A sharp upper bound on the spectral gap for graphene quantum dots

Abstract

The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected $C^3$-domains with infinite mass boundary conditions. This bound is given in terms of a conformal variation, explicit geometric quantities and of the first eigenvalue for the disk. Its proof relies on the min-max principle applied to the squares of these Dirac operators. A suitable test function is constructed by means of a conformal map. This general upper bound involves the norm of the derivative of the underlying conformal map in the Hardy space ${\mathcal{H}}^2(\mathbb{D})$. Then, we apply known estimates of this norm for convex and for nearly circular, star-shaped domains in order to get explicit geometric upper bounds on the eigenvalue. These bounds can be re-interpreted as reverse Faber-Krahn-type inequalities under adequate geometric constraints.

arXiv commentMathematical Physics, Analysis and Geometry (in press)

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• Self-adjointness of two-dimensional Dirac operators on domains
• Spectral gaps of Dirac operators describing graphene quantum dots
• Self-Adjointness of Dirac Operators with Infinite Mass Boundary Conditions in Sectors
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Cited by in corpus (8)

• Self-adjoint Dirac operators on domains in ${\mathbb{R}}^3$
• A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities
• Spectral properties of relativistic quantum waveguides
• 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
• A connection between quantum dot Dirac operators and $\overline{\partial }$-Robin Laplacians in the context of shape optimization problems