Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

1999/10/12 by David Damanik, Rowan Killip, Daniel Lenz · 5 citations
Materials Science · Mathematics · Physics and Astronomy · #Analytic and geometric function theory #Quasicrystal Structures and Properties #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:47B80 #msc:81Q10

paper · pdf · doi:10.1007/s002200000203

12 pages

arxiv created 1999/10/12 · openalex publication_date 2000/06/01 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely α-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.

Citations

• Uniform spectral properties of one-dimensional quasicrystals, II. The Lyapunov exponent
• Operators with Singular Continuous Spectrum: I. General Operators
• Hierarchical structures in Sturmian dynamical systems

Cited by

• The spectral characteristics of the Sturm Hamiltonian with eventually periodic type
• Spectral Properties of Schrödinger Operators With Pattern Sturmian Potentials
• The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
• Spectral and Quantum Dynamical Properties of the Weakly Coupled Fibonacci Hamiltonian
• DYNAMICAL BOUNDS FOR STURMIAN SCHRÖDINGER OPERATORS