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

Abstract

We prove that a linear d-dimensional Schr{ö}dinger equation on ${\mathbb{R}}^d$ with harmonic potential $|x{|}^2$ and small t-quasiperiodic potential $i\partial \_tu--\Delta u+|x{|}^{2}u+\epsilon V(t\omega ,x)u=0,x\in {\mathbb{R}}^d$ 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.

References in corpus (2)

• KAM for the quantum harmonic oscillator
• KAM for the Klein Gordon equation on ${\mathbb{S}}^d$

Cited by in corpus (12)

• On time dependent Schr{ö}dinger equations: global well-posedness and growth of Sobolev norms
• Reducibility of non-resonant transport equation on $T^d$ with unbounded perturbations
• Reducibility of 1-d Schrödinger equation with unbounded time quasiperiodic perturbations, III
• Long time stability of small finite gap solutions of the cubic Nonlinear Schrödinger equation on ${\mathbb{T}}^2$
• KAM for the Klein Gordon equation on ${\mathbb{S}}^d$
• KAM for quasi-linear forced hamiltonian NLS
• Growth of Sobolev Norms in 1-d Quantum Harmonic Oscillator with Polynomial Time Quasi-periodic Perturbation
• Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions on the Derivative of Perturbation Potentials
• A reducibility result for a class of linear wave equations on ${\mathbb{T}}^d$
• 1-d Quantum Harmonic Oscillator with Time Quasi-periodic Quadratic Perturbation: Reducibility and Growth of Sobolev Norms
• Reducibility of 1-d Schrödinger equation with unbounded oscillation perturbations
• Reducibility of relativistic Schrödinger equation with unbounded perturbations