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

TL;DR

This paper introduces the Supersymmetric Expansion Algorithm, a novel method combining logarithmic expansion and supersymmetric quantum mechanics to analytically solve non-exactly solvable Schrödinger equations, demonstrated on Hulthén and anharmonic potentials.

Contribution

The paper develops and applies a new analytical solution algorithm for Schrödinger's equation with complex potentials, extending supersymmetric quantum mechanics to non shape invariant cases.

Findings

Complete analytical solutions for Hulthén potential obtained.

Analytical solutions for one-dimensional anharmonic oscillator derived.

The method can handle non shape invariant potentials effectively.

Abstract

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.