黑洞微扰理论中的达布变换
The Darboux transformation in black hole perturbation theory
摘要：
常微分方程之间的达布变换是一种19世纪的技术，在量子理论中广泛使用，用于产生具有特定光谱特性的薛定谔方程的精确可解势。在本文中，我们表明，同样的变换出现在黑洞理论，有关，例如，Zerilli和Regge-Wheeler方程的轴向和极史瓦西扰动。变换揭示了这两个方程是等谱的，这是一个众所周知的结果，其方法已被多次重新引入不同的名称。我们强调的关键作用，所谓的代数特殊的解决方案发挥在黑洞达布理论，并表明，一个类似的关系之间存在的Kerr扰动的Erasekhar-Detweiler方程。最后，我们讨论了该方法的局限性时，处理长程电位和探索的可能性提供了一个广义的达布变换。
英文摘要：
The Darboux transformation between ordinary differential equations is a 19th century technique that has seen wide use in quantum theory for producing exactly solvable potentials for the Schrödinger equation with specific spectral properties. In this paper we show that the same transformation appears in black hole theory, relating, for instance, the Zerilli and Regge-Wheeler equations for axial and polar Schwarzschild perturbations. The transformation reveals these two equations to be isospectral, a well known result whose method has been repeatedly reintroduced under different names. We highlight the key role that the so-called algebraically special solutions play in the black hole Darboux theory and show that a similar relation exists between the Chandrasekhar-Detweiler equations for Kerr perturbations. Finally, we discuss the limitations of the method when dealing with long-range potentials and explore the possibilities offered by a generalised Darboux transformation.
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