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

We study the inverse boundary value problems for the Schrödinger equations with Yang-Mills potentials in a bounded domain Ω₀ ⊂ Rⁿ containing finite number of smooth obstacles Ω_j , 1 <= j <= r. We prove that the Dirichlet-to-Neumann operator on ∂Ω₀ 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 ∂Ω₀.