AI 中文总结
本文针对麦克斯韦传输本征值问题开展定量分析，建立全计数函数的魏尔渐近定律，证明表面局域模式受限计数函数的魏尔型上下界，首次定量表明表面局域本征函数是高频谱不可忽略的部分。
AI 中文摘要
本工作研究时谐电磁散射的表面局域传输本征函数。该现象的现有结果多为定性的，仅证明存在性而未量化分布。本文针对麦克斯韦传输本征值问题开展定量分析，涵盖TE模式与TM模式。我们首先建立了全计数函数的魏尔渐近定律，其随半径R呈三次方增长；随后证明了表面局域模式受限计数函数的魏尔型上下界，并表明二者共享三次方增长阶。据我们所知，这是同类问题的首个定量结果，证明表面局域本征函数构成高频谱中不可忽略的部分。
英文摘要
This work studies surface-localized transmission eigenfunctions for time-harmonic electromagnetic scattering. Prior results on this phenomenon are mostly qualitative, proving existence without quantifying distribution. Here we provide a quantitative analysis for the Maxwell transmission eigenvalue problem, covering both TE and TM modes. We first establish a Weyl asymptotic law for the full counting function, with cubic growth with respect to the radius $R$. We then prove Weyl-type upper and lower bounds for the counting functions restricted to surface-localized modes, and demonstrate that they share the same growth order three. To our knowledge, this is the first quantitative result of its kind, demonstrating that surface-localized eigenfunctions form a non-negligible fraction of the high-frequency spectrum.
