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Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials

  • Eugenia Malinnikova
  • Jiuyi Zhu
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Abstract

We study quantitative unique continuation for solutions of the Schrö\-din\-ger equation with complex-valued singular potentials V∈ Lt, t>d/2. We obtain a family of scale-invariant Lp→ Lq Carleman inequalities for the Laplacian and use them to prove the doubling inequalities with explicit dependence on the Lt-norm of the potential. These estimates are combined with multiscale arguments to obtain propagation of smallness from arbitrary measurable sets of positive measure. As an application of the main results, we prove an upper bound for the BMO norm for the logarithms of Dirichlet-Laplace eigenfunctions in simply connected planar Lipschitz domains.

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Cite

BibTeXRIS
BibTeX
@misc{malinnikova2026doubling,
  title = {{Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials}},
  author = {Eugenia Malinnikova and Jiuyi Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12192},
  eprint = {2609.12192},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12192},
}
RIS
TY  - UNPB
TI  - Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
AU  - Eugenia Malinnikova
AU  - Jiuyi Zhu
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.12192
UR  - https://arxiv.org/abs/2609.12192
AB  - We study quantitative unique continuation for solutions of the Schrö\-din\-ger equation with complex-valued singular potentials V∈ L^t, t>d/2. We obtain a family of scale-invariant L^p→ L^q Carleman inequalities for the Laplacian and use them to prove the doubling inequalities with explicit dependence on the L^t-norm of the potential. These estimates are combined with multiscale arguments to obtain propagation of smallness from arbitrary measurable sets of positive measure. As an application of the main results, we prove an upper bound for the BMO norm for the logarithms of Dirichlet-Laplace eigenfunctions in simply connected planar Lipschitz domains.
ER  -