In previous work [1] we proposed an improvement of the WKB-based semianalytic technique of Iyer and Will for calculation of the quasiormal modes of black holes by constructing the Padé approximants of the formal series for $ω^{2}.$ It has been demonstrated that (within the domain of applicability) the diagonal Padé transforms $\mathcal{P}_{6}^{6}$ and $\mathcal{P}_{7}^{6}$ are always in a very good agreement with the numerical results. In this paper we present a further extension of the method. We show that it is possible to reproduce many known numerical results with a great accuracy (or even exactly) if the Padé transforms are constructed from the perturbative series of a really high order. In our calculations the order depends on the problem but it never exceeds 700. For example, the frequencies of the gravitational mode $l=2,$ $n=0$ calculated with the aid of the Padé approximants and within the framework of the continued fractions method agree to 24 decimal places. The use of such a large number of terms is necessary as the stabilization of the quasinormal frequencies can be slow. Our results reveal some unexpected features of the WKB-based approximations and may shed some fresh light on the problem of overtones.
Quasinormal behavior of the D-dimensional Schwarzshild black hole and higher order WKB approach
研究了$D$维Schwarzshild黑洞的特征模.证明了复拟正规模的实部（代表实振荡频率）正比于维数与逆视界半径$\sim D r_{0}^{-1}$的乘积。导出了大多极数$l$和任意$D$的渐近公式。此外，本文还将计算QN模的WKB公式从程函近似的三阶推广到六阶。这为我们提供了一种准确且经济的方法来计算准正态频率。
An efficient higher-order WKB code for quasinormal modes and greybody factors
Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava（奥帕瓦西里西亚大学物理学院理论物理学与天体物理学研究中心）；Institute of Physics, Maria Curie-Skłodowska University（玛丽亚·居里-斯克隆多夫斯卡大学物理学院）；Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC (UFABC)（ABC联邦大学数学、计算与认知中心）
Quasinormal modes of four-dimensional regular black holes in quasi-topological gravity: Overtones' outburst via WKB method
Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava（奥帕瓦西里西亚大学物理研究所理论与天体物理学研究中心）；Department of Physics, Faculty of Science, University of Hradec Kralove（赫拉德茨克拉洛韦大学理学院物理系）
Quasinormal modes and greybody factors of the novel four dimensional Gauss-Bonnet black holes in asymptotically de Sitter space time: Scalar, Electromagnetic and Dirac perturbations
作为对WKB的改进，我们利用三阶WKB近似和Padé近似计算了最近提出的新型四维Gauss-Bonnet de Sitter黑洞在标量场、电磁场和Dirac场扰动下的低准正规模频率.计算了高斯-邦纳耦合$α$和宇宙学常数$Λ$对QNM频率实部和虚部的影响。我们还研究了上述背景下所有三种不同类型扰动的灰体因子和程函极限。
Quasinormal modes of rotating black holes beyond general relativity in the WKB approximation
Observatoire de Paris, Universite PSL（巴黎天文台，巴黎文理研究大学）；CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL（里斯本大学，技术研究所物理系与理论研究中心）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））；Johns Hopkins University（约翰斯·霍普金斯大学）
Resumming Kerr Quasinormal-Mode Frequencies: Accuracy and Breakdown Near Extremality
Illinois Center for Advanced Studies of the Universe & Department of Physics, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校宇宙高级研究中心与物理系）；Department of Physics, University of Virginia（弗吉尼亚大学物理系）
Cesàro convergence of the high-order WKB method and its applications to black-hole overtones and long-lived modes
Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava（奥帕瓦西里西亚大学物理学院理论物理与天体物理学研究中心）；Institute of Physics, Maria Curie-Skłodowska University（玛丽亚·居里-斯夸多夫斯卡大学物理学院）；Centro de Matemática, Computação e Cognição (CMCC), Universidade Federal do ABC (UFABC)（ABC联邦大学数学、计算与认知中心）
The Highly Real Quasinormal Modes of Schwarzschild-Anti De Sitter Black Holes
最近的一项研究表明，Schwarzschild-anti de Sitter黑洞的渐近拟正规模谱可能存在一个有趣的区域。在此渐近区域内，拟正规模态频率的实部趋于无穷大，而阻尼率趋于有限值。这些准正规模式的频率计算使用基于复坐标WKB方法的分析技术。在本文中，我们使用不同的分析技术来计算这样的准正规模式的频率。本文的结果进一步支持了这些模式存在的可能性。
Spectral instability of parametrized black hole quasinormal modes in the high-overtone limit via the exact WKB analysis
Osaka Metropolitan University（大阪公立大学）；Osaka Central Advanced Mathematical Institute (OCAMI)（大阪中央先进数学研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学中心交叉理论与数学科学中心）；Research Center for the Early Universe (RESCEU), Graduate School of Science, The University of Tokyo（东京大学理学研究科早期宇宙研究中心）；Department of Physics , Kyoto University（京都大学物理系）；Center for Gravitational Physics and Quantum Information,Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理中心引力物理与量子信息）；The Hakubi Center for Advanced Research, Kyoto University（京都大学白兔高级研究中心）
Scalar field quasinormal modes of noncommutative high dimensional Schwarzschild-Tangherlini black hole spacetime with smeared matter sources
本文研究了非对易$D$维Schwarzschild-Tangherlini黑洞时空的无质量标量拟正规模。利用WKB近似方法、渐近迭代法（AIM）和反势法（IPM），通过改变广义弥散物质分布和不同维数下的允许特征参数（$k$和$θ$），详细分析了无质量标量QNM的频率.研究发现，在非对易$D$维Schwarzschild黑洞周围的标量微扰QNM频率中，高阶WKB近似存在不收敛性.我们的数值结果也得到了AIM方法和IPM方法的验证，因此第三阶WKB方法的结果比高阶WKB方法的结果更可靠。在$4\leq D \leq7$的维数范围内，得到了标量QNM作为不同参数（非对易参数$θ$、涂抹物质分布参数$k$、多极数$l$和主节点数$n$）的函数.此外，我们还研究了非对易高维Schwarzschild-Tangherlini黑洞背景下标量场的动力学演化。
