Schrödinger Operators and the Zeros of Their Eigenfunctions

Abstract

In n-dimensional Euclidean space let us be given an infinitely differentiable real valued function V that is bounded below. We associate with the formal operator that sends a complex valued function ψ into −div(grad ψ) + V ψ a uniquely defined self adjoint operator which we will denote by −Δ + V.

If ψ ₀ is any eigenfunction of the self adjoint operator −Δ + V we prove that a necessary and sufficient condition for ψ ₀ to never equal zero is that the eigenspace to which ψ ₀ belongs contain a positive function. In this case the eigenspace must be one dimensional. The same result holds on any complete connected Riemannian manifold whose first Betti number is zero.