Exact solution for eigenfunction statistics at the center-of-band anomaly in the Anderson localization model

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An exact solution is found for the problem of the center-of-band (E=0) anomaly in the one-dimensional (1D) Anderson model of localization. By deriving and solving an equation for the generating function Φ(u,ϕ) we obtained an exact expression in quadratures for statistical moments Iq= |ψₑ(r)|²q of normalized wavefunctions ψₑ(r) which show violation of one-parameter scaling and emergence of an additional length scale at E≈ 0.