Geometric bounds for the magnetic Neumann eigenvalues in the plane

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Omega 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 Kroger 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 for a constant magnetic field, and the upper bound lambda(1) < max(x is an element of Omega)(-)|beta(x)| for a variable magnetic field when omega is simply connected. For smooth domains, we prove a lower bound of lambda t 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 -> infinity and consists of the semiclassical limit 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.(c) 2023 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).