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Article Dans Une Revue Annales de la Faculté des Sciences de Toulouse. Mathématiques. Année : 2010

Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

Résumé

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.
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Dates et versions

hal-00297207 , version 1 (15-07-2008)
hal-00297207 , version 2 (30-07-2008)
hal-00297207 , version 3 (23-10-2008)

Identifiants

  • HAL Id : hal-00297207 , version 3

Citer

Dorian Le Peutrec. Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2010, 19 (3-4), pp.735-809. ⟨hal-00297207v3⟩
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