  A spectral theory of linear operators on a rigged Hilbert space is applied to
Schr\"odinger operators with exponentially decaying potentials and dilation
analytic potentials. The theory of rigged Hilbert spaces provides a unified
approach to resonances (generalized eigenvalues) for both classes of potentials
without using any spectral deformation techniques. Generalized eigenvalues for
one dimensional Schr\"odinger operators (ordinary differential operators) are
investigated in detail. A certain holomorphic function $\mathbb{D}(\lambda)$ is
constructed so that $\mathbb{D}(\lambda) = 0$ if and only if $\lambda $ is a
generalized eigenvalue. It is proved that $\mathbb{D}(\lambda)$ is equivalent
to the analytic continuation of the Evans function. In particular, a new
formulation of the Evans function and its analytic continuation is given.
