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

Pavel M. Bleher, Denis V. Kosygin, Yakov G. Sinai

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We consider the Weyl asymptotic formula {Mathematical expression} 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)=R1/2θ(R), where θ(R) is an almost periodic function of the Besicovitch class B1, 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±t4 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.

Original languageAmerican English
Pages (from-to)375-403
Number of pages29
JournalCommunications In Mathematical Physics
Issue number2
StatePublished - Jun 1995

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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