Logarithmic Singularities in Two-Body, Bound-State Integral Equations

A logarithmic singularity is typically present in the kernels of two-body, bound-state integral equations after the two angular variables associated with threedimensional spherical coordinates are separated. The singularity occurs in the separated Schrödinger equation, the separated Bethe–Salpeter equation in the instantaneous approximation, and the partially separated Bethe–Salpeter equation. Problems integrating over the singularity have restricted the types of basis functions that have been used to obtain numerical solutions, making it particularly difficult to obtain bound-state solutions that decrease rapidly at both small and large momenta. Here integrals are evaluated analytically in the neighborhood of the singularity by expanding the integrands, excluding the singular kernels, either analytically or numerically in a Taylor series or a Maclaurin series. This technique makes possible the use of nonpolynomial basis functions that satisfy the boundary conditions, allowing the efficient calculation of all solutions.