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 | References | Similar Articles | Additional Information

Abstract: The spectral properties of the Carleman operator (the Hankel operator with the kernel ) are studied; in particular, an explicit formula for its resolvent is found. Then, perturbations are considered of the Carleman operator by Hankel operators with kernels decaying sufficiently rapidly as and not too singular at . The goal is to develop scattering theory for the pair , and to construct an expansion in eigenfunctions of the continuous spectrum of the Hankel operator . Also, it is proved that, under general assumptions, the singular continuous spectrum of the operator is empty and that its eigenvalues may accumulate only to the edge points 0 and in the spectrum of . Simple conditions are found for the finiteness of the total number of eigenvalues of the operator lying above the (continuous) spectrum of the Carleman operator , 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