  Spectral methods, thanks to the high accuracy and the possibility to use fast
algorithms, represent an effective way to approximate the Boltzmann collision
operator. On the other hand, the loss of some local invariants leads to the
wrong long time behavior. A way to overcome this drawback, without sacrificing
spectral accuracy, has been proposed recently with the construction of
equilibrium preserving spectral methods. Despite the ability to capture the
steady state with arbitrary accuracy, the theoretical properties of the method
have never been studied in details. In this paper, using the perturbation
argument developed by Filbet and Mouhot for the homogeneous Boltzmann equation,
we prove stability, convergence and spectrally accurate long time behavior of
the equilibrium preserving approach.
