Trace formulae for Schrödinger operators with singular interactions
aa r X i v : . [ m a t h . SP ] D ec Trace formulae for Schrödinger operators withsingular interactions
Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik ∗ Dedicated with great pleasure to our teacher, colleague and friendPavel Exner on the occasion of his 70th birthday.
Abstract.
Let Σ ⊂ R d be a C ∞ -smooth closed compact hypersurface, which splitsthe Euclidean space R d into two domains Ω ± . In this note self-adjoint Schrödingeroperators with δ and δ ′ -interactions supported on Σ are studied. For large enough m ∈ N the difference of m th powers of resolvents of such a Schrödinger operator and the freeLaplacian is known to belong to the trace class. We prove trace formulae, in which thetrace of the resolvent power difference in L ( R d ) is written in terms of Neumann-to-Dirichlet maps on the boundary space L (Σ). Primary 35P20; Secondary 35J10, 35P25,47B10, 47F05, 81U99.
Keywords.
Trace formula, delta interaction, Schrödinger operator, singular potential.
1. Introduction
This paper is strongly inspired by the work of Pavel Exner on Schrödinger operatorswith singular interactions of δ and δ ′ -type supported on hypersurfaces in R d . Suchoperators play an important role in mathematical physics, for instance, in nuclearphysics or solid state physics or in connection with photonic crystals or othernanostructures. In the case of a curve in R such models are also called “leakyquantum wires”. The first rigourous investigations of such operators started in thelate 1980s (see, e.g. [1, 14, 15]), and the interest in these operators grew steadily inthe last two decades. We refer the reader to the review paper [26], the monograph[30], the references therein and also to the more recent papers [6, 7, 10, 23, 24, 29,35, 44].Let Σ ⊂ R d , where d ≥
2, be a C ∞ -smooth closed compact hypersurfacewithout boundary, which naturally splits the Euclidean space R d into a boundeddomain Ω − and an exterior domain Ω + . Moreover, let α, ω ∈ L ∞ (Σ) be real-valuedfunctions. The Schrödinger operator H α, Σ with δ -interaction of strength α and theSchrödinger operator K ω, Σ with δ ′ -interaction of strength ω are formally given by − ∆ − αδ ( x − Σ) and − ∆ − ωδ ′ ( x − Σ) . (1.1) ∗ JB gratefully acknowledges financial support by the Austrian Science Fund (FWF): ProjectP 25162-N26. VL gratefully acknowledges financial support by the Czech Science Foundation(GAČR): Project 14-06818S.
Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik
We define these operators rigorously via quadratic forms; see Definition 1.1 below.Let us first fix some notation. Since the space L ( R d ) naturally decomposes as L ( R d ) = L (Ω + ) ⊕ L (Ω − ), we can write functions u ∈ L ( R d ) as u = u + ⊕ u − with u ± = u ↾ Ω ± ∈ L (Ω ± ). The L -based Sobolev spaces of order s ≥ R d and Ω ± are denoted by H s ( R d ) and H s (Ω ± ), respectively. Note that thehypersurface Σ coincides with the boundaries ∂ Ω ± of the domains Ω ± . Hence, forany u ∈ H ( R d ) and u ± ∈ H (Ω ± ) the traces u | Σ and u ± | Σ on Σ are well definedas functions in L (Σ). Further, for a function u ∈ H ( R d \ Σ) := H (Ω + ) ⊕ H (Ω − )we define its jump on Σ as [ u ] Σ := u + | Σ − u − | Σ .Let us now introduce the following quadratic forms that correspond to theformal expression in (1.1). According to [14, §2], [10, §3.4] and [6, Proposition 3.1],the symmetric quadratic forms h α, Σ [ u ] := k∇ u k − ( αu | Σ , u | Σ ) Σ , dom h α, Σ := H ( R d ) , k ω, Σ [ u ] := k∇ u + k + k∇ u − k − − ( ω [ u ] Σ , [ u ] Σ ) Σ , dom k ω, Σ := H ( R d \ Σ) , in L ( R d ) are closed, densely defined and bounded from below; here u = u + ⊕ u − with u ± as above, and k · k ± denotes the norm on L (Ω ± ; C d ). Definition 1.1.
