ar X iv : m at h / 05 05 46 8 v 2 [ m at h . A P ] 2 2 A ug 2 00 5 INSTABILITY FOR SEMI - CLASSICAL SCHRÖDINGER EQUATIONS

Using WKB methods for very small times, we prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. The main step of the analysis consists in reducing the problem to an ordinary differential equation. The solution to this o.d.e. is explicit, and the instability mechanism is due to the presence of the semi-classical parameter. For nonlin-ear equations, our approach also allows to consider the presence of an harmonic potential and/or weaker nonlinearities. As an application, we retrieve some ill-posedness properties for nonlinear Schrödinger equations.