Quantum-mechanical and semiclassical study of the collinear three-body Coulomb problem: Resonances

Abstract

This paper continues a quantum-mechanical (QM) and semiclassical (SC) study of the collinear three-body Coulomb problem [O. I. Tolstikhin and C. Namba, 70, 062721 (2004)], extending it to resonances. Our QM treatment is based on the theory of Siegert pseudostates (SPS) implemented in hyperspherical coordinates by means of the slow/smooth variable discretization method. A consistent formulation of this approach for arbitrary dimension of configuration space and number of open channels is given. Not only resonance energy and total width, but also partial widths are defined in terms of the SPS; for the cases of one and two open channels, they are expressed in terms of the SPS eigenvalues only. The SC results are obtained in the leading-order approximation from the asymptotic solution of the problem for $h\ensuremath{\rightarrow}0$, where $h$ is a dimensionless parameter that depends only on the masses of particles, varies in the interval $0\ensuremath{\leqslant}h\ensuremath{\leqslant}1$, and has the meaning of an effective Planck's constant for the motion in hyperradius. It is shown that resonance widths as functions of $1∕h$ oscillate with an exponentially decaying amplitude and almost constant period. The SC theory qualitatively explains such a behavior as a manifestation of interference effects in the nonadiabatic process of decay of the resonance state and provides surprisingly good quantitative results even for systems with $h\ensuremath{\sim}1$.