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

TL;DR

This paper investigates the spectral properties of non-self-adjoint extensions of symmetric operators using boundary triples and Weyl functions, providing conditions for spectral enclosures and applying results to various differential operators and quantum graphs.

Contribution

It introduces new spectral enclosure results for non-self-adjoint operator extensions using boundary triple techniques and applies these to differential operators and quantum graphs.

Findings

Spectral enclosures are established for non-self-adjoint extensions.

Conditions for sectoriality and resolvent set non-emptiness are provided.

Applications include elliptic operators, Schrödinger operators, and quantum graphs.

Abstract

The spectral properties of non-self-adjoint extensions $A_{[B]}$ 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 operator $B$. In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for $A_{[B]}$ to have a non-empty resolvent set are provided in terms of the parameter $B$ and 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 for $A_{[B]}$ 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\"odinger operators with $\delta$-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.