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Multichannel
scattering theory for Toeplitz operators with piecewise
continuous symbols
Alexander V. Sobolev and Dmitri Yafaev |
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Vol. 15 (2022), No. 6, 1457–1486
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Abstract |
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Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator , that is, by the behavior of for . 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 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 . 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. |
Keywords
Toeplitz operators, discontinuous symbols, spectral
classification, model operators, multichannel scattering,
wave operators
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Mathematical Subject Classification 2010
Primary: 47B35
Secondary: 47A40
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Milestones
Received: 4 October 2019
Revised: 18 January 2021
Accepted: 23 February 2021
Published: 10 November 2022
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Authors |
| Alexander V. Sobolev | |
| Department of Mathematics University College London London United Kingdom |
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| Dmitri Yafaev | |
| Université de Rennes I CNRS, IRMAR-UMR 6625, F-35000 Rennes France |
St. Petersburg State
University Saint Petersburg Russia |
