2017/07/25 by Habib, Georges, Kachmar, Ayman · 1 citation
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1707.07939
On a compact Riemannian manifold M with boundary, we give an estimate for the eigenvalues (λ_k(τ,α))_k of the magnetic Laplacian with the Robin boundary conditions. Here, τ is a positive number that defines the Robin condition and α is a real differential 1-form on M that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter τ and a lower bound of the Ricci curvature of M (see Theorem \refestimate1 and Corollary \refcorestimate). The main technique is to use the Bochner formula established in \citeELMP for the magnetic Laplacian and to integrate it over M (see Theorem \refbochnermagnetic1). In the last part, we compare the eigenvalues λ_k(τ,α) with the first eigenvalue λ_1(τ)=λ_1(τ,0) (i.e. without magnetic field) and the Neumann eigenvalues λ_k(0,α) (see Theorem \refthm:comp) using the min-max principle.