  We prove that if u is a bounded smooth function in the kernel of a
nonnegative Schrodinger operator $-L=-(\Delta +q)$ on a parabolic Riemannian
manifold M, then u is either identically zero or it has no zeros on M, and the
linear space of such functions is 1-dimensional. We obtain consequences for
orientable, complete stable surfaces with constant mean curvature
$H\in\mathbb{R}$ in homogeneous spaces $\mathbb{E}(\kappa,\tau)$ with four
dimensional isometry group. For instance, if M is an orientable, parabolic,
complete immersed surface with constant mean curvature H in
$\mathbb{H}^2\times\mathbb{R}$, then $|H|\leq 1/2$ and if equality holds, then
M is either an entire graph or a vertical horocylinder.
