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In mathematics and physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 1924. It describes the geodesic motion of a free particle on the non-compact Riemann surface where is the upper half-plane endowed with the Poincaré metric and is the modular group. It can be viewed as the motion on the fundamental domain of the modular group with the sides identified.
The system is notable in that it is an exactly solvable system that is strongly chaotic: it is not only ergodic, but is also strong mixing. As such, it is an example of an Anosov flow. Artin's paper used symbolic dynamics for analysis of the system.
The quantum mechanical version of Artin's billiard is also exactly solvable. The eigenvalue spectrum consists of a bound state and a continuous spectrum above the energy . The wave functions are given by Bessel functions.
The motion studied is that of a free particle sliding frictionlessly, namely, one having the Hamiltonian
where m is the mass of the particle, are the coordinates on the manifold, are the conjugate momenta:
In the case of the Artin billiards, the metric is given by the canonical Poincaré metric
on the upper half-plane. The non-compact Riemann surface is a symmetric space, and is defined as the quotient of the upper half-plane modulo the action of the elements of acting as Möbius transforms. The set
is a fundamental domain for this action.
- E. Artin, "Ein mechanisches System mit quasi-ergodischen Bahnen", Abh. Math. Sem. d. Hamburgischen Universität, 3 (1924) pp170-175.