Essential closures and AC spectra for reflectionless CMV, Jacobi, and Schrodinger operators revisited

摘要

We provide a concise, yet fairly complete discussion of the concept of essential closures of subsets of the real axis and their intimate connection with the topological support of absolutely continuous measures. As an elementary application of the notion of the essential closure of subsets of R we revisit the fact that CMV, Jacobi, and Schrodinger operators, reflectionless on a set epsilon of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure epsilon(-e) of the set epsilon (with uniform multiplicity two on epsilon). Though this result in the case of Schrodinger and Jacobi operators is known to experts, we feel it nicely illustrates the concept and usefulness of essential closures in the spectral theory of classes of reflectionless differential and difference operators.