Journal of Operator Theory

Volume 74, Issue 2, Fall 2015 pp. 417-455.

Spectral and scattering theory of self-adjoint Hankel operators with piecewise continuous symbols

Authors: Alexander Pushnitski (1) and Dmitri Yafaev (2)
Author institution: (1) Department of Mathematics, King's College London, Strand, London, WC2R~2LS, U.K.
(2) Department of Mathematics, University of Rennes-1, Campus Beaulieu, 35042, Rennes, France

Summary: We develop the spectral and scattering theory of self-adjoint Hankel operators $H$ with piecewise continuous symbols. In this case every jump of the symbol gives rise to a band of the absolutely continuous spectrum of $H$. We prove the existence of wave operators that relate simple `model'' (that is, explicitly diagonalizable) Hankel operators for each jump to the given Hankel operator $H$. We show that the set of all these wave operators is asymptotically complete. This determines the absolutely continuous part of $H$. We prove that the singular continuous spectrum of $H$ is empty and that its eigenvalues may accumulate only to `thresholds'' in the absolutely continuous spectrum. We also state all these results in terms of Hankel operators realized as matrix or integral operators.

DOI: http://dx.doi.org/10.7900/jot.2014aug11.2052
Keywords: Hankel operators, discontinuous symbols, model operators, multichannel scattering, wave operators, the absolutely continuous and singular spectra

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