Spectral asymptotics for two-dimensional Dirac operators in thin waveguides

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 $L^2(\mathbb{R})$ $${\mathcal{L}}_e:=-\frac{d^2}{d{s}^2}-\frac{{\kappa }^2}{{\pi }^2}$$ with a geometrically induced potential. The eigenvalues are shown to be at distance of order $\epsilon$ from the essential spectrum, where $2\epsilon$ 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.