GUTZWILLER’S OCTAGON AND THE TRIANGULAR BILLIARD T*(2,3,8) AS MODELS FOR THE QUANTIZATION OF CHAOTIC SYSTEMS BY SELBERG’S TRACE FORMULA

Abstract

Two strongly chaotic systems are investigated with respect to quantization rules based on Selberg’s trace formula. One of them results from the action of a particular strictly hyperbolic Fuchsian group on the Poincaré disk, leading to a compact Riemann surface of genus g=2. This Fuchsian group is denoted as Gutzwiller’s group. The other one is a billiard inside a hyperbolic triangle, which is generated by the operation of a reflection group denoted as T*(2,3,8). Since both groups belong to the class of arithmetical groups, their elements can be characterized explicitly as 2×2 matrices containing entries, which are algebraic numbers subject to a particular set of restrictions. In the case of Gutzwiller’s group this property can be used to determine the geodesic length spectrum of the associated dynamical system completely up to some cutoff length. For the triangular billiard T*(2,3,8) the geodesic length spectrum is calculated by building group elements as products of a suitable set of generators and separating a unique representative for each conjugacy class. The presence of reflections in T*(2,3,8) introduces additional classes of group elements besides the hyperbolic ones, which correspond to periodic orbits of the dynamical system. Due to different choices of boundary conditions along the edges of the fundamental domain of T*(2,3,8), several quantum mechanical systems are associated to one classical system. It has been observed, that these quantum mechanical systems can be divided into two classes according to the behavior of their spectral statistics. This peculiarity is examined from the point of view of classical quantities entering quantization rules. It can be traced back to a subtle influence of the boundary conditions, which introduces contributions from non-periodic orbits for one of the two classes.