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<p>In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5258_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> cos-type potentials, large coupling constants, and a fixed Diophantine frequency. We obtain the absolute continuity of the Lyapunov exponent. Moreover, we prove the Lyapunov exponent is <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5258_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(\frac{1}{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> </InlineEquation>-Hölder continuous. Furthermore, for any given <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5258_Article_IEq5.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">\(r\in (\frac{1}{2}, 1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5258_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">\((r-\epsilon )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>-</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Hölder continuity and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5258_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\((r+\epsilon )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Hölder continuity.</p>

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The Precise Regularity of the Lyapunov Exponent for \(C^2\) Cos-Type Quasiperiodic Schrödinger Cocycles with Large Couplings

  • 期刊论文

  • 发表日期：

    2025年3月5日

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  • Jiahao Xu,
  • Lingrui Ge,
  • Yiqian Wang

摘要

In this paper, we study the regularity of the Lyapunov exponent for quasiperiodic Schrödinger cocycles with \(C^2\) C 2 cos-type potentials, large coupling constants, and a fixed Diophantine frequency. We obtain the absolute continuity of the Lyapunov exponent. Moreover, we prove the Lyapunov exponent is \(\frac{1}{2}\) 1 2 -Hölder continuous. Furthermore, for any given \(r\in (\frac{1}{2}, 1)\) r ∈ ( 1 2 , 1 ) , we can find some energy in the spectrum where the local regularity of the Lyapunov exponent is between \((r-\epsilon )\) ( r - ϵ ) -Hölder continuity and \((r+\epsilon )\) ( r + ϵ ) -Hölder continuity.