Growth of Sobolev norms for completely resonant quantum Harmonic Oscillators in ℝ²

In this talk I present some results obtained in a recent work in collaboration with Beatrice Langella and Alberto Maspero. We consider time-dependently perturbed quantum harmonic oscillators in $\mathbb{R}^2$:

$$
i\partial_t u = \frac12(-\partial_{x_1}^2 - \partial_{x_2}^2 + x_1^2 + x_2^2 ) u + V(t,x,D)u, \quad x \in \mathbb{R}^2
$$

where $V(t,x,D)$ is a $2\pi$-periodic in time, selfadjoint pseudodifferential operator of degree zero. We identify sufficient conditions on 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 $2\pi$-time periodic symbols of order zero.

During the talk I first introduce the problem of growth of Sobolev norms for this class of equations, in order to motivate our result, and then I describe the main ingredients of our proof. The main difficulty is to find a conjugate operator $A$ for the resonant average of $V(t,x,D)$. We explicitly construct the symbol of the conjugate operator $A$, thus turning the problem into the study of a class of Hamiltonian systems and using techniques of differential geometry and dynamical systems.