An error analysis is presented for the spectral-Galerkin method to the
Helmholtz equation in 2-D and 3-D exterior domains. The problem in
unbounded domains is first reduced to a problem on a bounded domain
via the Dirichlet-to-Neumann operator, then a spectral-Galerkin method
is employed to approximate the reduced problem. The error analysis is
based on exploring delicate asymptotic behaviors of the Hankel
functions and on deriving {\it a priori} estimates with explicit
dependence on the wave number for both the continuous and discrete
problems. Explicit error bounds with respect to the wave number are
derived and some illustrative numerical examples are also presented.