Schrödinger Operators with Oblique Transmission Conditions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}^2$$\end{document}R2

In this paper we study the spectrum of self-adjoint Schrödinger operators in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\mathbb {R}^2)$$\end{document}L2(R2) with a new type of transmission conditions along a smooth closed curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma \subseteq \mathbb {R}^2$$\end{document}Σ⊆R2. Although these oblique transmission conditions are formally similar to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta '$$\end{document}δ′-conditions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document}Σ (instead of the normal derivative here the Wirtinger derivative is used) the spectral properties are significantly different: it turns out that for attractive interaction strengths the discrete spectrum is always unbounded from below. Besides this unexpected spectral effect we also identify the essential spectrum, and we prove a Krein-type resolvent formula and a Birman-Schwinger principle. Furthermore, we show that these Schrödinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document}δ-interactions justifying their usage as models in quantum mechanics.


Introduction
In many quantum mechanical applications one considers particles moving in an external potential field which is localized near a set Σ of measure zero.Such strongly localized fields can be modeled by singular potentials that are supported on Σ only; of particular importance in this regard are δ and δ ′ -interactions.To be more precise, assume that Σ splits R 2 into a bounded domain Ω + and an unbounded domain Ω − = R 2 \ Ω + , and consider the formal Schrödinger differential expressions (1.1) H δ,α = −∆ + αδ Σ and H δ ′ ,α = −∆ + αδ ′ Σ , α ∈ R.
These singular perturbations of the free Schrödinger operator −∆ are characterized by certain transmission conditions along the interface Σ for the functions in the operator domain.For δ-interactions one considers functions f : R 2 → C such that the restrictions f ± = f ↾ Ω ± satisfy the transmission conditions (1.2) while δ ′ -interactions are modeled by the transmission conditions here ∂ ν f ± is the normal derivative and ν = (ν 1 , ν 2 ) the unit normal vector field on Σ pointing outwards of Ω + .The spectra and resonances of the self-adjoint realizations associated with the formal expressions (1.1) in L 2 (R 2 ) are well understood, see, e.g., [8,9,12,13,15,16,17,18,24].In particular, the essential spectrum is given by [0, ∞) and the discrete spectrum consists of at most finitely many points for every interaction strength α < 0, while there is no negative spectrum if α ≥ 0.
In contrast to the transmission conditions (1.2) and (1.3) we are interested in a new type of transmission conditions of the form (1.4) ( where α ∈ R and ∂ z = 1 2 (∂ 1 + i∂ 2 ) is the Wirtinger derivative.In the sequel such jump conditions will be referred to as oblique transmission conditions.Note that the conditions (1.4) can be rewritten as (1.5) where ∂ t denotes the tangential derivative.Thus, on a formal level there is some analogy to the δ ′ -transmission conditions in (1.3), but it will turn out that the properties of the corresponding self-adjoint realization in L 2 (R 2 ) differ significantly from those of Schrödinger operators with δ ′ -interactions.
To make matters mathematically rigorous, assume that the curve Σ is the boundary of a bounded and simply connected C ∞ -domain Ω + with open complement Ω − = R 2 \ Ω + , denote the L 2 -based Sobolev space of first order by H 1 , let γ ± D : H 1 (Ω ± ) → L 2 (Σ) be the Dirichlet trace operators, and define for α ∈ R the Schrödinger operator with oblique transmission conditions by The next theorem is the main result in this paper.We discuss the spectral properties of the Schrödinger operators T α and, in particular, we show in item (ii) that for every α < 0 the operator T α is necessarily unbounded from below and the discrete spectrum in (−∞, 0) is infinite and accumulates to −∞.In items (iii) and (iv) we shall make use of the potential operator Ψ λ : L 2 (Σ) → L 2 (R 2 ) and the single layer boundary integral operator S(λ) : L 2 (Σ) → L 2 (Σ) defined in (2.2) and (2.4), respectively.
Theorem 1.1.For any α ∈ R the operator T α is self-adjoint in L 2 (R 2 ) and the essential spectrum is given by Furthermore, the following statements hold: ) is infinite, unbounded from below, and does not accumulate to 0. Moreover, for every fixed n ∈ N the n-th discrete eigenvalue λ n ∈ σ disc (T α ) (ordered non-increasingly) admits the asymptotic expansion where the dependence on n appears in the O(1)-term.(iii) For λ ∈ C \ [0, ∞) the Birman-Schwinger principle is valid: and the resolvent formula , where −∆ is the free Schrödinger operator defined on H 2 (R 2 ).
