Estimates for the Bounds of the Essential Spectrum of a 2 × 2 Operator Matrix

  • Tulkin Rasulov Faculty of Physics and Mathematics, Bukhara State University
  • Elyor B. Dilmurodov Bukhara branch of the Institute of Mathematics named after V.I.Romanovskiy M. Ikbol str. 11, 200100 Bukhara, Uzbekistan
Keywords: operator matrix, bosonic Fock space, annihilation and creation operators, the Faddeev equation, essential spectrum, lower and upper bounds

Abstract

We consider a 2 × 2 operator matrix Aμ, μ > 0, related with the lattice systems describing three particles in interaction, without conservation of the number of particles on a d-dimensional lattice. We obtain an analogue of the Faddeev type integral equation for the eigenfunctions of . We describe the two- and three-particle branches of the essential spectrum of via the spectrum of a family of generalized Friedrichs models. It is shown that the essential spectrum of consists of the union of at most three bounded closed intervals. We estimate the lower and upper bounds of the essential spectrum of with respect to the dimension d ∈ N of the torus Td and the coupling constant μ > 0.

Published
2020-07-17
How to Cite
Rasulov, T. and Dilmurodov, E. B. (2020) “Estimates for the Bounds of the Essential Spectrum of a 2 × 2 Operator Matrix”, Contemporary Mathematics, 1(4). doi: 10.37256/cm.142020409.