Existence condition of an eigenvalue of the three particle Schr¨odinger operator on a lattice
https://doi.org/10.26907/0021-3446-2023-9-3-19
Abstract
We consider the three-particle discrete Schrodinger operator Hµ,γ(К), К ϵТ3 associated to a system of three particles (two particle are fermions with mass 1 and third one is an another particle with mass m = 1/y < 1) interacting through zero range pairwise potential µ> 0 on the three-dimensional lattice Z3. It is proved that for γϵ(1, γ0) (γ0≈4,7655) the operator Hµ,γ(π), π=(π,π,π), has no eigenvalue and has only unique eigenvalue with multiplicity three for γ>γ0 lying right of the essential spectrum for sufficiently large µ.
About the Authors
J. I. Abdullaev
Samarkand State University
Uzbekistan
Uzbekistan
Janikul Ibragimovich Abdullaev
15 University blv., Samarkand, 140104
A. M. Khalkhuzhaev
V.I.Romanovskiy Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan
Uzbekistan
Uzbekistan
Ahmad Miyassarovich Khalkhuzhaev
15 University blv., Samarkand, 140104
T. H. Rasulov
Bukhara State University
Uzbekistan
Uzbekistan
Tulkin Husenovich Rasulov
11 M. Ikbol str., Bukhara, 200100
References
1. Reed M., and Simon B. Methods of Modern Mathematical Physics, Anal. of Operators 4 (Elsevier, 1978).
2. Лакаев С.Н., Муминов М.Э. Существенный и дискретный спектр трехчастичного оператора Шрёдингера на решетке, ТМФ, 135 (3), 478–503 (2003).
3. Efimov V. Energy levels arising from resonant two-body forces in a three-body system, Phys. Lett. B 33 (8), 563–564 (1970).
4. Albeverio S., Hoegh-Krohn R., Wu T.T. A class of exactly solvable three-body quantum mechanical problems and universal low energy behavior, Phys. Lett. A, 83 (3), 105–109 (1981).
5. Amado R.D., Noble J.V. Efimov’s effect: A new pathology of three-particle systems. I, Phys. Lett. B., 35 (1), 25–27 (1971).
6. Faddeev L.D., Merkuriev S.P. Quantum scattering theory for several particle systems (Kluwer Acad. Publ., 1993).
7. Яфаев Д.Р. К теории дискретного спектра трехчастичного оператора Шрёдингера, Матем. сб. 94 (136): 4 (8), 567–593 (1974).
8. Ovchinnikov Y.N., Sigal I.M. Number of bound states of three-body systems and Efimov’s effect, Ann. Phys. 123 (2), 274–295 (1989).
9. Sobolev A.V. The Efimov effect. Discrete spectrum asymptotics, Commun. Math. Phys. 156 (1), 101–126 (1993).
10. Tamura H. Asymptotics for the number of negative eigenvalues of three-body Schr¨odinger operators with Efimov effect, Spectral and Scattering Theory and Applications, Adv. Studies in Pure Math. 23, 311–322 (1994).
11. Лакаев С.Н., Абдуллаев Ж.И. Спектр разностного трехчастичного оператора Шрёдингера на решетке, Матем. заметки 71 (5), 686–696 (2002).
12. Минлос Р.А. Система трех квантовых частиц, взаимодействующих поточечно, УМН 69 (3 (417)), 145–172 (2014).
13. Лакаев С.Н., Халмухамедов А.Р., Халхужаев А.М. О связанных состояниях оператора Шрёдингера системы трех бозонов на решетке, ТМФ 188 (1), 36–48 (2016).
14. Lakaev S.N., Lakaev Sh.S. The existence of bound states in a system of three particles in an optical lattice, J. Phys. A: Math. Theor. 50 (33), 335202 (2017).
15. Lakaev S.N., Dell’Antonio G.F., Khalkhuzhaev A.M. Existence of an isolated band in a system of three particles in an optical lattice, J. Phys. A: Math. Theor., 49 (14), 145204 (2016).
16. Abbullaev J.I., Khalkhuzhaev A.M., Kuliev K.D. The existence of eigenvalues of Schr¨odinger operator on three dimensional lattice, Methods of Funct. Anal. and Topology, 28 (3), 189–208 (2022).
17. Albeverio S., Lakaev S.N., Muminov Z.I. Schr¨odinger Operators on lattices. The Efimov effect and discrete spectrum asymptotics, Ann. Henri Poincar´e 5 (4), 743–772 (2004).
18. Абдуллаев Ж.И., Икромов И.А. Конечность числа собственных значений двухчастичного оператора Шрёдингера на решетке, ТМФ 152 (3), 502–517 (2007).
19. Zucker I.J. 70+ Years of the Watson Integrals, J. Stat. Phys. 145, 591–612 (2011). [20] Pankov A.A. Lecture Notes on Schr¨odinger equations (Nova Sci., Publ. Inc., New York, 2007).
Review
For citations:
Abdullaev J.I., Khalkhuzhaev A.M., Rasulov T.H. Existence condition of an eigenvalue of the three particle Schr¨odinger operator on a lattice. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2023;(9):3-19. (In Russ.) https://doi.org/10.26907/0021-3446-2023-9-3-19
Views:
249
Email this article 