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

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}.