Resonances and residue operators for pseudo-Riemannian hyperbolic spaces.

For any pseudo-Riemannian hyperbolic space X over R , C , H or O , we show that the resolvent R (z) = (□ − z Id) − 1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators C c ∞ (X) → D ′ (X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D ′ (X) forms a representation of the isometry group of X , which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R (z) extends, and the residue representations can be infinite-dimensional. [ABSTRACT FROM AUTHOR]