Advertisement

Rate of Convergence for Eigenfunctions of Aharonov-Bohm Operators with a Moving Pole

  • Laura Abatangelo
  • Veronica FelliEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 22)

Abstract

We study the behavior of eigenfunctions for magnetic Aharonov-Bohm operators with half-integer circulation and Dirichlet boundary conditions in a planar domain. We prove a sharp estimate for the rate of convergence of eigenfunctions as the pole moves in the interior of the domain.

Keywords

Aharonov-Bohm potential Convergence of eigenfunctions Magnetic Schrödinger operators 

2010 AMS Classification

35J10 35Q40 35J75 

Notes

Acknowledgements

This paper is dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. The authors have been partially supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”. V. Felli has been partially supported by PRIN-2012-grant “Variational and perturbative aspects of nonlinear differential problems”.

References

  1. 1.
    Abatangelo, L., Felli, V.: Sharp asymptotic estimates for eigenvalues of Aharonov-Bohm operators with varying poles. Calc. Var. Partial Differ. Equ. 54, 3857–3903 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Abatangelo, L., Felli, V.: On the leading term of the eigenvalue variation for Aharonov-Bohm operators with a moving pole. SIAM J. Math. Anal. 48, 2843–2868 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Abatangelo, L., Felli, V., Terracini, S.: On the sharp effect of attaching a thin handle on the spectral rate of convergence. J. Funct. Anal. 266, 3632–3684 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Abatangelo, L., Felli, V., Noris, B., Nys, M.: Sharp boundary behavior of eigenvalues for Aharonov-Bohm operators with varying poles. J. Funct. Anal. 273, 2428–2487 (2017).CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bonnaillie-Noël, V., Helffer, B.: Numerical analysis of nodal sets for eigenvalues of Aharonov-Bohm Hamiltonians on the square with application to minimal partitions. Exp. Math. 20, 304–322 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bonnaillie-Noël, V., Helffer, B., Hoffmann-Ostenhof, T.: Aharonov-Bohm Hamiltonians, isospectrality and minimal partitions. J. Phys. A 42, 185203, 20 pp. (2009)Google Scholar
  7. 7.
    Bonnaillie-Noël, V., Helffer, B., Vial, G.: Numerical simulations for nodal domains and spectral minimal partitions. ESAIM Control Optim. Calc. Var. 16, 221–246 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Bonnaillie-Noël, V., Noris, B., Nys, M., Terracini, S.: On the eigenvalues of Aharonov-Bohm operators with varying poles. Anal. PDE 7, 1365–1395 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Felli, V., Ferrero, A., Terracini, S.: Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential. J. Eur. Math. Soc. 13, 119–174 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Helffer, B., Hoffmann-Ostenhof, T.: On a magnetic characterization of spectral minimal partitions. J. Eur. Math. Soc. 15, 2081–2092 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Helffer, B., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T., Owen, M.P.: Nodal sets for groundstates of Schrödinger operators with zero magnetic field in non-simply connected domains. Commun. Math. Phys. 202, 629–649 (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Laptev, A., Weidl, T.: Hardy inequalities for magnetic Dirichlet forms. In: Mathematical Results in Quantum Mechanics (Prague, 1998). Operator Theory, Analysis and Mathematical Physics, vol. 108, pp. 299–305. Birkhäuser, Basel (1999)Google Scholar
  13. 13.
    Léna, C.: Eigenvalues variations for Aharonov-Bohm operators. J. Math. Phys. 56, 011502 (2015). doi:10.1063/1.4905647CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Noris, B., Terracini, S.: Nodal sets of magnetic Schrödinger operators of Aharonov-Bohm type and energy minimizing partitions. Indiana Univ. Math. J. 59, 1361–1403 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Noris, B., Nys, M., Terracini, S.: On the eigenvalues of Aharonov-Bohm operators with varying poles: pole approaching the boundary of the domain. Commun. Math. Phys. 339, 1101–1146 (2015)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanoItaly
  2. 2.Dipartimento di Scienza dei MaterialiUniversità di Milano BicoccaMilanoItaly

Personalised recommendations