Scattering Expansion for Localization in One Dimension

We present a perturbative approach to a broad class of disordered systems in one spatial dimension. Considering a long chain of identically disordered scatterers, we expand in the reflection strength of any individual scatterer. This expansion accesses the full range of phase disorder from weak to strong. As an example application, we show analytically that in a discrete-time quantum walk, the localization length can depend non-monotonically on the strength of phase disorder (whereas expanding in weak disorder yields monotonic decrease). Returning to the general case, we obtain to all orders in the expansion a particular non-separable form for the joint probability distribution of the log-transmission and reflection phase. Furthermore, we show that for weak local reflection strength, a version of the scaling theory of localization holds: the joint distribution is determined by just three parameters.