Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs

Part of the book series: Springer INdAM 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Helffer, B., Hoffmann-Ostenhof, T.: On a magnetic characterization of spectral minimal partitions. J. Eur. Math. Soc. 15, 2081–2092 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Léna, C.: Eigenvalues variations for Aharonov-Bohm operators. J. Math. Phys. 56, 011502 (2015). doi:10.1063/1.4905647

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

Download references

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”.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Via Cozzi 55, 20125, Milano, Italy

    Laura Abatangelo

  2. Dipartimento di Scienza dei Materiali, Università di Milano Bicocca, Via Cozzi 55, 20125, Milano, Italy

    Veronica Felli

Authors
  1. Laura Abatangelo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Veronica Felli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Veronica Felli .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Pierluigi Colli

  2. Dipartimento di Matematica, Università di Bologna , Bologna, Italy

    Angelo Favini

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Elisabetta Rocca

  4. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    Giulio Schimperna

  5. für Angewandte Analysis und Stochastik (WIAS), Weierstraß-Institut , Berlin, Germany

    Jürgen Sprekels

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Abatangelo, L., Felli, V. (2017). Rate of Convergence for Eigenfunctions of Aharonov-Bohm Operators with a Moving Pole. In: Colli, P., Favini, A., Rocca, E., Schimperna, G., Sprekels, J. (eds) Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs. Springer INdAM Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-64489-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation