Generalized Eigenvalue and Resonance Problems

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Discretizing an eigenvalue equation produces many numbers, only some of which approximate the continuum spectrum. Boundary rows, polynomial linearization, finite endpoints, and roundoff can each generate plausible-looking artifacts. A resonance is therefore identified by a bundle of evidence—residual, boundary behavior, resolution and precision stability, branch continuity, and an independent method—not by its appearance in one matrix spectrum.

Required background. Ordinary Differential Equations supplies linear boundary problems, eigenvalues, and complex initial-value systems.

Helpful background. Formulations of General Relativity provides the continuum systems being discretized, while Partial Differential Equations and Green Functions supplies spectral domains, adjoints, and radiative boundary conditions.

Matrix pencils from differential operators

After basis expansion or collocation, a linear spectral problem often becomes

A_Nv=\lambda B_Nv.

If the differential equation depends quadratically or polynomially on frequency, it becomes $P_N(\lambda)v=0$ and may be enlarged into a linear pencil. For example, set $w=\lambda v$ to represent $(A_0+\lambda A_1+\lambda^2A_2)v=0$ as

\begin{pmatrix}0&I\\-A_0&-A_1\end{pmatrix} \begin{pmatrix}v\\w\end{pmatrix} =\lambda \begin{pmatrix}I&0\\0&A_2\end{pmatrix} \begin{pmatrix}v\\w\end{pmatrix}.

The first block row enforces $w=\lambda v$ and the second recovers the polynomial. For a regular polynomial problem, this companion linearization preserves its finite eigenvalues with algebraic multiplicities; singular leading coefficients can produce eigenvalues at infinity. Record the original polynomial residual and the auxiliary relation $w-\lambda v$ , as well as the enlarged-pencil residual. A singular polynomial with determinant identically zero requires separate constraint analysis.

For a candidate $(\lambda,v)$ , use a scale-invariant backward residual such as

\eta=\frac{\lVert A_Nv-\lambda B_Nv\rVert_2} {(\lVert A_N\rVert_2+\lvert\lambda\rvert\lVert B_N\rVert_2)\lVert v\rVert_2}.

A small $\eta$ says the pair solves the discrete pencil accurately; it does not prove that the pencil approximates the intended differential operator. The construction of generalized and polynomial collocation problems, including boundary-row treatment, is described in Jansen 2017, § 2.2, Eq. 2.9, and § 2.3, Eqs. 2.11–2.12.

An exact Dirichlet spectrum

Begin with

-u''=\lambda u, \qquad 0\lt x\lt \pi, \qquad u(0)=u(\pi)=0.

The exact eigenpairs are $\lambda_n=n^2$ and $u_n=\sin(nx)$ . Construct Chebyshev or other spectral matrices at $N=20,40,80$ . Homogeneous Dirichlet values may be eliminated, leaving only interior unknowns. If instead a boundary row is replaced by $u_0=0$ , its matching row in $B_N$ must be zero; setting $B_{00}=1$ would impose $(1-\lambda)u_0=0$ and create an unwanted eigenvalue at $\lambda=1$ . Solve the resulting pencil and separate any infinite eigenvalues from finite candidates. For each low mode report eigenvalue error, normalized matrix residual, an independently sampled differential residual, and boundary values. Compare eigenfunctions only after fixing a normalization and phase.

Resolved candidates approach $n^2$ as $N$ increases and remain stable when arithmetic precision rises. Modes that drift with $N$ , localize at endpoints, or have large continuous residuals are not established continuum modes at that resolution, even if the matrix residual is tiny. They may be discretization artifacts or under-resolved genuine modes; further refinement distinguishes the possibilities. This benchmark tests the full discretize–solve–filter pipeline before complex resonances enter.

Resonance boundary factors

A black-hole resonance problem adds frequency-dependent ingoing and outgoing behavior. The continuum equation and physical interpretation remain with black-hole perturbation theory; the computational task here is to factor known endpoint behavior, map the remaining regular function to a finite interval, and solve the resulting complex pencil. With time dependence $e^{-i\omega t}$ , stable modes have $\operatorname{Im}\omega\lt 0$ .

