Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 735-809.

This article follows the previous works [HeKlNi, HeNi] by Helffer-Klein-Nier and Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of Δ f,h (0) =-h 2 Δ+f(x) 2 -hΔf(x) are considered as the small parameter h>0 tends to 0. The function f is assumed to be a Morse function on some bounded domain Ω with boundary Ω. Neumann type boundary conditions are considered. With these boundary conditions, some possible simplifications in the Dirichlet problem studied in [HeNi] are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is here carried out.

Cet article est dans la continuation des travaux [HeKlNi, HeNi] de Helffer-Klein-Nier et Helffer-Nier sur l’étude de la métastabilité dans des processus de diffusions réversibles via une approche de Witten. Nous considérons encore ici les valeurs propres exponentionnellement petites d’une réalisation auto-adjointe de Δ f,h (0) =-h 2 Δ+f(x) 2 -hΔf(x) lorsque le paramètre h>0 tend vers 0. La fonction f est une fonction de Morse sur un domaine borné Ω de bord Ω. Des conditions au bord de type Neumann sont considérées ici. Avec ces conditions, certaines simplifications utilisées pour l’étude du problème de Dirichlet dans [HeNi] ne sont plus possibles. Un traitement plus fin des trois géométries intervenant dans le problème à bord (bord, métrique, fonction de Morse) est donc nécessaire.

DOI: 10.5802/afst.1265
Le Peutrec, D. 1

1 Laboratoire de Mathématiques, UMR-CNRS 8628, Université Paris-Sud 11, Bâtiment 425, 91405 Orsay, France
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Le Peutrec, D. Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. 3-4, pp. 735-809. doi : 10.5802/afst.1265. http://www.numdam.org/articles/10.5802/afst.1265/

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