Electron Levels in a One-Dimensional Random Lattice

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

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