Ladder Operators for Lam\'e Spheroconal Harmonic Polynomials

TL;DR

This paper introduces three sets of ladder operators for Lamé spheroconal harmonic polynomials, elucidating their actions and relationships in the context of asymmetric molecular rotations.

Contribution

It presents novel ladder operators for Lamé spheroconal harmonics, detailing their roles in manipulating quantum numbers and connecting different polynomial species.

Findings

Defined three sets of ladder operators and their actions.

Illustrated relationships among spheroconal harmonics.

Connected polynomial pairs with different quantum numbers and parities.

Abstract

Three sets of ladder operators in spheroconal coordinates and their respective actions on Lam\'e 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 Lam\'e 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.