Parabolic stable surfaces with constant mean curvature

We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\mathrm{\Delta }+q)$ 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 $\mathrm{\mid }H\mathrm{\mid }\le 1\mathrm{/}2$ and if equality holds, then M is either an entire graph or a vertical horocylinder.