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

Abstract

Let Γ be a simply connected unbounded C 2-hypersurface in ℝ n such that Γ divides ℝ n into two unbounded domains D ±. We consider the essential spectrum of Schrodinger operators on ℝ n with surface δΓ-interactions which can be written formally as $${H_\Gamma } = - \Delta + W - {\alpha _\Gamma }{\delta _{\Gamma ,}}$$ , where −Δ is the nonnegative Laplacian in ℝ n , W ∈ L ∞(ℝ n ) 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 H Γ as an unbounded operator A Γ in L 2(ℝ n ) generated by the Schrodinger 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.