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Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Alexander V. Sobolev and Dmitri Yafaev

Vol. 15 (2022), No. 6, 1457–1486

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 (iTt)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 (iTt)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.

Toeplitz operators, discontinuous symbols, spectral classification, model operators, multichannel scattering, wave operators
Mathematical Subject Classification 2010
Primary: 47B35
Secondary: 47A40
Received: 4 October 2019
Revised: 18 January 2021
Accepted: 23 February 2021
Published: 10 November 2022
Alexander V. Sobolev
Department of Mathematics
University College London
United Kingdom
Dmitri Yafaev
Université de Rennes I
CNRS, IRMAR-UMR 6625, F-35000
St. Petersburg State University
Saint Petersburg