arXiv 2201.064902022-01math.AP

哈密顿非线性Klein-Gordon方程中的能量转移、弱共振和费米黄金法则

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

Zhen Lei, Jie Liu, Zhaojie Yang

备注：: 34 pages

摘要：

本文研究了一类三维非线性Klein-Gordon方程，它是带位势的线性Klein-Gordon方程的Hamilton扰动.未扰动的动力系统具有频率为$ω$的束缚态，空间局部化和时间周期解。在量子力学中，亚稳态，持续时间比预期的更长，已经被观察到。这些亚稳态是束缚态在非线性费米黄金法则下不稳定的结果。在这项研究中，我们探讨了从束缚态到这些亚稳态的数学不稳定性机制。此外，当离散谱与Schördinger算符$H= -Δ+ V + m^2$的连续谱不接近时，即弱共振区$ σ_c(\sqrt{H}) = [m, \infty)$，$0< 3ω< m$，我们得到了从离散模到连续模的急剧能量转移率.这扩展了Soffer和Weinstein \cite{SW1999}对共振区$3ω> m$的工作，并证实了他们在\cite{SW1999}中的猜想。我们的证明依赖于Bambusi和Cuccagna \cite{BC}，广义费米黄金法则，以及某些加权色散估计的正规形式变换的更精细的版本。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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