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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 theasymptotic form of the band spectrum of $H_{\beta}$ as $\beta$ tends toinfinity. Furthermore, we prove the existence of the band gap of$\sigma(H_{\beta})$ for sufficiently large $\beta>0$. Finally, we also derivethe spectral behaviour for $\beta\to\infty$ in the case when $\Gamma$ isnon-periodic and asymptotically straight.Comment: LaTeX2e, 23 page

Band Gap of the Schrödinger Operator with a Strong δ-Interaction on a Periodic Curve