Affiliation: | a Institut de Mathématiques, Université de Neuchâtel, Emile-Argand 13, CH-2000, Neuchâtel, Switzerland b Institute of Mathematics, University of Wroclaw, Pl, Grundwaldzki 2/4, 50-384, Wroclaw, Poland c Laboratoire de Probabilité, Université Paul Sabatier, 118, rte de Narbonne, F-31062, Toulouse Cédex, France |
Abstract: | Harper's operator is the self-adjoint operator on defined by . 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 Hθ,: Hθ,≤ 2(1 + √2 + cos(2πθ)). We then prove this bound by reducing it to a problem on periodic Jacobi matrices, viewing Hθ, as the image of Hθ = Uθ + θ* + Vθ + Vθ* in a suitable representation of the rotation algebra Aθ. We also use powers of Hθ to obtain various upper and lower bounds on Hθ = maxHθ,. We show that “Fourier coefficients” of Hθk in Aθ have a combinatorial interpretation in terms of paths in the square lattice 2. This allows us to give some applications to asymptotics of lattice paths combinatorics. |