Triangular Billiards
Wang-Casati-Prosen counters
This work is motivated by an article by Wang, Casati, and Prosen [Phys. Rev. E, 89:042918, 2014] devoted to a study of ergodicity in two-dimensional irrational right-triangular billiards. Numerical results presented there suggest that generic irrational billiard angles, the irrational contribution to the velocity orientation remains localized as the system evolves, hence pointing to an absence of ergodicity. However, for Liouvillian angles, whose properties are similar to rational ones, ergodicity is restored.
We study an analogue of the Wang-Casati-Prosen counter for a 45:45:90 degree billiard---that is rational and, furthermore, integrable---and, expectedly, find no localization there[HMSJO23]. The counter is shown to grow logarithmically.
Quantun Chirikov criterion
Using a weakly distorted 45:45:90 degree billiard, we introduce a quantum analogue of the Chirikov criterion for the onset of chaos---an estimate preceeding a rigorous KAM analysis[YaHDHO22].
A weakly distorted 45:45:90 degree billiard
An example of two separate quantum nonlinear resonances, 2:1 (b) and 4:1 (c), separated by a KAM gap (a), at a weak deviation from 45 degrees, overlap and fuse into a single, broad eigenstate (d) that also comprises the already existing 1:0 and 3 : 2 resonances and newly emerged 4:3, 5:2, 6:1, and 5:4 ones, at a stronger distortion.
Quantum nonlinear resonances are depicted for the following p:q ratios: 1:0 (black), 2:1 (green), 3:2 (blue), 4:1 (orange), 4:3 (indigo), 5:2 (yellow), 6:1 (red), and 5: 4 (purple).
