Numerical investigation of regimes of Wigner and Shnirelman ergodicity in rough billiards

We study numerically regimes of Wigner and Shnirelman ergodicity [K. M. Fra hm and D. L Shepelyansky, Phys. Rev. Lett. 79,1833 (1997)] in rough half-ci rcular billiards. We show that in the regime of Wigner ergodicity eigenstat es are extended over the whole energy surface but have a strongly peaked no nergodic structure. In the regime of Shnirelman ergodicity the eigenstates are ergodically distributed along the energy surface. The Shannon width of the eigenstate distributions is calculated to estimate quantitatively their spreads. We show that in both regimes the amplitude distribution P(psi) is well approximated by a Gaussian distribution.