  We consider the semiclassical quantization of the Sinai billiard for disk
radii R small compared to the wave length 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 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 plus and minus
infinity within the range of validity of the diffractive approximation kR <<4.
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
whose sides obeys some constraints. For c=0 we find a good agreement of the
level spacing distribution with GOE, whereas the form factor and two-point
correlation function are similar but exhibit larger deviations. By varying the
parameter c from 0 to infinity the level statistics interpolates smoothly
between these limiting cases.

Chaotic Dynamics;Condensed Matter;Chaotic Dynamics