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Inverse problems for nonlinear magnetic Schrödinger equations on conformally transversally anisotropic manifolds

Katya Krupchyk and Gunther Uhlmann

Vol. 16 (2023), No. 8, 1825–1868
Abstract

We study the inverse boundary problem for a nonlinear magnetic Schrödinger operator on a conformally transversally anisotropic Riemannian manifold of dimension n 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 Schrödinger operator is still open in this generality.

Keywords
inverse boundary problem, nonlinear Schrödinger equation, conformally transversally anisotropic manifold, Gaussian beams
Mathematical Subject Classification
Primary: 35R30
Milestones
Received: 2 February 2021
Revised: 26 October 2021
Accepted: 4 February 2022
Published: 16 October 2023
Authors
Katya Krupchyk
Department of Mathematics
University of California
Irvine, CA
United States
Gunther Uhlmann
Department of Mathematics
University of Washington
Seattle, WA
United States

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