To illustrate the significance of Theorem 1.1 we show that Schrödinger operators with oblique transmission conditions arise naturally as non-relativistic limits of Dirac operators with electrostatic and Lorentz scalar δ-interactions.To motivate this, consider one-dimensional Dirac operators with δ ′ -interactions of strength α ∈ R supported in Σ = {0}.These are first order differential operators in L 2 (R) 2 and the singular interaction is modeled by transmission conditions for functions in the operator domain, which for sufficiently smooth where c > 0 is the speed of light.It is known that the associated self-adjoint Dirac operators converge in the non-relativistic limit to a Schrödinger operator with a δ ′ -interaction of strength α; cf.[2,19] and also [10,11] for generalizations.It is not difficult to see that (1.7) can be rewritten as the transmission conditions associated with a Dirac operator with a combination of an electrostatic and a Lorentz scalar δ-interaction of strengths η = − αc 2 2 and τ = αc 2 2 , respectively, as they were studied in dimension one recently in [7] and in higher space dimensions in, e.g., [3,5,6,7].
To find a counterpart of the above result in dimension two, consider a Dirac operator with electrostatic and Lorentz scalar δ-shell interactions of strength η and τ , respectively, supported on Σ, which is formally given by (1.8) here A 0 is the unperturbed Dirac operator, I 2 is the 2 × 2-identity matrix and σ 3 ∈ C 2×2 is given in (3.1).The differential expression A η,τ gives rise to a self-adjoint operator A η,τ in L 2 (R 2 ) 2 , see (3.3).If one chooses, as above, η = − αc 2 2 and τ = αc 2 2 and computes the non-relativistic limit, then instead of a Schrödinger operator with a δ ′ -interaction one gets the somewhat unexpected limit T α .Of course, this is compatible with the one-dimensional result described above, as the one-dimensional counterparts of (1.3) and (1.5) coincide, since there are no tangential derivatives in R.However, in higher dimensions Schrödinger operators with oblique transmission conditions should be viewed as the non-relativistic counterparts of Dirac operators with transmission conditions generalizing (1.7).Related results on nonrelativistic limits of three-dimensional Dirac operators with singular interactions can be found in [4,5,21].The precise result about the non-relativistic limit described above is stated in the following theorem and shown in Section 3.
where the convergence is in the operator norm and the convergence rate is O 1 c .Notations.Throughout this paper Ω + ⊆ R 2 is a bounded and simply connected C ∞ -domain and Ω − = R 2 \Ω + is the corresponding exterior domain with boundary Σ = ∂Ω − = ∂Ω + .The unit normal vector field on Σ pointing outwards of Ω + is denoted by ν.Moreover, for z ∈ C \ [0, ∞) we choose the square root √ z such that Im √ z > 0 holds.The modified Bessel function of order j ∈ N 0 is denoted by K j .For s ≥ 0 the spaces H s (R 2 ) n , H s (Ω ± ) n , and H s (Σ) n are the standard L 2 -based Sobolev spaces of C n -valued functions defined on R 2 , Ω ± , and Σ, respectively.If n = 1 we simply write H s (R 2 ), H s (Ω ± ), and H s (Σ).For negative s < 0 we define the spaces H s (R 2 ) n and H s (Σ) n as the anti-dual spaces of H −s (R 2 ) n and H −s (Σ) n , respectively.We denote the restrictions of functions f : R 2 → C n onto Ω ± by f ± ; in this sense we write The Dirichlet trace operators are denoted by γ ± D : H 1 (Ω ± ) → L 2 (Σ) and we shall write γ D : H 1 (R 2 ) → L 2 (Σ) for the trace on H 1 (R 2 ); sometimes these trace operators are also viewed as bounded mappings to H 1/2 (Σ).
For a Hilbert space H we write L(H) for the space of all everywhere defined, linear, and bounded operators on H. Furthermore, the domain, kernel, and range of a linear operator T from a Hilbert space G to H are denoted by dom T , ker T , and ran T , respectively.The resolvent set, the spectrum, the essential spectrum, the discrete spectrum, and the point spectrum of a self-adjoint operator T are denoted by ρ(T ), σ(T ), σ ess (T ), σ disc (T ), and σ p (T ).The eigenvalues of compact self-adjoint operators K ∈ L(H) are denoted by µ n (K) and are ordered by their absolute values.
