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

arXiv:0709.4151 · Journal of Experimental and Theoretical Physics Letters

Universal description of the rotational-vibrational spectrum of three particles with zero-range interactions

O. I. Kartavtsev, A. V. Malykh

Published Jan 1, 2008• 21 citations
arXivHTMLPDF

Abstract

A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and a third particle of mass m 1 in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio m/m 1 and the total angular momentum L. It is found that the number of vibrational states is determined by the functions L c(m/m 1) and L b(m/m 1). Explicitly, if the two-body scattering length is positive, the number of states is finite for L c(m/m 1) ≤ L ≤ L b(m/m 1), zero for L > L b(m/m 1), and infinite for L < L c(m/m 1). If the two-body scattering length is negative, the number of states is zero for L ≥ L c(m/m 1) and infinite for L < L c(m/m 1). For the finite number of vibrational states, all the binding energies are described by the universal function ɛLN(m/m 1) = (ξ, η), where ξ = (N − 1/2)/√L(L + 1), η = √m/[m 1 L(L + 1)], and N is the vibrational quantum number. This scaling dependence is in agreement with the numerical calculations for L > 2 and only slightly deviates from those for L = 1, 2. The universal description implies that the critical values L c(m/m 1) and L b(m/m 1) increase as 0.401 √m/m 1 and 0.563 √m/m 1, respectively, while the number of vibrational states for L ≥ L c(m/m 1) is within the range N ≤ N max ≈ 1.1√L(L + 1) + 1/2.

DOI
10.1134/s002136400722002x

Related papers

Low-energy three-body dynamics in binary quantum gases

O. I. Kartavtsev, A. V. Malykh

The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass $m$ and a particle of the mass $m_1$ with the zero-range two-body interaction in the states of the total angular momentum L=1 are considered. Using the boundary condition model for the s-wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependencies of the three-body binding energies on the mass ratio $m/m_1$ for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at $m/m_1 \\ge \\lambda_1 \\approx 8.17260$ and $m/m_1 \\ge \\lambda_2 \\approx 12.91743$, respectively. For $m/m_1 \\alt \\lambda_1$ and $m/m_1 \\alt \\lambda_2$, the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied and it is shown that a two-hump structure in the mass-ratio dependencies of the cross sections is connected with arising of the bound states.

•Oct 28, 2006•55 citations

Low-energy three-body dynamics in binary quantum gases

O I Kartavtsev, A V Malykh

The universal three-body dynamics in ultra-cold binary Fermi and Fermi–Bose mixtures is studied. Two identical fermions of mass m and a particle of mass m 1 with the zero-range two-body interaction in states of total angular momentum L = 1 are considered. Using the boundary condition model for the s -wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependences of the three-body binding energies on the mass-ratio m / m 1 for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m / m 1 ⩾ λ 1 ≈ 8.172 60 and m / m 1 ⩾ λ 2 ≈ 12.917 43, respectively. For m / m 1 ≲ λ 1 and m / m 1 ≲ λ 2 , the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied, and it is shown that a two-hump structure in the mass-ratio dependences of the cross sections is connected with the rise of the bound states.

arXiv:physics/0610261•Journal of Physics B Atomic Molecular and Optical Physics•Mar 15, 2007•125 citations

Three two-component fermions with contact interactions: correct formulation and energy spectrum

O. I. Kartavtsev, A. V. Malykh

Properties of two identical particles of mass $m$ and a distinct particle of mass $m_1$ in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum $L$ and parity $P$. For the unambiguous formulation of the problem in the interval $μ_r(L^P) < m/m_1 \le μ_c(L^P)$ ($μ_r(1^-) \approx 8.619$ and $μ_c(1^-) \approx 13.607$, $μ_r(2^+) \approx 32.948$ and $μ_c(2^+) \approx 38.630$,~etc.) in each $L^P$ sector an additional parameter $b$ determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on $m/m_1$ and $b$ in the sector of angular momentum and parity $L^P$ is calculated for $L \le 5$ and analysed with the aid of a simple model. A number of the bound states for each $L^P$ sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters $m/m_1$ and $b$.

arXiv:1904.04943•Apr 9, 2019•1 citation

Bound states and scattering lengths of three two-component particles with zero-range interactions under one-dimensional confinement

