Gaps in the spectrum of a periodic quantum graph with periodically distributed ${\delta }^{\prime }$-type interactions

Abstract

We consider a family of quantum graphs $\{(\Gamma ,{\mathcal{A}}_{\epsilon }){\}}_{\epsilon >0}$, where $\Gamma$ is a ${\mathbb{Z}}^n$-periodic metric graph and the periodic Hamiltonian ${\mathcal{A}}_{\epsilon }$ is defined by the operation $-{\epsilon }^{-1}\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}$ on the edges of $\Gamma$ and either ${\delta }^{\prime }$-type conditions or the Kirchhoff conditions at its vertices. Here $\epsilon >0$ is a small parameter. We show that the spectrum of ${\mathcal{A}}_{\epsilon }$ has at least $m$ gaps as $\epsilon \to 0$ ($m\in \mathbb{N}$ is a predefined number), moreover the location of these gaps can be nicely controlled via a suitable choice of the geometry of $\Gamma$ and of coupling constants involved in ${\delta }^{\prime }$-type conditions.

18 pages, 3 figures