Practical and algorithmical manifestations of quantum chaos

Quantum chaos manifests itself also in algorithmical complexity of methods, including the numerical ones, in solving the Schr\"odinger equation. In this contribution we address the problem of calculating the eigenenergies and the eigenstates by various numerical methods applied to 2-dim generic billiard systems. In particular we analyze the dependence of the accuracy (errors) on the density of discretization of the given numerical method. We do this for several different billiard shapes, especially for the Robnik billiard. We study the numerical error of the boundary integral method and the plane wave decomposition method as a function of the discretization parameter $b$ which by definition is the number of discretization nodes on the boundary per one de Broglie wavelength (arclength) interval. For boundary integral method, we discover that at each $\lambda$ the error scales as a power law $<|\Delta E|> = A b^{-\alpha}$, where $\alpha$ is a strong function of $\lambda$: In the KAM-like regime $0\le \lambda\le 1/4$ it is large and close to 3.5, but close to $\lambda = 1/4$ it changes almost discontinuously becoming hardly any larger than zero. This is because the billiard becomes nonconvex beginning at $\lambda=1/4$. For the plane wave decomposition method, we found at a fixed $b$ that the average absolute value of the error $<|\Delta E|>$ correlates strongly with the classical chaos, where the error $<|\Delta E|>$ does increase sharply with increasing classical chaos. This property is valid also for Bunimovich stadium and the Sinai billiard.