Asymptotic stability of ground states in 3D subcritical nonlinear Schroedinger equation

The talk will start with an overview of asymptotic stability results for NLS. By definition, the ground states are stable if a solution starting nearby decomposes into a part convergent to a ground state and a part radiating away. For ma ny years the theory covered only the supercritical (large power) nonlinearities mainly because a large power of a small perturbation a round a ground state is very small and easily dispersed by the Schroedinger type operator given by the linearization at the ground sta te. Recently results for critical nonlinearities have been obtained. The second part of the talk will focus on a new apprach which giv es asymptotic stability results even for subcritical (low power) nonlinearities. It relies on linearizing the equation along a one par ameter family of ground states. By continuously adapting the linearization to the actual evolution of the solution we are able to capt ure the correct effective potential induced by the nonlinearity into a time dependent Schroedinger type operator. The dispersive estim ates we prove for this operator allow us to control the remaining nonlinear terms and obtain the asymptotic stability results.