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→ 0 of the small eigenvalues of the Witten Laplacian, that is the operator associated with the quadratic form ψ∈ H¹₀(Ω)↦ h² ∫_Ω ∇ (e¹ʰᶠ ψ ) ²\, e- 2hf,where Ω=Ω∪ ∂ Ω is an oriented C^∞ compact and connected Riemannian manifold with non empty boundary ∂ Ω and f: Ω→ R is a C^∞ Morse function. The function f is allowed to admit critical points on ∂ Ω, which is the main novelty of this work in comparison with the existing literature.