Long-Range Perturbations of Bound States and Resonances

We consider the modification of the scattering matrix in multichannel potential-scattering theory by a long-range perturbation. The perturbation of "single-particle" energy levels (bound states or resonances) is discussed by means of Weinberg's eigenvalue analysis of potential scattering. A straightforward distinction between "single-particle" and "compound" states appears in this formalism, and it leads to a proof that there are no bound states embedded in the continuum. A simple approximation for the eigenvalues corresponding to single-particle levels is developed and applied to the calculation of the energy difference between the (${\mathrm{½}}^{+}$) first excited states of the ${\mathrm{C}}^{13}$-${\mathrm{N}}^{13}$ mirror pair.