Principal spectral rigidity implies subprincipal spectral rigidity

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.