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 0 of the small eigenvalues of the Witten Laplacian, that is, the operator associated with the quadratic form

ψ H01(Ω)h2Ω|(e1 hfψ)|2e2 hf,

where Ω ¯ = Ω Ω is an oriented C compact and connected Riemannian manifold with nonempty boundary Ω and f : Ω ¯ is a C Morse function. The function f is allowed to admit critical points on Ω, 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
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