Spectral enclosures for non-self-adjoint extensions of symmetric operators

摘要

The spectral properties of non-self-adjoint extensionsA[B]of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operatorB. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions forA[B]to have a non-empty resolvent set are provided in terms of the parameterBand the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures forA[B]are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schr?dinger operators withδ-potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings.