Spectral optimisation of Dirac rectangles

TL;DR

This paper investigates how the shape of rectangles affects the lowest positive eigenvalue of the Dirac operator, proposing bounds and symmetry-based methods to support the conjecture that squares minimize this eigenvalue.

Contribution

It introduces new bounds and a symmetry-based approach for optimizing Dirac eigenvalues on rectangles, addressing a conjecture without explicit solutions.

Findings

Partial optimization results via variational reformulation

New bounds for Dirac eigenvalues on rectangles

Symmetry-based conditions supporting the minimization conjecture

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.