Fermi's golden rule and <tex-math id="M1">H¹</tex-math> scattering for nonlinear Klein-Gordon equations with metastable states

In this paper, we explore the metastable states of nonlinear Klein-Gordon equations with potentials. These states come from instability of a bound state under a nonlinear Fermi's golden rule. In [16], Soffer and Weinstein studied the instability mechanism and obtained an anomalously slow-decaying rate 1/(1+t)¹⁴. Here we develop a new method to study the evolution of L²ₓ norm of solutions to Klein-Gordon equations. With this method, we prove a H¹ scattering result for Klein-Gordon equations with metastable states. By exploring the oscillations, with a dynamical system approach we also find a more robust and more intuitive way to derive the sharp decay rate 1/(1+t)¹⁴.