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

Abstract

We develop the spectral and scattering theory for 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 construct wave operators relating simple "model" (that is, explicitly diagonalizable) Hankel operators for each jump and 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 also prove that the singular continuous spectrum of $H$ is empty and that its eigenvalues may accumulate only to "thresholds" in the absolutely continuous spectrum. All these results are reformulated in terms of Hankel operators realized as matrix or integral operators.