Спектральный анализ двухчастичных гамильтонианов с короткодействующими взаимодействиями
https://doi.org/10.17586/2220-8054-2025-16-5-577-585
Аннотация
Мы анализируем спектральные характеристики решеточных операторов Шрёдингера Hγλμ(K), K∈(−π,π]3, которые представляют систему двух идентичных бозонов, расположенных на решетке Z3. Модель включает в себя взаимодействия на узлах решетки и взаимодействия ближайших соседей, параметризованные γ,λ,μ∈R. Наше исследование оператора Hγλμ(0) выявляет инвариантное подпространство, на котором его ограниченная форма Heaλμ(0) зависит исключительно от λ и μ. Чтобы прояснить механизмы рождения и уничтожения собственных значений для Heaλμ(0), мы определяем критический оператор. Впоследствии на плоскости, натянутой на λ и μ, разрабатывается подробный критерий, который включает: (i) вывод гладких критических кривых, которые отмечают наступление критичности для оператора, и (ii) доказательство точных условий существования именно α собственных значений ниже и β собственных значений выше существенного спектра, где α,β∈{0,1,2} и α+β≤2.
Об авторах
М. О Ахмадова
Samarkand State University
Узбекистан
Узбекистан
Ахмадова Мухайё Олимжон кизи
М. А. Азизова
Samarkand State University
Узбекистан
Узбекистан
Азизова Мукаммал Амриддин кизи
Список литературы
1. Mattis D. The few-body problem on a lattice. Rev. Mod. Phys., 58, P. 361–379, 1986.
2. Albeverio S., Lakaev S.N., Muminov Z.I. Schrodinger operators on lattices. The Efimov effect and discrete spectrum asymptotics. ¨ Ann. Henri Poincare´, 2004, 5, P. 743–772.
3. Albeverio S., Lakaev S.N., Makarov K.A., Muminov Z.I. The Threshold Effects for the Two-particle Hamiltonians on Lattices. Comm. Math. Phys., 2006, 262, P. 91–115.
4. Albeverio S., Lakaev S.N., Khalkhujaev A.M. Number of Eigenvalues of the Three-Particle Schrodinger Operators on Lattices. ¨ Markov Process. Relat. Fields, 2012, 18, P. 387–420.
5. Bach V., W. de Siqueira Pedra, Lakaev S.N. Bounds on the discrete spectrum of lattice Schrodinger operators. ¨ J. Math. Phys., 2017, 59(2), P. 022109.
6. P.A. Faria Da Veiga, Ioriatti L., O’Carroll M. Energy-momentum spectrum of some two-particle lattice Schrodinger Hamiltonians. ¨ Phys. Rev. E, 2002, 66, P. 016130.
7. Hiroshima F., Muminov Z., Kuljanov U. Threshold of discrete Schrodinger operators with delta-potentials on ¨ N-dimensional lattice. Linear and Multilinear Algebra, 2020, 68(11), P. 2267–2279.
8. Kholmatov Sh.Yu., Lakaev S.N., Almuratov F.M. On the spectrum of Schrodinger-type operators on two dimensional lattices. ¨ J. Math. Anal. Appl., 2022, 504(2), P. 126363.
9. Lakaev S.N. The Efimov’s effect of the three identical quantum particle on a lattice. Funct. Anal. Appl., 1993, 27, P. 15–28.
10. Lakaev S.N., Abdukhakimov S.Kh. Threshold effects in a two-fermion system on an optical lattice. Theoret. and Math. Phys., 2020, 203(2), P. 251–268.
11. Lakaev S.N., Kholmatov Sh.Yu., Khamidov Sh.I. Bose-Hubbard model with on-site and nearest-neighbor interactions; exactly solvable case. J. Phys. A: Math. Theor., 2021, 54, P. 245201.
12. Lakaev S.N., Ozdemir E. The existence and location of eigenvalues of the one particle Hamiltonians on lattices. ¨ Hacettepe J. Math. Stat., 2016, 45, P. 1693–1703.
13. Lakaev S.N., Khalkhuzhaev A.M., Lakaev Sh.S. Asymptotic behavior of an eigenvalue of the two-particle discrete Schrodinger operator. ¨ Theoret. and Math. Phys., 2012, 171(3), P. 438–451.
14. Motovilov A.K., Sandhas W., Belyaev V.B. Perturbation of a lattice spectral band by a nearby resonance. J. Math. Phys., 2001, 42, P. 2490–2506.
15. Muminov M.I., Khurramov A.M., Bozorov I.N. On eigenvalues and virtual levels of a two-particle Hamiltonian on a d-dimensional lattice. Nanosystems: Phys. Chem. Math., 2023, 14(3), P. 295–303.
16. Lakaev S.N., Kurbanov Sh.Kh., Alladustov Sh.U. Convergent Expansions of Eigenvalues of the Generalized Friedrichs Model with a Rank-One Perturbation. Complex Analysis and Operator Theory, 2021, 15, P. 121.
17. Faddeev L.D., Merkuriev S.P. Quantum Scattering Theory for Several Particle Systems. Kluwer Academic Publishers, Doderecht, 1993.
