A GENERAL STUDY OF WIGNER COEFFICIENTS OF SU(1,1)

The Wigner coefficients which couple any two irreducible unitary representations of the noncompact group (SU(1, 1) are derived by means of the second-order difference equation which defines them. It is found that whenever at least one of the three representations being coupled is discrete, then the two solutions are degenerate, but two linearly independent nontrivial solutions exist, in general, when all three representations are continuous. For this case two orthonormal solutions are found. Some elementary symmetries of the solutions are examined. The integral over the group manifold is regularized by means of a convergence factor in order to make all the irreducible unitary representation functions (the Bargmann functions) mutually orthonormal. This regularization is used for the investigation of the resolvent of the Laplace-Beltrami operator in the space of the Bargmann functions. © 1968.