On Landis' conjecture in the plane for real-valued potentials with decay
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917137296130048 |
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| author | Davey, Blair |
| author_facet | Davey, Blair |
| contents | We investigate the quantitative unique continuation properties of real-valued solutions to planar Schrödinger equations with potential functions that exhibit pointwise decay at infinity. That is, for equations of the form $-Δu + V u = 0$ in $\mathbb{R}^2$, where $|V(z)| \lesssim \langle z \rangle^{-N}$ for some $N > 0$, we prove that real-valued solutions satisfy exponential decay estimates with a rate that depends explicitly on $N$. Examples show that the estimates established here are essentially sharp. The case of $N = 0$ corresponds to the Landis conjecture, which was proved for real-valued solutions in the plane in [LMNN20], while the case of $N < 0$ was previously investigated by the author in [Dav24]. Here, the proof techniques rely on the ideas presented in [LMNN20] combined with conformal transformations and an iteration scheme. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Landis' conjecture in the plane for real-valued potentials with decay Davey, Blair Analysis of PDEs 35B60, 35J10 We investigate the quantitative unique continuation properties of real-valued solutions to planar Schrödinger equations with potential functions that exhibit pointwise decay at infinity. That is, for equations of the form $-Δu + V u = 0$ in $\mathbb{R}^2$, where $|V(z)| \lesssim \langle z \rangle^{-N}$ for some $N > 0$, we prove that real-valued solutions satisfy exponential decay estimates with a rate that depends explicitly on $N$. Examples show that the estimates established here are essentially sharp. The case of $N = 0$ corresponds to the Landis conjecture, which was proved for real-valued solutions in the plane in [LMNN20], while the case of $N < 0$ was previously investigated by the author in [Dav24]. Here, the proof techniques rely on the ideas presented in [LMNN20] combined with conformal transformations and an iteration scheme. |
| title | On Landis' conjecture in the plane for real-valued potentials with decay |
| topic | Analysis of PDEs 35B60, 35J10 |
| url | https://arxiv.org/abs/2512.09839 |