Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

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1 NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2012, 0 0, P Boundary triples for Schrödinger operators with singular interactions on hypersurfaces Jussi Behrndt Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria behrndt@tugraz.at Matthias Langer Department of Mathematics Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom m.langer@strath.ac.uk Vladimir Lotoreichik Department of Theoretical Physics, Nuclear Physics Institute CAS, Řež near Prague, Czech Republic PACS Tb, Db lotoreichik@ujf.cas.cz The self-adjoint Schrödinger operator A δ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L 2 R n. The aim of this note is to construct a boundary triple for S a self-adjoint parameter Θ δ,α in the boundary space L 2 C such that A δ,α corresponds to the boundary condition induced by Θ δ,α. As a consequence the well-developed theory of boundary triples their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula a description of the spectrum of A δ,α in terms of the Weyl function Θ δ,α. Keywords: Boundary triple, Weyl function, Schrödinger operator, singular potential, δ-interaction, hypersurface. 1. Introduction Boundary triples their Weyl functions are an efficient frequently used tool in extension theory of symmetric operators the spectral analysis of their self-adjoint extensions. Roughly speaking a boundary triple consists of two boundary mappings that satisfy an abstract second Green s identity a maximality condition. With the help of a boundary triple all self-adjoint extensions of a symmetric operator can be parameterized via abstract boundary conditions that involve a self-adjoint parameter in a boundary space. In addition, the spectral properties of these self-adjoint extensions can be described with the help of the Weyl function the corresponding boundary parameter. We refer the reader to [12 15, 28] Section 2 for more details on boundary triples their Weyl functions. The main objective of this note is to provide discuss boundary triples their Weyl functions for self-adjoint Schrödinger operators in L 2 R n with δ-interactions of strength α R supported on a compact smooth hypersurface C that separates R n into a

2 2 J. Behrndt, M. Langer V. Lotoreichik smooth bounded domain Ω i an unbounded smooth exterior domain Ω e. In an informal way such an operator is often written in the form A δ,α = αδ C, 1 where δ C denotes the δ-distribution supported on C. A precise definition of the self-adjoint operator A δ,α in terms of boundary or interface conditions is given at the beginning of Section 3 below; see also [8, 10] for an equivalent definition via quadratic forms. Schrödinger operators with δ-interactions are frequently used in mathematical physics to model interactions of quantum particles; we refer to the monographs [1] [23], to the review article [18] to [2, 3, 8, 16, 17, 19 22, 24, 25, 32, 33, 36, 37, 42] for a small selection of related papers on spectral analysis of such operators. Let A free be the usual self-adjoint realization of in L 2 R n let A δ,α be the self-adjoint operator with δ-interaction on C in 1. We consider the densely defined, closed symmetric operator S = A free A δ,α in L 2 R n its adjoint S, we construct a boundary triple {L 2 C, Γ 0, Γ 1 } for S a self-adjoint parameter Θ δ,α in L 2 C such that A free = S ker Γ 0 A δ,α = S kerγ 1 Θ δ,α Γ 0. Although it is clear from the general theory that such a boundary triple a self-adjoint parameter Θ δ,α exist, the construction is not trivial. Our idea here is based on a coupling of two boundary triples for elliptic PDEs which involve the Dirichlet-to-Neumann map as a regularization see [11, 29, 35], the restriction of this coupling to a suitable intermediate extension, certain transforms of boundary triples corresponding parameters. These efforts technical considerations are worthwhile for various reasons. In particular, if γ M denote the γ-field Weyl function corresponding to the boundary triple {L 2 C, Γ 0, Γ 1 } see Section 2 for more details, then it follows immediately from