Almost periodic Sturm-Liouville operators with Cantor homogeneous spectrum

Abstract

Being based on the infinite dimensional Jacobi inversion found earlier, we establish the direct generalization of the well-known properties of finite-band Sturm-Liouville operators in the case of operators with a homogeneous and, generally speaking, Cantor-type spectrum, and with pseudocontinuable Weyl functions. In our investigations the group of unimodular characters of the fundamental group of the resolvent set plays a role of the isospectral manifold of the operator. The generalized Abel map conjugates the nonlinear evolution of spectral data with a linear motion on this torus. In particular, the operators we consider turn out to be uniformly almost periodic.