Uniqueness for the inverse Schrödinger problem in the plane with $L^p$ potentials, $p>1$

Let $Ω\subset\mathbb R^2$ be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for $-Δ+V$ uniquely determines every complex-valued potential $V\in L^p(Ω)$, $p>1$, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range $p>4/3$ to all $p>1$. Th