Survey of the eigenfunctions of a billiard system between integrability and chaos

We study numerically the eigenfunctions and their Wigner phase space distributions of the two-dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by Robnik (1983). This system is a generic KAM system and displays a transition from integrability to almost ergodicity as the billiard shape changes. We clearly identify two classes of states: the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions, whilst the mixed type states were not found, in support of Percival's conjecture (1973). We confirm the existence of(extremely) intense scars in the classically chaotic regions, and demonstrate their association with classical periodic orbits. Three classes of scars are revealed: one-orbit scars, many-orbit-one-family scars (of statistically similar orbits in the homoclinic neighbourhood), and many-orbit-many-family scars. We argue that it is impossible to find an a priori semiclassical theory of individual eigenstates, but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates.

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