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Uniqueness and stability for inverse potential scattering with multiple wavenumbers

  • Dong Qiu
  • Xiang Xu
  • Yue Zhao
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Abstract

This paper is concerned with the inverse scattering problem of determining a real-valued potential for the three-dimensional Schrödinger equation from multi-wavenumber data. Under suitable a priori bounds, we establish local increasing stability from near-field Dirichlet data on a fixed sphere for polynomially decaying potentials, and global increasing stability from full far-field amplitudes for exponentially decaying potentials. The stability estimates consist of a noise term and a remainder term, with the latter decreasing as the upper measured wavenumber increases. We also obtain global uniqueness from exact near-field data under weaker regularity assumptions. The analysis combines analytic continuation, weighted resolvent estimates, and half-ray identities whose differences at chord endpoints cancel the unknown exterior contribution.

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Cite

BibTeXRIS
BibTeX
@misc{qiu2026uniqueness,
  title = {{Uniqueness and stability for inverse potential scattering with multiple wavenumbers}},
  author = {Dong Qiu and Xiang Xu and Yue Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14978},
  eprint = {2610.14978},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14978},
}
RIS
TY  - UNPB
TI  - Uniqueness and stability for inverse potential scattering with multiple wavenumbers
AU  - Dong Qiu
AU  - Xiang Xu
AU  - Yue Zhao
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14978
UR  - https://arxiv.org/abs/2610.14978
AB  - This paper is concerned with the inverse scattering problem of determining a real-valued potential for the three-dimensional Schrödinger equation from multi-wavenumber data. Under suitable a priori bounds, we establish local increasing stability from near-field Dirichlet data on a fixed sphere for polynomially decaying potentials, and global increasing stability from full far-field amplitudes for exponentially decaying potentials. The stability estimates consist of a noise term and a remainder term, with the latter decreasing as the upper measured wavenumber increases. We also obtain global uniqueness from exact near-field data under weaker regularity assumptions. The analysis combines analytic continuation, weighted resolvent estimates, and half-ray identities whose differences at chord endpoints cancel the unknown exterior contribution.
ER  -