TITLE: Cantor spectrum and Kotani eigenstates AUTHOR: Joaquim Puig INSTITUTION: Departament de Matematica Aplicada I, Universitat Politecnica de Catalunya Av. Diagonal 647, 08028 Barcelona, Spain ABSTRACT: In this note we consider Kotani eigenstates of one-dimensional Schr\"odinger operators with ergodic potential. We show that if the spectrum, restricted to an interval, has zero Lyapunov exponents and is a Cantor set, then for a residual subset of energies, Kotani eigenstates do not exist. In particular, we show that the quasi-periodic Schr\"odinger operators whose Schr\"odinger quasi-periodic cocycles are reducible for all energies have a limit band-type spectrum.