the general theory in [14,15] that the resolvent difference of A free A δ,α admits the representation A δ,α λ 1 A free λ 1 = γλ Θ δ,α Mλ 1 γ λ for all λ ρa δ,α belongs to some operator ideal in L 2 R n if only if the resolvent of Θ δ,α belongs to the analogous operator ideal in L 2 C; see Theorem 3.5. As a special case, Schatten von Neumann properties of the resolvent difference of A free A δ,α carry over to the resolvent of Θ δ,α, vice versa. Moreover, the spectral properties of A δ,α can be described with the help of the perturbation term Θ δ,α Mλ 1. We mention that in the context of the more general notion of quasi boundary triples their Weyl functions from [5,7] a similar approach as in this note closely related results can be found in [8,9]; we also refer to [26, 27, 29, 31, 35, 38 40] for other methods in extension theory of elliptic differential operators. 2. Boundary triples Weyl functions In this preparatory section we recall the notion of boundary triples, associated γ-fields Weyl functions, discuss some of their properties. For a more detailed exposition we refer the reader to [12 15, 28, 41]. In the following let H be a Hilbert space, let S be a densely defined, closed symmetric operator in H, let S be the adjoint operator. Definition 2.1. A triple {G, Γ 0, Γ 1 } is called a boundary triple for S if G is a Hilbert space Γ 0, Γ 1 : dom S G are linear mappings that satisfy the abstract second Green s identity S f, g H f, S g H = Γ 1 f, Γ 0 g G Γ 0 f, Γ 1 g G

3 Boundary triples for Schrödinger operators with δ-interactions on hypersurfaces 3 for all f, g dom S, the mapping Γ := Γ 0, Γ 1 : dom S G G is surjective. Recall that a boundary triple {G, Γ 0, Γ 1 } for S exists if only if the defect numbers of S coincide or, equivalently, S admits self-adjoint extensions in H. Moreover, a boundary triple is not unique except in the trivial case S = S. The following special observation will be used in Section 3: suppose that {G, Γ 0, Γ 1 } is a boundary triple for S let G be bounded self-adjoint operator in G; then {G, Γ 0, Γ 1}, where is also a boundary triple for S. mapping Γ 0 = Γ 1 I G 0 I Γ0 Γ 1, 2 Recall also that dom S = ker Γ 0 ker Γ 1 that the Θ A Θ := S { f dom S : Γf = Γ 0 f, Γ 1 f Θ } 3 establishes a bijective correspondence between the closed linear subspaces relations in G G the closed linear extensions A Θ S of S. In the case when Θ is the graph of an operator, the closed extension A Θ in 3 is given by A Θ = S kerγ 1 ΘΓ 0. 4 It is important to note that the identity A Θ = A Θ holds hence A Θ in 3 4 is self-adjoint in H if only if Θ is self-adjoint in G. It follows, in particular, that the extension A 0 = S ker Γ 0 5 is self-adjoint. This extension often plays the role of a fixed extension within the family of self-adjoint extensions of S. We also mention that Θ in 3 is an unbounded operator if only if the extensions A 0 A Θ are disjoint but not transversal, that is, S = A Θ A 0 A Θ + A 0 S, 6 where + denotes the sum of subspaces. Note that this appears only in the case when G is infinite-dimensional, that is, the defect numbers of S are both infinite. The next theorem can be found in [6]. Very roughly speaking it can be regarded as converse to the above considerations. Here the idea is to start with boundary mappings defined on the domain of some operator T that satisfy the abstract second Green s identity some additional conditions, to conclude that T coincides with the adjoint of the restriction of T to the intersection of the kernels of the boundary mappings. Theorem 2.2 will be used in the proof of Lemma 3.1. Theorem 2.2. Let T be a linear operator in H, let G be a Hilbert space assume that Γ 0, Γ 1 : dom T G are linear mappings that satisfy the following conditions: i There exists a self-adjoint restriction A 0 of T such that dom A 0 ker Γ 0 ; ii ranγ 0, Γ 1 = G G; iii for all f, g dom T the abstract Green s identity holds. T f, g H f, T g H = Γ 1 f, Γ 0 g G Γ 0 f, Γ 1 g G Then S := T ker Γ 0 ker Γ 1 is a densely defined, closed, symmetric operator in H such that S = T {G, Γ 0, Γ 1 } is a boundary triple for S with the property A 0 = S ker Γ 0.

