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Distribution of eigenvalues for the modular group

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Abstract

The two-point correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.

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Author notes

  1. E. Bogomolny

    Present address: L.D. Landau Institute of Theoretical Physics, 142432, Cherogolovka, Russia

  2. F. Leyvraz

    Present address: Instituto de Física, University of Mexico, Apdo. postal 20-364, 01000, Mexico City, Mexico

Authors and Affiliations

  1. Division de Physique Théorique, Unité de Recherche des Université Paris 11 et Paris 6, Associée au CNRS, Institut de Physique Nucléaire, F-91406, Orsay Cedex, France

    E. Bogomolny, F. Leyvraz & C. Schmit

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  1. E. Bogomolny
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  2. F. Leyvraz
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  3. C. Schmit
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Communicated by Ya. G. Sinai

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Bogomolny, E., Leyvraz, F. & Schmit, C. Distribution of eigenvalues for the modular group. Commun.Math. Phys. 176, 577–617 (1996). https://doi.org/10.1007/BF02099251

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  • DOI: https://doi.org/10.1007/BF02099251

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