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

TL;DR

This paper introduces a generalized Krein spectral function for even-dimensional conformally compact manifolds, linking it to the scattering operator's determinant, and applies it to hyperbolic quotients, Selberg zeta functions, and GJMS Laplacians.

Contribution

It defines a new spectral function for these manifolds, relates it to the scattering operator's determinant, and derives applications to Selberg zeta functions and conformal Laplacians.

Findings

Established a phase relation between the Krein spectral function and the scattering operator determinant.

Derived a functional equation for Selberg's Zeta function in this context.

Provided a Weyl type asymptotic for the Krein function.

Abstract

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.