Convergence to equilibrium of a linearized quantum Boltzmann equation for bosons at very low temperature

We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasipar-ticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the ex-istence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity