发表机构
Montana State University（蒙大拿州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究具有逆Hölder势的薛定谔方程，证明若解在无穷远处增长不超过由Agmon距离确定的阈值，则解必平凡，该结果属于Liouville型定理，与Landis猜想相关。
AI 中文摘要
本文研究了势函数属于逆Hölder类的广义薛定谔方程的解在无穷远处的唯一延拓性质。对于形式为$-div (A \nabla u) + V u = 0$在$\mathbb{R}^n$中的方程，其中$A$有界且椭圆，且$V \in RH_p$对某个$p \in [\frac n 2, \infty]$成立，我们证明：如果解在无穷远处增长不太快，则它必为平凡解。我们使用与$V$相关联的Agmon距离函数$d_V$来量化阈值增长率。更精确地，存在常数$\gamma_0 > 0$，使得若对某个$\gamma < \gamma_0$及所有$x \in \mathbb{R}^n$，有$|u(x)| \lesssim \exp(\gamma d_V(x, 0))$，则$u$必为平凡解。该结果可被解释为Liouville型定理，并与Landis猜想相关。我们的证明技巧受Z. Shen关于薛定谔算子基本解的指数衰减估计的启发，并涉及Fefferman-Phong不等式的应用。
英文摘要
In this article, we study unique continuation properties at infinity for solutions to generalized Schrödinger equations with potential functions that belong to the reverse Hölder class. For equations of the form $-div (A \nabla u) + V u = 0$ in $\mathbb{R}^n$, where $A$ is bounded and elliptic and $V \in RH_p$ for some $p \in [\frac n 2, \infty]$, we prove that if a solution doesn't grow too quickly at infinity, then it must be trivial. We use $d_V$, the Agmon distance function associated to $V$, to quantify the threshold growth rate. More precisely, there exists a constant $γ_0 > 0$ so that if $|u(x)| \lesssim \exp(γd_V(x, 0))$ for some $γ< γ_0$ and every $x \in \mathbb{R}^n$, then $u$ must be trivial. The result may be interpreted as a Liouville-type theorem and is related to Landis' conjecture. Our proof techniques are inspired by Z. Shen's exponential decay estimates for fundamental solutions of Schrödinger operators and involve the application of a Fefferman-Phong inequality.
Comments16 pages
