A.1 and [22] Proposition 2.1, we find that a ∗ ( f )F(H0 ) − a+∗ ( f )F(Hα ) = G(H0 )a ∗ ( f )F(H0 ) − G(Hα )a+∗ ( f )F(Hα ) = G(H0 )a ∗ ( f ) (F(H0 ) − F(Hα )) + G(H0 ) a ∗ ( f ) − a+∗ ( f ) F(Hα ) 3
+ (G(H0 ) − G(Hα )) a+∗ ( f )F(Hα ) = O(α 2 ) as α → 0.
Recall from the Introduction that H+α is the closure of the span of all vectors of the form a+∗ (h) α ,
h = (h 1 , . . . , h n ), where h i , ωh i ∈ L 2ω (R3 × {1, 2}),
(4.10)
and that P+α is the orthogonal projection onto H+α . Proof of Theorem 1.2. In the first two steps of this proof we shall establish (1.13) in the weak operator topology. Then we establish norm convergence to conclude the proof. Step 1. Suppose Hel ϕ = λϕ, n ∈ N and f = ( f 1 , . . . , f n ) with ε f 1 , . . . , ε f n ∈ C0∞ (R3 \{0}, C3 ). Then lim P+α ϕ ⊗ a ∗ ( f ) = ϕ ⊗ a ∗ ( f ) (4.11) α→0
and the analog statement holds for ϕ ⊗ . Since λ < 0 there exists F ∈ C0∞ (R) with F(λ) = 1 and supp(F) ⊆ (−∞, 0). Moreover P+α F(Hα ) = F(Hα ) by the hypothesis of Theorem 1.2 and because α ≥ 0 for all α ∈ R. Using, in addition, that 3
a+∗ ( f )F(Hα ) − a ∗ ( f )F(H0 ) = O(α 2 ), which we know from Lemma 4.5, we conclude that 3 P+α ϕ ⊗ a ∗ ( f ) = P+α a ∗ ( f )F(H0 )ϕ ⊗ = P+α a+∗ ( f )F(Hα )ϕ ⊗ + O(α 2 ) 3
3
= a+∗ ( f )F(Hα )ϕ ⊗ + O(α 2 ) = ϕ ⊗ a ∗ ( f ) + O(α 2 ). Step 1 implies that lim P+α = for all ∈ Ran1 pp (Hel ) ⊗ 1F .
α→0
Step 2. w − limα→0 P+α (1c (Hel ) ⊗ 1F ) = 0. Since P+α (1c (Hel ) ⊗ 1F ) ≤ 1 for all α ∈ R it suffices to show that lim a+∗ ( f ) α , P+α (1c (Hel ) ⊗ 1F )ϕ = 0
α→0
for all ϕ ∈ H and all f = ( f 1 , . . . , f n ) with ε f 1 , . . . , ε f n ∈ C0∞ (R3 \{0}). Since a+∗ ( f ) α ∈ Ran P+α , this follows from 3
a+∗ ( f ) α = a ∗ ( f ) 0 + O(α 2 ), which follows from Lemma 4.5 and Lemma A.5.
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From Step 1 and Step 2 it follows that w − lim P+α = 1 pp (Hel ) ⊗ 1F .
(4.12)
α→0
Since P+α and 1 pp (Hel ) ⊗ 1F are orthogonal projectors, we have α→0
P+α ϕ 2 = ϕ, P+α ϕ −→ ϕ, 1 pp (Hel ) ⊗ 1F ϕ = 1 pp (Hel ) ⊗ 1F ϕ 2 . Combined with (4.12) this proves the desired strong convergence.
5. Space-Time Analysis of the Ionization Process The purpose of this section is to connect our result with those of the previous papers [3,33], where expressions for the zeroth and first non-trivial order of the ionization probability were defined. We transcribe the definitions from [33] to our model and prove their equivalence to the definitions in this paper. Let FR := 1{|x|≥R} ⊗ 1F . Proposition 5.1. Let ε f 1 , . . . , ε f N ∈ C02 (R3 \{0}, C3 ). Then ∗ ( f τ ) α 2 = 0. lim lim sup sup FR a±
R→∞
α0
τ ∈R
(5.1)
Remarks. The left-hand side of Eq. (5.1) may be interpreted as the ionization probability to zeroth order in α [33]. Proposition 5.1 should be compared to Theorem 4.1 in [33]. Proof. As in the proof of Theorem 4.1, ∗ a± ( f τ ) α − a ∗ ( f τ ) α = i
±∞ 0
eis(Hα −E α ) [W, a ∗ ( f τ +s )] α ds,
3
where the integral is O(α 2 ) in norm, uniformly in τ , by Lemma 3.2. Hence it remains to show that lim lim sup sup FR a ∗ ( f τ ) α 2 = 0.
R→∞
α0
τ ∈R
(5.2)
To this end, we observe that, according to Lemma A.1, FR a ∗ ( f τ ) α 2 ≤ a ∗ ( f τ )2 α FR α ≤ C2N
N
l=1
1 fl 2ω (H f + 1) N α |x| α . R
This proves (5.2), because lim supα→0+ (H f + 1) N α and lim supα→0+ |x| α are finite by Lemma A.4 and by (A.5). Theorem 5.2. Let ε f 1 , . . . , ε f N ∈ C02 (R3 \{0}, C3 ), suppose σsc (Hel ) = ∅, and let τ (α) = α −β for some β ∈ (0, 23 ). Then 2 ∗ P (3) ( f ) = lim lim sup α −3 FR a− ( f τ (α) ) α − a ∗ ( f τ (α) ) α . (5.3) R→∞
α0
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Remarks. Equation (5.3) is to be compared with the expression defining Q (2) (A) in Eq. (1.9) from [33]: if we set g = α 3/2 and τ (g) = α −β in that equation, then Q (2) (A) coincides with the right-hand side of (5.3). Proof. From Proposition 3.5 we know that α 3/2 ∗ = Cα 3/2+β , a+ ( f τ (α) ) α − a ∗ ( f τ (α) ) α ≤ C τ (α) hence we may replace a ∗ ( f τ (α) ) α by a+∗ ( f τ (α) ) α for the proof of (5.3). From Theorem 4.1 we know that ∗ lim lim sup α −3 FR a− ( f τ (α) ) α − a+∗ ( f τ (α) ) α 2 R→∞
α0
= lim lim sup FR e−iτ (α)(H0 −E 0 ) ( f ) 2 R→∞
α0
= lim lim sup FR e−iτ Hel ⊗ 1F ( f ) 2 ,
(5.4)
R→∞ τ →∞
where ∞ ( f ) :=
2p(s)ϕel ⊗ [A(0, s), a ∗ ( f )] ds =
N
φl ⊗ ηl .
(5.5)
l=1
−∞
Explicit expressions for φl and ηl may be taken from the proof of Proposition 4.2, e.g., ηl = a ∗ ( f (l) ), but they are not needed here. From (5.5) it follows that FR e
−iτ Hel
N 1{|x|≥R} e−iτ Hel φl ⊗ ηl , ( f ) = l=1
where 1{|x|≥R} e−iτ Hel φl = (1 − 1{|x|
(5.6)
By the RAGE Theorem, see [32], Satz 12.8, lim sup 1{|x|≥R} eiτ Hel 1 pp (Hel )φl = 0,
(5.7)
lim 1{|x|
(5.8)
R→∞ τ →∞
τ →∞
From (5.4)–(5.8) it follows, that lim lim sup FR e−iτ H0 ( f ) 2 = (1ac (Hel ) ⊗ 1F )( f ) 2 = P (3) ( f ),
R→∞ τ →∞
by Eq. (4.6)
Acknowledgement. M.G. thanks Vadim Kostrykin for pointing out that it is advantageous to define ionization as the opposite of binding.
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A. Uniform Estimates In this Appendix we collect estimates used in the previous sections. Most of them are well-known for fixed α, but this is not sufficient for us: we need estimates holding uniformly for α in a neighborhood of α = 0. This forces us to review some of the derivations with special attention to the dependences on α. Lemma A.1. For every N ∈ N there is a finite constant C N such that for all h 1 , . . . , h N ∈ L 2ω (R3 × {1, 2}), N
a ∗ (h)(H f + 1)− 2 ≤ C N
N
h l ω ,
(A.1)
l=1 N
a ∗ (h 1 ) · · · a ∗ (h l−1 )A(αx)a ∗ (h l+1 ) · · · a ∗ (h N )(H f + 1)− 2 ≤ C N
N
h j ω .
j=1 j=l
(A.2) Proof. Both, (A.1) and (A.2) follow from Lemma 17 in [13]. We recall that A(αx) = a ∗ (Gx ) + a(Gx ), and we note that supx∈R3 Gx ω < ∞. Proposition A.2 ([15]). For every λ < e1 there exists a constant αλ > 0, such that for all n ∈ N, (A.3) sup |x|n 1(−∞,λ] (Hα ) < ∞. α≤αλ
Proposition A.3. There exists an α˜ > 0, such that: a) For all α ≤ α, ˜ E α := inf σ (Hα )
(A.4)
is a simple eigenvalue of Hα . In the following α denotes the unique normalized ground state of Hα whose phase is determined by α , 0 ≥ 0. b) For every n ∈ N, sup |x|n α < ∞.
(A.5)
α≤α˜
c) There exists a finite constant C, such that for all α ≤ α˜ and all k ∈ R3 \{0}, 3 |κ(k)| a(k) α ≤ α 2 C √ (1 + α|k|), |k| 3
|E 0 − E α | ≤ α 2 C, 3 2
(A.6) (A.7)
α − 0 ≤ α C.
(A.8)
sup [H n−1 , Hα ](Hα + i)−n+1 < ∞, f
(A.9)
sup H nf (Hα + i)−n < ∞,
(A.10)
sup Hel (Hα + i)−1 < ∞.
(A.11)
d) For every n ∈ N, α≤α˜
α≤α˜
α≤α˜
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Remark. Boundedness of [H n−1 , H ](H + i)−n and H nf (H + i)−n has previously been f established in [13], Lemma 5, for a class of Hamiltonians H that includes Hα . Yet, those results do not imply (A.9) and (A.10), and second, their proof is much more complicated than the proof of (A.9) and (A.10), because H in [13] is defined in terms of a Friedrichs’ extension. Proof. That E α = inf σ (Hα ) is an eigenvalue of Hα , for small α, was first shown in [2]. Its simplicity follows from (A.8), which holds for every normalized ground state vector α that satisfies the phase condition α , 0 ≥ 0. A proof of (A.8) may be found, e.g., in [15], Prop. 19, Steps 4 and 5. A weaker form of (A.6) is given in Lemma 20 of [15], but the proof there actually shows (A.6). Estimate (A.7) follows from Lemma 22 in [15] by choosing the infrared cutoff in this lemma larger than the UV-cutoff. Finally, (A.5) is a consequence of Proposition A.2 and (A.7). To prove (d) we set R0 := (H0 + i)−1 and Rα := (Hα + i)−1 . It is a simple exercise, using (A.1) and the boundedness of (H f + 1)1/2 pR0 , to show that W R0 = O(α 3/2 ) as α → 0. Hence we may assume that supα≤α˜ W R0 ≤ 1/2 after making α˜ smaller, if necessary. It follows that (H0 + i)Rα = (1 + W R0 )−1 ≤ (1 − W R0 )−1 ≤ 2
(A.12)
for all α ≤ α. ˜ Since Hel R0 and H f R0 are bounded operators, we have thus proven statement (d) for n = 1, (A.9) being trivial in this case. We now proceed by induction, assuming that (A.9) and (A.10) hold true for all positive integers smaller than or equal to a given n ≥ 1. To prove (A.10) for n replaced by (n + 1) we use that n n adlH f (W )H n−l [H nf , Hα ]Rαn = Rαn f l l=1 n n n adlH f (W )Rα H n−l = Rαn−1 − [H n−l f f , Hα ]Rα , (A.13) l l=1
where supα≤α˜ adlH f (W )Rα
< ∞ by (A.12), by explicit formulas for adlH f (W ) and by
the arguments above proving that W R0 = O(α 3/2 ). Hence supα≤α˜ [H nf , Hα ]Rαn < ∞ follows from (A.13) and from the induction hypothesis. Statement (A.10) with n n+1 = (H R )(H n R n )−H R [H n , H ]R n+1 , replaced by n+1 now follows from H n+1 f α f α α α f Rα f α f from the induction hypothesis, and from (A.9) with n replaced by n + 1, which we have just established. Lemma A.4. For all l, m ∈ N: sup (H f + 1)m α < ∞,
(A.14)
sup (H f + 1)m x l α < ∞,
(A.15)
sup p2 α < ∞,
(A.16)
sup (H f + 1) Hel α < ∞,
(A.17)
sup (H f + 1)m x l p α < ∞.
(A.18)
α≤α˜ α≤α˜
α≤α˜ m
α≤α˜ α≤α˜
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Proof. The statements (A.14) and (A.16) easily follow from (A.10), (A.11), and (A.7), because α = (Hα +i)−n α (E α +i)n ; note that p2 (Hel +i)−1 is bounded by assumption on V . To prove (A.15) we use that sup x l (H f + 1)m α 2 ≤ sup x 2l α · (H f + 1)2m α ,
α≤α˜
α≤α˜
where the right hand side is finite thanks to (A.5) and (A.14). To prove (A.17) we write Hel (H f + 1)m α = Hel (Hα + i)−1 (Hα + i)(H f + 1)m α
= Hel (Hα + i)−1 Hα , (H f + 1)m α +Hel (Hα + i)−1 (H f + 1)m α (E α + i). The vectors [Hα , (H f + 1)m ] α and (H f + 1)m α , and the operator Hel (Hα + i)−1 are bounded, uniformly in α ≤ α, ˜ by (A.9), (A.10) and (A.11). This proves (A.17). The statement (A.18) follows from (A.15) and (A.16) after moving both p’s to one side, and both factors x l to the other side of the inner product (H f + 1)m x l p α 2 = (H f + 1)m x l p α , (H f + 1)m x l p α . The following lemma improves upon (A.8). Lemma A.5. For each m ∈ N there is a finite constant K m , such that for all α ≤ α, ˜ 3
(Hel + i)(H f + 1)m ( α − 0 ) ≤ K m α 2 .
(A.19)
Proof. Let λ := (E 0 + e1 )/2. Thanks to (A.7) in Proposition A.3, we may assume that supα≤α˜ E α < λ by making α˜ smaller, if necessary. Pick g ∈ C0∞ (R) with suppg ⊂ (−∞, λ) and with g(E α ) = 1 for all α ≤ α. ˜ On the one hand, (Hel + i)(H f + 1)m g(H0 )( α − 0 ) ≤ (Hel + i)(H f + 1)m g(H0 ) α − 0 = O(α 3/2 ) by (A.8). On the other hand, (1 − g(H0 ))( α − 0 ) = (g(Hα ) − g(H0 )) α by construction of g. Hence it remains to prove that (Hel + i)(H f + 1)m (g(Hα ) − g(H0 )) α = O(α 3/2 ).
(A.20)
To do so, we use the Helffer-Sjöstrand functional calculus with a compactly supported almost analytic extension g˜ of g that satisfies an estimate |∂z¯ g(z)| ˜ ≤ C|y|2 . Here and henceforth z = x + i y with x, y ∈ R. It follows that (Hel + i)(H f + 1)m (g(Hα ) − g(H0 )) α ∂ g˜ 1 d xd y, =− (Hel +i)(H0 − z)−1 (H f +1)m W (Hα −z)−1 α π R2 ∂ z¯
(A.21)
where (H f + 1)m W =
m m l=0
l
adlH f (W )(H f + 1)m−l =: α 3/2 W˜ (m)(H f + 1)m .
(A.22)
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From the equations [H f , a ∗ (G x )] = a ∗ (ωG x ) and [H f , a(G x )] = −a(ωG x ) it is clear that the operator W˜ (m), defined by (A.22), is H0 -bounded. Hence we can estimate the norm of (A.21) from above by ∂ g˜ H0 + i 1 α 3/2 (Hel + i)(H0 + i)−1 d xd y π ∂ z¯ H0 − z |z − E α | × W˜ (m)(H0 + i)−1 (H0 + i)(H f + 1)m α .
(A.23)
The integral is finite by construction of g, ˜ because |z − E α |−1 ≤ |y|−1 , and because (H0 + i)(H0 − z)−1 ≤ 1 + (1 + |x|)/|y| by the spectral theorem. The last factor in (A.23) is bounded uniformly in α ≤ α˜ by (A.14) and (A.17) from Lemma A.4. This establishes (A.20) and thus concludes the proof of the lemma. Lemma A.6. Suppose that V satisfies the hypotheses in Sect. 2. Then (i) xϕel ∈ D(Hel ) and (Hel − E 0 )xϕel = −2∇ϕel . (ii) e−i Hel t xϕel ∈ D(|x|2 ) and there exists a constant C such that for all t ∈ R, |x|2 e−i Hel t xϕel ≤ C(1 + t 2 ). Proof. (i) For all γ ∈ C0∞ (R3 ) we have xHel γ = Hel xγ + 2∇γ , and hence Hel γ , xϕel = Hel xγ + 2∇γ , ϕel = γ , E 0 xϕel − 2∇ϕel . Since C0∞ (R3 ) is a core of Hel , we conclude that xϕel ∈ D(Hel ) and that Hel xϕel = E 0 xϕel − 2∇ϕ. (ii) Let ψ := xi ϕel for some i ∈ {1, 2, 3}. We shall only need that ψ ∈ D(|x|2 )∩ D(−) which follows from (i). By the fundamental theorem of calculus, in a weak sense eit Hel |x|2 e−it Hel ψ = x2 ψ +
t
eis Hel [i Hel , |x|2 ]e−is Hel ψds
0
t = |x|2 ψ + 2
eis Hel (x · p + p · x)e−is Hel ψds
0
t = |x| ψ +2t (x · p+p · x)ψ +2 2
s ds
0
dr eir Hel (4p2 −x · ∇V )e−ir Hel ψ.
(A.24)
0
Here ψ ∈ D(|x|2 ) ∩ D(−) ⊂ D(x · p + p · x) and e−ir Hel ψ ∈ D(Hel ) = D(−) because ψ ∈ D(Hel ) by part (i). Therefore assertion (ii) follows from (A.24) and from the hypotheses on V .
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References 1. Arendt, C., Dimitrovski, D., Briggs, J.S.: Electron detachment from negative ions by few-cycle laser pulses: Dependence on pulse duration. Phys. Rev. A (Atomic, Molecular, and Optical Physics) 76(2), 023423 (2007) 2. Bach, V., Fröhlich, J., Sigal, I.M.: Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field. Commun. Math. Phys. 207(2), 249–290 (1999) 3. Bach, V., Klopp, F., Zenk, H.: Mathematical analysis of the photoelectric effect. Adv. Theor. Math. Phys. 5(6), 969–999 (2001) 4. Brabec, T., Krausz, F.: Intense few-cycle laser fields: Frontiers of nonlinear optics. Rev. Mod. Phys. 72(2), 545–591 (2000) 5. Chen, T., Fröhlich, J., Pizzo, A.: Infraparticle scattering states in non-relativistic qed: I. the BlochNordsieck paradigm. Commun. Math. Phys. 294, 761–825 (2010) 6. Correggi, M., Dell’Antonio, G., Figari, R., Mantile, A.: Ionization for three dimensional time-dependent point interactions. Commun. Math. Phys. 257(1), 169–192 (2005) 7. Costin, O., Lebowitz, J.L., Stucchio, C.: Ionization in a 1-dimensional dipole model. Rev. Math. Phys. 20(7), 835–872 (2008) 8. Derezi´nski, J., Gérard, C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11(4), 383–450 (1999) 9. Derezi´nski, J., Gérard, C.: Scattering theory of classical and quantum N -particle systems. Berlin: Springer-Verlag, 1997 10. Enss, V., Kostrykin, V., Schrader, R.: Ionization of Rydberg atoms by an electric-field kick. Phys. Rev. A 50(2), 1578–1580 (1994) 11. Figueira de Morisson Faria, C., Fring, A., Schrader, R.: Analytical treatment of stabilization. Laser Phys. 9, 379 (1999) 12. Fring, A., Kostrykin, V., Schrader, R.: Ionization probabilities through ultra-intense fields in the extreme limit. J. Phys. A 30(24), 8599–8610 (1997) 13. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic electromagnetic fields in models of quantummechanical matter interacting with the quantized radiation field. Adv. Math. 164(2), 349–398 (2001) 14. Fröhlich, J., Griesemer, M., Schlein, B.: Asymptotic completeness for Rayleigh scattering. Ann. Henri Poincaré 3(1), 107–170 (2002) 15. Fröhlich, J., Griesemer, M., Sigal, I.M.: Spectral theory for the standard model of non-relativistic QED. Commun. Math. Phys. 283(3), 613–646 (2008) 16. Fröhlich, J., Merkli, M., Sigal, I.M.: Ionization of atoms in a thermal field. J. Stat. Phys. 116(1–4), 311–359 (2004) 17. Fröhlich, J., Merkli, M.: Thermal ionization. Math. Phys. Anal. Geom. 7(3), 239–287 (2004) 18. Gérard, C.: On the scattering theory of massless Nelson models. Rev. Math. Phys. 14(11), 1165–1280 (2002) 19. Griesemer, M.: Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics. J. Funct. Anal. 210(2), 321–340 (2004) 20. Griesemer, M.: Non-relativistic matter and quantized radiation. In: Derezinski, J., Siedentop, H. (eds.), Large Coulomb Systems, Volume 695 of Lect. Notes Phys., Berlin-Heidelberg-New York: Springer, 2006, pp. 217–248 21. Griesemer, M., Lieb, E.H., Loss, M.: Ground states in non-relativistic quantum electrodynamics. Invent. Math. 145(3), 557–595 (2001) 22. Griesemer, M., Zenk, H.: Asymptotic electromagnetic fields in non-relativistic qed: The problem of existence revisited. J. Math. Anal. Appl. 354(1), 339–346 (2009) 23. Hasler, D., Herbst, I.: On the self-adjointness and domain of Pauli-Fierz type Hamiltonians. Rev. Math. Phys. 20(7), 787–800 (2008) 24. Hiroshima, F.: Self-adjointness of the Pauli-Fierz Hamiltonian for arbitrary values of coupling constants. Ann. Henri Poincaré 3(1), 171–201 (2002) 25. Pizzo, A.: One-particle (improper) states in Nelson’s massless model. Ann. Henri Poincaré 4(3), 439–486 (2003) 26. Poerschke, T., Stolz, G.: On eigenfunction expansions and scattering theory. Math. Z. 212(3), 337–357 (1993) 27. Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1975 28. Reed, M., Simon, B.: Methods of modern mathematical physics. III. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1979 29. Rokhlenko, A., Lebowitz, J.L.: Ionization of a model atom by perturbations of the potential. J. Math. Phys. 41(6), 3511–3522 (2000) 30. Spohn, H.: Asymptotic completeness for Rayleigh scattering. J. Math. Phys. 38(5), 2281–2296 (1997)
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31. Spohn, H.: Dynamics of charged particles and their radiation field. Cambridge: Cambridge University Press, 2004 32. Weidmann, J.: Lineare Operatoren in Hilberträumen. Teil II. Mathematische Leitfäden. Stuttgart: B. G. Teubner, 2003 33. Zenk, H.: Ionization by quantized electromagnetic fields: the photoelectric effect. Rev. Math. Phys. 20(4), 367–406 (2008) Communicated by I.M. Sigal
Commun. Math. Phys. 300, 641–657 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1130-8
Communications in
Mathematical Physics
Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane Louis-Pierre Arguin1 , Michael Damron2 , C. M. Newman1 , D. L. Stein3 1 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
E-mail: [email protected]; [email protected]
2 Mathematics Department, Princeton University, Princeton, NJ 08544, USA.
E-mail: [email protected]
3 Courant Institute of Mathematical Sciences and Physics Department, New York University,
New York, NY 10012, USA. E-mail: [email protected] Received: 21 November 2009 / Accepted: 5 May 2010 Published online: 10 September 2010 – © Springer-Verlag 2010
Abstract: We consider the Edwards-Anderson Ising spin glass model on the half-plane Z × Z+ with zero external field and a wide range of choices, including mean zero Gaussian, for the common distribution of the collection J of i.i.d. nearest neighbor couplings. The infinite-volume joint distribution K(J, α) of couplings J and ground state pairs α with periodic (respectively, free) boundary conditions in the horizontal (respectively, vertical) coordinate is shown to exist without need for subsequence limits. Our main result is that for almost every J , the conditional distribution K(α | J ) is supported on a single ground state pair. 1. Introduction 1.1. Background. The problem of determining the number of distinct ground state pairs in realistic spin glass models remains of primary importance to understanding the nature of spin glasses [2,18]. It is a measure of the difficulty of the problem that, despite decades of effort, it remains unresolved on a mathematically rigorous or even an analytically but non-rigorous level for any nontrivial dimension. Of central interest is the Edwards-Anderson (EA) Ising model [3] on Zd . The model is defined by the Hamiltonian Jx y σ x σ y , (1) H J (σ ) = − x,y
where J denotes a specific realization of the couplings Jx y = Jx,y , the spins σx = ±1 and the sum is over nearest-neighbor pairs x, y only, with the sites x, y on the cubic lattice Zd . The Jx y ’s are independently chosen from a symmetric, continuous distribution with unbounded support, such as Gaussian with mean zero. Of course, for d = 1, the multiplicity of infinite-volume ground states is exactly two — i.e., a single ground state pair (GSP) of spin configurations related to each other by
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a global spin flip. At the opposite extreme, the infinite-range Sherrington-Kirkpatrick model [20], which is expected to possess a similar themodynamic structure to the EA model as d → ∞, is known to have (in an appropriate sense) an infinite number of GSP’s [2,7]. But for any nontrivial dimension — i.e., 2 ≤ d < ∞ — there are very few analytical results. One exception is the highly disordered model [10,11], in which a transition in the number of GSP’s from one at low dimensions to infinitely many at high dimensions is known (although only partially proved), with the probable crossover dimension predicted either as d = 8 [10] or d = 6 [4,5]. However, that model has an unusual (volume-dependent) coupling distribution, and should not be considered “realistic.” There have been efforts to solve the problem in two dimensions, whose special properties and simplifications might lend itself more readily to analytical approaches. Although over the past decade numerical simulations [8,19] have pointed toward a single pair of ground states in the EA model in two dimensions, mathematically the problem remains open, and the issue is not completely settled [6]. A partial result due to Newman and Stein [16,17], which we will make use of here, supports the conjecture of a single GSP for d = 2, but is not inconsistent with many GSP’s. In this paper, we provide the first rigorous result for an EA model in nontrivial dimension – on the half-plane. We begin with finite-volume measures corresponding to joint distributions of the couplings and ground states, and prove that these converge to a unique limit. More significantly, we also prove that the conditional distribution of the limiting measure (for almost every coupling realization) is supported on a single GSP. A technical tool that we will use to obtain these results is the metastate [1,12], which will be defined, reviewed and extended in Sect. 2. 1.2. Preliminaries. We hereafter restrict ourselves to the EA spin glass model on the half-plane. As in the general case, we assign i.i.d. random variables (the couplings, with product measure ν) to the nearest-neighbor edges of the upper half-plane, whose vertex set is H = {(m, m ) : m, m ∈ Z and m ≥ 0} and whose edge set we denote by E. Throughout the paper, we define the dual upper half-plane to be the graph with vertices H ∗ = {(x + 1/2, y − 1/2) : (x, y) ∈ H } (the dual vertices) and with all nearest-neighbor edges (the dual edges), except for those between dual vertices in the dual x-axis X ∗ = {(x + 1/2, −1/2) : x ∈ Z}. For each n, we consider the box n := [−n, n] × [0, 2n] and define the (random) energy on spin configurations σ ∈ {−1, +1}n by Hn (σ ) = − Jx y σ x σ y , (2) x,y∈E n
where the sum is over all nearest-neighbor edges E n with at least one endpoint in the box n . (We remark that we could as well consider rectangular boxes [−n, n] × [0, n ] and let n, n → ∞ independently of each other, with essentially no changes to our
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arguments.) We use periodic boundary conditions on the left and right sides of the box, and free boundary conditions on the top and bottom. In this case, the energy (2) has the symmetry σ → −σ . The relevant space is then {−1, +1}n modulo a global spinflip ˜ n . More generally, we will write ˜ A for the set of configurations and we denote it by A 2 {−1, +1} modulo spinflip for a subset A of Z . Because the coupling distribution is ˜ n with lowest continuous, there exists (with probability one) a unique element of energy. For each n, denote by αn,J the pair of spin configurations of least energy, i.e., the ground state pair (GSP) in n . The results in this paper concern limits of the GSP’s αn,J . For this, we need a defi˜ H is an infinite-volume GSP nition of a GSP in infinite volume. We will say that α ∈ if for each dual circuit C ∗ in H ∗ and for each path P ∗ in H ∗ which begins and ends in distinct dual vertices of the dual x-axis X ∗ , we have Jx y αx α y > 0 and Jx y αx α y > 0. (3) x,y∈C ∗
x,y∈P ∗
It is easy to see that with our choice of boundary conditions, this is equivalent to the more familiar characterization that α is an infinite-volume GSP if and only if Jx y αx α y > 0 for every finite set of vertices S. (4) x,y∈∂ S
Here, ∂ S refers to all edges which have one endpoint in S and one not in S. ˜ H if the For any edge x, y, we say that the coupling Jx y is satisfied in α ∈ inequality Jx y αx α y > 0 holds; otherwise we will call the coupling unsatisfied. When we say that the coupling at a dual edge in H ∗ is satisfied (unsatisfied), we mean that the coupling at the edge in the original lattice H (the perpendicular bisector edge which is dual to this dual edge) is satisfied (unsatisfied). The interface between configurations α ˜ H is the set of dual edges (and their endpoints, which are dual vertices) whose and β in couplings are satisfied in exactly one of the two configurations. The interface between α and β will be denoted α β and we will call each connected component of it a domain wall. 1.3. Main results. Recall the definition of the box n from Sect. 1.2. Let Kn be the joint distribution of the couplings and the corresponding GSP on n , using the boundary conditions listed in Sect. 1.2 (periodic on the left and right sides of the box and free on the top and bottom). The first theorem below states that the measures Kn converge in the sense of finite-dimensional distributions. The second theorem states that the limiting measure K is supported on a single GSP for almost every (a.e.) J . Theorem 1.1. The sequence of measures (Kn ) converges as n → ∞. The limiting measure K is supported on infinite-volume GSP’s; in other words, for a.e. coupling configuration J , the conditional distribution K( · | J ) is supported on GSP’s for that J . Theorem 1.2. The limiting measure K has the property that for a.e. J , the conditional distribution K( · | J ) is supported on only a single GSP. Our analysis will be based on the concept of metastate, developed in [1,9,12–15], and the excitation metastate, introduced in [17]. Theorem 2.2 below is stated in this framework and easily yields the above main results.
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2. Metastates and Outline of the Proof 2.1. Metastates on ground states. There are different, mathematically equivalent ways to define the metastate; the one that will be most convenient for our purposes is the first version, using joint distributions, due to Aizenman and Wehr [1]. The metastate concept provides both an appropriate setting for infinite-volume Gibbs states (or in our context, ground state pairs) for disordered systems and a useful tool for mathematical analysis. We present the definitions for boundary conditions such as periodic or free, where ground state spin configurations are considered modulo a global spinflip, though more general cases are similar. The treatment for Zd is the same as for H . ˜ we define a probability measure δα supported For any ⊂ H finite and fixed α ∈ ˜ , on α: for any α ∈ 1αx α y (αx α y ), δα (α ) := x,y x,y∈
where 1αx α y (αx α y ) = 1 if αx α y = αx α y and 0 otherwise. For the sequence n with n → H we recall that E n stands for the set of edges of ˜ × R En n and αn,J for the unique GSP on n . We consider the joint distribution on of the couplings together with the measure supported on αn,J : Kn := δαn,J νn (d J ),
(5)
where νn is the i.i.d. product measure for the couplings in n . A standard compactness argument leads to the existence along subsequences of a limiting measure K in the sense of finite-dimensional distributions. More precisely, for every subsequence, there exists a subsubsequence n k such that for any finite ⊂ H , m < ∞ and A any measurable ˜ × R Em , event in K(A) = lim Kn k (A). k→∞
The reader is referred to Lemma B.1 in [9] for more details. Moreover, by construction, the conditional of K given J is supported on GSP’s for that J since the property (4) is clearly preserved. Since the space {−1, +1} H is Polish, the distribution conditioned on the couplings exists for ν-almost all J , yielding the following definition of the metastate. Definition 2.1 (Metastate for Ground States). A metastate K J is a probability measure ˜ H obtained by conditioning a limit K of finite-volume measures (5) on the realizaon tion J of the couplings. In particular, for almost every J , it is supported on GSP’s for that J . Another construction of metastates, referred to as empirical metastates, consists of N taking subsequential limits of the empirical measures N1 n=1 δαn,J . It can be shown that there exist subsequences for which both constructions agree (more accurately, αn,J may need to be replaced by αm n ,J , where m n is increasing with n; see Appendix B of [9]). We can now present a precise version of our main result from which Theorems 1.1 and 1.2 follow. Theorem 2.2. Let α and β be two GSP’s sampled independently from metastates K J and KJ for the same realization J of the couplings. Then, for ν-almost all J , α = β with K J × KJ -probability one. Thus for almost every J there exists a unique metastate K J and it is supported on a single GSP.
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The theorem implies the existence of the limit K of finite-volume measures of the form (5) since it shows that every convergent subsequence has the same limit. 2.2. Outline of proof. The proof of Theorem 2.2 consists of two main parts. First we show that if there are two distinct GSP’s obtained from metastates for the same J , then their interface contains (with positive probability) infinitely many domain walls. We focus on the density properties of tethered domain walls, those that intersect the dual x-axis. The proof of the existence of tethered domain walls requires an extension of the metastate that includes excited states. These measures are discussed in Sect. 2.3. Second, we construct from two (possibly different) metastates for the half-plane H a measure μ∗ on pairs of GSP’s for the full plane. Under the assumption that there is more than one GSP for the half-plane metastates, we show that with positive probability, two ground states sampled independently from μ∗ have an interface which contains at least two distinct domain walls. This would lead to a contradiction by virtue of a theorem of [16,17] which prohibits the existence of more than one domain wall in the full plane. The theorem as stated in [16,17] does not apply to the measure μ∗ , since μ∗ is not a product of metastates constructed from finite-volume GSP’s for the full plane. However, we will give in Sect. 2.4 a more general version of that theorem which makes evident that the measure μ∗ satisfies all the necessary properties for the original proof of [16,17] to hold. 2.3. Excitation metastates. The excitation metastate is a probability measure on configurations of minimal energy where spins in some finite subset of H are specified. It includes the metastate constructed previously as a marginal. The measure we will use here is slightly different from the original definition in [17], although it contains essentially the same information. The main purpose of the excitation metastate is to express sufficient conditions for an infinite-volume state to become a GSP when finitely many couplings are modified as well as sufficient conditions for a GSP to lose this property (cf. Propositions 2.5, 2.6 and 2.7). We will apply this framework to prove Corollary 2.9, a general result on ground state interfaces which proves the existence of tethered domain walls. ˜ A . We suppose that n is large enough so that A ⊂ n . Let A ⊂ H be finite and η A ∈ ηA For a given coupling configuration J , we define the excited state αn,J as the element of ˜ n which minimizes the Hamiltonian Hn subject to the constraint that it equals η A on the set A. For any two pairs η A and ηA , we define the energy difference η
η
A A En,J (η A , ηA ) := Hn (αn,J ) − Hn (αn,J ).
η
(6)
A Clearly, the configuration η A for which αn,J is the ground state in n is determined by these energy differences. More generally, for any subset B ⊂ A and a given configuηA ration η B on B, the state of minimal energy among the states αn,J with configuration η B on B is determined by these energy differences. For example, finding the ground state corresponds to the case B = ∅. Similar quantities were introduced in [17], but the energy differences there were defined between an excited state and the ground state. We now use two excited states in the definition since the energy difference has then a natural decomposition as we shall see. Let J A be J restricted to couplings with both endpoints in A. We denote by H J A (·) the Hamiltonian like (2) but with the sum restricted to x, y ∈ A. We write the energy difference for the couplings with at most one endpoint in A,
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ηA ηA ext En,J (η A , ηA ) = Hn (αn,J ) − H J A (η A ) − Hn (αn,J ) − H J A (ηA ) . Letting h(η A , ηA , J A ) = H J A (η A ) − H J A (ηA ), we can rewrite the difference (6) as ext En,J (η A , ηA ) = En,J (η A , ηA ) + h(η A , ηA , J A ).
This decomposes the energy difference En,J into two pieces: the exterior energy difference and an interior term which depends only on variables inside of A. We can interpret the exterior energy difference as a sort of boundary condition. Once this term is known for all η A ’s and ηA ’s, the energy differences are determined for every J A through the functions h. Hence so are all excited states and excitation energies for subsets B ⊂ A. We now highlight four important properties following directly from the definitions: 1. 2. 3. 4.
η
A has the GSP property (4) in n \A. For any η A , αn,J ext (η , η ) + E ext (η , η ) = E ext (η , η ). For any η A , η A and ηA : En,J A A A A n,J A n,J A ext (η , η ) and α η A do not depend on J . For any η A and ηA , En,J A A n,J A ˜ B , let η∗ be the (almost surely) unique element of ˜ A such For B ⊂ A and η B ∈ A ext (η , η∗ ) + h(η , η∗ , J ) ≥ 0 for all η with ∗ that: i) η A = η B on B and ii) En,J A A A A A A η A = η B on B. Also define η ∗A similarly with η B replaced by ηB . Then,
η
η∗
∗
∗
ext ext ∗ B A αn,J = αn,J , En,J (η B , ηB ) = En,J (η A , η A )+h(η∗A , η A , J A )−h(η B , ηB , J B ). (7)
Since the above finite-volume relations use the variables in a fixed set of vertices, they naturally extend to the infinite volume. We introduce the joint measure on excited states and excitation energies in all finite subsets: ⎞ ⎛ ⎜ ⎜ Kn# := ⎜ ⎝
A⊂n ˜A η A ,η ∈
δ(α η A , E ext (η A ,η n,J
n,J
A ))
⎟ ⎟ ⎟ νn (d J ). ⎠
(8)
A
A compactness argument implies weak convergence along some subsequence of the measures Kn# to an infinite-volume measure K# that may depend on the choice of the subsequence. This leads to the definition of an excitation metastate for almost all J through conditioning. J Definition 2.3 (Excitation Metastate). An excitation metastate K#J for a realization
ηA of the couplings is a joint distribution on the collection α J , E Jext (η A , ηA ) A⊂H finite , η αJA
˜A η A ,ηA ∈
˜ H and E ext (η A , η ) ∈ R, obtained by conditioning on J a limit K# ∈ where J A of finite-volume measures of the form (8). The index # may be thought of as running over all the finite subsets A. We write K JA for the marginal of K#J on exterior excitation energies and excited states for the set A. We stress that K JA for A = ∅ (or A a singleton site) is simply the metastate on ground states. It is easily checked that the set of measures satisfying the above four properties is closed under taking convex combinations and limits. In particular, the infinite-volume measure K#J satisfies analogous properties.
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η Lemma 2.4. Let K#J be an excitation metastate on α J A , E Jext (η A , ηA ) A⊂H finite . For ˜A η A ,ηA ∈
any finite set A, η
1. with K#J -probability one, α J A is a GSP on H \A for any η A ; ext (η , η ) for any 2. with K#J -probability one, E Jext (η A , ηA ) + E Jext (ηA , ηA ) = En,J A A η A , η A and η A ; 3. K JA does not depend on J A ; η 4. if B ⊂ A, then with K#J -probability one, the variables (α J A , E Jext (η A , ηA ))η A ,ηA ηB ext and (α J , E J (η B , ηB ))η B ,ηB satisfy the equivalent of (7) with the subscript n removed. Using the above properties of the excitation metastate, we can study the GSP appearing in the metastate as a function of finitely many couplings. The basic procedure we use is as follows. Letting A be a finite subset of H , we first sample exterior excitation energies and excited states with K JA . Property 3 guarantees that this can be done independently of the couplings in A; therefore, we think of it as an “exterior” realization. We then use Property 4 to determine the GSP as a function of this realization and the “interior” realization – the couplings inside A. We start with the case where A = {x, y} consists of the endpoints of an edge b = x, y; in this setting we will use A and b interchangeably. There are then two possible ηb ’s (up to a joint spinflip): (+1, +1) and (−1, +1). To lighten notation we write ηb = +b when the spins at x and y have the same sign and ηb = −b when they are opposite. Keeping track of b will be helpful when dealing with more than one edge. The corresponding pairs of configurations sampled b from the excitation metastate will respectively be denoted by α +J b and α − J . η
Proposition 2.5. Let b = x, y be an edge and (α Jb , E Jext (ηb , ηb ))ηb ,ηb =±b be sampled from the excitation metastate KbJ . There exists C bJ ∈ R, independent of Jb , such b b that the GSP α J is α +J b for Jb > C bJ and is α − J for Jb < C J . Precisely, C bJ :=
1 E Jext (+b , −b ). 2
(9)
Proof. From Lemma 2.4, C bJ is independent of Jb . The same lemma allows us to determine α J by taking B = ∅ and A = {x, y} in (7). The values of h(+b , +b , Jb ) and h(−b , −b , Jb ) are both 0 and we have h(+b , −b , Jb ) = −h(−b , +b , Jb ) = −2Jb . This implies that η∗A = +b for Jb > 21 E Jext (+b , −b ), and η∗A = −b for Jb < 1 ext 2 E J (+b , −b ), implying (9). The statement of the proposition is illustrated in Fig. 1. We say that C bJ is the critical b ∗ value of the edge b. The critical contour is the set of dual edges α +J b α − J in H . This
+
−
Fig. 1. The GSP α J from the set {α J b , α J b } as a function of Jb . Jssat is the right-hand side of (10). The shaded region is where Jb is super-satisfied
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contour always goes through b and might be infinite. When Jb crosses C bJ from above b (resp. below) and the GSP α J changes, we say that it flips from α +J b to α − J (resp. from −b +b α J to α J ). We stress that even though the GSP flips when the critical value is crossed, the former minimizing GSP can possibly retain the ground state property (4). Hence α +J b b and α − J might simultaneously be GSP’s for J . The following gives sufficient conditions on Jb for only one of them to be a GSP. Proposition 2.6. If |Jx y | > min
⎧ ⎨ ⎩
z:z,x∈E,z= y
|Jx z |,
z:z,y∈E,z=x
⎫ ⎬
|Jyz | , ⎭
(10)
then αx α y = sgn Jx y for any GSP α. In particular, in the notation of Proposition 2.5, b exactly one of α +J b and α − J is a GSP when Jb satisfies (10). Proof. Whenever Jx y satisfies the inequality, we must have αx α y = sgn Jx y , otherwise (4) is violated either for S = {x} or for S = {y}. When the condition (10) holds, we say that the coupling Jx y is super-satisfied. The reader can verify that a dual edge whose coupling is super-satisfied cannot be in an interface between two GSP’s. Furthermore, such an edge cannot be in the critical contour of another edge. We shall later need the analogue of Proposition 2.5 for modifications of two couplings to prove the existence of tethered domain walls. Let b and e be two edges in H . We take A to be the set of vertices which are endpoints of b or e. We write ηb or ηe for the spin configuration up to a joint spinflip at the endpoints of the edge. Using the notation introduced in the one edge case, we have ηb = ±b , ηe = ±e . The excitation energies for two different configurations ηb , ηe and ηb , ηe on b and e then reads E Jext (ηb , ηe ; ηb , ηe ). We set for convenience C1 := 21 E Jext (+b , +e ; −b , +e ), C2 := 21 E Jext (+b , −e ; −b , −e ), C3 := 21 E Jext (+b , +e ; +b , −e ) and C4 := 21 E Jext (−b , +e ; −b , −e ). Throughout the paper, it is understood that the Ci ’s depend on J . By items 2 and 3 in Lemma 2.4 it is easily checked that C1 − C2 = C3 − C4 and that the Ci ’s are independent of Jb and Je . Moreover, taking A to be the endpoints of b or e and B to be the endpoints of just e, b ,+e we have, again by Lemma 2.4, that α +J e is chosen from α +J b ,+e and α − . Applying the J +b ,+e +e if Jb > C1 same argument as in the proof of Proposition 2.5, it follows that α J is α J b ,+e and is α − if J < C . Similarly, C (resp. C , C ) is the critical value for the state b 1 2 3 4 J +b −b −e α J (resp. α J , α J ). We can now use these four values to describe the GSP α J as a function of Jb and Je . η ,η
Proposition 2.7. Let b and e be edges of H and let (α Jb e , E Jext (ηb , ηe ; ηb , ηe ))ηb ,ηb =±b ;ηe ,ηe =±e be sampled from the excitation metastate K#J . There exists a criti-
2 cal set C b,e J ⊂ R independent of Jb and Je such that the GSP α J is constant for (Jb , Je ) in each of the four connected components of the complement of C b,e J (see Fig. 2). Moreover, C b,e is the union of straight lines and is one of three types determined as J follows:
• If C1 = C2 , then C b,e J is the union of the two lines {(C 1 , Je ) : Je ∈ R} and {(Jb , C 3 ) : Jb ∈ R}.
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η ,η
Fig. 2. The GSP α J (from the set {α Jb e : ηb = ±1, ηe = ±1}) as a function of (Jb , Je ). The cases C1 = C2 , C1 > C2 and C1 < C2 are depicted from left to right. The thick lines form the critical set C b,e J
• If C1 > C2 , then C b,e J is the union of the four rays {(Jb , C 3 ) : Jb > C 1 }, {(Jb , C 4 ) : Jb < C2 }, {(C1 , Je ) : Je > C3 } and {(C2 , Je ) : Je < C4 } and the line segment {(Jb , Je ) : Jb − Je = C1 − C3 for C2 < Jb < C1 , C4 < Je < C3 }. • If C1 < C2 , then C b,e J is the union of the four rays {(Jb , C 3 ) : Jb > C 2 }, {(Jb , C 4 ) : Jb < C1 }, {(C1 , Je ) : Je > C4 } and {(C2 , Je ) : Je < C3 } and the line segment {(Jb , Je ) : Jb + Je = C1 + C4 for C1 < Jb < C2 , C3 < Je < C4 }. Remark 2.8. The result implies that the region where Jb is between C1 and C2 corresponds exactly to the values of Jb for which b is in the critical contour of e, and vice-versa for the region where Je is between C3 and C4 . In particular, the middle square in the diagrams of Fig. 2 when C1 = C2 gives the values of (Jb , Je ) for which b and e share the same critical contour. In all three cases, C b,e J is the union of the two critical lines given by the graphs of C bJ as a function of Je and C eJ as a function of Jb . Proof. We prove the statement in the case C1 > C2 . The other cases are done the same way. It suffices to compute the value of C bJ as a function of Je and C eJ as a function of Jb . For a fixed Je , we know from Proposition 2.5 that the GSP α J takes the value α +J b ηb ηb ,ηe b or α − are known J depending on Jb . Since the exact values of Je for which α J = α J b from the Ci ’s computed above, C J can be derived explicitly. For instance, in the region −b ,+e b C4 < Je < C3 we have that α − and α +J b = α +J b ,−e . Therefore, J = αJ C bJ =
1 1 ext E J (+b , −e ; −b , +e ) + 2Je = Je + C1 − C3 , E Jext (+b , −b ) = 2 2
where we used item 4 in Lemma 2.4 for the second equality and item 2 for the third equality. A similar argument in the region Je > C3 and Je < C4 yields C bJ = C1 and e C bJ = C2 respectively. The picture for C b,e J is completed by computing C J as a function of Jb the same way. The excitation metastate is useful when investigating the interface between two GSP’s because questions about interfaces translate into questions about critical values. Let us write μ for the product measure on two excitation metastates joint with the distribution of the couplings:
μ := (K#J × K J# )ν(d J ).
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We use the notation α J for the states sampled from K J and β J for the states sampled from KJ . We denote by C eJ (α) and C eJ (β) the respective critical values for an edge e. To illustrate the connection between interfaces and critical values, consider the event that a fixed edge e belongs to the interface of two GSP’s α J and β J . Suppose that this occurs with positive probability when α J and β J are sampled from the two excitation metastates. In other words, suppose that μ(e ∈ α J β J ) = ν K#J × K# (11) J (e ∈ α J β J ) > 0. Using Proposition 2.5, we can show that the above is equivalent to the statement that C eJ (α) = C eJ (β) with positive probability:
(12) μ C eJ (α) = C eJ (β) > 0. This is intuitively clear from Fig. 1 and can be made precise as follows. We write the inner probability in (11) by first conditioning on the critical values and on the states that the GSP’s take as Je varies, which are the same as the ηe -excited states: ηe ηe e e e ∈ α β . μ(e ∈ α J β J ) = μ K#J × K# J J (α J , β J )ηe =±e , C J (α), C J (β) J (13) The inner conditioning essentially pins down two specific pictures of the form in Fig. 1 (one for α J and one for β J ). This conditional probability is a priori a function of (a) the coupling configuration J and (b) the choices for critical values and excited states. However, for almost every fixed choice of critical values and excited states, it can be viewed simply as a function of Je which is defined for all values of Je = C eJ (α), C eJ (β). By Proposition 2.5, it is equal to 1 when Je is between C eJ (α) and C eJ (β) and equal to 0 otherwise. Also, by Lemma 2.4, Je is independent of the variables on which we condition. The probability (13) can thus be computed by performing the integral over Je before the integral over all other variables. The result will be non-zero if and only if the critical values C eJ (α) and C eJ (β) differ with positive probability. This shows the equivalence of (11) and (12). The following corollary is another example where the connection between interfaces and critical values is fruitful in proving results about interfaces. It shows that a nonempty interface α J β J can always be modified to pass through a given edge. This is trivial in the plane, but not so in the half-plane where edges at different distances to the x-axis are not equivalent up to translation. Corollary 2.9. μ (α J β J = ∅) > 0 ⇐⇒ for any fixed edge b in H , μ (b ∈ α J β J ) > 0. Proof. The backward implication is obvious, so we prove the forward one and assume that the probability on the left is positive. We will first show that b μ α +J b α − (14) J ∩ α J β J = ∅ > 0. To do this, we consider a realization of the interface α J β J and make a coupling modification to force (14) to occur. Denote by x, y the edge dual to b. Let P be a path in the dual upper half-plane which connects x to a dual vertex in α J β J . Let ∂ P be the set of dual edges which are not in P but which have at least one endpoint in P. We
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Fig. 3. An illustration of the coupling modification argument used in the proof of Corollary 2.9. All edges are in the dual lattice. The edges crossed with diagonal line segments are super-satisfied. Since the critical contour of b (not pictured) cannot contain any crossed edges, it must intersect α J β J
now “super-satisfy” (cf. Proposition 2.6 and the discussion immediately following it) all dual edges in ∂ P which are not in the interface α J β J – see Fig. 3. (A small amount of care is needed in the choice of P to perform this procedure, namely that the set of lattice edges dual to those in ∂ P\α β cannot contain a circuit.) Precisely, notice that since these dual edges are not in the interface, we can modify their couplings one by one, away from their respective critical values in both states, beyond the super-satisfied threshold (10). Consequently neither α J nor β J will flip. By construction, the critical contour of b in each state α J and β J cannot contain any of these super-satisfied edges but it must contain b. Since the connected components of these contours which contain b are either loops or doubly-infinite paths, this means that they must intersect the interface. Therefore (14) holds. From (14), there exists a fixed edge e such that b μ e ∈ α +J b α − J ∩ α J β J > 0.
(15)
If e = b, the corollary is proved, so assume that e = b. Consider the excitation metastates conditioned on the critical sets C b,e J as well as the (ηb , ηe )-excited states: ηb ,ηe ηb ,ηe +b −b b,e , β J )ηb =±b ,ηe =±e , C b,e (α), C (β) . K#J ×K# J e ∈ α J α J ∩ α J β J (α J J J (16) We will now proceed like in the proof of the equivalence of (11) and (12). By Proposition 2.7 applied both to α J and to β J and with the help of Fig. 2, when we fix values for the critical sets and the excited states, we can view this conditional probability as a b,e function of the pair (Jb , Je ) except on the critical sets C b,e J (α) and C J (β). It is equal to 1 when (Jb , Je ) is in the region where both i) Je is between C3 (α J ) and C4 (α J ), since for those values, e is in the critical contour of b for the state α J (cf. Remark 2.8); and ii) Je is between C eJ (α) and C eJ (β), since then e ∈ α J β J . We shall temporarily overload notation and write C bJ for either α J or β J to denote the graph of C bJ as a function of Je
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and C eJ for the graph of C eJ as a function of Jb . If these two graphs coincide, we simply write C bJ = C eJ . We now write (15) as an expectation of the conditional probability (16). By independence between the pair (Jb , Je ) and the variables on which we condition, we may perform integration first over Jb and Je . If C3 (α J ) = C4 (α J ), the region for (Jb , Je ) described by i) and ii) has zero measure so it does not contribute to the expectation (15). Therefore the non-trivial contribution to the expectation comes from the realizations where C3 (α J ) = C4 (α J ). We claim then that in the region for (Jb , Je ) where i) is satisfied, C bJ (α) = C bJ (β) ⇒ C eJ (α) = C eJ (β).
(17)
To see this, note that in this region C eJ (α) coincides with C bJ (α) (they are both equal to a diagonal line segment). Now if C bJ (α) = C bJ (β), this further implies that C bJ (β) contains this same diagonal line segment, and therefore C bJ (β) = C eJ (β) in this region. The sequence of equalities yields that C eJ (α) = C eJ (β) and (17) is proved. However, the conclusion of (17) implies that the region described by i) and ii) has zero measure. The full expectation would then be zero, contradicting (15). We conclude that there must be a region of positive measure of the (Jb , Je )-plane for which ηb ,ηe ηb ,ηe b,e b,e b b (α C K#J × K# (α) = C (β) , β ) , C (α), C (β) ηb =±b ,ηe =±e J J J J J is non-zero. Integrating this first
with respect to Jb and Je and then with respect to the other variables, we see that μ C bJ (α) = C bJ (β) > 0, and the corollary follows after observing the equivalence of (11) and (12).
2.4. Uniqueness of the domain wall in the full plane. The framework of the excitation metastate developed in the last section did not rely heavily on the choice of the finitevolume measure Kn# in Eq. (8) or on the choice of underlying graph H . In fact, one only needs the four properties of the excitation metastate stated in Lemma 2.4. As we mentioned before, these are preserved when taking limits and convex combinations. These properties are in particular fulfilled when one constructs the excitation metastate in the full plane Z2 from empirical measures as in [16,17]. The proof in [16,17] that the interface between two GSP’s in Z2 contains at most one domain wall is based on the results derived in the last section from these properties. We thus may state a more general version for use in the proof of Theorem 1.1. Theorem 2.10 (Newman-Stein). Let ν be the law of J , iid couplings on the edges of Z2 , and let μ be a probability measure on J, (α η A , E ext (η A , ηA )) A,η A ,ηA , (β η A , ˜ A. E ext (η A , ηA )) A,η A ,ηA , where A runs over all finite subsets of Z2 and η A , ηA ∈ Suppose that the marginal of μ on J is ν and that μ is translation-invariant. Moreover, suppose that the conditional measure of μ given J on excited states α η A and β η A and corresponding exterior excitation energies satisfies the properties of Lemma 2.4 in Z2 . Then μ (α J β J = ∅ or α J β J is connected) = 1.
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3. Proofs 3.1. Tethered domain walls. We start with a general lemma about translation-invariant measures. Lemma 3.1. Suppose that μ is a measure on spin configurations in Z2 (or H ) which is invariant under horizontal translations. Let A be an event and for any x ∈ Z, let A x be the event horizontally translated by x. Then with probability one, either A x does not occur for any x or A x occurs for a set of x which has positive density in Z. Proof. Let μˆ be an ergodic component of the measure μ. If μ(A ˆ 0 ) > 0 then the ergodic theorem gives that with μ-probability ˆ one, the set of x such that A x occurs has positive density in Z (in fact, it has density μ(A ˆ 0 )). If, on the other hand, μ(A ˆ 0 ) = 0, then with μ-probability ˆ one, no A x will occur. This shows that the lemma holds if we replace μ by μ. ˆ Since this is true for each ergodic component, the lemma follows for μ. For this section, α J and β J will refer to two GSP’s in H , sampled independently from excitation metastates K#J and K J# for the same coupling configuration J . Recall the definition of μ as the joint distribution of (α, β, J ),
μ = (K#J × K J# )ν(d J ).
(18)
Note that the measure μ is horizontally translation-invariant (because of our choice of boundary conditions in constructing K#J and K J# ). Definition 3.2. A tethered domain wall is a domain wall which intersects the dual x-axis. Proposition 3.3. Suppose that μ(α J β J = ∅) > 0. Then the interface α J β J contains infinitely many tethered domain walls with positive μ-probability. Proof. By Corollary 2.9, with positive μ-probability we can find a dual edge incident to the dual x-axis which is in α J β J and therefore there exists at least one tethered domain wall. Using Lemma 3.1 with the event A = {the point (0, −1/2) is contained in a tethered domain wall}, we see that infinitely many points of the dual x-axis lie in tethered domain walls. We claim that these points must each lie in distinct domain walls. The assertion that there are infinitely many tethered domain walls will follow. If two of these points lie in the same domain wall then there must exist a path of dual edges connecting two of them which lies entirely in α J β J . But this is impossible since this path must have strictly negative energy in one of α J or β J , contradicting (4) (it cannot be zero since the coupling distribution is assumed continuous). We now investigate density properties of the tethered domain walls. For n ≥ 1 and k ≥ 0, define the set In,k = {(x, y) ∈ R2 : x ∈ [−n, n] and y = k − 1/2}, and let Nn,k be the number of distinct tethered domain walls intersecting In,k . Write Eμ for expectation w.r.t. μ.
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Proposition 3.4. The sequence (Eμ (Nn,k ))∞ n=1 is subadditive for fixed k ≥ 0. Therefore, lim Eμ (Nn,k )/n exists.
(19)
n→∞
Furthermore, under the assumption that μ(α J β J = ∅) > 0, there exists c > 0 such that for all n ≥ 1 and k ≥ 0, Eμ (Nn,k ) ≥ cn.
(20)
Proof. Subadditivity is a straightforward consequence of translation invariance of μ. Therefore we focus on showing (20). To this end, we first show that there exists a deterministic c1 > 0 such that with positive μ-probability, the following holds for all k ≥ 0: lim inf Nn,k /n > c1 .
(21)
n→∞
For the case k = 0, let α J β J be a domain wall configuration which has tethered domain walls. Using Lemma 3.1 with A as the event that the dual vertex (1/2, −1/2) is in a tethered domain wall, we see that the set of x ∈ Z such that (x + 1/2, −1/2) is in a tethered domain wall has positive density in Z. Each such dual vertex is in exactly one tethered domain wall by (3). Therefore there exists c2 > 0 (random) such that limn→∞ Nn,0 /n = c2 . By possibly decreasing c2 for some configurations, we may find a deterministic c1 > 0 such that with positive μ-probability, lim inf Nn,0 /n = lim Nn,0 /n > c1 . n→∞
n→∞
For the rest of the proof of (21), restrict α J β J to be a domain wall configuration for which this holds. By writing Nn,k /n = (Nn,k − Nn,0 )/n + Nn,0 /n, the relation (21) will hold in α J β J for all k ≥ 0 if we show that for all n ≥ 1, Nn,k − Nn,0 ≥ −2k.
(22)
To this end, notice that Nn,0 − Nn,k is no bigger than the number of tethered domain walls which intersect In,0 but do not intersect In,k . We estimate this number. Such a tethered domain wall must originate in the set In,0 and cannot intersect the top side of the box [−n, n] × [−1, k − 1] (i.e., the set [−n, n] × {k − 1}). Since the domain wall must leave this box, it must leave on either the left or right side. Therefore there exists some integer m ∈ [1, k − 1] and a dual vertex of the form (−n − 1/2, m − 3/2) or of the form (n + 1/2, m − 3/2) such that this vertex is in the tethered domain wall. However, as noted before, each dual vertex can be contained in at most one tethered domain wall. Therefore, the number of such tethered domain walls is at most 2k. This shows (22) and completes the proof of (21). We resume the proof of (20). It follows from (21) that there is a c3 > 0 such that for all k ≥ 0, lim Eμ (Nn,k /n) > c3 .
(23)
n→∞
Indeed, we use Fatou’s Lemma and (19) to see that lim Eμ (Nn,k /n) = lim inf Eμ (Nn,k /n) ≥ Eμ (lim inf Nn,k /n),
n→∞
n→∞
n→∞
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which is bounded away from 0 independently of k by (21). But now, subadditivity implies that for any k ≥ 0, Eμ (Nn,k )/n ≥ inf Eμ (Nn,k )/n = lim Eμ (Nn,k /n) > c3 . n→∞
n≥1
This proves (20). 3.2. Restoring vertical translation invariance. In the second half of the proof of Theorem 2.2, we create and study a measure μ∗ on excited states and excitation energies on all of Z2 . For this, let μ be the measure defined by (18). For any integer k ≥ 0 we define the shifted half-plane whose vertices are Hk = {(x, y − k) : (x, y) ∈ H }, and whose edges are defined similarly as those of H (see Sect. 1). We define the shifted measure μ(k) simply as the push-forward measure of μ through the map which translates the origin to the vertex (0, −k). Each μ(k) is a measure on coupling configurations for the edges of Hk and corresponding pairs of excited states and excitation energies for the finite subsets of Hk . We then define μ∗ (k) =
1 μ(i). k+1 k
(24)
i=0
By tightness properties (see, e.g., Sect. 2.3 and [17]) there is an increasing sequence (n k ) such that (μ∗ (n k )) converges to some measure μ∗ . This is a measure on coufor edges of Z2 and pairs of corresponding excitations, each of the form
plings ηA α J , E Jext (η A , ηA ) A⊂Z2 finite . It is easy to see that it is invariant under vertical as ˜A η A ,ηA ∈
well as horizontal translations in Z2 . Our aim is to use this measure to derive a contradiction from the assumption that there are multiple GSP’s for the half-plane - i.e., from μ(α J β J = ∅) > 0. The proposition below says that ground state interfaces produced from μ∗ would contain more than one domain wall. This fact along with Theorem 2.10 will give a contradiction. We denote by α ∗J and β ∗J the full-plane GSP’s sampled from μ∗ . Proposition 3.5. Suppose that μ(α J β J = ∅) > 0. Then with positive μ∗ -probability, the interface α ∗J β ∗J contains at least two domain walls. Remark 3.6. The proof below can easily be modified to show that with positive probability, α ∗J β ∗J contains infinitely many domain walls. However, since a consequence of Theorem 2.10 is that there is at most one domain wall, we need only show that there are at least two. Proof. Let An,k be the event that at least two tethered domain walls intersect the box [−n, n]×[k −n, k +n]. (We remark that for n > k this box extends below the half-space, but that does not affect our argument.) We first note that there exists n 0 such that for n > n0, μ(An,k ) > 0 uniformly in k ≥ 0.
(25)
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This follows from Proposition 3.4. To see this, we choose c > 0 from (20) such that for all k ≥ 0 and n ≥ 1, Eμ (Nn,k ) ≥ cn. We pick n 0 = 2/c to give Eμ (Nn 0 ,k ) ≥ n 0 c ≥ 2, from which (25) follows because
μ An 0 ,k ≥ μ Nn 0 ,k
Nn 0 ,k − 1 1 . ≥ 2 ≥ Eμ 1{Nn0 ,k ≥2} ≥ 2n 0 − 1 2n 0 − 1
We now finish the proof of Proposition 3.5. From (25), choose a, N > 0 such that for all k ≥ 0, we have μ(A N ,k ) ≥ a. For 0 < n < m and k ≥ 0, define Bm,n,k as the event that there are two dual vertices in the box [−n, n] × [k − n, k + n] which are in domain walls but such that there is no path connecting them, in the box [−m, m]×[k −m, k +m], which consists of only dual edges in domain walls. Since tethered domain walls do not intersect, we have Bm,n,k ⊃ An,k , and so (25) implies that for all M ≥ N and for all k ≥ 0, we have μ(B M,N ,k ) ≥ a. By construction of the measure μ∗ (recall the definition as the limit of (24)), we see that for any M ≥ N , μ∗ (B M,N ,0 ) ≥ a. By definition, the events Bm,n,k are decreasing in m, i.e., Bm+1,n,k ⊂ Bm,n,k . In particular, for n = N , B∞,n,k := limm→∞ Bm,n,k = ∩∞ m>n Bm,n,k satisfies μ∗ (B∞,N ,0 ) ≥ a.
(26)
We note that for any n, k, lim
m→∞
Bm,n,k ∩ {α ∗J β ∗J is connected} = ∅.
Hence, μ∗ (B∞,n,k ∩{α ∗J β ∗J is connected}) = μ∗
lim (Bm,n,k ∩{α ∗J β ∗J is connected}) = 0.
m→∞
(27) Combining (26) and (27), μ∗ ({α ∗J β ∗J is not connected}) ≥ μ∗ (B∞,N ,0 ∩ {α ∗J β ∗J is not connected}) = μ∗ (B∞,N ,0 ) ≥ a. This completes the proof. We can now use properties of the measure μ∗ to prove the main result, Theorem 2.2. Proof of Theorem 2.2. We first verify that the measure μ∗ satisfies the hypotheses of Theorem 2.10. By construction, it is easily seen to be translation invariant for vertical 2 . The conditional measure of μ∗ given J is a as well as for horizontal translations
η A of Z ext measure on pairs of excitations α J , E J (η A , ηA ) A,η ,η for all A ⊂ Z2 finite and A
A
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˜ A . It satisfies the four properties of Lemma 2.4. Indeed, these are satisfied by η A , ηA ∈ the conditional measure of μ∗ (k) for each k, since they are satisfied by the conditional measure of μ. This is so for the latter because it was contructed from finite-volume measures, each of which satisfied the properties, and they are carried over in the infinite-volume limit. By Theorem 2.10, the measure μ∗ is supported on pairs of GSP’s whose interface has at most one domain wall. This stands in contradiction to the result of Proposition 3.5, completing the proof. Acknowledgements. The research reported here was supported in part by NSF grants DMS-0604869 and OISE-0730136 and an NSF postdoctoral fellowship to M. Damron. L.-P. Arguin is grateful for the financial support and hospitality of Anton Bovier and the Hausdorff Center for Mathematics in Bonn during part of this work. All the authors thank the Centre de Recherches Mathématiques at the Université de Montréal for its hospitality during June 2009 when some of the work reported here was done during the workshop, Disordered Systems: Spin Glasses.
References 1. Aizenman, M., Wehr, J.: Rounding Effects of Quenched Randomness on First-Order Phase Transitions. Commun. Math. Phys. 130, 489–528 (1990) 2. Binder, K., Young, A.P.: Spin glasses: experimental facts, theoretical concepts, and open questions. Rev. Mod. Phys. 58, 801–976 (1986) 3. Edwards, S., Anderson, P.W.: Theory of spin glasses. J. Phys. F 5, 965–974 (1975) 4. Jackson, T.S., Read, N.: Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model. Phys. Rev. E 81, 021130-1–021130-16 (2010) 5. Jackson, T.S., Read, N.: Theory of minimum spanning trees. II. Exact graphical methods and perturbation expansion at the percolation threshold. Phys. Rev. E 81, 021131-1–021131-31 (2010) 6. Loebl, M.: Ground state incongruence in 2D spin glasses revisited. Elect. J. Comb. 11, R40 (2004) 7. Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. Singapore: World Scientific, 1987 8. Middleton, A.A.: Numerical investigation of the thermodynamic limit for ground states in models with quenched disorder. Phys. Rev. Lett. 83, 1672–1675 (1999) 9. Newman, C.: Topics in Disordered Systems. Birkhaüser, Basel, 1997 10. Newman, C.M., Stein, D.L.: Spin-glass model with dimension-dependent ground state multiplicity. Phys. Rev. Lett. 72, 2286–2289 (1994) 11. Newman, C.M., Stein, D.L.: Ground state structure in a highly disordered spin glass model. J. Stat. Phys. 82, 1113–1132 (1996) 12. Newman, C.M., Stein, D.L.: Spatial inhomogeneity and thermodynamic chaos. Phys. Rev. Lett. 76, 4821– 4824 (1996) 13. Newman, C.M., Stein, D.L.: Metastate approach to thermodynamic chaos. Phys. Rev. E 55, 5194–5211 (1997) 14. Newman, C.M., Stein, D.L.: Thermodynamic chaos and the structure of short-range spin glasses. In: Mathematics of Spin Glasses and Neural Networks, ed. Bovier, A., Picco, P., Boston: Birkhäuser, 1997, pp. 243–287 15. Newman, C.M., Stein, D.L.: Simplicity of state and overlap structure in finite volume realistic spin glasses. Phys. Rev. E 57, 1356–1366 (1998) 16. Newman, C.M., Stein, D.L.: Nature of ground state incongruence in two-dimensional spin glasses. Phys. Rev. Lett. 84, 3966–3969 (2000) 17. Newman, C.M., Stein, D.L.: Are there incongruent ground states in 2D Edwards-Anderson spin glasses? Commun. Math. Phys. 224(1), 205–218 (2001) 18. Newman, C.M., Stein, D.L.: Topical Review: Ordering and Broken Symmetry in Short-Ranged Spin Glasses. J. Phys.: Cond. Mat. 15, R1319–R1364 (2003) 19. Palassini, M., Young, A.P.: Evidence for a trivial ground-state structure in the two-dimensional Ising spin glass. Phys. Rev. B 60, R9919–R9922 (1999) 20. Sherrington, D., Kirkpatrick, S.: Solvable model of a spin glass. Phys. Rev. Lett. 35, 1792–1796 (1975) Communicated by H. Spohn
Commun. Math. Phys. 300, 659–671 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1124-6
Communications in
Mathematical Physics
Anderson Localization for a Supersymmetric Sigma Model M. Disertori1 , T. Spencer2 1 Laboratoire de Mathématiques Raphaël Salem, UMR CNRS 6085, Université de Rouen,
76801 St. Etienne du Rouvray, France. E-mail: [email protected]
2 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA. E-mail: [email protected]
Received: 1 December 2009 / Accepted: 26 May 2010 Published online: 16 September 2010 – © Springer-Verlag 2010
Abstract: We study a lattice sigma model which is expected to reflect the Anderson localization and delocalization transition for real symmetric band matrices in 3D. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. The existence of a diffusive phase in 3 dimensions was proved in Disertori et al. (Commun. Math. Phys., doi:10.1007/s00220-010-1117-5, 2009) [2] for low temperatures. Here we prove localization at high temperatures for any dimension d ≥ 1. Our analysis uses Ward identities coming from internal supersymmetry.
1. Introduction It is well known that the study of localization properties in a disordered material can be translated to the study of correlation functions in a lattice field theory, with an internal hyperbolic supersymmetry (SUSY), [5,6,8,9]. In the physics literature one usually assumes the sigma model approximation, which is believed to capture the essential features of the energy correlations and transport properties of the underlying quantum system. The SUSY field theories which are equivalent to the Anderson tight-binding model and random band matrices are difficult to analyze with mathematical rigor in more than one dimension. In this context, Zirnbauer introduced a lattice field model which may be thought of as a simplified version of one of Efetov’s nonlinear sigma models [3,10]. In Zirnbauer’s sigma model the field takes values in a target space H (2|2) which is a supermanifold extension of the hyperbolic plane. The model is expected to reflect the spectral properties of random band matrices, such as localization and diffusion, in any dimension. In [10] localization was established in a one dimensional chain by analyzing the transfer matrix. We refer to [2] for a historical introduction and motivations. More recently the existence of a ‘diffusive’ phase at low temperatures (β large) has been proved for the H (2|2) model in three or more dimensions, see [2]. For β small, a
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localized phase was expected. However, unlike conventional statistical mechanics models, the H (2|2) model has a noncompact hyperbolic symmetry and so high temperature expansions cannot be done in the usual way. In fact, it is known that the bosonic hyperbolic sigma model in 3D has no localized phase because its effective action is convex for all β > 0. On the other hand, numerical simulations [4] indicated that the SUSY hyperbolic sigma model has a phase transition for β < βc 0.038. In this paper we show that for any dimension d > 1 the H (2|2) model exhibits localization for β 1/2 ln β −1 ≤ 1/(2d − 1). Thus the sigma model approximation captures the physics of both localization and diffusion. Moreover, for a one dimensional chain we recover localization for all values of β. Localization is also expected in 2D (see [2] Sect. 1.4 and 4.3), for all values of β by both the renormalization group and by a simple saddle analysis. However, a rigorous proof is still missing for this case. The techniques employed in this work to prove localization are quite different from the ones used in [2] to prove extended states. The two papers can be read independently. The only common point is the use of supersymmetry to prove some identities. In the present case supersymmetry is applied only to prove that the partition function is normalized to 1. We refer to Sect. 4 and Appendix C in [2] for an introduction to supersymmetric Ward identities. 1.1. The model. Let be a finite subset in Zd and t j a real variable for each site j ∈ . We will consider periodic or Neumann boundary conditions. Accordingly for any two points x, y ∈ , |x − y| will denote the Euclidean distance on the lattice (Neumann bc) or the periodized lattice (periodic bc). We introduce the probability measure dt j ε ε (t) , dμε (t) = det D (1.1) √ e−F (∇t) e−M (t) × 2π j∈ where dt j is the Lebesgue measure, F is the kinetic part and M is the mass term: (cosh(t j − t j ) − 1), F (∇t) = β ( j j )∈
ε M (t)
=
(1.2)
ε j (cosh t j − 1).
j∈
We denoted by ( j j ) the nearest neighbor pairs | j matrix defined by ε ) = 0, (D ij ε) (D ij ε (D ) j j
ε (t) is a positive definite − j | = 1. D
|i − j| > 1,
= −β, |i − j| = 1, −t = β 2d + V j + ε j e j , i = j, etk −t j − 1 , Vj =
(1.3)
(1.4)
k,( jk)
and ε j ≥ 0 are regularizing parameters that are necessary to make the integral well defined. We remark that D looks like a random Schrödinger operator, ε D = −β + β Vˆ + εe−tˆ,
(1.5)
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where is the discrete Laplacian on (with suitable boundary conditions) and Vˆ +εe−tˆ is a local (t dependent) potential. The main difference is that the random variables V j ε ≥ 0 as a quadratic form. are strongly correlated instead of independent and D Now, for any function f (t) we will define its average by (1.6) f (t) = dμε (t) f (t). ε et corresponds to the symbol D Remark. Note that et D β, introduced in [2] Eq. (1.1). Let us set ⎧ |i − j| > 1 ⎨0 ti +t j |i − j| = 1 , Aiεj = eti Diεj et j = −βe (1.7) ⎩ β eti +ti + ε eti i = j i i ,(ii )
where A is the generator of a random walk in a random environment in [2] Eq. (1.1). For technical reasons the above representation is more convenient when we want to prove diffusion as in [2] while the other is more practical when we study localization (except in the proof of Theorem 2 where we will go back to the “diffusion” representation for a while). Origin of the model. In order to relate (1.1) to a hyperbolic, supersymmetric sigma model, we introduce a vector u j at each lattice point j ∈ with three bosonic components and two fermionic components u j = z j, x j, yj, ξj, ηj , (1.8) where ξ , η are odd elements and z, x y are even elements of a real Grassmann algebra (see [2] for more details). The scalar product is defined by (u, u ) = −zz + x x + yy + ξ η − ηξ ,
(u, u) = −z 2 + x 2 + y 2 + 2ξ η
(1.9)
and the action is S[u] =
1 2
β(u j − u j , u j − u j ) +
( j, j )∈
ε j (z j − 1).
(1.10)
j∈
Finally, the sigma model constraint is (u j , u j ) = −1
∀ j ∈ ⇒ z j = ± 1 + x 2j + y 2j + 2ξ j η j .
(1.11)
We choose the manifold corresponding to the + sign, so now in the action z j − 1 ≥ 0 for each j. Note that z is not really a number, but rather an even element of a Grassmann algebra so it may have fermionic components. We introduce horospherical coordinates −t 1 2 ¯ ¯ η = ψ, s + ψψ , y = s, ξ = ψ, x = sinh t − e (1.12) 2 ¯ ψ are odd elements of a real Grassmann algebra. where t and s are even elements and ψ, Then 1 2 ¯ s + ψψ (1.13) z = cosh t + e−t 2
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and the action becomes ¯ = S[t, s, ψ, ψ]
β(cosh(ti − t j ) − 1) +
(i j)∈
ε j (cosh t j − 1)
j∈
1 ¯ D ε ψ], (1.14) + [s; D ε s] + [ψ; 2
where D was defined in (1.3) above and [ f ; Dg] =: i, j∈ f i Di j f j . Finally integrating over the two Grassmann variables ψ ψ¯ and the variables s ∈ R we end up with the effective bosonic field theory defined in (1.1). Note that the change of coordinates (1.12) is different from the one introduced in Eq. (2.7) of [2]. There the covariance of the gaussian fields in the action is (1.7) (“diffusion” representation) and the measure (see (2.12) in [2] ) has a e−t factor. Here the covariance is (1.3) (“random Schrödinger” representation) and the factor e−t in the measure disappears. We can go back from (1.12) to (2.7) of [2] by performing the ¯ +t . change of coordinates: s → se+t , ψ → ψe+t , ψ¯ → ψe Symmetries of the model. It is clear by (1.10) that the action S[u] is invariant under transformations that do not modify the scalar product and the z component. These trasformations may mix the bosonic and fermionic components of the vector (supersymmetry). As a consequence, the final model (1.1) satisfies a large number of Ward identities. These are not transparent in (1.1) since the Grassmann variables have been integrated out. We refer to [2] for a more detailed exposition. In this work we need only the following result on the normalization. Normalization and choice of ε j . By internal supersymmetry (see [2] Sect. 4 and Eq. (5.1)) this measure is already normalized to 1 so the partition function is ε = dμε (t) = 1. (1.15) Z This identity is true regardless of the boundary conditions and the values of β or ε j as long as the integral is well defined. Since we consider β > 0 fixed we only need ε j to be non zero at one lattice point. In the following we will consider three cases. 1 for all j ∈ . The measure is translation invariant 1. Uniform pinning: ε j = ε ≤ || with periodic bc. The correlation function in this case has a divergent prefactor 1/ε in the localized regime. 2. Two pinnings: εx = ε y > 0 and ε j = 0 for all other points. This is the analog of inserting two electrical contacts in a metal sample. 3. One pinning at j = 0: ε0 > 0 and ε j = 0 for all other points. This is more suitable for an interpretation of the model as a random walk in a random environment. Our results suggest that the edge reinforced random walk (see [7]) will also localize when the reinforcement is strong. The observable. We will study the correlation function G x y := Dx−1 y,
(1.16)
where x, y can be any two points on the lattice such that both εx > 0 and ε y > 0. This observable does not give information on localization properties in the case of one pinning point. In such case a good observable to study is O j = e+t j /2 ,
(1.17)
Anderson Localization for a Supersymmetric Sigma Model
663 1/4
where j is any point in the lattice. This observable is analogous to xe of [7] where e is an edge ( j, j ).
in the notation
1.2. Main results.
Theorem 1. Let εx > 0, ε y > 0 and j∈ ε j ≤ 1. Then for all 0 < β < βc (βc defined below) the correlation function G x y (1.16) decays exponentially with the distance |x −y|. More precisely: |x−y| β(cd −1) G x y ≤ C0 εx−1 + ε−1 I e c , (1.18) β d y where cd = 2d − 1, C0 is a constant and ∞ dt −β(cosh t−1) . Iβ = β √ e 2π −∞ Finally βc is defined by: Iβ eβ(cd −1) cd < Iβc eβc (cd −1) cd = 1 ∀β < βc .
(1.19)
(1.20)
Our estimates hold uniformly in the volume. Remark 1. The integral Iβ is monotone (increasing) in β and satisfies Iβ < 1 ∀β > 0: ∞ ∞ β β Iβ = dt e−β(cosh t−1) < dt cosh(t/2) e−β(cosh t−1) = 1. 2π −∞ 2π −∞ (1.21) More precisely
Iβ ≤
√ (ln β −1 ) β
β < 0.15
ce
β >> 1
− β1
,
(1.22)
where c > 1 is a constant. Remark 2. The constraints εx > 0 and ε y > 0 exclude the case of one pinning. More 1 over j∈ ε j ≤ 1 implies ε ≤ || when j = ε is constant. The case of one pinning is covered by Theorem 2 below. Main consequence. For d = 1 the critical beta is βc = ∞ since Iβ eβ(cd −1) cd = Iβ < 1 ∀β > 0.
(1.23)
Therefore the correlation function decays exponentially for all values of β. On the other hand for d > 1 we obtain localization only for small β since βc < (2d − 1)−2 < 1. Theorem 2. Let ε0 = O(1) and ε j = 0 ∀ j = 0. Then for all 0 < β < βc (βc defined below), the field tx wants to be as negative as −|x|. More precisely Ox (defined in (1.17)) decays exponentially with the distance |x|, |x| Ox ≤ C0 Iβ eβ(cd −1) cd , (1.24) where cd , Iβ and βc are defined in Theorem 1 above and C0 is a constant. Our estimates hold uniformly in the volume.
