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The visualization of the angular probability distribution for the angular Teukolsky equation with m ≠ 0
International Journal of Quantum Chemistry ( IF 2.3 ) Pub Date : 2020-11-27 , DOI: 10.1002/qua.26546 Chang‐Yuan Chen 1 , Dong‐Sheng Sun 1 , Guo‐Hua Sun 2 , Xiao‐Hua Wang 1 , Yuan You 1 , Shi‐Hai Dong 3, 4
International Journal of Quantum Chemistry ( IF 2.3 ) Pub Date : 2020-11-27 , DOI: 10.1002/qua.26546 Chang‐Yuan Chen 1 , Dong‐Sheng Sun 1 , Guo‐Hua Sun 2 , Xiao‐Hua Wang 1 , Yuan You 1 , Shi‐Hai Dong 3, 4
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We present the exact solutions of the angular Teukolsky equation with m ≠ 0 given by a confluent Heun function. This equation is first transformed to a confluent Heun differential equation through some variable transformations. The Wronskian determinant, which is constructed by two linearly dependent solutions, is used to calculate the eigenvalues precisely. The normalized eigenfunctions can be obtained by substituting the calculated eigenvalues into the unnormalized eigenfunctions. The relations among the linearly dependent eigenfunctions are also discussed. When 
, the eigenvalues are approximately expressed as 
for small |c|2 but large l. The isosurface and contour visualizations of the angular probability distribution (APD) are presented for the cases of the real and complex values c2. It is found that the APD has obvious directionality, but the northern and southern hemispheres are always symmetrical regardless of the value of the parameter c2, which is real or imaginary.
中文翻译：
m≠0的角Teukolsky方程的角概率分布的可视化
我们 给出了由合流Heun函数给出的m ≠0的角Teukolsky方程的精确解。首先通过一些变量转换将该方程转换为合流的Heun微分方程。由两个线性相关解构成的Wronskian行列式用于精确计算特征值。可以通过将计算出的特征值代入未归一化的特征函数来获得归一化的特征函数。还讨论了线性相关特征函数之间的关系。当，特征值近似表示为对小| c | 2但大l。针对实值和复数值c 2的情况，给出了角概率分布（APD）的等值面和轮廓可视化。发现APD具有明显的方向性，但是北半球和南半球始终是对称的，而不管参数c 2的值是实还是虚。
更新日期：2020-11-27
, the eigenvalues are approximately expressed as for small |c|2 but large l. The isosurface and contour visualizations of the angular probability distribution (APD) are presented for the cases of the real and complex values c2. It is found that the APD has obvious directionality, but the northern and southern hemispheres are always symmetrical regardless of the value of the parameter c2, which is real or imaginary.
中文翻译：
m≠0的角Teukolsky方程的角概率分布的可视化
我们 给出了由合流Heun函数给出的m ≠0的角Teukolsky方程的精确解。首先通过一些变量转换将该方程转换为合流的Heun微分方程。由两个线性相关解构成的Wronskian行列式用于精确计算特征值。可以通过将计算出的特征值代入未归一化的特征函数来获得归一化的特征函数。还讨论了线性相关特征函数之间的关系。当
，特征值近似表示为对小| c | 2但大l。针对实值和复数值c 2的情况，给出了角概率分布（APD）的等值面和轮廓可视化。发现APD具有明显的方向性，但是北半球和南半球始终是对称的，而不管参数c 2的值是实还是虚。
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