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The planar Calderón problem for Lp potentials
Abstract
We prove that the Dirichlet-to-Neumann map associated to the Schrödinger operator −Δ+q on a planar bounded Lipschitz domain uniquely determines an arbitrary complex-valued potential q∈ Lp, p>1, whenever zero is not a Dirichlet eigenvalue. Our argument uses Nachman's d-bar equation only at sufficiently large complex frequencies, where exceptional points are absent. A new annular frequency estimate converts the first asymptotic coefficient of the complex geometric optics solutions into a pointwise bound for a primitive of the potential difference, reducing the inverse problem to a unique-continuation argument.
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BibTeX
@misc{feizmohammadi2026planar,
title = {{The planar Calderón problem for $L^p$ potentials}},
author = {Ali Feizmohammadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28047},
eprint = {2609.28047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28047},
}
RIS
TY - UNPB
TI - The planar Calderón problem for L^p potentials
AU - Ali Feizmohammadi
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28047
UR - https://arxiv.org/abs/2609.28047
AB - We prove that the Dirichlet-to-Neumann map associated to the Schrödinger operator −Δ+q on a planar bounded Lipschitz domain uniquely determines an arbitrary complex-valued potential q∈ L^p, p>1, whenever zero is not a Dirichlet eigenvalue. Our argument uses Nachman's d-bar equation only at sufficiently large complex frequencies, where exceptional points are absent. A new annular frequency estimate converts the first asymptotic coefficient of the complex geometric optics solutions into a pointwise bound for a primitive of the potential difference, reducing the inverse problem to a unique-continuation argument.
ER -