Root system of singular perturbations of the harmonic oscillator type operators

TL;DR

This paper studies how certain perturbations affect harmonic oscillator operators, showing that under specific conditions, the eigenvalues become simple and the eigenfunctions form a Riesz basis, ensuring stability of the spectral structure.

Contribution

It demonstrates that under local subordination, the eigenvalues of perturbed harmonic oscillator operators become simple and their root system forms a Riesz basis.

Findings

Eigenvalues become eventually simple after perturbation

Root system forms a Riesz basis under subordination

Spectral stability of perturbed operators

Abstract

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space H, i.e. of the self-adjoint operator with simple positive eigenvalues $\mu_k$ satisfying $\mu_{k+1}-\mu_k \geq \Delta >0$. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system forms a Riesz basis.