Spectra of soft ring graphs

TL;DR

This paper analyzes the spectral properties of a ring-shaped quantum wire with variable coupling, exploring how geometry, randomness, and magnetic fields influence eigenvalues, eigenfunctions, and localization phenomena.

Contribution

It derives conditions for the discrete spectrum of soft ring quantum wires with nonconstant coupling and examines effects of randomness and magnetic flux on spectral properties.

Findings

Random coupling induces localization in the spectrum.

Eigenvalues depend on ring geometry and coupling strength.

Magnetic flux influences persistent current behavior.

Abstract

We discuss of a ring-shaped soft quantum wire modeled by $\delta$ interaction supported by the ring of a generally nonconstant coupling strength. We derive condition which determines the discrete spectrum of such systems, and analyze the dependence of eigenvalues and eigenfunctions on the coupling and ring geometry. In particular, we illustrate that a random component in the coupling leads to a localization. The discrete spectrum is investigated also in the situation when the ring is placed into a homogeneous magnetic field or threaded by an Aharonov-Bohm flux and the system exhibits persistent currents.