### Abstract

The aim of this work in progress is the study of the generalized eigenfuinctions associated to singular (in particular singular continuous) spectrum for a Schrödinger operator on the half-line. This study reduces to the study of an ergodic matrix cocycle for which we develop an exact renormalization analysis. In particular, we get a complete description of the set of ergodic parameters for which the Lyapunov exponent of the cocycle exists and does not exist.