The role of boundary conditions in solving finite-energy, two-body, bound-state Bethe–Salpeter equations

The difficulties that typically prevent numerical solutions from being obtained to finite-energy, two-body, bound-state Bethe–Salpeter equations can often be overcome by expanding solutions in terms of basis functions that obey the boundary conditions. The method discussed here for solving the Bethe–Salpeter equation requires only that the equation can be Wick rotated and that the two angular variables associated with rotations in three-dimensional space can be separated, properties that are possessed by many Bethe–Salpeter equations including all two-body, bound-state Bethe–Salpeter equations in the ladder approximation. The efficacy of the method is demonstrated by calculating finite-energy solutions to the partially-separated Bethe–Salpeter equation describing the Wick–Cutkosky model when the constituents do not have equal masses.