Principal spectral rigidity implies subprincipal spectral rigidity

2025/03/25 by de Hoop, Maarten V., Ilmavirta, Joonas, Katsnelson, Vitaly
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.19866

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 ∇ ⋅ e^b ∇ on the unit ball B ⊂ ℝ³ , 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.

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