A mathematical method is developed which gives fairly generally the density of eigenstates for one-dimensional disordered systems. The method is applied first to a disordered linear chain of elastically coupled masses. The results for the energy spectrum are closely related to those obtained by Dyson.Then we consider the electronic energy-states in a one-dimensional disordered crystal, represented by a series of δ-function potentials of different strengths, randomly distributed.We solve the resulting functional equation explicitly in that case which corresponds to a uniform crystal with a small amount of impurities; that is, we find the shape of the impurity bands.
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Helmut Schmidt (1957) studied this question.