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

TL;DR

This paper investigates the spectral gaps of a family of periodic quantum graphs with delta'-type interactions, demonstrating controllable gap formation as the edge scaling parameter approaches zero.

Contribution

It introduces a method to control and predict spectral gaps in quantum graphs with delta'-type interactions by adjusting geometry and coupling constants.

Findings

At least m spectral gaps appear as epsilon approaches zero.

The location of spectral gaps can be precisely controlled.

The spectral gap structure depends on graph geometry and coupling constants.

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.