Growth of Sobolev norms for time dependent periodic Schr\"odinger equations with sublinear dispersion

Abstract

In this paper we consider Schr\"odinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 \pi \Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(\omega t)[u]$ where ${\cal V}(\omega t)$ is a quasi-periodic in time, self-adjoint pseudo-differential operator of the form ${\cal V}(\omega t) = V(\omega t, x) |D|^M + {\cal W}(\omega t)$, $0 < M \leq 1$, $|D| := \sqrt{- \partial_{xx}}$, $V$ is a smooth, quasi-periodic in time function and ${\cal W}$ is a quasi-periodic time-dependent pseudo-differential operator of order strictly smaller than $M$. Under suitable assumptions on $V$ and ${\cal W}$, we prove that if $\omega$ satisfies some non-resonance conditions, the solutions of the Schr\"odinger equation $\partial_t u = \ii {\cal V}(\omega t)[u]$ 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 $\ii {\cal V}(\omega t)$ 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 $\omega$.