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

In this article, we are interested in the exponentially small eigenvalues of the self adjoint realization of the semiclassical Witten Laplacian ∆ f,h, 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 formulas for) the smallest non zero eigenvalues of the self adjoint realization of ∆ f,h (acting on functions), presented in [HeKlNi], essentially contains all the necessary information to the treatment of the case of oriented surfaces, for p-forms. The function f is assumed to be a Morse function on Ω. MSC 2010: 58J37, 58J10, 81Q10, 58A10, 15A18.