Sharp estimates for the integrated density of states in Anderson tight-binding models

Abstract

Recent work [G. David, M. Filoche, and S. Mayboroda, arXiv:1909.10558 [Adv. Math. (to be published)]] has proved the existence of bounds from above and below for the integrated density of states (IDOS) of the Schr\"odinger operator throughout the spectrum, called the landscape law. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the landscape law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be approximated with high accuracy through a single formula involving a remarkably simple multiplicative energy shift. In 2D, the same idea applies but the prefactor has to be changed between the bottom and top parts of the spectrum.