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Algebra i Analiz, 2013, Volume 25, Issue 2, Pages 251–278 (Mi aa1332)  

This article is cited in 4 scientific papers (total in 4 papers)

Research Papers

Spectral and scattering theory for perturbations of the Carleman operator

D. R. Yafaev

IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France

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.

Keywords: Hankel operators, resolvent kernels, absolutely continuous spectrum, eigenfunctions, wave operators, scattering matrix, resonances, discrete spectrum, total number of eigenvalues.

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English version:
St. Petersburg Mathematical Journal, 2014, 25:2, 339–359

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Received: 20.09.2012
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Citation: D. R. Yafaev, “Spectral and scattering theory for perturbations of the Carleman operator”, Algebra i Analiz, 25:2 (2013), 251–278; St. Petersburg Math. J., 25:2 (2014), 339–359

Citation in format AMSBIB
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\vol 25
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\pages 251--278
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\pages 339--359
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. D. R. Yafaev, “On finite rank Hankel operators”, J. Funct. Anal., 268:7 (2015), 1808–1839  crossref  mathscinet  zmath  isi  scopus
    2. A. Pushnitski, D. Yafaev, “Spectral and scattering theory of self-adjoint Hankel operators with piecewise continuous symbols”, J. Operator Theory, 74:2 (2015), 417–455  crossref  mathscinet  zmath  isi  elib  scopus
    3. D. R. Yafaev, “Criteria for Hankel operators to be sign-definite”, Anal. PDE, 8:1 (2015), 183–221  crossref  mathscinet  zmath  isi  scopus
    4. D. R. Yafaev, “Spectral and scattering theory for differential and Hankel operators”, Adv. Math., 308 (2017), 713–766  crossref  mathscinet  zmath  isi  elib  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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