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

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.

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Cited by in corpus (14)

• Quantitative inductive estimates for Green's functions of non-self-adjoint matrices
• Full measure reducibility and localization for Jacobi operators: a topological criterion
• Upper bounds on the spectral gaps of quasi-periodic Schrödinger operators with Liouville frequencies
• Exact dynamical decay rate for the almost Mathieu operator
• Upper bounds on transport exponents for long range operators
• Twelve Tales in Mathematical Physics: An Expanded Heinemann Prize Lecture
• Power law logarithmic bounds of moments for long range operators in arbitrary dimension
• Ballistic transport for one-dimensional quasiperiodic Schrödinger operators
• Almost Mathieu operators with completely resonant phases
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles
• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles
• Inhomogeneous Diophantine approximation in the coprime setting
• Arithmetic version of Anderson localization via reducibility
• Arithmetic version of anderson localization for quasiperiodic Schrödinger operators with even cosine type potentials