Principal spectral rigidity implies subprincipal spectral rigidity

TL;DR

This work addresses recovering both a radial metric component and an additional coefficient from a single Dirichlet spectrum of a perturbed Laplace–Beltrami operator on the unit ball. The authors separate the roles of the principal symbol and lower-order terms, proving leading-order spectral rigidity that determines the wave speed (via the principal symbol) and applying Kato perturbation alongside density of squared eigenfunctions to recover the density coefficient. They show that, under suitable boundary and geometric conditions, the entire pair $(a,b)$ is spectrally rigid: $a$ is fixed by the spectrum, and any admissible variation of $b$ must be flat. The results extend the scope of inverse spectral rigidity to multi-parameter recovery on manifolds with boundary and have potential implications for geophysical and planetary interior modeling where radial symmetry is a reasonable approximation.

Abstract

We study the inverse spectral problem of jointly recovering a radially symmetric Riemannian metric and an additional coefficient from the Dirichlet spectrum of a perturbed Laplace-Beltrami operator on a bounded domain. Specifically, we consider the elliptic operator \[ L_{a,b} := e^{a-b} \nabla \cdot e^b \nabla \] on the unit ball $ B \subset \mathbb{R}^3 $, where the scalar functions $ a = a(|x|) $ and $ b = b(|x|) $ are spherically symmetric and satisfy certain geometric conditions. While the function $ a $ influences the principal symbol of $ L $, the function $ b $ appears in its first-order terms. We investigate the extent to which the Dirichlet eigenvalues of $ L_{a,b} $ uniquely determine the pair $ (a, b) $ and establish spectral rigidity results under suitable assumptions.

Theorems & Definitions (10)

• Remark 1
• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• Theorem 5
• proof
• proof : Proof of Theorem \ref{['t: intro main theorem acoustic)']}
• proof