Let H α, Σ and K ω, Σ be the self-adjoint operators in L ( R d ) corre-sponding to the forms h α, Σ and k ω, Σ , respectively, via the first representation the-orem ([37, Theorem VI.2.1]) . Moreover, set H free := H , Σ ( α ≡ and K N := K , Σ ( ω ≡ . The operator H α, Σ is called Schrödinger operator with δ -interaction of strength α supported on Σ; the operator K ω, Σ is called Schrödinger operator with δ ′ -interaction of strength ω supported on Σ. The operator H free is the usual freeLaplacian on R d , and K N is the orthogonal sum of the standard Neumann Lapla-cians on Ω + and Ω − . Let us mention that the operators H α, Σ and K ω, Σ can alsobe introduced via interface conditions at the hypersurface Σ; see, e.g. [10].The aim of this paper is to derive trace formulae for H α, Σ and K ω, Σ . Accordingto [10] for m ∈ N the resolvent power differences( H α, Σ − λ ) − m − ( H free − λ ) − m , m > d − , λ ∈ ρ ( H α, Σ ) , ( K ω, Σ − λ ) − m − ( H free − λ ) − m , m > d − , λ ∈ ρ ( K ω, Σ ) , (1.2)are in the trace class. Their traces as functions of λ or as functions of interactionstrengths are expected to encode a lot of information on the operators H α, Σ and K ω, Σ themselves and on the shape of Σ. Such non-trivial connections have beenobserved in various other settings in the classical papers [18, 36, 42] and morerecently in, e.g. [5, 34, 38, 39, 46].The main results of the paper (see Theorems 1.2 and 1.3) are formulae thatexpress the traces of the resolvent power differences in (1.2) in terms of traces We point out that, in the case of invertible ω , not ω itself, but its inverse is frequently calledthe strength of the δ ′ -interaction. race formulae for Schrödinger operators with singular interactions L (Σ).These operator-valued functions are, in turn, expressed in terms of Neumann-to-Dirichlet maps on Ω ± corresponding to the differential expression − ∆ − λ and interms of the coupling functions α , ω . Trace formulae of this kind are useful (see,e.g. [19, 20, 32]) in connection with the estimation of the spectral shift function. We first recall some notions that are needed in order to formulate the main resultsof this paper. For a compact operator K in a Hilbert space H we define its singularvalues s k ( K ), k = 1 , , . . . , as the eigenvalues of the non-negative compact operator | K | = ( K ∗ K ) / ≥ H ordered in non-decreasing way and with multiplicitiestaken into account. If P ∞ k =1 s k ( K ) < ∞ , we say that K belongs to the trace class and define its trace as Tr K := ∞ X k =1 λ k ( K ) , where λ k ( K ) are the eigenvalues of K repeated with their algebraic multiplicities .Note also that the series in the definition of the trace converges absolutely.Let us also define some auxiliary maps associated with partial differential equa-tions. For the sake of brevity, we introduce the spaces H / (Ω ± ) := (cid:8) u ± ∈ H / (Ω ± ) : ∆ u ± ∈ L (Ω ± ) (cid:9) . (1.3)For any u ± ∈ H / (Ω ± ) its Neumann trace ∂ ν ± u ± | Σ exists as a function in L (Σ);see, e.g. [43, §2.7.3]. For every λ ∈ C \ R + (where R + := [0 , ∞ )) and every ϕ ∈ L (Σ) the boundary value problems − ∆ u ± = λu ± in Ω ± ,∂ ν ± u ± (cid:12)(cid:12) Σ = ϕ on Σ , have unique solutions u λ, ± ( ϕ ) ∈ H / (Ω ± ); see, e.g. [43, §2.7.3]. The operator-valued functions λ M ± ( λ ), λ ∈ C \ R + , are then defined as M ± ( λ ) : L (Σ) → L (Σ) , M ± ( λ ) ϕ := u λ, ± ( ϕ ) | Σ . For fixed λ ∈ C \ R + the operators M ± ( λ ) are the Neumann-to-Dirichlet maps forthe differential expression − ∆ − λ on the domains Ω ± . The operators M ± ( λ ) arecompact and injective and their inverses are called Dirichlet-to-Neumann maps .Recently, there has been a considerable growth of interest in the investigation ofthese maps (see, e.g. [3, 4, 25]), in particular also with the aim to derive spectralproperties of the corresponding partial differential operators (see, e.g. [2, 13, 31]).Further, we define the following operator-valued functions λ f M ( λ ) , c M ( λ ), λ ∈ C \ R + , by f M ( λ ) := (cid:0) M + ( λ ) − + M − ( λ ) − (cid:1) − , c M ( λ ) := M + ( λ ) + M − ( λ ) . (1.4) Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik
We should mention that for every λ ∈ C \ R + the operator M + ( λ ) − + M − ( λ ) − is invertible and therefore f M ( λ ) is well defined. Moreover, f M ( λ ) and c M ( λ ) arecompact operators in L (Σ) for every λ ∈ C \ R + ; see [10, Propositions 3.2 and 3.8].It is worth mentioning that f M ( λ ) and the inverse of c M ( λ ) appear naturally in thetheory of boundary integral operators. They are used in the treatment of partialdifferential equations from both analytical [45] and computational [48] viewpoints.The operator-valued function f M ( · ) was successfully applied to the spectral analysisof the operator H α, Σ in quite a few papers; see, e.g. [27, 28, 30, 41] and the surveypaper [26]. In the first main result of this note weobtain a trace formula for the resolvent power difference of the operators H α, Σ and H free . Theorem 1.2.
Let the self-adjoint operators H free and H α, Σ with α ∈ L ∞ (Σ; R ) beas in Definition 1.1, and let the operator-valued function f M be as in (1.4) . Thenfor all m ∈ N such that m > d − and all λ ∈ ρ ( H α, Σ ) the resolvent power difference e D α,m ( λ ) := ( H α, Σ − λ ) − m − ( H free − λ ) − m belongs to the trace class, and its trace can be expressed as Tr (cid:0) e D α,m ( λ ) (cid:1) = 1( m − d m − d λ m − (cid:16)(cid:0) I − α f M ( λ ) (cid:1) − α f M ′ ( λ ) (cid:17)! . In the second main result of this note we obtain trace formulae for the resolventpower differences of the pairs of operators { K ω, Σ , K N } and { K ω, Σ , H free } . Theorem 1.3.
