Quantization of generic chaotic 3D billiard with smooth boundary II: structure of high-lying eigenstates

Abstract

This is the first survey of highly excited eigenstates of a chaotic 3D billiard. We introduce a strongly chaotic 3D billiard with a smooth boundary and we manage to calculate accurate eigenstates with sequential number (of a 48-fold desymmetrized billiard) about 45,000. Besides the brute-force calculation of 3D wavefunctions we propose and illustrate another two representations of eigenstates of quantum 3D billiards: (i) normal derivative of a wavefunction over the boundary surface, and (ii) ray - angular momentum representation. The majority of eigenstates is found to be more or less uniformly extended over the entire energy surface, as expected, but there is also a fraction of strongly localized - scarred eigenstates which are localized either (i) on to classical periodic orbits or (ii) on to planes which carry (2+2)-dim classically invariant manifolds, although the classical dynamics is strongly chaotic and non-diffusive.