Schrödinger Operators and the Zeros of Their Eigenfunctions
- First Online:
- Cite this article as:
- Schwartzman, S. Commun. Math. Phys. (2011) 306: 187. doi:10.1007/s00220-011-1272-3
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) + Vψ a uniquely defined self adjoint operator which we will denote by −Δ + V.
If ψ0 is any eigenfunction of the self adjoint operator −Δ + V we prove that a necessary and sufficient condition for ψ0 to never equal zero is that the eigenspace to which ψ0 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.