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On reducibility of Quantum Harmonic Oscillator on $\mathbb{R}^d$ with quasiperiodic in time potential (1603.07455v1)

Published 24 Mar 2016 in math.AP and math.DS

Abstract: We prove that a linear d-dimensional Schr{\"o}dinger equation on $\mathbb{R}d$ with harmonic potential $|x|2$ and small t-quasiperiodic potential $i\partial_t u -- \Delta u + |x|2 u + \epsilon V (t\omega, x)u = 0, x \in \mathbb{R}d$ reduces to an autonomous system for most values of the frequency vector $\omega \in \mathbb{R}n$. As a consequence any solution of such a linear PDE is almost periodic in time and remains bounded in all Sobolev norms.

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