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

Abstract

We study an inverse scattering problem at fixed energy for radial magnetic Schrödinger operators on $\mathbb {R}^2 {{\setminus }} B(0,r_0)$, 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 $\mathcal {C}$ of smooth, radial and compactly supported electric potentials and magnetic fields denoted by V and B, respectively. If (V, B) and $(\tilde{V},\tilde{B})$ are two couples belonging to $\mathcal {C}$, we then show that if the corresponding phase shifts $\delta _l$ and $\tilde{\delta }_l$ (i.e., the scattering data at fixed energy) coincide for all $l \in \mathcal {L}$, where $\mathcal {L} \subset \mathbb {N}^{\star }$ satisfies the Müntz condition $\sum _{l \in \mathcal {L}} \frac{1}{l} = + \infty$, then $V(x) = \tilde{V}(x)$ and $B(x) = \tilde{B}(x)$ 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.