Spectral asymptotics of harmonic oscillator perturbed by bounded potential

Details Spectral asymptotics of harmonic oscillator perturbed by bounded potential Spectral asymptotics of harmonic oscillator perturbed by bounded potential

2003-12-24

arxiv.org/abs/math-ph/0312066v1

Annales Henri Poincare, V.6, No.4, pp.747-789,2005.

Physics

Mathematical Physics

LaTeX, 39 pages, 2 postscript figures

Scientific paper

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

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