Zeroes of Eigenfunctions of Schrödinger Operators after Schwartzman

Abstract

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 $-{\Delta }_g+V$ 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.