Some Exact Results on Electron Energy Levels in Certain One-Dimensional Random Potentials

The integrated density of states $\mathfrak{N}\mathrm{}\left(E\right)\mathrm{}$ for a particle in a set of arbitrarily located repulsive $\delta$-function potentials is studied. Upper and lower bounds for $\mathfrak{N}\mathrm{}\left(E\right)\mathrm{}$ are given. For the completely disordered chain we derive the asymptotic form $\mathfrak{N}\mathrm{}\left(E\right)\mathrm{}\sim a{e}^{-\frac{b}{\sqrt{E}}}$, $E\to 0^{+}$. For a chain with short-range order a simple proof of the existence of forbidden gaps based on the present method is briefly sketched.