Periodic quantum graphs with predefined spectral gaps

TL;DR

This paper demonstrates how to engineer spectral gaps in periodic quantum graphs by choosing vertex conditions, allowing precise control over the spectrum's structure as the edge length parameter varies.

Contribution

It introduces a method to control and predict spectral gaps in quantum graphs through vertex condition adjustments, especially in the small parameter limit.

Findings

Spectral gaps can be made to appear at desired locations.

Asymptotic behavior of gaps is fully controllable via vertex conditions.

Exact placement of spectral gap edges can be achieved for small parameters.

Abstract

Let $\Gamma$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $\Gamma$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on its edges; here $\varepsilon>0$ is a small parameter. Let $m\in\mathbb{N}$. We show that under a proper choice of vertex conditions the spectrum $\sigma(\mathcal{H}^\varepsilon)$ of $\mathcal{H}^\varepsilon$ has at least $m$ gaps as $\varepsilon$ is small enough. We demonstrate that the asymptotic behavior of these gaps and the asymptotic behavior of the bottom of $\sigma(\mathcal{H}^\varepsilon)$ as $\varepsilon\to 0$ can be completely controlled through a suitable choice of coupling constants standing in those vertex conditions. We also show how to ensure for fixed (small enough) $\varepsilon$ the precise coincidence of the left endpoints of the first $m$ spectral gaps with predefined numbers.