<p>In this paper, we consider the Hölder continuity of the integrated density of states (IDS). Applying Avila’s almost reducible result and KAM technique, we proved that there exists a dense subset of Liouvillean frequencies <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, for which the IDS of the analytic quasi-periodic Schrödinger operator is (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>-<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\log $</EquationSource> <EquationSource Format="MATHML"><math> <mo>log</mo> </math></EquationSource> </InlineEquation>)-Hölder continuous for any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\chi >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>χ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, provided that the subcritical strip of the operator satisfies <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$h_0 > 2\beta (\alpha ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>></mo> <mn>2</mn> <mi>β</mi> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We also proved the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\chi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>χ</mi> </math></EquationSource> </InlineEquation>-Hölder continuity of the IDS for a dense subset of Liouvillean frequencies for operators with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0< \chi < \frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>χ</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, if the subcritical strip satisfies <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$h_0 > \frac{8\beta }{ 1-2\chi }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>h</mi> <mn>0</mn> </msub> <mo>></mo> <mfrac> <mrow> <mn>8</mn> <mi>β</mi> </mrow> <mrow> <mn>1</mn> <mo>-</mo> <mn>2</mn> <mi>χ</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.</p>

Hölder Continuity of the Integrated Density of States and Lyapunov Exponent for Quasi-Periodic Schrödinger Operator with Liouvillean Frequency

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发表日期：

2025年3月28日

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Jing Wang

摘要

In this paper, we consider the Hölder continuity of the integrated density of states (IDS). Applying Avila’s almost reducible result and KAM technique, we proved that there exists a dense subset of Liouvillean frequencies $\alpha $ $α$ , for which the IDS of the analytic quasi-periodic Schrödinger operator is ( $\chi $ $χ$ - $\log $ $log$ )-Hölder continuous for any $\chi >1$ $χ > 1$ , provided that the subcritical strip of the operator satisfies $h_0 > 2\beta (\alpha ) $ $h_{0} > 2 β (α)$ . We also proved the $\chi $ $χ$ -Hölder continuity of the IDS for a dense subset of Liouvillean frequencies for operators with $0< \chi < \frac{1}{2}$ $0 < χ < \frac{1}{2}$ , if the subcritical strip satisfies $h_0 > \frac{8\beta }{ 1-2\chi }$ $h_{0} > \frac{8 β}{1 - 2 χ}$ .