On a problem in eigenvalue perturbation theory

TL;DR

This paper investigates eigenvalue perturbations of self-adjoint operators, demonstrating that the measure-zero property of eigenvalue persistence under perturbation fails without the nonnegativity condition, supported by explicit counterexamples.

Contribution

It provides a detailed proof of a known eigenvalue measure-zero result and constructs counterexamples showing the necessity of the nonnegativity assumption.

Findings

Eigenvalues of perturbed operators typically have measure zero persistence sets.

Counterexamples show nonnegativity of W is essential for the measure-zero property.

The paper clarifies conditions under which eigenvalue stability results hold.

Abstract

We consider additive perturbations of the type $K_t=K_0+tW$, $t\in [0,1]$, where $K_0$ and $W$ are self-adjoint operators in a separable Hilbert space $\mathcal{H}$ and $W$ is bounded. In addition, we assume that the range of $W$ is a generating (i.e., cyclic) subspace for $K_0$. If $\lambda_0$ is an eigenvalue of $K_0$, then under the additional assumption that $W$ is nonnegative, the Lebesgue measure of the set of all $t\in [0,1]$ for which $\lambda_0$ is an eigenvalue of $K_t$ is known to be zero. We recall this result with its proof and show by explicit counterexample that the nonnegativity assumption $W\geq 0$ cannot be removed.