Quantum jumps of saturation level rigidity and anomalous oscillations of level number variance in the semiclassical spectrum of a modified Kepler problem

We discover quantum Hall like jumps in the saturation spectral rigidity in the semiclassical spectrum of a modified Kepler problem as a function of the interval center. These jumps correspond to integer decreases of the radial winding numbers in classical periodic motion. We also discover and explain single harmonic dominated oscillations of the level number variance with the width of the energy interval. The level number variance becomes effectively zero for the interval widths defined by the frequency of the shortest periodic orbit. This signifies that there are virtually no variations from sample to sample in the number of levels on such intervals.