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

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.

Spectral Enclosures for Non-self-adjoint Discrete Schrödinger Operators

“…can be easily justified by usual arguments. In particular, it follows that the implication 2]. Moreover, the implication =⇒ in (2.3) remains valid even if the spectra are replaced by point spectra.…”

Boundary Value Problems, Weyl Functions, and Differential Operators

“…However, for a selection of papers in which boundary triplet techniques were applied to differential operators and other related problems we refer to [28,50,86,87,97,112,113,123,143,144,155,169,170,173,174,184,185,241,288,307,342,343,344,377,461,475,477,478,483,511,529,551,562,563,564,565,589,590,591,626,627,642,644,677,750,751]. For boundary triplets and similar techniques in the analysis of quantum graphs see [110,134,…”

“…Boundary triplets and their Weyl functions for symmetric operators have been further generalized in [103,105,109,110,115,233,236,237,246] by relaxing some of the conditions in the definition of a boundary triplet. Moreover, in the setting of dual pairs of operators (see [525]) boundary triplets have been introduced in [559,560] and applied, e.g., in [170,173,300,302,381]; they were specialized to the case of isometric operators in [561].…”

Absence of eigenvalues of non‐self‐adjoint Robin Laplacians on the half‐space

“…Proof. Even though the proof of this result follows a standard scheme (see, for example, [26] or [27]), we will provide it for the sake of completeness (see also [5,Remark 7.5 ii)] and [6,Theorem 3.5] for the proof of a slightly more general result in the case of bounded domains performed using the technique of boundary triples).…”