Aharonov-Bohm effect in one-channel weakly disordered rings

A new diagrammatic method, which is a reformulation of Berezinskii's techni que, is constructed to study the density of electronic states rho(epsilon, phi) of a one-channel weakly disordered ring, threaded by an external magne tic flux. The exact result obtained for the density of states shows an osci llation of rho(epsilon, phi) with a period of the flux quantum phi(0) = hc/ e. As the sample length (or the impurity concentration) is reduced, a trans ition takes place from the weak localization regime (L much greater than l) to the ballistic one (L I I). The analytical expression for the density of states shows the exact dependence of rho(epsilon, phi) on the ring's circu mference and on disorder strength for both regimes.