# Perturbing microscopic black holes inspired by noncommutativity

## Abstract

We probe into the instabilities of microscopic quantum black holes. For this purpose, we study the quasinormal modes (QNMs) for a massless scalar perturbation of the noncommutative geometry inspired Schwarzschild black hole. By means of a sixth order Wentzel–Kramers–Brillouin (WKB) approximation we show that the widely used WKB method does not converge in the critical cases where instabilities show up at the third order. By employing the inverted potential method, we demonstrate that the instabilities are an artifact of the WKB method. Finally, we discuss the usefulness of the asymptotic iteration method to find the QNMs.

## 1 Introduction

The problem of understanding the mechanisms underlying the final evolution of a black hole, which, in turn, is closely related to the problem of the central singularity and the emergence of a possible black hole remnant, triggered several studies in the last two decades [1, 2, 3, 4, 5]. Even though we do not have at the moment a final theory of quantum gravity able to offer a definite answer to the aforementioned problem, the existing candidate theories such as string theory, loop quantum gravity, and noncommutative geometry seem to share global features like the noncommutativity of the coordinates at a typical length \(\sqrt{\theta }\) less than \(10^{-18}\) m [6, 7, 8, 9], a new uncertainty principle including gravity effects [10, 11, 12], the avoidance of physical singularities [13, 14, 15], and black hole remnants [16, 17]. In particular, [18, 19, 20, 21] showed that noncommutativity can provide a cure for the central singularities afflicting the Schwarzschild and Reissner–Nordström metrics. The corresponding noncommutative counterparts of these metrics are derived by incorporating noncommutativity only in the matter source while the Einstein action is kept unchanged. As a result of this approach, the central singularity is replaced by a regular region (deSitter core) represented by a self-gravitating, droplet of anisotropic fluid. The price to be paid is that the radial and tangential pressures are always negative, a fact which is difficult to explain. This difficulty was overcome in [22] where by means of a nonlocal equation of state a noncommutative mini black hole model was developed in which the pressure is positive in the interior of the droplet. Furthermore, [23] gave the maximal singularity-free atlas for the noncommutative geometry inspired Schwarzschild metric [18]. This atlas describes an infinite lattice of asymptotically flat universes connected by black hole tunnels. The stability problem of the noncommutative Schwarzschild interior under massless scalar perturbations was attacked by [24, 25] leading to conclusions opposite to ours. It is worth mentioning that [26, 27] claimed to have derived the noncommutative counterpart of the Kerr metric. Despite the educated guess leading to this result, the authors did not perform any consistency check regarding the question whether or not such a solution satisfies all Einstein field equations coupled to an anisotropic source. In this context, we mention the work of [28] where it was shown that any noncommutative Kerr candidate metric, be it an educated guess or a particular solution, must satisfy a highly complicated system of partial differential equations (PDEs) represented by Eqs. (12)–(19) therein. Moreover, Theorem 1 and 2 in [28] give two independent proofs indicating that the metric found by [26, 27] can never satisfy all Einstein field equations and it may lead to contradictions.

It is of general interest to study the stability of microscopic quantum black holes. Moreover, as it will become apparent in the text, there exists indeed a hint of possible instabilities when the third order WKB approximation is used to determine the QNMs. The main question we address here is if these instabilities persist by increasing the order of the WKB or by employing different algorithmic methods.

