Growth of Sobolev norms for completely resonant quantum harmonic oscillators on R2

We consider time dependently perturbed quantum harmonic oscillators in R2: [Formula presented] where V(t,x,D) is a selfadjoint pseudodifferential operator of degree zero, 2π periodic in time. We identify sufficient conditions on the principal symbol of the potential V(t,x,D) that ensure existence of solutions exhibiting unbounded growth in time of their positive Sobolev norms and we show that the class of symbols satisfying such conditions is generic in the Fréchet space of classical 2π-time periodic symbols of order zero. To prove our result we apply the abstract Theorem of [46]: the main difficulty is to find a conjugate operator A for the resonant average of V(t,x,D). We construct explicitly the symbol of the conjugate operator A, called escape function, combining techniques from microlocal analysis, dynamical systems and contact topology.