On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential

Abstract

We consider the Cauchy problem for nonlinear Schrödinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the H1-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly – no matter how large it is, unless the initial datum decays sufficiently fast at infinity.