### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 074, 16 pages      arXiv:1210.4632      https://doi.org/10.3842/SIGMA.2012.074
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

### Ladder Operators for Lamé Spheroconal Harmonic Polynomials

Ricardo Méndez-Fragoso a and Eugenio Ley-Koo b
b) Instituto de Física, Universidad Nacional Autónoma de México, México

Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012

Abstract
Three sets of ladder operators in spheroconal coordinates and their respective actions on Lamé spheroconal harmonic polynomials are presented in this article. The polynomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lamé polynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers $n_1$ and $n_2$ counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components $\hat L_x$, $\hat L_y$, $\hat L_z$ of the angular momentum connect pairs of the four species of polynomials of a chosen kind and angular momentum. The third set of operators, the cartesian components $\hat p_x$, $\hat p_y$, $\hat p_z$ of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. Relationships among spheroconal harmonics at the levels of the three sets of operators are illustrated.

Key words: Lamé polynomials; spheroconal harmonics; ladder operators.

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