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 small eigenvalues Witten Laplacian, that is operator associated with quadratic form $$ \psi\in H^1_0(\Omega)\mapsto h^2 \int_\Omega \big \vert \nabla (e^{\frac 1hf} \psi\big )\big \vert^2\, e^{-\frac 2hf},$$where $\overline\Omega=\Omega\cup \partial \Omega$ an oriented $C^\infty$ compact and connected Riemannian manifold non empty boundary $\partial $f: \overline \Omega\to \mathbb R$ a Morse function. The function $f$ allowed to admit critical points on $ \Omega$, which main novelty work comparison existing literature.