Linking the exotic structure of ${}^{17}$C to its unbound mirror ${}^{17}$Na

The structure of ${}^{17}$C is used to define a nuclear interaction that, when used in a multichannel algebraic scattering theory for the $n+{}^{16}$C system, gives a credible definition of the (compound) excitation spectra. When couplings to the low-lying collective excitations of the ${}^{16}$C-core are taken into account, both sub-threshold and resonant states about the $n+{}^{16}$C threshold are found. Adding Coulomb potentials to that nuclear interaction, the method is used for the mirror system of $p+{}^{16}$Ne to specify the low-excitation spectrum of the particle unstable $^{17}$Na. We compare the results with those of a microscopic cluster model. A spectrum of low excitation resonant states in ${}^{17}$Na is found with some differences to that given by the microscopic-cluster model. The calculated resonance half-widths (for proton emission) range from $\sim 2$ to $\sim 672$ keV.