A new basis for the representations of the rotation group. Lamé and Heun polynomials

Abstract

The representation theory of the rotation group O(3) is developed in a new basis, consisting of eigenfunctions of the operator E = −4(L₁² + rL₂²), where 0 < r < 1 and L_i are generators. This basis $|\mathbf{J}\mathrm{\lambda 〉}$ is shown to be a unique nonequivalent alternative to the canonical basis (eigenfunctions of L₃). The functions $\mathrm{|J\lambda 〉}$ are constructed as linear combinations of canonical basis functions and are shown to fall into four symmetry classes, distinguished by their behavior under reflections of the inidividual space axes. Algebraic equations for the eigenvalues λ of E are derived. When realized in terms of functions on an O(3) sphere, the basis $\mathrm{|J\lambda 〉}$ consists of products of two Lamé polynomials, obtained by separating variables in the corresponding Laplace equation in elliptic coordinates. When realized in a space of functions of one complex variable, $\mathrm{|J\lambda 〉}$ are Heun polynomials. Applications of the new basis in elementary particle, nuclear, and molecular physics are pointed out, due in particular to the symmetric form of $\mathrm{|J\lambda 〉}$ as functions on a sphere and to the fact that they are the wavefunctions of an asymmetrical top.