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