We consider non-relativistic systems in quantum mechanics interacting through the Coulomb potential, and discuss the existence of bound states which are stable against spontaneous dissociation into smaller atoms or ions. We review the studies that have been made of specific mass configurations and also the properties of the domain of stability in the space of masses or inverse masses. These rigorous results are supplemented by numerical investigations using accurate variational methods. A section is devoted to systems of three arbitrary charges and another to molecules in a world with two space-dimensions.

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In view of current interest in the trapping of antihydrogen () atoms at very low temperatures, we have carried out a calculation of s-wave hydrogen–antihydrogen scattering at very low energies, using the Kohn variational method, taking into account rearrangement scattering into the three channels that contain positronium in its ground state and lie closest to threshold. We find that our values for the elastic cross section are in good agreement with the values obtained by Jonsell et al (Jonsell et al 2001 Phys. Rev. A 64 052712) using a distorted wave approximation. However, our values for the total rearrangement cross section are much larger than their values. In particular, the largest such cross section is for the N = 23 s-state of protonium and positronium in its ground state, a channel that was estimated to make a negligible contribution by Jonsell et al. As a consequence of our much larger values for the total rearrangement cross section, we predict that cooling of by cold H would be considerably less efficient than was found to be the case by Jonsell et al.

We present calculations of et-H2 scattering below the positronium formation threshold at 8.63 eV for the Xi, X : , II, and II s mmetries using the generalized Kohn method. Mixing of the two lowest partial waves IS allowed for in the Xi, X : and IIu symmetries, using a two channel K-matrix. Comparisons with Kohn calculations of the lowest partial waves of these symmetries show that mixing of partial waves has a relatively small effect on the contributions to the total scattering cross section at the energies considered. The IT, calculation includes only the lowest partial wave. As well as separable short-range correlation functions, the trial functions used in these calculations include Hylleraas-type functions containing the positron-electron distance as a linear factor and functions appropriate for taking into account long-range polarization of the hydrogen molecule. The sum of the contributions to the total scattering cross section from these calculations accounts for experimental values up to incident energies of about 5-6 eV, the main contribution coming from the Z: symmetry below -2 eV and the IIu symmetry above -2 eV. This is a major step forward towards our aim of carrying out accurate calculations for e+-H, scattering up to the positronium formation threshold. Inclusion of the Hylleraastype functions, which take into account short-range interactions between the positron and target electrons, is of great importance in achieving the agreement with experiment. Comparisons are made with recent R-matrix calculations.

We have carried out an analysis of singularities in Kohn variational calculations for low energy (e + − H 2 ) elastic scattering. Provided that a sufficiently accurate trial wavefunction is used, we argue that our implementation of the Kohn variational principle necessarily gives rise to singularities which are not spurious. We propose two approaches for optimizing a free parameter of the trial wavefunction in order to avoid anomalous behaviour in scattering phase shift calculations, the first of which is based on the existence of such singularities. The second approach is a more conventional optimization of the generalized Kohn method. Close agreement is observed between the results of the two optimization schemes; further, they give results which are seen to be effectively equivalent to those obtained with the complex Kohn method. The advantage of the first optimization scheme is that it does not require an explicit solution of the Kohn equations to be found. We give examples of anomalies which cannot be avoided using either optimization scheme but show that it is possible to avoid these anomalies by considering variations in the nonlinear parameters of the trial function.