J. Behrndt, M. Langer, V. Lotoreichik and J. Rohleder,
Spectral enclosures for non-self-adjoint extensions of symmetric operators,
J. Funct. Anal. 275 (2018), 1808–1888
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 Schrödinger operators with δ-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.