Cantor spectrum and singular continuity for a hierarchical hamiltonian

We study the spectrum of the HamiltonianH onl 2(Zopf) given by (Hpsgr)(n)=psgr(n+1)+psgr(n–1)+V(n)psgr(n) with the hierarchical (ultrametric) potentialV(2 m (2l+1))=lambda(1–R m )/(1–R), corresponding to 1-, 2-, and 3-dimensional Coulomb potentials for 0R<>R=1 andR>1, respectively, in a suitably chosen valuation metric. We prove that the spectrum is a Cantor set and gaps open at the eigenvaluese n (1)e n (2)<>e n (2 n –1) of the Dirichlet problemHpsgr=Epsgr, psgr(0)=psgr(2 n )=0,ngE1. In the gap opening ate n (k) the integrated density of states takes on the valuek/2 n . The spectrum is purely singular continuous forRgE1 when the potential is unbounded, and the Lyapunov exponent gamma vanishes in the spectrum. The spectrum is purely continuous forR<1 in="">sgr(H)cap[–2, 2] and gamma=0 here, but one cannot exclude the presence of eigenvalues near the border of the spectrum. We also propose an explicit formula for the Green's function.