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

In this paper we consider the linear, time-dependent quantum Harmonic Schrdinger equation i partial derivative(t)u = 1/2(-partial derivative(x)(2) + x(2 ))u + V(t,x,d)u,x epsilon 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 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.