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 〈ψₙ(x+(1/2s)ψₙ(x- 1) / 2 s)〉 and the probability distribution Pₙ({ψ}) of wave amplitude for normal modes ψₙ 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({Δ}E) of neighboring energy-eigenvalue separations, which is shown to be similar to a Wigner distribution for the eigenvalues of a random matrix.
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McDonald et al. (1988) studied this question.