Coupling Resonances and Spectral Properties of the Product of Resolvent and Perturbation

TL;DR

This paper investigates the spectral properties of the product of the resolvent of a self-adjoint operator and a perturbation, introducing coupling resonances and analyzing their complex-analytic behavior to understand spectral shifts.

Contribution

It introduces the concept of coupling resonances for complex parameters and provides necessary and sufficient conditions for their branching, extending previous real-valued analyses.

Findings

Characterization of eigenvalues of the perturbed operator product

Analysis of branching points of coupling resonances

Extension of spectral shift function analysis to complex domain

Abstract

Given a self-adjoint operator $H_0$ and a relatively compact self-adjoint perturbation $V$, we study in some detail the spectral properties of the product $(H_0-z)^{-1}V$. For some numbers $r_z,$ the eigenvalues of $(H_0 + s V - z)^{-1}V$ are $(s - r_z)^{-1}$, where $s$ is any complex number. We study the root spaces of the eigenvalues $(s - r_z)^{-1}$ and complex-analytic properties of the functions $r_z$ such as branching points. In particular, for a generic case, we give a variety of necessary and sufficient conditions for branching. The functions $r_z,$ called coupling resonances, are important in the spectral analysis of $H_0 + r V$ for any real number $r.$ For instance, they afford a description of the spectral shift function (SSF) of the pair $H_0$ and $V,$ as well as the absolutely continuous and singular parts of the SSF. A thorough study of real-valued coupling resonances $r_\lambda$ for real $\lambda$ outside of the essential spectrum was carried out in a recent work by the first author. Here we extend this study to the complex domain, motivated by the fact, which is well known in the case of a rank-one perturbation, that the behaviour of coupling resonances $r_z$ near the essential spectrum provides valuable information about the latter.