  We prove that the averaged scattering solutions to the Schr\"odinger equation
with short-range electromagnetic potentials $(V,A)$ where $V(x)=O(|x|^{-\rho}),
A(x)= O(|x|^{-\rho}), |x| \to \infty, \rho >1,$ are dense in the set of all
solutions to the Schr\"odinger equation that are in $L^2(K)$ where $K$ is any
connected bounded open set in $\ere^n,n\geq 2,$ with smooth boundary.
  We use this result to prove that if two short-range electromagnetic
potentials $(V_1,A_1)$ and $(V_2,A_2)$ in $\ere^n, n\geq 3,$ have the same
scattering matrix at a fixed positive energy and if the electric potentials
$V_j$ and the magnetic fields $ F_j:={\rm curl} A_j, j=1,2,$ coincide outside
of some ball they necessarily coincide everywhere.
  In a previous paper of Weder and Yafaev the case of electric potentials and
magnetic fields that are asymptotic sums of homogeneous terms at infinity was
studied. It was proven that all these terms can be uniquely reconstructed from
the singularities in the forward direction of the scattering amplitude at a
fixed positive energy.
  The combination of the new uniqueness result of this paper and the result of
Weder and Yafaev implies that the scattering matrix at a fixed positive energy
uniquely determines electric potentials and magnetic fields that are a finite
sum of homogeneous terms at infinity, or more generally, that are asymptotic
sums of homogeneous terms that actually converge, respectively, to the electric
potential and to the magnetic field.
