A Computational Study of the Quantization of Billiards with Mixed Dynamics

We examine the relationship between the spectrum of quantum mushroom billiards and the structure of their classical counterparts, which have mixed integrable-chaotic dynamics. Accordingly, we study the eigenvalues corre- sponding to eigenfunctions of the stationary Schrodinger equation with homogeneous Dirichlet boundary conditions on half-mushroom-shaped geometries for very high energies/wavenumbers using the "scaling method," a technique for eigenvalue/boundary value problems. We compute several thousand consecutive eigenvalues to obtain a cumulative energy level spacing distribution that, according to a conjecture of Berry and Robnik, can also be determined from the relative volumes of the integrable components of the phase space of the classical billiard. Our results suggest that the Berry-Robnik conjecture for the quantization of systems with mixed dynamics holds for quantum mushroom billiards with circular caps. This paper details a numerical study of the quantization of billiards with mixed dynamics. Such quantum billiards can be thought of as "particle in a box" models with corresponding classical Hamil- tonian (energy-conserving) billiards that can exhibit either chaotic or integrable dynamics (depending upon initial conditions). Quantum billiards are more than just a theoretical curiosity, however, as they have been implemented experimentally using microwaves in reflective cavities (22), cold atoms (13), and quantum dots (22). The particular problem in which we are interested is closely related to the question popularly known as "Can one hear the shape of a drum?" That is, can one uniquely describe the boundary of a planar region from its spectrum (i.e., its set of eigenvalues)? Here we investigate the relation of the classical billiard's dynamics to the cumulative nearest-neighbor spacing distribution (CNNSD) of the spectrum of its quantization. It is known that one type of statistical distribution (Poisson statistics) describes the CNNSD of systems displaying regular dynamics and another (Wigner statistics) describes the CNNSD of systems displaying chaotic dynamics. For intermediate cases (when the dynamics are mixed) in which the integrable and chaotic components of the classical dynamical system's phase space are well-separated, the Berry-Robnik conjecture states that the CNNSD of its quantization can be determined from the number and relative phase space volumes of its integrable components using a distribution that interpolates between Poisson and Wigner statistics.