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Overdamped QNM for Schwarzschild black holes

Michael Hitrik, Maciej Zworski

TL;DR

Addresses global counts of quasinormal modes for Schwarzschild and Schwarzschild–de Sitter black holes. Uses complex scaling, FBI transform, and analytic-symbol calculus to obtain a Bohr–Sommerfeld type quantization and an explicit QNM lattice described by the leading symbol G0. Proves a sharp lower bound $|QNM ∩ D(0,r)| ≥ c r^3$ and derives an exponentially accurate one-dimensional spectral description via G((2n+1)h;h). Shows that this framework yields accurate QNM values deep in the complex plane where standard numerics fail and explains the distorted lattice observed in computations.

Abstract

We prove that the number of quasinormal modes (QNM) for Schwarzschild and Schwarzschild-de Sitter black holes in a disc of radius $ r $ is bounded from below by $ c r^3 $. This shows that the recent upper bound by Jézéquel is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM and explains the emergence of a distorted lattice on which they lie. Our presentation gives a general result about exponentially accurate Bohr-Sommerfeld quantization rules for one dimensional problems. The description of QNM allows their accurate evaluation ``deep in the complex" where numerical methods break down due to pseudospectral effects.

Overdamped QNM for Schwarzschild black holes

TL;DR

Addresses global counts of quasinormal modes for Schwarzschild and Schwarzschild–de Sitter black holes. Uses complex scaling, FBI transform, and analytic-symbol calculus to obtain a Bohr–Sommerfeld type quantization and an explicit QNM lattice described by the leading symbol G0. Proves a sharp lower bound and derives an exponentially accurate one-dimensional spectral description via G((2n+1)h;h). Shows that this framework yields accurate QNM values deep in the complex plane where standard numerics fail and explains the distorted lattice observed in computations.

Abstract

We prove that the number of quasinormal modes (QNM) for Schwarzschild and Schwarzschild-de Sitter black holes in a disc of radius is bounded from below by . This shows that the recent upper bound by Jézéquel is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM and explains the emergence of a distorted lattice on which they lie. Our presentation gives a general result about exponentially accurate Bohr-Sommerfeld quantization rules for one dimensional problems. The description of QNM allows their accurate evaluation ``deep in the complex" where numerical methods break down due to pseudospectral effects.
Paper Structure (10 sections, 26 theorems, 340 equations, 2 figures)

This paper contains 10 sections, 26 theorems, 340 equations, 2 figures.

Table of Contents

  1. Introduction and statement of the results
  2. Regge--Wheeler potentials
  3. Complexified Regge--Wheeler potentials
  4. Complex scaling
  5. Resonance free regions
  6. Analytic symbols
  7. Review of pseudodifferential and Fourier integral operators
  8. Global pseudodifferential operators
  9. Local Fourier integral operators
  10. A general semiclassical result

Key Result

Theorem 1

For $0 < t \ll 1$ and $0 < \Lambda < 1/9m^2$, let $A_t ( r ) := \{ \lambda : 1 \leq |\lambda | \leq r , \arg \lambda > - t \}$. Then, there exist $c ( t, m, \Lambda ) > 0$, such that, as $r \to \infty$, When $\Lambda = 0$ the asymptotic equality in eq:defN should be replaced by $\geq$.

Figures (2)

  • Figure 1: A comparison of numerically computed QNM in jan for the case of $m = 1$, $\Lambda = 0$ and the leading term in the semiclas-sical expression for QNM \ref{['eq:defG']} given by \ref{['eq:G0']}. Even at large values of $h = (\ell + \frac{1}{2})^{-1} = \frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$, one sees a reasonable agreement as well as the emergence of the distorted lattice deeper in the complex (as opposed to the lattice in strips described in Sá Barreto--Zworski saz).
  • Figure 2: The top plot shows the spectrum of the rotated harmonic oscillator $-h^2 \partial_x^2 + i x^2$ used by Davies and Embree--Trefethen trem to illustrate spectral instability for non-normal pseudodifferential operators. The eigenvalues (explicitly given by $e^{\pi i /4 } h ( 2n + 1 ))$ are com-pared to eigenvalues computed numerically using the basis of the first 151 Hermite functions (eigenfunctions of $- h^2 \partial_x^2 + x^2$). This illustrates the fragility of eigenvalues "deep in the complex". The bottom plot shows a calculation of QNM for $m = 1$, $\Lambda = 0$ and $\ell = 20$ using the Mathe-matica code jan. We again see divergence from the (mathematically) established result (modulo the issues of the size of the region).

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof : Proof of \ref{['eq:hitsj']}
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Proposition \ref{['p:ave']}
  • ...and 35 more