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

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.

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Notes

  1. 1.

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

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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.

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Correspondence to Hynek Kovařík.

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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

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Keywords

  • Pauli operator
  • Dirichlet
  • Semiclassical
  • Flux effects

Mathematics Subject Classification

  • 35P15
  • 81Q05
  • 81Q20