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

Abstract

We consider the Weyl asymptotic formula

$$\# \left\{ {E_n \leqq R^2 } \right\} = \frac{{Area Q}}{{4\pi }}R^2 + n(R),$$

for eigenvalues of the Laplace-Beltrami operator on a two-dimensional torusQ with a Liouville metric which is in a sense the most general case of an integrable metric. We prove that if the surfaceQ is non-degenerate then the remainder termn(R) has the formn(R)=R ^1/2 θ(R), where θ(R) is an almost periodic function of the Besicovitch classB ¹, and the Fourier amplitudes and the Fourier frequencies of θ(R) can be expressed via lengths of closed geodesics onQ and other simple geometric characteristics of these geodesics. We prove then that if the surfaceQ is generic then the limit distribution of θ(R) has a densityp(t), which is an entire function oft possessing an asymptotics on a real line, logp(t)≈−C±t ⁴ ast→±∞. An explicit expression for the Fourier transform ofp(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.