Universal reflective-hierarchical structure of quasiperiodic eigenfunctions and sharp spectral transition in phase
Svetlana Jitomirskaya, Wencai Liu
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.
Topics & Concepts
Quasiperiodic functionMathematicsEigenfunctionDiophantine equationSpectrum (functional analysis)Exponential functionPhase transitionPure mathematicsMathematical analysisPhase (matter)Eigenvalues and eigenvectorsQuantum mechanicsPhysicsSpectral Theory in Mathematical PhysicsQuantum chaos and dynamical systemsAnalytic and geometric function theory