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<p>We obtain a perturbative proof of localization for quasiperiodic operators on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5280_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> <mi mathvariant="double-struck">Z</mi> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also prove that the spectra of such operators contain infinitely many gaps.</p>

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Perturbative Diagonalization and Spectral Gaps of Quasiperiodic Operators on \(\ell ^2(\mathbb Z^d)\) with Monotone Potentials

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月7日

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  • Ilya Kachkovskiy,
  • Leonid Parnovski,
  • Roman Shterenberg

摘要

We obtain a perturbative proof of localization for quasiperiodic operators on \(\ell ^2(\mathbb Z^d)\) ℓ 2 ( Z d ) with one-dimensional phase space and monotone sampling functions, in the regime of small hopping. The proof is based on an iterative scheme which can be considered as a local (in the energy and the phase) and convergent version of KAM-type diagonalization, whose result is a covariant family of uniformly localized eigenvalues and eigenvectors. We also prove that the spectra of such operators contain infinitely many gaps.