Inverse Scattering at Fixed Energy for Radial Magnetic Schrödinger Operators with Obstacle in Dimension Two

We study an inverse scattering problem at fixed energy for radial magnetic Schrödinger operators on R 2 \ B ( 0 , r 0 ) , where r 0 is a positive and arbitrarily small radius. We assume that the magnetic potential A satisfies a gauge condition, and we consider the class C of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B , respectively. If ( V , B ) and ( V ~ , B ~ ) are two couples belonging to C , we then show that if the corresponding phase shifts δ l and δ ~ l (i.e., the scattering data at fixed energy) coincide for all l ∈ L , where L ⊂ N ⋆ satisfies the Müntz condition ∑ l ∈ L 1 l = + ∞ , then V ( x ) = V ~ ( x ) and B ( x ) = B ~ ( x ) outside the obstacle B ( 0 , r 0 ) . The proof uses the complex angular momentum method and is close in spirit to the celebrated Borg–Marchenko uniqueness theorem.