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

Abstract

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 δ_Γ-interactions which can be written formally as

\({H_\Gamma } = - \Delta + W - {\alpha _\Gamma }{\delta _{\Gamma ,}}\)

, where −Δ is the nonnegative Laplacian in ℝⁿ, W ∈ L^∞(ℝⁿ) is a real-valued electric potential, δ_Γ is the Dirac δ-function with the support on the hypersurface Γ and α_Γ ∈ L^∞(Γ) is a real-valued coupling coefficient depending of the points of Γ. We realize H_Γ as an unbounded operator A_Γ in L₂(ℝⁿ) generated by the Schrödinger operator

\({H_\Gamma } = - \Delta + Won{\mathbb{R}^n}\backslash \Gamma \)

and Robin-type transmission conditions on the hypersurface Γ. We give a complete description of the essential spectrum of A_Γ in terms of the limit operators generated by A_Γ and the Robin transmission conditions.