The Spectral Shift Function for Non-Self-Adjoint Perturbations

TL;DR

This paper extends the spectral shift function concept to non-self-adjoint operators, analyzing spectral singularities and applying results to Schrödinger operators with complex potentials, advancing scattering theory understanding.

Contribution

It introduces a generalized spectral shift function for non-self-adjoint operators, extending trace formulas and analyzing spectral singularities with applications to complex Schrödinger operators.

Findings

SSF carries information on complex eigenvalues

Extended Lifshits-Krein trace formula to non-self-adjoint operators

Analyzed spectral singularities and their impact

Abstract

This paper is devoted to the definition and analysis of the spectral shift function (SSF) associated with non-self-adjoint perturbations of self-adjoint operators. Motivated by applications in scattering theory, we consider both trace-class and relatively trace-class perturbations. We extend the Lifshits-Kre__n trace formula to non-self-adjoint operators under suitable assumptions on the spectrum and the behavior of the resolvent. The role of spectral singularities is carefully analyzed, and we provide a generalization of the SSF using functional calculus. Finally, we apply our results to Schr{\"o}dinger operators with complex-valued short-range potentials in dimension three. Toy models illustrate properties that one might hope to extend to general cases. In particular, they suggest that the SSF carries information on the presence of complex eigenvalues.