Threshold analysis for a family of 2 × 2 operator matrices
https://doi.org/10.17586/2220-8054-2019-10-6-616-622
Abstract
We consider a family of 2 × 2 operator matrices Aµ(k), k ∈ T3 := (−π, π]3, µ > 0, acting in the direct sum of zeroand 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 := {k(1), ..., k(8)} ⊂ T3 and a critical value of the coupling constant µ to establish necessary and sufficient conditions for either z = 0 = min/ k∈T3 σess(Aµ(k)) ( or z = 27/2 = max/k∈T3 σess(Aµ(k)) is a threshold eigenvalue or a virtual level of Aµ(k(i)) for some k(i) ∈ Λ.
Keywords
operator matrices,
Hamiltonian,
generalized Friedrichs model,
zeroand one-particle subspaces of a Fock space,
threshold eigenvalues,
virtual levels,
annihilation and creation operators
Supporting agencies: The authors thank the anonymous referee for reading the manuscript carefully and for making valuable suggestions
About the Authors
T. H. Rasulov
Bukhara State University
Uzbekistan
Uzbekistan
Department of Mathematics, Faculty of Physics and Mathematic.
M. Ikbol str. 11, 200100 Bukhara
E. B. Dilmurodov
Bukhara State University
Uzbekistan
Uzbekistan
Department of Mathematics, Faculty of Physics and Mathematic.
M. Ikbol str. 11, 200100 Bukhara
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Review
For citations:
Rasulov T.H., Dilmurodov E.B. Threshold analysis for a family of 2 × 2 operator matrices. Nanosystems: Physics, Chemistry, Mathematics. 2019;10(6):616-622. https://doi.org/10.17586/2220-8054-2019-10-6-616-622
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