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

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