Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae

We consider the Weyl asymptotic formula 220_2005_Article_BF02108334_TeX2GIFE1.gif # left\{ {E_n ≤q R^2 } right\} = {Area Q}/{4π }R^2 + n(R), for eigenvalues of the Laplace-Beltrami operator on a two-dimensional torus Q with a Liouville metric which is in a sense the most general case of an integrable metric. We prove that if the surface Q is non-degenerate then the remainder term n(R) has the form n(R)=R ^1/2 θ(R), where θ( R) is an almost periodic function of the Besicovitch class B ¹, and the Fourier amplitudes and the Fourier frequencies of θ( 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 θ( R) has a density p(t), which is an entire function of t possessing an asymptotics on a real line, log p(t)≈-C±t ⁴ as t→±∞. An explicit expression for the Fourier transform of p(t) via Fourier amplitudes of θ( R) is also given. We obtain the analogue of the Guillemin-Duistermaat trace formula for the Liouville surfaces and discuss its accuracy.