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.
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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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Communicated by Jiaping Wang.
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Krejčiřík, D., Raymond, N. & Tušek, M. The Magnetic Laplacian in Shrinking Tubular Neighborhoods of Hypersurfaces. J Geom Anal 25, 2546–2564 (2015). https://doi.org/10.1007/s12220-014-9525-y
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DOI: https://doi.org/10.1007/s12220-014-9525-y