Approach to the Extended States Conjecture

We develop a rather explicit approach concerning the extended states conjecture for the discrete random Schrödinger operator, or more generally for the so-called Anderson-type Hamiltonian. Our work is based on deep mathematical results by Jak?i?–Last (Duke Math. J. 133(1):185–204, 2006). Concretely, we suggest two new directions of research: We provide a formula which may lead the way to a rigorous proof of the conjecture, and an implementation of the proposed approach which yields numerical evidence in favor of the conjecture being true for the discrete random Schrödinger operator in dimension two. 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. 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 Schrödinger operator.