Dynamical localization II with an application to the almost Mathieu operator (Q1308027)

Various ways are investigated to prove dynamical localization for Schrödinger operators, using spectral hypotheses. The dynamical localization always implies the absence of continuous spectrum, while the converse is not true. Even in case of the exponential decay of the eigenfunctions (Anderson's localization), some extra requirements are needed to secure the dynamical localization. The main result of the paper (Theorem 1.3) refers to the so-called weak uniform localization of eigenfunctions as the weakest restriction that entails the dynamical localization. That extends to all random and quasi-periodic operators for which multiscale arguments allow to establish the exponential localization.