Spectral shift via "lateral" perturbation

TL;DR

This paper investigates how a specific type of perturbation affects eigenvalues of a self-adjoint operator, revealing a critical point with a Morse index equal to the spectral shift, thus linking spectral changes to perturbation properties.

Contribution

It introduces a novel analysis of eigenvalue continuation under lateral perturbations, connecting spectral shift to the Morse index of a critical point in the perturbation space.

Findings

Eigenvalue as a function of perturbation has a critical point at the original perturbation.

The Morse index of this critical point equals the spectral shift.

Results extend to some non-positive perturbations.

Abstract

We consider a compact perturbation $H_0 = S + K_0^* K_0$ of a self-adjoint operator $S$ with an eigenvalue $\lambda^\circ$ below its essential spectrum and the corresponding eigenfunction $f$. The perturbation is assumed to be "along" the eigenfunction $f$, namely $K_0f=0$. The eigenvalue $\lambda^\circ$ belongs to the spectra of both $H_0$ and $S$. Let $S$ have $\sigma$ more eigenvalues below $\lambda^\circ$ than $H_0$; $\sigma$ is known as the spectral shift at $\lambda^\circ$. We now allow the perturbation to vary in a suitable operator space and study the continuation of the eigenvalue $\lambda^\circ$ in the spectrum of $H(K)=S + K^* K$. We show that the eigenvalue as a function of $K$ has a critical point at $K=K_0$ and the Morse index of this critical point is the spectral shift $\sigma$. A version of this theorem also holds for some non-positive perturbations.