4 4 J. Behrndt, M. Langer V. Lotoreichik In the following assume that S is a densely defined, closed, symmetric operator in H that {G, Γ 0, Γ 1 } is a boundary triple for S. Let A 0 = S ker Γ 0 be as in 5 observe that the following direct sum decomposition of dom S is valid: dom S = dom A 0 + kers λ = ker Γ 0 + kers λ, λ ρa 0. It follows, in particular, that Γ 0 kers λ is a bijective operator from kers λ onto G. The inverse is denoted by γλ = Γ 0 kers λ 1, λ ρa0 ; when viewed as a function λ γλ on ρa 0, we call γ the γ-field corresponding to the boundary triple {G, Γ 0, Γ 1 }. The Weyl function M associated with {G, Γ 0, Γ 1 } is defined by Mλ = Γ 1 γλ = Γ 1 Γ0 kers λ 1, λ ρa0. It can be shown that the values Mλ of the Weyl function M are bounded, everywhere defined operators in G, that M is a holomorphic function on ρa 0 with the properties Mλ = M λ that Im Mλ is uniformly positive for λ C +, i.e. M is an operatorvalued Nevanlinna or Riesz Herglotz function that is uniformly strict; see [13]. 3. Schrödinger operators with δ-interactions on hypersurfaces Let Ω i R n, n 2, be a bounded domain with C -smooth boundary C = Ω i let Ω e = R n \ Ω i be the corresponding exterior domain with the same C -smooth boundary Ω e = C. In the following f i C f e C denote the traces of functions in Ω i Ω e, respectively; if f i C = f e C, we also set f C := f i C = f e C. Moreover, νi f i C νe f e C denote the traces of their normal derivatives; here we agree that the normal vectors ν i ν e point outwards of the domains, so that, ν i = ν e. In the following, let α 0 be a real constant consider the Schrödinger operator with a δ-interaction of strength α supported on C defined by A δ,α f = f, { dom A δ,α = f = fi } H f 2 Ω i H 2 f i C = f e C, Ω e,. e αf C = νi f i C + νe f e C According to [8, Theorem 3.5 Theorem 3.6] the operator A δ,α is self-adjoint in L 2 R n corresponds to the densely defined, closed sesquilinear form a δ,α [f, g] = f, g L 2 R n n αf C, g C L 2 C, dom a δ,α = H 1 R n. Observe that the normal derivatives of the functions in dom A δ,α may have a jump at the interface C or, more precisely, that f dom A δ,α is contained in H 2 R n if only if νi f i C = νe f e C. We also recall that the essential spectrum of the operator A δ,α is [0, that the negative spectrum consists of finitely many eigenvalues of finite multiplicity; see [8, 10]. In the following we fix some point η such that 7 η ρa δ,α, 0. 8 In Proposition 3.3 below we specify a boundary triple {L 2 C, Γ 0, Γ 1 } for the adjoint of the densely defined, closed, symmetric operator Sf = f, dom S = { f H 2 R n : f C = 0 }, 9

5 Boundary triples for Schrödinger operators with δ-interactions on hypersurfaces 5 such that the free or unperturbed Schrödinger operator A free f = f, dom A free = H 2 R n, corresponds to the kernel of the first boundary mapping Γ 0. Note that the operator A δ,α in 7 is a self-adjoint extension of S that the defect numbers dimrans i are infinite. Hence the abstract considerations in Section 2 ensure that there exists a self-adjoint operator or relation Θ δ,α such that A δ,α = S kerγ 1 Θ δ,α Γ The parameter Θ δ,α further properties of the operator A δ,α will be discussed in Lemma 3.4 Theorem 3.5 below. Some further notations preparatory results are required before Proposition 3.3 can be stated proved. Consider the densely defined, closed, symmetric operators S i f i = f i, dom S i = H 2 0Ω i, S e f e = f e, dom S e = H 2 0Ω e, in L 2 Ω i L 2 Ω e, respectively. Their adjoints are given by the maximal operators S i f i = f i, dom S i = { f i L 2 Ω i : f i L 2 Ω i }, S e f e = f e, dom S e = { f e L 2 Ω e : f e L 2 Ω e } ; here the expressions f i f e are understood in the sense of distributions. It is important to note that H 2 Ω i H 2 Ω e are proper subsets of the maximal domains dom Si dom Se, respectively, that the symmetric operator S in 9 is an infinitedimensional extension of the orthogonal sum S i S e, which is also a symmetric operator in L 2 R n = L 2 Ω i L 2 Ω e. Recall from [29, 34] that the trace maps admit continuous extensions to the maximal domains equipped with the graph norms, dom S i f i f i C H 1/2 C, dom S e f e f e C H 1/2 C, dom S i f i νi f i C H 3/2 C, dom S e f e νe f e C H 3/2 C. Furthermore, consider the self-adjoint extensions A D i A D e of S i S e, respectively, corresponding to Dirichlet boundary conditions on C, A D i f i = f i, dom A D i = H 1 0Ω i H 2 Ω i, A D e f e = f e, dom A D e = H 1 0Ω e H 2 Ω e. Since A D i A D e are both non-negative, it is clear that η in 8 belongs to ρa D i ρa D e, hence we have the direct sum decompositions dom S i = dom A D i + kers i η = H 1 0Ω i H 2 Ω i + kers i η 11 dom S e = dom A D e + kers e η = H 1 0Ω e H 2 Ω e + kers e η. 12 We agree to decompose functions f i dom S i f e dom S e in the form f i = f D i + f η i f e = f D e + f η e, 13 where f D i dom A D i, f η i kers i η, f D e dom A D e, f η e kers e η.

