Schrodinger's equation on a one-dimensional lattice with weak disorder

The stationary Schrodinger equation on a one-dimensional lattice endowed with a random potential is considered. Specifically, the equation studied is u_n+1 + u_n-1 = (E - εV_n) U_n, where εV_n is the random potential at site n. When ε = 0, the band of allowed energies is given by E=2cosπr, |r| < 1, and only this band is considered. A singular perturbation expansion of the stationary probability density p(x,E,ε)of the random process X_n = U_n/U_n-1 is constructed in the limit of weak disorder (ε ≪ 1). The coefficients in the expansion are analytic functions of r for ε > 0. They contain internal layers at rational values of r, which were previously termed 'anomalies.' The expansion approximates p(x,E, ε) uniformly for all r inside the band, away from bandcenter (r=1/2) and bandedge (r=0). It is used to calculate the first term in the expansion of the Lyapunov exponent γ(E,ε), which determines the localization length of the wave function, thus confirming the Thouless formula for γ(E,ε) inside the band and the Kappus-Wegner formula in bandcenter. Band-center and band-edge expansions are constructed, which match the in-band limits, allowing a uniform approximation for the Lyapunov exponent in all regions of the energy band.