Small eigenvalues of the Witten Laplacian acting on p-forms on a surface

Abstract

In this article, we are interested in the exponentially small eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian associated with some Morse function, in the general framework of p-forms, on a connected compact Riemannian manifold without boundary. Our purpose is to notice that the knowledge of (the asymptotic formulae for) the smallest non-zero eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian acting on functions, presented by Helffer, Klein and Nier in Matematica Contemporanea 26 (2004), 41–85, essentially contains all the necessary information to the treatment of the case of oriented surfaces, for p-forms.