薄膜和自由面约束半无限系统上$O(n)$ $ϕ^4$模型精确大$n$解的逆散射理论方法
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
摘要：
由一对平面自由表面以分离$L$为界的带上的$O(n)$ $ϕ^4$模型可以在大$n$极限下根据自洽的一维薛定谔方程的本征值和本征函数精确求解。被认为是一个连续的版本，这个模型的标度限制。它表明，自洽势可以消除有利于散射数据通过适当的扩展方法的逆散射理论。与自洽势相关的散射数据（Jost函数）被确定为${L=\infty}$半无限的情况下，在标度制度的温度标度场$t=(T-T_c)/T_c$以上和以下的体积临界温度$T_c$的所有值。这些结果被用于结合半经典和边界算子展开和一个跟踪公式，以获得精确的分析结果的数量，如两点函数，两个多余的表面量的普遍振幅，与热奇异性的表面自由能的普遍振幅差，和潜在的系数。对$x\to-\infty$确定了自洽薛定谔方程的标度本征能和标度本征函数作为$x= t(L/ξ_+)^{1/ν}$的函数的渐近行为.此外，还精确地计算了剩余自由能和Casimir力的通用有限尺度标度函数$Θ(x)$和$\vartheta(x)$的渐近${x\to -\infty}$形式，使之满足$1/x$阶，包括它们的$x^{-1}\ln|x|$反常.
英文摘要：
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.
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