#### Vol. 14, No. 8, 2021

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Small eigenvalues of the Witten Laplacian with Dirichlet boundary conditions: the case with critical points on the boundary

### Dorian Le Peutrec and Boris Nectoux

Vol. 14 (2021), No. 8, 2595–2651
##### Abstract

We give sharp asymptotic equivalents in the limit $h\to 0$ of the small eigenvalues of the Witten Laplacian, that is, the operator associated with the quadratic form

$\psi \in {H}_{0}^{1}\left(\Omega \right)↦{h}^{2}{\int }_{\Omega }|\nabla \left({e}^{\frac{1}{h}f}\psi \right){|}^{2}{e}^{-\frac{2}{h}f},$

where $\overline{\Omega }=\Omega \cup \partial \Omega$ is an oriented ${C}^{\infty }$ compact and connected Riemannian manifold with nonempty boundary $\partial \Omega$ and $f:\overline{\Omega }\to ℝ$ is a ${C}^{\infty }$ Morse function. The function $f$ is allowed to admit critical points on $\partial \Omega$, which is the main novelty of this work in comparison with the existing literature.

##### Keywords
Witten Laplacian, overdamped Langevin dynamics, semiclassical analysis, metastability, spectral theory, Eyring–Kramers formulas
##### Mathematical Subject Classification 2010
Primary: 35P15, 35P20, 35Q82, 47F05