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

## TL;DR

This paper demonstrates that in a semiclassical quantum system with an anharmonic potential, Sobolev norms of solutions can grow logarithmically over time, using a constructed potential and initial data.

## Contribution

The authors construct specific potentials and initial states to show logarithmic Sobolev norm growth in semiclassical anharmonic oscillators, extending understanding of long-time quantum dynamics.

## Key findings

- Sobolev norms grow logarithmically over time
- Growth occurs for times up to logarithmic scale in inverse semiclassical parameter
- Method combines unbounded classical solutions with semiclassical approximation

## Abstract

We consider the semiclassical Schr\"odinger 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.