### Abstract

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)=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.

Original language | English (US) |
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Pages (from-to) | 375-403 |

Number of pages | 29 |

Journal | Communications In Mathematical Physics |

Volume | 170 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1995 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Communications In Mathematical Physics*,

*170*(2), 375-403. https://doi.org/10.1007/BF02108334