Spectral asymptotics for two-dimensional Dirac operators in thin waveguides

TL;DR

This paper analyzes the spectral behavior of two-dimensional Dirac operators in thin waveguides, revealing that eigenvalue splitting is governed by a one-dimensional Schrödinger operator with a curvature-induced potential, contrasting with non-relativistic models.

Contribution

It establishes a link between the eigenvalue splitting of Dirac operators in thin waveguides and a one-dimensional Schrödinger operator with a geometric potential, in the thin-width limit.

Findings

Eigenvalues are within order ε of the essential spectrum.

Eigenvalue splitting is driven by a 1D Schrödinger operator with curvature potential.

Contrast with non-relativistic models where eigenvalues are at finite distance.

Abstract

We consider the two-dimensional Dirac operator with infinite mass boundary conditions posed in a tubular neighborhood of a $C^4$-planar curve. Under generic assumptions on its curvature $\kappa$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schr\"odinger operator on $L^2(\mathbb R)$ \[ \mathcal{L}_e := -\frac{d^2}{ds^2} - \frac{\kappa^2}{\pi^2} \] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $\varepsilon$ from the essential spectrum, where $2\varepsilon$ is the width of the waveguide. This is in contrast with the non-relativistic counterpart of this model, for which they are known to be at a finite distance.