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 $ψ\in H^1_0(Ω)\mapsto h^2 \int_Ω\big \vert \nabla \big (e^{\frac 1hf} ψ\big )\big \vert^2\, e^{-\frac 2hf},$where $\overlineΩ=Ω\cup \partial Ω$ is an oriented $C^\infty$ compact and connected Riemannian manifold with non empty boundary $\partial Ω$ and $f: \overline Ω\to \mathbb R$ is a $C^\infty$ Morse function. The function $f$ is allowed to admit critical points on $\partial Ω$, which is the main novelty of this work in comparison with the existing literature.

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