DISTRIBUTION OF ENERGY-LEVELS OF QUANTUM FREE PARTICLE ON THE LIOUVILLE SURFACE AND TRACE FORMULAS

We consider the Weyl asymptotic formula #(E(n) less than or equal to R(2)) = Area Q/4 pi R2 + n(R), for eigenvalues of the Laplace-Beltrami operator on a two-dimensional torus e with a Liouville metric which is in a sense the most general case of an integrable metric. We prove that if the surface e is non-degenerate then the remainder term n(R) has the form n(R) = R(1/2)theta(R), where theta(R) is an almost periodic function of the Besicovitch class B-1, and the Fourier amplitudes and the Fourier frequencies of theta(R) can be expressed via lengths of closed geodesics on Q and other simple geometric characteristics of these geodesics. We prove then that if the surface Q is generic then the limit distribution of theta(R) has a density p(t), which is an entire function of t possessing an asymptotics on a real line, logp(t)similar to-C(+/-)t(4) as t -->+/-infinity. An explicit expression for the Fourier transform of p(t) via Fourier amplitudes of theta(R) is also given. We obtain the analogue of the Guillemin-Duistermaat trace formula for the Liouville surfaces and discuss its accuracy.