Exact electronic spectra and inverse localization lengths in one-dimensional random systems

Analytic continuations into the complex energy plane of Dyson-Schmidt type of equations for the calculation of the density of states are constructed for a random alloy model, a liquid metal and for a liquid alloy. In all these models the characteristic function follows from the solution of this equation. Its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions. The equations have been solved exactly for some distributions of the random variables. In the random alloy case the strengths of the delta-potentials have an exponential distribution. They may also have finite, exponentially distributed values with probability 0 ⪕ p ⪕ 1 and be infinite with probability q = 1 −p. In the liquid metal the liquid particles are assumed to behave like hard rods. This implies an exponential distribution of the distances between the particles. The common electronic potential may be arbitrary, but is assumed to vanish outside the rods. In the one-dimensional liquid alloy there is, apart from positional randomness of the liquid particles, a distribution of the strengths of the electronic delta-potentials. For Cauchy distributions an argument of Lloyd is extended to obtain the characteristic function from the one in the model with equal strengths. For the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above. For several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed.