Opuscula Math. 35, no. 3 (2015), 371-395
http://dx.doi.org/10.7494/OpMath.2015.35.3.371

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
• Communicated by P.A. Cojuhari.
• Revised: 2014-08-07.
• Accepted: 2014-09-07.
• Published online: 2014-12-15.