title: Local quantum ergodic conjecture.
abstract: The quantum ergodic conjecture equates the Wigner function for a typical eigenstate of a classically chaotic Hamiltonian with a δ function on the energy shell. This ensures the evaluation of classical ergodic expectations of simple observables, in agreement with Shnirelman's theorem, but this putative Wigner function violates several important requirements. Consequently, we transfer the conjecture to the Fourier transform of the Wigner function, that is, the chord function. We show that all the relevant consequences of the usual conjecture require only information contained within a small (Planck) volume around the origin of the phase space of chords: translations in ordinary phase space. Loci of complete orthogonality between a given eigenstate and its nearby translation are quite elusive for the Wigner function, but our local conjecture stipulates that their pattern should be universal for ergodic eigenstates of the same Hamiltonian lying within a classically narrow energy range. Our findings are supported by numerical evidence in a Hamiltonian exhibiting soft chaos. Heavily scarred eigenstates are remarkable counter-examples of the ergodic universal pattern.
authors.affiliation: Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany.
authors.forename: Eduardo
authors.initials: E
authors.lastname: Zambrano
authors.affiliation: Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, R.J., Brazil.
authors.forename: W P Karel
authors.initials: WP
authors.lastname: Zapfe
authors.affiliation: Centro Brasileiro de Pesquisas Fisicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, R.J., Brazil.
authors.forename: Alfredo M
authors.initials: AM
authors.lastname: Ozorio de Almeida