  We investigate the propagation of arbitrarily coupled scalar fields on the
$N$-dimensional hyperbolic space ${\mathbb H}^N$. Using the $\zeta$-function
regularization we compute exactly the one loop effective action. The vacuum
expectation value of quadratic field fluctuations and the one loop renormalized
stress tensor are then computed using the recently proposed direct
$\zeta$-function technique. Our computation tests the validity of this approach
in presence of a continuous spectrum. Our results apply as well to the
$N$-dimensional anti-de Sitter spacetime, whose appropriate euclidean section
is the hyperbolic space ${\mathbb H}^N$.
