Spectral flow inside essential spectrum II: resonance set and its structure

TL;DR

This paper extends the study of spectral flow and resonance sets inside the essential spectrum of self-adjoint operators, revealing geometric structures and criteria for tangency, with implications for perturbation theory.

Contribution

It introduces criteria for tangent vectors to the resonance set inside the essential spectrum and describes the geometric structure of the resonance set, including straight lines and higher-order perturbations.

Findings

Resonance sets contain many straight lines.

Existence of finite rank operators aligning with resonance lines.

Presence of transversal perturbations of order ≥ 2 inside the essential spectrum.

Abstract

This paper is a continuation of the study of spectral flow inside essential spectrum initiated in \cite{AzSFIES}. Given a point $\lambda$ outside the essential spectrum of a self-adjoint operator $H_0,$ the resonance set, $\mathcal R(\lambda),$ is an analytic variety which consists of self-adjoint relatively compact perturbations $H_0+V$ of $H_0,$ for which $\lambda$ is an eigenvalue. One may ask for criteria for the vector $V$ to be tangent to the resonance set. Such criteria were given in \cite{AzSFnRI}. In this paper we study similar criteria for the case of $\lambda$ inside the essential spectrum of $H_0.$ For the case $\lambda \in \sigma_{ess}(H_0)$ the resonance set is defined in terms of the well-known limiting absorption principle. Among the results of this paper is that the resonance set contains plenty of straight lines, moreover, given any regular relatively compact perturbation $V$ there exists a finite rank self-adjoint operator, $\tilde V,$ such that the straight line $H_0 + \mathbb R(V-\tilde V)$ belongs to the resonance set. Another result of this paper is that inside the essential spectrum there exist plenty of transversal to the resonance set perturbations $V$ which have order $\geq 2,$ in contrast to what happens outside the essential spectrum, \cite{AzSFnRI}.