On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator

Abstract

Harper's operator is the self-adjoint operator on l 2 ( Z) defined by H ϑ, ∅ ξ(n) = ξ(n + 1) + ξ(n − 1) + cos(2π(nθ + ∅))ξ(n) (ξ ϵ l 2 ( Z), n ϵ Z, θ, ∅ ϵ [0,1]) . We first show that the determination of the spectrum of the transition operator on the Cayley graph of the discrete Heisenberg group in its standard presentation, is equivalent to the following upper bond on the norm of H θ,∅ : ∥H θ,∅∥≤ 2(1 + √2 + cos(2πθ)). We then prove this bound by reducing it to a problem on periodic Jacobi matrices, viewing H θ,∅ as the image of H θ = U θ + θ ∗ + V θ + V θ ∗ in a suitable representation of the rotation algebra A θ . We also use powers of H θ to obtain various upper and lower bounds on ∥ H θ ∥ = max ∅∥ H θ,∅ ∥. We show that “Fourier coefficients” of H θ k in A θ have a combinatorial interpretation in terms of paths in the square lattice Z 2. This allows us to give some applications to asymptotics of lattice paths combinatorics.