The Magnetic Laplacian in Shrinking Tubular Neighborhoods 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 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.

The Adiabatic Limit of the Connection Laplacian

“…Similar results were obtained for the heat equation in thin tubes in Riemannian manifolds by Wittich [Wit07], and Kolb and Krejčiřík [KK14]. There are also some results for operators built from a non-trivial connection on a line bundle over the waveguide-manifold, which models an external magnetic field [EJK01, BEK05, EK05, BdOV13, KR14,KRT15]. Our work here will pave the way for the analysis of magnetic fields in the generalised waveguides considered in [HLT15].…”

Reduction of Dimension as a Consequence of Norm‐resolvent Convergence and Applications

“…Although the latter is also very natural it was hardly visible in existing literature due to many seemingly different technical steps as well as various ways how these quantities enter. As particular cases of the application of Theorem 1.1, we recover, in a short manner, known results for quantum waveguides (see for instance [3], [11], [9] or [10]) and cast a new light on Born-Oppenheimer type results (see [12], [17], [7] or [16,Sec. 6.2]).…”

“…where λ Σ k is the k-th eigenvalue of´∆ s`V psq. We recover a result in the spirit of [3,10]. It is defined for ψ P DompQ DR α q where DompQ DR α q " tψ P H 1 pΣˆp0, 1qq : ψps, 1q " 0 , B t ψps, 0q "´αψps, 0qu .…”

A lower bound to the spectral threshold in curved quantum layers

“…Remark 2. Note that (13) yields a Hardy-Poincaré-type inequality (16) for all φ ∈ W 1,2 0 ((−a, a)) and κ 1 , κ 2 ∈ [−1/a, 1/a], where the Hardy weight V (·; κ 1 , κ 2 ) is given by (14) and the Poincaré constant λ 1 (κ 1 , κ 2 ) interpolates between 0 and π 2 /(2a) 2 . An equivalent version of this inequality in weighted spaces follows from (5).…”

“…Notice that the function a − | · | has the meaning of the distance to the boundary of the one-dimensional domain (−a, a). Hardy inequalities with weights of type (14) have been recently considered for higher-dimensional domains in [4] (see also [2,Lem. 8]).…”