  We study the asymptotic linear stability of a two-parameter family of
solitary waves for the isothermal Euler-Poisson system. When the linearized
equations about the solitary waves are considered, the associated eigenvalue
problem in $L^2$ space has a zero eigenvalue embedded in the neutral spectrum,
i.e., there is no spectral gap. To resolve this issue, use is made of an
exponentially weighted $L^2$ norm so that the essential spectrum is strictly
shifted into the left-half plane, and this is closely related to the fact that
solitary waves exist in the super-ion-sonic regime. Furthermore, in a certain
long-wavelength scaling, we show that the Evans function for the Euler-Poisson
system converges to that for the Korteweg-de Vries (KdV) equation as an
amplitude parameter tends to zero, from which we deduce that the origin is the
only eigenvalue on its natural domain with algebraic multiplicity two. We also
show that the solitary waves are spectrally stable in $L^2$ space. Moreover, we
discuss (in)stability of large amplitude solitary waves.
