Threshold analysis for a family of 2×2 operator matrices
Tulkin H. RasulovDepartment of Mathematics, Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, UzbekistanElyor B. DilmurodovDepartment of Mathematics, Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, Uzbekistan
ABI
Abstract
We consider a family of 2 2 operator matrices A(k), k T 3 := (-, ] 3 , > 0, acting in the direct sum of zero-and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice Z 3 , interacting via annihilation and creation operators. We find a set := {k (1) , ..., k (8) } T 3 and a critical value of the coupling constant to establish necessary and sufficient conditions for either z = 0 = min kT 3 ess(A(k)) ( or z = 27/2 = max kT 3 ess(A(k)) is a threshold eigenvalue or a virtual level of A(k (i) ) for some k (i) .