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

In this paper we consider time dependent Schrodinger equations on the one-dimensional torus $\T := \R /(2 \pi \Z)$ of the form $\partial_t u = \ii {\cal V}(t)[u]$ where ${\cal V}(t)$ is a time dependent, self-adjoint pseudo-differential operator of the form ${\cal V}(t) = V(t, x) |D|^M + {\cal W}(t)$, $M > 1$, $|D| := \sqrt{- \partial_{xx}}$, $V$ is a smooth function uniformly bounded from below and ${\cal W}$ is a time-dependent pseudo-differential operator of order strictly smaller than $M$. We prove that the solutions of the Schrodinger 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.