The three-dimensional $O(n\to\infty)$ $\phi^4$ model on a strip with free boundary conditions: exact results for a nontrivial dimensional crossover

The $O(2)$ $\phi^4$ model on a 3D film of thickness $L$ with free boundaries is relevant for the explanation of the thinning of wetting layers of ${}^4$He caused by critical Casimir forces near and below the $\lambda$-transition. Just as its $O(n)$ analog, the model has long-range order below the bulk critical temperature $T_c$ if $L=\infty$, but remains disordered for all $T>0$ when $L<\infty$. A proper analysis of its scaling behavior near $T_c$ is challenging: it involves a nontrivial dimensional crossover in addition to bulk, boundary, and finite-size critical behaviors. The $n\to\infty$ limit of the model can be solved exactly in terms of the eigenvalues and eigenfunctions of a self-consistent Schr\"odinger equation whose potential $v(z)$ becomes singular at the boundary planes. Complementing recent numerically exact results, we derive various exact analytical results for series expansion coefficients of $v(z)$, its $L=\infty$ scattering data for all values $m\gtreqless 0$ of the termperature scaling field, and the low-temperature asymptotic behavior of the residual free energy and the Casimir force using a combination of boundary-operator and short-distance expansions, proper extensions of inverse scattering theory, new trace formulae, and semi-classical expansions.