Generic Continuous Spectrum for Ergodic Schr"odinger Operators
arXiv:0708.1263 · doi:10.1007/s00220-008-0537-y
Abstract
We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr"odinger operators have no eigenvalues in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.
14 pages
References in corpus (2)
Cited by in corpus (14)
- Schrödinger Operators with Dynamically Defined Potentials: A Survey
- Dynamics and spectral theory of quasi-periodic Schrödinger-type operators
- Some Characterizations of Domination
- Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts
- Almost Periodicity in Time of Solutions of the KdV Equation
- Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators
- Singular Density of States Measure for Subshift and Quasi-Periodic Schrödinger Operators
- Singular continuous spectrum and generic full spectral/packing dimension for unbounded quasiperiodic Schrödinger operators
- Continuous spectrum for a class of smooth mixing Schrödinger operators
- Orthogonal Polynomials on the Unit Circle with quasiperiodic Verblunsky Coefficients have generic purely singular continuous spectrum
- The Repetition Property for Sequences on Tori Generated by Polynomials or Skew-Shifts
- Generic Spectral Results for CMV Matrices with Dynamically Defined Verblunsky Coefficients
- Pinned Repetitions in Symbolic Flows: Preliminary Results
- Generic continuous spectrum for multi-dimensional quasi periodic Schrödinger operators with rough potentials