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.

arXiv comment11 pages; due to a gap in the proof in our previous version (see Remark 1), we obtain just partial results, by an alternative approach; version accepted for publication in Journal of Mathematical Physics

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Cited by in corpus (4)

• Eigenvalue curves for generalized MIT bag models
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• A connection between quantum dot Dirac operators and $\overline{\partial }$-Robin Laplacians in the context of shape optimization problems