On small energy stabilization in the NLS with a trapping potential

TL;DR

This paper studies the long-term behavior of small energy solutions to the nonlinear Schrödinger equation with a trapping potential, extending previous work to more general spectral conditions and linking the Fermi Golden Rule to Hamiltonian structure.

Contribution

It generalizes prior results by allowing generic spectra in the analysis of NLS with trapping potentials and interprets the Fermi Golden Rule within a Hamiltonian framework.

Findings

Extended the analysis to generic spectra

Connected Fermi Golden Rule to Hamiltonian structure

Described asymptotic behavior of small energy solutions

Abstract

We describe the asymptotic behavior of small energy solutions of an NLS with a trapping potential. In particular we generalize work of Soffer and Weinstein, and of Tsai et. al. The novelty is that we allow generic spectra associated to the potential. This is yet a new application of the idea to interpret the nonlinear Fermi Golden Rule as a consequence of the Hamiltonian structure.