Spectral flow inside essential spectrum V: on absorbing points of coupling resonances

TL;DR

This paper investigates the behavior of coupling resonance functions within the essential spectrum of self-adjoint operators, focusing on absorbing points where these functions diverge, providing new insights into spectral analysis.

Contribution

It introduces partial results on absorbing points of coupling resonances, expanding understanding of spectral properties in operator theory.

Findings

Identification of absorbing points where resonance functions diverge

Analysis of the behavior of coupling resonance functions near absorbing points

Extension of spectral analysis techniques to essential spectrum

Abstract

Let $H_0$ and $V$ be self-adjoint operators, such that $V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact $F.$ Coupling resonance functions, $r_j(z),$ of the pair $H_0$ and $V$ can be defined as $r_j(z) = - \sigma_j(z) ^{-1},$ where $\sigma_j(z)$ are eigenvalues of the compact-operator valued holomorphic function $F(H_0-z)^{-1} F^*J.$ Taken together, the functions $r_j(z)$ form an infinite-valued holomorphic function on the resolvent set of~$H_0.$ These functions contain a lot of information about the pair $H_0, V$ (this is well-known in the case of rank one $V$). A point $z_0$ of the resolvent set we call \emph{absorbing} if some $r_j(z)$ goes to $\infty$ as $z \to z_0$ along some half-interval. In this note I present some partial results concerning absorbing points of coupling resonances.