Finite-time equilibration bounds for continuous-spectrum quantum systems via spectral coarse graining

We investigate equilibration-on-average and effective equilibration for quantum systems evolving on infinite-dimensional Hilbert spaces under Hamiltonians with purely continuous spectrum. In this regime, the standard infinite-time dephasing picture underlying finite-dimensional equilibration theory ceases to be informative: the corresponding ergodic average becomes trivial in the relevant sense and therefore cannot serve as a physically meaningful equilibrium state. This makes equilibration intrinsically a finite-time phenomenon. Motivated by this obstruction, we introduce a framework based on finite-time ergodic averages and spectral coarse graining at finite resolution. Within this setting, we derive explicit upper bounds for equilibration-on-average of bounded observables under purely continuous-spectrum dynamics. Unlike their finite-dimensional counterparts, these bounds depend not only on the Hamiltonian and the initial state, but also on the observation time horizon and on the resolution scale of the spectral partition. We also obtain corresponding bounds for effective equilibration with respect to finite families of physically realistic POVM (Positive Operator-Valued Measure) measurements. Our results provide a quantitative formulation of equilibration for continuous-spectrum quantum systems in a regime where the usual infinite-time dephasing paradigm breaks down. Within the coarse-grained framework developed here, the resulting bounds depend explicitly on both the finite observation time and the chosen spectral resolution. An explicit example based on a free-particle Hamiltonian illustrates the mechanism and interpretation of the general bounds.