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

Conditions for discreteness of the spectrum to Schr\"odinger operator via non-increasing rearrangement, Lagrangian relaxation and perturbations