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

Gesztesy, Fritz and Makarov, Konstantin A. and Zinchenko, Maxim (2008) Essential closures and AC spectra for reflectionless CMV, Jacobi, and Schrödinger operators revisited. Acta Applicandae Mathematicae, 103 (3). pp. 315-339. ISSN 0167-8019. doi:10.1007/s10440-008-9238-y.

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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 ∈ of positive Lebesgue measure, have absolutely continuous spectrum on the essential closure ⋶^e of the set ∈ (with uniform multiplicity two on ∈). 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.

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Additional Information:© Springer 2008. Received: 13 January 2008 Accepted: 10 April 2008 Published online: 24 April 2008. We are indebted to Jonathan Breuer for helpful discussions on this topic.
Subject Keywords:absolutely continuous spectrum; reflectionless Jacobi; CMV; Schrödinger operators.
Issue or Number:3
Classification Code:Mathematics Subject Classification. 34B20, 34L05, 34L40, 34B24, 34B27, 47A10
Record Number:CaltechAUTHORS:20090810-122708702
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Official Citation:Gesztesy, F., Makarov, K.A. & Zinchenko, M. Acta Appl Math (2008) 103: 315.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:14926
Deposited By: Joy Painter
Deposited On:10 Aug 2009 22:54
Last Modified:08 Nov 2021 23:15

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