Electron Levels in a One-Dimensional Random Lattice

Abstract

Let the potential of a one-dimensional scalar particle be $V\left(x\right)=V_0{{\Sigma }_{-\infty }}^{\infty }\delta (x-x_j)$, $-\infty <x<\mathrm{}\infty$, where $V_0<0\mathrm{}$, and where the sequence ($x_j$) 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 $\psi \mathrm{}\left(x\right)\mathrm{}$ of the Schrödinger equation. The random variables $z_j=\frac{{\psi }^{\prime }(x_j-0)}{\psi \left(x_j\right)}$, $-\infty <j<\mathrm{}\infty$, constitute an ergodic stationary Markov process. The stationary density $T\left(z\right)\mathrm{}$ of the ($z_j$) satisfies a first-order linear differential-difference equation, and the node density is given (with probability 1) by ${\lim }_{z\to \infty }{z}^{2}T\left(z\right)\mathrm{}$ (Rice's formula). Numerical results are obtained by integrating the second-order linear differential equation satisfied by the Fourier transform of $T\left(z\right)\mathrm{}$.