On the (non) existence of superregular boson clouds around extremal Kerr black holes and its connection with number theory

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. The latter solutions have been reported recently both analytically and numerically. The superregular clouds cannot be obtained from the regular clouds around subextremal Kerr BH's ($|a|< M$) in the limit of extremality $(a\rightarrow M)$ as in this limit the radial derivatives of $\Psi$ at the horizon $r_H$ diverge when $r_H\rightarrow r_H^{\rm ext}:=M=a$, thus, such superregular clouds must be analyzed separately. We conclude that the superregular clouds, which are found in the {\it exact} extremal scenario ($a = M$), are not continuously connected with the regular ones in the limit of extremality $(a\rightarrow M)$. Remarkably, the spectrum leading to the existence of the radial part of the full solution of these superregular clouds (which obeys a Teukolsky equation) is given by the exact formula $M=a=\frac{1}{2\mu}\sqrt{m^2 + \left[-\kappa+\sqrt{\kappa^2+m^2}\,\right]^2}$, which depends on three (positive) integers: the principal number $n$, the {\it magnetic number} $m$, and an integer $j$, related with the {\it type} of regularity at the horizon. Here $\kappa= j +n$, and $\mu$ is the mass associated with $\Psi$. This spectrum depends implicitly on the {\it orbital} number $l$, an integer number that determines the existence of well behaved spheroidal harmonics which are associated with the angular part of the cloud solution. Since the separation constants that are obtained from the superregularity conditions in the radial part of the solution do {\it not} coincide in general with the standard separation constants required for the spheroidal harmonics to be well behaved on the axis of symmetry, we conclude that non-trivial boson clouds having such superregularity conditions cannot exist in the background of an exact extremal Kerr BH. The only exception to this conclusion is in the limit $n\rightarrow \infty$ and $m\ll n$. In such a large $n$ limit consistency in the separation constants leads to a quadratic Diophantine equation of Pell's type for the integer numbers $(l,m)$. Such Pell's equation can be readily solved using standard techniques. In that instance well behaved spheroidal harmonics are obtained, and thus, well behaved non-trivial superregular clouds can be computed. Of course, this situation, does not preclude the existence of other kind of smooth cloud solutions for any other $n$, not necessarily large (e.g. clouds with a non-integer $\kappa$) when using a better behaved coordinate system at the horizon (e.g. Wheeler's tortoise coordinate or proper radial distance).