Electron Levels in a One-Dimensional Random Lattice

Let the potential of a one-dimensional scalar particle be V(x)=V₀Σ_-∞^∞δ(x-xⱼ), -∞<x<∞, where V₀<0, and where the sequence (xⱼ) is random, with a Poisson distribution. The quantity of interest is a certain limiting level distribution, equal numerically to the node density of real solutions ψ(x) of the Schr\"odinger equation. The random variables zⱼ=ψ^'(xⱼ-0)ψ(xⱼ), -∞<j<∞, constitute an ergodic stationary Markov process. The stationary density $T(z)$ of the (zⱼ) satisfies a first-order linear differential-difference equation, and the node density is given (with probability 1) by lim_z→∞z²T(z) (Rice's formula). Numerical results are obtained by integrating the second-order linear differential equation satisfied by the Fourier transform of $T(z)$.