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 ${\rm d}$-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 ${\rm d}=1$ 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.

UDC: 517.984

MSC: 81Q10

Received August 4, 2014, published March 14, 2015

DOI: 10.17377/semi.2015.12.014