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 xi 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 conformal infinity (partial derivative(X) over bar, [h(0)]). We show that det S-X(lambda) is meromorphic in lambda is an element of C, with divisors given by resonance multiplicities and dimensions of kernels of GJMS conformal Laplacians (P-k)(k is an element of N) of (partial derivative(X) over bar, [h(0)]). Moreover xi(z) is proved to be the phase of det S-X(n/2 + iz) on the essential spectrum {z is an element of R+}. Applying this theory to convex co-compact quotients X = Gamma\Hn+1 of hyperbolic space Hn+1, we obtain the functional equation Z(lambda)/Z(n - lambda) = (det SHn+1 (lambda))(chi(X))/det S-X(lambda) for Selberg zeta function Z(lambda) of X, where chi(X) is the Euler characteristic of X. This describes the poles and zeros of Z(lambda), computes det P-k in term of Z(n/2 - k)/Z(n/2 + k) and implies a sharp Weyl asymptotic for xi(z).
