Opuscula Math. 35, no. 3 (2015), 371-395

Opuscula Mathematica

On the eigenvalues of a 2ˣ2 block operator matrix

Mukhiddin I. Muminov
Tulkin H. Rasulov

Abstract. A \(2\times2\) block operator matrix \({\mathbf H}\) acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of \(H_{22}\) (the second diagonal entry of \({\bf H}\)) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number \(N(z)\) of eigenvalues of \(H_{22}\) lying below \(z\lt0\), the following asymptotics is found \[\lim\limits_{z\to -0} N(z) |\log|z||^{-1}=\,{\mathcal U}_0 \quad (0\lt {\mathcal U}_0\lt \infty).\] Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of \({\mathbf H}\) is proved.

Keywords: block operator matrix, Fock space, discrete and essential spectra, Birman-Schwinger principle, the Efimov effect, discrete spectrum asymptotics, embedded eigenvalues.

Mathematics Subject Classification: 81Q10, 35P20, 47N50.

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  • Mukhiddin I. Muminov
  • Universiti Teknologi Malaysia (UTM), Faculty of Science, 81310 Skudai, Johor Bahru, Malaysia
  • Tulkin H. Rasulov
  • Bukhara State University, Faculty of Physics and Mathematics, 11 M. Ikbol Str., Bukhara, 200100, Uzbekistan
  • Received: 2014-05-26.
  • Revised: 2014-08-07.
  • Accepted: 2014-09-07.
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Cite this article as:
Mukhiddin I. Muminov, Tulkin H. Rasulov, On the eigenvalues of a 2ˣ2 block operator matrix, Opuscula Math. 35, no. 3 (2015), 371-395, http://dx.doi.org/10.7494/OpMath.2015.35.3.371

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