Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase

  • Svetlana Jitomirskaya

    University of California, Irvine, USA
  • Wencai Liu

    University of California, Irvine; Texas A&M University, College Station, USA
Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase cover
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Abstract

We prove sharp spectral transition in the arithmetics of phase between localization and singular continuous spectrum for Diophantine almost Mathieu operators. We also determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices throughout the localization region. This uncovers a universal structure in their behavior governed by the exponential phase resonances. The structure features a new type of hierarchy, where self-similarity holds upon alternating reflections.

Cite this article

Svetlana Jitomirskaya, Wencai Liu, Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase. J. Eur. Math. Soc. 26 (2024), no. 8, pp. 2797–2836

DOI 10.4171/JEMS/1325