  The $O(n)$ $\phi^4$ model on a strip bounded by a pair of planar free
surfaces at separation $L$ can be solved exactly in the large-$n$ limit in
terms of the eigenvalues and eigenfunctions of a self-consistent
one-dimensional Schr\"odinger equation. The scaling limit of a continuum
version of this model is considered. It is shown that the self-consistent
potential can be eliminated in favor of scattering data by means of
appropriately extended methods of inverse scattering theory. The scattering
data (Jost function) associated with the self-consistent potential are
determined for the ${L=\infty}$ semi-infinite case in the scaling regime for
all values of the temperature scaling field $t=(T-T_c)/T_c$ above and below the
bulk critical temperature $T_c$. These results are used in conjunction with
semiclassical and boundary-operator expansions and a trace formula to derive
exact analytical results for a number of quantities such as two-point
functions, universal amplitudes of two excess surface quantities, the universal
amplitude difference associated with the thermal singularity of the surface
free energy, and potential coefficients. The asymptotic behaviors of the scaled
eigenenergies and eigenfunctions of the self-consistent Schr\"odinger equation
as function of $x= t(L/\xi_+)^{1/\nu}$ are determined for $x\to-\infty$. In
addition, the asymptotic ${x\to -\infty}$ forms of the universal finite-size
scaling functions $\Theta(x)$ and $\vartheta(x)$ of the residual free energy
and the Casimir force are computed exactly to order $1/x$, including their
$x^{-1}\ln|x|$ anomalies.
