# Influence of the bound states in the Neumann Laplacian in a thin waveguide

## TL;DR

This paper investigates how the spectral properties of the Neumann Laplacian in a thin, twisted waveguide are affected by boundary states and oscillations, revealing the influence on the effective operator and spectral gaps.

## Contribution

It provides a detailed analysis of the impact of boundary states on the effective operator in thin waveguides and characterizes the spectrum and band gaps for periodic structures.

## Key findings

- Eigenvalues diverge due to transverse oscillations.
- Each divergent eigenvalue affects the effective operator differently.
- Identifies the spectrum and band gaps in periodic thin waveguides.

## Abstract

We study the Neumann Laplacian operator $-\Delta_\Omega^N$ restricted to a twisted waveguide $\Omega$. The goal is to find the effective operator when the diameter of $\Omega$ tends to zero. However, when $\Omega$ is "squeezed" there are divergent eigenvalues due to the transverse oscillations. We show that each one of these eigenvalues influences the action of the effective operator in a different way. In the case where $\Omega$ is periodic and sufficiently thin, we find information about the absolutely continuous spectrum of $-\Delta_\Omega^N$ and the existence and location of band gaps in its structure.