# 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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