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Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Sobolev, AV; Yafaev, D; (2022) Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols. Analysis and PDE , 15 (6) pp. 1457-1486. 10.2140/apde.2022.15.1457. Green open access

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Abstract

Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators T with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator T , that is, by the behavior of exp ( − i T t ) f for t → ± ∞ . It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of the symbol. On the mixed spectrum, these two types of the asymptotic evolution of exp ( − i T t ) f coexist. This classification is justified in the framework of scattering theory. We prove the existence of wave operators that relate the model operators with the Toeplitz operator T . The ranges of these wave operators are pairwise orthogonal, and their orthogonal sum exhausts the whole space; i.e., the set of these wave operators is asymptotically complete.

Type: Article
Title: Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols
Open access status: An open access version is available from UCL Discovery
DOI: 10.2140/apde.2022.15.1457
Publisher version: https://doi.org/10.2140/apde.2022.15.1457
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery-pp.ucl.ac.uk/id/eprint/10126488
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