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

Article type: Research Article

Authors: Montalto, Riccardo

Affiliations: University of Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland. E-mail: [email protected]

Abstract: 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.

Keywords: Growth of Sobolev norms, linear Schrödinger equations, pseudo-differential operators

DOI: 10.3233/ASY-181470

Journal: Asymptotic Analysis, vol. 108, no. 1-2, pp. 85-114, 2018