Norm estimates of almost Mathieu operators

Abstract

We estimate the norm of the almost Mathieu operator H θ , λ = U θ + U θ * + ( λ / 2 ) ( V θ + V θ * ) , regarded as an element in the rotation C * -algebra A θ = C * ( U θ , V θ unitaries : U θ V θ = e 2 π i θ V θ U θ ) . In the process, we prove for every λ ∈ R and θ ∈ [ 1 4 , 1 2 ] the inequality ∥ H θ , λ ∥ ⩽ 4 + λ 2 - 1 - 1 tan π θ 1 - 1 + cos 2 4 π θ 2 min { 4 , λ 2 } . This significantly improves the inequality ∥ H θ , 2 ∥ ⩽ 2 2 , θ ∈ [ 1 4 , 1 2 ] , conjectured by Béguin, Valette and Zuk.