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The Journal of Geometric Analysis

, Volume 25, Issue 4, pp 2546–2564 | Cite as

The Magnetic Laplacian in Shrinking Tubular Neighborhoods of Hypersurfaces

  • D. Krejčiřík
  • N. Raymond
  • M. Tušek
Article

Abstract

The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the Laplacian converges in a norm-resolvent sense to a Schrödinger operator on the limiting hypersurface whose electromagnetic potential is expressed in terms of principal curvatures and the projection of the ambient vector potential to the hypersurface. As an application, we obtain an effective approximation of bound-state energies and eigenfunctions in thin quantum layers.

Keywords

Curvature of hypersurfaces Effective potential Eigenvalue asymptotics 

Mathematics Subject Classification

35J25 35B25 35B40 58J50 81Q10 

Notes

Acknowledgments

Support by the Institute Mittag-Leffler (Djursholm, Sweden), where this paper was being prepared, is gratefully acknowledged. The work has been partially supported by the Project RVO61389005, and the Grants No. P203/11/0701 and No. GA13-11058S of the Czech Science Foundation (GAČR).

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Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsNuclear Physics Institute ASCRŘežCzech Republic
  2. 2.Institut de Recherche Mathématique de RennesUniversité de Rennes 1Rennes CedexFrance
  3. 3.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePrague 2Czech Republic

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