6 6 J. Behrndt, M. Langer V. Lotoreichik In the following we often make use of the operators ι = C + I 1 4 ι 1 = C + I 1 4, where C denotes the Laplace Beltrami operator on C. Both mappings ι ι 1 are regarded as isomorphisms ι : H s C H s 1 2 C ι 1 : H t C H t+ 1 2 C for s, t R, also as operators that establish the duality ιϕ, ι 1 ψ L 2 C = ϕ, ψ H 1/2 C H 1/2 C for ϕ H 1/2 C ψ H 1/2 C, when the spaces H 1/2 C H 1/2 C are equipped with the corresponding norms. Note also that ι 1 can be viewed as a bounded self-adjoint operator in L 2 C with ran ι 1 = H 1/2 C that ι with domain dom ι = H 1/2 C is an unbounded self-adjoint operator in L 2 C with 0 ρι. Now we have finally collected all necessary notation to state the first lemma. Lemma 3.1. Let S be the densely defined, closed, symmetric operator in 9. Then the adjoint S of S is given by Further, let S f = f, { dom S = f = fi f e dom S i dom S e, f i C = f e C }. Υ 0 f = ι 1 f C Υ 1 f = ι νi f D i C + νe f D e C for f = f i, f e dom S with f D i, f D e as in 13. Then {L 2 C, Υ 0, Υ 1 } is a boundary triple for S with the property A D i A D e = S ker Υ 0. Proof. The assertions in Lemma 3.1 will be proved with the help of Theorem 2.2. To this end set T f = f, { } fi dom T = f = dom S f i dom Se, f i C = f e C, e consider the boundary mappings Υ 0, Υ 1 : dom T L 2 C in 15. First of all note that item i in Theorem 2.2 is satisfied with the self-adjoint operator A 0 = A D i A D e since for any function f = f i, f e doma D i A D e H 2 Ω i H 2 Ω e one has f i dom Si, f e dom Se, f i C = f e C. In order to see that the mapping Υ0 : dom T L Υ 2 C L 2 C 16 1 is surjective let ϕ, ψ L 2 C. Since the Neumann trace map is surjective from H 2 Ω i H0Ω 1 i onto H 1/2 C from H 2 Ω e H0Ω 1 e onto H 1/2 C there exist fi D dom A D i fe D dom A D e such that νi fi D C = νe fe D C = 1 2 ι 1 ψ H 1/2 C. Next we choose f η i kersi η fe η kerse η such that f η i C = fe η C = ιϕ H 1/2 C, which is possible by the surjectivity of the trace map from the maximal domain onto H 1/2 C; cf. [29, 30, 34]. Now it is easy to see that f = fi D + f η i, f e D + fe η dom T satisfies Υ 0 f = ι 1 f C = ϕ Υ 1 f = ι νi f D i C + νe f D e C = ψ, 14 15

7 Boundary triples for Schrödinger operators with δ-interactions on hypersurfaces 7 hence the map 16 is onto. Next we verify that the abstract second Green s identity T f, g L 2 R n f, T g L 2 R n = Υ 1 f, Υ 0 g L 2 C Υ 0 f, Υ 1 g L 2 C, f, g dom T, 17 holds. For this it is useful to recall that Green s identity for f i = f D i + f η i g i = g D i + g η i yields S i f i, g i L 2 Ω i f i, S i g i L 2 Ω i = f i C, νi g D i C H 1/2 C H 1/2 C νi f D i C, g i C H 1/2 C H 1/2 C for f e = f D e + f η e g e = g D e + g η e in the analogous form 18 S e f e, g e L 2 Ω e f e, S e g e L 2 Ω e = f e C, νe g D e C H 1/2 C H 1/2 C νe f D e C, g e C H 1/2 C H 1/2 C. 19 Since T is a restriction of the orthogonal sum S i S e f i C = f e C, g