Quasinormal Mode Expansion

• Quasinormal mode expansion is a spectral decomposition that represents open, non-selfadjoint wave systems as resonant states with complex frequencies.
• It separates resonant contributions from nonresonant background terms using techniques like contour integration, hyperboloidal slicing, and Dyson equations for regularization.
• This approach facilitates rigorous computation of Green functions, scattering matrices, and decay rates in disciplines such as electromagnetics, acoustics, and black-hole perturbation theory.

Quasinormal mode expansion is a spectral representation for open, radiative, lossy, or otherwise non-selfadjoint wave systems in which the response is decomposed into resonant states with complex frequencies. In continuum mechanics, acoustics, electrodynamics, quantum theory, and black-hole perturbation theory, the central objects are quasinormal modes (QNMs), i.e. resonances or outgoing-wave solutions whose complex eigenfrequencies encode oscillation and damping. The expansion may target fields, Green functions, scattering matrices, or time-domain waveforms; depending on the setting, it also includes background terms, branch-cut contributions, or residual contours (Nicolet et al., 2024, Macedo et al., 2018).

1. Basic structure of the expansion

In electromagnetic resonators, a standard form of the expansion is the QNM representation of the transverse Green function inside the resonator,

$${\bf G}^{\rm T}({\bf r}_1,{\bf r}_2;\omega) = \sum_{\mu} \frac{\omega^2}{2\tilde\omega_{\mu} (\tilde\omega_{\mu}-\omega)}\, \tilde{\bf f}_{\mu}({\bf r}_1)\tilde{\bf f}_{\mu}({\bf r}_2),$$

where $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$ and $\tilde{\bf f}_\mu$ are the QNM fields. This form underlies calculations of spontaneous decay rates, Purcell factors, and propagators in arbitrarily shaped metallic nanoresonators, provided the modes are normalized with the generalized inner product appropriate to dispersive, lossy media (Ge et al., 2013).

In black-hole perturbation theory, the analogous object is an asymptotic expansion of the solution itself. On hyperboloidal slices, the solution may be written as

$$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$

where $s_n$ are the quasinormal frequencies, $\phi_n$ the QNM profiles, $\eta_n$ the excitation coefficients, and the integral encodes the branch cut or tail contribution. This formulation makes explicit that QNM expansions in black-hole problems are generally resonant expansions rather than ordinary orthogonal-mode decompositions (Macedo et al., 2018).

A common structural feature is that QNM expansions separate resonant contributions from nonresonant ones. In photonics this residual piece may be written as a background Green function or a residual contour term; in black-hole perturbations it appears as a branch cut, a tail contribution, or an exponentially decaying remainder. This suggests that “quasinormal mode expansion” names a family of related spectral constructions rather than a single universal formula.

2. Spectral and geometric formulations

A rigorous spectral formulation emerges most clearly when outgoing boundary conditions are turned into regularity conditions. In the black-hole setting, hyperboloidal slicing and conformal compactification replace standard Cauchy slices by slices extending from the horizon to future null infinity, and the compactified coordinate places the horizon and null infinity at finite locations. In the Reissner–Nordström case, the “minimal gauge” simplifies the metric functions and wave operators to polynomials in the compact coordinate, enabling a regular spectral analysis and a generalized coefficient algorithm for $(m+2)$-term recurrence relations (Macedo et al., 2018).

Within this hyperboloidal setting, the Keldysh scheme gives a resolvent expansion based on right eigenvectors $v_n$ and left eigenvectors or comodes $\alpha_n$,

$\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$0

The construction is explicitly bi-orthogonal and non-selfadjoint. Scalar products are not necessary for the resonant time series at null infinity, but they are required when one wants constant excitation coefficients in bulk expansions; the product $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$1, not $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$2 alone, is the invariant object (Besson et al., 2024).

An operator-theoretic generalization appears in dispersive wave problems. There the starting point is a holomorphic operator function $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$3, with nonlinear eigenvalue problem $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$4. Linear pencils, polynomial matrices, and rational matrices are treated within the same framework, and the resulting modal expansion of the resolvent is described as physically agnostic because it depends on operator structure rather than on whether the underlying problem is optical, acoustic, mechanical, or quantum (Nicolet et al., 2024).

These formulations clarify a central point: QNM expansion is not restricted to one normalization convention, one coordinate system, or one physical interpretation. It is a spectral decomposition of a non-Hermitian response operator, with geometric regularization, auxiliary fields, or contour methods supplying the structure needed to make the expansion well posed.

3. Green functions, scattering matrices, and resonant observables

For open electromagnetic resonators, the Green-function expansion is the most direct implementation of QNM theory. In metal nanoresonators, the Dyson equation extends the interior QNM expansion to points outside the resonator,

$\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$5

and yields a regularized field $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$6 that removes the exponential far-field divergence of bare QNMs. This permits closed-form expressions for the Purcell factor, a generalized effective mode volume, and an analytic Green function valid both near and far from the metal surface (Ge et al., 2013).

