Summary In our previous work (R. C. Assier and A. V. Shanin, Q. J. Mech. Appl. Math., 72, 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 article, 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.

中文翻译：

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四分之一平面的顶点格林函数：函数方程、加性交叉和 Lamé 函数之间的联系

总结在我们之前的工作（RC Assier 和 AV Shanin，QJ Mech. Appl. Math.，72，2019）中，我们在与四分之一平面的平面波衍射问题相关的两个复变量中给出了一个新的光谱公式。特别是，我们证明了未知谱函数满足关于其分支集的加性交叉条件。在本文中，我们研究了一类非常相似的谱问题，并展示了如何利用加性交叉来表达其在 Lamé 函数方面的解决方案。获得的解可以被认为是定制的顶点格林函数，其在近场中的行为与拉普拉斯-贝尔特拉米算子的特征值直接相关。