Spectral asymptotics for two-dimensional Dirac operators in thin waveguides

• URL: http://arxiv.org/abs/2207.08700v1
• Date: Mon, 18 Jul 2022 15:50:10 GMT
• Title: Spectral asymptotics for two-dimensional Dirac operators in thin waveguides
• Authors: William Borrelli and Nour Kerraoui and Thomas Ourmi\`eres-Bonafos
• Abstract summary: We prove that in the thin-width regime the splitting of the eigenvalues is driven by the one dimensional Schr"odinger operator on $L1(mathbb R)$ [ mathcalL_e := -fracd2ds2 - frackappa2pi2 ] with a geometrically induced potential. The eigenvalues are shown to be at distance of order $varepsilon$ from the essential spectrum, where $2varepsilon$ is the width
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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