Let $M$ denote physical mass, so $GM$ is its geometrized length. For the spin-2, $\ell=2$ , $n=0$ Schwarzschild gravitational mode, the Regge–Wheeler and Zerilli master equations are isospectral. With the stated Fourier convention, a reference target is

GM\,\omega=0.37367168-0.08896232i.

Leaver formulates the ingoing-horizon and outgoing-infinity problem and its continued-fraction spectrum in Leaver 1985, § 2, Eqs. 2–14 and Table 1; the displayed higher-precision Schwarzschild value and its $(\ell,m,n)=(2,2,0)$ identification are stated in Dias, Godazgar, and Santos 2022, Figure 4 and its surrounding text. State harmonic indices, overtone, Regge–Wheeler or Zerilli master variable, mass normalization, endpoint factors, compactification, and complex branch choices. Compute at $N=40,60,80$ , then repeat $N=80$ at 100-bit and 200-bit precision. Require stable digits in the frequency, decreasing differential and boundary residuals, and a mode function with the declared asymptotics. Jansen’s examples provide a useful resolution/precision cross-comparison and artifact-filtering pattern in Jansen 2017, § 3, Eqs. 3.1–3.2 and Figures 2–3; those equations are package calls, not a residual formulation.

Filters that discriminate artifacts

No single filter suffices. Cluster candidates across resolution by both frequency and eigenfunction overlap; raise precision; vary map scale and truncated endpoints; inspect coefficient decay; and evaluate the original differential operator off grid. A genuine isolated mode should persist under all modest changes. An independent continued-fraction, shooting, or time-domain extraction is stronger because it does not reuse the same collocation pencil.

Nonnormal pencils can have sensitive eigenvalues: a tiny backward residual may coexist with a large forward error. The replacement of spectral statements by pseudospectral counterparts is developed in Embree and Trefethen 2001, §§ 1–3, pp. 583–590. Estimate left and right eigenvector conditioning or a local pseudospectral response. Near multiple modes, report an invariant subspace rather than unstable individual eigenvectors.

The validity regime is limited to the declared spectral sheet and boundary asymptotics. Changing the sign of the Fourier convention, selecting the other square-root branch, or crossing a cut changes the resonance problem rather than refining it.

Environment, rerun, and failure criteria

Pin the numerical library, eigensolver, matrix balancing, basis and node ordering, boundary-row construction, domain map, arithmetic precision, and mode-matching tolerances. A clean rerun emits matrices or reproducible matrix builders, candidate spectra, normalized residuals, coefficient tails, resolution and precision sweeps, endpoint variations, and the independent comparison.

Reject wrong branches, spurious collocation modes, endpoint-localized vectors, singular pencils handled without explanation, and apparent convergence in frequency alone. Further failure modes are an eigenvalue stable at fixed precision but drifting when precision rises, a small pencil residual with a large differential residual, or agreement obtained by matching to the reference value before solving. The reference benchmark validates a workflow; it must not become an input root.

Check your understanding

A candidate frequency is unchanged from $N=40$ to $N=80$ and has matrix residual $10^{-13}$ , but it moves in the fourth digit when arithmetic precision is doubled. How should it be classified?

Solution

It is not yet established. Fixed-precision agreement can be a roundoff plateau in an ill-conditioned pencil. Estimate eigenvalue conditioning, repeat at still higher precision, inspect the off-grid differential and boundary residuals, vary the domain map, and compare with an independent method. Quote only the digits stable under those tests.

References

Dias, O. J. C., Godazgar, M., and Santos, J. E. (2022), “Eigenvalue Repulsions and Quasinormal Mode Spectra of Kerr–Newman: An Extended Study,” Journal of High Energy Physics 2022, 076. DOI. Open preprint.
Embree, M., and Trefethen, L. N. (2001), “Generalizing Eigenvalue Theorems to Pseudospectra Theorems,” SIAM Journal on Scientific Computing 23, 583–590. DOI.
Jansen, A. (2017), “Overdamped Modes in Schwarzschild–de Sitter and a Mathematica Package for the Numerical Computation of Quasinormal Modes,” European Physical Journal Plus 132, 546. arXiv.
Leaver, E. W. (1985), “An Analytic Representation for the Quasi-Normal Modes of Kerr Black Holes,” Proceedings of the Royal Society of London A 402, 285–298. DOI. Open PDF.