The boundary and scattering rigidity problems for simple MP-systems

We consider a compact Riemannian manifold with boundary, endowed with a magnetic field and a potential. This is called an $\mathcal{M}\mathcal{P}$-system. On simple $\mathcal{M}\mathcal{P}$-systems, we consider both the boundary rigidity problem and the scattering rigidity problem. Under the assumption of knowing the boundary action functions or the scattering relations for only one energy level, we prove that we can recover the $\mathcal{M}\mathcal{P}$-system up to a gauge. To obtain the results, we reduce the $\mathcal{M}\mathcal{P}$-system to the corresponding magnetic system and we apply the results from [DPSU07] on simple magnetic systems.