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.

LaTeX2e, 23 pages

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