The same modal philosophy extends from fields to the scattering matrix. If $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$7 are the complex eigenfrequencies and $\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$8 the far-field scattering eigenvectors of the QNMs, then the full scattering matrix can be reconstructed as

$\tilde{\omega}_\mu=\omega_\mu-i\gamma_\mu$9

This representation is general for any number of QNMs and arbitrary numbers of input and output ports, requires only the complex eigenfrequencies and the far-field properties of the modes, and avoids ad hoc nonresonant channels when the QNM set is complete (Alpeggiani et al., 2016).

A further development concerns quantities that are quadratic in the field rather than linear in it. For absorption, radiated power, or emission rates, naive field expansions generate cross-terms and divergence problems. By contour integration and regularization at complex-conjugated eigenfrequencies, the Riesz projection approach provides modal contributions to quadratic observables without unphysical divergence, and it remains valid both inside and outside the resonator (Betz et al., 2022).

Taken together, these formulations show that the target of the expansion need not be the field itself. Green functions, scattering amplitudes, decay rates, energy fluxes, and other observables can all be organized into QNM contributions, provided the proper analytic continuation or regularization is built in.

4. Dispersive media, far-field regularization, and coupled resonators

Time dispersion introduces a nonlinear frequency dependence into the eigenproblem. One route is a general matrix-level approach based on polynomial or rational operator pencils, with rational approximations of frequency-dependent material data enforcing causality and enabling direct use of nonlinear eigenvalue solvers such as those available in SLEPc (Nicolet et al., 2024). Another route is an auxiliary-field linearization: for materials with partial-fraction permittivity, auxiliary polarization fields convert Maxwell’s nonlinear eigenproblem into an augmented linear system $\tilde{\bf f}_\mu$0, after which the scattered field is expanded as $\tilde{\bf f}_\mu$1. This yields a general QNM expansion method for both discrete and periodic resonant structures and is used to analyze metal–dielectric–metal perfect absorbers with only a few leading modes (Ming, 2023).

Far-field quantities require an additional layer of regularization because bare QNMs diverge exponentially with distance. For the energy flux density,

$\tilde{\bf f}_\mu$2

analytic continuation $\tilde{\bf f}_\mu$3 and contour deformation give a Riesz-projection expansion in which each modal term combines outgoing and incoming factors, cancelling the far-field exponential divergence. The method was demonstrated for the angular resolved modal energy flux in the far field of a nanophotonic device (Binkowski et al., 2020).

For quadratic observables more generally, the same logic gives

$\tilde{\bf f}_\mu$4

where the residual term $\tilde{\bf f}_\mu$5 may be evaluated from a background contour or interpolated from real-frequency simulations. This formalism eliminates problematic cross-terms and extends modal analysis to physically relevant quadratic observables outside nanophotonic resonators; the implementation was incorporated into the RPExpand software (Betz et al., 2022).

Coupled-resonator systems introduce further complications, especially when cavities are spatially separated and finite retardation is important. A recursive Dyson construction builds the $\tilde{\bf f}_\mu$6-cavity Green function from single-cavity QNMs and regularized external fields, with no need to compute full-system QNMs and no nested multidimensional integrals. In the two-dimer example studied, the method reproduced the full numerical Green function at both near-wavelength and much larger separations (Fuchs et al., 14 Oct 2025). At the quantum level, a quantized QNM formulation for coupled lossy resonators shows that the electric-field operator expands in QNMs whose annihilation operators satisfy non-canonical commutation relations, and that non-diagonal QNM couplings are needed to capture interference effects missed by commonly adopted dissipative Jaynes–Cummings-type models (Ren et al., 2021).

5. Black-hole perturbation theory and asymptotic analytic expansions

In black-hole physics, QNM expansion appears both as a rigorous resonant decomposition and as a family of asymptotic approximation schemes. For generic spherically symmetric, asymptotically flat black holes described by the Rezzolla–Zhidenko parametrization, perturbations reduce to

$\tilde{\bf f}_\mu$7

Using the WKB approach together with an expansion in inverse multipole number, analytic formulas were derived for small deviations from Schwarzschild, and the resulting least-damped mode was shown to satisfy Hod’s proposal relating damping to the Hawking temperature in the relevant parameter regime (Dubinsky, 2024).

The same parametrized framework also supports a hierarchical viewpoint on observables. For “moderate black-hole geometries,” the dominant quasinormal frequencies depend mainly on the first few continued-fraction coefficients of the metric expansion, notably $\tilde{\bf f}_\mu$8, $\tilde{\bf f}_\mu$9, and $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$0, while higher coefficients matter only if the geometry changes strongly between the horizon and the radiation zone. This provides an effective truncated formalism for constraining deviations from Schwarzschild with low-lying QNMs (Konoplya et al., 2022).

