Wave chaos in the stadium: Statistical properties of short-wave solutions of the Helmholtz equation

We numerically investigate statistical properties of short-wavelength normal modes and the spectrum for the Helmholtz equation in a two-dimensional stadium-shaped region. As the geometrical optics rays within this boundary (billiards) are nonintegrable, this wave problem serves as a simple model for the study of quantum chaos. The local spatial correlation function 〈${\psi }_n$(x+(1/2s)${\mathrm{\psi }}_{\mathrm{n}}$(x- 1) / 2 s)〉 and the probability distribution $P_n$(ψ) of wave amplitude for normal modes ${\psi }_n$ are computed and compared with predictions based on semiclassical arguments applied to this nonintegrable Hamiltonian. The spectrum is analyzed in terms of the probability P(ΔE) of neighboring energy-eigenvalue separations, which is shown to be similar to a Wigner distribution for the eigenvalues of a random matrix.