najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2021-12-6-657-663najo-539Research ArticleMATHEMATICSМАТЕМАТИКАMonotonicity of the eigenvalues of the two-particle Schrӧdinger operator on a latticeМонотонность собственных значений двухчастичного оператора Шредингера на решеткеAbdullaevJ. I.AbdullaevJ. I.

Mirzo Ulugbek 81, Tashkent 100170;

University Boulevard 15, Samarkand 140104.

jabdullaev@mail.ruKhalkhuzhaevA. M.KhalkhuzhaevA. M.

Mirzo Ulugbek 81, Tashkent 100170;

University Boulevard 15, Samarkand 140104.

ahmad_x@mail.ruUsmonovL. S.UsmonovL. S.

University Boulevard 15, Samarkand 140104.

u.lochinbek@bk.ruInstitute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan; Samarkand State UniversityUzbekistanSamarkand State UniversityUzbekistan202107082025126657663Copyright © Abdullaev J.I., Khalkhuzhaev A.M., Usmonov L.S., 20252025Abdullaev J.I., Khalkhuzhaev A.M., Usmonov L.S.Abdullaev J.I., Khalkhuzhaev A.M., Usmonov L.S.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/539

We consider the two-particle Schrodinger operator¨ H(k), (k∈T3 ≡ (−π,π]3 is the total quasimomentum of a system of two particles) corresponding to the Hamiltonian of the two-particle system on the three-dimensional lattice Z3. It is proved that the number N(k) ≡ N(k(1),k(2),k(3)) of eigenvalues below the essential spectrum of the operator H(k) is nondecreasing function in each k(i) ∈ [0,π], i = 1,2,3. Under some additional conditions potential vˆ, the monotonicity of each eigenvalue zn(k) ≡ zn(k(1),k(2),k(3)) of the operator H(k) in k(i) ∈ [0,π] with other coordinates k being fixed is proved.

Рассмотрим двухчастичный оператор Шредингера H(k), (k∈T3≡(—π, π]3 — полный квазиимпульс системы двух частиц), соответствующий гамильтониану двухчастичной системы на трехчастичном размерной решетке Z3 Доказано, что число N(k)≡N(k(1), k(2), k(3)) собственных значений ниже существенного спектра оператора H(k) является неубывающей функцией при каждом k(i)≡[0, π], i = 1, 2, 3. При некоторых дополнительных условиях на потенциал vˆ, доказана монотонность каждого собственного значения zn(k)≡zn(k(1), k(2), k(3)) оператора H(k) в k(i)≡[0, π] при фиксированных остальных координатах k.

