2016/11/07 by Bruno Colbois, Colbois, Bruno, Alessandro Savo +1
Mathematics · Computer Science · #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.1611.01930
We consider a compact Riemannian manifold M endowed with a potential 1-form A and study the magnetic Laplacian associated with those data (with Neumann magnetic boundary condition if the bpoundary of M is not empty). We first establish a family of upper bounds for all the eigenvalues, compatible with the Weyl law. When the potential is a closed 1-form, we get a sharp upper bound for the first eigenvalue. In the second part, we consider only closed potentials, and we establish a sharp lower bound for the first eigenvalue when the manifold is a 2-dimensional Riemannian cylinder. The equality case characterizes the situation where the metric is a product. We also look at the case of doubly convex domains in the Euclidean plane.