Localization in One-Dimensional Disordered Systems

The basic idea that the convergence of the renormalized perturbation expression for the self-energy ${\Delta }_0$ at a given energy is equivalent to the localizability of the eigenstates, if any, at this energy is applied to one-dimensional random systems, namely, electrons in the tightbinding approximation and phonons. For nearest-neighbor interactions all eigenstates are localized. If second-nearest-neighbor interactions are present, the possibility of the existence of extended states remains; we have shown that existing theories are unable to give a definite answer to the problem in this case.