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Purely magnetic tunneling effect in two dimensions

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Abstract

The magnetic Schrödinger operator, with Neumann boundary condition, on a smooth, bounded, and simply connected domain \(\Omega \) of the Euclidean plane is considered in the semiclassical limit. When \(\Omega \) has a symmetry axis, the semiclassical splitting of the first two eigenvalues is analyzed. The first explicit tunneling formula in a pure magnetic field is established. The analysis is based on a pseudo-differential reduction to the boundary and the proof of the first known optimal purely magnetic Agmon estimates.

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Notes

  1. and actually of the conjugated operator \({\mathscr {N}}^{\varphi }_{\hbar , r}= e^{\varphi /\hbar ^{\frac{1}{2}}}{\mathscr {N}}_{\hbar , r}e^{-\varphi /\hbar ^{\frac{1}{2}}}\), where \(\varphi \) is an appropriate subsolution of the effective eikonal equation.

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Authors and Affiliations

  1. Département de mathématiques et applications, École normale supérieure, CNRS, Université PSL, 75005, Paris, France

    Virginie Bonnaillie-Noël

  2. LMJL - UMR6629, CNRS, Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322, Nantes Cedex 3, France

    Frédéric Hérau

  3. Laboratoire Angevin de Recherche en Mathématiques, LAREMA, UMR 6093, UNIV Angers, SFR Math-STIC, 2 boulevard Lavoisier, 49045, Angers Cedex 01, France

    Nicolas Raymond

Authors
  1. Virginie Bonnaillie-Noël
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  2. Frédéric Hérau
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  3. Nicolas Raymond
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Correspondence to Nicolas Raymond.

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N. R. and F. H. are deeply grateful to the Mittag-Leffler Institute where part of the ideas of this article were discussed. N. R. also thanks Bernard Helffer, Pierig Keraval and Johannes Sjöstrand for many stimulating discussions.

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Bonnaillie-Noël, V., Hérau, F. & Raymond, N. Purely magnetic tunneling effect in two dimensions. Invent. math. 227, 745–793 (2022). https://doi.org/10.1007/s00222-021-01073-x

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  • DOI: https://doi.org/10.1007/s00222-021-01073-x

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