Let the self-adjoint operators H free , K N and K ω, Σ with ω ∈ L ∞ (Σ; R ) be as in Definition 1.1, and let the operator-valued function c M be as in (1.4) . Thenthe following statements hold. (i) For all m ∈ N such that m > d − and all λ ∈ ρ ( K ω, Σ ) the resolvent powerdifference b E ω,m ( λ ) := ( K ω, Σ − λ ) − m − ( K N − λ ) − m belongs to the trace class, and its trace can be expressed as Tr (cid:0) b E ω,m ( λ ) (cid:1) = 1( m − d m − d λ m − (cid:16)(cid:0) I − ω c M ( λ ) (cid:1) − ω c M ′ ( λ ) (cid:17)! . (ii) For all m ∈ N such that m > d − and all λ ∈ ρ ( K ω, Σ ) the resolvent powerdifference b D ω,m ( λ ) := ( K ω, Σ − λ ) − m − ( H free − λ ) − m belongs to the trace class, and its trace can be expressed as Tr (cid:0) b D ω,m ( λ ) (cid:1) = 1( m − d m − d λ m − (cid:16)(cid:0) I − ω c M ( λ ) (cid:1) − c M ( λ ) − c M ′ ( λ ) (cid:17)! . race formulae for Schrödinger operators with singular interactions H free and K N isderived in Lemma 3.4. This trace formula is also of certain independent interest.We also mention that a similar strategy of proof was employed in our previouspaper [12] where we proved trace formulae for generalized Robin Laplacians.
2. Preliminaries
This section consists of five subsections. In Subsection 2.1 we recall the notion ofweak Schatten–von Neumann classes and their connection with the trace class, andin Subsection 2.2 we collect certain formulae that involve derivatives of holomor-phic operator-valued functions. Next, in Subsection 2.3 we recall the definitionsof quasi boundary triples and associated γ -fields and Weyl functions. Krein’sresolvent formulae and sufficient conditions for self-adjointness of extensions arediscussed in Subsection 2.4. Finally, in Subsection 2.5 we introduce specific quasiboundary triples, which are used to parameterize Schrödinger operators with sin-gular interactions from Definition 1.1. S p, ∞ -classes and the trace mapping. Let H and K be Hilbert spaces.Denote by S ∞ ( H , K ) the class of all compact operators K : H → K . Recall that,for p >
0, the weak Schatten–von Neumann ideal S p, ∞ ( H , K ) is defined by S p, ∞ ( H , K ) := n K ∈ S ∞ ( H , K ) : s k ( K ) = O (cid:0) k − /p (cid:1) , k → ∞ o . Often we just write S p, ∞ instead of S p, ∞ ( H , K ). For 0 < p ′ < p the inclusion S p, ∞ ⊂ S p ′ , ∞ (2.1)holds, and for s, t > S s , ∞ · S t , ∞ = S s + t , ∞ , (2.2)where a product of operator ideals is defined as the set of all products. We referthe reader to [33, §§III.7 and III.14] and [47, Chapter 2] for a detailed study ofthe classes S p, ∞ ; see also [11, Lemma 2.3]. If K ∈ S p, ∞ with p <
1, then K belongs to the trace class. It is well known (see, e.g. [33, §III.8]) that, for traceclass operators K , K , the operator K + K is also in the trace class, and thatTr( K + K ) = Tr K + Tr K . (2.3) Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik
Moreover, if K ∈ B ( H , K ) and K ∈ B ( K , H ) are such that both products K K and K K are in the trace class, thenTr( K K ) = Tr( K K ) . (2.4)The next useful lemma is a special case of [11, Lemma 4.7] and is based on theasymptotics of the eigenvalues of the Laplace–Beltrami operator. For a smoothcompact manifold Σ we denote the usual L -based Sobolev spaces by H r (Σ), r ≥ Lemma 2.1.
Let Σ be a ( d − -dimensional compact C ∞ -manifold without bound-ary, let K be a Hilbert space and let K ∈ B ( K , L (Σ)) with ran K ⊂ H r (Σ) , where r > . Then K is compact and K ∈ S d − r , ∞ . In the follow-ing we shall often use product rules for holomorphic operator-valued functions. Let H i , i = 1 , . . . ,
4, be Hilbert spaces, U a domain in C and let A : U → B ( H , H ), B : U → B ( H , H ), C : U → B ( H , H ) be holomorphic operator-valued func-tions. Then for λ ∈ U we have d m d λ m (cid:0) A ( λ ) B ( λ ) (cid:1) = X p + q = mp,q ≥ (cid:18) mp (cid:19) A ( p ) ( λ ) B ( q ) ( λ ) , (2.5a) d m d λ m (cid:0) A ( λ ) B ( λ ) C ( λ ) (cid:1) = X p + q + r = mp,q,r ≥ m ! p ! q ! r ! A ( p ) ( λ ) B ( q ) ( λ ) C ( r ) ( λ ) . (2.5b)If A ( λ ) − is invertible for every λ ∈ U , then relation (2.5a) implies the followingformula for the derivative of the inverse, dd λ (cid:0) A ( λ ) − (cid:1) = − A ( λ ) − A ′ ( λ ) A ( λ ) − . (2.6) γ -fields. We begin thissubsection by recalling the abstract concept of quasi boundary triples introducedin [8] as a generalization of the notion of (ordinary) boundary triples [16, 40]. Forthe theory of ordinary boundary triples and associated Weyl functions the readermay consult, e.g. [17, 21, 22]. Recent developments on quasi boundary triples andtheir applications to PDEs can be found in, e.g. [9, 11, 12, 13].
Definition 2.2.
