Metastable diffusions with degenerate drifts

We study the spectrum of the semiclassical Witten Laplacian Δ f associated to a smooth function f on ℝ d . We assume that f is a confining Morse–Bott function. Under this assumption we show that Δ 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.