Magnetic spectral bounds on starlike plane domains

TL;DR

This paper establishes sharp upper bounds for magnetic Laplacian energy levels on starlike plane domains, showing disks maximize these bounds under certain conditions, with a focus on magnetic flux and domain shape.

Contribution

It introduces a novel bound for magnetic Laplacian eigenvalues on starlike domains, highlighting the disk as the extremal shape under magnetic flux.

Findings

Disks maximize the sum of eigenvalues for concave increasing functions.

The scale factor G measures deviation from roundness, G=1 for disks.

Bounds are valid for both Dirichlet and Neumann boundary conditions.

Abstract

We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that $\sum_{j=1}^n \Phi \big( \lambda_j A/G \big)$ is maximal for a disk whenever $\Phi$ is concave increasing, $n \geq 1$, the domain has area $A$, and $\lambda_j$ is the $j$-th Dirichlet eigenvalue of the magnetic Laplacian $\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2$. Here the flux $\beta$ is constant, and the scale invariant factor $G$ penalizes deviations from roundness, meaning $G \geq 1$ for all domains and $G=1$ for disks.