The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces

David Krejcirik (1) , Nicolas Raymond (2) , Matej Tusek (3)

1 Nuclear Physics Institute ASCR
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 FJFI CTU - Faculty of Nuclear Sciences and Physical Engineering [Prague]

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.

Mots clés: curvature of hypersurfaces; effective potential; eigenvalue asymptotics

Domaines: Physique mathématique [math-ph]

David Krejcirik, Nicolas Raymond, Matej Tusek. The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces. Journal of Geometric Analysis, 2015, 25 (4), pp.2546-2564. ⟨10.1007/s12220-014-9525-y⟩. ⟨hal-00802483⟩