We study analytically and numerically the formation of bound states in the continuum (BICs) in quantum-mechanical and optical spatially symmetric multimode waveguide systems. The widely used Friedrich-Wintgen model predicts a BIC for two (nearly) degenerate eigenstates of the resonator. However, numerical calculations typically show that BICs may appear away from the degeneracy point (or avoided crossing region) in the energy-parameter space. From the two-mode point of view based on the Friedrich-Wintgen model, such BICs can be considered as accidental. In this paper, we go beyond the two-mode approximation and, appealing to the notion of intermode bound states, provide an illustrative procedure for deriving conditions for BIC formation within an arbitrary finite-mode approximation. In particular, we show that a three-mode approximation allows the description of a continuous transition of a BIC between different pairs of modes, which can be associated with it within the two-mode model. Also, a manifestation of exceptional points as (anti)resonance coalescence is discussed. Analytical conclusions are verified by the results of numerical simulations of two-dimensional quantum-mechanical and optical stubbed waveguides with confining impurities in the stub. Moreover, numerical simulations confirm the existence of BICs in optical subwavelength resonators, which were earlier predicted in quantum-mechanical waveguides.

中文翻译：

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多模共振、模间束缚态和波导连续体中的束缚态

我们分析和数值研究量子力学和光学空间对称多模波导系统中连续谱 (BIC) 束缚态的形成。广泛使用的 Friedrich-Wintgen 模型预测谐振器的两个（几乎）简并本征态的 BIC。然而，数值计算通常表明，BIC 可能会出现在能量参数空间中远离简并点（或避免的交叉区域）的位置。从基于弗里德里希-温特根模型的双模态来看，这样的 BIC 可以被认为是偶然的。在本文中，我们超越了双模近似，并利用模间束缚态的概念，提供了一个说明性程序，用于在任意有限模近似内推导 BIC 形成的条件。尤其是，我们表明，三模式近似允许描述 BIC 在不同模式对之间的连续转换，这可以在双模式模型中与之相关联。此外，讨论了作为（反）共振合并的异常点的表现。分析结论通过二维量子力学和光学短截波导的数值模拟结果得到验证，其中限制了短截线中的杂质。此外，数值模拟证实了 BIC 在光学亚波长谐振器中的存在，这是早先在量子力学波导中预测的。分析结论通过二维量子力学和光学短截波导的数值模拟结果得到验证，其中限制了短截线中的杂质。此外，数值模拟证实了 BIC 在光学亚波长谐振器中的存在，这是早先在量子力学波导中预测的。分析结论通过二维量子力学和光学短截波导的数值模拟结果得到验证，其中限制了短截线中的杂质。此外，数值模拟证实了 BIC 在光学亚波长谐振器中的存在，这是早先在量子力学波导中预测的。