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 ${}^{12}\mathrm{C}+\alpha$ system, extracting the asymptotic normalization constant of the $2^{+}$ subthreshold state $C_{12}$, from the $\ell =2$ 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 $\ell =0,2,4,6$ partial waves in a simplified potential model. A new supersymmetric inversion potential and existing models give $C_{12}=144.5\pm 8.5\times {10}^3{\text{fm}}^{-1∕2}$.

• Received 19 June 2003

DOI:https://doi.org/10.1103/PhysRevC.69.034601