Long time growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

We consider the semiclassical Schr"odinger equation on ${\mathbb{R}}^d$ given by $$\mathrm{i}\mathrm{\hslash }{\mathrm{\partial }}_t\psi =(-\frac{{\mathrm{\hslash }}^2}{2}\mathrm{\Delta }+W_l(x))\psi +V(t,x)\psi ,$$ where $W_l$ is an anharmonic trapping of the form $W_l(x)=\frac{1}{2l}{\sum }_{j=1}^d{x}_j^{2l}$, $l\ge 2$ is an integer and $\mathrm{\hslash }$ 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 ${\log }^{\frac{1}{2}}({\mathrm{\hslash }}^{-1})$. 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.