From chaotic to disordered systems - a periodic orbit approach

We apply periodic orbit theory to a quantum billiard on a torus with a vari able number (N) of randomly distributed circular scatterers. Provided the s catterers are much smaller than the wavelength they may be regarded as sour ces of pure s-wave diffraction. The relevant part of the spectral determina nt is due only to diffractive periodic orbits. We formulate this diffractiv e zeta function in terms of an N x N transfer matrix, which is transformed to real form. The determinant is shown to reproduce the full density of sta tes for generic configurations if N greater than or equal to 4. The zeros o f the determinant are computed numerically. We study the statistics exhibit ed by these spectra. The numerical results suggest that the spectra tend to GOE statistics as the number of scatterers increases for typical members o f the ensemble. A peculiar situation arises for configurations with four sc atterers and kR tuned to kR = y(0,1) approximate to 0.899, where the statis tics appears to be perfectly Poissonian.