论文标题
Vertex Green的四分之一平面功能。功能方程，添加剂交叉和Lamé功能之间的链接
Vertex Green's functions of a quarter-plane. Links between the functional equation, additive crossing and Lamé functions
论文作者
论文摘要
在我们以前的工作（Assier \＆Shanin，QJMAM，2019年）中，我们在两个复杂的变量中给出了新的光谱公式，这两个复杂变量与四分之一平面相关的是平面衍射问题。特别是，我们表明未知的光谱函数满足了其分支集合的添加剂交叉的条件。在本文中，我们研究了非常相似的光谱问题，并展示了如何利用添加剂交叉，以表达其在Lamé功能方面的解决方案。获得的解决方案可以被认为是量身定制的顶点绿色的功能，其在近场中的行为与Laplace-Beltrami操作员的特征值直接相关。这很重要，因为到目前为止，从未通过多变量复杂分析方法获得了四分之一平面尖端的正确近场行为。
In our previous work (Assier \& Shanin, QJMAM, 2019), we gave a new spectral formulation in two complex variables associated with the problem of plane-wave diffraction by a quarter-plane. In particular, we showed that the unknown spectral function satisfies a condition of additive crossing about its branch set. In this paper, we study a very similar class of spectral problem, and show how the additive crossing can be exploited in order to express its solution in terms of Lamé functions. The solutions obtained can be thought of as tailored vertex Green's functions whose behaviours in the near-field are directly related to the eigenvalues of the Laplace-Beltrami operator. This is important since the correct near-field behaviour at the tip of the quarter-plane had so far never been obtained via a multivariable complex analysis approach.
