ON THE SPECTRUM OF A RANDOM-WALK ON THE DISCRETE HEISENBERG-GROUP ANDTHE NORM OF HARPERS OPERATOR

Harper's operator is the self-adjoint operator on l(2)(Z) defined by N (theta,phi)xi(n) = xi(n + 1) + xi(n - 1) + 2 cos(2 pi(n theta + phi))x i(n) (xi is an element of l(2)(Z), n is an element of Z, theta, phi is an element of [0, 1]). We first show that the determination of the sp ectrum of the transition operator on the Cayley graph of the discrete Heisenberg group in its standard presentation, is equivalent to the fo llowing upper bound on the norm of H-theta,H-phi:\\H-theta,H-phi\\ les s than or equal to 2(1 + root 2 + cos(2 pi theta)). We then prove this bound by reducing it to a problem on periodic Jacobi matrices, viewin g H-theta,H-phi as the image of H-theta = U-theta + U-theta + V-theta + V-theta in a suitable representation of the rotation algebra A(the ta). We also use powers of H-theta to obtain various upper and lower b ounds on \\H-theta\\ = max(phi) \\H-theta,H-phi\\. We show that ''Four ier coefficients'' of H-theta(k) in A(theta) have a combinatorial inte rpretation in terms of paths in the square lattice Z(2). This allows u s to give some applications to asymptotics of lattice paths combinator ics.