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

## TL;DR

This paper establishes a precise upper bound on the first positive eigenvalue of Dirac operators in 2D domains, linking spectral properties to geometric and conformal map characteristics, with applications to convex and star-shaped domains.

## Contribution

It introduces a sharp upper bound on Dirac eigenvalues using conformal maps and Hardy space norms, providing explicit geometric estimates and reverse Faber-Krahn inequalities.

## Key findings

- Derived a conformal variation-based upper bound for eigenvalues.
- Applied bounds to convex and star-shaped domains for explicit estimates.
- Reinterpreted bounds as reverse Faber-Krahn inequalities.

## 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.