Let S be a closed, densely defined, symmetric operator in a Hilbertspace ( H , ( · , · ) H ) . A triple { G , Γ , Γ } is called a quasi boundary triple for S ∗ if ( G , ( · , · ) G ) is a Hilbert space, and for some linear operator T ⊂ S ∗ with T = S ∗ thefollowing assumptions are satisfied: (i) Γ , Γ : dom T → G are linear mappings, and the mapping Γ := (cid:0) Γ Γ (cid:1) hasdense range in G × G ; (ii) A := T ↾ ker Γ is a self-adjoint operator in H ; race formulae for Schrödinger operators with singular interactions for all f, g ∈ dom T the abstract Green identity holds: ( T f, g ) H − ( f, T g ) H = (Γ f, Γ g ) G − (Γ f, Γ g ) G . Next, we recall the definitions of the γ -field and the Weyl function associated witha quasi boundary triple { G , Γ , Γ } for S ∗ . Note that the decompositiondom T = dom A ˙+ ker( T − λ )holds for all λ ∈ ρ ( A ), so that Γ ↾ ker( T − λ ) is injective for all λ ∈ ρ ( A ). The(operator-valued) functions γ and M defined by γ ( λ ) := (cid:0) Γ ↾ ker( T − λ ) (cid:1) − and M ( λ ) := Γ γ ( λ ) , λ ∈ ρ ( A ) , are called the γ -field and the Weyl function corresponding to the quasi boundarytriple { G , Γ , Γ } . The adjoint of γ ( λ ) has the following representation: γ ( λ ) ∗ = Γ ( A − λ ) − , λ ∈ ρ ( A ); (2.7)see [8, Proposition 2.6 (ii)] and also [9, Proposition 7.5]. According to [8, Propo-sition 2.6] the operator-valued functions λ γ ( λ ), λ γ ( λ ) ∗ and λ M ( λ ) areholomorphic on ρ ( A ). Finally, we recall formulae for their derivatives: for k ∈ N , ϕ ∈ ran Γ and λ ∈ ρ ( A ) we have γ ( k ) ( λ ) ϕ = k !( A − λ ) − k γ ( λ ) ϕ, (2.8a) d k d λ k (cid:0) γ ( λ ) (cid:1) ∗ = k ! γ ( λ ) ∗ ( A − λ ) − k , (2.8b) M ( k ) ( λ ) ϕ = k ! γ ( λ ) ∗ ( A − λ ) − ( k − γ ( λ ) ϕ ; (2.8c)see [12, Lemma 2.4]. Inthis subsection we parameterize subfamilies of self-adjoint extensions via quasiboundary triples and provide a couple of useful Krein-type formulae for resolventdifferences of these extensions.The following hypothesis will be useful in the following.
Hypothesis 2.1.
Let S be a closed, densely defined, symmetric operator in aHilbert space H and let { G , Γ , Γ } be a quasi boundary triple for S ∗ such that ran Γ = G . Moreover, let γ and M be the associated γ -field and Weyl function,respectively. We remark that the quasi boundary triple { G , Γ , Γ } in Hypothesis 2.1 is alsoa generalized boundary triple in the sense of [22]. In this case the γ -field and theWeyl function associated with { G , Γ , Γ } are defined on the whole space G , andthe formulae (2.8a) and (2.8c) are valid for all ϕ ∈ G .Next, we state a Krein-type formula for the resolvent difference of A j := T ↾ ker Γ j , j = 0 , Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik
Proposition 2.3. [12, Theorem 2.5]
Assume that Hypothesis 2.1 is satisfied andthat A is self-adjoint in H . Then the formula ( A − λ ) − − ( A − λ ) − = γ ( λ ) M ( λ ) − γ ( λ ) ∗ holds for all λ ∈ ρ ( A ) ∩ ρ ( A ) . In the next proposition, we formulate a sufficient condition for self-adjointnessof the extension of A defined by A [ B ] := T ↾ ker( B Γ − Γ ) , and provide a Krein-type formula for the resolvent difference of A [ B ] and A . Proposition 2.4. [12, Theorem 2.6]
Assume that Hypothesis 2.1 is satisfied, that M ( λ ) ∈ S ∞ ( G ) for some λ ∈ ρ ( A ) , and that B ∈ B ( G ) is self-adjoint in G .Then the extension A [ B ] of A is self-adjoint in H , and the formula ( A [ B ] − λ ) − − ( A − λ ) − = γ ( λ ) (cid:0) I − BM ( λ ) (cid:1) − Bγ ( λ ) ∗ holds for all λ ∈ ρ ( A [ B ] ) ∩ ρ ( A ) . In this formula the middle term satisfies (cid:0) I − BM ( λ ) (cid:1) − ∈ B ( G ) for all λ ∈ ρ ( A [ B ] ) ∩ ρ ( A ) . We recall particularquasi boundary triples, which are used to parameterize the self-adjoint operatorsfrom Definition 1.1. Furthermore, we reformulate some of the abstract statementsfrom Subsections 2.3 and 2.4 for these quasi boundary triples.First, we introduce the subspace H / ( R d \ Σ) of L ( R d ) by H / ( R d \ Σ) := H / (Ω + ) ⊕ H / (Ω − ) , where H / (Ω ± ) are as in (1.3). Further, to shorten the notations, we also de-fine the jump of the normal derivative by [ ∂ ν u ] Σ := ∂ ν + u + | Σ + ∂ ν − u − | Σ for u ∈ H / ( R d \ Σ). Following the lines of [10, Section 3], we define the opera-tors e T and b T in L ( R d ) by e T u := ( − ∆ u + ) ⊕ ( − ∆ u − ) , dom e T := (cid:8) u ∈ H / ( R d \ Σ) : [ u ] Σ = 0 (cid:9) , b T u := ( − ∆ u + ) ⊕ ( − ∆ u − ) , dom b T := (cid:8) u ∈ H / ( R d \ Σ) : [ ∂ ν u ] Σ = 0 (cid:9) , and their restrictions e S and b S by e S := e T ↾ (cid:8) u ∈ H / ( R d \ Σ) : u ± | Σ = 0 , [ ∂ ν u ] Σ = 0 (cid:9) , b S := b T ↾ (cid:8) u ∈ H / ( R d \ Σ) : ∂ ν ± u ± | Σ = 0 , [ u ] Σ = 0 (cid:9) . race formulae for Schrödinger operators with singular interactions e S (respectively, b S ) is the restriction of H free to functions,whose Dirichlet trace (respectively, Neumann trace) vanishes on Σ. In particular,as a consequence of this identification we arrive at the inclusions dom e S, dom b S ⊂ H ( R d ). It can also be shown that the operators e S and b S are closed, denselydefined, and symmetric in L ( R d ) and that the closures of e T and b T coincide with e S ∗ and b S ∗ , respectively. Furthermore, we define the boundary mappings by e Γ , e Γ : dom e T → L (Σ) , e Γ u := [ ∂ ν u ] Σ , e Γ u := u | Σ , (2.9) b Γ , b Γ : dom b T → L (Σ) , b Γ u := ∂ ν + u + | Σ , b Γ u := [ u ] Σ . (2.10)The identities H free = e T ↾ ker e Γ = b T ↾ ker b Γ and K N = b T ↾ ker b Γ can be checked in a straightforward way. According to [10, Proposition 3.2 (i)]the triple e Π := { L (Σ) , e Γ , e Γ } is a quasi boundary triple for e S ∗ , and by [10,Proposition 3.8 (i)] the triple b Π := { L (Σ) , b Γ , b Γ } is a quasi boundary triple for b S ∗ . Definition 2.5.
