Translator Disclaimer
2020 Analysis of the Discrete Spectrum of the Family SPECTRUM of $3 \times 3$ Operator Matrices
Mukhiddin I. Muminov, Tulkin H. Rasulov, Nargiza A. Tosheva
Commun. Math. Anal. 23(1): 17-37 (2020).


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.


Download Citation

Mukhiddin I. Muminov. Tulkin H. Rasulov. Nargiza A. Tosheva. "Analysis of the Discrete Spectrum of the Family SPECTRUM of $3 \times 3$ Operator Matrices." Commun. Math. Anal. 23 (1) 17 - 37, 2020.


Published: 2020
First available in Project Euclid: 19 June 2020

MathSciNet: MR4103523

Primary: 81Q10
Secondary: 35P20 , 47N50

Keywords: annihilation and creation operators , Birman-Schwinger principle , bosonic Fock space , discrete spectrum asymptotics , Friedrichs model , operator matrix , the Efimov effect , zero energy resonance

Rights: Copyright © 2020 Mathematical Research Publishers


This article is only available to subscribers.
It is not available for individual sale.

Vol.23 • No. 1 • 2020
Back to Top