Method for Generating Additive Shape Invariant Potentials from an Euler Equation

Abstract

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 $\hslash$-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 $\hslash$ explicitly, and derive a new $\hslash$-dependent superpotential by expanding a Scarf superpotential.

1 figure, 4 tables, 18 pages

References in corpus (3)

• Infinitely many shape invariant potentials and new orthogonal polynomials
• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Generation of a Complete Set of Supersymmetric Shape Invariant Potentials from an Euler Equation

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• Shape Invariant Potentials in Supersymmetric Quantum Cosmology
• Analytical Solution of Two-Dimensional Scarf II Model by Means of SUSY Methods
• Symmetries and the compatibility condition for the new translational shape invariant potentials
• Supersymmetrical Separation of Variables for Scarf II Model: Partial Solvability
• Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution
• New Implicitly Solvable Potential Produced by Second Order Shape Invariance
• Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum
• Infinite families of shape invariant potentials with n parameters subject to translation