Lower Bounds for Non-Classical Eigenvalue Problems

Several lower bounds on the ground state energy of Schrödinger operators with non-confining potentials Vα(x)=∏j=1n|xj|αj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V_{\alpha}(x) = {\prod_{j=1}^{n}} |x_{j}|^{\alpha_{j}}}$$\end{document}, αj>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha_{j} > 0}$$\end{document}, on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb R}^{n}}$$\end{document} are obtained. These results lead to explicit (computable) strictly positive lower bounds and can be generalized to certain ‘product’-potentials.