GEOMETRY OF HIGH-LYING EIGENFUNCTIONS IN A PLANE BILLIARD SYSTEM HAVING MIXED-TYPE CLASSICAL DYNAMICS

In this paper we study the geometrical properties of the high-lying ei genfunctions (200 000 and above) which are deep in the semiclassical r egime. The system we are analysing is the billiard system inside the r egion defined by the quadratic (complex) conformal map w = z + lambda z(2) of the unit disc \z\ less than or equal to 1 as introduced by Rob nik (1983), with the shape parameter value lambda = 0.15, so that the billiard is still convex and has KAM-type classical dynamics, where re gular and irregular regions of classical motion coexist in the classic al phase space. By inspecting 100 and by showing 36 consecutive numeri cally calculated eigenfunctions we reach the following conclusions: (i ) Percival's (1973) conjectured classification in regular and irregula r states works well: the mixed-type states 'living' on regular and irr egular regions disappear in the semiclassical limit. (ii) The irregula r (chaotic) states can be strongly localized due to the slow classical diffusion, but become fully extended in the semiclassical limit when the break time becomes sufficiently large with respect to the classica l diffusion time. (iii) Almost all states can be clearly associated wi th some relevant classical object such as the invariant torus, cantoru s or periodic orbits. This paper is largely qualitative but deep in th e semiclassical limit and as such it is a prelude to our next paper wh ich is quantitative and numerically massive but at about ten times low er energies.