Quantum ergodicity for a class of non-generic systems

TL;DR

This paper investigates quantum ergodicity in systems with residual spectral degeneracy and energy gap resonance, showing that degeneracy has limited impact while resonance significantly affects ergodic behavior.

Contribution

It extends von Neumann's quantum ergodic theorem to systems with spectral degeneracy and resonance, highlighting the dominant role of resonance in ergodicity.

Findings

Degeneracy does not significantly alter ergodic conditions.

Resonance in energy gaps more strongly obstructs ergodicity.

Modified conditions are required for typicality in these systems.

Abstract

We examine quantum normal typicality and ergodicity properties for quantum systems whose dynamics are generated by Hamiltonians which have residual degeneracy in their spectrum and resonance in their energy gaps. Such systems can be considered atypical in the sense that degeneracy, which is usually a sign of symmetry, is naturally broken in typical systems due to stochastic perturbations. In particular, we prove a version of von Neumann's quantum ergodic theorem, where a modified condition needs to hold in order to have normal typicality and ergodicity. As a result, we show that degeneracy of spectrum does not considerably modify the condition of the theorem, whereas the existence of resonance is more dominant for obstructing ergodicity.