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 Schrödinger equations on the one-dimensional torus T:=R/(2πZ) of the form ∂tu=iV(t)[u] where V(t) is a time dependent, self-adjoint pseudo-differential operator of the form V(t)=V(t,x)|D|M+W(t), M>1, |D|:=−∂xx, V is a smooth function uniformly bounded from below and W is a time-dependent pseudo-differential operator of order strictly smaller than M. We prove that the solutions of the Schrödinger equation ∂tu=iV(t)[u] grow at most as tε, t→+∞ for any ε>0. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field iV(t) which uses Egorov type theorems and pseudo-differential calculus