Universal hierarchical structure of quasiperiodic eigenfunctions

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.1 According to [43], the fact that the Diophantine properties of the frequencies should play a role was first observed in [42].

Inhomogeneous Diophantine approximation in the coprime setting

“…Questions of this type have applications, among other things, to several areas of dynamical systems and to the spectral theory of quasiperiodic Schrödinger operators (e.g. [1,2,12,13]). The inhomogeneous problem above can be understood in the metric sense: a.e.…”

Anderson localization for one-frequency quasi-periodic block operators with long-range interactions

“…Klein proved a non-perturbative AL and generalized the result of [8]. For recent AL results, we refer the reader to [1,2,5,7,[19][20][21]23].…”

Anderson localization for multi-frequency quasi-periodic operators on $$\pmb {\mathbb {Z}}^{d}$$

“…There are now non-perturbative results on both small and high coupling sides ( [4,19,21] and references therein), global theory [1], and sharp arithmetic transitions and related universality (e.g. [2,[14][15][16]). However, if one increases either the dimension of the undelying torus (the number of frequencies) or, especially, the space dimension, the situation becomes significantly more complicated.…”