Anderson model with decaying randomness: Mobility edge (Q5926361)

One of the most outstanding problems in the theory of random Schrödinger operators is the question of the existence of continuous spectrum. It is believed on physical grounds that continuous spectrum should exist if the dimension is large enough (and most people believe that two is large enough), if the energy is large or the disorder is sufficiently weak. The value of the energy that separates pure point spectrum from the continuous part of the spectrum is called the mobility edge. While there are now a large number of mathematically rigorous results showing that pure point spectrum exists, and in particular that in dimension one no other spectrum exists, no mathematical results are available on the existence of continuous spectrum [except in the case of models defined on the Cayley tree, see \textit{A. Klein}, Math. Res. Lett. 1, No. 4, 399-407 (1994; Zbl 0835.58045)]. In the present paper the authors prove the existence of a mobility edge in a discrete Anderson model on the \(\nu\)-dimensional lattice with non-stationary potential. In fact, the main assumptions are decay to zero of the potential at infinity. The main result of the paper is that if \(\nu\geq 3\), the potential \(a_n\) decays faster than \(|n|^{-1-\varepsilon}\), \(\varepsilon>0\), and the probability distribution of random potential has mean zero and finite second moments, then the absolutely continuous part of the spectrum contains the interval \([-2\nu,2\nu]\). Under some further assumption on the distribution it is also shown that the complement of this interval belongs to the pure point spectrum.