  This is the third paper in a series analyzing the asymptotic distribution of
the phase shifts in the semiclassical limit. We analyze the distribution of
phase shifts, or equivalently, eigenvalues of the scattering matrix, $S_h(E)$,
for semiclassical Schr\"odinger operators on $\mathbb{R}^d$ which are
perturbations of the free Hamiltonian by a potential $V$ with polynomial decay.
Our assumption is that $V(x) \sim |x|^{-\alpha} v(\hat x)$ as $x \to \infty$,
for some $\alpha > d$, with corresponding derivative estimates. In the
semiclassical limit $h \to 0$, we show that the atomic measure on the unit
circle defined by these eigenvalues, after suitable scaling in $h$, tends to a
measure $\mu$ on $\mathbb{S}^1$. Moreover, $\mu$ is the pushforward from
$\mathbb{R}$ to $\mathbb{R} / 2 \pi \mathbb{Z} = \mathbb{S}^1$ of a homogeneous
distribution $\nu$ of order $\beta$ depending on the dimension $d$ and the rate
of decay $\alpha$ of the potential function. As a corollary we obtain an
asymptotic formula for the accumulation of phase shifts in a sector of
$\mathbb{S}^1$.
  The proof relies on an extension of results of the second author and Wunsch
on the classical Hamiltonian dynamics and semiclassical Poisson operator to the
class of potentials under consideration here.
