Schrödinger operators with a complex valued potential

Abstract

If M n is a compact Riemannian manifold for which H 1 (M n , Z) = 0, V is a continuous complex valued function whose imaginary part is of constant sign, and −Δψ + V ψ = 0 for some C 2 complex valued function ψ on M n , then either ψ vanishes somewhere or there is a constant c and an everywhere positive function F such that ψ = cF .