Spectral Rigidity and Eigenfunction Correlations at the Anderson Transition

Abstract

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 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 $< (δN)^{2} > = χ $ for $ >> 1$. The compressibility, $χ=η/2d$, is given ``exactly'' in terms of the multifractal exponent $η=d-D_2$ at the mobility edge in a $d$-dimensional system.