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

Abstract

In this work, 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 $$\psi \in H_0^1(\Omega )\mapsto h^2{\int }_{\Omega }|\nabla (e^{\frac{1}{h}f}\psi ){|}^2{e}^{-\frac{2}{h}f},$$where $\overline{\Omega }=\Omega \cup \partial \Omega$ is an oriented $C^{\infty }$ compact and connected Riemannian manifold with non empty boundary $\partial \Omega$ and $f:\overline{\Omega }\to \mathbb{R}$ 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.

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Cited by in corpus (4)

• Bar codes of persistent cohomology and Arrhenius law for p-forms
• An Eyring-Kramers law for slowly oscillating bistable diffusions
• Exit time and principal eigenvalue of non-reversible elliptic diffusions
• A Grushin problem for Bismut's hypoelliptic Laplacian