Band gap of the Schroedinger operator with a strong delta-interaction on a periodic curve

Abstract

In this paper we study the operator $H_{\beta}=-\Delta-\beta\delta(\cdot-\Gamma)$ in $L^{2}(\mathbb{R}^{2})$, where $\Gamma$ is a smooth periodic curve in $\mathbb{R}^{2}$. We obtain the asymptotic form of the band spectrum of $H_{\beta}$ as $\beta$ tends to infinity. Furthermore, we prove the existence of the band gap of $\sigma(H_{\beta})$ for sufficiently large $\beta>0$. Finally, we also derive the spectral behaviour for $\beta\to\infty$ in the case when $\Gamma$ is non-periodic and asymptotically straight.