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Spectral optimisation of Dirac rectangles

Abstract : We are concerned with the dependence of the lowest positive eigenvalue of the Dirac operator on the geometry of rectangles, subject to infinite-mass boundary conditions. We conjecture that the square is a global minimiser both under the area or perimeter constraints. Contrary to well-known non-relativistic analogues, we show that the present spectral problem does not admit explicit solutions. We prove partial optimisation results based on a variational reformulation and newly established lower and upper bounds to the Dirac eigenvalue. We also propose an alternative approach based on symmetries of rectangles and a non-convex minimisation problem; this implies a sufficient condition formulated in terms of a symmetry of the minimiser which guarantees the conjectured results.
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https://hal.archives-ouvertes.fr/hal-03491686
Contributor : Philippe Briet Connect in order to contact the contributor
Submitted on : Friday, December 17, 2021 - 7:51:40 PM
Last modification on : Sunday, June 26, 2022 - 3:26:52 AM

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Philippe Briet, David Krejcirik. Spectral optimisation of Dirac rectangles. Journal of Mathematical Physics, American Institute of Physics (AIP), 2022, 63, pp.013502. ⟨10.1063/5.0056278⟩. ⟨hal-03491686⟩

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