On the spectrum of the Schrödinger operator for a three-particle system on a lattice
https://doi.org/10.26907/2541-7746.2025.3.547-565
Abstract
A three-particle discrete Schrödinger operator Hµ,γ(K) :≡ Hµ,γ(K), K = (K, K, K) ∈ 𝕋3 , which is associated with a system of three particles (two fermions of mass 1 and one other particle of mass m = 1/γ ,) interacting via pairwise repulsive contact potentials µ > 0 on a three-dimensional lattice ℤ3 , was analyzed. Critical values of mass ratios γs(K) and γas(K) were determined such that the operator Hµ,γ(K) has no eigenvalues if γ ∈ (0, γs(K)), the operator Hµ,γ(K) has a single eigenvalue if γ ∈ (γs(K), γas(K)), and the operator Hµ,γ(K) has three eigenvalues lying to the right of the essential spectrum for sufficiently large µ > 0 if γ ∈ (γas(K), +∞).
About the Authors
A. M. Khalkhuzhaev
Samarkand State University named after Sharof Rashidov
Uzbekistan
Uzbekistan
Ahmad M. Khalkhuzhaev, Dr. Sci. (Physics and Mathematics), Full Professor, Faculty of Mathematics
Samarkand
Kh. G. Khayitova
Bukhara State University
Uzbekistan
Uzbekistan
Khilola G. Khayitova, Teaching Assistant
Bukhara
I. A. Khujamiyorov
Uzbek-Finnish Pedagogical Institute
Uzbekistan
Uzbekistan
Islom A. Khujamiyorov, Teaching Assistant
Samarkand
References
1. Abdullaev J.I., Khalkhuzhaev A.M., Khujamiyorov I.A. Existence сondition for the eigenvalue of a three-particle Schr¨odinger operator on a lattice. Russ. Math., 2023, vol. 67, no. 2, pp. 1–22. https://doi.org/10.3103/S1066369X23020019.
2. Efimov V. Energy levels arising from resonant two-body forces in a three-body system. Phys. Lett. B, 1970, vol. 33, no. 8, pp. 563–564. https://doi.org/10.1016/0370-2693(70)90349-7.
3. Ovchinnikov Yu.N., Sigal I.M. Number of bound states of three-particle systems and Efimov’s effect. Ann. Phys., 1989, vol. 123, no. 2, pp. 274–295. https://doi.org/10.1016/0003-4916(79)90339-7.
4. Sobolev A.V. The Efimov effect. Discrete spectrum asymptotics. Commun. Math. Phys., 1993, vol. 156, no. 1, pp. 101–126. https://doi.org/10.1007/BF02096734.
5. Tamura H. Asymptotics for the number of negative eigenvalues of three-body Schr¨odinger operators with Efimov effect. In: Yajima K. (Ed.) Spectral and Scattering Theory and Applications: Conf. on Spectral and Scattering Theory, Celebrating ShigeToshi Kuroda June 30–July 3, 1992. Ser.: Advanced Studies in Pure Mathematics. Vol. 23. Tokyo, Tokyo Inst. Technol., 1994, pp. 311–322. https://doi.org/10.2969/aspm/02310311.
6. Chadan K., Martin A. A sufficient condition for the existence of bound states in a potential without spherical symmetry. J. Phys. Lett., 1980, vol. 41, no. 9, pp. 205–206. https://doi.org/10.1051/jphyslet:01980004109020500.
7. Liu Y., Yu Y.-C., Chen S. Three-body bound states of two bosons and one impurity in one dimension. Phys. Rev. A, 2021, vol. 104, no. 3, art. 033303. https://doi.org/10.1103/PhysRevA.104.033303.
8. Kholmatov Sh.Yu., Muminov Z.I. Existence of bound states of N -body problem in an optical lattice. J. Phys. A: Math. Theor., 2018, vol. 51, no. 26, art. 265202. https://doi.org/10.1088/1751-8121/aac534.
9. Lakaev S.N., Dell’Antonio G.F., Khalkhuzhaev A.M. Existence of an isolated band of a system of three particles in an optical lattice. J. Phys. A: Math. Theor., 2016, vol. 49, no. 14, art. 145204. https://doi.org/10.1088/1751-8113/49/14/145204.
10. Lakaev S.N. On Efimov’s effect in a system of three identical quantum particles. Funct. Anal. Its Appl., 1993, vol. 27, no. 3, pp. 166–175. https://doi.org/10.1007/BF01087534.
11. Klaus M., Simon B. Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case. Ann. Phys., 1980, vol. 130, no. 2, pp. 251–281. https://doi.org/10.1016/0003-4916(80)90338-3.
12. Valiente M., Zinner N.T. Strongly Interacting Quantum Systems. Vol. 1: Few-body physics. Ser.: IOP Series in Quantum Technology. Bristol, IOP Publ., 2023. 218 p. https://doi.org/10.1088/978-0-7503-3087-9.
13. Valiente M., Petrosyan D., Saenz A. Three-body bound states in a lattice. Phys. Rev. A, 2010, vol. 81, no. 1, art. 011601(R). https://doi.org/10.1103/PhysRevA.81.011601.
14. Khalkhuzhaev A.M., Khujamiyorov I.A. On the discrete spectrum of the Schr¨odinger operator using the 2 + 1 fermionic trimer on the lattice. Nanosyst.: Phys., Chem., Math., 2023, vol. 14, no. 5, pp. 518–529. https://doi.org/10.17586/2220-8054-2023-14-5-518-529.
15. Khalkhuzhaev A.M. The essential spectrum of a three-particle discrete operator corresponding to a system of three fermions on a lattice. Russ. Math., 2017, vol. 61, no. 9, pp. 67–78. https://doi.org/10.3103/S1066369X17090080.
16. Dell’Antonio G.F., 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, vol. 44, no. 31, art. 315302. https://doi.org/10.1088/1751-8113/44/31/315302.
Review
For citations:
Khalkhuzhaev A.M., Khayitova Kh.G., Khujamiyorov I.A. On the spectrum of the Schrödinger operator for a three-particle system on a lattice. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2025;167(3):547-565. (In Russ.) https://doi.org/10.26907/2541-7746.2025.3.547-565
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