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We study the long time behavior of small (in $l^2$) solutions of discretenonlinear Schr\"odinger equations with potential. In particular, we areinterested in the case that the corresponding discrete Schr\"odinger operatorhas exactly two eigenvalues. We show that under the nondegeneracy condition ofFermi Golden Rule, all small solutions decompose into a nonlinear bound stateand dispersive wave. We further show the instability of excited states andgeneralized equipartition property.

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