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

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

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