While there is a vast literature devoted to the study of QNMs for a commutative Schwarzschild spacetime [29, 30, 31, 32, 33, 34, 35, 36, 37, 38], the same cannot be said for the noncommutative Schwarzschild manifold for which only some partial results are available in the literature. For instance, [39] analysed the asymptotic QNMs of a noncommutative geometry inspired Schwarzschild black hole under the assumption that the black hole mass *M* is much larger than \(M_e\), the mass of the extremal noncommutative Schwarzschild solution. At this point a comment is in order. From our Table 2 we can evince that the QNMs of a noncommutative Schwarzschild black hole already coincide with the corresponding ones of its classic counterpart when *M* is of the same order of magnitude as \(M_e\). Therefore, it is not surprising that [39] reached the conclusion that the asymptotic QNMs of a noncommutative Schwarzschild black hole in the regime of large masses remain proportional to \(\ln {3}\) as is the case for the classic Schwarzschild solution. Recently, [40] studied the QNMs of massless scalar field perturbations in a noncommutative-geometry-inspired Schwarzschild black hole spacetime by means of the third-order WKB approximation. The author considered the cases with \(\ell =1,2,3\) only and computed the QNMs for different values of the noncommutative parameter \(\theta \) in the interval \(0.01<\theta <0.2758\) using the third order WKB approximation. Since the parameter \(\sqrt{\theta }\) acts as a quantum of length, it is natural to assume \(\sqrt{\theta }=\ell _P\) where \(\ell _P=\sqrt{\hbar G/c^3}\approx 10^{-35}\) m is the Planck length. Then, the interval chosen in [40] over which the noncommutative parameter varies corresponds to a choice of the black hole mass in the interval \(3.6M_P<M<19M_P\) with \(M_P\) the Planck mass. In this particular range studied by the author and within the third order WKB, results for the QNMs were found to be stable. The misleading instabilities occurring in the third order WKB, occur in the range \(1.91M_P\le M\le 2.3897M_P\) as can be evinced from our Table 2. Extending the WKB calculations up to the sixth order reveals that the method is not convergent exactly when the presumed instabilities occur. This forces us to turn to another reliable method to pin down the nature of these instabilities. We choose the inverted potential method by which we can assure that the instabilities are due to inefficiency of the WKB in such cases. This case teaches us an important lesson for the determination of QNMs: a single algorithm is not always appropriate to uncover all QNMs. We strengthen this conclusion by investigating the asymptotic iterative method (AIM) and show its advantage over other methods commonly used.

The paper is organised as follows. In Sect. 2, we introduce a suitable rescaling of the noncommutative Schwarzschild manifold and derive the effective potential for a massless scalar field in the aforementioned geometry. Exploiting the short range property of this potential, we obtain a novel inequality linking the event horizon of a noncommutative Schwarzschild black hole with its mass parameter and the angular momentum quantum number of the scalar field. In Sect. 3, we use a third and sixth order WKB approximation to compute the QNMs. Section 4 discusses the inverted potential method followed by Sect. 5 devoted to the asymptotic iteration method.

## 2 The noncommutative Schwarzschild black hole

*M*is the total mass in the space-time manifold, \(\theta \) is a parameter encoding noncommutativity and having the dimension of a length squared, while \(\gamma (\cdot ,\cdot )\) is the incomplete lower gamma function. Taking into account that the incomplete and lower gamma functions are related by the formula [41]

*f*(

*r*) appearing in (1) can be written as the usual Schwarzschild counterpart plus a perturbation due to noncommutativity, i.e.

*f*has been obtained by using the following relations in [41], namely (Figs. 1, 2, 3)

*f*(

*x*), we expect that \({\mathcal {U}}_{eff}\) vanishes at the event horizon. Furthermore, in the non extreme case the event horizon is a simple zero of the above equation, while it becomes a zero of order two in the extreme case [23]. The differential equation (9) can be further simplified by introducing the tortoise coordinate

*f*(

*x*) is positive on

*I*. Moreover, with the help of 7.1.23 [41] it is not difficult to verify that asymptotically for \(x\rightarrow +\infty \) the tortoise coordinate exhibits the behaviour

*f*(

*x*) is increasing on the interval \((x_m,+\infty )\) with \(x_m\) denoting the minimum of

*f*(

*x*) (see dotted line in Fig. 2). Moreover, (16) yields

Normal modes for scalar perturbations of the noncommutative geometry inspired Schwarzschild metric. The third column represents the numerical values found by [50] in the case of a classic Schwarzschild black hole while the fourth columns reports the corresponding WKB results found by [43]. Here, \(\sigma =M\omega \) is the dimensionless frequency emerging after having expressed the radial coordinate in units of the black hole mass *M*, and \(\sigma _e\) denotes the QNM in the case of an extreme non-commutative geometry inspired Schwarzschild black hole

\(\ell \) | | \(\sigma _{L}\) | \(\sigma _{WKB}\) | \(\sigma _e\) | \(\sigma \), \(\mu =1.95\) | \(\sigma \), \(\mu =2.25\) | \(\sigma \), \(\mu =100\) |
---|---|---|---|---|---|---|---|

0 | 0 | \(0.1105-0.1049i\) | \(0.1046-0.1152i\) | \(0.0395+0.1367i\) | \(0.0397+0.1330i\) | \(0.0222-0.0944i\) | \(0.1046-0.1152i\) |

1 | \(0.0861-0.3481i\) | \(0.0892-0.3550i\) | \(0.1524+0.4582i\) | \(0.1555+0.4548i\) | \(0.0880+0.3947i\) | \(0.0892-0.3550i\) | |

