On the growth of Sobolev norms for a class of linear Schr\"odinger equations on the torus with superlinear dispersion

In this paper we consider time dependent Schr"odinger equations on the one-dimensional torus $\text{T}:=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ 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 $\mathcal{W}$ is a time-dependent pseudo-differential operator of order strictly smaller than $M$. We prove that the solutions of the Schr"odinger equation ${\mathrm{\partial }}_{t}u=\text{ii}\mathcal{V}(t)[u]$ grow at most as $t^{\text{e}}$, $t\to +\mathrm{\infty }$ for any $\text{e}>0$. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field $\text{ii}\mathcal{V}(t)$ which uses Egorov type theorems and pseudo-differential calculus.