Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

TY - JOUR

T1 - Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

AU - Frahm, Jan

AU - Spilioti, Polyxeni

PY - 2023/9

Y1 - 2023/9

N2 - For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId) −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.

AB - For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId) −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.

KW - Hyperbolic space

KW - Laplace–Beltrami operator

KW - Pseudo-Riemannian manifold

KW - Resolvent

KW - Resonances

UR - http://www.scopus.com/inward/record.url?scp=85162855703&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2303.04780

DO - 10.48550/arXiv.2303.04780

M3 - Tidsskriftartikel

SN - 0021-7824

VL - 177

SP - 178

EP - 197

JO - Journal de Mathematiques Pures et Appliquees

JF - Journal de Mathematiques Pures et Appliquees

ER -