Exact solutions for three identical bosons interacting via zero-range $s$-wave potentials are derived. The solutions are contour integrals over a product of hyperradial Bessel functions times angular functions weighted by a coefficient. The product function is a solution of the free-particle Schr\"odigner equation and the weight function is chosen to satisfy the zero-range boundary conditions. Scattering matrix elements for boson-dimer elastic scattering, breakup of a dimer into three particles and the time-reversed recombination process are derived. For vanishing total energy $E$, these quantities are given as closed-form functions of the two-body $s$-wave scattering length $a$ and a three-body renormalization constant ${R}_{0}$. The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on ${R}_{0}$ of the order of 2% are noted. Comparison with the hidden-crossing theory finds similar agreement with the functional dependence upon $a$ and ${R}_{0}$.
Exact solutions for three identical bosons interacting via zero-range s-wave potentials are derived. The solutions are contour integrals over a product of hyperradial Bessel functions times angular functions weighted by a coefficient. The product function is a solution of the free-particle Schr"odigner equation and the weight function is chosen to satisfy the zero-range boundary conditions. Scattering matrix elements for boson-dimer elastic scattering, breakup of a dimer into three particles and the time-reversed recombination process are derived. For vanishing total energy E, these quantities are given as closed-form functions of the two-body s-wave scattering length a and a three-body renormalization constant R0. The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on R0 of the order of 2% are noted. Comparison with the hidden-crossing theory finds similar agreement with the functional dependence upon a and R0.
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 $\ensuremath{\rho}$ is studied, and the universal logarithmic dependence $\ensuremath{\sim}{\mathrm{ln}}^{3}\phantom{\rule{0.2em}{0ex}}\ensuremath{\rho}$ in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the (2+1)-scattering length are calculated.
We study the wave function φ(3) of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length a. We derive asymptotic expansions of φ(3) in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as ln3(B/a) at large hyperradius B=(s12+s22+s32)/2. At order B−2ln−3(B/a), we identify a three-body parameter D with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, D acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of D in terms of φ(3) and the changes of the two-body and three-body potentials. We also study the effects of D and φ(3) on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.
We consider the three-boson problem with $\ensuremath{\delta}$-function interactions in one spatial dimension. Three different approaches are used to calculate the phase shifts, which we interpret in the context of the effective range expansion, for the scattering of one free particle off a bound pair. We first follow a procedure outlined by McGuire in order to obtain an analytic expression for the desired S-matrix element. This result is then compared to a variational calculation in the adiabatic hyperspherical representation, and to a numerical solution to the momentum-space Faddeev equations. We find excellent agreement with the exact phase shifts, and comment on some of the important features in the scattering and bound-state sectors. In particular, we find that the 1+2 scattering length is divergent, marking the presence of a zero-energy resonance which appears as a feature when the pairwise interactions are short range. Finally, we consider the introduction of a three-body interaction, and comment on the cutoff dependence of the coupling.
We use the zero-range approximation to study a system of two identical bosons interacting resonantly with a third particle. The method is derived from effective field theory. It reduces the three-body problem to an integral equation which we then solve numerically. We also develop an alternative approach which gives analytic solutions of the integral equation in coordinate representation in the limit of vanishing total energy. The atom-dimer scattering length, the rates of atom-dimer relaxation, and the three-body recombination to shallow and to deep molecular states are calculated either analytically or numerically with a well-controlled accuracy for various energies as functions of the mass ratio, scattering length, and three-body parameter. We discuss in detail the relative positions of the recombination loss peaks, which in the universal limit depend only on the mass ratio. Our results have implications for ongoing and future experiments on Bose-Bose and Bose-Fermi atomic mixtures.
The Schrödinger equation, in hyperspherical coordinates, is solved in closed form for a system of three particles on a line, interacting via pair delta functions. This is for the case of equal masses and potential strengths. The interactions are replaced by appropriate boundary conditions. This leads then to requiring the solution of a free-particle Schrödinger equation subject to these boundary conditions. A generalized Kontorovich–Lebedev transformation is used to write this solution as an integral involving a product of Bessel functions and pseudo-Sturmian functions. The coefficient of the product is obtained from a three-term recurrence relation, derived from the boundary condition. The contours of the Kontorovich–Lebedev representation are fixed by the asymptotic conditions. The scattering matrix is then derived from the exact solution of the recurrence relation. The wavefunctions that are obtained are shown to be equivalent to those derived by McGuire. The method can clearly be applied to a larger number of particles and hopefully might be useful for unequal masses and potentials.