Behaviour of adiabatic potentials in the scattering of three particles

Klemm, Anthony D.; Larsen, Sigurd Yves

doi:10.1007/bf01091702

Cited by 5 publications

(23 citation statements)

References 12 publications

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“…Among other results, one has been able to show that in the semi-classical limit, for repulsive forces, one recovers the known classical results from an eigenphase shift formulation [4]. See, also, the calculation of the third virial in two dimensions, for repulsive step potentials [5,6,7], and -as a bonus but important -a formulation of the second virial for particles subject to anisotropic forces [8], i.e., for a Helium atom and a Hydrogen molecule.…”

Section: Introductionsupporting

confidence: 53%

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A generalized Uhlenbeck and Beth formula for the third cluster coefficient

2016

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Relatively recently (A. Amaya-Tapia, S. Y. Larsen, M. Lassaut. Ann. Phys.,306 (2011) 406), we presented a formula for the evaluation of the third Bose fugacity coefficient -leading to the third virial coefficient -in terms of three-body eigenphase shifts, for particles subject to repulsive forces. An analytical calculation for a 1-dim. model, for which the result is known, confirmed the validity of this approach. We now extend the formalism to particles with attractive forces, and therefore must allow for the possibility that the particles have bound states. We thus obtain a true generalization of the famous formula of Uhlenbeck and Beth (G.E. Uhlenbeck, E. Beth. Physica,3 (1936) 729; E. Beth, G.E. Uhlenbeck. ibid, 4 (1937) 915) (and of Gropper (L. Gropper. Phys. Rev. 50 (1936) 963; ibid 51 (1937) 1108)) for the second virial. We illustrate our formalism by a calculation, in an adiabatic approximation, of the third cluster in one dimension, using McGuire's model as in our previous paper, but with attractive forces.

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Section: Introductionsupporting

confidence: 53%

“…For the third cluster in 2-dimension [5,7], analytical and numerical data considerations led to the vanishing of the leading part of the first Born contribution to the authors' equivalent to our G(k), Eq. (24).…”

Section: Perspectivementioning

confidence: 88%

A generalized Uhlenbeck and Beth formula for the third cluster coefficient

2016

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“…. Linear combinations of the basic harmonics can then be formed [3] to obtain irreducible bases, adapted to the symmetries of the physical problems [1,2].…”

Section: The Kl Hyperspherical Coordinate Systemmentioning

confidence: 99%

“…In order for the λ in the following equation, to be the same as that used by [1] and [2], we need to add a term 3/(4ρ 2 ). That is because the angular part of the Laplacian has eigenvalues −N (N + 2) and not the −[(N + 1) 2 − 1/4].…”

Section: The Adiabatic Basismentioning

confidence: 99%

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Analytical Expressions for a Hyperspherical Adiabatic Basis Suitable for a particular Three Particle Problem in 2 Dimensions

Lassaut,

Amaya-Tapia,

Klemm

et al. 2019

Preprint

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For a particular case of three-body scattering in two dimensions, and matching analytical expressions at a transition point, we obtain accurate solutions for the adiabatic basis and potential. We find analytical expressions for the respective, asymptotic, inverse logarithmic and inverse power potential behaviors, that arise as functions of the radial coordinate. The model that we consider is that of two particles interacting with a repulsive step potential, a third particle acting as a spectator. The model is simple but gives insight, as the 2-body interaction is long ranged in hyperspherical coordinates. The fully interacting 3-body problem is known, numerically, to yield similar behaviors that we can now begin to understand. That, clearly, is the ultimate aim.

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