Eigenvalue bounds of the Robin Laplacian with magnetic field
On a compact Riemannian manifold M with boundary, we give an estimate for the eigenvalues \((\lambda _k(\tau ,\alpha ))_k\) of the magnetic Laplacian with Robin boundary conditions. Here, \(\tau \) is a positive number that defines the Robin condition and \(\alpha \) 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 \(\tau \), and a lower bound of the Ricci curvature of M (see Theorem 1.3 and Corollary 1.5). The main technique is to use the Bochner formula established in Egidi et al. (Ricci curvature and eigenvalue estimates for the magentic Laplacian on manifolds, arXiv:1608.01955v1) for the magnetic Laplacian and to integrate it over M (see Theorem 1.2). In the last part, we compare the eigenvalues \(\lambda _k(\tau ,\alpha )\) with the first eigenvalue \(\lambda _1(\tau )=\lambda _1(\tau ,0)\) (i.e. without magnetic field) and the Neumann eigenvalues \(\lambda _k(0,\alpha )\) (see Theorem 1.6) using the min-max principle.
KeywordsMagnetic Laplacian Robin boundary conditions
Mathematics Subject Classification58J50 53C21 58J35
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The authors are indebted to Nicolas Ginoux, Norbert Peyerimhoff, and Alessandro Savo for their valuable comments on the paper. The first named author acknowledges the financial support of the Alexander von Humboldt Foundation. The authors are indebted to the anonymous referee who suggested the proof of Corollary 1.5.