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Annales Henri Poincaré

, Volume 19, Issue 10, pp 3089–3128 | Cite as

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

  • Damien Gobin
Article

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 (VB) 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.

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada

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