Beyond vacuum spacetimes, inverse-multipole expansions have been extended to matter-supported black holes. For a regular black hole sourced by a Dehnen-type dark-matter halo, the axial gravitational sector splits into inequivalent “up” and “down” channels, breaking isospectrality. A double expansion in the halo parameter $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$1 and $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$2 yields analytic formulas accurate to under $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$3 for $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$4 and to a few percent for $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$5 when compared with sixth-order WKB results including Padé resummation (Malik, 19 Mar 2026).

Rotating spacetimes motivate several distinct asymptotic schemes. In Kerr, a direct $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$6 expansion in the eikonal limit ties the leading QNM frequency to the orbital frequency of the relevant unstable null orbit and the leading damping to the associated Lyapunov exponent, while also producing spin-dependent $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$7 corrections for equatorial modes and explicit leading results for polar modes (Dolan, 2010). In Kerr–de Sitter, one approach derives a Bohr–Sommerfeld-type quantization condition for slowly rotating black holes and proves that wave-equation solutions admit a QNM expansion with exponentially decaying remainder (Dyatlov, 2011). Another maps the radial and angular perturbation equations to Heun equations and uses accessory-parameter expansion plus the isomonodromic $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$8-function to obtain double series for frequencies and angular eigenvalues in the rotation parameter $$\Phi(\tau, x^k) = \sum_{n=0}^{\infty} \eta_{n} \phi_n(x^k) e^{s_{n}\tau} + \int_{-\infty}^{0} \eta(s)\phi(x^k; s) e^{s\tau} ds,$$9 and the extremality parameter $s_n$0 (Novaes et al., 2018).

Large-dimension methods constitute a separate asymptotic regime. For static Einstein–(Anti-)de Sitter black holes, the inverse-dimensional expansion isolates decoupled modes localized in the near-horizon region, including the hydrodynamic sound and shear modes of AdS black branes and the Gregory–Laflamme instability frequencies of Ricci-flat branes. The expansion is explicitly asymptotic rather than convergent, but it gives analytic control over modes that capture black-hole-specific physics (Emparan et al., 2015).

6. Non-uniqueness, convergence, and limitations

A recurrent issue is that QNM expansion is rarely unique in a strong algebraic sense. In Lorentz-dispersive electromagnetism, the excitation coefficient $s_n$1 depends on how Maxwell’s equations are linearized and how the source term is split among auxiliary variables. Several formulas are valid simultaneously, all converging to the same scattered field in the complete basis, and the discrete numerical expansion naturally contains both QNMs and PML modes (Gras et al., 2019).

A related non-uniqueness arises in black-hole ringdown. Small modifications of the radial potential or boundary conditions can produce very different QNM spectra, and therefore different QNM “basis sets,” even though the prompt ringdown waveform remains stable. In the setting of Kerr and Kerr–de Sitter perturbations, such destabilized basis sets can improve convergence and even capture the late-time tail; near avoided crossings and exceptional points, large mode amplitudes destructively interfere so that the prompt waveform is not anomalously enhanced (Oshita et al., 27 Mar 2025).

Convergence is likewise subtle. The hyperboloidal Keldysh expansion is generally asymptotic rather than convergent, and its practical validity depends on time, regularity, and the growth of overtone excitation coefficients. The same framework nevertheless recovers Schwarzschild late power-law tails, studies the earliest time of validity of the resonant expansion, and uses $s_n$2-Sobolev transient-growth and pseudospectral tools to analyze non-modal effects (Besson et al., 2024). In large-$s_n$3 gravity, the $s_n$4 expansion is also asymptotic rather than convergent because decoupling fails nonperturbatively at finite $s_n$5 (Emparan et al., 2015).

Regularization remains indispensable whenever bare QNMs diverge spatially. In photonics, Dyson-regularized fields, contour-integral expansions for far-field quantities, and the use of scattering solutions at complex-conjugated frequencies resolve this issue and make expansions well behaved outside the resonator (Binkowski et al., 2020). In asymptotically anti-de Sitter black holes, a different limit appears: for Schwarzschild–AdS and massive scalar fields above the Breitenlohner–Freedman bound, the modes closest to the real axis have imaginary parts of size $s_n$6, and the real parts admit an asymptotic expansion whose first coefficients depend sensitively on spacetime dimension (Gannot, 2012).

These results delimit the scope of the method. QNM expansions can be rigorous, asymptotic, regularized, or basis-dependent; they may require residual terms, branch cuts, or PML sectors; and their convergence can improve or deteriorate under changes of geometry, dispersion model, or boundary conditions. The unifying principle is not exact modal completeness in a Hermitian sense, but the controlled extraction of resonant content from non-selfadjoint wave dynamics.