n the talk I will study the spectral properties of a class of
SturmLiouville-type operators on the real line where the derivatives are replaced by a q-difference operator which has been introduced in the context of orthogonal polynomials. Using the relation of this operator to a direct integral of doubly-infinite Jacobi matrices, one can construct examples for isolated pure point, dense pure point, purely absolutely continuous and purely singular continuous spectrum. I will show that the last two spectral types are generic for analytic coefficients and for a class of positive, uniformly continuous coefficients, respectively. A key ingredient in the proof is the so- called Wonderland theorem.
The talk is based on joint work with Malcolm Brown and Karl
Michael Schmidt.