ENERGY-LEVEL STATISTICS OF MODEL QUANTUM-SYSTEMS - UNIVERSALITY AND SCALING IN A LATTICE-POINT PROBLEM

We investigate the statistics of the number N(R, S) of lattice points, n is-an-element-of Z2, in an annular domain PI(R, w) = (R + w)A\RA, where R, w > 0. Here A is a fixed convex set with smooth boundary and w is chosen so that the area of PI(R, w) is S. The statistics comes from R being taken as random (with a smooth density) in some interval [c1 T, C2 T], c2 > c1 > 0. We find that in the limit T --> infinity the variance and distribution of DELTAN = N(R; S) - S depend strongly on how S grows with T. There is a saturation regime S/T --> infinity, as T --> infinity, in which the fluctuations in DELTAN coming from the two boundaries of PI are independent. Then there is a scaling regime, S/T --> z, 0 < z < infinity, in which the distribution depends on z in an almost periodic way going to a Gaussian as z --> 0. The variance in this limit approaches z for ''generic'' A, but can be larger for ''degenerate'' cases. The former behavior is what one would ''pect from the Poisson limit of a distribution for annuli of finite area.