Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
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DOI：
10.5802/afst.1265
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
D. L. Peutrec
中科院分区：
文献类型：
--
作者：
D. L. Peutrec
本文沿袭了Helffer-Klein-Nier和Helffer-Nier关于可逆扩散过程中的亚稳性的Witten复方法的工作.同样，$\Delta_{f，h}^{(0)}=-h^{2}\Delta+\Left|\nabla f(X)\right|^{2}-h\Delta f(X)，$的某些自伴实现的指数小本征值被认为是小参数$h&>；0$到$0$。假定函数$f$是某有界域$\Omega$上的Morse函数，边界为$\Partial\Omega$。考虑了Neumann型边界条件。有了这些边界条件，在Cite{HelNi1}中研究的Dirichlet问题中的一些可能的简化是不可能的。对边界问题中涉及的三种几何(边界、度量、Morse函数)进行了更精细的处理。
This article follows the previous works \cite{HKN} by Helffer-Klein-Nier and \cite{HelNi1} by Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of $\Delta_{f,h}^{(0)}=-h^{2}\Delta +\left|\nabla f(x)\right|^{2}-h\Delta f(x)\;,$ are considered as the small parameter $h>0$ goes to $0$. The function $f$ is assumed to be a Morse function on some bounded domain $\Omega$ with boundary $\partial\Omega$. Neumann type boundary conditions are considered. With these boundary conditions, some simplifications possible in the Dirichlet problem studied in \cite{HelNi1} are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is carried out.