Eigensystem of an $L^2$-perturbed harmonic oscillator is an unconditional basis

TL;DR

This paper proves that for certain complex-valued perturbations, the spectrum of a harmonic oscillator remains discrete and simple, and its eigenfunctions form an unconditional basis in L^2, extending understanding of spectral stability under perturbations.

Contribution

It establishes that the eigen- and associated functions form an unconditional basis for a class of perturbed harmonic oscillators with complex-valued L^p or bounded perturbations.

Findings

Spectrum remains discrete and eventually simple.

Eigen- and associated functions form an unconditional basis.

Results apply to perturbations with specific L^p and L^ conditions.

Abstract

We prove the following. For any complex valued $L^p$-function $b(x)$, $2 \leq p < \infty$ or $L^\infty$-function with the norm $\| b | L^{\infty}\| < 1$, the spectrum of a perturbed harmonic oscillator operator $L = -d^2/dx^2 + x^2 + b(x)$ in $L^2(\mathbb{R}^1)$ is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in $L^2(\mathbb{R})$.