Clarification of the relationship between bound and scattering states in quantum mechanics: Application to 12C+α

Using phase-equivalent supersymmetric partner potentials, a general result from the inverse problem in quantum scattering theory is illustrated, i.e., bound-state properties cannot be extracted from the phase shifts of a single partial wave, as a matter of principle. In particular, recent R-matrix analyses of the 12C+α system, extracting the asymptotic normalization constant of the 2+ subthreshold state C12, from the l=2 elastic-scattering phase shifts and bound-state energy, are shown to be unreliable. In contrast, this important constant in nuclear astrophysics can be deduced from the simultaneous analysis of the l=0,2,4,6 partial waves in a simplified potential model. A new supersymmetric inversion potential and existing models give C12=144.5±8.5x103 fm-1/2