On the number of eigenvalues of the family of operator matrices

• Main Authors: Muminov, Mukhiddin, Rasulov, T. H.
• Format: Article
• Published: St. Petersburg National Research University of Information Technologies, Mechanics and Optics 2014
• Subjects: QA Mathematics
• Online Access: http://eprints.utm.my/id/eprint/59855/; http://nanojournal.ifmo.ru/en/wp-content/uploads/2014/10/NPCM55P619.pdf
• Institution: Universiti Teknologi Malaysia

We consider the family of operator matrices H(K), K ∈ T3 := (−π; π]3 acting in the direct sum of zero-, one- and two-particle subspaces of the bosonic Fock space. We find a finite set Λ ⊂ T3 to establish the existence of infinitely many eigenvalues of H(K) for all K ∈ Λ when the associated Friedrichs model has a zero energy resonance. It is found that for every K ∈ Λ, the number N (K, z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim z→−0 N (K, z)| log |z||−1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ. Moreover, we show that for any K ∈ Λ the operator H(K) has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.