On Landis' conjecture in the plane for real-valued potentials with decay

2025/12/10 by Blair Davey, Davey, Blair
Mathematics · #35B60 #35J10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2512.09839

openalex publication_date 2025/12/10 · openalex created_date 2025/12/12 · openalex updated_date 2026/07/28

Abstract

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 ℝ², where |V(z)| \lesssim ⟨ z ⟩^-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.

Citations

• A frequency function approach to quantitative unique continuation for elliptic equations
• On Landis' conjecture in the plane for potentials with growth
• The Landis conjecture on exponential decay
• Improved quantitative unique continuation for complex-valued drift equations in the plane
• On Landis' conjecture in the plane for some equations with sign-changing\n potentials
• On Landis' conjecture in the plane when the potential has an exponentially decaying negative part
• Landis' conjecture for general second order elliptic equations with singular lower order terms in the plane
• The Landis Conjecture for variable coefficient second-order elliptic PDES
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• On Landis' conjecture in the plane
• A Meshkov-type construction for the borderline case
• Quantitative uniqueness estimates for the general second order elliptic equations
• On localization in the continuous Anderson-Bernoulli model in higher dimension

Related

• On Landis' conjecture in the plane for potentials with growth
• On Landis' conjecture in the plane when the potential has an exponentially decaying negative part
• On Landis' conjecture in the plane for some equations with sign-changing\n potentials
• The Landis conjecture with sharp rate of decay
• Quantitative unique continuation for real-valued solutions to second order elliptic equations in the plane