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M. Disertori, T. Spencer
Main consequence. For d = 1 the critical beta is βc = ∞ (see (1.23) above), therefore Ox decays exponentially for all values of β. On the other hand for d > 1 the result holds only for small β since βc < (2d − 1)−2 < 1. 2. Proof of Theorem 1 We want to estimate dt j ε ε (t) D −1 . G x y = dμε (t)G x y = √ e−F (∇t) e−M (t) × det D xy 2π j∈ (2.1) The proof is done in four steps. Step √ 1. We mix the observable G x y and a piece of the probability measure namely det D. The key identity is
ε (t) D −1 = det D xy
Dx−1 y
ε Dx−1 y det D (t)
(2.2)
(remember that Dx−1 y > 0). The first term is bounded by Dx−1 y ≤
1 1 + εx e−tx ε y e−t y
.
This is proved in Lemma 1. Inserting this in (2.1) we have tx /2 t y /2 e e −1 ε ε (t)] G x y ≤ dν (t) [Dx y det D , √ +√ εx εy
(2.3)
(2.4)
where ε (t) = dν
dt j ε √ e−F (∇t) e−M (t) . 2π j∈
(2.5)
Unlike the measure dμε given by (1.1), this measure is no longer normalized to 1. Step 2. We need to extract some decay in the distance |x − y|. This is hidden in Dx−1 y det D. By some combinatorial arguments (the proof is given in Lemma 2 ) we can write ε [Dx−1 y det D (t)] =
γx y
ε˜ β |γ | det D c, γ
(2.6)
where the sum ranges over non self intersecting paths γ made of nearest neighbor pairs in starting at x and ending at y. Let |γ |, denote the length of γ and let γ be the ε˜ is the matrix one corresponding set of lattice points and set cγ = \γ . Finally D c γ obtains by deleting the rows and columns corresponding to the lattice points j ∈ γ . It
Anderson Localization for a Supersymmetric Sigma Model
665
ε , but defined on the complement of γ , c and with modified is exactly like the matrix D γ masses:
Diε˜j = 0,
|i − j| > 1,
Diε˜j
|i − j| = 1,
= −β, γ Diiε˜ = β di + V˜i + ε˜ i e−ti
(2.7)
i = j,
γ
where i, j ∈ cγ , and di = #{k ∈ cγ |(k, i)} is the number of neighbors of i in cγ . Finally V˜i =
t −t ek i −1 ,
k∈cγ ,(ki)
ε˜ i = εi + β
(2.8)
e tk .
(2.9)
k∈γ ,(ki)
By combining (2.4) and (2.6) we have G x y ≤
β
|γ |/2
γx y
ε (t) dν
etx /2 et y /2 √ +√ εx εy
ε˜ . det D c γ
(2.10)
ε (t) defined in (2.5) can be factored as a measure on times Step 3. The measure dν γ a measure on the complement set cγ , ε ε ε −F∂γ (∇t) dν (t) = dν (t) dν , c (t) e γ γ
(2.11)
where
F∂γ (∇t) =
β (cosh(t j − tk ) − 1)
(2.12)
( j,k),k∈γ , j∈γ
describes the interaction between γ and cγ . Then the integral in (2.10) can be written as tx /2 t y /2 etx /2 et y /2 e e γ ε ε ε˜ = det D dνγ (t) √ + √ Z c (tγ ), dν (t) √ + √ c γ γ εx εy εx εy (2.13) where we defined γ Z c (tγ ) γ
= =
ε dν c (t) γ
j∈cγ
−F∂γ (∇t) ε˜ det D c e γ
dt j −Fcγ (∇t) −Mε cγ (t) −F∂γ (∇t) ε˜ det D e . e √ c e γ 2π
(2.14)
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M. Disertori, T. Spencer γ
Note that Z c (tγ ) is still a function of the t variables along the path {tk }k∈γ (they are γ
γ
not integrated). Now Z c (tγ ) is almost equal to the partition function γ ε˜ ε˜ ε˜ ε˜ dμ (t) = dν (t) det D 1 = Z c = c c c γ γ γ γ dt j −Fcγ (∇t) −Mε˜ cγ (t) ε˜ , = e det D e √ c γ 2π j∈c
(2.15)
γ
ε˜ = 1 by supersymmetry (see (1.15)). Comparing (2.14) and (2.15) we see where Z c γ that there are two main differences: ε in (2.14) depends on ε instead of ε˜ and is smaller than what it – the mass term M c γ should be ε ε˜ M c ≤ M c , γ γ
since M ε˜ contains additional mass terms; – the exponent in (2.14) contains the additional factor −F∂γ (∇t) coming from the kinetic interaction between points on γ and points on cγ . This last term is helping us since it makes the integral smaller. We will use it to recover the missing mass. This is done in Lemma 3 below. The key ingredient is a global translation on the t variables. The result is etx /2 et y /2 1 |γ |/2 β|∂γ | ε G x y ≤ e dνγ (t) √ + √ β e , (2.16) εx εy γ xy
where |∂γ | ≤ (2d − 2)|γ | + 2 is the number of points inside cγ on the boundary with γ . Step 4. We are left with an integral along the path γ . The integral in (2.16) is bounded by tx /2 dt j ε etx /2 et y /2 et y /2 −Fγ (∇t) −Mγ (t) e ε dνγ (t) √ + √ e = √ e √ +√ εx εy εx εy 2π j∈γ tα /2 dt j e ≤ √ e−Fγ (∇t) e−εα (cosh tα −1) √ εα 2π α=x,y j∈γ y I1x I1 |γ | = √ +√ (2.17) I2 , εx εy ε (t) except for ε (cosh t − 1) and for each t along the path with where we dropped M α α j γ j = α we used ∇t as a new variable. The resulting integral factors, ∞ dt 1 I1x = (2.18) √ et/2 e−εx (cosh t−1) = √ , εx 2π −∞ ∞ dt 1 I2 = (2.19) √ e−β(cosh t−1) = 1/2 Iβ , β 2π −∞
Anderson Localization for a Supersymmetric Sigma Model
667
√ y and Iβ was defined in (2.18). In the same way I1 = 1/ ε y . Inserting all this we have 1 1 β|∂γ | 1 G x y ≤ e + e (Iβ )|γ | εx ε y γ xy n 1 1 eβ(2d−2) Iβ cdn ≤ 2 e1+2β + εx ε y n≥|x−y| |x−y| 1 1 eβ(2d−2) Iβ cd ≤ C0 + , (2.20) εx ε y where cd = (2d − 1), C0 is a constant and the second inequality holds since the number of self-avoiding walks made of n steps is bounded by 2d(2d − 1)n < 2cdn and |∂γ | ≤ (2d − 2)|γ | + 2. Finally the sum over n is convergent since (eβ(2d−2) Iβ cd ) < 1. This concludes the proof of Theorem 1. 2.1. The lemmas. Lemma 1. The following inequality holds: Dx−1 y ≤
etx et y + . εx εy
(2.21)
Proof. By Cauchy-Schwartz inequality, −1 −1 −1 Dx y ≤ Dx−1 D −1 x yy ≤ D x x + D yy .
−t j f 2 for any f ∈ R we have Since ( f, D f ) ≥ j j εje j Dx−1 x ≤ Hence the result.
1 . εx e−tx
Lemma 2. For any invertible matrix M on we have the following identity: (−Mx j1 )(−M j1 j2 ) · · · (−M jm y ) det cγ M, [Mx−1 y detM] =
(2.22)
γx y =( j1 ,... jm )
where γ is any non self intersecting path starting at x and ending at y. Proof. This is a classical formula arising from the fact that every permutation can be decomposed as a product of cycles. One may derive it using the representation of the determinant as a sum over a gas of disjoint non self intersecting closed paths: det M = A(L 1 ) · · · A(L p ), L 1 ,...L p
where a loop L = ( j1 , . . . , jm ) is an ordered set of m distinct points and − (−M j1 j2 )(−M j2 j3 ) · · · (−M jm j1 ) m > 1 A(L) = . M j1 j1 m=1
(2.23)
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M. Disertori, T. Spencer
The sign (−1)m−1 is the number of pairs inside the loop that need to be exchanged ∂ in order to recover the trivial permutation. Now since [Mx−1 y detM] = ∂ M yx detM, the derivation selects only loops that contain the pair yx. The corresponding matrix element disappears and the loop becomes a path from x to y. The sign −1 from −M yx cancels the global −1 in front of the product. Lemma 3. For any configuration of {tk | k ∈ γ }, the conditioned partition function γ Z c (tγ ) given by (2.14) is bounded by γ
γ
Z c (tγ ) ≤ e
β
k∈γ
γ
∗
γ
dk (1−etk −t )
e
j∈cγ
∗
ε j (1−e−t )
≤ eβ|∂γ | e
j∈cγ
εj
,
(2.24)
where t ∗ is any real number satisfying t ∗ ≥ 0,
and
t ∗ ≥ tk
∀k ∈ γ ,
(2.25)
γ
and dk is the number of points nearest neighbor to k that do not belong to γ : γ
dk = #{ j ∈ γ | | j − k| = 1}. Proof. Before doing any bound we perform a global translation inside the integral: t j → t j + t∗
∀ j ∈ cγ .
(2.26)
Then inside the exponential we have: Fcγ (∇t) → Fcγ (∇t), ε M c (t) γ
→
ε M c (t γ
(2.27) ∗
+ t ),
F∂γ (∇t) → F∂γ (∇t + t ∗ ) =
(2.28) β (cosh(t j + t ∗ − tk ) − 1).
(2.29)
( j,k),k∈γ , j∈cγ
In the numerator we have
∗ ε˜ ε˜ e−t det D → det D , c c γ γ
so the only effect of the translation is to modify the mass ε˜ j defined in (2.9) at each lattice point. After the translation ⎡ ⎤ ε ∗ dt j ∗ −Fcγ (∇t) γ ε˜ e−t e−Mcγ (t+t ) e−F∂γ (∇t+t ∗ ) ⎣ ⎦ Z c = det D e √ c γ γ 2π j∈cγ ⎡ ⎤ ∗ ∗ ε˜ e−t ε˜ e−t(t)−M ε (t+t ∗ )−F (∇t+t ∗ ) dt j M ∂γ c −Fcγ (∇t) −Mcγ (t) −t ∗ cγ ε ˜ e γ ⎣ ⎦ = e det Dc e e √ γ 2π j∈cγ
sup E rr j ∗ ∗ ˜ e−t ε˜ e−t j∈cγ E rr j j∈cγ supt j E rr j = dμε (t) e ≤ Z e = e tj , c c γ γ j∈cγ
(2.30)
Anderson Localization for a Supersymmetric Sigma Model −t ∗
ε˜ e where we used Z c
γ
669
= 1 (see (1.15)) and we defined
Err j =
j∈cγ
−t ∗
ε˜ e M c γ
ε ∗ ∗ (t) − M c (t + t ) − F∂γ (∇t + t ) γ
(2.31)
∗
Err j = ε˜ j e−t (cosh t j − 1) − ε j (cosh(t j + t ∗ ) − 1) β (cosh(t j + t ∗ − tk ) − 1). −
(2.32)
k∈γ ,( j,k)
To conclude we shall prove that the right hand side of (2.30) is bounded by the rhs of (2.24):
e
supt j E rr j
≤ e
β
∗
γ
dk (1−etk −t )
k∈γ
e
j∈cγ
∗
ε j (1−e−t )
.
j∈cγ
We distinguish two cases. Case 1. When j is far from the path γ that is | j − k| > 1 for all k ∈ γ , then ε˜ j = ε j and we have ∗ Err j = ε j e−t (cosh t j − 1) − (cosh(t j + t ∗ ) − 1) ∗ ∗ = ε j et j sinh(−t ∗ ) + 1 − e−t ≤ ε j (1 − e−t ), (2.33) where the last inequality holds for t ∗ ≥ 0. Case 2. When
j is near to the path γ that is | j − k| = 1 for some k ∈ γ , then ε˜ j = ε j + k∈γ ,( j,k) βetk and we have ∗ Err j = ε j e−t (cosh t j − 1) − (cosh(t j + t ∗ ) − 1) ∗ +β etk −t (cosh t j − 1) − (cosh(t j + t ∗ − tk ) − 1) k∈γ ,( j,k)
∗ = ε j et j sinh(−t ∗ ) + 1 − e−t + β ≤ ε j (1 − e
−t ∗
)+β
∗ et j sinh(tk − t ∗ ) + 1 − etk −t
k∈γ ,( j,k) ∗
(1 − etk −t ),
(2.34)
k∈γ ,( j,k)
where the last inequality holds if t ∗ ≥ 0 and t ∗ ≥ tk ∀k ∈ γ . Finally ∗ ∗ Err j ≤ [ ε j (1 − e−t ) + β (1 − etk −t ) ] j∈cγ
k∈γ ,( j,k)
j∈cγ
≤β
∗ γ dk (1 − etk −t )
j∈cγ
k∈γ
This concludes the proof of the lemma.
+
∗
ε j (1 − e−t ).
(2.35)
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3. Proof of Theorem 2 The proof of Theorem 2 is almost identical to that of Theorem 1. This time there is no term Dx−1 y ensuring we can extract a path γ connecting x to y. On the other hand, since there is a pinning only at one position the matrix-tree theorem (see [1] for a simple proof and many references) applied to the “diffusion” representation A given in (1.7) of the matrix D gives t +t det Aε = ε0 et0 βe j j , (3.1) (−Aεj j ) = ε0 et0 j j ∈T
T
T ( j j )∈T
where the sum is over the spanning trees on made of nearest neighbor pairs (since A j j = 0 when | j − j | > 1). Therefore each term in the sum contains a path γ from 0 to x. Note that det A0 = 0 and the cofactors satisfy t +t C x0 (Aε ) = C x0 (A0 ) = βe j j (3.2) T ( j j )∈T
for all x ∈ . Using (3.2) and (3.1) we see that ε ε (Aε )−1 0x det A = C x0 (A ) =
det Aε . ε0 et0
Therefore ε0 et0 (Aε )−1 0x = 1 and ε −tx ε −tx det D ε = ε0 et0 (Aε )−1 (D ε )−1 0x det D = ε0 e 0x det D = ε0 e
(3.3)
γ0x
β |γ | det cγ D ε , (3.4)
(Aε )−1 0x
−tx e−t0 (D ε )−1 0x e
where = and in the last term we applied Lemma 2. Inserting this result in (1.1) we have ε ε etx /2 Ox = dμε (t) Ox = dν (t) det D √ ε tx /2 −tx /2 ε˜ = ε0 dν (t)e e β |γ | det D c ≤ ≤
√
ε0
γ0x
β |γ |/2
γ0x
e
β|∂γ |
γ
γ0x γ
ε (t) Z c (tγ ) ≤ dν γ
|γ | Iε0 Iβ
γ
≤ C0 (cd e
β(cd −1)
√
ε0
β |γ |/2
γ0x
ε (t) eβ|∂γ | dν γ
|x|
Iβ ) ,
(3.5)
γ
ε (t) was defined in (2.5), Z where dν c (tγ ) in (2.14), Iβ in (1.21) and the same defγ
ε˜ (see (2.7)). We inition holds for Iε . In the second line we used det cγ D = det D c γ
γ
used Lemma 1, Eq. (2.24) to bound Z c (tγ ). Finally the last inequality holds since γ
(cd ecd −1 Iβ ) < 1. This concludes the proof of Theorem 2.
Acknowledgements. It is our pleasure to thank A. Abdesselam for discussions and suggestions related to this paper. A special thanks to M. Zirnbauer who explained the model to us and shared his many insights.
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References 1. Abdesselam, A.: The Grassmann-Berezin calculus and theorems of the matrix-tree type. Adv. in Appl. Math. 33(1), 51–70 (2004) 2. Disertori, M., Spencer, T., Zirnbauer, M.R.: Quasi-diffusion in a 3d supersymmetric hyperbolic sigma model. Commun. Math. Phys. (2009). doi:10.1007/s00220-010-1117-5 3. Drunk, W., Fuchs, D., Zirnbauer, M.R.: Migdal-Kadanoff renormalization of a nonlinear supervector model with hyperbolic symmetry. Ann. Physik 1, 134–150 (1992) 4. Dupré, T.: On the localization transition in three dimensions: Monte-Carlo simulation of a non-linear σ -model. Phys. Rev. B 54(18), 2763–12774 (1996) 5. Efetov, K.B.: Supersymmetry and theory of disordered metals. Adv. Phys. 32, 874 (1983) 6. Efetov, K.B.: Supersymmetry in Disorder and Chaos. Cambridge: Cambridge University Press, 1997 7. Merkl, F., Rolles, S.W.W.: Linearly edge-reinforced random walks. Volume 48 of IMS Lecture NotesMonograph Series, 2006, pp. 66–77 8. Wegner, F.: The mobility edge problem: continuous symmetry and a conjecture. Z. Phys. B 35, 207–210 (1979) 9. Wegner, F., Schaefer, L.: Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and goldstone modes. Z. Phys. B 38, 113–126 (1980) 10. Zirnbauer, M.R.: Fourier analysis on a hyperbolic supermanifold with constant curvature. Commun. Math. Phys. 141, 503–522 (1991) Communicated by M. Salmhofer
Commun. Math. Phys. 300, 673–713 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1126-4
Communications in
Mathematical Physics
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I Geoffrey Mason1, , Michael P. Tuite2 1 Department of Mathematics, University of California, Santa Cruz, CA 95064, U.S.A.
E-mail: [email protected]
2 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway,
University Road, Galway, Ireland. E-mail: [email protected] Received: 1 December 2009 / Accepted: 30 May 2010 Published online: 10 September 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract: We define the partition and n-point functions for a vertex operator algebra on a genus two Riemann surface formed by sewing two tori together. We obtain closed formulas for the genus two partition function for the Heisenberg free bosonic string and for any pair of simple Heisenberg modules. We prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties for the Heisenberg and lattice vertex operator algebras and a continuous orbifolding of the rank two fermion vertex operator super algebra. We compute the genus two Heisenberg vector n-point function and show that the Virasoro vector one point function satisfies a genus two Ward identity for these theories. Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Genus Two Riemann Surface from Two Sewn Tori . . . . . . . . . . . . . . Graphical Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertex Operator Algebras and the Li-Zamolodchikov Metric . . . . . . . . . Partition and n-Point Functions for Vertex Operator Algebras on a Riemann Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The Heisenberg VOA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Heisenberg Modules, Lattice VOAs and Theta Series . . . . . . . . . . . . . 8. Appendix - A Product Formula . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
673 676 681 684 687 693 702 710 712
1. Introduction One of the most striking features of Vertex Operator Algebras (VOAs) or chiral conformal field theory is the occurrence of elliptic functions and modular forms, manifested Supported by the NSF, NSA, and the Committee on Research at the University of California, Santa Cruz.
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in the form of n -point correlation trace functions. This phenomenon has been present in string theory since the earliest days e.g. [GSW,P]. In mathematics it dates from the Conway-Norton conjectures [CN] proved by Borcherds ([B1,B2]), and Zhu’s important paper [Z1]. Physically, we are dealing with probability amplitudes corresponding to a complex torus (compact Riemann surface of genus one) inflicted with n punctures corresponding to local fields (vertex operators). For a VOA V = ⊕Vn , the most familiar correlation function is the 0-point function, also called the partition function or graded dimension Z V(1) (q) = q −c/24 dim Vn q n , (1) n
(c is the central charge). An example which motivates much of the present paper is that of a lattice theory VL associated to a positive-definite even lattice L. Then c is the rank of L and θ L (q) , (2) η(q)c for the Dedekind eta function η(q) = q 1/24 n (1 − q n ) and θ L (q) is the usual theta function of L. Both θ L (q) and η(q)c are (holomorphic) elliptic modular forms of weight c/2 on a certain congruence subgroup of S L(2, Z), so that Z VL is an elliptic modular function of weight zero on the same subgroup. It is widely expected that an analogous result holds for any rational vertex operator algebra, namely that Z V(1) (q) is a modular function of weight zero on a congruence subgroup of S L(2, Z). There are natural physical and mathematical reasons for wanting to extend this picture to Riemann surfaces of higher genus. In particular, we want to know if there are natural analogs of (1) and (2) for arbitrary rational vertex operator algebras and arbitrary genus, in which genus g Siegel modular forms occur. This is considerably more challenging than the case of genus one. Many, but not all, of the new difficulties that arise are already present at genus two, and it is this case that we are concerned with in the present paper and a companion paper [MT4]. Our goal, then, is this: given a vertex operator algebra V , to define the partition and n-point correlation function on a compact Riemann surface of genus two which is associated to V , and study their convergence and automorphic properties. An overview of aspects of this program is given in the Introduction to [MT2]. Brief discussions of some of our methods and results can also be found in [T,MT3 and MT6]. The study of genus two (and higher) partition functions and correlation functions has a long history in conformal field theory e.g. [EO,FS,DP,So1,So2,BK,Kn,GSW,P] and, indeed, these ideas have heavily influenced our approach. Likewise, in pure mathematics, other approaches based on algebraic geometry have been been developed to describe n-point correlation functions but not the partition function e.g. [TUY,KNTY,Z2,U]. Our approach is constructively based only on the properties of a VOA in the spirit of Zhu’s genus one theory [Z1] with no a priori assumptions made about the analytic or modular properties of partition or n-point functions. Rather, in our approach, these genus two objects are formally defined and are then proved to be analytic and modular in appropriate domains for the VOAs considered. In our approach, we define the genus two partition and n-point functions in terms of genus one data coming from the VOA V . There are two rather different ways to obtain a compact Riemann surface of genus two from surfaces of genus one - one may sew two separate tori together, or self-sew a torus (i.e. attach a handle). This is discussed at length (1)
Z VL (q) =
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
675
in [MT2] where we refer to these two schemes as the - and ρ -formalism respectively. In the present paper we concentrate solely on developing a theory of partition and n-point correlation functions in the -formalism. We discuss the corresponding theory in the ρ -formalism in a companion paper [MT4]. The -formalism developed in [MT2] is reviewed in Sect. 2 below. This is concerned with expressing a differential 2-form ω(2) (the normalized differential of the second kind) in terms of a pair of infinite matrices Ai , whose entries are quasi-modular forms associated with the two sewn tori. This allows us to obtain explicit expressions for genus two holomorphic one forms ν1 , ν2 and the period matrix Ω in terms of this genus one data. In particular, Ω is determined by a holomorphic map F
D −→ H2 ,
(3)
where for g ≥ 1, Hg denotes the genus g Siegel upper half-space. Then D ⊆ H1 × H1 × C is the domain consisting of triples (τ1 , τ2 , ) which correspond to a pair of complex tori of modulus τ1 , τ2 sewn together by identifying two annular regions via a sewing parameter . This sewing produces a compact Riemann surface of genus two, which assigns to each point of D the period matrix Ω of the sewn surface via the map F . In Sect. 3 we introduce some graph-theoretic technology which provides a convenient way of describing ω(2) , νi and Ω in terms of the -formalism. Similar graphical techniques are employed later on as a means of computing the genus two partition function and n-point functions for the free bosonic Heisenberg VOA and its modules. Section 4 is a brief review of some necessary background on VOA theory and the Li-Zamolodchikov or Li-Z metric. We assume throughout that the Li-Z metric is unique and invertible (which follows if V is simple [Li]). Section 5 develops a theory of n-point functions for VOAs on Riemann surfaces of genus 0, 1 and 2 motivated by ideas in conformal field theory. The Zhu theory [Z1] of genus one n-point functions is reformulated in this language in terms of the self-sewing of a Riemann sphere to obtain a torus. We give a formal definition of genus two n-point functions based on the given sewing formalism. We also emphasize the interpretation of n-point functions in terms of formal differential forms. The genus two partition function involves extending (3) to a diagram F
D −→ H2 ↓ C where the partition function maps D → C, and is defined purely in terms of genus one data coming from V . Explicitly, the genus two partition function of V is a priori a formal power series in the variables , q1 , q2 (where as usual, q = e2πiτ , etc.) given by (1) (2) (1) Z V (τ1 , τ2 , ) = n Z V (u, τ1 )Z V (u, ¯ τ2 ). (4) n≥0 (1)
u∈V[n]
Here, Z V (u, τ ) is a genus one 1-point function with u¯ the Li-Z metric dual of u. The precise meaning of (4) together with similar definitions for n-point functions, is given in Sect. 5. In Sects. 6 and 7 we investigate the case of the free bosonic Heisenberg VOA M and the expression corresponding to (4) for a pair of simple M-modules. This later case is
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G. Mason, M. P. Tuite
used to analyze lattice VOAs and the bosonized version of the rank two fermion Vertex Operator Super Algebra. We find in all these cases that (4) is a holomorphic function on D . It is natural to expect that this result holds in much wider generality. Section 6 is devoted to the Heisenberg VOA M. In this case, holomorphy depends on an interesting new formula for the genus two partition function. Namely, we prove (Theorem 5) by reinterpreting (4) in terms of certain graphical expansion, that (1)
(2)
Z M (τ1 , τ2 , ) =
(1)
Z M (τ1 )Z M (τ2 ) . det(I − A1 A2 )1/2
(5)
(1)
Here, the Ai are the infinite matrices of Sect. 2 and Z M (τi ) = 1/η(qi ). The infinite determinant that occurs in (5) was introduced and discussed at length in [MT2]. The results obtained there are important here, as are the explicit computations of genus one 1-point functions obtained in [MT1]. We also give in Sect. 6 a product formula for the infinite determinant (Theorem 6) which depends on the graphical interpretation of the entries of the Ai . The domain D admits the group G 0 = S L(2, Z) × S L(2, Z) as automorphisms (in fact, there is a larger automorphism group G that contains G 0 with index 2). We show (2) (cf. Theorem 8) that the partition function Z M (τ1 , τ2 , ) is an automorphic form of weight −1/2 on G. This is a bit imprecise in several ways: we have not explained here what the automorphy factor is, and in fact this is an interesting point because it depends on the map F . Similarly to the eta-function, there is a 24th root of unity, corresponding to a character of G, that intervenes in the functional equation. These properties of Z M (τ1 , τ2 , ) justify the idea that it should be thought of as the genus two analog of η(q)−1 in the -formalism. We conclude Sect. 6 by computing, by means of the graphical technique, the genus two n-point function for n Heisenberg vectors in terms of symmetric tensor products of the differential 2-form ω(2) in Theorem 10. This allows us to also find the Virasoro vector 1-point function in terms of the genus two projective connection. Section 7 is concerned with the genus two n-point function associated with a pair of Heisenberg simple modules. We obtain a closed formula for the partition function in Theorem 11 and the Heisenberg vector n-point function in terms of symmetric tensor products of ω(2) and νi in Theorem 13. We also derive a genus two Ward identity for the Virasoro vector 1-point function in Proposition 10. We apply these results in Theorem 14 to the case of a lattice VOA VL to find a natural genus two generalization of (2), namely (2)
Z VL (τ1 , τ2 , ) (2)
Z M (τ1 , τ2 , )
(2)
= θ L (Ω),
(6)
(2)
where θ L (Ω) is the genus two Siegel theta function of the lattice L. Similarly, the Virasoro 1-point function obeys a Ward identity. Finally, we consider the bosonized version of a continuous orbifolding of the rank two fermion vertex super algebra to find the partition function is expressed in terms of the genus two Riemann theta series. 2. Genus Two Riemann Surface from Two Sewn Tori In this section we review some of the main results of [MT2] relevant to the present work. We review one of the two separate constructions of a genus two Riemann surface
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
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discussed there based on a general sewing formalism due to Yamada [Y]. In this construction, which we refer to as the -formalism, we parameterize a genus two Riemann surface by sewing together two once-punctured tori. Then various genus two structures such as the period matrix Ω can be determined in terms of genus one data. In particular, Ω is described by an explicit formula which defines a holomorphic map from a specified domain D into the genus two Siegel upper half plane H2 . This map is equivariant under a suitable subgroup of Sp(4, Z). We also review the convergence and holomorphy of an infinite determinant that naturally arises and which plays a dominant rôle later on.
2.1. Some elliptic function theory. We begin with the definition of various modular and elliptic functions that permeate this work [MT1,MT2]. We define P2 (τ, z) = ℘ (τ, z) + E 2 (τ ) ∞ 1 = 2+ (k − 1)E k (τ )z k−2 , z
(7)
k=2
where τ ∈ H1 , the complex upper half-plane, and where ℘ (τ, z) is the Weierstrass function and E k (τ ) is equal to 0 for k odd, and for k even is the Eisenstein series E k (τ ) = E k (q) = −
2 Bk + σk−1 (n)q n . k! (k − 1)!
(8)
n≥1
k−1 , and B is a k th Here and below, we take q = exp(2πiτ ); σk−1 (n) = k d|n d Bernoulli number e.g. [Se]. If k ≥ 4 then E k (τ ) is a holomorphic modular form of weight k on S L(2, Z), whereas E 2 (τ ) is a quasi-modular form [KZ,MT2]. We define P1 (τ, z) by P1 (τ, z) =
1 − E k (τ )z k−1 . z
(9)
k≥2
d P1 we define elliptic functions Pk (τ, z) for k ≥ 3, Noting P2 = − dz
Pk (τ, z) =
(−1)k−1 d k−1 P1 (τ, z). (k − 1)! dz k−1
(10)
Define for k, l ≥ 1, (k + l − 1)! E k+l (τ ), (k − 1)!(l − 1)! (k + l − 1)! Pk+l (τ, z). D(k, l, z) = D(k, l, τ, z) = (−1)k+1 (k − 1)!(l − 1)! C(k, l) = C(k, l, τ ) = (−1)k+1
(11) (12)
The Dedekind eta-function is defined by η(τ ) = q 1/24
∞
(1 − q n ).
n=1
(13)
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G. Mason, M. P. Tuite
2.2. The -formalism for sewing two tori. Consider a compact Riemann surface S of genus 2 with canonical homology basis a1 , a2 , b1 , b2 . There exist two holomorphic 1-forms νi , i = 1, 2 which we may normalize by [FK] ν j = 2πiδi j . (14) ai
These forms can also be defined via the unique singular bilinear two form ω(2) , known as the normalized differential of the second kind. It is defined by the following properties [FK,Y]: ω(2) (x, y) = (
1 + regular terms)d xd y (x − y)2
for any local coordinates x, y, with normalization ω(2) (x, ·) = 0,
(15)
(16)
ai
for i = 1, 2. Using the Riemann bilinear relations, one finds that νi (x) = ω(2) (x, ·),
(17)
bi
with νi normalized as in (14). The genus 2 period matrix Ω is then defined by 1 Ωi j = νj 2πi bi
(18)
for i, j = 1, 2. One further finds that Ω ∈ H2 , the Siegel upper half plane (Fig. 1). We now review a general method due to Yamada [Y] and discussed at length in [MT2] for calculating ω(2) (x, y), νi (x) and Ωi j on the genus two Riemann surface formed by sewing together two tori Sa for a = 1, 2. We shall sometimes refer to S1 and S2 as the left and right torus respectively. Consider an oriented torus Sa = C/Λa with lattice Λa = 2πi(Zτa ⊕ Z) for τa ∈ H1 . For local coordinate z a ∈ C/Λa consider the closed disk |z a | ≤ ra which is contained in Sa provided ra < 21 D(qa ), where D(qa ) =
min
λ∈Λa ,λ =0
|λ|,
is the minimal lattice distance. Introduce a complex sewing parameter , where || ≤ r1r2 < 41 D(q1 )D(q2 ) and excise the disk {z a , |z a | ≤ ||/ra¯ } centered at z a = 0 to form a punctured torus Sˆa = Sa \{z a , |z a | ≤ ||/ra¯ },
(19)
where we use the convention 1 = 2, 2 = 1.
(20)
Aa = {z a , ||/ra¯ ≤ |z a | ≤ ra } ⊂ Sˆa ,
(21)
Defining the annulus
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
679
Fig. 1. Sewing Two Tori
we identify A1 with A2 via the sewing relation z 1 z 2 = .
(22)
The genus two Riemann surface is parameterized by the domain D = {(τ1 , τ2 , ) ∈ H1 ×H1 ×C | || <
1 D(q1 )D(q2 )}. 4
(23)
We next introduce the infinite dimensional matrix Aa (τa , ) = (Aa (k, l, τa , )) for k, l ≥ 1, where Aa (k, l, τa , ) =
(k+l)/2 C(k, l, τa ). √ kl
(24)
The matrices A1 , A2 play a dominant role both here and in our later discussion of the free bosonic VOA and its modules on a genus two Riemann surface. In particular, the matrix I − A1 A2 and det(I − A1 A2 ) (where I denotes the infinite identity matrix) play an important role where det(I − A1 A2 ) is defined by log det(I − A1 A2 ) = Tr log(I − A1 A2 ) 1 Tr((A1 A2 )n ). =− n
(25)
n≥1
One finds Theorem 1. (a) (op. cit., Proposition 1) The infinite matrix (I − A1 A2 )−1 = (A1 A2 )n ,
(26)
n≥0
is convergent for (τ1 , τ2 , ) ∈ D . (b) (op. cit., Theorem 2 & Proposition 3) det(I − A1 A2 ) is non-vanishing and holomorphic for (τ1 , τ2 , ) ∈ D . The bilinear two form ω(2) (x, y), the holomorphic one forms νi (x) and the period matrix Ωi j are given in terms of the matrices Aa and holomorphic one forms on the punctured torus Sˆa given by √ (27) aa (k, x) = k k/2 Pk+1 (τa , x)d x. Letting aa (x), aaT (x) denote the infinite row, respectively column vector with elements (27) we have:
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Theorem 2. (op. cit., Lemma 2, Proposition 1, Theorem 4) P (τ , x − y)d xd y + aa (x)Aa¯ (I − Aa Aa¯ )−1 aaT (y), x, y ∈ Sˆa , (2) ω (x, y) = 2 a x ∈ Sˆa , y ∈ Sˆa¯ . −aa (x)(I − Aa¯ Aa )−1 aaT¯ (y), (28) Applying (17) we then find (op. cit., Theorem 4) d x + 1/2 (aa (x)Aa¯ (I − Aa Aa¯ )−1 )(1), x ∈ Sˆa , νa (x) = x ∈ Sˆa¯ , − 1/2 (aa¯ (x)(I − Aa Aa¯ )−1 )(1),
(29)
where (1) refers to the (1)-entry of a vector. Furthermore applying (18) we have Theorem 3. (op. cit., Theorem 4) The -formalism determines a holomorphic map F : D → H2 , (τ1 , τ2 , ) → Ω(τ1 , τ2 , ),
(30)
where Ω = Ω(τ1 , τ2 , ) is given by 2πiΩ11 = 2πiτ1 + (A2 (I − A1 A2 )−1 )(1, 1), 2πiΩ22 = 2πiτ2 + (A1 (I − A2 A1 )−1 )(1, 1), 2πiΩ12 = −(I − A1 A2 )−1 (1, 1).
(31) (32) (33)
Here (1, 1) refers to the (1, 1)-entry of a matrix. D is preserved under the action of G (S L(2, Z) × S L(2, Z)) Z2 , the direct product of two copies of S L(2, Z) (the left and right torus modular groups) which are interchanged upon conjugation by an involution β as follows: a1 τ1 + b1 , τ2 , ), c1 τ1 + d1 c1 τ1 + d1 a2 τ2 + b2 γ2 .(τ1 , τ2 , ) = (τ1 , , ), (34) c2 τ2 + d2 c2 τ2 + d2 β.(τ1 , τ2 , ) = (τ2 , τ1 , ),
ai bi . There is a natural injection for (γ1 , γ2 ) ∈ S L(2, Z) × S L(2, Z) with γi = ci di G → Sp(4, Z) in which the two S L(2, Z) subgroups are mapped to ⎧⎡ ⎧⎡ ⎤⎫ ⎤⎫ a 0 b1 0 ⎪ 1 0 0 0 ⎪ ⎪ ⎪ ⎨ 1 ⎨ ⎬ ⎬ ⎢ 0 1 0 0⎥ ⎢ 0 a2 0 b2 ⎥ Γ1 = ⎣ , (35) , Γ2 = ⎣ ⎦ ⎦ ⎪ ⎪ ⎩ c1 0 d1 0 ⎪ ⎩ 0 0 1 0 ⎪ ⎭ ⎭ 0 0 0 1 0 c2 0 d2 γ1 .(τ1 , τ2 , ) = (
and the involution is mapped to ⎡
0 ⎢1 β=⎣ 0 0
1 0 0 0
0 0 0 1
⎤ 0 0⎥ . 1⎦ 0
(36)
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Fig. 2. Chequered Cycle
Thus as a subgroup of Sp(4, Z), G also has a natural action on the Siegel upper half A B plane H2 , where for γ = ∈ Sp(4, Z), C D γ .Ω= (AΩ + B)(CΩ + D)−1 .
(37)
One then finds Theorem 4. (op. cit., Theorem 5) F is equivariant with respect to the action of G, i.e. there is a commutative diagram for γ ∈ G, F
D → H2 γ ↓ ↓γ F
D → H2 3. Graphical Expansions 3.1. Rotationless and Chequered Cycles. We set up some notation and discuss certain types of labeled graphs. These arise directly from consideration of the terms that appear in the expressions for ω(2) (x, y), νi (x) and Ωi j reviewed in the last section, and will later play an important rôle in the analysis of genus two partition functions for vertex operator algebras. We introduce the notion of a chequered cycle as a (clockwise) oriented, labeled polygon L with 2n nodes for some integer n ≥ 0, and nodes labeled by arbitrary positive integers. Moreover, edges carry a label 1 or 2 which alternate as one moves around the polygon (Fig. 2). A chequered cycle is said to be rotationless when its graph admits no non-trivial rotations where a rotation is an orientation-preserving automorphism of the graph which preserves the node labels. (See the Appendix for more details.) 2
1
1
We call a node with label 1 distinguished if its abutting edges are of type −→ • −→. Set R = {isomorphism classes of rotationless chequered cycles}, R21 = {isomorphism classes of rotationless chequered cycles with a distinguished node}, L21 = {isomorphism classes of chequered cycles with a unique distinguished node}.
(38)
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Fig. 3. Chequered necklace
Let S be a commutative ring and S[t] the polynomial ring with coefficients in S. Let M1 and M2 be infinite matrices with (k, l)-entries Ma (k, l) = t k+l sa (k, l)
(39)
for a = 1, 2 and k, l ≥ 1, where sa (k, l) ∈ S. Given this data, we define a map, or weight function, ζ : {chequered cycles} −→ S[t] k
a
l
as follows: if L is a chequered cycle then L has edges E labeled as • −→ •. Then set ζ (E) = Ma (k, l) and ζ (L) = ζ (E), (40) where the product is taken over all edges of L. It is useful to also introduce a variation on the theme of chequered cycles namely oriented chequered necklaces. These are connected graphs with n ≥ 3 nodes, (n − 2) of which have valency 2 and two of which have valency 1 (these latter are the end nodes) together with an orientation, say from left to right. There is also a degenerate necklace N0 with a single node and no edges. As before, nodes are labeled with arbitrary positive integers and edges are labeled with an index 1 or 2 which alternate along the necklace. For such a necklace N , we define the weight function ζ (N ) as a product of edge weights as in (40), with ζ (N0 ) = 1. Among all chequered necklaces there is a distinguished set for which both end nodes are labeled by 1. There are four types of such chequered necklaces, which may be further distinguished by the labels of the two edges at the extreme left and right. Using the convention (20) we say that the chequered necklace of Fig. 3 is of type ab for a, b ∈ {1, 2}, and set Nab = {isomorphism classes of oriented chequered necklaces of type ab}, ζab = ζ (N ).
(41) (42)
N ∈Nab
3.2. Necklace graphical expansions for ω(2) , νi and Ωi j . We now apply the formalism of the previous Subsection to the expressions for ω(2) (x, y), νi (x) and Ωi j in the -formalism reviewed in Sect. 2. We begin with the period matrix Ωi j . Here the ring S is taken to be the product S1 × S2 , where for a = 1, 2, Sa is the ring of quasi-modular forms C[E 2 (τa ), E 4 (τa ), E 6 (τa )], and t = 1/2 . The matrices Ma are taken to be the Aa defined in (24). Thus sa (k, l) =
C(k, l, τa ) , √ kl
(43)
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I k
683
l
a
and for the edge E labeled as • −→ • we have ζ (E) = Aa (k, l).
(44)
Recalling the notation (42), we find Proposition 1. ([MT2], Prop. 4) For a = 1, 2, ζa¯ a¯ , 2πi
Ωaa = τa + Ωa a¯ = −
ζaa ¯ . 2πi
Furthermore, in the notation of Sect. 3.1 we have Proposition 2.
ζ12 = ζ21 =
(1 − ζ (L))−1 .
(45)
L∈R21
Beyond the intrinsic interest of this product formula, our main use of it will be to provide an alternate proof of Theorem 8 below. We therefore relegate the proof of Proposition 13 to the Appendix. We can similarly obtain necklace graphical expansions for the bilinear form ω(2) (x, y) and the holomorphic one forms νi (x). We introduce further distinguished valence one nodes labeled by 1, x for x ∈ Sˆa , the punctured torus (19). The set of edges {E} is augmented by edges with weights defined by: 1,x a 1,y ζ ( • −→ • ) = P2 (τa , x − y), x, y ∈ Sˆa , √ 1,x a k k a 1,x ζ ( • −→ •) = ζ (• −→ • ) = k k/2 Pk+1 (τa , x), x ∈ Sˆa ,
(46)
for elliptic functions (10). Similarly to (41) we consider chequered necklaces where one or both end points are 1, x-type labeled nodes. We thus define for x ∈ Sˆa and y ∈ Sˆb three isomorphism clas1,y x,y x,1 ses of oriented chequered necklaces denoted Nab , Nab and Nab with the following respective configurations 1,x
j
i
a
b
1
{ • −→ • . . . • −→ •}, 1
a
j
i
b
1,y
b
1,y
{• −→ • . . . • −→ • }, 1,x
a
j
i
{ • −→ • . . . • −→ • }. 1,y
(47) (48) (49)
x,y
x,1 , ζab and ζab denote the respective sum of the weights for each class. ComparLet ζab ing to (28) and (29) and applying (17) we find the following graphical expansions for the bilinear form ω(2) (x, y) and the holomorphic one forms νi (x):
Proposition 3. For a = 1, 2,
x, y ∈ Sˆa , x ∈ Sˆa , y ∈ Sˆa¯ , x,1 (1 + 1/2 ζaa )d x, x ∈ Sˆa , νa (x) = x,1 1/2 − ζ d x, x ∈ Sˆa¯ .
ω(2) (x, y) =
x,y
ζaa d xd y, x,y −ζa a¯ d xd y, aa ¯
(50) (51)
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4. Vertex Operator Algebras and the Li-Zamolodchikov Metric 4.1. Vertex operator algebras. We review some relevant aspects of vertex operator algebras ([FHL,FLM,Ka,LL,MN,MT6]). A vertex operator algebra (VOA) is a quadruple (V, Y, 1, ω) consisting of a Z-graded complex vector space V = n∈Z Vn , a linear map Y : V → (End V )[[z, z −1 ]], for formal parameter z, and a pair of distinguished vectors (states): the vacuum 1 ∈ V0 , and the conformal vector ω ∈ V2 . For each state v ∈ V the image under the Y map is the vertex operator Y (v, z) = v(n)z −n−1 , (52) n∈Z
with modes v(n) ∈ EndV , where Resz=0 z −1 Y (v, z)1 = v(−1)1 = v. Vertex operators satisfy the Jacobi identity or equivalently, operator locality or Borcherds’s identity for the modes (loc. cit.). The vertex operator for the conformal vector ω is defined as Y (w, z) = L(n)z −n−2 . n∈Z
The modes L(n) satisfy the Virasoro algebra of central charge c: [L(m), L(n)] = (m − n)L(m + n) + (m 3 − m)
c δm,−n . 12
We define the homogeneous space of weight k to be Vk = {v ∈ V |L(0)v = kv}, where we write wt(v) = k for v in Vk . Then as an operator on V we have v(n) : Vm → Vm+k−n−1 . In particular, the zero mode o(v) = v(wt(v) − 1) is a linear operator on Vm . A state v is said to be quasi-primary if L(1)v = 0 and primary if additionally L(2)v = 0. The subalgebra {L(−1), L(0), L(1)} generates a natural action on vertex operators associated with S L(2, C) Möbius transformations on z ([B1,DGM,FHL,Ka]). In particular, we note the inversion z → 1/z for which Y (v, z) → Y † (v, z) = Y (e z L(1) (−
1 L(0) 1 ) v, ). z2 z
(53)
Y † (v, z) is the adjoint vertex operator [FHL]. Under the dilatation z → az we have Y (v, z) → a L(0) Y (v, z)a −L(0) = Y (a L(0) v, az).
(54)
We also note ([BPZ,Z2]) that under a general origin-preserving conformal map z → w = φ(z), Y (v, z) → Y ((φ (z)) L(0) v, w),
(55)
for any primary vector v. We consider some particular VOAs, namely Heisenberg free boson and lattice VOAs. Consider an l-dimensional complex vector space (i.e., abelian Lie algebra) H equipped with a non-degenerate, symmetric, bilinear form ( , ) and a distinguished orthonormal
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
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basis a1 , a2 , . . . al . The corresponding affine Lie algebra is the Heisenberg Lie algebra ˆ = 0 and ˆ = H ⊗ C[t, t −1 ] ⊕ Ck with brackets [k, H] H [ai ⊗ t m , a j ⊗ t n ] = mδi, j δm,−n k.
(56)
Corresponding to an element λ in the dual space H∗ we consider the Fock space defined by the induced (Verma) module ˆ ⊗U (H⊗C[t]⊕Ck) C, M (λ) = U (H) where C is the 1-dimensional space annihilated by H ⊗ tC[t] and on which k acts as the identity and H ⊗ t 0 via the character λ; U denotes the universal enveloping algebra. There is a canonical identification of linear spaces M (λ) = S(H ⊗ t −1 C[t −1 ]), where S denotes the (graded) symmetric algebra. The Heisenberg free boson VOA M l corresponds to the case λ = 0. The Fock states v = a1 (−1)e1 .a1 (−2)e2 . . . a1 (−n)en . . . al (−1) f1 .al (−2) f2 . . . al (− p) f p .1, (57) for non-negative integers ei , . . . , f j form a basis of M l with ai (n) ≡ ai ⊗ t n . The vacuum 1 is canonically identified with the identity of M0 = C, while the weight 1 subspace M1 may be naturally identified with H. M l is a simple VOA of central charge l. Next we consider the case of a lattice vertex operator algebra VL associated to a positive-definite even lattice L (cf. [B1,FLM]). Thus L is a free abelian group of rank l equipped with a positive definite, integral bilinear form ( , ) : L ⊗ L → Z such that (α, α) is even for α ∈ L. Let H be the space C⊗Z L equipped with the C-linear extension of ( , ) to H ⊗ H and let M l be the corresponding Heisenberg VOA. The Fock space of the lattice theory may be described by the linear space VL = M l ⊗ C[L] = M l ⊗ eα , (58) α∈L
where C[L] denotes the group algebra of L with canonical basis eα , α ∈ L. M l may be identified with the subspace M l ⊗ e0 of VL , in which case M l is a subVOA of VL and the rightmost equation of (58) then displays the decomposition of VL into irreducible M l -modules. VL is a simple VOA of central charge l. Each 1 ⊗ eα ∈ VL is a primary state of weight 21 (α, α) with vertex operator (loc. cit.) Y (1 ⊗ eα , z) = Y− (1 ⊗ eα , z)Y+ (1 ⊗ eα , z)eα z α , α(±n) Y± (1 ⊗ eα , z) = exp(∓ z ∓n ). n
(59)
n>0
The operators eα ∈ C[L] obey eα eβ = (α, β)eα+β for 2-cocycle (α, β) satisfying (α, β)(β, α) = (−1)(α,β) .
(60)
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4.2. The Li-Zamolodchikov metric. A bilinear form , : V × V −→C is called invariant in case the following identity holds for all a, b, c ∈ V ([FHL]): Y (a, z)b, c = b, Y † (a, z)c,
(61)
with Y † (a, z) the adjoint operator (53). Remark 1. Note that a, b = Resw=0 w −1 Resz=0 z −1 Y (a, w)1, Y (b, z)1
= Resw=0 w −1 Resz=0 z −1 1, Y † (a, w)Y (b, z)1 = “1, Y (a, z = ∞)Y (b, z = 0)1”,
(62)
with w = 1/z, following (53). Thus the invariant bilinear form is equivalent to what is known as the (chiral) Zamolodchikov metric in Conformal Field Theory ([BPZ,P]). First note that any invariant bilinear form on V is necessarily symmetric by a theorem of [FHL]. Generally a VOA may have no non-zero invariant bilinear form, even if it is well-behaved in other ways. Examples where V is rational can be found in [DM]. Results of Li [Li] guarantee that if V0 is spanned by the vacuum vector 1 then the following hold: (a) V has at most one nonzero invariant bilinear form up to scalars; (b) if V has a nonzero invariant bilinear form , then the radical Rad , is the unique maximal ideal of V , and in particular V is simple if, and only if, , is non-degenerate. In this case, V is self-dual in the sense that V is isomorphic to the contragredient module V as a V -module. Conversely, if V is a self-dual VOA then it has a nondegenerate invariant bilinear form. All of the VOAs that occur in this paper satisfy these conditions, i.e., they are simple and self-dual with V0 = C1. Then if we normalize so that 1, 1 = 1 then , is unique and nondegenerate. We refer to this particular bilinear form as the Li-Zamolodchikov metric on V , or LiZ-metric for short. Remark 2. Uniqueness entails that the LiZ-metric on the tensor product V1 ⊗ V2 of a pair of simple VOAs satisfying the appropriate conditions is just the tensor product of the LiZ metrics on V1 and V2 . If a is a homogeneous, quasi-primary state, the component form of (61) reads a(n)b, c = (−1)wt(a) b, a(2wt(a) − n − 2)c.
(63)
In particular, since the conformal vector ω is quasi-primary of weight 2 we may take ω in place of a in (63) and obtain L(n)b, c = b, L(−n)c.
(64)
The case n = 0 of (64) shows that the homogeneous spaces Vn , Vm are orthogonal if n = m. Taking u = 1 and using a = a(−1)1 in (63) yields a, b = (−1)wt(a) 1, a(2wt(a) − 1)b,
(65)
for a quasi-primary, and this affords a practical way to compute the LiZ-metric. Consider the rank one Heisenberg (free boson) VOA M = M 1 generated by a weight one state a with (a, a) = 1. Then a, a = −1, a(1)a(−1)1 = −1. Using (56), it is straightforward to verify that in general the Fock basis consisting of vectors of the form v = a(−1)e1 . . . a(− p)e p .1,
(66)
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for non-negative integers {ei } is orthogonal with respect to the LiZ-metric, and that v, v =
(−i)ei ei !.
(67)
1≤i≤ p
This result generalizes in an obvious way for a rank l free boson VOA M l with Fock basis (57) following Remark 2. 5. Partition and n-Point Functions for Vertex Operator Algebras on a Riemann Surface In this section we consider the partition and n-point functions for a VOA on a Riemann surface of genus zero, one or two. Our definitions are based on sewing schemes for the given Riemann surface in terms of one or more surfaces of lower genus and are motivated by ideas in conformal field theory especially [FS,So1,P]. We assume throughout that V has a non-degenerate LiZ metric , . Then for any V basis {u (a) }, we may define the dual basis {u¯ (a) } with respect to the LiZ metric where u (a) , u¯ (b) = δab .
(68)
5.1. Genus zero case. We begin with the definition of the genus zero n-point function given by: (0)
Z V (v1 , z 1 ; . . . vn , z n ) = 1, Y (v1 , z 1 ) . . . Y (vn , z n )1,
(69) (0)
for v1 , . . . vn ∈ V . In particular, the genus zero partition (or 0-point) function is Z V = 1, 1 = 1. The genus zero n-point function is a rational function of z 1 , . . . z n , which we refer to as the insertion points, with possible poles at z i = z j , i = j determined from the locality of the vertex operators. Thus we may consider z 1 , . . . z n ∈ C ∪ {∞}, the (0) Riemann sphere, with Z V (v1 , z 1 ; . . . ; vn , z n ) evaluated for |z 1 | > |z 2 | > · · · > |z n | (e.g. [FHL,Z2,GG]). The n-point function has a canonical geometric interpretation for (0) primary vectors vi of L(0) weight wt(vi ). Then Z V (v1 , z 1 ; . . . ; vn , z n ) parameterizes a global meromorphic differential form on the Riemann sphere, (0)
(0)
FV (v1 , . . . vn ) = Z V (v1 , z 1 ; . . . ; vn , z n )
(dz i )wt(vi ) .
(70)
1≤i≤n (0)
It follows from (55) that FV is conformally invariant. This is the starting point of various algebraic-geometric approaches to n-point functions at higher genera e.g. [TUY,Z2]. However, it is important to note that the n-point function is intrinsically defined by its meromorphic pole structure in these approaches. Thus the partition or 0-point function is an undetermined overall normalization factor which is conventionally chosen to be unity (op. cit.). (0) It is instructive to consider FV in the context of a trivial sewing of two Riemann spheres parameterized by z 1 and z 2 to form another Riemann sphere as follows. For ra > 0, a = 1, 2, and a complex parameter satisfying || ≤ r1r2 , excise the open disks
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|z a | < ||ra−1 ¯ (recall convention (20)) and identify the annular regions ra ≥ |z a | ≥ −1 ||ra¯ via the sewing relation z 1 z 2 = .
(71)
Consider Z V(0) (v1 , x1 ; . . . vn , xn ) for quasi-primary vi with r1 ≥ |xi | ≥ ||r2−1 and let yi = /xi . Then for 0 ≤ k ≤ n − 1 we find from (68) that Y (vk+1 , xk+1 ) . . . Y (vn , xn )1 u, ¯ Y (vk+1 , xk+1 ) . . . Y (vn , xn )1u, =
(72)
r ≥0 u∈Vr
where the inner sum is taken over any basis for Vr . Thus (0)
Z V (v1 , x1 ; . . . vn , xn ) 1, Y (v1 , x1 ) . . . Y (vk , xk )uu, ¯ Y (vk+1 , xk+1 ) . . . Y (vn , xn )1. = r ≥0 u∈Vr
But (0)
1, Y (v1 , x1 ) . . . Y (vk , xk )u = Resz 1 =0 z 1−1 Z V (v1 , x1 ; . . . vk , xk ; u, z 1 ), and u, ¯ Y (vk+1 , xk+1 ) . . . Y (vn , xn )1 = 1, Y † (vn , xn ) . . . Y † (vk+1 , xk+1 )u ¯ L(0) † −L(0) L(0) † = 1, Y (vn , xn ) ... Y (vk+1 , xk+1 ) −L(0) L(0) u ¯ wt(vi ) −1 (0) r ¯ z2 ) (− 2 ) . = Resz 2 =0 z 2 Z V (vn , yn ; . . . vk+1 , yk+1 ; u, xi k+1≤i≤n The last equation holds since for quasiprimary states vi , the Mö bius transformation x → y = /x induces Y (vi , xi ) → L(0) Y † (vi , xi ) −L(0) = (−
wt(vi ) ) Y (vi , yi ). xi2
(73)
Thus we find: Proposition 4. For homogeneous quasiprimary states vi with the sewing scheme (71), we have FV(0) (v1 , . . . , vn ) (0) r Resz 1 =0 z 1−1 Z V (v1 , x1 ; . . . vk , xk ; u, z 1 ), = r ≥0
u∈Vr
(0) Resz 2 =0 z 2−1 Z V (vn , yn ; . . . vk+1 , yk+1 ; u, ¯ z2 )
1≤i≤k
(d xi )wt(vi )
(dyi )wt(vi ) ,
k+1≤i≤n
for any k, 0 ≤ k ≤ n − 1, i.e. the RHS is independent of the choice of Riemann sphere on which the insertion point of each state vi lies.
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5.2. Genus one case. We now consider genus one n-point functions defined in terms of a self-sewing of a Riemann sphere where punctures are located at the origin and the point at infinity [MT2]. Choose local coordinates z 1 = z in the neighborhood of the origin and z 2 = 1/z for z in the neighborhood of the point at infinity. For a = 1, 2 and ra > 0, identify the annular regions |q|ra−1 ¯ ≤ |z a | ≤ ra for complex q satisfying |q| ≤ r1r2 via the sewing relation z 1 z 2 = q, i.e. z = qz . Then it is straightforward to show that the annuli do not intersect for |q| < 1, and that q = exp(2πiτ ), where τ is the torus modular parameter (e.g. [MT2], Prop. 8). We define the genus one partition function by (1)
(1)
Z V (q) = Z V (τ ) qn Resz 2 =0 z 2−1 Resz 1 =0 z 1−1 1, Y † (u, z 2 )Y (u, ¯ z 1 )1, = q −c/24 n≥0
(74)
u∈Vn
where the inner sum is taken over any basis for Vn . The external factor of q −c/24 is (1) introduced in the usual way to enhance the modular properties of Z V (q) [Z1]. From (62) and (68 ) it follows that (1) dim Vn q n−c/24 = Tr V (q L(0)−c/24 ), (75) Z V (τ ) = n≥0
the standard graded trace definition. The genus one n-point function might similarly be defined by q r −c/24 Resz 2 =0 z 2−1 Resz 1 =0 z 1−1 1, Y † (u, z 2 )Y (v1 , x1 ) . . . Y (vn , xn )Y (u, ¯ z 1 )1 r ≥0
u∈Vr
= Tr V (Y (v1 , x1 ) . . . Y (vn , xn )q L(0)−c/24 ).
(76)
However, it is natural to consider the conformal map x = qz ≡ exp(z) in order to describe the elliptic properties of the n-point function [Z1]. Since from (55), for a primary state L(0) v, Y (v, w) → Y (qz v, qz ) under this conformal map, we are led to the following definition of the genus one n-point function (op. cit.): (1)
Z V (v1 , z 1 ; . . . vn , z n ; τ ) = Tr V (Y (qzL(0) v1 , qz 1 ) . . . Y (qzL(0) vn , qz n )q L(0)−c/24 ), n 1
(77)
which agrees with (76) for homogeneous primary states vi . Furthermore, for primary (1) vi of weight wt(vi ), Z V parameterizes a global meromorphic differential form on the torus (1) (1) FV (v1 , . . . vn ; τ ) = Z V (v1 , z 1 ; . . . vn , z n ; τ ) (dz i )wt(vi ) . (78) 1≤i≤n
Zhu introduced ([Z1]) a second VOA (V, Y [, ], 1, ω) ˜ which is isomorphic to (V, Y (, ), 1, ω). It has vertex operators Y [v, z] = v[n]z −n−1 = Y (qzL(0) v, qz − 1), (79) n∈Z
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and conformal vector ω˜ = ω −
c 24 1.
Let
Y [ω, ˜ z] =
L[n]z −n−2 ,
(80)
n∈Z
and write wt[v] = k if L[0]v = kv, V[k] = {v ∈ V |wt[v] = k}. Only primary vectors are homogeneous with respect to both L(0) and L[0] , in which case wt(v) = wt[v]. Similarly, we define the square bracket LiZ metric , sq which is invariant with respect to the square bracket adjoint. We denote 1-point functions by (1)
(1)
Z V (v, τ ) = Z V (v, z; τ ) = Tr V (o(v)q L(0)−c/24 ).
(81)
(1)
(Z V (v, τ ) is necessarily z independent.) Any n-point function can be expressed in terms of 1-point functions ([MT1], Lemma 3.1) as follows: (1)
Z V (v1 , z 1 ; . . . vn , z n ; τ ) = Z V(1) (Y [v1 , z 1 ] . . . Y [vn−1 , z n−1 ]Y [vn , z n ]1, τ ) =
(82)
Z V(1) (Y [v1 , z 1n ] . . . Y [vn−1 , z n−1n ]vn , τ ),
(83)
where z in = z i − z n . We may consider a trivial sewing of a torus with local coordinate z 1 to a Riemann sphere with local coordinate z 2 by identifying the annuli ra ≥ |z a | ≥ ||ra−1 ¯ via the (1) sewing relation z 1 z 2 = . Consider Z V (v1 , x1 ; . . . vn , xn ) for quasi-primary vi of L[0] weight wt[vi ], with r1 ≥ |xi | ≥ ||r2−1 , and let yi = /xi . Using (82), and employing the square bracket version of (72) with square bracket LiZ metric , sq , we have (1)
Z V (v1 , x1 ; . . . vn , xn ; τ ) (1) Z V (Y [v1 , x1 ] . . . Y [vk , xk ]u; τ )u, ¯ Y [vk+1 , xk+1 ] . . . Y [vn , xn ]1sq , = r ≥0 u∈V[r ]
where the inner sum is taken over any basis {u} of V[r ] , and {u} ¯ is the dual basis with respect to , sq . Now Z V(1) (Y [v1 , x1 ] . . . Y [vk , xk ]u; τ ) = Resz 1 =0 z 1−1 Z V(1) (v1 , x1 ; . . . vk , xk ; u, z 1 ; τ ). Using the isomorphism between the round and square bracket formalisms, we find as before that u, ¯ Y [vk+1 , xk+1 ] . . . Y [vn , xn ]1sq (0)
= r Resz 2 =0 z 2−1 Z V (vn , yn ; . . . vk+1 , yk+1 ; u, ¯ z2 )
k+1≤i≤n
We thus obtain a natural analogue of Proposition 4:
(−
wt[vi ] ) . xi2
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Proposition 5. For square bracket homogeneous quasiprimary states vi with the above sewing scheme, then we have FV(1) (v1 , . . . , vn ; τ ) (1) r Resz 1 =0 z 1−1 Z V (v1 , x1 ; . . . vk , xk ; u, z 1 ; τ ), = r ≥0
u∈V[r ]
Resz 2 =0 z 2−1 Z V(0) (vn , yn ; . . . vk+1 , yk+1 ; u, ¯ z2 )
(d xi )wt[vi ]
1≤i≤k
(dyi )wt[vi ] ,
k+1≤i≤n
and is independent of k = 0, 1, . . . n − 1, where the inner sum is taken over any basis {u} for V[r ] , {u} ¯ is the dual basis with respect to , sq . We note that all the above definitions can be naturally extended for any V -module N with vertex operators Y N (v, x), where the trace in (83) is taken over N and o(v) is replaced by o N (v) the Virasoro level preserving part of Y N (v, x). 5.3. Genus two case. Motivated by Proposition 5, we now discuss the formal definition of the genus two n-point function associated with the genus two -sewing scheme reviewed in Sect. 2.2. Recall that we sew together a pair of punctured tori Sˆa of (19) with modular parameters τa for a = 1, 2 via the sewing relation (22). We define the genus two n-point function for v1 , . . . vk inserted at x1 , . . . , xk ∈ Sˆ1 and vk+1 , . . . vn inserted at yk+1 , . . . , yn ∈ Sˆ2 for k = 0, 1, . . . n − 1 by (2)
Z V (v1 , x1 ; . . . vk , xk |vk+1 , yk+1 ; . . . vn , yn ; τ1 , τ2 , ) r Resz 1 =0 z 1−1 Z V(1) (v1 , x1 ; . . . vk , xk ; u, z 1 ; τ1 ) = r ≥0
u∈V[r ] (1)
¯ z 2 ; τ2 ), · Resz 2 =0 z 2−1 Z V (vn , yn ; . . . vk+1 , yk+1 ; u, (1) r Z V (Y [v1 , x1 ] . . . Y [vk , xk ]u, z 1 ; τ1 ) = r ≥0
·
u∈V[r ]
(1) Z V (Y [vn , yn ] . . . Y [vk+1 , yk+1 ]u, ¯ z 2 ; τ2 ),
(84)
where the inner sum is taken over any basis V[r ] and u¯ is the dual of u with respect to , sq . The last expression in (84) follows from (83). Remark 3. Following Remark 2 it is clear that the genus two n-point function on the tensor product V1 ⊗ V2 of a pair of simple VOAs is just the product of n-point functions on V1 and V2 . In this paper we mainly concentrate on the genus two partition function (i.e. the 0-point function) given by (1) (2) (1) Z V (τ1 , τ2 , ) = n Z V (u, τ1 )Z V (u, ¯ τ2 ). (85) n≥0
u∈V[n]
Some examples of n-point functions will also be computed. A general discussion of all genus two n-point functions for the Heisenberg VOA and its modules will appear elsewhere [MT5].
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Clearly the definition of the n-point function (84) depends on the choice of punctured torus on which the insertion points lie. However, by defining an associated formal differential form, we find the following genus two analogue of Propositions 4 and 5: Proposition 6. For xi ∈ Sˆ1 and yi ∈ Sˆ2 with xi yi = and square bracket homogeneous quasiprimary states vi , the formal differential form (2)
FV (v1 , . . . , vn ; τ1 , τ2 , ) ≡ Z V(2) (v1 , x1 ; . . . vk , xk |vk+1 , yk+1 ; . . . vn , yn ; τ1 , τ2 , ) (d xi )wt[vi ] (dyi )wt[vi ] , · 1≤i≤k
(86)
k+1≤i≤n
is independent of k = 0, 1, . . . n − 1. Proof. Consider the left torus contribution in the summand of (84) and expand Y [vk , xk ]u in a square bracket homogeneous basis: (1)
Z V (Y [v1 , x1 ] . . . Y [vk , xk ]u; τ1 ) (1) Z V (Y [v1 , x1 ] . . . Y [vk−1 , xk−1 ]w; τ1 )w, ¯ Y [vk , xk ]usq . = s≥0 w∈V[s]
But for quasi-primary vk and using (73) we find r w, ¯ Y [vk , xk ]usq = L[0] Y † [vk , xk ]w, ¯ usq = s (−
wt[vk ] ) Y [vk , yk ]w, ¯ usq , xk2
where xk yk = . Noting that (1) Z V (Y [vn , yn ] . . . Y [vk+1 , yk+1 ]u; ¯ τ2 )u, Y [vk , yk ]w ¯ sq r ≥0 u∈V[r ] (1)
= Z V (Y [vn , yn ] . . . Y [vk+1 , yk+1 ]Y [vk , yk ]w; ¯ τ2 ), we therefore find that (2)
Z V (v1 , x1 ; . . . vk , xk |vk+1 , yk+1 ; . . . vn , yn ; τ1 , τ2 , ) = (− 2 )wt[vk ] Z V(2) (v1 , x1 ; . . . vk−1 , xk−1 |vk , yk ; . . . vn , yn ; τ1 , τ2 , ). xk Hence (2)
Z V (v1 , x1 ; . . . vk , xk |vk+1 , yk+1 ; . . . vn , yn ; τ1 , τ2 , ) (d xi )wt[vi ] (dyi )wt[vi ] · 1≤i≤k k+1≤i≤n (2) = Z V (v1 , x1 ; . . . vk−1 , xk−1 |vk , yk ; . . . vn , yn ; τ1 , τ2 , ) (d xi )wt[vi ] (dyi )wt[vi ] . · 1≤i≤k−1 k≤i≤n
The result follows by repeated application of this identity.
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
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(2)
Remark 4. If Z V (τ1 , τ2 , ) is convergent on D , we conjecture that for primary states (2) v1 , . . . vn then FV (v1 , . . . , vn ; τ1 , τ2 , ) is a genus two global meromorphic form with possible poles only at coincident insertion points. Finally, note that all the above definitions can be naturally extended for any pair of V -modules N1 , N2 , where the left (right) 1-point function in (84) is considered for N1 (respectively N2 ). 6. The Heisenberg VOA In this section we compute closed formulas for the genus two partition function for the rank one Heisenberg VOA M and compute the n-point function for n Heisenberg vectors and the Virasoro vector 1-point function. We also discuss the modular properties of the partition function in some detail. (2)
6.1. The genus two partition function Z M (τ1 , τ2 , ). We wish to establish a closed formula for the genus two partition function Z (2) M (τ1 , τ2 , ) of (85) in terms of the infinite matrices A1 , A2 introduced in (24) of Sect. 2. Recalling the definition (25) we have: Theorem 5. Let M be the vertex operator algebra of one free boson. Then (2)
(1)
(1)
Z M (τ1 , τ2 , ) = Z M (τ1 )Z M (τ2 )(det(I − A1 A2 ))−1/2 ,
(87)
(1)
where Z M (τ ) = 1/η(τ ). Remark 5. From Remark 3 it follows that the genus two partition function for l free bosons M l is just the l th power of (87). (2)
Proof of Theorem. The genus two partition function Z M (τ1 , τ2 , ) of (85) is V basis independent. We choose the standard Fock vectors (in the square bracket formulation) v = a[−1]e1 . . . a[− p]e p 1.
(88)
Of course, these Fock vectors correspond in a natural 1-1 manner with unrestrictedpartitions, the state v(88) corresponding to a partition λ = {1e1 . . . p e p } with |λ| = i ei elements of n = 1≤i≤ p iei . We sometimes write v = v(λ) to indicate this correspondence. Furthermore, following (67), ⎞ ⎛ v(λ) = (−1)|λ| ⎝ i ei ei !⎠ v(λ). ¯ 1≤i≤ p
Thus with this diagonal basis we have (2)
Z M (τ1 , τ2 , ) =
λ={i ei }
(−1)|λ| (1) e iei Z (1) M (v(λ), τ1 )Z M (v(λ), τ2 ). ie ! i i i
(89)
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G. Mason, M. P. Tuite
Fig. 4. Two complete matchings
As discussed at length in [MT1], the partition λ may be thought of as a labeled set Φ = Φλ with ei elements labeled i. One of the main results of [MT1] (loc.cit. Corollary 1 and Eq. (53)) is that for even |λ|, (1) (1) Γ (φ), (90) Z M (v(λ), τ ) = Z M (τ ) φ∈F(Φλ )
with Γ (φ, τ ) = Γ (φ) =
C(r, s, τ ),
(91)
(r,s)
for C of (11), where φ ranges over the elements of F(Φλ ) (the fixed-point-free involutions in Σ(Φλ )) and (r, s) ranges over the orbits of φ on Φλ . If |λ| is odd then Z (1) M (v(λ), τ ) = 0. With this notation, (89) reads (2)
(1)
(1)
Z M (τ1 , τ2 , ) = Z M (τ1 )Z M (τ2 )
λ={i ei }
E(λ) e iei , i i i ei !
where λ ranges over all even |λ| unrestricted partitions and where we have set Γ1 (φ)Γ2 (ψ), E(λ) =
(92)
(93)
φ,ψ∈F(Φλ )
Γi (φ) = Γ (φ, τi ).
(94)
We now analyze the nature of the expression E(λ) more closely. This will lead us to the connection between Z (2) (τ1 , τ2 , ) and the chequered cycles discussed in Sect. 3.1. The idea is to use the technique employed in the proof of Proposition 4 of [MT1]. If we fix for a moment a partition λ then a pair of fixed-point-free involutions φ, ψ correspond (loc.cit.) to a pair of complete matchings μφ , μψ on the labeled set Φλ which we may represent pictorially as Fig. 4. Here, μφ is the matching with edges labeled 1, μψ the matching with edges labeled 2, and where we denote the (labeled) elements of Φλ by {r1 , s1 , . . . , rb , sb } = {s1 , t1 , . . . , sb , tb }. From this data we may create a chequered cycle in a natural way: starting with some node of Φλ , apply the involutions φ, ψ successively and repeatedly until the initial node is reached, using the complete matchings to generate a chequered cycle. The resulting chequered cycle corresponds to an orbit of ψφ considered as a cyclic subgroup of Σ(Φλ ). Repeat this process for each such orbit to obtain a chequered diagram D consisting of the union of the chequered cycles corresponding to all of the orbits of ψφ on
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
695
Fig. 5. Chequered diagram
Φλ . To illustrate, for the partition λ = {12 .2.32 .5} with matchings μφ = (13)(15)(23) and μψ = (11)(35)(23), the corresponding chequered diagram is shown in Fig. 5 Two chequered diagrams are isomorphic if there is a bijection on the nodes which preserves edges and labels of nodes and edges. If λ = {1e1 . . . p e p }, then Σ(Φλ ) acts on the chequered diagrams which have Φλ as an underlying set of labeled nodes. The Automorphism subgroup Aut(D), consisting of the elements of Σ(Φλ ) which preserves node labels, is isomorphic to Σe1 × · · · × Σe p . It induces all isomorphisms among these chequered diagrams. Of course |Aut(D)| = 1≤i≤ p ei !. We have almost established the first step in the proof of Theorem 5, namely Proposition 7. We have (2)
(1)
(1)
Z M (τ1 , τ2 , ) = Z M (τ1 )Z M (τ2 )
D
γ (D) , |Aut(D)|
(95)
where D ranges over isomorphism classes of chequered configurations and E(λ) γ (D) = e iei . i ii
(96)
Proposition 7 follows from what we have said together with (92). It is only necessary to point out that because the label subgroup induces all isomorphisms of chequered diagrams, when we sum over isomorphism classes of such diagrams in (92) the term i ei ! must be replaced by |Aut(D)|. Recalling the weights (40), we define ζ (D) = Π E ζ (E), where the product is taken over the edges E of D and ζ (E) is as in (44). Lemma 1. For all D we have ζ (D) = γ (D). {1e1
(97)
Proof. Let D be determined by a partition λ = ... and a pair of involutions φ, ψ ∈ F(Φλ ), and let (a, b), (r, s) range over the orbits of φ resp. ψ on Φλ . Then we find E(λ) (a,b) C(a, b, τ1 ) (r,s) C(r, s, τ2 ) e iei = e iei i i ii ii (a+b)/2 (r +s)/2 = C(r, s, τ2 ) C(a, b, τ1 ) √ √ rs ab (ab) (r s) = A1 (a, b) A2 (r, s) = ζ (D). (ab)
pe p }
(r s)
696
G. Mason, M. P. Tuite
We may represent a chequered diagram formally as a product m D= Li i
(98)
i
in case D is the disjoint union of unoriented chequered cycles L i with multiplicity m Aut(D) is isomorphic to the direct product of the groups Aut(L im i ) of order i . Then Aut(L m i ) = |Aut(L i )|m i m i ! so that i |Aut(D)| =
|Aut(L im i )|m i !.
i
Noting that the expression ζ (D) is multiplicative over disjoint unions of diagrams, we calculate D
ζ (L)k ζ (D) = |Aut(D)| |Aut(L)|k k! L k≥0
ζ (L) exp = |Aut(L)| L ζ (L) = exp , |Aut(L)| L
where L ranges over isomorphism classes of unoriented chequered cycles. Now Aut(L) is either a dihedral group of order 2r or a cyclic group of order r for some r ≥ 1, depending on whether L admits a reflection symmetry or not. If we now orient our cycles, say in a clockwise direction, then we can replace the previous sum over L by a sum over the set of (isomorphism classes of) oriented chequered cycles O to obtain ζ (D) 1 ζ (M) = exp . (99) |Aut(D)| 2 |Aut(M)| M∈O
D
Let O2n ⊂ O be denoted the set of oriented chequered cycles with 2n nodes. Then we have Lemma 2. Tr((A1 A2 )n ) =
M∈O2n
n ζ (M). |Aut(M)|
(100)
Proof. The contribution A1 (i 1 , i 2 )A2 (i 2 , i 3 ) . . . A2 (i 2n , i 1 ) to the left-hand-side of (100) is
ζ (M) for some M ∈ O2n with vertices i 1 , i 2 , . . . i 2n . Let σ = equal to the weight i 1 . . . i k . . . i 2n denote the order n permutation of the indices which generates i 3 . . . i k+2 . . . i 2 rotations of M. Then Aut(M) = σ m for some m = n/|Aut(M)|. Now sum over all i k to compute Tr((A1 A2 )n ), noting that for inequivalent M the weight ζ (M) occurs with multiplicity m. The lemma follows.
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We may now complete the proof of Theorem 5. From (99) and (100) we obtain ζ (D) 1 1 n = exp Tr( (A1 A2 ) ) |Aut(D)| 2 n n D
1 = exp(− Tr(log(1 − A1 A2 ))) 2 1 = det(exp(− (log(1 − A1 A2 )))) 2 = (det(1 − A1 A2 ))−1/2 . (2) Z M (τ1 , τ2 , )
We may also obtain a product formula for as follows. Recalling the notation (38), for each oriented chequered cycle M, Aut(M) is a cyclic group of order r for some r ≥ 1. Furthermore it is evident that there is a rotationless chequered cycle N with ζ (M) = ζ (N )r . Indeed, N may be obtained by taking a suitable consecutive sequence of n/r nodes of M, where n is the total number of nodes of M. We thus see that ζ (N )r ζ (M) = |Aut(M)| r M∈O N ∈R r ≥1 log(1 − ζ (N )). =− N ∈R
Then (99) implies
det(1 − A1 A2 ) =
(1 − ζ (N )),
(101)
N ∈R
and thus we obtain Theorem 6. Let M be the vertex operator algebra of one free boson. Then (2) Z M (τ1 , τ2 , ) =
(1)
(1)
Z M (τ1 )Z M (τ2 ) . 1/2 N ∈R (1 − ζ (N ))
(102)
6.2. Holomorphic and modular invariance properties. In Sect. 2.2 we reviewed the genus two -sewing formalism and introduced the domain D parameterizing the genus two surface. An immediate consequence of Theorem 5 and Theorem 1(b) is the following: (2)
Theorem 7. Z M (τ1 , τ2 , ) is holomorphic on the domain D . We next consider the automorphic properties of the genus two partition function with respect to the group G reviewed in Sect. 2.2. For two free bosons the genus one partition function is (1)
Z M 2 (τ ) =
1 . η(τ )2
(103)
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G. Mason, M. P. Tuite
Let χ be the character of S L(2, Z) defined by its action on η(τ )−2 , i.e. η(γ τ )−2 = χ (γ )η(τ )−2 (cτ + d)−1 ,
(104)
ab ∈ S L(2, Z). Recall (e.g. [Se]) that χ (γ ) is a twelfth root of unity. cd For a function f (τ ) on H1 , k ∈ Z and γ ∈ S L(2, Z), we define
where γ =
f (τ )|k γ = f (γ τ ) (cτ + d)−k ,
(105)
so that (1)
(1)
Z M 2 (τ )|−1 γ = χ (γ )Z M 2 (τ ).
(106)
The genus two partition function for two free bosons is Z (2) (τ , τ , ) = M2 1 2
1 η(τ1 )2 η(τ2 )2 det(I
− A1 A2 )
.
(107)
Analogously to (105), we define f (τ1 , τ2 , )|k γ = f (γ (τ1 , τ2 , )) det(CΩ + D)−k .
(108)
Here, the action of γ on the right-hand-side is as in (35). We have abused notation by adopting the following conventions in (108), which we continue to use below:
Ω = F (τ1 , τ2 , ), γ =
A B C D
∈ Sp(4, Z),
(109)
where F is as in Theorem 3, and γ is identified with an element of Sp(4, Z) via (35)-(36). Note that (108) defines a right action of G on functions f (τ1 , τ2 , ). We will establish the natural extension of (106) to the genus 2 case. To describe this, introduce the character χ (2) of G defined by χ (2) (γ1 γ2 β m ) = (−1)m χ (γ1 γ2 ),
γi ∈ Γi , i = 1, 2,
(notation as in (35), (36)). Thus χ (2) takes values which are twelfth roots of unity, and we have Theorem 8. If γ ∈ G then (2)
(2)
Z M 2 (τ1 , τ2 , )|−1 γ = χ (2) (γ )Z M 2 (τ1 , τ2 , ). Corollary 1. For the rank 24 Heisenberg VOA M 24 we have (2)
(2)
Z M 24 (τ1 , τ2 , )|−12 γ = Z M 24 (τ1 , τ2 , ), for γ ∈ G.
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
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Proof. We will give two different proofs of this result. Using the convention (109), we have to show that (2)
(2)
Z M 2 (γ (τ1 , τ2 , )) det(CΩ + D) = χ (2) (γ )Z M 2 (τ1 , τ2 , )
(110)
for γ ∈ G, and it is enough to do this for a generating set of G. If γ = β then the result is clear since det(CΩ + D) = χ (2) (β) = −1 and β exchanges τ1 and τ2 . So we may assume that γ = (γ1 , γ2 ) ∈ Γ1 × Γ2 . Our first proof utilizes the determinant formula (87) as follows. For γ1 ∈ Γ1 , define Aa (k, l, τa , ) = Aa (k, l, γ1 τa , c1 τ1+d1 ) following (35). We find from Sect. 4.4 of [MT2] that I − A1 A2 = I − A1 A2 − κΔA2 = (I − κ S).(I − A1 A2 ), c1 −1 where Δ(k, l) = δk1 δl1 , κ = − 2πi c1 τ1 +d1 and S(k, l) = δk1 (A2 (I − A1 A2 ) )(1, l). Since det(I − A1 A2 ) and det(I − A1 A2 ) are convergent on D we find
det(I − A1 A2 ) = det(I − κ S) det(I − A1 A2 ). +d1 which implies (110) for γ1 ∈ Γ1 . A similar But det(I − κ S) = 1 − κ S(1, 1) = cc11Ωτ11 1 +d1 proof applies for γ2 ∈ Γ2 . The second proof uses Proposition 2 together with (45), which tell us that (2)
Z M 2 (τ1 , τ2 , ) =
−2πiΩ12 (1 − ζ (L))−1 , η(τ1 )2 η(τ2 )2
(111)
R
where R = R\R21 . Now in general a term ζ (L) will not be invariant under the action of γ . This is because of the presence of quasi-modular terms Aa (1, 1) arising from E 2 (τa ). But it is clear from (35) and the definition (11) of C(k, l, τ ) together with its modular-invariance properties that if L ∈ R then such terms are absent and ζ (L) is invariant. So the product term in (111) is invariant under the action of γ . Next, we see from (35) that the expression η(τ1 )2 η(τ2 )2 is invariant under the action of γ up to a scalar χ (γ1 )χ (γ2 ) = χ (2) (γ ). This reduces the proof of (110) to showing that (γ1 , γ2 ) : Ω12 → Ω12 det(CΩ + D)−1 , and this is implicit in (37) upon applying Theorem 4. This completes the second proof of Theorem 8. Remark 6. An unusual feature of the formulas in Theorem 8 and Corollary 1 is that the definition of the automorphy factor det(CΩ + D) requires the map F : D → H2 . Thus although the automorphy factor resembles that of a Siegel modular form on H2 , the partition function is not a function on H2 but rather on D .
