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This article follows the previous works [HeKlNi, HeNi] by Helffer-Klein-Nier and Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of ${\Delta }_{f,h}^{\left(0\right)}=-h^2\Delta +{\left|\nabla f\left(x\right)\right|}^2-h\Delta f\left(x\right)$ are considered as the small parameter $h\>0$ tends 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 possible simplifications...