1 | 0 | \(0.2929-0.0977i\) | \(0.2911-0.0980i\) | \(0.2699-0.0744i\) | \(0.2727-0.0818i\) | \(0.2882-0.0975i\) | \(0.2911-0.0980i\) |

1 | \(0.2645-0.3063i\) | \(0.2622-0.3704i\) | \(0.1061-0.2684i\) | \(0.1256-0.2689i\) | \(0.2449-0.3059i\) | \(0.2622-0.3704i\) | |

2 | \(0.2295-0.5401i\) | \(0.2235-0.5268i\) | \(0.1023+0.5721i\) | \(0.0779+0.5589i\) | \(0.1771-0.5284i\) | \(0.2235-0.5268i\) | |

3 | \(0.2033-0.7883i\) | \(0.1737-0.7486i\) | \(0.3374+0.9027i\) | \(0.3072+0.8805i\) | \(0.0862-0.7594i\) | \(0.1737-0.7486i\) | |

2 | 0 | \(0.4836-0.0968i\) | \(0.4832-0.0968i\) | \(0.4756-0.0870i\) | \(0.4767-0.0889i\) | \(0.4819-0.0964i\) | \(0.4832-0.0968i\) |

1 | \(0.4639-0.2956i\) | \(0.4632-0.2958i\) | \(0.4136-0.2612i\) | \(0.4209-0.2669i\) | \(0.4561-0.2944i\) | \(0.4632-0.2958i\) | |

2 | \(0.4305-0.5086i\) | \(0.4317-0.5034i\) | \(0.2782-0.4540i\) | \(0.2999-0.4590i\) | \(0.4122-0.5015i\) | \(0.4317-0.5034i\) | |

3 | \(0.3939-0.7381i\) | \(0.3926-0.7159i\) | \(0.0918-0.6961i\) | \(0.1285-0.6908i\) | \(0.3544-0.7154i\) | \(0.3926-0.7159i\) |

Transition from unstable to stable modes in the cases \(\ell =0\), \(n=0,1\), and \(\ell =1\), \(n=2,3\), using the WKB approximation at the third order

\(\mu \) | \(\sigma ,~\ell =n=0\) | \(\mu \) | \(\sigma ,~\ell =0,~n=1\) | \(\mu \) | \(\sigma ,~\ell =1,~n=2\) | \(\mu \) | \(\sigma ,~\ell =1,~n=3\) |
---|---|---|---|---|---|---|---|

1.9100 | \(0.0396+0.1363i\) | 1.9100 | \(0.1529+0.4579i\) | 1.9100 | \(0.0997+0.5705i\) | 1.9100 | \(0.3343+0.9001i\) |

1.9500 | \(0.0396+0.1330i\) | 1.9500 | \(0.1555+0.4548i\) | 1.9500 | \(0.0779+0.5589i\) | 1.9500 | \(0.3072+0.8805i\) |

2.0000 | \(0.0380+0.1279i\) | 2.3800 | \(0.0063+0.3670i\) | 2.0465 | \(0.0003+0.5334i\) | 2.1000 | \(0.1261+0.7960i\) |

2.1958 | \(0.0000+0.0989i\) | 2.3890 | \(0.0003+0.3658i\) | 2.0467 | \(0.0000+0.5333i\) | 2.1800 | \(0.0052+0.7677i\) |

2.1959 | \(0.0000-0.0989i\) | 2.3895 | \(0.0001+0.3658i\) | 2.0468 | \(0.0000-0.5333i\) | 2.1835 | \(0.0002+0.7669i\) |

2.2000 | \(0.0016-0.0983i\) | 2.3897 | \(0.0000+0.3658i\) | 2.0500 | \(0.0030-0.5327i\) | 2.1836 | \(0.0000+0.7669i\) |

2.6000 | \(0.1057-0.1189i\) | 2.3898 | \(0.0000-0.3657i\) | 2.1000 | \(0.0517-0.5252i\) | 2.1837 | \(0.0001-0.7668i\) |

2.7500 | \(0.1097-0.1201i\) | 2.3900 | \(0.0001-0.3657i\) | 2.2000 | \(0.1422-0.5247i\) | 2.2500 | \(0.0862-0.7594i\) |

2.8000 | \(0.1095-0.1196i\) | 2.3950 | \(0.0034-0.3651i\) | 2.4500 | \(0.2398-0.5376i\) | 2.3500 | \(0.1706-0.7605i\) |