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6.3. Some genus two n-point functions. In this section we calculate some examples of genus two n-point functions for the rank one Heisenberg VOA M. A general analysis of all such functions will appear elsewhere [MT5]. We consider here the examples of the n-point function for the Heisenberg vector a and the 1-point function for the Virasoro vector ω. ˜ We find that the formal differential form (86) associated with the Heisenberg n-point function is described in terms of the global symmetric two form ω(2) [TUY], whereas the Virasoro 1-point function is described by the genus two projective connection [Gu]. These results illustrate the general conjecture made in Remark 4. We first consider the example of the Heisenberg vector 1-point function where a (1) is inserted at x on the left torus (say). Since Z M (Y [a, x]v; τ ) = 0 for a Fock vector (1) v = v(λ) for even |λ| and Z M (v; τ ) = 0 for odd |λ| [MT1] we find from (84) that (2) Z M (a, x|τ1 , τ2 , ) = 0. Consider next the 2-point function for two Heisenberg vectors inserted on the left torus at x1 , x2 ∈ Sˆ1 with (1) (2) (1) r Z M (Y [a, x1 ]Y [a, x2 ]v; τ1 )Z M (v; ¯ τ2 ). Z M (a, x1 ; a, x2 |τ1 , τ2 , ) = r ≥0
v∈M[r ]
(112) Following (86) of Proposition 6, we consider the associated formal differential form F (2) (a, a; τ1 , τ2 , ) for (112) and find that it is determined by the bilinear form ω(2) of (15): Theorem 9. The genus two Heisenberg vector 2-point function is (2) (a, a; τ1 , τ2 , ) = ω(2) Z (2) FM M (τ1 , τ2 , ).
(113)
Proof. The proof proceeds along the same lines as Theorem 5. As before, we let v(λ) denote a Heisenberg Fock vector (88) determined by an unrestricted partition λ = {1e1 . . . p e p } with label set Φλ . Define a label set for the three vectors a, a, v(λ) given by Φ = Φ1 ∪ Φ2 ∪ Φ3 for Φ1 , Φ2 = {1} and Φ3 = Φλ and let F(Φ) denote the set of fixed point free involutions on Φ. For φ = . . . (r s) . . . ∈ F(Φ), let Γ1 (x1 , x2 , φ) = (r,s) γ (r, s), where for r ∈ Φi and s ∈ Φ j , ⎧ ⎨ D(1, 1, x1 − x2 , τ1 ) = P2 (τ1 , x1 − x2 ), i = 1; j = 2 i = 1, 2; j = 3 γ (r, s) = D(1, s, xi , τ1 ) = s Ps+1 (τ1 , xi ), ⎩ C(r, s, τ ), i, j = 3, 1
(114)
for C, D of (11) and (12). Then following Corollary 1 of [MT1] we find for even |λ| that (1) (1) Γ1 (x1 , x2 , φ). Z M (Y [a, x1 ]Y [a, x2 ]v(λ), τ1 ) = Z M (τ1 ) φ∈F(Φ) (2)
Recalling that F (2) (a, a; τ1 , τ2 , ) = Z M (a, x1 ; a, x2 |τ1 , τ2 , )d x1 d x2 we then obtain the following analogue of (92): (1)
(1)
F (2) (a, a; τ1 , τ2 , ) = Z M (τ1 )Z M (τ2 )
E(x1 , x2 , λ) e iei d x1 d x2 , (115) i i i ei ! ei
λ={i }
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
701
where
E(x1 , x2 , λ) =
Γ1 (x1 , x2 , φ)Γ2 (ψ),
φ∈F(Φ), ψ∈F(Φλ )
with Γ2 (ψ) as before. The expression (115) can be interpreted as a sum of weights ζ (D) associated with isomorphism classes of chequered configurations D where, in this case, each configuration includes two distinguished valence one nodes of type 1, xi (see Sect. 3.2) corresponding to the label sets Φ1 , Φ2 = {1}. As before, ζ (D) = E ζ (E) for standard chequered edges E (44) augmented by the contributions for edges connected to the two valence one nodes with weights as in (46) (for a = 1). Then we find, as in Proposition 7, that (1) F (2) (a, a; τ1 , τ2 , ) = Z (1) M (τ1 )Z M (τ2 )
ζ (D) d x1 d x2 . i ei ! D
x1 ,x2 Each D can be decomposed into exactly one necklace configuration N of type N11 of (49) connecting the two distinguished nodes and a standard configuration Dˆ of the type ˆ Furthermore, if λ = {1e1 . . . p e p } appearing in Subsect. 6.1 so that ζ (D) = ζ (N )ζ ( D). is the subset of λ that labels Dˆ then the necklace contribution ζ (N ) occurs with multi . It follows that plicity i eei !! = |Aut(D)| ˆ i
|Aut( D)|
(1) F (2) (a, a; τ1 , τ2 , ) = Z (1) M (τ1 )Z M (τ2 )
Dˆ
ˆ ζ ( D) ˆ |Aut( D)|
ζ (N )d x1 d x2
x ,x N ∈N111 2
x1 ,x2 = Z (2) M (τ1 , τ2 , )ζ11 d x 1 d x 2 (2)
= Z M (τ1 , τ2 , )ω(2) (x1 , x2 ), using (50) of Proposition 3. Applying Proposition 6, the same two form arises for the other possible insertions of two Heisenberg vectors. Alternatively, a similar explicit calculation can be carried out in each case leading to the expressions for ω(2) described by (50). In a similar fashion one can generally show that the n-point function for n Heisenberg vectors vanishes for n odd and for n even is determined by the global symmetric meromorphic n form given by the symmetric (tensor) product Symn ω(2) = ω(2) (xr , xs ), (116) ψ (r,s)
where the sum is taken over the set of fixed point free involutions ψ = . . . (r s) . . . of the labels {1, . . . , n}. Then one finds Theorem 10. The genus two Heisenberg vector n-point function is given by the global symmetric meromorphic n-form (2)
(2)
F M (a, . . . , a; τ1 , τ2 , ) = Symn ω(2) Z M (τ1 , τ2 , ).
(117)
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Theorem 10 is in agreement with earlier results in [TUY] based on an assumed (2) analytic structure for the ratio F M (a, . . . , a; τ1 , τ2 , )/Z (2) M (τ1 , τ2 , ). Using the associativity property of a VOA, the genus two Heisenberg n -point function (117) is a generator of all genus two n -point functions for M in an analogous way to that described for genus one in [MT1]. This will be further developed elsewhere [MT5]. We illustrate this by computing the 1-point function for the Virasoro vector ω˜ = 21 a[−1]a. This is determined by the genus two projective connection defined by e.g. [Gu]
d xd y . (118) s (2) (x) = 6 lim ω(2) (x, y) − x→y (x − y)2 We then find Proposition 8. The genus two 1-point function for the Virasoro vector ω˜ is 1 (2) (2) s Z M (τ1 , τ2 , ). 12 Proof. Using the associativity property of a VOA we have [MT1] (2)
F M (ω; ˜ τ1 , τ2 , ) =
(1)
(119)
(1)
Z M (Y [a, x1 ]Y [a, x2 ]v; τ1 ) = Z M (Y [Y [a, x1 − x2 ]a, x2 ]v; τ1 ) (1)
=
Z M (v; τ1 ) (1) + 2Z M (Y [ω, ˜ x2 ]v; τ1 ) + · · · . (x1 − x2 )2
Hence using the Heisenberg 2-point function (112) we find (2) Z M (τ1 , τ2 , ) 1 (2) (2) d x1 d x2 F (ω; τ1 , τ2 , ) = lim Z M (a, x1 ; a, x2 |τ1 , τ2 , ) − x1 →x2 2 (x1 − x2 )2 =
1 (2) s (x1 )Z (2) M (τ1 , τ2 , ). 12
F (2) (ω; τ1 , τ2 , )
Notice that is not a global differential 2-form since forms under a general conformal transformation φ(x) ([Gu]) as
s (2) (x)
trans-
s (2) (φ(x)) = s (2) (x) − {φ; x}d x 2 , (120) ! 2 is the usual Schwarzian derivative. This property where {φ; x} = φφ − 23 φφ of the Virasoro 1-point function has previously been discussed many times in the physics and mathematics literature based on a variety of stronger assumptions e.g. [EO,TUY,FS,U,Z2]. 7. Heisenberg Modules, Lattice VOAs and Theta Series In this section we generalize the methods of Sect. 6 to compute the genus two partition function for a pair of Heisenberg modules. We consider the genus two n-point function for the Heisenberg vector and the Virasoro 1-point function. We apply these results to obtain closed formulas for the genus two partition function for a lattice VOA VL (in terms of the genus two Siegel theta function for L) and the ‘twisted’ genus two partition function for the Z -lattice VOA (in terms of the genus two Riemann theta function with characters). We finally derive a genus two Ward identity for the Virasoro 1-point function for these theories.
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7.1. Heisenberg modules. In this section we discuss the genus two partition function for a pair of simple Heisenberg modules M ⊗ eα1 and M ⊗ eα2 for α1 , α2 ∈ C. The partition function is then (1) (1) (τ1 , τ2 , ) = n Z M⊗eα1 (u, τ1 )Z M⊗eα2 (u, ¯ τ2 ), (121) Z α(2) 1 ,α2 n≥0
u∈M[n] (1)
where u ranges over any basis for M[n] . An explicit formula for Z M⊗eα (u, τ ) was given in [MT1] (Corollary 3 and Theorem 1). We are going to use these results, together with graphical techniques similar to those employed for free bosons in Sect. 6 to establish a closed formula for (121). Letting α.Ω.α = i, j=1,2 αi Ωi j α j , where Ωi j is the genus two period matrix we find Theorem 11. We have Z α(2) (τ1 , τ2 , ) = eiπ α.Ω.α Z (2) M (τ1 , τ2 , ). 1 ,α2
(122)
(2)
Z α1 ,α2 (τ1 , τ2 , ) is holomorphic on the domain D . Remark 7. This is a natural generalization of the genus one partition function relation 2 (1) (1) Z M⊗eα (τ ) = q α /2 Z M (τ ). Proof. Consider the Fock basis vectors v = v(λ) (cf. (88)) identified with partitions λ = {i ei } as in Sect. 6. Recall that λ defines a labeled set Φλ with ei nodes labeled i. It is useful to re-state Corollary 3 of [MT1] in the following form: 2 (1) (1) Z M⊗eα (v, τ ) = Z M (τ )q α /2 Γλ,α (φ). (123) φ
Here, φ ranges over the set of involutions Inv1 (Φλ ) = {φ ∈ Inv(Φλ )| p ∈ Fix(φ) ⇒ p has label 1}.
(124)
In words, φ is an involution in the symmetric group Σ(Φλ ) such that all fixed-points of φ carry the label 1. Note that this includes the fixed-point-free involutions, which were the only involutions which played a role in the case of free bosons. The main difference between the free bosonic VOA and its modules is the need to include additional involutions in the latter case. In particular, we note that permutations with |λ| odd can contribute in this case for λ = {i ei } with e1 odd. Finally, Γ (Ξ ), (125) Γλ,α (φ, τ ) = Γλ,α (φ) = Ξ
where Ξ ranges over the orbits (of length ≤ 2) of φ acting on Φλ and C(r, s, τ ), if Ξ = {r, s}, Γ (Ξ ) = α, if Ξ = {1}.
(126)
From (143)-(125) we get (τ1 , τ2 , ) Z α(2) 1 ,α2 (1)
(1)
= Z M (τ1 )Z M (τ2 )
(−1)|λ| E α ,α (λ) α 2 /2 α 2 /2 e1 2 q1 1 q2 2 iei , i i i ei ! ei
λ={i }
(127)
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where
E α1 ,α2 (λ) =
Γλ,α1 (φ, τ1 )Γλ,α2 (ψ, τ2 ).
(128)
φ,ψ∈Inv1 (Φλ )
(Compare with Eqs. (92)–(93).) Now we follow the proof of Proposition 7 to obtain an expression analogous to (95), namely (1) (τ1 , τ2 , ) = Z (1) Z α(2) M (τ1 )Z M (τ2 ) 1 ,α2
γα0 ,α (D) 1 2 D
|Aut(D)|
α 2 /2 α 2 /2
q1 1 q2 2 ,
(129)
the meaning of which we now enlarge upon. Compared to (95), the chequered diagrams D which occur in (129) are more general than before, in that they reflect the fact that the relevant involutions may now have fixed-points. Thus D is the union of its connected (as yet unoriented) components which are either chequered cycles as before or else chequered necklaces (see Sect. 3.2 ). Necklaces arise from orbits of the group ψφ on Φλ in which one of the nodes in the orbit is a fixed-point of φ or ψ. In that case the orbit will generally contain two such nodes which comprise the end nodes of the necklace. Note that these end nodes necessarily carry the label 1 (cf. (124)). There is degeneracy when both φ and ψ fix the node, in which case the degenerate necklace is obtained. Similarly to (96), the term γα01 ,α2 (D) in (129) is given by Γ (Ξ1 ) Ξ2 Γ (Ξ2 ) ie 0 |λ| Ξ1 i, e γα1 ,α2 (D) = (−1) (130) i i i where Ξ1 , Ξ2 range over the orbits of φ, ψ respectively on Φλ . As usual the summands in (129) are multiplicative over connected components of the chequered diagram. This applies, in particular, to the chequered cycles which occur, and these are independent of the lattice elements. As a result, (129) factors as a product of two expressions, the first a sum over diagrams consisting only of chequered cycles and the second a sum over diagrams consisting only of chequered necklaces. However, the first expression corresponds precisely to the genus two partition function for the free boson (Proposition 7). We thus obtain (2) Z α1 ,α2 (τ1 , τ2 , ) γα01 ,α2 (D N ) α12 /2 α22 /2 q = q2 , (131) (2) |Aut(D N )| 1 Z M (τ1 , τ2 , ) DN where here D N ranges over all chequered diagrams all of whose connected components are chequered necklaces. So Theorem 11 is reduced to establishing Proposition 9. We have eiπ α.Ω.α =
γα0 ,α (D N ) 1 2 DN
α 2 /2 α 2 /2 q1 1 q2 2 . N |Aut(D )|
(132)
We may apply the argument of (98) et. seq. to the inner sum in (132) to write it as an exponential expression exp{iπ(α12 τ1 + α22 τ2 ) +
γα0 ,α (N ) 1 2 N
|Aut(N )|
where N ranges over all unoriented chequered necklaces.
},
(133)
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Recall the isomorphism class Nab of oriented chequered necklaces of type ab as displayed in Fig. 3 of Sect. 3.2. Then (133) can be written as 1 0 γα1 ,α2 (Nab )}, (134) exp{iπ(α12 τ1 + α22 τ2 ) + 2 a,b∈{1,2} Nab
where here N ranges over oriented chequered necklaces of type ab. From (126) and (130) we see that the contribution of the end nodes to γα01 ,α2 (N ) is equal to αa¯ αb¯ for a type ab necklace. The remaining edge factors of γα01 ,α2 (N ) have product γ (N ) = ζ (N ) by Lemma 1. Finally, necklaces of type 11 and 22 arise from Fock vectors with an even number |λ| of permutation symbols whereas necklaces of type 12 and 21 arise from Fock vectors for odd |λ| leading to a further −1 contribution in (130) in these cases. Overall we find that γα01 ,α2 (Nab ) = (−1)a+b αa¯ αb¯ ζab , recalling ζab =
Nab N ∈Nab
ζ (N ). Hence (134) may be re-expressed as
α12 α2 (2πiτ1 + ζ22 ) + 2 (2πiτ2 + ζ11 ) − α1 α2 ζ21 }, (135) 2 2 where ζ12 = ζ21 . Expression (135) reproduces (145) on applying Proposition 1. (2) Finally we note from Theorems 3 and 7 that Z α1 ,α2 (τ1 , τ2 , ) is holomorphic on the domain D . This completes the proof of Theorem 11. exp{
7.2. Some genus two n-point functions. In this section we consider the genus two n-point functions for the Heisenberg vector a and the 1-point function for the Virasoro vector ω˜ for a pair of Heisenberg modules M ⊗ eαi . We again express each n-point function in terms of the associated formal differential form following (86) of Proposition 6. The results generalize those of Sect. 6.3. They are established by making use of similar methods, so that detailed proofs will not be given. We first consider the example of the Heisenberg vector a inserted on the left torus (2) (2) (say). Then Fα1 ,α2 (a; τ1 , τ2 , ) = Z α1 ,α2 (a, x1 |τ1 , τ2 , )d x1 is the corresponding differential form. Defining να = α1 ν1 + α2 ν2 , for holomorphic 1-forms νi , we find Theorem 12. The Heisenberg vector 1-point function for a pair of modules M⊗eα1 , M⊗ eα2 is Fα(2) (a; τ1 , τ2 , ) = να Z α(2) (τ1 , τ2 , ). 1 ,α2 1 ,α2
(136)
Proof. The proof proceeds along the same lines as Theorems 9 and 11. We find that ζ (D) (1) d x1 , (a; τ1 , τ2 , ) = Z (1) Fα(2) M (τ1 )Z M (τ2 ) 1 ,α2 i ei ! D
where the sum is taken over isomorphism classes of chequered configurations D where, in this case, each configuration includes one distinguished valence one node of type x1 ,1 1, x1 . Each D can be decomposed into exactly one necklace configuration of type N11 of (47), standard configurations of the type appearing in Theorem 5 and necklace contributions of type Nab of (41) as in Theorem 11. The result then follows on applying the graphical expansion for νi (x1 ) of (51).
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In a similar fashion one can generalize Theorem 10 concerning the n-point function for n Heisenberg vectors. This is determined by the global symmetric meromorphic n form given by a symmetric (tensor) product of να and ω(2) defined by ! ω(2) (xr , xs ) να (xt ), (137) Symn ω(2) , να = ψ (r,s)
(t)
where the sum is taken over the set of involutions ψ = . . . (r s) . . . (t) . . . of the labels {1, . . . , n}. Then one finds Theorem 13. The genus two Heisenberg vector n-point function for a pair of modules M ⊗ eα1 , M ⊗ eα2 is given by the global symmetric meromorphic n-form ! (2) Fα(2) ω Z α(2) (a, . . . , a; τ , τ , ) = Sym , ν . (138) 1 2 α n ,α 1 2 1 ,α2 Theorem 13 is a natural generalization of Corollary 4 of [MT1] concerning genus one n-point functions for a Heisenberg module. Similarly to Proposition 8 it follows that Proposition 10. The genus two 1-point function for a pair of modules M ⊗ eα1 , M ⊗ eα2 for the Virasoro vector ω˜ is
1 2 1 (2) (2) Z α(2) ν + s ˜ τ1 , τ2 , ) = (τ1 , τ2 , ). (139) Fα1 ,α2 (ω; 1 ,α2 2 α 12 Finally, let us introduce the differential operator [Fa,U] D=
1 2πi
1≤i≤ j≤2
νi ν j
∂ . ∂Ωi j
(140)
D maps differentiable functions on H2 to the space of holomorphic 2-forms (spanned by ν12 , ν22 , ν1 ν2 ) and is Sp(4, Z) invariant. It follows from Theorem 11 that (139) can be rewritten as a Ward identity
1 (2) iπ α.Ω.α (2) (2) e Fα1 ,α2 (ω; ˜ τ1 , τ2 , ) = Z M (τ1 , τ2 , ) D + s . (141) 12 Remark 8. In theoretical physics, a conformal Ward identity is an identity between different correlation functions following from conformal invariance e.g. [EO,DFMS]. Thus in (141) the Virasoro 1-point function is related to the normalized partition function (2) (2) Z α1 ,α2 /Z M = eiπ α.Ω.α . 7.3. Lattice VOAs. Let L be an even lattice of dimension l with VL the corresponding lattice VOA. The underlying Fock space is VL = M l ⊗ C[L] = ⊕α∈L M l ⊗ eα ,
(142)
where M l is the corresponding rank l Heisenberg free boson theory. We follow Sect. 4.1 and [MT1] concerning further notation for lattice theories.
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
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(2)
The general shape of Z VL (τ1 , τ2 , ) is as in (85). Note that the modes of a state u ⊗eα
map M l ⊗ eβ to M l ⊗ eα+β . Thus if α = 0 then Z V(1)L (u ⊗ eα , τ ) vanishes, and as a result we see that (1) (2) (1) Z VL (τ1 , τ2 , ) = n Z VL (u, τ1 )Z VL (u, ¯ τ2 ) n≥0
=
l u∈M[n]
α,β∈L n≥0
n
(1)
(1)
Z M l ⊗eα (u, τ1 )Z M l ⊗eβ (u, ¯ τ2 ).
(143)
l u∈M[n]
l . Viewing M l ⊗ eα as a simple module for M l we Here, u ranges over any basis for M[n] may employ Theorem 11 for each component to obtain
Theorem 14. We have (2)
(2)
(2)
Z VL (τ1 , τ2 , ) = Z M l (τ1 , τ2 , )θ L (Ω),
(144)
where θ L(2) (Ω) is the (genus two) Siegel theta function associated to L (e.g. [Fr]) (2) θ L (Ω) = exp(πi((α, α)Ω11 + 2(α, β)Ω12 + (β, β)Ω22 )). (145) α,β∈L
We can similarly compute n-point functions for n Heisenberg vectors a1 , . . . al using Theorem 13. We can also employ Proposition 10 and the Ward identity (141) to obtain the 1-point function for the Virasoro vector ω˜ = 21 i ai [−1]ai as follows: Proposition 11. The Virasoro 1-point function for a lattice VOA satisfies a genus two Ward identity
l (2) (2) (2) (2) θ L (Ω). ˜ τ1 , τ2 , ) = Z M l (τ1 , τ2 , ) D + s (146) FVL (ω; 12 The Ward identity (146) is reminiscent of some earlier results in physics and mathematics, e.g. [EO,KNTY]. (2) We briefly discuss the holomorphic and automorphic properties of Z VL (τ1 , τ2 , ) and (2)
˜ τ1 , τ2 , ). There is more that one can say here, but a fuller discussion must wait F L (ω; (2) for another time [MT5]. The function θ L (Ω) is a Siegel modular form of weight l/2 ([Fr]) for some subgroup of Sp(4, Z), in particular it is holomorphic on the Siegel upper half-space H2 . From Theorems 3, 7 and 14, we deduce (2)
Theorem 15. Z VL (τ1 , τ2 , ) is holomorphic on the domain D . (2)
We can obtain the automorphic properties of Z VL (τ1 , τ2 , ) in the same way using (2)
that for θ L (Ω) together with Theorem 8. Rather than do this explicitly, let us introduce a variation of the partition function, namely the normalized partition function (2) Zˆ VL (τ1 , τ2 , ) =
(2)
Z VL (τ1 , τ2 , ) (2)
Z M l (τ1 , τ2 , )
.
(147)
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G. Mason, M. P. Tuite
Bearing in mind the convention (109), what (144) says is that there is a commuting diagram of holomorphic maps F
D −→ H2 (2) (2) . ˆ θL Z VL C
(148)
Furthermore, the G-actions on the two functions in question are compatible. More precisely, if γ ∈ G then we have (2) (2) Zˆ VL (τ1 , τ2 , )|l/2 γ = Zˆ VL (γ (τ1 , τ2 , )) det(CΩ + D)−l/2
= θ L(2) (F (γ (τ1 , τ2 , ))) det(CΩ + D)−l/2 = = =
(2) θ L (γ (F (τ1 , τ2 , ))) det(CΩ θ L(2) (γ Ω) det(CΩ + D)−l/2 (2) θ L (Ω)|l/2 γ .
+ D)
−l/2
(from (148)) (from Theorem 4)
(from (109)) (149) (2)
For example, if the lattice L is unimodular as well as even then θ L is a Siegel modular form of weight l/2 on the full group Sp(4, Z). Then (149) informs us that Zˆ V(2)L (τ1 , τ2 , )|l/2 γ = Zˆ V(2)L (τ1 , τ2 , ), γ ∈ G, i.e. Zˆ V(2)L (τ1 , τ2 , ) is automorphic of weight l/2 with respect to the group G. Similar remarks may be made about the normalized Virasoro 1-point function defined by (2) Fˆ VL (ω; ˜ τ1 , τ2 , ) =
(2)
˜ τ1 , τ2 , ) FVL (ω; (2)
Z M l (τ1 , τ2 , )
,
(150)
which obeys the Ward identity
l (2) ˆ (2) (2) ˆ FVL (ω; ˜ τ1 , τ2 , ) = D + s Z VL (τ1 , τ2 , ). 12
(151)
Using the modular transformation properties of the projective connection (e.g. [Fa,U]) one finds that (151) enjoys the same modular properties as Zˆ V(2)L (τ1 , τ2 , ) i.e. Proposition 12. The normalized Virasoro 1-point function for a lattice VOA obeys
! l (2) ˆ (2) (τ1 , τ2 , )|l/2 γ , Z s Fˆ V(2) ( ω; ˜ τ , τ , )| γ = D + 1 2 l/2 VL L 12 for γ ∈ G.
(152)
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7.4. Rank two fermion vertex super algebra and the genus two riemann theta series. As a last application of Theorem 11, we briefly consider the rank two fermion Vertex Operator Super Algebra (VOSA) V = V (H, + 21 )2 . V can be decomposed in terms of a Heisenberg subVOA generated by a Heisenberg state a and irreducible modules M ⊗ em for m ∈ Z e.g. [Ka]. One can construct orbifold n-point functions for a pair g, h of commuting V automorphisms generated by a(0) [MTZ]. In particular, consider the 1-point function (which is non-vanishing only for u ∈ M) for a g-twisted sector for g = e−2πiλa(0) together with an automorphism h = e2πiμa(0) (for real λ, μ ) which can be expressed as (op. cit.) (1)
Z V ((g, h); u, τ ) = Tr V (ho(u)q L(0)+λ /2+λa(0)−1/24 ) e2πimμ Tr M⊗em+λ (o(u)q L(0)−1/24 ), = 2
(153)
m∈Z
utilizing the Heisenberg decomposition. In particular, the orbifold partition function is expressed in terms of the Jacobi theta series " # e−2πiλμ λ (1) ϑ (τ ), Z V ((g, h); τ ) = μ η(τ ) " # (154) 2 λ (τ ) = ϑ eiπ(m+λ) τ +2πi(m+λ)μ . μ m∈Z
Similarly to (121), it is natural to define the genus two orbifold partition function for a pair of gi -twisted sectors together with commuting automorphisms h i parameterized by λi , μi for i = 1, 2 with (1) n Z V ((g1 , h 1 ); u, τ1 )Z V(1) ((g2 , h 2 ); u, ¯ τ2 ), Z V(2) ((gi , h i ); τ1 , τ2 , ) = n≥0
u∈M[n]
(155) where u ranges over any basis for M[n] . A more detailed description of this and an alternative fermionic VOSA approach to this will be described elsewhere [TZ]. Here we decompose the genus one 1-point functions of (155) in terms of Heisenberg modules M ⊗ em i +λi to find, in the notation of (121), that (2) (2) Z V ((gi , h i ); τ1 , τ2 , ) = e2πim.μ Z m 1 +λ1 ,m 2 +λ2 (τ1 , τ2 , ), (156) m∈Z2
where here λ = (λ1 , λ2 ), μ = (μ1 , μ2 ) ∈ R2 and m = (m 1 , m 2 ) ∈ Z2 . Theorem 11 implies Theorem 16. We have (2) Z V ((gi , h i ); τ1 , τ2 , )
=e
−2πiλ.μ
(2) Z M (τ1 , τ2 , )θ (2)
" # λ (Ω), μ
for genus two Riemann theta function (e.g. [Mu]) " # (2) λ θ (Ω) = eiπ(m+λ).Ω.(m+λ)+2πi(m+λ).μ . μ
(157)
(158)
m∈Z2
As already described for lattice VOAs, one can similarly obtain a Ward identity for the Virasoro 1-point function analogous to (146) and (151) and analyze the modular properties of (157) and the Virasoro 1-point function under the action of G.
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8. Appendix - A Product Formula Here we continue the discussion initiated in Subsect. 3.1, with a view to proving Proposition 2. Consider a set of independent (non-commuting) variables xi indexed by the elements of a finite set I = {1, . . . , N }. The set of all distinct monomials xi1 . . . xin (n ≥ 0) may be considered as a basis for the tensor algebra associated with an N dimensional vector space. Call n the degree of the monomial xi1 . . . xin . Let ρ = ρn be the standard cyclic permutation which acts on monomials of degree n via ρ : xi1 . . . xin → xin xi1 . . . xin−1 . The rotation group of a given monomial x = xi1 . . . xin is the subgroup of ρn that leaves x invariant. Call x rotationless in case its rotation group is trivial. Let us say that two monomials x, y of degree n are equivalent in case y = ρnr (x) for some r ∈ Z , and denote the corresponding equivalence class by (x). We call these cycles. Note that equivalent monomials have the same rotation group, so we may meaningfully refer to the rotation group of a cycle. In particular, a rotationless cycle is a cycle whose representative monomials are themselves rotationless. Let Cn be the set of inequivalent cycles of degree n. It is convenient to identify a cycle (xi1 . . . xin ) with a cyclic labeled graph, that is, a graph with n vertices labeled xi1 , . . . , xin and with edges xi1 xi2 , . . . , xin xi1 . We will sometimes afflict the graph with one of the two possible orientations. Let M(I ) be the (multiplicative semigroup generated by) the rotationless cycles in the symbols xi , i ∈ I . There is an injection $ ι: Cn −→ M(I ) (159) n≥0
defined as follows. If (x) ∈ Cn has rotation group of order r then r |n and there is a rotationless monomial y such that x = y r . We then map (x) → (y)r . It is readily verified that this is well-defined. In this way, each cycle is mapped to a power of a rotationless cycle in M(I ). A typical element of M(I ) is uniquely expressible in the form f
f
f
p1 1 p2 2 . . . pk k ,
(160)
where p1 , . . . , pk are distinct rotationless cycles and f 1 , . . . , f k are non-negative integers. We call (160) the reduced form of an element in M(I ). A general element of M(I ) is then essentially a labeled graph, each of whose connected components are rotationless labeled cycles as discussed in Subsect. 3.1. Now consider a second finite set T together with a map F : T −→ I.
(161)
Thus elements of I label elements of T via the map F. F induces a natural map F : Σ(T ) −→ M(I ) from the symmetric group Σ(T ) as follows. For an element τ ∈ Σ(T ), write τ as a product of disjoint cycles τ = σ1 .σ2 . . .. We set F(τ ) = F(σ1 )F(σ2 ) . . ., so it suffices to define F(σ ) for a cycle σ = (s1 s2 . . .) with s1 , s2 , . . . ∈ T . In this case we set F(σ ) = ι((x F(s1 ) x F(s2 ) . . .)), where ι is as in (159). When written in the form (160), we call F(τ ) the reduced F-form of τ .
Free Bosonic Vertex Operator Algebras on Genus Two Riemann Surfaces I
711
For i ∈ I , let si = |F −1 (i)| be the number of elements in T with label i. So the number of elements in T is equal to i∈I si . We say that two elements τ1 , τ2 ∈ Σ(T ) are F-equivalent if they have the same reduced F-form, i.e. F(τ1 ) = F(τ2 ). We will show that each equivalence class contains the same number of elements. Precisely, Lemma 3. Each F-equivalence class containsprecisely i∈I si elements. In particular, the number of F-equivalence classes is |T |!/ i∈I si . Proof. An element τ ∈ Σ(T ) may be represented uniquely as
0 1 ··· M τ (0) τ (1) · · · τ (M) so that
F(τ ) =
F(0) F(1) · · · F(τ (0)) F(τ (1)) · · ·
F(M) F(τ (M))
with an obvious notation. Exactly si of the τ ( j) satisfy
so that there are follows.
F(τ ( j)) = xi
i∈I si
choices of τ which have a given image under F. The lemma
The next results employs notation introduced in Subsects. 3.1 and 3.2. Lemma 4. We have (I − M1 M2 )−1 (1, 1) = (1 −
As before, the left-hand-side of (162) means series with entries being quasi-modular forms. Proof of Lemma. We have (M1 M2 )n (1, 1) =
ζ (L))−1 .
(162)
L∈L21 n≥0 (M1 M2 )
n (1, 1). It is a certain power
M1 (1, k1 )M2 (k1 , k2 ) . . . M2 (k2n−1 , 1),
(163)
where the sum ranges over all choices of positive integers k1 , . . . , k2n−1 . Such a choice corresponds to a (isomorphism class of) chequered cycle L with 2n nodes and with at least one distinguished node, so that the left-hand-side of (162) is equal to ζ (L) L
summed over all such L. We can formally write L as a product L = L 1 L 2 . . . L p , where each L i ∈ L21 . This indicates that L has p distinguished nodes and that the L i are the edges of L between consecutive distinguished nodes, which can be naturally thought of as chequered cycles in L21 . Note that in the representation of L as such a product, the L i do not commute unless they are equal, moreover ζ is multiplicative. Then (I − M1 M2 )−1 (1, 1) = ζ (L 1 . . . L p ) = (1 − ζ (L))−1 , L i ∈L21
as required.
L∈L21
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G. Mason, M. P. Tuite
Proposition 13. We have (I − M1 M2 )−1 (1, 1) =
(1 − ζ (L))−1 .
(164)
L∈R21
Proof. By Lemma 4 we have (I − M1 M2 )−1 (1, 1) =
m(e1 , . . . , ek )ζ (L 1 )e1 . . . ζ (L k )ek ,
(165)
where the sum ranges over distinct elements L 1 , . . . L k of L21 and all k-tuples of nonnegative integers e1 , . . . , ek , and where the multiplicity is ( i ei )! . m(e1 , . . . , ek ) = i (ei !) Let S be the set consisting of ei copies of L i , 1 ≤ i ≤ k, let I be the integers between 1 and k, and let F : S −→ I be the obvious labelling map. A reduced F-form is then an element of M(I ), where the variables xi are now the L i . The free generators of M(I ), i.e. rotationless cycles in the xi , are naturally identified precisely with the elements of R21 , and Lemma 3 implies that each element of M(I ) corresponds to just one term under the summation in (165). Equation (164) follows immediately from this and the multiplicativity of ζ , and the proposition is proved. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Borcherds, R.E.: Vertex algebras, kac-moody algebras and the monster. Proc. Nat. Acad. Sc. 83, 3068–3071 (1986) [B2] Borcherds, R.E.: Monstrous moonshine and monstrous lie superalgebras. Inv. Math. 109, 405– 444 (1992) [BK] Belavin, A., Knizhnik, V.: Algebraic geometry and the geometry of strings. Phys. Lett. 168B, 201–206 (1986) [BPZ] Belavin, A., Polyakov, A., Zamolodchikov, A.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984) [CN] Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 12, 308–339 (1979) [DFMS] Di Francesco, P., Mathieu, P., Senechal, D.: Conformal Field Theory. Springer-Verlag, New York (1997) [DGM] Dolan, L., Goddard, P., Montague, P.: Conformal field theories, representations and lattice constructions. Commun. Math. Phys. 179, 61–120 (1996) [DM] Dong, C., Mason, G.: Shifted vertex operator algebras. Proc. Camb. Phil. Math. Soc. 141, 67–80 (2006) [DP] D’Hoker, E., Phong, D.H.: The geometry of string perturbation theory. Rev. Mod. Phys. 60, 917– 1065 (1988) [EO] Eguchi, T., Ooguri, H.: Conformal and current algebras on a general riemann surface. Nucl. Phys. B 282, 308–328 (1987) [Fa] Fay, J.: Theta functions on Riemann surfaces, Lecture Notes in Mathematics 352, Berlin-New York: Springer-Verlag, 1973 [FHL] Frenkel, I., Huang, Y., Lepowsky, J.: On axiomatic approaches to vertex operator algebras and modules. Mem. Amer. Math. Soc. 104 (1993) [FK] Farkas, H.M., Kra, I.: Riemann surfaces, New York: Springer-Verlag, 1980 [FLM] Frenkel, I., Lepowsky, J., Meurman,A.: Vertex operator algebras and the Monster, New York: Academic Press, 1988 [Fr] Freitag, E.: Siegelische modulfunktionen, Berlin and New York: Springer-Verlag, 1983
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Freidan, D., Shenker, S.: The analytic geometry of two dimensional conformal field theory. Nucl. Phys. B281, 509–545 (1987) [GG] Gaberdiel, M., Goddard, P.: Axiomatic conformal field theory. Commun. Math. Phys. 209, 549– 594 (2000) [GSW] Green, M., Schwartz, J., Witten, E.: Superstring theory Vol. 1, Cambridge: Cambridge University Press, 1987 [Gu] Gunning, R.C.: Lectures on Riemann surfaces . Princeton, NJ: Princeton Univ. Press, 1966 [Ka] Kac, V.: Vertex operator algebras for beginners. University Lecture Series, Vol. 10, Providence, RI: Amer. Math. Soc., 1998 [Kn] Knizhnik, V.G.: Multiloop amplitudes in the theory of quantum strings and complex geometry. Sov. Phys. Usp. 32, 945–971 (1989) [KNTY] Kawamoto, N., Namikawa, Y., Tsuchiya, A., Yamada, Y.: Geometric realization of conformal field theory on riemann surfaces. Commun. Math. Phys. 116, 247–308 (1988) [KZ] Kaneko, M., Zagier, D.: A generalized Jacobi theta function and quasimodular forms. In: The Moduli Space of Curves (Texel Island, 1994), Progr. in Math. 129, Boston: Birkhauser, 1995 [Li] Li, H.: Symmetric invariant bilinear forms on vertex operator algebras. J. Pure. Appl. Alg. 96, 279–297 (1994) [LL] Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations. Boston: Birkhäuser, 2004 [MN] Matsuo, A., Nagatomo, K,: Axioms for a vertex algebra and the locality of quantum fields. Math. Soc. Jap. Mem. 4 (1999) [MT1] Mason, G., Tuite, M.P.: Torus chiral n-point functions for free boson and lattice vertex operator algebras. Commun. Math. Phys. 235, 47–68 (2003) [MT2] Mason, G., Tuite, M.P.: On genus two riemann surfaces formed from sewn tori. Commun. Math. Phys. 270, 587–634 (2007) [MT3] Mason, G., Tuite, M.P.: Partition functions and chiral algebras. In: Lie algebras, vertex operator algebras and their applications (in honor of Jim Lepowsky and Robert L. Wilson), Contemporary Mathematics 442, Providence, RI: Amer. Math. Soc., 2007, pp. 401–410 [MT4] Mason, G., Tuite, M.P.: Free bosonic vertex operator algebras on genus two Riemann surfaces II. to appear [MT5] Mason, G., Tuite, M.P.: In preparation [MT6] Mason, G., Tuite, M.P.: Vertex operators and modular forms. In: Kirsten, K., Williams, F. (eds.) A Window into Zeta and Modular Physics, vol. 57, pp. 183–278. MSRI Publications, Cambridge University Press, Cambridge (2010) [MTZ] Mason, G., Tuite, M.P., Zuevsky, A.: Torus n-point functions for R-graded vertex operator superalgebras and continuous fermion orbifolds. Commun. Math. Phys. 283, 305–342 (2008) [Mu] Mumford, D.: Tata lectures on Theta I and II. Birkhäuser, Boston (1983) [P] Polchinski, J.: String theory. Volume I, Cambridge: Cambridge University Press, 1998 [Se] Serre, J-P.: A course in arithmetic. Springer-Verlag, Berlin (1978) [So1] Sonoda, H.: Sewing conformal field theories i. Nucl. Phys. B311, 401–416 (1988) [So2] Sonoda, H.: Sewing conformal field theories ii. Nucl. Phys. B311, 417–432 (1988) [T] Tuite, M.P.: Genus two meromorphic conformal field theory. CRM Proceedings and Lecture Notes 30, Providence, RI: Amer. Math. Soc., 2001, pp. 231–251 [TUY] Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. Adv. Stud. Pure. Math. 19, 459–566 (1989) [TZ] Tuite, M.P., Zuevsky, A.: Genus two partition and correlation functions for fermionic vertex operator superalgebras I, arXiv:1007.5203 [U] Ueno, K.: Introduction to conformal field theory with gauge symmetries. In: Geometry and Physics - Proceedings of the conference at Aarhus Univeristy, Aaarhus, Denmark, New York: Marcel Dekker, 1997 [Y] Yamada, A.: Precise variational formulas for abelian differentials. Kodai Math. J. 3, 114–143 (1980) [Z1] Zhu, Y.: Modular invariance of characters of vertex operator algebras. J. Amer. Math. Soc. 9, 237–302 (1996) [Z2] Zhu, Y.: Global vertex operators on riemann surfaces. Commun. Math. Phys. 165, 485–531 (1994) Communicated by Y. Kawahigashi
Commun. Math. Phys. 300, 715–739 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1125-5
Communications in
Mathematical Physics
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory M. Junge1 , C. Palazuelos1,2 , D. Pérez-García2 , I. Villanueva2 , M. M. Wolf3 1 Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois 61801-2975, USA. 2 Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain.
E-mail: [email protected]
3 Niels Bohr Institute, 2100 Copenhagen, Denmark
Received: 9 December 2009 / Accepted: 19 May 2010 Published online: 23 September 2010 – © Springer-Verlag 2010
Abstract: In this work we show that bipartite quantum states with Hilbert space √local n dimension n can violate a Bell inequality by a factor of order log2 n when observables with n possible outcomes are used. A central tool in the analysis is a close relation between this problem and operator space theory and, in particular, the very recent noncommutative L p embedding theory. As a consequence of this result, we obtain better Hilbert space dimension witnesses and quantum violations of Bell inequalities with better resistance to noise. 1. Introduction The fact that certain quantum correlations cannot be explained within any local classical theory is one of the most intriguing phenomena arising from quantum mechanics. It was discovered by Bell [3] as a way of testing the validity of Einstein-Podolski-Rosen’s belief that local hidden variable models are a possible underlying explanation of physical reality [21]. Bell realized that the innocent looking assumptions behind any local hidden variable theory lead to non-trivial restrictions on the strength of correlations. These constraints bear his name and are since called Bell inequalities [54]. Nowadays, the violation of Bell inequalities in quantum mechanics has become an indispensable tool in the modern development of Quantum Information and its applications cover a variety of areas: quantum cryptography, where it opens the possibility of getting unconditionally secure quantum key distribution [1,2,36,37]; entanglement detection, where it is the only way of experimentally detecting entanglement without a priori hypothesis on the behavior of the experiment; complexity theory, where it enriches the theory of multipartite interactive proof systems [5,13,14,19,24,30,31]; communication complexity (see the recent review [10]); Hilbert space dimension estimates [7,9,42,52,53]; etc. The violation of Bell inequalities also provides a natural way of quantifying the deviation from a local classical description. Unfortunately, computing the maximal violation for a given quantum state or Bell inequality turns out to be a daunting task except for very
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special cases. In [42] we uncovered a close connection between tripartite correlation Bell inequalities and the mathematical theory of operator spaces, developed since the 80’s as a noncommutative version of the classical Banach space theory. With these connections at hand, and with the wide tool-box of operator spaces, we were able to prove the existence of unbounded violations of tripartite correlation Bell inequalities. At the same time this resolved an open problem in pure mathematics related to Grothendieck’s famous fundamental theorem of the metric theory of tensor products. The relation of Grothendieck’s theorem with correlation Bell inequalities was long ago pointed out by Tsirelson [51]. In the present paper we show how operator spaces are again the appropriate language to deal with the general bipartite case, opening in this way an avenue for the understanding of general bipartite Bell inequalities. √ Then, using operator space techniques, we show how to get violations of log2nn , using n dimensional Hilbert spaces and n outputs. This almost closes the gap to the O(n) (resp. O(k 2 )) upper bound for such violations given in Proposition 2 (resp. in [18]). Again our techniques rely on probabilistic tools and use the classical random subspaces from Banach space theory which are now popular in signal processing, see [15]. The result in this paper implies the existence of better Hilbert space dimension witnesses and non-local quantum distributions with a higher resistance to noise –a desirable property when looking for loophole free Bell tests. Based on the results in [18], one can also obtains from our result new quantum-classical savings in communication complexity. 2. Statement of the Result We deal with the following scenario. Alice and Bob represent spatially separated observers which can choose among different observables labeled by x = 1, . . . , N in the case of Alice and y = 1, . . . , M in the case of Bob. The possible measurement outcomes are labeled by a = 1, . . . , K for Alice and b = 1, . . . , L for Bob. For simplicity we will always assume that M = N and K = L. We will refer to the observables x and y as inputs and call a and b outputs. The object under study is the probability distribution of a, b given x, y, that is, the set {P(ab|x y)}a,b,x,y . Being a probability distribution, {P(ab|x y)}a,b,x,y verifies – P(ab|x y) ≥ 0 for all x, y, a, b – ab P(ab|x y) = 1 for all x, y
(positivity), (normalization).
In addition, we recall that a probability distribution P = {P(ab|x y)}a,b,x,y is a) Non-signalling if
P(a, b|x, y) = P(b|y) is independent of x,
a
P(a, b|x, y) = P(a|x) is independent of y.
b
That is, Alice’s choice of inputs does not affect Bob’s marginal probability distribution and viceversa. This is physically motivated by the principle of Einstein locality which implies non-signalling if we assume that Alice and Bob are spacelike separated. We denote the set of non-signalling probability distributions by C (Fig. 1).
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
717
Fig. 1. {P(ab|x y)}a,b,x,y is the probability distribution of the measurement outcomes a, b,if Alice and Bob choose the observables labeled by x and y respectively
b) LHV (Local Hidden Variable) if P(a, b|x, y) =
Pω (a|x)Q ω (b|y)dP(ω)
(1)
for everyx, y, a, b, where (, , P) is a probability space, Pω (a|x) ≥ 0 for all a, x, ω, a Pω (a|x) = 1 for all x, ω and the analogous conditions for Q ω (b|y). We denote the set of LHV probability distributions by L. c) Quantum if there exist two Hilbert spaces H1 , H2 such that P(a, b|x, y) = tr (E xa ⊗ Fyb ρ)
(2)
for every x, y, a, b, where ρ ∈ B(H1 ⊗ H2 ) is a density operator and (E xa )x,a ⊂ B(H1 ), (Fyb ) y,b ⊂ B(H2 ) are two sets of operators representing POVM measure ments on Alice’s and Bob systems. That is, E xa ≥ 0 for every x, a, a E xa = 1 for every x, Fyb ≥ 0 for every y, b and b Fyb = 1 for every y. We denote the set of quantum probability distributions by Q. It is well known [18,51] that L Q C and that C contains exactly those elements of Aff(L) that are probability distributions. Here, N N Aff(L) = αi Pi : N ∈ N, Pi ∈ L, αi ∈ R, αi = 1 i=1
i=1
denotes the affine hull of the space L. Our aim is to quantify the maximum distance between L and the elements in Q. For that, we define the ‘largest Bell violation’ that a given P ∈ C may attain as [18] ν(P) = sup{M, P : M verifies |M, P | ≤ 1 for every P ∈ L} |M, P| , = sup sup M P ∈L |M, P | a,b N ,K where M = {Mx,y }x,y=1,a,b=1 is the “Bell inequality”1 acting on P by duality as a,b M, P = x,y,a,b P(a, b|x, y)Mx,y . 1 We are not using the linear constraints coming from the restriction that P is a probability distribution and, in addition, non-signalling. Our “Bell inequalities” are therefore redundant compared to the description presented in, for instance, [16]. However, all upper and lower bounds to the maximal possible violation of a Bell inequality given in this paper apply equally to the non-redundant case. To see it one notices that the linear constraints allow to map from a redundant to a non-redundant Bell inequality and viceversa without changing its violation.
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Thus, in order to measure how far the elements in Q can be from L, we are interested in computing the maximal possible Bell violation sup ν(P).
P∈Q
One can immediately see that ν(P) ≥ 1 and that ν(P) = 1 if and only if P ∈ L. For the first claim, it is enough to choose M constant. For the second, since L is a convex set, for all P ∈ C, P ∈ L, there exists M such that |M, P | ≤ 1 for all P ∈ L but |M, P| > 1. In Sect. 5, we will show that ν(P) measures the resistance of P to noise by proving 2 that ν(P) = π(P) − 1, where π(P) = inf{π : for all P ∈ L , π P + (1 − π )P ∈ L}. Notation. In the whole paper, given a real number x we write [x] to denote the smallest natural number p such that x ≤ p. Our main result states: Theorem 1. For every 2 ≤ n ∈ N and every 2 < q < ∞, there exists a bipartite q quantum probability distribution P with [n 2 ]n inputs per site, n + 1 outputs and Hilbert spaces of dimension 2n each such that 1
ν(P) ≥ D(q)n 2
− q2
,
where D(q) is a constant depending only on q. Actually, by the definition of ν, this result is equivalent to the following dual formulation Theorem 2. For every 2 ≤ n ∈ N and every 2 < q < ∞, we can find a Bell inequality q a,b )x,y,a,b , with x, y = 1, . . . , [n 2 ]n , a, b = 1 . . . , n + 1 such that M = (Mx,y 1 2 sup P∈Q |M, P| − ≥ D(q)n 2 q . sup P∈L |M, P|
Furthermore, the local Hilbert space dimension required to get this violation is at most 2n. It follows from the proof of Theorem 1 given in Sect. 9, that D(q) can be taken to be bigger than q12 . Then, making q = log n in Theorem 1 we obtain the following Corollary 1. For every 4 < n ∈ N there exists a bipartite quantum probability distribution P with [2 such that
log2 n 2
]n inputs, n + 1 outputs and Hilbert spaces each of dimension 2n ν(P)
√ n , log2 n
where denotes inequality up to a universal constant. An analogous consequence holds for Theorem 2.
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719
3. Upper Bounds We want to understand how close to optimality Theorem 1 is. In this direction, we present upper bounds to ν(P) depending on the number of outputs and the Hilbert space dimension. First, we have the following result from [18], showing a bound for ν(P) as a function of the number of outputs. Proposition 1. Independently of the Hilbert space dimension and the number of inputs, if P is a quantum probability distribution with k outputs then ν(P) = O(k 2 ). If we fix instead the Hilbert space dimension n, one can prove the following proposition. A proof is provided in Appendix 2. Proposition 2. Independently of the number of inputs and outputs, if P is a bipartite quantum probability distribution obtained with Hilbert spaces of local dimension n, then ν(P) = O(n). 4. Prior Bipartite Unbounded Violations As pointed out by Tsirelson [51], Grothendieck’s Theorem, which he himself called the fundamental theorem of the metric theory of tensor products, shows that we can not obtain unbounded violations in the case of correlation matrices. The first unbounded violations of Bell inequalities can be traced back to an application of the Raz parallel repetition theorem [48], which trivially ensures that the parallel repetition of the magic square game has a violation which grows with n inputs, n outputs and a Hilbert space of dimension n as n x for some x > 0. Similar results hold for any pseudo-telepathy game [6]. Even using the improved version of the Raz theorem given recently in [23,47], or the concentration theorem given in [47], the best nowadays −5 available lower bound using this technique seems to be not much better than (n 10 ). In [30], a spectacular improvement over this last quantity is made. They prove the existence, for each ν, of unique two provers one round games with n outputs and 2n /n inputs such that the quantum value of the game is larger than 1 − 54ν and the classical 1 one smaller than 2/n ν . This involves a violation of order (n 54 ). Their proof strongly relies on a deep result of Khot and Vishnoi in the context of complexity theory [32]. Also recently, I. Pitowsky [46] claims unbounded violations of bipartite Bell inequalities, but in a different sense. He restricts himself to an affine subspace of Aff(L) by forcing some probabilities to vanish. Thus, his unbounded violation does not lead to unbounded resistance to noise, unless the noise is assumed to belong to that subspace (which is unrealistic). √ Therefore, our log2nn violation with n outputs and local Hilbert space dimension n can be seen as an important improvement results. The price to pay is
to the previous the increase of the number of inputs to O [2
log2 n 2
]n . One can not rule out, however,
the possibility of obtaining similar bounds to ours pushing forward the previous kind of techniques.
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M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, M. M. Wolf
5. Resistance to Noise In the search for a loophole free Bell test, much has been written about non-locality2 in the presence of detector inefficiencies (see for instance [8,10–12,38,39,41]). This is modeled in [38] by adding an extra output ⊥ that means “no detection” in both Alice’s and Bob’s sides. If the detector efficiency is η, we then change the “perfect” probability distribution P = {P(ab|x y)}a,b,x,y by η2 P +(1−η2 )P , where P = {P (ab|x y)}a,b,x,y ∈ L is the local distribution defined by (1 − η2 )P (ab|x y) = η(1 − η)P(a|x)δb,⊥ + η(1 − η)δa,⊥ P(b|y) + (1 − η)2 δa,⊥ δb,⊥ . That is, we can interpret the inefficiency of the detector as a local noise added to the original probability distribution. The same happens with other classes of imperfections in the detectors: for instance if, with certain probability, the detector produces a random output instead of working properly. Therefore, in order to have non-local distributions even in the presence of noise, we fix P ∈ C and look at π(P) = inf{π ∈ [0, 1] : for all P ∈ L , π P + (1 − π )P ∈ L}.
(3)
It is then immediately seen that π(P) = 1 if and only if P ∈ L. The following proposition shows that this is “exactly” what we are estimating. Specifically, Proposition 3. For every P ∈ C, ν(P) =
2 − 1. π(P)
Proof. Let P ∈ C. We refer to [18] for the fact that ν(P) = inf{
I i=1
|αi | : P =
I
αi Pi , Pi ∈ L, αi ∈ R,
i=1
I
αi = 1}.
i=1
Dividing the αi ’s in positive and negative and using the convexity of L it is easy to see that indeed ν(P) = inf{α + β : P = α P − β P , P , P ∈ L, α, β ≥ 0, α − β = 1} = inf{2α − 1 : P = α P + (1 − α)P , P , P ∈ L, α ≥ 1} .
(4)
Let λ < π(P). By (3), for some P ∈ L, we have that λP + (1 − λ)P = P is again in L. This gives P = λ1 P − ( λ1 − 1)P and therefore, by Eq. (4), ν(P) ≤ λ2 − 1. Hence 2 − 1. ν(P) ≤ π(P) For the converse we use again Eq. (4) and start with λ = 2α − 1 ≥ 1 such that there exist P , P ∈ L giving P = α P +(1−α)P . This implies that P = α1 P +(1− α1 )P ∈ 2 2 L. By (3), π(P) ≥ α1 = λ+1 . That is, λ ≥ π(P) − 1. Since ν(P) is the infimum of such 2 λ’s we get that ν(P) ≥ π(P) − 1. 2 Non-locality means formally non-LHV probability distributions.
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
721
By our main result, this proves the existence of quantum probability distributions with n outputs and Hilbert spaces 2 ofdimension n which can withstand any local noise with relative strength 1 − log√n(n) (see the next section). It is interesting to note that, by Proposition 2, 1 − n1 is an upper bound for the maximal possible resistance to noise. However, if one restricts exclusively to the noise coming from inefficient detectors, one can obtain exponential resistance [38]. 6. Incomplete Probability Distributions We present here incomplete probability distributions. We need them for the statement and proof of Theorem 3. Our main result, Theorem 1, will follow as a corollary. We also use incomplete probability distributions to formalize the treatment given to noise in the previous section. We are interested in computing π(P) when we consider local probability distributions P with k + 1 outputs in Eq. (3), where k is the number of outputs of P. To this end we embed P into the space of probability distribution of k + 1 outputs just by adding the ˜ We denote Lk to the local corresponding 0’s and denoting the new distribution by P. distributions with k outputs (the other parameters are fixed). By Proposition 3, instead ˜ we can compute of π( P), ˜ = sup ν( P) M
˜ |M, P| . sup P ∈Lk+1 |M, P |
Of course, restricting with M’s which vanish on the index given by the extra output ˜ That is, we have ⊥ will give a lower bound for ν( P). ˜ ≥ sup ν( P) M
where P is now of the form P (a, b|x, y) =
|M, P| , sup P ∈Lk |M, P |
P(a|x, ω)Q(b|y, ω)dP(ω).
(5)
(, , P) is a probability space and for every λ, x (resp. y) (P(a|x, ω))ax (resp. (Q(b|y, ω))by ) is a sequence of positive numbers such that a P(a|x, ω) ≤ 1 (resp. b Q(b|y, ω) ≤ 1). We will say that such a P is an incomplete LVH probability distribution. In this section we deal with this kind of incomplete probability distributions and prove a generalization of our main result, Theorem 1, to this setting. This will formalize the claim stated in Sect. 5 concerning the existence of quantum probability distributions with n outputs and Hilbert spaces of dimension n which can withstand any local noise 2 with extra outputs and relative strength 1 − ( log√n(n) ). The rest of the paper is essentially devoted to prove the above mentioned generalization, from which Theorem 1 can be deduced. We say that P is an incomplete quantum probability distribution if there exist two Hilbert spaces H1 , H2 such that P(a, b|x, y) = tr (E xa ⊗ Fyb ρ)
(6)
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for every x, y, a, b, where ρ ∈ B(H1 ⊗ H2 ) is a density operator and (E xa )x,a ⊂ B(H1 ), (Fyb ) y,b ⊂ B(H2 ) are two sets of operators representing incomplete POVM measure ments on the Alice and Bob systems. That is, E xa ≥ 0 for every x, a, a E xa ≤ 1 for every x, Fyb ≥ 0 for every y, b and b E yb ≤ 1 for every y. We denote the set of incomplete quantum distributions by Qin and the set of incomplete LHV distributions by Lin . With these definitions at hand, we can introduce a,b N ,K Definition 1. Given a linear functional (Bell inequality) M = (Mx,y )x,y=1,a,b=1 , we define the Classical bound of M as the number
BC (M) = sup{|M, P| : P ∈ Lin } and the Quantum bound of M as B Q (M) = sup{|M, P| : P ∈ Qin }. We define the largest quantum violation of M as the positive number L V (M) =
B Q (M) . BC (M)
(7)
Remark 1. It is easy to see that BC (M) = 0 implies B Q (M) = 0 for every M. We will rule out these cases because they lack interest. The next lemma relates L V (M) to the analogous quantity for complete probability distributions. Therefore it allows us to prove Theorem 2 (and, thus, Theorem 1) from Theorem 3. a,b Lemma 1. Suppose we have a linear functional (Mx,y )x,y,a,b , x, y = 1, . . . , N , a, b = 1, . . . , K such that L V (M) = C. Then, there exists another linear functional a,b ( Mˆ x,y )x,y,a,b , x, y = 1, . . . , N , a, b = 1, . . . , K + 1 such that
ˆ P| sup P∈Q | M, = C. ˆ P| sup P∈L | M, K +1,b a,K +1 Proof. It is enough to define Mˆ as the extension of M for which Mˆ x,y = 0, Mˆ x,y = 0.
The generalization of Theorem 1 to this context is the following one. Theorem 3. For every 2 ≤ n ∈ N and every 2 < q < ∞, we can find a linear functional q a,b M = (Mx,y )x,y,a,b , x, y = 1, . . . , [n 2 ]n , a, b = 1 . . . , n such that 1
L V (M) D(q)n 2
− q2
.
The local Hilbert space dimension required to get this violation is at most 2n. Our next two results follow straightforwardly.
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
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a,b Corollary 2. For every 4 < n ∈ N we can find a linear functional M = (Mx,y )x,y,a,b ,
x, y = 1, . . . , [2
log2 n 2
]n , a, b = 1 . . . , n such that √ L V (M)
n
log2 n
.
The local Hilbert space dimension needed to get this violation is at most 2n. log2 n
Corollary 3. For all 4 < n ∈ N, there exists a probability distribution P with [2 2 ]n inputs, n + 1 outputs and Hilbert space dimension 2n which can withstand any local 2 √ n ). noise with extra outputs and relative strength 1 − ( log n 7. Bounds for the Hilbert Space Dimension The interest in testing the Hilbert space dimension started with a crucial observation made in [2]. In that paper, the authors observe that the standard security proofs for the BB84 protocol [33,49] assume a given dimension in the Hilbert space and they can fail if this assumption is dropped. In [9], motivated by that, the authors define the concept of “dimension witness” and show some examples in low dimensions. Since then, several contributions to the field have appeared with different approaches: Bell inequalities [7,52], quantum random access codes [53] or quantum evolutions [55]. We define Qd to be the distributions in Q with the extra restriction that the Hilbert spaces H1 , H2 appearing in the definition are d-dimensional. With this notation, a dimension witness for dimension d is simply a “Bell inequality” Md,n such that |Md,n , Pd | ≤ Cd for all Pd ∈ Qd , and for such that there exists P ∈ Qn with |Md,n , P| > Cd . In the case of binary outcomes, Briët, Buhrman and Toner [7] and Vertesi and Pal [52] have shown how to get dimension estimates for any dimension. However, in their case sup Md,n
sup Pn ∈Qn |Md,n , Pn | ∈ [1, K G ]. sup Pd ∈Qd |Md,n , Pd |
This means that the resolution of the considered witnesses is bounded by Grothendieck’s constant K G and indeed could vanish with increasing dimension. It would be therefore desirable to get sup Md,n
sup Pn ∈Qn |Md,n , Pn | sup Pd ∈Qd |Md,n , Pd |
−→ ∞.
n≥d→∞
(8)
For two outcomes this was shown to be possible in the tripartite case [42]. Our main theorem, together with Proposition 2 implies that Theorem 4. For any d, n we can define dimension estimates Md,n verifying sup Md,n
√ sup Pn ∈Qn |Md,n , Pn | n . = sup Pd ∈Qd |Md,n , Pd | log2 (n)d
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M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, M. M. Wolf
8. Mathematical Tools and Connections In this section we will introduce the basic notions about operator spaces which we will need along this work. We do recommend [20] and [43] for a much more complete reference. We will denote by Mn (resp. Mmn ) the space of complex n × n (resp. m × n) matrices. The theory of operator spaces was born with the work of Effros and Ruan in the 80’s, see for instance [20,43]. They characterized, in an abstract sense, the structure of the closed subspaces of B(H ), the space of bounded linear operators on a Hilbert space. Formally, an operator space is a complex vector space E and a sequence of norms · n in the space of E-valued matrices Mn (E) = Mn ⊗ E.3 which verify the following two properties: 1. For every n, m ∈ N, x ∈ Mm (E), a ∈ Mnm and b ∈ Mmn we have that axbn ≤ axm b. 2. For every n, m ∈ N, x ∈ Mn (E), y ∈ Mm (E), we have that
x 0
= max{xn , ym }.
0y n+m
Any C ∗ -algebra A has a natural operator space structure induced by its natural embedding j : A → B(H ). Indeed, it is enough to consider the sequence of norms on Mn ⊗ A defined by the embedding id ⊗ j : Mn ⊗ A → Mn ⊗ B(H ) = B(n2 ⊗ H ). In particular, k∞ has a natural operator space structure. To compute it we isometrically embed k∞ into the diagonal of Mk and then, given x = i Ai ⊗ ei ∈ Mn (k∞ ) = Mn ⊗ k∞ , we have
xn = Ai ⊗ |ii| = max Ai Mn . (9)
i i
Mnk
In order to attain a better understanding of the differences between the Banach space category and the operator space category, we need to look not only at the spaces, but also at the morphisms, that is, the operations which preserve the structure. We will have to consider now the so called completely bounded maps. They are linear maps u : E −→ F between operator spaces such that all the dilations u n = 1n ⊗u : Mn ⊗ E = Mn (E) −→ Mn ⊗ F = Mn (F) are bounded. The cb-norm of u is then defined as ucb = supn u n . We will call C B(E, F) the resulting normed space. It has a natural operator space structure induced by Mn (C B(E, F)) = C B(E, Mn (F)). We can analogously define the notion of a complete isomorphism/isometry (see [20,43]). The so called minimal tensor product of two operator spaces E ⊂ B(H ) and F ⊂ B(K ) is defined as the operator space E ⊗min F with the structure inherited from the induced embedding E ⊗ F ⊂ B(H ⊗ K ). In particular, Mn (E) = Mn ⊗min E for every operator space E. The tensor norm min in the category of operator spaces will play the role of the so called norm in the classical theory of tensor norms in Banach spaces [17]. In particular min is injective, in the sense that if E ⊂ X and F ⊂ Y completely isomorphic/isometric, then E ⊗min F ⊂ X ⊗min Y completely isomorphic/isometric. 3 We are identifying M (E) and M ⊗ E via the canonical isomorphism. n n
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
725
The analogue in the operator space category of the π tensor norm is the projective tensor norm, defined as u Mn (E⊗∧ F) = inf{α Mn,lm x Ml (E) y Mm (F) β Mlm,n : u = α(x ⊗ y)β}, where u = α(x ⊗ y)β means the matrix product αr,i p β jq,s |r s| ⊗ xi j ⊗ y pq ∈ Mn ⊗ E ⊗ F. u= r si j pq
Both tensor norms, ∧ and min, are associative and commutative and they share the duality relations which verify π and in the context of Banach spaces. In particular, for finite dimensional operator spaces we have the natural completely isometric identifications (E ⊗∧ F)∗ = C B 2 (E, F; C) = C B(E, F ∗ ) = E ∗ ⊗min F ∗ ,
(10)
where, given an operator space E, we define its dual operator space E ∗ via the identification Mn (E ∗ ) = C B(E, Mn ). Given a Banach space X , we can consider in it different operator space structures or, equivalently, different isometric embeddings of X into B(H ) which lead to different families of matrix norms. For example we may embed an n-dimensional Hilbert space as the column Cn = { αk |k0| : αk ∈ C} or row space Rn = { αk |0k| : αk ∈ C}. k
k
Let us note that † 1 1 Ai ⊗ ei Mm ⊗min Rn = Ai Ai† 2 , Ai ⊗ ei Mm ⊗min Cn = Ai Ai 2 . i
i
i
i
Using matrices of the form Ai = |i0| we deduce the well-known fact that this yields different matrix norms (see [43] for more details). The natural operator space structure on n1 is the one obtained by the duality (n∞ )∗ = n 1 and it can be seen that for every operator space X the space ∞ ⊗min X (resp. 1 ⊗∧ X ) coincides, as a Banach space, with ∞ ⊗ X = ∞ (X ) (resp. 1 ⊗π X = 1 (X )). Furthermore, for every operator space X , the natural operator space structure defined on ⊕n1 X (see [43]) allows us to identify completely isometrically this operator space with n1 ⊗∧ X via the natural identification. This operator space is denoted by n1 (X ). Analogous reasonings hold for the operator space n∞ (X ). Actually, by the comments above, it follows that (n∞ (X ))∗ = n1 (X ∗ ) (completely isometrically) for every finite dimensional operator space X . The operator space L p -embedding theory has been developed in the last years. Some of the most important results of classical Banach space theory, as well as probability theory and harmonic analysis, have found analogous versions in the noncommutative case [26,27,29]. As we did in a previous work [42], we will reduce the problem of separating the LHV from the Quantum probability distributions to the problem of separating the epsilon and
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M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, M. M. Wolf
the min norm on the tensor products of certain operator spaces (see Sect. 8.1). The noncommutative L p -embedding theory will allow us to find “good” subspaces where we can compute the above mentioned tensor norms. Our new tool in this paper are spaces constructed as sums and intersection in interpolation theory. These spaces already play an important role for embedding problems in operator space theory [27,28]. For fixed t > 0 and m ∈ N, we consider the operator m m space K (t; m ∞ , Rm + C m , 1 ) defined as C endowed with the following sequence of matrix norms: x Mn (K (t;m∞ ,Rm +Cm ,m1 )) =
inf
x=x1 +x2 +x3
{x1 Mn (m∞ ) +
√
tx2 Mn (Rm +Cm ) + tx3 Mn (m1 ) }.
As in classical interpolation theory [28, Lemmas 3.1, 3.5], one can determine the dual space: m ∗ −1 m m K (t; m ∞ , Rm + C m , 1 ) ∼ J (t ; 1 , Rm ∩ C m , ∞ ),
where
J (t −1 ; m 1,
Rm ∩ Cm , m ∞)
(11)
denotes the operator space given by
a Mn (J (t −1 ;m ,Rm ∩Cm ,m∞ )) 1
1
= max{a Mn (m1 ) , t − 2 a Mn (R m ∩C m ) , t −1 a Mn (m∞ ) }. Here, ∼ denotes a complete isomorphism up to a universal constant (in this case 16). The following result will be crucial in our work: Theorem 5 ([28], Theorem 3.6). Let (, μ) be a measure space such that μ() = n. Then, the application j : L 1 () + L r2 () + L c2 () + L ∞ () → L 1 (n ; n∞ ), defined by n 1 j ( f )(ω1 , · · · , ωn ) = n f (ωk )ek n k=1
is a complete embedding (with absolute constants). This result is stated in a much more general context in [28]. In Appendix A.2 we boil the rather heavy notation down to the result used here. 8.1. Connection to the “ min vs ε problem”. In this section we will connect the Classical (resp. Quantum) bounds of a given linear functional M (see Definition 1) to two natural tensor norms in the framework of classical Banach spaces and operator spaces. a,b N ,K )x,y=1,a,b=1 We associate a four dimensional matrix with coefficients M = (Mx,y with the corresponding tensor N ,K
a,b Mx,y (ex ⊗ ea ) ⊗ (e y ⊗ eb )
x,y=1,a,b=1 K ) ⊗ N ( K ). Our next result deals with the Classical considered as an element of 1N (∞ ∞ 1 bound BC (recall Definition 1 in Sect. 6):
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727
a,b N ,K Proposition 4. Given M = (Mx,y )x,y=1,a,b=1 , we have the following equivalence:
BC (M) ≤ M N ( K )⊗ N ( K ) ≤ 4BC (M). ∞
1
∞
1
Proof. By duality, it follows that ⎧ ⎫ ⎨ ⎬ a,b a,b a,b Mx,y Tx,y : Tx,y ∈ B∞ M N ( K )⊗ N ( K ) = sup N ( K )⊗ N ( K ) . π ∞ 1 ⎭ ∞ 1 1 1 ∞ ⎩ a,x,b,y
Since B∞ N ( K )⊗ N ( K ) is the convex hull of the set {x ⊗ y : x ∈ B N ( K ) , y ∈ B N ( K ) }, π ∞ 1 ∞ 1 ∞ 1 1 we have that ⎧ ⎫ ⎨ ⎬ a,b Mx,y Pω (x, a)Q ω (y, b)dP(ω) , M N ( K )⊗ N ( K ) = sup ∞ 1 1 ∞ ⎩ ⎭ a,x,b,y
where the sup is taken over all a) (, P) probability space, b) a=1,··· ,K |Pω (x, a)| ≤ 1 for every x = 1, · · · , N and every ω, c) b=1,··· ,K |Q ω (y, b)| ≤ 1 for every y = 1, · · · , N and every ω. Using this, the first inequality follows. For the second one it is enough to consider the positive and negative part of each Pω (x, a) and Q ω (y, b). Next we deal with the Quantum bound: a,b N ,K Theorem 6. Given M = (Mx,y )x,y=1,a,b=1 , we have the following equivalence:
B Q (M) ≤ M N ( K )⊗min N ( K ) ≤ 16B Q (M). 1
∞
∞
1
Before we prove the result, let us note that M N ( K )⊗min N ( K ) = sup{(u ⊗ v)(M) B(H )⊗min B(H ) }, 1
∞
1
∞
(12)
K ) → B(H ) which verify u ≤ where the sup is taken over all the operators u : 1N (∞ cb 1 (and the same for v). We will use the following lemma: Lemma 2. Let (Tn )n ⊂ B(H ) be a sequence of positive operators. Then n Tn B(H ) = n Tn ⊗en B(H )⊗min 1 , where we are considering the natural operator space structure on 1 .
Proof. It can be seen [43, Lemma 8. 9] that, for every sequence (an )n in B(H ), we have 1 1 an ⊗ en B(H )⊗min 1 = inf{ bn bn∗ 2 cn∗ cn 2 }, n
n
n
where the inf is taken over all possible decompositions an = bn cn . Now, if we take 1 bn = cn = (Tn ) 2 , we obtain Tn ⊗ en B(H )⊗min 1 ≤ Tn B(H ) . n
n
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M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, M. M. Wolf
On the other hand, it is known [43, Lemma 8. 9] that the norm of n Tn ⊗en B(H )⊗min 1 is equal to ⎧ ⎫
⎨ ⎬
sup Tn ⊗ Un : Un ∈ B(H ), Un Un∗ = Un∗ Un = 1 .
⎩ n ⎭ B(H )⊗min B(H )
Then, taking Un = 1 for every n, we get
Tn ≤ Tn ⊗ en B(H )⊗min 1 .
B(H )
n
n
Alternatively, this follows from the fact that the functional : 1 → C, ((tn )) = is a complete contraction.
n tn
The following remark will make the proof of Theorem 6 easier to read. Remark 2. Note that, using the isometric identification, K N N C B(1N (∞ ), B(H )) = ∞ (1K ) ⊗min B(H ) = ∞ (1K ⊗min B(H )),
(13)
we can deduce from the previous lemma that B Q (M) can be written in the form (12) similarly to M K (∞ N )⊗ K N , when u and v are restricted to operators that map the min (∞ ) 1
1
N ) to positive elements of B(H ). canonical basis of 1K (∞ Indeed, this is immediate from the two following facts. First, given a complete conK ) → B(H ) such that u(e ⊗ e ) = G a ∈ B(H )+ for every x = traction u : 1N (∞ x a x 1, . . . , N ; a = 1, . . . , K , we will have that, for every x,
K a=1
G ax =
K
G ax ⊗ ea B(H )⊗min k ≤ sup 1
a=1
x
K
G ax ⊗ ea B(H )⊗min k ≤ 1, 1
a=1
where we have used Lemma 2 in the first equality and Eq. (13) in the last inequality. K a implies that we a=1 G x being a positive element for every x, the above estimation K + such that a ⊂ B(H ) G have a sequence of operators (G ax )a=1,...,K a=1 x ≤ 1 for every x=1,...,N + x = 1, . . . , N . On the other hand, given a sequence (E xa )a=1,...,K x=1,...,N ⊂ B(H ) such that K N K a a=1 E x ≤ 1 for every x = 1, . . . , N , we can consider the operator u : 1 (∞ ) → a + B(H ) defined by u(ex ⊗ ea ) = E x ∈ B(H ) . Using again Lemma 2 and Eq. (13) we can see that
ucb = sup x
K
E xa ⊗ ea B(H )⊗min k = sup
a=1
1
x
K
E xa ≤ 1.
a=1
We can prove now Theorem 6. Proof of Theorem 6. The first inequality is just the previous remark. For the proof of the K ) → B(H ). We consecond inequality we consider a complete contraction u : 1N (∞ K → B(H ). We recall that according to Wittstock’s factorization sider now the u x : ∞ theorem [40, Theorem 8.5] every complete contraction u defined on a C ∗ -algebra A with
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
729
values in B(H ) can be decomposed as u(x) = V π(x)W , with V and W contractions, and π a ∗ -representation. Thus, we have u = u 1 − u 2 − i(u 3 − u 4 ), where u 1 (x) = 1/4(V + W ∗ )π(x)(V ∗ + W ), u 2 (x) = 1/4(V − W ∗ )π(x)(V ∗ − W ), u 3 (x) = 1/4(V − i W ∗ )π(x)(V ∗ + i W ), u 4 (x) = 1/4(V + i W ∗ )π(x)(V ∗ − i W ). Note that, for every i = 1, . . . , 4, u i is a completely positive contraction. We apply this observation to every component and we decompose u x = u 1x − u 2x − i(u 3x − u 4x ) as a linear combination of completely positive maps. Then, for every x and a a ) is an incomplete POVM (see also Eq. (13)). This leads we see that (u ix (ea ))a = (E x,i a to the constant 16 = 4 × 4 in the assertion. K −→ M , then u i : K −→ Remark 3. Note that following [40, Theorem 8.5], if u : ∞ n ∞ M2n .
The next corollary follows now from the previous two theorems: a,b N ,K Corollary 4. Given M = (Mx,y )x,y=1,a,b=1 , we have that
L V (M)
M N ( K )⊗min N ( K ) ∞
1
1
∞
M N ( K )⊗ N ( K ) 1
∞
1
,
∞
where denotes equality up to universal constants. (For L V (M) see Definition 1 in Sect. 6). 9. Proof of the Main Result We introduce some notation that will be useful in the proof. Remark 4. Given an operator space X , we construct an associated operator space X n as follows: Let I be the collection of all complete contractions v : X −→ Mn . Then, we can define a new operator space structure on the Banach space X considering the application j : X −→ ∞ (I, Mn ) defined by j (x) = ((v(x))v∈I . It is easy to see that Mn (X n ) = Mn (X ). For our purpose it is interesting to note that a X n ⊗min Y n =
sup
v:X −→Mn cb ≤1,w:Y −→Mn cb ≤1
(v ⊗ w)(a) Mn ⊗min Mn .
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Then, the result that we will prove is q
q
Theorem 7. Given 2 < q < ∞ and n ∈ N, take m such that n 2 ≤ m ≤ 2n 2 (for q n n instance m = [n 2 ]) and denote X = m 1 (∞ ). Then, we can find an element x ∈ X ⊗ X 1
of rank n such that x X ⊗ X ≤ D(q) and x X n ⊗min X n ≥ n 2
− q2
.
Theorem 3 follows now from Theorem 7, Corollary 4 and Remark 3. For reference purposes, we state next Chevet’s inequality, which will be used often in the following. For a proof see [34]. Lemma 3 (Chevet’s inequality). There exists a universal constant b such that for every Banach spaces E, F and every sequence (gs,t )s,t of independent normalized gaussian random variables, we have gs,t xs ⊗ yt E⊗ε F ≤ bw2 ((xs )s ; E) gt yt F + bw2 ((yt )t ; F) gs xs E , s,t
s
t
where (gt )t and (gs )s are also independent normalized gaussian random variables and, given a sequence (xs )s in a Banach space X , we use the notation w2 ((xs )s ; X ) for ⎧ ⎫ 1 2 ⎨ ⎬ xs ⊗ es X ⊗ 2 = sup |x ∗ (xs )|2 : x ∗ ∈ X ∗ , x ∗ ≤ 1 . w2 ((xs )s ; X ) = ⎩ s ⎭ s We can take b = 1 if the spaces are real, whereas b = 4 if they are complex. We will also need the following three technical lemmas. Lemma 4. Let 1 < q < ∞, n ≤ m and (gi j )i,n,m j=1 be a family of independent normal− q1
q
ized gaussian random variables. Consider X t = t Then,
n,m
q
g e ⊗ e : X −→ R ∩ C E ij j i n n t
i, j=1
m K (t; m ∞ , Rm +C m , 1 ) for t =
≤ Km
cb
where K is a universal constant and C(m, n) = 1 +
√
1− q1
n m.
1
n q C(m, n),
log(m) . n
Proof. It follows from Eq. (11) that q
a : X t −→ Rn cb 1
= a Rn (X t∗ ) ≤ t q max{a Rn (m1 ) , t
−1 2
a Rn (Rm ∩Cm ) , t −1 a Rn (m∞ ) },
(14)
where c is a universal constant. We have to estimate the three terms appearing in this maximum. Recall our use of for inequalities valid up to a universal constant. For the first term, we use the little Grothendieck theorem [25, p. 183], which says that there n m exists a constant k such that for every operator a : m ∞ −→ 2 we have a : ∞ −→ Rn cb ≤ ka (and the same for Cm ). Then, we invoke Chevet’s Inequality and obtain
n,m
n,m
√ √
gi j e j ⊗ ei E g j,i ei ⊗ e j ( m n + m) ≤ K 1 m. E
i, j=1
i, j=1
m n m Rn (1 )
1 ⊗ 2
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
For the second term, it is easy to see that
n,m
n,m
E gi j e j ⊗ ei = E gi j e j ⊗ ei
i, j=1
i, j=1
Rn (Rm ∩Cm )
731
√
nm.
n2 ⊗2 m 2
Finally, we will use Chevet’s Inequality again to estimate the last expression,
n,m
n,m
√
E gi j e j ⊗ ei = E gi j e j ⊗ ei n + log m,
i, j=1
i, j=1
n m m Rn (∞ )
2 ⊗ ∞
m
√ where we have used that E i=1 gi ei m log m [50, p. 15]. Let us insert the ∞ precise value of t = mn . Then we obtain 1
−1
Ea : X t −→ Rn cb ≤ ct q E[max{a Rn (m1 ) , t 2 a Rn (Rm ∩Cm ) , t −1 a Rn (m∞ ) }] n 1 n −1 n = cE[( ) q max{a Rn (m1 ) , ( ) 2 a Rn (Rm ∩Cm ) , ( )−1 a Rn (m∞ ) }] m m m n q1 n −1 n −1 2 m ≤ cE[( ) (a Rn (1 ) + ( ) a Rn (Rm ∩Cm ) + ( ) a Rn (m∞ ) )] m m m 1 √ log(m) n m 1− 1 1 ≤ K ( ) q (m + m + ( n + log m)( )) ≤ K m q n q (1 + ), m n n q
where K is a universal constant. Replacing Rn by Cn we find the same estimates. By the definition of the intersection Rn ∩ Cn we obtain the result. Lemma 5. There exists δ ∈ (0, 1/2) with the following property: Given natural numbers n ≤ m and a family of independent normalized gaussian random variables (gi j )i,n,m j=1 , n to m . Then, “with high we consider G = i,n,m g e ⊗ e as an operator from i j i j 2 2 j=1 probability”4 , there exists an operator v : Hn −→ n2 such that v m1 G ∗ G| Hn = 1 Hn and v ≤ 2, where we denote Hn = [δn] 2 . Proof. Let us denote G˜ =
√1 G. m
Chevet’s Inequality tells us that
√ √ ˜ ≤ a √1 ( n + m) ≤ C E[G] m for some universal constant C. On the other hand, it is known [35, p. 80] that √ ˜ 2] ≥ c n E[G for a universal constant c > √1 , where · 2 denotes the Hilbert-Schmidt norm. Thus, 2 we can choose constants (independent of n) 0 < c < C such that, with high probability, √ ˜ ≤ C, and G ˜ 2 ≥ c n. This is a straightforward consequence of the G verifies G concentration of measure phenomenon as stated, for instance, in [44, Theorem 4.7]. We 4 High probability means here that the probability tends to 1 exponentially fast as m → ∞.
