  For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the
space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for
semi-boundedness and discreteness of its spectrum.are obtained without
assumption that the potential $V({\mathbf x})$ is bounded below. By reduction
of the problem to investigation of existence of regular solutions for Riccati
PDE necessary conditions for discreteness of the spectrum of operator $H$ are
obtained under assumption that it is bounded below. These results are similar
to ones obtained by author in \cite{Zel} for the one-dimensional case.
Furthermore, sufficient conditions for the semi-boundedness and discreteness of
the spectrum of $H$ are obtained in terms of a non-increasing rearrangement,
mathematical expectation and standard deviation from the latter for positive
part $V_+({\mathbf x})$ of the potential $V({\mathbf x})$ on compact domains
that go to infinity, under certain restrictions for its negative part
$V_-({\mathbf x})$. Choosing in an optimal way the vector field associated with
difference between the potential $V({\mathbf x})$ and its mathematical
expectation on the balls that go to infinity, we obtain a condition for
semi-boundedness and discreteness of the spectrum for $H$ in terms of solutions
of Neumann problem for nonhomogeneous $d/(d-1)$-Laplace equation. This type of
optimization refers to a divergence constrained transportation problem.