Acknowledgement.We are indebted to the referee for a very careful reading of our manuscript and various helpful suggestions to improve the text.Jussi Behrndt and Markus Holzmann gratefully acknowledge financial support by the Austrian Science Fund (FWF): P33568-N.This publication is based upon work from COST Action CA 18232 MAT-DYN-NET, supported by COST (European Cooperation in Science and Technology), www.cost.eu.

Proof of Theorem 1.1
In this section the main result of this paper will be proved.For this, some families of integral operators are used.Define for and the operator Ψ λ : Moreover, for λ ∈ C \ [0, ∞) we make use of the single layer potential SL(λ) : ) and the single layer boundary integral operator S(λ) : It is known that SL(λ) and S(λ) are bounded and ran S(λ) ⊆ H 1 (Σ); cf.[25, Theorem 6.12 and Theorem 7.2].In particular, S(λ) gives rise to a compact operator in H s (Σ) for every s ∈ [0, 1].Furthermore, S(λ) is self-adjoint and positive for λ < 0 (see Step 1 in the proof of Proposition 2.2).Some properties of Ψ λ and S(λ) that are important in the proof of Theorem 1.1 are summarized in the following two propositions; cf.Appendix A for the proof of Proposition 2.1 and Proposition 2.2.
Proposition 2.1.Let λ ∈ C \ [0, ∞) and let Ψ λ be given by (2.2).Then is bounded and the following is true: where For λ < 0 denote by µ n (S(λ)) the discrete eigenvalues of the positive self-adjoint operator S(λ) order non-increasingly and with multiplicities taken into account.Proof of Theorem 1.1.
Step 1: We verify that and hence T α is well-defined.Moreover, as C ∞ 0 (R2 \Σ) ⊆ dom T α it is also clear that dom T α is dense.In order to show that T α is symmetric, we note that integration by parts in Ω ± yields for f, g ∈ dom T α (2.6) Now, consider (2.6) for f = g and add the equations for Ω + and Ω − .Then, using Since this holds for all f ∈ dom T α , we conclude that T α is symmetric.
Recall that T α is symmetric; cf.
Step 1.Hence, to show that T α is self-adjoint, it suffices to verify that ran( ) was arbitrary, we conclude that ran (T α − λ) = L 2 (R 2 ) and that T α is self-adjoint.Moreover, the resolvent formula in item (iv) follows from (2.8).
It remains to prove the asymptotic expansion in item (ii).By the above considerations λ n is determined as the unique solution of λµ n (S(λ) for α → 0 − and that the dependence on n appears in the O(1)-term.

Proof of Theorem 1.2
In this section we show that T α is the nonrelativistic limit of a family of Dirac operators with electrostatic and Lorentz scalar δ-shell potentials formally given by (1.8), whose interaction strengths are suitably scaled.First, we recall the rigorous definition of the operator A η,τ associated with (1.8), see [5,6,7] for details.Let (3.1) be the Pauli spin matrices and denote the 2 × 2 identity matrix by I 2 .Furthermore, for x = (x 1 , x 2 ) ∈ R 2 we will use the abbreviations We define Dirac operators with electrostatic and Lorentz scalar δ-shell interactions of strengths It is shown in [6,7] that A η,τ is self-adjoint in L 2 (R 2 ) 2 , whenever η 2 − τ 2 = 4c 2 , and as in [5] one sees that these operators are the self-adjoint realisations of the formal differential expression (1.8).In the above definition we are using units such that = 1 and consider the mass m = 1 2 , but we keep the speed of light c as a parameter for the discussion of the non-relativistic limit c → ∞.
Throughout this section we make use of the self-adjoint free Dirac operator A 0 , which coincides with A 0,0 given in (3.3) and which is defined on . With this function we define the two families of integral operators In the following lemma, which is a preparation for the proof of Theorem 1.2, we will use the matrices , and M 3 = 0 0 0 1 ; products of scalar operators and matrices are understood componentwise, e.g.
Then there exists a constant K > 0, depending only on λ and Σ, such that the estimates are valid for all sufficiently large c > 0.
Proof.We use a similar strategy as in the proof of [4, Proposition 5.2].In the following let λ ∈ C \ R be fixed.Then λ + c 2 2 ∈ C \ R and hence all operators in (3.6a)-(3.6d)are well-defined.One verifies by direct calculation that for sufficiently large c > 0 and all t ∈ [0, 1] hold.With the well-known asymptotic expansions of the modified Bessel functions and [1]) one shows that there exist constants K, κ, R > 0, depending only on λ, such that (3.8) , and sufficiently large c > 0.
