Spectral analysis in broken sheared waveguides

TL;DR

This paper investigates the spectral properties of the Dirichlet Laplacian in broken sheared waveguides, proving the existence, finiteness, and specific multiplicity of the discrete spectrum based on geometric configurations.

Contribution

It establishes the non-empty, finite nature of the discrete spectrum in broken sheared waveguides and identifies conditions for its multiplicity to be exactly one.

Findings

Discrete spectrum is non-empty and finite.

Specific geometries lead to a total multiplicity of one.

Spectral properties depend on the waveguide's geometry.

Abstract

Let $\Omega \subset \mathbb R^3$ be a broken sheared waveguide, i.e., it is built by translating a cross-section in a constant direction along a broken line in $\mathbb R^3$. We prove that the discrete spectrum of the Dirichlet Laplacian operator in $\Omega$ is non-empty and finite. Furthermore, we show a particular geometry for $\Omega$ which implies that the total multiplicity of the discrete spectrum is equals 1.