On the eigenvalues of a 2ˣ2 block operator matrix

• Main Authors: Mukhiddin I. Muminov, Tulkin H. Rasulov
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2015-01-01
• Series: Opuscula Mathematica
• Subjects: block operator matrix; Fock space; discrete and essential spectra; Birman-Schwinger principle; the Efimov effect; discrete spectrum asymptotics; embedded eigenvalues
• Online Access: http://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3523.pdf

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.