Absence of localisation in the almost Mathieu equation

Abstract

The author considers the discrete Schrodinger operator on Z with the potential lambda cos 2 pi ( alpha n+ theta ). This one-dimensional model occurs in the study of an electron in a two-dimensional periodic potential with a uniform magnetic field. First it is proved that for every alpha and for lambda <2 this operator has no eigenvalue (i.e. localised state). Furthermore at lambda =2, the eigenvalues (if they exist) belong to the set where the Lyapunov exponent vanishes, and the associated eigenvectors at in l2(Z) but are not summable.