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 Li are generators. This basis is shown to be a unique nonequivalent alternative to the canonical basis (eigenfunctions of L3). The functions 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 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, 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 as functions on a sphere and to the fact that they are the wavefunctions of an asymmetrical top.
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August 1973
Research Article|
August 01 1973
A new basis for the representations of the rotation group. Lamé and Heun polynomials
J. Patera;
J. Patera
Centre de Recherches Mathématiques, Université de Montréal, Montréal 101, P.Q., Canada
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P. Winternitz
P. Winternitz
Centre de Recherches Mathématiques, Université de Montréal, Montréal 101, P.Q., Canada
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J. Math. Phys. 14, 1130–1139 (1973)
Article history
Received:
January 01 1973
Citation
J. Patera, P. Winternitz; A new basis for the representations of the rotation group. Lamé and Heun polynomials. J. Math. Phys. 1 August 1973; 14 (8): 1130–1139. https://doi.org/10.1063/1.1666449
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