Generalized Krein Formula, Determinants, and Selberg Zeta Function in Even Dimension - Guillarmou, Colin (American journal of mathematics, 2009)
For a class of even dimensional asymptotically hyperbolic (AH) manifolds, we develop a generalized Birman-Krein theory to study scattering asymptotics and, when the curvature is constant, to analyze the Selberg zeta function. The main objects we construct for an AH manifold (X, g) are, on the one hand, a natural spectral function ξ for the Laplacian$\Delta _g $, which replaces the counting function of the eigenvalues in this infinite volume case, and on the other hand the determinant of the scattering operator$S_X (\lambda )$of$\Delta _g $on X. Both need to be defined through regularized functional: renormalized trace on the bulk X and regularized determinant on the conformai infinity ($(\partial {\bar X},[h_0 ])$). We show that det$S_X (\lambda )$is meromorphic in$\lambda \in \mathbb{C}$, with divisors given by resonance multiplicities and dimensions of kernels of GJMS conformal Laplacians ($(P_k )_{k \in \mathbb{N}} $of$(\partial {\bar X},[h_0 ])$)-Moreover ξ(z) is proved to be the phase of det$S_X (\frac{n} {2} + iz)$on the essential spectrum$\{ z \in \mathbb{R}^ + \} $. Applying this theory to convex co-compact quotients$X = \Gamma \backslash \mathbb{H}^{n + 1} $of hyperbolic space$\mathbb{H}^{n + 1} $, we obtain the functional equation$Z(\lambda )/Z(n - \lambda ) = (\det S_{\mathbb{H}^{n + 1} } (\lambda ))^{x(X)} /\det S_X (\lambda )$for Selberg zeta function$Z(\lambda )$of X, where X(X) is the Euler characteristic of X. This describes the poles and zeros of Z(λ), computes det$P_k $in term of$Z(\frac{n}{2} - k)/Z(\frac{n} {2} + k)$and implies a sharp Weyl asymptotic for ξ(z).