We consider the Aharonov-Bohm effect for the Schrodinger operator H=(-i delta x-A(x))2+V(x) and the related inverse problem in an exterior domain in R2 with Dirichlet boundary condition. We study the structure and asymptotics of generalized eigenfunctions and show that the scattering operator determines the domain and H up to gauge equivalence under the equal flux condition. We also show that the flux is determined by the scattering operator if the obstacle c is convex.