najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2024-15-4-438-447najo-34Research ArticleMATHEMATICSМАТЕМАТИКАThe point spectrum of the three-particle Schr ¨odinger operator for a system comprising two identical bosons and one fermion on Z.Точечный спектр трехчастичного оператора Шредингера для системы, состоящей из двух идентичных бозонов и одного фермиона на Zhttps://orcid.org/0000-0002-7201-6330МуминовЗ. Э.MuminovZ. I.

Zahriddin I. Muminov

100066, Tashkent;  100174, Tashkent

zimuminov@gmail.comhttps://orcid.org/0009-0000-6587-0021АктамоваВ. У.AktamovaV. U.

Vasila U. Aktamova

140103, Samarkand

vaktamova@mail.ruTashkent State University of Economics; Institute of Mathematics named after V.I.RomanovskyUzbekistanSamarkand Institute of Veterinary MedicineUzbekistan202431052025154438447Copyright © Muminov Z.I., Aktamova V.U., 20252025Муминов З.Э., Актамова В.У.Muminov Z.I., Aktamova V.U.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/34

We consider the Hamiltonian of a system of three quantum particles (two identical bosons and a fermion) on the one-dimensional lattice interacting by means of zero-range attractive or repulsive potentials. We investigate the point spectrum of the three-particle discrete Schrödinger operator H(K), K ∈ T which possesses infinitely many eigenvalues depending on repulsive or attractive interactions, under the assumption that the bosons in the system have infinite mass

Мы рассматриваем гамильтониан системы из трех квантовых частиц (двух идентичных бозонов и фермиона) на одномерной решетке, взаимодействующих посредством потенциалов притяжения или отталкивания нулевого диапазона. Мы исследуем точечный спектр трехчастичного дискретного оператора Шредингера , который обладает бесконечно большим числом собственных значений, зависящих от отталкивающих или притягивающих взаимодействий, в предположении, что бозоны в системе имеют бесконечную массу. 

