标题： 开放系统中的波：本征函数展开 显示英文标题

标题： Waves in Open Systems: Eigenfunction Expansions

摘要： 开放系统不是保守的，因为能量可以逃逸到外部。 单独的开放系统因此不是保守的。 结果，时间演化算符在通常意义上不是厄米特的，本征函数（空间和时间分离的解）不再是正常模态，而是准正常模态（QNMs），其频率$\omega$是复数。 QNMs 分析已成为研究开放系统的强大工具。 以往的研究大多是针对特定系统的，并使用少量 QNMs 提供近似描述。 在这里，我们回顾了旨在统一处理的最新进展。 该方法导致了一个与保守、厄米特系统非常相似的数学结构。 因此，许多用于后者的数学工具可以被直接转录。 重点放在 QNMs 形成向外波函数完整集的情况，这样原则上所有 QNMs 一起可以给出动力学的精确描述。 回顾了在微球光学和黑洞引力波中的应用，并概述了进一步发展的方向。 显示英文摘要

摘要： An open system is not conservative because energy can escape to the outside. An open system by itself is thus not conservative. As a result, the time-evolution operator is not hermitian in the usual sense and the eigenfunctions (factorized solutions in space and time) are no longer normal modes but quasinormal modes (QNMs) whose frequencies $\omega$ are complex. QNM analysis has been a powerful tool for investigating open systems. Previous studies have been mostly system specific, and use a few QNMs to provide approximate descriptions. Here we review recent developments which aim at a unifying treatment. The formulation leads to a mathematical structure in close analogy to that in conservative, hermitian systems. Many of the mathematical tools for the latter can hence be transcribed. Emphasis is placed on those cases in which the QNMs form a complete set for outgoing wavefunctions, so that in principle all the QNMs together give an exact description of the dynamics. Applications to optics in microspheres and to gravitational waves from black holes are reviewed, and directions for further development are outlined.