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arXiv · 2005.01922

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

Mukhiddin I. Muminov·Tulkin H. Rasulov·Nargiza A. Tosheva
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Abstract

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.

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Mukhiddin I. Muminov, Tulkin H. Rasulov, Nargiza A. Tosheva. 2020-05-05. Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices. https://arxiv.org/abs/2005.01922

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