2.9000 | \(0.1083-0.1183i\) | 2.4000 | \(0.0066-0.3646i\) | 2.5500 | \(0.2417-0.5358i\) | 2.6500 | \(0.2001-0.7552i\) |

3.1000 | \(0.1059-0.1161i\) | 2.4500 | \(0.0363-0.3611i\) | 2.6000 | \(0.2398-0.5343i\) | 2.8000 | \(0.1851-0.7510i\) |

3.1500 | \(0.1051-0.1158i\) | 2.5000 | \(0.0611-0.3601i\) | 2.9500 | \(0.2252-0.5274i\) | 2.9000 | \(0.1789-0.7496i\) |

3.2000 | \(0.1053-0.1156i\) | 3.0000 | \(0.0972-0.4566i\) | 3.1000 | \(0.2238-0.5269i\) | 3.0000 | \(0.1756-0.7489i\) |

3.3000 | \(0.1049-0.1153i\) | 3.4500 | \(0.0893-0.3549i\) | 3.2000 | \(0.2236-0.5268i\) | 3.1000 | \(0.1742-0.7487i\) |

4.0000 | \(0.1046-0.1152i\) | 4.0000 | \(0.0892-0.3550i\) | 3.2500 | \(0.2235-0.5268i\) | 3.2000 | \(0.1737-0.7486i\) |

QNMs at different orders of the WKB approximation

Order | 6 | 5 | 4 | 3 | 2 | Eikonal |
---|---|---|---|---|---|---|

\(\mu = 1.91\) | ||||||

| ||||||

0, 0 | \(1.61847 + 0.06772 i\) | \(0.36561 + 0.29978 i\) | \(0.21084 + 0.02561 i\) | \(0.03962 + 0.13627 i\) | \(0.11332 - 0.17275 i\) | \(0.18958 - 0.10326 i\) |

1, 0 | \(0.21155 - 0.30434 i\) | \( 0.36645 - 0.17569 i\) | \(0.32747 - 0.06178 i\) | \( 0.27024 - 0.07487 i\) | \( 0.28245 - 0.11116 i\) | \( 0.32879 - 0.09549 i\) |

2, 0 | \(0.45257 - 0.11220 i\) | \(0.48987 - 0.10365 i\) | \(0.48633 - 0.08538 i\) | \(0.47574 - 0.08728 i\) | \( 0.47800 - 0.09886 i\) | \(0.50595 - 0.09340 i\) |

2, 1 | \(0.38600 - 0.47357 i\) | \(0.54088 - 0.33797 i\) | \(0.47923 - 0.22658 i\) | \(0.41456 - 0.26193 i\) | \(0.44999 - 0.31504 i\) | \(0.55833 - 0.25391 i\) |

3, 0 | \(0.66888 - 0.09442 i\) | \(0.67504 - 0.09356 i\) | \(0.67447 - 0.08938 i\) | \(0.67099 - 0.08985 i\) | \(0.67178 - 0.09552 i\) | \(0.69172 - 0.09277i\) |

3, 1 | \(0.62869 - 0.30765 i\) | \(0.66875 - 0.28923 i\) | \(0.65638 - 0.25934 i\) | \(0.63424 - 0.26840 i\) | \(0.64709 - 0.29750 i\) | \(0.73394 - 0.26230 i\) |

3, 2 | \(0.59901 - 0.57927 i\) | \(0.69462 - 0.49954 i\) | \(0.62507 - 0.39719 i\) | \(0.55413 - 0.44803 i\) | \(0.61513 - 0.52160 i\) | \(0.79536 - 0.40340 i\) |

\(\mu = 2.7\) | ||||||

0, 0 | \(0.84966 + 0.06067 i\) | \(0.20661 + 0.24951 i\) | \(0.08130 - 0.16179 i\) | \(0.10936 - 0.12028 i\) | \(0.13227 - 0.14143 i\) | \( 0.18991 - 0.09851 i\) |

1, 0 | \(0.30052 - 0.08903 i\) | \(0.28808 - 0.09288 i\) | \(0.29024 - 0.09937 i\) | \(0.29148 - 0.09895 i\) | \(0.29456 - 0.10768 i\) | \(0.32944 - 0.09628 i\) |

2, 0 | \(0.48119 - 0.09741 i\) | \(0.48328 - 0.09699 i\) | \(0.48328 - 0.09700 i\) | \(0.48322 - 0.09701 i\) | \(0.48395 - 0.10057 i\) | \(0.50632 - 0.09612 i\) |

