The potential harmonic expansion method: M. Fabre de la Ripelle. Division de Physique Theorique, Institut de Physique Nucléaire, F-91406 Orsay Cedex, France

Abstract

We discuss three aspects of the analysis of the quasi-bound-state (QBS) or resonance-state effects on multichannel continuum wavefunctions for few-body atomic systems. Section 1 is devoted to a brief introductory account of the basic concept necessary for understanding the later sections. General properties of the scattering matrix S and a related time-delay matrix Q are discussed in Section 2. Both isolated and overlapping resonances are treated, starting from single-channel problems and extending the theory for multichannel cases. Useful relations are derived in detail on the basis of linear algebra. The concept of the resonance channel and nonresonance channel spaces, separated from each other, is introduced via the Q-matrix diagonalization. Outstanding examples of resonance analysis by the inspection of the eigenvalues of the Q matrix are presented. They include the extraction of resonances from a monotonically decreasing phase shift due to a strong background, the detection of a weak resonance, seemingly a nearly constant background behind a strong resonance, and the decomposition of several overlapping resonances into the individual components, some of which are difficult to identify by the conventional methods.Section 3 examines effective potentials for understanding QBSs. First, the information obtainable from the asymptotic long-range potentials is summarized. This includes the analysis of infinite series of QBSs supported by an attractive Coulomb or dipole potential, the latter appearing in charged-particle collisions with hydrogen-like atoms. The infinite series of dipole-supported QBSs is terminated to only a few resonances, sometimes to one or two, by correcting for relativistic and quantum electrodynamic effects. The dipole potential can also cause the background phase shift to diverge as the energy tends to a channel threshold from above, complicating a resonance structure.The hyperspherical coordinate approach is then briefly discussed. Accurate numerical calculations for three-body systems are possible by solving coupled-channel equations in terms of these coordinates. The resultant resonance structures can be visually and qualitatively interpreted by inspecting the adiabatic potentials with the hyperradius adopted as the adiabatic parameter. Sometimes the adiabatic correction or diabatic connection, or both are necessary for proper resonance analysis. By combining with the knowledge of the long-range potentials, resonances can often be ascribed to particular adiabatic or diabatic hyperspherical potentials out of an intertwined set of potential curves. In particular, when two or more adiabatic potentials exhibit strongly avoided crossings, some resonances appear as if they were associated with no hyperspherical potentials, even though a careful diabatic interpretation can clearly explain them. A shift in hyperspherical potentials with a continuous change in some parameter(s) in the system can result in a sudden change from a bound state to QBS or from a Feshbach to shape resonance without any essential change in the main part of the wavefunction.Section 4 analyzes the energy dependence of the cross sections near the threshold for the formation of a QBS. The Wigner threshold law breaks down close to the threshold since the threshold itself has an uncertainty due to the finite lifetime of the QBS. The modifications, due to Baz', for a blurred threshold are explained. Positron–atom collisions are discussed as a prototype, where a bound electron–positron pair (positronium, Ps) can be formed. It is a QBS since it annihilates eventually. Pair annihilation, or positron absorption, can also occur during the collision. The conventional theory leads to a divergent positron absorption cross section at the Ps formation threshold Eth, and to a Ps formation cross section starting from zero and rapidly increasing above Eth. Both of these processes have a common final channel of pair annihilation and should be treated collectively. This is possible by phenomenologically modeling pair annihilation as particle loss due to an imaginary potential in the Schrödinger equation. This approach proves the continuous and smooth transition from positron absorption to Ps formation across Eth, following the Baz' threshold law. Thus, the two processes are indistinguishable in the energy region of this transition.