Essential spectrum of Schrödinger operators with δ-interactions on unbounded hypersurfaces

Let Γ be a simply connected unbounded C2-hypersurface in ℝn such that Γ divides ℝn into two unbounded domains D±. We consider the essential spectrum of Schrödinger operators on ℝn with surface δΓ-interactions which can be written formally as HΓ=−Δ+W−αΓδΓ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_\Gamma } = - \Delta + W - {\alpha _\Gamma }{\delta _{\Gamma ,}}$$\end{document}, where −Δ is the nonnegative Laplacian in ℝn, W ∈ L∞(ℝn) is a real-valued electric potential, δΓ is the Dirac δ-function with the support on the hypersurface Γ and αΓ ∈ L∞(Γ) is a real-valued coupling coefficient depending of the points of Γ. We realize HΓ as an unbounded operator AΓ in L2(ℝn) generated by the Schrödinger operator HΓ=−Δ+Wonℝn\Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_\Gamma } = - \Delta + Won{\mathbb{R}^n}\backslash \Gamma $$\end{document} and Robin-type transmission conditions on the hypersurface Γ. We give a complete description of the essential spectrum of AΓ in terms of the limit operators generated by AΓ and the Robin transmission conditions.