  Let $A$ be a non-negative self-adjoint operator in a Hilbert space
$\mathcal{H}$ and $A_{0}$ be some densely defined closed restriction of
$A_{0}$, $A_{0}\subseteq A \neq A_{0}$. It is of interest to know whether $A$
is the unique non-negative self-adjoint extensions of $A_{0}$ in $\mathcal{H}$.
We give a natural criterion that this is the case and if it fails, we describe
all non-negative extensions of $A_{0}$. The obtained results are applied to
investigation of non-negative singular point perturbations of the Laplace and
poly-harmonic operators in $\mathbb{L}_{2}(\mathbf{R}_{n})$.
