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

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 $I_{q}=\langle |ψ_{E}({\bf r})|^{2q}\rangle$ of normalized wavefunctions $ψ_{E}({\bf r})$ which show violation of one-parameter scaling and emergence of an additional length scale at $E\approx 0$.

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