Analysis of the Discrete Spectrum of the Family SPECTRUM of $3 \times 3$ Operator Matrices

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 $z$, $z\lt0$, satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$ with $0\lt{\mathcal U}_0\lt\infty$, 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.