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

Pedro R. S. Antunes; Rafael D. Benguria; Vladimir Lotoreichik; Thomas Ourmières-Bonafos

doi:10.1007/s00220-021-03959-6

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arXiv · 2003.04061

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

Pedro R. S. Antunes·Rafael D. Benguria·Vladimir Lotoreichik·Thomas Ourmières-Bonafos

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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 Szeg\"o 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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BibTeX RIS

Pedro R. S. Antunes, Rafael D. Benguria, Vladimir Lotoreichik, Thomas Ourmières-Bonafos. 2020-03-09. A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalities. https://doi.org/10.1007/s00220-021-03959-6

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