двухчастичный оператор Шрёдингерапринцип Бирмана-Швингераполный квазиимпульсмонотонность собственных значенийtwo-particle Schrӧdinger operatorBirman-Schwinger principletotal quasimomentummonotonicity of the eigenvaluesReferencesBloch I., Dalibard J., and Nascimbene S. Quantum simulations with ultracold quantum gases, Nature Physics, 2012, 8, P. 267–276.Bloch I., Dalibard J., and Nascimbene S. Quantum simulations with ultracold quantum gases, Nature Physics, 2012, 8, P. 267–276.Jaksch D., Zoller P. The cold atom Hubbard toolbox, Annals of Physics, 2005, 315, P. 52-79.Jaksch D., Zoller P. The cold atom Hubbard toolbox, Annals of Physics, 2005, 315, P. 52-79.Lewenstein M., Sanpera A., Ahufinger V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-body Systems, 2012, Oxford University Press.Lewenstein M., Sanpera A., Ahufinger V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-body Systems, 2012, Oxford University Press.Gullans M., Tiecke T.G., Chang D.E., Feist J., Thompson J.D., Cirac J.I., Zoller P., Lukin M.D. Nanoplasmonic Lattices for Ultracold Atoms, Phys. Rev. Lett., 2012, 109, P. 235309.Gullans M., Tiecke T.G., Chang D.E., Feist J., Thompson J.D., Cirac J.I., Zoller P., Lukin M.D. Nanoplasmonic Lattices for Ultracold Atoms, Phys. Rev. Lett., 2012, 109, P. 235309.Hecht E. Optics. Addison-Wesley, Reading, MA, 1998.Hecht E. Optics. Addison-Wesley, Reading, MA, 1998.Murphy B., Hau L.V. Electro-optical nanotraps for neutral atoms, Phys. Rev. Lett., 2009, 102, P. 033003.Murphy B., Hau L.V. Electro-optical nanotraps for neutral atoms, Phys. Rev. Lett., 2009, 102, P. 033003.N.P. de Leon, Lukin M D., and Park H. Quantum plasmonic circuits, IEEE J. Sel. Top. Quantum Electron., 2012, 18, P. 1781–1791.N.P. de Leon, Lukin M D., and Park H. Quantum plasmonic circuits, IEEE J. Sel. Top. Quantum Electron., 2012, 18, P. 1781–1791.Mattis D.C. The few-body problem on a lattice. Rev. Modern Phys., 1986, 58, P. 361–379.Mattis D.C. The few-body problem on a lattice. Rev. Modern Phys., 1986, 58, P. 361–379.Mogilner A.I. Hamiltonians in solid-state physics as multiparticle discrete Schrodinger operators: Problems and results, in: Many Particle¨ Hamiltonians: Spectra and Scattering (Adv. Soviet Math., Vol. 5, R. A. Minlos, ed.), Amer. Math. Soc., Providence, R.I., 1991, P. 139–194.Mogilner A.I. Hamiltonians in solid-state physics as multiparticle discrete Schrodinger operators: Problems and results, in: Many Particle¨ Hamiltonians: Spectra and Scattering (Adv. Soviet Math., Vol. 5, R. A. Minlos, ed.), Amer. Math. Soc., Providence, R.I., 1991, P. 139–194.Graf G.M., Schenker D. 2-magnon scattering in the Heisenberg model, Ann. Inst. H. Poincare’ Phys. The’or., 1997, 67(1), P. 91–107.Graf G.M., Schenker D. 2-magnon scattering in the Heisenberg model, Ann. Inst. H. Poincare’ Phys. The’or., 1997, 67(1), P. 91–107.Muminov M.I., Ghoshal S.K. Spectral features of two-particle Schrodinger operator on d-dimensiional lattice.¨ Complex Analysis and Operator Theory, 2020, 14(1).Muminov M.I., Ghoshal S.K. Spectral features of two-particle Schrodinger operator on d-dimensiional lattice.¨ Complex Analysis and Operator Theory, 2020, 14(1).Lakaev S.N., Khalkhuzhaev A.M. The number of eigenvalues of the two-particle discrete Schrodinger operator.¨ Theor. Math. Phys., 2009, 158(2), P. 221–232.Lakaev S.N., Khalkhuzhaev A.M. The number of eigenvalues of the two-particle discrete Schrodinger operator.¨ Theor. Math. Phys., 2009, 158(2), P. 221–232.Abdullaev J.I., Lakaev S.N. Asymptotics of the discrete spectrum of the three-particle Schrodinger difference operator on lattice,¨ Theor. Math. Phys., 2003, 136(2), P. 1096–1109.Abdullaev J.I., Lakaev S.N. Asymptotics of the discrete spectrum of the three-particle Schrodinger difference operator on lattice,¨ Theor. Math. Phys., 2003, 136(2), P. 1096–1109.Albeverio S., Lakaev S.N., Makarov K.A., Muminov Z.I. The threshold effects for the two-particle Hamiltonians on lattices, Commun. Math. Phys., 2006, 262(1), P. 91–115.Albeverio S., Lakaev S.N., Makarov K.A., Muminov Z.I. The threshold effects for the two-particle Hamiltonians on lattices, Commun. Math. Phys., 2006, 262(1), P. 91–115.Lakaev S.N., Khalkhuzhaev A.M. Spectrum of the two-particle Schrodinger Operator on a lattice,¨ Theor. Math. Phys., 2008, 155(2), P. 753– 764.Lakaev S.N., Khalkhuzhaev A.M. Spectrum of the two-particle Schrodinger Operator on a lattice,¨ Theor. Math. Phys., 2008, 155(2), P. 753– 764.Lakaev S., Kholmatov Sh., Khamidov Sh. Bose-Hubbard models with on-site and nearest-neighbor interactions: Exactly solvable case. J. Phys. A: Math. Theor., 2021, 54, P. 245201(22).Lakaev S., Kholmatov Sh., Khamidov Sh. Bose-Hubbard models with on-site and nearest-neighbor interactions: Exactly solvable case. J. Phys. A: Math. Theor., 2021, 54, P. 245201(22).Muminov M., Lokman C. Finiteness of discrete spectrum of the two-particle Schrodinger operator on diamond lattices,¨ Nanosystems: physics, chemistry, mathematics, 2017, 8(3), P. 310–316.Muminov M., Lokman C. Finiteness of discrete spectrum of the two-particle Schrodinger operator on diamond lattices,¨ Nanosystems: physics, chemistry, mathematics, 2017, 8(3), P. 310–316.M. Reed and B. Simon, Methods of modern mathematical physics. IV: Analysis of operators. Academic Press, N.Y., 1979.M. Reed and B. Simon, Methods of modern mathematical physics. IV: Analysis of operators. Academic Press, N.Y., 1979.Pankov A.A. Lecture Notes on Schrodinger equations. Nova Science, New York, 2007.¨Pankov A.A. Lecture Notes on Schrodinger equations. Nova Science, New York, 2007.¨Abdullaev Zh.I., Kuliev K.D. Bound States of a two-boson system on a two-dimensional lattice, Theor. Math. Phys., 2016, 186(2), P. 231–250.Abdullaev Zh.I., Kuliev K.D. Bound States of a two-boson system on a two-dimensional lattice, Theor. Math. Phys., 2016, 186(2), P. 231–250.Abdullaev Zh.I. Bound states of a system of two fermions on a one-dimensional lattice. Theor. and Math. Phys., 2006, 147(1), P. 486–495.Abdullaev Zh.I. Bound states of a system of two fermions on a one-dimensional lattice. Theor. and Math. Phys., 2006, 147(1), P. 486–495.

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