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Sibirsk. Mat. Zh., 2011, Volume 52, Number 2, Pages 400–415 (Mi smj2206)  

This article is cited in 1 scientific paper (total in 1 paper)

On the number of eigenvalues of a matrix operator

T. Kh. Rasulov

Bukhara State University, Bukhara, Uzbekistan

Abstract: We consider a matrix operator $H$ in the Fock space. We prove the finiteness of the number of negative eigenvalues of $H$ if the corresponding generalized Friedrichs model has the zero eigenvalue ($0=\min\sigma_\mathrm{ess}(H)$). We also prove that $H$ has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of $H$ below $z$ as $z\to-0$.

Keywords: Efimov effect, Fock space, zero energy resonance, Hilbert–Schmidt class, Birman–Schwinger principle, discrete spectrum.

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English version:
Siberian Mathematical Journal, 2011, 52:2, 316–328

Bibliographic databases:

UDC: 517.984
Received: 15.04.2010

Citation: T. Kh. Rasulov, “On the number of eigenvalues of a matrix operator”, Sibirsk. Mat. Zh., 52:2 (2011), 400–415; Siberian Math. J., 52:2 (2011), 316–328

Citation in format AMSBIB
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\by T.~Kh.~Rasulov
\paper On the number of eigenvalues of a~matrix operator
\jour Sibirsk. Mat. Zh.
\yr 2011
\vol 52
\issue 2
\pages 400--415
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\transl
\jour Siberian Math. J.
\yr 2011
\vol 52
\issue 2
\pages 316--328
\crossref{https://doi.org/10.1134/S0037446611020157}
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    This publication is cited in the following articles:
    1. M. I. Muminov, T. H. Rasulov, “Infiniteness of the number of eigenvalues embedded in the essential spectrum of a $2\times2$ operator matrix”, Eurasian Math. J., 5:2 (2014), 60–77  mathnet
  • Сибирский математический журнал Siberian Mathematical Journal
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