najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-3-304-311najo-302Research ArticleMATHEMATICSМАТЕМАТИКАThe first Schur complement for a lattice spin-boson model with at most two photonsПервое дополнение Шура для решетчатой модели спин-бозон с не более чем двумя фотонамиhttps://orcid.org/0000-0002-2868-4390РасуловТ. Х.RasulovT.

Расулов Тулкин Хусенович

Бухара

Tulkin Rasulov – Faculty of Physics and Mathematics

M. Ikbol str. 11, 200100 Bukhara

t.h.rasulov@buxdu.uzhttps://orcid.org/0000-0002-5583-5684ДилмуродовЭ. Б.DilmurodovE.

Дилмуродов Элер Бахтиерович

Бухара

Elyor Dilmurodov – Faculty of Physics and Mathematics

M. Ikbol str. 11, 200100 Bukhara

e.b.dilmurodov@buxdu.uzБухарский государственный университетBukhara State UniversityБухарский государственный университет; Бухарское отделение института математики имени В.И.РомановскогоBukhara State University; Bukhara branch of the Institute of Mathematics named after V. I. Romanovskiy202330062025143304311Copyright © Rasulov T., Dilmurodov E., 20252025Расулов Т.Х., Дилмуродов Э.Б.Rasulov T., Dilmurodov E.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/302

In the present paper, we consider a lattice spin-boson model A2 with a fixed atom and at most two photons. We construct the first Schur complement S1(_) with spectral parameter _ corresponding to A2. We prove the Birman–Schwinger principle for A2 with respect to S1(_). We investigate an important properties of S1(_) related to the number of eigenvalues of A2 for all dimensions d of the torus Td and for any coupling constant _ > 0.

В настоящей работе рассматривается решетчатая модель спин-бозона ${\cal A}_2$ с фиксированным атомом и не более чем двумя фотонами. Построен первое дополнение Шура $S_1(\lambda)$ со спектральным параметром $\lambda$, соответствующим ${\cal A}_2$. Доказан принцип Бирмана-Швингера для ${\cal A}_2$ относительно $S_1(\lambda)$. Исследованы важные свойства $S_1(\lambda)$, связанные с количеством собственных значений ${\cal A}_2$ для всех размерностей ${\rm d}$ тора ${\Bbb T}^{\rm d}$ и для любой значение константы связи $\alpha>0$.

