Local Spectral Deformation

TL;DR

This paper introduces an analytic perturbation framework for eigenvalues embedded in the essential spectrum of self-adjoint operators, leveraging spectral deformation techniques under Mourre estimates to isolate eigenvalues for perturbation analysis.

Contribution

It develops a novel local spectral deformation method for embedded eigenvalues using analytic perturbation theory and Mourre estimates.

Findings

Eigenvalues can be locally deformed away from the essential spectrum.

Embedded eigenvalues become isolated under spectral deformation.

The method enables perturbation analysis of embedded eigenvalues.

Abstract

We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator $H$. We assume the existence of another self-adjoint operator $A$ for which the family $H_\theta = e^{\mathrm{i}\theta A} H e^{-\mathrm{i}\theta A}$ extends analytically from the real line to a strip in the complex plane. Assuming a Mourre estimate holds for $\mathrm{i}[H,A]$ in the vicinity of the eigenvalue, we prove that the essential spectrum is locally deformed away from the eigenvalue, leaving it isolated and thus permitting an application of Kato's analytic perturbation theory.