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This work is motivated by an article by Wang, Casati, and Prosen[Phys. Rev. E vol. 89, 042918 (2014)] devoted to a study of ergodicityin two-dimensional irrational right-triangular billiards. Numericalresults presented there suggest that these billiards are generally notergodic. However, they become ergodic when the billiard angle is equalto \pi/2 π / 2 times a Liouvillian irrational, morally a class of irrational numberswhich are well approximated by rationals. In particular, Wang etal. study a special integer counter that reflects the irrationalcontribution to the velocity orientation; they conjecture that thiscounter is localized in the generic case, but grows in the Liouvilliancase. We propose a generalization of the Wang-Casati-Prosen counter:this generalization allows to include rational billiards intoconsideration. We show that in the case of a 45°\!\!:\!45°\!\!:\!90° 45 ° : 45 ° : 90 ° billiard, the counter grows indefinitely, consistent with theLiouvillian scenario suggested by Wang et al. 

We consider a toy model for emergence of chaos in a quantum many-body short-range-interacting system: two one-dimensional hard-core particles in a box, with a small mass defect as a perturbation over an integrable system, the latter represented by two equal mass particles.To that system, we apply a quantum generalization of Chirikov's criterion for the onset of chaos, i.e. the criterion of overlapping resonances.There, classical nonlinear resonances translate almost automatically to the quantum language. Quantum mechanics intervenes at a later stage: the resonances occupying less than one Hamiltonian eigenstate are excluded from the chaos criterion. Resonances appear as contiguous patches of low purity unperturbed eigenstates, separated by the groups of undestroyed states-the quantum analogues of the classical KAM tori.