Spectral asymptotics of harmonic oscillator perturbed by bounded potential

Abstract

Consider the operator $ T=-{d^2dx^2}+x^2+q(x)$ in $L^2(\mathbb{R})$, where real functions $q$, $q'$ and $\int_0^xq(s)ds$ 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 $\mu_n = (2n+1) + (2\pi)^{-1}\int_{-\pi}^{\pi}q(\sqrt{2n+1}\sin\theta)d\theta + O(n^{-1/3})$.