Pavel Exner, Kazushi Yoshitomi
Band gap of the Schroedinger operator with a strong
delta-interaction on a periodic curve
(57K, LaTeX 2e)
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.