Geometric bounds for the magnetic Neumann eigenvalues in the plane

Abstract

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of $R^2$ 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 Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields $\beta =\beta \left(x\right)$ on a simply connected domain in a Riemannian surface.

In particular: we prove the upper bound ${\lambda }_1<\beta$ for a general plane domain for a constant magnetic field, and the upper bound ${\lambda }_1<{\max }_{x\in \bar{\Omega }}\left|\beta \right(x\left)\right|$ for a variable magnetic field when Ω is simply connected.

For smooth domains, we prove a lower bound of ${\lambda }_1$ depending only on the intensity of the magnetic field β 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 $\frac{2\pi k}{\left|\Omega \right|}$ 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.