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

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.

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Witten Laplacian, overdamped Langevin dynamics, semiclassical analysis, metastability, spectral theory, Eyring–Kramers formulas
Mathematical Subject Classification 2010
Primary: 35P15, 35P20, 35Q82, 47F05
Received: 10 December 2019
Revised: 3 May 2020
Accepted: 31 July 2020
Published: 19 December 2021
Dorian Le Peutrec
Institut Denis Poisson
Université d’Orléans
Université de Tours, CNRS
Boris Nectoux
Institut für Analysis und Scientific Computing
TU Wien