Theoretical and Mathematical Physics

Existence of three-particle bound states in optical lattice

Saidakhmat Norzhigitovich Lakaev, Sh. I. Khamidov, Собир Сахибжанович Улашов

Published Dec 1, 2025• 3 citations

Abstract

We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength $$\mu_1\in\mathbb{R}$$ , while the interaction between a fermion and one other particle is via an on-site potential with strength $${\mu_2\in\mathbb{R}}$$ . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum $$K\in\mathbb{T}^1$$ . Furthermore, we show that both the bound state $$f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)$$ and its corresponding eigenvalue $$E_{\mu_1\mu_2}(K)$$ depend holomorphically on the quasimomentum.

DOI: 10.1134/s0040577925120116

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Theoretical and Mathematical Physics

Existence of three-particle bound states in optical lattice

Saidakhmat Norzhigitovich Lakaev, Sh. I. Khamidov, Собир Сахибжанович Улашов

Published Dec 1, 2025• 3 citations

Publisher

Abstract

We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength

\mu_1\in\mathbb{R}

, while the interaction between a fermion and one other particle is via an on-site potential with strength

{\mu_2\in\mathbb{R}}

. We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum

K\in\mathbb{T}^1

. Furthermore, we show that both the bound state

f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)

and its corresponding eigenvalue

E_{\mu_1\mu_2}(K)

depend holomorphically on the quasimomentum.

DOI: 10.1134/s0040577925120116

Related papers

The existence of bound states in a system of three particles in an optical lattice

Saidakhmat Norzhigitovich Lakaev, Sh. S. Lakaev

arXiv:1602.01571Journal of Physics A Mathematical and TheoreticalJul 21, 201726 citations

The point spectrum of the three-particle Schrödinger operator for a system comprising two identical bosons and one fermion on Z

Zahriddin Ishkobilovich Muminov, Актамова Васила Уктамовна

Nanosystems Physics Chemistry MathematicsAug 30, 20243 citations

The existence of an isolated band in a system of three particles in an optical lattice

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arXiv:1512.01983Dec 7, 20150 citations

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Saidakhmat Norzhigitovich Lakaev, Gianfausto Dell’Antonio, Akhmad Miyassarovich Khalkhuzhaev

Journal of Physics A Mathematical and TheoreticalFeb 22, 201627 citations

Critical mass ratio and phase transition in three-particle lattice systems: comparison of bosonic and fermionic cases

J. I. Abdullaev, A.I. Eshniyоzov, M. V. Dolgopolov

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Nanosystems Physics Chemistry MathematicsApr 30, 20260 citations

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Y.-M. Wang, J.-Q. Liang

Repulsively interacting particles in a periodic potential can form bound composite objects with long lifetimes, as has been observed experimentally [Winkler et al., Nature (London) 441, 853 (2006)]. In this paper, a complete two-particle solution of one-dimensional periodical potential was derived in a strong interaction regime, where the on-site approximation of a two-body interaction in the Bose-Hubbard model (BHM) is extended to include the interaction of nearest neighbors, which results in atom-pair hopping. The energy spectrum of the bound pair is drastically reshaped due to the pair-hopping term, and complex eigenenergy corresponding to metastable states is also found that has not been discovered in the usual BHM. When the absolute value of a center-of-mass quasimomentum wave vector is greater than a critical value ( $|K|>{K}_{c}$ ), two bound-pair solutions are found. Furthermore, the spatial and momentum distributions of the bound pair displays a crossover from dark to bright soliton-like solutions in the extended BHM. Our results reduce to that of the usual BHM in the weak interaction case.Repulsively interacting particles in a periodic potential can form bound composite objects with long lifetimes, as has been observed experimentally [Winkler et al., Nature (London) 441, 853 (2006)]. In this paper, a complete two-particle solution of one-dimensional periodical potential was derived in a strong interaction regime, where the on-site approximation of a two-body interaction in the Bose-Hubbard model (BHM) is extended to include the interaction of nearest neighbors, which results in atom-pair hopping. The energy spectrum of the bound pair is drastically reshaped due to the pair-hopping term, and complex eigenenergy corresponding to metastable states is also found that has not been discovered in the usual BHM. When the absolute value of a center-of-mass quasimomentum wave vector is greater than a critical value ($|K|>{K}_{c}$), two bound-pair solutions are found. Furthermore, the spatial and momentum distributions of the bound pair displays a crossover from dark to bright soliton-like solutions in the extended BHM. Our results reduce to that of the usual BHM in the weak interaction case.

Physical Review AApr 15, 201028 citations

We consider a system of three particles consisting of two identical fermions and one other particle on a one-dimensional lattice. The fermions interact via a nearest-neighbor potential of strength

\mu_1\in\mathbb{R}

, while the interaction between a fermion and one other particle is via an on-site potential with strength

{\mu_2\in\mathbb{R}}

. We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum

K\in\mathbb{T}^1

. Furthermore, we show that both the bound state

f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)

and its corresponding eigenvalue

E_{\mu_1\mu_2}(K)

depend holomorphically on the quasimomentum.

We prove the existence of two-and three-particle bound states of the Schrödinger operators $h_μ(k),k\in \T^d$ and $H_μ(K),K\in \T^d$ associated to Hamiltonians

\mathrm{h}_μ

and

\mathrm{H}_μ

of a system of two and three identical bosons on the lattice

\Z^d, d=1,2

interacting via pairwise zero-range attractive

μ<0

or repulsive

μ>0

potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.

|K|>{K}_{c}

), two bound-pair solutions are found. Furthermore, the spatial and momentum distributions of the bound pair displays a crossover from dark to bright soliton-like solutions in the extended BHM. Our results reduce to that of the usual BHM in the weak interaction case.