In this article, we investigate the quantitative unique continuation properties of real-valued solutions to elliptic equations in the plane. Under a general set of assumptions on the operator, we establish quantitative forms of Landis’ conjecture. Of note, we prove a version of Landis’ conjecture for solutions to $−\Delta u + Vu = 0$, where $V$ is a bounded function whose negative part exhibits polynomial decay at infinity. The main mechanism behind the proofs is an order of vanishing estimate in combination with an iteration scheme. To prove the order of vanishing result, we present a new idea for constructing positive multipliers and use it reduce the equation to a Beltrami system. The resulting first-order equation is analyzed using the similarity principle and the Hadamard three-quasi-circle theorem.

中文翻译：

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关于具有势变势的某些方程在平面上的兰迪斯猜想

在本文中，我们研究了平面上椭圆方程实值解的定量唯一连续性质。在关于算子的一般假设下，我们建立了兰迪斯猜想的定量形式。值得注意的是，我们证明了Landis猜想的一个版本，用于解$ − \ Delta u + Vu = 0 $，其中$ V $是有界函数，其负部分在无穷大处表现出多项式衰减。证明背后的主要机制是与迭代方案结合的消失估计的顺序。为了证明消失的顺序，我们提出了构造正乘子的新思路，并将其用于将方程式简化为Beltrami系统。使用相似原理和Hadamard三拟圆定理分析所得的一阶方程。