  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.
