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Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi-Periodic Perturbations

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Abstract

We prove that the 1-d quantum harmonic oscillator is stable under spatially localized, time quasi-periodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by Enss-Veselic in their 1983 paper [EV] in the general quasi-periodic setting. The motivation of the present paper also comes from construction of quasi-periodic solutions for the corresponding nonlinear equation.

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Authors and Affiliations

  1. Departement de Mathematique, Universite Paris Sud, 91405, Orsay Cedex, France

    W. -M. Wang

  2. Department of Mathematics, University of Massachussetts, Amherst, Ma, 01003, USA

    W. -M. Wang

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  1. W. -M. Wang
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Correspondence to W. -M. Wang.

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Communicated by B. Simon

Partially supported by NSF grant DMS-05-03563.

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Wang, W.M. Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi-Periodic Perturbations. Commun. Math. Phys. 277, 459–496 (2008). https://doi.org/10.1007/s00220-007-0379-z

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