Résumé : This is a work with Yves Colin de Verdière, Charlotte Dietze and Maarten De Hoop, motivated by recent works by M. De Hoop on inverse problems for sound wave propagation in gas giant planets. On such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian manifold with a boundary whose metric blows up near the boundary. With appropriate variable changes, we can reduce the study of the Laplacian?Beltrami to that of a kind of sub-Riemannian Laplacian. In this talk, I will explain how to approach the spectral analysis of such operators, and in particular how to calculate WeylÕs law.

Keywords : Weyl law; singular metrics; heat asymptotic; gas giant