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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5406_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\ell ^2(\mathbb {Z}^d)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ℓ</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年8月6日

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  • Jing Wang,
  • Xu Xu,
  • Jiangong You,
  • Qi Zhou

摘要

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 \(\ell ^2(\mathbb {Z}^d)\) ℓ 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.