A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities

Abstract

We investigate spectral features of the Dirac operator with infinite mass boundary conditions in a smooth bounded domain of ${\mathbb{R}}^2$. Motivated by spectral geometric inequalities, we prove a non-linear variational formulation to characterize its principal eigenvalue. This characterization turns out to be very robust and allows for a simple proof of a Szegö type inequality as well as a new reformulation of a Faber-Krahn type inequality for this operator. The paper is complemented with strong numerical evidences supporting the existence of a Faber-Krahn type inequality.

arXiv comment34 pages, 4 figures

References in corpus (12)

• On the eigenvalues of operators with gaps. Application to Dirac operators
• Self-adjointness of two-dimensional Dirac operators on domains
• On the MIT bag model: self-adjointness and non-relativistic limit
• Spectral gaps of Dirac operators describing graphene quantum dots
• Two-dimensional Dirac operators with singular interactions supported on closed curves
• Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors
• Dirac operators with Lorentz scalar shell interactions
• Upper bounds for the first eigenvalue of the Dirac operator on surfaces
• Resolvent convergence to Dirac operators on planar domains
• A sharp upper bound on the spectral gap for graphene quantum dots
• The Hijazi inequality on manifolds with boundary
• Dirac operators on hypersurfaces as large mass limits

Cited by in corpus (6)

• Spectral optimisation of Dirac rectangles
• Eigenvalue curves for generalized MIT bag models
• Spectral inequality for Dirac right triangles
• Dirac operator spectrum in tubes and layers with a zigzag type boundary
• The $\overline{\partial }$-Robin Laplacian
• A connection between quantum dot Dirac operators and $\overline{\partial }$-Robin Laplacians in the context of shape optimization problems