Complex geometry of real hyperboloids

We investigate, using horospheres, the spectrum of the Laplace-Beltrami operator on pseudo hyperbolic spaces (hyperboloids), which, in general, is ultra hyperbolic. Real horospheres define only the continuous spectrum, but the discrete spectrum is connected with complex horospheres. We connect with them the horospherical Cauchy transform which acts at $\bar{\partial}$-cohomology. Correspondingly, for continuous spectrum we have an analogue of the Poisson problem with data on the real boundary but for the discrete spectrum - with cohomological data on the complex boundary.