Gang Zhou
Asymptotic stability of ground states of nonlinear Schr\"odinger
equations

Abstract: In this talk I will present our results on the asymptotic
stability of ground states of nonlinear Schr\"odinger equations. To
prove the desired result one has to prove the mass of the neutral modes
and the essential spectrum go to zero as time t goes to infinity. For
the linear problem the neutral modes and essential spectrum are
decoupled. When they are coupled to second order the results were
established by M.I.Weinstein, A. Soffer etc. In our problem they are
coupled at very high order. Hence a better understanding of normal form
transformation is required. We invented the notion of N-order Fermi
Golden rule and observe certain special algebraic structures of the
equation to overcome the difficulties.