Transfer matrices and transport for 1D Schrodinger operators with singular spectrum
Résumé
We provide a general lower bound on the dynamics of one-dimensional Schrodinger operators in terms of transfer matrices. In particular, if their norm does not grow faster than polynomially on a set of positive Lebesgue measure, then we prove nontrivial lower bounds on the transport exponents. These bounds hold regardless to the nature of the spectrum. We also develop some general analysis of wave-packets that enables one to characterize transport exponents for low and large moments. As an application of our general lower bounds, we study a Schrodinger operator with random decaying potential, providing a new example of operators with point spectrum and nontrivial quantum transport. We also investigate sparse potentials, as well as we revisit almost Mathieu as given by celebrated pathological example of Del Rio, Jitomirskaya, Last and Simon.