Supersymmetric Expansion Algorithm and complete analytical solution for the Hulth\'en and anharmonic potentials

An algorithm for providing analytical solutions to Schr\"{o}dinger's equation with non-exactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of the superymmetric quantum mechanics as extended toward non shape invariant potentials. The complete solution to a given Hamiltonian $H_{0}$ is obtained from the nodeless states of the Hamiltonian $H_{0}$ and of a set of supersymmetric partners $H_{1}, H_{2},..., H_{r}$. The nodeless states (dubbed "edge" states) are unique and in general can be ground or excited states. They are solved using the logarithmic expansion which yields an infinite systems of coupled first order hierarchical differential equations, converted later into algebraic equations with recurrence relations which can be solved order by order. We formulate the aforementioned scheme, termed to as "Supersymmetric Expansion Algorithm'' step by step and apply it to obtain for the first time the complete analytical solutions of the three dimensional Hulth\'en--, and the one-dimensional anharmonic oscillator potentials.