For the Schrödinger operator H=-Δ + V(x)· H = - Δ + V ( x ) · , acting in the space L₂(Rᵈ)\,(d≥ 3) L 2 ( R d ) ( d ≥ 3 ) , necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum are obtained without the assumption that the potential V(x) V ( x ) is bounded below. By reducing the problem to study the existence of regular solutions of the Riccati PDE, the necessary conditions for the discreteness of the spectrum of operator H are obtained under the assumption that it is bounded below. These results are similar to the ones obtained by the author in [26] 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 the positive part V₊(x) V + ( x ) of the potential V(x) V ( x ) on compact domains that go to infinity, under certain restrictions for its negative part V₋(x) V - ( x ) . Choosing optimally the vector field associated with the difference between the potential V(x) V ( 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 the Neumann problem for the nonhomogeneous $$d/(d-1)$$ d / ( d - 1 ) -Laplace equation. This type of optimization refers to a divergence constrained transportation problem.
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Leonid Zelenko (2024) studied this question.