Sharp spectral gap for degenerated Witten Laplacian

In this talk I am interested in formulas describing the low-lying eigenvalues of the Witten Laplacian $\Delta_V = -h^2\Delta + | V^{\prime} |^2 - h V^{\prime \prime}$. The case where $V$ is a Morse function has been largely studied and here I try to obtain similar results when $V$ has some degeneracy. In the end of the presentation I will also give an example of new behaviors that were not observed in the Morse case.