Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators
We consider the semiclassical Schrodinger equation on \mathbb R^d$ given by \mathrm{i} \hbar \partial_t \psi \left( \frac{\hbar^2}{2} \Delta W_l(x) \right)\psi V(t,x)\psi where $W_l$ is an anharmonic trapping of the form $W_l(x) \frac{1}{2l}\sum_{j=1}^d x_j^{2l} $l\geq 2$ is an integer and \hbar$ is a semiclassical small parameter. We construct a smooth potential $V(t,x) bounded in time with its derivatives, and an initial datum such that the Sobolev norms of the solution grow at a logarithmic speed for all times of order \log^\frac12}\hbar^ 1} The proof relies on two ingredients: first we construct an unbounded solution to a forced mechanical anharmonic oscillator, then we exploit semiclassical approximation with coherent states to obtain growth of Sobolev norms for the quantum system which are valid for semiclassical time scales.