On the Fučik point spectrum for Schrödinger operators on \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}

We investigate the Fučik point spectrum of the Schrödinger operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\lambda} = - \Delta + V_{\lambda}\, {\rm in}\, L^{2}({\mathbb{R}}^{N})$$\end{document} when the potential Vλ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that Sλ has essential spectrum with finitely many eigenvalues below the infimum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma_{\rm ess}(S_\lambda)$$\end{document}. We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schrödinger equations with jumping nonlinearity.