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

Perceval Desforges, Svitlana Mayboroda, Shiwen Zhang, Guy David, Douglas N. Arnold, Wei Wang, Marcel Filoche

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2 Scopus citations

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 Schrödinger 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.

Original languageEnglish (US)
Article number012207
JournalPhysical Review A
Volume104
Issue number1
DOIs
StatePublished - Jul 2021

Bibliographical note

Funding Information:
This work was supported by grants to several of the authors from the Simons Foundation (Grants No. 601937, D.N.A.; No. 601941, G.D.; No. 601944, M.F.; No. 563916, S.M.) and, in part, by the DMS NSF No. 1839077. The authors would like to thank Christophe Texier for insightful discussions.

Publisher Copyright:
© 2021 American Physical Society.

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