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A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities

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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 Szegő 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.

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Acknowledgements

R. D. Benguria, V. Lotoreichik and T. Ourmières-Bonafos are very grateful to the American Institute of Mathematics (AIM) for supporting their participation in the AIM workshop Shape optimization with surface interactions in 2019, where this project was initiated. The authors thank Lukas Schimmer and Jan Philip Solovej for pointing out a technical gap in an earlier version of this manuscript. T. Ourmières-Bonafos thanks Nicolas Raymond for pointing out that the projectors introduced in Definition 19 are named after the famous mathematician Gábor Szegő.

Funding

The work of R. D. Benguria has been partially supported by FONDECYT (Chile) projects 116-0856, and 120-1055.

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Authors and Affiliations

  1. Secção de Matemática, Departamento de Ciências e Tecnologia, Universidade Aberta, Palácio Ceia, 1269-001, Lisbon, Portugal

    Pedro R. S. Antunes

  2. Grupo de Física Matemática, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, 1749-016, Lisboa, Portugal

    Pedro R. S. Antunes

  3. Instituto de Física, Pontificia Universidad Católica de Chile, Avda. Vicuña Mackenna, 4860, Santiago, Chile

    Rafael D. Benguria

  4. Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068, Řež, Czech Republic

    Vladimir Lotoreichik

  5. CNRS, Centrale Marseille, I2M, Aix Marseille Univ, Marseille, France

    Thomas Ourmières-Bonafos

Authors
  1. Pedro R. S. Antunes
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  2. Rafael D. Benguria
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  3. Vladimir Lotoreichik
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  4. Thomas Ourmières-Bonafos
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Corresponding author

Correspondence to Thomas Ourmières-Bonafos.

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Communicated by R. Seiringer.

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Antunes, P.R.S., Benguria, R.D., Lotoreichik, V. et al. A Variational Formulation for Dirac Operators in Bounded Domains. Applications to Spectral Geometric Inequalities. Commun. Math. Phys. 386, 781–818 (2021). https://doi.org/10.1007/s00220-021-03959-6

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