RENORMALIZATION AND ENVELOPE FUNCTION FORMALISM FOR INCOMMENSURATE SYSTEMS

We study the localization-delocalization transition in one-dimensional incommensurate crystals both numerically and analytically. From the n umerical point of view we provide an implementation of the renormaliza tion method, which allows to process with high accuracy millions of si tes, (whenever necessary). From the analytic point of view we extend t he envelope function concepts to incommensurate potentials, smoothly v arying on lattice constant scale. The control of the transition is mad e by numerical calculation of the Lyapunov exponent: it presents surpr ising aspects of universality and simplicity, with plateaux, linear re gions and, at times, much more complicated behaviours. The envelope fu nction method, together with semi-analytic considerations, allows to u nderstand, in a number of situations, the presence of mobility edges, pseudo-mobility edges, and gaps in the energy spectrum.