Local quantum ergodic conjecture

TL;DR

This paper proposes a local version of the quantum ergodic conjecture using the chord function, suggesting universal patterns in eigenstates of chaotic systems, supported by numerical evidence.

Contribution

It introduces a local conjecture based on the chord function, refining the quantum ergodic conjecture for ergodic eigenstates within a narrow energy range.

Findings

Chord function captures local phase space information.

Universal orthogonality patterns are predicted for ergodic eigenstates.

Numerical evidence supports the local conjecture in a chaotic Hamiltonian.

Abstract

The Quantum Ergodic Conjecture equates the Wigner function for a typical eigenstate of a classically chaotic Hamiltonian with a delta-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.