Periodic orbit quantization of the Sinai billiard in the small scatterer limit

Abstract

We consider the semiclassical quantization of the Sinai billiard for disk radii R small compared with the wavelength 2 pi/k. Via the application of the periodic orbit theory of diffraction we derive the semiclassical spectral determinant. The limitations of the derived determinant are studied by comparing it to the exact Korringa-Kohn-Rostoker (KKR) determinant, which we generalize here for the A(1) subspace. With the help of the Ewald resummation method developed for the full KKR determinant we transfer the complex diffractive determinant to a real form. The real zeros of the determinant are the quantum eigenvalues in semiclassical approximation. The essential parameter is the strength of the scatterer c = J(0)(kR)/Y(0)(kR). Surprisingly, this can take any value between -infinity and +infinity within the range of validity of the diffractive approximation kR << 4, causing strong perturbation despite the smallness of the disk. We study the statistics exhibited by spectra for fixed values of c. It is Poissonian for c = +/-infinity, provided the disk is placed inside a rectangle with irrational squared side ratio. For c = 0 we find a good agreement of the revel spacing distribution with Gaussian orthogonal ensemble (GOE). The form factor and hue-point correlation function are also similar to GOE, but with larger deviations. By varying the parameter c from 0 to +/-infinity the level statistics interpolates smoothly between these limiting cases. Any of these transitional level statistics can thus be found in the spectrum of a Sinai billiard with sufficiently small R while we go from the quantum to the semiclassical limit k --> infinity.