  In this paper we consider time dependent Schr\"odinger 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 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 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.
