On the stability of dense point spectrum for self-adjoint operators

Abstract

Let A be a (random) self-adjoint operator with fixed orthonormal eigenvectors, but with independently distributed random eigenvalues. [Typically, for the eigenvalue distributions, A is considered to have a dense point spectrum almost surely (a.s.).] A class of perturbations {B} is exhibited such that A+B has only point spectrum a.s. Examples are also constructed, including a rank-one perturbation B, such that A+μB has no eigenvalues (for μ≠0) a.s., despite A having dense point spectrum a.s.