A Meshkov-type construction for the borderline case

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.