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arXiv 2004.12513gr-qcmath-phmath.MP
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特殊情况下角型Teukolsky方程的精确解

Exact solutions of the angular Teukolsky equation in particular cases

Chang-Yuan Chen, Yuan You, Xiao-Hua Wang, Fa-Lin Lu, Dong-Sheng Sun, Shi-Hai Dong

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6 pages, 2 tables and 5 figures
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摘要：

在这项工作中，我们提出了一个新的计划来解决角Teukolsky方程的特殊情况：$m=0, s=0$。我们首先将此方程转化为一个合流Heun微分方程，然后构造Wronskian行列式来计算特征值和归一化特征函数。我们发现，较大的$l$的特征值近似由$_{0}{A_{l0}} \approx [l(l + 1) - τ_{R}^2/2] - i\;τ_{I}^2/2$与任意的$τ^2=τ_R^2 + i\,τ_{I}^2$。对于$τ_R^2>0$，基态的角概率分布（APD）向南北极移动，而对于$τ_R^2\leq0$，则向赤道聚集。然而，我们也注意到，大角动量$l$的APD总是向北极和南极移动，无论选择$τ^2$。

英文摘要：

In this work, we propose a new scheme to solve the angular Teukolsky equation for the particular case: $m=0, s=0$. We first transform this equation to a confluent Heun differential equation and then construct the Wronskian determinant to calculate the eigenvalues and normalized eigenfunctions. We find that the eigenvalues for larger $l$ are approximately given by $_{0}{A_{l0}} \approx [l(l + 1) - τ_{R}^2/2] - i\;τ_{I}^2/2$ with an arbitrary $τ^2=τ_R^2 + i\,τ_{I}^2$. The angular probability distribution (APD) for the ground state moves towards the north and south poles for $τ_R^2>0$, but aggregates to the equator for $τ_R^2\leq0$. However, we also notice that the APD for large angular momentum $l$ always moves towards the north and south poles , regardless the choice of $τ^2$.

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