Statistics of anomalously localized states at the center of band E=0 in the one-dimensional Anderson localization model

We consider the distribution function P(vertical bar psi vertical bar(2)) of the eigenfunction amplitude at the center-of-band (E = 0) anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with vertical bar psi vertical bar(2) much larger than the inverse typical localization length l(0). Using the recently found solution for the generating function Phi(an)(u, phi) we obtain the ALS probability distribution P(vertical bar psi vertical bar(2)) at vertical bar psi vertical bar(2)l(0) >> 1. As an auxiliary preliminary step, we found the asymptotic form of the generating function Phi(an)(u, phi) at u >> 1 which can be used to compute other statistical properties at the center-of-band anomaly. We show that at moderately large values of vertical bar psi vertical bar(2)l(0), the probability of ALS at E = 0 is smaller than at energies away from the anomaly. However, at very large values of vertical bar psi vertical bar(2)l(0), the tendency is inverted: it is exponentially easier to create a very strongly localized state at E = 0 than at energies away from the anomaly. We also found the leading term in the behavior of P(vertical bar psi vertical bar 2) at small vertical bar psi vertical bar(2) << l(0)(-1) and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner.