Eigenfunctions and quantum transport with applications to trimmed Schrödinger operators

We provide a simple proof of dynamical delocalization, that is, time-increasing lower bounds on quantum transport for discrete, one-particle Schr & ouml;dinger operators on & ell;(2)(Z(d)), provided solutions to the Schr & ouml;dinger equation satisfy certain growth conditions. The proof is based on basic resolvent identities and the Combes-Thomas estimate on the exponential decay of the Green's function. As a consequence, we prove that generalized eigenfunctions for energies outside the spectrum of H must grow exponentially in some directions. We also prove that if H has any absolutely continuous spectrum, then the Schr & ouml;dinger operator exhibits dynamical delocalization. We apply the general result to Gamma-trimmed Schr & ouml;dinger operators, with periodic Gamma, and prove dynamical delocalization for these operators. These results also apply to the Gamma-trimmed Anderson model, providing a random, ergodic model exhibiting both dynamical localization in an energy interval and dynamical delocalization.