Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

This article follows the previous works [HKN] by Helffer-KleinNier and [HeNi1] by Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of ∆ f,h = −h∆ + |∇f(x)| − h∆f(x) , are considered as the small parameter h > 0 goes to 0. The function f is assumed to be a Morse function on some bounded domain Ω with boundary ∂Ω. Neumann type boundary conditions are considered. With these boundary conditions, some simplifications possible in the Dirichlet problem studied in [HeNi1] are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is carried out. MSC 2000: 58J37 (58J10 58J32 60J60 81Q10 81Q20)