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<p>We show that the number of quasinormal modes (QNM) for Schwarzschild and Schwarzschild–de Sitter black holes in a disc of radius <i>r</i> is bounded from below by <i>cr</i><sup>3</sup>, proving that the recent upper bound by Jézéquel [Anal. PDE <b>17</b>, 2024,] is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM, explaining the emergence of a distorted lattice and generalizing the lattice structure in strips described by Sá Barreto-Zworski [Math. Res. Lett. <b>4</b>, 1997] (see Fig. 1). As a by-product we obtain an exponentially accurate Bohr–Sommerfeld quantization rule for one dimensional problems. The resulting description of QNM allows their accurate evaluation “deep in the complex” where numerical methods break down due to pseudospectral effects (see Fig. 2).</p>

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Overdamped QNM for Schwarzschild Black Holes

  • 期刊论文

  • 发表日期：

    2025年10月31日

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  • Michael Hitrik,
  • Maciej Zworski

摘要

We show 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 cr3, proving that the recent upper bound by Jézéquel [Anal. PDE 17, 2024,] is sharp. The argument is an application of a spectral asymptotics result for non-self-adjoint operators which provides a finer description of QNM, explaining the emergence of a distorted lattice and generalizing the lattice structure in strips described by Sá Barreto-Zworski [Math. Res. Lett. 4, 1997] (see Fig. 1). As a by-product we obtain an exponentially accurate Bohr–Sommerfeld quantization rule for one dimensional problems. The resulting description of QNM allows their accurate evaluation “deep in the complex” where numerical methods break down due to pseudospectral effects (see Fig. 2).