# Spectral shift via “lateral” perturbation

### Gregory Berkolaiko

Texas A&M University, College Station, USA### Peter Kuchment

Texas A&M University, College Station, USA

## 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.

## Cite this article

Gregory Berkolaiko, Peter Kuchment, Spectral shift via “lateral” perturbation. J. Spectr. Theory 12 (2022), no. 1, pp. 83–104

DOI 10.4171/JST/395