Schrödinger’s Equation on a One-Dimensional Lattice with Weak Disorder

Abstract

The stationary Schrödinger 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 - \varepsilon V_n )u_n $, where$\varepsilon V_n $ is the random potential at site n . When $\varepsilon = 0$, the band of allowed energies is given by$E = 2\cos \pi r,\, | r | < 1$, and only this band is considered A singular perturbation expansion of the stationary probability density $p(x, E, \varepsilon)$ of the random process $X_n = u_n /u_{n - 1} $ is constructed in the limit of weak disorder $( \varepsilon \ll 1)$. The coefficients in the expansion are analytic functions of r for $\varepsilon > 0$. They contain internal layers at rational values of r, which were previously termed “anomalies.” The expansion approximates $p( x,E,\varepsilon )$ uniformly for all r inside the band, away from band-center $( r = \tfrac{1}{2} )$ and band-edge $(r = 0)$. It is used to calculate the first term in the expansion of the Lyapunov exponent $\gamma( E,\varepsilon )$, which determines the localization length of the wave function, thus confirming the Thouless formula for $\gamma ( E,\varepsilon )$ inside the band and the Kappus–Wegner formula in band-center. 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.