Spectral asymptotics of harmonic oscillator perturbed by bounded potential

Abstract

Consider the operator $T=-d^{2}d{x}^2+x^2+q(x)$ in $L^2(\mathbb{R})$, where real functions $q$, $q^{\prime }$ and ${\int }_0^{x}q(s)ds$ are bounded. In particular, $q$ is periodic or almost periodic. The spectrum of $T$ is purely discrete and consists of the simple eigenvalues $\{{\mu }_n{\}}_{n=0}^{\infty }$, ${\mu }_n<{\mu }_{n+1}$. We determine their asymptotics ${\mu }_n=(2n+1)+(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }q(\sqrt{2n+1}\sin \theta )d\theta +O(n^{-1/3})$.

LaTeX, 39 pages, 2 postscript figures