Next, with G λ+c 2 /2 defined by (3.4) we find be the integral kernel of the resolvent of the free Laplace operator; cf.[28,Chapter 7.5].Then holds, where the matrix-valued functions t 1 , t 2 , and t 3 are given by With (3.7) and (3.8) applied with t = 1 one finds that there exist constants k 1 , κ, R > 0, depending only on λ, such that for j ∈ {1, 3} and sufficiently large c > 0 one has To estimate t 2 , we use K ′ 0 = −K 1 and obtain with the fundamental theorem of calculus, (3.7), and (3.8) with a constant k 2 which depends only on λ.Thus, if we define This estimation for the integral kernel yields with the Schur test; cf.[4, Proposition A.3] for a similar argument, for all sufficiently large c > 0, which is the first claimed estimate (3.6a).
Next, we prove (3.6b).Recall that the integral kernel L λ of Ψ λ is given by (2.1).Using that σ 1 M 3 = M 2 , σ 2 M 3 = −iM 2 , and M 1 M 3 = 0, we obtain with (3.10) the decomposition Similar as above it can be shown that there exists a k 4 > 0, depending only on λ, such that for all j ∈ {1, 2, 3} to see the estimate for τ 2 one has to use (3.9).With the help of the Schur test the estimate (3.6b) follows (see also [4,Proposition A.4] for a similar argument); the constant k 4 depends in this case on λ and Σ.The estimate in (3.6c) follows by taking adjoints.
It remains to prove (3.6d).Taking M 3 σ • xM 3 = 0, which holds for any x ∈ R 2 , and (3.11) into account we obtain that holds for all x ∈ R 2 \ {0}.Using the dominated convergence theorem, one sees that holds for all f ∈ L 2 (Σ) 2 and x ∈ Σ, i.e. the integral does not have to be understood as principal value.Thus we obtain with the Schur test III.Example 2.4] that In this case, the constant K depends on λ and Σ.This yields (3.6d) and finishes the proof of this lemma.Now we are prepared to prove Theorem 1.2 and show that A −αc 2 /2,αc 2 /2 converges in the nonrelativistic limit to T α defined in (1.6).
To conclude, note that (3.12) and (3.13) yield ), we conclude that (2.5) is true.In particular, this and the properties of the single layer potential mentioned at the beginning of this appendix imply that is bounded and for ϕ ∈ H 1/2 (Σ) we have where ∂ t is the tangential derivative on Σ.As SL(λ)ϕ ∈ H 1 (R 2 ), one has the relation ∂ t (SL(λ)ϕ) + = ∂ t (SL(λ)ϕ) − and consequently with (A.1) This finishes the proof of (iii).
Then it follows from the monotonicity of λ which is a contradiction to the fact that M α,1 is at most countable.Therefore, the mapping (−∞, 0) ∋ λ → µ n (S(λ)) is continuous and strictly monotonically increasing for n ∈ N.
Next, we consider the limit of λµ n (S(λ)) for λ → −∞.For this purpose, results on Schrödinger operators with δ-interactions will be used.Define for α < 0 the sesquilinear form By [9,14] the form h δ,α is semi-bounded and closed, and one can show for the self-adjoint operator H δ,α , which is associated with h δ,α by the first representation theorem, that σ ess (H δ,α ) = [0, ∞), that its discrete spectrum σ disc (H δ,α ) is finite, and for λ ∈ (−∞, 0) one has that Recall that the eigenvalues µ n (S(λ)) are ordered non-increasingly with multiplicities taken into account.If we order the discrete eigenvalues of H δ,α in an increasing way then the strict monotonicity of λ → µ n (S(λ)) implies that the k-th discrete eigenvalue E k (α) (if it exists) satisfies the equation −1 = αµ k (S(E k (α))).
Let n ∈ N. Then by [14,Theorem 1] the operator H δ,α has at least n negative discrete eigenvalues (counted with multiplicities) if −α > 0 is sufficiently large, and the n-th discrete eigenvalue E n (α) of H δ,α admits the asymptotic expansion (A.9) Here H is a fixed semibounded differential operator on Σ that is independent of α and has purely discrete spectrum µ To show item (ii), we note first that by (A.7), (A.11), and the strict monotonicity and continuity of the mapping λ → λµ n (S(λ)) it is clear that for any a < 0 there is a unique solution λ n (a) of λµ n (S(λ)) = a.Let µ n (H) be as in (A.9), define the numbers k ± = ±(|µ n (H)| + 1) and let (A.12) with some functions f ± (a) = O(a −3 ) for large |a| > 0, where the latter representation holds due to a Taylor series expansion.Then one has