This article is cited in 3 papers

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

M. I. Muminov^a, T. H. Rasulov<sup>b

a</sup> 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.

MSC: 81Q10, 35P20, 47N50

Received: 13.10.2013

Language: English