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Eurasian Math. J., 2014, Volume 5, Number 2, Pages 60–77 (Mi emj157)  

Infiniteness of the number of eigenvalues embedded in the essential spectrum of a $2\times2$ operator matrix

M. I. Muminova, T. H. Rasulovb

a Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor Bahru, Malaysia
b Faculty of Physics and Mathematics, Bukhara State University, 11 M. Ikbol Str., 200100, Bukhara, Uzbekistan

Abstract: In the present paper a $2\times2$ block operator matrix $\mathbf H$ is considered as a bounded self-adjoint operator in the direct sum of two Hilbert spaces. The structure of the essential spectrum of $\mathbf H$ is studied. Under some natural conditions the infiniteness of the number of eigenvalues is proved, located inside, in the gap or below the bottom of the essential spectrum of $\mathbf H$.

Keywords and phrases: block operator matrix, bosonic Fock space, discrete and essential spectra, eigenvalues embedded in the essential spectrum, discrete spectrum asymptotics, Birman–Schwinger principle, Hilbert–Schmidt class.

Full text: PDF file (447 kB)
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MSC: 81Q10, 35P20, 47N50
Received: 13.10.2013
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Citation: 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

Citation in format AMSBIB
\Bibitem{MumRas14}
\by M.~I.~Muminov, T.~H.~Rasulov
\paper Infiniteness of the number of eigenvalues embedded in the essential spectrum of a~$2\times2$ operator matrix
\jour Eurasian Math. J.
\yr 2014
\vol 5
\issue 2
\pages 60--77
\mathnet{http://mi.mathnet.ru/emj157}


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