Anomaly-Free Variational Method for Inelastic Scattering

Analysis, given in an earlier paper, of standard variational methods for elastic scattering is extended to the case of several open channels. As in the case of elastic scattering, and contrary to widespread expectation, the spurious singularities inherent in the Kohn formalism are shown, for the general multichannel case, not to arise from the singularities of the linear inhomogeneous system of equations common to standard variational methods. The Kohn formula for elements of the $R$ or $K$ matrix, and its analog for the $R^{-1\mathrm{}}$ matrix, are shown to vary smoothly, without poles, as the energy parameter goes through eigenvalues of this system of equations. The spurious singularities arise from isolated zeroes of determinants that occur in the denominator of the Kohn formula and of its analog (the inverse Kohn formula) for $R^{-1\mathrm{}}$. The singularities in these formulas do not in general coincide, and a criterion is proposed for alternative use of these formulas, resulting in a procedure free of spurious singularities. This analysis is illustrated by calculations on a soluble two-channel-model problem. An incidental result of the present formalism is a proof that the approximate $R$ matrix given by the Kohn formula is symmetric and real if the basis functions used are real.