A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions

We discuss asymptotics of the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold. We obtain an asymptotic formula for the average of the number of states. Furthermore, when the operator has integer eigenvalues and the periodic orbits of period less than 2 pi of the classical mechanics form clean submanifolds of lower dimensions, we give an asymptotic formula for the number of states itself. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer. We use the Meinardus saddle point method of obtaining the asymptotics of the number of partitions, combined with a theorem due to Duistermaat-Guillemin and other authors on the singularities of the trace of the wave operators.