Georgia Institute of Technology College of Sciences

In this talk, we will discuss dispersive and local decay estimates for a class of non-self-adjoint Schrödinger operators that naturally arise from the linearization of nonlinear Schrödinger equations around a solitary wave. We review the spectral properties of these linearized operators, and discuss how threshold resonances may appear in their spectrum. In the presence of threshold resonances, it will be shown that the slow local decay rate can be pinned down to a finite rank operator corresponding to the threshold resonances. We will also discuss examples of non-self-adjoint operators that arise from linearizing around solitons in other contexts.