We are interested in a 2D propagation medium obtained from a localized perturbation of a reference homogeneous periodic medium. This reference medium is a " thick graph " , namely a thin structure (the thinness being characterized by a small parameter e > 0) whose limit (when e tends to 0) is a periodic graph. The perturbation consists in changing only the geometry of the reference medium by modifying the thickness of one of the lines of the reference medium. In the first part of this work, we proved that such a geometrical perturbation is able to produce localized eigenmodes (the propagation model under consideration is the scalar Helmholtz equation with Neumann boundary conditions). This amounts to solving an eigenvalue problem for the Laplace operator in an unbounded domain. We used a standard approach of analysis that consists in (1) find a formal limit of the eigenvalue problem when the small parameter tends to 0, here the formal limit is an eigenvalue problem for a second order differential operator along a graph; (2) proceed to an explicit calculation of the spectrum of the limit operator; (3) deduce the existence of eigenvalues as soon as the thickness of the ladder is small enough. The objective of the present work is to complement the previous one by constructing and justifying a high order asymptotic expansion of these eigenvalues (with respect to the small parameter e) using the method of matched asymptotic expansions. In particular, the obtained expansion can be used to compute a numerical approximation of the eigenvalues and of their associated eigenvectors. An algorithm to compute each term of the asymptotic expansion is proposed. Numerical experiments validate the theoretical results.

中文翻译：

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薄和无限梯形域中的陷阱模式。第 2 部分：渐近分析和数值应用

我们对从参考均匀周期介质的局部扰动获得的 2D 传播介质感兴趣。这个参考介质是一个“厚图”，即一个薄结构（薄的特征是一个小参数 e > 0），其极限（当 e 趋于 0 时）是一个周期图。扰动在于通过修改参考介质的线之一的厚度来仅改变参考介质的几何形状。在这项工作的第一部分，我们证明了这种几何扰动能够产生局部本征模式（所考虑的传播模型是具有 Neumann 边界条件的标量亥姆霍兹方程）。这相当于在无界域中解决拉普拉斯算子的特征值问题。我们使用了一种标准的分析方法，包括（1）当小参数趋于 0 时找到特征值问题的形式极限，这里的形式极限是沿图的二阶微分算子的特征值问题；(2) 进行极限算子频谱的显式计算；(3) 只要梯子的厚度足够小，就可以推导出特征值的存在。当前工作的目标是通过使用匹配渐近展开的方法构造和证明这些特征值的高阶渐近展开（相对于小参数 e）来补充前一个。特别是，获得的扩展可用于计算特征值及其相关特征向量的数值近似值。提出了一种计算渐近展开式每一项的算法。数值实验验证了理论结果。