  Within the theory of Siegert pseudostates, it is possible to accurately
calculate bound states and resonances. The energy continuum is replaced by a
discrete set of states. Many questions of interest in scattering theory can be
addressed within the framework of this formalism, thereby avoiding the need to
treat the energy continuum. For practical calculations it is important to know
whether a certain subset of Siegert pseudostates comprises a basis. This is a
nontrivial issue, because of the unusual orthogonality and overcompleteness
properties of Siegert pseudostates. Using analytical and numerical arguments,
it is shown that the subset of bound states and outgoing Siegert pseudostates
forms a basis. Time evolution in the context of Siegert pseudostates is also
investigated. From the Mittag-Leffler expansion of the outgoing-wave Green's
function, the time-dependent expansion of a wave packet in terms of Siegert
pseudostates is derived. In this expression, all Siegert pseudostates--bound,
antibound, outgoing, and incoming--are employed. Each of these evolves in time
in a nonexponential fashion. Numerical tests underline the accuracy of the
method.
