Low-Lying Energy Spectrum of a One-Dimensional Disordered System

The energy spectrum of a particle in the presence of random fixed finite-ranged repulsive potentials is investigated at low energy. The resulting asymptotically exact density of states is derived. It exhibits an exponential dependence on energy. This result is shown to be equivalent, for the case of $\delta$-function potentials, to that of Frisch and Lloyd (1960) in the limit of low energy.