  We study the Cauchy problem for the one-dimensional wave equation with an
inverse square potential. We derive dispersive estimates, energy estimates, and
estimates involving the scaling vector field, where the latter are obtained by
employing a vector field method on the "distorted" Fourier side. In addition,
we prove local energy decay estimates. Our results have immediate applications
in the context of geometric evolution problems. The theory developed in this
paper is fundamental for the proof of the co-dimension 1 stability of the
catenoid under the vanishing mean curvature flow in Minkowski space.
