Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

We consider the semiclassical Schrödinger equation on Rd given by iħ∂tψ=(− [Formula presented] Δ+Wl(x))ψ+V(t,x)ψ, where Wl is an anharmonic trapping of the form Wl(x)= [Formula presented] ∑j=1dxj2l, l≥2 is an integer and ħ 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 [Formula presented] ⁡(ħ−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.