2021/09/06 by Azamov, Nurulla · 1 citation
#47A40 #FOS: Mathematics #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2109.02258
This paper is a continuation of the study of spectral flow inside essential spectrum initiated in \citeAzSFIES. Given a point λ outside the essential spectrum of a self-adjoint operator H0, the resonance set, \mathcal R(λ), is an analytic variety which consists of self-adjoint relatively compact perturbations H0+V of H0, for which λ is an eigenvalue. One may ask for criteria for the vector V to be tangent to the resonance set. Such criteria were given in \citeAzSFnRI. In this paper we study similar criteria for the case of λ inside the essential spectrum of H0. For the case λ∈ σess(H0) 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, V, such that the straight line H0 + \mathbb R(V- 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 ≥ 2, in contrast to what happens outside the essential spectrum, \citeAzSFnRI.