Lie theory and separation of variables. 3. The equation f _tt-f_ss = γ²f

Kalninis has related the 11 coordinate systems in which variables separate in the equation f_tt - f_ss = γ²f to 11 symmetric quadratic operators L in the enveloping algebra of the Lie algebra of the pseudo-Euclidean group in the plane E(1,1). There are, up to equivalence, only 12 such operators and one of them, L_E, is not associated with a separation of variables. Corresponding to each faithful unitary irreducible representation of E(1,1) we compute the spectral resolution and matrix elements in an L basis for seven cases of interest and also give overlap functions between different bases: Of the remaining five operators three are related to Mathieu functions and two are related to exponential solutions corresponding to Cartesian type coordinates. We then use these results to derive addition and expansion theorems for special solutions of f_tt - f_ss = γ²f obtained via separation of variables, e.g., products of Bessel, Macdonald and Bessel, Airy and parabolic cylinder functions. The exceptional operator L_E is also treated in detail.