  In this note, we present the formalism to start a quantum analysis for the
recent billiard representation introduced by Damour, Henneaux and Nicolai in
the study of the cosmological singularity. In particular we use the theory of
Maass automorphic forms and recent mathematical results about arithmetical
dynamical systems. The predictions of the billiard model give precise
automorphic properties for the wave function (Maass-Hecke eigenform), the
asymptotic number of quantum states (Selberg asymptotics for PSL(2,Z)), the
distribution for the level spacing statistics (the Poissonian one) and the
absence of scarred states. The most interesting implication of this model is
perhaps that the discrete spectrum is fully embedded in the continuous one.
