Spectral asymptotics of harmonic oscillator perturbed by bounded potential

Klein, M.; Korotyaev, E.; Pokrovski, A.

Spectral asymptotics of harmonic oscillator perturbed by bounded potential

Authors

Klein, M.

Korotyaev, E.

Pokrovski, A.

Description

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 $\{μ_n\}_{n=0}^\infty$, $μ_n<μ_{n+1}$. We determine their asymptotics $μ_n = (2n+1) + (2π)^{-1}\int_{-π}^πq(\sqrt{2n+1}\sinθ)dθ+ O(n^{-1/3})$.
LaTeX, 39 pages, 2 postscript figures