Short-time effects on eigenstate structure in Sinai billiards and related systems

There is much latitude between the requirements of Schnirelman's theorem re garding the ergodicity of individual high-energy eigenstates of classically chaotic systems on the one hand, and the extreme requirements of random ma trix theory on the other. It seems likely that some eigenstate statistics a nd long-time transport behavior bear nonrandom imprints of the underlying c lassical dynamics while simultaneously obeying Schnirelman's theorem. Indee d this was shown earlier in the case of systems that approach classical erg odicity slowly, and is also realized in the scarring of eigenstates, even i n the (h) over bar --> 0 limit, along unstable periodic orbits and their ma nifolds. Here we demonstrate the nonrandom character of eigenstates of Sina i-like systems. We show that mixing between channels in Sinai systems is dr amatically deficient compared to random matrix theory predictions. The defi cit increases as \In (h) over bar\ for (h) over bar --> 0, and is due to th e vicinity of the measure zero set of orbits that never collide with the Si nai obstruction. Coarse graining to macroscopic scales recovers the Schnire lman result. Three systems are investigated here: a Sinai-type billiard, a quantum map that possesses the essential properties of the Sinai billiard, and a unitary map corresponding to a quasirandom Hamiltonian. Various wave function and long-time transport statistics are defined, theoretically inve stigated, and compared to numerical data.