Invariant-imbedding approach to resistance fluctuations in disordered one-dimensional conductors

The invariant-imbedding method provides a first-order differential equation for the complex reflection amplitude R(L) of a one-dimensional conductor of length L, in terms of the random potential V(L) at the edge of incidence for a particle of energy E. The Landauer formula is used to express the resistance ρ(L) and the conductance ${\rho }^{\mathrm{-}1}$(L) in terms of R(L) and to derive an exact second-order differential equation for ρ(L), from the above equation for R(L). This equation for ρ(L) emphasizes important aspects of the problem but has not been solved explicitly. However, two types of explicit solutions, referred to as (a) and (b), have been derived, starting from the differential equation for R(L).