SCHRODINGER-EQUATION ON A ONE-DIMENSIONAL LATTICE WITH WEAK DISORDER

The stationary Schrodinger equation on a one-dimensional lattice endow ed with a random potential is considered. Specifically, the equation s tudied is u(n+1) + u(n-1) = (E - epsilonV(n))u(n), where epsilonV(n) i s the random potential at site n. When epsilon = 0, the band of allowe d energies is given by E = 2 cospir, \r\ < 1, and only this band is co nsidered. A singular perturbation expansion of the stationary probabil ity density p(x, E, epsilon) of the random process X(n) = u(n)\u(n-1) is constructed in the limit of weak disorder (epsilon much less than 1 ). The coefficients in the expansion are analytic functions of r for e psilon > 0. They contain internal layers at rational values of r, whic h were previously termed ''anomalies.'' The expansion approximates p(x , E, epsilon) uniformly for all r inside the band, away from band-cent er (r = 1/2) and band-edge (r = 0). It is used to calculate the first term in the expansion of the Lyapunov exponent gamma(E, epsilon), whic h determines the localization length of the wave function, thus confir ming the Thouless formula for gamma(E, epsilon) inside the band and th e Kappus-Wegner formula in band-center. Band-center and band-edge expa nsions are constructed, which match the in-band limits, allowing a uni form approximation for the Lyapunov exponent in all regions of the ene rgy band.