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

Caro, Pedro; Rogers, Keith M.

Journées équations aux dérivées partielles (2018), Talk no. 7, 9 p.

For potentials $V\in L^{\infty }({ℝ}^2,ℝ)$ and $A\in W^{1,\infty }({ℝ}^2,{ℝ}^2)$ 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.

Classification: 35P25, 45Q05, 35J10

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