Darboux transformation in black hole perturbation theory
Darboux transformation in black hole perturbation theory
复制标题
黑洞微扰理论中的达布变换
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
D. Kennefick
中科院分区：
文献类型：
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作者：
K. Glampedakis;A. Johnson;D. Kennefick
常微分方程之间的达布变换是一项19世纪的技术，在量子理论中被广泛应用，用于为具有特定谱性质的薛定谔方程产生精确可解的势。在本文中，我们证明了同样的变换出现在黑洞理论中，例如，与轴向和极向施瓦西摄动的Zerilli和Regge-Wheeler方程有关。变换表明这两个方程是等谱的，这是一个众所周知的结果，它的方法以不同的名称被反复引入。我们强调了所谓的代数特殊解在黑洞达布理论中所起的关键作用，并表明在Kerr摄动的钱德拉塞卡-德维勒方程之间存在类似的关系。最后，我们讨论了该方法在处理长程势时的局限性，并探讨了广义达布变换提供的可能性。
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 Schrodinger 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.