Level statistics and eigenfunctions of pseudointegrable systems: dependence on energy and genus number.

Abstract

We study the level statistics (second half moment I0 and rigidity Delta(3)) and the eigenfunctions of pseudointegrable systems with rough boundaries of different genus numbers g. We find that the levels form energy intervals with a characteristic behavior of the level statistics and the eigenfunctions in each interval. At low enough energies, the boundary roughness is not resolved and accordingly the eigenfunctions are quite regular functions and the level statistics shows Poisson-like behavior. At higher energies, the level statistics of most systems moves from Poisson-like toward Wigner-like behavior with increasing g. On investigating the wave functions, we find many chaotic functions that can be described as a random superposition of regular wave functions. The amplitude distribution P(psi) of these chaotic functions was found to be Gaussian with the typical value of the localization volume V(loc) approximately equal 0.33. For systems with periodic boundaries we find several additional energy regimes, where I0 is relatively close to the Poisson limit. In these regimes, the eigenfunctions are either regular or localized functions, where P(psi) is close to the distribution of a sine or cosine function in the first case and strongly peaked in the second case. An interesting intermediate case between chaotic and localized eigenfunctions also appears.