On the spectrum of narrow Neumann waveguide with periodically distributed $\delta'$ traps

TL;DR

This paper investigates how the spectral properties of a narrow Neumann waveguide with periodically distributed  traps are influenced by geometry and scaling of  interactions, revealing the dominant role of trap geometry in spectral gaps.

Contribution

It provides a rigorous analysis of the spectral gap behavior in narrow waveguides with scaled   traps, linking spectral properties to geometric features.

Findings

The first spectral gap is determined solely by trap geometry in the limit.

Scaling of  interactions influences spectral properties significantly.

Spectral behavior is governed by geometric properties as the waveguide narrows.

Abstract

We analyze a family of singular Schr\"odinger operators describing a Neumann waveguide with a periodic array of singular traps of a $\delta'$ type. We show that in the limit when perpendicular size of the guide tends to zero and the $\delta'$ interactions are appropriately scaled, the first spectral gap is determined exclusively by geometric properties of the traps.