  We study the scattering theory for the Schr\"odinger and wave equations with
rough potentials in a scale of homogeneous Sobolev spaces. The first half of
the paper concerns with an inverse-square potential in both of subcritical and
critical constant cases, which is a particular model of scaling-critical
singular perturbations. In the subcritical case, the existence of the wave and
inverse wave operators defined on a range of homogeneous Sobolev spaces is
obtained. In particular, we have the scattering to a free solution in the
homogeneous energy space for both of the Schr\"odinger and wave equations. In
the critical case, it is shown that the solution is asymptotically a sum of a
$n$-dimensional free wave and a rescaled two-dimensional free wave. The second
half of the paper is concerned with a generalization to a class of strongly
singular decaying potentials. We provides a simple criterion in an abstract
framework to deduce the existence of wave operators defined on a homogeneous
Sobolev space from the existence of the standard ones defined on a base Hilbert
space.
