Quasinormal modes (QNMs)

Dictionary of QNMs

• Algebraically special / symmetry‑protected frequencies
• Asymptotic (high‑overtone) spacing
• Avoided crossing (level repulsion)
• Boundary conditions (outer)
• Branch cut (continuous spectrum)
• Branch point in momentum (hydro breakdown)
• Diffusive / hydrodynamic mode
• Eikonal / photon‑sphere limit
• Exceptional point / pole collision
• Extremal (zero‑damped) branch
• Hydrodynamic series (gradient expansion)
• Instability (Im ω > 0)
• Isospectrality
• Late‑time tail
• Mode mixing (coupling across sectors)
• Mode trajectory (parameter flow)
• Overtones (radial index n)
• Pole (simple) & residue (excitation factor)
• Pole pattern (ladders, clustering, alignment)
• Pole motion (global)
• Pole‑skipping (holographic special points)
• Pseudospectrum / non‑normal sensitivity
• Quasi‑bound states
• Quasinormal expansion & completeness caveat
• Ringdown phases (prompt → QNM → tail)
• Spectral gap
• Superradiance
• Universality by asymptotics
• Zero mode (marginal)
• Zero‑damped modes (ZDMs)
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Algebraically special / symmetry‑protected frequencies

• What: Special frequencies fixed by symmetry or factorization of the perturbation equations; these can yield exact—often purely imaginary—solutions.
• Why it matters: They pin spectral features and can mark parity transitions or constraints on amplification or scattering.
• Identify by: Persistence of a frequency under small deformations that preserve the symmetry; closed‑form solutions.
• See also: Isospectrality, Mode mixing. Back to top

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Asymptotic (high‑overtone) spacing

• What: For large radial overtone $n$, QNMs align into regular “ladders”: nearly fixed $\mathrm{Re}\,\omega$ with approximately uniform spacing in $\mathrm{Im}\,\omega$.
• Why it matters: Encodes near‑horizon physics and boundary conditions; guides extrapolation and checks of numerics.
• Identify by: Vertical stacks of poles with $|\mathrm{Im}\,\omega|\to\infty$; approach to constant $\mathrm{Re}\,\omega$ and linear spacing in $\mathrm{Im}\,\omega$.
• See also: Overtones, Eikonal / photon‑sphere. Back to top

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Avoided crossing (level repulsion)

• What: As a parameter changes, two modes approach but exchange character without an exact eigenvalue crossing.
• Why it matters: Reveals non‑Hermitian spectral structure; explains mode identity swaps in parameter scans.
• Identify by: Smooth exchange of eigenfunction content or residues while eigenvalues veer apart.
• See also: Exceptional point / pole collision, Mode trajectory, Mode mixing. Back to top

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Boundary conditions (outer)

• What: The spectrum is defined by “ingoing at the horizon” plus an outer condition:
  • Flat: outgoing at infinity → poles + branch cut.
  • AdS: reflecting/normalizable at the boundary → purely discrete poles (no frequency‑axis branch cut).
  • dS: outgoing toward the cosmological horizon → discrete poles; exponential late‑time decay.
• Why it matters: Determines whether tails are power‑law versus exponential and whether hydrodynamic $k$-dependent poles exist.
• See also: Branch cut, Universality by asymptotics. Back to top

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Branch cut (continuous spectrum)

• What: Non‑meromorphic part of the Green’s function, typically along the negative imaginary $\omega$-axis in flat asymptotics.
• Why it matters: Produces power‑law late‑time tails; QNM pole sums alone are incomplete when a cut is present.
• Identify by: Non‑isolated singularity near $\omega=0$; failure of pure exponential decay at late times.
• See also: Late‑time tail, Quasinormal expansion & completeness. Back to top

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Branch point in momentum (hydro breakdown)

• What: In momentum‑dependent spectra, hydrodynamic and gapped branches collide at complex $k_{\star}$, creating a square‑root branch point in $\omega(k)$.
• Why it matters: Sets the radius of convergence of the hydrodynamic expansion; beyond $|k_{\star}|$ hydrodynamic poles cease to exist on the principal sheet.
• Identify by: Analytic continuation in $k$ showing a pole collision and two‑sheet structure near $k_{\star}$.
• See also: Diffusive / hydrodynamic mode, Exceptional point / pole collision, Hydrodynamic series. Back to top

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Diffusive / hydrodynamic mode

