Geometric bounds for the magnetic Neumann eigenvalues in the plane

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.