  The optimal stopping problem for a Hunt processes on $\R$ is considered via
the representation theory of excessive functions. In particular, we focus on
infinite horizon (or perpetual) problems with one-sided structure, that is,
there exists a point $x^*$ such that the stopping region is of the form
$[x^*,+\infty)$. Corresponding results for two-sided problems are also
indicated. The main result is a spectral representation of the value function
in terms of the Green kernel of the process. Specializing in L\'evy processes,
we obtain, by applying the Wiener-Hopf factorization, a general representation
of the value function in terms of the maximum of the L\'evy process. To
illustrate the results, an explicit expression for the Green kernel of Brownian
motion with exponential jumps is computed and some optimal stopping problems
for Poisson process with positive exponential jumps and negative drift are
solved.
