We characterize the spectra of self-adjoint extensions of a symmetric operator with equal deficiency indices in terms of boundary values of their Weyl functions. A complete description is obtained for the point and absolutely continuous spectrum while for the singular continuous spectrum additional assumptions are needed. The results are illustrated by examples.

We show that there is a family Schrödinger operators with scaled potentials which approximates the δ ′ -interaction Hamiltonian in the normresolvent sense. This approximation, based on a formal scheme proposed by Cheon and Shigehara, has nontrivial convergence properties which are in several respects opposite to those of the Klauder phenomenon.

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.

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expressionWe investigate the absolutely continuous parts of different self-adjoint realizations of A. In particular, we show that Dirichlet and Neumann realizations, A D and A N , are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization.In addition, we prove that the absolutely continuous part A ac of any realization A is unitarily equivalent to A D provided that the resolvent difference ( A − i) −1 − (A D − i) −1 is compact. The abstract results are applied to elliptic differential expression in the half-space.Subject Classification: 34G10, 47E05, 47F05, 47A20, 47B25Keywords: Sturm-Liouville operators, operator potentials, elliptic partial differential operators, boundary value problems, self-adjoint extensions, unitary equivalence, direct sums of symmetric operators 1

Let S be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non-zero deficiency indices. Let J be an open subset of R. If there exists a self-adjoint extension S 0 of S such that J is contained in the resolvent set of S 0 and the associated Weyl function of the pair {S, S 0 } is monotone with respect to J, then for any selfadjoint operator R there exists a self-adjoint extension S such that the spectral parts S J and R J are unitarily equivalent. It is shown that for any extension S of S the absolutely continuous spectrum of S 0 is contained in that one of S. Moreover, for a wide class of extensions the absolutely continuous parts of S and S are even unitarily equivalent.