Spectral flow inside essential spectrum VI: on essentially singular points

TL;DR

This paper investigates the nature of essentially singular points within the essential spectrum of a self-adjoint operator, providing conditions for their identification and exploring their relation to eigenvalues of infinite multiplicity.

Contribution

It introduces new criteria to determine when a real number is essentially singular and analyzes their connection to infinite multiplicity eigenvalues.

Findings

Conditions for essential singularity identification

Relation between singular points and infinite multiplicity eigenvalues

Insights into spectral properties of self-adjoint operators

Abstract

Let $H_0$ be a self-adjoint operator on a Hilbert space $\mathcal H$ endowed with a rigging $F,$ which is a zero-kernel closed operator from $\mathcal H$ to another Hilbert space $\mathcal K$ such that the sandwiched resolvent $F (H_0 - z)^{-1}F^*$ is compact. Assume that $H_0$ obeys the limiting absorption principle (LAP) in the sense that the norm limit $F (H_0 - \lambda - i0)^{-1}F^*$ exists for a.e.~$\lambda.$ Numbers~$\lambda$ for which such limit exists we call $H_0$-regular. A number~$\lambda$ we call semi-regular, if the limit $F (H_0 + F^*JF - \lambda - i0)^{-1}F^*$ exists for at least one bounded self-adjoint operator $J$ on $\mathcal K;$ otherwise we call~$\lambda$ essentially singular. In this paper I discuss essentially singular points. In particular, I give different conditions which ensure that a real number~$\lambda$ is essentially singular, and discuss their relation to eigenvalues of infinite multiplicity which are known examples of essentially singular points.