Anomalous scaling and generalized Lyapunov exponents of the one-dimensional Anderson model

We introduce a family of localization lengths ξq (related to generalized Lyapunov exponents in a transfer-matrix approach) for the one-dimensional discrete Schr\"odinger equation with diagonal disorder. We show that, at the band edge of the pure system and with a random bounded potential with zero average, there is a q-dependent crossover in the scaling of ξq with the disorder amplitude {ε}: For {ε}{≤}{ε}{}(q){~}q^-6, ξq{∝}ε^-2/3; otherwise, ξq{∝}ε^-1/2. The limit therefore reproduces the scaling with exponent -(2/3), whereas deviations from this scaling law appear at each fixed {ε} if q is sufficiently large. These results involve a ``multifractal'' structure of the asymptotic decay of the wave functions.