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The threshold recombination coefficient for three bosons whose two-body scattering length a is positive and large compared with the nominal range of the potentials is shown to proceed via a crossing of hyperspherical adiabatic potential curves at R ഠ 2.6a. The hidden crossing theory gives a simple expression for the rate in terms of the scattering length and a phase D. The phase depends upon the details of the interaction, and varies from species to species, however an upper limit to the rate which depends only upon the scattering length emerges from the hidden crossing theory.One of the main issues in Bose-Einstein condensation is the systematics of three-body recombination rates. A Bose condensate of some atoms, say B, is not the lowest energy state but a metastable state, where the atoms are so far apart that they interact weakly. Such systems can fall into lower energy states, i.e., clusters of atoms, by recombination. Usually the main loss is due to recombination to dimers, B 2 . Such recombination cannot occur by simple two-body reactions. Something has to carry away the extra energy and momentum. At the very low energies of Bose condensates, this will usually be a third atom. The important process is thusAn experimental treatment of recombination in a Bose condensate is described by Inouye et al. in [1], where they have succeeded in adjusting the two-body scattering length, a, by adjusting the magnetic field near a Feshbach resonance. This follows the normal trends in the theories of Bose condensates, where the only parameter from the two-body potential is indeed the scattering length. Several papers have dealt theoretically with the connection between scattering length and the recombination rate [2,3] based upon numerical calculations for a series of similar potentials with different scattering length. Fedichev et al. [4] propose an a 4 law. The aim of this paper is to investigate this dependence further and to give a simple intuitive understanding of the fundamental process of recombination.We do this by approximating the two-body interaction by a zero-range potential which is defined by the scattering length alone. It has been known since 1935 that the corresponding three-body system will have infinitely many bound states [5]. The same problem occurs in conventional coupled channel calculations for three-body states where the same singularity occurs at the origin.Many attempts have been made to work around this problem by adding some sort of cutoff to the potential. Bedaque et al. have added a three-body interaction that acts as a renormalization of this singularity [6]. This paper shows that such cutoffs contribute to the transition probability as significantly as they do to binding energies and elastic atom-dimer scattering.We clarify the theory by treating the problem in hyperspherical coordinates and applying the hidden crossing theory [7]. Hyperspherical hidden crossing theory has only been applied to three-body systems with Coulomb interactions [8]; thus another aim of this paper is to show that...

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We review topics of current interest in the physics of electronic, atomic and molecular scattering in the vicinity of thresholds. Starting from phase space arguments, we discuss the modifications of the Wigner law that are required to deal with scattering by Coulomb, dipolar and dispersion potentials, as well as aspects of threshold behaviour observed in ultracold atomic collisions. We employ the tools of quantum defect and semiclassical theories to bring out the rich variety of threshold behaviours. The discussion is then turned to recent progress in understanding threshold behaviour of many-body break-ups into both charged and neutral species, including both Wannier double ionization and three-body recombination in ultracold gases. We emphasize the dominant role that hyperspherical coordinate methods have played in understanding these problems. We assess the effects of external fields on scattering, and the corresponding modification of phase space that alters the Wigner law. Threshold laws in low dimensions and examples of their applications to specific collision processes are discussed.