Klein's Paradox and the Relativistic $\delta$-shell Interaction in ${\mathbb{R}}^3$

Abstract

Under certain hypothesis of smallness of the regular potential $\mathbf{V}$, we prove that the Dirac operator in ${\mathbb{R}}^3$ coupled with a suitable re-scaling of $\mathbf{V}$ converges in the strong resolvent sense to the Hamiltonian coupled with a $\delta$-shell potential supported on $\Sigma$, a bounded $C^2$ surface. Nevertheless, the coupling constant depends non-linearly on the potential $\mathbf{V}$: the Klein's Paradox comes into play.

References in corpus (3)

• Chiral tunneling and the Klein paradox in graphene
• Leaky Quantum Graphs: A Review
• Approximation of Schrödinger operators with $\delta$-interactions supported on hypersurfaces

Cited by in corpus (9)

• Dirac operators with Lorentz scalar shell interactions
• Self-adjoint Dirac operators on domains in ${\mathbb{R}}^3$
• Spectral Properties of the Dirac Operator coupled with $\delta$-Shell Interactions
• Spectral Analysis of Dirac Operators with delta interactions supported on the boundaries of rough domains
• Two-dimensional Dirac operators with general $\delta$-shell interactions supported on a straight line
• Approximation of one-dimensional relativistic point interactions by regular potentials revised
• Eigenvalue curves for generalized MIT bag models
• Curvature contribution to the essential spectrum of Dirac operators with critical shell interactions
• Limiting absorption principle and scattering matrix for Dirac operators with $\delta$-shell interactions