Quantum waveguides with corners

TL;DR

This paper studies the spectral properties of V-shaped quantum waveguides, revealing how the opening angle influences bound states and eigenfunction behavior, with insights supported by numerical simulations.

Contribution

It provides a detailed analysis of how the opening of V-shaped waveguides affects their spectral properties and eigenfunctions, including the existence and behavior of bound states.

Findings

Finite bound states depend on the V opening

Number of bound states tends to infinity as opening approaches zero

Eigenfunctions exhibit self-similar concentration or spreading patterns

Abstract

The simplest modeling of planar quantum waveguides is the Dirichlet eigenproblem for the Laplace operator in unbounded open sets which are uniformly thin in one direction. Here we consider V-shaped guides. Their spectral properties depend essentially on a sole parameter, the opening of the V. The free energy band is a semi-infinite interval bounded from below. As soon as the V is not flat, there are bound states below the free energy band. There are a finite number of them, depending on the opening. This number tends to infinity as the opening tends to 0 (sharply bent V). In this situation, the eigenfunctions concentrate and become self-similar. In contrast, when the opening gets large (almost flat V), the eigenfunctions spread and enjoy a different self-similar structure. We explain all these facts and illustrate them by numerical simulations.