Black Hole Spectral Instability

• Black hole spectral instability is the sensitivity of quasinormal-mode spectra to minute perturbations in the metric or effective potential.
• It encompasses phenomena such as overtone migration, branch bifurcation, and exponential mode shifts revealed through pseudospectral analysis.
• Despite dramatic spectral changes, key observables like gravitational-wave ringdowns and greybody factors remain robust under small perturbations.

Black hole spectral instability is the structural sensitivity of a black hole’s quasinormal-mode (QNM) spectrum to small deformations of the metric, effective potential, or boundary structure in a non-Hermitian scattering problem. In this usage, “instability” refers primarily to the eigenvalue set itself: small perturbations can produce disproportionately large deformations of the QNM spectrum, even when individual modes remain damped and even when the early time-domain waveform remains nearly unchanged (Shen et al., 15 Jul 2025). The subject therefore sits at the interface of resonance theory, pseudospectra, scattering, and gravitational-wave ringdown. It includes both mathematically spectral phenomena—such as overtone migration, branch bifurcation, and fundamental-mode overtaking—and, in more restrictive cases, genuine dynamical instabilities with $\operatorname{Im}\omega>0$ (Wang et al., 14 Oct 2025).

1. Definition and spectral formulation

The standard starting point is the black-hole master equation

$$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$

which after separation $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$ becomes

$$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$

QNMs are defined by purely ingoing behavior at the horizon and purely outgoing behavior at infinity, and equivalently arise as poles of the frequency-domain Green’s function or zeros of the Wronskian $$W\left(\phi_{\ell}^\mathrm{in}, \phi_{\ell}^\mathrm{out}\right)$$ (Shen et al., 15 Jul 2025).

This spectral problem is intrinsically non-self-adjoint and non-normal. That feature underlies the distinction between the temporal stability of a given mode and the structural stability of the spectrum as a whole. In the convention

$$\psi \sim e^{-i\omega t}=e^{-i\omega_R t}e^{\omega_I t},$$

$\omega_I<0$ gives exponential decay and $\omega_I>0$ gives exponential growth. Spectral instability, however, need not mean $\omega_I>0$. In the ringdown context studied for hairy Einstein–Maxwell–scalar black holes, the relevant modes remain damped while their frequencies migrate strongly under parameter variation; the “instability” is instability of the spectrum as an eigenvalue set, not runaway time evolution (Wang et al., 14 Oct 2025).

The modern literature therefore treats black-hole spectral instability as a resonance-theoretic statement: small deformations of the operator can strongly affect the pole structure of the Green’s function, and hence the QNM spectrum, even when the observable early waveform is initially robust (Shen et al., 15 Jul 2025).

2. Mechanisms of spectral sensitivity

Early work already showed that minor potential modifications can qualitatively change the higher overtones. Staircase approximations to the Regge–Wheeler potential can reproduce the time-domain waveform while generating a markedly different high-overtone QNM spectrum, and continuous piecewise-linear approximations show the same phenomenon. Later pseudospectral analyses demonstrated that randomized, sinusoidal, and especially ultraviolet perturbations strongly deform QNM spectra of both Pöschl–Teller and Regge–Wheeler potentials, tying the effect to non-normality and loss of eigenvector orthogonality (Shen et al., 15 Jul 2025).

A particularly transparent mechanism is the addition of a small distant bump to the effective potential. For

$V(x)=V_0(x)+\epsilon V_b(x), \qquad \epsilon\ll 1,$

the susceptibility of a QNM can be written in the asymptotic regime as

$$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$0

so that

$$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$1

Because physical QNMs satisfy $$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$2, the shift grows exponentially with the bump displacement $$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$3. This is the generic “Type II” instability: no fixed smallness of $$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$4 guarantees a small $$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$5 uniformly in $$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$6 (Yang et al., 2024). Explicit computations with Schwarzschild bumps then showed that even the fundamental mode can be destabilized under generic perturbations, not only the asymptotic overtone sector (Cheung et al., 2021).

A related transfer-matrix description gives the exact two-barrier QNM condition

$$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$7

for a main black-hole barrier plus a small distant bump. The exponential factor encodes repeated round trips across the cavity between the two scatterers and makes the resonance condition exponentially delicate in the separation $$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$8. In that formulation, spectral fragility is a direct consequence of multiple scattering in a non-Hermitian open system (Ianniccari et al., 2024).

3. Spectral phenomenology: branches, overtaking, and remnant scales

The resulting spectral phenomenology is diverse. In the high-overtone regime, a single discontinuity or ultraviolet corrugation can rotate asymptotic QNM branches so that poles line up nearly parallel to the real axis, rather than following the usual black-hole asymptotic pattern. This is the regime most closely associated with echo-like late-time behavior (Shen et al., 15 Jul 2025).

