The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces
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 normresolvent sense to a Schroedinger 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 boundstate energies and eigenfunctions in thin quantum layers.
 Publication:

arXiv eprints
 Pub Date:
 March 2013
 DOI:
 10.48550/arXiv.1303.4753
 arXiv:
 arXiv:1303.4753
 Bibcode:
 2013arXiv1303.4753K
 Keywords:

 Mathematical Physics;
 Mathematics  Differential Geometry;
 Mathematics  Spectral Theory
 EPrint:
 J. Geom. Anal. 25 (2015), 25462564