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In this work, we give sharp asymptotic equivalents in the limit $h\to 0$ ofthe small eigenvalues of the Witten Laplacian, that is the operator associatedwith the quadratic form $$ \psi\in H^1_0(\Omega)\mapsto h^2 \int_\Omega \big\vert \nabla \big (e^{\frac 1hf} \psi\big )\big \vert^2\, e^{-\frac2hf},$$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 $C^\infty$ Morsefunction. The function $f$ is allowed to admit critical points on $ \partial\Omega$, which is the main novelty of this work in comparison with the existingliterature.

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