A Meshkov-type construction for the borderline case

Abstract

We construct functions $u:{\mathbb{R}}^2\to \mathbb{C}$ that satisfy an elliptic eigenvalue equation of the form $-\Delta u+W\cdot \nabla u+Vu=\lambda u$, where $\lambda \in \mathbb{C}$, and $V$ and $W$ satisfy $|V(x)|\lesssim <x{>}^{-N}$, and $|W(x)|\lesssim <x{>}^{-P}$, with $\min \{N,P\}=1/2$. For $|x|$ sufficiently large, these solutions satisfy $|u(x)|\lesssim \exp (-c|x|)$. In the author's previous work, examples of solutions over ${\mathbb{R}}^2$ were constructed for all $N,P$ such that $\min \{N,P\}\in [0,1/2)$. These solutions were shown to have the optimal rate of decay at infinity. A recent result of Lin and Wang shows that the constructions presented in this note for the borderline case of $\min \{N,P\}=1/2$ also have the optimal rate of decay at infinity.

arXiv comment15 pages