On the growth of Sobolev norms for a class of linear Schrödinger equations on the torus with superlinear dispersion

Abstract

In this paper we consider time dependent Schrödinger equations on the one-dimensional torus $\T := \R /(2 π\Z)$ of the form $\partial_t u = \ii {\cal V}(t)[u]$ where $\mathcal{V}(t)$ is a time dependent, self-adjoint pseudo-differential operator of the form $\mathcal{V}(t)=V(t,x)|D{|}^M+\mathcal{W}(t)$, $M>1$, $|D|:=\sqrt{-{\partial }_{xx}}$, $V$ is a smooth function uniformly bounded from below and $\mathcal{W}$ is a time-dependent pseudo-differential operator of order strictly smaller than $M$. We prove that the solutions of the Schrödinger equation $\partial_t u = \ii {\cal V}(t)[u]$ grow at most as $t^\e$, $t\to +\infty$ for any $\e > 0$. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field $\ii {\cal V}(t)$ which uses Egorov type theorems and pseudo-differential calculus.

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