Spectral gap of segments of periodic waveguides

TL;DR

This paper investigates the spectral properties of finite segments of periodic waveguides, revealing that the gap between the first two eigenvalues scales as the inverse square of the segment length.

Contribution

It provides a precise asymptotic behavior of the spectral gap for finite segments of periodic waveguides with Dirichlet boundary conditions.

Findings

Eigenvalue gap scales as L^{-2} for segment length L

Spectral analysis of finite periodic waveguide segments

Quantitative relationship between segment size and eigenvalue spacing

Abstract

We consider a periodic strip in the plane and the associated quantum waveguide with Dirichlet boundary conditions. We analyse finite segments of the waveguide consisting of $L$ periodicity cells, equipped with periodic boundary conditions at the ``new'' boundaries. Our main result is that the distance between the first and second eigenvalue of such a finite segment behaves like $L^{-2}$.