• Corpus ID: 218502170

Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices

@article{Muminov2020AnalysisOT,
  title={Analysis of the discrete spectrum of the family of \$3 \times 3\$ operator matrices},
  author={Mukhiddin I. Muminov and Tulkin Husenovich Rasulov and Nargiza A. Tosheva},
  journal={arXiv: Mathematical Physics},
  year={2020}
}

• M. Muminov, T. Rasulov, N. Tosheva
• Published 5 May 2020
• Mathematics
• arXiv: Mathematical Physics

We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in {\Bbb T}^3:=(-\pi; \pi]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $\Lambda \subset {\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bf H}(K)$ for all $K \in \Lambda$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K \in \Lambda,$ the number $N(K, z)$ of eigenvalues of ${\bf H}(K)$ lying on the left of $z,$ $z<0,$ satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$ with $0<{\mathcal U}_0<\infty,$ independently on the cardinality of $\Lambda.$ Moreover, we prove that for any $K \in \Lambda$ the operator ${\bf H}(K)$ 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.