оператор Шредингерадисперсионные функциипарные потенциалы нулевого диапазонадискретный спектрсущественный спектрSchr ¨odinger operatordispersion functionszero-range pair potentialsdiscrete spectrumessential spectrumThe authors acknowledge support from the Innovative Development Agency of the Republic of Uzbekistan (Grant No. FZ–20200929224), and they thank the anonymous referee for reading the manuscript carefully and for making valuable suggestions.ReferencesWinkler K. et al. Repulsively bound atom pairs in an optical lattice. Nature, 2006, 441, P. 853–56.Winkler K. et al. Repulsively bound atom pairs in an optical lattice. Nature, 2006, 441, P. 853–56.Bloch I. Ultracold quantum gases in optical lattices. Nat. Phys., 2005, 1, P. 23–30.Bloch I. Ultracold quantum gases in optical lattices. Nat. Phys., 2005, 1, P. 23–30.Jaksch D.and Zoller P. The cold atom Hubbard toolbox. Ann. of Phys., 2005, 315(1), P. 52–79.Jaksch D.and Zoller P. The cold atom Hubbard toolbox. Ann. of Phys., 2005, 315(1), P. 52–79.Thalhammer G. et al. Inducing an optical Feshbach resonance via stimulated Raman coupling. Phys.Rev. A., 2005, 71, P. 033403.Thalhammer G. et al. Inducing an optical Feshbach resonance via stimulated Raman coupling. Phys.Rev. A., 2005, 71, P. 033403.Hofstetter W., et al. High-temperature superfluidity of fermionic atoms in optical lattices. Phys. Rev. Lett., 2002, 89, P. 220407.Hofstetter W., et al. High-temperature superfluidity of fermionic atoms in optical lattices. Phys. Rev. Lett., 2002, 89, P. 220407.Lewenstein M. et al. Atomic Bose-Fermi mixtures in an optical lattice. Phys. Rev. Lett., 2004, 92, P. 050401.Lewenstein M. et al. Atomic Bose-Fermi mixtures in an optical lattice. Phys. Rev. Lett., 2004, 92, P. 050401.Efimov V.N. Energy levels arising from resonant two-body forces in a three-body system. Physics Letters., 1970, 33(8), P. 563–564.Efimov V.N. Energy levels arising from resonant two-body forces in a three-body system. Physics Letters., 1970, 33(8), P. 563–564.Yafaev D. On the theory of the discrete spectrum of the three-particle Schr¨odinger operator. Math. USSR-Sb., 1974, 23, P. 535–559.Yafaev D. On the theory of the discrete spectrum of the three-particle Schr¨odinger operator. Math. USSR-Sb., 1974, 23, P. 535–559.Lakaev S.N. The Efimov’s effect of a system of three identical quantum lattice particles. Funct. Anal. Its Appl., 1993, 27, P. 15–28.Lakaev S.N. The Efimov’s effect of a system of three identical quantum lattice particles. Funct. Anal. Its Appl., 1993, 27, P. 15–28.Lakaev S.N. and Muminov Z.I. The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schr¨odinger Operator. Funct. Anal. Appl., 2003, 37, P. 228–231.Lakaev S.N. and Muminov Z.I. The Asymptotics of the Number of Eigenvalues of a Three-Particle Lattice Schr¨odinger Operator. Funct. Anal. Appl., 2003, 37, P. 228–231.Abdullaev Zh.I. and Lakaev S.N. Asymptotics of the Discrete Spectrum of the Three-Particle Schr¨odinger Difference Operator on a Lattice. Theor. Math. Phys., 136, P. 1096–1109.Abdullaev Zh.I. and Lakaev S.N. Asymptotics of the Discrete Spectrum of the Three-Particle Schr¨odinger Difference Operator on a Lattice. Theor. Math. Phys., 136, P. 1096–1109.Albeverio S., Lakaev S.N., Muminov Z.I. “Schr¨odinger operators on lattices. The Efimov effect and discrete spectrum asymptotics. Ann. Inst. H. Poincar´e Phys. Theor., 2004, 5, P. 743–772.Albeverio S., Lakaev S.N., Muminov Z.I. “Schr¨odinger operators on lattices. The Efimov effect and discrete spectrum asymptotics. Ann. Inst. H. Poincar´e Phys. Theor., 2004, 5, P. 743–772.Dell’Antonio G., Muminov Z.I., Shermatova Y.M. On the number of eigenvalues of a model operator related to a system of three-particles on lattices. J. Phys. A: Math. Theor., 2011, 44, P. 315302.Dell’Antonio G., Muminov Z.I., Shermatova Y.M. On the number of eigenvalues of a model operator related to a system of three-particles on lattices. J. Phys. A: Math. Theor., 2011, 44, P. 315302.Khalkhuzhaev A.M., Abdullaev J.I., Boymurodov J.K. The Number of Eigenvalues of the Three-Particle Schr¨odinger Operator on Three Dimensional Lattice. Lob. J. Math., 2022, 43(12), P. 3486–3495.Khalkhuzhaev A.M., Abdullaev J.I., Boymurodov J.K. The Number of Eigenvalues of the Three-Particle Schr¨odinger Operator on Three Dimensional Lattice. Lob. J. Math., 2022, 43(12), P. 3486–3495.Muminov M., Aliev N. Discrete spectrum of a noncompact perturbation of a three-particle Schr¨odinger operator on a lattice. Theor. Math. Phys., 2015, 1823(3), P. 381–396.Muminov M., Aliev N. Discrete spectrum of a noncompact perturbation of a three-particle Schr¨odinger operator on a lattice. Theor. Math. Phys., 2015, 1823(3), P. 381–396.Muminov Z.I., Aliev N.M., Radjabov T. On the discrete spectrum of the three-particle Schr¨odinger operator on a two-dimensional lattice. Lob. J. Math., 2022, 43(11), P. 3239–3251.Muminov Z.I., Aliev N.M., Radjabov T. On the discrete spectrum of the three-particle Schr¨odinger operator on a two-dimensional lattice. Lob. J. Math., 2022, 43(11), P. 3239–3251.Aliev N.M. Asymtotic of the Discrete Spectrum of the Three-Particle Schrodinger Operator on a One-Dimensional Lattice. Lob. J. Math, 2023, 44(2), P. 491–501.Aliev N.M. Asymtotic of the Discrete Spectrum of the Three-Particle Schrodinger Operator on a One-Dimensional Lattice. Lob. J. Math, 2023, 44(2), P. 491–501.Komarov I.V., Ponomarev L.I., and Slavyanov S.Yu. Spheroidal and Coulomb Spheroidal Functions [in Russian]. Nauka, Moscow, 1976.Komarov I.V., Ponomarev L.I., and Slavyanov S.Yu. Spheroidal and Coulomb Spheroidal Functions [in Russian]. Nauka, Moscow, 1976.Rabinovich V.S., Roch S. The essential spectrum of Schr¨oinger operators on lattice. J. Phys. A.:Math. Gen., 2006, 39, P. 8377.Rabinovich V.S., Roch S. The essential spectrum of Schr¨oinger operators on lattice. J. Phys. A.:Math. Gen., 2006, 39, P. 8377.Albeverio S., Lakaev S., Muminov Z. On the structure of the essential spectrum for the three-particle Schr¨odinger operators on lattices. Math. Nachr., 2007, 280, P. 699–716.Albeverio S., Lakaev S., Muminov Z. On the structure of the essential spectrum for the three-particle Schr¨odinger operators on lattices. Math. Nachr., 2007, 280, P. 699–716.Kholmatov Sh.Yu., Muminov Z.I. The essential spectrum and bound states of N -body problem in an optical lattice. J. Phys. A: Math. Theor., 2018, 51, P. 265202.Kholmatov Sh.Yu., Muminov Z.I. The essential spectrum and bound states of N -body problem in an optical lattice. J. Phys. A: Math. Theor., 2018, 51, P. 265202.Muminov Z., Lakaev Sh. Aliev N. On the Essential Spectrum of Three-Particle Discrete Schr¨odinger Operators with Short-Range Potentials. Lob. J. Math., 2021, 42(6), P. 1304–1316.Muminov Z., Lakaev Sh. Aliev N. On the Essential Spectrum of Three-Particle Discrete Schr¨odinger Operators with Short-Range Potentials. Lob. J. Math., 2021, 42(6), P. 1304–1316.Lakaev S.N., Boltaev A.T. The Essential Spectrum of a Three Particle Schr¨odinger Operator on Lattices. Lob. J. Math., 2023, 44(3), P. 1176–1187.Lakaev S.N., Boltaev A.T. The Essential Spectrum of a Three Particle Schr¨odinger Operator on Lattices. Lob. J. Math., 2023, 44(3), P. 1176–1187.Jakubaba-Amundsen D.H. The HVZ Theorem for a Pseudo-Relativistic Operator. Ann. Henri Poincar´e, 2007, 8, P. 337–360.Jakubaba-Amundsen D.H. The HVZ Theorem for a Pseudo-Relativistic Operator. Ann. Henri Poincar´e, 2007, 8, P. 337–360.

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