The magnetic Laplacian in shrinking tubular neighbourhoods of hypersurfaces

TL;DR

This paper studies the behavior of the magnetic Laplacian in thin regions around hypersurfaces, showing it converges to a Schrödinger operator with curvature-dependent potentials, aiding quantum layer analysis.

Contribution

It introduces a limit analysis of the magnetic Laplacian in shrinking tubular neighborhoods, deriving an effective Schrödinger operator incorporating curvature and magnetic effects.

Findings

Convergence of the magnetic Laplacian to a Schrödinger operator in the thin limit

Explicit expression of the electromagnetic potential in terms of curvatures

Approximation of bound-state energies and eigenfunctions in quantum layers

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 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 bound-state energies and eigenfunctions in thin quantum layers.