On reducibility of Quantum Harmonic Oscillator on ℝ^d with quasiperiodic in time potential

2016/03/24 by Paturel, Eric, Grébert, Benoît
#Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.07455

Abstract

We prove that a linear d-dimensional Schrödinger equation on ℝ^d with harmonic potential |x|² and small t-quasiperiodic potential i∂_t u -- Δu + |x|² u + εV (tω, x)u = 0, x ∈ ℝ^d reduces to an autonomous system for most values of the frequency vector ω∈ ℝⁿ. 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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