About small eigenvalues of Witten Laplacian

We study the eigenvalues of the semiclassical Witten Laplacian ${\mathrm{\Delta }}_{\varphi }$ associated to a potential $\varphi$. We consider the case where the sequence of Arrhenius numbers $S_1\le \dots \le S_n$ associated to $\varphi$ is degenerated, that is the preceding inequality are not necessarily strict.