Absolute Continuity of the Integrated Density of States in the Localized Regime

Abstract

We establish the absolute continuity of the integrated density of states (IDS) for quasi-periodic Schrödinger operators with large trigonometric potentials and Diophantine frequencies. This partially solves Eliasson’s open problem in 2002. Furthermore, this result can be extended to a class of quasi-periodic long-range operators on \(\ell ^2(\mathbb {Z}^d)\). Our proof is based on stratified quantitative almost reducibility results of dual cocycles. Specifically, we prove that a generic analytic one-parameter family of cocycles, sufficiently close to constant coefficients, is reducible except for a zero Hausdorff dimension set of parameters. This result affirms Eliasson’s conjecture in 2017.