TY  - JOUR
ID  - Muminov2015
LB  - Muminov2015
AU  - Muminov, Mukhiddin I.
AU  - Rasulov, Tulkin H.
TI  - On the eigenvalues of a 2×2 block operator matrix
ST  - On the eigenvalues of a 2×2 block operator matrix
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2015
VL  - 35
IS  - 3
SP  - 371
EP  - 395
DO  - http://dx.doi.org/10.7494/OpMath.2015.35.3.371
UR  - http://dx.doi.org/10.7494/OpMath.2015.35.3.371
AB  - 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.
KW  - block operator matrix
KW  - Fock space
KW  - discrete and essential spectra
KW  - Birman-Schwinger principle
KW  - the Efimov effect
KW  - discrete spectrum asymptotics
KW  - embedded eigenvalues
SN  - 1232-9274
ER  -