  This work is a continuation of our previos paper, where for the Schr\"odinger
operator $H=-\Delta+ V(\e)\cdot$ $(V(\e)\ge 0)$, acting in the space
$L_2(\R^d)\,(d\ge 3)$, some sufficient conditions for discreteness of its
spectrum have been obtained on the base of well known Mazya -Shubin criterion
and an optimization problem for a set function, which is an
infinite-dimensional generalization of a binary linear programming problem. A
sufficient condition for discreteness of the spectrum is formulated in terms of
the non-increasing rearrangement of the potential $V(\e)$. Using the method of
Lagrangian relaxation for this optimization problem, we obtain a sufficient
condition for discreteness of the spectrum in terms of expectation and
deviation of the potential. By means of suitable perturbations of the potential
we obtain conditions for discreteness of the spectrum, covering potentials
which tend to infinity only on subsets of cubes, whose Lebesgue measures tend
to zero when the cubes go to infinity. Also the case where the operator $H$ is
defined in the space $L_2(\Omega)$ is considered ($\Omega$ is an open domain in
$\R^d$).
