SURVEY OF THE EIGENFUNCTIONS OF A BILLIARD SYSTEM BETWEEN INTEGRABILITY AND CHAOS

We study numerically the eigenfunctions and their Wigner phase space d istributions of the two-dimensional billiard system defined by the qua dratic conformal image of the unit disk as introduced by Robnik (1983) . This system is a generic KAM SyStem and displays a transition from i ntegrability to almost ergodicity as the billiard shape changes. We cl early 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 suppo rt of Percival's conjecture (1973). We confirm the existence of (extre mely) intense scars in the classically chaotic regions, and demonstrat e their association with classical periodic orbits. Three classes of s cars are revealed: one-orbit scars, many-orbit-one-family scars (of st atistically similar orbits in the homoclinic neighbourhood), and many- orbit-many-family scars. We argue that it is impossible to find an a p riori semiclassical theory of individual eigenstates, but do not deny the usefulness of general semiclassical arguments in analysing the col lective and statistical properties of eigenstates.