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Letters in Mathematical Physics

, Volume 109, Issue 7, pp 1533–1558 | Cite as

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

  • Bernard Helffer
  • Hynek KovaříkEmail author
  • Mikael P. Sundqvist
Article
  • 31 Downloads

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.

Keywords

Pauli operator Dirichlet Semiclassical Flux effects 

Mathematics Subject Classification

35P15 81Q05 81Q20 

Notes

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.

References

  1. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bovier, A., Gayrard, V., Klein, M.: Metastability in reversible diffusion processes II: precise asymptotics for small eigenvalues. JEMS 7(1), 69–99 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dauge, M.: Neumann and mixed problems on curvilinear polyhedra. Integral Equ. Oper. Theory. 15, 227–261 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Di Gesu, G., Le Peutrec, D., Lelièvre, T., Nectoux, B.: Sharp asymptotics of the first exit point density. arXiv:1706.08726 (2017)
  6. 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)Google Scholar
  7. 7.
    Ekholm, T., Kovařík, H., Portmann, F.: Estimates for the lowest eigenvalue of magnetic Laplacians. J. Math. Anal. Appl. 439(1), 330–346 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)Google Scholar
  9. 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)MathSciNetzbMATHGoogle Scholar
  10. 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)Google Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Helffer, B., Sjöstrand, J.: Algebraic Analysis. A proof of the Bott inequalities, vol. 1, pp. 171–183. Academic Press, Cambridge (1988)CrossRefzbMATHGoogle Scholar
  14. 14.
    Henrot, A., Pierre, M.: Variation et optimisation de formes–une analyse géométrique–Mathématiques et Applications, vol. 48. Springer, Berlin (2005)zbMATHGoogle Scholar
  15. 15.
    Luttrell, S.: https://mathematica.stackexchange.com/a/154435/21414. Accessed 18 Sept 2017
  16. 16.
    Michel, L.: About small eigenvalues of Witten Laplacians. arXiv:1702.01837 (2017)
  17. 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. 18.
    Sternberg, S.: On the structure of local homeomorphisms of Euclidean n-space, II. Am. J. Math. 80(3), 623–631 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    van den Berg, M., Bucur, D.: Sign changing solutions of Poisson’s equation. arXiv:1804.00903v1 (2018)
  20. 20.
    Witten, E.: Supersymmetry and Morse inequalities. J. Differ. Geom. 17, 661–692 (1982)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Jean LerayUniversité de NantesNantesFrance
  2. 2.Laboratoire de MathématiquesUniversité Paris-SudOrsayFrance
  3. 3.DICATAM, Sezione di MatematicaUniversità degli studi di BresciaBresciaItaly
  4. 4.Department of Mathematical SciencesLund UniversityLundSweden

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