Gaps in the spectrum of a periodic quantum graph with periodically distributed $δ'$-type interactions

Abstract: We consider a family of quantum graphs ${(\Gamma,\mathcal{A}\varepsilon)}{\varepsilon>0}$, where $\Gamma$ is a $\mathbb{Z}^n$-periodic metric graph and the periodic Hamiltonian $\mathcal{A}\varepsilon$ is defined by the operation $-\varepsilon^{-1} {\mathrm{d} ^2\over \mathrm{d} x^2}$ on the edges of $\Gamma$ and either $\delta'$-type conditions or the Kirchhoff conditions at its vertices. Here $\varepsilon>0$ is a small parameter. We show that the spectrum of $\mathcal{A}\varepsilon$ has at least $m$ gaps as $\varepsilon\to 0$ ($m\in\mathbb{N}$ 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 $\delta'$-type conditions.