najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2020-11-2-138-144najo-436Research ArticleMATHEMATICSМАТЕМАТИКАAnalysis of the spectrum of a 2×2 operator matrix. Discrete spectrum asymptoticsRasulovT. H.

M.Ikbol str. 11, 200100 Bukhara

rth@mail.ruDilmurodovE. B.

M.Ikbol str. 11, 200100 Bukhara

Faculty of Physics and Mathematics, Bukhara State UniversityUzbekistan202030072025112138–144Copyright © Rasulov T.H., Dilmurodov E.B., 20252025Rasulov T.H., Dilmurodov E.B.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/436

We consider a 2×2 operator matrix Aµ, µ > 0 related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analog of the Faddeev equation and its symmetric version for the eigenfunctions of Aµ. We describe the new branches of the essential spectrum of Aµ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of Aµ consists the union of at most three bounded closed intervals and their location is studied. For the critical value µ0 of the coupling constant µ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of Aµ. In this case, an asymptotic formula for the discrete spectrum of Aµ is found.

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