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PreprintCC BY 4.0arXiv

Uniqueness for the inverse Schrödinger problem in the plane with Lp potentials, p>1

  • Cătălin I. Cârstea
  • Jenn-Nan Wang
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Abstract

Let Ω⊂ℝ2 be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for −Δ+V uniquely determines every complex-valued potential V∈ Lp(Ω), p>1, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range p>4/3 to all p>1. The proof uses Bukhgeim's quadratic-phase solutions and an average of Alessandrini's identity over the phase center. After separating the Neumann-series tails, we show that each remaining Born term tends to zero. The fixed-order estimates combine bounds for the absolute kernels with oscillatory cancellation and a duality argument for the product of the two Cauchy transforms.

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Cite

BibTeXRIS
BibTeX
@misc{carstea2026uniqueness,
  title = {{Uniqueness for the inverse Schrödinger problem in the plane with $L^p$ potentials, $p>1$}},
  author = {Cătălin I. Cârstea and Jenn-Nan Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28046},
  eprint = {2609.28046},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28046},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniqueness for the inverse Schrödinger problem in the plane with L^p potentials, p>1
AU  - Cătălin I. Cârstea
AU  - Jenn-Nan Wang
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28046
UR  - https://arxiv.org/abs/2609.28046
AB  - Let Ω⊂ℝ^2 be a bounded smooth domain. We prove that the weak Dirichlet-to-Neumann map for −Δ+V uniquely determines every complex-valued potential V∈ L^p(Ω), p>1, provided that zero is not a Dirichlet eigenvalue. This extends the previously known range p>4/3 to all p>1. The proof uses Bukhgeim's quadratic-phase solutions and an average of Alessandrini's identity over the phase center. After separating the Neumann-series tails, we show that each remaining Born term tends to zero. The fixed-order estimates combine bounds for the absolute kernels with oscillatory cancellation and a duality argument for the product of the two Cauchy transforms.
ER  -