Sturmian expansion of the Coulomb t matrix

The convergence of the Sturmian expansion (i.e., the Weinberg series) for the Coulomb $t$ matrix at negative energies is discussed. It is pointed out that the series for the full three-dimensional $t$ matrix diverges, and that the series for the partial-wave $t$ matrix converges conditionally at best and sometimes diverges because of the long range of the Coulomb potential. The $t$ matrix is rigorously expressed as the limit of a power series at a point on its circle of convergence, with the power series itself at that point being just the Sturmian expansion. The possibility of finding alternative separable expansions with better convergence behavior is discussed, the conclusion being reached that the poor convergence is an inevitable result of the long range of the Coulomb potential.