Exact Results for the Scattering of Three Charged Particles

Abstract

It is notoriously difficult to obtain reliable results for quantum mechanical scattering problems. Since they involve complicated interference phenomena of waves any simple uncontrolled approximation is not worth more than the weather forecast. However, for two body problems with central forces the computer can be used to calculate the phase shifts so one may consider such problems as solved, and I will turn to the three body case. I shall study in particular three charged particles, p, μ^±, e^± in various combinations. Since between these particles at low energies the Coulomb potential is dominant the corresponding Schrödinger equation should give a realistic and mathematically managable description. The long range of the 1/r potential causes some difficulty for scattering theory but the situation is well understood⁽¹⁾. I shall concentrate on the simplest situation, namely scattering of a charged particle on a neutral atom with energy below ionisation threshold. There the 1/r potential is screened and does not make trouble. The existence and asymptotic completness of the Möller-operators for this problem can be shown⁽²⁾ so that scattering theory works in it usual form. Thus the stage is set for making a reliable calculation of the scattering length. Here the difficulty has the same origin as in the two body case, namely the evaluation of a term V 1/H V. 1/H can become arbitrarily large if there is a zero-energy bound state and one has to know how far one is from this situation. Since one can show that the system p μ⁻ e⁻ has no bound state⁽³⁾ and p e⁻ e⁻ has only one⁽⁴⁾ (below the continuum) with electronspins antiparallel, there is the hope that one should be able to give bounds on the scattering length in these situations.

Work supported in part by “Fonds zur Förderung der wissenschaftlichen Forschung in Österreich”, Projekt Nr. 3569.