Abstract Motivated by applications of recent interest related to propagation problems in the left-handed 2D inductor-capacitor metamaterial and standard 2D inductor-capacitor lattice with monochromatic inputs along the left and bottom boundary of a rectangular slab, we address the problem of wave diffraction on the 2D square lattice in a quadrant. The peculiar structure allows us to consider problems on half-plane, consequently, we study Dirichlet problems for the two-dimensional discrete Helmholtz equation in a half-plane and a quadrant. In view of the existence and uniqueness of solution, we provide new results for the real wave number k ∈ ( 0 , 2 2 ) ∖ { 2 } without passing to the complex wave number and derive an exact representation formula for solutions. For this purpose, we use the notion of the radiating solution and propose sufficient conditions for the given boundary data at infinity.

中文翻译：

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象限内二维晶格波的衍射问题

摘要 受与左手 2D 电感-电容器超材料和标准 2D 电感-电容器晶格中传播问题相关的应用的启发，单色输入沿矩形板的左侧和底部边界，我们解决了波衍射问题象限中的 2D 方格。奇特的结构使我们能够考虑半平面上的问题，因此，我们研究了半平面和象限内二维离散亥姆霍兹方程的狄利克雷问题。鉴于解的存在性和唯一性，我们对实波数k ∈ ( 0 , 2 2 ) ∖ { 2 } 提供了新的结果，而无需传递到复波数，并推导出解的精确表示公式。以此目的，