1993/07/02 by Michael Eisele, Dieter Mayer · 4 citations
Mathematics · Physics and Astronomy · #Arithmetic zeta function #Constant curvature #Curvature #Dynamical billiards #Eigenfunction #Eigenvalues and eigenvectors #Factorization #Geometric and Algebraic Topology #Geometry #Hamiltonian (control theory) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Operator (biology) #Physics #Prime zeta function #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Riemann zeta function #Zeta function regularization #chao-dyn #nlin.CD
paper · pdf · doi:10.1016/0167-2789(94)90070-1
published in Physica D Nonlinear Phenomena 70(4), 342-356 (Elsevier BV) · 18 pages, no figures
arxiv created 1993/07/02 · openalex publication_date 1994/02/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Dynamical zeta functions are expected to relate the Schrödinger operator's spectrum to the periodic orbits of the corresponding fully chaotic Hamiltonian system. The relationsship is exact in the case of surfaces of constant negative curvature. The recently found factorisation of the Selberg zeta function for the modular surface is known to correspond to a decomposition of the Schrödinger operator's eigenfunctions into two sets obeying different boundary condition on Artin's billiard. Here we express zeta functions for Artin's billiard in terms of generalized transfer operators, providing thereby a new dynamical proof of the above interpretation of the factorization formula. This dynamical proof is then extended to the Artin--Venkov--Zograf formula for finite coverings of the modular surface.