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

We consider a family of quantum graphs $\{(Γ,\mathcal{A}_\varepsilon)\}_{\varepsilon>0}$, where $Γ$ 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 $Γ$ and either $δ'$-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 $Γ$ and of coupling constants involved in $δ'$-type conditions.

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