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

This paper focuses on a class of nonlinear Klein-Gordon equations in three dimensions, which are Hamiltonian perturbations of the linear Klein-Gordon equation with potential. The unperturbed dynamical system has a bound state with frequency $\omega$, a spatially localized and time periodic solution. In quantum mechanics, metastable states, which last longer than expected, have been observed. These metastable states are a consequence of the instability of the bound state under the nonlinear Fermi's Golden Rule. In this study, we explore the underlying mathematical instability mechanism from the bound state to these metastable states. Besides, we derive the sharp energy transfer rate from discrete to continuum modes, when the discrete spectrum was not close to the continuous spectrum of the Sch\"ordinger operator $H= -\Delta + V + m^2$, i.e. weak resonance regime $ \sigma_c(\sqrt{H}) = [m, \infty)$, $0< 3\omega < m$. This extends the work of Soffer and Weinstein \cite{SW1999} for resonance regime $3\omega > m$ and confirms their conjecture in \cite{SW1999}. Our proof relies on a more refined version of normal form transformation of Bambusi and Cuccagna \cite{BC}, the generalized Fermi's Golden Rule, as well as certain weighted dispersive estimates.