Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae
Abstract
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 LaplaceBeltrami operator on a twodimensional 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 nondegenerate 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 GuilleminDuistermaat trace formula for the Liouville surfaces and discuss its accuracy.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 1995
 DOI:
 10.1007/BF02108334
 Bibcode:
 1995CMaPh.170..375B