Communications in Mathematical Physics

, Volume 212, Issue 1, pp 191–204

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

  • David Damanik
  • Rowan Killip
  • Daniel Lenz

DOI: 10.1007/s002200000203

Cite this article as:
Damanik, D., Killip, R. & Lenz, D. Comm Math Phys (2000) 212: 191. doi:10.1007/s002200000203

Abstract:

We study the spectral properties of one-dimensional whole-line Schrödinger operators, especially those with Sturmian potentials. Building upon the Jitomirskaya–Last extension of the Gilbert–Pearson theory of subordinacy, we demonstrate how to establish α-continuity of a whole-line operator from power-law bounds on the solutions on a half-line. However, we require that these bounds hold uniformly in the boundary condition.

We are able to prove these bounds for Sturmian potentials with rotation numbers of bounded density and arbitrary coupling constant. From this we establish purely α-continuous spectrum uniformly for all phases.

Our analysis also permits us to prove that the point spectrum is empty for all Sturmian potentials.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • David Damanik
    • 1
  • Rowan Killip
    • 1
  • Daniel Lenz
    • 2
  1. 1.Department of Mathematics 253–37, California Institute of Technology, Pasadena, CA 91125, USA.¶E-mail: damanik@its.caltech.edu; killip@its.caltech.eduUS
  2. 2.Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, 60054 Frankfurt, Germany.¶E-mail: dlenz@math.uni-frankfurt.deDE

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