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

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

$$\psi \in H_0^1\left(\mathrm{\Omega }\right)\mapsto h^2{\int }_{\mathrm{\Omega }}|\nabla \left(e^{\frac{1}{h}f}\psi \right){|}^2{e}^{-\frac{2}{h}f},$$

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