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define δ = that
c2 ˜ for the j th . We recall the notation s j (G) 2C 2
˜ 22 = c2 n ≤ G
n
singular value of G˜ and observe
˜ 2 ≤ s1 (G) ˜ 2 ([δn] − 1) + s[δn] (G) ˜ 2n ≤ s j (G)
j=1
c2 n ˜ 2 n. + s[δn] (G) 2
2 ˜ 2 . We may take c2 > 1 , so we have 1 ≤ s[δn] (G). ˜ Therefore, we find 0 < c2 ≤ s[δn] (G) 2 2 ˜ By the definition of the singular values of G, the above estimation says that we can invert the operator m1 G ∗ G : n2 −→ n2 on a “large” subspace of dimension kn = [δn]. n Thus, if we denote Hn = [δn] 2 , we know that there exists an operator vn : Hn −→ 2 such that vn m1 G ∗ G| Hn = 1 Hn and vn ≤ 2.
Before we prove the next lemma, let us observe the following remark. Remark 5. By the definition of the K -spaces and the standard interpolation equality m m [m ∞ , 1 ]1/q = q , it is clear that for every t > 0, m ∈ N and 1 < q < ∞ the map m m m id = id ◦ id : qm → K (t; m ∞ , 1 ) → K (t; ∞ , Rm + C m , 1 ) is a composition of two contractions (see for instance [4] for the first one), hence id is itself a contraction. Lemma 6. Given 2 < q < ∞, there exists a constant c(q) > 0 such that for every n ≤ 2
m q and every family of independent normalized gaussian random variables (gi j )i,n,m j=1 , we have
n,m
−1 q n
q E m gi j ei ⊗ e j : 2 −→ X t
≤ c(q)
i, j=1 (for every t > 0). q
Proof. Applying Chevet’s Inequality again for X t , we get
m
n,m
√ q m
gi j ei ⊗ e j E gjej E
+ nw2 ((e j ) j=1 ; X t ).
i, j=1
n
q q j=1 2 ⊗ X t
Xt
Hence, it suffices to show that q
id : m −→ X t ≤ A(q) and
2
m
1
≤ B(q)m q . E g e j j
j=1
q Xt
Both estimations follow easily using Remark 5. Indeed, the upper estimate follows, with A(q) = 1, from m id : m 2 −→ q ≤ 1.
In the same way, the next estimate follows from
m
m
g e ≤ E g e E j j j j
j=1
j=1
q
Xt
qm
1
≤ B(q)m q .
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
733
√ Remark 6. It is well known that B(q) ≤ √ C q, where C is a universal constant indepen dent of q. Thus, we have that c(q) ≤ C q. Using this, we can separate the epsilon and the min norm on a suitable subspace of n n m 1 (∞ ). q
q
Lemma 7. Given 2 < q < ∞ and n ∈ N, if we take n 2 ≤ m ≤ 2n 2 , there exists a q
1
q
matrix a ∈ X t ⊗ X t of rank n such that a ≤ D(q) and amin ≥ n 2 define t =
n m
and
q Xt
=t
− q1
K (t; m ∞,
− q2
, where we
Rm + Cm , m 1 ). q
q
q
Proof. Given n ∈ N and 2 < q < ∞, taking n 2 ≤ m ≤ 2n 2 , we define t and X t as in the statement of the theorem. Since t is considered fixed we may simplify the notation q and write X q = X t . Thanks to the three previous lemmas, we know that there exists a matrix G = (gi j (w))i,n,m j=1 such that 1
1
1) G ∗ : X q −→ Rn ∩ Cn cb ≤ C(q)m q n q . 2) There exist δ, vn and Hn as in Lemma 5. 1 3) G : n2 −→ X q ≤ c(q)m q . Observe that, due to the choice of m, the function C(n, m) = C(q) (in Lemma 4) only depends on q. Consider an arbitrary matrix a in Hn ⊗ Hn . Then, we have m
− q1
G⊗m
− q1
G(a) X q ⊗ X q ≤ c(q)2 a Hn ⊗ Hn .
On the other hand, we have 1 ∗ 1 G G) ⊗ v( G ∗ G)(a)n2 ⊗2 n2 m m 1 ∗ 1 = v( G G) ⊗ v( G ∗ G)(a) Rn ∩Cn ⊗min Rn ∩Cn m m
an2 ⊗2 n2 = v(
≤ vm
− q1
G ∗ 2cb (m
− q1
G⊗m
− q1
G)(a) X qn ⊗min X qn .
In the special case where a represents the identity on Hn we obtain √ 1 1 n ≤ δ − 2 kn = δ − 2 an2 ⊗2 n2 2
1
≤ 4δ − 2 C(q)2 n q (m
− q1
G⊗m
− q1
G)(a) X qn ⊗min X qn .
This leads to the two competing estimates m
− q1
G⊗m
− q1
G(a) X q ⊗ X q ≤ c(q)2 , 1 2 D −1 −1 − n2 q . (m q G ⊗ m q G)(a) X qn ⊗min X qn ≥ 2 C(q) Combining (15) and (16) yields the result. Remark 7. According to Remark 4, we have actually proved that a X qn ⊗min X qn ≥
1 2 D − n2 q. 2 C(q)
(15) (16)
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Remark 8. The constant in the previous theorem can be taken D(q) ≤ Cc(q)2 C(q)2 , where C is a universal constant which does not depend on q. Furthermore, we have seen √ in Remark 6 that c(q)2 q. It can be checked that C(q) ≤ 1 +
q log(n) . n
We can prove now Theorem 7. Proof. By Theorem 5, for every measure space (, μ) such that μ() = k < ∞, k k we have that L 1 () + L 2R () + L C 2 () + L ∞ () completely embeds into L 1 ( ; ∞ ). Furthermore, the complete embedding j : L 1 () + L r2 () + L c2 () + L ∞ () → L 1 (k ; k∞ ) can be specifically written. Indeed, consider the measure space (, μ), where = {1, . . . , m} and μ(i) = t = mn for every i = 1, . . . , m. Then, μ() = mt = n. But it is easy to see that for this measure space, the operator space L 1 () + L r2 () + L c2 () + m L ∞ () is exactly the operator space K (t; m ∞ , Rm + C m , 1 ). Thus, we have a comm m pletely isomorphic embedding of K (t; ∞ , Rm + Cm , 1 ) into L 1 (n , n∞ ). Note that n n the difference between L 1 (n , n∞ ) and X = m 1 (∞ ) is just the normalization in the L 1 -norm and hence the spaces are completely isometrically isomorphic. Thus, it will be −1 −1 enough to consider the completely isomorphic embedding j˜ = r ◦ t q j from t q K t ˜ into X and to take the element x = ( j˜ ⊗ j)(a) ∈ X ⊗ X , where a is the same element as in Lemma 7. We invoke Remark 7 and the fact that the formal identity map id : X → X n is completely contractive. This yields the difference for the min and ε norm claimed in the assertion. Remark 9. It follows from Remark 8 that we can take D(q) ≤ q 2 (actually, this estimate is not tight). Then, for a fixed dimension n, just taking q = log(n), we obtain √ x X n ⊗min X n n ≥ (17) x X ⊗ X log(n)2 with X = [2 1
log2 n 2 ]n
(n∞ ).
Remark 10. We have the following interesting alternatives: either a) for every subspace F ⊂ L 1 (∞ ) 2 ⊗ε F = R + C ⊗min F, or b) there exists a subspace F ⊂ L 1 (∞ ) such that 2 ⊗ε F = R + C ⊗min F. In case a), it follows easily from John’s theorem [45] that for every rank n tensor a ∈ F1 ⊗ F2 that √ amin ≤ C naε . This means our estimate (17) for a rank n tensor is optimal up to the logarithmic factor. However, in case b) there are violations of Bell’s inequality involving POVM’s only for Alice or Bob, but not both. To wrap this up we could formulate it as follows. Either there are asymmetric Bell violations which are of simpler nature than everything discovered so far, or our estimates are best possible. It would certainly be interesting to know which of these alternatives holds true.
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
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Acknowledgements. The authors are grateful to the organizers of the Operator Structures in Quantum Information Workshop, held in Toronto during July 6-10, 2009, where part of this work was developed. M. Junge is partially supported by the NSF grant DMS-0901457. C. Palazuelos, D. Perez-Garcia and I. Villanueva are partially supported by EU grant QUEVADIS and by Spanish grants I-MATH, MTM2008-01366 and QUITEMAD. M.M. Wolf acknowledges support by QUANTOP, QUEVADIS, COQUIT and the Danish Natural Science Research Council(FNU).
A. Some Proofs A.1. Proof of Proposition 2. The result is based on the fact that the norm of the identity id : Mn ⊗ Mn → Mn ⊗min Mn is ≤ n (actually it is exactly n). Indeed, using that dcb (Rn , min(n2 )) = dcb (Cn , min(n2 )) =
√
n,
it is easy to see that dcb (Mn , min(Mn )) = n. The result follows now trivially from the fact min(Mn ) ⊗min Mn = Mn ⊗ Mn . a,b Let us take then a Bell inequality M = {Mx,y }x,y,a,b and a quantum probability distribution P. By the previous estimation, we have
y a,b x
|M, P| ≤ Mx,y E a ⊗ Fb
a,b,x,y
Mn ⊗min Mn
y a,b x ≤ n Mx,y E a ⊗ Fb .
a,b,x,y
Mn ⊗ε Mn
Now, this is exactly the same as ⎧ ⎫ ⎨ ⎬ y a,b sup Mx,y tr(E ax ρ1 ) tr(Fb ρ2 ) : ρ1 , ρ2 ∈ B S1n . ⎩ ⎭ a,b,x,y
(18)
But it is well known that every ρ ∈ B S1n can be written as ρ = ρ11 + iρ12 with ρ1i self adjoint elements in B S1n for i = 1, 2. Then, y a,b Mx,y tr(E ax ρ1 ) tr(Fb ρ2 )| : ρ1 , ρ2 ∈ B S1n and self adjoint }. (18) ≤ 4 sup{| a,b,x,y
But ρ1 can be written as ρ1 = nj=1 δ j | f j f j | with (| f j ) j an orthonormal basis of n2 , and nj=1 |δ j | ≤ 1 (and the same for ρ2 ). Then, for every pair of selfadjoint ρ1 , ρ2 we have y a,b x Mx,y tr(E a ρ1 ) tr(Fb ρ2 ) a,b,x,y ⎧ ⎫ ⎨ ⎬ y a,b ≤ sup Mx,y u|E ax |uv|Fb |v : |u, |v ∈ Sn2 , ⎩ ⎭ a,b,x,y
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M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva, M. M. Wolf
which is bounded above by sup P ∈L |M, P |. Therefore, we have |M, P| n sup |M, P |. P ∈L
A.2. Explanation of [[28], Theorem 3.6]. Suppose we have a probability space (, μ) and k ∈ N. We may consider the particular case of (Theorem 3.6, [28]) in which A = Mk ⊗min L ∞ (n ), M = Mk ⊗min L ∞ (), N = Mk , the conditional expectation EN : M → N is defined by EN = 1 ⊗ · dμ and K = C. The algebras (M)k≥1 ’s form a system of independent symmetric copies of M over N (see ([28], Example 1), which is a stronger condition than the one appearing in (Theorem 3.6, [28]). We start with the easy case L 1 (A, n∞ ) = L 1 (Mk ⊗ L ∞ (n ), n∞ ) = S1k (L 1 (n ), n∞ ). Let us turn to the more complicated K space n K1,∞ (M, EN ) = n L 1 (M) + L s1 (M, EN ) +
√ r √ n L 1 (M, EN ) + n L c1 (M, EN ).
Here we refer to definition before ([28], Lemma 3.5) x L s1 (M,EN ) = inf a L 2 (Mk ) y Mk ⊗min L ∞ () b L 2 (Mk ) = x S k (L ∞ ()) . x=ayb
1
Hence L s1 (M) = S1k (L 1 ()) as predicted. For the column term we have x L c1 (M,EN ) = inf a L 2 (M) y Mk ⊗L ∞ () b L 2 (Mk ) x=ayb
= inf a L 2 (M) b L 2 (Mk ) . x=ab
Given such a factorization x = ab we see that 2 1/2 ∗ ( |x| dμ) = b ( a ∗ adμ)b21/2 1/2 ≤ b L 2 (Mk ) tr (a ∗ a)dμ1 = b2 a2 .
This shows x S k (L r ()) ≤ inf ab. Conversely, for x ∈ S1k (L r2 ()) = Rk ⊗h 2 1 L r2 () ⊗h Ck we deduce from the definition of the Haagerup tensor product that we can find a factorization x = ba such that b ∈ Rk ⊗h L r2 () ⊗h Rk and a ∈ L 2 (Mk ). Note however, that b Rk ⊗h L r2 ()⊗h Rk = b L 2 (,S k ) = b L 2 (M) . 2
Thus we have in fact x S k (L r ()) = x L c1 (M,EN ) . 1
2
Interchanging rows and columns yields the missing estimate. Theorem 5 follows now easily. Suppose we have a measure space (, μ) such that μ() = n. Then, we consider (, μ) ˆ = (, μn ) and, thus, √ √ i : n L 1 (, μ) ˆ + n L r2 (, μ) ˆ + n L c2 (, μ) ˆ + L ∞ (, μ) ˆ → L 1 (n , ⊗n μ; ˆ n∞ ),
Unbounded Violations of Bipartite Bell Inequalities via Operator Space Theory
737
is a completely embedding. But it is obvious that √ √ n L 1 (, μ) ˆ = L 1 (, μ), n L r2 (, μ) ˆ = L r2 (, μ), n L c2 (, μ) ˆ = L c2 (, μ), L ∞ (, μ) ˆ = L ∞ (, μ) and n n L 1 (n , ⊗n μ; ˆ n∞ ) = L 1 (n , ⊗n μ; n∞ ). Therefore, j=
i : L 1 (, μ) + L r2 (, μ) + L c2 (, μ) + L ∞ (, μ) → L 1 (n , ⊗n μ; n∞ ) nn
is a complete embedding (with absolute constants).
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Commun. Math. Phys. 300, 741–763 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1093-9
Communications in
Mathematical Physics
Rieffel Deformation of Group Coactions P. Kasprzak, Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark Received: 8 January 2010 / Accepted: 18 February 2010 Published online: 10 July 2010 – © The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract: Let G be a locally compact group, ⊂ G an abelian subgroup and let ˆ Let B be a C∗ -algebra and B ∈ Mor be a continuous 2-cocycle on the dual group . (B, B ⊗ C0 (G)) a continuous right coaction. Using Rieffel deformation, we can con˜ struct a quantum group (C0 (G)⊗ , ) and the deformed C∗ -algebra B . The aim of this paper is to present a construction of the continuous coaction B of the quantum ˜ ⊗ , ) on B . The transition from the coaction B to its deformed group (C0 (G) counterpart is nontrivial in the sense that B B contains complete information about B . In order to illustrate our construction we apply it to the action of the Lorentz group on the Minkowski space obtaining a C∗ -algebraic quantum Minkowski space.
Contents 1. 2. 3.
Introduction . . . . . . . . . . . . . . . . . . Rieffel Deformation via Crossed Products . . Technicalities . . . . . . . . . . . . . . . . . . 3.1 Technicalities of the Rieffel Deformation . 3.2 Product of ( p, q)-commuting Pairs . . . . 4. Rieffel Deformation of Group Coactions . . . 5. Quantum Minkowski Space . . . . . . . . . . 5.1 Generators of C0 (H) . . . . . . . . . . 5.2 Coaction of G on C0 (H) . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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Supported by the Marie Curie Research Training Network Non-Commutative Geometry MRTN-CT2006-031962 and by Geometry and Symmetry of Quantum Spaces, PIRSES-GA-2008-230836. On leave from: Department of Mathematical Methods in Physics, Faculty of Physics, Warsaw University, Warszawa, Poland. E-mail: [email protected]
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1. Introduction The Rieffel deformation is already a well established method of deforming C∗ -algebras. In his original approach M. Rieffel starts from deformation data (A, ρ, J ) which consists of a C∗ -algebra A, an Rn action ρ on A and a skew symmetric matrix J : Rn → Rn . Using these data M. Rieffel was able to deform the original product on the algebra of ρ-smooth elements: A∞ ⊂ A. The deformed C∗ -algebra A J is defined as a C∗ -algebraic completion of A∞ considered as an algebra with this deformed product. Our recent approach to the Rieffel deformation (see [1]) is based on the observation that deformation data (A, ρ, J ) enable us to deform an Rn -product structure emerging in the crossed product construction: (A Rn , λ, ρ). ˆ Namely, using a skew-symmetric matrix J we may deform a dual action obtaining ρˆ J . The Rieffel deformation A J is defined as the Landstad algebra of the deformed Rn -product (A Rn , λ, ρˆ J ). This second approach generalizes to the cases when the deformation data consist of (A, ρ, ), where A is a C∗ -algebra equipped with an action ρ of an abelian group and is ˆ The concise account of the deformation a continuous 2-cocycle on the dual group . procedure is the subject of Sect. 2 (for the definition of a Landstad algebra we refer to point 3 of that section). The Rieffel deformation which was originally developed to deform C∗ -algebras can be also used for deforming locally compact groups (see [4,10]). Namely, let G be a locally ˆ Using compact group, a closed abelian subgroup and a continuous 2-cocycle on . the action of 2 on C0 (G) given by the left and the right shifts and defining a particular ˜ ⊗ on ˆ 2 we may perform the Rieffel deformation of C0 (G). This leads to 2-cocycle ˜ ∗ a C -algebra C0 (G)⊗ , which can be shown to carry the structure of a locally compact ˜ quantum group. We shall denote this quantum group by G = (C0 (G)⊗ , ). ∗ Let B be a C -algebra equipped with a continuous coaction B ∈ Mor(B, B ⊗ C0 (G)). The coaction B corresponds to an action β : G → Aut(B) of G on B. Restricting this action to the subgroup ⊂ G we get an action of on B. This lets us construct the deformed C∗ -algebra B . The aim of this paper is to show that the coaction B ∈ Mor(B, B ⊗ C0 (G)) can be naturally deformed to a coaction B ∈ ˜ is nontrivial in the sense Mor(B , B ⊗ C0 (G)⊗ ). The transition from B to B that B contains all the information about B so it is possible to reconstruct B out of B. A particular case of the construction presented in our paper has been discussed in [11]. In this paper J. Varilly treats the situation where there is given a pair ⊂ K ⊂ G of closed subgroups with being abelian and G compact. It was then shown that it is possible to perform a covariant deformation of the C∗ -algebra C0 (K \G). The resulting C∗ -algebra C0 (K \G) is equipped with a continuous ergodic coaction of the quantum group G . In this specific situation the difficulties that one encounters in general do not manifest themselves. In the final section of our paper we apply our deformation procedure to the action of the Lorentz group (more precisely of S L(2, C)) on the Minkowski space M. We base the deformation procedure on the subgroup ⊂ S L(2, C) consisting of the diagonal matrices. The quantum Lorentz group that we obtain was already described in [1]. In this paper we focus on the quantum Minkowski space describing it in terms of the generators of the C∗ -algebra C0 (M) which satisfy ( p, q)-type commutation relations. In particular, we give a description of the twisted coaction M in terms of its action on the generators.
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Throughout the paper we will freely use the language of C∗ -algebras and the theory of locally compact quantum groups. For the notions of multipliers, affiliated elements, algebras generated by a family of affiliated elements and the category of C∗ -algebras we refer the reader to [12 and 15]. In particular the morphism of C∗ -algebras will always be Woronowicz morphisms (see 0.4 [15]). For the theory of locally compact quantum groups we refer to [2 and 5]. For any subset X of a Banach space B, [X ] ⊂ B will denote the closed linear span of X . 2. Rieffel Deformation via Crossed Products Throughout the paper we shall use the crossed products approach to the Rieffel deformation. For the detailed treatment of the subject we refer to [1]. In what follows we shall give a concise account of the deformation procedure. The deformation data (A, ρ, ) consist of a C∗ -algebra A, an action ρ of an abelian group and a continuous 2-cocycle ˆ The deformation procedure consists of the following steps: on the dual group . 1. Let B be the crossed product C∗ -algebra B = A and let (B, λ, ρ) ˆ be the -product structure on this crossed product, i.e. λ : → M(B) is the representation of implementing the action ρ and ρˆ is the dual action on B. 2. Let λ ∈ Mor(C∗ (), B) be the morphism corresponding to the representation ˆ be the family of functions given by λ ∈ Rep(, B) and let γˆ ∈ M(C0 ()) ˆ with γˆ (γˆ ) = (γˆ , γˆ ). Applying λ to γˆ (note the identification of C0 () C∗ () via the Fourier transform), we get a ρ-projective ˆ 1-cocycle Uγˆ ∈ M(B). Using Uγˆ we define the deformed dual action ρˆ : ˆ → Aut(B) by the formula: ρˆγˆ (b) = Uγ∗ˆ ρˆγˆ (b)Uγˆ , for any γˆ ∈ ˆ and b ∈ B.
3. The deformed C∗ -algebra A is defined as the Landstad algebra of the deformed -product (B, λ, ρˆ ): ⎧ ⎫ 1. ρˆ (b) = b ⎪ ⎪ ⎪ ⎪ γ ˆ ⎨ ⎬ ∗ aλ ∈ M(B) 2. The map γ
→ λ γ γ . A = b ∈ M(B) ⎪ ⎪ is norm-continuous ⎪ ⎪ ⎩ ⎭ ∗ 3. λ(x)a, aλ(x) ∈ B for any x ∈ C ()
Remark 2.1. In the course of this paper we shall use the functorial properties of the Rieffel deformation (see Sect. 3.2, [1]). Let (A, ρ, ) and (A , ρ , ) be deformation data and let π ∈ Mor(A, A ) be a covariant morphism: π(ργ (a)) = ργ (π(a)) for any γ ∈ and a ∈ A. Then there exists a morphism π ∈ Mor(A , A ) that sends A ⊂ M(A ) to A ⊂ M(A ) by means of π and which is identity on C∗ () ⊂ M(A ) ∩ M(A ). Restricting π to A we get a morphism π ∈ Mor(A , A ). 3. Technicalities This section is divided into two parts. The first part deals with some specific situations encountered while performing the Rieffel deformation of C∗ -algebras. The results derived here will be needed in a construction of the deformed coaction B . In the second part we shall construct the product of ( p, q)-commuting elements affiliated with a C∗ -algebra. This construction will be useful in the description of the quantum Minkowski space which is the subject of the last section.
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3.1. Technicalities of the Rieffel Deformation. Let ρ be an action of 2 on a C∗ -algebra ˆ Using A, such that ργ ,γ = id for any γ ∈ . Let be a continuous 2-cocycle on . ˜ we may define a 2-cocycle : ˜ γˆ1 , γˆ2 ) = (−γˆ1 , −γˆ2 ). (
(1)
The 2-cocycle on ˆ 2 that we shall use in the deformation procedure is the tensor product: ˜ The family of unitaries Uγˆ ,γˆ ∈ M(A 2 ) is obtained by taking the images of ⊗ . 1 2 ˜ γˆ ∈ M(C0 ()⊗C ˆ ˆ under the morphism λ ∈ Mor(C0 ()⊗C ˆ ˆ A 2 ): γˆ1 ⊗ 0 ()) 0 (), 2 ˜ γˆ ) ∈ M(A 2 ). Uγˆ1 ,γˆ2 = λ(γˆ1 ⊗ 2 ˜ γˆ see the previous section. The deformed dual action For the definition of γˆ1 and 2 ˜
ρˆ ⊗ is defined by the formula ˜
(b) = Uγ∗ˆ1 ,γˆ2 ρˆγˆ1 ,γˆ2 (b)Uγˆ1 ,γˆ2 ρˆγ⊗ ˆ ,γˆ 1
2
for any b ∈ A 2 . ˜
Lemma 3.1. Let (A 2 , λ, ρˆ ⊗ ) be the 2 -product introduced above. Then the ˜ Landstad algebra A⊗ is isomorphic with A. Proof. Let us introduce an auxiliary function : ˆ 2 → T given by (γˆ1 , γˆ2 ) = (γˆ1 , −γˆ1 − γˆ2 ). Using the 2-cocycle property of one can prove that: ˜ v (y)(−x − y, −v)(u, −x − y) (x + u, y + v) = (x, y)u (x)
(2)
ˆ Using the morphism λ ∈ Mor(C∗ ( 2 ), A 2 ) we define a for any x, y, u, v ∈ . unitary element ϒ = λ( ). The above formula may be expressed in terms of ϒ and Uγˆ1 ,γˆ2 , ρˆγˆ1 ,γˆ2 (ϒ) = Uγˆ1 ,γˆ2 ϒ Z γˆ1 ,γˆ2 ,
(3)
where Z γˆ1 ,γˆ2 ∈ M(A 2 ) corresponds to the last two factors in the product on the right hand side of (2). Let us note that the equality ργ ,γ (a) = a implies that λγ ,γ is in the center of the M(A 2 ). Using this fact one can conclude that λ(( f )) is in the center ˆ In particular, Z γˆ ,γˆ ∈ M(A 2 ) is central for of M(A 2 ) for any f ∈ M(C0 ()). 1 2 ˆ any γˆ1 , γˆ2 ∈ . Let us move on to the main part of the proof. For any a ∈ A we define π(a) = ˜ ϒaϒ ∗ ∈ M(A 2 ). Our aim is to show that π(a) ∈ A⊗ . In order to do that we must check the three Landstad conditions for an element π(a) ∈ M(A 2 ). ˜
1. The invariance of π(a) under the twisted dual action ρˆ ⊗ can be checked as follows: ˜
(ϒaϒ ∗ ) = Uγ∗ˆ1 ,γˆ2 ρˆγˆ1 ,γˆ2 (ϒaϒ ∗ ) Uγˆ1 ,γˆ2 ρˆγ⊗ ˆ ,γˆ 1
2
= ϒ Z γˆ1 ,γˆ2 a Z γ∗ˆ1 ,γˆ2 ϒ ∗ = ϒaϒ ∗ = π(a),
where in the last line we used Eq. (3) and the fact that Z is a central element.
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2. In order to check the second Landstad condition for π(a) we note that λγ1 ,γ2 π(a) λ∗γ1 ,γ2 = π(λγ1 ,γ2 aλ∗γ1 ,γ2 ). This equality, together with the second Landstad condition for a ∈ A, shows that the map 2 (γ1 , γ2 ) → λγ1 ,γ2 π(a)λ∗γ1 ,γ2 ∈ M(A 2 ) is norm continuous. 3. Using the third Landstad condition for a ∈ A we get xπ(a)y = (xϒ)a(ϒ ∗ y) ∈ A 2 for any x, y ∈ C∗ ( 2 ). ˜
The above reasoning shows that π(A) ⊂ A⊗ . In order to prove the opposite inclu˜ ¯ as a 2-cocycle and use Lemma 3.5 of sion we have to switch A and A⊗ , take [1].
Now let us pass to the second specific situation that may be encountered while performing Rieffel deformation. Assume that we are given two commuting actions α and β of locally compact abelian groups 1 and 2 on a C∗ -algebra A: αγ1 ◦ βγ2 (a) = βγ2 ◦ αγ1 (a) for any a ∈ A, γ1 ∈ 1 and γ2 ∈ 2 . Fixing γ2 in the above formula we see that the automorphism βγ2 is α-covariant. Now let 1 and 2 be continuous 2-cocycles on ˆ 1 and ˆ 2 respectively. Using Proposition 3.8 of [1] we get the deformed automorphism: βγ21 ∈ Aut(A1 ). By the functorial properties of the Rieffel deformation (see Sect. 3.2 1 of [1]) it follows that γ → βγ1 ∈ Aut(A 1 ) is an action of 2 on A . The continuity of that action demands an additional reasoning. To check that for any a ∈ A1 the map 2 γ → βγ1 (a) ∈ A1 is norm continuous, we have to invoke the averaging map E1 : D(E1 ) → M(A 1 ) (see Remark 2.5, [1]). Let a ∈ A and f 1 , f 2 ∈ C0 (ˆ 1 ) be functions of compact support. The elements of the form E1 ( f 1 a f 2 ) constitute a dense subset of A1 and the map A a → E1 ( f 1 a f 2 ) ∈ A1
(4)
is norm continuous. Using the definition of β 1 (see Remark 2.1) we get:
βγ21 E1 ( f 1 a f 2 ) = E1 ( f 1 βγ2 (a) f 2 ). The above formula, the continuity of map (4) and the continuity of the 2 -action β, together imply that β 1 is a continuous 2 -action. A similar reasoning, with the roles of 1 and 2 reversed, leads to a continuous 1 -action α 2 on A2 . There arises the natural question concerning the relation between A1 and A2 . Before answering it let us note that the two commuting actions α and β, give rise to an action α × β of 1 × 2 on A. Tensoring 1 and 2 we get a 2-cocycle on ˆ 1 × ˆ 2 . A Fubini type theorem for the averaging maps E1 , E2 and E1 ⊗2 and a functorial gymnastics enables us to prove the following lemma. Lemma 3.2. Let α, β and α × β be actions on A of 1 , 2 and 1 × 2 respectively, as introduced above. Let β 1 be the 2 -action on A1 and α 2 be the 1 -action on A2 . Then (A1 )2 = (A2 )1 = A1 ⊗1 .
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Remark 3.3. Let α, β be as in the above lemma. Let γ ∈ 2 and βγ = id. Using the functorial properties of the Rieffel deformation we get βγ1 = id = id. As the last instance of this subsection, let us consider a situation in which we have a C∗ -algebra A acted on by a group and assume that B and C are subalgebras of M(A) such that [BC] = A. Assume also that ρ can be restricted to continuous actions on B and on C. Note that the embedding of B into M(A) is a morphism in the sense of Woronowicz: A ⊃ [B A] ⊃ [BC A] = [A A] = A. Similarly, the embedding of C into M(A) is a morphism. Using Proposition 3.8 of [1] we get the embeddings of B and C into M(A ). Our aim is to show that A = [B C ]. Lemma 3.4. Let A, B and C be C∗ -algebras introduced above. Then A = [B C ]. Proof. The proof of this lemma is an application of Lemma 2.6 of paper [1]. The usage of this lemma will be legitimate if the equality [C∗ ()B C C∗ ()] = A holds, which we check below [C∗ ()B C C∗ ()] = [C∗ ()BC C∗ ()] = [C∗ ()A C∗ ()] = A .
3.2. Product of ( p, q)-commuting Pairs. Let ( p, q) be a pair of positive numbers and let H be a Hilbert space. The notion of a ( p, q)-commuting pair of normal operators R, S, acting on H was introduced in [14]. The fact that the product R ◦ S is a densely defined, closable operator acting on H follows from Proposition 0.1, [14]. Let A be a C∗ -algebra. The notion of a ( p, q)-commuting pair (R, S) of elements affiliated with A was introduced in Definition 5.7, [1]. In what follows we shall analyze R ◦ S, showing that it is a densely defined operator acting on the Banach space A and that its closure RS is affiliated with A. The proof for p = 1 with an additional spectral condition imposed on R and S was given in Lemma 2.15, [8]. The z-transform of an element T affiliated with A will be denoted by z(T ): 1
z(T ) = T (1 + T ∗ T ) 2 ∈ M(A). For any s ∈ C, the z-transform of sT ∈ Aη will be denoted by z s (T ). For notational convenience, we shall define a ( p 2 , q 2 )-commuting pair of normal elements. Definition 3.5. Let A be a C∗ -algebra and let (R, S) be a pair of normal elements affiliated with A. We say that (R, S) is a ( p 2 , q 2 )-commuting pair if 1. z(R)z(S ∗ ) = z pq (S ∗ )z q/ p (R), 2. z q/ p (R)z(S) = z pq (S)z(R). The set of all ( p 2 , q 2 )-commuting pairs of normal elements affiliated with a C∗ -algebra A is denoted by D p2 ,q 2 (A). Theorem 3.6. Let A be a C∗ -algebra and let (R, S) be a ( p 2 , q 2 )-commuting pair of elements affiliated with A. Then the operator R ◦ S : D(R ◦ S) → A is closable and its closure RS is affiliated with A.
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Proof. The proof of our theorem is based on Theorem 2.3 of [15], which describes a correspondence between elements affiliated with A and a subset of 2 × 2 matrices of elements of M(A). Let Q ∈ M2 (C) ⊗ M(A):
d −c∗ . Q= b a∗ The affiliated element T η A is related with Q by the following correspondence. The first column of Q provides information about T ∈ Aη , in the sense that d A is a core of T and T d x = bx for any x ∈ A. The second column of Q provides information about T ∗ ∈ Aη in the sense that, a ∗ A is a core of T ∗ and T ∗ a ∗ x = c∗ x. The consistency condition ab = cd reflects the defining property of the ∗-operation. In order for T and T ∗ to be densely defined it is necessary that d A and a ∗ A are dense in A. Finally, to ensure that such a matrix does in fact define an affiliated element T , one needs to check that the image of Q as a map acting on the Hilbert module A2 = A ⊕ A is dense in A2 : ||·|| Q A2 = A2 . Let us move on to the main part of the proof, which was inspired by the proof of Theorem 6.1 of [6]. Let (R, S) be a ( p 2 , q 2 ) commuting pair of elements affiliated with A. The matrix Q, which will let us define RS, has the form: Q=
1 1 1 − z p/q (R)∗ z p/q (R) 2 (1 − z(S)∗ z(S)) 2 −z(S)∗ z(R)∗ . 1 1 z(R)z(S) (1 − z(R)∗ z(R)) 2 (1 − z pq (S)∗ z pq (S)) 2
The only notrivial condition to check is that Q A2 is dense in A2 . In order to do that let us consider the matrix Q Q ∗ ∈ M2 (C) ⊗ M(A). We start with a computation of the left upper corner of Q Q ∗
(Q Q ∗ )11 = 1 − z p/q (R)∗ z p/q (R) (1 − z(S)∗ z(S)) + z(S)∗ z(R)∗ z(R)z(S)
= 1 − z p/q (R)∗ z p/q (R) (1 − z(S)∗ z(S)) + z p/q (R)∗ z p/q (R)z(S)∗ z(S), where in the last equality we used Definition 3.5 to commute z(S)∗ with z(R)∗ z(R). Similarly, we compute the right bottom corner of Q Q ∗ : (Q Q ∗ )22 = (1 − z(R)∗ z(R))(1 − z pq (S)∗ z pq (S)) + z(R)∗ z(R)z pq (S)∗ z pq (S). The matrix elements (Q Q ∗ )12 = (Q Q ∗ )∗21 turn out to be 0:
1 1 (Q Q ∗ )12 = 1 − z p/q (R)∗ z p/q (R) 2 (1 − z(S)∗ z(S)) 2 z(S)∗ z(R)∗ 1
1
−z(S)∗ z(R)∗ (1 − z(R)z(R)∗ ) 2 (1 − z pq (S)z pq (S)∗ ) 2 = 0. To show that the above difference is zero we used the following two identities: 1
1
(1 − z(S)∗ z(S)) 2 z(R)∗ = z(R)∗ (1 − z pq (S)z pq (S)∗ ) 2 ,
1 1 z(S)∗ (1 − z(R)z(R)∗ ) 2 = z(S)∗ 1 − z p/q (R)∗ z p/q (R) 2 , which immediately follows from Definition 3.5. Let us note that (Q Q ∗ )11 =
1 + |R|2 |S|2 . (1 + |R|2 )(1 + |S|2 )
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In particular the right ideal generated by (Q Q ∗ )11 is dense in A : (Q Q ∗ )11 A = A. The same concerns (Q Q ∗ )22 . From the fact that Q Q ∗ is a diagonal matrix we can see ||·|| that Q Q ∗ A2 = A2 , which implies that Q A2 is a dense subset of A2 . This in turns shows that Q satisfies the assumptions of Theorem 2.3 of paper [15]. We conclude that Q defines an affiliated element T η A such that the set
1 − z p/q (R)∗ z p/q (R)
1 2
1
(1 − z(S)∗ z(S)) 2 A
(5)
is a core of T and
1 1 T 1 − z p/q (R)∗ z p/q (R) 2 (1 − z(S)∗ z(S)) 2 a = z(R)z(S)a. Finally, using point 2 of Theorem 2.3 of [15] one can check that the set D(R ◦ S) = {a ∈ D(S) : Sa ∈ D(R)} is a subset of D(T ) and T |D(R◦S) = R ◦ S, hence we may conclude that R ◦ S is a closeable operator acting on A and (R ◦ S)cl ⊂ T . In order to get the opposite inclusion it is enough to note that the core (5) of T is contained in D(R ◦ S).
Theorem 3.7. Let (R, S) be a ( p 2 , q 2 )-commuting pair of normal elements affiliated with a C∗ -algebra A. Then RS = p 2 S R, RS ∗ = q 2 S ∗ R. Proof. In order to prove the first relation let us observe that for any positive number t > 0 and any affiliated element X η A we have 1
1
(1 − z t (X )∗ z t (X )) 2 A = (1 − z(X )∗ z(X )) 2 A. This shows that the set 1
1
(1 − z(R)∗ z(R)) 2 (1 − z(S)∗ z(S)) 2 A is a joint core for RS and S R. Using the ( p 2 , q 2 )-commutation relations for the pair (R, S) we get p z q/ p (R)z(S), q 1 1 1 z pq (S)z(R). S R(1 − z(R)∗ z(R)) 2 (1 − z(S)∗ z(S)) 2 = pq 1
1
RS(1 − z(R)∗ z(R)) 2 (1 − z(S)∗ z(S)) 2 =
From Definition 3.5 we see that the right hand sides of the above equations are equal up to a multiplicative constant p 2 . This ends the proof of the equality RS = p 2 S R. The second equality RS ∗ = q 2 S ∗ R can be proved in a similar way.
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4. Rieffel Deformation of Group Coactions In this section we shall describe the Rieffel deformation of continuous group coactions. We adopt the following definition. Definition 4.1. Let (A, ) be a locally compact quantum group and let B be a C∗ -algebra. A morphism B ∈ Mor(B, B ⊗ A) is said to be a continuous right coaction of (A, ) on B if (1) ( B ⊗ ι) B = (ι ⊗ ) B ; (2) [(1 ⊗ A) B (B)] = B ⊗ A. There is a one to one correspondence between the continuous coaction B ∈ Mor(B, B⊗ C0 (G)) of (C0 (G), ) and the continuous action of G on B. The action β : G → Aut(B) that corresponds to B is defined as follows. Let χg ∈ Mor(C0 (G), C) be the character associated with a group element g ∈ G. We define βg ∈ Aut(B) by the following formula: βg (b) = (ι ⊗ χg ) B (b). It is easy to check that this indeed defines a continuous action of G on B. In order to perform the Rieffel deformation of B let us assume that G contains an ˆ abelian subgroup ⊂ G, and let be a continuous 2-cocycle on the dual group . Restricting β : G → Aut(B) to the subgroup ⊂ G we get an action of on B, which shall be denoted by α : → Aut(B). Let μ and ν be the actions of on C0 (G) given by the left and the right shifts along : μγ ( f )(g) = f (γ −1 g), (νγ f )(g) = f (gγ ) for and g ∈ G, γ ∈ and f ∈ C0 (G). Using the deformation data (B, α, ) we may ˜ ⊗ ) (for the notation μ × ν construct the C∗ -algebra B and using (C0 (G), μ × ν, ˜ see Lemma 3.2) we may construct the quantum group G = (C0 (G)⊗ , ). Let us move on to the construction of the coaction B of G on B . In order to do that we define an auxiliary C∗ -algebra: D = [ B (B)(1 ⊗ C0 (\G)] ⊂ M(B ⊗ C0 (G)), where we treat C0 (\G) as a subalgebra of M(C0 (G)). The embedding D ⊂ M(B ⊗ C0 (G)) is non-degenerate: [D(B ⊗ C0 (G))] = B ⊗ C0 (G). For the ease of reference we describe other properties of D in the following lemma. Lemma 4.2. Let D be the C∗ -algebra defined above. Then: (1) the 3 -action α ⊗ (μ × ν) on M(B ⊗ C0 (G)) restricts to a continuous action on D; (2) the image of the coaction B ∈ Mor(B, B ⊗ C0 (G)) is contained in M(D) and B ∈ Mor(B, D); (3) the embedding of C0 (\G) ⊂ M(D) given by C0 (\G) f → (1 ⊗ f ) ∈ M(D) is a Woronowicz morphism.
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Proof. The only point which is not obvious is the first one. To prove it let us note that D is isomorphic with B ⊗ C0 (\G), where the isomorphism ∈ Mor(B ⊗ C0 (\G), D) is given by (b ⊗ f ) = B (b)(1 ⊗ f ) ∈ D for any b ∈ B and f ∈ C0 (\G). Furthermore, it may be checked that (αγ1 ⊗ (μγ2 × νγ3 )) = (αγ1 γ −1 ⊗ νγ3 )
(6)
2
for any γ1 , γ2 , γ3 ∈ . This shows that the 3 -action α ⊗ (μ × ν) on M(B ⊗ C0 (G)) restricts to a continuous action on D ⊂ M(B ⊗ C0 (G)).
˜
⊗ ). By Theorem Let us move on to the construction of B ∈ Mor(B , B ⊗ C0 (G) 3.8 the intertwinig properties of B
B (αγ (b)) = (id ⊗ νγ ) B (b) enable us to define B ∈ Mor(B , D ) (we wish to keep the symbol B for a morphism which will be defined later). Consider the 2 -action α ⊗ μ on D. It has the following properties: (1) αγ ⊗ μγ = id, (2) id ⊗ ν and α ⊗ μ mutually commute. The first equality follows from (6), whereas the second one is obvious. Using Lemma 3.2 ˜ we see that the Rieffel deformation D ⊗⊗ obtained by the 3 -action introduced in ˜ point 1 of Proposition 4.2, is isomorphic with (D )⊗ . This in turn, by Lemma 3.1 and Remark 3.3 is isomorphic with D . Therefore, composing the morphism B ∈ ˜ Mor(B , D ) with the isomorphism D ∼ = D ⊗⊗ of Lemma 3.1 we may define the morphism: ˜
⊗⊗ ∗ ) : B ∈ Mor(B , D B (a) = ϒ B (a)ϒ .
(7)
Finally, the embedding ι ∈ Mor(D, B ⊗C0 (G)) is 3 -covariant (see Lemma 4.2), which ˜ ˜ by Remark 2.1 of [1] gives a morphism ι⊗⊗ that embeds D ⊗⊗ into M(B ⊗ ˜ C0 (G)⊗ ). Composing B with this morphism we define B ∈ Mor(B , B ⊗ ˜ C0 (G)⊗ ). Note that the construction of B is obtained by the ϒ twist of B (see Eq. (7)). Fur thermore, B extends naturally to the morphism B ∈ Mor(B , B ⊗ (C0 (G) )). Noting that B contains the information about B , we see that the transition B B is nontrivial: using B and we may recover B . ˜
⊗ ) defined above is a Theorem 4.3. The morphism B ∈ Mor(B , B ⊗ C0 (G) right coaction of G on B : ( B ⊗ id) B = (id ⊗ ) B .
(8)
Proof. It follows from the construction above that the morphism B ∈ Mor(B , B ⊗ ˜ ⊗ ) is a restriction of the crossed product morphisms Adϒ ◦ B ∈ Mor(B C0 (G) , B ⊗ C0 (G) 2 ). Similarly, the comultiplication is a restriction of the crossed product morphism Adϒ ◦ ∈ Mor(C0 (G) 2 , C0 (G) 2 ⊗ C0 (G) 2 )
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(see Theorem 3.8 [1]). For the purpose of this proof these crossed product morphisms will also be denoted by B and respectively. We shall prove the coaction identity (8) on the level of crossed products, which implies the same equality on the level of the deformed algebras. From the fact B = [B C∗ ()] we see that it is enough to check (8) on B and C∗ () separately. Let λγ ∈ C∗ () be a unitary generator of C∗ (). In order to check that (8) holds on C∗ () it is enough to note that ( B ⊗ id) B (λγ ) = (1 ⊗ 1 ⊗ λe,γ ) = (id ⊗ ) B (λγ ).
(9)
Now for any b ∈ B ⊂ M(B ) we have ∗ ( B ⊗ id) B (b) = ( B ⊗ id)(ϒ B (b)ϒ ) ∗ ∗ = ϒ12 ϒ23 (( B ⊗ id) B (b))ϒ23 ϒ12 .
On the other hand: ∗ (id ⊗ ) B (b) = (id ⊗ )(ϒ B (b)ϒ ) ∗ ∗ = ϒ12 ϒ23 ((id ⊗ ) B (b))ϒ23 ϒ12 . Using the coaction equation for B we get ( B ⊗ id) B (b) = (id ⊗ ) B (b) for any b ∈ B. This together with (9) ends the proof.
Theorem 4.4. The coaction B of G on B defined above is continuous.
Proof. Let us first note that using Lemma 3.4 and point (3) of Lemma 4.2 we get [ B (B) (1 ⊗ C0 (\G) )] = D . It can be checked that B (B) = B (B ), where B denotes the Rieffel deformation of B treated as a morphism from B to D (see the paragraph following Lemma 4.2). ˜ Applying the isomorphism D → D ⊗⊗ of Lemma 3.1 we get ˜
˜
⊗ [ )] = D ⊗⊗ . B (B )(1 ⊗ C0 (\G) ˜
˜
Moreover, the embedding of C0 (\G)⊗ into M(C0 (G)⊗ ) is non-degenerate, hence we see that ˜
˜
˜
⊗ ⊗ [ )] = [ )(1 ⊗ C0 (G)⊗ )] B (B )(1 ⊗ C0 (G) B (B )(1 ⊗ C0 (\G) ˜
˜
= [D ⊗⊗ (1 ⊗ C0 (G)⊗ )].
(10)
The continuity of B : [ B (B)(1 ⊗ C0 (G))] = [B ⊗ C0 (G)] shows that [D(1 ⊗ C0 (G))] = [D B (B)(1 ⊗ C0 (G))] = B ⊗ C0 (G). In the first equality above we used the non-degeneracy of B ∈ Mor(B, D) : [D B (B)] = D. Applying Lemma 3.4 to D and 1 ⊗ C0 (G) we see that ˜
˜
˜
D ⊗⊗ (1 ⊗ C0 (G)⊗ ) = B ⊗ C0 (G)⊗ . This together with Eq. (10) ends the proof.
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5. Quantum Minkowski Space In this section we shall apply the Rieffel deformation to the action of the Lorentz group G (more precisely of S L(2, C)) on the Minkowski space M, obtaining a C∗ -algebraic quantum Minkowski space as a result. For a survey on the quantum Minkowski spaces on a purely algebraic level we refer to [7]. There exists a natural extension of our construction to an action of a quantum Poincaré group, which gives a C∗ -algebraic realization of a family of the quantum Minkowski spaces described in [7]. As usual, we shall identify M with the set H of 2 × 2 hermitian matrices:
x0 + x3 x1 + i x2 ∈ H. M (x0 , x1 , x2 , x3 ) → x1 − i x2 x0 − x3 Under this identification the right action of G on H is given by: H × G (h, g) → g ∗ hg ∈ H. To perform the Rieffel deformation we use the subgroup ⊂ G consisting of the diagonal matrices: z
e 0 = : z ∈ C . (11) 0 e−z Our choice of is the same as in [1]. We shall pull back the actions of to the actions of C, which is possible due to the morphism σ : C → given by z
e 0 C z → σ (z) = ∈ . (12) 0 e−z To be more precise, if α : → Aut(A) is an action of on a C∗ -algebra A, then the formula C z → ασ (z) ∈ Aut(A) defines an action of C on A. It can be shown that all of the constructions of this paper can be performed in the case where we use a continuous group homomorphism σ : → G instead of a pure embedding ⊂ G. The benefits of pulling back the actions of to the actions of C are related to the self-duality of C and the simple forms of continuous 2-cocycles on C. The duality that we shall use in this paper is established by the following bicharacter on C: C2 (z 1 , z 2 ) → exp(i(z 1 z 2 )) ∈ T.
(13)
We shall use the 2-cocycle on C of the form C2 (z 1 , z 2 ) → (z 1 , z 2 ) = exp(−is(z 1 z 2 )) ∈ T,
(14)
where s ∈ R is the deformation parameter. Note that differs from the 2-cocycle used in the example presented in the paper [1] by the sign in the exponent. This is related to some sign mistakes that we found in [1] during the preparation of the example for this paper. Some further inconsistencies which the reader may have noticed are due to the fact that we have corrected the mistakes of [1]. Once and have been fixed, we can perform the Rieffel deformation of C0 (G) and ˜ C0 (H). The analysis of the quantum group G = (C0 (G)⊗ , ) was undertaken in [1]. In what follows we shall give a concise description of G in terms of the generators ˜ ˆ γˆ , δˆ η C0 (G)⊗ . α, ˆ β,
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Let ρ be the C2 -action on C0 (G) given by: ρz 1 ,z 2 ( f )(g) = f (σ (z 1 )−1 gσ (z 2 )), ˜ ⊗ , one can deform where σ is the morphism defined by (12). Using a 2-cocycle ˜ the standard C2 -structure on the crossed product, obtaining (C0 (G) C2 , λ, ρˆ ⊗ ). It ˆ may be checked that our choice of the 2-cocycle and the way that we identify C and C (see Eq. (13)) lead to the following formula for the deformed dual action: ˜
ρˆz⊗ (b) = λ−s z¯ 1 ,s z¯ 2 ρˆz 1 ,z 2 (b)λ∗−s z¯ 1 ,s z¯ 2 . 1 ,z 2
(15)
Applying the morphism λ ∈ Mor(C0 (C2 ), C0 (G) C2 ) to ∈ M(C0 (C2 )) we get a unitary element U = λ() ∈ M(C0 (G) C2 ). Using U and the coordinate funcˆ γˆ , δˆ affiliated with tions α, β, γ , δ affiliated with C0 (G) we define four elements α, ˆ β, C0 (G) C2 : αˆ = U ∗ αU, βˆ = UβU ∗ , γˆ = U γ U ∗ , δˆ = U ∗ δU.
(16)
The main results of Sect. 5 of [1] are contained in the following ˆ γˆ , δˆ be the elements affiliated with C0 (G) C2 introduced Theorem 5.1. Let α, ˆ β, above. Then ˜ ˆ γˆ , δˆ are affiliated with C0 (G)⊗ 1. α, ˆ β, and they generate it. 2. They satisfy the following commutation relations: αˆ βˆ = βˆ α, ˆ ˆ ˆ αˆ δ = δ α, ˆ αˆ γˆ = γˆ α, ˆ ˆ βˆ γˆ = γˆ β, ˆ βˆ δˆ = δˆβ, γˆ δˆ = δˆγˆ , αˆ δˆ = 1 + βˆ γˆ , αˆ αˆ ∗ αˆ βˆ ∗ αˆ γˆ ∗ αˆ δˆ∗
= = = =
αˆ ∗ α, ˆ ∗ ˆ t β α, ˆ t −1 γˆ ∗ α, ˆ δˆ∗ α, ˆ
ˆ βˆ βˆ ∗ = βˆ ∗ β, ∗ ∗ ˆ γˆ γˆ ∗ = γˆ ∗ γˆ , ˆ β γˆ = γˆ β, ∗ −1 ∗ ˆ ˆ γˆ δˆ∗ = t δˆ∗ γˆ , ˆ ˆ β δ = t δ β,
ˆ δˆδˆ∗ = δˆ∗ δ,
where t in these relations and the deformation parameter s (see (14)) are related by t = e−8s . 3. The action of on the generators is given by: (α) ˆ = αˆ ⊗ αˆ + βˆ ⊗ γˆ , ˆ ˆ (β) = αˆ ⊗ βˆ + βˆ ⊗ δ, (γˆ ) = γˆ ⊗ αˆ + δˆ ⊗ γˆ , ˆ = γˆ ⊗ βˆ + δˆ ⊗ δ. ˆ (δ)
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Some comments on this theorem are necessary. In point 1 we used the fact the embed˜ ding C0 (G)⊗ ⊂ M(C0 (G) C2 ) is the Woronowicz morphism and it extends to the ˜ embedding of affiliated elements (C0 (G)⊗ )η ⊂ (C0 (G) C2 )η (see [15]). The commutation relations in point 2 are to be understood in the sense of ( p, q)-commuting pairs for appropriate p and q (see Definition 3.5). In particular, by the results of Sect. 3.2 the above commutation relations may be understood literally - all elements in these relations ˜ exist as elements affiliated with C0 (G)⊗ . The sums used in point 3 denotes the sums ˜ ˜ of strongly commuting normal elements affiliated with C0 (G)⊗ ⊗ C0 (G)⊗ . The summation operation, which in general cannot be defined for a pair of affiliated elements, ˜ ˜ in this case gives rise to normal elements affiliated with C0 (G)⊗ ⊗ C0 (G)⊗ . 5.1. Generators of C0 (H) . Let us move on to the analysis of the C∗ -algebra C0 (H) . It is defined as the Landstad algebra of the C-product (C0 (H)C, λ, ρˆ ). The deformed action ρˆ is given by: ρˆz (b) = λs z¯ ρˆz (b)λ∗s z¯ , for any z ∈ C and b ∈ C0 (H) C (compare with (15)). Let x, y, w ∈ C0 (H)η be the matrix coefficient functions on the set of hermitian matrices:
x w : x, y ∈ R, w ∈ C . (17) H= w¯ y It is obvious that x, y and w generate C0 (H) in the sense of Woronowicz. Our next objective is to introduce three elements x, ˆ yˆ , wˆ generating C0 (H) . In order to do that let us introduce a function : C z → (z) = exp(−i 2s (z 2 )) ∈ T and a unitary element V ∈ M(C0 (H) C), which is the image of ∈ M(C0 (C)) under λ ∈ Mor(C0 (C), C0 (H) C): V = λ(). Using V and the coordinate functions x, y, w ∈ C0 (H)η ⊂ (C0 (H) C)η we define x, ˆ yˆ , wˆ as elements affiliated with C0 (H) C: xˆ = e−2s V x V ∗ ,
yˆ = e−2s V yV ∗ , wˆ = e2s V ∗ wV.
(18)
The multiplicative factors e±2s are introduced to get a nice formula for the coaction of the quantum Lorentz group G on C0 (H) . Theorem 5.2. Let x, ˆ yˆ and wˆ be the elements affiliated with C0 (H) C defined above. Then x, ˆ yˆ and wˆ are affiliated with C0 (H) and they generate it. Proof. The proof follows the same line as the proof of the respective theorem concerning ˆ γˆ , δˆ given in [1]. Let us check that xˆ is ρˆ -invariant: elements α, ˆ β, ρˆw (x) ˆ = e−2s λs w¯ ρˆw (V x V ∗ )λ∗s w¯ = e−2s ρˆw (V ) λs w¯ xλ∗s w¯ ρˆw (V )∗ .
(19)
In order to calculate ρˆw (V ) we use the fact that λ ∈ Mor(C0 (C), C0 (H)C) intertwines the dual action ρˆw with the shift action of C on C0 (C). It is easy to see that (z + w) = (z)(w) exp(−is(zw)).
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ˆ (see Eq. (13)) enable us to see that This formula and the way that we identify C with C ρˆw (V ) = V (w)λ−sw . We may now substitute the above equality into (19) to obtain ρˆw (x) ˆ = e−2s V λ−sw+s w¯ xλ∗sw+s w¯ V ∗ .
(20)
Finally, using the fact that λz implements the action ρz we get λz xλ∗z = e z+¯z x. In particular λ−sw+s w¯ xλ∗sw+s w¯ = x, which substituted into (20) gives ρˆw (x) ˆ = x. ˆ The ρˆ -invariance of xˆ ∈ (C0 (H) C)η is a necessary condition to prove that xˆ is affiliated with C0 (H) , but it is not sufficient. To this end let us define a morphism π : C0 (R) → M(C0 (H) C): π( f ) = f (x). ˆ Obviously, π( f ) is ρˆ -invariant for any f . Furthermore, the map C z → λz π( f )λ∗z ∈ M(C0 (H) C) is norm continuous. The latter statement follows from the following computation: λz π( f )λ∗z = λz f (x)λ ˆ ∗z = V λz f (e−2s x)λ∗z V ∗ = V f (e−2s e z+¯z x)V ∗ . Hence we see that π( f ) satisfies the sufficient conditions to be an element of M(C0 (H) ). Let us now show that π ∈ Mor(C0 (R), C0 (H) ). In order to do that we have to check the nondegeneracy [π(C0 (R)) C0 (H) ] = C0 (H) . Invoking Lemma 2.6 of [1] it follows from the equality [π(C0 (R)) C0 (H) C∗ (C)] = C0 (H) , which we prove as follows: [π(C0 (R)) C0 (H) C∗ (C)] = [π(C0 (R)) C0 (H) ] = [π(C0 (R)) C∗ () C0 (H)] = [V { f (x)| f ∈ C0 (R)}V ∗ C∗ () C0 (H)] = [V { f (x)| f ∈ C0 (R)} C∗ () C0 (H)] = [V { f (x)| f ∈ C0 (R)} C0 (H) ] = [V C0 (H) ] = C0 (H) . In the fifth equality we used the fact that x ∈ (C0 (H) )η while in the third and the sixth equality we used the unitarity of V . We see that π ∈ Mor(C0 (R), C0 (H) ), hence xˆ = π(id) is affiliated with C0 (H) . In a similar way one can also prove that yˆ and wˆ are affiliated with C0 (H) . Our next objective is to show that x, ˆ yˆ and wˆ generate C0 (H) in the sense of Woronowicz. This follows from the fact that the subset of M(C0 (H) ) given by { f 1 (x) ˆ f 2 ( yˆ ) f 3 (w)| ˆ f 1 , f 2 ∈ C0 (R), f 3 ∈ C0 (C)} is in fact a linearly dense subset of C0 (H) . To prove this density we use the same arguments that were used in the proof of Theorem 5.5 of [1].
Let us move on to the analysis of the commutation relations for x, ˆ yˆ and w. ˆ It is easy to see that xˆ and yˆ strongly commute. We shall show that the relations between xˆ and wˆ and between yˆ and wˆ are of the ( p, q)-type in the sense of Definition 3.5.
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Theorem 5.3. Let x, ˆ yˆ , wˆ be the generators of C0 (H) introduced above and t = e−8s , where s ∈ R is the deformation parameter that specifies the 2-cocycle (14). Then (x, ˆ w) ˆ and ( yˆ , w) ˆ are respectively a (t −1 , t) and (t, t −1 )-commuting pair of normal elements affiliated with C0 (H) . Proof. Let us introduce an affiliated element T ∈ (C0 (H) C)η such that λz = exp(i(zT )). Using the fact that λ implements the action of C on C(H) we get: e z¯ −z w = exp(i(zT ))w exp(−i(zT )),
(21)
e z¯ −z w = exp(i(zT ∗ ))w exp(−i(zT ∗ )).
(22)
In particular, the affiliated element (T − T ∗ ) ∈ (C0 (H) C)η strongly commutes with w ∈ (C0 (H) C)η . The unitary V (see (18)) can be expressed by T : V = exp(−i 2s (T 2 )). Using this we see that: s s wˆ = e2s exp i (T 2 ) w exp −i (T 2 ) 2s 2 s = e2s exp i (T (T − T ∗ )) w exp −i (T (T − T ∗ )) 2 2 2s ∗ 2s = e exp(−sT + sT )w = e w exp(−2is(T )). In the second equality we used the fact that (T T ∗ ) = 0 and in the third equality we used Eq. (21) with z replaced by 2s (T − T ∗ ) (this is legitimate since T − T ∗ and w strongly commute). The product w exp(−is(T )) is well defined due to the fact that w and T − T ∗ strongly commute (see Theorem 3.6). Using the above considerations, it is easy to check equality ˆ exp(−2is(T ))z(x) ˆ exp(2is(T )) = z e−4s (x); we see that: z(w)z( ˆ x) ˆ = = = = = =
ˆ z e2s (w) exp(−2is(T ))z(x) z e2s (w)z e−4s (x) ˆ exp(−2is(T )) z e2s (w)V z e−6s (x)V ∗ exp(−2is(T )) V z e2s (w) exp(−2is(T ))z e−6s (x)V ∗ exp(−2is(T )) V z e2s (w)z e−10s (x)V ∗ exp(−4is(T )) V z e−10s (x)V ∗ z e2s (w) exp(−2is(T )) = z e−8s (x)z( ˆ w). ˆ
This shows that xˆ and wˆ satisfy the second identity of Definition 3.5 of a (t, t −1 )-commuting pair. Using the fact that xˆ is self-adjoint and taking the adjoint of the above calculation we may see that (x, ˆ w) ˆ is in fact an example of a (t, t −1 )-commuting pair of normal elements. A similar reasoning shows that the pair ( yˆ , w) ˆ is an example of a (t −1 , t)-commuting pair.
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5.2. Coaction of G on C0 (H) . Let G be the quantum Lorentz group described in Theorem 5.1. From the results of Sect. 4 we know that there exists a continuous right coaction H of G on C0 (H) . The aim of this section is to describe H in terms of η its action on generators x, ˆ yˆ , wˆ ∈ (C0 (H) ) . The coaction H of (C0 (G), ) on C0 (H) when applied to generators x, y, w ∈ C0 (H)η gives H (x) = x ⊗ α ∗ α + w ⊗ α ∗ γ + w ∗ ⊗ γ ∗ α + y ⊗ γ ∗ γ , H (y) = x ⊗ β ∗ β + w ⊗ β ∗ δ + w ∗ ⊗ δ ∗ β + y ⊗ δ ∗ δ, H (w) = x ⊗ α ∗ β + w ⊗ α ∗ δ + w ∗ ⊗ γ ∗ β + y ⊗ γ ∗ δ.
(23)
In what follows we shall show that in the case of H the only change is that one has to add a hat over each affiliated element above. Theorem 5.4. Let G be the quantum group described in Theorem 5.1, C0 (H) be the C∗ -algebra described in Theorem 5.2 and H be the coaction of G on C0 (H) described in the beginning of Sect. 5.2. The action of ˆ w, ˆ yˆ ∈ H on the generators x, (C0 (H) )η is given by ˆ = xˆ ⊗ αˆ ∗ αˆ + wˆ ⊗ αˆ ∗ γˆ + wˆ ∗ ⊗ γˆ ∗ αˆ + yˆ ⊗ γˆ ∗ γˆ , H ( x) ˆ (w) ˆ = xˆ ⊗ αˆ ∗ βˆ + wˆ ⊗ αˆ ∗ δˆ + wˆ ∗ ⊗ γˆ ∗ βˆ + yˆ ⊗ γˆ ∗ δ, H
ˆ ˆ ∗ ˆ ˆ ⊗ βˆ ∗ δˆ + wˆ ∗ ⊗ δˆ∗ βˆ + yˆ ⊗ δˆ∗ δ, H ( yˆ ) = xˆ ⊗ β β + w where on the right-hand side of each of these equalities we have the sums of strongly ˜ commuting elements affiliated with C0 (H) ⊗ C0 (G)⊗ . Proof. In the course of the proof we shall use the affiliated element T ∈ (C0 (H) C)η such that λz = exp(i(zT )) (see the proof of Theorem 5.3). We shall also use Tl , Tr ∈ (C0 (G) C2 )η such that λz 1 ,z 2 = exp(i(z 1 Tl + z 2 Tr )). The generators x, ˆ yˆ , wˆ of C0 (H) may be expressed in terms of T and the coordinates x, y, w ∈ C0 (H)η : s s (24) xˆ = e−2s exp −i (T 2 ) x exp i (T 2 ) , 2s 2s (25) yˆ = e−2s exp −i (T 2 ) y exp i (T 2 ) , s 2 s2 (26) wˆ = e2s exp i (T 2 ) w exp −i (T 2 ) . 2 2 ˜
ˆ γˆ , δˆ of C0 (G)⊗ may be expressed in terms of Similarly, the generators α, ˆ β, and α, β, γ , δ:
αˆ = exp −is(Tr Tl∗ ) α exp is(Tr Tl∗ ) ,
βˆ = exp is(Tr Tl∗ ) β exp −is(Tr Tl∗ ) ,
γˆ = exp is(Tr Tl∗ ) γ exp −is(Tr Tl∗ ) ,
δˆ = exp −is(Tr Tl∗ ) δ exp is(Tr Tl∗ ) .
Tl , Tr (27) (28) (29) (30)
Let us move on to the proof of the equality ˆ = xˆ ⊗ αˆ ∗ αˆ + wˆ ⊗ αˆ ∗ γˆ + wˆ ∗ ⊗ γˆ ∗ αˆ + yˆ ⊗ γˆ ∗ γˆ . H ( x)
(31)
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From the definition of H (see Sect. 4) we get: ˆ = e−2s Y (x ⊗ α ∗ α + w ⊗ α ∗ γ + w ∗ ⊗ γ ∗ α + y ⊗ γ ∗ γ )Y ∗ , H ( x)
(32)
where Y ∈ M(C0 (H) C ⊗ C0 (G) C2 ) is the unitary element given by s Y = exp −i (1 ⊗ Tr2 + 2T ⊗ Tl∗ ) . 2 In what follows we shall analyze the four terms appearing in (32), proving that they are equal to the corresponding four terms appearing on the right side of (31). Let us first show that e−2s Y (x ⊗ α ∗ α)Y ∗ = xˆ ⊗ αˆ ∗ α. ˆ
(33)
Substituting the formula (24) for xˆ and the formula (27) for αˆ in Eq. (33) we get an equivalent form of (33): X (x ⊗ α ∗ α)X ∗ = (x ⊗ α ∗ α), where the unitary element X ∈ M(C0 (H) C ⊗ C0 (G) C2 ) is given by s exp i (T 2 ⊗ 1 − 2T ⊗ Tl∗ − 1 ⊗ (Tr2 − 2Tr Tl∗ )) . 2 It is easy to check that T 2 ⊗ 1 − 2T ⊗ Tl∗ − 1 ⊗ (Tr2 − 2Tr Tl∗ ) = (T ⊗ 1 − 1 ⊗ Tl∗ )2 − 1 ⊗ (Tr − Tl∗ )2 . (34) The element T ⊗ 1 − 1 ⊗ Tl∗ strongly commutes with x ⊗ α ∗ α, which follows from the equality:
exp(i(zT )) ⊗ exp(−i(zTl∗ )) (x ⊗ α ∗ α) (exp(−i(zT )) (35) ⊗ exp(i(zTl∗ )) = (x ⊗ α ∗ α). Similarly, the element 1 ⊗ (Tr − Tl∗ ) strongly commutes with x ⊗ α ∗ α and using Eq. (34) we see that (33) is satisfied. Our next objective is to prove that: e−2s Y (w ⊗ α ∗ γ )Y ∗ = wˆ ⊗ αˆ ∗ γˆ .
(36)
To this end, let us note that 2T ⊗ Tl∗ = (T ⊗ 1 + 1 ⊗ Tl∗ )2 − T 2 ⊗ 1 − 1 ⊗ Tl∗2 . It can be checked that T ⊗ 1 + 1 ⊗ Tl∗ strongly commutes with w ⊗ α ∗ γ , hence Eq. (36) is equivalent to e−2s Y (w ⊗ α ∗ γ )Y ∗ = wˆ ⊗ αˆ ∗ γˆ , where Y = exp −i 2s (1 ⊗ Tr2 − 1 ⊗ Tl∗2 − T 2 ⊗ 1) . Using formula (26) for wˆ we see that s e−2s Y (w ⊗ α ∗ γ )Y ∗ = wˆ ⊗ exp(−4s) exp −i (Tr2 − Tl∗2 ) 2 s ∗ 2 ∗2 α γ exp i (Tr − Tl ) . 2
Rieffel Deformation of Group Coactions
Therefore to show (36) we have to check that s s exp(−4s) exp −i (Tr2 − Tl∗2 ) α ∗ γ exp i (Tr2 − Tl∗2 ) = αˆ ∗ γˆ . 2 2
759
(37)
The identity exp(i(zTl∗ ))α ∗ γ exp(−i(zTl∗ )) = exp(z − z¯ )Tl with z replaced by 2s (Tl∗ − Tl ) (this is legitimated by the strong commutativity of α ∗ γ and Tl∗ − Tl ) shows that s s exp i (Tl∗2 ) α ∗ γ exp −i (Tl∗2 ) = exp(sTl∗ − sTl )α ∗ γ . (38) 2 2 Similarly, using the identity exp(i(zTr ))α ∗ γ exp(−i(zTr )) = exp(z + z¯ )Tl with z replaced by − 2s (Tr∗ +Tr ) (which in turn is legitimated by the strong commutativity of α ∗ γ and Tr∗ + Tr ) we get s s exp −i (Tr2 ) α ∗ γ exp i (Tr2 ) = exp(−sTr∗ − sTr )α ∗ γ . (39) 2 2 Equations (38) and (39) together show that the left-hand side of (37) is given by exp(−4s) exp(s(Tl∗ − Tl − Tr∗ − Tr ))α ∗ γ . This is equal to the right hand side of (37), because αˆ ∗ γˆ = exp(−s(Tl + Tr ))α ∗ exp(s(Tl∗ − Tr∗ ))γ = exp(−4s) exp(s(Tl∗ − Tl − Tr∗ − Tr ))α ∗ γ .
(40)
To prove the above equality we first use the identity αˆ = exp(−is(Tr Tl∗ ))α exp(is(Tr Tl∗ )) = exp(−sTl∗ − sTr∗ )α
(41)
which in turn follows from the equality exp(i(zTr )α exp(−i(zTr ) = exp(z)α with z replaced by −sTl∗ − sTr∗ (as the reader may expect, one has to invoke at this moment the strong commutativity of α and Tl +Tr to legitimate this argument). Similarly, we can show that γˆ = exp(sTl∗ − sTr∗ )γ . Finally, in the second equality of (40) we have used the fact α ∗ exp(s(Tl∗ − Tr∗ )) = exp(−4s) exp(s(Tl∗ − Tr∗ ))α ∗ , which can be proved using the framework of ( p, q)-commuting pairs of elements affiliated with C0 (G) C2 . This ends the proof of (36), which by taking the adjoint also shows that ˆ e−2s Y (w ∗ ⊗ γ ∗ α)Y ∗ = wˆ ∗ ⊗ γˆ ∗ α.
(42)
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Let us now check that e−2s Y (y ⊗ γ ∗ γ )Y ∗ = yˆ ⊗ γˆ ∗ γˆ .
(43)
The reasoning is similar to the one which proved (33). We begin by substituting formula (25) for yˆ and (29) for γˆ in Eq. (43) to obtain the equivalent equality: X (y ⊗ γ ∗ γ )X ∗ = (y ⊗ γ ∗ γ ).
(44)
In this case X ∈ M(C0 (H) C ⊗ C0 (G) C2 ) is given by s exp i (T 2 ⊗ 1 − 2T ⊗ Tl∗ − 1 ⊗ (Tr2 + 2Tr Tl∗ )) . 2 Let us note that T 2 ⊗ 1 − 2T ⊗ Tl∗ − 1 ⊗ (Tr2 + 2Tr Tl∗ ) = (T ⊗ 1 − 1 ⊗ Tl∗ )2 − 1 ⊗ (Tr + Tl∗ )2 . Equation (44) follows from the fact that both operators T ⊗ 1 − 1 ⊗ Tl∗ and 1 ⊗ (Tr + Tl∗ ) strongly commute with y ⊗ γ ∗ γ . Equation (33) together with (36), (42) and (43) proves (31). It follows from construction that the sum on the right is the sum of strongly com˜ muting, normal elements affiliated with C0 (H) ⊗ C0 (G)⊗ . Let us move on to the proof of the second equality of our theorem: ˆ ˆ = xˆ ⊗ αˆ ∗ βˆ + wˆ ⊗ αˆ ∗ δˆ + wˆ ∗ ⊗ γˆ ∗ βˆ + yˆ ⊗ γˆ ∗ δ. H (w)
(45)
By the definition of H (see Sect. 4) it may be seen that ˆ = e2s Y (x ⊗ α ∗ β + w ⊗ α ∗ δ + w ∗ ⊗ γ ∗ β + y ⊗ γ ∗ δ)Y ∗ , H (w)
(46)
where in this case Y ∈ M(C0 (H)C⊗C0 (G)C2 ) is a unitary element of the following form: s Y = exp i (1 ⊗ Tr2 − 2T ⊗ Tl∗ ) . (47) 2 As before, we shall analyze the four terms appearing on the right hand side of Eq. (46). In order to show that e2s Y (x ⊗ α ∗ β)Y ∗ = xˆ ⊗ αˆ ∗ βˆ
(48)
− 2T ⊗ Tl∗ = (T ⊗ 1 − 1 ⊗ Tl∗ )2 − T 2 ⊗ 1 − 1 ⊗ Tl∗2 .
(49)
we use the equality:
The fact that T ⊗ 1 − 1 ⊗ Tl∗ strongly commutes with x ⊗ α ∗ β leads to the following equality (see also (37)): s s ˆ exp(4s) exp i (Tr2 − Tl∗2 ) α ∗ β exp −i (Tr2 − Tl∗2 ) = αˆ ∗ β, (50) 2 2 which we check below. Using exp(i(zTr ))α ∗ β exp(−i(zTr )) = e z¯ −z α ∗ β
Rieffel Deformation of Group Coactions
with z replaced by 2s (Tr − Tr∗ ) we get s s exp i (Tr2 ) α ∗ β exp −i (Tr2 ) = exp(sTr∗ − sTr )α ∗ β. 2 2 Similarly, we may show that s s exp −i (Tl∗2 ) α ∗ β exp i (Tl∗2 ) = exp(−sTl − sTl∗ )α ∗ β. 2 2
761
(51)
(52)
Equations (51) and (52) show that the left hand side of (50) is equal to e4s exp(sTr∗ − sTr − sTl∗ − sTl )α ∗ β.
(53)
Equation (41) gives αˆ ∗ = exp(−sTl − sTr )α ∗ . Similarly, using (28) we can check that βˆ = exp(sTr∗ − sTl∗ )β. The above two equalities give: αˆ ∗ βˆ = exp(−sTl − sTr )α ∗ exp(sTr∗ −sTl∗ )β = e4s exp(sTr∗ −sTr −sTl∗ − sTl )α ∗ β, (54) where in the final step we used α ∗ exp(sTr∗ − sTl∗ ) = e4s exp(sTr∗ − sTl∗ )α ∗ . The last formula can be proved in the framework of ( p, q)-commuting pairs of elements affiliated with C0 (G) C2 (see Theorem 3.7). Using (53) and (54) we get (48). Our next objective is to prove that ˆ e2s Y (w ⊗ α ∗ δ)Y ∗ = wˆ ⊗ αˆ ∗ δ.
(55)
Inserting formula (26), (27) and (30) for w, ˆ αˆ and δˆ respectively we see that the above equality is equivalent with the following one Y (w ⊗ α ∗ δ)Y ∗ = (w ⊗ α ∗ δ),
(56)
where Y ∈ M(C0 (H) C ⊗ C0 (G) C2 ) is a unitary element given by s Y = exp i (1 ⊗ Tr2 − 2T ⊗ Tl∗ − T 2 ⊗ 1 + 2 ⊗ Tr Tl∗ ) . 2 It is easy to see that 1 ⊗ Tr2 − 2T ⊗ Tl∗ −T 2 ⊗ 1 + 2 ⊗ Tr Tl∗ = (1 ⊗ Tr + 1 ⊗ Tl∗ )2 − (T ⊗ 1 + 1 ⊗ Tl∗ )2 . Equation (56) follows from the observation that both elements 1 ⊗ (Tr + Tl∗ ) and (T ⊗ 1 + 1 ⊗ Tl∗ ) strongly commute with (w ⊗ α ∗ δ). The proof of the equality e2s Y (w ∗ ⊗ γ ∗ β)Y ∗ = wˆ ∗ ⊗ γˆ ∗ βˆ
(57)
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is similar to the proof of (55). It is based on the observation that the element 1 ⊗ Tr2 − 1 ⊗ 2Tr Tl∗ − T 2 ⊗ 1 − 2T ⊗ Tl∗ = 1 ⊗ (Tr − Tl∗ )2 − (T ⊗ 1 + 1 ⊗ Tl∗ )2 strongly commutes with w ∗ ⊗ γ ∗ β. Finally, let us check that ˆ e2s Y (y ⊗ γ ∗ δ)Y ∗ = yˆ ⊗ γˆ ∗ δ, where Y is given by (47). The following reasoning is similar to the one which proved (48). Using (49) together with the strong commutativity of T ⊗ 1 − 1 ⊗ Tl∗ and y ⊗ γ ∗ δ we get an equivalent formula, which is s s ˆ exp(4s) exp i (Tr2 − Tl∗2 ) γ ∗ δ exp −i (Tr2 − Tl∗2 ) = γˆ ∗ δ. (58) 2 2 It can be shown (compare with (51) and (52)) that s s exp i (Tr2 ) γ ∗ δ exp −i (Tr2 ) = exp(sTr∗ − sTr )γ ∗ δ, 2 2 s s exp −i (Tl∗2 ) γ ∗ δ exp i (Tl∗2 ) = exp(sTl + sTl∗ )γ ∗ δ. 2 2
(59)
(60)
These two identities show that the left hand side of (58) is given by e4s exp(sTl + sTl∗ + sTr∗ − sTr )γ ∗ δ and moreover, it is equal to the right hand side by the following computation (compare with (54)): γˆ ∗ δˆ = exp(sTl − sTr )γ ∗ exp(sTr∗ + sTl∗ )δ = e4s exp(sTl + sTl∗ + sTr∗ − sTr )γ ∗ δ. In the second equality we used the fact that γ ∗ exp(sTr∗ + sTl∗ ) = e4s exp(sTr∗ + sTl∗ )γ ∗ , which can be proved in the framework of ( p, q)-commuting pairs of elements affiliated with C0 (G) C2 (see Theorem 3.7). The equalities (48), (55), (57) and (58) together with (46) imply (45). The proof of the equality ˆ ˆ ∗ ˆ ˆ ⊗ βˆ ∗ δˆ + wˆ ∗ ⊗ δˆ∗ βˆ + yˆ ⊗ δˆ∗ δ, H ( yˆ ) = xˆ ⊗ β β + w is very similar to the proof of (31) and is left to the reader.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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References 1. Kasprzak, P.: Rieffel deformation via crossed products. J. Funct. Anal. 257(5), 1288–1332 (2009) 2. Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. Ec. Norm. Sup. 33(4), 837– 934 (2000) 3. Landstad, M.B.: Duality theory for covariant systems. Trans. AMS 248(2), 223–267 (1979) 4. Landstad, M.B.: Quantizations arising from abelian subgroups. Internat. J. Math. 5(6), 897–936 (1994) 5. Masuda, T., Nakagami, Y., Woronowicz, S.L.: A C∗ -algebraic framework for quantum groups. Internat. J. Math. 14(9), 903–1001 (2003) 6. Napiórkowski, K., Woronowicz, S.L.: Operator theory in the C ∗ -algebra framework. Rep. Math. Phys. 31(3), 353–371 (1992) 7. Podle´s, P.: Quantum Minkowski spaces. AIP Conf. Proc. 453, 97–106 (1998) 8. Pusz, W., Woronowicz, S.L.: A Quantum G L(2, C) group at roots of unity. Rep. Math. Phys. 47, 431– 462 (2001) 9. Rieffel, M.A.: Deformation quantization for action of Rd . Mem. Am. Math. Soc. 106(506) (1993) 10. Rieffel, M.A.: Non-compact quantum groups associated with abelian subgroups. Commun. Math. Phys. 171(1), 181–201 (1995) 11. Varilly, J.C.: Quantum symmetry groups of noncommutative spheres. Commun. Math. Phys. 221(3), 511–523 (2001) 12. Woronowicz, S.L.: C∗ -algebras generated by unbounded elements. Rev. Math. Phys. 7(3), 481–521 (1995) 13. Woronowicz, S.L.: From multiplicative unitaries to quantum groups. Internat. J. Math. 7(1), 127–149 (1996) 14. Woronowicz, S.L.: Operator equalities related to the quantum E(2) group. Commun. Math. Phys. 144(2), 417–428 (1992) 15. Woronowicz, S.L.: Unbounded elements affiliated with C∗ -algebras and non-compact quantum groups. Commun. in Math. Phys. 136(2), 399–432 (1991) Communicated by Y. Kawahigashi
Commun. Math. Phys. 300, 765–788 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1128-2
Communications in
Mathematical Physics
Relative Equilibria in Continuous Stellar Dynamics Juan Campos1 , Manuel del Pino2 , Jean Dolbeault1 1 Ceremade (UMR CNRS n◦ 7534), Université Paris-Dauphine, Place de Lattre de Tassigny,
75775 Paris Cédex 16, France. E-mail: [email protected]; [email protected]
2 Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago,
Chile. E-mail: [email protected] Received: 27 January 2010 / Accepted: 14 May 2010 Published online: 11 September 2010 – © The Author(s) 2010
Abstract: We study a three dimensional continuous model of gravitating matter rotating at constant angular velocity. In the rotating reference frame, by a finite dimensional reduction, we prove the existence of non-radial stationary solutions whose supports are made of an arbitrarily large number of disjoint compact sets, in the low angular velocity and large scale limit. At first order, the solutions behave like point particles, thus making the link with the relative equilibria in N -body dynamics. 1. Introduction and Statement of the Main Results We consider the Vlasov-Poisson system ⎧ ∂ f + v · ∇x f − ∇x φ · ∇v f = 0 ⎪ ⎨ t 1 ⎪ ⎩φ = − ∗ ρ, ρ := f dv 4π | · | R3
(1)
which models the dynamics of a cloud of particles moving under the action of a mean field gravitational potential φ solving the Poisson equation: φ = ρ. Kinetic models like system (1) are typically used to describe gaseous stars or globular clusters. Here f = f (t, x, v) is the so-called distribution function, a nonnegative function in L ∞ (R, L 1 (R3 × R3 )) depending on time t ∈ R, position x ∈ R3 and velocity v ∈ R3 , which represents a density of particles in the phase space, R3 × R3 . The function ρ is the spatial density function and depends only on t and x. The total mass is conserved and hence f (t, x, v) d x dv = ρ(t, x) d x = M R 3 ×R 3
R3
does not depend on t. © 2010 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.
