发表机构
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley; Department of Mathematics, Rutgers University; Department of Mathematics and Statistics, McGill University（德克萨斯大学里奥格兰德河谷分校数学与统计科学学院; 罗格斯大学数学系; 麦吉尔大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过计算机辅助证明，严格验证了非线性光学中二次谐波产生模型在低频区域存在广义透射特征值，其特征函数具有爆破解行为，并提供了开源算法实现。
AI 中文摘要
本文针对高阶谐波产生背景下非线性光学中出现的广义透射特征值的存在性，给出了严格的计算机辅助证明，这一结果在Cakoni、Hovsepyan、Lassas和Vogelius的论文（SIAM J. Math. Anal. 57 (2025), 1370-1405）中被猜想。分析针对一维非线性介质进行，在该介质中问题简化为满足非标准边界条件的非线性齐次常微分方程组。这些特征值对应于探测频率$\omega$，对于这些频率存在一个非平凡的入射$\omega$波，使得产生的二次谐波场在非线性介质的紧致支撑之外不持续存在，从而使其非线性特性对外部观察者不可检测。基于早期的数值证据，我们证明在低频区域存在广义透射特征值，其相关特征函数在频率趋于零时表现出爆破解行为。该证明结合了分析论证与验证数值方法，采用Newton-Kantorovich框架和区间算术来严格控制近似误差。计算机辅助证明所依据的算法实现可在GitHub仓库中获取，网址为this http URL，作者为Dominic Blanco（2026年）。
英文摘要
This paper provides a rigorous computer-assisted proof of the existence of generalized transmission eigenvalues arising in nonlinear optics in the context of high-order harmonic generation, a result conjectured in Cakoni, Hovsepyan, Lassas, and Vogelius, SIAM J. Math. Anal. 57 (2025), 1370-1405. The analysis is carried out for a one-dimensional nonlinear medium, where the problem reduces to a coupled system of nonlinear homogeneous ordinary differential equations subject to nonstandard boundary conditions. These eigenvalues correspond to probing frequencies $ω$ for which there exists a nontrivial incident $ω$-wave such that the second-harmonic field generated does not persist outside the compact support of the nonlinear medium, thereby rendering its nonlinear properties undetectable to an external observer. Building on earlier numerical evidence, we establish that, in the low-frequency regime, there exist generalized transmission eigenvalues whose associated eigenfunctions exhibit blow-up behavior as the frequency tends to zero. The proof combines analytical arguments with validated numerics, employing a Newton-Kantorovich framework together with interval arithmetic to rigorously control approximation errors. The algorithmic implementation underlying the computer-assisted proof is made available in the GitHub repository TransmissionEigenvalues.jl by Dominic Blanco (2026).
