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Spectral Asymptotics of the Harmonic Oscillator Perturbed by Bounded Potentials

Authors:	Markus Klein Evgeny Korotyaev Alexis Pokrovski

Institution:	(1) Institut für Mathematik, Universität Potsdam, Germany;(2) Institut für Mathematik, Humboldt Universität zu Berlin, Rudower Chaussee 25, D-12489 Berlin, Germany;(3) Institute for Physics, St.Petersburg State University, Russia

Abstract:	Consider the operator $T = - \frac{{d^2 }}{{dx^2 }} + x^2 + q(x)$ in $L^2 \left( \mathbb{R} \right),$ where q is a real function with q′ and $\int_0^x {q(s)\,ds}$ bounded. The spectrum of T is purely discrete and consists of simple eigenvalues. We determine their asymptotics $\mu _n = (2n + 1) + (2\pi )^{ - 1} \int_{ - \pi }^\pi {q\left( {\sqrt {2n + 1} \sin \theta } \right)d\theta + O\left( {n^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 3}} \right. \kern-\nulldelimiterspace} 3}} } \right)}$ and we extend these results for complex q.Communicated by Bernard Helffersubmitted 23/04/04, accepted 26/10/04

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