i C = g e C for f, g dom T we conclude from that T f, g L 2 R n f, T g L 2 R n = S i f i, g i L 2 Ω i f i, S i g i L 2 Ω i + S e f e, g e L 2 Ω e f e, S e g e L 2 Ω e = f i C, νi g D i C H 1/2 C H 1/2 C νi f D i C, g i C H 1/2 C H 1/2 C + f e C, νe g D e C H 1/2 C H 1/2 C νe f D e C, g e C H 1/2 C H 1/2 C = f C, νi g D i C + νe g D e C H 1/2 C H 1/2 C νi f D i C + νe f D e C, g C H 1/2 C H 1/2 C = ι 1 f C, ι νi g D i C + νe g D e C L 2 C ι νi f D i C + νe f D e C, ι 1 g C L 2 C = ι νi f D i C + νe f D e C, ι 1 g C L 2 C ι 1 f C, ι νi g D i C + νe g D e C L 2 C = Υ 1 f, Υ 0 g L 2 C Υ 0 f, Υ 1 g L 2 C holds. Thus 17 is shown item iii in Theorem 2.2 is satisfied. Hence Theorem 2.2 implies that the symmetric operator Ŝ := T ker Υ 0 ker Υ 1 20 is densely defined, closed its adjoint coincides with T. We show that Ŝ coincides with the symmetric operator S in 9. Note first that Theorem 2.2 also implies that A D i A D e = T ker Υ Both operators, S Ŝ, are restrictions of the operator in 21. Let now f = f i, f e doma D i A D e = ker Υ 0. For such f we have f ker Υ 1 νi f i C + νe f e C = 0 f H 2 R n f dom S. Thus Ŝ = S. Now the remaining statements in Lemma 3.1 follow immediately from Theorem 2.2. Next we specify the Weyl function N the γ-field ζ corresponding to the boundary triple {L 2 C, Υ 0, Υ 1 } in Lemma 3.1. It is clear from the definition of Υ 0 that the γ-field acts as follows: ζλ : L 2 C L 2 R n, ϕ f λ, λ ρa D i ρa D e = C \ [0,, where f λ = f i,λ, f e,λ H 2 Ω i H 2 Ω e satisfies f i,λ = λf i,λ, f e,λ = λf e,λ f i,λ C = f e,λ C = ιϕ.

8 8 J. Behrndt, M. Langer V. Lotoreichik In order to specify the Weyl function N we recall the definition of the Dirichlet-to-Neumann maps D i λ D e λ associated with the Laplacians on Ω i Ω e, respectively. Note first that for ϕ, ψ H 1/2 C λ ρa D i µ ρa D e the boundary value problems f i = λf i, f i C = ϕ f e = µf e, f e C = ψ admit unique solutions f i,λ dom S i f e,µ dom S e. Hence the operators D i, 1/2 λf i,λ C = νi f i,λ C, dom D i, 1/2 λ = H 1/2 C, 22 D e, 1/2 µf e,µ C = νe f e,λ C, dom D e, 1/2 µ = H 1/2 C, 23 are well defined, map H 1/2 C into H 3/2 C. We have used the index 1/2 in the definition of the Dirichlet-to-Neumann maps in to indicate that their domain is H 1/2 C. For the following it is important that the restrictions D i λf i,λ C = νi f i,λ C, D e µf e,µ C = νe f e,µ C, dom D i λ = H 1 C, dom D e µ = H 1 C, of D i, 1/2 λ D e, 1/2 µ to H 1 C are densely defined, closed, unbounded operators in L 2 C that satisfy D i λ = D i λ D e µ = D e µ for all λ ρa D i for all µ ρa D e, respectively. For λ ρa D i ρa D e = C \ [0, it is convenient to introduce the operators E 1/2 λ := D i, 1/2 λ + D e, 1/2 λ Eλ := D i λ + D e λ. 24 Furthermore, the restrictions of D i, 1/2 λ D e, 1/2 µ to H 3/2 C will be used. These restrictions are denoted by D i,3/2 λ D e,3/2 µ, respectively; they map H 3/2 C into H 1/2 C, as above the index 3/2 is