[#P11152] Minimal Normal-Mode Couplings for Recovering Band-Limited 3D Earth Structure
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Problem. Let a spherically symmetric, non-rotating, elastic and isotropic reference Earth have normal-mode multiplets K=(n,\ell,\mathrm{type}), with angular eigenspaces V_\ell of dimension 2\ell+1. Let a small perturbation of one scalar material parameter be band-limited in angle to degrees 0\le s\le S and expanded in a fixed P-dimensional radial basis as \[ h(r,\Omega)=\sum_{s=0}^S\sum_{t=-s}^s\sum_{a=1}^P x_{a,s,t}\,\psi_a(r)Y_s^t(\Omega). \] For a selected finite family E of self- and cross-coupled normal-mode pairs (K,K'), assume first-order quasi-degenerate perturbation theory gives the full coupling matrices H^{KK'}, after known rotation and hydrostatic-ellipticity terms have been removed. These matrices depend linearly on x through radial sensitivity integrals and Gaunt/Wigner-3j angular coefficients.
Characterize generic injectivity and stable invertibility of the resulting finite-dimensional measurement operator A_E:x\mapsto\{H^{KK'}\}_{(K,K')\in E}. Give necessary and sufficient angular coverage conditions in terms of \ell,\ell' and Wigner-3j selection rules, and explicit sufficient radial-rank conditions for full recovery through angular degree S and radial rank P. Determine or sharply bound the minimum number and parity pattern of multiplet pairs needed to make all angular degrees 1,\ldots,S observable. In particular, prove that self-couplings alone cannot recover odd-degree scalar heterogeneity and that odd-degree recovery requires at least one opposite-parity pair \ell+\ell' odd. Derive a quantitative stability estimate \(\|x\|\le C\|A_E x\|\) with C controlled by the smallest singular values of the angular Wigner blocks and radial sensitivity matrices, and characterize how weak or near-degenerate mode couplings degrade C. The target is a finite-dimensional identifiability and minimal-experiment theorem for Earth normal-mode coupling, not the classical computation of splitting functions or a new tomography algorithm.
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“Minimal Normal-Mode Couplings for Recovering Band-Limited 3D Earth Structure.” TheoremDB. P11152. Problem statement; statement identity tdbc1:19abe2bc5a36e186b41289da086ecaac4defafe16141f6b0cd43c0a0568a2614; statement text SHA-256 49f224e5c7fb8da90f262573b49ccc10c6a93456986709a4bd0f1121b3bfa2d1. https://theoremdb.org/statement/?ref=P11152
@misc{theoremdb-problem-49f224e5c7fb8da90f262573b49ccc10c6a93456986709a4bd0f1121b3bfa2d1,
title = {{Minimal Normal-Mode Couplings for Recovering Band-Limited 3D Earth Structure}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:19abe2bc5a36e186b41289da086ecaac4defafe16141f6b0cd43c0a0568a2614; statement text SHA-256 49f224e5c7fb8da90f262573b49ccc10c6a93456986709a4bd0f1121b3bfa2d1},
url = {https://theoremdb.org/statement/?ref=P11152}
}Plain text: Built Markdown snapshot
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