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Spectral and scattering theory for perturbations of the Carleman operator


Author: D. R. Yafaev
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
Journal: St. Petersburg Math. J. 25 (2014), 339-359
MSC (2010): Primary 47A40; Secondary 47B25
DOI: https://doi.org/10.1090/S1061-0022-2014-01294-0
Published electronically: March 12, 2014
MathSciNet review: 3114858
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Abstract: The spectral properties of the Carleman operator (the Hankel operator with the kernel $ h_{0}(t)=t^{-1}$) are studied; in particular, an explicit formula for its resolvent is found. Then, perturbations are considered of the Carleman operator $ H_{0}$ by Hankel operators $ V$ with kernels $ v(t)$ decaying sufficiently rapidly as $ t\to \infty $ and not too singular at $ t=0$. The goal is to develop scattering theory for the pair $ H_{0}$, $ H=H_{0} +V $ and to construct an expansion in eigenfunctions of the continuous spectrum of the Hankel operator $ H$. Also, it is proved that, under general assumptions, the singular continuous spectrum of the operator $ H$ is empty and that its eigenvalues may accumulate only to the edge points 0 and $ \pi $ in the spectrum of $ H_{0}$. Simple conditions are found for the finiteness of the total number of eigenvalues of the operator $ H$ lying above the (continuous) spectrum of the Carleman operator $ H_{0}$, and an explicit estimate of this number is obtained. The theory constructed is somewhat analogous to the theory of one-dimensional differential operators.


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Additional Information

D. R. Yafaev
Affiliation: IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France
Email: yafaev@univ-rennes1.fr

DOI: https://doi.org/10.1090/S1061-0022-2014-01294-0
Keywords: Hankel operators, resolvent kernels, absolutely continuous spectrum, eigenfunctions, wave operators, scattering matrix, resonances, discrete spectrum, total number of eigenvalues
Received by editor(s): September 20, 2012
Published electronically: March 12, 2014
Dedicated: In memory of Vladimir Savel’evich Buslaev
Article copyright: © Copyright 2014 American Mathematical Society