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It is shown that the thermodynamic properties for a random impurity system may be expressed in terms of an "effective energy" which is a function of momentum alone. A perturbation expression for the effective energy is given, which requires a "regularization" (rule for going around singularities) process. The method of regularization is analyzed in detail and it is shown that the effective energy levels can be made real by some special classes of regularization. The method can be extended to the case of disordered binary alloys, where all effective energies are obtained as solutions to a determinantal equation which assumes dual symmetry between the two constituents. The density of states in such a system is easily obtained from the set of effective energies. In a sense, the result for binary alloys represents a generalization of the "coherent potential" often used, but is rigorously true.
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