Schrödinger operators with a complex valued potential

If $M_n$ is a compact Riemannian manifold for which $H^1(M_n,\mathbb{Z})=0$, $V$ is a continuous complex valued function whose imaginary part is of constant sign, and $-\Delta \psi +V\psi =0$ for some $C^2$ complex valued function $\psi$ on $M_n$, then either $\psi$ vanishes somewhere or there is a constant $c$ and an everywhere positive function $F$ such that $\psi =cF$.