Generic Transporters for the Linear Time-Dependent Quantum Harmonic Oscillator on ℝ

Abstract

Abstract In this paper we consider the linear, time-dependent quantum Harmonic Schrdinger equation ${\textrm {i}} \partial _t u= \frac {1}{2} ( - \partial _x^2 + x^2) u + V(t, x, D)u$, $x \in \mathbb {R}$, where $V(t,x,D)$ is classical pseudodifferential operator of order 0, self-adjoint, and $2\pi $ periodic in time. We give sufficient conditions on the principal symbol of $V(t,x,D)$ ensuring the existence of solutions displaying infinite time growth of Sobolev norms. These conditions are generic in the Fréchet 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.