2022/01/17 by Zhen Lei, Jie Liu, Lei, Zhen +3
Computer Science · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Nonlinear Photonic Systems #Quantum optics and atomic interactions
paper · pdf · doi:10.48550/arxiv.2201.06490
openalex publication_date 2022/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ω, 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 Schördinger operator H= -Δ+ V + m2, i.e. weak resonance regime σc(√(H)) = [m, ∞), 0< 3ω< m. This extends the work of Soffer and Weinstein \citeSW1999 for resonance regime 3ω> m and confirms their conjecture in \citeSW1999. Our proof relies on a more refined version of normal form transformation of Bambusi and Cuccagna \citeBC, the generalized Fermi's Golden Rule, as well as certain weighted dispersive estimates.