Conditions for Semi-Boundedness and Discreteness of the Spectrum to Schrödinger Operator and Some Nonlinear PDEs

For the Schrödinger operator H=-Δ+V(x)·, acting in the space L₂(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) 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) of the potential V(x) on compact domains that go to infinity, under certain restrictions for its negative part V_-(x). Choosing optimally the vector field associated with the difference between the potential 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)-Laplace equation. This type of optimization refers to a divergence constrained transportation problem.