Stability of the gapless pure point spectrum of self-adjoint operators

TL;DR

This paper establishes explicit conditions under which the pure-point spectrum of a self-adjoint operator remains stable under bounded perturbations, focusing on operators with eigenvalues accumulating at finite points.

Contribution

It provides new explicit criteria for the spectral stability of self-adjoint operators with pure-point spectra under bounded perturbations.

Findings

Spectral stability is guaranteed under certain eigenvalue and perturbation conditions.

Conditions are applicable to operators with eigenvalues accumulating at finite points.

Results contribute to understanding the robustness of spectral properties in quantum mechanics.

Abstract

We consider a self-adjoint operator $T$ on a separable Hilbert space, with pure-point and simple spectrum with accumulations at finite points. Explicit conditions are stated on the eigenvalues of $T$ and on the bounded perturbation $V$ ensuring the global stability of the spectral nature of $T+V$.