Inverse problems for the Bakry-Émery Laplacian on manifolds with boundary -- uniqueness and non-uniqueness

We study the questions of uniqueness and non-uniqueness for a pair of closely related inverse problems for the Bakry-Émery Laplacian $-{\mathrm{\Delta }}_{\mathcal{E}}$ on a smooth compact and oriented Riemannian manifold with boundary $(\bar{M},g)$, endowed with a volume form $\mathfrak{m}=e^{-V}{\omega }_g$. These consist in recovering the Taylor coefficients of metric $g$ and weight $V$ along the boundary of $\bar{M}$ from the knowledge of a pair of operators that can be viewed as geometrically natural Dirichlet-to-Neumann maps associated to $-{\mathrm{\Delta }}_{\mathcal{E}}$.