A quantum-mechanical three-body problem for two identical fermions of mass m and a distinct particle of mass m 1 in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval μ r < m / m 1 ≤ μ c ( μ r ≈ 8.619 and μ c ≈ 13.607 ) 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. The dependence of the bound-state energies on m/m 1 and b in the sector of angular momentum and parity L P = 1 − is calculated and analysed with the aid of a simple model.
A quantum-mechanical three-body problem for two identical fermions of mass m and a distinct particle of mass m 1 in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval μ r < m / m 1 ≤ μ c ( μ r ≈ 8.619 and μ c ≈ 13.607 ) 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. The dependence of the bound-state energies on m/m 1 and b in the sector of angular momentum and parity L P = 1 − is calculated and analysed with the aid of a simple model.
Properties of two identical particles of mass m and a distinct particle of mass m1 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 μr(LP)<m/m1≤μc(LP) (μr(1−)≈8.619 and μc(1−)≈13.607, μr(2+)≈32.948 and μc(2+)≈38.630,~etc.) in each LP 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/m1 and b in the sector of angular momentum and parity LP is calculated for L≤5 and analysed with the aid of a simple model. A number of the bound states for each LP sector is analysed and presented in the form of `phase diagrams' in the plane of two parameters m/m1 and b.
Overview of the recent advances in description of the few two-component fermions is presented. Themodel of zero-range interaction is generally considered to discuss the principal aspects of the few-body dynamics. Particular attention is paid to detailed description of two identical fermions of mass m and a distinct particle of mass m 1: it turns out that two L P = 1− three-body bound states emerge if mass ratio m/m 1 increases up to the critical value µ c ≈ 13.607, abovewhich the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, lowdimensional results, the four-body energy spectrum, crossover of the energy spectra for m/m 1 near µ c , and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.
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.
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 m1 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/m1 for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at m/m1gelambda1approx8.17260 and m/m1gelambda2approx12.91743, respectively. For m/m1altlambda1 and m/m1altlambda2, 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.
We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for ↑ and ↓ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.
We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for ↑ and ↓ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93 , 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.