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arXiv · 2207.08700

Spectral asymptotics for two-dimensional Dirac operators in thin waveguides

William Borrelli·Nour Kerraoui·Thomas Ourmières-Bonafos
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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 $κ$, we prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schrödinger operator on $L^2(\mathbb R)$ \[ \mathcal{L}_e := -\frac{d^2}{ds^2} - \frac{κ^2}{π^2} \] 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.

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William Borrelli, Nour Kerraoui, Thomas Ourmières-Bonafos. 2022-07-18. Spectral asymptotics for two-dimensional Dirac operators in thin waveguides. https://arxiv.org/abs/2207.08700

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