Selberg’s ζ function and the quantization of chaos

Abstract

We study Artin’s billiard, an extremely chatoic system defined on the fundamental domain of the modular group PSL(2,openZ), and show that its quantum energy levels are given exactly by the nontrivial zeros of a certain Selberg ζ function expressed as an Euler product over the classical periodic orbits. We demonstrate that at least the first 73 energy levels can be determined by using only a finite number of periodic orbits.