On the spectrum of spatially disordered systems

The spectral density of states in systems where the atoms are completely randomly distributed in space with the concentration c is studied with the locator expansion of Matsubara and Toyozawa in the case of low concentration beta =c alpha -3 <<1 where alpha -1 is a characteristic length of the exponentially decreasing transfer integral. It is demonstrated that the nature of the spectrum in the vicinity of the level of an isolated atom is closely connected with the degree of localisation of the eigenstates. The limits of validity of the Lifshitz model, that predicts a minimum in the middle of the band, and of the Matsubara and Toyozawa approximation are established. The estimate beta c approximately=9.4*10 -4 for the concentration beta c of the Anderson transition in this spatially disordered system is obtained by the application of the bond percolation theory to the Lifshitz model.