Zeroes of Eigenfunctions of Schr\"odinger Operators after Schwartzman

Consider a complete, connected, smooth, oriented Riemannian manifold $(M,g)$ with boundary, such that the first Betti number vanishes. Sol Schwartzman proved that for Schr\"odinger operators of the form $-\Delta_g + V$ where $\Im(V)$ is signed, if $f: M\to\mathbb{C}$ is a non-vanishing element of its kernel, then $f$ has constant phase. The proof relied on dynamical systems methods applied to the gradient flow of the phase of $f$. In this manuscript we provide a more direct PDE argument that proves strengthened versions of the same facts.