Approximation-scheme independence of admissible billiard trajectories

Determine whether, or in what precise sense, the notion of an admissible billiard trajectory defined through approximation Hamiltonians is independent of the chosen approximation scheme.

Background

The lower-bound argument defines admissible billiard trajectories as limits of periodic trajectories for approximation Hamiltonians. For smooth domains the paper fixes a particular approximation scheme, while for nonsmooth domains it fixes specified approximating families. The authors explicitly leave unresolved whether the resulting admissibility notion depends on these choices, an issue relevant to the intrinsicness and general applicability of the billiard-based capacity bound.