A New Method for Quasinormal Modes From Bound States and Homotopy deformations

Technical Summary: A New Method for Quasinormal Modes From Bound States and Homotopy Deformations

Problem Statement
Quasinormal modes (QNMs) characterize the oscillations of perturbed black holes and encode geometric information crucial for gravitational-wave astronomy and tests of general relativity. While established numerical methods like Leaver's continued fraction technique offer high precision, they lack analytical intuition. Conversely, analytical methods like the Wentzel-Kramers-Brillouin (WKB) approximation often lose accuracy for low multipole numbers or high overtones.

A specific analytical approach, pioneered by Mashhoon, relates QNM frequencies to the bound state spectra of an inverted potential via analytic continuation. However, this traditional method is limited by its reliance on closed-form analytical solutions for bound states, restricting its application to specific potentials (e.g., Pöschl-Teller) that only approximate the true black hole potential. Recent numerical extensions by Völkel allowed for the use of non-analytic potentials but still faced limitations in accuracy for higher overtones.

Methodology
The authors propose a new numerical method that maps the QNM problem to a bound state problem using a single real scaling parameter, $\alpha$, rather than transforming model parameters.

1. Coordinate Transformation and Parameterization: Starting from the radial master equation for a spherically symmetric black hole, the authors introduce a coordinate scaling $-ix \to \alpha x$. This transforms the QNM equation into a standard bound state eigenvalue problem:
   $$\frac{d^2}{dx^2} \Psi(x) + [E - V_{\text{inv}}(\alpha x, P)] \Psi(x) = 0$$
   where $E \equiv -\omega^2$ and $V_{\text{inv}}$ is the inverted potential.
2. Analytic Continuation: The QNM frequencies $\omega_n$ are recovered by analytically continuing the bound state eigenvalues $E_n(\alpha)$ from real values of $\alpha$ to the target point $\alpha = -i$.
3. Numerical Implementation:
   • Bound state eigenvalues $E_n(\alpha)$ are computed numerically on a uniform grid of real $\alpha$ using the shooting method.
   • The data is fitted using Taylor expansions and Padé approximants to construct a numerical representation of $E_n(\alpha)$.
   • The analytic continuation is performed to obtain $\omega_n = \sqrt{-E_n(\alpha = -i)}$.
4. Singularity Analysis: To understand the limitations of the method, the authors analyze the pole distribution of the Padé approximants in the complex $\alpha$-plane. This identifies the singularities of $E_n(\alpha)$, which determine the radius of convergence for the analytic continuation.
5. Homotopy Deformation: To address convergence failures for high overtones, the authors introduce a homotopy deformation to the inverted potential:
   $$V_\lambda(\alpha x) = H(\alpha, \lambda)V_{\text{inv}}(\alpha x)$$
   where $H(\alpha, \lambda) = (\lambda - 1)\alpha^2 + \lambda$. This parameter $\lambda$ deepens the potential well for real $\alpha$ while ensuring $H(-i, \lambda) = 1$, thereby restoring the exact QNM problem at the target point. This deformation shifts the singularities of $E_n(\alpha)$ in the complex plane.

Key Results
The method was applied to scalar, electromagnetic, and axial gravitational perturbations of the Schwarzschild black hole using the exact Regge-Wheeler potential.

• Accuracy for Low Overtones: For low-lying modes (specifically $n \leq \ell$), the method yields high-accuracy QNM frequencies. For gravitational perturbations with $\ell=2$, the Padé approximant results for $n=0, 1, 2$ show relative errors on the order of $10^{-7}$ to $10^{-9}$ compared to Leaver's reference values.
• Limitation for High Overtones: For higher overtones ($n \geq 3$ for $\ell=2$), the Taylor expansion diverges, and the accuracy of Padé approximants degrades.
• Origin of Limitation: The pole analysis reveals that for higher overtones, the singularities of $E_n(\alpha)$ lie within the circle of analytic continuation centered at the expansion point $\alpha_0 = 0.5$ and extending to the target $\alpha = -i$. This obstruction prevents the convergence of the series.
• Success of Homotopy Deformation: By increasing the homotopy parameter $\lambda$, the singularities are shifted outward, crossing the boundary of the analytic continuation circle. This allows the method to reliably compute QNM frequencies for previously inaccessible modes (e.g., $n=3$ and $n=4$). The deformed method produces results consistent with reference values, improving upon the non-deformed Padé results.

Significance and Claims
The paper claims several technical and conceptual contributions:

1. Unified Framework: Unlike traditional methods requiring specific parameter transformations for each potential model, this method provides a unified treatment using a single scaling parameter $\alpha$, applicable to exact potentials without approximation.
2. Direct Application to Exact Potentials: This is the first numerical calculation of Schwarzschild QNMs directly from the exact Regge-Wheeler potential using a bound-state-to-QNM mapping, avoiding the need for approximate potentials (such as Pöschl-Teller) used in previous analytical attempts.
3. Diagnostic Tool: The use of Padé approximants to map the singularity structure of the eigenvalues provides a direct explanation for the breakdown of analytic continuation methods at high overtones.
4. Remediation Strategy: The proposed homotopy deformation offers a practical mechanism to overcome analytic obstructions, enabling the reliable computation of additional high overtone modes.

The authors note that while the method works for $n \leq 4$ with deformation, singularities for $n > 4$ cannot be effectively pushed outward with the current deformation scheme, indicating a remaining limit. They suggest future work could explore conformal mappings or extensions to rotating (Kerr) black holes.