Delocalization for the 3-D discrete random Schroedinger operator at weak disorder

We apply a recently developed approach (Liaw 2013) to study the existence of extended states for the three dimensional discrete random Schroedinger operator at small disorder. The conclusion of delocalization at small disorder agrees with other numerical and experimental observations. Further the work furnishes a verification of the numerical approach and its implementation. Not being based on scaling theory, this method eliminates problems due to boundary conditions, common to previous numerical methods in the field. At the same time, as with any numerical experiment, one cannot exclude finite-size effects with complete certainty. Our work can be thought of as a new and quite different use of Lanczos' algorithm; a posteriori tests to show that the orthogonality loss is very small. We numerically track the "bulk distribution" (here: the distribution of where we most likely find an electron) of a wave packet initially located at the origin, after iterative application of the discrete random Schroedinger operator.