The case of QNM instability analysed in [1] and related works is one in which (at least part of) the already existing BH QNM spectrum is displaced by a disproportionately large amount due to a small perturbation to the operator L . The smallness of this perturbation is defined quantitatively through the energy norm [19], which has a natural physical interpretation. The overall instability to any perturbation of L is captured by the pseudospectrum in this norm, which is defined as σ ϵ L = λ ∈ C : ‖ R L λ ‖ E > 1 / ϵ , where ‖ ⋅ ‖ E indicates the energy norm of the operator, defined from the product u 1 , u 2 E = ϕ 1 ψ 1 , ϕ 2 ψ 2 E = 1 2 ∫ 0 1 w χ ψ ̄ 1 ψ 2 + p χ ∂ χ ϕ ̄ 1 ∂ χ ϕ 2 + q χ ϕ ̄ 1 ϕ 2 d χ ,

An equivalent definition is the one which directly relates the level sets of the pseudospectrum to the space of possible new eigenvalue positions after a perturbation, σ ϵ L = λ ∈ C , ∃ δ L , ‖ δ L ‖ < ϵ : λ ∈ σ L + δ L . (1)

Note that this second definition involves any perturbation to L which has a small energy norm, including ones which can potentially be related to a physical modification of the environment of the black hole, but also ones which completely change the nature of the operator (e.g., changing the structure of the derivatives). That said, it was shown in [1] that the instability is in fact triggered by physical perturbations, encoded in the addition of a perturbation function δ V to the effective potential, without disturbing the structure of the differential part of the operator. Additionally, it was shown that the degree to which the spectrum is destabilised depends strongly on the “high wave-number” content of the perturbation, that is, the sharpness of the variation of δ V in r .

While the particular choices for the perturbations δ V used in ref. [1] may not correspond to the addition of classically reasonable matter content to the system [22], they are a proof of principle which shows that whatever the perturbation may be, as long as it has a large enough gradient in r , it will trigger the instability. Ref. [10] in fact explicitly shows the relation between the magnitude of the derivatives of δ V and the rate of displacement of the QNMs in a specific example, further solidifying this result.

The above-mentioned stability of the fundamental mode was also one of the key results, which can directly be related to the fact that gravitational wave observations of compact object collisions which result in a black hole as an end state appear to contain a part which matches well with a fundamental-mode-dominated ringdown [32].

The second type of “instability” is due to the emergence of new mode branches. It is important to understand that the characteristics of the spectrum depend strongly on the shape of the potential V . For the axial gravitational case, the potential has a single barrier with a peak close to the photon sphere, from which it decreases to zero exponentially (in r * ) towards the horizon and polynomially towards infinity. The corresponding “barrier top” modes are not very long-lived (in terms of the characteristic scale of the problem). However, seemingly small perturbations can lead to a qualitative change in the shape of the potential, such as the addition of a well which goes below the asymptotic values, or of a second barrier (or bump). The former can lead to the presence of bound states, while the latter to slowly decaying “shape resonances” [25].

Some examples of such qualitative modifications to the potential in the context of QNMs are the double barrier model in [33], or some of the models explored in [22], such as the addition of a perturbatively small mass parameter. As the new families of modes that these modifications introduce can have a slower decay than the fundamental “barrier top” QNM, the new fundamental mode and first overtones can be said to have been displaced disproportionately to the size of the perturbation, even if the original modes (which are no longer the fundamental and first overtones) happen to still be present in the new spectrum with only a slight displacement. In other words, what can occur is that the label of “fundamental” and of the overtone numbers may jump to modes in the new branch (according to the usual assignment of these labels), rather than the old modes being displaced. This can also happen by changing the parameters of a problem such that QNMs in two different branches which are already present can switch roles as the fundamental mode, such as in the overtaking of the fundamental oscillatory mode by a de Sitter mode discussed in [12].

It is important to note that this by itself is distinct from the usual definition of (perturbative) spectral instability discussed above, in which already existing modes are displaced by large distances in the complex plane. It is also interesting that for a spectrally unstable system such as the case of QNMs, adding, say, a bump to the effective potential, can lead to a combination of both of the above effects: the emergence of new long-lived modes, as well as the large displacement of (some of) the already existing modes. This is precisely the case in the example below, for which the perturbed potential goes from having a single barrier to a double barrier, the latter being akin to the case dubbed a “well on an island” in ref. [25].

Let us begin with a particularly illustrative example of a perturbation of this type, which will be the centre-point of this analysis. We set the units to the characteristic scale of the problem by taking 2 M = 1 , and we set a (seemingly) small amplitude for the bump a = 0.005 , a position for the peak at r b = 25 and a width s = 4 . The spectrum of axial ℓ = 2 modes with this perturbation is shown in Figure 2. We see that there is indeed a new branch of modes, some of which decay more slowly than the unperturbed BH fundamental mode. In this sense, the distance between the old and new fundamental mode does indeed seem quite large compared to the size of the perturbation, as discussed in ref. [20]. However, it is also clear form Figure 2 that the spectrum after the perturbation contains a mode which coincides with the unperturbed fundamental one (in fact it is only ∼ 1 0 − 3 away), implying that this mode was actually stable under the perturbation.

