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 $\mathbb{T}:=\mathbb{R}\mathrm{/}\mathrm{(}2\mathit{\pi }\mathbb{Z}\mathrm{)}$ of the form ${\partial }_{\mathit{t}}\mathit{u}=\mathrm{i}\mathcal{V}\mathrm{(}\mathit{t}\mathrm{)}[\mathit{u}]$ where $\mathcal{V}\mathrm{(}\mathit{t}\mathrm{)}$ is a time dependent, self-adjoint pseudo-differential operator of the form $\mathcal{V}\mathrm{(}\mathit{t}\mathrm{)}=\mathit{V}\mathrm{(}\mathit{t}\mathrm{,}\mathit{x}\mathrm{)}|\mathit{D}{|}^{\mathit{M}}+\mathcal{W}\mathrm{(}\mathit{t}\mathrm{)}$ , $\mathit{M}\mathrm{>}1$ , $|\mathit{D}|:=\sqrt{-{\partial }_{\mathit{x}\mathit{x}}}$ , 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 ${\partial }_{\mathit{t}}\mathit{u}=\mathrm{i}\mathcal{V}\mathrm{(}\mathit{t}\mathrm{)}[\mathit{u}]$ grow at most as ${\mathit{t}}^{\mathit{\epsilon }}$ , $\mathit{t}\to +\infty$ for any $\mathit{\epsilon }\mathrm{>}0$ . The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field $\mathrm{i}\mathcal{V}\mathrm{(}\mathit{t}\mathrm{)}$ which uses Egorov type theorems and pseudo-differential calculus.