Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and Theoretical

Existence of an isolated band in a system of three particles in an optical lattice

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

Published Feb 22, 2016• 27 citations
Publisher

Abstract

Abstract We prove the existence of two- and three-particle bound states of the Schrödinger operators and associated to Hamiltonians and of a system of two and three identical bosons on the lattice interacting via pairwise zero-range attractive or repulsive potentials. As a consequence, we show the existence of an isolated band in the two- and three-bosonic systems in an optical lattice.

DOI
10.1088/1751-8113/49/14/145204

Related papers

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

Gianfausto Dell’Antonio, Saidakhmat Norzhigitovich Lakaev, Khalkhuzhaev, Ahmad M.

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.

arXiv:1512.01983•Dec 7, 2015•0 citations

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

Saidakhmat Norzhigitovich Lakaev, Sh. S. Lakaev

Abstract We consider the hamiltonian H μ , μ ∈ R of a system of three-particles (two identical fermions and one different particle) moving on the lattice Z d , d = 1 , 2 interacting through repulsive ( μ > 0 ) or attractive ( μ < 0 ) zero-range pairwise potential μ v . We prove for any μ ≠ 0 the existence of bound states of the discrete three-particle Schrödinger operator H μ ( K ) , K ∈ T d being the three-particle quasi-momentum, associated to the hamiltonian H μ .

arXiv:1602.01571•Journal of Physics A Mathematical and Theoretical•Jul 21, 2017•26 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

We consider the Hamiltonian of a system of three quantum particles (two identical bosons and a fermion) on the one-dimensional lattice interacting by means of zero-range attractive or repulsive potentials. We investigate the point spectrum of the three-particle discrete Schrödinger operator H(K), K ∈ T which possesses infinitely many eigenvalues depending on repulsive or attractive interactions, under the assumption that the bosons in the system have infinite mass

Nanosystems Physics Chemistry Mathematics•Aug 30, 2024•3 citations

Existence of three-particle bound states in optical lattice

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

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.

Theoretical and Mathematical Physics•Dec 1, 2025•3 citations

The existence of many-particle bound states despite a pair interaction with positive scattering length

Bernhard Baumgartner

Examples of bound states are presented, for a system of three identical particles as well as for a system of particles on a lattice. The particles may be bosons as well as fermions. The interaction between them is given by a pair potential, which does not allow two-particle bound states, has positive scattering length and is not too different from realistic interatomic potentials.

Journal of Physics A Mathematical and General•Nov 21, 1997•13 citations

EXOTIC FEW-BODY BOUND STATES IN A LATTICE

D. PETROSYAN, M. VALIENTE

Strongly-interacting ultra-cold atoms in tight-binding optical lattice potentials provide an ideal platform to realize the fundamental Hubbard model. Here, after outlining the elementary single particle solution, we review and expand our recent work on complete characterization of the bound and scattering states of two and three bosonic atoms in a one-dimensional optical lattice. In the case of two atoms, there is a family of interaction-bound "dimer" states of co-localized particles that exists invariantly for either attractive or repulsive on-site interaction, with the energy below or above the two-particle scattering continuum, respectively. Adding then the third particle -- "monomer" -- we find that, apart from the simple strongly-bound "trimer" corresponding to all three particles occupying the same lattice site, there are two peculiar families of weakly-bound trimers with energies below and above the monomer-dimer scattering continuum, the corresponding binding mechanism being an effective particle exchange interaction.

arXiv:1002.2778•Jun 1, 2010•2 citations
Abstract We prove the existence of two- and three-particle bound states of the Schrödinger operators and associated to Hamiltonians and of a system of two and three identical bosons on the lattice interacting via pairwise zero-range attractive or repulsive potentials. As a consequence, we show the existence of an isolated band in the two- and three-bosonic systems in an optical lattice.
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 hμ\mathrm{h}_μhμ​ and Hμ\mathrm{H}_μHμ​ of a system of two and three identical bosons on the lattice Zd,d=1,2\Z^d, d=1,2Zd,d=1,2 interacting via pairwise zero-range attractive μ<0μ<0μ<0 or repulsive μ>0μ>0μ>0 potentials. As a consequence we show the existence of an isolated band in the two and three bosonic systems in an optical lattice.
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 μ1∈R\mu_1\in\mathbb{R}μ1​∈R , while the interaction between a fermion and one other particle is via an on-site potential with strength μ2∈R{\mu_2\in\mathbb{R}}μ2​∈R . We establish existence of bound states of the associated three-body lattice Schrödinger operator for all values of the total quasimomentum K∈T1K\in\mathbb{T}^1K∈T1 . Furthermore, we show that both the bound state fμ1μ2(K; ⋅ , ⋅ )f_{\mu_1\mu_2}(K;\,{\cdot}\,{,}\,{\cdot}\,)fμ1​μ2​​(K;⋅,⋅) and its corresponding eigenvalue Eμ1μ2(K)E_{\mu_1\mu_2}(K)Eμ1​μ2​​(K) depend holomorphically on the quasimomentum.
Strongly-interacting ultra-cold atoms in tight-binding optical lattice potentials provide an ideal platform to realize the fundamental Hubbard model. Here, after outlining the elementary single particle solution, we review and expand our recent work on complete characterization of the bound and scattering states of two and three bosonic atoms in a one-dimensional optical lattice. In the case of two atoms, there is a family of interaction-bound "dimer" states of co-localized particles that exists invariantly for either attractive or repulsive on-site interaction, with the energy below or above the two-particle scattering continuum, respectively. Adding then the third particle -- "monomer" -- we find that, apart from the simple strongly-bound "trimer" corresponding to all three particles occupying the same lattice site, there are two peculiar families of weakly-bound trimers with energies below and above the monomer-dimer scattering continuum, the corresponding binding mechanism being an effective particle exchange interaction.