Sums of magnetic eigenvalues are maximal on rotationally symmetric domains

TL;DR

This paper proves that among various symmetric domains, the sum of the first n magnetic eigenvalues is maximized on shapes like equilateral triangles, squares, and disks, with results valid for multiple boundary conditions.

Contribution

It establishes new maximality results for sums of magnetic eigenvalues on symmetric domains, extending known spectral optimization principles.

Findings

Equilateral triangle maximizes eigenvalue sum among triangles.

Square maximizes among parallelograms.

Disk maximizes among ellipses.

Abstract

The sum of the first n energy levels of the planar Laplacian with constant magnetic field of given total flux is shown to be maximal among triangles for the equilateral triangle, under normalization of the ratio (moment of inertia)/(area)^3 on the domain. The result holds for both Dirichlet and Neumann boundary conditions, with an analogue for Robin (or de Gennes) boundary conditions too. The square similarly maximizes the eigenvalue sum among parallelograms, and the disk maximizes among ellipses. More generally, a domain with rotational symmetry will maximize the magnetic eigenvalue sum among all linear images of that domain. These results are new even for the ground state energy (n=1).