We consider integrals of products of Bessel functions and of spherical Bessel functions, combined with a Gaussian factor guaranteeing convergence at infinity. These integrals arise in wave and quantum mechanical scattering problems of open systems containing cylindrical or spherical scatterers, particularly when those problems are considered in the framework of complex resonant modes. Explicit representations are obtained for the integrals, building on those in the 1992 paper by McPhedran, Dawes and Scott. Attention is paid to those sums with a distributive part arising as the Gaussian tends towards the unit function. In this limit, orthogonality and normalisability of complex modes are investigated.

中文翻译：

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“轻柔地杀死三重”：复杂共振态的解析积分

我们考虑贝塞尔函数和球贝塞尔函数的乘积积分，并结合一个高斯因数，以保证无穷大处的收敛。这些积分出现在包含圆柱或球形散射体的开放系统的波和量子机械散射问题中，特别是在复杂共振模式的框架中考虑这些问题时。在1992年McPhedran，Dawes和Scott的论文的基础上，获得了积分的显式表示。当高斯趋向于单位函数时，应注意那些具有分布部分的和。在此极限下，研究了复模的正交性和归一化。