Numerical Optimization of Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field

TL;DR

This paper develops a numerical method to optimize the first seven eigenvalues of the magnetic Dirichlet Laplacian under a constant magnetic field, revealing that disks minimize eigenvalues when magnetic flux exceeds eigenvalue index.

Contribution

It introduces a gradient descent approach combined with the Method of Fundamental solutions for eigenvalue optimization in magnetic Laplacian problems, demonstrating disk minimizers under certain conditions.

Findings

Minimizers are disks when magnetic flux exceeds eigenvalue index.

The method effectively computes eigenvalues for various magnetic field strengths.

Numerical results support theoretical conjectures about eigenvalue minimizers.

Abstract

We present numerical minimizers for the first seven eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field in a wide range of field strengths. Adapting an approach by Antunes and Freitas, we use gradient descent for the minimization procedure together with the Method of Fundamental solutions for eigenvalue computation. Remarkably, we observe that when the magnetic flux exceeds the index of the target eigenvalue, the minimizer is always a disk.