  We study the microlocal properties of the scattering matrix associated to the
semiclassical Schr\"odinger operator $P=h^2\Delta_X+V$ on a Riemannian manifold
with an infinite cylindrical end. The scattering matrix at $E=1$ is a linear
operator $S=S_h$ defined on a Hilbert subspace of $L^2(Y)$ that parameterizes
the continuous spectrum of $P$ at energy $1$. Here $Y$ is the cross section of
the end of $X$, which is not necessarily connected. We show that, under certain
assumptions, microlocally $S$ is a Fourier integral operator associated to the
graph of the scattering map $\kappa:\mathcal{D}_{\kappa}\to T^*Y$, with
$\mathcal{D}_\kappa\subset T^*Y$. The scattering map $\kappa$ and its domain
$\mathcal{D}_\kappa$ are determined by the Hamilton flow of the principal
symbol of $P$. As an application we prove that, under additional hypotheses on
the scattering map, the eigenvalues of the associated unitary scattering matrix
are equidistributed on the unit circle.
