Relative stability of singular spectrum

TL;DR

This paper introduces a theorem showing that the singular spectrum of a self-adjoint operator can be relatively stable under certain conditions, especially when the limiting absorption principle holds, providing a new tool for spectral analysis.

Contribution

The paper presents a novel theorem demonstrating the relative stability of the singular spectrum under the limiting absorption principle, contrasting with its known volatility under perturbations.

Findings

Singular spectrum stability is linked to the limiting absorption principle.

The theorem offers a method to disprove the limiting absorption principle.

Provides insights into spectral stability under specific conditions.

Abstract

There are classical theorems of analysis which, given certain conditions on a perturbation, assert stability of the essential and absolutely continuous components of the spectrum of a self-adjoint operator. Whereas the singular component is known to be highly volatile under the weakest of all possible perturbations, -- rank one. In this note I announce a theorem which asserts that, nevertheless, the singular component of spectrum in an open interval is in a sense \emph{relatively stable} provided the limiting absorption principle (LAP) holds in the interval. One of the benefits of this result is the provision of a method for disproving LAP where it is suspected to fail.