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The $O(n)$ $\phi^4$ model on a strip bounded by a pair of planar freesurfaces at separation $L$ can be solved exactly in the large-$n$ limit interms of the eigenvalues and eigenfunctions of a self-consistentone-dimensional Schr\"odinger equation. The scaling limit of a continuumversion of this model is considered. It is shown that the self-consistentpotential can be eliminated in favor of scattering data by means ofappropriately extended methods of inverse scattering theory. The scatteringdata (Jost function) associated with the self-consistent potential aredetermined for the ${L=\infty}$ semi-infinite case in the scaling regime forall values of the temperature scaling field $t=(T-T_c)/T_c$ above and below thebulk critical temperature $T_c$. These results are used in conjunction withsemiclassical and boundary-operator expansions and a trace formula to deriveexact analytical results for a number of quantities such as two-pointfunctions, universal amplitudes of two excess surface quantities, the universalamplitude difference associated with the thermal singularity of the surfacefree energy, and potential coefficients. The asymptotic behaviors of the scaledeigenenergies and eigenfunctions of the self-consistent Schr\"odinger equationas function of $x= t(L/\xi_+)^{1/\nu}$ are determined for $x\to-\infty$. Inaddition, the asymptotic ${x\to -\infty}$ forms of the universal finite-sizescaling functions $\Theta(x)$ and $\vartheta(x)$ of the residual free energyand the Casimir force are computed exactly to order $1/x$, including their$x^{-1}\ln|x|$ anomalies.

physical review. e, statistical, nonlinear and soft matter physics

Inverse scattering-theory approach to the exact large-$n$ solutions of  $O(n)$ $\phi^4$ models on films and semi-infinite systems bounded by free  surfaces