Unique determination of the electric potential in the presence of a fixed magnetic potential in the plane

Abstract

For potentials $V\in L^{\infty }({\mathbb{R}}^2,\mathbb{R})$ and $A\in W^{1,\infty }({\mathbb{R}}^2,{\mathbb{R}}^2)$ with compact support, we consider the Schrödinger equation $-(\nabla +iA{)}^{2}u+Vu=k^{2}u$ with fixed positive energy $k^2$. Under a mild additional regularity hypothesis, and with fixed magnetic potential $A$, we show that the scattering solutions uniquely determine the electric potential $V$. For this we develop the method of Bukhgeim for the purely electric Schrödinger equation.

9 pages; Proceedings of Journées Équations aux Dérivées Partielles; references added