Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time-dependent perturbation

Abstract

We prove a reducibility result for a quantum harmonic oscillator in arbitrary dimension with arbitrary frequencies perturbed by a linear operator which is a polynomial of degree two in xj, −i∂j with coefficients which depend quasiperiodically on time.

Stability and instability properties of rotating Bose–Einstein condensates

“…There exponential growth naturally occurs in the case of (even only partially) repulsive harmonic potentials. We finally mention that very recently a somewhat similar instability phenomenon for linear Schrödinger equations with quadratic time-dependent Hamiltonian has been established in [3].…”

Reducibility of 1-d Schrödinger equation with unbounded time quasiperiodic perturbations. III

“…The problem of reducibility of equations of the form of (1.1) has a long history, and the main results have been obtained in [Com87, DŠ96, DLŠV02, Kuk93, BG01, LY10, Bam18, Bam17] (see [Bam17] for a more detailed history). We also mention that our result is limited to the one dimensional case, while some results on this problems in more then one dimension have been recently obtained [GP16,BGMR18,Mon17b,FGMP18]. We also recall that related techniques have been used in order to get a control on the growth of Sobolev norms in [BGMR17,Mon18].…”

Spectral stability and semiclassical measures for renormalized KAM systems

“…al. in [9], in which they obtain reducibility for a class of perturbations of the quantum harmonic oscillator.…”