On the Number of Electron Levels in a One-Dimensional Random Lattice

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Abstract
Let the potential of a one‐dimensional scalar particle be V(x) = V0 Σ∞i=−∞ δ(x — xi), — ∞ < x < ∞, where V0 < 0, and where the sequence (xi) is random, with a Poisson distribution. This paper investigates analytically the number N of electron levels per atom below energy E = —ℏ2κ2/2m, when 0 < n/κ0 « 1 and 0 < κ/κ0 < 2, where n is the expected density of atoms and κ0 = —mV0/ℏ2. The region κ/κ0 ≈ 1, with n/κ0 small, is of considerable interest, and some previous numerical computations have been inaccurate in this region. Explicit bounds on N−1 may be written down which give the asymptotic behavior of N, as κ0/n → ∞, for 0 < κ/κ0 < 1 and 1 < κ/κ0 < (2 — δ), δ > 0.

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