Green's Functions for a Particle in a One-Dimensional Random Potential

A method has been found for exactly calculating the spectral density A (k, E) for a particle in a one-dimensional random potential, when the potential at each point is statistically independent of the potential at all other points. Generalizations of this method can also be used to find the phonon Green's functions for a chain of atoms of random mass, or to find various two-particle functions, such as the electrical conductivity of a system of noninteracting electrons in a random potential. Two functions of the position x on the line are defined, which depend on the particular potential configuration and on the parameters k and E. The spectral density is expressed in terms of the probability distribution of these functions when x is at the right-hand end of the line. The distribution is known at the left-hand end, and the values of the functions, as x moves from left to right, form a Markoff process. One can therefore obtain the spectral density by solving the equation of motion for the probability distribution. Further simplification is possible because it is sufficient to know the first two moments of the joint distribution with respect to one variable, and because in the limit of an infinite line, only the asymptotic form of these moments is necessary. The spectral density requires the solution of a pair of differential or integral equations in one variable, while two-particle functions involve similar equations in two variables. Calculations have been carried out for the spectral density of a Schr\"odinger particle in a "white-Gaussian-noise" potential and of a particle confined to fixed lattice sites in a random thermal-deformation potential.