O. I. Kartavtsev, A. V. Malykh, S. A. Sofianos

The universal three-body dynamics in ultracold binary gases confined to one-dimensional motion is studied. The three-body binding energies and the (2 + 1)-scattering lengths are calculated for two identical particles of mass m and a different particle of mass m 1, whose interaction is described in the low-energy limit by zero-range potentials. The critical values of the mass ratio m/m 1 at which three-body states occur and the (2 + 1)-scattering length vanishes are determined for both zero and infinite interaction strength λ1 of the identical particles. A number of exact results are listed and asymptotic dependences for both m/m 1 → ∞ and λ1 → −∞ are derived. Combining the numerical and analytic results, we deduce a schematic diagram showing the number of three-body bound states and the sign of the (2 + 1)-scattering length in the plane of the mass ratio and the interaction-strength ratio. The results provide a description of the homogeneous and mixed phases of atoms and molecules in dilute binary quantum gases.

arXiv:0808.2704•Journal of Experimental and Theoretical Physics•Mar 1, 2009•41 citations

Scaling and universality in two dimensions: three-body bound states with short-ranged interactions

F F Bellotti, T. Frederico, M T Yamashita +3 more

The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as a function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22, we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio, and we map out an energy–mass phase diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.

arXiv:1107.3017•Journal of Physics B Atomic Molecular and Optical Physics•Sep 26, 2011•29 citations

Three-body scattering area for particles with infinite or zero scattering length in two dimensions

Junjie Liang, Shina Tan

We derive the asymptotic expansions of the wave function of three particles having equal mass with finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero energy and zero orbital angular momentum, from which a three-body parameter $D$ is defined. The dimension of $D$ is length squared, and we call $D$ three-body scattering area. We find that the ground-state energy per particle of a zero-temperature dilute Bose gas with these interactions is approximately $\frac{{\ensuremath{\hbar}}^{2}D}{6m}{\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density of the bosons and $m$ is the mass of each boson. Such a zero-temperature Bose gas is stable at $D>0$ in the thermodynamic limit, and metastable at $D<0$ in the isotropic harmonic trap if the number of bosons is less than ${N}_{\mathrm{cr}}\ensuremath{\approx}3.6413\sqrt{\frac{\ensuremath{\hbar}}{m\ensuremath{\omega}|D|}\phantom{{}^{1}}}$, where $\ensuremath{\omega}$ is the angular frequency of the harmonic trap. If the two-body interaction supports bound states, $D$ typically acquires a negative imaginary part, and we find the relation between this imaginary part and the amplitudes of the pair-boson production processes. We derive a formula for the three-body recombination rate constant of the many-boson system in terms of the imaginary part of $D$.