used to indicate that their domain is H 3/2 C. Lemma 3.2. Let S be the symmetric operator in 9, let {L 2 C, Υ 0, Υ 1 } be the boundary triple for S in Lemma 3.1 fix η as in 8. For λ ρa D i ρa D e = C \ [0, the operators E 1/2 λ in 24 have the property ran E 1/2 λ E 1/2 η H 1/2 C 25 the Weyl function corresponding to the boundary triple {L 2 C, Υ 0, Υ 1 } is given by Nλ = ι E 1/2 λ E 1/2 η ι, λ C \ [0,. Proof. Let λ ρa D i ρa D e let f λ = f i,λ, f e,λ kers λ. Then f i,λ C = f e,λ C according to we have f i,λ = f D i,λ + f η i,λ f e,λ = f D e,λ + f η e,λ,

9 Boundary triples for Schrödinger operators with δ-interactions on hypersurfaces 9 where fi,λ D dom AD i, f η i,λ kers i η, fe,λ D dom AD e f η e,λ kers e η. Hence it follows with the help of f i,λ C = f η i,λ C f e,λ C = f η e,λ C, the definition of the Dirichletto-Neumann maps that E 1/2 λ E 1/2 η ιυ 0 f λ = D i, 1/2 λ D i, 1/2 η + D e, 1/2 λ D e, 1/2 η f λ C = D i, 1/2 λf i,λ C D i, 1/2 ηf η i,λ C + D e, 1/2 λf e,λ C D e, 1/2 ηf η e,λ C = νi f i,λ C νi f η i,λ C + νe f e,λ C νe f η e,λ C 26 = νi fi,λ f η i,λ C + νe fe,λ f η e,λ C = νi f D i,λ C + νe f D e,λ C hence ι E 1/2 λ E 1/2 η ιυ 0 f λ = ι νi f D i,λ C + νe f D e,λ C = Υ1 f λ. Also the inclusion 25 follows from 26 since f D i,λ H2 Ω i f D e,λ H2 Ω e, hence νi f D i,λ C + νe f D e,λ C H 1/2 C in 26, for any ϕ dom E 1/2 λ = dom E 1/2 η = H 1/2 C there exists f λ = f i,λ, f e,λ kers λ f η = f i,η, f e,η kers η such that f i,λ C = f e,λ C = ϕ = f i,η C = f e,η C. In the following proposition we provide a boundary triple for S such that the operator A free corresponds to the first boundary mapping. For this we modify the boundary triple in Lemma 3.1 in a suitable way. Proposition 3.3. Let S be the densely defined, closed, symmetric operator in 9 with adjoint S in 14, let Eη = ιeηι. Then Eη 1 is a bounded self-adjoint operator in L 2 C {L 2 C, Γ 0, Γ 1 }, where Γ 0 f = ι 1 f C + Eη 1 ι νi f D i C + νe f D e C Γ 1 f = ι νi f D i C + νe f D e C, 27 is a boundary triple for S with the property A free = S ker Γ 0. For λ C \ [0, the Weyl function corresponding to {L 2 C, Γ 0, Γ 1 } is given by Mλ = ι E 1/2 λ E 1/2 η ι I + Eη 1 ι E 1/2 λ E 1/2 η ι. Proof. First we show that Eη 1 = ι 1 Eη 1 ι 1 is a bounded self-adjoint operator in L 2 C. Observe that Eη = D i η + D e η is injective. In fact, assume that Eηϕ = 0 for some ϕ H 1 C, ϕ 0. Then there exists f η = f η i, f η e kers η such that f η C = ϕ hence 0 = Eηϕ = Eηf η C = D i ηf η i C + D e ηf η e C = νi f η i C + νe f η e C. 28 Together with f η i C = f η e C this implies that f η dom A free hence kera free η {0}. This is impossible as η < 0. Thus Eη is injective. It follows from [8, Proposition 3.2 iii] that Eη is surjective. Hence Eη 1 is a bounded self-adjoint operator in L 2 C. Since ι 1 is also a bounded self-adjoint operator in L 2 C, it is clear that Eη 1 = ι 1 Eη 1 ι 1 is a bounded self-adjoint operator in L 2 C.

