The Clebsch–Gordan coefficients of the three-dimensional Lorentz algebra in the parabolic basis

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Abstract
Starting from the oscillator representation of the three-dimensional Lorentz algebra so(2,1), we build a Lie algebra of second-order differential operators which realizes all series of self-adjoint irreducible representations. The choice of the common self-adjoint extention over a two-chart function space determines whether they lead to single- or multivalued representations over the corresponding Lie group. The diagonal operator defining the basis is the parabolic subgroup generator. The direct product of two such algebras allows for the calculation of all Clebsch–Gordan coefficients explicitly, as solutions of Schrödinger equations for Pöschl–Teller potentials over one (𝒟×𝒟), two (𝒟×𝒞), or three (𝒞×𝒞) charts. All coefficients are given in terms of up to two 2F1 hypergeometric functions.

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