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

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

2001-06-25

arxiv.org/abs/math-ph/0106025v1

Ann. H. Poincare 2 (2001), 1139-1158

Physics

Mathematical Physics

LaTeX2e, 23 pages

Scientific paper

10.1007/s00023-001-8605-2

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.

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