Resonances for the Laplacian: the cases BC2 and C2 (except SO0(p,2) with p>2 odd)

J. Hilgert (1) , A. Pasquale (2) , T. Przebinda (3)

1 University of Paderborn
2 IECL - Institut Élie Cartan de Lorraine
3 Department of Mathematics [Norman]

Let $X=G/K$ be a Riemannian symmetric space of the noncompact type and restricted root system $BC_2$ or $C_2$ (except $G=SO_0(p,2)$ with $p>2$ odd). The analysis of the meromorphic continuation of the resolvent of the Laplacian of $X$ is reduced from the analysis of the same problem for a direct product of two isomorphic rank-one Riemannian symmetric spaces of the noncompact type which are not isomorphic to real hyperbolic spaces. We prove that the resolvent of the Laplacian of $X$ can be lifted to a meromorphic function on a Riemann surface which is a branched covering of the complex plane. Its poles, that is the resonances of the Laplacian, are explicitly located on this Riemann surface. The residue operators at the resonances have finite rank. Their images are finite direct sums of finite-dimensional irreducible spherical representations of $G$.

Mots clés: Resonances; resolvent; Laplacian; Riemannian symmetric spaces of the noncompact type; rank two; BC; direct products

Domaines: Mathématiques [math]

J. Hilgert, A. Pasquale, T. Przebinda. Resonances for the Laplacian: the cases BC2 and C2 (except SO0(p,2) with p>2 odd). Geometric methods in physics, Birkhäuser, pp.159-182, 2016, Trends in Mathematics, 978-3-319-31756-4. ⟨10.1007/978-3-319-31756-4_15⟩. ⟨hal-01232037⟩