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

TL;DR

This paper develops a comprehensive spectral and scattering theory for self-adjoint Hankel operators with piecewise continuous symbols, revealing how symbol jumps influence the spectrum and establishing the asymptotic completeness of wave operators.

Contribution

It introduces a detailed spectral analysis for Hankel operators with piecewise continuous symbols, including explicit construction of wave operators and spectral characterization.

Findings

Jump discontinuities in symbols create bands of absolutely continuous spectrum.

Wave operators are asymptotically complete, linking model and actual Hankel operators.

Singular continuous spectrum is absent; eigenvalues only accumulate at spectrum thresholds.

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.