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.