2023/10/29 by Leonid Zelenko, Zelenko, Leonid
Engineering · Mathematics · #35P05 #47B25 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #FOS: Physical sciences #Material Science and Thermodynamics #Mathematical Physics (math-ph) #Primary 47F05 #Secondary 81Q10 #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2310.19101
openalex publication_date 2023/10/29 · openalex created_date 2023/11/01 · openalex updated_date 2026/07/28
For Schrödinger operator H=-Δ+ V(\mathbf x)⋅, acting in the space L2(\mathbb Rd) (d≥ 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 \citeZel 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.