Generic transporters for the linear time dependent quantum Harmonic oscillator on $\mathbb R$

TL;DR

This paper investigates the stability of solutions to the time-dependent quantum harmonic oscillator with pseudodifferential perturbations, showing that generic perturbations lead to solutions with unbounded Sobolev norm growth over time.

Contribution

It establishes generic conditions on pseudodifferential symbols that cause instability and unbounded Sobolev norm growth in the quantum harmonic oscillator.

Findings

Existence of solutions with infinite Sobolev norm growth under generic perturbations

Conditions on the principal symbol ensuring instability are generic in the symbol space

Use of pseudodifferential normal form and Mourre's theory to prove local energy decay

Abstract

In this paper we consider the linear, time dependent quantum Harmonic Schr\"odinger equation $i \partial_t u= \frac{1}{2} ( - \partial_x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb R$, where $V(t,x,D)$ is classical pseudodifferential operator of order 0, selfadjoint, and $2\pi$ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of weakly turbulent solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Frechet space of symbols. This shows that generic, classical pseudodifferential, $2\pi$-periodic perturbations provoke unstable dynamics. The proof builds on the results of [36] and it is based on pseudodifferential normal form and local energy decay estimates. These last are proved exploiting Mourre's positive commutator theory.