Approximate Solution of a Sturm-Liouville System Using Nonorthogonal Expansions: Application to α-α Nuclear Scattering

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Abstract
A stationary principle for the Sturm‐Liouville system with inhomogeneous boundary conditions leads to an approximate solution in which nonorthogonal expansions are used. It is shown that if the nonorthogonal functions are chosen to be the eigenfunctions of a similar Sturm‐Liouville equation, then the numerical solution of the boundary value problem no longer involves an integration of a second derivative, but only the more simple integration of a function. As an illustration, formulas for calculating the nuclear phase shifts of a Schrödinger equation with a long‐range Coulomb potential are presented in the notation of R‐matrix theory. The numerical convergence of the method is investigated for α‐α nuclear scattering from real potentials that have a repulsive core followed by a short‐range attraction (Ali and Bodmer). Generalizations of the method have been applied in nuclear physics to coupled differential equations and to eigenvalue problems.