Gordon-type arguments in the spectral theory of one-dimensional quasicrystals

Abstract

We review the recent developments in the spectral theory of discrete one-dimensional Schrödinger operators with potentials generated by substitutions and circle maps. We discuss how occurrences of local repetitive structures allow for estimates of generalized eigenfunctions. Among the recent applications of this general approach are almost sure and uniform results on the absence of eigenvalues as well as continuity of the spectral measures with respect to Hausdorff measures.

28 pages

References in corpus (5)

• Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrodinger operators
• Uniform spectral properties of one-dimensional quasicrystals, III. $\alpha$-continuity
• Trace maps, invariants, and some of their applications
• A note on palindromicity
• Half-line eigenfunction estimates and singular continuous spectrum of zero Lebesgue measure

Cited by in corpus (30)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• Uniform spectral properties of one-dimensional quasicrystals, III. $\alpha$-continuity
• The Fibonacci Hamiltonian
• Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals
• Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian
• Spectral and topological properties of a family of generalised Thue-Morse sequences
• A note on palindromicity
• Continuum Schrödinger Operators Associated With Aperiodic Subshifts
• Generalized Inverse Participation Numbers in Metallic-Mean Quasiperiodic Systems
• Energy spectra and eigenstates of quasiperiodic tight-binding Hamiltonians
• Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices
• Schrödinger Operators Defined by Interval Exchange Transformations
• Observation of Log-Periodic Oscillations in the Quantum Dynamics of Electrons on the One-Dimensional Fibonacci Quasicrystal
• Substitution Dynamical Systems: Characterization of Linear Repetitivity and Applications
• Uniform spectral properties of one-dimensional quasicrystals, IV. Quasi-Sturmian potentials
• On eigenvalue bounds for a general class of Sturm-Liouville operators
• A quantitative version of Gordon's Theorem for Jacobi and Sturm-Liouville operators
• Dynamical upper bounds on wavepacket spreading
• The Spectrum and the Spectral Type of the Off-Diagonal Fibonacci Operator
• Pinned Repetitions in Symbolic Flows: Preliminary Results
• Arnoux-Rauzy Subshifts: Linear Recurrence, Powers, and Palindromes
• Dynamical Upper Bounds for One-Dimensional Quasicrystals
• Log-dimensional spectral properties of one-dimensional quasicrystals
• Uniform ergodic theorems on subshifts over a finite alphabet
• Uniform dynamical bounds for the Fibonacci Hamiltonian
• The Spectrum of the Weakly Coupled Fibonacci Hamiltonian
• The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian
• A Condition of Boshernitzan and Uniform Convergence in the Multiplicative Ergodic Theorem
• Factor versus palindromic complexity of uniformly recurrent infinite words
• Localization for Discrete One Dimensional Random Word Models