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 ∂ M at the rate 1/xα, for α∈ (0,2), where x ∈ C∞(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 Δg 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 Δg that is stable under truncations. Using this, we study the behavior of the resolvent of Δg 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 $(ε, x)$.
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BibTeX
@misc{yin2026transition,
title = {{A transition calculus for 0-pseudodifferential operators with applications to gas giant geometry}},
author = {Andy Yin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04191},
eprint = {2611.04191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04191},
}
RIS
TY - UNPB
TI - A transition calculus for 0-pseudodifferential operators with applications to gas giant geometry
AU - Andy Yin
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04191
UR - https://arxiv.org/abs/2611.04191
AB - A gas giant Riemannian metric g on a compact n-dimensional manifold-with-boundary M blows up at the boundary ∂ M at the rate 1/x^α, for α∈ (0,2), where x ∈ C^∞(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 $(ε, x)$.
ER -