Inverse problems for Schrodinger equations with Yang-Mills potentials in domains with obstacles and the Aharonov-Bohm effect

We study the inverse boundary value problems for the Schr"{o}dinger equations with Yang-Mills potentials in a bounded domain ${\mathrm{\Omega }}_0\subset {\mathbb{R}}^n$ containing finite number of smooth obstacles ${\mathrm{\Omega }}_j,1\le j\le r$. We prove that the Dirichlet-to-Neumann operator on $\mathrm{\partial }{\mathrm{\Omega }}_0$ determines the gauge equivalence class of the Yang-Mills potentials. We also prove that the metric tensor can be recovered up to a diffeomorphism that is identity on $\mathrm{\partial }{\mathrm{\Omega }}_0$.