Spectral asymptotics of harmonic oscillator perturbed by bounded potential

TL;DR

This paper analyzes the spectral asymptotics of a harmonic oscillator with a bounded potential, providing precise eigenvalue asymptotics for operators with periodic or almost periodic potentials.

Contribution

It derives explicit asymptotic formulas for eigenvalues of the perturbed harmonic oscillator with bounded, periodic or almost periodic potentials.

Findings

Eigenvalues are asymptotically close to (2n+1) plus a potential-dependent integral.

The eigenvalue asymptotics include an explicit correction term involving the potential.

The remainder term in the asymptotics is of order O(n^{-1/3}).

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 $\mu_n = (2n+1) + (2\pi)^{-1}\int_{-\pi}^{\pi}q(\sqrt{2n+1}\sin\theta)d\theta + O(n^{-1/3})$.