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.


Magnetic Laplacian Robin boundary conditions 

Mathematics Subject Classification

58J50 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.


  1. 1.
    B. Colbois and A. Savo, Eigenvalue bounds for the magnetic Laplacian, arXiv:1611.01930v1.
  2. 2.
    M. Egidi, S. Liu, F. Münch, and N. Peyerimhoff, Ricci curvature and eigenvalue estimates for the magentic Laplacian on manifolds, arXiv:1608.01955v1.
  3. 3.
    L. Erdös, Rayleigh-type isoperimetric inequality with a homogeneous magnetic field, Calc. Var. Partial Differential Equations 4 (1996), 283-292.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    R. L. Frank, A. Laptev, and S. Molchanov, Eigenvalue estimates for magnetic Schrödinger operators in domains, Proc. Amer. Math. Soc. 136 (2008), 4245-4255.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    R. C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J. 26 (1977), 459-472.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    I. Shigekawa, Eigenvalue problems for the Schrödinger operator with the magnetic field on a compact Riemannian manifold, J. Funct. Anal. 75 (1987), 92-127.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. A. Shubin, Discrete magnetic Laplacian, Comm. Math. Phys. 164 (1994), 259-275.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    M. A. Shubin, Essential self-adjointness for semi-bounded magnetic Schrödinger operators on non-compact manifolds, J. Funct. Anal. 186 (2001), 92-116.MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences IILebanese UniversityFanar-MatnLebanon
  2. 2.Department of Mathematics, Faculty of Sciences VLebanese UniversityNabatiehLebanon

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