2, 1 | \(0.45450 - 0.29924 i\) | \(0.46001 - 0.29566 i\) | \(0.46161 - 0.29816 i\) | \(0.46369 - 0.29682 i\) | \(0.47186 - 0.30943 i\) | \(0.56110 - 0.26022 i\) |

3, 0 | \(0.67489 - 0.09662 i\) | \(0.67521 - 0.09657 i\) | \(0.67521 - 0.09660 i\) | \(0.67519 - 0.09660 i\) | \(0.67546 - 0.09846 i\) | \(0.69173 - 0.09615 i\) |

3, 1 | \(0.65835 - 0.29288 i\) | \(0.65936 - 0.29243 i\) | \(0.65969 - 0.29320 i\) | \(0.66048 - 0.29285 i\) | \(0.66392 - 0.30052 i\) | \(0.73662 - 0.27086 i\) |

3, 2 | \(0.62898 - 0.49442 i\) | \(0.62721 - 0.49581 i\) | \(0.63020 - 0.49960 i\) | \(0.63546 - 0.49546 i\) | \(0.64884 - 0.51250 i\) | \(0.80100 - 0.41515 i\) |

## 3 A WKB approach

We compute the quasinormal mode frequencies of a noncommutative geometry inspired Schwarzschild black hole in the presence of a massless scalar field by means of the WKB (Wentzel-Kramers-Brillouin) approximation carried to sixth order beyond the eikonal approximation [47]. On the way, we discover some shortcomings of the method, especially when applied to the noncommutative Schwarzschild case. In principle, the WKB approximation already at the third order is found to give reasonably accurate results [43, 48, 49] in literature. However, in the noncommutative case under consideration, it can lead to misleading results. In order to bring forth the above point, we shall first present the WKB results at third order and then go over to show the limitations even at the sixth order of WKB.

*A*and \({\widehat{\phi }}\) functions to be determined. Substituting (26) into (25) gives

Table 1 shows that the extreme noncommutative geometry inspired Schwarzschild black hole is unstable, that is a positive imaginary part of the QNM leads to exponential growth of a scalar perturbation. Using different boundary conditions in the context of black plus mirror, [51] also finds similar instability behaviour for small Kerr-AdS black holes. Furthermore, the non extreme case also exhibits instabilities when the mass parameter \(\mu \) slightly exceed the corresponding mass parameter of an extreme Schwarzschild black hole. Table 2 also shows that for fixed \(\ell \) but increasing *n* the corresponding QNM of the noncommutative geometry inspired Schwarzschild black hole approaches its classical counterpart already for small values of the mass parameter. These results at third order WKB suggest that the effects due to noncommutative geometry are relevant for microscopic black holes but they can be neglected when dealing with black holes of astrophysical interest.

Normal modes for scalar perturbations of the noncommutative geometry inspired Schwarzschild metric. The third and fourth columns represent the numerical values found by [50] and [43], respectively, for a classic Schwarzschild black hole. Here, \(\sigma _{PT}\) is the dimensionless frequency computed according to (36), and \(\sigma _{e,PT}\) denotes the QNM in the case of an extreme non-commutative geometry inspired Schwarzschild black hole

\(\ell \) | | \(\sigma _{L}\) | \(\sigma _{WKB}\) | \(\sigma _{e,PT}\) | \(\sigma _{PT}\), \(\mu =1.95\) | \(\sigma _{PT}\), \(\mu =2.25\) | \(\sigma _{PT}\), \(\mu =100\) |
---|---|---|---|---|---|---|---|

0 | 0 | \(0.1105-0.1049i\) | \(0.1046-0.1152i\) | \(0.1003-0.1232i\) | \(0.1022-0.1225i\) | \(0.1103-0.1184i\) | \(0.1148-0.1148i\) |

1 | \(0.0861-0.3481i\) | \(0.0892-0.3550i\) | \(0.1003-0.3697i\) | \(0.1022-0.3675i\) | \(0.1103-0.3552i\) | \(0.1148-0.3444i\) | |

1 | 0 | \(0.2929-0.0977i\) | \(0.2911-0.0980i\) | \(0.2983-0.0997i\) | \(0.2983-0.1002i\) | \(0.2983-0.1009i\) | \(0.2985-0.1006i\) |

1 | \(0.2645-0.3063i\) | \(0.2622-0.3704i\) | \(0.2983-0.2991i\) | \(0.2983-0.3007i\) | \(0.2983-0.3028i\) | \(0.2985-0.3019i\) | |

