A note on the Fermi Golden Rule constant for the pure power NLS

TL;DR

This paper rigorously proves that the Fermi Golden Rule coefficient, crucial for understanding the stability of ground states in certain nonlinear Schrödinger equations, is nonzero, supporting the stability analysis.

Contribution

It provides a detailed proof confirming the nonzero value of the Fermi Golden Rule coefficient for the pure power NLS with exponents close to 3.

Findings

Fermi Golden Rule coefficient is nonzero for the considered NLS.

Supports the asymptotic stability of ground states in the studied regime.

Enhances understanding of stability mechanisms in nonlinear Schrödinger equations.

Abstract

We provide a detailed proof that the Nonlinear Fermi Golden Rule coefficient that appears in our recent proof of the asymptotic stability of ground states for the pure power Nonlinear Schr\"odinger equations in $\mathbb{R}$ with exponent $0<|p-3|\ll 1$ is nonzero.