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Method for Generating Additive Shape Invariant Potentials from an Euler Equation

In the supersymmetric quantum mechanics formalism, the shape invariance condition provides a sufficient constraint to make a quantum mechanical problem solvable; i.e., we can determine its eigenvalues and eigenfunctions algebraically. Since shape invariance relates superpotentials and their derivatives at two different values of the parameter $a$, it is a non-local condition in the coordinate-parameter $(x, a)$ space. We transform the shape invariance condition for additive shape invariant superpotentials into two local partial differential equations. One of these equations is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow. The second equation provides the constraint that helps us determine unique solutions. We solve these equations to generate the set of all known $\hbar$-independent shape invariant superpotentials and show that there are no others. We then develop an algorithm for generating additive shape invariant superpotentials including those that depend on $\hbar$ explicitly, and derive a new $\hbar$-dependent superpotential by expanding a Scarf superpotential.