Further analysis of electronic states in the muffin-tin model of a disordered alloy

Several new results on the evaluation of the spectral function $A(\overrightarrow{\mathrm{k}},E)$ and average density of states $\rho \mathrm{}\left(E\right)\mathrm{}$ in a realistic model of a substitutional binary alloy are presented. This work is primarily concerned with the conditions under which a given approach can be relied upon to yield a non-negative spectrum. In the average $t$ matrix and coherent-potential approximations, a link is established between the sign of $A(\overrightarrow{\mathrm{k}},E)$ and the Argand diagram for the energy-shell matrix elements ${\tau }_L$ of the effective atomic scattering operators: If ${\tau }_L$ lies within the unitarity circle, then $A(\overrightarrow{\mathrm{k}},E)>\mathrm{}0\mathrm{}$. The connection between this result and our earlier work on the optical theorem is shown to require a renormalization of certain free-electron singularities. Finally, an independent proof is outlined, based on the Lloyd equations, that the coherent potential density of states is always non-negative.