Cantor spectrum and singular continuity for a hierarchical Hamiltonian (Q1263011)

The authors consider a hierarchical Hamiltonian H on \(\ell_ 2({\mathbb{Z}})\) given by one-dimensional discrete Schrödinger operator. They prove that the corresponding spectrum is a nowhere dense set without isolated points that gaps open at the eigenvalues of a special Dirichlet problem. There are discussed sufficient conditions for the purely singular continuity and for the purely continuity of the spectrum. It is pointed out the existence of the Lyapunov exponent and discussed a renormation property and the asymptotic behaviour of the wave functions. The paper describes a formal ``explicit'' solution for the Green's functions and suggests a device for a rigorous verification.