Inverse problems for nonlinear magnetic Schr\"odinger equations on conformally transversally anisotropic manifolds

TL;DR

This paper proves that, under certain conditions, the nonlinear magnetic and electric potentials in a Schr"odinger equation on a special class of manifolds can be uniquely identified from boundary measurements, advancing inverse boundary problem theory.

Contribution

It establishes unique determination of nonlinear magnetic and electric potentials on conformally transversally anisotropic manifolds, extending inverse problem results beyond linear cases.

Findings

Unique determination of nonlinear potentials from boundary data.

No assumptions on the transversal manifold are needed.

Advances inverse boundary problem for nonlinear equations.

Abstract

We study the inverse boundary problem for a nonlinear magnetic Schr\"odinger operator on a conformally transversally anisotropic Riemannian manifold of dimension $n\ge 3$. Under suitable assumptions on the nonlinearity, we show that the knowledge of the Dirichlet-to-Neumann map on the boundary of the manifold determines the nonlinear magnetic and electric potentials uniquely. No assumptions on the transversal manifold are made in this result, whereas the corresponding inverse boundary problem for the linear magnetic Schr\"odinger operator is still open in this generality.