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arXiv:nucl-th/0504049 · Physical Review A

Three bosons in one dimension with short-range interactions: Zero-range potentials

Nirav P. Mehta, James R. Shepard

Published Sep 28, 2005• 16 citations
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Abstract

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.

DOI
10.1103/physreva.72.032728

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We consider the three-boson problem with \ensuremathδ\ensuremath{\delta}\ensuremathδ-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 SSS-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+21+21+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 study the wave function φ(3)φ^{(3)}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 aaa. We derive asymptotic expansions of φ(3)φ^{(3)}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 ln⁡3(B/a)\ln^3(B/a)ln3(B/a) at large hyperradius B=(s12+s22+s32)/2B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}B=(s12​+s22​+s32​)/2​. At order B−2ln⁡−3(B/a)B^{-2}\ln^{-3}(B/a)B−2ln−3(B/a), we identify a three-body parameter DDD 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, DDD 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 DDD in terms of φ(3)φ^{(3)}φ(3) and the changes of the two-body and three-body potentials. We also study the effects of DDD and φ(3)φ^{(3)}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.
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We use the hyperspherical adiabatic expansion method to formulate the angular Faddeev-like equation for N identical bosons including effects of two-body correlations. Eigenvalues and eigenfunctions determine the effective potentials in the coupled set of radial equations. The zero-range approximation is implemented on the free solutions as a boundary condition obtained by using the effective-range expansion of the two-body phase shift. This results in a new angular eigenvalue equation expressed as a transcendental equation in the index of the Jacobi functions corresponding to solutions for non-interacting particles. The collapse for attractive potentials is avoided by including scattering length, effective range and shape-parameter terms in the effective-range expansion of the phase shift. Correspondingly the hyperradial dependence in the eigenvalue equation is either a polynomial of first, second, or fourth order. Two-body correlations are included. The arithmetic eigenvalue equation allows derivation of analytic properties such as for example simple scaling relations for the particle number dependence. Excellent agreement with finite-range numerical results is obtained. The couplings between adiabatic channels are discussed. Including only the lowest adiabatic potential is shown to be sufficient for most applications on dilute systems.
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We study the three-body problem in one dimension for both zero- and finite-range interactions using the adiabatic hyperspherical approach. Particular emphasis is placed on the threshold laws for recombination, which are derived for all combinations of the parity and exchange symmetries. For bosons, we provide a numerical demonstration of several universal features that appear in the three-body system, and discuss how certain universal features in three dimensions are different in one dimension. We show that the probability for inelastic processes vanishes as the range of the pairwise interaction is taken to zero and demonstrate numerically that the recombination threshold law manifests itself for large scattering length.
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