10 10 J. Behrndt, M. Langer V. Lotoreichik Now let {L 2 C, Υ 0, Υ 1 } be the boundary triple in Lemma 3.1. Note that the boundary mappings Γ 0 Γ 1 in 27 satisfy Γ0 I Eη 1 Υ0 =. Γ 1 0 I Υ 1 Hence it follows that {L 2 C, Γ 0, Γ 1 } is a boundary triple for S ; see 2 in Section 2. Let again N denote the Weyl function corresponding to the boundary triple {L 2 C, Υ 0, Υ 1 }. It is not difficult to see that the Weyl function corresponding to {L 2 C, Γ 0, Γ 1 } is given by Mλ = Nλ I Eη 1 Nλ 1, λ C \ [0,. Hence the form of the Weyl function M follows from Lemma 3.2. It remains to show that A free = S ker Γ 0 holds. Assume that for some f dom S we have ι 1 f C + Eη 1 ι νi f D i C + νe f D e C = As ker E 1/2 η = {0} this can be seen as in 28, this is equivalent to Further, since E 1/2 ηf C + νi f D i C + νe f D e C = 0. E 1/2 ηf C = D i, 1/2 ηf η i C + D e, 1/2 ηf η e C = νi f η i C + νe f η e C holds for f decomposed as in 13, we conclude that 29 is equivalent to νi f η i C + νe f η e C + νi f D i C + νe f D e C = 0, which in turn is equivalent to νi f i C + νe f e C = 0. Therefore f ker Γ 0 if only if f dom A free. Our next goal is to identify the self-adjoint parameter Θ δ,α such that 10 holds with the boundary triple in Proposition 3.3. Lemma 3.4. Let S be the densely defined, closed, symmetric operator in 9 with adjoint S in 14, let {L 2 C, Γ 0, Γ 1 } be the boundary triple in Proposition 3.3. Then Θ δ,α = ι D i,3/2 η + D e,3/2 η α ι I Eη 1 ι D i,3/2 η + D e,3/2 η α 1 ι is an unbounded self-adjoint operator in L 2 C such that the Schrödinger operator A δ,α in 7 corresponds to Θ δ,α, that is, A δ,α = S kerγ 1 Θ δ,α Γ Proof. We make use of the fact that the boundary triple {L 2 C, Υ 0, Υ 1 } in Lemma 3.1 the boundary triple {L 2 C, Γ 0, Γ 1 } in Proposition 3.3 are related via we also make use of the operator Our first task is to show that Γ 0 = Υ 0 Eη 1 Υ 1 Γ 1 = Υ 1, 31 Λ δ,α = ι D i,3/2 η + D e,3/2 η α ι, dom Λ δ,α = H 2 C. 32 A δ,α = S kerυ 1 Λ δ,α Υ 0 33 holds. In fact, f kerυ 1 Λ δ,α Υ 0 if only if f dom S ι D i,3/2 η + D e,3/2 η α f C = ι νi f D i C + νe f D e C,

11 Boundary triples for Schrödinger operators with δ-interactions on hypersurfaces 11 where f C dom D i,3/2 η = dom D e,3/2 η = H 3/2 C, together with elliptic regularity, also implies that f = f i, f e with f i H 2 Ω i f e H 2 Ω e. With f decomposed as in 13 we have Di,3/2 η + D e,3/2 η α f C = νi f η i C + νe f η e C αf C. Therefore f kerυ 1 Λ δ,α Υ 0 if only if f = f i, f e dom S with f i H 2 Ω i f e H 2 Ω e νi f η i C + νe f η e C αf C = νi f D i C + νe f D e C, the latter can be rewritten in the form νi f i C + νe f e C = αf C. We have shown 33, as A δ,α is a self-adjoint operator in L 2 R n, it follows that Λ δ,α in 32 is an unbounded self-adjoint operator in L 2 C. Next we consider the operator Θ δ,α = Λ δ,α I Eη 1 Λ δ,α 1 34 on its natural domain; note that keri Eη 1 Λ δ,α = {0} as otherwise Eηϕ = Λ δ,α ϕ for some non-trivial ϕ H 2 C, which is a contradiction to α 0. Now assume that f kerγ 1 Θ δ,α Γ 0. Then yield Υ 1 f Λ δ,α Υ 0 f = Γ 1 f Λ δ,α Γ0 + Eη 1 Υ 1 f = Γ 1 f Λ δ,α Γ0 + Eη 1 Γ 1 f = Γ 1 f Λ δ,α Γ0 + Eη 1 Θ δ,α Γ 0 f = Γ 1 f Λ δ,α I + Eη 1 Λ δ,α I Eη 1 Λ δ,α 1 Γ0 f = Γ 1 f Λ δ,α I Eη 1 Λ δ,α 1Γ0 f = Γ 1 f Θ δ,α Γ 0 f = 0 hence f kerυ 1 Λ δ,α Υ 0. The converse inclusion is shown in the same way therefore kerγ 1 Θ δ,α Γ 0 = kerυ 1 Λ δ,α Υ 0 thus the extensions S kerγ 1 Θ δ,α Γ 0 S kerυ 1 Λ δ,α Υ 0 coincide. Therefore 33 implies 30. Since A δ,α is self-adjoint in L 2 R n, it also follows from 30 that Θ δ,α is self-adjoint in L 2 C. Moreover, as S in 9 coincides with the intersection of A free A δ,α, that is, A free A δ,α are disjoint, since A free A δ,α are not transversal, one concludes that Θ δ,α is an unbounded operator in L 2 C; cf. 6. We are now able to obtain some immediate important consequences from the previous considerations, well-known results for boundary triples Weyl functions [14, 15] the resolvent estimates in [4, 8]. Theorem 3.5. Let S be the densely defined, closed, symmetric operator in 9 with adjoint S in 14, let {L 2 C, Γ 0, Γ 1 } be the boundary triple in Proposition 3.3 with A free = S ker Γ 0,

