A Practical Method to Solve Cut-off Coulomb Problems in the Momentum Space -- Application to the Lippmann-Schwinger Resonating-Group Method and the pd Elastic Scattering --

Abstract

A practical method to solve cut-off Coulomb problems of two-cluster systems in the momentum space is given. When a sharply cut-off Coulomb force with a cut-off radius $\rho$ is introduced at the level of constituent particles, two-cluster direct potential of the Coulomb force becomes in general a local screened Coulomb potential. The asymptotic Hamiltonian yields two types of asymptotic waves; one is an approximate Coulomb wave with $\rho$ in the middle-range region, and the other a free (no-Coulomb) wave in the longest-range region. The constant Wronskians of this Hamiltonian can be calculated in either region. We can evaluate the Coulomb-modified nuclear phase shifts for the screened Coulomb problem, using the matching condition proposed by Vincent and Phatak for the sharply cut-off Coulomb problem. We apply this method first to an exactly solvable model of the $\alpha \alpha$ scattering with the Ali-Bodmer potential and confirm that a complete solution is obtained with a finite $\rho$. The stability of nuclear phase shifts with respect to the change of $\rho$ in some appropriate range is demonstrated in the $\alpha \alpha$ resonating-group method (RGM) using the Minnesota three-range force. An application to the pd elastic scattering is also discussed.