Weyl formulae for some singular metrics with application to acoustic modes in gas giants

TL;DR

This paper derives Weyl law asymptotics for the Laplace-Beltrami operator on manifolds with singular metrics, motivated by acoustic wave studies in gas giants, revealing dimension-dependent spectral properties.

Contribution

It provides the first spectral analysis and Weyl law for Laplace-Beltrami operators with metrics blowing up near boundaries, relevant to planetary acoustics.

Findings

Weyl law derived for singular metrics with boundary blow-up

Exponents depend on Hausdorff dimension, exceeding topological dimension in supercritical cases

Spectral properties linked to acoustic modes in gas giant planets

Abstract

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.