A transition calculus for 0-pseudodifferential operators with applications to gas giant geometry

Abstract

A gas giant Riemannian metric $g$ on a compact $n$-dimensional manifold-with-boundary $M$ blows up at the boundary $\partial M$ at the rate $1/x^α$, for $α\in (0,2)$, where $x \in C^\infty(M)$ is a boundary-defining function. Such metrics arise in the modeling of acoustic waves in gaseous planets. De Hoop-Ilmavirta-Kykkänen-Mazzeo [arXiv:2403.05475] showed that, if $α> 2/n$, the Laplace-Beltrami operator $Δ_g$ is essentially self-adjoint and has discrete spectrum. Moreover, its spectrum is stable under truncations of $M$. The proof leverages the 0-calculus of Mazzeo-Melrose. We extend this theory by introducing a 0-transition calculus for studying families of 0-operators. We demonstrate a parametrix construction for $Δ_g$ that is stable under truncations. Using this, we study the behavior of the resolvent of $Δ_g$ under truncation and extend the characterization of the spectrum obtained by de Hoop-Ilmavirta-Kykkänen-Mazzeo. In particular, we prove that (on appropriate functional-analytic domains) simple eigenvalues admit asymptotic expansions in the truncation parameter $ε$ and that the corresponding eigenfunctions admit joint asymptotic expansions in $(ε, x)$.