A note on the semiclassical measure at singular points of the boundary of the Bunimovich stadium

TL;DR

This paper clarifies Hassell's argument on semiclassical measures in Bunimovich stadiums, extending the notion of non-gliding points to domains with less regular boundaries.

Contribution

It adapts the concept of non-gliding points to non-$C^2$ boundaries, clarifying the semiclassical measure analysis at boundary singularities.

Findings

Existence of semiclassical measures with positive mass on bouncing ball trajectories

Extension of non-gliding point notion to less regular boundaries

Clarification of Hassell's argument at boundary singularities

Abstract

An argument by Hassell proving the existence of a Bunimovich stadium for which there are semiclassical measures giving positive mass to the submanifold of bouncing ball trajectories uses a notion of non-gliding points. However, this notion is defined only for domains with $C^2$-boundaries. The purpose of this note is to clarify the argument.