Spectral properties of spiral-shaped quantum waveguides

TL;DR

This paper studies the spectral properties of quantum particles confined in spiral-shaped waveguides, revealing how geometry influences the existence of bound states and the nature of the spectrum.

Contribution

It provides a detailed spectral analysis of spiral waveguides, including Archimedean spirals and modifications, highlighting the geometric factors affecting bound states and spectrum types.

Findings

Discrete spectrum is empty for Archimedean spirals due to curvature.

Bound states exist in spiral waveguides with sufficiently large central cavities.

Spectral properties depend on whether the spiral region is expanding or shrinking.

Abstract

We investigate properties of a particle confined to a hard-wall spiral-shaped region. As a case study we analyze in detail the Archimedean spiral for which the spectrum above the continuum threshold is absolutely continuous away from the thresholds. The subtle difference between the radial and perpendicular width implies, however, that in contrast to `less curved' waveguides, the discrete spectrum is empty in this case. We also discuss modifications such a multi-arm Archimedean spirals and spiral waveguides with a central cavity; in the latter case bound state already exist if the cavity exceeds a critical size. For more general spiral regions the spectral nature depends on whether they are `expanding' or `shrinking'. The most interesting situation occurs in the asymptotically Archimedean case where the existence of bound states depends on the direction from which the asymptotics is reached.