New Implicitly Solvable Potential Produced by Second Order Shape Invariance

TL;DR

This paper generalizes a recent method to higher-order SUSY quantum mechanics, constructing a new shape invariant potential with explicit solutions and spectrum depending on boundary conditions, advancing the understanding of solvable quantum systems.

Contribution

It introduces a novel second-order shape invariant potential in higher-order SUSY QM, expanding the class of analytically solvable quantum models.

Findings

Constructed a new singular shape invariant potential at the origin.

Explicitly solved the Schrödinger equation for the new potential.

Derived the energy spectrum implicitly via a transcendental equation.

Abstract

The procedure proposed recently by J.Bougie, A.Gangopadhyaya and J.V.Mallow to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schr\"odinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions.