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arXiv:nucl-th/9508024 · Physical Review A

Weakly bound three-body systems with no bound subsystems

J. Goy, J. M. Richard, Sonia Fleck

Published Nov 1, 1995• 43 citations
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Abstract

We investigate the domain of coupling constants which achieve binding for a three-body system while none of the two-body subsystems is bound. We derive some general properties of the shape of the domain and rigorous upper bounds on its size, using a Hall-Post decomposition of the Hamiltonian. Numerical illustrations are provided in the case of a Yukawa potential, using a simple variational method.

DOI
10.1103/physreva.52.3511

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We investigate the domain of coupling constants which achieve binding for a three-body system while none of the two-body subsystems is bound. We derive some general properties of the shape of the domain and rigorous upper bounds on its size, using a Hall-Post decomposition of the Hamiltonian. Numerical illustrations are provided in the case of a Yukawa potential, using a simple variational method.
We estimate the ratio R=g3/g2{R=g}_{3}{/g}_{2}R=g3​/g2​ of the critical coupling constants g2{g}_{2}g2​ and g3{g}_{3}g3​ that are required to achieve binding of two or three bosons, respectively, with a short-range interaction, and examine how this ratio depends on the shape of the potential. Simple monotonous potentials give $R\ensuremath{\simeq}0.8.$ A wide repulsive core pushes this ratio close to R=1.R=1.R=1. On the other hand, for an attractive well protected by an external repulsive barrier, the ratio approaches the rigorous lower bound R=2/3.R=2/3.R=2/3. We also present results for N=4N=4N=4 bosons, sketch the extension to N>4,N>4,N>4, and discuss various consequences.
A method to study weakly bound three-body quantum systems in two dimensions is formulated in coordinate space for short-range potentials. Occurrences of spatially extended structures (halos) are investigated. Borromean systems are shown to exist in two dimensions for a certain class of potentials. An extensive numerical investigation shows that a weakly bound two-body state gives rise to two weakly bound three-body states, a reminiscence of the Efimov effect in three dimensions. The properties of these two states in the weak binding limit turn out to be universal.
Abstract We study a heavy–heavy–light three-body system confined to one space dimension in the regime where an excited state in the heavy–light subsystems becomes weakly bound. The associated two-body system is characterized by (i) the structure of the weakly-bound excited heavy–light state and (ii) the presence of deeply-bound heavy–light states. The consequences of these aspects for the behavior of the three-body system are analyzed. We find a strong indication for universal behavior of both three-body binding energies and wave functions for different weakly-bound excited states in the heavy–light subsystems.
A method for systematically improving Hall–Post–Stenschke lower bounds to the bound state eigenvalues of three-body Schrödinger equations is given. The improved bounds are obtained by solving coupled one variable integral equations; the bounds get better as the number of coupled equations is increased. The method generates explicit wave functions which can be used to obtain complementary upper bounds via the Rayleigh–Ritz variational method. Either identical or nonidentical particles can be handled. The method is illustrated by calculations for three identical particles bound by Hooke’s law forces. A brief discussion of extensions to more than three particles is given.
Alternative lower bounds for the binding energy of a quantum-mechanical system of interacting particles are presented. These bounds are expressed in terms of two-particle quantities and improve the conventional bounds of the Hall-Post type. They are constructed by considering not only the energy in the two-particle system, but also the structure of the pair wave function. We apply the formal results to various numerical examples, and show that in some cases dramatic improvement over the existing bounds is reached.
Variational principles for lower bounds to the energy, or free energy for T > 0°, of many-body systems are obtained in a form requiring density matrix minimization subject to certain model restrictions. The latter restrict the domain in which the density matrices can vary, and only utilize the energy—or free energy—for the model Hamiltonian HM. Increasingly accurate bounds are obtained as the model system begins to resemble the system of interest, and the behavior of the error as H − HM approaches zero is shown by two examples based upon the Ising model. Coupling the lower bound principle for the free energy with the standard Gibbs-Bogoliubov upper bound principle results in bounds on generalized susceptibility as well.