• What: Poles tied to conservation laws with $\omega(k)\to 0$ as $k\to 0$ (e.g., diffusion $\omega=-iDk^2+\cdots$, sound $\omega=\pm c_s k - i\Gamma k^2+\cdots$).
• Where: Systems with continuous momentum (planar horizons); requires translation or gauge symmetry.
• Identify by: Vanishing frequency and residue that couples to conserved densities or currents as $k\to 0$.
• See also: Hydrodynamic series, Branch point in momentum. Back to top

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Eikonal / photon‑sphere limit

• What: High‑$\ell$ QNMs governed by unstable null orbits: $\mathrm{Re}\,\omega \approx \ell\,\Omega_c$, $\mathrm{Im}\,\omega \approx -(n+\tfrac{1}{2})\lambda$.
• Why it matters: Direct link between wave dynamics and light‑ring geometry; informs high‑$\ell$ asymptotics.
• See also: Asymptotic spacing, Pole pattern. Back to top

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Exceptional point / pole collision

• What: Two (or more) poles coalesce into a double root as a control parameter is tuned; beyond it, solutions live on multiple Riemann sheets.
• Why it matters: Generates square‑root non‑analyticity; residues can blow up; mode identities swap.
• Identify by: Coincident eigenvalues with defective eigenvector structure; square‑root behavior of $\omega(\lambda)$.
• See also: Avoided crossing, Zero mode, Branch point in momentum. Back to top

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Extremal (zero‑damped) branch

• What: As an extremality parameter is approached, families with $\mathrm{Im}\,\omega\to 0^-$ emerge while $\mathrm{Re}\,\omega$ locks to a horizon‑set frequency.
• Why it matters: Ultra‑long‑lived ringing; signals singular dynamics of the extremal limit.
• See also: Zero‑damped modes, Superradiance, Spectral gap. Back to top

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Hydrodynamic series (gradient expansion)

• What: Low‑$\omega$, low‑$k$ expansion of correlators; transport coefficients read off from the lowest poles.
• Limits: Breaks down at the nearest complex‑$k$ singularity (often a pole collision).
• See also: Diffusive / hydrodynamic mode, Branch point in momentum. Back to top

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Instability (Im ω > 0)

• What: Poles in the upper‑half plane signal linear instability (exponential growth).
• Mechanisms: Superradiance plus confinement; long‑wavelength instabilities of extended horizons; near‑horizon AdS$_2$ BF‑bound violations.
• Identify by: Parameter where $\mathrm{Im}\,\omega$ crosses zero; characterize stable ↔ unstable branches.
• See also: Zero mode, Superradiance, Exceptional point. Back to top

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Isospectrality

• What: Distinct perturbation sectors share identical spectra due to a Darboux or supersymmetric transformation.
• Fragility: Small symmetry breaking (rotation, charge, anisotropy, or dimension) typically lifts it.
• See also: Algebraically special, Mode mixing. Back to top

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Late‑time tail

• What: Post‑ringdown decay: power‑law when a branch cut contributes (flat asymptotics) and exponential when spectra are purely discrete (AdS or dS).
• Variants: Massive or threshold effects yield oscillatory tails at special frequencies.
• See also: Branch cut, Quasinormal expansion & completeness. Back to top

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Mode mixing (coupling across sectors)

• What: Rotation, charge, or anisotropy mixes angular or spin‑weight sectors so eigenmodes are hybrids.
• Why it matters: Drives avoided crossings and identity exchanges; changes selection rules for excitation.
• See also: Avoided crossing, Isospectrality. Back to top

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Mode trajectory (parameter flow)

• What: Path of a single pole $\omega(\lambda)$ in the complex plane as a control parameter $\lambda$ varies (spin, charge, $k$, temperature, …).
• Why it matters: Reveals monotonic trends, spirals near extremality, and proximity to collisions.
• See also: Pole motion (global), Avoided crossing. Back to top

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Overtones (radial index $n$)

• What: For fixed angular or quantum labels, $n=0$ (fundamental) is least damped; $n\geq 1$ decay faster.
• Asymptotics: High‑$n$ modes approach the asymptotic spacing; finite‑$n$ can show non‑monotonic parameter trends.
• See also: Pole pattern, Spectral gap. Back to top

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Pole (simple) & residue (excitation factor)

• What: Each QNM is a simple pole of the retarded Green’s function with residue $\mathcal{R}$ setting excitation weight.
• Nuance: Near exceptional points, residues can be large or ill‑conditioned; bi‑orthogonal normalization matters.
• See also: Quasinormal expansion & completeness, Pseudospectrum. Back to top

