Spectral properties of relativistic quantum waveguides

TL;DR

This paper analyzes the spectral properties of the Dirac operator in relativistic quantum waveguides, deriving effective models and conditions for bound states based on geometric and physical parameters.

Contribution

It introduces a spectral analysis framework for the Dirac operator in curved waveguides, including effective Hamiltonians and bound state criteria in the relativistic setting.

Findings

Location of the essential spectrum for the Dirac operator

Derivation of an effective Hamiltonian in the thin-strip limit

Geometric conditions for the existence of bound states

Abstract

We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip limit. We also investigate the existence of bound states in the non-relativistic limit and give a geometric quantitative condition for the bound states to exist.