Recovery of $L^p$-potential in the plane

TL;DR

This paper addresses the inverse problem of recovering an $L^p$-potential in the plane for the Schrödinger equation using the $ar{ ext{d}}$-method, providing new estimates that ensure uniqueness without smallness assumptions.

Contribution

It introduces a novel estimate on the Faddeev Green function, guaranteeing the absence of exceptional points near zero and infinity for $L^p$-potentials, enabling potential recovery from boundary data.

Findings

Unique recovery of $L^p$-potentials from boundary measurements.

New estimate on Faddeev Green function eliminates exceptional points.

Recovery holds without smallness or non-exceptional assumptions.

Abstract

An inverse problem for the two-dimensional Schrodinger equation with $L^p_{com}$-potential, $p>1$, is considered. Using the $\overline{\partial}$-method, the potential is recovered from the Dirichlet-to-Neumann map on the boundary of a domain containing the support of the potential. We do not assume that the potential is small or that the Faddeev scattering problem does not have exceptional points. The paper contains a new estimate on the Faddeev Green function that immediately implies the absence of exceptional points near the origin and infinity when $v\in L^p_{com}$.