Metastable diffusions with degenerate drifts

We study the spectrum of the semiclassical Witten Laplacian ${\mathrm{\Delta }}_f$ associated to a smooth function $f$ on ${\mathbb{R}}^d$. We assume that $f$ is a confining Morse–Bott function. Under this assumption we show that ${\mathrm{\Delta }}_f$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.