Let e γ , f M and b γ , c M be the γ -fields and the Weyl functions of thequasi boundary triples e Π and b Π , respectively.Remark . The definitions of the operator-valued functions f M and c M as Neumann-to-Dirichlet maps in (1.4) and as Weyl functions of the quasi boundary triples e Πand b Π are equivalent; see [10, Propositions 3.2 (iii) and 3.8 (iii)].
Remark . According to [10, Propositions 3.2 (ii) and 3.8 (ii)], for any ϕ ∈ L (Σ)both transmission boundary value problems − ∆ u = λu in R d \ Σ , [ u ] Σ = 0 on Σ , [ ∂ ν u ] Σ = ϕ on Σ , − ∆ u = λu in R d \ Σ ,∂ ν + u + | Σ = ϕ on Σ ,∂ ν − u − | Σ = − ϕ on Σ , have unique solutions e u ( ϕ ) , b u ( ϕ ) ∈ H / ( R d \ Σ). Moreover, the operator-valuedfunctions e γ and b γ satisfy e γ ( λ ) ϕ = e u λ ( ϕ ) and b γ ( λ ) ϕ = b u λ ( ϕ ) for ϕ ∈ L (Σ) and λ ∈ C \ R + .Thanks to (2.7) the adjoints of e γ ( λ ) and b γ ( λ ) can be expressed as e γ ( λ ) ∗ = e Γ ( H free − λ ) − and b γ ( λ ) ∗ = b Γ ( K N − λ ) − (2.11)for λ ∈ C \ R + . We also remark that, by [10, Propositions 3.2 (iii) and 3.8 (iii)], wehave ran f M ( λ ) = ran c M ( λ ) = H (Σ) , λ ∈ C \ R + . (2.12)According to Proposition 2.3, the formula( K N − λ ) − − ( H free − λ ) − = b γ ( λ ) c M ( λ ) − b γ ( λ ) ∗ , (2.13)0 Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik holds for all λ ∈ C \ R + . Since the operators of multiplication with α and ω arebounded and self-adjoint in L (Σ), by Proposition 2.4 the extensions e T ↾ ker( α e Γ − e Γ ) and b T ↾ ker( ω b Γ − b Γ )are self-adjoint in L ( R d ). In a way similar to [10], one can check that these re-strictions coincide with H α, Σ and K ω, Σ , respectively. Moreover, by Proposition 2.4,the formulae( H α, Σ − λ ) − − ( H free − λ ) − = e γ ( λ ) (cid:0) I − α f M ( λ ) (cid:1) − α e γ ( λ ) ∗ , (2.14a)( K ω, Σ − λ ) − − ( K N − λ ) − = b γ ( λ ) (cid:0) I − ω c M ( λ ) (cid:1) − ω b γ ( λ ) ∗ , (2.14b)hold for all λ ∈ ρ ( H α, Σ ) and all λ ∈ ρ ( K ω, Σ ), respectively. In these formulae themiddle terms on the right-hand sides satisfy (cid:0) I − α f M ( λ ) (cid:1) − , (cid:0) I − ω c M ( λ ) (cid:1) − ∈ B ( L (Σ)) (2.15)for λ in the respective resolvent sets.
3. Proofs of the main results
In this section we prove the main results of the paper: the trace formulae forthe Schrödinger operators with singular interactions. Theorems 1.2 and 1.3 areproved in Subsections 3.1 and 3.2, respectively. Throughout this section we usethe notations R ( λ ) := ( H free − λ ) − and R N ( λ ) := ( K N − λ ) − . To prove Theorem 1.2 we need an auxiliary lemma.
Lemma 3.1.
