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Spin-weighted spheroidal harmonics play a central role in the mathematical description of diverse physical phenomena, including black-hole perturbation theory and wave scattering.
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In the context of black-hole perturbation theory, the parameter c c stands for a ω a\omega , where a a is the angular momentum per unit mass of the black hole and ω \omega is the (conserved) frequency of the perturbation mode [ 2 ]
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We shall assume without loss of generality that c > 0 c>0 . For c < 0 c<0 one should simply replace { s , m } → { − s , − m } \{s,m\}\to\{-s,-m\} in the final answer [see Eqs. ( 16
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H. Yang, D. A. Nichols, F. Zhang, A. Zimmerman, Z. Zhang and Y. Chen, arXiv:1207.4253
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Note that in the quantum-mechanical terminology − U -U stands for 2 m ℏ 2 ( E − V ) {{2m}\over{\hbar^{2}}}(E-V) , where E , V E,V , and m m are the total energy, potential energy, and mass of the particle, respectively
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The effective potential U ( θ ) U(\theta) becomes symmetric (invariant under the transformation θ → π − θ \theta\to\pi-\theta ) in the scalar ( s = 0 s=0 ) case
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Higher order corrections to the asymptotic eigenvalues [see formula ( 17
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