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Mathematics > Spectral Theory
Title: Absolute continuity of the integrated density of states in the localized regime
(Submitted on 30 Apr 2023 (v1), last revised 6 May 2023 (this version, v2))
Abstract: We establish the absolute continuity of the integrated density of states (IDS) for quasi-periodic Schr\"odinger operators with a large trigonometric potential and Diophantine frequency. 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 $\ell^2(\Z^d)$. 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.
Submission history
From: Zhou Qi [view email][v1] Sun, 30 Apr 2023 11:42:42 GMT (62kb)
[v2] Sat, 6 May 2023 15:03:26 GMT (61kb)
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