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

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 Schrödinger equations in $\mathbb{R}$ with exponent $0<\mathrm{\mid }p-3\mathrm{\mid }\ll 1$ is nonzero.