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

Abstract For the Schrödinger operator H=- + V ({x}) H = - Δ + V (x) ·, acting in the space L₂ ({R}ᵈ) \, (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.