# Inverse scattering at fixed energy for radial magnetic Schr{\"o}dinger operators with obstacle in dimension two

## TL;DR

This paper proves that the electric and magnetic potentials outside an obstacle in a 2D space can be uniquely determined from fixed-energy phase shift data, using complex angular momentum methods, extending inverse scattering theory.

## Contribution

It establishes a uniqueness result for inverse scattering at fixed energy for radial magnetic Schrödinger operators in two dimensions, considering obstacle effects and using the M{"u}ntz condition.

## Key findings

- Unique determination of potentials from phase shifts
- Extension of inverse scattering theory to magnetic operators
- Application of complex angular momentum method

## Abstract

We study an inverse scattering problem at fixed energy for radial magnetic Schr{\"o}dinger 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 (\tilde{V} , \tilde{B}) are two couples belonging to 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 $\in$ L, where L $\subset$ N^$\star$ satisfies the M{\"u}ntz condition \sum\_{l$\in$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 use the Complex Angular Momentum method and is close in spirit to the celebrated B{\"o}rg-Marchenko uniqueness Theorem.