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

Jussi Behrndt, Matthias Langer, Vladimir Lotoreichik, Jonathan Rohleder

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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.
Original languageEnglish
Pages (from-to)1808-1888
Number of pages81
JournalJournal of Functional Analysis
Volume275
Issue number7
Early online date3 May 2018
DOIs
Publication statusPublished - 1 Oct 2018

Keywords

  • non-self-adjoint extension
  • spectral enclosure
  • differential operator
  • Weyl function

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