18. Efimov V.N. Weakly Bound States of Three Resonantly Interacting Particles. Yad. Fiz., 1970, 12, P. 1080. [Sov. J. Nucl. Phys., 1970, 12, P. 589].
19. Ovchinnikov Y.N., Sigal I.N. Number of bound states of three-body systems and Efimov’s effect. Ann. Phys., 1979, 123(2), P. 274–295.
20. Sobolev A.V. The Efimov effect. Discrete spectrum asymptotics. Commun. Math. Phys., 1993, 156(1), P. 101–126.
21. Tamura H. The Efimov effect of three-body Schrodinger operators. ¨ J. Funct. Anal., 1991, 95(2), P. 433–459.
22. Yafaev D.R. On the theory of the discrete spectrum of the three-particle Schrodinger operator. ¨ Mat. Sb., 1974, 94(136), P. 567–593.
23. Dell’Antonio G., Muminov Z.I., Shermatova V.M. On the number of eigenvalues of a model operator related to a system of three particles on lattices. Journal of Physics A: Mathematical and Theoretical, 2011, 44, P. 315302.
24. Bagmutov A.S., Popov I.Y. (2020). Window-coupled nanolayers: window shape influence on one-particle and two-particle eigenstates. Nanosystems: Physics, Chemistry, Mathematics, 11(6), P. 636–641.
25. Bloch I. Ultracold quantum gases in optical lattices. Nat. Phys., 2005, 1, P. 23–30.
26. Winkler K., Thalhammer G., Lang F., Grimm R., J. Hecker Denschlag, Daley A.J., Kantia A.n, Buchler H.P., Zoller P. Repulsively bound atom ¨ pairs in an optical lattice. Nature, 2006, 441, P. 853–856.
27. Jaksch D., Bruder C., Cirac J., Gardiner C.W., Zoller P. Cold bosonic atoms in optical lattices. Phys. Rev. Lett., 1998, 81, P. 3108–3111.
28. Jaksch D., Zoller P. The cold atom Hubbard toolbox. Ann. Phys., 2005, 315, P. 52–79.
29. Lewenstein M., Sanpera A., Ahufinger V. Ultracold Atoms in Optical Lattices: Simulating Quantum Many-body Systems. Oxford University Press, Oxford, 2012.
30. Ospelkaus C., Ospelkaus S., Humbert L., Ernst P., Sengstock K., Bongs K. Ultracold heteronuclear molecules in a 3d optical lattice. Phys. Rev. Lett., 2006, 97, P. 120402.
31. Reed M., Simon B. Methods of Modern Mathematical Physics. III: Scattering Theory. Academic Press, N.Y., 1978.
32. Lakaev S.N., Bozorov I.N. The number of bound states of a one-particle Hamiltonian on a three-dimensional lattice. Theoret. and Math. Phys., 2009, 158, P. 360–376.
33. Lakaev S.N., Alladustova I.U. The exact number of eigenvalues of the discrete Schrodinger operators in one-dimensional case. ¨ Lobachevskii J. Math., 2021, 42, P. 1294–1303.
34. Akhmadova M.O., Alladustova I.U., Lakaev S.N. On the Number and Locations of Eigenvalues of the Discrete Schrodinger Operator on a Lattice. ¨ Lobachevskii Journal of Mathematics, 2023, 44, P. 1091–1099.
35. Lakaev S.N., Akhmadova M.O. The number and location of eigenvalues for the two-particle Schrodinger operators on lattices. ¨ Complex Analysis and Operator Theory, 2023, 17, P. 104.
36. Lakaev S.N., Motovilov A.K., Abdukhakimov S.Kh. Two-fermion lattice Hamiltonian with first and second nearest-neighboring-site interactions. J. Phys. A: Math. Theor., 2023, 56, P. 315202.
37. Reed M., Simon B. Methods of Modern Mathematical Physics. IV: Analyses of Operators. Academic Press, N.Y., 1978.
38. Lippmann B.A., Schwinger J. Variational principles for scattering processes. I. Phys. Rev., 1950, 79, P. 469.
39. Kato T. Perturbation Theory for Linear Operators (2nd ed.). Springer-Verlag, 1995.
40. Lakaev S.N., Khamidov Sh.I., Akhmadova M.O. Number of bound states of the Hamiltonian of a lattice two-boson system with interactions up to the next neighbouring sites. Lobachevskii Journal of Mathematics, 2024, 45(12), P. 6409–6420.
Дополнительные файлы
Рецензия
Для цитирования:
Ахмадова М.О., Азизова М.А. Спектральный анализ двухчастичных гамильтонианов с короткодействующими взаимодействиями. Наносистемы: физика, химия, математика. 2025;16(5):577-585. https://doi.org/10.17586/2220-8054-2025-16-5-577-585
For citation:
Akhmadova M.O., Azizova M.A. Spectral analysis of two-particle Hamiltonians with short-range interactions. Nanosystems: Physics, Chemistry, Mathematics. 2025;16(5):577-585. https://doi.org/10.17586/2220-8054-2025-16-5-577-585
Просмотров:
16
Послать статью по эл. почте 