Approximation methods in the context of the bound-state Bethe-Salpeter equation

Numerical approximation schemes of the Wick-rotated scalar Bethe-Salpeter equation are discussed for general local potentials with special emphasis on mesh-point methods. Convergence properties are obtained by considering the analytic properties of the kernel. To this end, the four-dimensional partial wave equations are formulated in a new representation-independent way. The close relationship of variational and mesh-point methods is demonstrated and the difficulties which arise if singular potentials are introduced are discussed. For marginal singular potentials those difficulties are overcome in a new way by redefining the corresponding two-particle Green's function. Numerical examples for this case are given.