Asymptotic stability of soliton for discrete nonlinear Schrödinger equation on one-dimensional lattice

In this paper we prove the asymptotic stability of solitons for a discrete nonlinear Schrödinger equation near the anti-continuous limit. Our novel insight is that the analysis of linearized operator, usually non-symmetric, can be reduced to a study of simple self-adjoint operator almost like the free discrete Laplacian restricted on the space of odd functions.