Norm resolvent convergence in parameter regimes not covered by the current results

Determine whether generalized norm resolvent convergence of the Laplace–Beltrami operators holds for manifolds obtained by attaching many small handles (wormholes) in scaling regimes for the handle length and density that lie outside the sufficient conditions established in this work. Concretely, for dimensions m ≥ 2, investigate whether such convergence (in the sense of quasi-unitary equivalence) occurs when the handle length scales as ℓ_ε = ε^λ and the uniform cover distance of handle attachment points scales as η_ε = r0 ε^α (with the analogous logarithmic variant in m = 2), in those (α, λ) parameter regions not encompassed by the proven fading and adhering cases.

Background

The paper studies Laplace–Beltrami operators on Riemannian manifolds perturbed by attaching many thin, short handles (wormholes). Two primary regimes are analyzed: fading handles, where the limit operator is the unperturbed Laplacian on the original manifold, and adhering handles, where densely distributed short handles identify two isometric regions, yielding a limit operator acting on functions equal on the identified sets.

The analysis provides generalised norm resolvent convergence under explicit scaling assumptions that relate the handle radius ε, the handle length ℓε = ε^λ, and the uniform cover distance ηε ≈ r0 ε^α (with a logarithmic variant for m = 2). These conditions define ranges in the (α, λ)-plane where convergence to the respective limit operators is established.

In comparing with related results and mapping out parameter regions, the authors note gaps—parts of the (α, λ)-plane not covered by their theorems. They explicitly state that it is unknown whether generalised norm resolvent convergence holds in these uncovered parameter regions, leaving open the question of convergence (and potentially the form of the limit operator) beyond the established sufficient conditions.