Distribution of Eigenvalues for the Modular Group

E. Bogomolny 1, F. Leyvraz 1, 2, C. Schmit 1

Communications in Mathematical Physics 176 (1996) 577-617

The two-point correlation function 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 ion 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.

  • 1. Division de Physique Théorique, IPN,
    Université Paris XI - Paris Sud
  • 2. Instituto de Fisica,
    University of Mexico