Weak-field limit for magnetic Dirichlet-to-Neumann operators on general unbounded domains

Establish the weak-magnetic-field (b→0+) limit of the magnetic Dirichlet-to-Neumann operator for general unbounded domains with compact boundary in the plane. Specifically, for a general unbounded domain Ω⊂R^2, determine whether the magnetic Dirichlet-to-Neumann operator Λ_{bA,Ω} associated with a magnetic potential A generating a weak magnetic field converges in operator norm on L^2(∂Ω) to the corresponding zero-field Dirichlet-to-Neumann operator as b→0+, and derive sharp convergence rates and leading asymptotics.

Background

The paper proves precise weak-field limits and rates for exterior domains in the special case of the unit disk, both with and without an Aharonov–Bohm flux. However, the authors point out that extending such results to general unbounded domains is not presently available.

They note related progress on weak magnetic fields for Laplace operators in exterior domains, but emphasize that the weak-field limit for the magnetic Dirichlet-to-Neumann operator itself in general unbounded geometries remains unresolved.