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Pole pattern (ladders, clustering, alignment)

• What: Global organization: vertical ladders (fixed $\mathrm{Re}\,\omega$), clustering near special frequencies (e.g., bounds), or alignment toward a branch cut.
• Why it matters: Diagnoses scattering barriers and near‑horizon or near‑boundary structures.
• See also: Eikonal / photon‑sphere, Asymptotic spacing. Back to top

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Pole motion (global)

• What: Collective behavior as many poles shift, merge, or disappear when parameters change.
• Why it matters: Reveals phase transitions or topology changes in the spectral Riemann surface.
• See also: Mode trajectory, Exceptional point. Back to top

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Pole‑skipping (holographic special points)

• What: Points $(\omega_{\star},k_{\star})$ where both numerator and denominator vanish so the retarded Green’s function is not uniquely defined; poles “skip.”
• Why it matters: Tied to many‑body chaos data (Lyapunov exponent, butterfly velocity) in planar horizons.
• See also: Diffusive / hydrodynamic mode, Hydrodynamic series. Back to top

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Pseudospectrum / non‑normal sensitivity

• What: Non‑normal operators can have spectra highly sensitive to small perturbations; large transient growth despite modal stability.
• Why it matters: Explains numerical sensitivity near collisions and strong but short‑lived responses.
• See also: Pole (simple) & residue, Exceptional point. Back to top

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Quasi‑bound states

• What: Long‑lived, weakly leaky resonances due to extra trapping (mass terms, double‑barrier structures, or external potentials).
• Why it matters: Frequencies approach thresholds with exponentially small $|\mathrm{Im}\,\omega|$; can mimic near‑stable modes.
• See also: Superradiance, Spectral gap. Back to top

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Quasinormal expansion & completeness caveat

• What: Time‑domain signal = prompt response → finite QNM sum → tail. With a branch cut, poles are not a complete basis.
• AdS/dS: Purely discrete spectra ⇒ late time is dominated by the least damped pole.
• See also: Ringdown phases, Late‑time tail. Back to top

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Ringdown phases (prompt → QNM → tail)

• What:
  • Prompt: direct propagation, initial‑data dependent.
  • QNM: exponentially damped oscillations from poles.
  • Tail: branch‑cut‑controlled decay (or exponential if discrete‑only).
• Why it matters: Guides windowing and fitting strategies in data analysis.
• See also: Quasinormal expansion & completeness, Spectral gap. Back to top

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Spectral gap

• What: Minimal $|\mathrm{Im}\,\omega|$ over all decaying modes; sets the longest relaxation time.
• Why it matters: Controls late‑time decay scale; in AdS, bounds thermalization rates.
• See also: Overtones, Extremal branch. Back to top

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Superradiance

• What: Amplification extracting rotational or electrostatic energy when a kinematic inequality holds (e.g., $\omega<m\Omega_H$ or $\omega<q\Phi_H$).
• Outcomes: Without confinement → reduced damping; with confinement (mirror, AdS, mass) → Instability.
• See also: Zero‑damped modes, Exceptional point. Back to top

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Universality by asymptotics

• Flat: poles plus branch cut; power‑law tails.
• AdS: discrete poles; exponential decay; momentum‑dependent hydrodynamics.
• dS: discrete poles between horizons; exponential decay set by the slowest mode.
• See also: Boundary conditions (outer), Late‑time tail. Back to top

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Zero mode (marginal)

• What: $\omega=0$ stationary perturbation at a threshold (onset of instability or bifurcation to a new solution branch).
• Why it matters: Separates stable and unstable spectra; often seeds new “hairy” branches.
• See also: Instability, Exceptional point. Back to top

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Zero‑damped modes (ZDMs)

• What: Families with $\mathrm{Im}\,\omega\to 0^-$ as extremality is approached; $\mathrm{Re}\,\omega$ tends to a horizon value.
• Why it matters: Dominate ultra‑late‑time behavior near extremality; indicate singular extremal dynamics.
• See also: Extremal branch, Superradiance, Spectral gap. Back to top

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How to use this page

• Observe each pole with: complex $\omega$, labels (angular/momentum if applicable), overtone $n$, residue, asymptotics class, and control parameters.
• Classify using ≥1 of the entries above; add secondary tags as needed.
• Track motion if parameters vary (trajectory snapshots, collisions, gaps).
• Flag novelty if the behavior does not fit any existing entry (see submission below).

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Email: renphysics@adscft.org

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