Lie Theory and Separation of Variables. I: Parabolic Cylinder Coordinates

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Winternitz and co-workers have characterized the parabolic cylinder function solutions of the reduced wave equation in two variables as eigenfunctions of a quadratic operator $E = MP_2 + P_2 M$ in the enveloping algebra of the Lie algebra of the Euclidean group in the plane. Here we study the representation theory of the Euclidean and pseudo-Euclidean groups in an E-basis and use the results to derive a number of new addition and expansion theorems for products of parabolic cylinder functions.