Weighted Time Averages and Weak Convergence to Equilibrium in Quantum Integrable Systems

This paper establishes a natural quantum counterpart of weak equilibration for statistical ensembles in integrable systems. For quantum systems with pure point spectrum, single-time expectation values under unitary evolution are typically quasiperiodic, and hence generally do not admit a pointwise limit as $t\to \mathrm{\infty }$. To overcome this difficulty, we introduce a weighted time-averaging procedure and prove that the resulting averaged dynamics converge to the diagonal (dephased) equilibrium state. We further illustrate and validate the theoretical result through a three-spin quantum integrable model.