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We argue about the (non) existence of {\it superregular} scalar clouds (i.e., bound states of a massive and complex-valued scalar field $\Psi$) around exact {\it extremal} ($a = M$) Kerr black holes (BH's) possessing {\it bounded radial derivatives at the horizon} (in Boyer-Lindquist coordinates) as opposed to similar cloud solutions that exist but with unbounded derivatives in the same coordinate system.
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The coefficient μ \mu has dimensions 1/length. The actual coefficient with mass units is given by μ p := ℏ c μ / G \mu_{p}:=\hbar c\mu/G , and in natural units ( G G = c c = ℏ \hbar = 1) both coincide
Cited in the paper.
We remarked a typo in the original paper [ 4 ] : the dimensionless radial coordinate z z is defined by z = 2 M ( r / r H ext − 1 ) μ 2 − m 2 4 z=2M(r/r_{H}^{\rm ext}-1)\sqrt{\mu^{2}-\frac{m^{2}}{4}} , and thus, a factor 1 / M 2 1/M^{2} is missing within the square root. This factor is important in order for the numerical solutions to match the actual exact solution. The actual expression should read as Eq.( 20
Cited in the paper.
For β > 1 / 2 \beta>1/2 the radial function vanishes at the horizon ( z = 0 z=0 ) and if in addition β ≠ k / 2 \beta\not=k/2 the radial function has unbounded gradients. In our previous analysis [ 1 ] we considered the near extremal case with unbounded gradients at the horizon, but assuming R H = 1 R_{H}=1 . Therefore in order to compare with Hod’s analytic solution ( 28
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One can write the hydrogen-atom radial wave function that is regular at r = 0 r=0 in terms of a confluent hypergeometric function with a β = l + 1 / 2 \beta=l+1/2 having exactly the same form as Hod’s exact solution ( 17
Cited in the paper.
Actually, Hod’s Eq. (22) reads M μ ± = m 2 + m 3 16 n 2 − m 3 16 n 3 ( 1 ± 1 ) + 𝒪 ( n − 4 ) M\mu_{\pm}=\frac{m}{2}+\frac{m^{3}}{16n^{2}}-\frac{m^{3}}{16n^{3}}(1\pm 1)+{\cal O}(n^{-4}) . Thus the subleading terms beyond the second one do not coincide with Eq. ( 31
Cited in the paper.