Stabilization of small solutions of discrete NLS with potential having two eigenvalues

We study the long time behavior of small (in $l^2$) solutions of discrete nonlinear Schr\"odinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schr\"odinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.