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

## TL;DR

This paper proves that, with a fixed magnetic potential, the electric potential in a Schrödinger equation can be uniquely identified from scattering data in the plane, extending Bukhgeim's method to include magnetic effects.

## Contribution

It establishes unique determination of the electric potential given a fixed magnetic potential, using an adapted Bukhgeim method for the Schrödinger equation.

## Key findings

- Unique determination of electric potential from scattering data.
- Extension of Bukhgeim's method to magnetic Schrödinger equations.
- Results hold under mild regularity assumptions.

## 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 Schr\"odinger equation $-(\nabla +iA)^2 u+Vu=k^2u$ 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 Schr\"odinger equation.