Positive Solutions of Schrödinger Equations in Product form and Martin Compactifications of the Plane, II

We determine the structure of all positive solutions to a Schrödinger equation (−Δ + V1(x1) + V2(x2))u = 0 on ℝ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb R^{2}$\end{document}, where real potentials Vj, j = 1,2, satisfy Vj∈L11={V;(1+|t|)V(t)∈L1(ℝ)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V_{j}\in {L^{1}_{1}} =\{V; (1+|t|)V(t)\in L^{1}(\mathbb R)\}$\end{document}. We also treat the case where V1∈L11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V_{1}\in {L^{1}_{1}}$\end{document} and V2 belongs to a wide class of functions including model potentials V2(t) = |t|a, a > 0. We show that non-minimal Martin boundary points appear generically. On analysis of asymptotics of the Green functions of the Schrödinger equations, the Jost solutions of one dimensional Schrödinger equations with potential functions in L11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${L_{1}^{1}}$\end{document} play a central role.