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

Abstract

We consider the family of $3\times 3$ operator matrices $\mathbf{H}(K),$ $K\in {\mathbb{T}}^3:=(-\pi ;\pi {]}^3$ 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 $\Lambda \subset {\mathbb{T}}^3$ to prove the existence of infinitely many eigenvalues of $\mathbf{H}(K)$ 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 $\mathbf{H}(K)$ lying on the left of $z,$ $z<0,$ satisfies the asymptotic relation $\underset{z\to -0}{\lim }N(K,z)|\log |z|{|}^{-1}={\mathcal{U}}_0$ with $0<{\mathcal{U}}_0<\infty ,$ independently on the cardinality of $\Lambda .$ Moreover, we prove that for any $K\in \Lambda$ the operator $\mathbf{H}(K)$ 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.

18 pages

References in corpus (1)

• On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case