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The first equation in (1) is the Vlasov equation, also known as the collisionless Boltzmann equation in the astrophysical literature; see [5]. It is obtained by writing that the mass is transported by the flow of Newton’s equations, when the gravitational field is computed as a mean field potential. Reciprocally, the dynamics of discrete particle systems can be formally recovered by considering empirical distributions, namely measure valued solutions made of a sum of Dirac masses, and neglecting the self-consistent gravitational terms associated to the interaction of each Dirac mass with itself. It is also possible to relate (1) with discrete systems as follows. Consider the case of N gaseous spheres, far away from each other, in such a way that they weakly interact through gravitation. In terms of system (1), such a solution should be represented by a distribution function f , whose space density ρ is compactly supported, with several nearly spherical components. At large scale, the location of these spheres is governed at leading order by the N -body gravitational problem. The purpose of this paper is to unveil this link by constructing a special class of solutions: we will build time-periodic, non radially symmetric solutions, which generalize to kinetic equations the notion of relative equilibria for the discrete N -body problem. Such solutions have a planar solid motion of rotation around an axis which contains the center of gravity of the system, so that the centrifugal force counter-balances the attraction due to gravitation. Let us give some details. Consider N point particles with masses m j , located at points x j (t) ∈ R3 and assume that their dynamics is governed by Newton’s gravitational equations N d2x j m j m k xk − x j = , dt 2 4π |xk − x j |3
mj
j = 1, . . . N .
(2)
j=k=1
Let us write x ∈ R3 as x = (x , x 3 ) ∈ R2 × R ≈ C × R where, using complex notations, x = (x 1 , x 2 ) ≈ x 1 + i x 2 and rewrite system (2) in coordinates relative to a reference frame rotating at a constant velocity ω > 0 around the x 3 -axis. This amounts to carry out the change of variables x = (ei ω t z , z 3 ), z = z 1 + i z 2 . In terms of the coordinates (z , z 3 ), system (2) then reads N d2z j m k zk − z j 2 = + ω (z , 0) + 2 ω i j dt 2 4π |z k − z j |3
dz j dt
,0 ,
j = 1, . . . N .
(3)
j=k=1
We consider solutions which are stationary in the rotating frame, namely constant solutions (z 1 , . . . z N ) of system (3). Clearly all z j ’s have their third component with the same value, which we assume zero. Hence, we have that z k = (ξk , 0), ξk ∈ C, where the ξk ’s are constants and satisfy the system of equations N m k ξk − ξ j + ω2 ξ j = 0, 4π |ξk − ξ j |3
k= j=1
j = 1, . . . N .
(4)
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In the original reference frame, the solution of (2) obeys a rigid motion of rotation around the center of mass, with constant angular velocity ω. This solution is known as a relative equilibrium, thus taking the form x ωj (t) = (ei ω t ξ j , 0), ξ j ∈ C,
j = 1, . . . N .
System (4) has a variational formulation. In fact a vector (ξ1 , . . . ξ N ) solves (4) if and only if it is a critical point of the function Vmω (ξ1 , . . . ξ N )
N N 1 m j mk ω2 := m j |ξ j |2 . + 8π |ξk − ξ j | 2 j=k=1
j=1
Here m denotes (m j ) Nj=1 . A further simplification is achieved by considering the scaling ξ j = ω−2/3 ζ j , Vmω (ξ1 , . . . ξ N ) = ω2/3 Vm (ζ1 , . . . ζ N ),
(5)
where Vm (ζ1 , . . . ζ N ) :=
N N 1 1 m j mk + m j |ζ j |2 . 8π |ζk − ζ j | 2 j=k=1
j=1
This function has in general many critical points, which are all relative equilibria. For instance, Vm clearly has a global minimum point. Our aim is to construct solutions of gravitational models in continuum mechanics based on the theory of relative equilibria. We have the following result. Theorem 1. Given masses m j , j = 1, . . . N , and any sufficiently small ω > 0, there exists a solution f ω (t, x, v) of Eq. (1) which is 2π ω -periodic in time and whose spatial density takes the form ρ(t, x) :=
R3
f ω dv =
N
ρ j (x − x ωj (t)) + o(1).
i=1
Here o(1) means that the remainder term uniformly converges to 0 as ω → 0+ and identically vanishes away from ∪ Nj=1 B R (x ωj (t)), for some R > 0, independent of ω. The functions ρ j (y) are non-negative, radially
symmetric, non-increasing, compactly supported functions, independent of ω, with R3 ρ j (y) dy = m j and the points x ωj (t) are such that x ωj (t) = ω−2/3 (ei ω t ζ jω , 0), ζ jω ∈ C,
j = 1, . . . N
and lim Vm (ζ1ω , . . . ζ Nω ) = min Vm ,
ω→0+
CN
lim ∇Vm (ζ1ω , . . . ζ Nω ) = 0.
ω→0+
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The solution of Theorem 1 has a spatial density which is nearly spherically symmetric on each component of its support and these ball-like components rotate at constant, very small, angular velocity around the x 3 -axis. The radii of these balls are very small compared with their distance to the axis. We shall call such a solution a relative equilibrium of (1), by extension of the discrete notion. The construction provides much more accurate information on the solution. In particular, the building blocks ρ j are obtained as minimizers of an explicit reduced free energy functional, under suitable mass constraints. It is also natural to consider other discrete relative equilibria, namely critical points of the energy Vm that may or may not be globally minimizing, and ask whether associated relative equilibria of system (1) exist. There are plenty of relative equilibria of the N -body problem. For instance, if all masses m j are equal to some m ∗ > 0, a critical point is found by locating the ζ j ’s at the vertices of a regular polygon: ζ j = r e2 i π( j−1)/N ,
j = 1, . . . N ,
(6)
where r is such that N −1 d aN m∗ 1 2 1 1 + r = 0 with a N := √ , √ dr 4π r 2 2 j=1 1 − cos (2π j/N ) i.e. r = (a N m ∗ /(4π ))1/3 . This configuration is called the Lagrange solution, see [35]. The counterpart in terms of continuum mechanics goes as follows. Theorem 2. Let (ζ1 , . . . ζ N ) be a regular polygon, namely with ζ j given by (6), and assume that all masses are equal. Then there exists a solution f ω exactly as in Theorem 1, but with limω→0+ (ζ1ω , . . . ζ Nω ) = (ζ1 , . . . ζ N ). Further examples of relative equilibria in the N -body problem can be obtained for instance by setting N − 1 point particles of the same mass at the vertices of a regular polygon centered at the origin, then adding one more point particle at the center (not necessarily with the same mass), and finally adjusting the radius. Another family of solutions, known as the Euler–Moulton solutions is constituted by arrays of aligned points. Critical points of the functional Vm are always degenerate because of their invariance under rotations: for any α ∈ R we have Vm (ζ1 , . . . ζ N ) = Vm (ei α ζ1 , . . . ei α ζ N ). Let ζ¯ = (ζ¯1 , . . . ζ¯ N ) be a critical point of Vm with ζ¯ = 0. After a uniquely defined rotation, we may assume that ζ¯ 2 = 0. Moreover, we have a critical point of the function of 2N − 1 real variables, V˜ m (ζ1 , . . . ζ 1 , . . . ζ N ) := Vm (ζ1 , . . . (ζ 1 , 0), . . . ζ N ). We shall say that a critical point of Vm is non-degenerate up to rotations if the matrix D 2 V˜ m (ζ¯1 , . . . ζ¯ 1 , . . . ζ¯ N ) is non-singular. This property is clearly independent of the choice of . Palmore in [30–34] has obtained classification results for the relative equilibria. In particular, it turns out that for almost every choice of masses m j , all critical points of the functional Vm are non-degenerate up to rotations. Moreover, in such a case there exist at least [2 N −1 (N − 2) + 1] (N − 2) ! such distinct critical points. Many other results on relative equilibria are available in the literature. We have collected some of them in Appendix A with a list of relevant references. These results have a counterpart in terms of relative equilibria of system (1).
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Theorem 3. Let (ζ1 , . . . ζ N ) be a non-degenerate critical point of Vm up to rotations. Then there exists a solution f ω as in Theorem 1, which satisfies, as in Theorem 2, limω→0+ (ζ1ω , . . . ζ Nω ) = (ζ1 , . . . ζ N ). This paper is organized as follows. In the next section, we explain how the search for relative equilibria for the Vlasov-Poisson system can be reduced to the study of critical points of a functional acting on the gravitational potential. The construction of these critical points is detailed in Sect. 3. Sections 4 and 5 are respectively devoted to the linearization of the problem around a superposition of solutions of the problem with zero angular velocity, and to the existence of a solution of a nonlinear problem with appropriate orthogonality constraints depending on parameters (ξ j ) Nj=1 related to the location of the N components of the support of the spatial density. Solving the original problem amounts to make all corresponding Lagrange multipliers equal to zero, which is equivalent to find a critical point of a function depending on (ξ j ) Nj=1 : this is the variational reduction described in Sect. 6. The proof of Theorems 1, 2 and 3 is given in Sect. 7 while known results on relative equilibria for the N -body, discrete problem are summarized in Appendix A. 2. The Setup Guided by the representation (3) of the N -body problem in a rotating frame, we change variables in Eq. (1), replacing x = (x , x 3 ) and v = (v , v 3 ) respectively by (ei ω t x , x 3 ) and (i ω x + ei ω t v , v 3 ). Written in these new coordinates, Problem (1) becomes ⎧ ∂f 2 ⎪ ⎪ ∂t + v · ∇x f − ∇x U · ∇v f − ω x · ∇v f + 2 ω i v · ∇v f = 0, ⎨ 1 ⎪ ⎪ ∗ ρ, ρ := f dv. ⎩U = − 4π | · | R3
(7)
The last two terms in the equation take into account the centrifugal and Coriolis force effects. System (7) can be regarded as the continuous version of problem (3). Accordingly, a relative equilibrium of System (1) will simply correspond to a stationary state of (7). Such stationary solutions of (7) can be found by considering for instance critical points of the free energy functional
1 1 F[ f ] := |v|2 −ω2 |x |2 f d x dv− β( f ) d x dv + |∇U |2 d x 3 3 3 3 3 2 2 R ×R R ×R R for some arbitrary convex function β, under the mass constraint f d x dv = M. R 3 ×R 3
A typical example of such a function is β( f ) =
1 q−1 q f κq q
(8)
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for some q ∈ (1, ∞) and some positive constant κq , to be fixed later. An additional restriction, q > 9/7, will come from the variational setting. The corresponding solution is known as the solution of the polytropic gas model, see [3–5,40,44]. When dealing with stationary solutions, it is not very difficult to rewrite
the problem in terms of the potential. A critical point of F under the mass constraint R3 ×R3 f d x dv = M is indeed given in terms of U by 1 1 (9) f (x, v) = γ λ + |v|2 + U (x) − ω2 |x |2 , 2 2 where γ is, up to a sign, an appropriate generalized inverse of β . In case (8), γ (s) = 1/(q−1) , where s+ = (s + |s|)/2 denotes the positive part of s. The parameter κq−1 (−s)+ λ stands for the Lagrange multiplier associated to the mass constraint, at least if f has a single connected component. At this point, one should mention that the analysis is not exactly as simple as written above. Identity (9) indeed holds only component by component of the support of the solution, if this support has more than one connected component, and the Lagrange multipliers have to be defined for each component. The fact that M U (x) ∼ − |x|→∞ 4π |x| is dominated by − 21 ω2 |x |2 as |x | → ∞ is also a serious cause of trouble, which clearly discards the possibility that the free energy functional can be bounded from below if ω = 0. This issue has been studied in [9], in the case of the so-called flat systems. Finding a stationary solution in the rotating frame amounts to solving a non-linear Poisson equation, namely 1 2 2 if x ∈ supp(ρ) (10) U = g λ + U (x) − ω |x | 2 and U = 0 otherwise, where g is defined by 1 g(μ) := γ μ + |v|2 dv. 2 R3 Hence, the problem can also be reduced to look for a critical point of the functional 1 1 J [U ] := |∇U |2 d x + G λ + U (x) − ω2 |x |2 d x − λ ρ d x, 2 R3 2 ∪i K i ∪i K i where λ = λ[x, U ] is now a functional which is constant with respect to x, with value λi , on each connected component K i of the support of ρ(x) = g (λ[x, U ] + U (x)− 1 2 2 , x ∈ ∪i K i and implicitly determined by the condition ω |x | 2 1 g λi + U (x) − ω2 |x |2 d x = m i . 2 Ki N By G, we denote a primitive of g and the total mass is M = i=1 m i . Hence we can rewrite J as N 1 1 J [U ] = |∇U |2 d x + G λi + U (x) − ω2 |x |2 d x − m i λi . (11) 2 R3 2 Ki i=1
Multiple Components Configurations in Continuous Stellar Dynamics
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We may also observe that critical points of F correspond to critical points of the reduced free energy functional 1 1 2 2 |∇U |2 d x G[ρ] := h(ρ) − ω |x | ρ d x − 2 2 R3 R3
ρ acting on the spatial densities if h(ρ) = 0 g −1 (−s) ds. Also notice that, using the same function γ as in (9), to each distribution function f , we can associate a local equilibrium, or local Gibbs state, 1 G f (x, v) = γ μ(ρ(x)) + |v|2 , 2 where μ is such that g(μ) = ρ. This identity defines μ = μ(ρ) = g −1 (ρ) as a function of ρ. Furthermore, by convexity, it follows that F[ f ] ≥ F[G f ] = G[ρ] if ρ(x) = R3 f (x, v) dv, with equality if f is a local Gibbs state. See [11] for more details. Summarizing, the heuristics are now as follows. The various components K i of the support of the spatial density ρ of a critical point are assumed to be far away from each other so that the dynamics of their center of mass is described by the N -body point particles system, at first order. On each component K i , the solution is a perturbation of an isolated minimizer of the free energy functional F (without angular rotation) under the constraint that the mass is equal to m i . In the spatial density picture, on K i , the solution is a perturbation of a minimizer of the reduced free energy functional G. To further simplify the presentation of our results, we shall focus on the model of 1 polytropic gases corresponding to (8). In such a case, with p := q−1 + 23 , g is given by p
g(μ) = (−μ)+ if the constant κq is fixed so that κq = 4π
1 √ +∞ √ 2 0 t (1+t) q−1 dt
3
= (2π ) 2
q ( q−1 ) q ( 23 + q−1 )
. For com-
pactness reasons, we shall further restrict p to be subcritical, so that the range covered by our aproach is p ∈ 3/2, 5). Free energy functionals have been very much studied over the last years, not only to characterize special stationary states, but also because they provide a framework to deal with orbital stability, which is a fundamental issue in the mechanics of gravita
tion. The use of a free energy functional, whose entropy part, R3 ×R3 β( f ) d x dv is sometimes also called the Casimir energy functional, goes back to the work of V.I. Arnold (see [1,2,45]). The variational characterization of special stationary solutions and their orbital stability have been studied by Y. Guo and G. Rein in a series of papers [16–19,36–39] and by many other authors, see for instance [9,10,22–25,40,41,44]. The main drawback of such approaches is that stationary solutions which are characterized by these techniques are in some sense trivial: radial, with a single simply connected component support. Here we use a different approach to construct the solutions, which goes back to [13] in the context of Schrödinger equations. We are not aware of attempts to use dimensional reduction coupled to power-law non-linearities and Poisson force fields except in the similar case of a nonlinear Schrödinger equation with power law nonlinearity and repulsive Coulomb forces (see [8]), or in the case of an attractive Hartree-Fock model (see [21]). Technically, our results turn out to be closely related to the ones in [6,7].
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Compared to previous results on gravitational systems, the main interest of our approach is to provide a much richer set of solutions, which is definitely of interest in astrophysics for describing complex patterns like binary gaseous stars or even more complex objects. The need of such an improvement was pointed out for instance in [20]. An earlier attempt in this direction has been done in the framework of Wasserstein’s distance and mass transport theory in [27]. The point of this paper is that we can take advantage of the knowledge of special solutions of the N -body problem to produce solutions of the corresponding problem in continuum mechanics, which are still reminiscent of the discrete system. 3. Construction of Relative Equilibria 3.1. Some notations. We denote by x = (x , x 3 ) ∈ R2 × R a generic point in R3 . We may reformulate Problem (10) in terms of the potential u = − U as follows. Given N positive numbers λ1 , . . . λ N and a small positive parameter ω, we consider the problem of finding N non-empty, compact, disjoint, connected subsets K i of R3 , i = 1, 2 . . . N , and a positive solution u of the problem −u =
N i=1
ρi
1 2 2 p in R , ρi := u − λi + ω |x | χi , 2 + 3
lim u(x) = 0,
|x|→∞
(12) (13)
where χi denotes the characteristic function of K i . We define the mass and the center of mass associated to each component by 1 ω ω m i := ρi d x and xi := x ρi d x. m i R3 R3 In our construction, when ω → 0, the sets K i are asymptotically balls centered around xi . It is crucial to localize the support of ρi since u − λi + 21 ω2 |x |2 is always positive for large values of |x |. We shall find a solution of (12) as a critical point u of the functional J [u] =
p+1 N 1 1 1 u − λi + ω2 |x |2 |∇u|2 d x − χi d x, 3 2 R3 p+1 2 + i=1 R
(14) p+1
1 so that −u is a critical point of J in (11) in the case (8), namely G(−s) = p+1 s+ . Heuristically, our method goes as follows. We first consider the so-called basic cell problem: we characterize the solution with a single component support, when ω = 0 and then build an ansatz by considering approximate solutions made of the superposition of basic cell solutions located close to relative equilibrium points, when they are far apart from each other. This can be done using the scaling invariance, in the low angular velocity limit ω → 0+ . The proof of our main results will be given in Sects. 4–7. It relies on a dimensional reduction of the variational problem: we shall prove that for a well N 5− p λi e∗ − Vmω (ξ1 , . . . ξ N ) + o(1) for some constant e∗ , up to chosen u, J [u] = i=1 o(1) terms, which are uniform in ω > 0, small. Hence finding a critical point for J will be reduced to look for a critical point of Vmω as a function of (ξ1 , . . . ξ N ).
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3.2. The basic cell problem. Let us consider the following problem: p
− w = (w − 1)+ in R3 .
(15)
Lemma 1. Under the condition lim|x|→∞ w(x) = 0, Eq. (15) has a unique solution, up to translations, which is positive and radially symmetric if p ∈ (1, 5). Proof. Since p is subcritical, it is well known that the problem −Z = Z p in B1 (0) with homogeneous Dirichlet boundary conditions, Z = 0, on ∂ B1 (0), has a unique positive solution, which is also radially symmetric (see [15]). For any R > 0, the function Z R (x) := R −2/( p−1) Z (x/R) is the unique radial, positive solution of p
−Z R = Z R in B R (0) with homogeneous Dirichlet boundary conditions on ∂ B R (0). According to [14,15], any positive solution of (15) is radially symmetric, up to translations. Finding such a solution w of (15) is equivalent to finding numbers R > 0 and m ∗ > 0 such that the function, defined by pieces as w = Z R + 1 in B R and w(x) = m ∗ /(4π |x|) for any x ∈ R3 such that |x| > R, is of class C 1 . These numbers are therefore uniquely determined by w(R − ) = 1 =
m∗ m∗ − 2 −1 = w(R + ) , w (R − ) = R p−1 Z (1) = − = w (R + ) , 4π R 4π R 2
which uniquely determines the solution of (15).
Now let us consider the slightly more general problem −w λ = (w λ − λ)+ in R3 p
with lim|x|→∞ w λ (x) = 0. For any λ > 0, it is straightforward to check that it has a unique radial solution given by
w λ (x) = λ w λ( p−1)/2 x ∀x ∈ R3 . Let us observe, for later reference, that R3
(w
λ
p − λ)+
dx = λ
(3− p)/2
(w − 1)+ d x =: λ(3− p)/2 m ∗ . p
R3
Moreover, w λ is given by w λ (x) =
m∗ λ(3− p)/2 ∀x ∈ R3 such that |x| > R λ−( p−1)/2 . 4π |x|
(16)
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3.3. The ansatz. We consider now a first approximation of a solution of (12)–(13), built as a superposition of the radially symmetric functions w λi translated to points ξi , i = 1, . . . N in R2 × {0}, far away from each other: wi (x) := w λi (x − ξi ), Wξ :=
N
wi .
i=1
Recall that we are given the masses m 1 , . . . m N . We choose, according to formula (16), the positive numbers λi so that p (wi − λi )+ d x = m i for all i = 1, . . . N . R3
By ξ we denote the array (ξ1 , ξ2 , . . . ξ N ). We shall assume in what follows that the points ξi are such that for a large, fixed μ > 0, and all small ω > 0 we have |ξi | < μ ω−2/3 , |ξi − ξ j | > μ−1 ω−2/3 .
(17)
Equivalently, |ζi | < μ , |ζi − ζ j | > μ−1
where
ζi := ω−2/3 ξi .
(18)
We look for a solution of (12) of the form u = Wξ + φ for a convenient choice of the points ξi , where φ is globally uniformly small when compared with Wξ . For this purpose, we consider a fixed number R > 1 such that supp (w λi − λi )+ ⊂ B R−1 (0) ∀i = 1, 2 . . . N and define the functions χ (x) =
⎧ ⎨1 if |x| < R ⎩0 if |x| ≥ R
and χi (x) = χ (x − ξi ).
Thus we want to find a solution to the problem (Wξ + φ) +
N i=1
Wξ − λi + φ +
1 2 2 ω |x | 2
p χi = 0 in R3 +
with lim|x|→∞ φ(x) = 0, that is we want to solve the problem φ +
p−1 N 1 p Wξ − λi + ω2 |x |2 χi φ = −E − N[φ], 2 + i=1
(19)
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where E := Wξ + N[φ] :=
N
N
Wξ − λi +
i=1
1 2 2 ω |x | 2
p χi , +
p p 1 2 2 1 ω |x | + φ − Wξ − λi + ω2 |x |2 2 2 + + i=1 p−1 1 − p Wξ − λi + ω2 |x |2 φ χi . 2 + Wξ − λi +
4. A Linear Theory The purpose of this section is to develop a solvability theory for the operator N 1 2 2 p−1 L[φ] = φ + p Wξ − λi + ω |x | χi φ. 2 + i=1
To this end we introduce the norms N N 4 φ∗ = sup |x − ξi | + 1 |φ(x)|, h∗∗ = sup |x − ξi | + 1 |h(x)|. x∈R3
x∈R3
i=1
i=1
We want to solve problems of the form L[φ] = h with h and φ having the above norms finite. Rather than solving this problem directly, we consider a projected problem of the form L[φ] = h +
N 3
ci j Z i j χi ,
(20)
i=1 j=1
lim φ(x) = 0,
|x|→∞
where Z i j := ∂x j wi , subject to orthogonality conditions φ Z i j χi d x = 0 ∀i = 1, 2 . . . N , j = 1, 2, 3. R3
(21)
(22)
Equation (20) involves the coefficients ci j as Lagrange multipliers associated to the constraints (22). If we can L[ψ] = h and L[Yi j ] = Z i j , and if
solve the equations we define ci j such that R3 ψ Z i j d x + ci j R3 Yi j Z i j d x = 0, then we observe that φ = ψ + i, j ci j Yi j solves (20) and satisfies (22). However, for existence, we will rather reformulate the question as a constrained variational problem; see Eq. (25) below. Lemma 2. Assume that (17) holds. Given h with h∗∗ < +∞, Problem (20)–(22) has a unique solution φ =: T[h] and there exists a positive constant C, which is independent of ξ such that, for ω > 0 small enough, φ∗ ≤ C h∗∗ .
(23)
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Proof. In order to solve (20)–(22), we first establish (23) as an a priori estimate. Assume by contradiction the existence of sequences ωn → 0, ξin satisfying (17) for ω = ωn , of functions φn , h n and of constants cinj for which φn ∗ = 1,
lim h n ∗∗ = 0,
n→∞
φn Z i j χi d x = 0 ∀i, j and L[φn ] = h n +
R3
N 3
cinj Z i j χi .
i=1 j=1
Testing the equation against Z k , we obtain, after an integration by parts, N
p
R3
i=1
=
Wξ n
R3
1 − λi + ωn2 |x |2 2
h n Z k d x +
N 3
p−1
p−1 χi − (wk − λk )+
φn Z k d x
+
cinj
i=1 j=1
R3
Z k Z i j χi d x.
(24)
p−1
Here we have used Z k = p (wk − λk )+ Z k . The integrals in the sum can be estimated as follows: |Z k Z i j χi | d x = |∂x w λk (x) ∂x j w λi (x + ξin − ξkn )| d x R3 B(0,R) C 4/3 |∂x w λk (x)| ≤ n − ξ n |2 d x = O(ωn ) |x + ξ B(0,R) i k for some generic constant C > 0 which will change from line to line. Now we turn our attention to the left-hand side of (24). Since p − 1 > 0, we first notice that
N 1 2 2 p−1 p−1 Wξ n − λi + ωn |x | χi − (wk − λk )+ |φn Z k | d x 2 R3 + i=1
1 2 2 p−1 χi |φn Z k | d x ωn |x | 2 R3 i=1,i=k + ⎛ ⎞ N 1 ⎝ ωn2 |x |2 + wi ⎠ χk |φn Z k | d x. +C R3 2
≤
N
Wξ n − λi +
i=1,i=k
It is not hard to check that
N
R3 i=1,i=k
Wξ n − λi + ⎛
≤ C φn ∗ ⎝
1 2
ωn2 |x |2
+
χi |φn Z k | d x ⎞
N i=1,i=k
p−1
|∂x w λk (x + ξi − ξk )| d x ⎠ = O(ωn ) 4/3
B(0,R)
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and
777
⎞ N 1 ⎝ ωn2 |x |2 + wi ⎠ χk |φn Z k | d x R3 2 i=1,i=k ⎛ ⎞ N λk | |∂ w x ⎠ d x = O(ωn2/3 ). ≤ C φn ∗ ⎝ωn2 + B(0,R) |x + ξk − ξi |
⎛
i=1,i=k
Summarizing, we have found that, for each k = 1, 2 . . . N , 2/3 4/3 n O(ωn ) φn ∗ = h n Z k d x + ck |Z k |2 χk d x + O(ωn ) R3
R3
|cinj |,
(i, j)=(k, )
2/3
n = O(ω from which we deduce that ck n ) + O(h n ∗∗ ) → 0 for all k, . We may indeed notice that for ωn small enough, the above equations define an almost diagonal system, so that the coefficients ci j are uniquely determined. Let us prove that limn→∞ φn L ∞ (R3 ) = 0. If not, since φn ∗ = 1, we may assume that there is an index i and a sufficiently large number R > 0 for which
lim inf φn L ∞ (B R (ξi )) > 0. n→∞
Using elliptic estimates, and defining ψn (x) = φn (ξin + x), we may assume that ψn uniformly converges in the C 1 sense over compact subsets of R3 to a bounded, non-trivial solution ψ of the equation p−1 ψ + p w λi − λi + ψ = 0, ψ ∂x j w λi χ d x = 0 ∀ j = 1, 2, 3. R3
According to [14, Lemma 5], ψ must be a linear combination of the functions ∂x j w λi , j = 1, 2, 3. The latter orthogonality conditions yield ψ ≡ 0. This is a contradiction and the claim is proven. Finally, let h˜ n := h n +
N 3
cinj Z i j χi .
i=1 j=1
Then we have that k
2/3 |h˜ n (x)| ≤ O(ωn ) + O(h n ∗∗ ) i=1
1 , 1 + |x − ξin |4
and hence φ˜ n , the unique solution in R3 of φ˜ n = h˜ n ,
lim φ˜ n (x) = 0,
|x|→∞
satisfies k
2/3 |φ˜ n (x)| ≤ O(ωn ) + O(h n ∗∗ ) i=1
1 . |x − ξin |
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Now, since φn − φ˜ n is harmonic in R3 \ ∪i B R (ξin ), it tends to zero as |x| → ∞ and gets uniformly small on the boundary of this set. By the maximum principle, we get the estimate k
2/3 |φn (x)| ≤ O(ωn ) + O(h n ∗∗ ) i=1
1 on R3 \∪i B R (ξin ). |x − ξin |
This shows that limn→∞ φn ∗ = 0, a contradiction with φn ∗ = 1, and (23) follows. Now, for existence issues, we observe that problem (20)-(22) can be set up in variational form in the Hilbert space 1,2 3 H = φ ∈ D (R ) : φ Z i j χi d x = 0 ∀i = 1, 2 . . . N , j = 1, 2, 3 R3
endowed with the inner product φ, ψ = R3 ∇ψ · ∇φ d x, as p−1 N 1 ∇φ · ∇ψ d x − p Wξ − λi + ω2 |x |2 χi φ ψ d x 2 R3 R3 i=1 + ψ h dx = 0 + R3
(25)
for all ψ ∈ H. Since the potential defined by the second term of the above equality is compactly supported and h decays sufficiently fast, this equation takes the form ˜ where K is a compact linear operator of H. The equation for h˜ = 0 has φ + K[φ] = h, just the trivial solution in view of estimate (23). Fredholm’s alternative thus applies to yield existence. This concludes the proof of Lemma 2. Notice that the convergence in the norm · ∗∗ -norm guarantees that there is no issue with the localization of the support of the components of the spatial density. We conclude this section with some considerations on the differentiability of the solution with respect to the parameter ξ . Let us assume that h = h(·, ξ ) defines a continuous operator into the space of functions with finite · ∗∗ -norm. We also assume that ∂ξ h(·, ξ )∗∗ < +∞. Let φ = φ(·, ξ ) be the unique solution of Problem (20)–(22) for that right hand side, with corresponding constants ci j (ξ ). Then φ is differentiable in ξ . Moreover ∂ξ φ can be decomposed as di j Z i j χ j , ∂ξ φ = ψ + ij
where ψ solves L[ψ] = ∂ξ h −
N i=1
+
p ∂ξ
1 Wξ − λi + ω2 |x |2 2
p−1 χi
φ
+
N 3 ci j ∂ξ Z i j χi + bi j Z i j χi , i=1 j=1
lim ψ(x) = 0,
|x|→∞
R3
ψ Z i j χi d x = 0 ∀i = 1, 2 . . . N , j = 1, 2, 3,
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N 3 and the constants di j are chosen so that η := i=1 j=1 di j Z i j satisfies χi j Z i j η d x = − ∂ξ χi j Z i j φ d x ∀i = 1, 2 . . . N , j = 1, 2, 3. R3
R3
Lemma 3. With the same notations and conditions as in Lemma 2, we have ∂ξ φ(·, ξ )∗ ≤ C h(·, ξ )∗∗ + ∂ξ h(·, ξ )∗∗ . 5. The Projected Nonlinear Problem Next we want to solve a projected version of the nonlinear problem (19) using Lemma 2. Thus we consider the problem of finding φ with φ∗ < +∞, the solution of L[φ] = −E − N[φ] +
N 3
ci j Z i j χi
(26)
i=1 j=1
lim φ(x) = 0,
(27)
|x|→+∞
where the coefficients ci j are Lagrange multipliers associated to the orthogonality conditions φ Z i j χi d x = 0 ∀i = 1, 2 . . . N , j = 1, 2, 3. (28) R3
In other words, we look for a critical point of the functional J defined by (14) under the constraints (28). For this purpose, we first have to measure the error E. We recall that E = Wξ +
N
Wξ − λi +
i=1
=
N i=1
=
N i=1
⎡⎛
⎣⎝wi + ⎡
j=i
1 2 2 ω |x | 2
p χi +
⎞p ⎤ 1 2 2⎠ p w j − λi + ω |x | − (wi − λi )+ ⎦ χi 2 +
⎛ ⎞⎤ p−1 ⎛ ⎞ 1 1 ⎝ p ⎣wi − λi + t ⎝ w j + ω2 |x |2 ⎠⎦ w j + ω2 |x |2 ⎠ χi 2 2 j=i
+
j=i
for some function t taking values in (0, 1). It follows that ⎡ ⎤ N N 1 1 ⎣ + ω2 |ξi |2 ⎦ χi ≤ C ω2/3 |E| ≤ C χi , |ξi − ξ j | 2 i=1
j=i
i=1
from which we deduce the estimate E∗∗ ≤ C ω2/3 .
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As for the operator N[φ], we easily check that for φ∗ ≤ 1, | N[φ] | ≤ C
N
|φ|γ χi with γ := min{ p, 2},
i=1
which implies γ
N[φ] ∗∗ ≤ C φ∗ . Let T be the linear operator defined in Lemma 2. Equation (26) can be rewritten as φ = A[φ] := −T E + N[φ] . Clearly the operator A maps the region $ # B = φ ∈ L ∞ (R3 ) : φ∗ ≤ K ω2/3 into itself if the constant K is fixed, large enough. It is straightforward to check that N[φ] satisfies in this region a Lipschitz property of the form N[φ1 ] − N[φ2 ] ∗∗ ≤ κω φ1 − φ2 ∗ for some positive κω such that limω→0 κω = 0, and hence existence of a unique fixed point φ of A in B immediately follows for ω small enough. We have then solved the projected nonlinear problem. Since the error E is even with respect to the variable x 3 , uniqueness of the solution of (26)–(28) implies that this symmetry is also valid for φ itself, and besides, the numbers ci3 are automatically all zero. Summarizing, we have proven the following result. Lemma 4. Assume that ξ = (ξ1 , ξ2 , . . . ξ N ) ∈ R2N is given and satisfies (17). Then Problem (26)–(28) has a unique solution φξ which depends continuously on ξ and ω for the ∗ -norm and satisfies φξ ∗ ≤ C ω2/3 for some positive C, which is independent of ω, small enough. Besides, the numbers ci3 are all equal to zero for i = 1, 2 . . . N . It is important to mention that φξ also defines a continuously differentiable operator in its parameter. Indeed, combining its fixed point characterization with the implicit function theorem and the result of Lemma 3, we find in fact that ∂ξ φξ ∗ ≤ C ω2/3 . We leave the details to the reader. With the complex notation of Sect. 1, let us consider the rotation ei α of an angle α around the x 3 -axis and let ei α ξ = (ei α ξ1 , . . . ei α ξ N ). By construction, there is a rotational symmetry around the x 3 -axis, which is reflected at the level of Problem (26)–(28) as follows. Lemma 5. Consider the solution φ found in Lemma 4. For any α ∈ R and any (x , x 3 ) ∈ C × R, we have that φei α ξ (x , x 3 ) = φξ (e−i α x , x 3 ). The proof is a direct consequence of uniqueness and rotation invariance of the equation satisfied by φξ .
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6. The Variational Reduction We consider the functional J defined in (14). Our goal is to find a critical point satisfying (28), of the form u = Wξ + φξ . We estimate J [Wξ ] by computing first 2 N N 2 |∇Wξ | d x = ∇wi d x = |∇wi |2 d x + ∇wi · ∇w j d x. R3 R3 R3 R3 i=1
i=1
i= j
The last term of the right hand side can be estimated by p ∇wi · ∇w j d x = − wi w j d x = (wi − λi )+ w j d x R3 R3 R3 λ p = w i − λi + w λ j (x + ξi − ξ j ) d x 3 R λ p mj = d x. w i − λi + 3 4π |x + ξi − ξ j | R If we Taylor expand x → |x + ξi − ξ j |−1 around x = 0, we obtain by (18) λ p mj dx w i − λi + 4π |x + ξi − ξ j | R3 λ p m j (ξi −ξ j )·x 1 2 |x|2 ) d x = w i − λi + + O(ω − |ξi −ξ j |3 4π |ξi −ξ j | B(0,R) mi m j + O(ω4/3 ), = 4π |ξi − ξ j |
p (3− p)/2 . Next we find that where m i = R3 w λi − λi + d x = m ∗ λi ⎛ ⎞ p+1 1 ⎝wi + w j − λi + ω2 |x |2 ⎠ χi d x 2 R3 j=i + p+1 = (wi − λi )+ d x R3 ⎛ ⎞ 1 2 2⎠ p⎝ + ( p + 1) (wi − λi )+ w j + ω |x | d x + O(ω4/3 ) 2 R3 j=i ⎛ ⎞ mi m j 1 p+1 = + ω2 m i |ξi |2 ⎠ + O(ω4/3 ). (wi − λi )+ d x + ( p + 1) ⎝ 4π |ξi − ξ j | 2 R3 j=i
Let us define
1 1 p+1 |∇w|2 d x − (w − 1)+ d x. 2 R3 p + 1 R3 Combining the above estimates, we obtain that e∗ :=
J [Wξ ] =
N i=1
5− p
λi
e∗ − Vmω (ξ1 , . . . ξ N ) + O(ω4/3 ),
(29)
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where Vmω (ξ ) =
mi m j i= j 8π |ξi −ξ j |
+
1 2
ω2
N
i=1 m i
|ξi |2 has been introduced in Sect. 1.
Here the O(ω4/3 ) term is uniform as ω → 0 on the set of ξ satisfying the constraints (17). This approximation is also uniform in the C 1 sense. Indeed, we directly check that ∇ξ J [Wξ ] = −∇ξ Vmω (ξ ) + O(ω4/3 ).
(30)
According to (5), we have Vmω (ξ ) = ω2/3 Vm (ζ ) for ζ = ω2/3 ξ . We get a solution of Problem (12)–(13) as soon as all constants ci j are equal to zero in (26). Lemma 6. With the above notations, ci j = 0 for all i = 1, 2 . . . N , j = 1, 2, 3 if and only if ξ is a critical point of the functional ξ → (ξ ) := J [Wξ + φξ ]. Proof. We have already noticed in Lemma 4 that the numbers ci3 are all equal to zero. On the other hand, we have that ∂ξi j = DJ [Wξ + φξ ] · ∂ξi j (Wξ + φξ ) = ck ∂ξi j (Wξ + φξ ) Z k χ d x
= ci j
R3
|Z i j |2 χi d x
⎛ +⎝
k,
R3
⎞
ck ⎠ O(ω2/3 ).
(31)
(k, )=(i, j)
From here the assertion of the lemma readily follows, provided that ω is sufficiently small. Remark 1. An important observation that follows from the rotation invariance of the equation is the following. Assume that the point ξ is such that ξ = (ξ 1 , 0) = (0, 0) for some ∈ {1, . . . N }. Then if ∂ξk j (ξ ) = 0 for all k = 1, . . . N , j = 1, 2, (k, j) = ( , 2), it follows that actually ξ is a critical point of . Indeed, differentiating in α the relation (ei α ξ ) = (ξ ) we get 0=
N
∂ξk (ξ ) · i ξk = ∂ξ (ξ ) · i ξ = − ξ 1 ∂ξ 2 (ξ ),
k=1
and the result follows. 7. Proofs of Theorems 1–3 Let us consider the solution φξ of (26)–(28), i.e. of the problem L[φξ ] = −E − N[φξ ] +
N 3
ci j (ξ ) Z i j χi
i=1 j=1
lim φξ (x) = 0
|x|→+∞
given by Lemma 4. We will then get a solution of Problem (12)–(13), of the desired form u = Wξ + φξ , inducing the ones for Theorems 1 and 3, if we can adjust ξ in such a way that ci j (ξ ) = 0
for all i = 1, 2 . . . N , j = 1, 2, 3.
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According to Lemma 6, this is equivalent to finding a critical point of the functional (ξ ) := J [Wξ + φξ ]. We expand this functional as follows: J [Wξ ] = J [Wξ +φξ ]− DJ [Wξ +φξ ] · φξ +
1 2
1 0
D 2 J [Wξ +(1−t) φξ ] · (φξ , φξ ) dt.
By definition of φξ we have that DJ [Wξ + φξ ] · φξ = 0. On the other hand, using Lemma 4, we check directly, out of the definition of φξ , that D 2 J [Wξ + (1 − t) φξ ] · (φξ , φξ ) = O(ω4/3 ) uniformly on points ξi satisfying constraints (17). Hence, from expansion (29) we obtain that (ξ ) =
N
5− p
λi
e∗ − Vmω (ξ ) + O(ω4/3 ).
i=1
We claim that this expansion also holds in the C 1 sense. Let us first observe that E ∂ξ Wξ d x = ∇ξ J [Wξ ] and ∂ξi j Wξ = Z i j . R3
Then, testing Eq. (26) against Z i j , we see that N[φ] Z i j + L[Z i j ] φ d x = − E Zi j d x + ck R3
R3
Next we observe that % % % L[Z i j ] % = O(ω2/3 ) and ∗∗
R3
k
R3
Z i j Z k χi d x.
Z i j Z k χi d x = O(ω2/3 ) if (i, j) = (k, ).
By Lemma 4, φξ ∗ = O(ω2/3 ), and so we get
1 + O(ω2/3 ) ci j = O(ω4/3 ) + ∂ξi j J [Wξ ]. Hence, according to relation (31), we obtain
I 3N + O(ω2/3 ) ∇ξ (ξ ) = ∇ξ J [Wξ ] + O(ω4/3 ), where ∇ξ J [Wξ ] has been computed in (30). Summarizing, we have found that ∇ξ (ξ ) = −∇ξ Vmω (ξ1 , . . . ξ N ) + O(ω2/3 ). Therefore, ξ = ω2/3 ζ 'with ζ = (ζ1 , . . . ζ N ) and defining (ζ ) := (ξ ) on & setting 2N Bμ := ζ ∈ R : (18) holds , we have shown the following result. Proposition 1. With the above notations, we have that (ζ ) =
N
5− p
λi
e∗ − ω2/3 Vm (ζ ) + O(ω4/3 ),
i=1
∇(ζ ) = − ∇Vm (ζ ) + O(ω2/3 ), uniformly on ζ satisfying (18). Here the terms O(·) are continuous functions of ζ defined on Bμ .
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7.1. Proof of Theorem 1. If μ > 0 is fixed large enough, we have that inf Vm < inf Vm .
Bμ
∂ Bμ
Fixing such a μ, we get from Proposition 1 that, for all sufficiently small ω, sup > sup , Bμ
∂ Bμ
so that the functional has a maximum value somewhere in ω2/3 Bμ , which is close to a maximum value of Vmω . This value is achieved at critical point of , and hence a solution with the desired features exists. The construction is concluded. 7.2. Proof of Theorem 2. When (ζ1 , . . . ζ N ) is a regular polygon with ζ j given by (6) and all masses are equal, the system is invariant under the rotation defined by
x = (x 1 , x 2 , x 3 ) ≈ ((x 1 + i x 2 ), x 3 ) → e2 i π/N (x 1 + i x 2 ), x 3 =: R N x. )* + ( )* + ( ∈R3
∈C×R
We can therefore pass to the quotient with respect to this group of invariance and look for solutions u which are invariant under the action of R N and moreover symmetric with respect to the reflections (x 1 , x 2 , x 3 ) → (x 1 , −x 2 , x 3 ) and (x 1 , x 2 , x 3 ) → (x 1 , x 2 , −x 3 ). Here we assume that (ζ1 , . . . ζ N ) is contained in the plane {x 3 = 0} and ζ1 = (r, 0, 0). Altogether this amounts to look for critical points of the functional p+1 1 1 1 u − λ1 + ω2 |x |2 J1 [u] = |∇u|2 d x − χ1 d x, 2 1 p + 1 1 2 + π π <θ < N } and u ∈ where 1 = {x = (x , x 3 ) ∈ R3 ≈ C × R : x = r ei θ s.t. − N 1 H () is invariant under the above two reflections and such that ∇u·n = 0 on ∂\{0}×R p+1 and χ1 is the characteristic function of the support of ρ1 = u − λ1 + 21 ω2 |x |2 + χ1 in 1 . Here n = n(x) denotes the unit outgoing normal vector at x ∈ ∂1 . With J defined by (14), it is straightforward to see that J [u] = N J1 [u] if u is extended to R3 by assuming that u(R N x) = u(x). With these notations, we find that aN m∗ 1 2 + r . Vm (ζ1 , . . . ζ N ) = N m ∗ 4π r 2
The proof goes the same as for Theorem 1. Because of the symmetry assumptions, c1 j = 0 if j = 2 or 3. Details are left to the reader. 7.3. Proof of Theorem 3. We look for a critical point of the functional of Proposition 1 in a neighborhood of a critical point ζ of Vm , which is nondegenerate up to rotations. With no loss of generality, we may assume that ζ1 = 0, ζ12 = 0 and denote by V˜ m the restriction of Vm to (R × {0}) × (R2 ) N −1 ζ . Similarly, we denote by ˜ the restriction of to (R × {0}) × (R2 ) N −1 . By assumption, ζ is a non-degenerate critical point of V˜ m , i.e. an isolated zero of ∇ V˜ m . Besides, its local degree is non-zero. It follows that on an arbitrarily small
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neighborhood of that point, the degree for ∇ ˜ is non-zero for all sufficiently small ω. Hence there exists a zero ζ ω ∈ (R × {0}) × (R2 ) N −1 of ∇ ˜ as close to é1ζ as we wish. From the rotation invariance, it follows that ζ ω is also a critical point of . The proof of Theorem 3 is concluded. Acknowledgements. We thank the referee for a thorough reading of the paper and useful suggestions that led to an improvement of the original version. This project is part of the scientific program of the MathAmSud network NAPDE and of the ANR funded research projectCBDif-Fr. It has also been supported by grants Fondecyt 1070389, Fondo Basal CMM and Anillo ACT 125.
Appendix A. Facts on Relative Equilibria In this Appendix we have collected some results on the N -body problem introduced in Sect. 1 which are of interest for the proofs of Theorems 1-3, with a list of relevant references. Non-degeneracy of relative equilibria in a standard form. Relative equilibria are by definition critical points of the function Vm : R2N → R defined by Vm (ζ1 , ζ2 , . . . ζ N ) =
N N 1 1 mi m j + m i |ζi |2 . 8π |ζ j − ζi | 2 i= j=1
i=1
Here we assume that N ≥ 2, and m i > 0, i = 1, . . . N are given parameters. Following Smale in [42], we can rewrite this problem as follows. Let us consider the (2N − 3)-dimensional manifold , N 1 2N 2 Sm := q = (q1 , . . . q N ) ∈ R : m i (qi , |qi | ) = (0, 1) , qi = q j if i = j . 2 i=1
The problem of finding critical points of the functional Um (q1 , . . . q N ) =
N 1 mi m j 8π |q j − qi | i= j=1
on Sm is equivalent to that of relative equilibria; see for instance [12]. Let us give some details. Let q¯ be a critical point of Um on Sm . Then by definition, there are Lagrange multipliers λ ∈ R and μ ∈ R2 for which N 1 mi m j − (q¯i − q¯ j ) = λ m j q¯ j + m j μ ∀ j = 1, . . . N . 8π |q¯i − q¯ j |3 i= j
First, adding in j the above relations and using that M = Nj=1 m j > 0 we obtain that μ = 0. Second, taking the scalar product of R2 against q¯ j and then adding in j, we easily obtain that Um (q) ¯ = λ. From here it follows that the point ζ¯ = λ1/3 q¯ is a critical point of the functional Vm , hence a relative equilibrium. With the reparametrization of R2N given by ζ (α, p, q) = (ζ1 , . . . ζ N ) = (α q1 + p, . . . α q N + p), (α, p, q) ∈ R × R2 × Sm ,
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the Hessian matrix of Vm at the critical point ζ¯ = ζ (λ1/3 , 0, q) ¯ found above is represented as the block matrix ⎛ ⎞ 2 ⎠, M I2 D 2 Vm (ζ¯ ) = ⎝ λ−1/3 D S2mUm (q) ¯ where I2 is the 2 × 2 identity matrix and D S2m represents the second covariant derivative on Sm . Reciprocally, we check that a critical point ζ¯ = (ζ j ) Nj=1 of Vm necessarily satis−1/2
fies Nj=1 m j ζ j = 0. Defining q¯ = 21 Nj=1 m j |ζ j |2 ζ¯ , we readily check that q¯ is a critical point of Um in Sm . Any rotation ei α q¯ of a critical point q¯ of Um on Sm is also a critical point. We say that two such critical points are equivalent in Sm . Let us denote by Sm the quotient manifold of Sm by this equivalence relation. On Sm , critical points of the potential Um yield critical points of Um on Sm and hence equivalence classes of critical points ei α ζ for Vm using the reparametrization. A critical point q˜ of Um on Sm is said to be non-degenerate if the second variation of Um at q˜ is non-singular. Let us assume that q˜ = 0, with either = 1, or = 2 if q˜1 = 0. Then there is a unique representative q¯ of this class of equivalence for which q¯ 2 = 0. It is a routine verification to check that q¯ is then a critical point of Um on the (2N − 4)-dimensional manifold Sm := {q ∈ Sm : q = 0 as above , q 2 = 0} . Moreover, the second derivative of Um on Sm at q˜ is non-degenerate if and only if DS2 mUm (q) ¯ is non-singular. Because of the expression of D 2 Vm (ζ¯ ), we see that ζ¯ is non-degenerate as a critical point of Vm on the space of ζ ∈ R2N with q 2 = 0, which is the notion of non-degeneracy up to rotations of a relative equilibrium that we have used in this paper. Finally we define the index of a non-degenerate relative equilibrium ¯ ζ¯ as the number of negative eigenvalues of DS2 mUm (q). Some results on classification of relative equilibria. For simplicity, we will assume that masses are all different: for any i, j = 1, . . . N , if m i = m j , then i = j. This is the generic case. The cases N = 2, 3 are well known; see for instance [28]. For N = 2, the only class of critical points is such that 1/3 M |ζ1 − ζ2 | = and m 1 ζ1 + m 2 ζ2 = 0 with M = m 1 + m 2 . 4π For N = 3, there are two types of solutions, the Lagrange and the Euler solutions. The Lagrange solutions are such that their center of mass is fixed at the origin, the masses are located at the vertices of an equilateral triangle, and the distance between each point is (M/(4π ))1/3 with M = m 1 + m 2 + m 3 . They give rise to two classes of solutions corresponding to the two orientations of the triangle when labeled by the masses. The Euler solutions are made of aligned points and provide three classes of critical points, one for each ordering of the masses on the line. In the case N ≥ 4, the classes of solutions for which all points are collinear still exist (see [29]) and are known as the Moulton solutions. But the configuration of relative
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equilibria where all particles are located at the vertices of a regular N -polygon exists if and only if all masses are equal; see [12,26,35,43,46]. Various classification results which have been obtained by Palmore are summarized below. Theorem 4 ([30–34]). We have the following multiplicity results: (a) For N ≥ 3, the index of a relative equilibrium is always greater than or equal to N − 2. This bound is achieved by Moulton’s solutions. (b) For N ≥ 3, there are at least μi (N ) := Ni (N − 1 − i) (N − 2) ! distinct relative equilibria in Sm of index 2N − 4 − i if Um is a Morse function. As a consequence, there are at least N −2
μi (N ) = [2 N −1 (N − 2) + 1] (N − 2) !
i=0
distinct relative equilibria in Sm if Um is a Morse function. (c) For every N ≥ 3 and for almost all masses m ∈ R+N , Um is a Morse function. (d) There are only finitely many classes of relative equilibria for every N ≥ 3 and for N ∈ RN . almost all masses m = (m i )i=1 + References 1. Arnol d, V.I.: On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid. Dokl. Akad. Nauk SSSR 162, 975–978 (1965) 2. Arnol d, V.I.: An a priori estimate in the theory of hydrodynamic stability. Izv. Vysš. Uˇcebn. Zaved. Mat. 1966, 3–5 (1966) 3. Batt, J., Faltenbacher, W., Horst, E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rat. Mech. Anal. 93, 159–183 (1986) 4. Batt, J., Pfaffelmoser, K.: On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden-Fowler equation. Math. Meth. Appl. Sci. 10, 499–516 (1988) 5. Binney, J., Tremaine, S.: Galactic dynamics. Princeton, NJ: Princeton University Press, 1987 6. Dancer, E.N., Yan, S.: On the superlinear Lazer-McKenna conjecture. J. Diff. Eqs. 210, 317–351 (2005) 7. Dancer, E.N., Yan, S.: On the superlinear Lazer-McKenna conjecture. II. Comm. Part. Diff. Eqs. 30, 1331–1358 (2005) 8. D’Aprile, T., Wei, J.: Layered solutions for a semilinear elliptic system in a ball. J. Diff. Eqs. 226, 269–294 (2006) 9. Dolbeault, J., Fernández, J.: Localized minimizers of flat rotating gravitational systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 25, 1043–1071 (2008) 10. Dolbeault, J., Fernández, J., Sánchez, Ó.: Stability for the gravitational Vlasov–Poisson system in dimension two. Comm. Part. Diff. Eqs. 31, 1425–1449 (2006) 11. Dolbeault, J., Markowich, P., Oelz, D., Schmeiser, C.: Non linear diffusions as limit of kinetic equations with relaxation collision kernels. Arch. Rat. Mech. Anal. 186, 133–158 (2007) 12. Elmabsout, B.: Sur l’existence de certaines configurations d’équilibre relatif dans le problème des n corps. Cel. Mech. 41, 131–151 (1987) 13. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986) 14. Flucher, M., Wei, J.: Asymptotic shape and location of small cores in elliptic free-boundary problems. Math. Z. 228, 683–703 (1998) 15. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68, 209–243 (1979) 16. Guo, Y., Rein, G.: Existence and stability of Camm type steady states in galactic dynamics. Indiana Univ. Math. J. 48, 1237–1255 (1999) 17. Guo, Y., Rein, G.: Stable steady states in stellar dynamics. Arch. Rat. Mech. Anal. 147, 225–243 (1999) 18. Guo, Y., Rein, G.: Isotropic steady states in galactic dynamics. Commun. Math. Phys. 219, 607–629 (2001) 19. Guo, Y., Rein, G.: Stable models of elliptical galaxies. Mon. Not. R. Astro. Soc. 344(4), 1296–1306 (2003)
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20. Guo, Y., Rein, G.: A non-variational approach to nonlinear stability in stellar dynamics applied to the King model. Commun. Math. Phys. 271, 489–509 (2007) 21. Krieger, J., Martel, Y., Raphaël, P.: Two soliton solutions to the three dimensional gravitational Hartree equation. Comm. Pure Appl. Math. 62, 1501–1550 (2009) 22. Lemou, M., Méhats, F., Raphaël, P.: Orbital stability and singularity formation for Vlasov-Poisson systems. C. R. Math. Acad. Sci. Paris 341, 269–274 (2005) 23. Lemou, M., Méhats, F., Raphaël, P.: The orbital stability of the ground states and the singularity formation for the gravitational Vlasov-Poisson system. Arch. Rat. Mech. Anal. 189, 425–468 (2008) 24. Lemou, M., Méhats, F., Raphaël, P.: Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system. J. Amer. Math. Soc. 21, 1019–1063 (2008) 25. Lemou, M., Méhats, F., Raphaël, P.: Structure of the linearized gravitational Vlasov-Poisson system close to a polytropic ground state. SIAM J. Math. Anal. 39, 1711–1739 (2008) 26. MacMillan, W., Bartky, W.: Permanent configurations in the problem of four bodies. Trans. Amer. Math. Soc. 34, 838–875 (1932) 27. McCann, R.J.: Stable rotating binary stars and fluid in a tube. Houston J. Math. 32, 603–631 (2006) 28. Meyer, K., Hall, H.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Vol. 90 of Applied Mathematical Sciences, Berlin-Heidelberg-New York: Springer-Verlag, 1992 29. Moulton, F.R.: The straight line solutions of the problem of n bodies. Ann. of Math. 12(2), 1–17 (1910) 30. Palmore, J.I.: Classifying relative equilibria. II. Bull. Amer. Math. Soc. 81, 489–491 (1975) 31. Palmore, J.I.: Classifying relative equilibria. III. Lett. Math. Phys. 1, 71–73 (1975) 32. Palmore, J.I.: New relative equilibria of the n-body problem. Lett. Math. Phys. 1, 119–123 (1975) 33. Palmore, J.I.: Minimally classifying relative equilibria. Lett. Math. Phys. 1, 395–399 (1975) 34. Palmore, J.I.: Measure of degenerate relative equilibria. I. Ann. of Math. 104(2), 421–429 (1976) 35. Perko, L.M., Walter, E.L.: Regular polygon solutions of the N -body problem. Proc. Amer. Math. Soc. 94, 301–309 (1985) 36. Rein, G.: Flat steady states in stellar dynamics – Existence and stability. Commun. Math. Phys. 205, 229– 247 (1999) 37. Rein, G.: Reduction and a concentration-compactness principle for energy-Casimir functionals. SIAM J. Math. Anal. 33, 896–912 (2001) 38. Rein, G.: Non-linear stability of gaseous stars. Arch. Rat. Mech. Anal. 168, 115–130 (2003) 39. Rein, G.: Nonlinear stability of newtonian galaxies and stars from a mathematical perspective. Ann. New York Acad. Sci. 1045, 103–119 (2005) 40. Sánchez, Ó., Soler, J.: Orbital stability for polytropic galaxies. Ann. Inst. H. Poincaré Anal. Non Linéaire 23, 781–802 (2006) 41. Schaeffer, J.: Steady states in galactic dynamics. Arch. Rat. Mech. Anal. 172, 1–19 (2004) 42. Smale, S.: Topology and mechanics II. Invent. Math 11, 45–64 (1970) 43. Williams, W.: Permanent configurations in the problem of five bodies. Trans. Amer. Math. Soc. 44, 562– 579 (1938) 44. Wolansky, G.: On nonlinear stability of polytropic galaxies. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 15–48 (1999) 45. Wolansky, G., Ghil, M.: An extension of Arnol’d’s second stability theorem for the Euler equations. Phys. D 94, 161–167 (1996) 46. Xie, Z., Zhang, S.: A simpler proof of regular polygon solutions of the N -body problem. Phys. Lett. A 227, 156–158 (2000) Communicated by H. Spohn
Commun. Math. Phys. 300, 789–833 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1102-z
Communications in
Mathematical Physics
Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes Jian Qiu, Maxim Zabzine Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden. E-mail: [email protected] Received: 2 February 2010 / Accepted: 22 March 2010 Published online: 13 August 2010 – © Springer-Verlag 2010
Abstract: We investigate the generic 3D topological field theory within the AKSZ-BV framework. We use the Batalin-Vilkovisky (BV) formalism to construct explicitly cocycles of the Lie algebra of formal Hamiltonian vector fields and we argue that the perturbative partition function gives rise to secondary characteristic classes. We investigate a toy model which is an odd analogue of Chern-Simons theory, and we give some explicit computation of two point functions and show that its perturbation theory is identical to the Chern-Simons theory. We give a concrete example of the homomorphism taking Lie algebra cocycles to Q-characteristic classes, and we reinterpret the Rozansky-Witten model in this light. Contents 1. 2. 3. 4.
5.
6. 7.
Introduction . . . . . . . . . . . . . . . . . BV Formalism . . . . . . . . . . . . . . . . Background Material . . . . . . . . . . . . . 3.1 Characteristic classes for flat bundles . . 3.2 Lie algebra/graph cohomology . . . . . 3D AKSZ Topological Field Theory . . . . . 4.1 Construction of AKSZ model . . . . . . 4.2 Formal properties of correlators . . . . . 4.3 Formal properties of perturbation theory Perturbative Expansion of the AKSZ Model 5.1 Gauge fixing . . . . . . . . . . . . . . . 5.2 Feynman rules . . . . . . . . . . . . . . 5.3 The properties of correlators . . . . . . 5.4 Partition function . . . . . . . . . . . . Example 1: Q-Equivariant Bundle . . . . . . Example 2: Odd Chern-Simons Theory . . .
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1. Introduction Topological field theory (TFT) is a well-developed subject spreading across physics and mathematics. TFT can be viewed as a very powerful machine for producing the topological invariants. If one looks at TFT from the point of view of path integral, then one should deal with the appropriate gauge symmetries and thus with BRST formalism. The usual way of constructing a topological field theory is that one proposes a set of BRST transformations for a set of fields, and then writes down an action which is usually a BRST-exact term plus perhaps some additions of topological nature (e.g., such as the pull back of the Kähler form of the target manifold). Apart from the insight required to come up with a reasonable BRST rule, one is constantly faced with the problem that the BRST transformation closes only on-shell, and the problem of determination of observables, etc. Thus dealing with all these issues is somewhat ad hoc. The Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) construction [1] is an elegant and powerful tool to engineer the topological field theories in various dimensions within the BatalinVilkovisky (BV) formalism. Many problems are all avoided with one single ingenious stroke of the AKSZ construction. Its beauty lies in that it converts the finding of the BRST transformation rules to a purely geometrical problem, namely, one seeks the so-called Q-structure on a target manifold. The Q-structure is by definition an odd nilpotent vector field. This does not seem much improvement so far, but with the unifying language of graded manifolds (GrMfld), the possible Q-structures are well understood. For example, on a degree 1 GrMfld, a Q-structure encodes the data of a Lie algebroid. Thus the BRST rule will be related to the Lie algebroid differential for the target manifold, e.g. see [4] for the construction of a whole gamut of topological models. The AKSZ construction is done naturally within the BV formalism, which then clarifies the problem of on-shell closure of BRST transformation and at the same time gives geometrical interpretation to the otherwise unilluminating routine of gauge fixing. On the other hand, in physics we are equipped with the handy tool of path integral which, albeit being totally formal, allows one to manipulate the formalisms conveniently. And it is no new phenomenon that one could use a topological field theory and path integral to produce non-trivial mathematical results. In this work we offer the systematic study of the perturbative AKSZ-BV topological theories. Moreover we suggest the interpretation of the perturbative correlators and partition function in these theories. In particular we concentrate our attention on 3-dimensional (3D) theories. The present work is heavily influenced by several pieces of work along this direction. First the Chern-Simons perturbation theory [2], where the evaluation of the partition function led to the physical construction of invariants of 3-manifolds. Later Kontsevich [12] exposed the connection between the Feynman integral and graph (co)homology (namely the Feynman integral gives a cocycle in the graph complex); and thereby the construction of the low dimension topological invariants. Another piece of inspiration came from the works of Schwarz [22] and Lazarev and Hamilton [9], especially the latter, who used the tool of BV path integral to furnish a proof of the claim made by
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Kontsevich. Their proof made an excursion of first showing that the path integral is a cocycle in the cohomology of the Lie algebra of Hamiltonian vector fields. Since the latter is proven to be isomorphic to the graph cohomology, one can first send a graph chain to an element in the Lie algebra chain complex, then evaluate this chain in the path integral giving the desired cochain. We will show that all these ideas arise naturally within the AKSZ-BV framework. Indeed the BV path integral always gives rise to certain cocyles and the perturbative theory offers a concrete way of calculating them. Although we look mainly at 3D AKSZ models, many ideas can be extended to other AKSZ theories. Being furnished with a cocycle coming from the BV path integral one is led naturally to construct some characteristic classes using the Chern-Weil homomorphism. Now instead of plugging in the curvature two form to an invariant polynomial of Lie algebra, we plug in a flat connection into a cocycle. This is exactly what happens when we calculate the partition function of the AKSZ theory. One purpose of this work is to clarify what exactly the perturbation theory of these AKSZ models is computing. The partition function for such models turns out to be the (hopefully non-vanishing) characteristic classes of the relevant Q-(super)manifold. In the work by Lyakhovich, Mosman and Sharapov [14], they are able to use graph cohomology1 to find three infinite series of characteristic classes of any Q-manifold. Especially, their B,C series depend on the properties of the homological vector Q alone and survive even for a flat manifold. In a nutshell, due to the observation L Q ∂i ∂ j Q k = 0, where Q i ∂i is a homological vector field, they show if one plugs the second Taylor coefficient of Q i into certain graphs made of 3-valent vertices, out comes some Q-characteristic classes.2 This version of the characteristic classes for the flat bundle is again tied to a second construction of graph cycle (except they are using it backwards) by Kontsevich. The construction is intuitive; one obtains a graph cocycle by plugging into the vertices the Taylor coefficient of the Hamiltonian lift of Q and connecting edges using the symplectic form. We shall show that this is indeed what happens when one evaluates the partition function for the AKSZ models. For such a model the interaction term is just the Hamiltonian lift of Q, and for anyone who knows anything about perturbation theory in physics, the evaluation of the Feynman diagrams are just about plugging the Taylor coefficients of the interaction terms. The article is organized as follows, the BV formalism is reviewed in Sect. 2. We also show that the quantum observables form a closed algebra and the path integral gives a cocycle in Lie algebra cohomology of formal Hamiltonian vector fields generated by these observables. In Sect. 3 we review some relevant background material. The characteristic classes of flat bundles are recalled and we discuss the scenario in which they can arise in the path integral. The isomorphism between Lie algebra (co)homology and graph (co)homology is sketched without any claim for rigor. We give the construction of the AKSZ model in Sect. 4, in particular, the free theory gives a cocycle in Lie algebra cohomology of formal Hamiltonian vector fields of the target space. To do serious perturbation calculation, one needs to gauge fix the model; this is the topic of Sect. 5. There we present the set of Feynman rules and we investigate the perturbative partition function. We claim that the partition function corresponds to a characteristic class of appropriate flat bundle. Sections 6–8 are dealing with different examples of 3D AKSZ models. In 1 Their graph complex is slightly different from what we consider and it is isomorphic to the cohomology of the Lie algebra of formal vector fields vanishing at the origin. 2 This gives their B,C series of invariants; their A series come from two valent graphs and requires the vanishing of the Pontryagin class.
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Sect. 6 we consider the AKSZ model associated to the Q-equivariant vector bundle. In Sect. 7 we examine the 3D AKSZ model constructed on a flat symplectic space R2m , and we show that it is a kind of odd analogue of Chern-Simons perturbation theory and the weight functions associated with the diagrams are identical to Chern-Simons and the Rozansky-Witten model. Finally as a grand finale Sect. 8, we put all the ingredients together and reformulate the Rozansky-Witten model in the light of Lie algebra cohomology and the characteristic classes of flat bundles. At the end of the paper there are two appendices which contain some technical calculations relevant for the paper. 2. BV Formalism We give the essential facts about BV formalism in this section and show that the standard manipulations in the BV framework allow us to interpret the path integral as a certain cocycle. The original BV formalism was for the supermanifolds, namely manifolds with Z2 grading, yet the formalism may be carried to Z-graded manifolds making some of the results stronger. A degree n graded manifold is by definition locally parameterized by coordinates of degrees 0 up to n. And these coordinate patches are glued together through degree preserving transition functions (for more details on the graded manifolds, see [23 and 17]). An example of such a manifold is: T [1]M; the notation being: M is an ordinary manifold, T [1] means that we take the total space of the tangent bundle of M and we assign the fiber coordinate degree 1. This is an odd manifold since the highest coordinate degree is 1. An example of even graded manifold is T ∗ [2]T [1]M, locally, we have x μ as the coordinate of M, the coordinate v μ parameterizing the fiber of T [1]M is given degree 1, the coordinates dual to x μ , v μ are Pμ and qμ of degree − deg(x) + 2 = 2 and − deg(v) + 2 = 1 respectively. The advantage of using graded manifolds instead of supermanifolds is that degrees of these coordinates eventually correspond to the ghost number in a physical theory. The BV manifold is a manifold where the space of functions is equipped with the structure of BV algebra which is defined as the Gerstenhaber algebra (odd Poisson algebra) together with an odd Laplacian. Simply speaking the BV manifold is a manifold equipped with an odd symplectic form. The archetypical example of such spaces is of the form T ∗ [−1]M, where M itself is allowed to be a graded manifold. The reason for the degree −1 shift is to make the BRST transformation of ghost number +1 in the end. For definiteness, let us take the coordinate of M as x and that of the fiber x + , then the canonical symplectic form of the BV space is just ω = d x ∧ d x + . If M has dimension n, then a Lagrangian submanifold (LagSubMfld) L is a dimension n submanifold of the BV space such that ω|L = 0. Suppose that a volume form μ(x) is given for M, then we have also a volume form for T ∗ [−1]M which is μ2 (x)∧n d x + ∧n d x. With the density μ(x) we can define a Laplacian ≡
∂ 1 ∂ 2 μ (x) + , μ2 (x) ∂ x ∂x
which can be checked to satisfy 2 = 0. The key fact of the BV formalism [21] is the statement that the integral of a function f over a LagSubMfld is invariant under continuous deformation of the LagSubMfld provided f satisfies f = 0; and the integral of -exact functions gives zero. This statement is just the Stokes theorem in disguise [24]. By Fourier transforming the odd degree coordinates in T ∗ [−1]M (namely, exchanging the coordinate and its dual momentum),
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the Laplacian becomes the de Rham differential d over the degree even submanifold of T ∗ [−1]M. And the integration of functions over LagSubMfld is reformulated as integration of forms along submanifolds. In contrast to d, is not a derivation (does not obey the Leibnitz rule), in fact, when acting on a product of functions, it gives ( f g) = ( f )g + (−)| f | f (g) + (−)| f | { f, g},
(1)
where {·, ·} is the odd Poisson bracket corresponding to the odd symplectic form ω. We are going to explore the consequence of (1). The usual use of BV formalism is in the quantization of the gauge system: suppose one has an action S satisfying e−S = 0, then one seeks a suitable L such that the restriction of S to L has a non-degenerate quadratic term. The choice of the LagSubMfld is the choice of the gauge fixing condition, and due to e−S = 0, the end result should not depend on the choice of gauge fixing. Having chosen L, one then inserts operators O with (Oe−S ) = 0 into the path integral and obtain the expectation value of O. It is usually stated that the path integral is a homomorphism sending elements of H (T ∗ [−1]M, q ) (q ≡ e S e−S = − {S, }) to the number fields. Due to the fact that is not a derivation, there is no ring structure defined for the cohomology group H (T ∗ [−1]M, q ). This point of view is of course correct, however, it misses some rich structure innate in the BV formalism. In fact the cohomology group of is quite boring, as can always be Fourier transformed into a de Rham differential. One of the purposes of this paper is to elaborate some results in the paper by Schwartz [22] and by Hamilton and Lazarev [9]. The first crucial observation made by Schwartz is that the quantum observables (namely functions satisfying q f = 0) form a closed algebra under the Poisson bracket, more concretely, by using (1), { f, g} = (−1)| f | ( f g) − (−1)| f | ( f )g − f (g) = (−1)| f | (q ( f g) + {S, f g}) − (−1)| f | {S, f }g − f {S, g} = (−1)| f | q ( f g),
(2)
hence the bracket quantity { f, g} remains closed under q . But the bracket here does not yield a super Lie algebra structure for the quantum observables: the difference between the two is a shift in the assignment of the degree. More concretely, the Poisson bracket appearing here is odd and obeys { f, g} = −(−1)(| f |+1)(|g|+1) {g, f }, while for a super Lie algebra we would like to have graded anti-commutativity or { f, g} = −(−1)| f ||g| {g, f }. So a shift of the degree by 1 solves the problem. This shift can be achieved by considering the Lie algebra of Hamiltonian vector fields generated by the observables instead. If ω is the symplectic form of the BV space, then the Hamiltonian vector field generated by a function is defined such that LX f g ≡ { f, g}, where g is any function on the BV space and X f = ιd f ω−1 . Since ω has degree −1, deg X f = deg f + 1. We have the relation [X f , Xg ] = X{ f,g} , note the degree shift converts the Gerstenhaber algebra on the right hand side to the super Lie algebra on the left hand side. The Hamiltonian vector fields X f are in one to one correspondence with Hamiltonians f modulo constants. Thus we can fix all functions to vanish at a given point to remove this ambiguity. The Chevalley-Eilenberg (CE) complex of the Lie algebra of such Hamiltonian vector fields at degree n is spanned by the n-chain, cn = X f0 ∧ · · · ∧ X fn .
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The boundary operator for such a chain is the conventional one ∂ X f0 ∧ X f1 , . . . , X fn = sgni j (−1)| fi | X{ fi , f j } ∧ X f0 ∧ · · · ∧ X fi ∧ · · · ∧ X f j ∧ · · · ∧ X fn , i< j
where the sgn is the Koszul sign factor (−1)(| f0 |+···+| fi−1 |)| fi |+(| f0 |+···+| f j−1 |)| f j |−| fi || f j | , which accounts for the minuses caused by moving X fi and X f j to the front. Here we make a remark about the convention of graded (anti)-commutativity. One can either understand X f ∧ Xg as a graded anti-commutative, i.e. X f ∧ Xg = −1 × (−1)|X f ||Xg | Xg ∧ X f . Another point of view is to shift the degree X f up by 1 and call it graded commutative: X f ∧ Xg = (−1)(|X f |+1)(|Xg |+1) Xg ∧ X f . The two views make no difference so long as X f has degree 0, yet in working with graded manifolds, the latter is more advantageous, for then all the commutation relations are controlled by the degree. In the above Koszul sign, we used the latter convention, therefore deg X f = deg f − 1 + 1 (−1 because the symplectic form has degree −1) and X f ∧ Xg = (−1)| f ||g| Xg ∧ X f . The cochains of the CE complex are just the dual of the chains cn : cn → R. The differential δ for the cochain is induced from ∂ through δcn (cn+1 ) = cn (∂cn+1 ). These definitions fit neatly into the BV framework as follows. Consider all functions f i which satisfy q f i = 0, then the corresponding Hamiltonian vector fields X fi give rise to a closed Lie algebra Aq since [X fi , X f j ] = (−1)| fi | Xq ( fi f j ) . We can construct the n-chains and boundary operator for Aq in the way described above. Using the property (2) we can prove the following identity: q ( f 0 f 1 . . . f n ) = sgni j (−1)| fi | { f i , f j } f 0 . . . fi . . . f j . . . fn . (3) i< j
In BV context we have a naturally defined cochain, which is evaluated on X f0 ∧ X f1 ∧ . . . ∧ X fn according to the following expression: n c (X f0 ∧ X f1 ∧ · · · ∧ X fn ) ≡ f 0 f 1 . . . f n e−S ∈ R. (4) L
One can check easily that it is a multilinear functional with the correct graded commutativity properties. This cochain defined through the path integral is in fact a cocycle. This is shown by using the definition of the coboundary operator and the relation (3) δcn (X f0 ∧ X f1 ∧ · · · ∧ X fn+1 ) = cn ∂(X f0 ∧ X f1 ∧ · · · ∧ X fn+1 ) = q ( f 0 f 1 . . . f n+1 )e−S L
=
( f 0 f 1 . . . f n+1 e−S ) = 0,
L
where in the last step we used the fact that the integral of any -exact function is zero.
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We would like to emphasize that the cochain thus defined does depend on the choice of the Lagrangian submanifold. Although each f i obeys q ( f i ) = 0, q ( f 0 . . . f n ) = 0 n in general, so the Stokes theorem does not apply. Hence we denote the cochain by cL and we study the L dependence next. By Schwarz’s explicit construction, every L is locally embedded in the BV space as T ∗ [−1]M = T ∗ [−1]L; the simplest L, namely M itself, is such an example. If we denote the coordinates of L as x a and xa+ that of the transverse direction to L (L is given by x + = 0 locally). Then any small deformation is parameterized as ∂ xa+ = a (x). ∂x The function only depends on x and may be regarded as the generating function for the canonical transformation going from L to L + δL. Locally, the Laplacian is = ∂x a ∂xa+ , so = 0 trivially. Now → ← − − ∂ ∂ − ) f 0 f 1 . . . f n e−S = a + f 0 f 1 . . . f n e−S ( ∂ x ∂ xa L+δ L L L = {, f 0 f 1 . . . f n e−S } L
=−
( f 0 f 1 . . . f n e−S ) + ( f 0 f 1 . . . f n e−S )
L
=−
q ( f 0 f 1 . . . f n )e−S .
L
If we define a new (n − 1)-cochain by c˜
n−1
(X f0 ∧ X f1 ∧ · · · ∧ X fn−1 ) ≡ −
L
( f 0 f 1 . . . f n−1 )e−S ,
this cochain is not closed, however we have
n−1 n n ∂(X f0 ∧ X f1 ∧ · · · ∧ X fn ) (cL +δ L − cL )(X f 0 ∧ X f 1 ∧ · · · ∧ X f n ) = c˜L = δ c˜n−1 (X f0 ∧ X f1 ∧ · · · ∧ X fn ). cn
(5) (6)
by a Our observation is thus: the change of the LagSubMfld changes the cochain coboundary δ c˜n−1 . Thus for any choice of L, the path integral gives a representative of the class in the cohomology of the Lie algebra of the quantum observables. Yet two choices of L that are not homotopic to each other will produce different classes in the cohomology. So far our discussion has been formal, and may only be applied properly to a finite dimensional BV manifold. While for most cases of interest to physics, the BV space is the space of mappings and hence of infinite dimension, one usually does not have a well defined Laplacian, and the condition q f = 0 can at best be realized formally. Another drawback is that the relevant Lie algebra cohomology is on the space of mappings, while we quite often would like to ask questions about the properties of the target manifold alone, and the formalism developed above becomes unwieldy. In Sect. 4 we will set up a 3D topological field theory that focuses on the Lie algebra cohomology of Hamiltonian vector fields of the target manifold. The discussion there is along the lines of [9]. But before we do so, we have to digress a little for some other background material.
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3. Background Material In this section, we review the relevant background material. We recall the idea behind the construction of characteristic classes of flat bundles. We hint on the application of this construction within BV formalism. We also review the necessary facts concerning Lie algebra homology of formal Hamiltonian vector fields and its relation to the graph homology.
3.1. Characteristic classes for flat bundles. Consider the principal bundle P over base M with structure group G, P ←−−−− G ⏐ ⏐
(7)
M. If we choose the connection A on P with curvature R, then R is a Lie algebra valued 2-form on M. The procedure we are familiar with is to take an invariant polynomial of the generators of the Lie algebra g (usually a trace or a determinant), and plug in the curvature 2-form R. The Chern-Weil theorem guarantees that the resulting form is a closed form and so we have the mapping C[g∗ ] AdG → H 2k (M, R). This is the standard construction of the classical characteristic classes for the principle bundles. A flat bundle is a principal bundle equipped with a connection whose curvature vanishes identically, a flat connection. Thus, by the Chern-Weil theory all characteristic classes vanish and it may appear that the flat bundle is close to a trivial bundle. However, it is far from being true. Let us sketch the main idea behind the construction of the characteristic classes for flat bundles, which are also called secondary characteristic classes. Now we use the connection rather than the curvature. For the Lie algebra g there is the CE cochain complex c• = ∧• g∗ with the standard CE differential. Instead of invariant polynomials, take any cocycle cq in this complex and plug in the connection, resulting in a differential form on the bundle P given by A
cn −→ cn (A, . . . , A) ∈ n+1 (P).
(8)
n+1
This mapping from the cochain complex to the differential forms on P does not yet send cochain differential to the de Rham differential. To mend this, one must require the connection to be flat, i.e. it satisfies the Maurer-Cartan equation d A + A ∧ A = 0. To make it look more familiar, we pick a basis t a for the Lie algebra g and we can write the flatness condition as 1 (d Aa )(t a ) + (Ab ∧ Ac )[t b , t c ] = 0, 2
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where [, ] is Lie bracket for g. The last identity makes it clear that the flatness condition qualifies the mapping (8) as a differential graded map, for dcn (A, . . . , A) = d(Aa0 ∧ · · · ∧ Aan )cn (t a0 , . . . , t an ) 1 =− (−1)i Aa0 ∧ · · · ∧ Ab ∧ Ac ∧ · · · ∧ Aan cn (t a0 , . . . , [t b , t c ], t an ) 2 i
i
i
1 = − Aa0 ∧ · · · ∧ Aan+1 (δcn )(t a0 , . . . , t an+1 ). 2
(9)
Moreover, if cn is a cocycle in the CE complex, the map (8) gives us a closed form on P. Thus the flat connection induces the map of the cohomology groups A
s
H n (g, R) −→ H n+1 (P, R) −→ H n+1 (M, R),
(10)
where the last step involves the choice of the section s (or trivialization of P). The above map does not change if we choose another trivialization of P in the same homotopy class of trivializations. This is the construction of the secondary characteristic classes. This theory can be applied to the case of infinite dimensional algebras (groups) as well and it plays the central role in the characteristic classes of foliations. For further details about the characteristic classes of the flat bundles the reader may consult the book by Morita [15]. The flat connections appear a lot in physics. Let us discuss the relevant setup in which we generalize this slightly to include not just the Lie algebra valued differential forms but a general Q-structure. Recall a Q-structure is a degree one vector field satisfying Q 2 = 0. As a Q-structure is a natural generalization of the de Rham differential, the Q-equivariant fiber bundles are the generalization of flat bundles in the following way. π Given any fiber bundle E → M, suppose there is Q structure over a graded manifold M and a Q˜ over the total space E, which is also graded manifold. The Q-equivariantness says π∗ Q˜ = Q. In a local coordinate such Q˜ can be written as (taking e I as the coordinates of the fiber) ∂ ˜ Q(x, e) = Q(x) + A I (x, e) I , ∂e where A I is a vector field along a fiber. Q˜ 2 = 0 implies that A satisfies the Cartan-Maurer equation 1 Q A + [A, A] = 0, 2
(11)
where [, ] stands for the Lie bracket of vector fields along the fiber. Thus in this setup the Lie algebra g can be identified with the algebra of formal vector fields along the fiber. By using the construction analogous to (9) one obtains Q-closed functions by evaluating the A on the cocycle of this infinite dimensional algebra of g. These Q-closed functions are the characteristic classes for the Q-structure. As the Q-structure includes a wide variety of differentials such as the de Rham, Doubeault, Chevalley-Eilenberg, Poisson-Lichnerowicz, etc., we have a more uniform way of investigating the characteristic classes associated with these structures.
