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 ≥ λ 1 ≈ 8.17260 and m/m 1 ≥ λ 2 ≈ 12.91743, 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 dependencies of the cross sections is connected with arising of the bound states.

Universal low-energy properties are studied for three identical bosons confined in two dimensions. The short-range pairwise interaction in the low-energy limit is described by means of the boundary condition model. The wave function is expanded in a set of eigenfunctions on the hypersphere, and the system of hyperradial equations is used to obtain analytical and numerical results. Within the framework of this method, exact analytical expressions are derived for the eigenpotentials and the coupling terms of hyperradial equations. The derivation of the coupling terms is generally applicable to a variety of three-body problems provided the interaction is described by the boundary condition model. The asymptotic form of the total wave function at a small and a large hyperradius is studied, and the universal logarithmic dependence ϳln 3 in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the ͑2+1͒-scattering length are calculated.

A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and the 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

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

The 0 + states of 12 C are considered within the framework of the microscopic three-α-cluster model. The main attention is paid to accurate calculation of the width of the extremely narrow near-threshold 0 + 2 state which plays a key role in stellar nucleosynthesis. It is shown that the

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.

Overview of the recent advances in description of the few two-component fermions is presented.The model 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, above which the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, low-dimensional 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.PACS numbers:

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 interval619 and µ c (1 − ) ≈ 13.607, µ r (2 + ) ≈ 32.948 and µ c (2 + ) ≈ 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 ≤ 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.