A Sharp Upper Bound on the Spectral Gap for Graphene Quantum Dots

  • Vladimir LotoreichikEmail author
  • Thomas Ourmières-Bonafos


The main result of this paper is a sharp upper bound on the first positive eigenvalue of Dirac operators in two dimensional simply connected C3-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.


Dirac operator Infinite mass boundary conditions Lowest eigenvalue Shape optimization 

Mathematics Subject Classification (2010)

35P15 58J50 



The authors have benefited a lot from fruitful discussions with Leonid Kovalev, Konstantin Pankrashkin, and Loïc Le Treust. TOB is grateful for the stimulating research stay and the hospitality of the Nuclear Physics Institute of Czech Republic where this project has been initiated. VL is grateful for the possibility to have in 2018 two inspiring research stays at the University Paris-Sud where a large part of the work was done.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Vladimir Lotoreichik
    • 1
    Email author
  • Thomas Ourmières-Bonafos
    • 2
  1. 1.Department of Theoretical Physics, Nuclear Physics InstituteCzech Academy of SciencesŘežCzech Republic
  2. 2.CNRS & Université Paris-DauphinePSL Research University, CEREMADEParisFrance

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