Abstract. Self-adjoint Schrödinger operators with δ and δ ′ -potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman-Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten-von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ ′ -potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

In this paper the spectral properties of Dirac operators Aη with electrostatic δ-shell interactions of constant strength η supported on compact smooth surfaces in R 3 are studied. Making use of boundary triple techniques a Krein type resolvent formula and a Birman-Schwinger principle are obtained. With the help of these tools some spectral, scattering, and asymptotic properties of Aη are investigated. In particular, it turns out that the discrete spectrum of Aη inside the gap of the essential spectrum is finite, the difference of the third powers of the resolvents of Aη and the free Dirac operator A 0 is trace class, and in the nonrelativistic limit Aη converges in the norm resolvent sense to a Schrödinger operator with an electric δ-potential of strength η.2010 Mathematics Subject Classification. Primary 81Q10; Secondary 35Q40. Key words and phrases. Dirac operator; existence and completeness of wave operators; finite discrete spectrum; nonrelativistic limit; quasi boundary triple; self-adjoint extension; shell interaction. 1 corresponding Birman-Schwinger principle; for more details and additional results see Theorem 4.4.Theorem 1.1. Let η ∈ R \ {±2c} and let A η be the Dirac operator with an electrostatic δ-shell interaction of strength η. Then the essential spectrum is given byand the discrete spectrum in the gap (−mc 2 , mc 2 ) is finite, that is,The next result on the trace class property of the difference of the third powers of the resolvents of A η and A 0 has important consequences for mathematical scattering theory. In particular, it follows that the wave operators for the scattering system {A η , A 0 } exist and are complete and that the absolutely continuous parts of A η and A 0 are unitarily equivalent. For more details see Theorem 4.6, where also a trace formula in terms of the Weyl function and its derivatives is provided.Theorem 1.2. Let η ∈ R\{±2c}, let A η be the Dirac operator with an electrostatic δ-shell interaction and let λ ∈ ρ(A η ) ∩ ρ(A 0 ). Then the operatorbelongs to the trace class ideal.

Abstract. The notion of quasi boundary triples and their Weyl functions is an abstract concept to treat spectral and boundary value problems for elliptic partial differential equations. In the present paper the abstract notion is further developed, and general theorems on resolvent differences belonging to operator ideals are proved. The results are applied to second order elliptic differential operators on bounded and exterior domains, and to partial differential operators with δ and δ ′ -potentials supported on hypersurfaces.

Abstract. We investigate Schrödinger operators with δ and δ ′ -interactions supported on hypersurfaces, which separate the Euclidean space into finitely many bounded and unbounded Lipschitz domains. It turns out that the combinatorial properties of the partition and the spectral properties of the corresponding operators are related. As the main result we prove an operator inequality for the Schrödinger operators with δ and δ ′ -interactions which is based on an optimal colouring and involves the chromatic number of the partition. This inequality implies various relations for the spectra of the Schrödinger operators and, in particular, it allows to transform known results for Schrödinger operators with δ-interactions to Schrödinger operators with δ ′ -interactions.

We show that a Schrödinger operator Aδ,α with a δ‐interaction of strength α supported on a bounded or unbounded C2‐hypersurface Σ⊂double-struckRd,d≥2, can be approximated in the norm resolvent sense by a family of Hamiltonians with suitably scaled regular potentials. The differential operator Aδ,α with a singular interaction is regarded as a self‐adjoint realization of the formal differential expression −Δ−αfalse⟨δΣ,·false⟩δnormalΣ, where α:Σ→R is an arbitrary bounded measurable function. We discuss also some spectral consequences of this approximation result.