Localisation and marginal dimensionality

Abstract

The renormalised perturbation series (RPS) of self-energy is transformed into a simple continued fraction through a rearrangement procedure that incorporates into the denominators of the continued fraction all the contributions from closed polygonal paths. Convergence of RPS implies the convergence of the continued fraction. This results is used to prove that apart from one dimension no other marginal dimension exists for Anderson localisation in the sense of absence of either extended states or localised states.