This is therefore a case in which it is the qualitative change in the shape of the potential has lead to the appearance of new long-lived modes, while part of the old spectrum has remained stable, in this case only the BH fundamental mode. From the first overtone onwards, the BH spectrum is in fact destabilised, much like it is in some of the cases studied in ref. [1].

The behaviour of the new fundamental mode depends strongly on the parameters of the gaussian bump a , r b and s , and while a detailed analysis of this dependence is not within the scope of this work (see [20, 22] for a quantitative analysis of part of the parameter space), we will make some general remarks regarding the behaviour we have observed from a few spectra.

• Increasing the amplitude a tends to decrease the imaginary part of the fundamental mode, as the modes trapped between the two peaks (the bump and the light-ring peak) need to tunnel out of a larger barrier to decay. Conversely, if a is made smaller, the imaginary part increases. At around a = 1 0 − 5 the longest-lived of these new modes is no longer the fundamental one, as its imaginary part is larger than that of the BH mode.

It is also worth noting that while we identify these longer-lived modes as a new branch due to the qualitative change in the potential and the stability of the BH fundamental mode, from the numerical results alone it is not clear where exactly this new branch becomes entwined with the perturbed BH overtones. To identify which modes go to infinity and which to BH overtones in the zero perturbation limit, a more detailed study which traces the migration of individual overtones would be required.

The perturbation operator being added to L can be written as δ L = 0 0 δ q w I 0 , where δ q = | g ′ | δ V . In order to give a physical measure of the size of this perturbation, the energy norm of δ L can be computed. Contrary to what might be expected from the small a parameter, the energy norm of the example case used for Figure 2 is actually quite significant: in units of the horizon scale, it is approximately 0.33, on the very high end of what can reasonably be considered a “perturbation”. The reason for this apparent discrepancy between the intended smallness in the choice of a and the large energetic contribution of this perturbation lies in the simple fact that the energy measure comes from an integral related to the full three dimensional space of constant time slices [19], rather than just the one dimension of the wave problem (although in the end it simplifies to the latter). It therefore encodes the fact that a perturbation at a large radius would require a thick shell of this same radius, the size and matter content of which would scale with r 2 . This is indeed the scaling we can observe in the left plot of Figure 3, where the energy norm is calculated as a function of r b (with all other parameters remaining the same), and fitted to a parabola.

Therefore, the increased destabilisation of the BH QNMs (effectively, the lowering of the mode branches seen in Figure 2) for a larger r b which was commented above can be related to precisely this increase of the energy norm. The dependence of this energy on the parameters a and s is just as predictable: an increase in both these parameters leads to a proportional (linear) increase in the energy norm.

This example has shown the importance of quantifying the size of perturbations added to the problem, since, for instance, the increase of the energy contained in perturbations at larger radii is something that could easily have been overlooked otherwise. However, using the energy norm in particular, while having many advantages [19], may not be the most adequate choice in some respects. One particular issue, raised in [13], is the fact that the energy norm of the resolvent operator, used to calculate the pseudospectrum, is not well behaved in a large part of the complex plane, which includes the vicinity of most (if not all) QNMs. Numerically, this norm tends to a divergence in the limit of infinite grid point number N in most of the upper half of the complex plane. Since the issue in that case stems from the presence of additional eigenmodes of a lower regularity class [30], and not simply from a numerical problem, it is likely to be a generic property of other setups as well.

One may then ask whether this issue extends to calculating the energy norm of other operators as well, particularly that of δ L , since the pseudospectrum can equivalently be defined from its norm (albeit for a very large set of perturbations). Fortunately, it appears that the norm of this operator actually does have a good convergent behaviour. The second plot in Figure 3 shows a convergence test in a representative example. The result is clear: the convergence is in fact exponential. Such convergence was previously observed for other quantities computed with this discretisation scheme, such as the spectrum itself (see fig. 8 of [1]), but had thus far not been tested for energy norms of operators, except for the case of the non-convergent resolvent norm in [13].

The convergence of ‖ δ L ‖ in fact confirms that the issue with the resolvent norm studied in [13] goes beyond the particular numerical implementation. It also gives an appealing potential alternative approach to calculating the pseudospectrum by exploring a sufficiently large space of perturbation operators and applying Equation 1; [9, 34]. However, there would be two issues with such an approach. First, it would be computationally very expensive to attempt to span a “full” space of perturbation operators δ L . This would not be a critical impediment, at least for a small numerical resolution N . However, the second and most crucial issue is the fact that the result would differ depending on the resolution, as a higher N could capture perturbations with a higher wave-number content, which would destabilise the spectrum ever more strongly. It is not clear that the limit of operators δ L with the same energy norm but with ever higher gradients in r (which would need a correspondingly higher N to be resolved) would lead to a convergent definition of the pseudospectrum, or if this issue would turn out to be equivalent to the non-convergence observed in the resolvent approach.

Testing whether this claim is true, while absolutely crucial, goes beyond the scope of the present work. If it were indeed proven true, then a consistent definition of a QNM pseudospectrum would require a modification of the scheme summarised above. One example of such a modification would be the use of norms with higher order spatial derivatives, as introduced in [30], and applied to the pseudospectral calculation in [13] (see also [35]). However, a reasonable physical interpretation of such norms and their associated stability analyses would need to be devised.