论文标题
汉密尔顿非线性klein-gordon方程中的能源传递，弱共振和费米的黄金法则
Energy transfer, weak resonance, and Fermi's golden rule in Hamiltonian nonlinear Klein-Gordon equations
论文作者
论文摘要
本文重点介绍了三个维度的一类非线性klein-gordon方程，这是具有潜力的线性klein-gordon方程的哈密顿扰动。不受干扰的动态系统具有限制状态，其频率$ω$是空间定位和时间周期性解决方案。在量子力学中，观察到的亚稳态状态持续时间比预期的要长。这些亚稳态是在非线性费米的黄金法则下结束国家不稳定的结果。 在这项研究中，我们探索了从结合状态到这些亚稳态状态的潜在数学不稳定性机制。此外，当离散频谱不接近Schördingeroperator $ h =-Δ + V + m^2 $，即弱共振状态$σ_c（\ sqrt {h}）= [m，\ fty）$ 0 <3时这扩展了Soffer和Weinstein \ cite {SW1999}的共鸣式$3Ω> m $的工作，并在\ cite {sw1999}中确认了他们的猜想。我们的证明依赖于Bambusi和Cuccagna \ Cite {Bc}的正常形式转换的更精致版本，该版本是广义的Fermi的黄金法则，以及某些加权分散估计。
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 $H= -Δ+ V + m^2$, i.e. weak resonance regime $ σ_c(\sqrt{H}) = [m, \infty)$, $0< 3ω< m$. This extends the work of Soffer and Weinstein \cite{SW1999} for resonance regime $3ω> m$ 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.
