Growth of Sobolev norms for time dependent periodic Schrödinger equations with sublinear dispersion

Abstract

In this paper we consider Schrödinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 π\Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(ωt)[u]$ where $\mathcal{V}(\omega t)$ is a quasi-periodic in time, self-adjoint pseudo-differential operator of the form $\mathcal{V}(\omega t)=V(\omega t,x)|D{|}^M+\mathcal{W}(\omega t)$, $0<M\le 1$, $|D|:=\sqrt{-{\partial }_{xx}}$, $V$ is a smooth, quasi-periodic in time function and $\mathcal{W}$ is a quasi-periodic time-dependent pseudo-differential operator of order strictly smaller than $M$. Under suitable assumptions on $V$ and $\mathcal{W}$, we prove that if $\omega$ satisfies some non-resonance conditions, the solutions of the Schrödinger equation $\partial_t u = \ii {\cal V}(ωt)[u]$ grow at most as $t^{\eta }$, $t\to +\infty$ for any $\eta >0$. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field $\ii {\cal V}(ωt)$ which uses Egorov type theorems and pseudo-differential calculus. The {\it homological equations} arising in the reduction procedure involve both time and space derivatives, since the dispersion relation is sublinear. Such equations can be solved by imposing some Melnikov non-resonance conditions on the frequency vector $\omega$.

arXiv commentarXiv admin note: text overlap with arXiv:1706.09704

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