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

Le Peutrec, Dorian; Nectoux, Boris

doi:10.2140/apde.2021.14.2595

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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

DOI: 10.2140/apde.2021.14.2595

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

ψ \in H_{0}^{1} (Ω) \mapsto h^{2} \int_{Ω} | \nabla (e^{\frac{1}{h} f} ψ) |^{2} e^{- \frac{2}{h} f},

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

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Keywords

Witten Laplacian, overdamped Langevin dynamics, semiclassical analysis, metastability, spectral theory, Eyring–Kramers formulas

Mathematical Subject Classification 2010

Primary: 35P15, 35P20, 35Q82, 47F05

Milestones

Received: 10 December 2019

Revised: 3 May 2020

Accepted: 31 July 2020

Published: 19 December 2021

Authors

Dorian Le Peutrec
Institut Denis Poisson Université d’Orléans Université de Tours, CNRS Orléans France

Boris Nectoux
Institut für Analysis und Scientific Computing TU Wien Wien Austria

Vol. 14, No. 8, 2021