In the low-lying sector, several distinct effects occur. Tiny Gaussian or Pöschl–Teller bumps can force the fundamental mode to trace an outward spiral in the complex-frequency plane and can trigger overtaking by newly induced modes, so that the mode continuously connected to the original Schwarzschild fundamental ceases to be the true least-damped mode (Cheung et al., 2021). The overtaking phenomenon is a recurring signature of spectral instability because it changes the identity of the “fundamental” QNM discontinuously.

A separate mechanism arises when the effective potential develops more than one relevant scale. In static, spherically symmetric hairy black holes of the Einstein–Maxwell–scalar model, the scalar effective potential can develop a main peak near the photon-sphere region, a secondary outer barrier, and a potential valley between them. The spectrum then splits into two families: photon-sphere-associated “peak modes,” which are comparatively spectrally stable, and “off-peak modes,” which migrate strongly under variation of $$\frac{\partial^2}{\partial t^2}\psi_{\ell}(t, r_*)+\left(-\frac{\partial^2}{\partial r_*^2}+V_\mathrm{eff}\right)\psi_{\ell}(t, r_*)=0 ,$$9. The central result is that the off-peak family can persist even after the potential well has vanished, as a remnant branch inherited from the nearby double-peaked regime (Wang et al., 14 Oct 2025).

Mechanism	Spectral signature	Representative paper
Ultraviolet or discontinuous perturbation	Asymptotic poles nearly parallel to the real axis	(Shen et al., 15 Jul 2025)
Small distant bump	Exponential migration, spiral motion, Type II instability	(Yang et al., 2024)
Fundamental-mode destabilization	Overtaking and reordered least-damped mode	(Cheung et al., 2021)
Double-peaked or remnant outer scale	Coexisting peak and off-peak families	(Wang et al., 14 Oct 2025)
Near-horizon non-positive deformation	New purely imaginary branch	(Ma et al., 1 Jun 2026)

This range of mechanisms makes spectral instability a structural phenomenon rather than a single pathology. The decisive ingredients are extra reflection, extra cavity length, extra scale, or an altered asymptotic matching problem.

4. Frequency domain, time domain, and echoes

A central result of the subject is that large spectral changes do not automatically imply large early waveform changes. The causal resolution is clearest in the Green-function analysis of a distant bump. For internal initial data, the perturbed-minus-unperturbed response splits into wave packets arriving at $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$0, $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$1, $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$2, and $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$3. The components carrying the new unstable QNMs only arrive after the extra round trip to the bump, so before that delay the waveform remains close to the unperturbed black-hole response (Yang et al., 2024).

This yields a quantitative statement: for $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$4, the difference between perturbed and unperturbed waveforms in the early ringdown phase is proportional to $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$5 (Yang et al., 2024). The exact QNM spectrum may already have reorganized, but the prompt signal is still governed mainly by the original light-ring barrier. The transfer-matrix approach expresses the same point by showing that the prompt ringdown and the greybody factor receive only parametrically small corrections even when the exact pole spectrum has been drastically reshaped (Ianniccari et al., 2024).

The same separation appears in the coexistence of peak and off-peak modes. In the hairy Einstein–Maxwell–scalar example, time-domain fits for $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$6 recover only the first three peak modes in early windows, while the leading off-peak mode has an amplitude roughly $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$7 smaller than the peak-mode amplitudes. For $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$8, the off-peak contribution is less suppressed but still subdominant, with weak-fit amplitudes reduced by about $$\psi_{\ell}(t,r_*)=e^{-i\omega t}\phi_{\ell}(\omega,r_*)$$9 to $$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$0 relative to the peak-family amplitudes (Wang et al., 14 Oct 2025). The spectrum is fragile, but the prompt ringdown is selected by excitation amplitudes and causality.

Echoes form the complementary late-time manifestation. In one class of models, two reflective/scattering regions create a cavity and the echo period is roughly twice the cavity length in tortoise coordinate. In another, even a single discontinuity acts as a reflector. When asymptotic QNMs become nearly evenly spaced along the real axis,

$$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$1

the spacing sets the echo delay time $$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$2 (Shen et al., 15 Jul 2025). Spectral instability in the high-overtone sector and late-time echo phenomenology are therefore closely linked.

5. Spectral instability versus genuine dynamical instability

Although much of the literature concerns damped but spectrally fragile modes, some settings do produce genuine dynamical instabilities with $$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$3. A clean example is the charged, massive scalar field on the electrically charged Ayón-Beato–García regular black hole. There the quasibound-state spectrum satisfies

$$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$4

in an overlap region where charged superradiance and bound-state confinement coexist. The fastest instability reported is

$$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$5

for the monopole mode $$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$6, near $$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$7 and $$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$8 (Dolan et al., 2024).

