The abstract Birman-Schwinger principle and spectral stability

TL;DR

This paper extends the Birman-Schwinger principle to analyze spectral stability of self-adjoint operators under small non-self-adjoint perturbations, providing new insights for Schrödinger operators in hyperbolic space.

Contribution

It generalizes classical spectral stability results by incorporating non-self-adjoint perturbations and applies these to Schrödinger operators in hyperbolic space.

Findings

Extended Birman-Schwinger principles for non-self-adjoint perturbations

Revisited spectral stability results for Euclidean Schrödinger and Dirac operators

Established new spectral stability results for Schrödinger operators in hyperbolic space

Abstract

We discuss abstract Birman-Schwinger principles to study spectra of self-adjoint operators subject to small non-self-adjoint perturbations in a factorised form. In particular, we extend and in part improve a classical result by Kato which ensures spectral stability. As an application, we revisit known results for Schr\"odinger and Dirac operators in Euclidean spaces and establish new results for Schr\"odinger operators in three-dimensional hyperbolic space.