Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian

This article follows the previous works \cite{HKN} by Helffer-Klein-Nier and \cite{HelNi1} 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 $\Delta_{f,h}^{(0)}=-h^{2}\Delta +\left|\nabla f(x)\right|^{2}-h\Delta f(x)\;,$ 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 $\Omega$ with boundary $\partial\Omega$. Neumann type boundary conditions are considered. With these boundary conditions, some simplifications possible in the Dirichlet problem studied in \cite{HelNi1} are no more possible. A finer treatment of the three geometries involved in the boundary problem (boundary, metric, Morse function) is carried out.