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 $-\Delta u + W \cdot \nabla u + V u = \lambda u$, where $\lambda \in \mathbb C$, and $V$ and $W$ satisfy $| V ({x} ) | \lesssim \langle {x} \rangle ^{-N}$, and $ | W ({x})| \lesssim \langle{x} \rangle ^{-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.