Energy transfer, weak resonance, and Fermi’s golden rule in Hamiltonian nonlinear Klein-Gordon equations

We consider a class of nonlinear Klein–Gordon equations in R3, viewed as Hamiltonian perturbations of the linear Klein–Gordon equation with potential. The linear operator admits a bound state with frequency ω, corresponding to a spatially localized, time-periodic solution. In the weakly resonant regime 3ω<m, we establish the explicit rate of energy transfer from the discrete mode to the continuous spectrum. This extends the classical result of Soffer–Weinstein (1999), who analyzed the resonant regime ω<m<3ω via Fermi’s Golden Rule (FGR), and the subsequent generalization by Bambusi–Cuccagna for multiple bound states. Our approach combines a sharp version of Birkhoff normal form, yielding a generalized FGR in the non-resonant setting, with weighted dispersive estimates that control nonlinear interactions over long times. These tools allow us to derive precise asymptotic behavior and quantify metastability of the bound state. The result provides a rigorous mechanism for nonlinear energy leakage in the absence of strong resonance.