Physics of Atomic Nuclei

Differential equations in momentum space for the three-body problem in the case of pointlike pair interactions

Ф. М. Пеньков, Werner Sandhas

Published Apr 1, 2014• 0 citations

Abstract

Corrects schemes for solving equations of three-body dynamics for systems governed by a zerorange two-body interaction are considered. Correlations between spectral features of a three-boson system are obtained. The results are compared with the results obtained by calculating the spectra and scattering lengths in the system of three helium atoms with realistic two-body interaction potentials.

DOI: 10.1134/s1063778814040085

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Physics of Atomic Nuclei

Differential equations in momentum space for the three-body problem in the case of pointlike pair interactions

Ф. М. Пеньков, Werner Sandhas

Published Apr 1, 2014• 0 citations

Publisher

Abstract

DOI: 10.1134/s1063778814040085

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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

\ln^3(B/a)

at large hyperradius

B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}

. At order

B^{-2}\ln^{-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.