Perturbed ladder-operator method: An algebraic recursive solution of the perturbed Coulomb eigenequation

The perturbed ladder-operator method is applied to the solution of the Coulomb eigenequation perturbed by an anharmonic potential, i.e., with a total potential U(x)=-m(m+1)/x²-2q/x +b₁x+b₂{x}²$+${b}₃x³+... . This method is an extension of the Schr\"odinger-Infeld-Hull factorization method within the perturbative scheme. The introduction of specific basis functions for the finite-difference solution of the factorazibility condition, together with the use of the symmetry properties of the Bernoulli polynomials, allows a straightforward determination of analytical expressions of the perturbed Coulomb ladder functions and eigenvalues. Some illustrative examples showing the capabilities of the method are given; particularly, analytical expressions of the linear, quadratic, and cubic Stark shifts are quickly derived.