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

Abstract

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

arXiv comment28 pages