Conditions for semi-boundedness and discreteness of the spectrum to Schr\"odinger operator and some nonlinear PDEs

Abstract

For Schr\"odinger operator $H=-\Delta+ V({\mathbf x})\cdot$, acting in the space $L_2(\mathbb R^d)\,(d\ge 3)$, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum.are obtained without assumption that the potential $V({\mathbf x})$ is bounded below. By reduction of the problem to investigation of existence of regular solutions for Riccati PDE necessary conditions for discreteness of the spectrum of operator $H$ are obtained under assumption that it is bounded below. These results are similar to ones obtained by author in \cite{Zel} 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 positive part $V_+({\mathbf x})$ of the potential $V({\mathbf x})$ on compact domains that go to infinity, under certain restrictions for its negative part $V_-({\mathbf x})$. Choosing in an optimal way the vector field associated with difference between the potential $V({\mathbf x})$ 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 Neumann problem for nonhomogeneous $d/(d-1)$-Laplace equation. This type of optimization refers to a divergence constrained transportation problem.