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

Abstract

For the Schrödinger operator H=-Δ+V(x)·\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=-\\Delta + V({{\ extbf{x}}})\\cdot $$\\end{document}, acting in the space L2(Rd)(d≥3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2({{\ extbf{R}}}^d)\\,(d\\ge 3)$$\\end{document}, necessary and sufficient conditions for semi-boundedness and discreteness of its spectrum are obtained without the assumption that the potential V(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V({{\ extbf{x}}})$$\\end{document} 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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_+({{\ extbf{x}}})$$\\end{document} of the potential V(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V({{\ extbf{x}}})$$\\end{document} on compact domains that go to infinity, under certain restrictions for its negative part V-(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_-({{\ extbf{x}}})$$\\end{document}. Choosing optimally the vector field associated with the difference between the potential V(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V({{\ extbf{x}}})$$\\end{document} 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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d/(d-1)$$\\end{document}-Laplace equation. This type of optimization refers to a divergence constrained transportation problem.