New Implicitly Solvable Potential Produced by Second Order Shape Invariance

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.