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There is an immediate application of these ideas in the BV path integral framework. Recall from Sect. 2 that n f0 . . . fn (12) c (X f0 ∧ · · · ∧ X fn ) = L
defines the cocycle for the Lie algebra of divergenceless Hamiltonian vector fields (i.e., f i = 0) on BV space. Consider the BV action S which satisfies S = 0. Suppose that the action also depends on some extra parameters and that there exists another odd differential Q acting on those parameters, such that 1 Q S + {S, S} = 0. 2
(13)
This is a quite typical setup in BV theory. Equation (13) appears as a consequence of the classical master equation and the extra parameters can originate from the zero modes of the theory, for example. Now let us evaluate the partition function of this BV theory, Z= L
e−S =
∞ (−1)n n=0
n!
cn−1 (X S ∧ · · · ∧ X S ),
(14)
where cn (X S ∧ · · · ∧ X S ) is a cocycle since S = 0 and it is now a function of the extra parameters. We can show easily that this function is annihilated by Q, 1 Qcn (X S ∧ · · · ∧ X S ) = − cn (∂(X S ∧ · · · X S ) = 0, 2
(15)
where we used the property (13). The most important example where this situation can arise is of course when we have a bundle structure whose fiber is equipped with an odd symplectic form and the extra parameter is the coordinate of the base. Then the relation (13) is nothing but the Q-equivariantness condition (11), namely the Q-structure on the base is lifted to Q˜ = Q + {S, ·} in the total space. Within this picture the partition function Z gives rise to a Q characteristic class (the concrete representative depends on the choice of L). Although the present argument is formal, we will argue later that this is a generic feature of 3D TFTs.
3.2. Lie algebra/graph cohomology. In this subsection we review briefly the algebra of formal Hamiltonian vector fields. We will use these materials in the next sections. Consider the vector space R2m equipped with the canonical symplectic structure. Let Ham02m be the Lie algebra of formal (polynomial) Hamiltonian vector fields over R2m preserving the origin; let Ham12m consist of those elements of Ham02m whose Taylor expansion starts from the quadratic term and finally sp(2m, R) are those elements whose coefficients are linear. If one chooses to talk about the Hamiltonian function instead, then sp(2m, R) corresponds to quadratic polynomials, Ham12m corresponds to cubic or higher polynomials. Let C• (Ham02m ) be the Chevalley-Eilenberg complex of Ham02m and sp(2m, R) acts on this complex through the adjoint action. We shall consider
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the sp(2m, R) coinvariants3 of the complex C• (Ham12m ). If we denote such coinvariants as C• (Ham02m , sp(2m, R)), then we have the isomorphism due to Kontsevich [11] that H• (Ham02m , sp(2m, R)) ∼ H• (G),
(16)
where G is the (undecorated) graph complex. The reason for ’modding’ out the sp(2m, R) subgroup will become clear once we consider this isomorphism from the path integral point of view. The same isomorphism (16) can be generalized to the superspace R2m|k with the even symplectic structure, see [8]. We use here the same conventions as in the previous section. However we are interested in a different Lie algebra now. We use X f to denote a Hamiltonian vector field generated by f over R2m with the canonical symplectic structure. The CE complex will be spanned by the exterior product of the form cn = X f0 ∧ · · · ∧ X fn . The Chevalley-Eilenberg boundary operator is ∂cn =
(−1)i+ j+1 [X fi , X f j ] ∧ X f0 · · · ∧ X fi · · · ∧ X f j · · · ∧ X fn .
i< j
By using the relation [X f , Xg ] = X{ f,g} , we can abbreviate X f0 ∧ · · · ∧ X fn as ( f 0 , . . . , f n ),
(17)
and the boundary operator by ∂( f 0 , · · · f n ) =
(−1)i+ j+1 ({ f i , f j }, f 0 , . . . fˆi , . . . fˆj , . . . f n ).
(18)
i< j
Apart from the little details, the mapping in (16) is easy to understand. Take the Euclid ean space R2m equipped with the standard symplectic structure μ<ν μν d x μ ∧ d x ν . The function f ’s are all polynomials on R2m , so a given chain corresponds to a sum cn =
(m0 , m1 , . . . , mn ),
where mi are all monomials. An l th order monomial will correspond to an l-valent vertex in the graph. For every propagator connecting leg μ and ν one incorporates a factor μν into the coefficient of the graph
(19) 3 The coinvariants, in contrast to the invariants, are the largest quotient of C (Ham0 ) on which sp(2m, R) • 2m acts trivially, or simply speaking, the orbits of the sp(2m, R) action.
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The Poisson bracket between two monomials of degree p and q produces a sum of monomials of order p + q − 2, so ∂ σ γ (. . . , (x μ x ν x σ ), (x γ x ρ x λ ), . . .) = · · · + σ γ (. . . , {x μ x ν x σ , x γ x ρ x λ }, . . .) + · · · = · · · + (. . . , x μ x ν x ρ x λ , . . .) + . . . , where we have only focused on the propagator σ γ while assuming the legs μνρλ are connected to other parts of the graph in a certain way. In the graph language the boundary operator acts as
So the boundary operator acts on a graph by deleting one propagator. This is exactly the differential for the graph complex. We have omitted lots of details, especially those concerning how to work out the sign factors and the orientation of the graph; the reader may see [8] for a full treatment. The similar construction can be applied to the superspace R2m|k with even symplectic structure. In what follows we use the Greek letters for even case R2m , while upper case Latin letters are for the supercase R2m|k . 4. 3D AKSZ Topological Field Theory The Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) construction [1] allows one to produce a variety of topological σ -models in a rather canonical fashion. The AKSZ approach uses mapping space between a source supermanifold (graded manifold) and target supermanifold (graded manifold). The source and target manifolds are equipped with additional structures. In this section we define the 3D AKSZ model and explore the geometrical meaning of the correlation functions in the AKSZ model on a 3D source manifold; the result turns out related to the Lie algebra cohomology of Hamiltonian vector fields on the target space. In particular we make some remarks on the perturbative theory. The correlators of the two 3D AKSZ models we know, namely the Chern-Simons theory and the Rozansky-Witten model, both fit this description. 4.1. Construction of AKSZ model. The general construction of the AKSZ models is standard by now. We shall be brief and concentrate our attention only on 3D models. For more details the reader may consult [1,3,18]. Consider an even symplectic supermanifold M with the local coordinates X A and |A| which denote the degree X A . Suppose the even symplectic form on the target manifold is , and it can be written locally as the differential of the Liouville form = d. Assume we are in a Darboux coordinate, then = X A AB d X B . We will refer to M as a target. Also let us consider the three dimensional manifold 3 . We are interested in the odd tangent bundle T [1]3 , where ξ is the bosonic coordinate of 3 and θ is the odd fiber coordinate of T [1]3 . We will refer to 3 as the source. In the following discussion we have chosen the source manifold 3 to be S 3 or more generally a rational homology sphere.
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Consider the mapping space Maps(T [1]3 , M) and denote the mapping by X A (ξ, θ ). The even symplectic form on M induces an odd symplectic form in the space of mappings Maps(T [1]3 , M) according to 1 ω= 2
d 6 z AB δ X A δ X B ,
(20)
T [1]3
where we write ξ, θ collectively as z and d 6 z ≡ d 3 θ d 3 ξ . Note that each X is a superfield, hence can be expanded into components 1 1 X(ξ, θ ) = X (ξ ) + θ a X (ξ )a + θ b θ a X (ξ )ab + θ c θ b θ a X (ξ )abc . 2 3! The components correspond to forms of different degrees on 3 . When we do not want to spell out all the indices, we will write X (0) = X (ξ ); X (1) = X (ξ )a dξ a ; X (2) =
X (ξ )ab dξ a ∧ dξ b . . . .
a
We may obtain the odd symplectic form written in components by integrating out d 3 θ in (20) (details left to the Appendix) ω=−
A B A B . d 3 ξ AB δ X (3) ∧ δ X (0) + δ X (1) ∧ δ X (2)
(21)
3
The BV space Maps(T [1]3 , M) is infinite dimensional and there is no well defined measure for the path integral, we shall use the naive one vol = ∧top d X (0) ∧top d X (1) ∧top d X (2) ∧top d X (3) . With this volume form we have a naive odd Laplacian
≡
−1 AB
d ξ( 3
3
)
(−1)
|A||B|
δ
δ
A (ξ ) δ X B (ξ ) δ X (3) (0)
+
δ
δ
A (ξ ) δ X B (ξ ) δ X (1) (2)
.
Note that the odd Laplacian is the restriction of a distribution on 3 ×3 to the diagonal, and is a singular object in this infinite dimensional context. Thus it should be understood as the limit of a suitably regularized expression (see the Appendix). The odd Laplacian has a number of formal properties:
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J. Qiu, M. Zabzine
d 6 z f (X(z)) = 0,
T [1]3
d z 1 f (X(z 1 ))
T [1]3
d 6 z { f (X(z)), g(X(z))}, T [1]3
T [1]3
= (−1) f
d 6 z 2 g(X(z 2 ))
6
d 6 z f (X(z)),
T [1]3
d 6 z g(X(z)) = −
T [1]3
d 6 z { f (X(z)), g(X(z))} .
T [1]3
(22) We refer the reader to the Appendix for further details. Let us choose an odd function on M which satisfies {, } = 0 with respect to the even symplectic structure on M. Then the AKSZ construction gives the standard BV action S = Skin + Sint = d 6 z X A AB D X B + X ∗ , (23) T [1]3
D ≡ θ ∂a , a
where the first term involving the Liouville form is called the kinetic term. The kinetic term is independent of the concrete choice of the Liouville form. X ∗ is the pullback of through X to the space of mappings. It serves as the interaction term. We often write the pull back X ∗ simply as . One can check easily that S satisfies the classical master equation {S, S} = 0 with respect to {S, S} = − d 6 z D( A D X A + ) + X ∗ {, } = − d 6 z {, } = 0, T [1]3
T [1]3
where the first term drops because it is a total derivative and 3 has no boundary. So the only requirement is merely {, } = 0. In the expression above and in further discussion we use the same notations for the bracket on M and for the BV bracket on Maps(T [1]3 , M). Hopefully it is not confusing since it can be understood from the context which bracket is used. Thus the action (23) formally satisfies the quantum master equation e−S = 0, by using the first property of in (22) and {S, S} = 0. For further reference the component form of the kinetic term and the interaction terms are A B A B Skin = d 3 ξ AB −X (1) , (24) ∧ d X (1) + X (2) ∧ d X (0) 3
Sint = 3
1 A A A B B C d 3 ξ X (3) (∂ A )+ X (2) ∧ X (1) (∂ B ∂ A )+ X (1) ∧ X (1) ∧ X (1) (∂C ∂ B ∂ A ). 6
In the present discussion we keep in mind only Z2 -grading. The construction can be refined to Z-grading with the source and with the target being graded manifolds equipped with the extra structure [18]; the main example is given by the Courant sigma model.
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The Chern-Simons theory is a special case of the Courant sigma model with the target being M = g[1], where g is a metric Lie algebra. In principle we will allow the BV theory to depend on extra free parameters, e.g. may depend on the parameters other than the coordinates of M.
4.2. Formal properties of correlators. In this subsection we apply the formal observation from Sect. 2 to 3D AKSZ theory constructed in the previous subsection. The simple observation is that a certain subalgebra of quantum observables can be mapped to a specific subalgebra of Hamiltonian vector fields on M. Thus the corresponding correlator can be interpreted entirely in terms of target space geometry. Consider the objects of the form F ≡ d 6 z f (X(z)). According to the property (22) the BV bracket between {F i , F j } gets mapped to the even bracket { f i , f j } on M. Moreover F’s are quantum observables if they satisfy 0 = e−S
d z f (X(z)) = e 6
T [1]3
−S
d 6 z D f (X(z)) + X ∗ {, f (X )}.
T [1]3
Thus the only requirement on F to be a quantum observable is that the corresponding f commutes with on M, i.e. {, f } = 0. The observables of this type form a closed algebra and we define the correlator using the path integral F 0 F 1 . . . F n ≡
d z 0 f 0 (z 0 ) d z 1 f 1 (z 1 ) · · · 6
6
d z n f n (z n ) e−S , (25) 6
L
where f (z) is the short hand of f (X(z)). Repeating the formal argument from Sect. 2 and using the fact that the F’s algebra is mapped to the f ’s algebra on M we arrive at the conclusion that the correlator F 0 F 1 . . . F n = cn (X f0 ∧ X f1 ∧ · · · ∧ X fn )
(26)
corresponds to the cocycle of the Lie algebra of Hamiltonian vector fields X f which commute with X . We once again remark that this way of writing the correlation function (25) agrees with the graded commutativity on the left-hand side, namely4 X f ∧ Xg = +(−1)( f −2+1)(g−2+1) Xg ∧ X f (−2 because the symplectic form has degree 2). Because the integration measure d 6 z carries −3 degree, so
d 6 z 1 f (z 1 )
T [1]3
T [1]3
d 6 z 2 g(z 2 ) = (−1)( f −3)(g−3)
d 6 z 1 g(z 1 ) T [1]3
d 6 z 2 f (z 2 ).
T [1]3
One may compare this to the situation of Sect. 2 where the degree shift is due to the odd symplectic form while here it is due to the degree of the measure for the source. 4 To avoid ugly expressions we adopt the simple notation for degree, namely deg f = | f | = f . Since the degree is essential only for signs, it appears in the expressions like (−1) f and thus there should be no confusion.
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Moreover using the standard BV manipulations and the correspondence of BV algebra of F’s with the algebra on M, one can easily check that our correlator is a cocycle δcn−1 (X f0 ∧ X f1 ∧ · · · ∧ X fn ) = =
q (F 0 F 1 · · · F n ) e−S
L
(−1) fi +( fi +3)( f0 +··· fi−1 +3i)+( f j +3)( f0 +··· f j−1 +3 j)+( fi +3)( f j +3)
L i< j
j . . . F n e−S = cn−1 (∂(X f0 ∧ X f1 ∧ · · · ∧ X fn )), ×F { fi , f j } F 0 . . . Fi . . . F where everything matches including the signs.
4.3. Formal properties of perturbation theory. Our argument so far was quite formal and we would like to convert it into the concrete calculation with precise properties. For this we will have to resolve the perturbation theory. Before defining the precise Feynman rules, let us make a few comments about the expected properties of the correlators in the perturbative theory. We can repeat the formal argument from the previous subsection with = 0. The correlators are now
d 6 z 0 f 0 (z 0 ) . . .
d 6 z n f n (z n ) e−Skin = cn (X f0 ∧ X f1 ∧ · · · ∧ X fn ), (27)
L
which should be cocycles for the Lie algebra of Hamiltonian vector fields on M. We can give some general remarks about the structure of the correlation function. Assuming that X A are the Darboux coordinates of the target space, then the kinetic term is X A AB D X B . If this gives a non-degenerate quadratic term when restricted to L, then we can invert it and obtain the propagator. In the model we have the propagator will basically consist of −1 and the inversion of the de Rham operator G(z 1 , z 2 ) (inverting the de Rham operator requires one to choose L carefully, more of this later). Applying the Wick theorem, the correlation function (27) is represented by the Feynman diagrams (graphs), and the end result is written schematically as cn (X f0 ∧ X f1 ∧ · · · ∧ X fn ) = F 0 F 1 . . . F n =
b I ,
(28)
where we sum over the graphs . Concretely, any graph gives a particular way of routing the propagators. Since every insertion F i is integrated with the measure d 6 z i , we have an integration of these propagators over the configuration space T [1]3 × · · · × T [1]3 . This integral thus associates a graph with a number which is called the weight function b . While I corresponds to the combination of derivatives of f ’s (vertices) contracted by −1 (edges) in the way prescribed by , the essential property of sum b I is that it should give a cocycle for the Hamiltonian vector fields on M. In the next section we are going to discuss the concrete prescription behind formula (28).
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5. Perturbative Expansion of the AKSZ Model In this section we construct the perturbation theory for the 3D AKSZ model constructed in the previous section. From now on we assume that M is a super(graded) vector space R2m|k equipped with the canonical even symplectic form. This assumption is not essential and it is done for clarity of the argument. For the general supermanifold M we will have to apply the exponential map in order to map the problem to the vector space and keep the covariance. The example of this full covariant construction will be given when we discuss the Rozansky-Witten theory in Sect. 8.
5.1. Gauge fixing. We continue to discuss the AKSZ model of the previous section. In general it may be tricky to pick a Lagrangian subspace L such that the restriction of the action to L has a non-degenerate quadratic term. In our case we expand out the kinetic term from the action (23), A B A B Skin = d 3 ξ AB −X (1) , ∧ d X (1) + X (2) ∧ d X (0) 3
where we have assumed that we are in a Darboux coordinate so that is a constant. So the kinetic term to be inverted is the de Rham differential, which has infinitely many zero modes. It is an intricate game trying to find a set of constraints upon the component fields such that we are able to invert d. However, a brutal gauge fixing is possible and works for all models at the cost of explicit covariance on M, and this is why we discuss the case when M is a vector space. To this end, we introduce a metric on the source manifold 3 . Using this metric we can use the Hodge decomposition to break any differential form on 3 into three parts, ω = ωh + ωe + ωc = ωh + dτ + d † λ, where h, e, c stand for harmonic, exact and co-exact respectively and d † is the adjoint of d. The three parts are mutually orthogonal under the following non-degenerate pairing: (ω1 , ω2 ) ≡ ω1 ∧ ∗ω2 . 3
Since all the components of a superfield X A are some differential forms on 3 , we can decompose them likewise X (Ap) = (X (Ap) )h + (X (Ap) )e + (X (Ap) )c . The troublemaker is the exact part, since they are annihilated by d and infinite in number. Our choice for LagSubMfld L will be to simply stay clear of these exact parts. More concretely, we first decompose the symplectic form (21) into A h B h A e B c ω ∼ d 3 x AB δ(X (3) ) ∧ δ(X (0) ) + δ(X (3) ) ∧ δ(X (0) ) 3
A h B h A e B c A c A e + δ(X (1) ) ∧ δ(X (2) ) + δ(X (1) ) ∧ δ(X (2) ) + δ(X (1) ) ∧ δ(X (2) ) .
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Moreover using the integration by parts there are no ee and cc combinations and the harmonic part is decoupled from the rest. Since we are on a rational homology 3-sphere A )h = (X A )h = 0. We make there are no harmonic terms for the 1 and 2 forms, i.e. (X (1) (2) the following gauge choice: A e A e ) = 0, (X (2) ) = 0, (X (1)
A X (3) = 0,
where we put to zero both harmonic and exact parts of 3-forms. For further discussion A )h = x which do not appear in the kinetic term we adopt the following notations (X (0) 0 and thus correspond to the zero modes. We will not perform the integral over zero modes! We will treat the zero modes as formal parameters and the integral will be performed only over co-exact fields, c c c F 0 . . . F n e−Skin = D X (0) D X (1) D X (2) F 0 . . . F n e−Skin , (29) L
thus the correlator F 0 . . . F n is a function of x0 ∈ M. The observable F ≡ d 3 ξ d 3 θ f (X(ξ, θ )) 3
=
A A B d 3 ξ X (3) ∂ A f (X (0) ) + X (2) ∧ X (1) ∂ B ∂ A f (X (0) )
3
1 A B C + X (1) ∧ X (1) ∧ X (1) ∂C ∂ B ∂ A f (X (0) ) 6
upon the gauge fixing and Taylor expansion becomes ∞ 1 1 c 3 c k 3+k 3 c c c k 2+k d ξ X (2) X (1) (X (0) ) ∂ f (x0 )+ (X (1) ) (X (0) ) ∂ f (x0 ) , F= k! 6 k=0
(30)
3
where we suppressed all indices and wedges. Now we have to contract the co-exact fields according to the Wick theorem. Using the propagator which is proportional to −1 the c are contracted to X c and the fields X c are contracted to X c , namely fields X (1) (1) (2) (0) A B X (2) (ξ1 )X (0) (ξ2 ) = (−1 ) AB G 0 (ξ1 , ξ2 ), A B X (1) (ξ1 )X (1) (ξ2 )
−1 AB
= (
)
G (ξ1 , ξ2 ), 1
(31) (32)
where we suppressed the superscript c on the fields. Since the observable (30) is composite in terms of elementary fields, we may have to use the point splitting procedure and study the tadpole contributions, if they are there. Equivalently we may work with the superfields and develop the perturbation theory entirely in terms of superfields, thus avoiding the components. For this we have to introduce the adjoint of D which can be written as D † = ∇ a ∂θ a . The superfield admits the Hodge decomposition with respect to D and D † . Thus the gauge fixing corresponds to setting to zero the exact part of the superfield. We leave the explicit formulas for the propogators for later discussion in Sect. 7. Now we would like to concentrate on the general features of the Feynman rules.
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5.2. Feynman rules. The current gauge fixing is very explicit, allowing us to sharpen some features of the perturbation expansion. The main consequence of this gauge fixing is that every diagram will have an even number of vertices, all of which are 3-valent and there are no tadpoles. As discussed in the previous subsection, the Feynman rules are defined by the (k +2)and (k + 3)-valent vertices (30) and the propogators (31, 32). The first observation is that there will be no 2-valent vertices since c c d 3 ξ X (2) X (1) ∂ 2 f (x 0 ) = 0 (33) 3
is identically zero5 . For the next observation it can be useful to think about the Feynman rules in terms of superfields. Let us look at the correlator F 0 . . . F n e−Skin = d 6 z 0 f 0 (z 0 ) · · · d 6 z n f n (z n ) e−Skin , L
L
where f i (z i ) = f i (X(z i )). We have a total of (n+1) d 3 θ . The propagator X(z 1 )X(z 2 ) is quadratic in θ ’s. Since we have to saturate all θ -integration there is the following relation between the number V of vertices and the number P of propagators: 3V = 2P.
(34)
Equivalently we can make a similar argument within the component form of the perturbation theory. The integration over the configuration space 3 × · · · × 3 requires a 3(n + 1)-form. The propagator is a 2-form6 on 3 × 3 since according to (24) we have propagators between X (1) , X (1) and between X (0) , X (2) . Thus to absorb all integration we have to require (34). The property (34) says that in order for the diagram to be non-zero there should be 3-valent vertices on the average. For example, if there is 4-valent vertex then it should be accompanied by 2-valent vertex. However we have argued that 2-valent vertices vanish identically. Therefore we can conclude that only 3-valent vertices contribute A B C d 3 ξ X (0) X (2) ∧ X (1) ∂C ∂ B ∂ A f (x0 ), (35) 3
1 6
A B C d 3 ξ X (1) ∧ X (1) ∧ X (1) ∂C ∂ B ∂ A f (x0 ).
(36)
3
Since only 3-valent graphs contribute, we need the even number of 3-valent vertices to contract all legs. Therefore only graphs with the even number of vertices give non-zero contribution. Now we have to discuss the tadpoles, the situation when the vertex leg is 5 wonder that this argument is too rough. At best we can say that 2-valent vertex 3 Onec may c ∂ 2 f (x ) is a surface term and due to the singularities in the propagators there may be nond ξ X (2) X (1) 0 trivial contributions of 2-valent vertices. However one may perform more careful analysis taking into account the possible singularities of the propagator and arrive at the same conclusion that 2-valent vertices do not contribute. 6 Strictly speaking the correlator is 2-form on ( × − diagonal). However the singularity of the 3 3 propagator along the diagonal is not enough to spoil the argument.
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contracted with another leg from the same vertex. The tadpoles contain the following contribution: ∂ A ∂ B ∂C f (x0 )(−1 ) AB , which is identically zero due to the fact that we contract a (graded) symmetric combination ∂ A ∂ B with a (graded) antisymmetric (−1 ) AB . Thus all tadpoles are automatically zero in the theory7 .
Thus the correlator F 0 F 1 . . . F n has the form b I , where ’s are all 3-valent graphs with (n +1) vertices. The number b is an integral of the collection of propagators G over 3 ×· · ·×3 dictated by the graph , while I is a collection of third derivatives ∂ 3 f i (x0 ) contracted by −1 in the way dictated by the 3-valent graph . Thus I is a function of zero modes x0 . The explicit example of the calculation of the correlator is presented later. 5.3. The properties of correlators. In the previous subsection we discussed the calculation of the correlators in the perturbative theory corresponding to 3D AKSZ models. Now we would like to go back to our formal BV arguments about the properties of the correlators, see Subsects. 4.2 and 4.3. We want to understand if those arguments are applicable to the perturbative theory, maybe with possible refinements. The perturbative correlator associated with the collection of functions f 0 , f 1 , . . . , f n on target M is defined as follows: cn (X f0 ∧ X f1 ∧ · · · ∧ X fn ) = F 0 F 1 . . . F n = b I (x0 ), (37)
and it depends on zero modes parametrized by M itself. We choose not to integrate over zero modes due to the fact that they do not enter the perturbative theory. Moreover quite often the integration over zero modes is either not well-defined or even when it is defined we may miss some interesting structures if we perform the integral right away. From the formal BV arguments we expect that (37) is a cocycle of the Lie algebra of Hamiltonian vector fields on M. However now cn is a function on M itself. Thus we are dealing with the cochain cn taking values in the function on M and one would expect that the differential δ should be modified. The natural modification looks like: (δcn )(X f0 ∧ · · · X fn+1 ) = (−1)li j cn (X{ fi , f j } ∧ · · · X fi ∧ · · · X f j ∧ · · · X fn+1 ) i< j
− (−1) pi f i (x0 ), cn (X f0 ∧ · · · X fi ∧ · · · X fn+1 ) , (38) i
where we assume that (n + 1) is even and the following sign conventions are valid: li j = f i +( f i + 3)( f 0 + · · · f i−1 + i3)+( f j + 3)( f 0 + · · · f j−1 + j3)+( f i + 3)( f j + 3), pi = ( f i + 3)( f 0 + · · · f i−1 + i3). Indeed the formula (38) can be derived from the first principle with the BV framework. We have to treat carefully the contribution of “zero modes” to the odd Laplacian operator and a regularization of the odd Laplacian is required in order to make some of the 7 It is important to stress that on we can do systematically the point-splitting regularization by a picking 3 nowhere vanishing vector field.
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manipulations well-defined. It all can be done and we present the BV derivation of the formula (38) in the Appendix. The final claim is that the perturbative correlator is a cocycle with the values in functions on M: b I (x0 ) = 0, (39) (δcn )(X f0 ∧ · · · X fn+1 ) = δ
where δ is defined by formula (38). In order to avoid the discussion of the covariance on M we consider the case when M = R2m|k with even constant symplectic structure . In this case the correlator (37) is automatically osp(2m|k, R) invariant since our model has global osp(2m|k, R) symmetry. We believe that property (39) is true for any functions f i on R2m|k with the constant even symplectic structure (see Subsect. 7 for some simple explicit check). Now if we fix the point x0 (e.g, choose it to be the origin x0 = 0) and consider the polynomial functions f i with property f i (x0 ) = ∂ f i (x0 ) = 0, namely members of Ham02m|k , then n the last term (38) disappears and
we can regard c as a cocycle with the values in R. Since the construction of cn as b I is osp(2m|k, R)-invariant we get that cn is the representative of the relative cohomology class H • (Ham02m|k , osp(2m|k, R), R). Another observation by Kontsevich which is natural within the present BV context is that the following cochain in the graph complex:
b ∗
is closed. The rough proof of this goes as follows. We take any graph chain in G• and find the corresponding chain in the CE complex C• (Ham02m|k ) ≡ ∧• Ham02m|k according to Sect. 3. Evaluating this CE chain in the path integral results in the number b , which shows b is a graph cocycle. For a more careful proof of this statement see reference [9]. We can illustrate this argument by the following example. Recall from Eq. (19) the correspondence between the graph complex and the CE complex C• (Ham02m|k ). Suppose we are given the following 3-valent graph with two vertices:
(40)
To construct a cochain mapping this graph to the number field, one can first map it to an element of CE complex −→ AB C D K L Xx A x C x K ∧ Xx L x D x B , and then evaluate this Lie algebra chain in the path integral according to Eq. (25). If one defines the two point correlator as X A (z 1 )X B (x2 ) ≡ AB G(z 1 , z 2 ),
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then the result of evaluating the Lie algebra chain is AB C D K L c1 Xx A x C x K ∧ Xx L x D x B ∼
d 6 z 1 d 6 z 2 G(z 1 , z 2 )3 ≡ b .
T [1]3
Since the mapping between the graph complex and the CE chain complex is an isomorphism, and moreover the path integral is a cocycle in the CE cochain complex, we conclude that b ∗ is a cocycle. 5.4. Partition function. In this subsection we study the perturbative partition function for the action (23) with the interaction term. We want to apply general ideas about the characteristic classes of flat bundles reviewed in Subsect. 3.1 to the perturbative partition function of 3D AKSZ models. Using the gauge fixing and Feynman rules from the previous subsections the partition function has the following expansion: Z (x0 ) =
∞ (−1)n n=0
n!
cn−1 (X ∧ · · · ∧ X ),
(41)
is a cocycle evaluated at , cn (X ∧ · · · ∧ X ) ≡ ∫d 6 z 0 (z 0 ) · · · ∫d 6 z n (z n )e−Skin = b c (x0 ),
where
cn
L
(42)
with c (x0 ) constructed by contracting ∂ 3 (x0 ) with −1 according to the graph . We stress that we do not perform the integral over zero modes which are parametrized by the target M and thus our partition function is a function of x0 . However in the Chern-Simons theory is cubic in fields and thus ∂ 3 (x0 ) are constants and we end up with the constant partition function, in this particular example. Since the function on M satisfies {, } = 0 there is homological vector field Q A (x0 ) = AB ∂ B , which squares to zero (i.e., the Lie bracket Q A ∂ A Q B = 0). Thus there is a natural differential acting on the functions on M and we can define a cohomology group H Q (M). We would like to argue that Q · cn = 0 and as a result Q · Z (x0 ) = 0. Therefore the partition function Z (x0 ) can be understood as some sort of Q-characteristic class, i.e. the element of H Q (M). Let us start from a very elementary proof of this fact. As we argued before cn (x0 ) is a cocycle with respect to the differential δ defined by the formula (38). Thus we have the following chain of relations: Q · cn = {(x0 ), cn } ∼ cn (X{,} ∧ X ∧ · · · ∧ X ) = 0,
(43)
where we used δcn = 0 and {, } = 0. Although this derivation is correct and simple, it misses some important geometrical aspects. Now we will move on to a more elaborate argument, but with clear geometrical meaning. For simplicity, we take the target space M to be a vector space and identify its tangent space with the manifold itself, T M = M × M, where both are equipped with a symplectic structure and a bracket. Let us try to compute the partition function Z (x0 ) of the AKSZ model. The standard method is to split any field into a classical part and a fluctuation part X = x0 + ξ . x0 is treated as the background while the fluctuation ξ is
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taken to parameterize the fiber of the tangent bundle of M at x0 . For a general curved manifold the simple splitting x0 + ξ does not make sense; one has to use the exponential map to identify a neighborhood of the tangent bundle with the neighborhood of x0 . This is what we do in Sect. 8. From the master equation {, } = 0, we want to derive some sort of Cartan-Maurer equation for the bundle T M. We Taylor expand {, } = 0 around x0 in powers of ξ , and get a series of equations 0 = 2(∂C ∂ A )(−1 ) AB ∂ B , 0 = (∂ A )(−1 ) AB ∂C1 ∂C2 ∂ B + (∂C1 ∂ A )(−1 ) AB ∂C2 ∂ B , ··· 2 0 = (∂ A )(−1 ) AB ∂C1 · · · ∂Cn ∂ B n! n−1 1 n (∂C1 · · · ∂C p ∂ A )(−1 ) AB ∂C p+1 · · · ∂Cn ∂ B . + p n!
(44)
p=1
If we define as =
∞ 1 (∂C · · · ∂Cn (x0 ))ξ C1 · · · ξ Cn , n! 1 n=2
the series of equations except the first one can be packaged into the following compact form: (∂ A (x0 ))(−1 ) AB
∂ 1 (x0 ) + { , }ξ = 0, B 2 ∂ x0
where the bracket is written as {, }ξ to stress that it is the bracket of the fiber of T M. This is then our favorite Cartan-Maurer equation, Q B (x0 )
∂ 1 (x0 ) + { , }ξ = 0. B 2 ∂ x0
(45)
Now we regard (x0 , ξ ) as a function on T M = M × M. Since the zero term in expansion (x0 ) will not be joined by the propagators, the connected diagrams in cn from (41) will be given by ∫d 6 z 0 (z 0 ) · · · ∫d 6 z n (z n )e−Skin , (46) L
which can be regarded as a cocyle of Lie algebra formal Hamiltonian vector fields along the fiber T M (i.e., ξ -direction). By construction (x0 , 0) = 0 and ∂ξ (x0 , 0) = 0 and thus the expression (46) is annihilated by δ as defined in (38), but without the last term. From (45) can be thought of as a flat connection on bundle M × M with the Lie algebra being a Lie algebra of formal vector fields Ham02m|k and the base differential being Q. The perturbative calculation can be understood as plugging the flat connection into a cocycle cn and obtaining the characteristic class in H Q (M) of the flat bundle M ×M. Thus according to the general discussion around Eq. (15), the correlator · · · will be annihilated by Q A (x0 ). But it may be helpful to understand this
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in a concrete context. If we adopt the particular gauge fixing given in Sect. 5, then x0 is h , ξ is the non-harmonic naturally taken to be the harmonic 0-form part of the fields X (0) e c part ξ ∼ X + X . Recall from the previous discussion, the perturbation expansion picks up only the part of that is cubic in ξ , and the correlator · · · is given by tri-valent graphs. Since the higher powers of do not enter the computation, we should be able to understand the invariance of the correlator under Q in a direct way. Indeed, using the third equation of (44), we have 1 Q(∂ 3 (x0 )ξ 3 ) = − {∂ 2 ξ 2 , ∂ 3 ξ 3 }ξ , 6 and recall that the quadratic Hamiltonian functions generate the osp rotations, so Q will act on a correlator ∫∂ 3 ξ 3 · · · ∫∂ 3 ξ 3 as a rotation in the ξ space. But the correlator clearly has an invariance under such rotations. The homomorphism induced by (45) gives us a mapping H • (Ham02m|k , osp(2m|k, R), R) −→ H Q (M), which offers a better explanation for (43). We have already explained the modding out of osp(2m|k, R). Taking the Lie algebra to be the formal Hamiltonian vector fields has two advantages, first it has more stable cohomology classes than, say, the Lie algebra of formal vector fields preserving zero (in which case, there is only three infinite series of cocycles, see [6,14]). A larger number of cocylces means more characteristic classes after plugging the flat connection into cocycles. Secondly any foliation with a symplectic structure transverse to the leaf naturally gives rise to a flat connection taking values in the Lie algebra of Hamiltonian vector fields [7]. Indeed any symplectic graded(super)manifold M with the nilpotent Hamiltonian will give rise to the flat connection on T M upon using the exponential map. To summarize, since the partition function of the AKSZ model will consist of a series of (42), each of which is invariant under the homological vector field Q, the AKSZ model calculates the characteristic class associated to Q. If the integration over zero modes is well-defined then one can calculate the characteristic number for the corresponding class. This is a possible way of producing characteristic classes associated with a Q-structure using the path integral for 3D AKSZ models and this agrees with the prescription given in [14], where the authors showed that basically all that matters are the 3-valent graphs. Finally let us make comments about the partition function. There is a dual construction of graph cycles as follows. Suppose we have an odd Hamiltonian function satisfying {, } = 0, and is at least quadratic in its Taylor expansion. We take the cubic or higher Taylor coefficients as vertices, then we follow the graph and connect the vertices together using −1 , take the graph as in (40) (denoting ∂ A · · · ∂ B |x=0 as A···B )
−→ AB C D K L AC K B DL ≡ c ,
and the chain c is a graph cycle. Now we have seen two dual constructions of the graph (co)cycle, when we compute the partition function of a general AKSZ model with the action (23), each term in the perturbation expansion is of the form Z AK S Z =
∞ 1 ∫d 6 z 1 (z 1 ) · · · ∫d 6 z 1 (z n )e−S ; (−1)n n! n=0
L
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as complicated as the integral might be, it can always be written as b ∗ , c = b c ,
which realizes the pairing of the two dual constructions. 6. Example 1: Q-Equivariant Bundle In this section, we give a slightly more concrete example of the general discussion above, preparing the way for the Rozansky-Witten model. We consider an example of the Q-equivariant vector bundle E → M (see the discussion around Eq. (11)) and we set up the corresponding 3D AKSZ model which provides the realization of the characteristic classes described in Sect. 3. We assume that M = T [1]M with M being the usual smooth manifold and the fiber of E has a symplectic structure AB , and we assign to it degree 2. The Lie algebra of the structure group of the bundle is the Lie algebra of formal Hamiltonian vector fields. Suppose for definiteness, the Q-structure of the base is the de Rham differential and it is lifted to a Q-structure Q˜ acting on the total space E. This action preserves AB so Q˜ can be written as ∂ Q˜ = v μ μ + v μ {Aμ (X, e), ·} , ∂X for some function Aμ of the total space E. Of course, we need ∂[μ Aν] + {Aμ , Aν } = 0 to ensure Q˜ 2 = 0. ˜ We do this within the To set up an AKSZ model we need a Hamiltonian lift of Q. minimal symplectic realization of E. Denote the local coordinate of this symplectic space to be X μ , Pμ , v μ , qμ , e A , where e A is the fiber coordinate of E and v μ is that of T [1]M, etc. The degree assignment is 0 2 1 1 0 respectively. This big space has the symplectic structure 1 ω = δ Pμ ∧ δ X μ + δqμ ∧ δv μ + AB δe A ∧ δe B , 2 ˜ with deg = 2. Then we can lift Q into a Hamiltonian function = Pμ v μ + v μ Aμ . Using as the interaction term, we have the standard 3D AKSZ action 1 S= d 6 z P μ D X μ + q μ Dv μ + e A AB De B + P μ v μ + v μ Aμ (X, e) . 2
(47)
T [1]3
We adopt the same gauge fixing by breaking every field into the harmonic and nonharmonic parts and setting to zero all the exact fields. Since q c only appears in the kinetic term q c Dv c , we can integrate it out enforcing Dv c = 0, and hence v c = 0, i.e. v is harmonic. We are then left with 1 S= d 6 z ( P μ )c D(X μ )c + (e A )c AB D(e B )c 2 T [1]3
+ (v μ )h ( P μ )h + (v μ )h Aμ (X h + X c , ec + eh ) .
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J. Qiu, M. Zabzine
We integrate out P c enforcing X c = 0: 1 c 6 c h h μ h h c h e De + v P + (v ) Aμ (X , e + e ) . S= d z 2 T [1]3
Remember we are regarding the harmonic fields as parameters rather than dynamical variables; we can for example put eh , P h to zero, and reduce the action down to its minimal ingredients 1 c 6 c μ h h c e De + (v ) Aμ (X , e ) . S= d z 2 T [1]3
It is then clear from the earlier discussion that the perturbation expansion will have the Taylor coefficients of (v μ )h Aμ (X h , ec ) as interaction vertices and the propagators will h and v h only, contract ec ’s together. The result of the path integral is a function of X (0) (0) which is just a differential form on M. This form is closed by construction, and is the secondary characteristic class. 7. Example 2: Odd Chern-Simons Theory Here we present a toy model on a vector space that reproduces the same weight function b as in Chern-Simons and Rozansky-Witten theory. This model also provides us with cocycles in Lie algebra cohomology, setting the stage for Sect. 8.
7.1. Chern-Simons theory. In this subsection we briefly recall some well-known perturbative aspects of the Chern-Simons theory. The 3D Chern-Simons theory is defined for any metric Lie algebra g by the following classical action: k 2 γ β β SC S = (48) d 3 ξ ηαβ Aα(1) ∧ d A(1) + f αβγ Aα(1) ∧ A(1) ∧ A(1) , 4π 3 3
where A(1) is a connection 1-form on 3 , η is the metric and f is the structure constant on g. The Chern-Simons theory can be embedded into the BV-framework through the AKSZ action 2 6 α β d z ηαβ A D A + f αβγ A A A , (49) 3 T [1]3
where A is a degree 1 superfield valued in a Lie algebra understood as T [1]3 −→ g[1], and the odd symplectic structure is T [1]3
d 6 zηαβ δ Aα ∧ δ Aβ .
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The naive gauge fixing of (49) with A(3) = A(2) = 0 leads to the action (48) which is not suited for perturbative theory. We have to resolve to the gauge fixing we have discussed by setting the exact parts of the fields to zero. Namely we get 2 SC S = d 3 ξ Tr Ac(1) ∧ d Ac(1) + Ac(2) ∧ d Ac(0) + Ac(1) Ac(1) Ac(1) + Ac(2) [Ac(1) , Ac(0) ] , 3 3
(50) where for the sake of clarity we suppressed η and f . One can recognize in this action A(1) as the connection 1-form, A(0) as the ghost c and A(2) as the anti-ghost d † c, ¯ while the Lagrange multiplier appearing in the standard gauge fixing in [2] has been integrated out here, forcing every field to be co-exact. Therefore our gauge fixing is equivalent to the gauge fixing used in [2] for the Chern-Simons action (expanded around a trivial connection). If we look at the correlators in the perturbation theory then according to the BVargument we have to get a cocycle of the Lie algebra of formal Hamiltonian vector fields on g[1] with the even symplectic structure given by metric η. The functions on g[1] are ∧• g∗ and thus the perturbation theory gives us the map cn : ∧• g∗ ⊗ ∧• g∗ ⊗ · · · ⊗ ∧• g∗ −→ R, which is a cocycle with respect to the differential δ defined previously. But of course calculating the Lie algebra cocycles is hardly the principle use of Chern-Simons theory. 7.2. Odd Chern-Simons theory. We take a 2m dimensional vector space M = R2m equipped with the standard symplectic structure . The BV model will be T [1]3 → M. The action is the free action 1 d 6 z X μ μν D X ν , S= 2 T [1]3
where we assign formally the symplectic form grading 2 to match the degree as is done in [16]. The model is interesting because if we perform a naive gauge fixing by setting the 2- and 3-form components of the superfield X μ to zero, we get a component action 1 μ ν , d 3 ξ μν −X (1) ∧ d X (1) S= 2 3
which can be compared to the free part of Chern-Simons action (48). The only difference between this model and the Chern-Simons one is that the symmetric metric ηαβ is replaced with the anti-symmetric symplectic form μν , while the even 1-form Aα(1) is μ replaced with the odd 1-forms X (1) . The similarity does not stop here, as we go on to look at their perturbation expansion. We will refer to this new theory as odd Chern-Simons theory. We use the gauge fixing of the previous section. The resulting action is μ μ ν c ν c d 3 ξ μν (X (2) )c ∧ d(X (0) ) − (X (1) )c ∧ d(X (1) ) , 3
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which again can be compared to the free part of the gauge fixed Chern-Simons action (50). We want to discuss the perturbative theory for the odd Chern-Simons model. From the above consideration, it is not surprising that, as far as the weight function b is concerned, these two models (odd and even Chern-Simons) are equivalent. And what is more, although the Rozansky-Witten model, being an AKSZ model was gauge fixed slightly differently, also produces the same weight function. Next we look at a two point function and show the total agreement between the odd CS model, the CS model and the Rozansky-Witten model. The Green’s function may be worked out in a conventional manner, we first insert sources for the fields and compute the partition function Z [J ] = D X exp ∫d 3 ξ X (2) d X (0) − X (1) d X (1) + J(1) X (2) + J(2) X (1) + J(3) X (0) . Complete the square for the action (watch out, J(2) is odd, J(1) , J(3) are even) 1 1 S[J ] = d 3 ξ (X (2) − J(3) −1 )(d X (0) + −1 J(1) ) − J(1) −1 J(3) d d 1 1 1 1 J(2) −1 )(d X (1) − −1 J(2) ) − J(2) −1 J(2) . − (X (1) + 2d 2 4 d So the partition function is 1 exp Z [J ] =
1 −1 1 −1 1 d ξ −J(1) J(3) − J(2) J(2) , d 4 d 3
where we used to denote the 1-loop determinant factor. Note that the absolute value of it is the Ray-Singer torsion and it is independent of the metric on 3 . The phase of is much more delicate; according to [25], a gravitational Chern-Simons term must be added to the phase factor of to restore the metric independence. The Green’s functions for J(2) , J(3) are 1 J3 (u) = ∗du dv G(u, v) ∗ J3 (v)g 1/2 , d 1 J2 (u) = dv Habc (u, v)Jbc (v)g 1/2 , d √ where G(u, v) is just the scalar Green’s function satisfying ∇u2 G(u, v) = δ(u, v)/ g. We do not have the Green’s function for J(2) over arbitrary 3 , so we write it as Habc (u, v) in general. But in flat 3 , the Green’s function is G(u, v)d † J(2) . The correlators are obtained by varying Z [J ] with respect to the source: 0 (u, v) = X ab (u), X (v) = abd (−1 )G ab
∂ ∂ Z [J ] ∂ Jd (u) ∂ J(3) (v)
= −g 1/2 (u)∂u c G(u, v) cab , ∂ ∂ 1 1 (−1 )G ab (u, v) = X a (u), X b (v) = acd be f Z [J ] 4 ∂ Jcd (u) ∂ Je f (v) = Hacd (u, v)g 1/2 bcd ,
(51)
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817
where we suppressed the target space indices. We can assemble the Green’s function into the superfield form X(u, θ ), X(v, η) =
1 b a 0 1 1 0 θ θ G ab (u, v) − θ a ηb G ab (u, v) + ηb ηa G ab (v, u). 2 2
1 is given by c ∂ G(u, v), so the super Green’s Note in the limit 3 is flat, the G ab ab c function becomes
1 X(u, θ ), X(v, η) = − θ b θ a abc ∂c G(u, v) − θ a ηb abc ∂c G(u, v) 2 1 − ηb ηa abc ∂c G(u, v) ∼ (θ − η)2 , 2 hence there can not be two propagators between two vertices in the flat limit. This is important for this accounts for the improved short distance behavior. Since one propagator blows up with the square inverse of distance between two insertions, so there will be a naive UV divergence when three propagators connect two insertions. Yet, as two insertions become coincident, the flat space propagator dominates and we just saw the mitigation of such UV behavior. In fact ref.[2] proved the finiteness of the perturbation expansion. Now we try to compute the two point function, c1 (X f1 ∧ X f2 ) = ∫d 6 z f 1 ∫d 6 z f 2 e−S . L
There is only one 3-valent diagram
in order to have enough θ ’s to satisfy the Grassmann integral. The correlator is formally written as c1 (X f1 ∧ X f2 ) = b · (−1 )μ1 ν1 (−1 )μ2 ν2 (−1 )μ3 ν3 (∂μ1 ∂μ2 ∂μ3 f 1 )(∂ν1 ∂ν2 ∂ν3 f 2 ), (52) where b is the weight function8 1 3 1 0 1 0 b = d 3 ud 3 θ d 3 vd 3 η θ b θ a G ab (u, v)−θ a ηb G ab (u, v)+ ηb ηa G ab (v, u) 2 2 6 1 = − d 3 ud 3 θ d 3 vd 3 η (θ a θ b G 0 (u, v))(θ a ηb G ab (u, v))(θ a θ b G 0 (u, v)) 4 1 (u, v))3 + (θ a ηb G ab 6 0 = − d 3 ud 3 v G ab (u, v)G 1cd (u, v)G 0e f (v, u) 4 1 (u, v)G 1cd (u, v)G 1e f (u, v) eabc ede f + G ab 8 Our convention is that eabc , e abc = 1/geabc . abc = abc is the Levi-Civita symbol, while
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J. Qiu, M. Zabzine
d 3 ud 3 v 6g 1/2 (u)∂uc G(u, v)G 1cd (u, v)∂vd G(v, u)g 1/2 (v) 1 (u, v)G 1cd (u, v)G 1e f (u, v)eabc ede f . +G ab
=−
We observe that this is the same weight function appearing in the CS perturbation theory [2] and in the Rozansky-Witten model [19]. And we expect the agreement9 to continue even for larger diagrams. The metric dependence of b can be addressed in the same way as for the Chern-Simons theory and thus we leave this issue aside. For this simple correlator (52) we may check explicitly that it is a cocycle (δc1 )(X f1 ∧ X f2 ∧ X f3 ) = c1 (X{ f1 , f2 } ∧ X f3 ) − c1 (X{ f1 , f3 } ∧ X f2 ) +c1 (X{ f2 , f3 } ∧ X f1 ) − { f 1 , c1 (X f2 ∧ X f3 )} +{ f 2 , c1 (X f1 ∧ X f3 )} + { f 3 , c1 (X f1 ∧ X f2 )} = 0, where magically all derivatives cancel out for any functions f 1 , f 2 , f 3 . Some comments must be made regarding the subtlety of the (source) metric independence. The original model is written down without any metric and hence is classically invariant under any orientation preserving diffeomorphism of 3 . The metric only comes in through the gauge fixing we use, and so the correlator should not depend on the gauge choice by the discussion of Sect. 2 (known as the the Ward-identity in gauge theory). However, caution is needed when making such assertions, for the Ward-identity may be spoiled by the quantum correction and we already saw that the 1-loop determinant has an anomalous dependence on the metric. Even when a correlator is finite accidentally, like the two point function above, one cannot conclude based on finiteness its gauge invariance, indeed [2] showed that this two loop diagram suffers a similar anomaly as the determinant factor. Another somewhat remote example for this is the 1-loop light by light scattering in QED. The diagrams when computed in 4D are finite accidentally, yet the result is not gauge invariant. The usual wisdom for the gauge theory is that the question hangs upon whether one possesses a gauge invariant regulator; if yes, then one can subtract the divergence in a manner preserving the symmetry in question and the symmetry is anomaly free. For the CS model, the common practice is that when one integrates over the configuration space, which is copies of T [1]3 , one carefully subtracts the diagonals because these correspond to the singular configuration where two insertions are coincident. As a result one is led to consider certain compactification of the configuration space, but this is out of the scope of this paper. 8. Example 3: Reinterpreting Rozansky-Witten Model The Rozansky-Witten (RW) model was introduced in [19] and it gives rise to the Rozansky-Witten invariants, see [20] for a nice review of these invariants. The authors of [19] constructed the model by writing down a set of BRST rules associated to a hyperKähler manifold. They also pointed out the similarity between their model and the Chern-Simons model and went as far as calling it the odd Chern-Simons model. Yet one important difference between the two is that the perturbation expansion of the RW model stops at finite order while that of the CS does not. The reason is basically due to the need to saturate the zero modes. This feature did not fail to catch the attention of Kontsevich, who then pointed out that the RW model is an AKSZ model with parameters and the 9 Indeed we expect the same agreement for b for the generic theory with the target R2m|k .
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819
model gives the characteristic classes of the holomorphic foliation. We can understand this from the discussion of Sect. 6. In particular, the RW model is a special case of the model (47). The ’parameters’ alluded to in [13] are the harmonic fields in (47). At the same time, Kapranov [10] interpreted the RW model from the point of view of Atiyahclass. In this section, we will investigate this model from the field theory perspective and try to endow physical embodiment to the works [10,13]. The RW model is also an AKSZ model. The space of fields is T [1]3 −→ T ∗ [2]T ∗ [1]M, where M is a HyperKähler manifold10 . The symplectic form for the target space T ∗ [2]T ∗ [1]M is given by 1 ω = δ Pμ δ X μ + δv μ δqμ + i j δ X i δ X j , 2 where is the holomorphic symplectic 2-form for M. We use labels μ, ν, . . . for real ¯ j¯ , . . . for complex coordinates. The kinetic term of the AKSZ coordinates and i, j, i, action is the standard one determined by the symplectic form above, 1 d 6 z P μ D X μ + X i i j D X j + q μ Dv μ . (53) Skin = 2 T [1]3
The interaction term is the one corresponding to the Doubeault differential, ¯ d 6 z P i¯ v i . Sint =
(54)
T [1]3
It is possible to find a LagSubMfld, and restriction of this action to it gives the RW model [16], ¯ j j¯ i i i S RW = gi j¯ d X (0) ∧ ∗d X (0) + gi j¯ X (1) ∧ ∗d ∇ v j − i j X (1) ∧ d ∇ X (1) 1 ¯ j l k¯ − Rk k¯ i j X (1) ∧ li X (1) ∧ X (1) v k . 3
(55)
The first line of the action gives a non-degenerate kinetic term. This construction of the RW model is of course correct, but when we are only interested in the computation of invariants of holomorphic foliation, we can strip down the extraneous parts of the model and make the geometrical meaning more pronounced. As far as the RW invariant is concerned, all we need is a cocycle in the cohomology of the Lie algebra of formal Hamiltonian vector fields and a Hamiltonian function to plug in. At a given point in M, the holomorphic tangent bundle is identified as C2m (dimR M = 4m) and equipped with the symplectic form i j . First recall that a tangent vector induces a flow; when we have a flat connection, we may fix the flow to be the geodesic flow and unambiguously identify a neighborhood of the origin in the tangent space of x 0 with a neighborhood of x 0 in M. In this way we obtain the so called normal coordinates. Any function in M defined in a neighborhood 10 The construction can be relaxed to the case of a holomorphic symplectic manifold.
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of x 0 may be pulled back to the normal coordinate system. We denote the geodesic flow induced by the vector ξ as exp∗ , so 1 μ ν ρ 1 ν ρ σ 1 μ μ μ κ X μ (x0 , ξ ) = x0 +ξ μ − νρ ξ ξ − ξ ξ ξ ∂ν ρσ + ξ ν ξ ρ ξ σ κν ρσ + · · · , (56) 2 6 3 1 exp∗ φ(X ) = φ(X (x0 , ξ )) = φ(x0 ) + ξ μ ∂μ φ(x0 ) + ξ μ ξ ν ∇μ ∂ν φ(x0 ) + · · · , 2 where we assumed that is some flat connection. Of course, the exponential map requires just a connection, but for the sake of further discussion we assume in (56) that is flat. As a mnemonic, the pullback of the function φ by exp is just the Taylor expansion of φ(x0 + ξ ) around x0 , except that one uses a covariant derivative rather than an ordinary derivative. The same remark applies to the tensors as well, for example 1 exp∗ φμ d X μ = φμ (x0 )dξ μ + ξ ν ∇ν φμ (x0 )dξ μ + ξ ν ξ ρ ∇ν ∇ρ φμ (x0 )dξ μ + · · · . 2 Now we take the base coordinate x0 and also v as fixed parameters. The target space for the AKSZ model is (1,0),∞
M = TM
(x0 ),
where T (1,0),∞ denotes the formal neighborhood of the zero section of the holomorphic tangent bundle at x0 . Due to the Kähler property, the Levi-Civita connection has either totally holomorphic or totally anti-holomorphic indices, and the curvature Ri j×× = 0, namely ijk can be regarded as the flat connection. Of course, we now only have the holomorphic half of the tangent bundle, so the geodesic flow is understood as the analytical continuation away from the real one. For all practical purposes, we just understand the geodesic flow as given by the formal Taylor expansion as in (56). Now the target space will be parameterized by the coordinates ξ i in the formal neighborhood of x0 . We have originally a holomorphic symplectic form d X i i j d X j , which we pull back to the ξ coordinate exp∗ d X i i j d X j = dξ i i j dξ j + ξ k (∇k i j )dξ i dξ j + · · · = dξ i i j (x0 )dξ j .
(57)
We can now set up a free AKSZ theory as in Sect. 7. The odd symplectic structure is given by 1 ω= d 6 z δξ i i j (x0 )δξ j . 2 T [1]3
We stress that now i j is constant in the ξ space and equal to i j (x0 ). The action is 1 S= d 6 z ξ i i j (x0 )Dξ j . (58) 2 T [1]3
The path integral provides us with the desired cocycle, and we will evaluate the correlator of the particular function ¯
¯
= v i i¯ = v i
∞ n=0
where
1 Rn ξ i1 · · · ξ in+3 , (n + 3)! ii¯ 1 ···in+3
Rii¯n ···i 1 n+3
= ∇i1 · · · ∇in Rii¯
j n+1 i n+3
jin+2 ;
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821
note that due to the Kähler property as well as the covariant constancy and holomorphy of , all the holomorphic indices in R n are symmetric11 . We show in the Appendix that ¯ + 1 {, } = 0, ∂ 2
(59)
where the Poisson bracket is with respect to in the ξ -direction. Without specifying which gauge fixing, we still know that the path integral gives a cocycle with respect to . The correlator is a function of extra parameters x0 , v, cq (X ∧ · · · ∧ X ) = f (x0 , v), which can be regarded as an anti-holomorphic form on M. In fact this is an element in H∂¯• (M), for the Dolbeault differential acts on cq as the differential in the graph cohomology ¯ q (X ∧ · · · ∧ X ) = − 1 cq X ∧ · · · ∧ X{,} ∧ · · · ∧ X ∂c 2 ∼ cq (∂(X ∧ · · · ∧ X )) = 0. Furthermore, since the space of ξ now has a flat structure C2m , we can apply the naive gauge fixing by setting exact forms to zero as in Sect. 7. This gauge fixing automatically keeps only tri-valent graphs. These graphs are given by contracting the ξ ’s in ¯ v i Rii¯ l k l j ξ i ξ j ξ k with −1 . This exactly reproduces the Rozansky-Witten invariants. ¯ The ∂-closedness is automatic due to ∂¯ R = 0. We pointed out before that the cocycle given by the path integral does depend on the choice of the LagSubMfld, or the gauge fixing; it is reasonable to expect that a drastically different gauge choice shall produce a different class in H∂¯• due to the abundance of cocycles in the graph cohomology. The above construction grasps the main feature of the Rozansky Witten model, yet we would like to incorporate also the extra parameters x0 , v into the theory and furthermore justify the definition of . In particular we show that the RW model fits the general description of the AKSZ model for flat bundles of Sect. 6. We first complexify M by taking two copies M × M; the second one is equipped with
the opposite complex structure as the first one. So the diagonal embedding M → M × M gives the real slice. The picture here can also be studied in the light of holomorphic foliation [13]. We try to motivate the analogy between our problem and the holomorphic foliation with a few words, though this analogy is not strictly necessary for the rest of the paper, so for the first reading, the reader may jump over the next five paragraphs. We label the two copies of M by the holomorphic and anti-holomorphic coordinate respectively, i.e, a point ( p, q) ∈ M × M is parameterized by ¯
(X i ( p), X i (q)), ∀( p, q) ∈ M × M. We can take the second factor of the product as the leaf space of the foliation while the first factor is the transverse direction. The holomorphic geodesic exponential map amounts to the following change of coordinates: ¯
¯
(X i ( p), X i (q)) → (expξ X i (q), X i (q)). 11 Please notice our convention for R n ¯ 1 ···i n+3 , where the superscript n is not a holomorphic index! ii
(60)
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For a foliation with constant co-dimension, we have a principle bundle structure. The fiber is (rather abstractly) all possible ways of identifying the transverse space at the neighborhood of a point with the Euclidean space C2m . The structure group is by definition isomorphic to the fiber and can be taken as the group of formal diffeomorphisms of the transverse direction. Now that there is a symplectic structure i j in the transverse direction, the relevant structure group should become the group of formal symplectomorphisms. The previously defined exponential map offers one way of such identification, namely at point q; the point p in the transverse space is identified with ξ ∈ C2n through X i ( p) = expξ X i (q). So the exponential map is a section of the principle bundle. This principle bundle has a flat connection. Here we follow the work of Fuks [7]. Let the foliation be determined by a system of 2n 1-forms θ i , which means the leaf is the null-space of these forms. By the Frobenius theorem for an integrable system of 1-forms, the differential dθ i is dθ i = γ ij ∧ θ j , where γ is again some 1-forms, and can be thought of as the connection of the principle bundle. This connection is flat only along the leaf: (dγ ij −γ ik γ kj )θ j = 0. But this in turn implies dγ ij − γ ik γ kj = γ ijk θ k for some 1-form γ ijk . One can carry on this procedure and obtain a collection of such γ ’s, and out of these one can construct a connection ∂ ∂ i = i i = θ i + γ ji η j + γ jk η j ηk + · · · , ∂η ∂ηi where η is some formal variable. This connection now takes value in the Lie algebra of formal vector fields in the η-space. This connection is flat in all directions. Next we apply the above machinery to bear uponour problem. Due to the mapping ¯
∗
Eq. 60, the previous x0 (q) is identified as X i (q) = X i (q) . The holomorphic foliation is determined by 2n 1-forms d X i , because the leaf is clearly the null space of these 1-forms. It will be shown in the Appendix that the pullback of these 1-forms exp∗ξ d X i ¯
is the linear combination of the system θ i = dξ i − d X i {i¯ , ξ i }. According to the above recipe, we differentiate the 1-forms and it turns out that ¯
¯
γ ij ∼ d X i ∂ξ j {i¯ , ξ i }; γ ijk ∼ d X i ∂ξ j ∂ξ k {i¯ , ξ i }; · · · . And the flat connection for the foliation is given by n=∞ 1 ¯ (η∂ξ )n {i¯ , ξ i } ∂ηi , = dξ i − d X i n! n=0
d − = 0. This connection is flat in all (including the ξ ) directions after applying Eq. 59. Now we have a flat principle bundle defined over M × M, but we can restrict it to the diagonal (given by ξ = 0). The pullback of the connection, for which we use the same symbol, is ¯
= −d X i
n=∞ n=0
1 (η∂ξ )n {i¯ , ξ i }∂ηi . ξ =0 n!
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¯
This obviously is just d X i {i¯ , ξ i } with all the ξ ’s replaced with η. Finally, we see that it is in this way Eq. 59 is interpreted as the flat connection of holomorphic foliation and the correlator of the RW model gives rise to the characteristic class of the holomorphic foliation. Back from our digression, the key observation by Kapranov [10] is the following, exp ¯ ¯ ¯ under the mapping T (1,0),∞ → M × M, (ξ, X i ) → (expξ (X i )∗ , X i ), the differential ∂¯ is pulled back as exp {i¯ , ·}, ∂¯i¯ → 0, ∂¯i¯ . (61) This motivates (59) because the Dolbeault differential on the rhs is nilpotent. It is also instructive to prove this relation explicitly which we do in the Appendix. On the space M × M, we can construct the GrMfld, M = M × T (0,1) [2]T (0,1) [1]M, ¯
¯
parameterized by (X i , Pi¯ , qi¯ , v i , X i ). It has the even symplectic form 1 ¯ ¯ (62) ω = δ Pi¯ ∧ δ X i + δqi¯ ∧ δv i + i j δ X i ∧ δ X j . 2 With this data one can set up the standard AKSZ model, whose homological vector field ¯ is ∂–the rhs of Eq. 61, 1 ¯ ¯ ¯ S = d 6 z P i¯ D X i + q i¯ Dv i + X i i j D X j + P i¯ v i . (63) 2 This action is a truncation of the AKSZ action (53)+(54) and it still reduces to the gauge fixed RW action (55) along the lines presented in [16]. (1,0)∞ While for the manifold TM one has the GrMfld, M = T (1,0)∞ M ⊕ T (0,1) [2]T (0,1) [1]M, ¯
¯
parameterized by (ξ i , Pi¯ , qi¯ , v i , X i ). We want to pull the model (63) on M × T (0,1) [2]T (0,1) [1]M back to this target space. The required change of variable is ¯ (x0 = (X i )∗ ) ξ i ∂ i −ξ i ξ j ikj ∂ξ k i 1 exp x0 , ξ i → xi0 + ξ i − ijk ξ j ξ k + · · · = e x0 2 ¯ ¯ ¯ ¯ Pi¯ , qi¯ , v i , X i → Pi¯ , qi¯ , v i , X i .
The same calculation that led to (61) shows ∂ X i (ξ ) ¯ δ X i = δξ j − δ X i {i¯ , ξ j }) . ∂ξ j The holomorphic symplectic form is pulled back according to (using Eq. (57)) ¯
¯
exp∗ δ X i i j δ X j = (δξ i − δ X i {i¯ , ξ i })(exp∗ξ i j )(δξ j − δ X j { j¯ , ξ j }) ¯
¯
¯
= δξ i i j δξ j − 2δ X i {i¯ , ξ i }i j δξ j + δ X i {i¯ , ξ i }i j δ X j { j¯ , ξ j } ¯ ∂i¯ δξ j ∂ξ j
= δξ i i j δξ j − 2δ X i
¯
= δξ i i j δξ j − 2δ X i δi¯ .
¯
¯
− δ X i δ X j {i¯ , j¯ }
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So the symplectic form Eq. 62 is pulled back to 1 ¯ ¯ ¯ ∗ exp ω = d 6 z δ P i¯ ∧ δ X i + δq i¯ δv i + δξ i i j δξ j − δ X i δi¯ . 2 The action Eq. 63 is pulled back as 1 ¯ ¯ ¯ exp∗ S = d 6 z ( P i¯ + i¯ )D X i + ξ i i j Dξ j + q i¯ Dv i + P i¯ v i . 2 ¯
Since now the momentum dual to X i is Pi¯ + i¯ , it is proper that we changed the variable P˜i¯ := Pi¯ + i¯ , 1 ¯ ¯ ¯ ¯ ∗ exp S = d 6 z P˜ i¯ D X i + ξ i i j Dξ j + q i¯ Dv i + P˜ i¯ v i − i¯ v i . 2 ¯ ¯ ¯ ¯ ¯ Note that P˜i¯ v i − i¯ v i generates the vector field −(v i ∂¯i¯ + v i {i¯ , ·}) on functions of X i i and ξ , which is the lhs of Eq. 61. This model is totally in line with the general picture for the model given by (47). ¯ ¯ We can perform the partial gauge fixing in the P˜i¯ , X i , qi¯ , v i sector by setting P˜i¯ = qi¯ = 0. Then we are left with the action with only ξ as dynamical variables, relegating ¯ X i¯ and v i as extra parameters. This gives back the odd Chern-Simons model (58) with an interaction term .
9. Summary In this paper, we have explained the idea of using the AKSZ-BV path integral as a construction of cocycles and that its relation to the graph cohomology is nothing but the Feynman integrals and the standard Wick theorem. We took the construction of [9] and put it into a concrete physical system. In particular, we discussed how to deal with zero modes which must exist in any realistic field theory. This leads to the embodiment of Kontsevich’s idea of applying a homomorphism to a cocycle of a Lie algebra cohomology to obtain secondary Chern-Simons type invariants (characteristic classes of flat bundles). Thus we conclude that the AKSZ construction of TFT is powerful not only at the classical level, it also offers a very unified perturbative treatment of the corresponding TFTs. We constructed the odd Chern-Simons theory over the target R2n and showed that its perturbation expansion is identical to that of the Chern-Simons theory, in particular, we obtained an identical weight function for each given graph. We did this for the Rozansky-Witten model painstakingly, and showed from the field theory perspective that this model fits the picture painted by Kontsevich, namely, it is a model associated with a flat bundle related to the holomorphic foliation. The further issues include of course applying the presented ideas for the general AKSZ model for different algebroids and foliations and construct explicitly characteristic classes and invariants. One can naturally associate 3D AKSZ models to Courant algebroids and Lie algebroids. Thus applying the ideas presented in this work one may hope to obtain interesting characteristic classes for these algebroids. The main complication in the treatment of these models would be the application of the exponential map carefully, or in other words, performing the covariant Taylor expansions.
Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes
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Another interesting issue would to apply the formal BV arguments from Sect. 2 to a wide class of quantum observables. In this paper we concentrate our attention on observables which are written as a full integral over a source manifold. The BV algebra of these observables can be mapped to the algebra of functions on the target space. However we may look at the quantum observables which are integrals over cycles on the source (or even full Wilson loops). One has to embed these wide classes of observables into the BV framework and calculate the corresponding BV algebra generated by them. The path integral evaluations of those observables should still give rise to a cocycle for some Lie algebra. We hope to return to this idea in the future. Acknowledgement. The research of M.Z. was supported by VR-grant 621-2008-4273. The authors would like to thank Francesco Bonechi for helpful discussions.
A. Brackets of Even and Odd Type In this section we fix the sign conventions of the symplectic form, Poisson bracket and odd Laplacian, etc. These signs are important for the perturbation theory, it is worth the effort. The degree n symplectic form = AB d X A ∧ d X B , |A| + |B| = n, A
where |A| = deg X A and |B| = deg X B . We always assume that AB is a constant for simplicity, and it satisfies AB = (−1)(|A|+1)(|B|+1) B A , matching the graded commutativity d X A d X B = (−1)(|A|+1)(|B|+1) d X B d X A . We take the odd symplectic form as the starting point. This gives rise to an odd Poisson bracket, which can be induced from the odd Laplacian according to formula (1). We fix the convention for this odd Laplacian first, and from there we derive the conventions of other brackets. The reason is that whether or not a Laplacian annihilates some function is crucial for the discussion of Sect. 4. We assume that AB is a constant for simplicity, and fix the following:
AB d X A ∧ d X B ⇒ =
A
(−1 ) AB
A
∂ ∂ . A ∂X ∂XB
(64)
One may check that satisfies (1), with the Poisson bracket given by { f, g} =
← − (−1 ) AB ( f ∂ A )∂ B g.
(65)
A,B
← − Note that our convention for the right derivative is X B ∂ A = (−1)|A| δ AB . The bracket (65) is derived for the odd case, but we take this as the definition of the Poisson bracket, both for even and odd. This bracket satisfies { f, g} = −(−1)( f +n)(g+n) {g, f }. Suppose we have now a deg n symplectic GrMfld M, with symplectic form . Because we give degree 1 to δ, as a result, δθ = −θ δ and we dispense with the ∧. In the AKSZ construction, we build a TFT on a dimension n + 1 source manifold n+1 with M as
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the target. For such degree n GrMfld, we can form the degree −1 symplectic form over Maps(T [1]n+1 , M) by ω= d 2(n+1) z AB δ X A (z)δ X B (z) , (66) A
T [1]n+1
where d 2(n+1) z = d n+1 ξ d n+1 θ and X stands for a map from T [1]n+1 to M. This form has the desired degree −1 because the measure carries degree −(n + 1). We may obtain the odd symplectic form written in component fields by integrating out d n+1 θ . Assume AB is a constant, the symplectic form (66) can be rewritten as B ω= (−1) A(n+1− p)+ p d n+1 ξ AB δ X (Ap) δ X (n+1− p) . p
The Laplacian according to the rule (64) is (−1) A(n+1− p)+ p+(A+1)(B+1) d n+1 ξ(−1 ) AB = p,A
δ
δ
δ X (Ap) (ξ )
B δ X (n+1− p) (ξ )
.
This naive form of the Laplacian must be improved, otherwise, when it hits a local functional, it will produce δ(0). Nevertheless, if we proceed and investigate d 2(n+1) z f (X(x)) = 0, we find the following formal expression: d 2(n+1) z f (X(z)) n + 1 p |A||B|+n+1 −1 AB (−1) =4 (−1) ( ) d n+1 ξ ∂ B ∂ A f (X (ξ ))δ(0). p p A
Note that this sum formally vanishes for any n + 1, n+1 n+1 (−1) p = (1 − 1)n+1 = 0. p p=0
A better definition of with regularization12 , which is suited both for separating the subtlety from zero modes and for renormalization is the following. We expand any pform on the source manifold into eigen-modes of the self-adjoint operator = {d † , d}, where d is the de Rham differential and d † is its adjoint, X ( p) = X I p ψ I p ; ψ I p = λ2I p ψ I p ; ψ I p (x)i1 ···i p (∗ψ I p (y))i p+1 ···in+1 = i1 ···in+1 δ(x − y). Ip
After changing variables from X (Ap) to X IAp , the original Laplacian becomes =
A
(−1) Ap+ p(n+1)+AB (−1 ) AB
∂ ∂ . ∂ X IAn+1− p ∂ X IBp
12 We use the regularization of odd Laplacian which is similar to the one discussed in [5].
Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes
827
To regularize this expression, one inserts the factor exp (− 2 λ2I p ) in the summation. This regularization is commonly known as the heat kernel regularization. We denote ∂ ∂ (− 2 λ2I p ) = (−1) Ap+ p(n+1)+AB e (−1 ) AB . A B ∂ X ∂ X In+1− p Ip A
What happens here is that we are effectively replacing the original Laplacian with δ = (−1) A(n+1− p)+ p+AB+n+1 d n+1 ξ1 d n+1 ξ2 K p (ξ1 , ξ2 , ) A δ X ( p) (ξ1 ) p,A
δ B δ X (n+1− p) (ξ2 )
,
K p (ξ1 , ξ2 , ) =
e
− 2 λ2I p
ψ I p (ξ1 ) ∧ ψ In+1− p (ξ2 ).
Ip
For small , the heat kernel K asymptotes to K p (ξ1 , ξ2 , ) ∼ −(n+1) e
|ξ1 −ξ2 |2 4 2
,
which is just the smeared delta function, and we recover the original definition of in the limit → 0. It turns out that acts on a full integral as d 2(n+1) z f (X(z)) = (−1) p (−1 ) AB d n+1 ξ K p (ξ, ξ, )∂ A ∂ B f (X (ξ )). p,A
Suppose that ∂ A ∂ B f (X (ξ )) is a constant, then the sum over p gives nothing but the index of the de Rham operator on n+1 , 2 p (−1) d n+1 ξ K p (ξ, ξ, ) = Tr[e− (−1) p ] = χ (n+1 ), p
while for odd (n + 1), which is the main interest of this paper, the sum can be reshuffled into d 2(n+1) z f (X(z)) p −1 AB = (−1) ( ) (67) d n+1 ξ K p (ξ, ξ, )∂ A ∂ B f (X (ξ )), p≤(n/2),A,B
where now sum is taken over all A and B. We used the fact that if ψ I p is an eigenfunction of with the eigenvalue λ2I p , then ∗ψ I p is an eigenfunction of with the same
eigenvalue. The expression (67) vanishes for nonzero since (−1 ) AB ∂ A ∂ B f = 0 due to contraction of (graded)symmetric with a (graded)anti-symmetric and this happens only when n + 1 is even. Here we can see the crucial difference between even and odd dimension theory. We would now like to investigate the relation between the Poisson bracket on the targetspace M and the induced odd bracket in the mapping space, in particular whether ( f g) = { f, g} is true. After a lengthy but straightforward calculation, we obtain ∫d 2(n+1) z 1 f (X(z 1 ))∫d 2(n+1) z 2 g(X(z 2 )) = (−1) f ∫d 2(n+1) z{ f (X(z)), g(X(z))}.
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J. Qiu, M. Zabzine
From this we can get ∫d 2(n+1) z f (X(z)), ∫d 2(n+1) z g(X(z)) = (−1)n+1 ∫d 2(n+1) z { f (X(z)), g(X(z))}. Furthermore, in the text we quite often treat the harmonic modes (i.e., the eigenfunctions with λ2I p = 0) as parameters of the theory, and the path integral is taken only for the non-harmonic fields. If we denote the Laplacian of the non-harmonic fields by , the Ward identity is given by (· · · ) = 0, L
where L is Lagrangian only in the non-harmonic sector. Thus we should investigate what is the bracket induced by the . To do this, we do not need the regularization above. We restrict ourselves to the case of the rational homology sphere for simplicity. Then in the mode sum of (67), we need to exclude two modes √ g 1 h ψ0h = √ ; ψn+1 = √ i1 ···in+1 . vol vol As a result we obtain ∫d 2(n+1) z 1 f (X(z 1 ))∫d 2(n+1) z 2 g(X(z 2 )) = (−1) f ∫d 2(n+1) z{ f (X(z)), g(X(z))} 1 ← − h − (−1 ) AB (−1)n f +A(n+1) √ ∫ψn+1 (ξ ) f (X (ξ )) ∂ A ∫d 2(n+1) z∂ B g(X(z)) vol A,B (−1 ) AB (−1) f (n+1)+g+g f +AB+n+1 − A,B
× √
1 vol
← − h ∫ψn+1 (ξ )g(X (ξ )) ∂ A ∫d 2(n+1) z∂ B f (X(z)).
The first term is the usual one, while the last two are due to the exclusion of the zero modes. To make sense of the formula, we have to resort to the explicit gauge fixing and the Feynmann rules we introduced in Sect. 5. Thus from now on n + 1 = 3. Our claim is that we can do the following replacement in the path integral: 1 h ( √ ∫ψ3 (ξ ) f (X (ξ )) · · · = f (x0 ) · · · , vol L
L
as long as all the other insertions are of the form d 6 zg(X). The reasoning is, suppose that · · · consists of q insertions, thenthe number of propagators routed amongst themselves is # = 3q/2. If the insertion d 3 ξ ψ3h (ξ ) f (X (ξ )) was connected to the rest of the diagram with p ≥ 2 propagators, then # will be reduced to # ≤ (q − 2 p)/3, forcing a 2-valent vertex to appear somewhere and it vanishes within our rules.
Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes
829
With this consideration, the second term of the previous formula under the path integral becomes − (−1)n f f (x0 ), · · · , L
where the Poisson bracket is now taken over x0 . The path integral over the non-harmonic fields produces a function of x0 , accordingly the path integral should be interpreted now as a cochain of the CE complex of formal Hamiltonian vector fields of M, taking values in C ∞ (M). So the differential of such cochains must be modified correspondingly, and this new differential is induced by : (δcq )(X f0 ∧ · · · X fq+1 ) (−1) fi +( fi +n+1)( f0 +··· fi−1 +i(n+1))+( f j +n+1)( f0 +··· f j−1 + j (n+1))+( fi +n+1)( f j +n+1) = i< j q
×c (X{ fi , f j } ∧ · · · X fi ∧ · · · X f j ∧ · · · X fq+1 ) f i n+( f i +n+1)( f 0 +··· f i−1 +i(n+1))+( f i +n) deg cq (−1) −
i
× f i (x0 ), cq (X f0 ∧ · · · X fi ∧ · · · X fq+1 ) .
(68)
In fact, this formula is completely in accordance with the de Rham differential dω(X 0 , . . . , X q ) = −
(−1)i+ j+1 ω([X i , X j ], X 0 , . . . , Xˆ i , . . . , Xˆ j , . . . , X q )
i< j
+
(−1)i X i ω(X 0 , . . . Xˆ i , . . . X q ).
i
Because of the formal relation
L
· · · = 0, the path integral is a cocycle for the
modified differential. What is not expected in (68) is perhaps the sudden appearance of ( f i + n) deg cq . When one derives the formula of the differential using , one is restricted to n + 1 = 3 and q = 2k − 1, so the factor ( f i + n) deg cq = ( f i + 2)6k is invisible. But for a general CE differential for general degree n bracket, this factor is needed in order for δ 2 = 0. If one wishes to check this point for himself, he will find the following useful: { f, g} = −(−1)( f +n)(g+n) {g, f }, {{ f, g}, h} + {{g, h}, f }(−1)(g+h)( f +n) + {{h, f }, g}(−1)(g+ f )(h+n) = 0. Lastly, we also check that for an odd symplectic form, the Hamiltonian vector fields satisfy [X f , Xg ] ≡ X f Xg − (−1)(| f |+1)(|g|+1) Xg X f = X{ f,g} . We do this in Darboux coordinates for simplicity. The symplectic form is ω = δxa+ δx a , where x + is odd, x is even,
830
J. Qiu, M. Zabzine
← − ← − ∂ ∂ ∂ ∂ } + { f, g } + − (−1)(| f |+1)(|g|+1) ( f ↔ g) + a a ∂ xa ∂ x ∂ x ∂ xa ← − ← − ∂ ∂ ∂ ∂ = X{ f,g} + (−1)|g| { f + , g} a − { f a , g} + ∂ xa ∂x ∂x ∂ xa ← − ← − ∂ ∂ ∂ ∂ (| f |+1)(|g|+1) (| f |+1)(|g|+1) −(−1) {g, f + } a − (−1) {g, f a } + ∂ xa ∂ x ∂ x ∂ xa = X{ f,g} ,
[X f , Xg ] = { f, g
where we have used { f, g} = −(−1)(| f |+1)(|g|+1) {g, f }. B. L ∞ Structure from HyperKähler Manifold In this Appendix we present some explicit formulas about L ∞ -structure for the hyperKähler manifold. The idea was presented in [10], but we could not read off the explicit numerical factors from this work. Therefore we present our own derivation of these relations. All expressions are written in complex coordinates. For a hyperKähler manifold, the three indices i, j, k are totally symmetric in (R)ii¯ jk = Rii¯ l k l j and ∂[ j¯ (R)i]i ¯ jk = 0. If we define Ril¯n
1 ···ln+3
≡ ∇l1 · · · ∇ln (R)il¯ n+1 ln+2 ln+3 ,
then apply the covariant derivatives to 0 = ∂[ j¯ (R)i]i ¯ jk , 1 ∇l ···l ∂¯ ¯ (R)i]l ¯ n+1 ln+2 ln+3 + perm in li (n + 3)! 1 n [ j n−1 (n − k + 2) 1 n−k−1 m ∇l1 ···lk (R[ j|l + = ∂¯ ¯ R¯n ···l +perm in li . ¯ k+1 lk+2 )Ri]m···l ¯ 1 n+3 n+3 (n + 3)! (n + 3)! [ j i]l
0=
k=1
So we get n ∂¯[ j¯ Ri]l ¯
1 ···ln+3
=−
n−1 (n − k + 2) k=0
(n + 3)!
m ∇l1 ···lk (R[ j|l ¯
k+1 lk+2
n−k−1 )Ri]m···l ¯
n+3
+ perm in li .
