Gap control by singular Schr\"odinger operators in a periodically structured metamaterial

TL;DR

This paper studies how to control spectral gaps in a family of periodic Schrödinger operators with delta-prime interactions on surfaces, showing that the number and size of gaps can be precisely managed through geometric and interaction parameters.

Contribution

It demonstrates that the spectral gaps of these operators can be fully controlled in the limit by adjusting surface configurations and interaction strengths.

Findings

At most m spectral gaps can appear in the limit.

The behavior of the first m gaps can be completely controlled.

Spectral gap control is achieved through geometric and interaction parameter tuning.

Abstract

We consider a family $\{\mathcal{H}^\varepsilon\}_{\varepsilon>0}$ of $\varepsilon\mathbb{Z}^n$-periodic Schr\"odinger operators with $\delta'$-interactions supported on a lattice of closed compact surfaces; within a minimal period cell one has $m\in\mathbb{N}$ surfaces. We show that in the limit when $\varepsilon\to 0$ and the interactions strengths are appropriately scaled, $\mathcal{H}^\varepsilon$ has at most $m$ gaps within finite intervals, and moreover, the limiting behavior of the first $m$ gaps can be completely controlled through a suitable choice of those surfaces and of the interactions strengths.