Supersymmetry, shape invariance and the Legendre equations

Abstract

In three space dimensions, when a physical system possesses spherical symmetry, the dynamical equations automatically lead to the Legendre and the associated Legendre equations, with the respective orthogonal polynomials as their standard solutions. This is a very general and important result and appears in many problems in physics (for example, the multipole expansion etc). We study these equations from an operator point of view, much like the harmonic oscillator, and show that there is an underlying shape invariance symmetry in these systems responsible for their solubility. We bring out various interesting features resulting from this analysis from the shape invariance point of view.

4 pages, 1 figure; to appear in PLB

References in corpus (6)

• Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry
• Algebraic Approach to Shape Invariance
• Generation of a Complete Set of Supersymmetric Shape Invariant Potentials from an Euler Equation
• Infinitely many shape invariant potentials and cubic identities of the Laguerre and Jacobi polynomials
• The structure of supersymmetry in $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics
• Reconstruction of field theory from excitation spectra of defects

Cited by in corpus (7)

• Shape Invariant Potentials in Supersymmetric Quantum Cosmology
• Supersymmetry and Shape Invariance of exceptional orthogonal polynomials
• Generalized raising and lowering operators for supersymmetric quantum mechanics
• A Relation Between $Z_3$-Graded Symmetry and Shape Invariant Supersymmetric Systems
• Supersymmetric Quantum Mechanics, Engineered Hierarchies of Integrable Potentials, and the Generalised Laguerre Polynomials
• Supersymmetry, shape invariance and the hypergeometric equation
• Description of the associated Legendre functions in analogy to the state space of the free electromagnetic field