The rhs can be worked out explicitly rhs =
n−1 k k (n − k + 2) k=0 p=0
=
(n + 3)!
n−1 k k (n − k + 2) k=0 p=0
=
p
n−1 n−1 p=0 k= p
p
(n + 3)!
m ∇l1 ···l p (R[ j|l ¯
k+1 lk+2
p
R[ j|l ¯
1 ···l p lk+1 lk+2 m
n−k−1 )∇l p+1 ···lk Ri]m···l + perm in li ¯ n+3
n− p−1 p+1 ···lk n···ln+3
mn Ri]l ¯
+ perm in li
k!(n − k + 2) p mn n− p−1 R[ j|l ¯ 1 ···l p+2 m Ri]l ¯ p+3 ···ln+3 m + perm in li . (n + 3)! p!(k − p)!
Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes
831
The summation of factorials can be worked out as follows: n k= p
k! k! (k + 1)( − p) = lim = lim (k − p)! →0 (k − p + )! →0 (k − p + + 1)( − p) n
n
k= p
1 →0 ( − p)
= lim
n 1
k= p
x k (1 − x)− p−1
k= p 0
1 1 = lim (x p − x n+1 )(1 − x)− p−2 →0 ( − p) 0 ( p + 1)( − p − 1) (n + 2)( − p − 1) 1 − = lim →0 ( − p) () ( − p + n + 1) (n + 1)! = . (n − p)!( p + 1) This leads to n−1 n−1 p=0 k= p
n−1 k!(n − k + 2) 2p + n + 5 = (n + 3)! p!(k − p)! ( p + 2)!(n − p − 1)!(n + 1)(n + 2)(n + 3) p=0
=
n−1 1 1 , 2 ( p + 2)!(n − p + 1)! p=0
where we take the average between p ↔ n − p − 1 for the last step. The final result is n ∂¯[ j¯ Ri]l ¯
1 ···ln+3
=−
n−1 1 1 p mn n− p−1 R[ j|l ¯ 1 ···l p+2 m Ri]l ¯ p+3 ···ln+3 n + perm in li . 2 ( p + 2)!(n − p + 1)! p=0
Introducing the formal variable ξ i which transforms as a vector we define i¯ (x, ξ ) ≡
∞ n=0
1 Rn (x)ξ l1 · · · ξ ln+3 , (n + 3)! il¯ 1 ···ln+3
(69)
which satisfies the key identity 1 ∂¯[ j¯ i] ¯ = − {[ j¯ , i] ¯ }, 2
(70)
where {, } stands for the Poisson bracket in the ξ -direction with respect to i j (x). Equation (70) is the flatness condition and hence there is an L ∞ structure defined for a hyperKähler manifold. Next we show that ∂¯i¯ = exp−1∗ (∂¯i¯ + {i¯ , ·}) exp∗ .
(71) (1,0)∞
It should be understood that the right ∂¯ acts on the base of TM on the second factor of M × M.
and the left one acts
832
J. Qiu, M. Zabzine ¯
We need to show (∂i¯ + {i¯ , ·}) expξ X i = 0 on the diagonal (where X i = (X i )∗ ), and this is equivalent to showing e−ξ ·∇ (∂i¯ + {i¯ , ·})eξ ·∇ X i = 0; ∇i :=
∂ ∂ − ξ j ikj k . ∂ Xi ∂ξ
By using the formula e−A Be A = e−[A,·] B, and the flatness property Ri j×× = 0, we can show for example e−ξ ·∇ ∂i¯ eξ ·∇ = ∂i¯ +
∞ (−1)n n=1
n!
ξ i1 · · · ξ in ξ in+1 ∇i1 · · · ∇in−1 Rii¯ k i
n n+1
− (n − 1)∇i1 · · · ∇in−2 Rii¯
k n−1 i n
∇k .
Let us agree to write ξ n ∇ n−2 Ri¯k := ξ i1 · · · ξ in ∇i1 · · · ∇in−2 Rii¯ e−ξ ·∇ ξ m+2 ∇ m Ri¯k ∂ξ k eξ ·∇ =
∞ (−1)n n=0
n!
∂ξ k
k n−1 i n
, then
ξ n+m+2 ∇ n+m Ri¯k ∂ξ k − nξ n+m+1 ∇ n+m−1 Ri¯k ∇k .
And combining the two, e
−ξ ·∇
∞ ξ ·∇ −ξ ·∇ ∂¯i¯ + {i¯ , ·} e =e ∂¯i¯ + m=0
1 m+2 m k ξ ∇ Ri¯ ∂ξ k eξ ·∇ (m + 2)!
∞ (−1)n n+1 n−1 k ξ ∇ = ∂¯i¯ + Ri¯ ∂ξ k . (n + 1)! n=1
This shows clearly that (∂i¯ + {i¯ , ·}) expξ X i = 0. References 1. Alexandrov, M., Kontsevich, M., Schwartz, A., Zaboronsky, O.: The Geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A 12, 1405 (1997) 2. Axelrod, S., Singer, I.M.: Chern-Simons perturbation theory. In: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991), River Edge, NJ: World Sci. Publ., 1992, pp. 3–45 3. Cattaneo, A.S., Felder, G.: On the AKSZ formulation of the Poisson sigma model. Lett. Math. Phys. 56, 163 (2001) 4. Cattaneo, A.S., Qiu, J., Zabzine, M.: 2D and 3D topological field theories for generalized complex geometry. http://arxiv.org/abs/0911.0993v1 [hep-th], 2009 5. Costello, K.J.: Renormalisation and the Batalin-Vilkovisky formalism. http://arxiv.org/abs/0706.1533v3 [math.QA], 2007 6. Fuks, D.B.: Stable cohomologies of a Lie algebra of formal vector fields with tensor coefficients. Funkts. Anal. i Priloz. 17(4), 62 (1983) 7. Fuks, D.B.: Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations. J. Math. Sci. 11, 6 (1979) 8. Hamilton, A., Lazarev, A.: Characteristic classes of A∞ algebras. J. Homotopy Relat. Struct. 3(1), 65 (2008) 9. Hamilton, A., Lazarev, A.: Graph cohomology classes in the Batalin-Vilkovisky formalism. J. Geom. Phys. 59, 555 (2009)
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10. Kapranov, M.: Rozansky-Witten invariants via Atiyah classes. Compositio Math. 115(1), 71–113 (1999) 11. Kontsevich, M.: Formal (non)-commutative symplectic geometry. The Gelfand Mathematical Seminars, 1990–1992, Basel: Birkhäuser, 1993, pp. 173–187 12. Kontsevich, M.: Feynman diagrams and low-dimensional topology. In: First European Congress of Mathematics, 1992, Paris, Volume II, Progress in Mathematics 120, Basel: Birkhäuser, 1994, pp. 97–121 13. Kontsevich, M.: Rozansky-Witten invariants via formal geometry. Compositio Math. 115, 115 (1999) 14. Lyakhovich, S.L., Mosman, E.A., Sharapov, A.A.: Characteristic classes of Q-manifolds: classification and applications. http://arxiv.org/abs/0906.0466v2 [math-ph], 2010 15. Morita, S.: Geometry of Characteristic Classes. Iwanami Series in Modern Mathematics, Providence, RI: American Mathematical Society, 2001, xiv+185 pp 16. Qiu, J., Zabzine, M.: On the AKSZ formulation of the Rozansky-Witten theory and beyond. JHEP 0909, 024 (2009) 17. Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids. In: Quantization, Poisson Brackets and Beyond, Theodore Voronov (ed.), Contemp. Math, Vol. 315, Providence, RI: Amer. Math. Soc., 2002 18. Roytenberg, D.: AKSZ-BV formalism and Courant algebroid-induced topological field theories. Lett. Math. Phys. 79, 143 (2007) 19. Rozansky, L., Witten, E.: Hyper-Kähler geometry and invariants of three-manifolds. Selecta Math. 3, 401 (1997) 20. Sawon, J.: Rozansky-Witten Invariants of Hyperkähler Manifolds. PhD thesis, Oxford 1999 21. Schwarz, A.S.: Geometry of Batalin-Vilkovisky quantization. Commun. Math. Phys. 155, 249 (1993) 22. Schwarz, A.S.: Quantum observables, Lie algebra homology and TQFT. Lett. Math. Phys. 49, 115 (1999) 23. Voronov, T.: Graded manifolds and Drinfeld doubles for Lie bialgebroids. In: Quantization, Poisson Brackets and Beyond, Theodore Voronov (ed.), Contemp. Math, Vol. 315, Providence, RI: Amer. Math. Soc., 2002 24. Witten, E.: A note on the antibracket formalism. Mod. Phys. Lett. A 5, 487 (1990) 25. Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989) Communicated by A. Kapustin
Commun. Math. Phys. 300, 835–876 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1127-3
Communications in
Mathematical Physics
Reiterated Ergodic Algebras and Applications Gabriel Nguetseng1 , Mamadou Sango2 , Jean Louis Woukeng2,3 1 Department of Mathematics, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon.
E-mail: [email protected]
2 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa.
E-mail: [email protected]
3 Department of Mathematics and Computer Science, University of Dschang, P.O. Box 67, Dschang,
Cameroon. E-mail: [email protected]; [email protected] Received: 13 February 2010 / Accepted: 17 April 2010 Published online: 6 September 2010 – © Springer-Verlag 2010
Dedicated to the memory of I.M. Gelfand (1913–2009) Abstract: We redefine the homogenization algebras without requiring the separability assumption. We show that this enables one to treat more complicated homogenization problems than those solved by the previous theory. In particular we exhibit an example of algebra which, contrary to the algebra of almost periodic functions, induces no homogenization algebra. We prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized Reynolds type equations and to some nonlinear hyperbolic equations. 1. Introduction A homogenization algebra (H -algebra, in short) is any separable Banach subalgebra of the algebra of bounded continuous complex-valued functions which contains the constants, is closed under complex conjugation and whose elements possess a mean value. This concept has been the starting point of a recent theory developed by the first author [26]: the deterministic homogenization theory. The basic result of this theory is stated as follows. Theorem 1.1. Any bounded sequence (u ε )ε∈E in L p () (1 < p ≤ ∞ and E being an ordinary sequence) admits a subsequence which is weakly R-convergent. The proof of this result is based fundamentally on the separability of the H -algebra A considered. In 2002, in the framework of the theory of algebras with mean value introduced by Zhikov and Krivenko [33], Casado and Gayte [11] proved Theorem 1.1 without resorting to the separability assumption. However, their proof seems not to take into account the sequence (u ε )ε∈E that is complex-valued. Indeed it routinely relies on the following result. Theorem 1.2. Let X be a subspace (not necessarily closed) of a reflexive Banach space Y and let f n : X → R be a sequence of linear functionals (not necessarily continuous).
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G. Nguetseng, M. Sango, J. L. Woukeng
Assume there exists a constant c > 0 such that lim sup f n (x) ≤ c x for all x ∈ X.
(1.1)
n
Then there exist a subsequence ( f n k )k of ( f n ) and a functional f ∈ Y such that limk f n k (x) = f (x) for all x ∈ X . A thorough analysis of the proof of this result allows to realize that there is implicitly a certain positive assumption on the functionals f n , which makes the result inapplicable in its present form, even for a sequence (u ε )ε∈E (as in Theorem 1.1) with real values. But this difficulty is quickly bypassed when replacing in (1.1), f n (x) by | f n (x)|. Here we show that if in Theorem 1.2 we replace “ f n (x)” by “| f n (x)| ” and “ f n : X → R” by “ f n : X → C”, we obtain the conclusion of this theorem without much change to the proof presented by Casado and Gayte. The result we get is of a very wide applicability for both nonlinear problems (whose operators are generally real coefficients) and linear problems (whose operators are generally complex coefficients). More precisely, in this paper we revisit the theory of H -algebras. We show that the separability assumption is unnecessary. This allows one to identify a large class of algebras to solve homogenization problems. We show in particular that the algebra of weakly almost periodic functions induces (apart from the algebra of perturbed almost periodic functions) no H -algebra, as has always been the case for the algebra of almost periodic functions. The paper is organized as follows. In Sect. 2 we state the simplified form of H -algebra (without the separability assumption) which we call the H -supralgebra. We show that all the existing properties of the framework of H -algebras are still valid in our context. In Sect. 3 we collect and prove a few general compactness results. Finally in Sects. 4 and 5, we apply the results of the earlier sections to the resolution of two homogenization problems: the one related to the generalized Reynolds’ type equations and the other associated to a nonlinear hyperbolic equation. All that is done in the framework of reiterated homogenization theory. Unless otherwise specified, vector spaces throughout are assumed to be complex vector spaces, and scalar functions are assumed to take complex values. We shall always assume that the numerical space Rm (integer m ≥ 1) and its open sets are each provided with the Lebesgue measure d x = d x1 . . . d xm , also denoted by λ. For those concepts regarding integration theory, we refer to [6,7].
2. Homogenization Supralgebras 2.1. Homogenization supralgebras. We use a new concept of homogenization algebras. It is more general than the previous one by the first author because we do not need the algebra to be separable. Before we go any further, we need to give some preliminaries. Let H = (Hε )ε>0 be either of the following two actions of R∗+ (the multiplicative group of positive real numbers) on the numerical space Rd (d = N or m), defined as follows: x (x ∈ R N ), ε1 x Hε (x) = (x ∈ Rm ), ε2 Hε (x) =
(2.1) (2.2)
Ergodic Algebras and Applications
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where ε1 and ε2 are two well-separated functions of ε tending to zero with ε, that is, 0 < ε1 , ε2 , ε2 /ε1 → 0 as ε → 0. For given ε > 0 , let u ε (x) = u(Hε (x)) (x ∈ Rd ). For u ∈ L 1loc (Rdy ) (as usual, Rdy denotes the numerical space Rd of variables y = p (y1 , . . . , yd )), u ε lies in L 1loc (Rdx ). More generally, if u lies in L loc (Rd ) (resp. L p (Rd )), ε 1 ≤ p < +∞, then so also does u . A function u ∈ B(Rdy ) (the C *-algebra of bounded continuous complex functions on d R y ) is said to have a mean value for H, if there exists a complex number M(u) such that u ε → M(u) in L ∞ (Rdx )-weak ∗ as ε → 0. The complex number M(u) is called the mean value of u (for H). It is evident that this defines a mapping M which is a positive linear form (on the space of functions u ∈ B(Rdy ) with mean value) attaining the value 1 on the constant function 1 and verifying the inequality |M(u)| ≤ u∞ ≡ sup y∈Rd |u(y)| for all such u’s. The mapping M is called the mean value on Rd for H; see [27] for further details. It is also a fact, as the characteristic function of all relatively compact sets in Rd lies in L 1 (Rd ), that 1 u(y)dy, (2.3) M(u) = lim R→+∞ |B R | B R where B R stands for the bounded open ball in Rd with radius R, and |B R | denotes its Lebesgue measure. Expression (2.3) also works for u ∈ L 1loc (Rd ) provided that the above limit makes sense. Definition 2.1. By a homogenization supralgebra (or H - supralgebra, in short) on Rd for H is meant any closed subalgebra of B(Rd ) which contains the constants, is closed under complex conjugation and whose elements possess a mean value for H. From the above definition we see that the concept of H -supralgebra is more general than those of H -algebra [26] and of algebra with mean value [33]. In fact any separable H -supralgebra is an H -algebra while any algebra with mean value is an H -supralgebra as any uniformly continuous function is continuous. Let A be an H -supralgebra on Rd (for H). Clearly A (with the sup norm) is a commutative C*-algebra with identity (the involution here is the usual one of complex conjugation). We denote by (A) the spectrum of A and by G the Gelfand transformation on A. We endow (A) with the relative weak∗ topology on A . A thorough analysis of the results of [26] together with their proofs reveals that all these results are still valid in our context. In particular we have the following result whose proof can be found in [26]. Theorem 2.1. Let A be an H -supralgebra on Rdy . Then (i) The spectrum (A) is a compact space and the Gelfand transformation G is an isometric ∗-isomorphism identifying A with C((A)) (the complex continuous functions on (A)) as C*-algebras. (ii) The mean value M considered as defined on A is representable by some probability measure β (called the M-measure for A) as follows: M(u) = G(u)dβ for any u ∈ A. (A)
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The notion of a product H -supralgebra will be very useful in this study. To this end we firstly define the product action H∗ of the preceding two actions (2.1) and (2.2) by H∗ = (Hε∗ )ε>0 : x x ((x, x ) ∈ R N × Rm ). Hε∗ (x, x ) = , ε1 ε2 In the sequel, action (2.2) will be denoted by H = (Hε )ε>0 , that is, Hε (x) = x/ε2 (x ∈ Rm ). Now, if A1 (resp. A2 ) is a H -supralgebra on R N (resp. Rm ) for H (resp. H ), ( N and m being two positive integers) then we define the product H -supralgebra of A1 and A2 to be the closure in B(R N ×Rm ) of the tensor product A1 ⊗ A2 = { finite u i ⊗vi : u i ∈ A1 and vi ∈ A2 }. This defines an H -supralgebra on R N × Rm for H∗ , denoted by A1 A2 . To enhance the comprehension of the notion of a product H -supralgebra, let A y and A z be two H -supralgebras on R yN and Rm z respectively, N and m being two positive N +m . We integers. Let A = A y A z be their product, which is an H -supralgebra on R y,z have the following result. N +m ), we define f ∈ B(Rm ) Theorem 2.2. Let A y , A z and A be as above. For f ∈ B(R y,z y z z N and f ∈ B(R y ) by
f y (z) = f z (y) = f (y, z) for (y, z) ∈ R yN × Rm z and put B f = { f z : z ∈ Rm }, C f = { f y : y ∈ R N }. Then B f ⊂ A y and C f ⊂ A z for every f ∈ A. Also for f ∈ A both B f and C f are relatively compact in A y and in A z respectively (in the sup norm topology). Proof. Let f ∈ A. Since A y ⊗ A z is dense in A in the sup norm topology it suffices to check the first part of this theorem for f = u ⊗ v with u ∈ A y and v ∈ A z . But for such an f this is straightforward since we have f y = u(y)v ∈ A z and f z = uv(z) ∈ A y . So f y ∈ A z and f z ∈ A y for every f ∈ A, that is, C f ⊂ A z and B f ⊂ A y . Now assume f ∈ A, and let’s show that C f is relatively compact in A z (note that this is the same as showing C f is relatively compact in B(Rm z )). The proof for B f will be identical to the next one for C f and thus will be omitted. First and foremost let us recall that [26] (A) = (A y ) × (A z ). With this in mind let C f = { f s ∈ C((A z )) : s ∈ (A y )}, where fs = f (s, ·), s ∈ (A y ) with f = G( f ), G being the Gelfand transformation on A, and so that f s (r ) = f (s, r ) for (s, r ) ∈ (A y ) × (A z ). Then C f is compact as the image under the continuous mapping s → f s , of the compact set (A y ). The continuity of this mapping follows from an elementary property of the product topology and the compactness of (A z ). Indeed, let s0 ∈ (A y ), and let η > 0 be freely fixed. Let us exhibit a neighborhood V of s0 such that f s0 ∞ = sup fs − f (s, r ) − f (s0 , r ) < η for all s ∈ V. r ∈(A z )
Ergodic Algebras and Applications
To do this, fix freely r0 open neighborhoods Vs0 f (s, r ) −
839
∈ (A z ); since f is continuous at (s0 , r0 ) there exists some and Wr0 of s0 and r0 respectively such that η whenever (s, r ) ∈ Vs0 × Wr0 . f (s0 , r0 ) < 2
The collection {Wr : r ∈ (A z )} is an open cover of the compact set (A z ), so there n W . Set V = ∩n V , where V is an exist r1 , . . . , rn ∈ (A z ) such that (A z ) ⊂ ∪i=1 ri i i=1 i open neighborhood of s0 such that η whenever (s, r ) ∈ Vi × Wri . f (s, r ) − f (s0 , ri ) < (2.4) 2 Then V is an open neighborhood of s0 . Let s ∈ V , and let r ∈ (A z ); there exists i 0 such that r ∈ Wri0 . Thus, f (s, r ) − f (s0 , ri0 ) + f (s0 , ri0 ) − f (s0 , r ) f (s, r ) − f (s0 , r ) ≤ < η (because of (2.4)), i.e.,
f (s, r ) − f (s0 , r ) < η
for all r ∈ (A z ) and all s ∈ V,
which shows the continuity of this mapping. Now, since C f is compact and C f ⊂ G −1 (C f ) (G being here the Gelfand transformation on A z ) one gets at once the last part of the proposition, that is C f is relatively compact in A z , and hence the result. This concludes the proof. We denote by A P(Rdy ) the space of all Bohr almost periodic complex functions defined as the algebra of functions on Rd that are uniformly approximated by finite linear combinations of functions in the set {γk : k ∈ Rd }, where γk (y) = exp(2iπ k · y) (y ∈ Rd ). It is known that A P(Rdy ) is an H -supralgebra called the almost periodic H -supralgebra on Rd . As a first consequence of the preceding proposition we have the following Proposition 2.3. Let A y = A P(R yN ) and A z = A P(Rm z ) be two almost periodic H -supralgebras. Then A ≡ A y A z = A P(R yN × Rm ). z Proof. The result is a consequence of the following fact: A function f ∈ A P(R yN ×Rm z ) is in A if and only if either C f or B f is relatively compact (in the sup norm topology). One can also easily show that Cper (Y ) Cper (Z ) = Cper (Y × Z ), where Y = (0, 1) N and Z = (0, 1)m . This follows from the identification Cper (Y ) = C(T N ), where T N is the N -torus in R N . Similarly we have Cper (Y ) A P(Rm z ) Cper (T ) = Cper (Y × T ; A P(Rm z )), where T = (0, 1). Other examples of product H -supralgebras can be given. Next, the partial derivative of index i (1 ≤ i ≤ d) on (A) is defined to be the mapping ∂i = G ◦ ∂/∂ yi ◦ G −1 (usual composition) of D1 ((A)) = {ϕ ∈ C((A)) : G −1 (ϕ) ∈ A1 } into C((A)), where A1 = {ψ ∈ C 1 (Rd ) : ψ, ∂ψ/∂ yi ∈ A (1 ≤ i ≤ d)}. Higher order derivatives are defined analogously. At the present time, let A∞ be the space |α| of ψ ∈ C ∞ (Rdy ) such that D αy ψ = α∂1 ψ αd ∈ A for every α = (α1 , . . . , αd ) ∈ Nd , ∂ y1 ···∂ yd
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G. Nguetseng, M. Sango, J. L. Woukeng
and let D((A)) = {ϕ ∈ C((A)) : G −1 (ϕ) ∈ A∞ }. Endowed with a suitable locally convex topology (see [26]), A∞ (resp. D((A))) is a Fréchet space and further, G viewed as defined on A∞ is a topological isomorphism of A∞ onto D((A)). Analogously to the space D (Rd ), we now define the space of distributions on (A) to be the space of all continuous linear forms on D((A)). We denote it by D ((A)) and we endow it with the strong dual topology. It is an easy exercise to see that if A∞ is dense in A (this is the case when, e.g., A is translation invariant and moreover each element of A is uniformly continuous; see [32, Prop. 2.3] for the justification) then L p ((A)) (1 ≤ p ≤ ∞) is a subspace of D ((A)) (with continuous embedding), so that one may define the Sobolev spaces on (A) as follows: W 1, p ((A)) = {u ∈ L p ((A)) : ∂i u ∈ L p ((A)) (1 ≤ i ≤ d)} (1 ≤ p < ∞), where the derivative ∂i u is taken in the distribution sense on (A). We equip W 1, p ((A)) with the norm p
||u||W 1, p ((A)) = ||u|| L p ((A)) +
d
1p p
||∂i u|| L p ((A))
u ∈ W 1, p ((A)) ,
i=1
1 ≤ p < ∞, which makes it a Banach space. To that space are attached some other spaces such as W 1, p ((A))/C = {u ∈ W 1, p ((A)) : (A) udβ = 0} and its separated completion 1, p
W# ((A)); we refer to [26] for a documented presentation of these spaces. 2.2. The generalized Besicovitch spaces. We can define the generalized Besicovitch spaces associated to a H -supralgebra. The notations are those of the preceding subsection. Let A be a H -supralgebra on Rm (integer m ≥ 1). Let 1 ≤ p < ∞. If u ∈ A, then |u| p ∈ A with G(|u| p ) = |G(u)| p . Hence the limit lim R→+∞ |B1R | B R |u(y)| p dy exists and we have 1 p p |u(y)| dy = M(|u| ) = |G(u)| p dβ. lim R→+∞ |B R | B R (A) Hence, for u ∈ A, put 1/ p u p = M(|u| p ) . p
This defines a seminorm on A with which A is not complete. We denote by B A the p completion of A with respect to · p . B A is a Fréchet space, and an argument due to p p Besicovitch [3] states that B A is a complete subspace of L loc (Rmy ). We have the following properties that can be achieved using the theory of the completion; see e.g. [8]. Proposition 2.4. The following hold true: p
(i) A is dense in B A ; (ii) If F is a Banach space then any continuous linear mapping l from A to F extends p by continuity to a unique continuous linear mapping L, of B A into F.
Ergodic Algebras and Applications
841 q
p
Now, let 1 ≤ p ≤ q < ∞. Obviously we have B A ⊂ B A , so that one may naturally define the space B ∞ A as follows: ⎧ ⎫ ⎨ ⎬ p B A : sup f p < ∞ . B∞ A =⎩f ∈ ⎭ 1≤ p<∞ 1≤ p<∞
We endow B ∞ A with the seminorm [ f ]∞ = sup1≤ p<∞ f p , which makes it a Fréchet space. Next, thanks to the preceding proposition, the following properties are worth noting: (1) The Gelfand transformation G : A → C((A)) extends by continuity to a unique p continuous linear mapping, still denoted by G, of B A into L p ((A)). Furthermore p if u ∈ B A ∩ L ∞ (Rmy ), then G(u) ∈ L ∞ ((A)) and G(u) L ∞ ((A)) ≤ u L ∞ (Rmy ) . (2) The mean value M viewed as defined on A, extends by continuity to apositive conp tinuous linear form (still denoted by M) on B A satisfying M(u) = (A) G(u)dβ p p (u ∈ B A ). Furthermore, M(τa u) = M(u) for each u ∈ B A and all a ∈ Rm , where m τa u(y) = u(y − a) for almost all y ∈ R . (3) Let 1 ≤ p, q, r < ∞ be such that 1p + q1 = r1 ≤ 1. The usual multiplication p q A× A → A; (u, v) → uv, extends by continuity to a bilinear form B A × B A → B rA with p
q
uvr ≤ u p vq for (u, v) ∈ B A × B A . The following result will be of great interest in this work. Proposition 2.5. Let A be a H -supralgebra on Rm . Assume each element of A is uniformly continuous and moreover A is translation invariant (i.e. τa u = u(· + a) ∈ A p for all u ∈ A and all a ∈ Rm ). Then A∞ is dense in B A . Proof. Indeed by [32, Prop. 2.3] A∞ is dense in A. The result follows therefore by [part (i) of] Proposition 2.4. p
Now, let u ∈ B A (1 ≤ p < ∞); then |u| p ∈ B 1A (this is easily seen) and so, by part p p (2) above one has M(|u| p ) = (A) |G(u)| p dβ = G(u) L p ((A)) . Thus for u ∈ B A we 1/ p have u p = M(|u| p ) , and u p = 0 if and only if G(u) = 0. Unfortunately, the p mapping G (defined on B A ) is not in general injective. So let N = K er G (the kernel of G) and let p
p
B A = B A /N . Endowed with the norm p
u + N B p = u p (u ∈ B A ), A
p
B A is a Banach space with the following property. p
Theorem 2.6. The mapping G : B A → L p ((A)) induces an isometric isomorphism p G1 of B A onto L p ((A)).
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G. Nguetseng, M. Sango, J. L. Woukeng p
Proof. Since G is an isometry (indeed G(u) L p ((A)) = u p for u ∈ B A ) it suffices to show that G is surjective so that, using the first isomorphism theorem, G will induce p an algebraic isomorphism G1 : B A → L p ((A)) which will be moreover a topological isomorphism. p So, first and foremost, G(B A ) is dense in L p ((A)): this comes on one hand from the density of C((A)) = G(A) in L p ((A)) and on the other hand from the inclup sion C((A)) ⊂ G(B A ) ⊂ L p ((A)). Now, let v ∈ L p ((A)). Then there exists a p sequence (u n )n ⊂ B A such that G(u n ) → v in L p ((A)) as n → ∞. But we have u n − u n p = G(u n ) − G(u n ) L p ((A)) → 0 as n, n → ∞. Thus (u n )n is a Cauchy p p p sequence in B A . As B A is complete there exists a function u ∈ B A such that u n → u in p B A as n → ∞, hence G(u n ) → G(u) in L p ((A)) as n → ∞. Whence v = G(u). This shows that G is surjective. Therefore, by the first isomorphism theorem, the mapping p p G1 : B A = B A /N → L p ((A)) defined by G1 (u + N ) = G(u)
p
for u ∈ B A
is an algebraic isomorphism. But G1 is a topological isometric isomorphism since p
G1 (u + N ) L p ((A)) = u p = u + N B p
for u ∈ B A .
A
This completes the proof.
As a first consequence of the preceding theorem one can also define the mean value p of u + N (for each u ∈ B A ) as follows: 1 M1 (u + N ) = M(u), so that M1 (u + N ) = lim u(y)dy. r →+∞ |Br | B r One crucial result that can be derived from the preceding theorem is the following Corollary 2.7. The following hold true: p
(i) The spaces B A are reflexive for 1 < p < ∞;
p
(ii) The topological dual of the space B A (1 ≤ p < ∞) is the space B A ( p = p/( p − 1)), the duality being given by u + N , v + N p p = M(uv) = G1 (u + N )G1 (v + N )dβ p
B A ,B A
p
for u ∈ B A and v ∈
(A)
p BA.
This result is easily proven by using the properties of L p -spaces and the above isometric isomorphism. p
p
Remark 2.1. The space B A is the separated completion of B A and the canonical mapping p p p p of B A into B A is just the canonical surjection of B A onto B A ; see once more [8] for the theory of completion. Another definition which will be of great interest in the ensuing sections is Definition 2.2. An H -supralgebra A on R N is ergodic if for every u ∈ B 1A such that u − u(· + a)1 = 0 for every a ∈ R N we have u − M(u)1 = 0.
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An equivalent property stated by Casado and Gayte [10] is given in the following proposition. Proposition 2.8. [10] An H -supralgebra A on R N is ergodic if and only if 1 = 0 for all u ∈ B p , 1 ≤ p < ∞. (2.5) u(· + y)dy − M(u) lim A r →+∞ |Br | B r p The following result provides us with a few examples of ergodic H -supralgebras (see the next section for its application). Lemma 2.9. Let A be an H -supralgebra on R N with the following property: 1 lim u(x + y)d x = M(u) uniformly with respect to y. r →+∞ |Br | B r
(2.6)
Then A is ergodic. p
Proof. As A is dense in B A , it suffices to check (2.5) for u ∈ A. Let η > 0 be freely fixed. For such u’s, according to (2.6 ) there exists some r0 > 0 such that 1 u(x + y)d x − M(u) ≤ η for r > r0 |B | r Br and all y ∈ Rm . This leads at once to p 1 u(x + y)d x − M(u) ≤ η p M y |Br | Br The ergodicity of A follows thereby.
for r > r0 .
In order to simplify the presentation of the paper we will from now on, use the same letter u (if there is no danger of confusion) to denote the equivalence class of an element p p p p u ∈ B A . The symbol will denote the canonical mapping of B A onto B A = B A /N . p Our goal here is to define another space attached to B A . Let u ∈ L p ((A)), and let 1 ≤ i ≤ m. We know that ∂i u ∈ D ((A)) exists and is defined by ∂i u, ϕ = − u, ∂i ϕ
for any ϕ ∈ D((A)). p
If we assume further that ∂i u ∈ L p ((A)), then there exists a unique u i ∈ B A such that ∂i u = G1 (u i ). This leads to the following definition. p
Definition 2.3. By a formal derivative of index 1 ≤ i ≤ m, of a function u ∈ B A is p meant the unique element ∂u/∂ yi of B A (if it exists) such that (2.7) G1 ∂u/∂ yi = ∂i G1 (u). 1, p
p
p
Remark 2.2. For u ∈ B A (that is the space of u ∈ B A such that D y u ∈ (B A )m ) we have ∂u ∂u ∂ =G = ∂i G (u) = ∂i G1 ((u)) = (by definition) G1 G1 ((u)) , ∂ yi ∂ yi ∂ yi
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G. Nguetseng, M. Sango, J. L. Woukeng
hence
∂u ∂ yi
=
∂ ((u)), ∂ yi
or equivalently, ◦
∂ ∂ = ◦ ∂ yi ∂ yi
Now, set (for 1 ≤ p < ∞) 1, p BA
= u∈
p BA
1, p
on B A .
(2.8)
∂u p : ∈ B A , for 1 ≤ i ≤ m . ∂ yi
1, p
We endow B A with the norm ⎡
uB1, p A
p ⎤1/ p m
∂u ⎦ p 1, p = ⎣u p + (u ∈ B A ), ∂ yi i=1
p
1, p
which makes it a Banach space with the property that the restriction of G1 to B A 1, p is an isometric isomorphism of B A onto W 1, p ((A)). However we will be mostly 1, p 1, p 1, p concerned with the subspace B A /C of B A consisting of functions u ∈ B A with M1 (u) ≡ M(u) = 0. Equipped with the seminorm ⎡ p ⎤1/ p m
∂u ⎦ 1, p uB1, p /C = D y u p := ⎣ (u ∈ B A /C), ∂ yi A i=1
p
1, p
where D y = (∂/∂ yi )1≤i≤m , B A /C is a locally convex topological space which is in 1, p general nonseparate and noncomplete. We denote by B# A the separated completion of 1, p 1, p B A /C with respect to ·B1, p /C , and by J1 the canonical mapping of B A /C into A
1, p
B# A . By the theory of completion of the uniform spaces [8] it is a fact that the mapping 1, p p ∂/∂ yi : B A /C → B A extends by continuity to a unique continuous linear mapping 1, p p still denoted by ∂/∂ yi : B# A → B A and satisfying ∂ ∂ ◦ J1 = ∂ yi ∂ yi
and
1, p uB1, p = D y u p (u ∈ B# A ),
(2.9)
#A
1, p
where D y = (∂/∂ yi )1≤i≤m . Since G1 is an isometric isomorphism of B A onto 1, p W 1, p ((A)) we have by (2.7) that the restriction of G1 to B A /C sends isometri1, p cally and isomorphically B A /C onto W 1, p ((A))/C. So by [8] there exists a unique 1, p 1, p isometric isomorphism G 1 : B# A → W# ((A)) such that G 1 ◦ J1 = J ◦ G1
(2.10)
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and ∂i ◦ G 1 = G1 ◦
∂ (1 ≤ i ≤ m). ∂ yi
(2.11)
We recall that J is the canonical mapping of W 1, p ((A))/C into its separated comple1, p 1, p 1, p tion W# ((A)) while J1 is the canonical mapping of B A /C into B# A . Furthermore, 1, p 1, p as J1 (B A /C) is dense in B# A (this is classical), it follows that if A∞ is dense in A, 1, p then (J1 ◦ )(A∞ /C) is dense in B# A , where A∞ /C = {u ∈ A∞ : M(u) = 0}. 3. The R-Convergence In this subsection we rewrite the definition of the multiscale convergence. Thanks to p p the identification G1 (B A ) = L p ((A)) we will merely work on the spaces B A . In all N that follows is an open subset of R (integer N ≥ 1) and A = A y A z is an H -supralgebra on R yN × RzN , A y and A z being H -supralgebras on R yN and RzN respectively. We use the same letter G to denote the Gelfand transformation on A y , A z and A, as well. Points in (A y ) (resp. (A z )) are denoted by s (resp. r ). We still denote by M the mean value on R N for H and for H , and on R2N for H∗ as well. The compact space (A y ) (resp. (A z )) is equipped with the M-measure β y (resp. βz ) for A y (resp. A z ). It is fundamental to recall that we have (A) = (A y ) × (A z ) and further the M-measure for A, with which (A) is equipped, is precisely the product measure β = β y ⊗ βz (see [26]). We can now introduce the concept of weak R-convergence which is a generalization of that of multiscale convergence [1] (see also [24] ). Definition 3.1. A sequence (u ε )ε∈E ⊂ L p () (1 ≤ p < ∞) is said to weakly p R-converge in L p () to some u 0 ∈ L p (; B A ) if as E ε → 0, we have x x dx → u ε (x) f x, , u 0 (x, s, r ) f (x, s, r )d xdβ (3.1) ε1 ε2 ×(A) u 0 = G1 ◦ u 0 and f = G1 ◦ ( ◦ f ) = for every f ∈ L p (; A) (1/ p = 1 − 1/ p), where G◦ f .
Notation. We express this by writing u ε → u 0 in L p ()-weak R. The above convergence result (3.1) strongly relies on the following convergence property (see [32, Prop. 2.4] for the proof) : For each ψ in A we have, as ε → 0, ψ ε → M(ψ)
in L ∞ (RxN )-weak ∗,
where ψ ε is defined in an obvious way by ψ ε (x) = ψ(x/ε1 , x/ε2 ) for x ∈ R N , and M is the mean value on R2N = R N × R N for the product action H∗ . Remark 3.1. The letter “R” stands for reiteratively. The above definition is more accurate p than the previous one in [25]; this is due to the equality G1 (BA ) = L p ((A)). Indeed one immediately sees that the right-hand side of ( 3.1) is equal to M(u 0 (x, ·, ·) f (x, ·, ·))d x. In particular when A = Cper (Y ) Cper (Z ) one is led at once to the convergence result x x dx → u ε (x) f x, , u 0 (x, y, z) f (x, y, z)dzdyd x, ε1 ε2 Y Z
846
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where u 0 ∈ L p ( × Y × Z ), which is the original definition of the multiscale convergence [1]. Moreover the uniqueness of the limit u 0 is ensured, and it is also a fact that the weak R-convergence in L p implies the weak convergence in L p . Throughout the paper the letter E will denote any ordinary sequence E = (εn ) (integers n ≥ 0) with 0 < εn ≤ 1 and εn → 0 as n → ∞. Such a sequence will be termed a fundamental sequence. The following result is the starting point of all the compactness results involved in this paper. Theorem 3.1. Any bounded sequence (u ε )ε∈E in L p () (where E is a fundamental sequence and 1 < p < ∞) admits a subsequence which is weakly R-convergent in L p (). Its proof relies on the following result. Proposition 3.2. Let X be a subspace (not necessarily closed) of a reflexive Banach space Y and let f n : X → C be a sequence of linear functionals (not necessarily continuous). Assume there exists a constant c > 0 such that lim sup | f n (x)| ≤ c x for all x ∈ X.
(3.2)
n
Then there exist a subsequence ( f n k )k of ( f n ) and a functional f ∈ Y such that limk f n k (x) = f (x) for all x ∈ X . Proof. We follow closely the outlines of the proof of [11, Theorem 2.1]. As pointed out in [11], since every reflexive Banach space has an equivalent smooth norm (see [21]), we assume without loss of generality that the norm in Y is smooth, that is, for every y ∈ Y with y = 1, there exists a unique f ∈ Y such that f Y = 1 and f, y = 1. With this in mind, we will proceed in two steps. Step 1. First and foremost, let us show that there exist a subsequence ( f n k )k of ( f n )n , a constant α ≥ 0 and a sequence (z j ) j ⊂ X such that z j = 1 ( j ∈ N∗ ), (3.3) (3.4) lim sup f n (x) ≤ α x (x ∈ X ), k
k
1 lim f n k (z j ) ≥ α − ( j ∈ N∗ ), k j lim f n k (z j ) exists. k
(3.5) (3.6)
But arguing exactly as in the first part of the proof of [11, Theorem 2.1] we get the existence of some subsequence ( f n 1 )k of ( f n )n and of α ≥ 0 and (z j ) j ⊂ X such that k (3.3)–(3.5) hold. Now fix j ≥ 1; as the sequence ( f n 1 (z j ))k is bounded, there exists a k subsequence ( f n 1 ( j) (z j ))k of ( f n 1 (z j ))k such that limk f n 1 ( j) (z j ) exists. By a diagonal k
k
k
process we construct a subsequence (n k )k of all (n 1k ( j))k such that (3.3 )–(3.6) hold. Step 2. We show now that there exists f ∈ Y such that ( f n k )k and f satisfy limk f n k (x) = f, x for all x ∈ X . Let us note that if α = 0 then one is immediately led to the result with f = 0. So we can assume that α > 0. By (3.3) and the reflexivity of Y , one gets the existence of a subsequence of (z j ) j (still denoted by (z j ) j ) and of some z 0 ∈ Y such that z j → z0
in Y -weak,
(3.7)
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whence, by (3.3) and the convergence result (3.7) we have z 0 ≤ 1. Now, let x ∈ X be freely fixed. As ( f n k (x))k is bounded, there exists a subsequence ( f n k ( j) (x)) j of ( f n k (x))k such that lim f n k ( j) (x) exists.
(3.8)
j
It is to be noted that the sequence (n k ( j)) j depends upon x. One deduces from (3.6) that for each s in the linear span of the set {x} ∪ {z j : j ∈ N∗ } (denoted below by S), the limit lim j f n k ( j) (s) exists (use also the linearity of f n k ( j) ). Therefore we can define the linear mapping f : S → C by f (s) = lim f n k ( j) (s) j
for s ∈ S.
Then by (3.4) we have, for every s ∈ S, | f (s)| = lim f n k ( j) (s) ≤ lim sup f n k (s) ≤ α s , j
k
from which we deduce that f ∈ S and f S ≤ α. On the other hand, by (3.3) and (3.5) we have f (z j ) ≥ α − 1 = α − 1 z j , (3.9) j j hence f S ≥ α − 1j for each j. Passing to the limit as j → ∞ we get f S ≥ α, and so f S = α. The Hahn-Banach theorem yields the existence of an extension of f to some element f of Y satisfying f Y = α. From (3.4 )-(3.5) and (3.7) we are ! " led to f , z 0 = α, and so we deduce on one hand that α ≤ f Y z 0 = α z 0 , that hand, the existence of a unique θ ∈ (−π, π ] such that !is z 0 " ≥ 1,iθand on the 1other f , z 0 = αe . Set g = α e−iθ f ; then we have g ∈ Y and moreover g, z 0 = 1, gY = 1, z 0 = 1.
(3.10)
Since the norm in Y is smooth, (3.10) uniquely determines g (and so f ). Therefore ! " in (3.8) the convergence result lim j f n k ( j) (x) = f , x does not depend upon the subsequence (n k ( j)) j of (n k )k , but on the whole sequence (n k )k . The result follows thereby. We can now prove Theorem 3.1.
Proof of Theorem 3.1. Let Y = L p ( × (A)), X = L p (; C((A))). Let us define the mapping L ε by Lε( f ) = uε f ε d x ( f ∈ L p (; C((A))) = G(L p (; A))).
Then
f L p (×(A)) lim sup L ε ( f ) ≤ c ε
for all f ∈ L p (; C((A))).
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Indeed one has the inequality |L ε ( f )| ≤ c f ε L p () and thus, as ε → 0, f ε L p () → f L p (×(A)) (see e.g., [26]). We therefore apply Proposition 3.2 with the above notations to get the existence of a subsequence E of E and of a unique v0 ∈ L p ( × (A)) such that uε f ε d x → v0 (x, s, r ) f (x, s, r )d xdβ for all f ∈ L p (; C((A))).
×(A)
p
But v0 = G1 ◦ u 0 , where u 0 ∈ L p (; B A ), and so the result follows. Remark 3.2. Theorem 3.1 was proved in [11] by using Theorem 1.2 and the following result: • Let V be a vector space and M ⊂ V a subspace of V . Then there exists a linear mapping h : V /M → V such that h(v) ∈ v for every v ∈ V /M. Here we rely only on Proposition 3.2 to prove it. Thus our proof is shorter than the one of [11]. The next result can be proven as in [11, Theorem 4.10]. Theorem 3.3. Any uniformly integrable sequence (u ε )ε∈E in L 1 () admits a subsequence which is weakly R-convergent in L 1 (). We recall that a sequence (u ε )ε>0 in L 1 () is said to be uniformly integrable if (u ε )ε>0 is bounded in L 1 () and further supε>0 X |u ε | d x → 0 as |X | → 0 (X being an integrable set in with |X | denoting the Lebesgue measure of X ). Now we assume in the sequel that the H -supralgebras A y and A z are translation invariant and moreover each of their elements is uniformly continuous. The next result requires some preliminaries. Let a = (a1 , a2 ) ∈ R N × R N . Since A is translation invariant, the translation operator τa : A → A extends by continuity to a unique translation p p operator still denoted by τa : B A → B A (1 ≤ p < ∞). Indeed τa is bijective and p τa u p = u p since M(|τa u| ) = M(τa |u| p ) = M(|u| p ) for all u ∈ A. Besides, as each element of A is uniformly continuous, the group of unitary operators {τa }a∈R N ×R N p p thus defined is strongly continuous, i.e. τa u → u in B A as |a| → 0 for all u ∈ B A . Moreover M(τa u) = M(u)
p
for all u ∈ B A and any a ∈ R N × R N .
(3.11)
With this in mind, we begin with the following preliminary result. Lemma 3.4. Assume the H -supralgebras A y and A z are translation invariant and moreover that each element of A y and A z is uniformly continuous. Let (u ε )ε∈E be a uniformly integrable sequence in L 1 () which weakly R-converges towards u 0 ∈ L 1 (; B 1A ). Assume that the sequences (vε )ε∈E and (wε )ε∈E defined below by vε (x) = u ε (x + ε1 ρ)dρ and wε (x) = u ε (x + ε2 ρ)dρ (x ∈ ) BR
BR
are well defined. Then, as E ε → 0, wε → w0
in L 1 ()-weak R,
(3.12)
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where w0 is defined by w0 (x, y, z) = B R u 0 (x, y, z+ρ)dρ for (x, y, z) ∈ ×R N ×R N . Moreover, for each ϕ ∈ D() and each f ∈ A y we have, as E ε → 0, x dx → vε (x)ϕ(x) f v0 (x, s, r ) f (s)ϕ(x)dβd x, (3.13) ε1 (A) where v0 (x, y, z) = B R u 0 (x, y + ρ, z)dρ. Remark 3.3. (1) One particular case where the sequences (vε )ε∈E and (wε )ε∈E make sense is the case when u ε ∈ L p (). Indeed u ε can be viewed as its zero-extension off , so that u ε ∈ L p (R N ). (2) Assume Lemma 3.4 holds. Then as E ε → 0 we have 1 1 u ε (x + y)dy → (3.14) w0 in L 1 ()-weak R. Bε R B |B | R ε2 R 2 The above convergence result will be of particular interest in the next result. Proof of Lemma 3.4. We will just show (3.12) since the way of showing (3.13) will be similar to the one of (3.12). So let ϕ ∈ D(), f ∈ A y and g ∈ A z . One has x x u ε (x + ε2 ρ)dρ ϕ(x) f g dx ε1 ε2 BR x x g d x dρ. u ε (x + ε2 ρ)ϕ(x) f = ε1 ε2 BR In view of the Lebesgue dominated convergence theorem, (3.12) will be checked as soon as we will show that for each fixed ρ ∈ R N one has, as E ε → 0, x x g dx → u ε (x +ε2 ρ)ϕ(x) f τ(0,−ρ) u 0 (x, s, r )ϕ(x) f (s) g (r )dβd x. ε1 ε2 (A) First of all, let us begin by noticing that since G1 is a bounded linear operator of B 1A into L 1 ((A)) we have the following property: u 0 (x, ·, · + ρ)dρ = G1 (u 0 (x, ·, · + ρ))dρ, G1 BR
BR
where u 0 is as above. So let a ∈ R N , ϕ , f and g be as above. One has x x g dx u ε (x − ε2 a)ϕ(x) f ε ε 1 2 x ε2 x = u ε (x)ϕ(x + ε2 a) f + a g + a dx ε1 ε1 ε2 −ε2 a x ε2 x = u ε (x)ϕ(x + ε2 a) f + a g + a dx ε ε ε 1 1 2 x ε2 x u ε (x)ϕ(x + ε2 a) f + a g + a dx − ε1 ε1 ε2 \(−ε2 a) x ε2 x + u ε (x)ϕ(x + ε2 a) f + a g + a dx ε1 ε1 ε2 (−ε2 a)\ = (I ) − (I I ) + (I I I ).
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As for (I ) we have x ε2 x dx u ε (x)ϕ(x) f + a (τ−a g) (I ) = ε1 ε1 ε2 x ε2 x dx + u ε (x)[ϕ(x + ε2 a) − ϕ(x)] f + a (τ−a g) ε1 ε1 ε2 = (I1 ) + (I2 ). But x (τ−a g) dx ε 2 # $ x ε2 x x f dx + u ε (x)ϕ(x)(τ−a g) + a − f ε ε ε ε 2 1 1 1 = (I1 ) + (I2 ).
(I1 ) =
u ε (x)ϕ(x) f
x ε1
The H -supralgebra A z being translation invariant, we have τ−a g ∈ A z and so, (I1 ) → u 0 (x, s, r )ϕ(x) f (s) τ−a g(r )dβd x as E ε → 0. (A)
But (A)
u 0 (x, s, r )ϕ(x) f (s) τ−a g(r )dβ = M(u 0 (x, ·, ·) f (τ−a g)) = M(τ(0,−a) [τ(0,a) u 0 (x, ·, ·) f g]) = M(τ(0,a) u 0 (x, ·, ·) f g) (see (3.11)) τ f (s) g (r )dβ. = (0,a) u 0 (x, s, r ) (A)
Note that here we have identified u 0 (x, ·, ·) ∈ B 1A with its representant still denoted by u 0 (x, ·, ·) ∈ B 1A so that M1 (u 0 (x, ·, ·)) = M(u 0 (x, ·, ·)) , u 0 (x, ·, ·) on the left-hand side of the above equality is an equivalence class whereas u 0 (x, ·, ·) on the right-hand side is its representant. We claim that (I2 ) → 0 as E ε → 0. In fact let c > ϕ∞ g∞ |u ε (x)| d x be a positive constant independent of ε. Let η > 0 be freely fixed. As f is uniformly continuous, there exists some α > 0 such that for all y, y ∈ R N we have f (y) − f (y ) < η/c whenever y − y < α. Let γ0 > 0 be such that (ε2 /ε1 ) |a| < α for all E ε ≤ γ0 . Then f x + ε2 a − f x < η/c for all x ∈ R N , E ε ≤ γ0 , ε1 ε1 ε1 hence (I2 ) < η for all E ε ≤ γ0 , and our claim is justified. We have just shown that (I1 ) → 0 as E ε → 0. Besides we have |(I2 )| ≤ ε2 |a| (sup |Dϕ(x)|) f ∞ g∞ |u ε (x)| d x x∈
Ergodic Algebras and Applications
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and so (I2 ) → 0 as E ε → 0. On the other hand, the uniform integrability of (u ε )ε∈E and the inequality x x ε2 |u ε (x)| |ϕ(x + ε2 a)| f + a g + a d x ε1 ε1 ε2 (−ε2 a) |u ε (x)| d x ≤ ϕ∞ f ∞ g∞ (−ε2 a)
(where BC denotes the symmetric difference between the two sets B and C) ensure that (I I ) and (I I I ) go towards 0 as E ε → 0. The lemma follows thereby. We are now able to state and prove the next compactness result. It will be of capital interest in the next section. Theorem 3.5. Let 1 < p < ∞. Let be an open subset in R N . Let A = A y A z be an H -supralgebra where A y (resp. A z ) is an ergodic H -supralgebra on R yN (resp. RzN ). Assume that A y and A z are translation invariant, and moreover that their elements 1, p are uniformly continuous. Finally, let (u ε )ε∈E be a bounded sequence in W0 (). 1, p There exist a subsequence E from E and a triple u = (u 0 , u 1 , u 2 ) ∈ W0 () × 1, p p 1, p L p (; B# A y ) × L p (; B A y (R yN ; B# Az )) such that, as E ε → 0 , 1, p
uε → u0
in W0 ()-weak
(3.15)
and ∂u ε ∂u 0 ∂u 1 ∂u 2 → + + ∂x j ∂ x j ∂ y j ∂z j
in L p ()-weak R (1 ≤ j ≤ N ). 1, p
Proof. By the reflexivity of the space W0 () and also using Proposition 3.2, there 1, p exist a subsequence E from E and, a function u 0 ∈ W0 () and a vector funcp tion w = (wi )1≤i≤N ∈ L p (; (B A ) N ) such that, as E ε → 0, we have (3.15) p ε and ∂u ∂ xi → wi in L ()-weak R (1 ≤ i ≤ N ). It remains to check that there 1, p
p
1, p
exist two functions u 1 ∈ L p (; B# A y ) and u 2 ∈ L p (; B A y (R yN ; B# Az )) such that ∂u 1 ∂u 2 0 wi = ∂u ∂ xi + ∂ yi + ∂z i (1 ≤ i ≤ N ). For that purpose, let R > 0 be freely fixed. Let Bε2 R denote the open ball in R N with radius ε2 R centered at the origin. By the equalities 1 1 1 1 u ε (x + ρ)dρ = u ε (x) − (u ε (x) − u ε (x + ρ)) dρ Bε R B ε2 ε2 Bε2 R Bε2 R ε2 R 2 1 1 = (u ε (x) − u ε (x + ε2 ρ)) dρ ε2 |B R | B R 1 1 dρ Du ε (x + tε2 ρ) · ρdt, =− |B R | B R 0
(the dot denoting the usual Euclidean inner product in R N ) we deduce from the bound1, p edness of (u ε )ε∈E in W0 () that the sequence (z εR )ε∈E defined by 1 1 R z ε (x) = u ε (x + ρ)dρ (x ∈ , ε ∈ E ) u ε (x) − Bε R B ε2 2
ε2 R
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is bounded in L p (). It is important to note that the function z εR is well defined since u ε % ε , v denoting and Du ε can be naturally extended off by 0 and hence verify D u ε = Du the zero-extension off of any function v ∈ L p (). Arguing as above one also shows that the sequence (vεR )ε∈E defined by 1 1 R vε (x) = u ε (x + ρ)dρ (x ∈ ) u ε (x) − Bε R B ε1 ε1 R 1 is well defined and bounded in L p (). Hence because of Theorem 3.1 there exist a p subsequence from E not relabeled and two functions V R and Z R in L p (; B A ) such that, as E ε → 0, vεR → V R
in L p ()-weak R
(3.16)
z εR → Z R
in L p ()-weak R.
(3.17)
and
As (z εR )ε∈E is bounded in L p () we have that εk z εR → 0
in L p ()
as E ε → 0
for k = 1, 2.
∞ Now, for ϕ ∈ D(), f ∈ A∞ y and g ∈ A z we have ∂u ε ∂u ε 1 x x (x) − (x + ρ)dρ ϕ(x) f ( )g( )d x ε1 ε2 Bε2 R Bε2 R ∂ xi ∂ xi x x ∂ϕ =− ε2 z εR (x) f ( )g( ) (x)d x ε ε 1 2 ∂ xi ε2 R x ∂f x z ε (x)ϕ(x)g( ) ( )d x − ε2 ∂ yi ε1 ε1 x ∂g x − z εR (x)ϕ(x) f ( ) ( )d x. ε1 ∂z i ε2
(3.18)
(3.19)
Passing to the limit in (3.19) (as E ε → 0) using conjointly (3.18), (3.17) and Remark 3.3 (see (3.14) therein) one gets 1 G1 wi (x, ·, ·) − wi (x, ·, · + ρ)dρ (s, r )ϕ(x) f (s) g (r )d xdβ |B R | B R ×(A) =− z R (x, s, r )ϕ(x) f (s)∂i g (r )d xdβ, ×(A)
the derivative ∂i in front of g being the partial derivative of index i with respect to (A z ). Therefore, because of the arbitrariness of ϕ, f and g we arrive at 1 ∂i wi (x, ·, · + ρ)dρ a.e. in x ∈ . Z R (x, ·, ·) = G1 wi (x, ·, ·) − |B R | B R But ∂i Z R (x, ·, ·) = ∂i G1 (Z R (x, ·, ·)) = G1 ∂ Z R /∂z i (x, ·, ·) , hence, for 1 ≤ i ≤ N , ∂ZR 1 (x, y, ·) = wi (x, y, ·) − wi (x, y, · + ρ)dρ a.e. in (x, y) ∈ × R N |B R | B R ∂z i
Ergodic Algebras and Applications
853 p
(recall that G1 is an isomorphism of B 1A onto L 1 ((A)) which carries over B A onto L p ((A)) isomorphically and isometrically). Set f R = Z R − Mz (Z R ), where here: p p p p Z R (x, ·, ·) ∈ B A is viewed as its representant in B A = B A y (R yN ; B Az ), and Mz is the mean value operator on RzN with respect to z. Then Mz ( f R ) = 0 and moreover D z f R = p p p p D z Z R so that f R ∈ L p (; B A y (R yN ; B Az )) with ∂ f R /∂z i ∈ L p (; B A y (R yN ; B Az )), that is, p
1, p
f R ∈ L p (; B A y (R yN ; B Az /C)). 1, p
So let g R = J1z ◦ f R , where J1z denotes the canonical mapping of B Az /C into its 1, p
p
1, p
separated completion B# Az . Then g R ∈ L p (; B A y (R yN ; B# Az )) and moreover ∂ gR 1 (x, ·, ·) = wi (x, ·, ·) − |B R | ∂z i
wi (x, ·, · + ρ)dρ (1 ≤ i ≤ N ), BR
since ∂∂zg Ri (x, ·, ·) = ∂∂zf Ri (x, ·, ·) = ∂∂zZ iR (x, ·, ·). Now, we also view wi (x, ·, ·) as its p representant in B A . With this, we have g R (x, y, ·) − g R (x, y, ·)B1, p ≤ D z g R (x, y, ·) − w(x, y, ·) + Mz (w(x, y, ·)) p # Az + D z g R (x, y, ·)−w(x, y, ·)+ Mz (w(x, y, ·)) . p
(3.20) But
D z g R (x, y, ·) − w(x, y, ·) + Mz (w(x, y, ·)) p 1 = w(x, y, · + ρ)dρ − Mz (w(x, y, ·)) . |B R | B R p
Therefore, since the algebra A z is ergodic, the right-hand side (and hence the left-hand side) of (3.20) goes to zero when R, R → +∞. Thus the sequence (g R (x, y, ·)) R>0 1, p is a Cauchy sequence in the Banach space B# Az , whence the existence of a unique 1, p
u 2 (x, y, ·) ∈ B# Az such that 1, p
g R (x, y, ·) → u 2 (x, y, ·)
in B# Az
as R → +∞,
that is D z g R (x, y, ·) → D z u 2 (x, y, ·)
p
in (B Az ) N
as R → +∞.
Once again the ergodicity of A z and the uniqueness of the limit leads at once to D z u 2 (x, y, ·) = w(x, y, ·) − Mz (w(x, y, ·)) p
a.e. (x, y) ∈ × R N , 1, p
hence the existence of u 2 : → B A y (R yN , B# Az ), x → u 2 (x, ·, ·), lying in p
1, p
L p (; B A y (R yN ; B# Az )) such that D z u 2 = w − Mz (w).
(3.21)
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G. Nguetseng, M. Sango, J. L. Woukeng
Next, returning to (3.19) and taking (in the left-hand side of the equality) there g ≡ 1 and replacing Bε2 R by Bε1 R , and finally proceeding as we have done to get u 2 (using this 1, p time the convergence result (3.13)), we get the existence of an unique u 1 ∈ L p (; B# A y ) such that D y u 1 (x, ·) = Mz (w(x, ·, ·) − M y (w(x, ·, ·))) = Mz (w(x, ·, ·)) − Mz (M y (w(x, ·, ·))).
(3.22)
Thus w(x, ·, ·) = Mz (w(x, ·, ·)) + D z u 2 (x, ·, ·) by (3.21) = Mz (M y (w(x, ·, ·))) + D y u 1 (x, ·) + D z u 2 (x, ·, ·)
by (3.22).
Finally by the convergence result Du ε → w in L p () N -weak R as E ε → 0, it easily follows that Du ε → Mz (M y (w)) in L p () N -weak as E ε → 0, so that by (3.15) we get Mz (M y (w)) = Du 0 . Whence w = Du 0 + D y u 1 + D z u 2 and the proof is complete. The preceding result easily generalizes to a finite number of separated scales as follows. Let (A yi )1≤i≤n be n ergodic H -supralgebras that are translation invariant and whose elements are uniformly continuous. For 1 ≤ i ≤ n, set RiyN1 ,...,yi = R yN1 × . . . × R yNi
ij=1 A y j = A y1 . . . A yi .
and
For f ∈ L p (; ij=1 A y j ) one defines the trace function f ε (ε > 0) by x x ε (x ∈ ), f (x) = f x, , . . . , ε1 εi where the scales εi satisfy the following relation: εi+1 /εi → 0 as ε → 0 for 1 ≤ i ≤ n − 1. We have the following result (in the above notations). 1, p
Theorem 3.6. Let 1 < p < ∞. Let (u ε )ε∈E be a bounded sequence in W0 () ( an open subset in R N ). There exist a subsequence E from E and a (n + 1)-tuple u = &n−1 p 1, p 1, p p L (; B A y ... A y (RiyN1 ,...,yi ; (u 0 , u 1 , . . . , u n ) ∈ W0 () × L p (; B# A y ) × i=1 1, p
B# A y
i+1
)) such that, as E ε → 0, uε → u0
1
1
i
1, p
in W0 ()-weak,
Du ε → Du 0 + D y1 u 1 + · · · + D yn u n
in L p () N -weak R.
Theorem 3.6 can be proven inductively. In any case its proof is copied on that of Theorem 3.5. We begin by finding u 0 , next u n , and inductively, u n−1 ,…, u 1 . This is left to the reader. The R-convergence method can also apply to time dependent functions. To see this, let Aτ be a further H -supralgebra on Rτ . Set A = A y A z Aτ , where A y and A z are as at the beginning of this section. Points in (Aτ ) are denoted by s0 . We know that (A) = (A y ) × (A z ) × (Aτ ) and further if βτ (resp. β) is the M-measure for Aτ (resp. A) then β = β y ⊗ βz ⊗ βτ , β y and βz as at the beginning of the current section. Now, let 0 < T < ∞ and let be an open subset in R N . Set Q = × (0, T ). We have the following time-dependent version of the R-convergence.
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Definition 3.2. A sequence (u ε )ε∈E ⊂ L p (Q) (1 ≤ p < ∞) is said to weakly p R -converge in L p (Q) to some u 0 ∈ L p (Q; B A ) if as E ε → 0, we have x x t u ε (x, t) f x, t, , , u 0 (x, t, s, r, s0 ) f (x, t, s, r, s0 )dxdtdβ d xdt → ε 1 ε2 ε3 Q Q×(A)
for every f ∈ L p (Q; A) (1/ p = 1 − 1/ p ). In Definition 3.2 we assume ε1 and ε2 to be well-separated (i.e., ε2 /ε1 → 0 as ε → 0) and ε3 to be any sequence of positive real numbers such that ε3 → 0 when ε → 0. We recall here that, as before, u 0 = G1 ◦ u 0 and f = G ◦ f , G1 is the isometric isomorphism p p sending B A onto L ((A)) and G, the Gelfand transformation on A. The homologue of Theorems 3.5 and 3.6 is valid mutatis mutandis in this context. In order to state this, we need further preliminaries. For 1 < p < ∞, set
V p = {v ∈ L p (0, T ; W0 ()) : v = ∂v/∂t ∈ L p (0, T ; W −1, p ())}, a Banach space with the norm vV p = v L p (0,T ;W 1, p ()) + v L p (0,T ;W −1, p ()) 0 (v ∈ V p ) with the further property that V p (for p ≥ 2) is continuously embedded in the space C([0, T ]; L 2 ()) and compactly embedded in the space L 2 (Q). With all that in mind, we have the following important result. 1, p
Theorem 3.7. Let 1 < p < ∞. Let be an open subset in R N . Let A = A y A z Aτ , where A y (resp. A z ) is an ergodic H -supralgebra on R yN (resp. RzN ). Assume that A y and A z are translation invariant and moreover that their elements are uniformly continuous. Finally, let (u ε )ε∈E be a bounded sequence in V p . There exist a subsep 1, p quence E from E and a triple u = (u 0 , u 1 , u 2 ) ∈ V p × L p (Q; B Aτ (Rτ ; B# A y )) × N +1 ; B L p (Q; B A y Aτ (R y,τ # A z )) such that, as E ε → 0, p
1, p
uε → u0
in V p -weak
and ∂u ε ∂u 0 ∂u 1 ∂u 2 → + + ∂x j ∂ x j ∂ y j ∂z j
in L p (Q)-weak R (1 ≤ j ≤ N ).
Proof. The proof of this theorem follows the same outlines as the proof of Theorem 3.5, except that instead of the sequences (z εR ) and (vεR ) we take here the following one: 1 1 R u ε (x + ρ, t)dρ ((x, t) ∈ Q) u ε (x, t) − z 1,ε (x, t) = Bε R B ε2 ε2 R 2 and 1 R v1,ε (x, t) = ε1
1
u ε (x, t) − Bε R 1
Bε1 R
u ε (x + ρ, t)dρ
((x, t) ∈ Q),
which also make sense, the rest remaining unchanged. Whence the result.
p
Remark 3.4. Theorem 3.7 is still valid when replacing V by
W p = {v ∈ L p (0, T ; W0 ()) : v = ∂ 2 v/∂t 2 ∈ L p (0, T ; W −1, p ())}. 1, p
We give below a few examples of H -supralgebras which satisfy hypotheses of Theorems 3.5, 3.6 and 3.7.
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3.1. Example 3.1: Periodic setting. Let A y = Cper (Y ) (Y = (0, 1) N ) be the algebra of Y -periodic continuous functions on R yN . It is classically known that A y is an ergodic H -supralgebra so that Theorems 3.5 , 3.6 and 3.7 apply with A = A y A z (A y = A z ), A yi = A y for 1 ≤ i ≤ n, and A = A y A z Aτ = Cper (T ) (Aτ = Cper (T ), T = (0, 1)), respectively. Also for any H -supralgebra Aτ on Rτ , Theorem 3.7 holds for A = Cper (Y ) Cper (Y ) Aτ . 3.2. Example 3.2: Almost periodic setting. Let A P(R N ) be the algebra of Bohr continuous almost periodic functions R N . We recall that a function u ∈ B(R N ) is in A P(R N ) if the set of translates {τa u : a ∈ R N } is relatively compact in B(R N ). An argument due to Bohr [5] specifies that u ∈ A P(R N ) if and only if u may be uniformly approximated by finite linear combinations of functions in the set {γk : k ∈ R N }, where γk (y) = exp(2iπ k · y) (y ∈ R N ). It is also a classical result that A y is an ergodic H -supralgebra; see e.g. [16,33]. Therefore Theorems 3.5, 3.6 and 3.7 apply with A y = A P(R yN ), A z = A P(RzN ), A yi = A P(R yNi ) and Aτ any H -supralgebra on Rτ . Now, let R be a countable subgroup of R N . We denote by A PR (R yN ) the space of those functions in A P(R yN ) that can be uniformly approximated by finite linear combinations in the set {γk : k ∈ R}. Then A y = A PR (R yN ) is an H -algebra [26] . We have the following Proposition 3.8. The H -algebra A y = A PR (R yN ) is an ergodic H -algebra. Proof. As A PR (R yN ) ⊂ A P(R yN ), the ergodicity of A y is clear. Thus our only concern here is to show that A y is translation invariant. To this end, we know by [20, Theorem 4.1.4] that (A y ) is a compact Abelian topological group additively written. With this in mind, let the mapping j : R yN → (A y ) be given by j (y) = δ y
(the Dirac mass at y ∈ R N ).
Then thanks to [29] j is a group homomorphism with the following properties: (a) (b)
j is continuous; j (R yN ) is dense in (A y ).
So let a ∈ R N , and let u ∈ A y . It is a fact that the function τ j (a) G(u), defined by τ j (a) G(u)(s) = G(u)(s − j (a)) (s ∈ (A y )), lies in C((A y )). On the other hand we have, for each y ∈ R N , ! " τ j (a) G(u)( j (y)) = G(u)( j (y) − j (a)) = G(u)( j (y − a)) = δ y−a , u ! " = u(y − a) = (τa u)(y) = δ y , τa u = G(τa u)( j (y)). By the density of j (R yN ) in (A y ) and by the continuity of G it follows that τ j (a) G(u) = G(τa u). It turns out that τa u ∈ A y because by (3.23), G(τa u) ∈ C((A y )) = G(A y ).
(3.23)
The above result allows us to see that the conclusions of all three preceding theorems still hold with A PR (R N ) in place of A P(R N ).
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3.3. Example 3.3: The convergence at infinity setting. Let B∞ (R N ) denote the space of all continuous functions on R N that converge at infinity, that is of all u ∈ B(R N ) such that lim|y|→∞ u(y) ∈ C. The space B∞ (R N ) is an H -algebra on R N [26]. It is an easy exercise to see that B∞ (R N ) is an ergodic H -supralgebra. Moreover by [16] any u ∈ B∞ (R N ) is uniformly continuous and in addition, the mean value of a function u is given by M(u) = lim|y|→∞ u(y). It is also a fact that B∞ (R N ) is translation invariant. Therefore we have the conclusions of Theorems 3.5, 3.6 and 3.7 with A y = B∞ (R N ) = A z = A yi (1 ≤ i ≤ n) and for any Aτ as in the preceding subsection. 3.4. Example 3.4: The weakly almost periodic setting. We begin with the notion of weakly almost periodicity due to Eberlein [16] (see also [13,14]). Definition 3.3. A continuous function u on R N is weakly almost periodic if the set of translates {τa u : a ∈ R N } is relatively weakly compact in B(R yN ). We denote by W A P(R yN ) the set of all weakly almost periodic functions on R yN ; W A P(R yN ) is a vector space over C. Endowed with the norm sup topology, W A P(R yN ) is a Banach algebra with the usual multiplication. The space W A P(R yN ) is sometimes called the space of Eberlein’s functions. As examples of Eberlein’s functions we have the continuous Bohr almost periodic functions, the continuous functions vanishing at infinity, the positive definite functions (hence Fourier-Stieltjes transforms); see [16] for more details. The following properties are worth mentioning (see [16] for details): (P1) W A P(R yN ) is a translation invariant C ∗ -subalgebra of B(R yN ). (P2) A weakly almost periodic function is uniformly continuous and bounded. (P3) A weakly almost periodic function possesses a mean value with 1 M(u) = lim u(y + a)dy, r →+∞ |Br | B r the convergence being uniform in a ∈ R N . (P4) If u, v ∈ W A P(R yN ) the convolution defined by the mean value w(y) = (u ∗v) (y) = Mz (u(y − z)v(z)) = Mz (u(z)v(y − z)) is an usual Bohr almost periodic function, where Mz stands for the mean value with respect to z. (P5) ([17, Theorem 1]) Every u ∈ W A P(R yN ) admits the unique decomposition u = v + w, v being a Bohr almost periodic function and w a continuous function of quadratic mean value zero: M(|w|2 ) = 0. Property (P5) above, known as a decomposition theorem, is crucial in the definition of the weak almost periodic algebra. Indeed let W0 (R yN ) denote the subset of W A P(R yN ) consisting of elements with quadratic mean value zero. One easily observes that the set of bounded continuous functions on R N of quadratic mean value zero is a complete vector subspace of the algebra of bounded continuous functions on R N . Hence W0 (R yN ) is a complete vector subspace of W A P(R yN ). Property (P5) is stated as follows: W A P(R yN ) is a direct sum of the two spaces A P(R yN ) and W0 (R yN ): W A P(R yN ) = A P(R yN ) ⊕ W0 (R yN ).
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G. Nguetseng, M. Sango, J. L. Woukeng
Another representation of the space W0 (R yN ) is given by de Leeuw and Glicksberg [15]: W0 (R yN ) = {u ∈ W A P(R yN ) : M(|u|) = 0}. With this in mind, let R be a subgroup of R N (not necessarily countable). Set A = A PR (R yN ) ⊕ W0 (R yN ),
(3.24)
where A PR (R yN ) is defined in the preceding subsection. Note that in the case when R = R N then we have A = W A P(R yN ). The following holds true. Proposition 3.9. The vector space A =A PR (R yN)⊕W0 (R yN ) is an ergodic H -supralgebra which is translation invariant and whose elements are uniformly continuous. Definition 3.4. The algebra A defined by (3.24) is called the weakly almost periodic H -supralgebra represented by the subgroup R. A is denoted by W A PR (R yN ) when R = R N . Proof of Proposition 3.9. (1) The ergodicity of A is a consequence of Lemma 2.9 and of property (P3). (2) Thanks to the properties (P1)-(P4) our only concern here is to check that A is a Banach subalgebra of W A P(R yN ). To this end, because of the decomposition theorem any u ∈ W A P(R yN ) uniquely expresses u = f + g with f ∈ A P(R yN ) and g ∈ W0 (R yN ). Set ψ(u) = f. This defines a mapping ψ : W A P(R yN ) → A P(R yN ). (a) Claim: ψ is continuous. ' In fact let F = W A P(R N ), P = A P(R N ) and G = W0 (R N ). Then F = P G, P and G are closed in F, hence P × G is a Banach space and further, the mapping i : P × G → F defined by i(x, y) = x + y is a one-to-one continuous mapping, hence, in view of the open mapping theorem, i is an isomorphism (we endow P × G with the Hilbertian norm (x, ' y) = (x2∞ + y2∞ )1/2 ). But ψ = p1 ◦ i, where p1 is the natural projection of P G onto P (which is naturally continuous), hence ψ N N is continuous, as claimed. (b) Now as A PR (R ' ) is a Nclosed subspace of A P(R ),N it −1 N N turns out that ψ (A PR (R )) = A PR (R ) W0 (R ) = A is closed in W A P(R ). Moreover A is a Banach algebra. Indeed this will be accomplished if we can check that ψ( f g) = ψ( f )ψ(g) for all f, g ∈ W A P(R N ). So let f = u + v and g = u 1 + v1 with u, u 1 ∈ A P(R N ) and v, v1 ∈ W0 (R N ); then f g = uu 1 + u 1 v + uv1 + vv1 with uu 1 ∈ A P(R N ). But u 1 v+uv1 +vv1 ∈ W0 (R N ), since |u 1 v| ≤ c |v| (c = u∞ ), whence M(|u 1 v|) ≤ cM(|v|) = 0, i.e., M(|v|) = 0 as M is a positive linear form on W A P(R N ); M(|uv1 |) = 0 and M(|vv1 |) = 0 by the same argument. It turns out that ψ( f g) = uu 1 = ψ( f )ψ(g), so that ψ is a homomorphism of algebras. This being so, let f, g ∈ A; then ψ( f ) and ψ(g) lie in the algebra A PR (R N ), hence ψ( f g) = ψ( f )ψ(g) ∈ A PR (R N ), that is f g ∈ A. Therefore A is a closed subalgebra of W A P(R N ), that is A is an ergodic algebra on R N . We have the same conclusion as in the preceding examples with A y = W A PR (R N ) = A z = A yi . Also, since any H -supralgebra of all the preceding examples is a subalgebra of W A P(R N ), the conclusions of Theorems 3.5, 3.6 and 3.7 follow if we take instead
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of W A PR (R N ), any of the algebras of the above examples. In particular, Theorem 3.5 holds with A = W A PR (R yN ) A P(RzN ). The following result will allow us to see that weakly almost periodic H -supralgebras are not H -algebras. Theorem 3.10. The algebra W0 (Rm ) is nonseparable. For the proof of this theorem we need the following lemma. Lemma 3.11. Let G and H be two Banach spaces, and let ϕ be a surjective continuous linear mapping of G onto H . If G is separable then so also is H . Proof. Let (an )n≥1 be a countable dense set in G, and let c be a positive constant such that ϕ(g) H ≤ c gG for all g ∈ G, where ·G and · H denote respectively the norms in G and in H . Set B = (ϕ(an ))n≥1 ⊂ H , and let us show that B is dense in H . For that, let h ∈ H and let ε > 0 befreely fixed; There h = ϕ(g). let finally g ∈G be such that exists some n 0 ≥ 1 such that g − an 0 G < ε/c, hence h − ϕ(an 0 ) H ≤ c g − an 0 G < ε. Thus B is a countable dense subset of H , and the proof is completed. Proof of Theorem 3.10. Set G = W0 (Rm ) , H = W0 (Rm )/C0 (Rm ) (C0 (Rm ) denotes the algebra of all continuous functions on Rm that vanish at infinity) and ϕ the natural (canonical) homomorphism of G onto H . Assume G is separable (with the sup norm topology), then according to the above lemma, H is separable. But we know by [14, Theorem 4.6] that the quotient space H contains a linear isometric copy of ∞ and hence is nonseparable. This contradicts our assumption, and hence G is nonseparable. Now let R be a subgroup of Rm (countable or not). Since W0 (Rm ) ⊂ W A P(Rm ) we conclude by Theorem 3.10 that the ergodic H -supralgebras W A PR (Rm ) are not H -algebras because they are not separable with respect to the sup norm topology. So we have in hand an example of an H -supralgebra which contrary to the algebra A P(Rm ), induces no H -algebra, say W A P(Rm ). This is therefore sufficient to show that the concept of H -algebras cannot handle weakly almost periodic homogenization problems. This is a significant contribution to the theory. Now, let C0 (R yN ) denote the space of all continuous functions on R yN that vanish at infinity. It is well known that A y = A P(R yN ) + C0 (R yN ) is a proper subalgebra of the algebra of weakly almost periodic functions on R yN . A y is an H -supralgebra called the algebra of perturbed almost periodic functions. It generates the H -algebras as follows. Considering some countable subgroup R of R yN we denote by A PR (R yN ) the space of those functions in A P(R yN ) that can be uniformly approximated by finite linear combinations in the set {γk : k ∈ R}; it is known that A PR (R yN ) is an H -algebra. If we set A y,R = A PR (R yN ) + C0 (R yN ), then, as C0 (R yN ) is a separable subalgebra of B(R yN ), we define by this a further H -algebra [26]. So A y generates H -algebras. Besides, as seen by Theorem 3.10, even though R is a countable subgroup of R yN , A PR (R yN ) + W0 (R yN ) is never separable, so that A y above is probably the bigger subalgebra of W A P(R yN ) that generates H -algebras. Before passing to the applications, we need to clarify our contribution so far in this section. Theorems 3.5, 3.6 and 3.7 are new in a general framework. In the periodic setting, that is when A y = Cper (Y ), A z = Cper (Z ) and Aτ = Cper (T ) (Y = Z = (0, 1) N and T = (0, 1)), Theorem 3.5, although very often used, but rarely justified, has been rigorously proved for the first time in [25] . Beyond the periodic setting, and in some
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G. Nguetseng, M. Sango, J. L. Woukeng
particular cases related to the theory of H -algebras, the same result was shown in [25] for p = 2. As discussed above, Theorem 3.5 generalizes all the compactness results contained in [25]. It is to be noted that till now, the work [25] is the sole published work where such an undertaking is done. Regarding Theorems 3.6 and 3.7, it is the first time that such results are rigorously proved, even in the periodic environments. It is important to note that the proofs of all the above theorems rely systematically on the ergodicity key property of the H -supralgebra under consideration. 4. Application to the Reiterated Homogenization of Nonlinear Degenerated Reynolds-Type Equations In this section we study the reiterated homogenization problem for nonlinear degenerated Reynolds-type equations. More precisely let 1 < p < ∞ be fixed and let the functions (x, y, z, μ, λ) → a(x, y, z, μ, λ) and (x, y, z, μ, λ) → a0 (x, y, z, μ, λ) satisfy the following conditions: For each fixed x ∈ and (μ, λ) ∈ R × R N , the functions a(x, ·, ·, μ, λ) and a0 (x, ·, ·, μ, λ) are measurable; a(x, y, z, μ, 0) = 0 a.e. in y, z ∈ R N and for all (x, μ) ∈ × R. There are five constants c0 , c1 , c2 > 0, 0 < α ≤ min(1, p − 1) and δ ≥ max(2, p), and a continuity modulus ν (i.e., a nondecreasing continuous function on [0, +∞) such that ν(0) = 0, ν(r ) > 0 if r > 0, and ν(r ) = 1
(4.1) (4.2)
if r > 1) such that for almost all y ∈ R N and for all z ∈ R N : (i) a(x, y, z, μ, λ) − a(x, y, z, μ, λ ) · λ − λ δ ≥ c0 (1 + |μ| + |λ| + λ ) p−δ λ − λ . (ii) a0 (x, y, z, μ, λ)μ ≥ 0.
