Geometrical Realization of Beutler-Fano formulas appearing in eigenphase shifts and time delays in multichannel scattering
Recently, we showed that eigenphase shifts and eigentime delays near a resonance for a system of one discrete state and two continua are functionals of the Beutler-Fano formula using appropriate dimensionless energy units and line profile indices and identified parameters responsible for the avoided crossing of eigenphase shifts and eigentime delays and also identified parameters responsible for the eigentime delays due to a change in frame transformation. In this paper, the geometrical realization of the Beutler-Fano formulas is considered in the three-dimensional Liouville space spanned by the Pauli matrices, where dynamic operators are vectors. Vectors corresponding to the background scattering matrix, the S matrix, and the time delay matrix Q form a spherical triangle whose vertex and edge angles are parameters pertaining to the frame transformations among eigenchannels of those matrices and eigenphase shifts of the scattering matrices and the phase shift due to a resonance scattering. The cotangent laws of the spherical triangle yield Beutler-Fano resonance formulas appearing in eigenphase shifts and time delays. Duality holding for the spherical triangle explains the symmetry observed in the relations among parameters and provides a systematic way of defining conjugate dynamic parameters. The spherical triangle also shows the rule of combining the channel-channel couplings in the background scattering with the resonant interaction to give the avoided crossing interactions in the curves of eigenphase shifts as functions of eneryg. The theory developed in the previous and present papers is applied to the vibrational predissociation of triatomic van der Waals molecules.