Probability of the level number in a given energy interval

Abstract

The statistics of the energy spectra for random electron systems are investigated near the metal-insulator transition. The fluctuations of the number of one-electron states in a finite energy interval are considered. The total probability of the level number is shown to be scale-invariant. The results are obtained at the critical point with and without time-reversal symmetry. The existence of the critical statistics is a prominent signature of the localization transition.