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 Schrödinger operators on ℓ2(Zd), provided solutions to the Schrödinger 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 Schrödinger operator exhibits dynamical delocalization. We apply the general result to Γ-trimmed Schrödinger operators, with periodic Γ, and prove dynamical delocalization for these operators. These results also apply to the Γ-trimmed Anderson model, providing a random, ergodic model exhibiting both dynamical localization in an energy interval and dynamical delocalization.