Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

Le Peutrec, D.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 735-809.

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 ${\Delta }_{f,h}^{\left(0\right)}=-h^2\Delta +{\left|\nabla f\left(x\right)\right|}^2-h\Delta f\left(x\right)$ lorsque le paramètre $h>0$ tend vers $0$. La fonction $f$ est une fonction de Morse sur un domaine borné $\Omega$ de bord $\partial \Omega$. 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.

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 ${\Delta }_{f,h}^{\left(0\right)}=-h^2\Delta +{\left|\nabla f\left(x\right)\right|}^2-h\Delta f\left(x\right)$ 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 $\Omega$ with boundary $\partial \Omega$. 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.

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