Metastability Results for a Class of Linear Boltzmann Equations

We consider a semiclassical linear Boltzmann model with a non-local collision operator. We provide sharp spectral asymptotics for the small spectrum in the low temperature regime from which we deduce the rate of return to equilibrium as well as a metastability result. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.