Classical and quantum mechanics of a strongly chaotic billiard system

Abstract

We study the hyperbola billiard, a strongly chaotic system whose classical dynamics is the free motion of a particle within the region D = l{( x, y)| x≥0∧ y≥0∧ y≤1/ x} r with elastic reflections on the boundary ∂D. The corresponding quantum mechanical problem is to determine the bound state energies as eigenvalues of the Dirichlet Laplacian on D. It is shown that the classical periodic orbits of the hyperbola billiard can be effectively enumerated by a ternary code. Combining this code with an extremum principle, we are able to determine with high precision more than 500 000 primitive periodic orbits together with their lengths, multiplicities and Lyapunov exponents. The statistical properties of the length spectrum of the periodic orbits are found to be consistent with a random walk model, which in turn predicts asymptotically an exponential proliferation of long periodic orbits and leads to a novel formula for the topological entropy τ, whose value turns out to be approximately 0.6. The periodic orbits are used for a quantitative test of Gutzwiller's periodic-orbit theory, which plays the role of a semiclassical quantization rule. We find that the predictions of the periodic-orbit theory for the Gaussian level density agree at low energies surprisingly well with the “true” results obtained from a numerical solution of the Schrödinger equation.