Physical Review Letters
In supersymmetric quantum mechanics, shape invariance is a sufficient condition for solvability. We show that all conventional additive shape-invariant superpotentials that are independent of ħ can be generated from two partial differential equations. One of these is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow, and it is closed by the other. We solve these equations, generate the set of all conventional shape-invariant superpotentials, and show that there are no others in this category. We then develop an algorithm for generating all additive shape-invariant superpotentials including those that depend on ħ explicitly.
Bougie, Jonathan, Asim Gangopadhyaya, and Jeffry V. Mallow. “Generation of a Complete Set of Additive Shape-Invariant Potentials from an Euler Equation.” Physical Review Letters 105, no. 21 (November 19, 2010): 210402. doi:10.1103/PhysRevLett.105.210402.
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