Unique determination of a magnetic Schr\"odinger operator with unbounded magnetic potential from boundary data

TL;DR

This paper proves the unique determination of magnetic and electric potentials in a Schrödinger operator from boundary data, even with unbounded magnetic potentials, advancing inverse boundary value problem theory.

Contribution

It extends uniqueness results to cases with unbounded magnetic potentials, assuming smallness in specific Sobolev spaces, unlike previous boundedness assumptions.

Findings

Magnetic field and electric potential are uniquely determined by boundary data.

Results hold for unbounded magnetic potentials in certain Sobolev spaces.

Advances inverse boundary value problem understanding for less regular potentials.

Abstract

We consider the Gel'fand-Calder\'on problem for a Schr\"odinger operator of the form $-(\nabla + iA)^2 + q$, defined on a ball $B$ in $\mathbb R^3$. We assume that the magnetic potential $A$ is small in $W^{s,3}$ for some $s>0$, and that the electric potential $q$ is in $W^{-1,3}$. We show that, under these assumptions, the magnetic field $\operatorname{curl} A$ and the potential $q$ are both determined by the Dirichlet-Neumann relation at the boundary $\partial B$. The assumption on $q$ is critical with respect to homogeneity, and the assumption on $A$ is nearly critical. Previous uniqueness theorems of this type have assumed either that both $A$ and $q$ are bounded or that $A$ is zero.