Density of states of tight-binding disordered systems

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Abstract
Rigorous upper and lower bounds to the integrated density of states of systems described within the tight-binding scheme are calculated using a Lagrange multipliers formalism which is valid for both equality and inequality constraints. Upper and lower bounds to the density of states are calculated as well; in this case, the constraints are the first few moments of the density of states and a bounded-below derivative of the density of states function. The bounds have been calculated for an oversimplified model of a binary alloy and for disordered magnetic insulators described within the Hubbard-model framework. Since the bounds are rather narrow, a great deal of information can be obtained from them. Upper and lower bounds are obtained to the density of states at the band tail energies in the antiferromagnetic arrangement of spins case. In the other cases studied, only upper bounds are obtained at these energies. Rigorous averages of the density of states and of the integrated density of states have also been obtained in terms of the moments of the density of states.