(iii) |a0 (x, y, z, μ, λ)| + |a(x, y, z, μ, λ)| ≤ c1 1 + |μ| p−1 + |λ| p−1 . (iv) a0 (x, y, z, μ, λ) − a0 (x , y, z, μ , λ ) + a(x, y, z, μ, λ) − a(x , y, z, μ , λ ) p−1 p−1 ≤ ν(x − x + μ − μ )(1 + |μ| p−1 + |λ| p−1 + μ + λ ) p−1−α α λ − λ + c2 1 + |μ| + |λ| + μ + λ
(4.3)
for all x, x ∈ , all (μ, λ), (μ , λ ) ∈ R × R N , where the dot denotes the usual Euclidean inner product in R N and | · | the associated norm, being a bounded open set in R N . We consider the boundary value problem x x x x 1, p u ε ∈ W0 () : − div a x, , , u ε , Du ε + a0 x, , , u ε , Du ε ε1 ε2 ε1 ε2 x x in (4.4) = − div b x, , ε1 ε2
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with b ∈ L p (, B(R yN × RzN )) ( p = p/( p − 1)), where we assume that the scales ε1 and ε2 are well-separated as in Sect. 3. This problem admits (at least) a solution u ε ; see e.g., [22, Chap. 2] or [4]. It is known that under some conditions, the above problem possesses a unique solution (see e.g., [2,9,30]) so that we will assume the uniqueness of u ε for each fixed ε > 0. The main advantage of considering this problem lies in its application in hydrodynamic. To fix ideas, the two-dimensional model of the Reynolds equation that describes the flow in a thin film of viscous fluid that is enclosed between two rigid surfaces in relative motion reads as follows: 3 3 h ∂ pε ∂ h ∂ pε v ∂h ∂ + = in , (4.5) ∂ x1 12μ0 ∂ x1 ∂ x2 12μ0 ∂ x2 2 ∂ x1 where pε is the unknown pressure distribution, is the bearing domain, h : → R stands for the thickness of the film, μ0 denotes the fluid viscosity (assumed to be constant), and v is the constant speed of the moving surface. In order to study the effects of the multiscale surface roughness with well-separated wavelengths ε1 and ε2 , on the pressure solution pε , we assume that the film thickness function h ε is given by h ε (x) = h(x, x/ε1 , x/ε2 ), where h : × R N × R N → R is assumed to be continuous and behave realistically with respect to its second and third occurrences y = x/ε1 and z = x/ε2 , respectively (it could be assumed to be either periodic in both y and z, or almost periodic in y and weakly almost periodic in z). It is also assumed that h satisfies 0 < c ≤ h ≤ d where c and d are constants. Assuming the boundary condition pε = 0 on ∂, Eq. (4.5) is equivalent to x x x x −div A x, , Du ε = −div b x, , in , ε1 ε2 ε1 ε2 where A(x) =
(h(x, y, z))3 0 0 (h(x, y, z))3 and u ε = pε ,
,
b(x, y, z) = (6μ0 vh(x, y, z), 0)
which leads to (4.4) with a(x, y, μ, λ) = A(x, y, z)λ and a0 = 0. We obtain by this a good approximation for Newtonian fluids. However in the case of non-Newtonian fluids the above equation is not a suitable one. It will therefore be convenient to rewrite (4.5) in such a way that it captures nonNewtonian effects. In that direction He [18] suggested the following Reynolds-type equation for a one-dimensional incompressible non-Newtonian viscous fluid: d h 3 dpε kh 5 dpε 3 v dh + in = (a, b) ( − ∞ < a < b < ∞), (4.6) = d x 12μ0 d x 320 d x 2 dx where μ0 is the zero shear rate viscosity and k ≥ 0 denotes a nonlinear factor accounting for non-Newtonian effects that arise from the Rabinowitsch constitutive relation. Equation (4.6) therefore reads: d x x d x x du ε , a x, , , u ε , = b x, , dx ε1 ε2 dx dx ε1 ε2
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where a(x, y, z, μ, λ) = (h(x, y, z))3 λ+k1 (h(x, y, z))5 λ3 and b(x, y, z) = k2 h(x, y, z) with k1 = 3kμ0 /80 and k2 = 6μ0 v, u ε = pε . We will therefore take p = 4, α = 1 and δ = 4. As seen above the two examples presented here are rather particular cases of the general model problem x x x x in , (4.7) − div a x, , , Du ε = −div b x, , ε1 ε2 ε1 ε2 the first one being linear and the second, nonlinear. Equation (4.7 ) has been considered for the first time by Kwame [19] under the periodicity hypothesis on the coefficients a and b (with respect to y = x/ε1 and z = x/ε2 , where in [19], they have taken ε1 = ε and ε2 = ε2 ). To the best of our knowledge, this is the first time that the homogenization problem for (4.4) is envisaged, even in the periodic setting. The difficulty is twofold. Firstly, in order to use the results of the earlier sections of this work, one needs the sequence (u ε )ε of solutions to (4.4) to be bounded in the Sobolev space W 1, p (). However it is well-known that due to the presence of the term a0 (x, x/ε1 , x/ε2 , u ε , Du ε ), the sequence (u ε )ε is generally unbounded; see [4]. To overcome this difficulty and simplify the presentation of this section, we make assumption (ii) in (4.3) which really plays no role in the process of the existence of solutions to (4.4). Secondly, once more because of the term a0 (x, x/ε1 , x/ε2 , u ε , Du ε ), the operator under consideration here is non-monotone so that the determination of the limiting problem is quite another matter; see the proof of Theorem 4.4. It is also worth mentioning that the operator in (4.4) is degenerated, so that the problem to be solved here is one of reiterated homogenization of the nonlinear degenerated non-monotone Reynolds-type equation. That is why, in order to have a sequence, we need to the coefficients a and a0 fulfill some conditions (that we do not specify here) so that, to each ε is associated a unique solution u ε to (4.4). We will therefore assume that problem (4.4) has a unique solution associated to each fixed ε > 0, in the above sense. Another difficulty underlined is that the right-hand side of (4.4) depends upon ε and rather weakly converges in W −1, p (), contrary to what is usually considered in the literature. We refer to [32] for more details about the following functions: x → ai (x, x/ε1 , x/ε2 , u ε (x), Du ε (x)) of into R as element of L p (), x → ai (x, x/ε1 , x/ε2 , ψ0 (x, x/ε1 , x/ε2 ), (x, x/ε1 , x/ε2 )) (for (ψ0 , ) ∈ C(; B(R yN × RzN ) N +1 ) of into R denoted by aiε (·, ψ0ε , ε ) as an element of L ∞ (), (x, y, z) → × R yN × RzN into R as an element of C(; L ∞ (R yN × RzN )), and other preliminary results. Now, let A = A y A z be an H -supralgebra on R yN × RzN , A y and A z be two ergodic H -supralgebras. Our goal here is to investigate the limiting behaviour of (u ε )ε>0 (the sequence of solutions to (4.4 ) under the assumptions
ai (x, ·, ·, μ, λ) ∈
p BA
b ∈ L p (; A),
(4.8)
for all (x, μ, λ) ∈ × R × R (0 ≤ i ≤ N ), N
(4.9)
where ai , for 1 ≤ i ≤ N , denotes the i th component of the function a, and p = p/( p−1) with 1 < p < ∞. The following result will be of particular interest in the homogenization process (see [32] for the proof). Proposition 4.1. Assume (4.9) holds. Then, for every (ψ0 , ) ∈ (A) N +1 the function p (y, z) → ai (x, y, z, ψ0 (y, z), (y, z)) denoted by ai (x, ·, ·, ψ0 , ), lies in B A .
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Now, assume (4.9) holds. Then for (ψ0 , ) ∈ C(; (A) N +1 ) it is an easy matter to see, using Proposition 4.1, that the function (x, y, z) → ai (x, y, z, ψ0 (x, y, z), (x, y, z)), p ,∞ p ,∞ p = B A ∩ L ∞ (R yN × RzN ). denoted by ai (·, ψ0 , ), lies in C(; B A ), where B A p ,∞
Since property (3.1) in Definition 3.1 still holds for f in C(; B A following convergence result: aiε (·, ψ0ε , ε ) → ai (·, ψ0 , )
) we have the
in L p ()-weak R as ε → 0 (0 ≤ i ≤ N ). (4.10)
Moreover the following result will be very useful in this section. Proposition 4.2. Let a(·, ψ0 , ) = (ai (·, ψ0 , ))1≤i≤N . The mapping (ψ0 , ) → p (a0 (·, ψ0 , ), a(·, ψ0 , )) of C(; (A) N +1 ) into L p (; B A ) N +1 , extends by continuity p
p
to a unique mapping still denoted by (a0 , a), of L p (; (B A ) N +1 ) into L p (; B A ) N +1 such that (i)
(a(·, u, v) − a(·, u, w)) · (v − w) ≥ c0 (1 + |u| + |v| + |w|) p−δ |v − w|δ a.e. in × R yN × RzN , p−1 p−1 ai (·, u, v) p p + v p p p ≤ c1 1 + u p L (;B A )
L (;B A )
ai (·, u, v) − ai (·, u, w) ≤
L (;(B A ) N )
,
p
L p (;B A ) p−2 c2 1 + |u| + |v| + |w| L p (;B p ) v A
− w L p (;(B p ) N ) , A
ai (x, y, z, u, w) − ai (x , y, z, v, w) ≤ ν(x − x + |u − v|) 1 + |u| p−1 + |v| p−1 + |w| p−1 a.e. in × R yN × RzN
for all u, v ∈ L p (; B A ), v, w ∈ L p (; (B A ) N ) and all x, x ∈ , where the constant c1 depends only on c1 and on . p
p
Proof. It is immediate that for (ψ0 , ) ∈ C(; (A) N +1 ), the function ai (·, ψ0 , ) verifies properties of the same type as in (4.3), so that arguing as in the proof of [31, Prop. 3.1] we get the result. As a consequence of the convergence result (4.10) we have the following result whose proof is quite similar to that of [32, Cor. 3.9]. Proposition 4.3. Let 0 ≤ i ≤ N . For ψ0 ∈ D(), ψ1 ∈ D() ⊗ A∞ y and ψ2 ∈ D() ⊗ A∞ define the function ε ∈ D() (ε > 0) by ε = ψ0 + ε1 ψ1ε + ε2 ψ2ε ,
(4.11)
i.e., ε (x) = ψ0 (x) + ε1 ψ1 (x, x/ε1 ) + ε2 ψ2 (x, x/ε1 , x/ε2 ) for x ∈ . Let (u ε )ε∈E be a sequence in L p () such that u ε → u 0 in L p () as E ε → 0, where u 0 ∈ L p (). Then, as E ε → 0, one has
(i) aiε (·, u ε , Dε ) → ai (·, u 0 , Dψ0 + D y ψ1 + Dz ψ2 ) in L p ()-weak R. Moreover, if (vε )ε∈E is a sequence in L p () such that vε → v0 in L p ()-weak R as p E ε → 0 where v0 ∈ L p (; B A ), then, as E ε → 0, 1 + ∂r ψ 2 ) (ii) aiε (·, u ε , Dε )vε d x → bi (·, u 0 , Dψ0 + ∂s ψ v0 d xdβ,
×(A)
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1 = (∂ j ψ 1 )1≤ j≤N with ∂ j ψ 1 = ∂ j ◦ ψ 1 , the partial derivative ∂ j being here where ∂s ψ 2 = (∂ j ψ 2 )1≤ j≤N with taken on (A) = (A y ) × (A z ) with respect to (A y ); ∂r ψ 2 = ∂ j ◦ ψ 2 , ∂ j being taken on (A) with respect to (A z ); the functions ψ 1 and ∂jψ ψ2 are viewed as defined on with values in D((A)). We recall that the algebra A = A y A z is as stated earlier in this section. For 1, p 1, p 1, p p 1, p 1 < p < ∞ we put F0 = W0 () × L p (; B# A y )) × L p (; B A y (R yN ; B# Az )).We 1, p
endow F0
with the norm
uF1, p = 0
N #
Dx u 0 p + D y u 1 p p i i L () L (;B
Ay
i=1
$ Dz u2 p p , + i ) L (;B ) A
1, p
u = (u 0 , u 1 , u 2 ) ∈ F0 . In this norm, F0 is a Banach space admitting F0∞ = D() × [D() ⊗ (J1 ◦ y ) z ∞ ∞ (A∞ y /C)] × [D() ⊗ y (A y ) ⊗ (J1 ◦ z )(A z /C)] as a dense subspace (see the end 1, p
y
ζ
p
of Sect. 3) where, for ζ = y, z, J1 (resp. ζ ) denotes the canonical mapping of B Aζ /C p
1, p
p
(resp. B Aζ ) into its separated completion B# Aζ (resp. B Aζ ). With this in mind, we have the following homogenization result. Theorem 4.4. Let 1 < p < ∞. Assume (4.9) and (4.8) hold with A = A y A z , where A y and A z are ergodic H -supralgebras on R N . Moreover we assume that A y and A z are translation invariant and that each of their elements is uniformly continuous. For each real ε > 0, let u ε be the unique solution of (4.4). There exists a subsequence of {ε}, still denoted by ε, such that, as ε → 0, uε → u0 ∂u ε ∂u 0 ∂u 1 ∂u 2 → + + ∂ xi ∂ xi ∂ yi ∂z i 1, p
where u = (u 0 , u 1 , u 2 ) ∈ F0
1, p
in W0 ()-weak,
(4.12)
in L p ()-weak R (1 ≤ i ≤ N ),
(4.13)
is solution of the variational equation
×(A)
=
a (·, u 0 , Du) · Dvd xdβ +
b(x, s, r ) · d( ivvd xdβ
×(A)
a0 (·, u 0 , Du)v0 d xdβ 1, p
for all v = (v0 , v1 , v2 ) ∈ F0
(4.14)
1 + ∂r w 2 = G1N (Dw0 + D y w1 + D z w2 ) for w = (w0 , w1 , w2 ) ∈ with Dw = Dw0 + ∂s w 1, p F0 , where ∂s w 1 = (∂i w 1 )1≤i≤N , ∂i w 1 = G1 (∂w1 /∂ yi ) (the same definition for ∂r w 2 ), d( ivv = divv0 + div y v1 +d( ivz v2 with d( iv y v1 = G1 (div y v1 ) (the same definition for d( ivz v2 ), N p div y v1 = i=1 ∂v1 /∂ yi , G1 being the isometric isomorphism of B A y onto L p ((A y )) p
and of B Az onto L p ((A z )) as well. Moreover u 1 and u 2 are unique and any weak 1, p
R-limit point in W0 () of (u ε )ε>0 is a solution to problem (4.14).
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Proof. First of all, it is evident that due to the properties of the mappings a and a0 (see in particular the inequality (ii) in (4.3)), there exists a constant c > 0 such that c0 u ε
p−1 1, p W0 ()
≤ div bε W −1, p () .
(4.15)
Thus, as the sequence (div bε )ε>0 is bounded in W −1, p (), it results from (4.15) that the 1, p sequence (u ε )ε>0 is bounded in W0 (). Therefore the sequences (a ε (·, u ε , Du ε ))ε>0 and (a0ε (·, u ε , Du ε ))ε>0 are bounded in L p () N and L p (), respectively. Thus, given a fundamental sequence E, Theorem 3.5 guarantees the existence of a subsequence E 1, p extracted from E and a triplet u = (u 0 , u 1 , u 2 ) ∈ F0 such that (4.12)-(4.13) hold when E ε → 0. The next part of the proof is to show that u solves Eq. (4.14). For y that purpose, let = (ψ0 , (J1 ◦ y )(ψ1 ), (J1z ◦ z )(ψ2 )) ∈ F0∞ with ψ0 ∈ D(), ∞ ∞ ∞ ψ1 ∈ D() ⊗ (A y /C) and ψ2 ∈ D() ⊗ (A∞ y ) ⊗ (A z /C), where A y /C = {ψ ∈ ∞ N ∞ ∞ A y : M(ψ) = 0} (M the mean value on R y for H), A z /C = {ψ ∈ A z : M(ψ) = 0} (M the mean value on RzN for H ). Define ε (ε > 0 ) as in (4.11). Then ε ∈ D() and further, in view of [part (i) of] (4.3) we have 0 ≤ (a ε (·, u ε , Du ε ) − a ε (·, u ε , Dε )) · (Du ε − Dε )d x,
or, 0≤
a0ε (·, u ε , Du ε )(ε − u ε )d x + bε div( u ε − ε )d x ε − a (·, u ε , Dε ) · (Du ε − Dε )d x.
(4.16)
Because of the boundedness of the sequence a0ε (·, u ε , Du ε ) ε>0 in L p (), there exist
p
a subsequence from E not relabeled, and a function χ ∈ L p (; B A ) such that, as E ε → 0 , a0ε (·, u ε , Du ε ) → χ
in L p ()-weak R.
(4.17)
On the other hand we have ε ε b div( u ε − ε )d x = b div u ε − bε ( div ψ0 + (div y ψ1 )ε + (divz ψ2 )ε )d x − bε (ε1 (div y ψ1 )ε + ε2 (divz ψ2 )ε )d x,
and, as E ε → 0 we have by using (4.13), bε div( u ε − ε )d x →
×(A) 1, p
( − )d xdβ. b div(u
(4.18)
Next, using the compactness of the embedding W0 () → L p (), we have ε −u ε → ψ0 − u 0 in L p () as E ε → 0. Therefore passing to the limit in (4.16) using
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(4.12)–(4.13), the above convergence results ( 4.17)–(4.18) together with [part (ii) of] Proposition 4.3 , we get ( − )d xdβ a (·, u 0 , D) · D(u − )d xdβ + 0≤− b div(u ×(A) ×(A) + χ (ψ0 − u 0 )d xdβ, (4.19) ×(A)
ranging over F0∞ , and hence over F0 too (by a density argument). Taking in (4.19) 1, p the particular functions = u − tv with t > 0 and v = (v0 , v1 , v2 ) ∈ F0 , then dividing both sides of the obtained inequality by t, and finally letting t → 0, we get a (·, u 0 , Du) · Dvd xdβ + χ v0 d xdβ = b d( ivvd xdβ 1, p
×(A)
×(A)
for all v = (v0 , v1 , v2 ) ∈
1, p F0 .
(4.20)
Now, arguing exactly as in the proof of [32, Theorem 3.10] we get χ = a0 (·, u 0 , Du) so that u solve Eq. (4.14) as claimed. The next point to check is to show that u 1 and u 2 are unique. For that, we take in (4.20) the function v = (v0 , v1 , v2 ) with v0 = 0 and v1 = 0; then for each fixed (x, s) ∈ × (A), u 2 (x, s, ·) is a solution to a (x, s, r, u 0 (x), Du 0 (x) + ∂s u 1 (x, s) + ∂r u 2 (x, s, r )) · ∂r w dβz (A z ) 1, p = (4.21) b(x, s, r ) div z wdβz for all w ∈ B# A z . (A z )
So, let (x, s, ρ, ξ ) ∈ × (A y ) × R × R N be freely fixed, and let π = π(x, s, ρ, ξ ) ∈ 1, p B# Az be defined by the cell equation, 1, p a (x, s, ·, ρ, ξ +∂r π ) · ∂r w dβz = b(x, s, ·)div z wdβz for all w ∈ B# A z . (A z )
(A z )
(4.22)
Since the linear functional w → (Az ) it b(x, s, ·) div z wdβz is continuous on follows by [22, Chap. 2] that Eq. (4.22) admits at least a solution. But this solution is unique; indeed if π1 and π2 are solutions to (4.22), then we have ( a (x, s, ·, ρ, ξ + ∂r π1 ) − a (x, s, ·, ρ, ξ + ∂r π2 )) · ∂r w dβz = 0 1, p B# Az ,
(A z )
1, p
for all w ∈ B# Az . Taking in particular w = π1 − π2 in the above equation it follows by [part (ii) of] Proposition 4.2 that ∂r π1 = ∂r π2 and hence D z π1 = D z π2 . We deduce that π1 = π2 since 1, p u 1 (x, s), they belong to B# Az . Now, taking in (4.22) ρ = u 0 (x) and ξ = Du 0 (x) + ∂s and comparing the resultant equation with (4.21), we get by the uniqueness of the soluπ (x, s, u 0 (x), Du 0 (x) + ∂s u 1 (x, s)), equivalently, or tion of (4.22) that u 2 (x, s, ·) = u 2 (x, y, ·) = π(x, y, u 0 (x), Du 0 (x) + D y u 1 (x, y)) for a.e. (x, y) ∈ × R N . This shows the uniqueness of u 2 . The same process shows the uniqueness of u 1 . This completes the proof of the theorem.
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One can work out some homogenization problems related to problem (4.4 ), (4.8) and (4.9). In particular one can solve: (P)1 The periodic homogenization problem stated as follows: For each fixed 0 ≤ i ≤ N and any (x, μ, λ) ∈ × R × R N , the function (y, z) → ai (x, y, z, μ, λ), and the function (y, z) → b(x, y, z), are Y -periodic in y ∈ R N and Z -periodic in z ∈ R N , where Y = Z = (0, 1) N . Here we get the homogenization of (4.4) with A = Cper (Y ) Cper (Z ). (P)2 The almost periodic homogenization problem stated as follows: p
ai (x, ·, ·, μ, λ) ∈ B A P (R yN × RzN ) for any (x, μ, λ) ∈ × R N +1 , 0 ≤ i ≤ N ; b(x, ·, ·) ∈ A P(R yN × RzN ) for a.e. x ∈ , which yields the homogenization of (4.4) with A = A P(R yN ) A P(RzN ) = A P(R yN × RzN ). (P)3 The weakly almost periodic homogenization problem I: p
p
ai (x, ·, ·, μ, λ) ∈ BW A P (R yN ; B A P (RzN )) for any (x, μ, λ) ∈ × R N +1 , 0 ≤ i ≤ N ; b(x, ·, ·) ∈ A P(R yN × RzN ) for a.e. x ∈ ,
which leads to the homogenization of (4.4) with A = W A P(R yN ) A P(RzN ). (P)4 The weakly almost periodic homogenization problem II: p
p
ai (x, ·, ·, μ, λ) ∈ B A P (R yN ; BW A P (RzN )) for any (x, μ, λ) ∈ × R N +1 , 0 ≤ i ≤ N ; b(x, ·, ·) ∈ A P(R yN ; W A P(RzN )) for a.e. x ∈ ,
which yields the homogenization of (4.4) with A = A P(R yN ) W A P(RzN ). We recall here that A P(R yN ; W A P(RzN )) is the space of all almost periodic continuous functions of R yN into W A P(RzN ). (P)5 The fully weakly almost periodic homogenization problem III: p
p
ai (x, ·, ·, μ, λ) ∈ BW A P (R yN ; BW A P (RzN )) for any (x, μ, λ) ∈ × R N +1 , 0 ≤ i ≤ N ; b(x, ·, ·) ∈ W A P(R yN ) W A P(RzN ) for a.e. x ∈ .
Here the suitable H -supralgebra is A = W A P(R yN ) W A P(RzN ). (P)6 The homogenization problem in the Fourier-Stieltjes algebra. We first need to define the Fourier-Stieltjes algebra F S(R N ) on R N . Definition 4.1. The Fourier-Stieltjes algebra on R N is defined as the closure in B(R N ) of the space ) * F S∗ (R N ) = f : R N → R, f (x) = exp(i x · y)dν(y) for some ν ∈ M∗ (R N ) , RN
where M∗ (R N ) denotes the space of complex valued measures ν with finite total variation: |ν| (R N ) < ∞. We denote it by F S(R N ).
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Since by [16] any function in F S∗ (R N ) is a weakly almost periodic continuous function, we have that F S(R N ) ⊂ W A P(R N ). Moreover thanks to [13, Theorem 4.5] F S(R N ) is a proper subalgebra of W A P(R N ). As F S(R N ) is an ergodic algebra which is translation invariant (this is easily seen: indeed F S∗ (R N ) is translation invariant) we see that hypotheses of Theorem 3.5 are satisfied with any algebra A = F S(R yN ) A z , A z being any ergodic H -supralgebra on RzN which is translation invariant and whose elements are uniformly continuous. With all this in mind, one can solve the homogenization problem for (4.4) under the assumptions that p
p
ai (x, ·, ·, μ, λ) ∈ B F S (R yN ; B F S (RzN )) for any (x, μ, λ) ∈ × R N +1 , 0 ≤ i ≤ N ; b(x, ·, ·) ∈ F S(R yN ) F S(RzN ) for a.e. x ∈ . 5. Application to the Reiterated Homogenization of a Nonlinear Hyperbolic Equation 5.1. Setting of the problem. The theory of hyperbolic equations is a multifaceted subject, with applications in fluid dynamics, electromagnetic waves, the general theory of relativity, etc. The problem of asymptotic behaviour for hyperbolic equations is well known in the literature. However, the situation for the nonlinear hyperbolic equations with the nonlinear term depending on a microscopic time variable has received much less discussion, so that the study of this problem seems worthwhile. We focus our attention on hyperbolic equations with a nonlinear dissipative term. More precisely, our aim here is to study the reiterated homogenization problem for the following strongly nonlinear dissipative hyperbolic equation
x x ∂ 2uε x x t in Q = × (0, T ), , , Du + h −div A , , u , Du ε ε ε = f ∂t 2 ε ε2 ε ε2 εk u ε = 0 on ∂ × (0, T ), (5.1) ∂u ε u ε (x, 0) = (x, 0) = 0 in , ∂t where ε is a small positive parameter intended to tend to zero, is a bounded open set in R N with C 2 boundary (integer N ≥ 1), T and k are positive real numbers with 0 < k ≤ 1/2, f ∈ L 2 (Q), and the functions (y, z, τ, μ, λ) → h(y, z, τ, μ, λ) from R N ×R N ×R×R×R N to R and (y, z) → A(y, z) = (ai j (y, z))1≤i, j≤N from R N ×R N 2 2 to R N , lie respectively in C 1 (R2N +1 × R × R N ) and in C 1 (R N × R N ) N , and satisfy: There exists a constant γ satisfying γ > 0 if N = 1, 2, and 0 < γ < 2/(N − 2) if N ≥ 3, and there exist some positive constants c0 , c1 , c2 , c3 , c4 and c such that, for all (y, z, τ ) ∈ R2N +1 , (i) |h(y, z, τ, μ, λ)| ≤ c0 (1 + |μ|γ +1 + |λ|), γ (ii) h(y, z, τ, μ, λ)η ≥ c1 |μ| μη − c2 (1 + |η| |λ|), ∂h (y, z, τ, μ, λ) ≤ c0 (1 + |μ|γ ) , (iii) ∂μ (iv) |D λ h(y, z, τ, μ, λ)| ≤ c, , λ ) (η − η ) (v) h(y, z, τ, μ, λ) − h(y, z, τ, μ γ ≥ −c3 |μ|γ + μ μ − μ η − η − c4 η − η λ − λ , for all η, η ∈ R and all (μ, λ), (μ , λ ) ∈ R × R N ;
(5.2)
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ai j = a ji (1 ≤ i, j ≤ N ) and there is a positive constant α such that N
ai j (y, z)ξ j ξi ≥ α |ξ |2 for all ξ = (ξi ) ∈ R N and for all y, z ∈ R N .
(5.3)
i, j=1
Under hypotheses (5.2) and (5.3) the problem of existence and uniqueness of a solution to (5.1) has been discussed in [12] where the author derived the existence of a unique u ε ∈ C([0, T ]; H01 ()) ∩ C 1 ([0, T ]; L 2 ()) solution to (5.1). Moreover, in view of [part (i) of] (5.2), the functions (x, t) → h(x/ε, x/ε2 , t/εk , u ε (x, t), Du ε (x, t)) (denoted below by h ε (·, ·, ·, u ε , Du ε )) lies in L 2 (Q) so that, because of the leading equation in (5.1), we have ∂ 2 u ε /∂t 2 ∈ L 2 (0, T ; H −1 ()). Therefore the solution u ε to (5.1) lies in the space ) * ∂ 2v p 1 2 −1 V = v ∈ L (0, T ; H0 ()) : v = 2 ∈ L (0, T ; H ()) ∂t (where p = 2(γ + 1)), a Banach space with the norm vV = v L p (0,T ;H 1 ()) + v L 2 (0,T ;H −1 ()) (v ∈ V ). 0
Furthermore, since 1 < p < 2N /(N − 2) (for N ≥ 3) we have by the RellichKondratchov theorem, the compactness of the continuous embedding H01 () → L p (). Whence by [22, p.58, Theorem 5.1], V is compactly embedded in L p (Q). In addition (see [23, p. 23]) V is continuously embedded in C(0, T ; L 2 ()) and in C 1 (0, T ; H −1 ()), in such a way that the existence of u ε (0) and u ε (0) is justified. Thus u ε belongs to the space V0 = {v ∈ V : v(0) = v (0) = 0}, a Banach space with the relative topology on V . To the best of our knowledge, the reiterated homogenization problem for (5.1) has never been considered, even in the periodic setting. Here we consider this problem in the general framework of ergodic algebras. The choice of the parameter k to be in (0, 1/2] is just to simplify the presentation. The current section is organized as follows. Subsect. 5.2 deals with some preliminary results and in Subsect. 5.3 we state and prove the main homogenization result of the section. Finally, we present some concrete applications of our result. We note by the way that an example of a nontrivial function h satisfying ( 5.2) is the function h(y, z, τ, μ, λ) = |μ|γ μ+g(y, z, τ )
N
sin λi , where g ∈ C 1 (R N × R N × R) with
i=1
g ≥ 0 and g with all its derivatives are bounded. 5.2. Preliminary results. Let A y , A z and Aτ be three H -supralgebras which are translation invariant and each of their elements is uniformly continuous. We assume that A y and A z are ergodic. Points in (A y ) (resp. (A z ), (Aτ )) are denoted by s (resp. r, s0 ). The compact space (A y ) (resp. (A z ), (Aτ )) is equipped with the M-measure β y (resp. βz , βτ ), for A y (resp. A z , Aτ ). We have (A) = (A y )×(A z )× (Aτ ) (Cartesian product) and the M-measure for A, with which (A) is equipped, is
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precisely the product measure β = β y ⊗ βz ⊗ βτ ; the last equality follows in an obvious way by density of A y ⊗ A z ⊗ Aτ in A and by the Fubini’s theorem. The main objective of this section is to analyze the asymptotic behaviour of (u ε )ε>0 (the sequence of solutions to (5.1)) under the following assumptions: ai j ∈ A y A z
for all 1 ≤ i ≤ N ,
h(·, ·, ·, μ, λ) ∈ A = A y A z Aτ
(5.4)
for all (μ, λ) ∈ R × R . N
(5.5)
From the second assumption (5.5) we derive the following result whose proof is evident: h(·, ·, ·, ψ0 , ) ∈ A
for all (ψ0 , ) ∈ A × (A) N ,
(5.6)
where h(·, ·, ·, ψ0 , ) stands for the function (y, z, τ ) → h(y, z, τ, ψ0 (y, z, τ ), (y, z, τ )) of R N × R N × R into R. With this in mind we have the following result. Proposition 5.1. Let ∈ (D() ⊗ A) N . Let (u ε )ε∈E (E a fundamental sequence) be a sequence in L p (Q) such that u ε → u 0 in L p (Q) as E ε → 0, where u 0 ∈ L p (Q) with p = 2(γ + 1), and γ being given by (5.2). Assume (5.5) holds. Then, as E ε → 0, h ε (·, ·, ·, u ε , ε ) → h(·, ·, ·, u 0 , ) in L 2 (Q)-weak R, where h ε (·, ·, ·, u ε , ε ) stands for the obvious function (x, t) → h(x/ε, x/ε2 , t/εk , u ε (x, t), (x, t, x/ε, x/ε2 , t/εk )) of Q into R. Proof. One can easily see that the function (x, t, y, z, τ ) → h(y, z, τ, u 0 (x, t), (y, z, τ )), lies in L 2 (Q; A). Thus h ε (·, ·, ·, u 0 , ε ) → h(·, ·, ·, u 0 , )
in L 2 (Q)-weak R.
The proposition will therefore be checked once we show that h ε (·, ·, ·, u ε , ε ) − h ε (·, ·, ·, u 0 , ε ) → 0 in L 2 (Q)-weak R. To this end, set aε = h ε (·, ·, ·, u ε , ε ) − h ε (·, ·, ·, u 0 , ε ) (ε ∈ E). Since the sequence (u ε )εE is bounded in L p () (indeed it strongly converges in L p ()) one deduces from [part (i) of] (5.2) that the sequence (aε )ε∈E is bounded in L 2 (Q). Thus, there exist a subsequence E from E and a function χ ∈ L 2 (Q; B 2A ) such that aε → χ
in L 2 (Q)-weak R as E ε → 0.
(5.7)
Now, owing to [part (v) of] (5.2) we have, for each fixed g ∈ L 2 (Q; A), |u ε |γ + |u 0 |γ |u ε − u 0 | g ε d xdt − aε g ε d xdt ≤ c3 Q Q
γ γ ≤ c3 u ε L p (Q) + u 0 L p (Q) u ε − u 0 L p (Q) g ε L 2 (Q) . Passing to the limit (as E ε → 0) in the above inequality using (5.7), we arrive at − χ g d xdtdβ ≤ 0. Q×(A)
The above inequality being true for any arbitrary g, we get χ g d xdtdβ = 0 for all g ∈ L 2 (Q; A), Q×(A)
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which means that χ = 0, where χ = G1 ◦ χ , G1 being defined in Sect. 2. It follows that χ = 0, and so, any R-convergent subsequence of (aε )ε∈E R-converges towards 0, whence the whole sequence (aε )ε∈E R -converges towards 0. The result follows thereby. Following the same way of reasoning as in the proof of Proposition 5.1, we have the next corollary whose proof is omitted. Corollary 5.2. Let ε be defined by x t x x t 2 ((x, t) ∈ Q), ε (x, t) = ψ0 (x, t)+εψ1 x, t, , k +ε ψ2 x, t, , 2 , k ε ε ε ε ε (5.8) ∞ ∞ where ψ0 ∈ D(Q), ψ1 ∈ D(Q) ⊗ (A∞ y Aτ ) and ψ2 ∈ D(Q) ⊗ A . Let (u ε )ε∈E be as in Proposition 5.1. Then, as E ε → 0,
h ε (·, ·, ·, u ε , Dε ) → h(·, ·, ·, u 0 , Dψ0 + D y ψ1 + Dz ψ2 ) in L 2 (Q)-weak R. 1,2 2 2 N +1 Now, let V = V0 × L 2 (Q; B 2Aτ (Rτ ; B#1,2 A y )) × L (Q; B A y Aτ (R y,τ ; B# A z )). V is a Banach space (with an obvious norm) which admits F0∞ = D(Q)×(D(Q)⊗[τ (A∞ τ )⊗ y ∞ ) ⊗ (A∞ ) ⊗ (J z ◦ )(A∞ /C)]) as a dense (J1 ◦ y )(A∞ /C)]) × (D(Q) ⊗ [ (A y τ z y y τ z 1 subspace. For u = (u 0 , u 1 , u 2 ) ∈ V we put Di u = ∂u 0 /∂ xi + ∂i u 1 + ∂i u 2 (1 ≤ i ≤ N ) and Du = Du 0 + ∂s u 1 + ∂r u 2 = (Di u)1≤i≤N , and we define the bilinear form a Q on V as follows:
a Q (u, v) =
N
i, j=1
ai j D j uDi vd xdtdβ (u, v ∈ V). Q×(A)
We recall that ai j ∈ C((A y ) × (A z )) ( ai j = G(ai j ), G denoting here the Gelfand transformation on A y A z ) with ai j = a ji , such that the bilinear form a Q is continuous on V and is coercive. With all that in mind, we consider the following variational problem:
T 0
u = (u 0 , u 1 , u 2 ) ∈ V, u 0 (t), v0 (t) dt + a Q (u, v) + h(u 0 , Du)v0 d xdt =
!
"
Q
f v0 d xdt (5.9) Q
for all v = (v0 , v1 , v2 ) ∈ V, where h(u 0 , Du) = (A) h(s, r, s0 , u 0 , Du)dβ. Using Gronwall’s inequality, one can show that Eq. (5.9) admits at most a solution.
5.3. Homogenization result. Before se set our homogenization result, it seems important to note that if in Theorem 3.7 we replace the space V p by V0 , then the conclusion of this theorem still holds mutatis mutandis. We are now in a position to state and prove the main homogenization result of this section.
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Theorem 5.3. Assume (5.4) and (5.5) hold with A = A y A z Aτ , where A y and A z are ergodic H -supralgebras on R N and Aτ is and H -supralgebra on R. Moreover we assume that A y , A z and Aτ are translation invariant and that each of their elements is uniformly continuous. For each positive real number ε, let u ε be the unique solution to (5.1). Let p = 2(γ + 1). Then, as ε → 0, uε → u0 ∂ 2u
∂ 2u
ε
in L p (0, T ; H01 ())-weak,
→ in L 2 (0, T ; H −1 ())-weak, ∂t 2 ∂t 2 ∂u 0 ∂u 1 ∂u 2 ∂u ε → + + in L 2 (Q)-weak R (1 ≤ j ≤ N ), ∂x j ∂ x j ∂ y j ∂z j 0
(5.10) (5.11) (5.12)
where the vector u = (u 0 , u 1 , u 2 ) is the unique solution to (5.9). Proof. It is an easy exercise to see that the sequence (u ε )ε>0 is bounded in V (and hence in V0 ) and so the sequence (h ε (·, ·, ·, u ε , Du ε ))ε>0 is bounded in L 2 (Q). So let E be a fundamental sequence. By Theorem 3.7, there exist a subsequence E from E and a 1,2 2 2 N +1 vector u = (u 0 , u 1 , u 2 ) ∈ V0 × L 2 (Q; B 2Aτ (Rτ ; B#1,2 A y ))× L (Q; B A y Aτ (R y,τ ; B# A z )) such that (5.10)-(5.12) hold with E ε → 0. Because of the uniqueness of the solution to problem (5.9), this will hold for any E, and finally for 0 < ε → 0. It therefore remains to verify that u = (u 0 , u 1 , u 2 ) solves the variational equation in (5.9). To this end we have " ! f (t)vd x, v ∈ H01 (), u ε (t), v +a ε (u ε , v)+ h ε (·, ·, ·, u ε , Du ε )vd x =
(5.13) where a ε (u, v) =
N
i, j=1
aiεj
∂u ∂v dx ∂ x j ∂ xi
(u, v ∈ H01 ())
with aiεj (x) = ai j (x/ε, x/ε2 ) (x ∈ ). Taking in (5.13) the function v = ε (·, t) (t ∈ (0, T )) with ε as in (5.8), and integrating the resultant equation over (0, T ), we get T ∂ 2 ε ε uε d xdt + a (u ε (t), ε (·, t))dt + h ε (·, ·, ·, u ε , Du ε )ε d xdt ∂t 2 Q Q 0 = f ε d xdt. (5.14) Q
Let us pass to the limit in (5.14). To this end, we will evaluate the limit, as E ε → 0, of each term involved in (5.14). For the first term on the left-hand side of the above equation, we have 2 ε ∂ ψ1 ∂ 2 ε ∂ 2 ψ0 1−2k uε d xdt = u d xdt + ε u d xdt ε ε 2 2 ∂t ∂t ∂τ 2 Q Q Q 2 ε 2 ε ∂ ψ1 ∂ ψ1 1−k uε d xdt + ε uε d xdt + 2ε ∂τ ∂t ∂t 2 Q Q
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2 ε ε ∂ 2 ψ2 ∂ ψ2 2−2k + 2ε uε d xdt + ε uε d xdt ∂τ ∂t ∂τ 2 Q Q 2 ε ∂ ψ2 uε d xdt. (5.15) + ε2 ∂t 2 Q
2−k
Then by using the fact that 0 < k ≤ 1/2, we get, as E ε → 0, ∂ 2 ε ∂ 2 ψ0 uε d xdt → u 0 2 d xdt. 2 ∂t ∂t Q Q Whence, as E ε → 0, T ! " u ε (t), ε (·, t) dt → 0
0
T
!
" u 0 (t), ψ0 (·, t) dt.
As for the second term on the left-hand side of (5.14), one shows by a classical argument that, as E ε → 0, T a ε (u ε (t), ε (·, t))dt → a Q (u, ), 0
where = (ψ0 , ψ1 , ψ2 ) ∈ F0∞ . Since the sequence (h ε (·, ·, ·, u ε , Du ε ))ε∈E is bounded in L 2 (Q) , there exist a function χ ∈ L 2 (Q; B 2A ) and a subsequence from E not relabeled, such that, as E ε → 0, h ε (·, ·, ·, u ε , Du ε ) → χ
in L 2 (Q)-weak R. But ε → ψ0 in L 2 (Q) as E ε → 0, whence, as E ε → 0, Q h ε (·, ·, ·, u ε , Du ε ) ψ0 d xdtdβ. Lastly, as E ε → 0, one has Q f ε d xdt → ε d xdt → Q×(A) χ Q f ψ0 d xdt. We therefore obtain the variational problem
T 0
!
u 0 (t), ψ0 (·, t)
"
dt + a Q (u, ) +
χ ψ0 d xdtdβ = Q×(A)
and hence, in view of the density of F0∞ in V, T ! " u 0 (t), v0 (t) dt + a Q (u, v) +
Q×Y ×Z 2
0
f ψ0 d xdt, Q
χ v0 d xdtd ydτ dζ =
f v0 d xdt Q
for all v = (v0 , v1 ) ∈ V.
dβ, or equivalently The last point to check is to verify that h(u 0 , Du) = (A) χ χ v0 d xdtdβ = h(s, r, s0 , u 0 , Du)v0 d xdtdβ (v0 ∈ V0 ). Q×(A)
Q×(A)
To this end, we use [point (v) of] (5.2) to get the inequality ε h (·, ·, ·, u ε , Du ε ) − h ε (·, ·, ·, u ε , Dε ) (u ε − ε )d xdt Q |u ε − ε | |Du ε − Dε | d xdt ≥ 0, + c4 Q
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or more precisely,
Q
h ε (·, ·, ·, u ε , Du ε ) − h ε (·, ·, ·, u ε , Dε ) (u ε − ε )d xdt
+c4 u ε − ε L 2 (Q) Du ε − Dε L 2 (Q) N ≥ 0.
(5.16)
First and foremost, since the embedding V → L 2 (Q) is compact (this is a classical result) we have in view of (5.10)–(5.11) that uε → u0
in L 2 (Q) as E ε → 0,
and hence u ε − ε L 2 (Q) → u 0 − ψ0 L 2 (Q)
when E ε → 0.
Next, let us show that, as E ε → 0, Du ε − Dε L 2 (Q) N → Du − D L 2 (Q×(A)) N . But if we proceed exactly as in the proof of [28, Theorem 2.2] we reach the above result. On the other hand, (5.10)–(5.11) together with the compactness of the continuous embedding V → L p (Q) yields the convergence result u ε → u 0 in L p (Q) as E ε → 0. Therefore taking the limit as E ε → 0 of both sides of ( 5.16) we get, using Corollary 5.2 together with all the convergence results obtained above,
χ − h(s, r, s0 , u 0 , D) (u 0 − ψ0 )d xdtdβ
Q×(A)
+ c4 u 0 − ψ0 L 2 (Q) Du − D L 2 (Q×(A)) N ≥ 0.
(5.17)
Inequality (5.17) still holds for ∈ F10 , hence if we take in (5.17) the particular function = u − r v with r > 0 fixed and v ∈ F10 , we are led (after simplification by r ) to
χ − h(s, r, s0 , u 0 − r v0 , Du − r Dv) v0 d xdtdβ
Q×(A)
+ r c4 v0 L 2 (Q) Dv L 2 (Q×(A)) N ≥ 0. On letting r → 0 (using the Lebesgue theorem), we arrive at
χ − h(s, r, s0 , u 0 , Du) v0 d xdtdβ ≥ 0,
Q×(A)
whence, by changing v0 into −v0 , we finally obtain
χ − h(s, r, s0 , u 0 , Du) v0 d xdtdβ = 0.
Q×(A)
This concludes the proof.
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One can also work out some homogenization problems associated to the problem (5.1), (5.4) and (5.5). Especially we can consider the periodic homogenization problem (A = Cper (Y ) Cper (Z ) Cper (T ), T = (0, 1)), the almost periodic homogenization problem (either A = A P(R yN ) A P(RzN ) A P(Rτ ) or A = A P(R yN ) A P(RzN ) Aτ , Aτ any translation invariant H -supralgebra in which elements are uniformly continuous), the weakly almost periodic homogenization problem (either A = W A P(R yN ) A P(RzN ) Aτ or A = A P(R yN ) W A P(RzN ) Aτ or A = W A P(R yN ) W A P(RzN ) Aτ , Aτ being like above). Other homogenization problems can be considered as the one leading to A = (A P(R yN ) + C0 (R yN )) A P(RzN ) Aτ , where C0 (R yN ) denotes the space of all continuous functions that vanish at infinity. It is well known that A y = A P(R yN ) + C0 (R yN ) is a proper subalgebra of the algebra of weakly almost periodic functions on R yN . Acknowledgements. JL Woukeng acknowledges the support of the University of Pretoria through a postdoctoral fellowship. M Sango and JL Woukeng are supported by a Focus Area Grant from the National Research Foundation of South Africa.
References 1. Allaire, G., Briane, M.: Multi-scale convergence and reiterated homogenization. Proc. Roy. Soc. Edinb. Sect. A. 126, 297–342 (1996) 2. Barles, G., Murat, F.: Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions. Arch. Rat. Mech. Anal. 133, 77–101 (1995) 3. Besicovitch, A.S.: Almost periodic functions. Cambridge: Dover Publications, 1954 4. Boccardo, L., Murat, F., Puel, J.P.: Existence de solutions non born ées pour certaines équations quasilinéaires. Port. Math. 41, 507–534 (1982) 5. Bohr, H.: Almost periodic functions. New York: Chelsea, 1947 6. Bourbaki, N.: Intégration. Chap. 1–4. Paris: Hermann, 1966 7. Bourbaki, N.: Intégration. Chap. 5. Paris: Hermann, 1967 8. Bourbaki, N.: Topologie générale. Chap. 1–4. Paris: Hermann, 1971 9. Carrillo, J., Wittbold, P.: Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems. J. Diff. Eq. 156, 93–121 (1999) 10. Casado Diaz, J., Gayte, I.: A derivation theory for generalized Besicovitch spaces and its application for partial differential equations. Proc. R. Soc. Edinb. A 132, 283–315 (2002) 11. Casado Diaz, J., Gayte, I.: The two-scale convergence method applied to generalized Besicovitch spaces. Proc. R. Soc. Lond. A 458, 2925–2946 (2002) 12. Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients. Electron. J. Diff. Eq. 1998, 1–21 (1998) 13. Chou, C.: Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups. Trans. Amer. Math. Soc. 274, 141–157 (1982) 14. Chou, C.: Weakly almost periodic functions and almost convergent functions on a group. Trans. Amer. Math. Soc. 206, 175–200 (1975) 15. De Leeuw, K., Glicksberg, I.: Applications to almost periodic compactifications. Acta Math. 105, 63–97 (1961) 16. Eberlein, W.F.: Abstract ergodic theorems and weak almost periodic functions. Trans. Amer. Math. Soc. 67, 217–240 (1949) 17. Eberlein, W.F.: A note on Fourier-Stieltjes transforms. Proc. Amer. Math. Soc. 6, 310–313 (1955) 18. He, J.-H.: Variational principle for non-Newtonian lubrication: Rabinowitsch fluid model. Appl. Math. Comp. 157, 281–286 (2004) 19. Kwame Essel, E.: Homogenization of Reynolds equations and of some parabolic problems via Rothes method. PhD thesis, Luleå University of Technology, Sweden, 2008 20. Larsen, R.: Banach algebras. New York: Marcel Dekker, 1973 21. Lindenstrauss, J.: On non-separable reflexive Banach spaces. Bull. Amer. Math. Soc. 72, 967–970 (1966) 22. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod, 1969 23. Lions, J.L., Magenes, E.: Problèmes aux limites non homog ènes et applications. Vol 1. Paris: Dunod, 1968
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24. Lukkassen, D., Nguetseng, G., Wall, P.: Two scale convergence. Int. J. Pure Appl. Math. 1, 35–86 (2002) 25. Lukkassen, D., Nguetseng, G., Nnang, H., Wall, P.: Reiterated homogenization of nonlinear elliptic operators in a general deterministic setting. J. Funct. Spaces Appl. 7, 121–152 (2009) 26. Nguetseng, G.: Homogenization structures and applications I. Z. Anal. Anwen. 22, 73–107 (2003) 27. Nguetseng, G.: Mean value on locally compact abelian groups. Acta Sci. Math. 69, 203–221 (2003) 28. Nguetseng, G.: Deterministic homogenization of a semilinear elliptic equation. Math Reports 8, 167–195 (2006) 29. Nguetseng, G.: Almost periodic homogenization: asymptotic analysis of a second order elliptic equation. Preprint 30. Rakotoson, J.M.: Uniqueness of renormalized solutions in a T -set for the L 1 -data problem and the link between various formulations. Indiana Univ. Math. J. 43, 685–702 (1994) 31. Woukeng, J.L.: Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales. Ann. Mat. Pura Appl. 189(3), 357–379 (2010) 32. Woukeng, J.L.: Homogenization of nonlinear degenerate non-monotone elliptic operators in domains perforated with tiny holes. Acta Appl. Math. (2009) doi:10.1007/s10440-009-9552-z, 2009 33. Zhikov, V.V., Krivenko, E.V.: Homogenization of singularly perturbed elliptic operators. Matem. Zametki. 33, 571–582 (1983) (English transl.: Math. Notes 33, 294–300 (1983)) Communicated by P. Constantin
Commun. Math. Phys. 300, 877–888 (2010) Digital Object Identifier (DOI) 10.1007/s00220-010-1135-3
Communications in
Mathematical Physics
The Spectral Action for Dirac Operators with Skew-Symmetric Torsion Florian Hanisch, Frank Pfäffle, Christoph A. Stephan Institut für Mathematik, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany. E-mail: [email protected]; [email protected]; [email protected] Received: 30 March 2010 / Accepted: 1 June 2010 Published online: 22 September 2010 – © Springer-Verlag 2010
Abstract: We derive a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional manifolds with totally anti-symmetric torsion. We find that the torsion becomes dynamical and couples to the traceless part of the Riemann curvature tensor. Finally we deduce the Lagrangian for the Standard Model of particle physics in the presence of torsion from the Chamseddine-Connes Dirac operator. 1. Introduction In classical relativity one presumes that the gravitational degrees of freedom are encoded only in the choice of the metric and one can therefore restrict to Levi-Civita connections. This is usually justified by the Einstein-Cartan theory because the critical points of the corresponding action are space-times with torsion free connections. If we want to incorporate the other forces of Nature (i.e. the electro-weak and the strong force) and the known fermions (Leptons and Quarks) we have to consider unified theories. A geometrical approach to such a unification is offered by Connes’ noncommutative geometry [1–4]. The natural action to consider in this framework is given by the spectral action principle [5]. In the language of noncommutative geometry any geometry is encoded by a spectral triple which consists of an algebra, a module over this algebra and a first order differential operator. Here one should picture one of the simplest examples given by compact Riemannian spin manifolds: the algebra is formed by the smooth functions, the module is given by the square-integrable spinor fields and the first order differential operator is the classical Dirac operator. To reconstruct the manifold one needs further ingredients (a real structure, a chirality operator) [1–4]. We will omit any details here, since the only relevant objects for this article turn out to be classical twisted Dirac operators. In combination with product geometries based on spectral triples for Riemannian manifolds and finite geometries one even finds a conceptual explanation for the Standard Model of particle physics [6–8]. Still, the relevant Dirac operators are classical twisted
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Dirac operators and the module consists of twisted spinor fields. For these so-called almost-commutative geometries, the spectral action precisely predicts the Lagrangian of the corresponding Einstein-Yang-Mills-Higgs model. It has also successfully predicted scale invariant Lagrangians with dilaton fields [9], and quantum gravity boundary terms [10] in the case of manifolds with boundary. In this paper we take the spectral geometric point of view, in the sense that we are only dealing with classical objects where the noncommutative input resides in the exact structure of the twist bundle. We expand the known calculations of the spectral action (with and without the standard model) by implementing skew-symmetric torsion into the relevant twisted Dirac operators. As a result we find that in this context the Einstein-Cartan theory does not apply. One finds an action that suggests critical points with nonzero torsion which might even be dynamical. 2. The Spectral Action for Pure Gravity with Torsion In this first section we concern ourselves only with the gravitational part of the spectral action for a 4-dimensional closed Riemannian manifold M equipped with a spin structure. Let us now briefly recall the basic notions of connections and Dirac operators with torsion. Each connection on the tangent bundle of a manifold can be written as a sum of the Levi-Cevita connection ∇ LC and a (2, 1)-tensor field A, i.e. ∇ X Y = ∇ XLC Y + A(X, Y ). For such a general connection ∇ the torsion (3,0)-tensor is by definition T (X, Y, Z ) = ∇ X Y − ∇Y X − [X, Y ], Z . The connection ∇ is compatible with the Riemannian metric ·, · and has the same geodesics as ∇ LC if and only if A(X, Y, Z ) = A(X, Y ), Z is totally anti-symmetric. We will only consider this case. Then the torsion of ∇ is given by T = 2 A, and hence 1 ∇ X Y = ∇ XLC Y + T (X, Y, ·)# . 2
(1)
Note that (X, Y ) → T (X, Y, ·)# equals the vector valued torsion 2-form of E. Cartan, which is defined as the exterior covariant derivative of the soldering form. Since we assume that the manifold carries a spin structure, we can consider spinor fields ψ and the spin connection induced by ∇ can be expressed as 1 ∇ X ψ = ∇ XLC ψ + (X T ) · ψ. 4
(2)
Here (X T )· denotes Clifford multiplication1 by the 2-form T (X, ·, ·). This spin connection yields a Dirac operator D which one can write as Dψ = i ei · ∇ei ψ, for any orthonormal frame ei . From [11, Thm. 6.2] we deduce the Bochner form of the square of this Dirac operator: 1 9 3 D 2 = + dT + R − T02 , 4 4 8
(3)
where is the Laplacian associated to the spin connection X ψ = ∇ XLC ψ + 3 (X T ) · ψ, ∇ 4
(4)
1 Here we use the Clifford relations X · Y + Y · X = −2X, Y for tangent vectors X, Y . Any 2-form X ∧ Y acts as 21 X · Y · on the spinor module.
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dT is the exterior differential of the 3-form T , R is the scalar curvature of the Riemannian manifold (in our convention spheres have positive curvature, i.e. R = 12 for the 4-dimensional sphere) and T02 = 16 i,n j=1 T (ei , e j , ·)# 2 . For the Dirac operator D we will calculate the bosonic part of the spectral action. It is defined to be the number of Eigenvalues of D in the interval [−, ] with ∈ R+ . In [5] it is expressed as 2 D I = tr F . 2 Here tr denotes the operator trace in the Hilbert space of L 2 -spinor fields, and F : R+ → R+ is a cut-off function with support in the interval [0, +1] which is constant near the origin. Here we follow the notation of [12]. For t → 0 one has the heat trace asymptotics [13] 2 t n−2 a2n (D 2 ). tr e−t D ∼ n≥0
One uses the Seeley-deWitt coefficients a2n (D 2 ) and t = −2 to obtain an asymptotics for the spectral action [5,14] 2 D I = tr F ∼ 4 F4 a0 (D 2 ) + 2 F2 a2 (D 2 ) + 0 F0 a4 (D 2 ) as → ∞ (5) 2 ∞ with the first three moments of the cut-off function which are given by F4 = 0 s · ∞ F(s) ds, F2 = 0 F(s) ds and F0 = F(0). Note that these moments are independent of the geometry of the manifold. Setting E = − 43 dT − 41 R + 98 T02 , we get D 2 = − E from (3). We use [13, Thm. 4.1.6] to obtain the first three coefficients of the heat trace asymptotics:
1 a0 (D 2 ) = dvol, (6) 4π 2 M
1 a2 (D 2 ) = (6 tr(E) + 4R) dvol, 96π 2 M
1 2 tr 60 E + 60R E + 180E (7) + 30 a4 (D 2 ) = i j i j 5760π 2 M +48 LC R + 20R 2 − 8Ric2 + 8Riem2 dvol. Here Ric and Riem denote the Ricci curvature and the Riemannian curvature tensors ei ∇ e j ∇ [ei ,e j ] is the curvature of ∇. e j − ∇ ei − ∇ of the metric, and i j = ∇ 2 We evaluate the ai (D ) above and take into account that tr(dT ) = 0 due to Clifford relations and cyclicity of the trace. For the first two coefficients we get
9 2 1 1 1 2 a0 (D 2 ) = dvol, a (D ) = − T R dvol. 2 4π 2 M 16π 2 M 2 0 3 For a4 (D 2 ) we use that tr(E) = LC tr(E) and LC R vanish after integration over the closed manifold M:
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F. Hanisch, F. Pfäffle, C. A. Stephan
a4 (D 2 ) =
1 16π 2
M
1 2 1 1 R − Ric2 + Riem2 72 45 45
3 9 81 2 2 1 2 2 − RT0 + dT + (T0 ) + tr(i j i j ) dvol. 8 8 32 12 i, j
Similar calculations have been done in [15–19]. For the curvature i j of the connection we proceed by computing ∇ 1 3 i j = R(ei , e j )ea , eb + a(∇T )(ei , e j , ea , eb ) 4 8 a,b 9 + T (ei , ec , ea )T (e j , ec , eb ) − T (e j , ec , ea )T (ei , ec , eb ) ea eb , 16 c where a(∇T ) denotes the anti-symmetrisation in the first two entries of ∇T , i.e. a(∇T )(ei , e j , ea , eb ) = ∇ei T (e j , ea , eb ) − ∇e j T (ei , ea , eb ). Using the identity tr(ek el es et ) = 4(δls δkt − δlt δks ) we obtain 1 3 R(ei , e j )ea , eb + a(∇T )(ei , e j , ea , eb ) tr(i j i j ) = −8 4 8 i = j i, j
a,b
2 9 + c(T )(ei , e j , ea , eb ) , 8 where c(T )(ei , e j , ea , eb ) = c T (ei , ec , ea )T (e j , ec , eb ). This term equals the square of a norm in the space of (4, 0)-tensors. We use representation theory [20, Chap. 4] of O(4) to decompose these tensors into irreducible components. We note that c(T ) is a formal curvature tensor after interchanging the second and the third entry and hence we may write Riem, c(T ) ∈ R ⊕ Sym20 ⊕ Weyl. Here we consider R ⊕ Sym20 ⊕ Weyl ⊂ Sym2 (2 ). Furthermore a : 1 ⊗ 3 → a(1 ⊗ 3 ) ⊂ 2 ⊗ 2 is an isomorphism of O(4)-representations. The image splits into a(1 ⊗ 3 ) = 4 ⊕ 2 ⊕ Sym20 , where 4 ⊂ Sym2 (2 ) and 2 ⊕ Sym20 ⊂ 2 (2 ). As the above decompositions are orthogonal, we conclude that c(T ) ⊥ a(∇T ) and Riem ⊥ a(∇T ) in the space of (4, 0)-tensors. After identification by the above √ isomorphisms, this yields that the norm of the 4 -component of a(∇T ) equals 6dT and norm of the 2 -component is 2d ∗ T . The norm of the remaining component in Sym20 is denoted by 2sym 20 (∇T ). We compute: 1 1 3 9 3 tr(i j i j ) = − Riem2 − d ∗ T 2 − dT 2 − sym 20 (∇T )2 12 24 8 16 8 i, j
− where we abbreviate P(T ) :=
27 3 c(T )2 − P(T ), 32 8
R(ei , e j )ea , eb T (ei , ec , ea )T (e j , ec , eb ).
i = j a,b,c
Once more we insert the Ricci decomposition of the curvature tensor into the scalar curvature, the traceless Ricci tensor and the Weyl tensor W and obtain
The Spectral Action for Dirac Operators with Skew-Symmetric Torsion
P(T ) = −RT02 − +
881
Ric(e j , ea )T (ei , ec , ea )T (e j , ec , ei )
i = j a,c
W (ei , e j )ea , eb T (ei , ec , ea )T (e j , ec , eb ).
i = j a,b,c
Finally we obtain for the fourth heat coefficient with R(T ) = P(T ) + RT02 ,
1 2 1 1 7 81 27 R − Ric2 − Riem2 + (T02 )2 − c(T )2 a4 (D 2 ) = 2 16π M 72 45 360 32 32 9 3 3 3 + dT 2 − d ∗ T 2 − sym 20 (∇T )2 − R(T ) dvol. 16 8 8 8 Neglecting the term a4 in the spectral action, we would obtain the classical EinsteinCartan-action which has only torsion free critical points upon variation of metric and torsion 3-forms. A similar action functional for Dirac operators with totally anti-symmetric torsion has already been considered in [21,22], where the authors used the Wodzicki residue as the bosonic action. This gives an action involving only the second Seeley-deWitt coefficient. Considering the full spectral action (5), which also includes a4 (D 2 ), we observe that the term R(T ) seems to couple torsion and the trace free component of the curvature tensor. Therefore we expect critical points of the spectral action with non-zero torsion. Furthermore, due to the derivative terms of T , the torsion becomes dynamical. 3. The Spectral Action for the Standard Model with Torsion In the noncommutative approach to the Standard Model of particle physics, the fermionic Hilbert space is the product space of the Hilbert space of L 2 -sections in the spinor bundle S and a finite dimensional Hilbert space H f (called the finite or internal Hilbert space). The specific particle model is encoded in H f . The other important ingredient is a generalised Dirac operator D acting in the Hilbert space H = L 2 (M, S) ⊗ H f , where we follow the notation in [23].
S M = ∇ ⊗ idH + On the twisted spinor bundle S ⊗ H f one considers a connection ∇ f id S ⊗ ∇ H f , where ∇ is a connection with skew-symmetric torsion as in (2) and ∇ H f is a covariant derivative in the trivial bundle H f induced by gauge fields.2 The associated
S M
S M is called D ∇ Dirac operator to ∇ . The generalised Dirac operator of the Standard Model D contains the Higgs boson, Yukawa couplings, neutrino masses and the CKMmatrix encoded in a field of endomorphisms of H f . We follow the conventions of Chamseddine and Connes [5–8] and define D for sections ψ ⊗ χ ∈ H as
S M
D (ψ ⊗ χ ) = D ∇
(ψ ⊗ χ ) + γ5 ψ ⊗ χ ,
(8)
S M
where γ5 = e0 e1 e2 e3 is the volume element and D ∇ is the twisted Dirac operator (compare (18) in the Appendix). We note that D is required to be a self-adjoint operator, consistent with the axioms of noncommutative geometry [1–4]. From this one gets 2 The Clifford multiplication by a tangent vector X acts as X · (ψ ⊗ χ ) = (X · ψ) ⊗ χ . Note that the
S M is compatible with the Clifford multiplication. twisted connection ∇
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F. Hanisch, F. Pfäffle, C. A. Stephan
restrictions on , in particular it has to be self-adjoint and compatible with the real structure J and the chirality operator. We choose the same as Chamseddine and Connes [5–8] since the torsion does not effect these relations. The bosonic part of the Lagrangian of the Standard Model is obtained by replacing D by D in (5). First we need to calculate the Bochner formula for the square of D . We use the results from the Appendix (e.g. the definition of ∇ (16) and the Bochner
S M formula for D ∇ (17)) and get the following Bochner formula: n S M2
2 D (γ5 ei · ∇ei ψ) ⊗ χ + (ei · γ5 ∇ei ψ) ⊗ χ (ψ ⊗ χ ) = D ∇ (ψ ⊗ χ )+ i=1
+
n
H H γ5 ei · ψ ⊗ ∇ei f χ +ei ·γ5 ψ ⊗ ∇ei f ( χ ) +(γ5 )2 ψ ⊗ ( 2 )χ
i=1
S M 2
= D∇
(ψ ⊗ χ ) −
n
H
γ5 ei · ψ ⊗ [∇ei f , ] χ + ψ ⊗ ( 2 )χ
i=1 ∇
= (ψ ⊗ χ ) − E (ψ ⊗ χ ),
(9)
where the potential is defined as ⊗ χ) + E (ψ ⊗ χ ) = E(ψ
n
H
γ5 ei · ψ ⊗ [∇ei f , ] χ − ψ ⊗ ( 2 )χ
(10)
i=1
as in (19). with E We denote the trace on H and on H f as Tr and tr f , respectively (both pointwise and in the L 2 -sense). As above tr is the trace for the spinorial part S. From (10) 2 one obtains the trace Tr(E ) = rank Hf · tr(E) − 4 tr f ( ), since the endomorphism H i = j ei · e j · ψ ⊗ i j χ in (19) is skew-symmetric and hence traceless. 2 ) we also need to calculate For the Seeley-deWitt coefficient a4 (D
H H 1 ei · e j · ek · e · ψ ⊗ i j f k f χ (E )2 (ψ ⊗ χ ) = E 2 ψ ⊗ χ + · 4 i = j k =
+
n
H
H
γ5 ei · γ5 e j · ψ ⊗ [∇ei f , ] [∇e j f , ]χ + ψ ⊗ ( 4 ) χ − 2 E ψ ⊗ ( 2 ) χ
i, j=1
+
n H H E ei · e j · ψ ⊗ i j χ + (E γ5 ei · ψ + γ5 ei · E ψ) ⊗ [∇ei f , ] χ
i = j
i=1
H 1 H + · ei · e j · γ5 ek · ψ ⊗ i j f [∇ek f , ] χ 2 i = j k
H 1 H + · γ5 ek · ei · e j · ψ ⊗ [∇ek f , ] i j f χ 2 i = j k
The Spectral Action for Dirac Operators with Skew-Symmetric Torsion
883
H 1 H ei · e j · ψ ⊗ (i j f ( 2 ) + ( 2 ) i j f ) χ − · 2 −
i = j n
H H γ5 ei · ψ ⊗ ( 2 ) [∇ei f , ] + [∇ei f , ] ( 2 ) χ .
(11)
i=1
Only the first five summands on the right-hand side contribute to the trace of (E )2 . In four dimensions E consists of two summands proportional to the identity and one summand proportional to γ5 . Therefore the endomorphism defined by the sixth and seventh summand of (11) ψ ⊗ χ →
n H E ei · e j · ψ ⊗ iHj χ + (E γ5 ei · ψ + γ5 ei · E ψ) ⊗ [∇ei f , ] χ
i = j
i=1
is traceless due to Clifford relations and cyclicity of the trace. The trace of the endomorphism given by the remaining summands of (11) vanishes due to Clifford relations without employing that the dimension of the manifold is four (in other dimensions γ5 is then the volume element). Thus we find for the trace Hf 2 Hf f ) = rankHf · tr(E2 ) + tr f (H , ]2 ) Tr(E ij ij ) + 4 · tr f ([∇
+ 4 · tr f ( 4 ) − 2 · tr(E) · tr f ( 2 ).
(12)
2 ) is the trace of the squared curvature The last ingredient we need to calculate a4 (D tensor i∇j , see (20), associated to the connection in the Bochner formula (17). We note
H
that Tr(i j ⊗ i j f ) = 0 and therefore we find H H H Tr i∇j i∇j = Tr (i j i j ) ⊗ 1H f + 14 ⊗ (i j f i j f ) + 2 i j ⊗ i j f i, j
i, j
= rank Hf ·
H H tr ij ij + 4 · tr f ij f ij f .
i,j
i,j
We choose the finite space H f according to the construction of the noncommutative Standard Model [5,1–4,6–8], i.e. rankHf = 96 and ∇ H f is the appropriate covariant derivative associated to the Standard Model gauge group U (1)Y × SU (2)w × SU (3)c . Inserting the above results into the spectral action (5) we obtain for the bosonic Lagrangian of the Standard Model coupled to gravity and torsion: 2 D 2 2 2 Ibos. = Tr F ) + 2 F2 a2 (D ) + 0 F0 a4 (D ) + O(−2 ) = 4 F4 a0 (D 2
4 F4 2 F2 = Tr(1rankH ) dvol + Tr (6E + R) dvol 16π 2 M 96π 2 M
0 F0 ∇ ∇ 2 Tr 60R E + + 180E + 30 i j i j 5760π 2 M +20R 2 − 8Ric2 + 8Riem2 dvol + O(−2 )
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F. Hanisch, F. Pfäffle, C. A. Stephan
244 F4 = dvol π2 M
2 F2 2 96 6tr(E) + 4R − 4tr + ( ) dvol f 96π 2 M
F0 96 60Rtr(E) + 180tr(E 2 ) + 30tr(i j i j ) + 2 5760π M +80R 2 − 32Ric2 + 32Riem2 H
H
+300 tr f (i j f i j f ) + 720 tr f ([∇ H f , ]2 ) +720 tr f ( 4 )−360 tr(E)·tr f ( 2 ) dvol +O(−2 ). To cast this action into a more familiar form we use the standard formulas to H H express tr f (i j f i j f ) in terms of the norms of the gauge field strengths G2 = i μνi , F2 = α μνα , B2 = μν i μ,ν,i G μν G μ,ν,α Fμν F μν Bμν B . Here G μν is α the curvature of the SU (3)c -connection with coupling g3 , Fμν is the curvature of the SU (2)w - connection with coupling g2 and Bμν is the curvature of the U (1)Y -connection with coupling g1 . We also calculate the traces of the Higgs endomorphisms 2 and 4 explicitly in terms of the Higgs doublet ϕ and obtain tr f ( 2 ) = 4a |ϕ|2 + 2c and tr f ( 4 ) = 4b |ϕ|4 + 8e|ϕ|2 + 2d. The coefficients a, b, c, d and e are traces of the 3 × 3 Yukawa matrices for the quarks (ku and kd ), the leptons (ke and kν ) and the Majorana mass matrix for the right-handed neutrinos (kν R ) given by a = tr 3 (3|ku |2 + 3|kd |2 + |ke |2 + |kν |2 ), b = tr 3 (3|ku |4 + 3|kd |4 + |ke |4 + |kν |4 ), c = tr 3 (|kν R |2 ), d = tr 3 (|kν R |4 ), e = tr 3 (|kν |2 |kν R |2 ). We conclude that the spectral action principle predicts the following form of the bosonic Lagrangian for the Standard model in the presence of skew-symmetric torsion: Ibos. =
244 F4 2 F2 1 2 2 27 T c dvol dvol + − 2R − a|ϕ| − 0 π2 π2 M 2 M
1 2 F0 4 7 243 2 2 27 R − Ric2 − Riem2 + (T0 ) − c(T )2 + 2 2π M 6 15 30 8 4 27 9 9 9 + dT 2 − d ∗ T 2 − sym 20 (∇T )2 − R(T ) 4 2 2 2 5 + g32 G2 + g22 F2 + g12 B2 3 1 1 1 2 4 + a|Dν ϕ| + b|ϕ| + 2e|ϕ|2 + d + R a|ϕ|2 + c 2 6 2 9 2 1 − T0 a|ϕ|2 + c dvol + O(−2 ). (13) 4 2
The Spectral Action for Dirac Operators with Skew-Symmetric Torsion
885
Here we were able to use the standard results from the torsion free case, see [12, p. 22] or [6–8,23]. As in the pure gravity+torsion case the torsion becomes dynamical and couples only with the trace free part of the Riemann curvature tensor. In presence of the Standard Model fields we obtained essentially one new term (apart from the usual suspects) coupling the torsion to the Higgs field
9a F0 Inew = − T 2 |ϕ|2 dvol. (14) 8π 2 M 0 This is another amazing feature of the spectral action principle: it supports the interpretation of the Higgs field as the gravitational field of the internal space in the noncommutative product geometry [1–4]. The full Standard Model action is given by 2 D 1 I S M = Tr F with ∈ H, (15) + J , D 2 2 where the fermionic action 21 J , D contains a coupling between torsion and the fermions, and J is the real structure of the spectral triple. This action takes care of the fermion doubling problem, compare [12, (5.9)]. 4. Conclusions We have calculated the spectral action for Dirac operators arising from geometries with skew-symmetric torsion and their twisted version originating in the noncommutative approach to the standard model. In both cases we find that torsion couples to the trace free part of the Riemann curvature tensor and in the latter case to the Higgs boson of the standard model. Furthermore the torsion becomes dynamical due to derivative terms in the action. Now one certainly has to wonder about possible experimental signatures of these new phenomena, both on local scales (Earth and the solar system) and cosmological scales. We may assume for the moment that the Schwarzschild metric is a good approximation to the gravitational field of the Earth (even for the “new” spectral action with torsion). The curvature of the Schwarzschild metric is non-zero and hence torsion for the corresponding critical point of the action seems probable. Then we would expect effects on freely falling particles or atoms with different spins. This might lead to measurable effects in atom interferometry experiments. The cosmological consequences are much more speculative. It has been noted [24,25] that torsion induces four-fermion interactions which in turn may provide a possible solution to the problem of the enormously large cosmological constant c ∼ ∼ 1017 GeV predicted by the spectral action (without torsion) [26–28]. In this framework also a natural mechanism for inflation appears naturally. To obtain more rigourous results it will be necessary to investigate the Euler-Lagrange equations of the spectral action with torsion. It would be interesting to find exact solutions with non-vanishing torsion and compare them with the known solutions of Einstein’s equations. Acknowledgements. We gratefully acknowledge funding of this work by the Deutsche Forschungsgemeinschaft in particular the SFB 647 “Raum-Zeit-Materie”.
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F. Hanisch, F. Pfäffle, C. A. Stephan
Appendix: Bochner Formula for Twisted Dirac Operators in Presence of Torsion We consider a closed Riemannian spin manifold M of dimension n and a connection ∇ on the tangent bundle which is compatible with the metric and has totally anti-symmetric torsion T . By ∇ we also denote the induced connection on the spinor bundle S of M which can be expressed as in (2). Given a vector bundle H over M with connection ∇ H
= ∇ ⊗ idH + id S ⊗ ∇ H and the associated twisted we consider the tensor connection ∇
∇ Dirac operator D acting on sections of S ⊗ H. We define another tensor connection ⊗ idH + id S ⊗ ∇ H , ∇=∇
(16)
is the spin connection from (4), and we claim the Bochner formula where ∇ 2
D ∇ = ∇ − E,
(17)
denotes an endomorwhere ∇ denotes the horizontal Laplacian associated to ∇ and E phism field of S ⊗ H which still has to be determined. To that end we fix an arbitrary point p ∈ M, and we choose a local orthonormal basis of vector fields e1 , . . . , en with ∇ei = 0 in p (for any i). From (1) we get that ∇eLC ei = ∇ei ei = 0 in p (for any i) and the Lie bracket [ei , e j ] = ∇eLC e j − ∇eLC ei = i i j # −T (ei , e j , ·) in p (for any i, j). For any section ψ of S and any section χ of H we get in p:
D ∇ (ψ ⊗ χ ) =
n ei · ∇ei ψ ⊗ χ + (ei · ψ) ⊗ ∇eHi χ , i=1
D∇
2
n e j · ei · ψ ⊗ ∇eHj ∇eHi χ (ψ ⊗ χ ) = D 2 ψ ⊗ χ +
i, j=1
+
n
e j · ei · ∇ei ψ ⊗ ∇eHj χ + e j · ei · ∇e j ψ ⊗ ∇eHi χ
i, j=1
n n = D2ψ ⊗ χ + e j ·ei ·ψ ⊗∇eHj ∇eHi χ − 2· ∇ei ψ ⊗ ∇eHi χ , i, j=1
i=1
(18) where the last equation holds due to Clifford relations. In p we As before, let denote the Laplacian associated to the spin connection ∇. obtain for the Laplacian associated to ∇: ∇ (ψ ⊗ χ ) = (ψ) ⊗ χ −
n
ψ ⊗ ∇eHi ∇eHi χ − 2 ·
i=1
= (ψ) ⊗ χ +
n
−
i=1
ei ψ ⊗ ∇eH χ ∇ i
i=1
ei · ei · ψ ⊗ ∇eHi ∇eHi χ − 2 ·
i=1 n
n
((ei T ) · ψ) ⊗ ∇eHi χ ,
n i=1
∇ei ψ ⊗ ∇eHi χ
The Spectral Action for Dirac Operators with Skew-Symmetric Torsion
887
where we have used (2) and (4). In p we observe n n H ei · e j · ψ ⊗ ∇[e ei · e j · ψ ⊗ ∇TH(e ,e χ = − i ,e j ] i
i, j=1
=−
j ,·)
#
χ
i, j=1 n n ei · e j · ψ ⊗∇TH(ei ,e j ,ek )ek χ = − T (ei , e j , ek ) ei · e j · ψ ⊗∇eHk χ i, j,k=1 n
= −2 ·
i, j,k=1
((ek T ) · ψ) ⊗ ∇eHk χ .
k=1
Putting all this together we get in p that 2
D ∇ (ψ ⊗ χ ) − ∇ (ψ ⊗ χ ) 1 H ei · e j · ψ ⊗ ∇eHi ∇eHj χ − · ei ·e j ·ψ ⊗ ∇[e = D 2 ψ − ψ ⊗χ + χ i ,e j ] 2 i = j i = j 1 = D 2 ψ − ψ ⊗ χ − · ei · e j · ψ ⊗ H i j χ , 2 i = j
H is the curvature endomorphism of the twist where iHj = ∇eHi ∇eHj − ∇eHj ∇eHi − ∇[e i ,e j ] in the bundle H. Taking (3) into account we can identify the endomorphism field E following Bochner formula (17) as 1 9 2 3 1 E (ψ ⊗ χ ) = (− dT − R + T0 ) ψ ⊗ χ + · ei · e j · ψ ⊗ iHj χ 4 4 8 2 i = j H 1 = (E ψ) ⊗ χ + · ei · e j · ψ ⊗ i j χ . (19) 2 i = j
⊗ idH + id S ⊗ ∇ H is Finally we remark that the curvature of the connection ∇ = ∇ given by ∇ i j (ψ ⊗ χ ) = i j ψ ⊗ χ + ψ ⊗ iHj χ , (20) where i j is the curvature of the spin connection ∇. References 1. Connes, A.: Noncommutative Geometry. Academic Press, London-NewYork (1994) 2. Connes, A.: Noncommutative geometry and reality. J. Math. Phys. 36, 6194 (1995) 3. Connes, A.: Gravity coupled with matter and the foundation of noncommutative geometry. Commun. Math. Phys. 155, 109 (1996) 4. Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. http://www. alainconnes.org/docs/bookwebfinal.pdf, 2007 5. Chamseddine, A., Connes, A.: The spectral action principle. Commun. Math. Phys. 182, 155 (1996) 6. Chamseddine, A., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. 11, 991 (2007)
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7. Chamseddine, A., Connes, A.: Why the Standard Model. J. Geom. Phys. 58, 38 (2008) 8. Chamseddine, A., Connes, A.: Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model. Phys. Rev. Lett. 99, 191601 (2007) 9. Chamseddine, A., Connes, A.: Scale invariance in the spectral action. J. Math. Phys. 47, 063504 (2006) 10. Chamseddine, A., Connes, A.: Quantum Gravity Boundary Terms from Spectral Action. Phys. Rev. Lett. 99, 071302 (2007) 11. Agricola, I., Friedrich, T.: On the holonomy of connections with skew-symmetric torsion. Math. Ann. 328, 711 (2004) 12. Chamseddine, A., Connes, A.: Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I. http://arxiv.org/abs/1004.0464v1 [hep-th], 2010 13. Gilkey, P.B.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (second edition), Boca Raton, FL: CRC Press, 1995 14. Nest, R., Vogt, E., Werner, W.: Spectral Action and the Connes-Chamseddine Model. In: Scheck, F., Upmeier, H., Werner, W. (eds.), Lecture Notes in Physics 596, Berlin-Heidelberg-NewYork: Springer, 2002, p.109 15. Goldthorpe, W.H.: Spectral Geometry and SO(4) Gravity in a Riemann-Cartan Spacetime. Nucl. Phys. B170, 307 (1980) 16. Obukhov, Yu.N.: Spectral Geometry Of The Riemann-Cartan Space-Time. Nucl. Phys. B212, 237 (1983) 17. Buchbinder, I.L., Odintsov, S.D., Shapiro, I.L.: Nonsingular cosmological model with torsion induced by vacuum quantum effects. Phys. Lett. B162, 92 (1985) 18. Grensing, G.: Induced Gravity For Nonzero Torsion. Phys. Lett. B169, 333 (1986) 19. Cognola, G., Zerbini, S.: Seeley-deWitt Coefficents in a Riemann-Cartan Manifold. Phys. Lett. B214, 70 (1988) 20. Salamon, S.: Riemannian geometry and holonomy groups. Pitman Research Notes in Mathematics Series 201; New York: Wiley & Sons, Inc., 1989 21. Kalau, W., Walze, M.: Gravity, noncommutative geometry and the Wodzicki residue. J. Geom. Phys. 16, 327 (1995) 22. Ackermann, T., Tolksdorf, J.: The Generalized Lichnerowicz formula and analysis of Dirac operators. http://arxiv.org/abs/hep-th/9503153v1, 1995 23. Iochum, B., Kastler, D., Schücker, T.: On the universal Chamseddine-Connes action. I. Details of the action computation. J. Math. Phys. 38(10), 4929 (1997) 24. Perez, A., Rovelli, C.: Physical effects of the Immirzi parameter. Phys. Rev. D73, 044013 (2006) 25. Mercuri, S.: Fermions in the Ashtekar-Barbero connection formalism for arbitrary values of the Immirzi parameter. Phys. Rev. D73, 084016 (2006) 26. Alexander, S., Vaid, D.: Gravity induced chiral condensate formation and the cosmological constant. http://arxiv.org/abs/hep-th/0609066v1, 2006; 27. Alexander, S., Vaid, D.: A Fine tuning free resolution of the cosmological constant problem. http://arxiv. org/abs/hep-th/0702064v2, 2007; 28. Alexander, S., Biswas, T.: Cosmological BCS mechanism and the Big Bang Singularity. Phys. Rev. D80, 023501 (2009) Communicated by A. Connes
