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

TL;DR

This paper analyzes the spectral properties of a Schrödinger operator with a strong delta interaction supported on a periodic curve in two dimensions, revealing asymptotic band structure and the existence of spectral gaps for large interaction strength.

Contribution

It provides the asymptotic form of the band spectrum and proves the existence of spectral gaps for large delta-interaction strength on periodic curves.

Findings

Asymptotic form of the band spectrum as beta tends to infinity

Existence of spectral gaps for sufficiently large beta

Spectral behavior for non-periodic, asymptotically straight curves

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.