Geometric bounds for the magnetic Neumann eigenvalues in the plane
Bruno Colbois, Corentin Léna, Luigi Provenzano, Alessandro Savo · arXiv (Cornell University) · 2023
We consider the eigenvalues of the magnetic Laplacian on a bounded domain $Ω$ of $\mathbb R^2$ with uniform magnetic field $β>0$ and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy $λ_1$ 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 $β=β(x)$ on a simply connected domain in a Riemannian surface. In particular: we prove the upper bound $λ_1<β$ for a general plane domain, and the upper bound $λ_1<\sup_{x\inΩ}|β(x)|$ for a variable magnetic field when $Ω$ is simply connected. For smooth domains, we prove a lower bound of $λ_1$ 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 $\dfrac{2πk}{|Ω|}$ 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 $λ_1$ is always small.