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

For the Schrödinger operator \(H=-\Delta + V({{\textbf{x}}})\cdot \), acting in the space \(L_2({{\textbf{R}}}^d)\,(d\ge 3)\), necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum are obtained without the assumption that the potential \(V({{\textbf{x}}})\) 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 \(V_+({{\textbf{x}}})\) of the potential \(V({{\textbf{x}}})\) on compact domains that go to infinity, under certain restrictions for its negative part \(V_-({{\textbf{x}}})\). Choosing optimally the vector field associated with the difference between the potential \(V({{\textbf{x}}})\) 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.