Generalized Krein formula and determinants for even dimensional Poincare-Einstein manifolds

Details Generalized Krein formula and determinants for even dimensional Poincare-Einstein manifolds Generalized Krein formula and determinants for even dimensional Poincare-Einstein manifolds

2005-12-08

arxiv.org/abs/math/0512173v3

Mathematics

Spectral Theory

39 pages. Exposition improved. Weyl asymptotic precised

Scientific paper

For a class of even dimensional conformally compact manifolds (X,g), we define a generalized Krein spectral function by applying a renormalized trace functional to the spectral measure of the Laplacian. We then show that this is the phase of the Kontsevich-Vishik determinant det S(s) of the scattering operator S(s) of (the Laplacian of) g and we analyze the divisors of det(S(s)). As an application for convex co-compact hyperbolic quotients, we obtain a functional equation for Selberg's Zeta function Z(s), we express the determinant of the GJMS conformal Laplacians of the conformal infinity of (X,g) in term of particular values of Z(s), and we show a sharp Weyl type asymptotic for the Krein function.

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