  We compute the graviton Kaluza-Klein spectrum on a gravity-dilaton background
with a naked singularity for all possible boundary conditions at the
singularity which are consistent with unitary evolution. We apply methods from
non-relativistic quantum mechanics with singular Schr\"{o}dinger potentials. In
general the spectrum contains a tachyon, a sign of instability. Only for a
particular boundary condition at the singularity is the spectrum free of
tachyons. In this case the lowest-lying graviton mode is massless. We argue
that this result will also hold for other backgrounds with similar geometry
near the curvature singularity. We complete our study with a brief discussion
on radion perturbations and the Higgs mechanism on this singular background.
