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 $$L_{a,b}:=e^{a-b}\mathrm{\nabla }\cdot e^b\mathrm{\nabla }$$ on the unit ball $B\subset {\mathbb{R}}^3$, where the scalar functions $a=a(\mathrm{\mid }x\mathrm{\mid })$ and $b=b(\mathrm{\mid }x\mathrm{\mid })$ 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.