Let the γ -field e γ and the Weyl function f M be as in Definition 2.5.Then for every λ ∈ C \ R + and every k ∈ N the following relations hold: (i) e γ ( k ) ( λ ) , d k d λ k e γ ( λ ) ∗ ∈ S d − k +3 / , ∞ ; (ii) f M ( k ) ( λ ) ∈ S d − k +1 , ∞ .Proof. (i) Let λ ∈ C \ R + and k ∈ N . First, we observe that ran( R ( λ ) k ) ⊂ H k ( R d ). By the trace theorem we have u | Σ ∈ H s − / (Σ) for every u ∈ H s ( R d )with s > /
2. Hence, we obtain from (2.11) thatran (cid:0)e γ ( λ ) ∗ R ( λ ) k (cid:1) ⊂ H k +3 / (Σ) . Thus Lemma 2.1 with K = L ( R d ) and r = 2 k + 3 / e γ ( λ ) ∗ R ( λ ) k ∈ S d − k +3 / , ∞ . (3.1)By taking the adjoint in (3.1) and replacing λ by λ we obtain R ( λ ) k e γ ( λ ) ∈ S d − k +3 / , ∞ . (3.2) race formulae for Schrödinger operators with singular interactions e γ ( k ) ( λ ) , d k d λ k e γ ( λ ) ∗ ∈ S d − k +3 / , ∞ .(ii) For k = 0 we observe that by (2.12) we have ran f M ( λ ) = H (Σ). Therefore,Lemma 2.1 with K = L (Σ) and r = 1 implies that f M ( λ ) ∈ S d − , ∞ . For k ≥ f M ( k ) ( λ ) = k ! e γ ( λ ) ∗ R ( λ ) k − e γ ( λ ) ∈ S d − k − / , ∞ · S d − / , ∞ = S d − k +1 , ∞ , where we applied (3.1), (3.2) and (2.2). Proof of Theorem 1.2.
In order to shorten notation and to avoid the distinction ofseveral cases, we set A r := ( S d − r , ∞ (cid:0) L (Σ) (cid:1) if r > , B (cid:0) L (Σ) (cid:1) if r = 0 . It follows from (2.2) and the fact that S p, ∞ ( L (Σ)) is an ideal in B ( L (Σ)) for p > A r · A r = A r + r , r , r ≥ . (3.3)The remainder of the proof is divided into two steps. Step 1.
Let α ∈ L ∞ (Σ; R ) and set e T ( λ ) := (cid:0) I − α f M ( λ ) (cid:1) − , λ ∈ ρ ( H α, Σ ) , where e T ( λ ) ∈ B ( L (Σ)) by (2.15). Next, we show that e T ( k ) ( λ ) ∈ A k +1 , k ∈ N , (3.4)by induction. Relation (2.6) implies that e T ′ ( λ ) = e T ( λ ) α f M ′ ( λ ) e T ( λ ) , (3.5)which is in A by Lemma 3.1 (ii). Let m ∈ N and assume that (3.4) is true forevery k = 1 , . . . , m , which implies, in particular, that e T ( k ) ( λ ) ∈ A k , k = 0 , . . . , m. (3.6)Then e T ( m +1) ( λ ) = d m d λ m (cid:16) e T ( λ ) α f M ′ ( λ ) e T ( λ ) (cid:17) = X p + q + r = mp,q,r ≥ m ! p ! q ! r ! e T ( p ) ( λ ) α f M ( q +1) ( λ ) e T ( r ) ( λ )2 Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik by (3.5) and (2.5b). Relation (3.6), the boundedness of α , Lemma 3.1 (ii) and (3.3)imply that e T ( p ) ( λ ) α f M ( q +1) ( λ ) e T ( r ) ( λ ) ∈ A p · A q +1)+1 · A r = A m +1)+1 , since p + q + r = m . This shows (3.4) for k = m + 1 and hence, by induction, forall k ∈ N . Since e T ( λ ) ∈ B ( L (Σ)), we have, in particular, e T ( k ) ( λ ) ∈ A k , k ∈ N , λ ∈ ρ ( H α, Σ ) . (3.7) Step 2.
By taking derivatives we obtain from (2.14a) that, for m ∈ N ,( m − e D α,m ( λ ) = d m − d λ m − (cid:0) e D α, ( λ ) (cid:1) = d m − d λ m − (cid:16)e γ ( λ ) e T ( λ ) α e γ ( λ ) ∗ (cid:17) = X p + q + r = m − p,q,r ≥ ( m − p ! q ! r ! e γ ( p ) ( λ ) e T ( q ) ( λ ) α d r d λ r e γ ( λ ) ∗ . (3.8)By Lemma 3.1 (i) and (3.7), each term in the sum satisfies e γ ( p ) ( λ ) e T ( q ) ( λ ) α d r d λ r e γ ( λ ) ∗ ∈ A p +3 / · A q · A r +3 / = A m +1 = S d − m +1 , ∞ . (3.9)If m ∈ N is such that m > d − , then d − m +1 < m − (cid:0) e D α,m ( λ ) (cid:1) = Tr X p + q + r = m − p,q,r ≥ ( m − p ! q ! r ! e γ ( p ) ( λ ) e T ( q ) ( λ ) α d r d λ r e γ ( λ ) ∗ ! = X p + q + r = m − p,q,r ≥ ( m − p ! q ! r ! Tr (cid:16)e γ ( p ) ( λ ) e T ( q ) ( λ ) α d r d λ r e γ ( λ ) ∗ (cid:17) = X p + q + r = m − p,q,r ≥ ( m − p ! q ! r ! Tr (cid:18) e T ( q ) ( λ ) α (cid:16) d r d λ r e γ ( λ ) ∗ (cid:17)e γ ( p ) ( λ ) (cid:19) = Tr X p + q + r = m − p,q,r ≥ ( m − p ! q ! r ! e T ( q ) ( λ ) α (cid:16) d r d λ r e γ ( λ ) ∗ (cid:17)e γ ( p ) ( λ ) ! = Tr (cid:18) d m − d λ m − (cid:16) e T ( λ ) α e γ ( λ ) ∗ e γ ( λ ) (cid:17)(cid:19) = Tr (cid:18) d m − d λ m − (cid:16) e T ( λ ) α f M ′ ( λ ) (cid:17)(cid:19) , which finishes the proof. race formulae for Schrödinger operators with singular interactions First, we need three preparatory lemmas. Theproof of the first of them is completely analogous to the proof of Lemma 3.1 andis therefore omitted.
