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

✍ Scribed by Cédric Béguin; Alain Valette; Andrzej Zuk

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 908 KB
• Volume: 21
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(96)00024-1

✦ Synopsis

Harper's operator is the self-adjoint operator on 12(7/) defined by Ho,o~(n) = ~(n + 1) + se(n -1 ) + 2 cos(2zr(n0 + (p))~(n) (~ ~ l 2 (7/), n 6 7/, 0, (p c [0, I ]). 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 bound on the norm of Ho.O: II n0.~ [I -< 2( 1 + x/~ + cos(2zr0)). We then prove this bound by reducing it to a problem on periodic Jacobi matrices, viewing Ho,o as the image of 11o = Uo + U~ + Vo + V~ in a suitable representation of the rotation algebra .,40. We also use powers of Ho to obtain various upper and lower bounds on II H~ II = max~ II/-/o4,11. We show that "Fourier coefficients" of H0 k in .Ao have a combinatorial interpretation in terms of paths in the square lattice 7/2. This allows us to give some applications to asymptotics of lattice paths combinatorics.