решетчатая модель спин-бозонадополнение Шурабозонное пространство Фокасущественный спектрчисло собственных значенийпринцип Бирмана-Швингераlattice spin-boson modelSchur complementbosonic Fock spaceessential spectrumnumber of eigenvaluesBirman–Schwinger principleThe authors thank the anonymous referee for reading the manuscript carefully and for making valuable suggestions. Dedicated to the memory of Prof. R. A. Minlos and Prof. H. Neidhardt.ReferencesLeggett A.J., Chakravarty S., Dorsey A.T., Fisher M.P.A., Garg A., ZwergerW. Dynamics of the dissipative two-state system. Rev. Mod. Phys., 59, (Jan 1987), P. 1–85.Leggett A.J., Chakravarty S., Dorsey A.T., Fisher M.P.A., Garg A., ZwergerW. Dynamics of the dissipative two-state system. Rev. Mod. Phys., 59, (Jan 1987), P. 1–85.Hübner M., Spohn H. Radiative decay: nonperturbative approaches. Rev. Math. Phys., 1995, 7(3), P. 363–387.Hübner M., Spohn H. Radiative decay: nonperturbative approaches. Rev. Math. Phys., 1995, 7(3), P. 363–387.Hübner M., Spohn H. Spectral properties of spin-boson Hamiltonian. Ann. Inst. Henri Poincar´e, 1995, 62(3), P. 289–323.Hübner M., Spohn H. Spectral properties of spin-boson Hamiltonian. Ann. Inst. Henri Poincar´e, 1995, 62(3), P. 289–323.Minlos R.A., Spohn H. The three-body problem in radioactive decay: the case of one atom and at most two photons. Topics in Statistical and Theoretical Physics, American Mathematical Society Translations–Series 2, 1996, 177, P. 159–193.Minlos R.A., Spohn H. The three-body problem in radioactive decay: the case of one atom and at most two photons. Topics in Statistical and Theoretical Physics, American Mathematical Society Translations–Series 2, 1996, 177, P. 159–193.Spohn H. Ground states of the spin-boson Hamiltonian. Comm. Math. Phys., 1989, 123, P. 277–304.Spohn H. Ground states of the spin-boson Hamiltonian. Comm. Math. Phys., 1989, 123, P. 277–304.Zhukov Yu., Minlos R. Spectrum and scattering in a “spin-boson” model with not more than three photons. Theor. Math. Phys., 1995, 103(1), P. 398–411.Zhukov Yu., Minlos R. Spectrum and scattering in a “spin-boson” model with not more than three photons. Theor. Math. Phys., 1995, 103(1), P. 398–411.Muminov M., Neidhardt H., Rasulov T. On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case. Journal of Mathematical Physics, 2015, 56, P. 053507.Muminov M., Neidhardt H., Rasulov T. On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case. Journal of Mathematical Physics, 2015, 56, P. 053507.Rasulov T.Kh. Branches of the essential spectrum of the lattice spin-boson model with at most two photons. Theor. Math. Phys., 2016, 186(2), P. 251–267.Rasulov T.Kh. Branches of the essential spectrum of the lattice spin-boson model with at most two photons. Theor. Math. Phys., 2016, 186(2), P. 251–267.Lakaev S.N., Rasulov T.Kh. A model in the theory of perturbations of the essential spectrum of multi-particle operators. Math. Notes., 2003, 73(4), P. 521–528.Lakaev S.N., Rasulov T.Kh. A model in the theory of perturbations of the essential spectrum of multi-particle operators. Math. Notes., 2003, 73(4), P. 521–528.Tretter C. Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London, 2008.Tretter C. Spectral Theory of Block Operator Matrices and Applications. Imperial College Press, London, 2008.Lakaev S.N. Some spectral properties of the generalized Friedrichs model. J. Soviet Math., 1989, 45(6), P. 1540–1565.Lakaev S.N. Some spectral properties of the generalized Friedrichs model. J. Soviet Math., 1989, 45(6), P. 1540–1565.Muminov M.I., Rasulov T.H. The Faddeev equation and essential spectrum of a Hamiltonian in Fock space. Methods Func. Anal. Topology, 2011, 17(1), P. 47–57.Muminov M.I., Rasulov T.H. The Faddeev equation and essential spectrum of a Hamiltonian in Fock space. Methods Func. Anal. Topology, 2011, 17(1), P. 47–57.Rasulov T.Kh. Study of the essential spectrum of a matrix operator. Theor. Math. Phys., 2010, 164(1), P. 883–895.Rasulov T.Kh. Study of the essential spectrum of a matrix operator. Theor. Math. Phys., 2010, 164(1), P. 883–895.Zhang F. The Schur Complement and Its Applications. Numerical Methods and Algorithms. Vol. 4. Springer, Berlin, 2005.Zhang F. The Schur Complement and Its Applications. Numerical Methods and Algorithms. Vol. 4. Springer, Berlin, 2005.Haynsworth E.V. On the Schur complement. Basel Mathematical Notes, BNB 20, 17 pages, 1968.Haynsworth E.V. On the Schur complement. Basel Mathematical Notes, BNB 20, 17 pages, 1968.Muminov M.I., Rasulov T.Kh. An eigenvalue multiplicity formula for the Schur complement of a 3 x 3 block operator matrix. Siberian Math. J., 2015, 56(4), P. 878–895.Muminov M.I., Rasulov T.Kh. An eigenvalue multiplicity formula for the Schur complement of a 3 x 3 block operator matrix. Siberian Math. J., 2015, 56(4), P. 878–895.

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