A Meshkov-type construction for the borderline case

TL;DR

This paper constructs solutions to a specific elliptic eigenvalue problem in two dimensions at the critical decay rate, demonstrating they decay exponentially and are optimal in the borderline case.

Contribution

It extends previous constructions to the borderline case where decay rates are minimal, confirming optimal decay behavior for these solutions.

Findings

Solutions decay exponentially at infinity

Constructed solutions are optimal in decay rate

Extends prior work to the borderline case

Abstract

We construct functions $u: \mathbb{R}^2 \to \mathbb{C}$ that satisfy an elliptic eigenvalue equation of the form $-\Delta u + W \cdot \nabla u + V u = \lambda u$, 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.