Essential Closures and AC Spectra for Reflectionless CMV, Jacobi, and Schrödinger Operators Revisited

• Published in: Acta applicandae mathematicae Vol. 103; no. 3; pp. 315 - 339
• Main Authors: Gesztesy, Fritz, Makarov, Konstantin A., Zinchenko, Maxim
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-09-2008; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Closures; Computational Mathematics and Numerical Analysis; Cytomegalovirus; Joints; Mathematics; Mathematics and Statistics; Operators; Partial Differential Equations; Probability Theory and Stochastic Processes; Schroedinger equation; Spectra; Spectral theory; Texts; Topology; 34L40; Absolutely continuous spectrum; 34B27; 34B24; 34L05; 34B20; 47A10; Reflectionless Jacobi; CMV; Schrödinger operators
• ISSN: 0167-8019; 1572-9036
• DOI: 10.1007/s10440-008-9238-y

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
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.