12 12 J. Behrndt, M. Langer V. Lotoreichik let γ M be the γ-field Weyl function corresponding to {L 2 C, Γ 0, Γ 1 }. Furthermore, let Θ δ,α be as in Lemma 3.4 so that A δ,α = S kerγ 1 Θ δ,α Γ 0. Then the following assertions hold for all λ / [0, : i λ σ p A δ,α if only if 0 σ p Θ δ,α Mλ; ii λ ρa δ,α if only if 0 ρθ δ,α Mλ; iii for all λ ρa δ,α the resolvent formula A δ,α λ 1 A free λ 1 = γλ Θ δ,α Mλ 1 γ λ is valid, the resolvent difference of A δ,α A free belongs to the Schatten von Neumann ideal S p L 2 R n for all p > n 1; 3 iv for all ξ ρθ δ,α the operator Θ δ,α ξ 1 belongs to the Schatten von Neumann ideal S p L 2 C for all p > n 1. 3 Acknowledgements J. Behrndt V. Lotoreichik gratefully acknowledge financial support by the Austrian Science Fund FWF: Project P N26. J. Behrndt also wishes to thank Professor Igor Popov for the pleasant fruitful research stay at the ITMO University in St. Petersburg in September V. Lotoreichik was also supported by the Czech Science Foundation GAČR under the project S. References [1] S. Albeverio, F. Gesztesy, R. Hoegh-Krohn H. Holden, Solvable Models in Quantum Mechanics. With an appendix by Pavel Exner. AMS Chelsea Publishing, [2] S. Albeverio, A. Kostenko, M. Malamud H. Neidhardt, Spherical Schrödinger operators with δ-type interactions, J. Math. Phys , , 24 pp. [3] J. Behrndt, P. Exner V. Lotoreichik, Schrödinger operators with δ δ interactions on Lipschitz surfaces chromatic numbers of associated partitions, Rev. Math. Phys , , 43 pp. [4] J. Behrndt, G. Grubb, M. Langer V. Lotoreichik, Spectral asymptotics for resolvent differences of elliptic operators with δ δ -interactions on hypersurfaces, J. Spectral Theory , [5] J. Behrndt M. Langer, Boundary value problems for elliptic partial differential operators on bounded domains, J. Funct. Anal , [6] J. Behrndt M. Langer, On the adjoint of a symmetric operator, J. Lond. Math. Soc , [7] J. Behrndt M. Langer, Elliptic operators, Dirichlet-to-Neumann maps quasi boundary triples, in: Operator Methods for Boundary Value Problems, London Math. Soc. Lecture Note Series, vol. 404, 2012, pp [8] J. Behrndt, M. Langer V. Lotoreichik, Schrödinger operators with δ δ -potentials supported on hypersurfaces, Ann. Henri Poincaré , [9] J. Behrndt, M. Langer V. Lotoreichik, Spectral estimates for resolvent differences of self-adjoint elliptic operators, Integral Equations Operator Theory , [10] J. F. Brasche, P. Exner, Yu. A. Kuperin P. Šeba, Schrödinger operators with singular interactions, J. Math. Anal. Appl , [11] B. M. Brown, G. Grubb I. G. Wood, M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems, Math. Nachr , [12] J. Brüning, V. Geyler K. Pankrashkin, Spectra of self-adjoint extensions applications to solvable Schrödinger operators, Rev. Math. Phys , [13] V. A. Derkach, S. Hassi, M. M. Malamud H. de Snoo, Boundary relations their Weyl families, Trans. Amer. Math. Soc , [14] V. A. Derkach M. M. Malamud, Generalized resolvents the boundary value problems for Hermitian operators with gaps, J. Funct. Anal , 1 95.

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