Norm estimates of almost Mathieu operators

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,Vθunitaries:UθVθ=e2πiθVθUθ). In the process, we prove for every λ∈R and θ∈[14,12] the inequality ∥Hθ,λ∥⩽4+λ2-1-1tanπθ1-1+cos24πθ2min{4,λ2}.This significantly improves the inequality ∥Hθ,2∥⩽22, θ∈[14,12], conjectured by Béguin, Valette and Zuk.