Short-Time Effects on Eigenstate Structure in Sinai Billiards and Related Systems

Abstract

There is much latitude between the requirements of Schnirelman's theorem regarding the ergodicity of individual high-energy eigenstates of classically chaotic systems on the one hand, and the extreme requirements of random matrix theory on the other. It seems likely that some eigenstate statistics and long-time transport behavior bear nonrandom imprints of the underlying classical dynamics while simultaneously obeying Schnirelman's theorem. Indeed this was shown earlier in the case of systems which approach classical ergodicity slowly, and is also realized in the scarring of eigenstates, even in the $\hbar \to 0$ limit, along unstable periodic orbits and their manifolds. Here we demonstrate the nonrandom character of eigenstates of Sinai-like systems. We show that mixing between channels in Sinai systems is dramatically deficient compared to random matrix theory predictions. The deficit {\it increases} as $|\log \hbar|$ for $\hbar\to 0$, and is due to the vicinity of the measure zero set of orbits which never collide with the Sinai obstruction. Coarse graining to macroscopic scales recovers the Schnirelman result. Three systems are investigated here: a Sinai-type billiard, a quantum map which possesses the essential properties of the Sinai billiard, and a unitary map corresponding to a quasirandom Hamiltonian. Various wavefunction and long-time transport statistics are defined, theoretically investigated, and compared to numerical data.