  We consider the linear relaxation Boltzmann equation in a semiclassical
framework. We construct a family of sharp quasimodes for the associated
operator which yields sharp spectral asymptotics for its small spectrum in the
low temperature regime. We deduce some information on the long time behavior of
the solutions with a sharp estimate on the return to equilibrium as well as a
quantitative metastability result. The main novelty is that the collision
operator is a pseudo-differential operator in the critical class S^1/2 and that
its action on the gaussian quasimodes yields a superposition of exponentials.
