For a symmetric operator or relation A with infinite deficiency indices in a Hilbert space we develop an abstract framework for the description of symmetric and self-adjoint extensions A_Θ of A as restrictions of an operator or relation T which is a core of the adjoint A^*. This concept is applied to second order elliptic partial differential operators on smooth bounded domains, and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

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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.

The notion of quasi boundary triples and their Weyl functions is reviewed and applied to self-adjointness and spectral problems for a class of elliptic, formally symmetric, second order partial differential expressions with variable coefficients on bounded domains.

Abstract:For a scattering system {A Θ , A 0 } consisting of selfadjoint extensions A Θ and A 0 of a symmetric operator A with finite deficiency indices, the scattering matrix {S Θ (λ)} and a spectral shift function ξ Θ are calculated in terms of the Weyl function associated with the boundary triplet for A * and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar and matrix potentials, to Dirac operators and to Schrödinger operators with point interactions.