Lemma 3.2.
Let the γ -field b γ and the Weyl function c M be as in Definition 2.5.Then for every λ ∈ C \ R + and every k ∈ N the following relations hold: (i) b γ ( k ) ( λ ) , d k d λ k b γ ( λ ) ∗ ∈ S d − k +3 / , ∞ ; (ii) c M ( k ) ( λ ) ∈ S d − k +1 , ∞ . Lemma 3.3.
Let the γ -field b γ and the Weyl function c M be as in Definition 2.5.Then for all s ≥ , and all λ ∈ C \ R + the following statements hold: (i) ran (cid:0)b γ ( λ ) ∗ ↾ H s ( R d ) (cid:1) ⊂ H s + (Σ) ; (ii) ran (cid:0) c M ( λ ) ↾ H s (Σ) (cid:1) = H s +1 (Σ) .Proof. (i) According to (2.11) we have b γ ( λ ) ∗ = b Γ R N ( λ ) . Employing the regularity shift property [45, Theorem 4.20] and the trace theorem[45, Theorem 3.37] we conclude thatran (cid:0)b γ ( λ ) ∗ ↾ H s ( R d ) (cid:1) ⊂ H s + (Σ)holds for all s ≥ H s ( R d \ Σ) := H s (Ω + ) ⊕ H s (Ω − ). It follows from thedecomposition dom b T = dom K N ∔ ker( b T − λ ), λ ∈ C \ R + , and the properties ofthe Neumann trace [43, §2.7.3] that the restriction of the mapping b Γ toker( b T − λ ) ∩ H s + ( R d \ Σ)is a bijection onto H s (Σ) for s ≥
0. This, together with the definition of the γ -field,implies thatran (cid:0)b γ ( λ ) ↾ H s (Σ) (cid:1) = ker( b T − λ ) ∩ H s + ( R d \ Σ) ⊂ H s + ( R d \ Σ) . Hence, it follows from the definition of c M ( λ ), the definition of b Γ in (2.10) and thetrace theorem that ran( c M ( λ ) ↾ H s (Σ)) ⊂ H s +1 (Σ) . To verify the opposite inclusion, let ψ ∈ H s +1 (Σ). The decomposition dom b T =dom H free ∔ ker( b T − λ ), λ ∈ C \ R + implies that there exists a function f λ ∈ ker( b T − λ ) ∩ H s + ( R d \ Σ) such that b Γ f λ = ψ . Thus, b Γ f λ = ϕ ∈ H s (Σ) and c M ( λ ) ϕ = ψ, that is, H s +1 (Σ) ⊂ ran (cid:0) c M ( λ ) ↾ H s (Σ) (cid:1) , and the assertion is shown.4 Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik
Lemma 3.4.
Let the self-adjoint operators H free and K N be as in Definition 1.1,and let the operator-valued function c M be as in (1.4) . Then for all m ∈ N suchthat m > d − and all λ ∈ C \ R + the resolvent power difference b D m ( λ ) := ( K N − λ ) − m − ( H free − λ ) − m belongs to the trace class, and its trace can be expressed as Tr (cid:0) b D m ( λ ) (cid:1) = 1( m − (cid:18) d m − d λ m − (cid:0) c M ( λ ) − c M ′ ( λ ) (cid:1)(cid:19) . Proof.
The proof is divided into three steps.
Step 1.
Let us introduce the operator-valued function S ( λ ) := c M ( λ ) − b γ ( λ ) ∗ , λ ∈ C \ R + . Note that the product is well defined since by Lemma 3.3 (i)ran (cid:0)b γ ( λ ) ∗ (cid:1) ⊂ H (Σ) = dom (cid:0) c M ( λ ) − (cid:1) . The closed graph theorem implies that S ( λ ) ∈ B ( L ( R d ) , L (Σ)) for all λ ∈ C \ R + .Next we prove the following smoothing property for the derivatives of S :ran (cid:0) S ( k ) ( λ ) ↾ H s ( R d ) (cid:1) ⊂ H s +2 k +1 / (Σ) , s ≥ , k ∈ N , (3.10)by induction. Since, by Lemma 3.3 (i), b γ ( λ ) ∗ maps H s ( R d ) into H s +3 / (Σ) forall s ≥ c M ( λ ) − maps H s +3 / (Σ) into H s +1 / (Σ) by Lemma 3.3 (ii), rela-tion (3.10) is true for k = 0. Now let l ∈ N and assume that (3.10) is true forevery k = 0 , , . . . , l . It follows from (2.5a), (2.6), (2.8b), (2.8c) and (2.13) that for λ ∈ C \ R + , S ′ ( λ ) = dd λ (cid:0) c M ( λ ) − (cid:1)b γ ( λ ) ∗ + c M ( λ ) − dd λ b γ ( λ ) ∗ = − c M ( λ ) − c M ′ ( λ ) c M ( λ ) − b γ ( λ ) ∗ + c M ( λ ) − b γ ( λ ) ∗ R N ( λ )= − c M ( λ ) − b γ ( λ ) ∗ b γ ( λ ) c M ( λ ) − b γ ( λ ) ∗ + c M ( λ ) − b γ ( λ ) ∗ R N ( λ )= S ( λ ) (cid:2) R N ( λ ) − b γ ( λ ) c M ( λ ) − b γ ( λ ) ∗ (cid:3) = S ( λ ) R ( λ ) . Hence, with the help of (2.5a) we obtain S ( l +1) ( λ ) = d l d λ l (cid:16) S ( λ ) R ( λ ) (cid:17) = X p + q = lp,q ≥ (cid:18) lp (cid:19) S ( p ) ( λ ) R ( q ) ( λ )= X p + q = lp,q ≥ l ! p ! S ( p ) ( λ ) R ( λ ) q +1 . (3.11) race formulae for Schrödinger operators with singular interactions R ( λ ),we deduce that, for p, q ≥ p + q = l ,ran( S ( p ) ( λ ) R ( λ ) q +1 ↾ H s ( R d )) ⊂ ran( S ( p ) ( λ ) ↾ H s +2( q +1) ( R d )) ⊂ H s +2( p + q +1)+1 / (Σ) = H s +2( l +1)+1 / (Σ) , which shows (3.10) for k = l + 1 and hence, by induction, for all k ∈ N . Therefore,an application of Lemma 2.1 with K = L (Σ) and r = 2 k + 1 / S ( k ) ( λ ) ∈ S d − k +1 / , ∞ , k ∈ N , λ ∈ C \ R + . (3.12) Step 2.
