Non-symmetric perturbations of self-adjoint operators

TL;DR

This paper analyzes how non-symmetric perturbations affect the spectrum of self-adjoint operators, providing stability results and resolvent estimates that extend classical theories with broad applications.

Contribution

It establishes new stability theorems for spectral gaps under non-symmetric perturbations, including cases with large relative bounds and structured perturbations.

Findings

Extended classical perturbation results by Kato and Gohberg/Krein.

Proved stability of spectral gaps with resolvent estimates.

Applied results to Dirac operators, periodic systems, and dissipative Hamiltonians.

Abstract

We investigate the effect of non-symmetric relatively bounded perturbations on the spectrum of self-adjoint operators. In particular, we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates. These results extend, and improve, classical perturbation results by Kato and by Gohberg/Krein. Further, we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator; in the latter case, we can even allow for perturbations with relative bound $\ge 1$. The generality of our results is illustrated by several applications, massive and massless Dirac operators, point-coupled periodic systems, and two-channel Hamiltonians with dissipation.