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

On the eigenvalues of a 2×2 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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About the authors ▼
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
About the article ▼
Communicated by P.A. Cojuhari.
Received: 2014-05-26.
Revised: 2014-08-07.
Accepted: 2014-09-07.
Published online: 2014-12-15.
Cite this article as ▼
Mukhiddin I. Muminov, Tulkin H. Rasulov, On the eigenvalues of a 2×2 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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