Uniformity in n of small-field asymptotics for exterior magnetic Steklov eigenvalues on the disk

Determine whether the small-magnetic-field asymptotic expansions of the magnetic Steklov eigenvalues for the Dirichlet-to-Neumann operator on the exterior of the unit disk with constant magnetic field are uniform in the Fourier mode index n. Concretely, for the eigenvalues \check{\lambda}_n(b) defined by the exterior problem with magnetic potential A(x,y)=b(-y,x), ascertain whether the expansions \check{\lambda}_1(b)=1+b\log b+O(b) and, for n\ge 2, \check{\lambda}_n(b)=n-\tfrac{n}{n-1}b+O(b^2) as b\to 0^+ admit error terms O(b) and O(b^2) whose bounds are independent of n (for n\ge 2), and precisely characterize any permissible ranges of n for which such uniformity holds.

Background

In Subsection 4.3 the authors prove a weak-field limit for the magnetic Dirichlet-to-Neumann operator \check{\Lambda}(b) on the exterior of the unit disk, establishing operator-norm convergence to the non-magnetic map as b\to 0^+. Using small-z expansions of the confluent hypergeometric function U, they then provide more precise small-b asymptotics for each fixed Fourier mode n: \check{\lambda}_1(b)−1=b\log b+O(b) and, for n\ge 2, \check{\lambda}_n(b)−n=−\tfrac{n}{n−1}b+O(b^2).

They explicitly note that they do not know whether these asymptotics can be made uniform in n, i.e., whether the error terms O(b) and O(b^2) can be controlled independently of n (for n\ge 2). Establishing such uniformity would sharpen the weak-field analysis and clarify the interplay between small b and high angular modes.