Another mechanism arises from localized non-positive near-horizon deformations of the effective potential. In that case a new purely imaginary mode $$\frac{d^2\phi_{\ell}(\omega, r_*)}{dr_*^2}+[\omega^2-V_\mathrm{eff}]\phi_{\ell}(\omega, r_*) = 0 .$$9 is created; as the deformation is moved toward the horizon, $$W\left(\phi_{\ell}^\mathrm{in}, \phi_{\ell}^\mathrm{out}\right)$$0 increases, reaches zero at a critical location, and then enters the upper half-plane. The threshold scales as

$$W\left(\phi_{\ell}^\mathrm{in}, \phi_{\ell}^\mathrm{out}\right)$$1

so sufficiently near-horizon localized deformations can trigger exponential growth (Ma et al., 1 Jun 2026).

The general spectral theory of such dynamical instabilities differs from ordinary QNM damping. In an indefinite-norm setting, unstable modes are square integrable, have vanishing conserved norm, and appear in conjugate pairs or quartets. On an infinite domain they emerge from the reservoir of outgoing quasi-normal modes as poles cross from the lower to the upper half-plane (Coutant et al., 2016). That framework clarifies why “spectral instability” and “dynamical instability” are related but distinct notions: the former concerns structural sensitivity of the resonance set, the latter the existence of growing solutions.

6. Stable observables and the search for robustness

One response to QNM fragility has been to identify observables that are more stable than the QNM frequencies themselves. Real-frequency scattering quantities are the most developed case. The greybody factor is

$$W\left(\phi_{\ell}^\mathrm{in}, \phi_{\ell}^\mathrm{out}\right)$$2

and recent analyses found that it remains largely unchanged under small metric perturbations until rather high frequencies. This suggests that scattering observables may encode black-hole parameters more robustly than QNM frequencies do (Shen et al., 15 Jul 2025).

The complex-angular-momentum counterpart of QNMs, the Regge poles, is also spectrally unstable under small perturbations, but the reconstructed observables remain stable. For a Schwarzschild black hole with a small distant Gaussian bump, the perturbed Regge-pole spectrum reorganizes into multiple branches, yet the scattering amplitude and absorption cross section satisfy

$$W\left(\phi_{\ell}^\mathrm{in}, \phi_{\ell}^\mathrm{out}\right)$$3

and can be reconstructed accurately from the unstable Regge-pole data themselves (Torres, 2023). The effect is therefore not a trivial low-overtone dominance; it is a collective compensation among unstable poles and residues.

The review literature extends this “stable observables” program to Regge poles, absorption cross sections, greybody factors, and perhaps reflectionless modes. The first few Regge poles are reported to be much more stable than the low-lying QNMs, and even when high-order Regge poles bifurcate, the associated scattering observables stay close to the unperturbed ones because the unstable contributions remain subleading in the relevant regime (Shen et al., 15 Jul 2025).

7. Physical significance, caveats, and scope

The physical significance of black hole spectral instability is neither uniformly catastrophic nor negligible. Some constructions that generate dramatic spectral changes rely on exotic matter, extreme spacetimes, or qualitatively large geometric changes disguised as “small” perturbations. Yet physically compelling mechanisms remain: massive fields generate quasi-bound states; charged black holes exhibit asymptotic spectral changes for arbitrarily small charge; matter-coupled systems and de Sitter scales introduce new slow families; and continued-fraction deformations such as the BH3 class can display overtone instability while preserving several basic physical constraints (Cardoso et al., 2024).

Near-extremal spacetimes introduce an additional nuance. Using the bilinear-form framework appropriate to QNM perturbation theory, near-extremal Kerr zero-damping modes can suffer order-unity fractional changes in decay rates under fine-tuned energetically infinitesimal perturbations, but the decay rates remain positive. By contrast, the near-extremal Reissner–Nordström–de Sitter modes relevant to strong cosmic censorship are stable under energetically infinitesimal perturbations (Yang et al., 2022). The correct notion of “small perturbation” is therefore norm-dependent and physically nontrivial.

Several open problems remain explicit in the literature. One is the extension from test fields to full coupled perturbation systems, already emphasized for the Einstein–Maxwell–scalar example (Wang et al., 14 Oct 2025). Another is the relation between migrated low-lying modes, echo branches, and continuous perturbations in the Green-function framework (Shen et al., 15 Jul 2025). A third is observational: whether future detectors can extract overtone structure accurately enough for spectral instability to become a practical limitation or a new diagnostic.

A plausible synthesis is that black-hole spectroscopy contains a built-in separation of roles. The exact QNM spectrum is structurally fragile because it belongs to a non-self-adjoint resonance problem, but the prompt ringdown and some real-frequency scattering quantities are often much more robust because causality, excitation amplitudes, and collective cancellations suppress the spectrally unstable sector (Wang et al., 14 Oct 2025). Black hole spectral instability is therefore best understood not as a single claim that “ringdown is unstable,” but as a layered statement about which spectral objects are fragile, which observables are stable, and under what physical perturbations the distinction breaks down.