Sharp spectral gap for some degenerate Witten Laplacians

We consider Witten Laplacians associated to some non-Morse potentials. We prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the semiclassical regime and quantify the spectral gap separating these eigenvalues from the rest of the spectrum. The main ingredient is the construction of sharp degenerate Gaussian quasimodes through an adaptation of the WKB method.