Expressions describing electron eigenstates in a one-dimensional liquid chain of weak wells dagger

Algebraic expressions are found for the density of electron eigenstates, degree of localization and probability distribution of the phase in a liquid chain of δ-function potential wells of strength -2κ 0 ( h 2 /2 m ), for positive electron energies h 2 κ 2 /2 m . A `diffusion approximation' is used, in which the probability distribution of the spacings enters only through the mean u and the standard deviation σ u . The approximation is valid provided mainly that the wells are weak (κ 0 /κ << 1) and the disorder small (κ u σ << 1). Excellent agreement is obtained with the numerical results of Borland and Bird. The energy gap, when still present, is `eroded' by a fraction of order (σ 2 κ 3 u 2 /κ 0 ) 2/3 . This `gap' is really a `quasi-gap', in which the density of states falls off exponentially, its dominant factor being exp{-θ|κ-κ c | 3/2 /κ 3/2 κ 0 1/2 σ 2 u 1/2 where 4/3 × 2 1/2