NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2019, 10 (6), P. 616–622
Threshold analysis for a family of 2×2 operator matrices
T. H. Rasulov – Department of Mathematics, Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, Uzbekistan; rth@mail.ru
E. B. Dilmurodov – Department of Mathematics, Faculty of Physics and Mathematics, Bukhara State University, M. Ikbol str. 11, 200100 Bukhara, Uzbekistan; elyor.dilmurodov@mail.ru
We consider a family of 2×2 operator matrices Aμ(k); k∈T3:= (-π,π]3, μ> 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 Z3; interacting via annihilation and creation operators. We find a set Λ:= {k1, ….. ,k8}⊂T3 and a critical value of the coupling constant μ to establish necessary and sufficient conditions for either z = 0 = min σess(Aμ(k)), k∈T3 ( or z = 27/2 = max σess(Aμ(k)), k∈T3) is a threshold eigenvalue or a virtual level of (Aμ(k(i)) for some k(i)∈Λ.
Keywords: operator matrices, Hamiltonian, generalized Friedrichs model, zero- and one-particle subspaces of a Fock space, threshold eigenvalues,
virtual levels, annihilation and creation operators.
DOI 10.17586/2220-8054-2019-10-6-616-622