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<p>This paper is motivated by recent works on inverse problems for acoustic wave propagation in the interior of gas giant planets. In such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with boundary whose metric blows up near the boundary. Here, the spectral analysis of the corresponding Laplace–Beltrami operator is presented, and the Weyl law is derived. The involved exponents depend on the Hausdorff dimension which, in the supercritical case, is larger than the topological dimension.</p>

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Weyl Formulae for Some Singular Metrics with Application to Acoustic Modes in Gas Giants

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Yves Colin de Verdière,
  • Charlotte Dietze,
  • Maarten V. de Hoop,
  • Emmanuel Trélat

摘要

This paper is motivated by recent works on inverse problems for acoustic wave propagation in the interior of gas giant planets. In such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with boundary whose metric blows up near the boundary. Here, the spectral analysis of the corresponding Laplace–Beltrami operator is presented, and the Weyl law is derived. The involved exponents depend on the Hausdorff dimension which, in the supercritical case, is larger than the topological dimension.