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IL NUOVO CIMENTO VOL. 112 B, N. 6 Giugno 1997 NOTE BREVI

Gegenbauer polynomials and supersymmetric quantum mechanics

H. ROSU(1)(2)(*) and J. R. GUZMA´N(1)

(1_{) Instituto de Física, IFUG - Apdo Postal E-143, 37150 León, Gto, México}

(2_{) Institute of Gravitation and Space Sciences - Magurele-Bucharest, Romania}

(ricevuto il 23 Gennaio 1997; approvato il 29 Aprile 1997)

Summary. — We show how the simple supersymmetric approach recently used by Dutt, Gangopadhyaya, and Sukhatme (Am. J. Phys., 65 (1997) 400) for spherical harmonics can be applied to Gegenbauer polynomials.

PACS 02.30.Gp – Special functions.

PACS 02.30.Jr – Partial differential equations. PACS 03.65 – Quantum mechanics.

Recently, Dutt, Gangopadhyaya and Sukhatme [1] put at work supersymmetric quantum mechanics and the concept of shape invariance to derive properties of spherical harmonics in a simple way. In this note, we apply their scheme to Gegenbauer polynomials Cpq(x), which are polynomial solutions of the ultraspherical equation

( 1 2x2

) y 92 (2q11) xy 81p(p12q) y40 . (1)

This equation can be put in the following associated Legendre form: ( 1 2x2

) y 922(m 811) xy 81 (n 82m 8)(n 81m 811) y40 (2)

by the substitutions p 4n 82m 8 and q4m 811/2. Since in eq. (2) we want y to be an associated Legendre function, q should be half-integer. To eq. (2) one can apply the change of function v 4 (12x2)m 8 /2y to obtain the self-adjoint form of the associated Legendre equation ( 1 2x2 ) v 922xv 81

y

z

or in spherical polar coordinates d2_v du2 1 cot u dv du 1

y

z

H.ROSUandJ.R.GUZMA´N 942

From now on, we can apply the scheme of [1], i.e. obtaining the Schrödinger form of eq. (4) by the change of variable u 42tan21_(ez_{) leading to}

2d 2 v dz2 2 [n 8 (n 8 1 1 ) sech 2 z] v 42m 82v . (5)

This is a well-known shape-invariant, exactly solvable Schrödinger equation, for which the supersymmetric techniques are straightforward [2]. Thus, it is well known that the energy eigenvalues of the potential V(z) 42n 8(n 811) sech2_{z are E}

n4 2(n 8 2 n)2

(n 40, 1, 2, R, N), where N is the number of bound states in the potential well, and is equal to the largest integer contained in n 8. The eigenfunctions vn(z ; n 8) are obtained

by applying the factorization (creation) operators A†(z ; ai) 4 (2dOdz1aitanh z), where

ai4 n 8 2 i, onto the ground-state wave function v0(z ; an) 4sechn 82nz f sechq 21/2z, i.e.

vn(z ; n 8)BA†(z ; n 8) A†(z ; n 821) A†(z ; n 822) R A†(z ; n 82n11) sechn 82nz .

(6)

In our case n 4n 82m 8fp, and therefore we are dealing with associated Legendre functions of the type vnfPn 8m 8( tanh z). The original Gegenbauer polynomials can be

written formally as Cpq(x) P (12tanh2z)( 1 22q)/4Pp 1q21/2q 21/2 ( tanh z); the proportionality is

used since we are not going into the normalization problem here. Although this relationship appears to be known, see [3], we found it in a simple way.

It is also known that the n 8(n 811) sech2_{z potential is reflectionless if and only if}

n 8 is an integer. Since p should be an integer, the reflectionless property requires again a half-integer q.

Gegenbauer polynomials are connecting higher-dimensional spherical harmonics to the usual two-dimensional ones. In the theory of hyperspherical harmonics they play a role which is analogous to the role played by Legendre polynomials in the usual three-dimensional space [4]. Therefore, we believe our discussion to be useful in such a context as well.

* * *

The work was supported in part by the CONACyT Project No. 4868-E9406 and a CONACyT undergraduate fellowship.

R E F E R E N C E S

[1] DUTTR., GANGOPADHYAYAA. and SUKHATMEU. P., Am. J. Phys., 65 (1997) 400. [2] DUTTR., KHAREA. and SUKHATMEU. P., Am. J. Phys., 56 (1988) 163.

[3] WANGZ. X. and GUOD. R., Special Functions (World Scientific, Singapore) 1989, Exercise 47, p. 291.

[4] AVERYJ., Hyperspherical Harmonics: Applications in Quantum Theory (Kluwer Academic Publishers, Dordrecht) 1989.