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

The $O(n)$ $ϕ^4$ 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 Schrödinger 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 Schrödinger equation as function of $x= t(L/ξ_+)^{1/ν}$ are determined for $x\to-\infty$. In addition, the asymptotic ${x\to -\infty}$ forms of the universal finite-size scaling functions $Θ(x)$ 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.