TRACE FORMULAS FOR ARITHMETICAL SYSTEMS

For quantum problems on the pseudosphere generated by arithmetic groups there exist special trace formulas, called trace formulas for Hecke operators, which permit the reconstruction of wave. functions from the knowledge of periodic orbits. After a short discussion of this subject we present the Hecke operators' trace formulas for the Dirichlet problem on the modular billiard; which is a prototype of arithmetical systems. The results of numerical computations for these semiclassical-type relations are in good agreement with the directly computed eigenfunctions.