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In this paper we consider time dependent Schr\"odinger equations on theone-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-adjointpseudo-differential operator of the form ${\cal V}(t) = V(t, x) |D|^M + {\calW}(t)$, $M > 1$, $|D| := \sqrt{- \partial_{xx}}$, $V$ is a smooth functionuniformly bounded from below and ${\cal W}$ is a time-dependentpseudo-differential operator of order strictly smaller than $M$. We prove thatthe solutions of the Schr\"odinger 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 basedon a reduction to constant coefficients up to smoothing remainders of thevector field $\ii {\cal V}(t)$ which uses Egorov type theorems andpseudo-differential calculus.

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