# Eigenvalue bounds of the Robin Laplacian with magnetic field

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1 Lebanese University 2 Lebanese University

Abstract : 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 the 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 those estimates in terms of the mean curvature of the boundary, the parameter $\tau$ and a lower bound of the Ricci curvature of $M$ see Theorems 1.3 and 1.5. The main technique is to use the Bochner formula established in 3 for the magnetic Laplacian and to integrate it over $M$ see Theorem 1.2. As a direct application, we find the standard Lichnerowicz estimates for both the Neumann and Dirichlet Laplacian, when the parameter $\tau$ tends to $0$ or to $\infty$. In the last part, we compare the eigenvalues $\lambda k \tau, \alpha$ with the first eigenvalue $\lambda 1 \tau, 0$ i.e. without magnetic field and the Neumann eigenvalues $\lambda k 0, \alpha$ see Theorem 1.7 using the min-max principle.

Autor: Georges Habib - Ayman Kachmar -

Fuente: https://hal.archives-ouvertes.fr/

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