$R$-matrix theory with level-dependent boundary condition parameters

I present a new formalism of the $R$-matrix theory where the formal parameters for the resonance energies and widths are identical to the observed values. By allowing the boundary condition parameters to vary from level to level, the freedom required to adjust the formal parameters for the pole positions to the observed values is obtained. The basis of the resulting theory becomes nonorthogonal, and I describe the procedure to construct a consistent $R$-matrix theory with such a nonorthogonal basis. And by adjusting the normalization of the states that form the basis, the formal parameters for the reduced decay widths also become the same as those observed, leaving no formal parameters that are different from the observed ones. A demonstration of the developed theory to the elastic ${}^{12}\mathrm{C}+p$ scattering data is presented.

Figure 1

${}^{12}\mathrm{C}+p$ elastic scattering cross-sections (in mb/sr) with respect to the incident proton energy in the laboratory frame for the center-of-mass angle $\theta =89.1^{\circ }$ (left) and $146.9^{\circ }$ (right). The experimental data are from Ref. [10] (closed circles).