arXiv:2402.02202•Physical Review A•Jun 24, 2024•2 citations
A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and a third particle of mass m 1 in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio m/m 1 and the total angular momentum L. It is found that the number of vibrational states is determined by the functions L c(m/m 1) and L b(m/m 1). Explicitly, if the two-body scattering length is positive, the number of states is finite for L c(m/m 1) ≤ L ≤ L b(m/m 1), zero for L > L b(m/m 1), and infinite for L < L c(m/m 1). If the two-body scattering length is negative, the number of states is zero for L ≥ L c(m/m 1) and infinite for L < L c(m/m 1). For the finite number of vibrational states, all the binding energies are described by the universal function ɛLN(m/m 1) = (ξ, η), where ξ = (N − 1/2)/√L(L + 1), η = √m/[m 1 L(L + 1)], and N is the vibrational quantum number. This scaling dependence is in agreement with the numerical calculations for L > 2 and only slightly deviates from those for L = 1, 2. The universal description implies that the critical values L c(m/m 1) and L b(m/m 1) increase as 0.401 √m/m 1 and 0.563 √m/m 1, respectively, while the number of vibrational states for L ≥ L c(m/m 1) is within the range N ≤ N max ≈ 1.1√L(L + 1) + 1/2.
The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass mmm and a particle of the mass m1m_1m1​ with the zero-range two-body interaction in the states of the total angular momentum L=1 are considered. Using the boundary condition model for the s-wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependencies of the three-body binding energies on the mass ratio m/m1m/m_1m/m1​ for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m/m1gelambda1approx8.17260m/m_1 \\ge \\lambda_1 \\approx 8.17260m/m1​gelambda1​approx8.17260 and m/m1gelambda2approx12.91743m/m_1 \\ge \\lambda_2 \\approx 12.91743m/m1​gelambda2​approx12.91743, respectively. For m/m1altlambda1m/m_1 \\alt \\lambda_1m/m1​altlambda1​ and m/m1altlambda2m/m_1 \\alt \\lambda_2m/m1​altlambda2​, the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied and it is shown that a two-hump structure in the mass-ratio dependencies of the cross sections is connected with arising of the bound states.
The universal three-body dynamics in ultra-cold binary Fermi and Fermi–Bose mixtures is studied. Two identical fermions of mass m and a particle of mass m 1 with the zero-range two-body interaction in states of total angular momentum L = 1 are considered. Using the boundary condition model for the s -wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependences of the three-body binding energies on the mass-ratio m / m 1 for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m / m 1 ⩾ λ 1 ≈ 8.172 60 and m / m 1 ⩾ λ 2 ≈ 12.917 43, respectively. For m / m 1 ≲ λ 1 and m / m 1 ≲ λ 2 , the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied, and it is shown that a two-hump structure in the mass-ratio dependences of the cross sections is connected with the rise of the bound states.
Properties of two identical particles of mass mmm and a distinct particle of mass m1m_1m1​ in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum LLL and parity PPP. For the unambiguous formulation of the problem in the interval μr(LP)<m/m1≤μc(LP)μ_r(L^P) < m/m_1 \le μ_c(L^P)μr​(LP)<m/m1​≤μc​(LP) (μr(1−)≈8.619μ_r(1^-) \approx 8.619μr​(1−)≈8.619 and μc(1−)≈13.607μ_c(1^-) \approx 13.607μc​(1−)≈13.607, μr(2+)≈32.948μ_r(2^+) \approx 32.948μr​(2+)≈32.948 and μc(2+)≈38.630μ_c(2^+) \approx 38.630μc​(2+)≈38.630,~etc.) in each LPL^PLP sector an additional parameter bbb determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on m/m1m/m_1m/m1​ and bbb in the sector of angular momentum and parity LPL^PLP is calculated for L≤5L \le 5L≤5 and analysed with the aid of a simple model. A number of the bound states for each LPL^PLP sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters m/m1m/m_1m/m1​ and bbb.
The universal three-body dynamics in ultracold binary gases confined to one-dimensional motion is studied. The three-body binding energies and the (2 + 1)-scattering lengths are calculated for two identical particles of mass m and a different particle of mass m 1, whose interaction is described in the low-energy limit by zero-range potentials. The critical values of the mass ratio m/m 1 at which three-body states occur and the (2 + 1)-scattering length vanishes are determined for both zero and infinite interaction strength λ1 of the identical particles. A number of exact results are listed and asymptotic dependences for both m/m 1 → ∞ and λ1 → −∞ are derived. Combining the numerical and analytic results, we deduce a schematic diagram showing the number of three-body bound states and the sign of the (2 + 1)-scattering length in the plane of the mass ratio and the interaction-strength ratio. The results provide a description of the homogeneous and mixed phases of atoms and molecules in dilute binary quantum gases.
The momentum space zero-range model is used to investigate universal properties of three interacting particles confined to two dimensions. The pertinent equations are first formulated for a system of two identical and one distinct particle and the two different two-body subsystems are characterized by two-body energies and masses. The three-body energy in units of one of the two-body energies is a universal function of the other two-body energy and the mass ratio. We derive convenient analytical formulae for calculations of the three-body energy as a function of these two independent parameters and exhibit the results as universal curves. In particular, we show that the three-body system can have any number of stable bound states. When the mass ratio of the distinct to identical particles is greater than 0.22, we find that at most two stable bound states exist, while for two heavy and one light mass an increasing number of bound states is possible. The specific number of stable bound states depends on the ratio of two-body bound state energies and on the mass ratio, and we map out an energy–mass phase diagram of the number of stable bound states. Realizable systems of both fermions and bosons are discussed in this framework.
We derive the asymptotic expansions of the wave function of three particles having equal mass with finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero energy and zero orbital angular momentum, from which a three-body parameter DDD is defined. The dimension of DDD is length squared, and we call DDD three-body scattering area. We find that the ground-state energy per particle of a zero-temperature dilute Bose gas with these interactions is approximately $\frac{{\ensuremath{\hbar}}^{2}D}{6m}{\ensuremath{\rho}}^{2}$, where $\ensuremath{\rho}$ is the number density of the bosons and mmm is the mass of each boson. Such a zero-temperature Bose gas is stable at D>0D>0D>0 in the thermodynamic limit, and metastable at D<0D<0D<0 in the isotropic harmonic trap if the number of bosons is less than ${N}_{\mathrm{cr}}\ensuremath{\approx}3.6413\sqrt{\frac{\ensuremath{\hbar}}{m\ensuremath{\omega}|D|}\phantom{{}^{1}}}$, where $\ensuremath{\omega}$ is the angular frequency of the harmonic trap. If the two-body interaction supports bound states, DDD typically acquires a negative imaginary part, and we find the relation between this imaginary part and the amplitudes of the pair-boson production processes. We derive a formula for the three-body recombination rate constant of the many-boson system in terms of the imaginary part of DDD.