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We consider a compact Riemannian manifold M endowed with a potential 1-form Aand study the magnetic Laplacian associated with those data (with Neumannmagnetic boundary condition if the bpoundary of M is not empty). We firstestablish a family of upper bounds for all the eigenvalues, compatible with theWeyl law. When the potential is a closed 1-form, we get a sharp upper bound forthe first eigenvalue. In the second part, we consider only closed potentials,and we establish a sharp lower bound for the first eigenvalue when the manifoldis a 2-dimensional Riemannian cylinder. The equality case characterizes thesituation where the metric is a product. We also look at the case of doublyconvex domains in the Euclidean plane.

Eigenvalue bounds for the magnetic Laplacian