  We consider semiclassical Schr\"odinger operators on the real line of the
form $$H(\hbar)=-\hbar^2 \frac{d^2}{dx^2}+V(\cdot;\hbar)$$ with $\hbar>0$
small. The potential $V$ is assumed to be smooth, positive and exponentially
decaying towards infinity. We establish semiclassical global representations of
Jost solutions $f_\pm(\cdot,E;\hbar)$ with error terms that are uniformly
controlled for small $E$ and $\hbar$, and construct the scattering matrix as
well as the semiclassical spectral measure associated to $H(\hbar)$. This is
crucial in order to obtain decay bounds for the corresponding wave and
Schr\"odinger flows. As an application we consider the wave equation on a
Schwarzschild background for large angular momenta $\ell$ where the role of the
small parameter $\hbar$ is played by $\ell^{-1}$. It follows from the results
in this paper and \cite{DSS2}, that the decay bounds obtained in \cite{DSS1},
\cite{DS} for individual angular momenta $\ell$ can be summed to yield the
sharp $t^{-3}$ decay for data without symmetry assumptions.
