Norm estimates of almost Mathieu operators

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Abstract

We estimate the norm of the almost Mathieu operator Hθ,λ = Uθ + Uθ* + (λ/2)(Vθ + Vθ*), regarded as an element in the rotation C*-algebra Aθ=C*(Uθ, Vθ unitaries : Uθ Vθ = e2πiθVθUθ). In the process, we prove for every λ ∈ ℝ and θ ∈ [1/4, 1/2] the inequality A formula is presented. This significantly improves the inequality ∥Hθ,2∥ ≤ 2 2, θ ∈ [1/4, 1/2], conjectured by Béguin, Valette and Zuk.

Original languageEnglish (US)
Pages (from-to)76-96
Number of pages21
JournalJournal of Functional Analysis
Volume220
Issue number1
DOIs
StatePublished - Mar 1 2005

Keywords

  • Almost Mathieu operators
  • Norms
  • Rotation C*-algebras

ASJC Scopus subject areas

  • Analysis

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