Essential Spectrum of Schr\"{o}dinger Operators

V. S. Rabinovich

Matematičeskie zametki, Tome 102 (2017) no. 5, pp. 761-774

Résumé

Let $\Gamma$ be a simply connected unbounded $C^{2}$-hypersurface in $\mathbb{R}^{n}$ such that $\Gamma$ divides $\mathbb{R}^{n}$ into two unbounded domains $D^{\pm}$. We consider the essential spectrum of Schrödinger operators on $\mathbb{R}^{n}$ with surface $\delta_{\Gamma}$-interactions which can be written formally as $$ H_{\Gamma}=-\Delta+W-\alpha_{\Gamma}\delta_{\Gamma}, $$ where $-\Delta$ is the nonnegative Laplacian in $\mathbb{R}^{n}$, $W\in L^{\infty}(\mathbb{R}^{n})$ is a real-valued electric potential, $\delta_{\Gamma}$ is the Dirac $\delta$-function with the support on the hypersurface $\Gamma$ and $\alpha_{\Gamma}\in L^{\infty}(\Gamma)$ is a real-valued coupling coefficient depending of the points of $\Gamma$. We realize $H_{\Gamma}$ as an unbounded operator $\mathcal{A}_{\Gamma}$ in $L^{2}(\mathbb{R}^{n})$ generated by the Schrödinger operator $$ H_{\Gamma}=-\Delta+W\qquad \text{on}\quad \mathbb{R}^{n}\setminus\Gamma $$ and Robin-type transmission conditions on the hypersurface $\Gamma$. We give a complete description of the essential spectrum of $\mathcal{A}_{\Gamma}$ in terms of the limit operators generated by $A_{\Gamma}$ and the Robin transmission conditions.

Détail

Keywords: surface $\delta$-interaction, self-adjoint realization, Robin transmission conditions, limit operators, essential spectra.