Spectral optimization 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 minimizer under both the area or perimeter constraints. Contrary to the well-known non-relativistic analogs, we show that the present spectral problem does not admit explicit solutions. We prove partial optimization 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 minimization problem; this implies a sufficient condition formulated in terms of a symmetry of the minimizer which guarantees the conjectured results.