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# On the finiteness of the discrete spectrum of a 3x3 operator matrix

### Abstract

An operator matrix $H$ associated with a lattice system describing three particles in interactions, without conservation of the number of particles, is considered. The structure of the essential spectrum of $H$ is described by the spectra of two families of the generalized Friedrichs models. A symmetric version of the Weinberg equation for eigenvectors of $H$ is obtained. The conditions which guarantee the finiteness of the number of discrete eigenvalues located below the bottom of the three-particle branch of the essential spectrum of $H$ is found.

Key words: Operator matrix, bosonic Fock space, annihilation and creation operators, generalized Friedrichs model, essential and discrete spectra, Weinberg equation, continuity in the uniform operator topology.

### Article Information

 Title On the finiteness of the discrete spectrum of a 3x3 operator matrix Source Methods Funct. Anal. Topology, Vol. 22 (2016), no. 1, 48-61 MathSciNet MR3522861 zbMATH 06630282 Milestones Received 01/02/2014 Copyright The Author(s) 2016 (CC BY-SA)

### Authors Information

Tulkin H. Rasulov
Faculty of Physics and Mathematics, Bukhara State University, 11 M. Ikbol str., Bukhara, 200100, Uzbekistan

### Citation Example

Tulkin H. Rasulov, On the finiteness of the discrete spectrum of a 3x3 operator matrix, Methods Funct. Anal. Topology 22 (2016), no. 1, 48-61.

### BibTex

@article {MFAT844,
AUTHOR = {Rasulov, Tulkin H.},
TITLE = {On the finiteness of the discrete spectrum of a
3x3 operator matrix},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {1},
PAGES = {48-61},
ISSN = {1029-3531},
MRNUMBER = {MR3522861},
ZBLNUMBER = {06630282},
URL = {http://mfat.imath.kiev.ua/article/?id=844},
}

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