Supersymmetric Expansion Algorithm and Complete Analytical Solution for the Hulthén and Anharmonic Potentials

Abstract

Abstract An algorithm for providing analytical solutions to Schrödinger’s equation with nonexactly solvable potentials is elaborated. It represents a symbiosis between the logarithmic expansion method and the techniques of supersymmetric 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 system 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 the “Supersymmetric Expansion Algorithm,” step by step and apply it to obtain for the first time the complete analytical solutions of the 3D Hulthén, and the 1D anharmonic, oscillator potentials.