  Let $H$ be a Schr\"odinger operator defined on a noncompact Riemannian
manifold $\Omega$, and let $W\in L^\infty(\Omega;\mathbb{R})$. Suppose that the
operator $H+W$ is critical in $\Omega$, and let $\varphi$ be the corresponding
Agmon ground state. We prove that if $u$ is a generalized eigenfunction of $H$
satisfying $|u|\leq \varphi$ in $\Omega$, then the corresponding eigenvalue is
in the spectrum of $H$. The conclusion also holds true if for some $K\Subset
\Omega$ the operator $H$ admits a positive solution in
$\tilde{\Omega}=\Omega\setminus K$, and $|u|\leq \psi$ in $\tilde{\Omega}$,
where $\psi$ is a positive solution of minimal growth in a neighborhood of
infinity in $\Omega$.
  Under natural assumptions, this result holds true also in the context of
infinite graphs, and Dirichlet forms.