Using Krein’s formula in (2.13) and (2.5a) we obtain that, for m ∈ N and λ ∈ C \ R + , b D m ( λ ) = 1( m − · d m − d λ m − (cid:0) b D ( λ ) (cid:1) = 1( m − · d m − d λ m − (cid:0)b γ ( λ ) S ( λ ) (cid:1) = 1( m − X p + q = m − p,q ≥ (cid:18) m − p (cid:19)b γ ( p ) ( λ ) S ( q ) ( λ ) . (3.13)By Lemma 3.2 (i), (3.12) and (2.2) we have b γ ( p ) ( λ ) S ( q ) ( λ ) ∈ S d − p +3 / , ∞ · S d − q +1 / , ∞ = S d − p + q )+2 , ∞ = S d − m , ∞ (3.14)for p, q with p + q = m − Step 3. If m > d − , then d − m < S ( q ) ( λ ) b γ ( p ) ( λ ). Hence,the resolvent power difference b D m ( λ ) is a trace class operator, and we can applythe trace to (3.13) and use (2.3), (2.4) and (2.8c) to obtain( m − (cid:0) b D m ( λ ) (cid:1) = Tr X p + q = m − p,q ≥ (cid:18) m − p (cid:19)b γ ( p ) ( λ ) S ( q ) ( λ ) ! = X p + q = m − p,q ≥ (cid:18) m − p (cid:19) Tr (cid:16)b γ ( p ) ( λ ) S ( q ) ( λ ) (cid:17) = X p + q = m − p,q ≥ (cid:18) m − p (cid:19) Tr (cid:16) S ( q ) ( λ ) b γ ( p ) ( λ ) (cid:17) = Tr X p + q = m − p,q ≥ (cid:18) m − p (cid:19) S ( q ) ( λ ) b γ ( p ) ( λ ) ! = Tr (cid:18) d m − d λ m − (cid:16) S ( λ ) b γ ( λ ) (cid:17)(cid:19) Jussi Behrndt, Matthias Langer and Vladimir Lotoreichik = Tr (cid:18) d m − d λ m − (cid:16) c M ( λ ) − b γ ( λ ) ∗ b γ ( λ ) (cid:17)(cid:19) = Tr (cid:18) d m − d λ m − (cid:16) c M ( λ ) − c M ′ ( λ ) (cid:17)(cid:19) , which finishes the proof. Proof of Theorem 1.3. (i) The proof of this statement is fully analogous to theproof of Theorem 1.2. One has to replace in the argument H free , α , H α, Σ , f M , e γ ,by K N , ω , K ω, Σ , c M , b γ , respectively, Moreover, Krein’s resolvent formula is used in(2.14b) instead of Krein’s formula in (2.14a) and Lemma 3.2 instead of Lemma 3.1.(ii) By item (i) of this theorem and by Lemma 3.4, for every m ∈ N such that m > d − and every λ ∈ ρ ( K ω, Σ ) both operators b D m ( λ ) and b D ω,m ( λ ) belong tothe trace class. In view of the identity b E ω,m ( λ ) = b D m ( λ ) + b D ω,m ( λ ), we infer that b E ω,m ( λ ) is also in the trace class. Using the formula (2.3) we haveTr( b E ω,m ( λ )) = Tr( b D ω,m ( λ )) + Tr( b D m ( λ )) . Combining the trace formula in (i) of this theorem and the trace formula inLemma 3.4 we obtainTr( b E ω,m ( λ )) = 1( m − (cid:18) d m − d λ m − (cid:16)(cid:0) I − ω c M ( λ ) (cid:1) − ω c M ′ ( λ ) + c M ( λ ) − c M ′ ( λ ) (cid:17)(cid:19) = 1( m − (cid:18) d m − d λ m − (cid:16)(cid:0) I − ω c M ( λ ) (cid:1) − c M ( λ ) − c M ′ ( λ ) (cid:19) , which finishes the proof. References [1] J.-P. Antoine, F. Gesztesy and J. Shabani, Exactly solvable models of sphere inter-actions in quantum mechanics.
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