Potentials for which the Radial Schr\"odinger Equation can be solved

TL;DR

This paper identifies a subclass of potentials allowing explicit solutions to the radial Schrödinger equation at zero energy, expanding the class of solvable potentials by combining different analytical approaches.

Contribution

It introduces a simpler subclass of potentials for explicit solutions and demonstrates how combining approaches broadens the class of solvable potentials.

Findings

Explicit solutions for a subclass of potentials at zero energy.

Combination of approaches yields a larger class of solvable potentials.

Results are explicit and easily verifiable.

Abstract

In a previous paper$^1$, submitted to Journal of Physics A -- we presented an infinite class of potentials for which the radial Schr\"odinger equation at zero energy can be solved explicitely. For part of them, the angular momentum must be zero, but for the other part (also infinite), one can have any angular momentum. In the present paper, we study a simple subclass (also infinite) of the whole class for which the solution of the Schr\"odinger equation is simpler than in the general case. This subclass is obtained by combining another approach together with the general approach of the previous paper. Once this is achieved, one can then see that one can in fact combine the two approaches in full generality, and obtain a much larger class of potentials than the class found in ref. $^1$ We mention here that our results are explicit, and when exhibited, one can check in a straightforward manner their validity.