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- Title
Essential Closures and AC Spectra for Reflectionless CMV, Jacobi, and Schrödinger Operators Revisited.
- Authors
Gesztesy, Fritz; Makarov, Konstantin; Zinchenko, Maxim
- Abstract
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 Schrödinger operators, reflectionless on a set ℰ of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure ${\overline{\mathcal{E}}}^{e}$ of the set ℰ (with uniform multiplicity two on ℰ). Though this result in the case of Schrödinger 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.
- Subjects
SPECTRAL theory; MEASURE theory; SELFADJOINT operators; JACOBI operators; SCHRODINGER operator
- Publication
Acta Applicandae Mathematicae, 2008, Vol 103, Issue 3, p315
- ISSN
0167-8019
- Publication type
Article
- DOI
10.1007/s10440-008-9238-y