Numerical optimisation of Dirac eigenvalues

TL;DR

This work addresses spectral isoperimetric questions for the Dirac operator with infinite-mass boundary conditions in the plane, focusing on the first three positive eigenvalues and their ratios under area and perimeter constraints. It develops a robust numerical framework based on the Method of Fundamental Solutions to solve the Dirac problem via a Helmholtz reformulation, and uses the Subspace Angle Technique to accurately locate eigenvalues. Key findings include a mass-dependent upper bound for the first eigenvalue on rectangles that appears to extend to general quadrilaterals, numerical confirmation that the second eigenvalue is minimised by two disjoint disks, and evidence that the third eigenvalue is not disk- or square-minimised in general, with several relativistic conjectures for higher polygons. These results extend classical nonrelativistic isoperimetric insights into the Dirac setting and provide a practical computational roadmap to guide future rigorous proofs in relativistic spectral geometry.

Abstract

Motivated by relativistic materials, we develop a numerical scheme to support existing or state new conjectures in the spectral optimisation of eigenvalues of the Dirac operator, subject to infinite-mass boundary conditions. We study the optimality of the regular polygon (respectively, disk) among all polygons of a given number of sides (respectively, arbitrary sets), subject to area or perimeter constraints. We consider the three lowest positive eigenvalues and their ratios. Roughly, we find results analogous to known or expected for the Dirichlet Laplacian, except for the third eigenvalue which does not need to be minimised by the regular polygon (respectively, the disk) for all masses. In addition to the numerical results, a new, mass-dependent upper bound to the lowest eigenvalue in rectangles is proved and its extension to arbitrary quadrilaterals is conjectured.

Figures (26)

• Figure 1: Absolute values of the first eigenfunction components of $\mathbb{D}$ for $m=0$.
• Figure 2: Behaviour of the first five eigenvalues for rectangles with unit area, width $a$ and $m=1$.
• Figure 3: Behaviour of the first five eigenvalues for rectangles with unit area, width $a$ and $m=5$.
• Figure 4: Behaviour of the first five eigenvalues for rectangles with perimeter equal to 4, width $a$ and $m=1$.
• Figure 5: Behaviour of the first five eigenvalues for rectangles with perimeter equal to 4, width $a$ and $m=5$.

Theorems & Definitions (11)

• Conjecture 3.1: briet-krej
• Theorem 3.2: briet-krej
• Theorem 3.5
• proof : Proof of Theorem \ref{['Thm']}
• Conjecture 3.17
• Definition 4.1
• Conjecture 4.2
• Conjecture 5.1
• Conjecture 6.5
• Conjecture 6.6