A Resonance Problem in Relaxation of Ground States of Nonlinear Schrodinger Equations

TL;DR

This paper investigates the impact of resonance on the relaxation of ground states in nonlinear Schrödinger equations, revealing how resonances affect normal form transformations and the Fermi Golden Rule.

Contribution

It extends previous non-resonance results to include resonance cases, providing a unified analysis and new insights into cancellations caused by small denominators.

Findings

Resonance introduces small denominators affecting normal form transformations.

Cancellations occur when denominators are zero or small, leading to small numerators.

Uniform recovery of key results under resonance conditions.

Abstract

In this paper we consider a resonance problem, in a generic regime, in the consideration of relaxation of ground states of semilinear Schrodinger equations. Different from previous results, our consideration includes the presence of resonance, resulted by overlaps of frequencies of different states. All the known key results, proved under non-resonance conditions, have been recovered uniformly. These are achieved by better understandings of normal form transformation and Fermi Golden rule. Especially, we find that if certain denominators are zeros (or small), resulted by the presence of resonances (or close to it), then cancellations between terms make the corresponding numerators proportionally small.