and supersymmetric quantum mechanics

Standard and non-standard associated Legendre equations and supersymmetric quantum mechanics Abstract. A simple supersymmetric approach recently used by Dutt, Gan-gopadhyaya, and Sukhatme (hep-th/9611087, Am. J. Phys., to appear) for spherical harmonics is generalized to Gegenbauer and Jacobi equations. The coordinate transformation procedure is well known to the authors writing on supersymmetric quantum mechanics (see section 5 in [1]). Essentially, one starts with a one-dimensional Schrödinger equation and tries to obtain a new one by a coordinate transformation. In a recent work, Dutt, Gangopad-hyaya and Sukhatme (DGS) [2] used a coordinate transformation to recast the associated Legendre equation in the Schrödinger form and then employed the concept of shape invariance to derive properties of spherical harmonics in a simple way. In the following, after presenting the case of spherical harmonics, we generalize the DGS scheme to Gegenbauer and Jacobi equations. 1. Spherical harmonics The equation for the associated Legendre polynomials d 2 y dθ 2 + cotθ dy dθ + [l(l + 1) − m 2 sin 2 θ ]y = 0 (1) can be transformed into a Schrödinger eigenvalue equation by a mapping function θ = f (z) that can be found from the condition of putting to nought the coefficient of the first derivative. The result is θ ≡ f = 2arctan(e z). This mapping is equivalent to the replacement sin θ = sechz and cos θ = −tanhz. The range of the variable z is the full real line −∞ < z < ∞. We notice that θ(z) = π 2 + gd(z), where gd(z) is the so-called Gudermannian or hyperbolic amplitude function [3]. The associated Legendre equation is transformed in one of the best known shape invariant, exactly solvable Schrödinger equation − d 2 v dz 2 − [l(l + 1)sech 2 z]v = −m 2 v (2) for which the algebraic supersymmetric scheme can be readily applied. Since l is always an integer for the common spherical harmonics, Eq. (2) is moreover a reflectionless one. The energy eigenvalues are known to be of the type E n = −(l − n) 2 , with n = 0, 1, 2...