2 | \(0.2295-0.5401i\) | \(0.2235-0.5268i\) | \(0.2983-0.4985i\) | \(0.2983-0.5012i\) | \(0.2983-0.5048i\) | \(0.2985-0.5032i\) | |

3 | \(0.2033-0.7883i\) | \(0.1737-0.7486i\) | \(0.2983-0.6979i\) | \(0.2983-0.7017i\) | \(0.2983-0.7067i\) | \(0.2985-0.7045i\) | |

2 | 0 | \(0.4836-0.0968i\) | \(0.4832-0.0968i\) | \(0.4881-0.0949i\) | \(0.4878-0.0957i\) | \(0.4873-0.0977i\) | \(0.4873-0.0979i\) |

1 | \(0.4639-0.2956i\) | \(0.4632-0.2958i\) | \(0.4881-0.2847i\) | \(0.4878-0.2873i\) | \(0.4873-0.2932i\) | \(0.4873-0.2937i\) | |

2 | \(0.4305-0.5086i\) | \(0.4317-0.5034i\) | \(0.4881-0.4745i\) | \(0.4878-0.4788i\) | \(0.4873-0.4887i\) | \(0.4873-0.4895i\) | |

3 | \(0.3939-0.7381i\) | \(0.3926-0.7159i\) | \(0.4881-0.6643i\) | \(0.4878-0.6704i\) | \(0.4873-0.6842i\) | \(0.4873-0.6853i\) |

Fundamental normal modes for massless scalar perturbations of the classic Schwarzschild metric. The second column reports the fundamental quasi-normal frequencies obtained by means of the improved AIM while the third column displays the exact fundamental QNMs computed using formula (52) in [57]

\(\ell \) | \(\sigma \), \(n=0\) | \(\text{ Exact }\) |
---|---|---|

0 | \(0.1250-0.1250i\) | \(0.1250-0.1250i\) |

1 | \(0.2795-0.1250i\) | \(0.2795-0.1250i\) |

2 | \(0.4506-0.1250i\) | \(0.4506-0.1250i\) |

## 4 The inverted potential method

*f*(

*x*) given by (10). Taking into account that the bound states of the Pöschl-Teller potential are given by [52, 53, 54, 55]

*l*and \(\mu = 1.91\). The PT potential reproduces the effective potential quite well in the peak region in all 3 cases.

In Table 4, we present some of the QNMs of a noncommutative geometry inspired Schwarzschild black hole computed according to the method described above. According to the method used, the imaginary part of the QNMs do not exhibit any sign flip. Furthermore, it is gratifying to observe that for large values of the parameter \(\mu \), the numerical values of the QNMs in the tenth column of Table 4 coincide with the quasinormal frequencies computed by [52, 53, 54, 55] for the case of the scalar perturbations of the classic Schwarzschild black hole.

## 5 QNMs by the asymptotic iteration method (AIM)

From the above exposition it becomes evident that it is important to choose an appropriate method to determine the QNMs. It is worthwhile to look for other similar cases where conventionally used methods fail.

*n*times with respect to the independent variable, which produces the following equation

## 6 Conclusions

Instabilities of microscopic black holes are an interesting area of research as certain examples in the literature show. The existence of such instabilities could possibly hint toward the inappropriateness of models of microscopic black holes. We choose here, as an example, the case of a Schwarzschild black hole inspired by noncommutative geometry. Indeed one of our motivation to probe into this case is a hint of instability revealed at the third order of the WKB method. It is then obligatory to pursue further the nature of such a possibility, either by assuring the convergence of the WKB method or, in case the convergence fails, choosing another appropriate algorithm to determine the QNMs. We show by going up to the sixth order of WKB, that the convergence of the WKB method is not guaranteed for the cases under discussion. The inverted potential method is then a more suitable alternative which reveals that these instabilities were only an artifact of the third order WKB. Furthermore, we show how the asymptotic iterative method confirms the existence of a new branch of analytical QNMs recently discovered in [57].

## Notes

### Acknowledgements

The authors would like to thank the anonymous referee for his/her enlightening comments on the manuscript and his/her valuable suggestions. The authors are grateful to R. A. Konoplya for providing the code to evaluate the QNMs using the WKB method. One of the authors (N. G. K.) thanks the Faculty of Science, Universidad de los Andes, Colombia for financial support through grant no. P18.160322.001-17.

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