Regularization of a three-body problem with zero range potentials

D. V. Fedorov, A. S. Jensen

Published Jan 1, 2001• 28 citations

Abstract

We propose a coordinate-space regularization of the three-body problem with zero-range potentials. We include the effective range and the shape parameter in the boundary condition of the zero-range potential. The proposed extended zero-range model is tested against atomic helium trimers and is shown to provide an adequate quantitative description of these systems.

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Regularization of a three-body problem with zero range potentials

D. V. Fedorov, A. S. Jensen

Published Jan 1, 2001• 28 citations

OpenAlex

Abstract

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D V Fedorov, A S Jensen

arXiv:quant-ph/0106039Journal of Physics A Mathematical and GeneralJul 20, 200145 citations

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Esben Nielsen, D. V. Fedorov, A. S. Jensen

The Faddeev equations for the atomic helium-trimer systems are solved numerically with high accuracy both for the most sophisticated realistic potentials available and for simple phenomenological potentials. An efficient numerical procedure is described. The large-distance asymptotic behaviour, crucial for weakly bound three-body systems, is described almost analytically for arbitrary potentials. The Efimov effect is especially considered. The geometric structures of the bound states are quantitatively investigated. The accuracy of the schematic models and previous computations is comparable, i.e. within 20% for the spatially extended states and within 40% for the smaller -trimer ground state.The Faddeev equations for the atomic helium-trimer systems are solved numerically with high accuracy both for the most sophisticated realistic potentials available and for simple phenomenological potentials. An efficient numerical procedure is described. The large-distance asymptotic behaviour, crucial for weakly bound three-body systems, is described almost analytically for arbitrary potentials. The Efimov effect is especially considered. The geometric structures of the bound states are quantitatively investigated. The accuracy of the schematic models and previous computations is comparable, i.e. within 20% for the spatially extended states and within 40% for the smaller -trimer ground state.

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\sigma < {{\sigma }_{c}}

, the additional boundary conditions of two types should be imposed at the triple-collision point for

{{\sigma }_{c}} \leqslant \sigma < {{\sigma }_{e}}

and

{{\sigma }_{e}} < \sigma < {{\sigma }_{r}}

, and the problem is regularized for

\sigma \geqslant {{\sigma }_{r}}

. Critical values

{{\sigma }_{c}} < {{\sigma }_{e}} < {{\sigma }_{r}}

separating different alternatives are determined both for a two-component three-body system and for three identical bosons.

The two-body scattering length a for

{}^{4}\mathrm{He}

atoms is much larger than their effective range

{r}_{s}.

As a consequence, low-energy few-body observables have universal characteristics that are independent of the interaction potential. Universality implies that, up to corrections suppressed by

{r}_{s}/a,

all low-energy three-body observables are determined by a and a three-body parameter ${\ensuremath{\Lambda}}_{*}.$ We give simple expressions in terms of a and ${\ensuremath{\Lambda}}_{*}$ for the trimer binding-energy equation, the atom-dimer scattering phase shifts, and the rate for three-body recombination at threshold. We determine ${\ensuremath{\Lambda}}_{*}$ for several

{}^{4}\mathrm{He}

potentials from the calculated binding energy of the excited state of the trimer and use it to obtain the universality predictions for the other low-energy observables. We also use the calculated values for one potential to estimate the effective range corrections for the other potentials.