Classical and quantum mechanics of a strongly chaotic billiard system
Periodic orbits for the hyperbola billiard (a prominent chaotic conservative dynamical system) are studied numerically using an extremum principle and Newton's method. More than 50000 primitive periods are obtained. The length spectrum behaves as if it would result from a one- dimensional random-walk process. Most strikingly, the length spacings seem to follow a Poisson distribution and the stochastic model agrees well with the asymptotic law \(e^{\ell \tau}/\ell \tau\) for the number of (primitive) periodic orbits of length less than \(\ell\). The topological entropy \(\tau\) (close to 0.6) may be expressed in terms of the parameters of the random-walk model. Finally, the classical system is compared with the corresponding quantum system: the periodic-orbit theory due to Gutzwiller evaluated with 36 terms agrees at low energies with the Gaussian level density obtained from a numerical solution of the Schrödinger equation.
scientific article; zbMATH DE number 4159693