Bounds on the integrated density of electronic states for disordered Hamiltonians

A direct and mathematically simple method for obtaining upper and lower bounds on the integrated density of states (IDOS) for both the Schrödinger and the tight-binding lattice models is presented. The method is applied to the one-band tight-binding Hamiltonian with diagonal disorder on a d-dimensional lattice (d≥2). The asymptotic behavior near the lower end of the spectrum depends on the continuity properties of the distribution function of the site energies. If P(ε=0)>0—the site energies may take the value 0 with nonvanishing probability (binary-alloy model)—the IDOS has a Lifschitz behavior, lnσ(W)∼$W^{\mathrm{-}d/2\mathrm{}}$ as W→0. For distribution functions continuous near the origin (Anderson model), lnσ(W)∼$W^{\mathrm{-}d/2\mathrm{}}$ lnW.