Zeroes of Eigenfunctions of Schrödinger 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 Schrödinger operators of the form -Δg + V where (V) is signed, if f: M 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.