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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).

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.

Citation

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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.

Information

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

MathSciNet: MR4103523

Subjects:
Primary: 81Q10
Secondary: 35P20, 47N50

Rights: Copyright © 2020 Mathematical Research Publishers

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Vol.23 • No. 1 • 2020
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