Classical and quantum chaos in bean- and peanut-shaped billiards

The geometry of a billiard boundary fundamentally governs its dynamics, ranging from integrable to mixed and fully chaotic regimes. Bean- and peanut-shaped billiards have varying curvature with both focusing and defocusing walls without a neutral segments. Particle dynamics inside these billiards show a strong correlation between classical and quantum dynamics in the chaotic regime also. This fundamental observation comes from our study of classical tools like Lyapunov exponent, Poincar\'e sections, flow trajectories in phase space and quantum tools that includes both statistical and dynamical measures. Statistical indicators include nearest-neighbour spacing distributions, level-spacing ratios, and the spectral staircase function, while dynamical measures include out-of-time-order correlators and spectral complexity. The dynamics in both of these billiard systems also exhibit eigenfunction scarring, an unexpected phenomenon observed in chaotic systems. Overall, our results provide a unified perspective on billiard systems with non-uniform curvature.