Spectral rigidity and eigenfunction correlations at the Anderson transition

The statistics of energy levels for a disordered conductor are considered in the critical energy window near the mobility edge. It is shown that, if the critical wave functions are multifractal, the one-dimensional gas of levels on the energy axis is compressible, in the sense that the variance of the level number in an interval is [(delta N)(2)]similar to chi[N] for [N] much greater than 1. The compressibility, chi=eta/2d, is given exactly in terms of the multifractal exponent eta=d-D-2 at the mobility edge in d-dimensional system. (C) 1996 American Institute of Physics