  We examine the spectral properties of single and multiple matter-wave dark
solitons in Bose-Einstein condensates confined in parabolic traps, where the
scattering length is periodically modulated. In addition to the large-density
limit picture previously established for homogeneous nonlinearities, we explore
a perturbative analysis in the vicinity of the linear limit, which provides
good agreement with the observed spectral modes. Between these two analytically
tractable limits, we use numerical computations to fill in the relevant
intermediate regime. We find that the scattering length modulation can cause a
variety of features absent for homogeneous nonlinearities. Among them, we note
the potential oscillatory instability even of the single dark soliton, the
potential absence of instabilities in the immediate vicinity of the linear
limit for two dark solitons, and the existence of an exponential instability
associated with the in-phase motion of three dark solitons.
