Published Paper

Inserted: 19 jul 2022
Last Updated: 23 dec 2023

Journal: Quantum Mathematics I - Springer Indam Series
Year: 2023
Doi: https://doi.org/10.1007/978-981-99-5894-8

To appear on "INdAM Quantum Meeting 2022" proceedings.

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 Schrödinger 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.