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 Sib. Èlektron. Mat. Izv., 2015, Volume 12, Pages 168–184 (Mi semr577)

Computational mathematics

On the spectrum of a three-particle model operator on a lattice with non-local potentials

T. Kh. Rasulov, Z. D. Rasulova

Bukhara State University, Muhammad Igbol, 11, 705018 Bukhara, Uzbekistan

Abstract: A model operator $H$ associated to a system of three particles on a $d$-dimensional lattice that interact via non-local potentials is considered. The channel operators are identified. An analogue of the Faddeev equation for the eigenfunctions of $H$ is constructed and the spectrum of $H$ is described. The location of the essential spectrum of $H$ is described by the spectrum of channel operators. It is shown that the essential spectrum of $H$ consists the union of at most $2n+1$ bounded closed intervals, where $n$ is the rank of the kernel of non-local interaction operators. The upper bound of the spectrum of $H$ is found. The lower bound of the essential spectrum of $H$ for the case $d=1$ is estimated.

Keywords: model operator, discrete Schrödinger operator, non-local interaction operators, Hubbard model, channel operator, Hilbert–Schmidt class, Faddeev equation, essential and discrete spectrum.

DOI: https://doi.org/10.17377/semi.2015.12.014

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UDC: 517.984
MSC: 81Q10
Received August 4, 2014, published March 14, 2015

Citation: T. Kh. Rasulov, Z. D. Rasulova, “On the spectrum of a three-particle model operator on a lattice with non-local potentials”, Sib. Èlektron. Mat. Izv., 12 (2015), 168–184

Citation in format AMSBIB
\Bibitem{RasRas15} \by T.~Kh.~Rasulov, Z.~D.~Rasulova \paper On the spectrum of a three-particle model operator on a lattice with non-local potentials \jour Sib. \Elektron. Mat. Izv. \yr 2015 \vol 12 \pages 168--184 \mathnet{http://mi.mathnet.ru/semr577} \crossref{https://doi.org/10.17377/semi.2015.12.014} `

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