najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2019-10-6-616-622najo-752Research ArticleMATHEMATICSМАТЕМАТИКАThreshold analysis for a family of 2 × 2 operator matricesThreshold analysis for a family of 2 × 2 operator matricesRasulovT. H.RasulovT. H.

Department of Mathematics, Faculty of Physics and Mathematic.

M. Ikbol str. 11, 200100 Bukhara

rth@mail.ruDilmurodovЕ. В.DilmurodovE. B.

Department of Mathematics, Faculty of Physics and Mathematic.

M. Ikbol str. 11, 200100 Bukhara

 

elyor.dilmurodov@mail.ruBukhara State UniversityBukhara State University201913082025106616622Copyright © Rasulov T.H., Dilmurodov E.B., 20252025Rasulov T.H., Dilmurodov Е.В.Rasulov T.H., Dilmurodov E.B.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/752

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.

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.

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The authors declare that there are no conflicts of interest present.