In the paper [The existence of unbounded oscillating trajectories in a problem of billiards][1] (1962) Leontovich proved that under bell-like curve (it must be zero at  $\pm\infty$) each trajectory oscillates, i.e. it crosses y-axis infinitely often. Also he proved that among all trajectories do exist finite and infinite ones.


  [1]: https://zbmath.org/?q=an:0124.05002