关于复值内部传输本征值轨迹的研究
On trajectories of complex-valued interior transmission eigenvalues
AI总结：
本文研究复值内部传输本征值轨迹的性质，发现其与狄利克雷本征值存在一一对应关系，并提出对一般单连通散射体的猜想。
摘要：
本文研究了由介质折射率参数化内部传输问题的复值本征值轨迹的性质。我们的理论分析表明，在单位圆盘的情况下，与实轴的唯一交点以及当折射率趋于无穷大时的唯一轨迹极限点都是拉普拉斯算子的狄利克雷本征值。补充的数值实验还揭示了狄利克雷本征值与复值内部传输本征值轨迹之间存在一一对应关系。我们还检查了其他散射体，发现可以得到类似的数值观察结果。我们对一般单连通散射体提出一个猜想来总结我们的结果。
英文摘要：
This paper investigates properties of complex-valued eigenvalue trajectories for the interior transmission problem parametrized by the index of refraction for homogeneous media. Our theoretical analysis for the unit disk shows that the only intersection points with the real axis, as well as the unique trajectorial limit points as the refractive index tends to infinity, are Dirichlet eigenvalues of the Laplacian. Complementing numerical experiments even give rise to an underlying one-to-one correspondence between Dirichlet eigenvalues of the Laplacian and complex-valued interior transmission eigenvalue trajectories. We also examine other scatterers than the disk for which similar numerical observations can be made. We summarize our results in a conjecture for general simply-connected scatterers.
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