Skip to main content
SpringerLink
Log in

On the semiclassical analysis of the ground state energy of the Dirichlet Pauli operator III: magnetic fields that change sign

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the semiclassical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm–Kovařík–Portmann and Helffer–Sundqvist for the asymptotics of the ground state energy in the semiclassical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semiclassical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disk are discussed. Various natural conjectures are disproved, and this leaves the research of an optimal result in the general case still open.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. The condition reads: \( 2 -\frac{2}{p} < \frac{\pi }{\omega }\) where \(\omega \) is the maximal aperture of the corners.

References

  1. Barbaroux, J.-M., Le Treust, L., Raymond, N., Stockmeyer, E.: On the semi-classical spectrum of the Dirichlet-Pauli operator. arXiv:1804.00903v1 (October 2018)

  2. Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes I: sharp asymptotics for capacities and exit times. JEMS 6(4), 399–424 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues. JEMS 7(1), 69–99 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Dauge, M.: Neumann and mixed problems on curvilinear polyhedra. Integral Equ. Oper. Theory. 15, 227–261 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Di Gesu, G., Le Peutrec, D., Lelièvre, T., Nectoux, B.: Sharp asymptotics of the first exit point density. arXiv:1706.08726 (2017)

  6. Di Gesu, G., Le Peutrec, D., Lelièvre, T., Nectoux, B.: The exit from a metastable state: concentration of the exit point on the low energy saddle points (In preparation)

  7. Ekholm, T., Kovařík, H., Portmann, F.: Estimates for the lowest eigenvalue of magnetic Laplacians. J. Math. Anal. Appl. 439(1), 330–346 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. Translated from the Russian by Joseph Szuecs. 2nd ed. Grundlehren der Mathematischen Wissenschaften. 260. New York (1998)

  9. Helffer, B., Klein, M., Nier, F.: Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Matematica Contemporanea 26, 41–85 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Helffer, B., Nier, F.: Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary. Mém. Soc. Math. Fr. (N.S.) No. 105 (2006)

  11. Helffer, B., Sundqvist, M.Persson: On the semi-classical analysis of the Dirichlet Pauli operator. J. Math. Anal. Appl. 449(1), 138–153 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Helffer, B., Sundqvist, M.Persson: On the semi-classical analysis of the groundstate energy of the Dirichlet Pauli operator in non-simply connected domains. J. Math. Sci. 226(4), 531–544 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Helffer, B., Sjöstrand, J.: Algebraic Analysis. A proof of the Bott inequalities, vol. 1, pp. 171–183. Academic Press, Cambridge (1988)

    Book  MATH  Google Scholar 

  14. Henrot, A., Pierre, M.: Variation et optimisation de formes–une analyse géométrique–Mathématiques et Applications, vol. 48. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  15. Luttrell, S.: https://mathematica.stackexchange.com/a/154435/21414. Accessed 18 Sept 2017

  16. Michel, L.: About small eigenvalues of Witten Laplacians. arXiv:1702.01837 (2017)

  17. Nectoux, B.: Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit. arXiv:1710.07510 (2017)

  18. Sternberg, S.: On the structure of local homeomorphisms of Euclidean n-space, II. Am. J. Math. 80(3), 623–631 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  19. van den Berg, M., Bucur, D.: Sign changing solutions of Poisson’s equation. arXiv:1804.00903v1 (2018)

  20. Witten, E.: Supersymmetry and Morse inequalities. J. Differ. Geom. 17, 661–692 (1982)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank M. Dauge for useful discussions about the paper [4]. B. H. would like to thank D. Le Peutrec for discussions around his work with G. Di Gesu, T. Lelièvre and B. Nectoux and N. Raymond for discussions on [1]. H. K. has been partially supported by Gruppo Nazionale per Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The support of MIUR-PRIN2010-11 Grant for the project “Calcolo delle variazioni” (H. K.), is also gratefully acknowledged.

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 rue de la Houssinière, 44322, Nantes, France

    Bernard Helffer

  2. Laboratoire de Mathématiques, Université Paris-Sud, Orsay, France

    Bernard Helffer

  3. DICATAM, Sezione di Matematica, Università degli studi di Brescia, Brescia, Italy

    Hynek Kovařík

  4. Department of Mathematical Sciences, Lund University, Box 118, 221 00, Lund, Sweden

    Mikael P. Sundqvist

Authors
  1. Bernard Helffer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hynek Kovařík
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikael P. Sundqvist
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hynek Kovařík.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Helffer, B., Kovařík, H. & Sundqvist, M.P. On the semiclassical analysis of the ground state energy of the Dirichlet Pauli operator III: magnetic fields that change sign. Lett Math Phys 109, 1533–1558 (2019). https://doi.org/10.1007/s11005-018-01153-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-018-01153-9

Keywords

Mathematics Subject Classification

Search

Navigation