Convergence of the non-relativistic and relativistic R-matrix expansions at the reaction volume boundary

The convergence of the non-relativistic and relativistic fixed-boundary-condition R-matrix expansions at the reaction volume boundary is discussed. It is shown that in the non-relativistic case the expansion of the wavefunction converges to this function at the boundary. In the relativistic case, however, the expansion does not generally converge to the wavefunction at the reaction surface because the set of relativistic basis functions spanning the interior of the reaction volume is incomplete on the surface. With one exception, this fact has not been recognized before and has been a source of errors in previous presentations of the relativistic R-matrix method. We present a corrected derivation of the R-matrix theory for the Dirac equation and generalize it to the multi-channel case. We also explain why, in spite of being based on an incorrect theory, the results of electron-atom and electron-ion scattering calculations performed so far within the framework of the relativistic R-matrix method were not afflicted by these errors.