TY - JOUR

T1 - Multichannel Scattering Theory for Toeplitz operators with piecewise continuous symbols

AU - Sobolev, Alexander V.

AU - Yafaev, Dmitri

N1 - Alexander V. Sobolev. Dmitri Yafaev. "Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols." Anal. PDE 15 (6) 1457 - 1486, 2022. https://doi.org/10.2140/apde.2022.15.1457 Publisher Copyright: © 2022 Mathematical Sciences Publishers

PY - 2022

Y1 - 2022

N2 - 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(−iT 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(−iT 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.

AB - 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(−iT 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(−iT 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.

KW - discontinuous symbols

KW - model operators

KW - multichannel scattering

KW - spectral classification

KW - Toeplitz operators

KW - Wave operators

KW - Multichannel scattering

KW - Discontinuous symbols

KW - Wave operators.

KW - Model operators

KW - Spectral classification

UR - https://projecteuclid.org/journals/analysis-and-pde/volume-15/issue-6/Multichannel-scattering-theory-for-Toeplitz-operators-with-piecewise-continuous-symbols/10.2140/apde.2022.15.1457.short

UR - http://www.scopus.com/inward/record.url?scp=85142238342&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/40f117c6-888c-3805-a914-018a117484b9/

U2 - 10.2140/apde.2022.15.1457

DO - 10.2140/apde.2022.15.1457

M3 - Article

VL - 15

SP - 1457

EP - 1486

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 6

ER -