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Let the potential of a one-dimensional scalar particle be V (x) =V₀{-}^ (x-x₉), -<x<, where V₀<0, and where the sequence (x₉) 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 (x) of the Schr\"odinger equation. The random variables z₉={^' (x₉-0) } ({x₉) }, -<j<, constitute an ergodic stationary Markov process. The stationary density T (z) of the (z₉) satisfies a first-order linear differential-difference equation, and the node density is given (with probability 1) by limₙz^2T (z) (Rice's formula). Numerical results are obtained by integrating the second-order linear differential equation satisfied by the Fourier transform of T (z).
Frisch et al. (Tue,) studied this question.