Geometric bounds for the magnetic Neumann eigenvalues in the plane

TL;DR

This paper derives bounds and semiclassical estimates for the magnetic Neumann eigenvalues in planar domains, highlighting the influence of magnetic field variations and domain topology on the eigenvalues.

Contribution

It provides new upper and lower bounds for the first eigenvalue of the magnetic Laplacian, including semiclassical estimates and topological insights.

Findings

Proves <eta for general plane domains.

Establishes <\u221f_{x\u2208}|eta(x)| for simply connected domains with variable magnetic fields.

Shows eigenvalue bounds depend on domain topology, with examples illustrating this effect.

Abstract

We consider the eigenvalues of the magnetic Laplacian on a bounded domain $\Omega$ of $\mathbb R^2$ with uniform magnetic field $\beta>0$ and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy $\lambda_1$ and we provide semiclassical estimates in the spirit of Kr\"oger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields $\beta=\beta(x)$ on a simply connected domain in a Riemannian surface. In particular: we prove the upper bound $\lambda_1<\beta$ for a general plane domain, and the upper bound $\lambda_1<\sup_{x\in\Omega}|\beta(x)|$ for a variable magnetic field when $\Omega$ is simply connected. For smooth domains, we prove a lower bound of $\lambda_1$ depending only on the intensity of the magnetic field $\beta$ and the rolling radius of the domain. The estimates on the Riesz mean imply an upper bound for the averages of the first $k$ eigenvalues which is sharp when $k\to\infty$ and consists of the semiclassical limit $\dfrac{2\pi k}{|\Omega|}$ plus an oscillating term. We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which $\lambda_1$ is always small.