SPECTRAL FUNCTIONS AND ZETA-FUNCTIONS IN HYPERBOLIC SPACES

The spectral function (also known as the Plancherel measure), which gi ves the spectral distribution of the eigenvalues of the Laplace-Beltra mi operator, is calculated for a field of arbitrary integer spin (i.e. , for a symmetric traceless and divergence-free tensor field) on the N -dimensional real hyperbolic space (H(N)). In odd dimensions the spect ral function mu(lambda) is analytic in the complex lambda plane, while in even dimensions it is a meromorphic function with simple poles on the imaginary axis, as in the scalar case. For N even a simple relatio n between the residues of mu(lambda) at these poles and the (discrete) degeneracies of the Laplacian on the N sphere (S(N)) is established. A similar relation between mu(lambda) at discrete imaginary values of lambda and the degeneracies on S(N) is found to hold for N odd. These relations are generalizations of known results for the scalar field. T he zeta functions for fields of integer spin on H(N) are written down. Then a relation between the integer-spin zeta functions on H(N) and S (N) is obtained. Applications of the zeta functions presented here to quantum field theory of integer spin in anti-de Sitter space-time are pointed out.